CINXE.COM
Hooke's law - Wikipedia
<!DOCTYPE html> <html class="client-nojs vector-feature-language-in-header-enabled vector-feature-language-in-main-page-header-disabled vector-feature-page-tools-pinned-disabled vector-feature-toc-pinned-clientpref-1 vector-feature-main-menu-pinned-disabled vector-feature-limited-width-clientpref-1 vector-feature-limited-width-content-enabled vector-feature-custom-font-size-clientpref-1 vector-feature-appearance-pinned-clientpref-1 vector-feature-night-mode-enabled skin-theme-clientpref-day vector-sticky-header-enabled vector-toc-available" lang="en" dir="ltr"> <head> <meta charset="UTF-8"> <title>Hooke's law - Wikipedia</title> <script>(function(){var className="client-js vector-feature-language-in-header-enabled vector-feature-language-in-main-page-header-disabled vector-feature-page-tools-pinned-disabled vector-feature-toc-pinned-clientpref-1 vector-feature-main-menu-pinned-disabled vector-feature-limited-width-clientpref-1 vector-feature-limited-width-content-enabled vector-feature-custom-font-size-clientpref-1 vector-feature-appearance-pinned-clientpref-1 vector-feature-night-mode-enabled skin-theme-clientpref-day vector-sticky-header-enabled vector-toc-available";var cookie=document.cookie.match(/(?:^|; )enwikimwclientpreferences=([^;]+)/);if(cookie){cookie[1].split('%2C').forEach(function(pref){className=className.replace(new RegExp('(^| )'+pref.replace(/-clientpref-\w+$|[^\w-]+/g,'')+'-clientpref-\\w+( |$)'),'$1'+pref+'$2');});}document.documentElement.className=className;}());RLCONF={"wgBreakFrames":false,"wgSeparatorTransformTable":["",""],"wgDigitTransformTable":["",""],"wgDefaultDateFormat":"dmy", "wgMonthNames":["","January","February","March","April","May","June","July","August","September","October","November","December"],"wgRequestId":"0334e230-528d-4ff8-8afb-a5cb7906f142","wgCanonicalNamespace":"","wgCanonicalSpecialPageName":false,"wgNamespaceNumber":0,"wgPageName":"Hooke's_law","wgTitle":"Hooke's law","wgCurRevisionId":1261498479,"wgRevisionId":1261498479,"wgArticleId":229553,"wgIsArticle":true,"wgIsRedirect":false,"wgAction":"view","wgUserName":null,"wgUserGroups":["*"],"wgCategories":["Articles containing Latin-language text","Articles with short description","Short description is different from Wikidata","Wikipedia indefinitely semi-protected pages","Articles lacking in-text citations from July 2017","All articles lacking in-text citations","Use dmy dates from October 2017","Pages using sidebar with the child parameter","Wikipedia articles needing page number citations from November 2016","1676 in science","Elasticity (physics)","Solid mechanics","Springs (mechanical)" ,"Structural analysis"],"wgPageViewLanguage":"en","wgPageContentLanguage":"en","wgPageContentModel":"wikitext","wgRelevantPageName":"Hooke's_law","wgRelevantArticleId":229553,"wgIsProbablyEditable":false,"wgRelevantPageIsProbablyEditable":false,"wgRestrictionEdit":["autoconfirmed"],"wgRestrictionMove":["autoconfirmed"],"wgNoticeProject":"wikipedia","wgCiteReferencePreviewsActive":false,"wgFlaggedRevsParams":{"tags":{"status":{"levels":1}}},"wgMediaViewerOnClick":true,"wgMediaViewerEnabledByDefault":true,"wgPopupsFlags":0,"wgVisualEditor":{"pageLanguageCode":"en","pageLanguageDir":"ltr","pageVariantFallbacks":"en"},"wgMFDisplayWikibaseDescriptions":{"search":true,"watchlist":true,"tagline":false,"nearby":true},"wgWMESchemaEditAttemptStepOversample":false,"wgWMEPageLength":60000,"wgEditSubmitButtonLabelPublish":true,"wgULSPosition":"interlanguage","wgULSisCompactLinksEnabled":false,"wgVector2022LanguageInHeader":true,"wgULSisLanguageSelectorEmpty":false,"wgWikibaseItemId":"Q170282", "wgCheckUserClientHintsHeadersJsApi":["brands","architecture","bitness","fullVersionList","mobile","model","platform","platformVersion"],"GEHomepageSuggestedEditsEnableTopics":true,"wgGETopicsMatchModeEnabled":false,"wgGEStructuredTaskRejectionReasonTextInputEnabled":false,"wgGELevelingUpEnabledForUser":false};RLSTATE={"ext.globalCssJs.user.styles":"ready","site.styles":"ready","user.styles":"ready","ext.globalCssJs.user":"ready","user":"ready","user.options":"loading","ext.math.styles":"ready","ext.cite.styles":"ready","skins.vector.search.codex.styles":"ready","skins.vector.styles":"ready","skins.vector.icons":"ready","jquery.makeCollapsible.styles":"ready","ext.wikimediamessages.styles":"ready","ext.visualEditor.desktopArticleTarget.noscript":"ready","ext.uls.interlanguage":"ready","wikibase.client.init":"ready","ext.wikimediaBadges":"ready"};RLPAGEMODULES=["ext.cite.ux-enhancements","ext.math.popup","mediawiki.page.media","site","mediawiki.page.ready","jquery.makeCollapsible", "mediawiki.toc","skins.vector.js","ext.centralNotice.geoIP","ext.centralNotice.startUp","ext.gadget.ReferenceTooltips","ext.gadget.switcher","ext.urlShortener.toolbar","ext.centralauth.centralautologin","mmv.bootstrap","ext.popups","ext.visualEditor.desktopArticleTarget.init","ext.visualEditor.targetLoader","ext.echo.centralauth","ext.eventLogging","ext.wikimediaEvents","ext.navigationTiming","ext.uls.interface","ext.cx.eventlogging.campaigns","ext.cx.uls.quick.actions","wikibase.client.vector-2022","ext.checkUser.clientHints","ext.growthExperiments.SuggestedEditSession"];</script> <script>(RLQ=window.RLQ||[]).push(function(){mw.loader.impl(function(){return["user.options@12s5i",function($,jQuery,require,module){mw.user.tokens.set({"patrolToken":"+\\","watchToken":"+\\","csrfToken":"+\\"}); }];});});</script> <link rel="stylesheet" href="/w/load.php?lang=en&modules=ext.cite.styles%7Cext.math.styles%7Cext.uls.interlanguage%7Cext.visualEditor.desktopArticleTarget.noscript%7Cext.wikimediaBadges%7Cext.wikimediamessages.styles%7Cjquery.makeCollapsible.styles%7Cskins.vector.icons%2Cstyles%7Cskins.vector.search.codex.styles%7Cwikibase.client.init&only=styles&skin=vector-2022"> <script async="" src="/w/load.php?lang=en&modules=startup&only=scripts&raw=1&skin=vector-2022"></script> <meta name="ResourceLoaderDynamicStyles" content=""> <link rel="stylesheet" href="/w/load.php?lang=en&modules=site.styles&only=styles&skin=vector-2022"> <meta name="generator" content="MediaWiki 1.44.0-wmf.15"> <meta name="referrer" content="origin"> <meta name="referrer" content="origin-when-cross-origin"> <meta name="robots" content="max-image-preview:standard"> <meta name="format-detection" content="telephone=no"> <meta property="og:image" content="https://upload.wikimedia.org/wikipedia/commons/thumb/f/fc/Hookes-law-springs.png/1200px-Hookes-law-springs.png"> <meta property="og:image:width" content="1200"> <meta property="og:image:height" content="1248"> <meta property="og:image" content="https://upload.wikimedia.org/wikipedia/commons/thumb/f/fc/Hookes-law-springs.png/800px-Hookes-law-springs.png"> <meta property="og:image:width" content="800"> <meta property="og:image:height" content="832"> <meta property="og:image" content="https://upload.wikimedia.org/wikipedia/commons/thumb/f/fc/Hookes-law-springs.png/640px-Hookes-law-springs.png"> <meta property="og:image:width" content="640"> <meta property="og:image:height" content="666"> <meta name="viewport" content="width=1120"> <meta property="og:title" content="Hooke's law - Wikipedia"> <meta property="og:type" content="website"> <link rel="preconnect" href="//upload.wikimedia.org"> <link rel="alternate" media="only screen and (max-width: 640px)" href="//en.m.wikipedia.org/wiki/Hooke%27s_law"> <link rel="apple-touch-icon" href="/static/apple-touch/wikipedia.png"> <link rel="icon" href="/static/favicon/wikipedia.ico"> <link rel="search" type="application/opensearchdescription+xml" href="/w/rest.php/v1/search" title="Wikipedia (en)"> <link rel="EditURI" type="application/rsd+xml" href="//en.wikipedia.org/w/api.php?action=rsd"> <link rel="canonical" href="https://en.wikipedia.org/wiki/Hooke%27s_law"> <link rel="license" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en"> <link rel="alternate" type="application/atom+xml" title="Wikipedia Atom feed" href="/w/index.php?title=Special:RecentChanges&feed=atom"> <link rel="dns-prefetch" href="//meta.wikimedia.org" /> <link rel="dns-prefetch" href="login.wikimedia.org"> </head> <body class="skin--responsive skin-vector skin-vector-search-vue mediawiki ltr sitedir-ltr mw-hide-empty-elt ns-0 ns-subject page-Hooke_s_law rootpage-Hooke_s_law skin-vector-2022 action-view"><a class="mw-jump-link" href="#bodyContent">Jump to content</a> <div class="vector-header-container"> <header class="vector-header mw-header"> <div class="vector-header-start"> <nav class="vector-main-menu-landmark" aria-label="Site"> <div id="vector-main-menu-dropdown" class="vector-dropdown vector-main-menu-dropdown vector-button-flush-left vector-button-flush-right" title="Main menu" > <input type="checkbox" id="vector-main-menu-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-main-menu-dropdown" class="vector-dropdown-checkbox " aria-label="Main menu" > <label id="vector-main-menu-dropdown-label" for="vector-main-menu-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-menu mw-ui-icon-wikimedia-menu"></span> <span class="vector-dropdown-label-text">Main menu</span> </label> <div class="vector-dropdown-content"> <div id="vector-main-menu-unpinned-container" class="vector-unpinned-container"> <div id="vector-main-menu" class="vector-main-menu vector-pinnable-element"> <div class="vector-pinnable-header vector-main-menu-pinnable-header vector-pinnable-header-unpinned" data-feature-name="main-menu-pinned" data-pinnable-element-id="vector-main-menu" data-pinned-container-id="vector-main-menu-pinned-container" data-unpinned-container-id="vector-main-menu-unpinned-container" > <div class="vector-pinnable-header-label">Main menu</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-main-menu.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-main-menu.unpin">hide</button> </div> <div id="p-navigation" class="vector-menu mw-portlet mw-portlet-navigation" > <div class="vector-menu-heading"> Navigation </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="n-mainpage-description" class="mw-list-item"><a href="/wiki/Main_Page" title="Visit the main page [z]" accesskey="z"><span>Main page</span></a></li><li id="n-contents" class="mw-list-item"><a href="/wiki/Wikipedia:Contents" title="Guides to browsing Wikipedia"><span>Contents</span></a></li><li id="n-currentevents" class="mw-list-item"><a href="/wiki/Portal:Current_events" title="Articles related to current events"><span>Current events</span></a></li><li id="n-randompage" class="mw-list-item"><a href="/wiki/Special:Random" title="Visit a randomly selected article [x]" accesskey="x"><span>Random article</span></a></li><li id="n-aboutsite" class="mw-list-item"><a href="/wiki/Wikipedia:About" title="Learn about Wikipedia and how it works"><span>About Wikipedia</span></a></li><li id="n-contactpage" class="mw-list-item"><a href="//en.wikipedia.org/wiki/Wikipedia:Contact_us" title="How to contact Wikipedia"><span>Contact us</span></a></li> </ul> </div> </div> <div id="p-interaction" class="vector-menu mw-portlet mw-portlet-interaction" > <div class="vector-menu-heading"> Contribute </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="n-help" class="mw-list-item"><a href="/wiki/Help:Contents" title="Guidance on how to use and edit Wikipedia"><span>Help</span></a></li><li id="n-introduction" class="mw-list-item"><a href="/wiki/Help:Introduction" title="Learn how to edit Wikipedia"><span>Learn to edit</span></a></li><li id="n-portal" class="mw-list-item"><a href="/wiki/Wikipedia:Community_portal" title="The hub for editors"><span>Community portal</span></a></li><li id="n-recentchanges" class="mw-list-item"><a href="/wiki/Special:RecentChanges" title="A list of recent changes to Wikipedia [r]" accesskey="r"><span>Recent changes</span></a></li><li id="n-upload" class="mw-list-item"><a href="/wiki/Wikipedia:File_upload_wizard" title="Add images or other media for use on Wikipedia"><span>Upload file</span></a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> <a href="/wiki/Main_Page" class="mw-logo"> <img class="mw-logo-icon" src="/static/images/icons/wikipedia.png" alt="" aria-hidden="true" height="50" width="50"> <span class="mw-logo-container skin-invert"> <img class="mw-logo-wordmark" alt="Wikipedia" src="/static/images/mobile/copyright/wikipedia-wordmark-en.svg" style="width: 7.5em; height: 1.125em;"> <img class="mw-logo-tagline" alt="The Free Encyclopedia" src="/static/images/mobile/copyright/wikipedia-tagline-en.svg" width="117" height="13" style="width: 7.3125em; height: 0.8125em;"> </span> </a> </div> <div class="vector-header-end"> <div id="p-search" role="search" class="vector-search-box-vue vector-search-box-collapses vector-search-box-show-thumbnail vector-search-box-auto-expand-width vector-search-box"> <a href="/wiki/Special:Search" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only search-toggle" title="Search Wikipedia [f]" accesskey="f"><span class="vector-icon mw-ui-icon-search mw-ui-icon-wikimedia-search"></span> <span>Search</span> </a> <div class="vector-typeahead-search-container"> <div class="cdx-typeahead-search cdx-typeahead-search--show-thumbnail cdx-typeahead-search--auto-expand-width"> <form action="/w/index.php" id="searchform" class="cdx-search-input cdx-search-input--has-end-button"> <div id="simpleSearch" class="cdx-search-input__input-wrapper" data-search-loc="header-moved"> <div class="cdx-text-input cdx-text-input--has-start-icon"> <input class="cdx-text-input__input" type="search" name="search" placeholder="Search Wikipedia" aria-label="Search Wikipedia" autocapitalize="sentences" title="Search Wikipedia [f]" accesskey="f" id="searchInput" > <span class="cdx-text-input__icon cdx-text-input__start-icon"></span> </div> <input type="hidden" name="title" value="Special:Search"> </div> <button class="cdx-button cdx-search-input__end-button">Search</button> </form> </div> </div> </div> <nav class="vector-user-links vector-user-links-wide" aria-label="Personal tools"> <div class="vector-user-links-main"> <div id="p-vector-user-menu-preferences" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <div id="p-vector-user-menu-userpage" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <nav class="vector-appearance-landmark" aria-label="Appearance"> <div id="vector-appearance-dropdown" class="vector-dropdown " title="Change the appearance of the page's font size, width, and color" > <input type="checkbox" id="vector-appearance-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-appearance-dropdown" class="vector-dropdown-checkbox " aria-label="Appearance" > <label id="vector-appearance-dropdown-label" for="vector-appearance-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-appearance mw-ui-icon-wikimedia-appearance"></span> <span class="vector-dropdown-label-text">Appearance</span> </label> <div class="vector-dropdown-content"> <div id="vector-appearance-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <div id="p-vector-user-menu-notifications" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <div id="p-vector-user-menu-overflow" class="vector-menu mw-portlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-sitesupport-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="https://donate.wikimedia.org/?wmf_source=donate&wmf_medium=sidebar&wmf_campaign=en.wikipedia.org&uselang=en" class=""><span>Donate</span></a> </li> <li id="pt-createaccount-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=Special:CreateAccount&returnto=Hooke%27s+law" title="You are encouraged to create an account and log in; however, it is not mandatory" class=""><span>Create account</span></a> </li> <li id="pt-login-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=Special:UserLogin&returnto=Hooke%27s+law" title="You're encouraged to log in; however, it's not mandatory. [o]" accesskey="o" class=""><span>Log in</span></a> </li> </ul> </div> </div> </div> <div id="vector-user-links-dropdown" class="vector-dropdown vector-user-menu vector-button-flush-right vector-user-menu-logged-out" title="Log in and more options" > <input type="checkbox" id="vector-user-links-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-user-links-dropdown" class="vector-dropdown-checkbox " aria-label="Personal tools" > <label id="vector-user-links-dropdown-label" for="vector-user-links-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-ellipsis mw-ui-icon-wikimedia-ellipsis"></span> <span class="vector-dropdown-label-text">Personal tools</span> </label> <div class="vector-dropdown-content"> <div id="p-personal" class="vector-menu mw-portlet mw-portlet-personal user-links-collapsible-item" title="User menu" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-sitesupport" class="user-links-collapsible-item mw-list-item"><a href="https://donate.wikimedia.org/?wmf_source=donate&wmf_medium=sidebar&wmf_campaign=en.wikipedia.org&uselang=en"><span>Donate</span></a></li><li id="pt-createaccount" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Special:CreateAccount&returnto=Hooke%27s+law" title="You are encouraged to create an account and log in; however, it is not mandatory"><span class="vector-icon mw-ui-icon-userAdd mw-ui-icon-wikimedia-userAdd"></span> <span>Create account</span></a></li><li id="pt-login" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Special:UserLogin&returnto=Hooke%27s+law" title="You're encouraged to log in; however, it's not mandatory. [o]" accesskey="o"><span class="vector-icon mw-ui-icon-logIn mw-ui-icon-wikimedia-logIn"></span> <span>Log in</span></a></li> </ul> </div> </div> <div id="p-user-menu-anon-editor" class="vector-menu mw-portlet mw-portlet-user-menu-anon-editor" > <div class="vector-menu-heading"> Pages for logged out editors <a href="/wiki/Help:Introduction" aria-label="Learn more about editing"><span>learn more</span></a> </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-anoncontribs" class="mw-list-item"><a href="/wiki/Special:MyContributions" title="A list of edits made from this IP address [y]" accesskey="y"><span>Contributions</span></a></li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/Special:MyTalk" title="Discussion about edits from this IP address [n]" accesskey="n"><span>Talk</span></a></li> </ul> </div> </div> </div> </div> </nav> </div> </header> </div> <div class="mw-page-container"> <div class="mw-page-container-inner"> <div class="vector-sitenotice-container"> <div id="siteNotice"><!-- CentralNotice --></div> </div> <div class="vector-column-start"> <div class="vector-main-menu-container"> <div id="mw-navigation"> <nav id="mw-panel" class="vector-main-menu-landmark" aria-label="Site"> <div id="vector-main-menu-pinned-container" class="vector-pinned-container"> </div> </nav> </div> </div> <div class="vector-sticky-pinned-container"> <nav id="mw-panel-toc" aria-label="Contents" data-event-name="ui.sidebar-toc" class="mw-table-of-contents-container vector-toc-landmark"> <div id="vector-toc-pinned-container" class="vector-pinned-container"> <div id="vector-toc" class="vector-toc vector-pinnable-element"> <div class="vector-pinnable-header vector-toc-pinnable-header vector-pinnable-header-pinned" data-feature-name="toc-pinned" data-pinnable-element-id="vector-toc" > <h2 class="vector-pinnable-header-label">Contents</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">hide</button> </div> <ul class="vector-toc-contents" id="mw-panel-toc-list"> <li id="toc-mw-content-text" class="vector-toc-list-item vector-toc-level-1"> <a href="#" class="vector-toc-link"> <div class="vector-toc-text">(Top)</div> </a> </li> <li id="toc-Formal_definition" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Formal_definition"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Formal definition</span> </div> </a> <button aria-controls="toc-Formal_definition-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Formal definition subsection</span> </button> <ul id="toc-Formal_definition-sublist" class="vector-toc-list"> <li id="toc-Linear_springs" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Linear_springs"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>Linear springs</span> </div> </a> <ul id="toc-Linear_springs-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Torsional_springs" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Torsional_springs"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.2</span> <span>Torsional springs</span> </div> </a> <ul id="toc-Torsional_springs-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-General_"scalar"_springs" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#General_"scalar"_springs"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.3</span> <span>General "scalar" springs</span> </div> </a> <ul id="toc-General_"scalar"_springs-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Vector_formulation" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Vector_formulation"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.4</span> <span>Vector formulation</span> </div> </a> <ul id="toc-Vector_formulation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-General_tensor_form" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#General_tensor_form"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.5</span> <span>General tensor form</span> </div> </a> <ul id="toc-General_tensor_form-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Hooke's_law_for_continuous_media" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Hooke's_law_for_continuous_media"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.6</span> <span>Hooke's law for continuous media</span> </div> </a> <ul id="toc-Hooke's_law_for_continuous_media-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Analogous_laws" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Analogous_laws"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Analogous laws</span> </div> </a> <ul id="toc-Analogous_laws-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Units_of_measurement" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Units_of_measurement"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Units of measurement</span> </div> </a> <ul id="toc-Units_of_measurement-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-General_application_to_elastic_materials" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#General_application_to_elastic_materials"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>General application to elastic materials</span> </div> </a> <ul id="toc-General_application_to_elastic_materials-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Derived_formulae" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Derived_formulae"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Derived formulae</span> </div> </a> <button aria-controls="toc-Derived_formulae-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Derived formulae subsection</span> </button> <ul id="toc-Derived_formulae-sublist" class="vector-toc-list"> <li id="toc-Tensional_stress_of_a_uniform_bar" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Tensional_stress_of_a_uniform_bar"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1</span> <span>Tensional stress of a uniform bar</span> </div> </a> <ul id="toc-Tensional_stress_of_a_uniform_bar-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Spring_energy" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Spring_energy"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.2</span> <span>Spring energy</span> </div> </a> <ul id="toc-Spring_energy-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Relaxed_force_constants_(generalized_compliance_constants)" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Relaxed_force_constants_(generalized_compliance_constants)"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.3</span> <span>Relaxed force constants (generalized compliance constants)</span> </div> </a> <ul id="toc-Relaxed_force_constants_(generalized_compliance_constants)-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Harmonic_oscillator" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Harmonic_oscillator"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.4</span> <span>Harmonic oscillator</span> </div> </a> <ul id="toc-Harmonic_oscillator-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Rotation_in_gravity-free_space" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Rotation_in_gravity-free_space"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.5</span> <span>Rotation in gravity-free space</span> </div> </a> <ul id="toc-Rotation_in_gravity-free_space-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Linear_elasticity_theory_for_continuous_media" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Linear_elasticity_theory_for_continuous_media"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Linear elasticity theory for continuous media</span> </div> </a> <button aria-controls="toc-Linear_elasticity_theory_for_continuous_media-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Linear elasticity theory for continuous media subsection</span> </button> <ul id="toc-Linear_elasticity_theory_for_continuous_media-sublist" class="vector-toc-list"> <li id="toc-Isotropic_materials" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Isotropic_materials"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.1</span> <span>Isotropic materials</span> </div> </a> <ul id="toc-Isotropic_materials-sublist" class="vector-toc-list"> <li id="toc-Plane_stress" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Plane_stress"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.1.1</span> <span>Plane stress</span> </div> </a> <ul id="toc-Plane_stress-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Plane_strain" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Plane_strain"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.1.2</span> <span>Plane strain</span> </div> </a> <ul id="toc-Plane_strain-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Anisotropic_materials" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Anisotropic_materials"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.2</span> <span>Anisotropic materials</span> </div> </a> <ul id="toc-Anisotropic_materials-sublist" class="vector-toc-list"> <li id="toc-Matrix_representation_(stiffness_tensor)" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Matrix_representation_(stiffness_tensor)"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.2.1</span> <span>Matrix representation (stiffness tensor)</span> </div> </a> <ul id="toc-Matrix_representation_(stiffness_tensor)-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Change_of_coordinate_system" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Change_of_coordinate_system"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.2.2</span> <span>Change of coordinate system</span> </div> </a> <ul id="toc-Change_of_coordinate_system-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Orthotropic_materials" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Orthotropic_materials"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.2.3</span> <span>Orthotropic materials</span> </div> </a> <ul id="toc-Orthotropic_materials-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Transversely_isotropic_materials" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Transversely_isotropic_materials"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.2.4</span> <span>Transversely isotropic materials</span> </div> </a> <ul id="toc-Transversely_isotropic_materials-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Universal_elastic_anisotropy_index" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Universal_elastic_anisotropy_index"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.2.5</span> <span>Universal elastic anisotropy index</span> </div> </a> <ul id="toc-Universal_elastic_anisotropy_index-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </li> <li id="toc-Thermodynamic_basis" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Thermodynamic_basis"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Thermodynamic basis</span> </div> </a> <ul id="toc-Thermodynamic_basis-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>Notes</span> </div> </a> <ul id="toc-Notes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">11</span> <span>External links</span> </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" title="Table of Contents" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Toggle the table of contents</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Hooke's law</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. Available in 69 languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-69" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">69 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-als mw-list-item"><a href="https://als.wikipedia.org/wiki/Hookesches_Gesetz" title="Hookesches Gesetz – Alemannic" lang="gsw" hreflang="gsw" data-title="Hookesches Gesetz" data-language-autonym="Alemannisch" data-language-local-name="Alemannic" class="interlanguage-link-target"><span>Alemannisch</span></a></li><li class="interlanguage-link interwiki-am mw-list-item"><a href="https://am.wikipedia.org/wiki/%E1%8B%A8%E1%88%81%E1%8A%AD_%E1%88%85%E1%8C%8D" title="የሁክ ህግ – Amharic" lang="am" hreflang="am" data-title="የሁክ ህግ" data-language-autonym="አማርኛ" data-language-local-name="Amharic" class="interlanguage-link-target"><span>አማርኛ</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%82%D8%A7%D9%86%D9%88%D9%86_%D9%87%D9%88%D9%83" title="قانون هوك – Arabic" lang="ar" hreflang="ar" data-title="قانون هوك" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Llei_d%27elasticid%C3%A1_de_Hooke" title="Llei d'elasticidá de Hooke – Asturian" lang="ast" hreflang="ast" data-title="Llei d'elasticidá de Hooke" data-language-autonym="Asturianu" data-language-local-name="Asturian" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/Huk_qanunu" title="Huk qanunu – Azerbaijani" lang="az" hreflang="az" data-title="Huk qanunu" data-language-autonym="Azərbaycanca" data-language-local-name="Azerbaijani" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%B9%E0%A7%81%E0%A6%95%E0%A7%87%E0%A6%B0_%E0%A6%B8%E0%A7%82%E0%A6%A4%E0%A7%8D%E0%A6%B0" title="হুকের সূত্র – Bangla" lang="bn" hreflang="bn" data-title="হুকের সূত্র" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%97%D0%B0%D0%BA%D0%BE%D0%BD_%D0%93%D1%83%D0%BA%D0%B0" title="Закон Гука – Belarusian" lang="be" hreflang="be" data-title="Закон Гука" data-language-autonym="Беларуская" data-language-local-name="Belarusian" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%97%D0%B0%D0%BA%D0%BE%D0%BD_%D0%BD%D0%B0_%D0%A5%D1%83%D0%BA" title="Закон на Хук – Bulgarian" lang="bg" hreflang="bg" data-title="Закон на Хук" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Hookeov_zakon" title="Hookeov zakon – Bosnian" lang="bs" hreflang="bs" data-title="Hookeov zakon" data-language-autonym="Bosanski" data-language-local-name="Bosnian" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Llei_de_Hooke" title="Llei de Hooke – Catalan" lang="ca" hreflang="ca" data-title="Llei de Hooke" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%93%D1%83%D0%BA_%D1%81%D0%B0%D0%BA%D0%BA%D1%83%D0%BD%C4%95" title="Гук саккунĕ – Chuvash" lang="cv" hreflang="cv" data-title="Гук саккунĕ" data-language-autonym="Чӑвашла" data-language-local-name="Chuvash" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Hook%C5%AFv_z%C3%A1kon" title="Hookův zákon – Czech" lang="cs" hreflang="cs" data-title="Hookův zákon" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Hookes_lov" title="Hookes lov – Danish" lang="da" hreflang="da" data-title="Hookes lov" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Hookesches_Gesetz" title="Hookesches Gesetz – German" lang="de" hreflang="de" data-title="Hookesches Gesetz" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Hooke%27i_seadus" title="Hooke'i seadus – Estonian" lang="et" hreflang="et" data-title="Hooke'i seadus" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%9D%CF%8C%CE%BC%CE%BF%CF%82_%CF%84%CE%BF%CF%85_%CE%A7%CE%BF%CF%85%CE%BA" title="Νόμος του Χουκ – Greek" lang="el" hreflang="el" data-title="Νόμος του Χουκ" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Ley_de_elasticidad_de_Hooke" title="Ley de elasticidad de Hooke – Spanish" lang="es" hreflang="es" data-title="Ley de elasticidad de Hooke" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Le%C4%9Do_de_Hooke" title="Leĝo de Hooke – Esperanto" lang="eo" hreflang="eo" data-title="Leĝo de Hooke" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Hookeren_elastikotasun_legea" title="Hookeren elastikotasun legea – Basque" lang="eu" hreflang="eu" data-title="Hookeren elastikotasun legea" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%82%D8%A7%D9%86%D9%88%D9%86_%D9%87%D9%88%DA%A9" title="قانون هوک – Persian" lang="fa" hreflang="fa" data-title="قانون هوک" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Loi_de_Hooke" title="Loi de Hooke – French" lang="fr" hreflang="fr" data-title="Loi de Hooke" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/Dl%C3%AD_Hooke" title="Dlí Hooke – Irish" lang="ga" hreflang="ga" data-title="Dlí Hooke" data-language-autonym="Gaeilge" data-language-local-name="Irish" class="interlanguage-link-target"><span>Gaeilge</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Lei_de_Hooke" title="Lei de Hooke – Galician" lang="gl" hreflang="gl" data-title="Lei de Hooke" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%ED%9B%85%EC%9D%98_%EB%B2%95%EC%B9%99" title="훅의 법칙 – Korean" lang="ko" hreflang="ko" data-title="훅의 법칙" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D5%80%D5%B8%D6%82%D5%AF%D5%AB_%D6%85%D6%80%D5%A5%D5%B6%D6%84" title="Հուկի օրենք – Armenian" lang="hy" hreflang="hy" data-title="Հուկի օրենք" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%B9%E0%A5%81%E0%A4%95_%E0%A4%95%E0%A4%BE_%E0%A4%A8%E0%A4%BF%E0%A4%AF%E0%A4%AE" title="हुक का नियम – Hindi" lang="hi" hreflang="hi" data-title="हुक का नियम" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Hookeov_zakon" title="Hookeov zakon – Croatian" lang="hr" hreflang="hr" data-title="Hookeov zakon" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Hukum_Hooke" title="Hukum Hooke – Indonesian" lang="id" hreflang="id" data-title="Hukum Hooke" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Legge_di_Hooke" title="Legge di Hooke – Italian" lang="it" hreflang="it" data-title="Legge di Hooke" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%97%D7%95%D7%A7_%D7%94%D7%95%D7%A7" title="חוק הוק – Hebrew" lang="he" hreflang="he" data-title="חוק הוק" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-kn mw-list-item"><a href="https://kn.wikipedia.org/wiki/%E0%B2%B9%E0%B3%81%E0%B2%95%E0%B3%8D%E2%80%8C%E0%B2%A8_%E0%B2%A8%E0%B2%BF%E0%B2%AF%E0%B2%AE" title="ಹುಕ್ನ ನಿಯಮ – Kannada" lang="kn" hreflang="kn" data-title="ಹುಕ್ನ ನಿಯಮ" data-language-autonym="ಕನ್ನಡ" data-language-local-name="Kannada" class="interlanguage-link-target"><span>ಕನ್ನಡ</span></a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%B0%E1%83%A3%E1%83%99%E1%83%98%E1%83%A1_%E1%83%99%E1%83%90%E1%83%9C%E1%83%9D%E1%83%9C%E1%83%98" title="ჰუკის კანონი – Georgian" lang="ka" hreflang="ka" data-title="ჰუკის კანონი" data-language-autonym="ქართული" data-language-local-name="Georgian" class="interlanguage-link-target"><span>ქართული</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%93%D1%83%D0%BA_%D0%B7%D0%B0%D2%A3%D1%8B" title="Гук заңы – Kazakh" lang="kk" hreflang="kk" data-title="Гук заңы" data-language-autonym="Қазақша" data-language-local-name="Kazakh" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Lex_Hookiana" title="Lex Hookiana – Latin" lang="la" hreflang="la" data-title="Lex Hookiana" data-language-autonym="Latina" data-language-local-name="Latin" class="interlanguage-link-target"><span>Latina</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Huka_likums" title="Huka likums – Latvian" lang="lv" hreflang="lv" data-title="Huka likums" data-language-autonym="Latviešu" data-language-local-name="Latvian" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Huko_d%C4%97snis" title="Huko dėsnis – Lithuanian" lang="lt" hreflang="lt" data-title="Huko dėsnis" data-language-autonym="Lietuvių" data-language-local-name="Lithuanian" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Hooke-t%C3%B6rv%C3%A9ny" title="Hooke-törvény – Hungarian" lang="hu" hreflang="hu" data-title="Hooke-törvény" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%A5%D1%83%D0%BA%D0%BE%D0%B2_%D0%B7%D0%B0%D0%BA%D0%BE%D0%BD" title="Хуков закон – Macedonian" lang="mk" hreflang="mk" data-title="Хуков закон" data-language-autonym="Македонски" data-language-local-name="Macedonian" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-mt mw-list-item"><a href="https://mt.wikipedia.org/wiki/Li%C4%A1i_ta%27_Hooke" title="Liġi ta' Hooke – Maltese" lang="mt" hreflang="mt" data-title="Liġi ta' Hooke" data-language-autonym="Malti" data-language-local-name="Maltese" class="interlanguage-link-target"><span>Malti</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Hukum_Hooke" title="Hukum Hooke – Malay" lang="ms" hreflang="ms" data-title="Hukum Hooke" data-language-autonym="Bahasa Melayu" data-language-local-name="Malay" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-mn mw-list-item"><a href="https://mn.wikipedia.org/wiki/%D0%93%D1%83%D0%BA%D0%B8%D0%B9%D0%BD_%D1%85%D1%83%D1%83%D0%BB%D1%8C" title="Гукийн хууль – Mongolian" lang="mn" hreflang="mn" data-title="Гукийн хууль" data-language-autonym="Монгол" data-language-local-name="Mongolian" class="interlanguage-link-target"><span>Монгол</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Wet_van_Hooke" title="Wet van Hooke – Dutch" lang="nl" hreflang="nl" data-title="Wet van Hooke" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E3%83%95%E3%83%83%E3%82%AF%E3%81%AE%E6%B3%95%E5%89%87" title="フックの法則 – Japanese" lang="ja" hreflang="ja" data-title="フックの法則" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Hookes_lov" title="Hookes lov – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Hookes lov" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Hooke-lova" title="Hooke-lova – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Hooke-lova" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-oc mw-list-item"><a href="https://oc.wikipedia.org/wiki/L%C3%A8i_de_Hooke" title="Lèi de Hooke – Occitan" lang="oc" hreflang="oc" data-title="Lèi de Hooke" data-language-autonym="Occitan" data-language-local-name="Occitan" class="interlanguage-link-target"><span>Occitan</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Hooke_qonuni" title="Hooke qonuni – Uzbek" lang="uz" hreflang="uz" data-title="Hooke qonuni" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="Uzbek" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-km mw-list-item"><a href="https://km.wikipedia.org/wiki/%E1%9E%85%E1%9F%92%E1%9E%94%E1%9E%B6%E1%9E%94%E1%9F%8B%E1%9E%A0%E1%9F%8A%E1%9E%BC%E1%9E%80" title="ច្បាប់ហ៊ូក – Khmer" lang="km" hreflang="km" data-title="ច្បាប់ហ៊ូក" data-language-autonym="ភាសាខ្មែរ" data-language-local-name="Khmer" class="interlanguage-link-target"><span>ភាសាខ្មែរ</span></a></li><li class="interlanguage-link interwiki-pms mw-list-item"><a href="https://pms.wikipedia.org/wiki/Lej_%C3%ABd_Hooke" title="Lej ëd Hooke – Piedmontese" lang="pms" hreflang="pms" data-title="Lej ëd Hooke" data-language-autonym="Piemontèis" data-language-local-name="Piedmontese" class="interlanguage-link-target"><span>Piemontèis</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Prawo_Hooke%E2%80%99a" title="Prawo Hooke’a – Polish" lang="pl" hreflang="pl" data-title="Prawo Hooke’a" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Lei_de_Hooke" title="Lei de Hooke – Portuguese" lang="pt" hreflang="pt" data-title="Lei de Hooke" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Legea_lui_Hooke" title="Legea lui Hooke – Romanian" lang="ro" hreflang="ro" data-title="Legea lui Hooke" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%97%D0%B0%D0%BA%D0%BE%D0%BD_%D0%93%D1%83%D0%BA%D0%B0" title="Закон Гука – Russian" lang="ru" hreflang="ru" data-title="Закон Гука" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-si mw-list-item"><a href="https://si.wikipedia.org/wiki/%E0%B7%84%E0%B7%94%E0%B6%9A%E0%B7%8A%E0%B6%9C%E0%B7%9A_%E0%B6%B1%E0%B7%92%E0%B6%BA%E0%B6%B8%E0%B6%BA" title="හුක්ගේ නියමය – Sinhala" lang="si" hreflang="si" data-title="හුක්ගේ නියමය" data-language-autonym="සිංහල" data-language-local-name="Sinhala" class="interlanguage-link-target"><span>සිංහල</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Hooke%27s_law" title="Hooke's law – Simple English" lang="en-simple" hreflang="en-simple" data-title="Hooke's law" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Hookov_z%C3%A1kon" title="Hookov zákon – Slovak" lang="sk" hreflang="sk" data-title="Hookov zákon" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Hookov_zakon" title="Hookov zakon – Slovenian" lang="sl" hreflang="sl" data-title="Hookov zakon" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%A5%D1%83%D0%BA%D0%BE%D0%B2_%D0%B7%D0%B0%D0%BA%D0%BE%D0%BD" title="Хуков закон – Serbian" lang="sr" hreflang="sr" data-title="Хуков закон" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Hookeov_zakon" title="Hookeov zakon – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Hookeov zakon" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="Serbo-Croatian" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Hooken_laki" title="Hooken laki – Finnish" lang="fi" hreflang="fi" data-title="Hooken laki" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Hookes_lag" title="Hookes lag – Swedish" lang="sv" hreflang="sv" data-title="Hookes lag" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%8A%E0%AE%95%E0%AF%8D%E0%AE%95%E0%AE%BF%E0%AE%A9%E0%AF%8D_%E0%AE%B5%E0%AE%BF%E0%AE%A4%E0%AE%BF" title="ஊக்கின் விதி – Tamil" lang="ta" hreflang="ta" data-title="ஊக்கின் விதி" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%81%E0%B8%8E%E0%B8%82%E0%B8%AD%E0%B8%87%E0%B8%AE%E0%B8%B8%E0%B8%81" title="กฎของฮุก – Thai" lang="th" hreflang="th" data-title="กฎของฮุก" data-language-autonym="ไทย" data-language-local-name="Thai" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Hooke_yasas%C4%B1" title="Hooke yasası – Turkish" lang="tr" hreflang="tr" data-title="Hooke yasası" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%97%D0%B0%D0%BA%D0%BE%D0%BD_%D0%93%D1%83%D0%BA%D0%B0" title="Закон Гука – Ukrainian" lang="uk" hreflang="uk" data-title="Закон Гука" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/%C4%90%E1%BB%8Bnh_lu%E1%BA%ADt_Hooke" title="Định luật Hooke – Vietnamese" lang="vi" hreflang="vi" data-title="Định luật Hooke" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E8%83%A1%E5%85%8B%E5%AE%9A%E5%BE%8B" title="胡克定律 – Wu" lang="wuu" hreflang="wuu" data-title="胡克定律" data-language-autonym="吴语" data-language-local-name="Wu" class="interlanguage-link-target"><span>吴语</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E8%83%A1%E5%85%8B%E5%AE%9A%E5%BE%8B" title="胡克定律 – Cantonese" lang="yue" hreflang="yue" data-title="胡克定律" data-language-autonym="粵語" data-language-local-name="Cantonese" class="interlanguage-link-target"><span>粵語</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E8%83%A1%E5%85%8B%E5%AE%9A%E5%BE%8B" title="胡克定律 – Chinese" lang="zh" hreflang="zh" data-title="胡克定律" data-language-autonym="中文" data-language-local-name="Chinese" class="interlanguage-link-target"><span>中文</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q170282#sitelinks-wikipedia" title="Edit interlanguage links" class="wbc-editpage">Edit links</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="Namespaces"> <div id="p-associated-pages" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-associated-pages" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-nstab-main" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/Hooke%27s_law" title="View the content page [c]" accesskey="c"><span>Article</span></a></li><li id="ca-talk" class="vector-tab-noicon mw-list-item"><a href="/wiki/Talk:Hooke%27s_law" rel="discussion" title="Discuss improvements to the content page [t]" accesskey="t"><span>Talk</span></a></li> </ul> </div> </div> <div id="vector-variants-dropdown" class="vector-dropdown emptyPortlet" > <input type="checkbox" id="vector-variants-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-variants-dropdown" class="vector-dropdown-checkbox " aria-label="Change language variant" > <label id="vector-variants-dropdown-label" for="vector-variants-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span class="vector-dropdown-label-text">English</span> </label> <div class="vector-dropdown-content"> <div id="p-variants" class="vector-menu mw-portlet mw-portlet-variants emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> </div> </div> </nav> </div> <div id="right-navigation" class="vector-collapsible"> <nav aria-label="Views"> <div id="p-views" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-views" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-view" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/Hooke%27s_law"><span>Read</span></a></li><li id="ca-viewsource" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Hooke%27s_law&action=edit" title="This page is protected. You can view its source [e]" accesskey="e"><span>View source</span></a></li><li id="ca-history" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Hooke%27s_law&action=history" title="Past revisions of this page [h]" accesskey="h"><span>View history</span></a></li> </ul> </div> </div> </nav> <nav class="vector-page-tools-landmark" aria-label="Page tools"> <div id="vector-page-tools-dropdown" class="vector-dropdown vector-page-tools-dropdown" > <input type="checkbox" id="vector-page-tools-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-tools-dropdown" class="vector-dropdown-checkbox " aria-label="Tools" > <label id="vector-page-tools-dropdown-label" for="vector-page-tools-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span class="vector-dropdown-label-text">Tools</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-tools-unpinned-container" class="vector-unpinned-container"> <div id="vector-page-tools" class="vector-page-tools vector-pinnable-element"> <div class="vector-pinnable-header vector-page-tools-pinnable-header vector-pinnable-header-unpinned" data-feature-name="page-tools-pinned" data-pinnable-element-id="vector-page-tools" data-pinned-container-id="vector-page-tools-pinned-container" data-unpinned-container-id="vector-page-tools-unpinned-container" > <div class="vector-pinnable-header-label">Tools</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-page-tools.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-page-tools.unpin">hide</button> </div> <div id="p-cactions" class="vector-menu mw-portlet mw-portlet-cactions emptyPortlet vector-has-collapsible-items" title="More options" > <div class="vector-menu-heading"> Actions </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-more-view" class="selected vector-more-collapsible-item mw-list-item"><a href="/wiki/Hooke%27s_law"><span>Read</span></a></li><li id="ca-more-viewsource" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Hooke%27s_law&action=edit"><span>View source</span></a></li><li id="ca-more-history" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Hooke%27s_law&action=history"><span>View history</span></a></li> </ul> </div> </div> <div id="p-tb" class="vector-menu mw-portlet mw-portlet-tb" > <div class="vector-menu-heading"> General </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-whatlinkshere" class="mw-list-item"><a href="/wiki/Special:WhatLinksHere/Hooke%27s_law" title="List of all English Wikipedia pages containing links to this page [j]" accesskey="j"><span>What links here</span></a></li><li id="t-recentchangeslinked" class="mw-list-item"><a href="/wiki/Special:RecentChangesLinked/Hooke%27s_law" rel="nofollow" title="Recent changes in pages linked from this page [k]" accesskey="k"><span>Related changes</span></a></li><li id="t-upload" class="mw-list-item"><a href="//en.wikipedia.org/wiki/Wikipedia:File_Upload_Wizard" title="Upload files [u]" accesskey="u"><span>Upload file</span></a></li><li id="t-specialpages" class="mw-list-item"><a href="/wiki/Special:SpecialPages" title="A list of all special pages [q]" accesskey="q"><span>Special pages</span></a></li><li id="t-permalink" class="mw-list-item"><a href="/w/index.php?title=Hooke%27s_law&oldid=1261498479" title="Permanent link to this revision of this page"><span>Permanent link</span></a></li><li id="t-info" class="mw-list-item"><a href="/w/index.php?title=Hooke%27s_law&action=info" title="More information about this page"><span>Page information</span></a></li><li id="t-cite" class="mw-list-item"><a href="/w/index.php?title=Special:CiteThisPage&page=Hooke%27s_law&id=1261498479&wpFormIdentifier=titleform" title="Information on how to cite this page"><span>Cite this page</span></a></li><li id="t-urlshortener" class="mw-list-item"><a href="/w/index.php?title=Special:UrlShortener&url=https%3A%2F%2Fen.wikipedia.org%2Fwiki%2FHooke%2527s_law"><span>Get shortened URL</span></a></li><li id="t-urlshortener-qrcode" class="mw-list-item"><a href="/w/index.php?title=Special:QrCode&url=https%3A%2F%2Fen.wikipedia.org%2Fwiki%2FHooke%2527s_law"><span>Download QR code</span></a></li> </ul> </div> </div> <div id="p-coll-print_export" class="vector-menu mw-portlet mw-portlet-coll-print_export" > <div class="vector-menu-heading"> Print/export </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="coll-download-as-rl" class="mw-list-item"><a href="/w/index.php?title=Special:DownloadAsPdf&page=Hooke%27s_law&action=show-download-screen" title="Download this page as a PDF file"><span>Download as PDF</span></a></li><li id="t-print" class="mw-list-item"><a href="/w/index.php?title=Hooke%27s_law&printable=yes" title="Printable version of this page [p]" accesskey="p"><span>Printable version</span></a></li> </ul> </div> </div> <div id="p-wikibase-otherprojects" class="vector-menu mw-portlet mw-portlet-wikibase-otherprojects" > <div class="vector-menu-heading"> In other projects </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="wb-otherproject-link wb-otherproject-commons mw-list-item"><a href="https://commons.wikimedia.org/wiki/Category:Hooke%27s_law" hreflang="en"><span>Wikimedia Commons</span></a></li><li id="t-wikibase" class="wb-otherproject-link wb-otherproject-wikibase-dataitem mw-list-item"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q170282" title="Structured data on this page hosted by Wikidata [g]" accesskey="g"><span>Wikidata item</span></a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> </div> </div> </div> <div class="vector-column-end"> <div class="vector-sticky-pinned-container"> <nav class="vector-page-tools-landmark" aria-label="Page tools"> <div id="vector-page-tools-pinned-container" class="vector-pinned-container"> </div> </nav> <nav class="vector-appearance-landmark" aria-label="Appearance"> <div id="vector-appearance-pinned-container" class="vector-pinned-container"> <div id="vector-appearance" class="vector-appearance vector-pinnable-element"> <div class="vector-pinnable-header vector-appearance-pinnable-header vector-pinnable-header-pinned" data-feature-name="appearance-pinned" data-pinnable-element-id="vector-appearance" data-pinned-container-id="vector-appearance-pinned-container" data-unpinned-container-id="vector-appearance-unpinned-container" > <div class="vector-pinnable-header-label">Appearance</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">hide</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> <div id="mw-indicator-pp-default" class="mw-indicator"><div class="mw-parser-output"><span typeof="mw:File"><a href="/wiki/Wikipedia:Protection_policy#semi" title="This article is semi-protected."><img alt="Page semi-protected" src="//upload.wikimedia.org/wikipedia/en/thumb/1/1b/Semi-protection-shackle.svg/20px-Semi-protection-shackle.svg.png" decoding="async" width="20" height="20" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/1/1b/Semi-protection-shackle.svg/30px-Semi-protection-shackle.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/1/1b/Semi-protection-shackle.svg/40px-Semi-protection-shackle.svg.png 2x" data-file-width="512" data-file-height="512" /></a></span></div></div> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Physical law: force needed to deform a spring scales linearly with distance</div> <p class="mw-empty-elt"> </p> <style data-mw-deduplicate="TemplateStyles:r1251242444">.mw-parser-output .ambox{border:1px solid #a2a9b1;border-left:10px solid #36c;background-color:#fbfbfb;box-sizing:border-box}.mw-parser-output .ambox+link+.ambox,.mw-parser-output .ambox+link+style+.ambox,.mw-parser-output .ambox+link+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+style+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+link+.ambox{margin-top:-1px}html body.mediawiki .mw-parser-output .ambox.mbox-small-left{margin:4px 1em 4px 0;overflow:hidden;width:238px;border-collapse:collapse;font-size:88%;line-height:1.25em}.mw-parser-output .ambox-speedy{border-left:10px solid #b32424;background-color:#fee7e6}.mw-parser-output .ambox-delete{border-left:10px solid #b32424}.mw-parser-output .ambox-content{border-left:10px solid #f28500}.mw-parser-output .ambox-style{border-left:10px solid #fc3}.mw-parser-output .ambox-move{border-left:10px solid #9932cc}.mw-parser-output .ambox-protection{border-left:10px solid #a2a9b1}.mw-parser-output .ambox .mbox-text{border:none;padding:0.25em 0.5em;width:100%}.mw-parser-output .ambox .mbox-image{border:none;padding:2px 0 2px 0.5em;text-align:center}.mw-parser-output .ambox .mbox-imageright{border:none;padding:2px 0.5em 2px 0;text-align:center}.mw-parser-output .ambox .mbox-empty-cell{border:none;padding:0;width:1px}.mw-parser-output .ambox .mbox-image-div{width:52px}@media(min-width:720px){.mw-parser-output .ambox{margin:0 10%}}@media print{body.ns-0 .mw-parser-output .ambox{display:none!important}}</style><table class="box-More_footnotes_needed plainlinks metadata ambox ambox-style ambox-More_footnotes_needed" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a4/Text_document_with_red_question_mark.svg/40px-Text_document_with_red_question_mark.svg.png" decoding="async" width="40" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a4/Text_document_with_red_question_mark.svg/60px-Text_document_with_red_question_mark.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a4/Text_document_with_red_question_mark.svg/80px-Text_document_with_red_question_mark.svg.png 2x" data-file-width="48" data-file-height="48" /></span></span></div></td><td class="mbox-text"><div class="mbox-text-span">This article includes a list of <a href="/wiki/Wikipedia:Citing_sources#General_references" title="Wikipedia:Citing sources">general references</a>, but <b>it lacks sufficient corresponding <a href="/wiki/Wikipedia:Citing_sources#Inline_citations" title="Wikipedia:Citing sources">inline citations</a></b>.<span class="hide-when-compact"> Please help to <a href="/wiki/Wikipedia:WikiProject_Reliability" title="Wikipedia:WikiProject Reliability">improve</a> this article by <a href="/wiki/Wikipedia:When_to_cite" title="Wikipedia:When to cite">introducing</a> more precise citations.</span> <span class="date-container"><i>(<span class="date">July 2017</span>)</i></span><span class="hide-when-compact"><i> (<small><a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span></div></td></tr></tbody></table> <p class="mw-empty-elt"> </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Hookes-law-springs.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fc/Hookes-law-springs.png/220px-Hookes-law-springs.png" decoding="async" width="220" height="229" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fc/Hookes-law-springs.png/330px-Hookes-law-springs.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fc/Hookes-law-springs.png/440px-Hookes-law-springs.png 2x" data-file-width="1269" data-file-height="1320" /></a><figcaption>Hooke's law: the force is proportional to the extension</figcaption></figure> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Manometer_anim_02.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/10/Manometer_anim_02.gif/220px-Manometer_anim_02.gif" decoding="async" width="220" height="225" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/1/10/Manometer_anim_02.gif 1.5x" data-file-width="277" data-file-height="283" /></a><figcaption><a href="/wiki/Bourdon_tube" class="mw-redirect" title="Bourdon tube">Bourdon tubes</a> are based on Hooke's law. The force created by gas <a href="/wiki/Pressure" title="Pressure">pressure</a> inside the coiled metal tube above unwinds it by an amount proportional to the pressure.</figcaption></figure> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Balancier_avec_ressort_spiral.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Balancier_avec_ressort_spiral.png/220px-Balancier_avec_ressort_spiral.png" decoding="async" width="220" height="127" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Balancier_avec_ressort_spiral.png/330px-Balancier_avec_ressort_spiral.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Balancier_avec_ressort_spiral.png/440px-Balancier_avec_ressort_spiral.png 2x" data-file-width="748" data-file-height="431" /></a><figcaption>The <a href="/wiki/Balance_wheel" title="Balance wheel">balance wheel</a> at the core of many mechanical clocks and watches depends on Hooke's law. Since the torque generated by the coiled spring is proportional to the angle turned by the wheel, its oscillations have a nearly constant period.</figcaption></figure> <style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul dl,.mw-parser-output .hlist ul ol,.mw-parser-output .hlist ul ul{display:inline}.mw-parser-output .hlist .mw-empty-li{display:none}.mw-parser-output .hlist dt::after{content:": "}.mw-parser-output .hlist dd::after,.mw-parser-output .hlist li::after{content:" · ";font-weight:bold}.mw-parser-output .hlist dd:last-child::after,.mw-parser-output .hlist dt:last-child::after,.mw-parser-output .hlist li:last-child::after{content:none}.mw-parser-output .hlist dd dd:first-child::before,.mw-parser-output .hlist dd dt:first-child::before,.mw-parser-output .hlist dd li:first-child::before,.mw-parser-output .hlist dt dd:first-child::before,.mw-parser-output .hlist dt dt:first-child::before,.mw-parser-output .hlist dt li:first-child::before,.mw-parser-output .hlist li dd:first-child::before,.mw-parser-output .hlist li dt:first-child::before,.mw-parser-output .hlist li li:first-child::before{content:" (";font-weight:normal}.mw-parser-output .hlist dd dd:last-child::after,.mw-parser-output .hlist dd dt:last-child::after,.mw-parser-output .hlist dd li:last-child::after,.mw-parser-output .hlist dt dd:last-child::after,.mw-parser-output .hlist dt dt:last-child::after,.mw-parser-output .hlist dt li:last-child::after,.mw-parser-output .hlist li dd:last-child::after,.mw-parser-output .hlist li dt:last-child::after,.mw-parser-output .hlist li li:last-child::after{content:")";font-weight:normal}.mw-parser-output .hlist ol{counter-reset:listitem}.mw-parser-output .hlist ol>li{counter-increment:listitem}.mw-parser-output .hlist ol>li::before{content:" "counter(listitem)"\a0 "}.mw-parser-output .hlist dd ol>li:first-child::before,.mw-parser-output .hlist dt ol>li:first-child::before,.mw-parser-output .hlist li ol>li:first-child::before{content:" ("counter(listitem)"\a0 "}</style><style data-mw-deduplicate="TemplateStyles:r1126788409">.mw-parser-output .plainlist ol,.mw-parser-output .plainlist ul{line-height:inherit;list-style:none;margin:0;padding:0}.mw-parser-output .plainlist ol li,.mw-parser-output .plainlist ul li{margin-bottom:0}</style><style data-mw-deduplicate="TemplateStyles:r1246091330">.mw-parser-output .sidebar{width:22em;float:right;clear:right;margin:0.5em 0 1em 1em;background:var(--background-color-neutral-subtle,#f8f9fa);border:1px solid var(--border-color-base,#a2a9b1);padding:0.2em;text-align:center;line-height:1.4em;font-size:88%;border-collapse:collapse;display:table}body.skin-minerva .mw-parser-output .sidebar{display:table!important;float:right!important;margin:0.5em 0 1em 1em!important}.mw-parser-output .sidebar-subgroup{width:100%;margin:0;border-spacing:0}.mw-parser-output .sidebar-left{float:left;clear:left;margin:0.5em 1em 1em 0}.mw-parser-output .sidebar-none{float:none;clear:both;margin:0.5em 1em 1em 0}.mw-parser-output .sidebar-outer-title{padding:0 0.4em 0.2em;font-size:125%;line-height:1.2em;font-weight:bold}.mw-parser-output .sidebar-top-image{padding:0.4em}.mw-parser-output .sidebar-top-caption,.mw-parser-output .sidebar-pretitle-with-top-image,.mw-parser-output .sidebar-caption{padding:0.2em 0.4em 0;line-height:1.2em}.mw-parser-output .sidebar-pretitle{padding:0.4em 0.4em 0;line-height:1.2em}.mw-parser-output .sidebar-title,.mw-parser-output .sidebar-title-with-pretitle{padding:0.2em 0.8em;font-size:145%;line-height:1.2em}.mw-parser-output .sidebar-title-with-pretitle{padding:0.1em 0.4em}.mw-parser-output .sidebar-image{padding:0.2em 0.4em 0.4em}.mw-parser-output .sidebar-heading{padding:0.1em 0.4em}.mw-parser-output .sidebar-content{padding:0 0.5em 0.4em}.mw-parser-output .sidebar-content-with-subgroup{padding:0.1em 0.4em 0.2em}.mw-parser-output .sidebar-above,.mw-parser-output .sidebar-below{padding:0.3em 0.8em;font-weight:bold}.mw-parser-output .sidebar-collapse .sidebar-above,.mw-parser-output .sidebar-collapse .sidebar-below{border-top:1px solid #aaa;border-bottom:1px solid #aaa}.mw-parser-output .sidebar-navbar{text-align:right;font-size:115%;padding:0 0.4em 0.4em}.mw-parser-output .sidebar-list-title{padding:0 0.4em;text-align:left;font-weight:bold;line-height:1.6em;font-size:105%}.mw-parser-output .sidebar-list-title-c{padding:0 0.4em;text-align:center;margin:0 3.3em}@media(max-width:640px){body.mediawiki .mw-parser-output .sidebar{width:100%!important;clear:both;float:none!important;margin-left:0!important;margin-right:0!important}}body.skin--responsive .mw-parser-output .sidebar a>img{max-width:none!important}@media screen{html.skin-theme-clientpref-night .mw-parser-output .sidebar:not(.notheme) .sidebar-list-title,html.skin-theme-clientpref-night .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle{background:transparent!important}html.skin-theme-clientpref-night .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle a{color:var(--color-progressive)!important}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-list-title,html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle{background:transparent!important}html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle a{color:var(--color-progressive)!important}}@media print{body.ns-0 .mw-parser-output .sidebar{display:none!important}}</style><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1246091330"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1246091330"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1246091330"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1246091330"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1246091330"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1246091330"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><table class="sidebar sidebar-collapse nomobile nowraplinks plainlist"><tbody><tr><td class="sidebar-pretitle">Part of a series on</td></tr><tr><th class="sidebar-title-with-pretitle"><a href="/wiki/Continuum_mechanics" title="Continuum mechanics">Continuum mechanics</a></th></tr><tr><td class="sidebar-image"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J=-D{\frac {d\varphi }{dx}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>J</mi> <mo>=</mo> <mo>−<!-- − --></mo> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>φ<!-- φ --></mi> </mrow> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle J=-D{\frac {d\varphi }{dx}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1856f88def2056f28ed27c7d31180a6240820ea6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:11.874ex; height:5.509ex;" alt="{\displaystyle J=-D{\frac {d\varphi }{dx}}}"></span><div class="sidebar-caption"><a href="/wiki/Fick%27s_laws_of_diffusion" title="Fick's laws of diffusion">Fick's laws of diffusion</a></div></td></tr><tr><td class="sidebar-content-with-subgroup"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:transparent;border-top:1px solid #aaa;text-align:center;;color: var(--color-base)">Laws</div><div class="sidebar-list-content mw-collapsible-content"><table class="sidebar-subgroup"><tbody><tr><th class="sidebar-heading" style="font-style:italic;font-weight:normal;"> Conservations</th></tr><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Conservation_of_mass" title="Conservation of mass">Mass</a></li> <li><a href="/wiki/Conservation_of_momentum" class="mw-redirect" title="Conservation of momentum">Momentum</a></li> <li><a href="/wiki/Conservation_of_energy" title="Conservation of energy">Energy</a></li></ul></td> </tr><tr><th class="sidebar-heading" style="font-style:italic;font-weight:normal;"> Inequalities</th></tr><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Clausius%E2%80%93Duhem_inequality" title="Clausius–Duhem inequality">Clausius–Duhem (entropy)</a></li></ul></td> </tr></tbody></table></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible"><div class="sidebar-list-title" style="background:transparent;border-top:1px solid #aaa;text-align:center;;color: var(--color-base)"><a href="/wiki/Solid_mechanics" title="Solid mechanics">Solid mechanics</a></div><div class="sidebar-list-content mw-collapsible-content"><div class="hlist"> <ul><li><a href="/wiki/Deformation_(physics)" title="Deformation (physics)">Deformation</a></li> <li><a href="/wiki/Elasticity_(physics)" title="Elasticity (physics)">Elasticity</a> <ul><li><a href="/wiki/Linear_elasticity" title="Linear elasticity">linear</a></li></ul></li> <li><a href="/wiki/Plasticity_(physics)" title="Plasticity (physics)">Plasticity</a></li> <li><a class="mw-selflink selflink">Hooke's law</a></li> <li><a href="/wiki/Stress_(mechanics)" title="Stress (mechanics)">Stress</a></li> <li><a href="/wiki/Strain_(mechanics)" title="Strain (mechanics)">Strain</a> <ul><li><a href="/wiki/Finite_strain_theory" title="Finite strain theory">Finite strain</a></li> <li><a href="/wiki/Infinitesimal_strain_theory" title="Infinitesimal strain theory">Infinitesimal strain</a></li></ul></li> <li><a href="/wiki/Compatibility_(mechanics)" title="Compatibility (mechanics)">Compatibility</a></li> <li><a href="/wiki/Bending" title="Bending">Bending</a></li> <li><a href="/wiki/Contact_mechanics" title="Contact mechanics">Contact mechanics</a> <ul><li><a href="/wiki/Frictional_contact_mechanics" title="Frictional contact mechanics">frictional</a></li></ul></li> <li><a href="/wiki/Material_failure_theory" title="Material failure theory">Material failure theory</a></li> <li><a href="/wiki/Fracture_mechanics" title="Fracture mechanics">Fracture mechanics</a></li></ul> </div></div></div></td> </tr><tr><td class="sidebar-content-with-subgroup"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:transparent;border-top:1px solid #aaa;text-align:center;;color: var(--color-base)"><a href="/wiki/Fluid_mechanics" title="Fluid mechanics">Fluid mechanics</a></div><div class="sidebar-list-content mw-collapsible-content"><table class="sidebar-subgroup"><tbody><tr><th class="sidebar-heading" style="font-style:italic;"> <a href="/wiki/Fluid" title="Fluid">Fluids</a></th></tr><tr><td class="sidebar-content"> <div class="wraplinks"> <ul><li><a href="/wiki/Hydrostatics" title="Hydrostatics">Statics</a> <b>·</b> <a href="/wiki/Fluid_dynamics" title="Fluid dynamics">Dynamics</a></li> <li><a href="/wiki/Archimedes%27_principle" title="Archimedes' principle">Archimedes' principle</a> <b>·</b> <a href="/wiki/Bernoulli%27s_principle" title="Bernoulli's principle">Bernoulli's principle</a></li> <li><a href="/wiki/Navier%E2%80%93Stokes_equations" title="Navier–Stokes equations">Navier–Stokes equations</a></li> <li><a href="/wiki/Hagen%E2%80%93Poiseuille_equation" title="Hagen–Poiseuille equation">Poiseuille equation</a> <b>·</b> <a href="/wiki/Pascal%27s_law" title="Pascal's law">Pascal's law</a></li> <li><a href="/wiki/Viscosity" title="Viscosity">Viscosity</a> <ul><li>(<a href="/wiki/Newtonian_fluid" title="Newtonian fluid">Newtonian</a> <b>·</b> <a href="/wiki/Non-Newtonian_fluid" title="Non-Newtonian fluid">non-Newtonian</a>)</li></ul></li> <li><a href="/wiki/Buoyancy" title="Buoyancy">Buoyancy</a> <b>·</b> <a href="/wiki/Mixing_(process_engineering)" title="Mixing (process engineering)">Mixing</a> <b>·</b> <a href="/wiki/Pressure" title="Pressure">Pressure</a></li></ul> </div></td> </tr><tr><th class="sidebar-heading" style="font-style:italic;"> <a href="/wiki/Liquid" title="Liquid">Liquids</a></th></tr><tr><td class="sidebar-content"> <div class="hlist"> <ul><li><a href="/wiki/Adhesion" title="Adhesion">Adhesion</a></li> <li><a href="/wiki/Capillary_action" title="Capillary action">Capillary action</a></li> <li><a href="/wiki/Chromatography" title="Chromatography">Chromatography</a></li> <li><a href="/wiki/Cohesion_(chemistry)" title="Cohesion (chemistry)">Cohesion (chemistry)</a></li> <li><a href="/wiki/Surface_tension" title="Surface tension">Surface tension</a></li></ul> </div></td> </tr><tr><th class="sidebar-heading" style="font-style:italic;"> <a href="/wiki/Gas" title="Gas">Gases</a></th></tr><tr><td class="sidebar-content"> <div class="hlist"> <ul><li><a href="/wiki/Atmosphere" title="Atmosphere">Atmosphere</a></li> <li><a href="/wiki/Boyle%27s_law" title="Boyle's law">Boyle's law</a></li> <li><a href="/wiki/Charles%27s_law" title="Charles's law">Charles's law</a></li> <li><a href="/wiki/Combined_gas_law" class="mw-redirect" title="Combined gas law">Combined gas law</a></li> <li><a href="/wiki/Fick%27s_law" class="mw-redirect" title="Fick's law">Fick's law</a></li> <li><a href="/wiki/Gay-Lussac%27s_law" title="Gay-Lussac's law">Gay-Lussac's law</a></li> <li><a href="/wiki/Graham%27s_law" title="Graham's law">Graham's law</a></li></ul> </div></td> </tr><tr><th class="sidebar-heading" style="font-style:italic;"> <a href="/wiki/Plasma_(physics)" title="Plasma (physics)">Plasma</a></th></tr></tbody></table></div></div></td> </tr><tr><td class="sidebar-content-with-subgroup"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:transparent;border-top:1px solid #aaa;text-align:center;;color: var(--color-base)"><a href="/wiki/Rheology" title="Rheology">Rheology</a></div><div class="sidebar-list-content mw-collapsible-content"><table class="sidebar-subgroup"><tbody><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Viscoelasticity" title="Viscoelasticity">Viscoelasticity</a></li> <li><a href="/wiki/Rheometry" title="Rheometry">Rheometry</a></li> <li><a href="/wiki/Rheometer" title="Rheometer">Rheometer</a></li></ul></td> </tr><tr><th class="sidebar-heading" style="font-style:italic;"> <a href="/wiki/Smart_fluid" title="Smart fluid">Smart fluids</a></th></tr><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Electrorheological_fluid" title="Electrorheological fluid">Electrorheological</a></li> <li><a href="/wiki/Magnetorheological_fluid" title="Magnetorheological fluid">Magnetorheological</a></li> <li><a href="/wiki/Ferrofluid" title="Ferrofluid">Ferrofluids</a></li></ul></td> </tr></tbody></table></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:transparent;border-top:1px solid #aaa;text-align:center;;color: var(--color-base)">Scientists</div><div class="sidebar-list-content mw-collapsible-content"><div class="hlist"> <ul><li><a href="/wiki/Daniel_Bernoulli" title="Daniel Bernoulli">Bernoulli</a></li> <li><a href="/wiki/Robert_Boyle" title="Robert Boyle">Boyle</a></li> <li><a href="/wiki/Augustin-Louis_Cauchy" title="Augustin-Louis Cauchy">Cauchy</a></li> <li><a href="/wiki/Jacques_Charles" title="Jacques Charles">Charles</a></li> <li><a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Euler</a></li> <li><a href="/wiki/Adolf_Eugen_Fick" title="Adolf Eugen Fick">Fick</a></li> <li><a href="/wiki/Joseph_Louis_Gay-Lussac" title="Joseph Louis Gay-Lussac">Gay-Lussac</a></li> <li><a href="/wiki/Thomas_Graham_(chemist)" title="Thomas Graham (chemist)">Graham</a></li> <li><a href="/wiki/Robert_Hooke" title="Robert Hooke">Hooke</a></li> <li><a href="/wiki/Isaac_Newton" title="Isaac Newton">Newton</a></li> <li><a href="/wiki/Claude-Louis_Navier" title="Claude-Louis Navier">Navier</a></li> <li><a href="/wiki/Walter_Noll" title="Walter Noll">Noll</a></li> <li><a href="/wiki/Blaise_Pascal" title="Blaise Pascal">Pascal</a></li> <li><a href="/wiki/Sir_George_Stokes,_1st_Baronet" title="Sir George Stokes, 1st Baronet">Stokes</a></li> <li><a href="/wiki/Clifford_Truesdell" title="Clifford Truesdell">Truesdell</a></li></ul> </div></div></div></td> </tr><tr><td class="sidebar-navbar"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Continuum_mechanics" title="Template:Continuum mechanics"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Continuum_mechanics" title="Template talk:Continuum mechanics"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Continuum_mechanics" title="Special:EditPage/Template:Continuum mechanics"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> <p>In <a href="/wiki/Physics" title="Physics">physics</a>, <b>Hooke's law</b> is an <a href="/wiki/Empirical_law" class="mw-redirect" title="Empirical law">empirical law</a> which states that the <a href="/wiki/Force" title="Force">force</a> (<span class="texhtml mvar" style="font-style:italic;">F</span>) needed to extend or compress a <a href="/wiki/Spring_(device)" title="Spring (device)">spring</a> by some distance (<span class="texhtml mvar" style="font-style:italic;">x</span>) <a href="/wiki/Proportionality_(mathematics)#Direct_proportionality" title="Proportionality (mathematics)">scales linearly</a> with respect to that distance—that is, <span class="nowrap"><span class="texhtml"><i>F<sub>s</sub></i> = <i>kx</i></span>,</span> where <span class="texhtml mvar" style="font-style:italic;">k</span> is a constant factor characteristic of the spring (i.e., its <a href="/wiki/Stiffness" title="Stiffness">stiffness</a>), and <span class="texhtml mvar" style="font-style:italic;">x</span> is small compared to the total possible deformation of the spring. The law is named after 17th-century British physicist <a href="/wiki/Robert_Hooke" title="Robert Hooke">Robert Hooke</a>. He first stated the law in 1676 as a Latin <a href="/wiki/Anagram" title="Anagram">anagram</a>.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> He published the solution of his anagram in 1678<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> as: <span title="Latin-language text"><i lang="la">ut tensio, sic vis</i></span> ("as the extension, so the force" or "the extension is proportional to the force"). Hooke states in the 1678 work that he was aware of the law since 1660. </p><p>Hooke's equation holds (to some extent) in many other situations where an <a href="/wiki/Elasticity_(physics)" title="Elasticity (physics)">elastic</a> body is <a href="/wiki/Deformation_(physics)" title="Deformation (physics)">deformed</a>, such as wind blowing on a tall building, and a musician plucking a <a href="/wiki/String_(music)" title="String (music)">string</a> of a guitar. An elastic body or material for which this equation can be assumed is said to be <a href="/wiki/Linear_elasticity" title="Linear elasticity">linear-elastic</a> or <b>Hookean</b>. </p><p>Hooke's law is only a <a href="/wiki/Taylor_series" title="Taylor series">first-order linear approximation</a> to the real response of springs and other elastic bodies to applied forces. It must eventually fail once the forces exceed some limit, since no material can be compressed beyond a certain minimum size, or stretched beyond a maximum size, without some permanent deformation or change of state. Many materials will noticeably deviate from Hooke's law well before those <a href="/wiki/Elastic_limit" class="mw-redirect" title="Elastic limit">elastic limits</a> are reached. </p><p>On the other hand, Hooke's law is an accurate approximation for most solid bodies, as long as the forces and deformations are small enough. For this reason, Hooke's law is extensively used in all branches of science and engineering, and is the foundation of many disciplines such as <a href="/wiki/Seismology" title="Seismology">seismology</a>, <a href="/wiki/Molecular_mechanics" title="Molecular mechanics">molecular mechanics</a> and <a href="/wiki/Acoustics" title="Acoustics">acoustics</a>. It is also the fundamental principle behind the <a href="/wiki/Spring_scale" title="Spring scale">spring scale</a>, the <a href="/wiki/Manometer" class="mw-redirect" title="Manometer">manometer</a>, the <a href="/wiki/Galvanometer" title="Galvanometer">galvanometer</a>, and the <a href="/wiki/Balance_wheel" title="Balance wheel">balance wheel</a> of the <a href="/wiki/Mechanical_clock" class="mw-redirect" title="Mechanical clock">mechanical clock</a>. </p><p>The modern <a href="/wiki/Theory_of_elasticity" class="mw-redirect" title="Theory of elasticity">theory of elasticity</a> generalizes Hooke's law to say that the <a href="/wiki/Deformation_(mechanics)" class="mw-redirect" title="Deformation (mechanics)">strain</a> (deformation) of an elastic object or material is proportional to the <a href="/wiki/Stress_(mechanics)" title="Stress (mechanics)">stress</a> applied to it. However, since general stresses and strains may have multiple independent components, the "proportionality factor" may no longer be just a single real number, but rather a <a href="/wiki/Linear_map" title="Linear map">linear map</a> (a <a href="/wiki/Tensor" title="Tensor">tensor</a>) that can be represented by a <a href="/wiki/Matrix_(mathematics)" title="Matrix (mathematics)">matrix</a> of real numbers. </p><p>In this general form, Hooke's law makes it possible to deduce the relation between strain and stress for complex objects in terms of intrinsic properties of the materials they are made of. For example, one can deduce that a <a href="/wiki/Homogeneous" class="mw-redirect" title="Homogeneous">homogeneous</a> rod with uniform <a href="/wiki/Cross_section_(geometry)" title="Cross section (geometry)">cross section</a> will behave like a simple spring when stretched, with a stiffness <span class="texhtml mvar" style="font-style:italic;">k</span> directly proportional to its cross-section area and inversely proportional to its length. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Formal_definition">Formal definition</h2></div> <div class="mw-heading mw-heading3"><h3 id="Linear_springs">Linear springs</h3></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Spring-elongation-and-forces.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/7a/Spring-elongation-and-forces.svg/220px-Spring-elongation-and-forces.svg.png" decoding="async" width="220" height="151" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/7a/Spring-elongation-and-forces.svg/330px-Spring-elongation-and-forces.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/7a/Spring-elongation-and-forces.svg/440px-Spring-elongation-and-forces.svg.png 2x" data-file-width="283" data-file-height="194" /></a><figcaption>Elongation and compression of a spring</figcaption></figure> <p>Consider a simple <a href="/wiki/Helix" title="Helix">helical</a> spring that has one end attached to some fixed object, while the free end is being pulled by a force whose magnitude is <span class="texhtml mvar" style="font-style:italic;">F<sub>s</sub></span>. Suppose that the spring has reached a state of <a href="/wiki/Mechanical_equilibrium" title="Mechanical equilibrium">equilibrium</a>, where its length is not changing anymore. Let <span class="texhtml mvar" style="font-style:italic;">x</span> be the amount by which the free end of the spring was displaced from its "relaxed" position (when it is not being stretched). Hooke's law states that <span class="mwe-math-element" data-qid="Q170282"><a href="/w/index.php?title=Special:MathWikibase&qid=Q170282" style="color:inherit;"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{s}=kx}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <mo>=</mo> <mi>k</mi> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{s}=kx}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4167538e8bec1d328457ce9fa31029ba698b3df5" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.137ex; height:2.509ex;" alt="{\displaystyle F_{s}=kx}"></a></span> or, equivalently, <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x={\frac {F_{s}}{k}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <mi>k</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x={\frac {F_{s}}{k}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b018fcc1be0a63f72e02b7c05348a1a336b517ff" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:7.762ex; height:5.343ex;" alt="{\displaystyle x={\frac {F_{s}}{k}}}"></span> where <span class="texhtml mvar" style="font-style:italic;">k</span> is a positive real number, characteristic of the spring. A spring with spaces between the coils can be compressed, and the same formula holds for compression, with <span class="texhtml mvar" style="font-style:italic;">F<sub>s</sub></span> and <span class="texhtml mvar" style="font-style:italic;">x</span> both negative in that case.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Hooke%27s_Law_wikipedia.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f2/Hooke%27s_Law_wikipedia.png/220px-Hooke%27s_Law_wikipedia.png" decoding="async" width="220" height="134" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f2/Hooke%27s_Law_wikipedia.png/330px-Hooke%27s_Law_wikipedia.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f2/Hooke%27s_Law_wikipedia.png/440px-Hooke%27s_Law_wikipedia.png 2x" data-file-width="1781" data-file-height="1086" /></a><figcaption>Graphical derivation</figcaption></figure> <p>According to this formula, the <a href="/wiki/Graph_of_a_function" title="Graph of a function">graph</a> of the applied force <span class="texhtml mvar" style="font-style:italic;">F<sub>s</sub></span> as a function of the displacement <span class="texhtml mvar" style="font-style:italic;">x</span> will be a straight line passing through the <a href="/wiki/Cartesian_coordinates" class="mw-redirect" title="Cartesian coordinates">origin</a>, whose <a href="/wiki/Slope" title="Slope">slope</a> is <span class="texhtml mvar" style="font-style:italic;">k</span>. </p><p>Hooke's law for a spring is also stated under the convention that <span class="texhtml mvar" style="font-style:italic;">F<sub>s</sub></span> is the <a href="/wiki/Restoring_force" title="Restoring force">restoring force</a> exerted by the spring on whatever is pulling its free end. In that case, the equation becomes <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{s}=-kx}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <mo>=</mo> <mo>−<!-- − --></mo> <mi>k</mi> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{s}=-kx}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/66774554c5cee5c22c57b0950c06ded0d36da720" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.945ex; height:2.509ex;" alt="{\displaystyle F_{s}=-kx}"></span> since the direction of the restoring force is opposite to that of the displacement. </p> <div class="mw-heading mw-heading3"><h3 id="Torsional_springs">Torsional springs</h3></div> <p>The <a href="/wiki/Torsional" class="mw-redirect" title="Torsional">torsional</a> analog of Hooke's law applies to <a href="/wiki/Torsional_spring" class="mw-redirect" title="Torsional spring">torsional springs</a>. It states that the torque (τ) required to rotate an object is directly proportional to the angular displacement (θ) from the equilibrium position. It describes the relationship between the torque applied to an object and the resulting angular <a href="/wiki/Deformation_(physics)" title="Deformation (physics)">deformation</a> due to torsion. Mathematically, it can be expressed as: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau =-k\theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>τ<!-- τ --></mi> <mo>=</mo> <mo>−<!-- − --></mo> <mi>k</mi> <mi>θ<!-- θ --></mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tau =-k\theta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8cc8e2824b20855edfb221b9f859e6625a055afb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:8.41ex; height:2.343ex;" alt="{\displaystyle \tau =-k\theta }"></span></dd></dl> <p>Where: </p> <ul><li>τ is the <a href="/wiki/Torque" title="Torque">torque</a> measured in Newton-meters or N·m.</li> <li>k is the <a href="/wiki/Torsional_constant" class="mw-redirect" title="Torsional constant">torsional constant</a> (measured in N·m/radian), which characterizes the stiffness of the torsional spring or the resistance to angular displacement.</li> <li>θ is the <a href="/wiki/Angular_displacement" title="Angular displacement">angular displacement</a> (measured in radians) from the equilibrium position.</li></ul> <p>Just as in the linear case, this law shows that the torque is proportional to the angular displacement, and the negative sign indicates that the torque acts in a direction opposite to the angular displacement, providing a restoring force to bring the system back to equilibrium. </p> <div class="mw-heading mw-heading3"><h3 id="General_"scalar"_springs"><span id="General_.22scalar.22_springs"></span>General "scalar" springs</h3></div> <p>Hooke's spring law usually applies to any elastic object, of arbitrary complexity, as long as both the deformation and the stress can be expressed by a single number that can be both positive and negative. </p><p>For example, when a block of rubber attached to two parallel plates is deformed by <a href="/wiki/Simple_shear" title="Simple shear">shearing</a>, rather than stretching or compression, the shearing force <span class="texhtml"><i>F<sub>s</sub></i></span> and the sideways displacement of the plates <span class="texhtml mvar" style="font-style:italic;">x</span> obey Hooke's law (for small enough deformations). </p><p>Hooke's law also applies when a straight steel bar or concrete beam (like the one used in buildings), supported at both ends, is bent by a weight <span class="texhtml mvar" style="font-style:italic;">F</span> placed at some intermediate point. The displacement <span class="texhtml mvar" style="font-style:italic;">x</span> in this case is the deviation of the beam, measured in the transversal direction, relative to its unloaded shape. </p> <div class="mw-heading mw-heading3"><h3 id="Vector_formulation">Vector formulation</h3></div> <p>In the case of a helical spring that is stretched or compressed along its <a href="/wiki/Axial_symmetry" title="Axial symmetry">axis</a>, the applied (or restoring) force and the resulting elongation or compression have the same direction (which is the direction of said axis). Therefore, if <span class="texhtml mvar" style="font-style:italic;">F<sub>s</sub></span> and <span class="texhtml mvar" style="font-style:italic;">x</span> are defined as <a href="/wiki/Vector_(mathematics)" class="mw-redirect" title="Vector (mathematics)">vectors</a>, Hooke's <a href="/wiki/Equation" title="Equation">equation</a> still holds and says that the force vector is the <a href="/wiki/Displacement_(vector)" class="mw-redirect" title="Displacement (vector)">elongation vector</a> multiplied by a fixed <a href="/wiki/Scalar_(mathematics)" title="Scalar (mathematics)">scalar</a>. </p> <div class="mw-heading mw-heading3"><h3 id="General_tensor_form">General tensor form</h3></div> <p>Some elastic bodies will deform in one direction when subjected to a force with a different direction. One example is a horizontal wood beam with non-square rectangular cross section that is bent by a transverse load that is neither vertical nor horizontal. In such cases, the <i>magnitude</i> of the displacement <span class="texhtml mvar" style="font-style:italic;">x</span> will be proportional to the magnitude of the force <span class="texhtml mvar" style="font-style:italic;">F<sub>s</sub></span>, as long as the direction of the latter remains the same (and its value is not too large); so the scalar version of Hooke's law <span class="texhtml"><i>F<sub>s</sub></i> = −<i>kx</i></span> will hold. However, the force and displacement <i>vectors</i> will not be scalar multiples of each other, since they have different directions. Moreover, the ratio <span class="texhtml mvar" style="font-style:italic;">k</span> between their magnitudes will depend on the direction of the vector <span class="texhtml mvar" style="font-style:italic;">F<sub>s</sub></span>. </p><p>Yet, in such cases there is often a fixed <a href="/wiki/Linear_map" title="Linear map">linear relation</a> between the force and deformation vectors, as long as they are small enough. Namely, there is a <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">function</a> <span class="texhtml mvar" style="font-style:italic;"><b>κ</b></span> from vectors to vectors, such that <span class="texhtml"><b>F</b> = <i><b>κ</b></i>(<b>X</b>)</span>, and <span class="texhtml"><i><b>κ</b></i>(<i>α</i><b>X</b><sub>1</sub> + <i>β</i><b>X</b><sub>2</sub>) = <i>α<b>κ</b></i>(<b>X</b><sub>1</sub>) + <i>β<b>κ</b></i>(<b>X</b><sub>2</sub>)</span> for any real numbers <span class="texhtml mvar" style="font-style:italic;">α</span>, <span class="texhtml mvar" style="font-style:italic;">β</span> and any displacement vectors <span class="texhtml"><b>X</b><sub>1</sub></span>, <span class="texhtml"><b>X</b><sub>2</sub></span>. Such a function is called a (second-order) <a href="/wiki/Tensor" title="Tensor">tensor</a>. </p><p>With respect to an arbitrary <a href="/wiki/Cartesian_coordinates" class="mw-redirect" title="Cartesian coordinates">Cartesian coordinate system</a>, the force and displacement vectors can be represented by 3 × 1 <a href="/wiki/Matrix_(mathematics)" title="Matrix (mathematics)">matrices</a> of real numbers. Then the tensor <span class="texhtml"><b>κ</b></span> connecting them can be represented by a 3 × 3 matrix <span class="texhtml mvar" style="font-style:italic;"><b>κ</b></span> of real coefficients, that, when <a href="/wiki/Matrix_product" class="mw-redirect" title="Matrix product">multiplied</a> by the displacement vector, gives the force vector: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {F} \,=\,{\begin{bmatrix}F_{1}\\F_{2}\\F_{3}\end{bmatrix}}\,=\,{\begin{bmatrix}\kappa _{11}&\kappa _{12}&\kappa _{13}\\\kappa _{21}&\kappa _{22}&\kappa _{23}\\\kappa _{31}&\kappa _{32}&\kappa _{33}\end{bmatrix}}{\begin{bmatrix}X_{1}\\X_{2}\\X_{3}\end{bmatrix}}\,=\,{\boldsymbol {\kappa }}\mathbf {X} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mspace width="thinmathspace" /> <mo>=</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mspace width="thinmathspace" /> <mo>=</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>κ<!-- κ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>11</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>κ<!-- κ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>κ<!-- κ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>13</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>κ<!-- κ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>21</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>κ<!-- κ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>22</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>κ<!-- κ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>23</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>κ<!-- κ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>31</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>κ<!-- κ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>32</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>κ<!-- κ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>33</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mspace width="thinmathspace" /> <mo>=</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">κ<!-- κ --></mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {F} \,=\,{\begin{bmatrix}F_{1}\\F_{2}\\F_{3}\end{bmatrix}}\,=\,{\begin{bmatrix}\kappa _{11}&\kappa _{12}&\kappa _{13}\\\kappa _{21}&\kappa _{22}&\kappa _{23}\\\kappa _{31}&\kappa _{32}&\kappa _{33}\end{bmatrix}}{\begin{bmatrix}X_{1}\\X_{2}\\X_{3}\end{bmatrix}}\,=\,{\boldsymbol {\kappa }}\mathbf {X} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/95b7979bd85019dc82561399f5884c3a6fc19281" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:48.247ex; height:9.176ex;" alt="{\displaystyle \mathbf {F} \,=\,{\begin{bmatrix}F_{1}\\F_{2}\\F_{3}\end{bmatrix}}\,=\,{\begin{bmatrix}\kappa _{11}&\kappa _{12}&\kappa _{13}\\\kappa _{21}&\kappa _{22}&\kappa _{23}\\\kappa _{31}&\kappa _{32}&\kappa _{33}\end{bmatrix}}{\begin{bmatrix}X_{1}\\X_{2}\\X_{3}\end{bmatrix}}\,=\,{\boldsymbol {\kappa }}\mathbf {X} }"></span> </p><p>That is, <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{i}=\kappa _{i1}X_{1}+\kappa _{i2}X_{2}+\kappa _{i3}X_{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>κ<!-- κ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mn>1</mn> </mrow> </msub> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>κ<!-- κ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mn>2</mn> </mrow> </msub> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>κ<!-- κ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mn>3</mn> </mrow> </msub> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{i}=\kappa _{i1}X_{1}+\kappa _{i2}X_{2}+\kappa _{i3}X_{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4a53fddc103d3723d1e9e6566de58f54f3699861" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:28.891ex; height:2.509ex;" alt="{\displaystyle F_{i}=\kappa _{i1}X_{1}+\kappa _{i2}X_{2}+\kappa _{i3}X_{3}}"></span> for <span class="texhtml"><i>i</i> = 1, 2, 3</span>. Therefore, Hooke's law <span class="texhtml"><b>F</b> = <b><i>κ</i>X</b></span> can be said to hold also when <span class="texhtml"><b>X</b></span> and <span class="texhtml"><b>F</b></span> are vectors with variable directions, except that the stiffness of the object is a tensor <span class="texhtml mvar" style="font-style:italic;"><b>κ</b></span>, rather than a single real number <span class="texhtml mvar" style="font-style:italic;">k</span>. </p> <div class="mw-heading mw-heading3"><h3 id="Hooke's_law_for_continuous_media"><span id="Hooke.27s_law_for_continuous_media"></span>Hooke's law for continuous media</h3></div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Linear_elasticity" title="Linear elasticity">Linear elasticity</a></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Hookes_law_nanoscale.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/ad/Hookes_law_nanoscale.jpg/290px-Hookes_law_nanoscale.jpg" decoding="async" width="290" height="240" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/ad/Hookes_law_nanoscale.jpg/435px-Hookes_law_nanoscale.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/ad/Hookes_law_nanoscale.jpg/580px-Hookes_law_nanoscale.jpg 2x" data-file-width="926" data-file-height="765" /></a><figcaption>(a) Schematic of a polymer nanospring. The coil radius, R, pitch, P, length of the spring, L, and the number of turns, N, are 2.5 μm, 2.0 μm, 13 μm, and 4, respectively. Electron micrographs of the nanospring, before loading (b-e), stretched (f), compressed (g), bent (h), and recovered (i). All scale bars are 2 μm. The spring followed a linear response against applied force, demonstrating the validity of Hooke's law at the nanoscale.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup></figcaption></figure> <p>The stresses and strains of the material inside a <a href="/wiki/Continuum_mechanics" title="Continuum mechanics">continuous</a> elastic material (such as a block of rubber, the wall of a <a href="/wiki/Boiler" title="Boiler">boiler</a>, or a steel bar) are connected by a linear relationship that is mathematically similar to Hooke's spring law, and is often referred to by that name. </p><p>However, the strain state in a solid medium around some point cannot be described by a single vector. The same parcel of material, no matter how small, can be compressed, stretched, and sheared at the same time, along different directions. Likewise, the stresses in that parcel can be at once pushing, pulling, and shearing. </p><p>In order to capture this complexity, the relevant state of the medium around a point must be represented by two-second-order tensors, the <a href="/wiki/Strain_tensor" class="mw-redirect" title="Strain tensor">strain tensor</a> <span class="texhtml"><b>ε</b></span> (in lieu of the displacement <span class="texhtml"><b>X</b></span>) and the <a href="/wiki/Cauchy_stress_tensor" title="Cauchy stress tensor">stress tensor</a> <span class="texhtml"><b>σ</b></span> (replacing the restoring force <span class="texhtml"><b>F</b></span>). The analogue of Hooke's spring law for continuous media is then <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {\sigma }}=\mathbf {c} {\boldsymbol {\varepsilon }},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">σ<!-- σ --></mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">c</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ε<!-- ε --></mi> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {\sigma }}=\mathbf {c} {\boldsymbol {\varepsilon }},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7309557504a1624ac107631d2b2ccf94c4351e36" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.758ex; height:2.009ex;" alt="{\displaystyle {\boldsymbol {\sigma }}=\mathbf {c} {\boldsymbol {\varepsilon }},}"></span> where <span class="texhtml"><b>c</b></span> is a fourth-order tensor (that is, a linear map between second-order tensors) usually called the <a href="/wiki/Stiffness_tensor" class="mw-redirect" title="Stiffness tensor">stiffness tensor</a> or <a href="/wiki/Elasticity_tensor" title="Elasticity tensor">elasticity tensor</a>. One may also write it as <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {\varepsilon }}=\mathbf {s} {\boldsymbol {\sigma }},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ε<!-- ε --></mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">s</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">σ<!-- σ --></mi> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {\varepsilon }}=\mathbf {s} {\boldsymbol {\sigma }},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f15adc8c963aa3468d152cc47c83bfcf9c534bc5" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.625ex; height:2.009ex;" alt="{\displaystyle {\boldsymbol {\varepsilon }}=\mathbf {s} {\boldsymbol {\sigma }},}"></span> where the tensor <span class="texhtml"><b>s</b></span>, called the <a href="/wiki/Stiffness_tensor" class="mw-redirect" title="Stiffness tensor">compliance tensor</a>, represents the inverse of said linear map. </p><p>In a Cartesian coordinate system, the stress and strain tensors can be represented by 3 × 3 matrices </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {\varepsilon }}\,=\,{\begin{bmatrix}\varepsilon _{11}&\varepsilon _{12}&\varepsilon _{13}\\\varepsilon _{21}&\varepsilon _{22}&\varepsilon _{23}\\\varepsilon _{31}&\varepsilon _{32}&\varepsilon _{33}\end{bmatrix}}\,;\qquad {\boldsymbol {\sigma }}\,=\,{\begin{bmatrix}\sigma _{11}&\sigma _{12}&\sigma _{13}\\\sigma _{21}&\sigma _{22}&\sigma _{23}\\\sigma _{31}&\sigma _{32}&\sigma _{33}\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ε<!-- ε --></mi> </mrow> <mspace width="thinmathspace" /> <mo>=</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>ε<!-- ε --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>11</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>ε<!-- ε --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>ε<!-- ε --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>13</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>ε<!-- ε --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>21</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>ε<!-- ε --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>22</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>ε<!-- ε --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>23</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>ε<!-- ε --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>31</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>ε<!-- ε --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>32</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>ε<!-- ε --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>33</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mspace width="thinmathspace" /> <mo>;</mo> <mspace width="2em" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">σ<!-- σ --></mi> </mrow> <mspace width="thinmathspace" /> <mo>=</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>σ<!-- σ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>11</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>σ<!-- σ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>σ<!-- σ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>13</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>σ<!-- σ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>21</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>σ<!-- σ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>22</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>σ<!-- σ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>23</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>σ<!-- σ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>31</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>σ<!-- σ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>32</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>σ<!-- σ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>33</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {\varepsilon }}\,=\,{\begin{bmatrix}\varepsilon _{11}&\varepsilon _{12}&\varepsilon _{13}\\\varepsilon _{21}&\varepsilon _{22}&\varepsilon _{23}\\\varepsilon _{31}&\varepsilon _{32}&\varepsilon _{33}\end{bmatrix}}\,;\qquad {\boldsymbol {\sigma }}\,=\,{\begin{bmatrix}\sigma _{11}&\sigma _{12}&\sigma _{13}\\\sigma _{21}&\sigma _{22}&\sigma _{23}\\\sigma _{31}&\sigma _{32}&\sigma _{33}\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8562f2cdb30e833333c2b3cd985d93c287d0fe93" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:52.12ex; height:9.176ex;" alt="{\displaystyle {\boldsymbol {\varepsilon }}\,=\,{\begin{bmatrix}\varepsilon _{11}&\varepsilon _{12}&\varepsilon _{13}\\\varepsilon _{21}&\varepsilon _{22}&\varepsilon _{23}\\\varepsilon _{31}&\varepsilon _{32}&\varepsilon _{33}\end{bmatrix}}\,;\qquad {\boldsymbol {\sigma }}\,=\,{\begin{bmatrix}\sigma _{11}&\sigma _{12}&\sigma _{13}\\\sigma _{21}&\sigma _{22}&\sigma _{23}\\\sigma _{31}&\sigma _{32}&\sigma _{33}\end{bmatrix}}}"></span> </p><p>Being a linear mapping between the nine numbers <span class="texhtml"><i>σ<sub>ij</sub></i></span> and the nine numbers <span class="texhtml"><i>ε<sub>kl</sub></i></span>, the stiffness tensor <span class="texhtml"><b>c</b></span> is represented by a matrix of <span class="texhtml">3 × 3 × 3 × 3 = 81</span> real numbers <span class="texhtml"><i>c<sub>ijkl</sub></i></span>. Hooke's law then says that <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{ij}=\sum _{k=1}^{3}\sum _{l=1}^{3}c_{ijkl}\varepsilon _{kl}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>σ<!-- σ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>=</mo> <munderover> <mo>∑<!-- ∑ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </munderover> <munderover> <mo>∑<!-- ∑ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </munderover> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> <mi>k</mi> <mi>l</mi> </mrow> </msub> <msub> <mi>ε<!-- ε --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mi>l</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{ij}=\sum _{k=1}^{3}\sum _{l=1}^{3}c_{ijkl}\varepsilon _{kl}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a1aa9fd0a6cd8e5d41500573ce39160985174a1" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:19.88ex; height:7.176ex;" alt="{\displaystyle \sigma _{ij}=\sum _{k=1}^{3}\sum _{l=1}^{3}c_{ijkl}\varepsilon _{kl}}"></span> where <span class="texhtml"><i>i</i>,<i>j</i> = 1,2,3</span>. </p><p>All three tensors generally vary from point to point inside the medium, and may vary with time as well. The strain tensor <span class="texhtml"><b>ε</b></span> merely specifies the displacement of the medium particles in the neighborhood of the point, while the stress tensor <span class="texhtml"><b>σ</b></span> specifies the forces that neighboring parcels of the medium are exerting on each other. Therefore, they are independent of the composition and physical state of the material. The stiffness tensor <span class="texhtml"><b>c</b></span>, on the other hand, is a property of the material, and often depends on physical state variables such as temperature, <a href="/wiki/Pressure" title="Pressure">pressure</a>, and <a href="/wiki/Microstructure" title="Microstructure">microstructure</a>. </p><p>Due to the inherent symmetries of <span class="texhtml"><b>σ</b></span>, <span class="texhtml"><b>ε</b></span>, and <span class="texhtml"><b>c</b></span>, only 21 elastic coefficients of the latter are independent.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> This number can be further reduced by the symmetry of the material: 9 for an <a href="/wiki/Orthorhombic_crystal_system" title="Orthorhombic crystal system">orthorhombic</a> crystal, 5 for an <a href="/wiki/Hexagonal_crystal_family" title="Hexagonal crystal family">hexagonal</a> structure, and 3 for a <a href="/wiki/Cubic_crystal_system" title="Cubic crystal system">cubic</a> symmetry.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> For <a href="/wiki/Isotropic" class="mw-redirect" title="Isotropic">isotropic</a> media (which have the same physical properties in any direction), <span class="texhtml"><b>c</b></span> can be reduced to only two independent numbers, the <a href="/wiki/Bulk_modulus" title="Bulk modulus">bulk modulus</a> <span class="texhtml mvar" style="font-style:italic;">K</span> and the <a href="/wiki/Shear_modulus" title="Shear modulus">shear modulus</a> <span class="texhtml mvar" style="font-style:italic;">G</span>, that quantify the material's resistance to changes in volume and to shearing deformations, respectively. </p> <div class="mw-heading mw-heading2"><h2 id="Analogous_laws">Analogous laws</h2></div> <p>Since Hooke's law is a simple proportionality between two quantities, its formulas and consequences are mathematically similar to those of many other physical laws, such as those describing the motion of <a href="/wiki/Fluid" title="Fluid">fluids</a>, or the <a href="/wiki/Ionic_polarization" class="mw-redirect" title="Ionic polarization">polarization</a> of a <a href="/wiki/Dielectric" title="Dielectric">dielectric</a> by an <a href="/wiki/Electric_field" title="Electric field">electric field</a>. </p><p>In particular, the tensor equation <span class="texhtml"><b>σ</b> = <b>cε</b></span> relating elastic stresses to strains is entirely similar to the equation <span class="texhtml"><b>τ</b> = <b>με̇</b></span> relating the <a href="/wiki/Viscous_stress_tensor" title="Viscous stress tensor">viscous stress tensor</a> <span class="texhtml"><b>τ</b></span> and the <a href="/wiki/Strain_rate_tensor" class="mw-redirect" title="Strain rate tensor">strain rate tensor</a> <span class="texhtml"><b>ε̇</b></span> in flows of <a href="/wiki/Viscosity" title="Viscosity">viscous</a> fluids; although the former pertains to <a href="/wiki/Statics" title="Statics">static</a> stresses (related to <i>amount</i> of deformation) while the latter pertains to <a href="/wiki/Dynamics_(physics)" class="mw-redirect" title="Dynamics (physics)">dynamical</a> stresses (related to the <i>rate</i> of deformation). </p> <div class="mw-heading mw-heading2"><h2 id="Units_of_measurement">Units of measurement</h2></div> <p>In <a href="/wiki/International_System_of_Units" title="International System of Units">SI units</a>, displacements are measured in meters (m), and forces in <a href="/wiki/Newton_(unit)" title="Newton (unit)">newtons</a> (N or kg·m/s<sup>2</sup>). Therefore, the spring constant <span class="texhtml mvar" style="font-style:italic;">k</span>, and each element of the tensor <span class="texhtml"><b>κ</b></span>, is measured in newtons per meter (N/m), or kilograms per second squared (kg/s<sup>2</sup>). </p><p>For continuous media, each element of the stress tensor <span class="texhtml"><b>σ</b></span> is a force divided by an area; it is therefore measured in units of pressure, namely <a href="/wiki/Pascal_(unit)" title="Pascal (unit)">pascals</a> (Pa, or N/m<sup>2</sup>, or kg/(m·s<sup>2</sup>). The elements of the strain tensor <span class="texhtml"><b>ε</b></span> are <a href="/wiki/Dimensionless" class="mw-redirect" title="Dimensionless">dimensionless</a> (displacements divided by distances). Therefore, the entries of <span class="texhtml mvar" style="font-style:italic;">c<sub>ijkl</sub></span> are also expressed in units of pressure. </p> <div class="mw-heading mw-heading2"><h2 id="General_application_to_elastic_materials">General application to elastic materials</h2></div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Stress_v_strain_A36_2.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f1/Stress_v_strain_A36_2.svg/300px-Stress_v_strain_A36_2.svg.png" decoding="async" width="300" height="310" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f1/Stress_v_strain_A36_2.svg/450px-Stress_v_strain_A36_2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f1/Stress_v_strain_A36_2.svg/600px-Stress_v_strain_A36_2.svg.png 2x" data-file-width="777" data-file-height="804" /></a><figcaption> <a href="/wiki/Stress%E2%80%93strain_curve" title="Stress–strain curve">Stress–strain curve</a> for low-carbon steel, showing the relationship between the <a href="/wiki/Stress_(mechanics)" title="Stress (mechanics)">stress</a> (force per unit area) and <a href="/wiki/Deformation_(mechanics)" class="mw-redirect" title="Deformation (mechanics)">strain</a> (resulting compression/stretching, known as deformation). Hooke's law is only valid for the portion of the curve between the origin and the yield point (2). <div><ol style="margin-left:0;"><li style="list-style-position:inside; white-space:nowrap;"><a href="/wiki/Ultimate_tensile_strength" title="Ultimate tensile strength">Ultimate strength</a></li><li style="list-style-position:inside; white-space:nowrap;"><a href="/wiki/Yield_(engineering)" title="Yield (engineering)">Yield strength (yield point)</a></li><li style="list-style-position:inside; white-space:nowrap;">Rupture</li><li style="list-style-position:inside; white-space:nowrap;"><a href="/wiki/Strain_hardening" class="mw-redirect" title="Strain hardening">Strain hardening</a> region</li><li style="list-style-position:inside; white-space:nowrap;"><a href="/wiki/Necking_(engineering)" title="Necking (engineering)">Necking</a> region</li></ol></div> <div><ol style="margin-left:0; margin-top:0; list-style-type:upper-alpha;"><li style="list-style-position:inside; white-space:nowrap;">Apparent stress (<i>F</i>/<i>A</i><sub>0</sub>)</li><li style="list-style-position:inside; white-space:nowrap;">Actual stress (<i>F</i>/<i>A</i>)</li></ol></div> </figcaption></figure> <p>Objects that quickly regain their original shape after being deformed by a force, with the molecules or atoms of their material returning to the initial state of stable equilibrium, often obey Hooke's law. </p><p>Hooke's law only holds for some materials under certain loading conditions. Steel exhibits linear-elastic behavior in most engineering applications; Hooke's law is valid for it throughout its <b>elastic range</b> (i.e., for stresses below the <a href="/wiki/Yield_(engineering)" title="Yield (engineering)">yield strength</a>). For some other materials, such as aluminium, Hooke's law is only valid for a portion of the elastic range. For these materials a <a href="/wiki/Proportional_limit" class="mw-redirect" title="Proportional limit">proportional limit</a> stress is defined, below which the errors associated with the linear approximation are negligible. </p><p>Rubber is generally regarded as a "non-Hookean" material because its elasticity is stress dependent and sensitive to temperature and loading rate. </p><p>Generalizations of Hooke's law for the case of <a href="/wiki/Finite_strain_theory" title="Finite strain theory">large deformations</a> is provided by models of <a href="/wiki/Neo-Hookean_solid" title="Neo-Hookean solid">neo-Hookean solids</a> and <a href="/wiki/Mooney%E2%80%93Rivlin_solid" title="Mooney–Rivlin solid">Mooney–Rivlin solids</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Derived_formulae">Derived formulae</h2></div> <div class="mw-heading mw-heading3"><h3 id="Tensional_stress_of_a_uniform_bar">Tensional stress of a uniform bar</h3></div> <p>A rod of any <a href="/wiki/Elasticity_(physics)" title="Elasticity (physics)">elastic</a> material may be viewed as a linear <a href="/wiki/Spring_(device)" title="Spring (device)">spring</a>. The rod has length <span class="texhtml mvar" style="font-style:italic;">L</span> and cross-sectional area <span class="texhtml mvar" style="font-style:italic;">A</span>. Its <a href="/wiki/Tensile_stress" class="mw-redirect" title="Tensile stress">tensile stress</a> <span class="texhtml mvar" style="font-style:italic;">σ</span> is linearly proportional to its fractional extension or strain <span class="texhtml mvar" style="font-style:italic;">ε</span> by the <a href="/wiki/Modulus_of_elasticity" class="mw-redirect" title="Modulus of elasticity">modulus of elasticity</a> <span class="texhtml mvar" style="font-style:italic;">E</span>: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma =E\varepsilon .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>σ<!-- σ --></mi> <mo>=</mo> <mi>E</mi> <mi>ε<!-- ε --></mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma =E\varepsilon .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/14bdaba7a5058e54763101d6b7540d944f0f276c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.934ex; height:2.176ex;" alt="{\displaystyle \sigma =E\varepsilon .}"></span> </p><p>The modulus of elasticity may often be considered constant. In turn, <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon ={\frac {\Delta L}{L}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ε<!-- ε --></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">Δ<!-- Δ --></mi> <mi>L</mi> </mrow> <mi>L</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon ={\frac {\Delta L}{L}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5bb5cadd9a7504b7ce19dc37b5ffd1e002f2b33c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:8.537ex; height:5.343ex;" alt="{\displaystyle \varepsilon ={\frac {\Delta L}{L}}}"></span> (that is, the fractional change in length), and since <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma ={\frac {F}{A}}\,,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>σ<!-- σ --></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>F</mi> <mi>A</mi> </mfrac> </mrow> <mspace width="thinmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma ={\frac {F}{A}}\,,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/19f4991f5ac3a0de52148dd4d2d6247c28527e51" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:8.041ex; height:5.343ex;" alt="{\displaystyle \sigma ={\frac {F}{A}}\,,}"></span> it follows that: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon ={\frac {\sigma }{E}}={\frac {F}{AE}}\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ε<!-- ε --></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>σ<!-- σ --></mi> <mi>E</mi> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>F</mi> <mrow> <mi>A</mi> <mi>E</mi> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon ={\frac {\sigma }{E}}={\frac {F}{AE}}\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c0da521b168499d3bc64a9ba851bc6759c8798b0" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:15.281ex; height:5.343ex;" alt="{\displaystyle \varepsilon ={\frac {\sigma }{E}}={\frac {F}{AE}}\,.}"></span> </p><p>The change in length may be expressed as </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta L=\varepsilon L={\frac {FL}{AE}}\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ<!-- Δ --></mi> <mi>L</mi> <mo>=</mo> <mi>ε<!-- ε --></mi> <mi>L</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>F</mi> <mi>L</mi> </mrow> <mrow> <mi>A</mi> <mi>E</mi> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta L=\varepsilon L={\frac {FL}{AE}}\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ff804ad0c5458c9a4dc333c729ecca4037f3a1ce" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:17.771ex; height:5.343ex;" alt="{\displaystyle \Delta L=\varepsilon L={\frac {FL}{AE}}\,.}"></span> </p> <div class="mw-heading mw-heading3"><h3 id="Spring_energy">Spring energy</h3></div> <p>The potential energy <span class="texhtml"><i>U</i><sub>el</sub>(<i>x</i>)</span> stored in a spring is given by <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U_{\mathrm {el} }(x)={\tfrac {1}{2}}kx^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">l</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mi>k</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U_{\mathrm {el} }(x)={\tfrac {1}{2}}kx^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/40931a3eaa783f9762a73e61acb223dadf9c69a9" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:14.498ex; height:3.509ex;" alt="{\displaystyle U_{\mathrm {el} }(x)={\tfrac {1}{2}}kx^{2}}"></span> which comes from adding up the energy it takes to incrementally compress the spring. That is, the integral of force over displacement. Since the external force has the same general direction as the displacement, the potential energy of a spring is always non-negative. Substituting <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=F/k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=F/k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f8e8c649a0af91d581fdc720bbe6d99746956fee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.543ex; height:2.843ex;" alt="{\displaystyle x=F/k}"></span> gives <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U_{\mathrm {el} }(F)={\frac {F^{2}}{2k}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">l</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mi>F</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mn>2</mn> <mi>k</mi> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U_{\mathrm {el} }(F)={\frac {F^{2}}{2k}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d92ea961f8be0b34ad1159171c95e51b6b32555" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:14.008ex; height:5.843ex;" alt="{\displaystyle U_{\mathrm {el} }(F)={\frac {F^{2}}{2k}}.}"></span> </p><p>This potential <span class="texhtml"><i>U</i><sub>el</sub></span> can be visualized as a <a href="/wiki/Parabola" title="Parabola">parabola</a> on the <span class="texhtml mvar" style="font-style:italic;">Ux</span>-plane such that <span class="texhtml"><i>U</i><sub>el</sub>(<i>x</i>) = <style data-mw-deduplicate="TemplateStyles:r1214402035">.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}</style><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span><i>kx</i><sup>2</sup></span>. As the spring is stretched in the positive <span class="texhtml mvar" style="font-style:italic;">x</span>-direction, the potential energy increases parabolically (the same thing happens as the spring is compressed). Since the change in potential energy changes at a constant rate: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {d^{2}U_{\mathrm {el} }}{dx^{2}}}=k\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">l</mi> </mrow> </mrow> </msub> </mrow> <mrow> <mi>d</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>=</mo> <mi>k</mi> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {d^{2}U_{\mathrm {el} }}{dx^{2}}}=k\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2eceec3c55f3bfe5600f35b638606899b3dc9e12" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:11.459ex; height:6.009ex;" alt="{\displaystyle {\frac {d^{2}U_{\mathrm {el} }}{dx^{2}}}=k\,.}"></span> Note that the change in the change in <span class="texhtml mvar" style="font-style:italic;">U</span> is constant even when the displacement and acceleration are zero. </p> <div class="mw-heading mw-heading3"><h3 id="Relaxed_force_constants_(generalized_compliance_constants)"><span id="Relaxed_force_constants_.28generalized_compliance_constants.29"></span>Relaxed force constants (generalized compliance constants)</h3></div> <p>Relaxed force constants (the inverse of generalized <a href="/wiki/Compliance_Constants" class="mw-redirect" title="Compliance Constants">compliance constants</a>) are uniquely defined for molecular systems, in contradistinction to the usual "rigid" force constants, and thus their use allows meaningful correlations to be made between force fields calculated for <a href="/wiki/Reactant" class="mw-redirect" title="Reactant">reactants</a>, <a href="/wiki/Transition_state" title="Transition state">transition states</a>, and products of a <a href="/wiki/Chemical_reaction" title="Chemical reaction">chemical reaction</a>. Just as the <a href="/wiki/Potential_energy" title="Potential energy">potential energy</a> can be written as a quadratic form in the internal coordinates, so it can also be written in terms of generalized forces. The resulting coefficients are termed <a href="/wiki/Compliance_constant" class="mw-redirect" title="Compliance constant">compliance constants</a>. A direct method exists for calculating the compliance constant for any internal coordinate of a molecule, without the need to do the normal mode analysis.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> The suitability of relaxed force constants (inverse compliance constants) as <a href="/wiki/Covalent_bond" title="Covalent bond">covalent bond</a> strength descriptors was demonstrated as early as 1980. Recently, the suitability as non-covalent bond strength descriptors was demonstrated too.<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Harmonic_oscillator">Harmonic oscillator</h3></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Harmonic_oscillator" title="Harmonic oscillator">Harmonic oscillator</a></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Mass-spring-system.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/5a/Mass-spring-system.png/170px-Mass-spring-system.png" decoding="async" width="170" height="280" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/5a/Mass-spring-system.png/255px-Mass-spring-system.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/5a/Mass-spring-system.png/340px-Mass-spring-system.png 2x" data-file-width="1030" data-file-height="1699" /></a><figcaption>A mass suspended by a spring is the classical example of a harmonic oscillator</figcaption></figure> <p>A mass <span class="texhtml mvar" style="font-style:italic;">m</span> attached to the end of a spring is a classic example of a <a href="/wiki/Harmonic_oscillator" title="Harmonic oscillator">harmonic oscillator</a>. By pulling slightly on the mass and then releasing it, the system will be set in <a href="/wiki/Sine_wave" title="Sine wave">sinusoidal</a> oscillating motion about the equilibrium position. To the extent that the spring obeys Hooke's law, and that one can neglect <a href="/wiki/Friction" title="Friction">friction</a> and the mass of the spring, the amplitude of the oscillation will remain constant; and its <a href="/wiki/Frequency" title="Frequency">frequency</a> <span class="texhtml mvar" style="font-style:italic;">f</span> will be independent of its amplitude, determined only by the mass and the stiffness of the spring: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f={\frac {1}{2\pi }}{\sqrt {\frac {k}{m}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>π<!-- π --></mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mi>k</mi> <mi>m</mi> </mfrac> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f={\frac {1}{2\pi }}{\sqrt {\frac {k}{m}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/828dc0fac93d162af7251eccedf8c506efb93151" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:12.908ex; height:6.176ex;" alt="{\displaystyle f={\frac {1}{2\pi }}{\sqrt {\frac {k}{m}}}}"></span> This phenomenon made possible the construction of accurate <a href="/wiki/Mechanical_clock" class="mw-redirect" title="Mechanical clock">mechanical clocks</a> and watches that could be carried on ships and people's pockets. </p> <div class="mw-heading mw-heading3"><h3 id="Rotation_in_gravity-free_space">Rotation in gravity-free space</h3></div> <p>If the mass <span class="texhtml mvar" style="font-style:italic;">m</span> were attached to a spring with force constant <span class="texhtml mvar" style="font-style:italic;">k</span> and rotating in free space, the spring tension (<span class="texhtml"><i>F</i><sub>t</sub></span>) would supply the required <a href="/wiki/Centripetal_force" title="Centripetal force">centripetal force</a> (<span class="texhtml"><i>F</i><sub>c</sub></span>): </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{\mathrm {t} }=kx\,;\qquad F_{\mathrm {c} }=m\omega ^{2}r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">t</mi> </mrow> </mrow> </msub> <mo>=</mo> <mi>k</mi> <mi>x</mi> <mspace width="thinmathspace" /> <mo>;</mo> <mspace width="2em" /> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">c</mi> </mrow> </mrow> </msub> <mo>=</mo> <mi>m</mi> <msup> <mi>ω<!-- ω --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{\mathrm {t} }=kx\,;\qquad F_{\mathrm {c} }=m\omega ^{2}r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1000729020923ad8593e2992c31346d25ab9d9a3" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:25.217ex; height:3.009ex;" alt="{\displaystyle F_{\mathrm {t} }=kx\,;\qquad F_{\mathrm {c} }=m\omega ^{2}r}"></span> Since <span class="texhtml"><i>F</i><sub>t</sub> = <i>F</i><sub>c</sub></span> and <span class="texhtml"><i>x</i> = <i>r</i></span>, then: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k=m\omega ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>=</mo> <mi>m</mi> <msup> <mi>ω<!-- ω --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k=m\omega ^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba9cfe124acf6867c7d3e751208f2a27aa7245e5" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.85ex; height:2.676ex;" alt="{\displaystyle k=m\omega ^{2}}"></span> Given that <span class="texhtml"><i>ω</i> = 2π<i>f</i></span>, this leads to the same frequency equation as above: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f={\frac {1}{2\pi }}{\sqrt {\frac {k}{m}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>π<!-- π --></mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mi>k</mi> <mi>m</mi> </mfrac> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f={\frac {1}{2\pi }}{\sqrt {\frac {k}{m}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/828dc0fac93d162af7251eccedf8c506efb93151" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:12.908ex; height:6.176ex;" alt="{\displaystyle f={\frac {1}{2\pi }}{\sqrt {\frac {k}{m}}}}"></span> </p> <div class="mw-heading mw-heading2"><h2 id="Linear_elasticity_theory_for_continuous_media">Linear elasticity theory for continuous media</h2></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Elasticity_tensor" title="Elasticity tensor">Elasticity tensor</a></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Note: the <a href="/wiki/Einstein_summation_convention" class="mw-redirect" title="Einstein summation convention">Einstein summation convention</a> of summing on repeated indices is used below.</div> <div class="mw-heading mw-heading3"><h3 id="Isotropic_materials">Isotropic materials</h3></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">For an analogous development for viscous fluids, see <a href="/wiki/Viscosity" title="Viscosity">Viscosity</a>.</div> <p>Isotropic materials are characterized by properties which are independent of direction in space. Physical equations involving isotropic materials must therefore be independent of the coordinate system chosen to represent them. The strain tensor is a symmetric tensor. Since the <a href="/wiki/Trace_(linear_algebra)" title="Trace (linear algebra)">trace</a> of any tensor is independent of any coordinate system, the most complete coordinate-free decomposition of a symmetric tensor is to represent it as the sum of a constant tensor and a traceless symmetric tensor.<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> Thus in <a href="/wiki/Ricci_calculus" title="Ricci calculus">index notation</a>: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon _{ij}=\left({\tfrac {1}{3}}\varepsilon _{kk}\delta _{ij}\right)+\left(\varepsilon _{ij}-{\tfrac {1}{3}}\varepsilon _{kk}\delta _{ij}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ε<!-- ε --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mstyle> </mrow> <msub> <mi>ε<!-- ε --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mi>k</mi> </mrow> </msub> <msub> <mi>δ<!-- δ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mrow> <mo>(</mo> <mrow> <msub> <mi>ε<!-- ε --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>−<!-- − --></mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mstyle> </mrow> <msub> <mi>ε<!-- ε --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mi>k</mi> </mrow> </msub> <msub> <mi>δ<!-- δ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon _{ij}=\left({\tfrac {1}{3}}\varepsilon _{kk}\delta _{ij}\right)+\left(\varepsilon _{ij}-{\tfrac {1}{3}}\varepsilon _{kk}\delta _{ij}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fea2ce3c2eca8e47b46f9c6b462720872e95bb71" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:33.844ex; height:4.843ex;" alt="{\displaystyle \varepsilon _{ij}=\left({\tfrac {1}{3}}\varepsilon _{kk}\delta _{ij}\right)+\left(\varepsilon _{ij}-{\tfrac {1}{3}}\varepsilon _{kk}\delta _{ij}\right)}"></span> where <span class="texhtml mvar" style="font-style:italic;">δ<sub>ij</sub></span> is the <a href="/wiki/Kronecker_delta" title="Kronecker delta">Kronecker delta</a>. In direct tensor notation: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {\varepsilon }}=\operatorname {vol} ({\boldsymbol {\varepsilon }})+\operatorname {dev} ({\boldsymbol {\varepsilon }})\,;\qquad \operatorname {vol} ({\boldsymbol {\varepsilon }})={\tfrac {1}{3}}\operatorname {tr} ({\boldsymbol {\varepsilon }})~\mathbf {I} \,;\qquad \operatorname {dev} ({\boldsymbol {\varepsilon }})={\boldsymbol {\varepsilon }}-\operatorname {vol} ({\boldsymbol {\varepsilon }})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ε<!-- ε --></mi> </mrow> <mo>=</mo> <mi>vol</mi> <mo>⁡<!-- --></mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ε<!-- ε --></mi> </mrow> <mo stretchy="false">)</mo> <mo>+</mo> <mi>dev</mi> <mo>⁡<!-- --></mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ε<!-- ε --></mi> </mrow> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mo>;</mo> <mspace width="2em" /> <mi>vol</mi> <mo>⁡<!-- --></mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ε<!-- ε --></mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mstyle> </mrow> <mi>tr</mi> <mo>⁡<!-- --></mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ε<!-- ε --></mi> </mrow> <mo stretchy="false">)</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">I</mi> </mrow> <mspace width="thinmathspace" /> <mo>;</mo> <mspace width="2em" /> <mi>dev</mi> <mo>⁡<!-- --></mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ε<!-- ε --></mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ε<!-- ε --></mi> </mrow> <mo>−<!-- − --></mo> <mi>vol</mi> <mo>⁡<!-- --></mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ε<!-- ε --></mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {\varepsilon }}=\operatorname {vol} ({\boldsymbol {\varepsilon }})+\operatorname {dev} ({\boldsymbol {\varepsilon }})\,;\qquad \operatorname {vol} ({\boldsymbol {\varepsilon }})={\tfrac {1}{3}}\operatorname {tr} ({\boldsymbol {\varepsilon }})~\mathbf {I} \,;\qquad \operatorname {dev} ({\boldsymbol {\varepsilon }})={\boldsymbol {\varepsilon }}-\operatorname {vol} ({\boldsymbol {\varepsilon }})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fee1ce9130b2d4cb000e3d2dcd96b2ebfa172902" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:69.474ex; height:3.676ex;" alt="{\displaystyle {\boldsymbol {\varepsilon }}=\operatorname {vol} ({\boldsymbol {\varepsilon }})+\operatorname {dev} ({\boldsymbol {\varepsilon }})\,;\qquad \operatorname {vol} ({\boldsymbol {\varepsilon }})={\tfrac {1}{3}}\operatorname {tr} ({\boldsymbol {\varepsilon }})~\mathbf {I} \,;\qquad \operatorname {dev} ({\boldsymbol {\varepsilon }})={\boldsymbol {\varepsilon }}-\operatorname {vol} ({\boldsymbol {\varepsilon }})}"></span> </p><p>where <span class="texhtml"><b>I</b></span> is the second-order identity tensor. </p><p>The first term on the right is the constant tensor, also known as the <b>volumetric strain tensor</b>, and the second term is the traceless symmetric tensor, also known as the <b>deviatoric strain tensor</b> or <a href="/w/index.php?title=Shear_tensor&action=edit&redlink=1" class="new" title="Shear tensor (page does not exist)">shear tensor</a>. </p><p>The most general form of Hooke's law for isotropic materials may now be written as a linear combination of these two tensors: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{ij}=3K\left({\tfrac {1}{3}}\varepsilon _{kk}\delta _{ij}\right)+2G\left(\varepsilon _{ij}-{\tfrac {1}{3}}\varepsilon _{kk}\delta _{ij}\right)\,;\qquad {\boldsymbol {\sigma }}=3K\operatorname {vol} ({\boldsymbol {\varepsilon }})+2G\operatorname {dev} ({\boldsymbol {\varepsilon }})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>σ<!-- σ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>=</mo> <mn>3</mn> <mi>K</mi> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mstyle> </mrow> <msub> <mi>ε<!-- ε --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mi>k</mi> </mrow> </msub> <msub> <mi>δ<!-- δ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mn>2</mn> <mi>G</mi> <mrow> <mo>(</mo> <mrow> <msub> <mi>ε<!-- ε --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>−<!-- − --></mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mstyle> </mrow> <msub> <mi>ε<!-- ε --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mi>k</mi> </mrow> </msub> <msub> <mi>δ<!-- δ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mspace width="thinmathspace" /> <mo>;</mo> <mspace width="2em" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">σ<!-- σ --></mi> </mrow> <mo>=</mo> <mn>3</mn> <mi>K</mi> <mi>vol</mi> <mo>⁡<!-- --></mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ε<!-- ε --></mi> </mrow> <mo stretchy="false">)</mo> <mo>+</mo> <mn>2</mn> <mi>G</mi> <mi>dev</mi> <mo>⁡<!-- --></mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ε<!-- ε --></mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{ij}=3K\left({\tfrac {1}{3}}\varepsilon _{kk}\delta _{ij}\right)+2G\left(\varepsilon _{ij}-{\tfrac {1}{3}}\varepsilon _{kk}\delta _{ij}\right)\,;\qquad {\boldsymbol {\sigma }}=3K\operatorname {vol} ({\boldsymbol {\varepsilon }})+2G\operatorname {dev} ({\boldsymbol {\varepsilon }})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2be31e2b0563edcf34240c2fdd210a917af075b5" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:74.726ex; height:4.843ex;" alt="{\displaystyle \sigma _{ij}=3K\left({\tfrac {1}{3}}\varepsilon _{kk}\delta _{ij}\right)+2G\left(\varepsilon _{ij}-{\tfrac {1}{3}}\varepsilon _{kk}\delta _{ij}\right)\,;\qquad {\boldsymbol {\sigma }}=3K\operatorname {vol} ({\boldsymbol {\varepsilon }})+2G\operatorname {dev} ({\boldsymbol {\varepsilon }})}"></span> where <span class="texhtml mvar" style="font-style:italic;">K</span> is the <a href="/wiki/Bulk_modulus" title="Bulk modulus">bulk modulus</a> and <span class="texhtml mvar" style="font-style:italic;">G</span> is the <a href="/wiki/Shear_modulus" title="Shear modulus">shear modulus</a>. </p><p>Using the relationships between the <a href="/wiki/Elastic_modulus" title="Elastic modulus">elastic moduli</a>, these equations may also be expressed in various other ways. A common form of Hooke's law for isotropic materials, expressed in direct tensor notation, is <sup id="cite_ref-Simo98_11-0" class="reference"><a href="#cite_note-Simo98-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {\sigma }}=\lambda \operatorname {tr} ({\boldsymbol {\varepsilon }})\mathbf {I} +2\mu {\boldsymbol {\varepsilon }}={\mathsf {c}}:{\boldsymbol {\varepsilon }}\,;\qquad {\mathsf {c}}=\lambda \mathbf {I} \otimes \mathbf {I} +2\mu {\mathsf {I}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">σ<!-- σ --></mi> </mrow> <mo>=</mo> <mi>λ<!-- λ --></mi> <mi>tr</mi> <mo>⁡<!-- --></mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ε<!-- ε --></mi> </mrow> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">I</mi> </mrow> <mo>+</mo> <mn>2</mn> <mi>μ<!-- μ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ε<!-- ε --></mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">c</mi> </mrow> </mrow> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ε<!-- ε --></mi> </mrow> <mspace width="thinmathspace" /> <mo>;</mo> <mspace width="2em" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">c</mi> </mrow> </mrow> <mo>=</mo> <mi>λ<!-- λ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">I</mi> </mrow> <mo>⊗<!-- ⊗ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">I</mi> </mrow> <mo>+</mo> <mn>2</mn> <mi>μ<!-- μ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">I</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {\sigma }}=\lambda \operatorname {tr} ({\boldsymbol {\varepsilon }})\mathbf {I} +2\mu {\boldsymbol {\varepsilon }}={\mathsf {c}}:{\boldsymbol {\varepsilon }}\,;\qquad {\mathsf {c}}=\lambda \mathbf {I} \otimes \mathbf {I} +2\mu {\mathsf {I}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f074d855776ea73dae96db89f1a63a879c70eef0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:48.708ex; height:2.843ex;" alt="{\displaystyle {\boldsymbol {\sigma }}=\lambda \operatorname {tr} ({\boldsymbol {\varepsilon }})\mathbf {I} +2\mu {\boldsymbol {\varepsilon }}={\mathsf {c}}:{\boldsymbol {\varepsilon }}\,;\qquad {\mathsf {c}}=\lambda \mathbf {I} \otimes \mathbf {I} +2\mu {\mathsf {I}}}"></span> where <span class="texhtml"><i>λ</i> = <i>K</i> − <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">2</span><span class="sr-only">/</span><span class="den">3</span></span>⁠</span><i>G</i> = <i>c</i><sub>1111</sub> − 2<i>c</i><sub>1212</sub></span> and <span class="texhtml"><i>μ</i> = <i>G</i> = <i>c</i><sub>1212</sub></span> are the <a href="/wiki/Lam%C3%A9_constants" class="mw-redirect" title="Lamé constants">Lamé constants</a>, <span class="texhtml"><b>I</b></span> is the second-rank identity tensor, and <b>I</b> is the symmetric part of the fourth-rank identity tensor. In index notation: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{ij}=\lambda \varepsilon _{kk}~\delta _{ij}+2\mu \varepsilon _{ij}=c_{ijkl}\varepsilon _{kl}\,;\qquad c_{ijkl}=\lambda \delta _{ij}\delta _{kl}+\mu \left(\delta _{ik}\delta _{jl}+\delta _{il}\delta _{jk}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>σ<!-- σ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>=</mo> <mi>λ<!-- λ --></mi> <msub> <mi>ε<!-- ε --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mi>k</mi> </mrow> </msub> <mtext> </mtext> <msub> <mi>δ<!-- δ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>+</mo> <mn>2</mn> <mi>μ<!-- μ --></mi> <msub> <mi>ε<!-- ε --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> <mi>k</mi> <mi>l</mi> </mrow> </msub> <msub> <mi>ε<!-- ε --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mi>l</mi> </mrow> </msub> <mspace width="thinmathspace" /> <mo>;</mo> <mspace width="2em" /> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> <mi>k</mi> <mi>l</mi> </mrow> </msub> <mo>=</mo> <mi>λ<!-- λ --></mi> <msub> <mi>δ<!-- δ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <msub> <mi>δ<!-- δ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mi>l</mi> </mrow> </msub> <mo>+</mo> <mi>μ<!-- μ --></mi> <mrow> <mo>(</mo> <mrow> <msub> <mi>δ<!-- δ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>k</mi> </mrow> </msub> <msub> <mi>δ<!-- δ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mi>l</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>δ<!-- δ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>l</mi> </mrow> </msub> <msub> <mi>δ<!-- δ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mi>k</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{ij}=\lambda \varepsilon _{kk}~\delta _{ij}+2\mu \varepsilon _{ij}=c_{ijkl}\varepsilon _{kl}\,;\qquad c_{ijkl}=\lambda \delta _{ij}\delta _{kl}+\mu \left(\delta _{ik}\delta _{jl}+\delta _{il}\delta _{jk}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aba5089d1f53e3cc2ae9cbdbdd72df1f49245e60" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:69.926ex; height:3.009ex;" alt="{\displaystyle \sigma _{ij}=\lambda \varepsilon _{kk}~\delta _{ij}+2\mu \varepsilon _{ij}=c_{ijkl}\varepsilon _{kl}\,;\qquad c_{ijkl}=\lambda \delta _{ij}\delta _{kl}+\mu \left(\delta _{ik}\delta _{jl}+\delta _{il}\delta _{jk}\right)}"></span> </p><p>The inverse relationship is<sup id="cite_ref-Milton02_12-0" class="reference"><a href="#cite_note-Milton02-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {\varepsilon }}={\frac {1}{2\mu }}{\boldsymbol {\sigma }}-{\frac {\lambda }{2\mu (3\lambda +2\mu )}}\operatorname {tr} ({\boldsymbol {\sigma }})\mathbf {I} ={\frac {1}{2G}}{\boldsymbol {\sigma }}+\left({\frac {1}{9K}}-{\frac {1}{6G}}\right)\operatorname {tr} ({\boldsymbol {\sigma }})\mathbf {I} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ε<!-- ε --></mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>μ<!-- μ --></mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">σ<!-- σ --></mi> </mrow> <mo>−<!-- − --></mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>λ<!-- λ --></mi> <mrow> <mn>2</mn> <mi>μ<!-- μ --></mi> <mo stretchy="false">(</mo> <mn>3</mn> <mi>λ<!-- λ --></mi> <mo>+</mo> <mn>2</mn> <mi>μ<!-- μ --></mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mi>tr</mi> <mo>⁡<!-- --></mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">σ<!-- σ --></mi> </mrow> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">I</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>G</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">σ<!-- σ --></mi> </mrow> <mo>+</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>9</mn> <mi>K</mi> </mrow> </mfrac> </mrow> <mo>−<!-- − --></mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>6</mn> <mi>G</mi> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mi>tr</mi> <mo>⁡<!-- --></mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">σ<!-- σ --></mi> </mrow> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">I</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {\varepsilon }}={\frac {1}{2\mu }}{\boldsymbol {\sigma }}-{\frac {\lambda }{2\mu (3\lambda +2\mu )}}\operatorname {tr} ({\boldsymbol {\sigma }})\mathbf {I} ={\frac {1}{2G}}{\boldsymbol {\sigma }}+\left({\frac {1}{9K}}-{\frac {1}{6G}}\right)\operatorname {tr} ({\boldsymbol {\sigma }})\mathbf {I} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9ec441cf1a0472885bc3e2ed137317a577a14c76" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:64.047ex; height:6.343ex;" alt="{\displaystyle {\boldsymbol {\varepsilon }}={\frac {1}{2\mu }}{\boldsymbol {\sigma }}-{\frac {\lambda }{2\mu (3\lambda +2\mu )}}\operatorname {tr} ({\boldsymbol {\sigma }})\mathbf {I} ={\frac {1}{2G}}{\boldsymbol {\sigma }}+\left({\frac {1}{9K}}-{\frac {1}{6G}}\right)\operatorname {tr} ({\boldsymbol {\sigma }})\mathbf {I} }"></span> </p><p>Therefore, the compliance tensor in the relation <span class="texhtml"><b>ε</b> = <b>s</b> : <b>σ</b></span> is </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {s}}=-{\frac {\lambda }{2\mu (3\lambda +2\mu )}}\mathbf {I} \otimes \mathbf {I} +{\frac {1}{2\mu }}{\mathsf {I}}=\left({\frac {1}{9K}}-{\frac {1}{6G}}\right)\mathbf {I} \otimes \mathbf {I} +{\frac {1}{2G}}{\mathsf {I}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">s</mi> </mrow> </mrow> <mo>=</mo> <mo>−<!-- − --></mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>λ<!-- λ --></mi> <mrow> <mn>2</mn> <mi>μ<!-- μ --></mi> <mo stretchy="false">(</mo> <mn>3</mn> <mi>λ<!-- λ --></mi> <mo>+</mo> <mn>2</mn> <mi>μ<!-- μ --></mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">I</mi> </mrow> <mo>⊗<!-- ⊗ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">I</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>μ<!-- μ --></mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">I</mi> </mrow> </mrow> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>9</mn> <mi>K</mi> </mrow> </mfrac> </mrow> <mo>−<!-- − --></mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>6</mn> <mi>G</mi> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">I</mi> </mrow> <mo>⊗<!-- ⊗ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">I</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>G</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">I</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {s}}=-{\frac {\lambda }{2\mu (3\lambda +2\mu )}}\mathbf {I} \otimes \mathbf {I} +{\frac {1}{2\mu }}{\mathsf {I}}=\left({\frac {1}{9K}}-{\frac {1}{6G}}\right)\mathbf {I} \otimes \mathbf {I} +{\frac {1}{2G}}{\mathsf {I}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8a1509695234d82344aa7f005653827cd960c08d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:60.502ex; height:6.343ex;" alt="{\displaystyle {\mathsf {s}}=-{\frac {\lambda }{2\mu (3\lambda +2\mu )}}\mathbf {I} \otimes \mathbf {I} +{\frac {1}{2\mu }}{\mathsf {I}}=\left({\frac {1}{9K}}-{\frac {1}{6G}}\right)\mathbf {I} \otimes \mathbf {I} +{\frac {1}{2G}}{\mathsf {I}}}"></span> </p><p>In terms of <a href="/wiki/Young%27s_modulus" title="Young's modulus">Young's modulus</a> and <a href="/wiki/Poisson%27s_ratio" title="Poisson's ratio">Poisson's ratio</a>, Hooke's law for isotropic materials can then be expressed as </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon _{ij}={\frac {1}{E}}{\big (}\sigma _{ij}-\nu (\sigma _{kk}\delta _{ij}-\sigma _{ij}){\big )}\,;\qquad {\boldsymbol {\varepsilon }}={\frac {1}{E}}{\big (}{\boldsymbol {\sigma }}-\nu (\operatorname {tr} ({\boldsymbol {\sigma }})\mathbf {I} -{\boldsymbol {\sigma }}){\big )}={\frac {1+\nu }{E}}{\boldsymbol {\sigma }}-{\frac {\nu }{E}}\operatorname {tr} ({\boldsymbol {\sigma }})\mathbf {I} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ε<!-- ε --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>E</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <msub> <mi>σ<!-- σ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>−<!-- − --></mo> <mi>ν<!-- ν --></mi> <mo stretchy="false">(</mo> <msub> <mi>σ<!-- σ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mi>k</mi> </mrow> </msub> <msub> <mi>δ<!-- δ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>−<!-- − --></mo> <msub> <mi>σ<!-- σ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mspace width="thinmathspace" /> <mo>;</mo> <mspace width="2em" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ε<!-- ε --></mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>E</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">σ<!-- σ --></mi> </mrow> <mo>−<!-- − --></mo> <mi>ν<!-- ν --></mi> <mo stretchy="false">(</mo> <mi>tr</mi> <mo>⁡<!-- --></mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">σ<!-- σ --></mi> </mrow> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">I</mi> </mrow> <mo>−<!-- − --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">σ<!-- σ --></mi> </mrow> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>+</mo> <mi>ν<!-- ν --></mi> </mrow> <mi>E</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">σ<!-- σ --></mi> </mrow> <mo>−<!-- − --></mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ν<!-- ν --></mi> <mi>E</mi> </mfrac> </mrow> <mi>tr</mi> <mo>⁡<!-- --></mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">σ<!-- σ --></mi> </mrow> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">I</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon _{ij}={\frac {1}{E}}{\big (}\sigma _{ij}-\nu (\sigma _{kk}\delta _{ij}-\sigma _{ij}){\big )}\,;\qquad {\boldsymbol {\varepsilon }}={\frac {1}{E}}{\big (}{\boldsymbol {\sigma }}-\nu (\operatorname {tr} ({\boldsymbol {\sigma }})\mathbf {I} -{\boldsymbol {\sigma }}){\big )}={\frac {1+\nu }{E}}{\boldsymbol {\sigma }}-{\frac {\nu }{E}}\operatorname {tr} ({\boldsymbol {\sigma }})\mathbf {I} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3b18fe3b6a5872b16c61f9f6ce980c235bcae38" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:86.632ex; height:5.176ex;" alt="{\displaystyle \varepsilon _{ij}={\frac {1}{E}}{\big (}\sigma _{ij}-\nu (\sigma _{kk}\delta _{ij}-\sigma _{ij}){\big )}\,;\qquad {\boldsymbol {\varepsilon }}={\frac {1}{E}}{\big (}{\boldsymbol {\sigma }}-\nu (\operatorname {tr} ({\boldsymbol {\sigma }})\mathbf {I} -{\boldsymbol {\sigma }}){\big )}={\frac {1+\nu }{E}}{\boldsymbol {\sigma }}-{\frac {\nu }{E}}\operatorname {tr} ({\boldsymbol {\sigma }})\mathbf {I} }"></span> </p><p>This is the form in which the strain is expressed in terms of the stress tensor in engineering. The expression in expanded form is <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\varepsilon _{11}&={\frac {1}{E}}{\big (}\sigma _{11}-\nu (\sigma _{22}+\sigma _{33}){\big )}\\\varepsilon _{22}&={\frac {1}{E}}{\big (}\sigma _{22}-\nu (\sigma _{11}+\sigma _{33}){\big )}\\\varepsilon _{33}&={\frac {1}{E}}{\big (}\sigma _{33}-\nu (\sigma _{11}+\sigma _{22}){\big )}\\\varepsilon _{12}&={\frac {1}{2G}}\sigma _{12}\,;\qquad \varepsilon _{13}={\frac {1}{2G}}\sigma _{13}\,;\qquad \varepsilon _{23}={\frac {1}{2G}}\sigma _{23}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>ε<!-- ε --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>11</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>E</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <msub> <mi>σ<!-- σ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>11</mn> </mrow> </msub> <mo>−<!-- − --></mo> <mi>ν<!-- ν --></mi> <mo stretchy="false">(</mo> <msub> <mi>σ<!-- σ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>22</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>σ<!-- σ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>33</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <msub> <mi>ε<!-- ε --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>22</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>E</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <msub> <mi>σ<!-- σ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>22</mn> </mrow> </msub> <mo>−<!-- − --></mo> <mi>ν<!-- ν --></mi> <mo stretchy="false">(</mo> <msub> <mi>σ<!-- σ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>11</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>σ<!-- σ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>33</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <msub> <mi>ε<!-- ε --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>33</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>E</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <msub> <mi>σ<!-- σ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>33</mn> </mrow> </msub> <mo>−<!-- − --></mo> <mi>ν<!-- ν --></mi> <mo stretchy="false">(</mo> <msub> <mi>σ<!-- σ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>11</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>σ<!-- σ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>22</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <msub> <mi>ε<!-- ε --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>G</mi> </mrow> </mfrac> </mrow> <msub> <mi>σ<!-- σ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> <mspace width="thinmathspace" /> <mo>;</mo> <mspace width="2em" /> <msub> <mi>ε<!-- ε --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>13</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>G</mi> </mrow> </mfrac> </mrow> <msub> <mi>σ<!-- σ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>13</mn> </mrow> </msub> <mspace width="thinmathspace" /> <mo>;</mo> <mspace width="2em" /> <msub> <mi>ε<!-- ε --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>23</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>G</mi> </mrow> </mfrac> </mrow> <msub> <mi>σ<!-- σ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>23</mn> </mrow> </msub> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\varepsilon _{11}&={\frac {1}{E}}{\big (}\sigma _{11}-\nu (\sigma _{22}+\sigma _{33}){\big )}\\\varepsilon _{22}&={\frac {1}{E}}{\big (}\sigma _{22}-\nu (\sigma _{11}+\sigma _{33}){\big )}\\\varepsilon _{33}&={\frac {1}{E}}{\big (}\sigma _{33}-\nu (\sigma _{11}+\sigma _{22}){\big )}\\\varepsilon _{12}&={\frac {1}{2G}}\sigma _{12}\,;\qquad \varepsilon _{13}={\frac {1}{2G}}\sigma _{13}\,;\qquad \varepsilon _{23}={\frac {1}{2G}}\sigma _{23}\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/21d3802614ca3978ceb67ef819147c73337845fd" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -10.338ex; width:52.145ex; height:21.843ex;" alt="{\displaystyle {\begin{aligned}\varepsilon _{11}&={\frac {1}{E}}{\big (}\sigma _{11}-\nu (\sigma _{22}+\sigma _{33}){\big )}\\\varepsilon _{22}&={\frac {1}{E}}{\big (}\sigma _{22}-\nu (\sigma _{11}+\sigma _{33}){\big )}\\\varepsilon _{33}&={\frac {1}{E}}{\big (}\sigma _{33}-\nu (\sigma _{11}+\sigma _{22}){\big )}\\\varepsilon _{12}&={\frac {1}{2G}}\sigma _{12}\,;\qquad \varepsilon _{13}={\frac {1}{2G}}\sigma _{13}\,;\qquad \varepsilon _{23}={\frac {1}{2G}}\sigma _{23}\end{aligned}}}"></span> where <span class="texhtml mvar" style="font-style:italic;">E</span> is <a href="/wiki/Young%27s_modulus" title="Young's modulus">Young's modulus</a> and <span class="texhtml mvar" style="font-style:italic;">ν</span> is <a href="/wiki/Poisson%27s_ratio" title="Poisson's ratio">Poisson's ratio</a>. (See <a href="/wiki/3-D_elasticity" class="mw-redirect" title="3-D elasticity">3-D elasticity</a>). </p> <style data-mw-deduplicate="TemplateStyles:r1174254338">.mw-parser-output .math_proof{border:thin solid #aaa;margin:1em 2em;padding:0.5em 1em 0.4em}@media(max-width:500px){.mw-parser-output .math_proof{margin:1em 0;padding:0.5em 0.5em 0.4em}}</style><div class="math_proof" style=""><strong>Derivation of Hooke's law in three dimensions</strong> <p>The three-dimensional form of Hooke's law can be derived using Poisson's ratio and the one-dimensional form of Hooke's law as follows. Consider the strain and stress relation as a superposition of two effects: stretching in direction of the load (1) and shrinking (caused by the load) in perpendicular directions (2 and 3), <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\varepsilon _{1}'&={\frac {1}{E}}\sigma _{1}\,,\\\varepsilon _{2}'&=-{\frac {\nu }{E}}\sigma _{1}\,,\\\varepsilon _{3}'&=-{\frac {\nu }{E}}\sigma _{1}\,,\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msubsup> <mi>ε<!-- ε --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mo>′</mo> </msubsup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>E</mi> </mfrac> </mrow> <msub> <mi>σ<!-- σ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mspace width="thinmathspace" /> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>ε<!-- ε --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mo>′</mo> </msubsup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo>−<!-- − --></mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ν<!-- ν --></mi> <mi>E</mi> </mfrac> </mrow> <msub> <mi>σ<!-- σ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mspace width="thinmathspace" /> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>ε<!-- ε --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> <mo>′</mo> </msubsup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo>−<!-- − --></mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ν<!-- ν --></mi> <mi>E</mi> </mfrac> </mrow> <msub> <mi>σ<!-- σ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mspace width="thinmathspace" /> <mo>,</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\varepsilon _{1}'&={\frac {1}{E}}\sigma _{1}\,,\\\varepsilon _{2}'&=-{\frac {\nu }{E}}\sigma _{1}\,,\\\varepsilon _{3}'&=-{\frac {\nu }{E}}\sigma _{1}\,,\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6a5fb11a34a137dbdbe53d2abdcde231df427cfe" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.72ex; margin-bottom: -0.285ex; width:13.823ex; height:15.176ex;" alt="{\displaystyle {\begin{aligned}\varepsilon _{1}'&={\frac {1}{E}}\sigma _{1}\,,\\\varepsilon _{2}'&=-{\frac {\nu }{E}}\sigma _{1}\,,\\\varepsilon _{3}'&=-{\frac {\nu }{E}}\sigma _{1}\,,\end{aligned}}}"></span> where <span class="texhtml mvar" style="font-style:italic;">ν</span> is Poisson's ratio and <span class="texhtml mvar" style="font-style:italic;">E</span> is Young's modulus. </p><p>We get similar equations to the loads in directions 2 and 3, <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\varepsilon _{1}''&=-{\frac {\nu }{E}}\sigma _{2}\,,\\\varepsilon _{2}''&={\frac {1}{E}}\sigma _{2}\,,\\\varepsilon _{3}''&=-{\frac {\nu }{E}}\sigma _{2}\,,\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msubsup> <mi>ε<!-- ε --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mo>″</mo> </msubsup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo>−<!-- − --></mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ν<!-- ν --></mi> <mi>E</mi> </mfrac> </mrow> <msub> <mi>σ<!-- σ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mspace width="thinmathspace" /> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>ε<!-- ε --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mo>″</mo> </msubsup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>E</mi> </mfrac> </mrow> <msub> <mi>σ<!-- σ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mspace width="thinmathspace" /> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>ε<!-- ε --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> <mo>″</mo> </msubsup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo>−<!-- − --></mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ν<!-- ν --></mi> <mi>E</mi> </mfrac> </mrow> <msub> <mi>σ<!-- σ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mspace width="thinmathspace" /> <mo>,</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\varepsilon _{1}''&=-{\frac {\nu }{E}}\sigma _{2}\,,\\\varepsilon _{2}''&={\frac {1}{E}}\sigma _{2}\,,\\\varepsilon _{3}''&=-{\frac {\nu }{E}}\sigma _{2}\,,\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3bc96098a06cfdf5036e680d3f2a6f25bd2202b6" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.72ex; margin-bottom: -0.285ex; width:13.906ex; height:15.176ex;" alt="{\displaystyle {\begin{aligned}\varepsilon _{1}''&=-{\frac {\nu }{E}}\sigma _{2}\,,\\\varepsilon _{2}''&={\frac {1}{E}}\sigma _{2}\,,\\\varepsilon _{3}''&=-{\frac {\nu }{E}}\sigma _{2}\,,\end{aligned}}}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\varepsilon _{1}'''&=-{\frac {\nu }{E}}\sigma _{3}\,,\\\varepsilon _{2}'''&=-{\frac {\nu }{E}}\sigma _{3}\,,\\\varepsilon _{3}'''&={\frac {1}{E}}\sigma _{3}\,.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msubsup> <mi>ε<!-- ε --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mo>‴</mo> </msubsup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo>−<!-- − --></mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ν<!-- ν --></mi> <mi>E</mi> </mfrac> </mrow> <msub> <mi>σ<!-- σ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mspace width="thinmathspace" /> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>ε<!-- ε --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mo>‴</mo> </msubsup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo>−<!-- − --></mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ν<!-- ν --></mi> <mi>E</mi> </mfrac> </mrow> <msub> <mi>σ<!-- σ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mspace width="thinmathspace" /> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>ε<!-- ε --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> <mo>‴</mo> </msubsup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>E</mi> </mfrac> </mrow> <msub> <mi>σ<!-- σ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mspace width="thinmathspace" /> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\varepsilon _{1}'''&=-{\frac {\nu }{E}}\sigma _{3}\,,\\\varepsilon _{2}'''&=-{\frac {\nu }{E}}\sigma _{3}\,,\\\varepsilon _{3}'''&={\frac {1}{E}}\sigma _{3}\,.\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/348e23d47f9d4a1e3865a24be97ad7401b2d724d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.72ex; margin-bottom: -0.285ex; width:14.358ex; height:15.176ex;" alt="{\displaystyle {\begin{aligned}\varepsilon _{1}'''&=-{\frac {\nu }{E}}\sigma _{3}\,,\\\varepsilon _{2}'''&=-{\frac {\nu }{E}}\sigma _{3}\,,\\\varepsilon _{3}'''&={\frac {1}{E}}\sigma _{3}\,.\end{aligned}}}"></span> </p><p>Summing the three cases together (<span class="texhtml"><i>ε<sub>i</sub></i> = <i>ε<sub>i</sub></i>′ + <i>ε<sub>i</sub></i>″ + <i>ε<sub>i</sub></i>‴</span>) we get <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\varepsilon _{1}&={\frac {1}{E}}{\big (}\sigma _{1}-\nu (\sigma _{2}+\sigma _{3}){\big )}\,,\\\varepsilon _{2}&={\frac {1}{E}}{\big (}\sigma _{2}-\nu (\sigma _{1}+\sigma _{3}){\big )}\,,\\\varepsilon _{3}&={\frac {1}{E}}{\big (}\sigma _{3}-\nu (\sigma _{1}+\sigma _{2}){\big )}\,,\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>ε<!-- ε --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>E</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <msub> <mi>σ<!-- σ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−<!-- − --></mo> <mi>ν<!-- ν --></mi> <mo stretchy="false">(</mo> <msub> <mi>σ<!-- σ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>σ<!-- σ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mspace width="thinmathspace" /> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>ε<!-- ε --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>E</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <msub> <mi>σ<!-- σ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−<!-- − --></mo> <mi>ν<!-- ν --></mi> <mo stretchy="false">(</mo> <msub> <mi>σ<!-- σ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>σ<!-- σ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mspace width="thinmathspace" /> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>ε<!-- ε --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>E</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <msub> <mi>σ<!-- σ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>−<!-- − --></mo> <mi>ν<!-- ν --></mi> <mo stretchy="false">(</mo> <msub> <mi>σ<!-- σ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>σ<!-- σ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mspace width="thinmathspace" /> <mo>,</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\varepsilon _{1}&={\frac {1}{E}}{\big (}\sigma _{1}-\nu (\sigma _{2}+\sigma _{3}){\big )}\,,\\\varepsilon _{2}&={\frac {1}{E}}{\big (}\sigma _{2}-\nu (\sigma _{1}+\sigma _{3}){\big )}\,,\\\varepsilon _{3}&={\frac {1}{E}}{\big (}\sigma _{3}-\nu (\sigma _{1}+\sigma _{2}){\big )}\,,\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ef0124ad64c7284f2c2356416d1418937c5c357f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.24ex; margin-bottom: -0.264ex; width:27.63ex; height:16.176ex;" alt="{\displaystyle {\begin{aligned}\varepsilon _{1}&={\frac {1}{E}}{\big (}\sigma _{1}-\nu (\sigma _{2}+\sigma _{3}){\big )}\,,\\\varepsilon _{2}&={\frac {1}{E}}{\big (}\sigma _{2}-\nu (\sigma _{1}+\sigma _{3}){\big )}\,,\\\varepsilon _{3}&={\frac {1}{E}}{\big (}\sigma _{3}-\nu (\sigma _{1}+\sigma _{2}){\big )}\,,\end{aligned}}}"></span> or by adding and subtracting one <span class="texhtml mvar" style="font-style:italic;">νσ</span> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\varepsilon _{1}&={\frac {1}{E}}{\big (}(1+\nu )\sigma _{1}-\nu (\sigma _{1}+\sigma _{2}+\sigma _{3}){\big )}\,,\\\varepsilon _{2}&={\frac {1}{E}}{\big (}(1+\nu )\sigma _{2}-\nu (\sigma _{1}+\sigma _{2}+\sigma _{3}){\big )}\,,\\\varepsilon _{3}&={\frac {1}{E}}{\big (}(1+\nu )\sigma _{3}-\nu (\sigma _{1}+\sigma _{2}+\sigma _{3}){\big )}\,,\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>ε<!-- ε --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>E</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>ν<!-- ν --></mi> <mo stretchy="false">)</mo> <msub> <mi>σ<!-- σ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−<!-- − --></mo> <mi>ν<!-- ν --></mi> <mo stretchy="false">(</mo> <msub> <mi>σ<!-- σ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>σ<!-- σ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>σ<!-- σ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mspace width="thinmathspace" /> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>ε<!-- ε --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>E</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>ν<!-- ν --></mi> <mo stretchy="false">)</mo> <msub> <mi>σ<!-- σ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−<!-- − --></mo> <mi>ν<!-- ν --></mi> <mo stretchy="false">(</mo> <msub> <mi>σ<!-- σ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>σ<!-- σ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>σ<!-- σ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mspace width="thinmathspace" /> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>ε<!-- ε --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>E</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>ν<!-- ν --></mi> <mo stretchy="false">)</mo> <msub> <mi>σ<!-- σ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>−<!-- − --></mo> <mi>ν<!-- ν --></mi> <mo stretchy="false">(</mo> <msub> <mi>σ<!-- σ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>σ<!-- σ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>σ<!-- σ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mspace width="thinmathspace" /> <mo>,</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\varepsilon _{1}&={\frac {1}{E}}{\big (}(1+\nu )\sigma _{1}-\nu (\sigma _{1}+\sigma _{2}+\sigma _{3}){\big )}\,,\\\varepsilon _{2}&={\frac {1}{E}}{\big (}(1+\nu )\sigma _{2}-\nu (\sigma _{1}+\sigma _{2}+\sigma _{3}){\big )}\,,\\\varepsilon _{3}&={\frac {1}{E}}{\big (}(1+\nu )\sigma _{3}-\nu (\sigma _{1}+\sigma _{2}+\sigma _{3}){\big )}\,,\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24903477d0abcf97bbfc6a6c1f6f8c2387cd4c4d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.24ex; margin-bottom: -0.264ex; width:39.896ex; height:16.176ex;" alt="{\displaystyle {\begin{aligned}\varepsilon _{1}&={\frac {1}{E}}{\big (}(1+\nu )\sigma _{1}-\nu (\sigma _{1}+\sigma _{2}+\sigma _{3}){\big )}\,,\\\varepsilon _{2}&={\frac {1}{E}}{\big (}(1+\nu )\sigma _{2}-\nu (\sigma _{1}+\sigma _{2}+\sigma _{3}){\big )}\,,\\\varepsilon _{3}&={\frac {1}{E}}{\big (}(1+\nu )\sigma _{3}-\nu (\sigma _{1}+\sigma _{2}+\sigma _{3}){\big )}\,,\end{aligned}}}"></span> and further we get by solving <span class="texhtml"><i>σ</i><sub>1</sub></span> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{1}={\frac {E}{1+\nu }}\varepsilon _{1}+{\frac {\nu }{1+\nu }}(\sigma _{1}+\sigma _{2}+\sigma _{3})\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>σ<!-- σ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>E</mi> <mrow> <mn>1</mn> <mo>+</mo> <mi>ν<!-- ν --></mi> </mrow> </mfrac> </mrow> <msub> <mi>ε<!-- ε --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ν<!-- ν --></mi> <mrow> <mn>1</mn> <mo>+</mo> <mi>ν<!-- ν --></mi> </mrow> </mfrac> </mrow> <mo stretchy="false">(</mo> <msub> <mi>σ<!-- σ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>σ<!-- σ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>σ<!-- σ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{1}={\frac {E}{1+\nu }}\varepsilon _{1}+{\frac {\nu }{1+\nu }}(\sigma _{1}+\sigma _{2}+\sigma _{3})\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7a722060a5eada5740c476f16d566bfa34a05761" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:38.269ex; height:5.343ex;" alt="{\displaystyle \sigma _{1}={\frac {E}{1+\nu }}\varepsilon _{1}+{\frac {\nu }{1+\nu }}(\sigma _{1}+\sigma _{2}+\sigma _{3})\,.}"></span> </p><p>Calculating the sum <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\varepsilon _{1}+\varepsilon _{2}+\varepsilon _{3}&={\frac {1}{E}}{\big (}(1+\nu )(\sigma _{1}+\sigma _{2}+\sigma _{3})-3\nu (\sigma _{1}+\sigma _{2}+\sigma _{3}){\big )}={\frac {1-2\nu }{E}}(\sigma _{1}+\sigma _{2}+\sigma _{3})\\\sigma _{1}+\sigma _{2}+\sigma _{3}&={\frac {E}{1-2\nu }}(\varepsilon _{1}+\varepsilon _{2}+\varepsilon _{3})\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>ε<!-- ε --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>ε<!-- ε --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>ε<!-- ε --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>E</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>ν<!-- ν --></mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msub> <mi>σ<!-- σ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>σ<!-- σ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>σ<!-- σ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>−<!-- − --></mo> <mn>3</mn> <mi>ν<!-- ν --></mi> <mo stretchy="false">(</mo> <msub> <mi>σ<!-- σ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>σ<!-- σ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>σ<!-- σ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>−<!-- − --></mo> <mn>2</mn> <mi>ν<!-- ν --></mi> </mrow> <mi>E</mi> </mfrac> </mrow> <mo stretchy="false">(</mo> <msub> <mi>σ<!-- σ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>σ<!-- σ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>σ<!-- σ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>σ<!-- σ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>σ<!-- σ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>σ<!-- σ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>E</mi> <mrow> <mn>1</mn> <mo>−<!-- − --></mo> <mn>2</mn> <mi>ν<!-- ν --></mi> </mrow> </mfrac> </mrow> <mo stretchy="false">(</mo> <msub> <mi>ε<!-- ε --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>ε<!-- ε --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>ε<!-- ε --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\varepsilon _{1}+\varepsilon _{2}+\varepsilon _{3}&={\frac {1}{E}}{\big (}(1+\nu )(\sigma _{1}+\sigma _{2}+\sigma _{3})-3\nu (\sigma _{1}+\sigma _{2}+\sigma _{3}){\big )}={\frac {1-2\nu }{E}}(\sigma _{1}+\sigma _{2}+\sigma _{3})\\\sigma _{1}+\sigma _{2}+\sigma _{3}&={\frac {E}{1-2\nu }}(\varepsilon _{1}+\varepsilon _{2}+\varepsilon _{3})\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8709b11e9fdcb2fe9ad2d5d7f5170c5e8c75fd7d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.612ex; margin-bottom: -0.226ex; width:87.933ex; height:10.843ex;" alt="{\displaystyle {\begin{aligned}\varepsilon _{1}+\varepsilon _{2}+\varepsilon _{3}&={\frac {1}{E}}{\big (}(1+\nu )(\sigma _{1}+\sigma _{2}+\sigma _{3})-3\nu (\sigma _{1}+\sigma _{2}+\sigma _{3}){\big )}={\frac {1-2\nu }{E}}(\sigma _{1}+\sigma _{2}+\sigma _{3})\\\sigma _{1}+\sigma _{2}+\sigma _{3}&={\frac {E}{1-2\nu }}(\varepsilon _{1}+\varepsilon _{2}+\varepsilon _{3})\end{aligned}}}"></span> and substituting it to the equation solved for <span class="texhtml"><i>σ</i><sub>1</sub></span> gives <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\sigma _{1}&={\frac {E}{1+\nu }}\varepsilon _{1}+{\frac {E\nu }{(1+\nu )(1-2\nu )}}(\varepsilon _{1}+\varepsilon _{2}+\varepsilon _{3})\\&=2\mu \varepsilon _{1}+\lambda (\varepsilon _{1}+\varepsilon _{2}+\varepsilon _{3})\,,\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>σ<!-- σ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>E</mi> <mrow> <mn>1</mn> <mo>+</mo> <mi>ν<!-- ν --></mi> </mrow> </mfrac> </mrow> <msub> <mi>ε<!-- ε --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>E</mi> <mi>ν<!-- ν --></mi> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>ν<!-- ν --></mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−<!-- − --></mo> <mn>2</mn> <mi>ν<!-- ν --></mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo stretchy="false">(</mo> <msub> <mi>ε<!-- ε --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>ε<!-- ε --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>ε<!-- ε --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mn>2</mn> <mi>μ<!-- μ --></mi> <msub> <mi>ε<!-- ε --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mi>λ<!-- λ --></mi> <mo stretchy="false">(</mo> <msub> <mi>ε<!-- ε --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>ε<!-- ε --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>ε<!-- ε --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mo>,</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\sigma _{1}&={\frac {E}{1+\nu }}\varepsilon _{1}+{\frac {E\nu }{(1+\nu )(1-2\nu )}}(\varepsilon _{1}+\varepsilon _{2}+\varepsilon _{3})\\&=2\mu \varepsilon _{1}+\lambda (\varepsilon _{1}+\varepsilon _{2}+\varepsilon _{3})\,,\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/070bf2130c8a2d8bb373c0712e5f4b6e8e2e5f38" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.751ex; margin-bottom: -0.254ex; width:47.271ex; height:9.176ex;" alt="{\displaystyle {\begin{aligned}\sigma _{1}&={\frac {E}{1+\nu }}\varepsilon _{1}+{\frac {E\nu }{(1+\nu )(1-2\nu )}}(\varepsilon _{1}+\varepsilon _{2}+\varepsilon _{3})\\&=2\mu \varepsilon _{1}+\lambda (\varepsilon _{1}+\varepsilon _{2}+\varepsilon _{3})\,,\end{aligned}}}"></span> where <span class="texhtml mvar" style="font-style:italic;">μ</span> and <span class="texhtml mvar" style="font-style:italic;">λ</span> are the <a href="/wiki/Lam%C3%A9_parameters" title="Lamé parameters">Lamé parameters</a>. </p><p>Similar treatment of directions 2 and 3 gives the Hooke's law in three dimensions. </p> </div> <p>In matrix form, Hooke's law for isotropic materials can be written as <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}\varepsilon _{11}\\\varepsilon _{22}\\\varepsilon _{33}\\2\varepsilon _{23}\\2\varepsilon _{13}\\2\varepsilon _{12}\end{bmatrix}}\,=\,{\begin{bmatrix}\varepsilon _{11}\\\varepsilon _{22}\\\varepsilon _{33}\\\gamma _{23}\\\gamma _{13}\\\gamma _{12}\end{bmatrix}}\,=\,{\frac {1}{E}}{\begin{bmatrix}1&-\nu &-\nu &0&0&0\\-\nu &1&-\nu &0&0&0\\-\nu &-\nu &1&0&0&0\\0&0&0&2+2\nu &0&0\\0&0&0&0&2+2\nu &0\\0&0&0&0&0&2+2\nu \end{bmatrix}}{\begin{bmatrix}\sigma _{11}\\\sigma _{22}\\\sigma _{33}\\\sigma _{23}\\\sigma _{13}\\\sigma _{12}\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>ε<!-- ε --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>11</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>ε<!-- ε --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>22</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>ε<!-- ε --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>33</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mn>2</mn> <msub> <mi>ε<!-- ε --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>23</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mn>2</mn> <msub> <mi>ε<!-- ε --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>13</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mn>2</mn> <msub> <mi>ε<!-- ε --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mspace width="thinmathspace" /> <mo>=</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>ε<!-- ε --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>11</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>ε<!-- ε --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>22</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>ε<!-- ε --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>33</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>γ<!-- γ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>23</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>γ<!-- γ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>13</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>γ<!-- γ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mspace width="thinmathspace" /> <mo>=</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>E</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mo>−<!-- − --></mo> <mi>ν<!-- ν --></mi> </mtd> <mtd> <mo>−<!-- − --></mo> <mi>ν<!-- ν --></mi> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mo>−<!-- − --></mo> <mi>ν<!-- ν --></mi> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mo>−<!-- − --></mo> <mi>ν<!-- ν --></mi> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mo>−<!-- − --></mo> <mi>ν<!-- ν --></mi> </mtd> <mtd> <mo>−<!-- − --></mo> <mi>ν<!-- ν --></mi> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>2</mn> <mo>+</mo> <mn>2</mn> <mi>ν<!-- ν --></mi> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>2</mn> <mo>+</mo> <mn>2</mn> <mi>ν<!-- ν --></mi> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>2</mn> <mo>+</mo> <mn>2</mn> <mi>ν<!-- ν --></mi> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>σ<!-- σ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>11</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>σ<!-- σ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>22</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>σ<!-- σ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>33</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>σ<!-- σ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>23</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>σ<!-- σ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>13</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>σ<!-- σ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}\varepsilon _{11}\\\varepsilon _{22}\\\varepsilon _{33}\\2\varepsilon _{23}\\2\varepsilon _{13}\\2\varepsilon _{12}\end{bmatrix}}\,=\,{\begin{bmatrix}\varepsilon _{11}\\\varepsilon _{22}\\\varepsilon _{33}\\\gamma _{23}\\\gamma _{13}\\\gamma _{12}\end{bmatrix}}\,=\,{\frac {1}{E}}{\begin{bmatrix}1&-\nu &-\nu &0&0&0\\-\nu &1&-\nu &0&0&0\\-\nu &-\nu &1&0&0&0\\0&0&0&2+2\nu &0&0\\0&0&0&0&2+2\nu &0\\0&0&0&0&0&2+2\nu \end{bmatrix}}{\begin{bmatrix}\sigma _{11}\\\sigma _{22}\\\sigma _{33}\\\sigma _{23}\\\sigma _{13}\\\sigma _{12}\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72b4991dc0a94ed2fb0b40d6b84cf6da219e81ef" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -9.005ex; width:76.098ex; height:19.176ex;" alt="{\displaystyle {\begin{bmatrix}\varepsilon _{11}\\\varepsilon _{22}\\\varepsilon _{33}\\2\varepsilon _{23}\\2\varepsilon _{13}\\2\varepsilon _{12}\end{bmatrix}}\,=\,{\begin{bmatrix}\varepsilon _{11}\\\varepsilon _{22}\\\varepsilon _{33}\\\gamma _{23}\\\gamma _{13}\\\gamma _{12}\end{bmatrix}}\,=\,{\frac {1}{E}}{\begin{bmatrix}1&-\nu &-\nu &0&0&0\\-\nu &1&-\nu &0&0&0\\-\nu &-\nu &1&0&0&0\\0&0&0&2+2\nu &0&0\\0&0&0&0&2+2\nu &0\\0&0&0&0&0&2+2\nu \end{bmatrix}}{\begin{bmatrix}\sigma _{11}\\\sigma _{22}\\\sigma _{33}\\\sigma _{23}\\\sigma _{13}\\\sigma _{12}\end{bmatrix}}}"></span> where <span class="texhtml"><i>γ<sub>ij</sub></i> = 2<i>ε<sub>ij</sub></i></span> is the <b>engineering shear strain</b>. The inverse relation may be written as <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}\sigma _{11}\\\sigma _{22}\\\sigma _{33}\\\sigma _{23}\\\sigma _{13}\\\sigma _{12}\end{bmatrix}}\,=\,{\frac {E}{(1+\nu )(1-2\nu )}}{\begin{bmatrix}1-\nu &\nu &\nu &0&0&0\\\nu &1-\nu &\nu &0&0&0\\\nu &\nu &1-\nu &0&0&0\\0&0&0&{\frac {1-2\nu }{2}}&0&0\\0&0&0&0&{\frac {1-2\nu }{2}}&0\\0&0&0&0&0&{\frac {1-2\nu }{2}}\end{bmatrix}}{\begin{bmatrix}\varepsilon _{11}\\\varepsilon _{22}\\\varepsilon _{33}\\2\varepsilon _{23}\\2\varepsilon _{13}\\2\varepsilon _{12}\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>σ<!-- σ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>11</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>σ<!-- σ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>22</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>σ<!-- σ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>33</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>σ<!-- σ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>23</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>σ<!-- σ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>13</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>σ<!-- σ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mspace width="thinmathspace" /> <mo>=</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>E</mi> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>ν<!-- ν --></mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−<!-- − --></mo> <mn>2</mn> <mi>ν<!-- ν --></mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> <mo>−<!-- − --></mo> <mi>ν<!-- ν --></mi> </mtd> <mtd> <mi>ν<!-- ν --></mi> </mtd> <mtd> <mi>ν<!-- ν --></mi> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mi>ν<!-- ν --></mi> </mtd> <mtd> <mn>1</mn> <mo>−<!-- − --></mo> <mi>ν<!-- ν --></mi> </mtd> <mtd> <mi>ν<!-- ν --></mi> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mi>ν<!-- ν --></mi> </mtd> <mtd> <mi>ν<!-- ν --></mi> </mtd> <mtd> <mn>1</mn> <mo>−<!-- − --></mo> <mi>ν<!-- ν --></mi> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>−<!-- − --></mo> <mn>2</mn> <mi>ν<!-- ν --></mi> </mrow> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>−<!-- − --></mo> <mn>2</mn> <mi>ν<!-- ν --></mi> </mrow> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>−<!-- − --></mo> <mn>2</mn> <mi>ν<!-- ν --></mi> </mrow> <mn>2</mn> </mfrac> </mrow> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>ε<!-- ε --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>11</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>ε<!-- ε --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>22</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>ε<!-- ε --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>33</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mn>2</mn> <msub> <mi>ε<!-- ε --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>23</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mn>2</mn> <msub> <mi>ε<!-- ε --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>13</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mn>2</mn> <msub> <mi>ε<!-- ε --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}\sigma _{11}\\\sigma _{22}\\\sigma _{33}\\\sigma _{23}\\\sigma _{13}\\\sigma _{12}\end{bmatrix}}\,=\,{\frac {E}{(1+\nu )(1-2\nu )}}{\begin{bmatrix}1-\nu &\nu &\nu &0&0&0\\\nu &1-\nu &\nu &0&0&0\\\nu &\nu &1-\nu &0&0&0\\0&0&0&{\frac {1-2\nu }{2}}&0&0\\0&0&0&0&{\frac {1-2\nu }{2}}&0\\0&0&0&0&0&{\frac {1-2\nu }{2}}\end{bmatrix}}{\begin{bmatrix}\varepsilon _{11}\\\varepsilon _{22}\\\varepsilon _{33}\\2\varepsilon _{23}\\2\varepsilon _{13}\\2\varepsilon _{12}\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bec60df69abfdd446aeffe0010ffec26c275afbc" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -10.338ex; width:80.05ex; height:21.843ex;" alt="{\displaystyle {\begin{bmatrix}\sigma _{11}\\\sigma _{22}\\\sigma _{33}\\\sigma _{23}\\\sigma _{13}\\\sigma _{12}\end{bmatrix}}\,=\,{\frac {E}{(1+\nu )(1-2\nu )}}{\begin{bmatrix}1-\nu &\nu &\nu &0&0&0\\\nu &1-\nu &\nu &0&0&0\\\nu &\nu &1-\nu &0&0&0\\0&0&0&{\frac {1-2\nu }{2}}&0&0\\0&0&0&0&{\frac {1-2\nu }{2}}&0\\0&0&0&0&0&{\frac {1-2\nu }{2}}\end{bmatrix}}{\begin{bmatrix}\varepsilon _{11}\\\varepsilon _{22}\\\varepsilon _{33}\\2\varepsilon _{23}\\2\varepsilon _{13}\\2\varepsilon _{12}\end{bmatrix}}}"></span> which can be simplified thanks to the Lamé constants: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}\sigma _{11}\\\sigma _{22}\\\sigma _{33}\\\sigma _{23}\\\sigma _{13}\\\sigma _{12}\end{bmatrix}}\,=\,{\begin{bmatrix}2\mu +\lambda &\lambda &\lambda &0&0&0\\\lambda &2\mu +\lambda &\lambda &0&0&0\\\lambda &\lambda &2\mu +\lambda &0&0&0\\0&0&0&\mu &0&0\\0&0&0&0&\mu &0\\0&0&0&0&0&\mu \end{bmatrix}}{\begin{bmatrix}\varepsilon _{11}\\\varepsilon _{22}\\\varepsilon _{33}\\2\varepsilon _{23}\\2\varepsilon _{13}\\2\varepsilon _{12}\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>σ<!-- σ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>11</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>σ<!-- σ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>22</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>σ<!-- σ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>33</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>σ<!-- σ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>23</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>σ<!-- σ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>13</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>σ<!-- σ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mspace width="thinmathspace" /> <mo>=</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>2</mn> <mi>μ<!-- μ --></mi> <mo>+</mo> <mi>λ<!-- λ --></mi> </mtd> <mtd> <mi>λ<!-- λ --></mi> </mtd> <mtd> <mi>λ<!-- λ --></mi> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mi>λ<!-- λ --></mi> </mtd> <mtd> <mn>2</mn> <mi>μ<!-- μ --></mi> <mo>+</mo> <mi>λ<!-- λ --></mi> </mtd> <mtd> <mi>λ<!-- λ --></mi> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mi>λ<!-- λ --></mi> </mtd> <mtd> <mi>λ<!-- λ --></mi> </mtd> <mtd> <mn>2</mn> <mi>μ<!-- μ --></mi> <mo>+</mo> <mi>λ<!-- λ --></mi> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mi>μ<!-- μ --></mi> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mi>μ<!-- μ --></mi> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mi>μ<!-- μ --></mi> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>ε<!-- ε --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>11</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>ε<!-- ε --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>22</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>ε<!-- ε --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>33</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mn>2</mn> <msub> <mi>ε<!-- ε --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>23</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mn>2</mn> <msub> <mi>ε<!-- ε --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>13</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mn>2</mn> <msub> <mi>ε<!-- ε --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}\sigma _{11}\\\sigma _{22}\\\sigma _{33}\\\sigma _{23}\\\sigma _{13}\\\sigma _{12}\end{bmatrix}}\,=\,{\begin{bmatrix}2\mu +\lambda &\lambda &\lambda &0&0&0\\\lambda &2\mu +\lambda &\lambda &0&0&0\\\lambda &\lambda &2\mu +\lambda &0&0&0\\0&0&0&\mu &0&0\\0&0&0&0&\mu &0\\0&0&0&0&0&\mu \end{bmatrix}}{\begin{bmatrix}\varepsilon _{11}\\\varepsilon _{22}\\\varepsilon _{33}\\2\varepsilon _{23}\\2\varepsilon _{13}\\2\varepsilon _{12}\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e79b75812560c7cc8ebd81e9a37da4355aced62b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -9.005ex; width:58.852ex; height:19.176ex;" alt="{\displaystyle {\begin{bmatrix}\sigma _{11}\\\sigma _{22}\\\sigma _{33}\\\sigma _{23}\\\sigma _{13}\\\sigma _{12}\end{bmatrix}}\,=\,{\begin{bmatrix}2\mu +\lambda &\lambda &\lambda &0&0&0\\\lambda &2\mu +\lambda &\lambda &0&0&0\\\lambda &\lambda &2\mu +\lambda &0&0&0\\0&0&0&\mu &0&0\\0&0&0&0&\mu &0\\0&0&0&0&0&\mu \end{bmatrix}}{\begin{bmatrix}\varepsilon _{11}\\\varepsilon _{22}\\\varepsilon _{33}\\2\varepsilon _{23}\\2\varepsilon _{13}\\2\varepsilon _{12}\end{bmatrix}}}"></span> In vector notation this becomes <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}\sigma _{11}&\sigma _{12}&\sigma _{13}\\\sigma _{12}&\sigma _{22}&\sigma _{23}\\\sigma _{13}&\sigma _{23}&\sigma _{33}\end{bmatrix}}\,=\,2\mu {\begin{bmatrix}\varepsilon _{11}&\varepsilon _{12}&\varepsilon _{13}\\\varepsilon _{12}&\varepsilon _{22}&\varepsilon _{23}\\\varepsilon _{13}&\varepsilon _{23}&\varepsilon _{33}\end{bmatrix}}+\lambda \mathbf {I} \left(\varepsilon _{11}+\varepsilon _{22}+\varepsilon _{33}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>σ<!-- σ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>11</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>σ<!-- σ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>σ<!-- σ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>13</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>σ<!-- σ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>σ<!-- σ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>22</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>σ<!-- σ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>23</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>σ<!-- σ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>13</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>σ<!-- σ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>23</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>σ<!-- σ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>33</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mspace width="thinmathspace" /> <mo>=</mo> <mspace width="thinmathspace" /> <mn>2</mn> <mi>μ<!-- μ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>ε<!-- ε --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>11</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>ε<!-- ε --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>ε<!-- ε --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>13</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>ε<!-- ε --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>ε<!-- ε --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>22</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>ε<!-- ε --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>23</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>ε<!-- ε --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>13</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>ε<!-- ε --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>23</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>ε<!-- ε --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>33</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>+</mo> <mi>λ<!-- λ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">I</mi> </mrow> <mrow> <mo>(</mo> <mrow> <msub> <mi>ε<!-- ε --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>11</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>ε<!-- ε --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>22</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>ε<!-- ε --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>33</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}\sigma _{11}&\sigma _{12}&\sigma _{13}\\\sigma _{12}&\sigma _{22}&\sigma _{23}\\\sigma _{13}&\sigma _{23}&\sigma _{33}\end{bmatrix}}\,=\,2\mu {\begin{bmatrix}\varepsilon _{11}&\varepsilon _{12}&\varepsilon _{13}\\\varepsilon _{12}&\varepsilon _{22}&\varepsilon _{23}\\\varepsilon _{13}&\varepsilon _{23}&\varepsilon _{33}\end{bmatrix}}+\lambda \mathbf {I} \left(\varepsilon _{11}+\varepsilon _{22}+\varepsilon _{33}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c8c6f6548cfa88d24a7c55fbc572618ee5121584" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:63.887ex; height:9.176ex;" alt="{\displaystyle {\begin{bmatrix}\sigma _{11}&\sigma _{12}&\sigma _{13}\\\sigma _{12}&\sigma _{22}&\sigma _{23}\\\sigma _{13}&\sigma _{23}&\sigma _{33}\end{bmatrix}}\,=\,2\mu {\begin{bmatrix}\varepsilon _{11}&\varepsilon _{12}&\varepsilon _{13}\\\varepsilon _{12}&\varepsilon _{22}&\varepsilon _{23}\\\varepsilon _{13}&\varepsilon _{23}&\varepsilon _{33}\end{bmatrix}}+\lambda \mathbf {I} \left(\varepsilon _{11}+\varepsilon _{22}+\varepsilon _{33}\right)}"></span> where <span class="texhtml"><b>I</b></span> is the identity tensor. </p> <div class="mw-heading mw-heading4"><h4 id="Plane_stress">Plane stress</h4></div> <p>Under <a href="/wiki/Plane_stress#Plane_stress" title="Plane stress">plane stress</a> conditions, <span class="texhtml"><i>σ</i><sub>31</sub> = <i>σ</i><sub>13</sub> = <i>σ</i><sub>32</sub> = <i>σ</i><sub>23</sub> = <i>σ</i><sub>33</sub> = 0</span>. In that case Hooke's law takes the form <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}\sigma _{11}\\\sigma _{22}\\\sigma _{12}\end{bmatrix}}\,=\,{\frac {E}{1-\nu ^{2}}}{\begin{bmatrix}1&\nu &0\\\nu &1&0\\0&0&{\frac {1-\nu }{2}}\end{bmatrix}}{\begin{bmatrix}\varepsilon _{11}\\\varepsilon _{22}\\2\varepsilon _{12}\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>σ<!-- σ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>11</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>σ<!-- σ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>22</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>σ<!-- σ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mspace width="thinmathspace" /> <mo>=</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>E</mi> <mrow> <mn>1</mn> <mo>−<!-- − --></mo> <msup> <mi>ν<!-- ν --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mi>ν<!-- ν --></mi> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mi>ν<!-- ν --></mi> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>−<!-- − --></mo> <mi>ν<!-- ν --></mi> </mrow> <mn>2</mn> </mfrac> </mrow> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>ε<!-- ε --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>11</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>ε<!-- ε --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>22</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mn>2</mn> <msub> <mi>ε<!-- ε --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}\sigma _{11}\\\sigma _{22}\\\sigma _{12}\end{bmatrix}}\,=\,{\frac {E}{1-\nu ^{2}}}{\begin{bmatrix}1&\nu &0\\\nu &1&0\\0&0&{\frac {1-\nu }{2}}\end{bmatrix}}{\begin{bmatrix}\varepsilon _{11}\\\varepsilon _{22}\\2\varepsilon _{12}\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d0e8b746a4f4942077f39a6da33f9b57b1c4d13" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.287ex; margin-bottom: -0.217ex; width:40.823ex; height:10.176ex;" alt="{\displaystyle {\begin{bmatrix}\sigma _{11}\\\sigma _{22}\\\sigma _{12}\end{bmatrix}}\,=\,{\frac {E}{1-\nu ^{2}}}{\begin{bmatrix}1&\nu &0\\\nu &1&0\\0&0&{\frac {1-\nu }{2}}\end{bmatrix}}{\begin{bmatrix}\varepsilon _{11}\\\varepsilon _{22}\\2\varepsilon _{12}\end{bmatrix}}}"></span> </p><p>In vector notation this becomes <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}\sigma _{11}&\sigma _{12}\\\sigma _{12}&\sigma _{22}\end{bmatrix}}\,=\,{\frac {E}{1-\nu ^{2}}}\left((1-\nu ){\begin{bmatrix}\varepsilon _{11}&\varepsilon _{12}\\\varepsilon _{12}&\varepsilon _{22}\end{bmatrix}}+\nu \mathbf {I} \left(\varepsilon _{11}+\varepsilon _{22}\right)\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>σ<!-- σ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>11</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>σ<!-- σ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>σ<!-- σ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>σ<!-- σ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>22</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mspace width="thinmathspace" /> <mo>=</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>E</mi> <mrow> <mn>1</mn> <mo>−<!-- − --></mo> <msup> <mi>ν<!-- ν --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−<!-- − --></mo> <mi>ν<!-- ν --></mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>ε<!-- ε --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>11</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>ε<!-- ε --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>ε<!-- ε --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>ε<!-- ε --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>22</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>+</mo> <mi>ν<!-- ν --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">I</mi> </mrow> <mrow> <mo>(</mo> <mrow> <msub> <mi>ε<!-- ε --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>11</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>ε<!-- ε --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>22</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}\sigma _{11}&\sigma _{12}\\\sigma _{12}&\sigma _{22}\end{bmatrix}}\,=\,{\frac {E}{1-\nu ^{2}}}\left((1-\nu ){\begin{bmatrix}\varepsilon _{11}&\varepsilon _{12}\\\varepsilon _{12}&\varepsilon _{22}\end{bmatrix}}+\nu \mathbf {I} \left(\varepsilon _{11}+\varepsilon _{22}\right)\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4a7d1cae454cd1d405d37b0541763d522cbe155f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:61.303ex; height:6.176ex;" alt="{\displaystyle {\begin{bmatrix}\sigma _{11}&\sigma _{12}\\\sigma _{12}&\sigma _{22}\end{bmatrix}}\,=\,{\frac {E}{1-\nu ^{2}}}\left((1-\nu ){\begin{bmatrix}\varepsilon _{11}&\varepsilon _{12}\\\varepsilon _{12}&\varepsilon _{22}\end{bmatrix}}+\nu \mathbf {I} \left(\varepsilon _{11}+\varepsilon _{22}\right)\right)}"></span> </p><p>The inverse relation is usually written in the reduced form <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}\varepsilon _{11}\\\varepsilon _{22}\\2\varepsilon _{12}\end{bmatrix}}\,=\,{\frac {1}{E}}{\begin{bmatrix}1&-\nu &0\\-\nu &1&0\\0&0&2+2\nu \end{bmatrix}}{\begin{bmatrix}\sigma _{11}\\\sigma _{22}\\\sigma _{12}\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>ε<!-- ε --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>11</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>ε<!-- ε --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>22</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mn>2</mn> <msub> <mi>ε<!-- ε --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mspace width="thinmathspace" /> <mo>=</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>E</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mo>−<!-- − --></mo> <mi>ν<!-- ν --></mi> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mo>−<!-- − --></mo> <mi>ν<!-- ν --></mi> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>2</mn> <mo>+</mo> <mn>2</mn> <mi>ν<!-- ν --></mi> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>σ<!-- σ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>11</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>σ<!-- σ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>22</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>σ<!-- σ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}\varepsilon _{11}\\\varepsilon _{22}\\2\varepsilon _{12}\end{bmatrix}}\,=\,{\frac {1}{E}}{\begin{bmatrix}1&-\nu &0\\-\nu &1&0\\0&0&2+2\nu \end{bmatrix}}{\begin{bmatrix}\sigma _{11}\\\sigma _{22}\\\sigma _{12}\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/072cc100b40a1af10003059e125d28e88a63c9de" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:42.49ex; height:9.176ex;" alt="{\displaystyle {\begin{bmatrix}\varepsilon _{11}\\\varepsilon _{22}\\2\varepsilon _{12}\end{bmatrix}}\,=\,{\frac {1}{E}}{\begin{bmatrix}1&-\nu &0\\-\nu &1&0\\0&0&2+2\nu \end{bmatrix}}{\begin{bmatrix}\sigma _{11}\\\sigma _{22}\\\sigma _{12}\end{bmatrix}}}"></span> </p> <div class="mw-heading mw-heading4"><h4 id="Plane_strain">Plane strain</h4></div> <p>Under <a href="/wiki/Infinitesimal_strain_theory#Plane_strain" title="Infinitesimal strain theory">plane strain</a> conditions, <span class="texhtml"><i>ε</i><sub>31</sub> = <i>ε</i><sub>13</sub> = <i>ε</i><sub>32</sub> = <i>ε</i><sub>23</sub> = <i>ε</i><sub>33</sub> = 0</span>. In this case Hooke's law takes the form <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}\sigma _{11}\\\sigma _{22}\\\sigma _{12}\end{bmatrix}}\,=\,{\frac {E}{(1+\nu )(1-2\nu )}}{\begin{bmatrix}1-\nu &\nu &0\\\nu &1-\nu &0\\0&0&{\frac {1-2\nu }{2}}\end{bmatrix}}{\begin{bmatrix}\varepsilon _{11}\\\varepsilon _{22}\\2\varepsilon _{12}\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>σ<!-- σ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>11</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>σ<!-- σ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>22</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>σ<!-- σ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mspace width="thinmathspace" /> <mo>=</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>E</mi> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>ν<!-- ν --></mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−<!-- − --></mo> <mn>2</mn> <mi>ν<!-- ν --></mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> <mo>−<!-- − --></mo> <mi>ν<!-- ν --></mi> </mtd> <mtd> <mi>ν<!-- ν --></mi> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mi>ν<!-- ν --></mi> </mtd> <mtd> <mn>1</mn> <mo>−<!-- − --></mo> <mi>ν<!-- ν --></mi> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>−<!-- − --></mo> <mn>2</mn> <mi>ν<!-- ν --></mi> </mrow> <mn>2</mn> </mfrac> </mrow> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>ε<!-- ε --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>11</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>ε<!-- ε --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>22</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mn>2</mn> <msub> <mi>ε<!-- ε --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}\sigma _{11}\\\sigma _{22}\\\sigma _{12}\end{bmatrix}}\,=\,{\frac {E}{(1+\nu )(1-2\nu )}}{\begin{bmatrix}1-\nu &\nu &0\\\nu &1-\nu &0\\0&0&{\frac {1-2\nu }{2}}\end{bmatrix}}{\begin{bmatrix}\varepsilon _{11}\\\varepsilon _{22}\\2\varepsilon _{12}\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2e9bce2e0f46ff4df7a617675ec7a10ea5d07eec" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.287ex; margin-bottom: -0.217ex; width:58.587ex; height:10.176ex;" alt="{\displaystyle {\begin{bmatrix}\sigma _{11}\\\sigma _{22}\\\sigma _{12}\end{bmatrix}}\,=\,{\frac {E}{(1+\nu )(1-2\nu )}}{\begin{bmatrix}1-\nu &\nu &0\\\nu &1-\nu &0\\0&0&{\frac {1-2\nu }{2}}\end{bmatrix}}{\begin{bmatrix}\varepsilon _{11}\\\varepsilon _{22}\\2\varepsilon _{12}\end{bmatrix}}}"></span> </p> <div class="mw-heading mw-heading3"><h3 id="Anisotropic_materials">Anisotropic materials</h3></div> <p>The symmetry of the <a href="/wiki/Stress_(physics)" class="mw-redirect" title="Stress (physics)">Cauchy stress tensor</a> (<span class="texhtml"><i>σ<sub>ij</sub></i> = <i>σ<sub>ji</sub></i></span>) and the generalized Hooke's laws (<span class="texhtml"><i>σ<sub>ij</sub></i> = <i>c<sub>ijkl</sub>ε<sub>kl</sub></i></span>) implies that <span class="texhtml"><i>c<sub>ijkl</sub></i> = <i>c<sub>jikl</sub></i></span>. Similarly, the symmetry of the <a href="/wiki/Infinitesimal_strain_theory" title="Infinitesimal strain theory">infinitesimal strain tensor</a> implies that <span class="texhtml"><i>c<sub>ijkl</sub></i> = <i>c<sub>ijlk</sub></i></span>. These symmetries are called the <b>minor symmetries</b> of the stiffness tensor <b>c</b>. This reduces the number of elastic constants from 81 to 36. </p><p>If in addition, since the displacement gradient and the Cauchy stress are work conjugate, the stress–strain relation can be derived from a strain energy density functional (<span class="texhtml mvar" style="font-style:italic;">U</span>), then <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{ij}={\frac {\partial U}{\partial \varepsilon _{ij}}}\quad \implies \quad c_{ijkl}={\frac {\partial ^{2}U}{\partial \varepsilon _{ij}\partial \varepsilon _{kl}}}\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>σ<!-- σ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂<!-- ∂ --></mi> <mi>U</mi> </mrow> <mrow> <mi mathvariant="normal">∂<!-- ∂ --></mi> <msub> <mi>ε<!-- ε --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mspace width="1em" /> <mspace width="thickmathspace" /> <mo stretchy="false">⟹<!-- ⟹ --></mo> <mspace width="thickmathspace" /> <mspace width="1em" /> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> <mi>k</mi> <mi>l</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi mathvariant="normal">∂<!-- ∂ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>U</mi> </mrow> <mrow> <mi mathvariant="normal">∂<!-- ∂ --></mi> <msub> <mi>ε<!-- ε --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mi mathvariant="normal">∂<!-- ∂ --></mi> <msub> <mi>ε<!-- ε --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mi>l</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{ij}={\frac {\partial U}{\partial \varepsilon _{ij}}}\quad \implies \quad c_{ijkl}={\frac {\partial ^{2}U}{\partial \varepsilon _{ij}\partial \varepsilon _{kl}}}\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1438bea40aae9173c744d21f2e13f0a97f381021" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:38.308ex; height:6.509ex;" alt="{\displaystyle \sigma _{ij}={\frac {\partial U}{\partial \varepsilon _{ij}}}\quad \implies \quad c_{ijkl}={\frac {\partial ^{2}U}{\partial \varepsilon _{ij}\partial \varepsilon _{kl}}}\,.}"></span> The arbitrariness of the order of differentiation implies that <span class="texhtml"><i>c<sub>ijkl</sub></i> = <i>c<sub>klij</sub></i></span>. These are called the <b>major symmetries</b> of the stiffness tensor. This reduces the number of elastic constants from 36 to 21. The major and minor symmetries indicate that the stiffness tensor has only 21 independent components. </p> <div class="mw-heading mw-heading4"><h4 id="Matrix_representation_(stiffness_tensor)"><span id="Matrix_representation_.28stiffness_tensor.29"></span>Matrix representation (stiffness tensor)</h4></div> <p>It is often useful to express the anisotropic form of Hooke's law in matrix notation, also called <a href="/wiki/Voigt_notation" title="Voigt notation">Voigt notation</a>. To do this we take advantage of the symmetry of the stress and strain tensors and express them as six-dimensional vectors in an orthonormal coordinate system (<span class="texhtml"><b>e</b><sub>1</sub>,<b>e</b><sub>2</sub>,<b>e</b><sub>3</sub></span>) as <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [{\boldsymbol {\sigma }}]\,=\,{\begin{bmatrix}\sigma _{11}\\\sigma _{22}\\\sigma _{33}\\\sigma _{23}\\\sigma _{13}\\\sigma _{12}\end{bmatrix}}\,\equiv \,{\begin{bmatrix}\sigma _{1}\\\sigma _{2}\\\sigma _{3}\\\sigma _{4}\\\sigma _{5}\\\sigma _{6}\end{bmatrix}}\,;\qquad [{\boldsymbol {\varepsilon }}]\,=\,{\begin{bmatrix}\varepsilon _{11}\\\varepsilon _{22}\\\varepsilon _{33}\\2\varepsilon _{23}\\2\varepsilon _{13}\\2\varepsilon _{12}\end{bmatrix}}\,\equiv \,{\begin{bmatrix}\varepsilon _{1}\\\varepsilon _{2}\\\varepsilon _{3}\\\varepsilon _{4}\\\varepsilon _{5}\\\varepsilon _{6}\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">σ<!-- σ --></mi> </mrow> <mo stretchy="false">]</mo> <mspace width="thinmathspace" /> <mo>=</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>σ<!-- σ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>11</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>σ<!-- σ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>22</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>σ<!-- σ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>33</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>σ<!-- σ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>23</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>σ<!-- σ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>13</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>σ<!-- σ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mspace width="thinmathspace" /> <mo>≡<!-- ≡ --></mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>σ<!-- σ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>σ<!-- σ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>σ<!-- σ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>σ<!-- σ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>σ<!-- σ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>σ<!-- σ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mspace width="thinmathspace" /> <mo>;</mo> <mspace width="2em" /> <mo stretchy="false">[</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ε<!-- ε --></mi> </mrow> <mo stretchy="false">]</mo> <mspace width="thinmathspace" /> <mo>=</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>ε<!-- ε --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>11</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>ε<!-- ε --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>22</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>ε<!-- ε --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>33</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mn>2</mn> <msub> <mi>ε<!-- ε --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>23</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mn>2</mn> <msub> <mi>ε<!-- ε --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>13</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mn>2</mn> <msub> <mi>ε<!-- ε --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mspace width="thinmathspace" /> <mo>≡<!-- ≡ --></mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>ε<!-- ε --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>ε<!-- ε --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>ε<!-- ε --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>ε<!-- ε --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>ε<!-- ε --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>ε<!-- ε --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [{\boldsymbol {\sigma }}]\,=\,{\begin{bmatrix}\sigma _{11}\\\sigma _{22}\\\sigma _{33}\\\sigma _{23}\\\sigma _{13}\\\sigma _{12}\end{bmatrix}}\,\equiv \,{\begin{bmatrix}\sigma _{1}\\\sigma _{2}\\\sigma _{3}\\\sigma _{4}\\\sigma _{5}\\\sigma _{6}\end{bmatrix}}\,;\qquad [{\boldsymbol {\varepsilon }}]\,=\,{\begin{bmatrix}\varepsilon _{11}\\\varepsilon _{22}\\\varepsilon _{33}\\2\varepsilon _{23}\\2\varepsilon _{13}\\2\varepsilon _{12}\end{bmatrix}}\,\equiv \,{\begin{bmatrix}\varepsilon _{1}\\\varepsilon _{2}\\\varepsilon _{3}\\\varepsilon _{4}\\\varepsilon _{5}\\\varepsilon _{6}\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/99d84c34fc9efc62922b42a33f888656c62d794b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -9.005ex; width:54.222ex; height:19.176ex;" alt="{\displaystyle [{\boldsymbol {\sigma }}]\,=\,{\begin{bmatrix}\sigma _{11}\\\sigma _{22}\\\sigma _{33}\\\sigma _{23}\\\sigma _{13}\\\sigma _{12}\end{bmatrix}}\,\equiv \,{\begin{bmatrix}\sigma _{1}\\\sigma _{2}\\\sigma _{3}\\\sigma _{4}\\\sigma _{5}\\\sigma _{6}\end{bmatrix}}\,;\qquad [{\boldsymbol {\varepsilon }}]\,=\,{\begin{bmatrix}\varepsilon _{11}\\\varepsilon _{22}\\\varepsilon _{33}\\2\varepsilon _{23}\\2\varepsilon _{13}\\2\varepsilon _{12}\end{bmatrix}}\,\equiv \,{\begin{bmatrix}\varepsilon _{1}\\\varepsilon _{2}\\\varepsilon _{3}\\\varepsilon _{4}\\\varepsilon _{5}\\\varepsilon _{6}\end{bmatrix}}}"></span> Then the stiffness tensor (<b>c</b>) can be expressed as <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [{\mathsf {c}}]\,=\,{\begin{bmatrix}c_{1111}&c_{1122}&c_{1133}&c_{1123}&c_{1131}&c_{1112}\\c_{2211}&c_{2222}&c_{2233}&c_{2223}&c_{2231}&c_{2212}\\c_{3311}&c_{3322}&c_{3333}&c_{3323}&c_{3331}&c_{3312}\\c_{2311}&c_{2322}&c_{2333}&c_{2323}&c_{2331}&c_{2312}\\c_{3111}&c_{3122}&c_{3133}&c_{3123}&c_{3131}&c_{3112}\\c_{1211}&c_{1222}&c_{1233}&c_{1223}&c_{1231}&c_{1212}\end{bmatrix}}\,\equiv \,{\begin{bmatrix}C_{11}&C_{12}&C_{13}&C_{14}&C_{15}&C_{16}\\C_{12}&C_{22}&C_{23}&C_{24}&C_{25}&C_{26}\\C_{13}&C_{23}&C_{33}&C_{34}&C_{35}&C_{36}\\C_{14}&C_{24}&C_{34}&C_{44}&C_{45}&C_{46}\\C_{15}&C_{25}&C_{35}&C_{45}&C_{55}&C_{56}\\C_{16}&C_{26}&C_{36}&C_{46}&C_{56}&C_{66}\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">c</mi> </mrow> </mrow> <mo stretchy="false">]</mo> <mspace width="thinmathspace" /> <mo>=</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1111</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1122</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1133</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1123</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1131</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1112</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2211</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2222</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2233</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2223</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2231</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2212</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3311</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3322</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3333</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3323</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3331</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3312</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2311</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2322</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2333</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2323</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2331</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2312</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3111</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3122</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3133</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3123</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3131</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3112</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1211</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1222</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1233</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1223</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1231</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1212</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mspace width="thinmathspace" /> <mo>≡<!-- ≡ --></mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>11</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>13</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>14</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>15</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>16</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>22</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>23</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>24</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>25</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>26</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>13</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>23</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>33</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>34</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>35</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>36</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>14</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>24</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>34</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>44</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>45</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>46</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>15</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>25</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>35</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>45</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>55</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>56</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>16</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>26</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>36</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>46</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>56</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>66</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [{\mathsf {c}}]\,=\,{\begin{bmatrix}c_{1111}&c_{1122}&c_{1133}&c_{1123}&c_{1131}&c_{1112}\\c_{2211}&c_{2222}&c_{2233}&c_{2223}&c_{2231}&c_{2212}\\c_{3311}&c_{3322}&c_{3333}&c_{3323}&c_{3331}&c_{3312}\\c_{2311}&c_{2322}&c_{2333}&c_{2323}&c_{2331}&c_{2312}\\c_{3111}&c_{3122}&c_{3133}&c_{3123}&c_{3131}&c_{3112}\\c_{1211}&c_{1222}&c_{1233}&c_{1223}&c_{1231}&c_{1212}\end{bmatrix}}\,\equiv \,{\begin{bmatrix}C_{11}&C_{12}&C_{13}&C_{14}&C_{15}&C_{16}\\C_{12}&C_{22}&C_{23}&C_{24}&C_{25}&C_{26}\\C_{13}&C_{23}&C_{33}&C_{34}&C_{35}&C_{36}\\C_{14}&C_{24}&C_{34}&C_{44}&C_{45}&C_{46}\\C_{15}&C_{25}&C_{35}&C_{45}&C_{55}&C_{56}\\C_{16}&C_{26}&C_{36}&C_{46}&C_{56}&C_{66}\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/85c8bf05adff9dcaec56f4863dd039fae5986a79" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -9.005ex; width:89.392ex; height:19.176ex;" alt="{\displaystyle [{\mathsf {c}}]\,=\,{\begin{bmatrix}c_{1111}&c_{1122}&c_{1133}&c_{1123}&c_{1131}&c_{1112}\\c_{2211}&c_{2222}&c_{2233}&c_{2223}&c_{2231}&c_{2212}\\c_{3311}&c_{3322}&c_{3333}&c_{3323}&c_{3331}&c_{3312}\\c_{2311}&c_{2322}&c_{2333}&c_{2323}&c_{2331}&c_{2312}\\c_{3111}&c_{3122}&c_{3133}&c_{3123}&c_{3131}&c_{3112}\\c_{1211}&c_{1222}&c_{1233}&c_{1223}&c_{1231}&c_{1212}\end{bmatrix}}\,\equiv \,{\begin{bmatrix}C_{11}&C_{12}&C_{13}&C_{14}&C_{15}&C_{16}\\C_{12}&C_{22}&C_{23}&C_{24}&C_{25}&C_{26}\\C_{13}&C_{23}&C_{33}&C_{34}&C_{35}&C_{36}\\C_{14}&C_{24}&C_{34}&C_{44}&C_{45}&C_{46}\\C_{15}&C_{25}&C_{35}&C_{45}&C_{55}&C_{56}\\C_{16}&C_{26}&C_{36}&C_{46}&C_{56}&C_{66}\end{bmatrix}}}"></span> </p><p>and Hooke's law is written as </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [{\boldsymbol {\sigma }}]=[{\mathsf {C}}][{\boldsymbol {\varepsilon }}]\qquad {\text{or}}\qquad \sigma _{i}=C_{ij}\varepsilon _{j}\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">σ<!-- σ --></mi> </mrow> <mo stretchy="false">]</mo> <mo>=</mo> <mo stretchy="false">[</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">C</mi> </mrow> </mrow> <mo stretchy="false">]</mo> <mo stretchy="false">[</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ε<!-- ε --></mi> </mrow> <mo stretchy="false">]</mo> <mspace width="2em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>or</mtext> </mrow> <mspace width="2em" /> <msub> <mi>σ<!-- σ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <msub> <mi>ε<!-- ε --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [{\boldsymbol {\sigma }}]=[{\mathsf {C}}][{\boldsymbol {\varepsilon }}]\qquad {\text{or}}\qquad \sigma _{i}=C_{ij}\varepsilon _{j}\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f0315b5cfc25f83e499fadf8ce4921e11340f8e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:34.045ex; height:3.009ex;" alt="{\displaystyle [{\boldsymbol {\sigma }}]=[{\mathsf {C}}][{\boldsymbol {\varepsilon }}]\qquad {\text{or}}\qquad \sigma _{i}=C_{ij}\varepsilon _{j}\,.}"></span> Similarly the compliance tensor (<b>s</b>) can be written as <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [{\mathsf {s}}]\,=\,{\begin{bmatrix}s_{1111}&s_{1122}&s_{1133}&2s_{1123}&2s_{1131}&2s_{1112}\\s_{2211}&s_{2222}&s_{2233}&2s_{2223}&2s_{2231}&2s_{2212}\\s_{3311}&s_{3322}&s_{3333}&2s_{3323}&2s_{3331}&2s_{3312}\\2s_{2311}&2s_{2322}&2s_{2333}&4s_{2323}&4s_{2331}&4s_{2312}\\2s_{3111}&2s_{3122}&2s_{3133}&4s_{3123}&4s_{3131}&4s_{3112}\\2s_{1211}&2s_{1222}&2s_{1233}&4s_{1223}&4s_{1231}&4s_{1212}\end{bmatrix}}\,\equiv \,{\begin{bmatrix}S_{11}&S_{12}&S_{13}&S_{14}&S_{15}&S_{16}\\S_{12}&S_{22}&S_{23}&S_{24}&S_{25}&S_{26}\\S_{13}&S_{23}&S_{33}&S_{34}&S_{35}&S_{36}\\S_{14}&S_{24}&S_{34}&S_{44}&S_{45}&S_{46}\\S_{15}&S_{25}&S_{35}&S_{45}&S_{55}&S_{56}\\S_{16}&S_{26}&S_{36}&S_{46}&S_{56}&S_{66}\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">s</mi> </mrow> </mrow> <mo stretchy="false">]</mo> <mspace width="thinmathspace" /> <mo>=</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1111</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1122</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1133</mn> </mrow> </msub> </mtd> <mtd> <mn>2</mn> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1123</mn> </mrow> </msub> </mtd> <mtd> <mn>2</mn> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1131</mn> </mrow> </msub> </mtd> <mtd> <mn>2</mn> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1112</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2211</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2222</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2233</mn> </mrow> </msub> </mtd> <mtd> <mn>2</mn> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2223</mn> </mrow> </msub> </mtd> <mtd> <mn>2</mn> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2231</mn> </mrow> </msub> </mtd> <mtd> <mn>2</mn> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2212</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3311</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3322</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3333</mn> </mrow> </msub> </mtd> <mtd> <mn>2</mn> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3323</mn> </mrow> </msub> </mtd> <mtd> <mn>2</mn> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3331</mn> </mrow> </msub> </mtd> <mtd> <mn>2</mn> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3312</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mn>2</mn> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2311</mn> </mrow> </msub> </mtd> <mtd> <mn>2</mn> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2322</mn> </mrow> </msub> </mtd> <mtd> <mn>2</mn> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2333</mn> </mrow> </msub> </mtd> <mtd> <mn>4</mn> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2323</mn> </mrow> </msub> </mtd> <mtd> <mn>4</mn> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2331</mn> </mrow> </msub> </mtd> <mtd> <mn>4</mn> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2312</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mn>2</mn> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3111</mn> </mrow> </msub> </mtd> <mtd> <mn>2</mn> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3122</mn> </mrow> </msub> </mtd> <mtd> <mn>2</mn> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3133</mn> </mrow> </msub> </mtd> <mtd> <mn>4</mn> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3123</mn> </mrow> </msub> </mtd> <mtd> <mn>4</mn> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3131</mn> </mrow> </msub> </mtd> <mtd> <mn>4</mn> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3112</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mn>2</mn> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1211</mn> </mrow> </msub> </mtd> <mtd> <mn>2</mn> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1222</mn> </mrow> </msub> </mtd> <mtd> <mn>2</mn> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1233</mn> </mrow> </msub> </mtd> <mtd> <mn>4</mn> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1223</mn> </mrow> </msub> </mtd> <mtd> <mn>4</mn> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1231</mn> </mrow> </msub> </mtd> <mtd> <mn>4</mn> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1212</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mspace width="thinmathspace" /> <mo>≡<!-- ≡ --></mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>11</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>13</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>14</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>15</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>16</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>22</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>23</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>24</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>25</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>26</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>13</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>23</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>33</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>34</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>35</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>36</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>14</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>24</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>34</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>44</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>45</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>46</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>15</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>25</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>35</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>45</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>55</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>56</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>16</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>26</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>36</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>46</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>56</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>66</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [{\mathsf {s}}]\,=\,{\begin{bmatrix}s_{1111}&s_{1122}&s_{1133}&2s_{1123}&2s_{1131}&2s_{1112}\\s_{2211}&s_{2222}&s_{2233}&2s_{2223}&2s_{2231}&2s_{2212}\\s_{3311}&s_{3322}&s_{3333}&2s_{3323}&2s_{3331}&2s_{3312}\\2s_{2311}&2s_{2322}&2s_{2333}&4s_{2323}&4s_{2331}&4s_{2312}\\2s_{3111}&2s_{3122}&2s_{3133}&4s_{3123}&4s_{3131}&4s_{3112}\\2s_{1211}&2s_{1222}&2s_{1233}&4s_{1223}&4s_{1231}&4s_{1212}\end{bmatrix}}\,\equiv \,{\begin{bmatrix}S_{11}&S_{12}&S_{13}&S_{14}&S_{15}&S_{16}\\S_{12}&S_{22}&S_{23}&S_{24}&S_{25}&S_{26}\\S_{13}&S_{23}&S_{33}&S_{34}&S_{35}&S_{36}\\S_{14}&S_{24}&S_{34}&S_{44}&S_{45}&S_{46}\\S_{15}&S_{25}&S_{35}&S_{45}&S_{55}&S_{56}\\S_{16}&S_{26}&S_{36}&S_{46}&S_{56}&S_{66}\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/34760b2d8ef86f720051aebe5a45a65b312bcab6" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -9.005ex; width:95.305ex; height:19.176ex;" alt="{\displaystyle [{\mathsf {s}}]\,=\,{\begin{bmatrix}s_{1111}&s_{1122}&s_{1133}&2s_{1123}&2s_{1131}&2s_{1112}\\s_{2211}&s_{2222}&s_{2233}&2s_{2223}&2s_{2231}&2s_{2212}\\s_{3311}&s_{3322}&s_{3333}&2s_{3323}&2s_{3331}&2s_{3312}\\2s_{2311}&2s_{2322}&2s_{2333}&4s_{2323}&4s_{2331}&4s_{2312}\\2s_{3111}&2s_{3122}&2s_{3133}&4s_{3123}&4s_{3131}&4s_{3112}\\2s_{1211}&2s_{1222}&2s_{1233}&4s_{1223}&4s_{1231}&4s_{1212}\end{bmatrix}}\,\equiv \,{\begin{bmatrix}S_{11}&S_{12}&S_{13}&S_{14}&S_{15}&S_{16}\\S_{12}&S_{22}&S_{23}&S_{24}&S_{25}&S_{26}\\S_{13}&S_{23}&S_{33}&S_{34}&S_{35}&S_{36}\\S_{14}&S_{24}&S_{34}&S_{44}&S_{45}&S_{46}\\S_{15}&S_{25}&S_{35}&S_{45}&S_{55}&S_{56}\\S_{16}&S_{26}&S_{36}&S_{46}&S_{56}&S_{66}\end{bmatrix}}}"></span> </p> <div class="mw-heading mw-heading4"><h4 id="Change_of_coordinate_system">Change of coordinate system</h4></div> <p>If a linear elastic material is rotated from a reference configuration to another, then the material is symmetric with respect to the rotation if the components of the stiffness tensor in the rotated configuration are related to the components in the reference configuration by the relation<sup id="cite_ref-Slaughter_13-0" class="reference"><a href="#cite_note-Slaughter-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{pqrs}=l_{pi}l_{qj}l_{rk}l_{sl}c_{ijkl}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mi>q</mi> <mi>r</mi> <mi>s</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mi>i</mi> </mrow> </msub> <msub> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> <mi>j</mi> </mrow> </msub> <msub> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> <mi>k</mi> </mrow> </msub> <msub> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> <mi>l</mi> </mrow> </msub> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> <mi>k</mi> <mi>l</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c_{pqrs}=l_{pi}l_{qj}l_{rk}l_{sl}c_{ijkl}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6849505c35a5dabe32c34eefd94bdebf9771dc13" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:20.654ex; height:2.843ex;" alt="{\displaystyle c_{pqrs}=l_{pi}l_{qj}l_{rk}l_{sl}c_{ijkl}}"></span> where <span class="texhtml mvar" style="font-style:italic;">l<sub>ab</sub></span> are the components of an <a href="/wiki/Orthogonal_matrix" title="Orthogonal matrix">orthogonal rotation matrix</a> <span class="texhtml">[<i>L</i>]</span>. The same relation also holds for inversions. </p><p>In matrix notation, if the transformed basis (rotated or inverted) is related to the reference basis by </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [\mathbf {e} _{i}']=[L][\mathbf {e} _{i}]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mo>′</mo> </msubsup> <mo stretchy="false">]</mo> <mo>=</mo> <mo stretchy="false">[</mo> <mi>L</mi> <mo stretchy="false">]</mo> <mo stretchy="false">[</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [\mathbf {e} _{i}']=[L][\mathbf {e} _{i}]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/213d0bb55cc1da894c855871790e09d78635c17b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.612ex; height:3.009ex;" alt="{\displaystyle [\mathbf {e} _{i}']=[L][\mathbf {e} _{i}]}"></span> </p><p>then </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{ij}\varepsilon _{i}\varepsilon _{j}=C_{ij}'\varepsilon '_{i}\varepsilon '_{j}\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <msub> <mi>ε<!-- ε --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>ε<!-- ε --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>=</mo> <msubsup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> <mo>′</mo> </msubsup> <msubsup> <mi>ε<!-- ε --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mo>′</mo> </msubsup> <msubsup> <mi>ε<!-- ε --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> <mo>′</mo> </msubsup> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{ij}\varepsilon _{i}\varepsilon _{j}=C_{ij}'\varepsilon '_{i}\varepsilon '_{j}\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b855f531b6b05ad146321f7b82da8f4c44631056" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:18.163ex; height:3.176ex;" alt="{\displaystyle C_{ij}\varepsilon _{i}\varepsilon _{j}=C_{ij}'\varepsilon '_{i}\varepsilon '_{j}\,.}"></span> In addition, if the material is symmetric with respect to the transformation <span class="texhtml">[<i>L</i>]</span> then <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{ij}=C'_{ij}\quad \implies \quad C_{ij}(\varepsilon _{i}\varepsilon _{j}-\varepsilon '_{i}\varepsilon '_{j})=0\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>=</mo> <msubsup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> <mo>′</mo> </msubsup> <mspace width="1em" /> <mspace width="thickmathspace" /> <mo stretchy="false">⟹<!-- ⟹ --></mo> <mspace width="thickmathspace" /> <mspace width="1em" /> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>ε<!-- ε --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>ε<!-- ε --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>−<!-- − --></mo> <msubsup> <mi>ε<!-- ε --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mo>′</mo> </msubsup> <msubsup> <mi>ε<!-- ε --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> <mo>′</mo> </msubsup> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{ij}=C'_{ij}\quad \implies \quad C_{ij}(\varepsilon _{i}\varepsilon _{j}-\varepsilon '_{i}\varepsilon '_{j})=0\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/547dc39ea492ca5d9ee3f5e7bd16409625e284eb" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:41.244ex; height:3.343ex;" alt="{\displaystyle C_{ij}=C'_{ij}\quad \implies \quad C_{ij}(\varepsilon _{i}\varepsilon _{j}-\varepsilon '_{i}\varepsilon '_{j})=0\,.}"></span> </p> <div class="mw-heading mw-heading4"><h4 id="Orthotropic_materials">Orthotropic materials</h4></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Orthotropic_material" title="Orthotropic material">Orthotropic material</a></div> <p><a href="/wiki/Orthotropic_material" title="Orthotropic material">Orthotropic materials</a> have three <a href="/wiki/Orthogonal" class="mw-redirect" title="Orthogonal">orthogonal</a> <a href="/wiki/Plane_of_symmetry" class="mw-redirect" title="Plane of symmetry">planes of symmetry</a>. If the basis vectors (<span class="texhtml"><b>e</b><sub>1</sub>,<b>e</b><sub>2</sub>,<b>e</b><sub>3</sub></span>) are normals to the planes of symmetry then the coordinate transformation relations imply that </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}\sigma _{1}\\\sigma _{2}\\\sigma _{3}\\\sigma _{4}\\\sigma _{5}\\\sigma _{6}\end{bmatrix}}\,=\,{\begin{bmatrix}C_{11}&C_{12}&C_{13}&0&0&0\\C_{12}&C_{22}&C_{23}&0&0&0\\C_{13}&C_{23}&C_{33}&0&0&0\\0&0&0&C_{44}&0&0\\0&0&0&0&C_{55}&0\\0&0&0&0&0&C_{66}\end{bmatrix}}{\begin{bmatrix}\varepsilon _{1}\\\varepsilon _{2}\\\varepsilon _{3}\\\varepsilon _{4}\\\varepsilon _{5}\\\varepsilon _{6}\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>σ<!-- σ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>σ<!-- σ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>σ<!-- σ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>σ<!-- σ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>σ<!-- σ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>σ<!-- σ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mspace width="thinmathspace" /> <mo>=</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>11</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>13</mn> </mrow> </msub> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>22</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>23</mn> </mrow> </msub> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>13</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>23</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>33</mn> </mrow> </msub> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>44</mn> </mrow> </msub> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>55</mn> </mrow> </msub> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>66</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>ε<!-- ε --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>ε<!-- ε --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>ε<!-- ε --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>ε<!-- ε --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>ε<!-- ε --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>ε<!-- ε --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}\sigma _{1}\\\sigma _{2}\\\sigma _{3}\\\sigma _{4}\\\sigma _{5}\\\sigma _{6}\end{bmatrix}}\,=\,{\begin{bmatrix}C_{11}&C_{12}&C_{13}&0&0&0\\C_{12}&C_{22}&C_{23}&0&0&0\\C_{13}&C_{23}&C_{33}&0&0&0\\0&0&0&C_{44}&0&0\\0&0&0&0&C_{55}&0\\0&0&0&0&0&C_{66}\end{bmatrix}}{\begin{bmatrix}\varepsilon _{1}\\\varepsilon _{2}\\\varepsilon _{3}\\\varepsilon _{4}\\\varepsilon _{5}\\\varepsilon _{6}\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9ac25d50b680d76d994a6077c483bff4ba6a86ed" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -9.005ex; width:52.79ex; height:19.176ex;" alt="{\displaystyle {\begin{bmatrix}\sigma _{1}\\\sigma _{2}\\\sigma _{3}\\\sigma _{4}\\\sigma _{5}\\\sigma _{6}\end{bmatrix}}\,=\,{\begin{bmatrix}C_{11}&C_{12}&C_{13}&0&0&0\\C_{12}&C_{22}&C_{23}&0&0&0\\C_{13}&C_{23}&C_{33}&0&0&0\\0&0&0&C_{44}&0&0\\0&0&0&0&C_{55}&0\\0&0&0&0&0&C_{66}\end{bmatrix}}{\begin{bmatrix}\varepsilon _{1}\\\varepsilon _{2}\\\varepsilon _{3}\\\varepsilon _{4}\\\varepsilon _{5}\\\varepsilon _{6}\end{bmatrix}}}"></span> The inverse of this relation is commonly written as<sup id="cite_ref-Boresi_14-0" class="reference"><a href="#cite_note-Boresi-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup><sup class="noprint Inline-Template" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citing_sources" title="Wikipedia:Citing sources"><span title="This citation requires a reference to the specific page or range of pages in which the material appears. (November 2016)">page needed</span></a></i>]</sup> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}\varepsilon _{xx}\\\varepsilon _{yy}\\\varepsilon _{zz}\\2\varepsilon _{yz}\\2\varepsilon _{zx}\\2\varepsilon _{xy}\end{bmatrix}}\,=\,{\begin{bmatrix}{\frac {1}{E_{x}}}&-{\frac {\nu _{yx}}{E_{y}}}&-{\frac {\nu _{zx}}{E_{z}}}&0&0&0\\-{\frac {\nu _{xy}}{E_{x}}}&{\frac {1}{E_{y}}}&-{\frac {\nu _{zy}}{E_{z}}}&0&0&0\\-{\frac {\nu _{xz}}{E_{x}}}&-{\frac {\nu _{yz}}{E_{y}}}&{\frac {1}{E_{z}}}&0&0&0\\0&0&0&{\frac {1}{G_{yz}}}&0&0\\0&0&0&0&{\frac {1}{G_{zx}}}&0\\0&0&0&0&0&{\frac {1}{G_{xy}}}\\\end{bmatrix}}{\begin{bmatrix}\sigma _{xx}\\\sigma _{yy}\\\sigma _{zz}\\\sigma _{yz}\\\sigma _{zx}\\\sigma _{xy}\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>ε<!-- ε --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>x</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>ε<!-- ε --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> <mi>y</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>ε<!-- ε --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> <mi>z</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mn>2</mn> <msub> <mi>ε<!-- ε --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> <mi>z</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mn>2</mn> <msub> <mi>ε<!-- ε --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> <mi>x</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mn>2</mn> <msub> <mi>ε<!-- ε --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>y</mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mspace width="thinmathspace" /> <mo>=</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mfrac> </mrow> </mtd> <mtd> <mo>−<!-- − --></mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>ν<!-- ν --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> <mi>x</mi> </mrow> </msub> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mfrac> </mrow> </mtd> <mtd> <mo>−<!-- − --></mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>ν<!-- ν --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> <mi>x</mi> </mrow> </msub> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mfrac> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mo>−<!-- − --></mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>ν<!-- ν --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>y</mi> </mrow> </msub> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mfrac> </mrow> </mtd> <mtd> <mo>−<!-- − --></mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>ν<!-- ν --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> <mi>y</mi> </mrow> </msub> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mfrac> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mo>−<!-- − --></mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>ν<!-- ν --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>z</mi> </mrow> </msub> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mfrac> </mrow> </mtd> <mtd> <mo>−<!-- − --></mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>ν<!-- ν --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> <mi>z</mi> </mrow> </msub> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mfrac> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msub> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> <mi>z</mi> </mrow> </msub> </mfrac> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msub> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> <mi>x</mi> </mrow> </msub> </mfrac> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msub> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>y</mi> </mrow> </msub> </mfrac> </mrow> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>σ<!-- σ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>x</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>σ<!-- σ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> <mi>y</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>σ<!-- σ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> <mi>z</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>σ<!-- σ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> <mi>z</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>σ<!-- σ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> <mi>x</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>σ<!-- σ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>y</mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}\varepsilon _{xx}\\\varepsilon _{yy}\\\varepsilon _{zz}\\2\varepsilon _{yz}\\2\varepsilon _{zx}\\2\varepsilon _{xy}\end{bmatrix}}\,=\,{\begin{bmatrix}{\frac {1}{E_{x}}}&-{\frac {\nu _{yx}}{E_{y}}}&-{\frac {\nu _{zx}}{E_{z}}}&0&0&0\\-{\frac {\nu _{xy}}{E_{x}}}&{\frac {1}{E_{y}}}&-{\frac {\nu _{zy}}{E_{z}}}&0&0&0\\-{\frac {\nu _{xz}}{E_{x}}}&-{\frac {\nu _{yz}}{E_{y}}}&{\frac {1}{E_{z}}}&0&0&0\\0&0&0&{\frac {1}{G_{yz}}}&0&0\\0&0&0&0&{\frac {1}{G_{zx}}}&0\\0&0&0&0&0&{\frac {1}{G_{xy}}}\\\end{bmatrix}}{\begin{bmatrix}\sigma _{xx}\\\sigma _{yy}\\\sigma _{zz}\\\sigma _{yz}\\\sigma _{zx}\\\sigma _{xy}\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d34c35f7208728d527b680bd474e323d9129a56f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -13.005ex; width:60.8ex; height:27.176ex;" alt="{\displaystyle {\begin{bmatrix}\varepsilon _{xx}\\\varepsilon _{yy}\\\varepsilon _{zz}\\2\varepsilon _{yz}\\2\varepsilon _{zx}\\2\varepsilon _{xy}\end{bmatrix}}\,=\,{\begin{bmatrix}{\frac {1}{E_{x}}}&-{\frac {\nu _{yx}}{E_{y}}}&-{\frac {\nu _{zx}}{E_{z}}}&0&0&0\\-{\frac {\nu _{xy}}{E_{x}}}&{\frac {1}{E_{y}}}&-{\frac {\nu _{zy}}{E_{z}}}&0&0&0\\-{\frac {\nu _{xz}}{E_{x}}}&-{\frac {\nu _{yz}}{E_{y}}}&{\frac {1}{E_{z}}}&0&0&0\\0&0&0&{\frac {1}{G_{yz}}}&0&0\\0&0&0&0&{\frac {1}{G_{zx}}}&0\\0&0&0&0&0&{\frac {1}{G_{xy}}}\\\end{bmatrix}}{\begin{bmatrix}\sigma _{xx}\\\sigma _{yy}\\\sigma _{zz}\\\sigma _{yz}\\\sigma _{zx}\\\sigma _{xy}\end{bmatrix}}}"></span> where </p> <ul><li><span class="texhtml mvar" style="font-style:italic;">E<sub>i</sub></span> is the <a href="/wiki/Young%27s_modulus" title="Young's modulus">Young's modulus</a> along axis <span class="texhtml mvar" style="font-style:italic;">i</span></li> <li><span class="texhtml mvar" style="font-style:italic;">G<sub>ij</sub></span> is the <a href="/wiki/Shear_modulus" title="Shear modulus">shear modulus</a> in direction <span class="texhtml mvar" style="font-style:italic;">j</span> on the plane whose normal is in direction <span class="texhtml mvar" style="font-style:italic;">i</span></li> <li><span class="texhtml mvar" style="font-style:italic;">ν<sub>ij</sub></span> is the <a href="/wiki/Poisson%27s_ratio" title="Poisson's ratio">Poisson's ratio</a> that corresponds to a contraction in direction <span class="texhtml mvar" style="font-style:italic;">j</span> when an extension is applied in direction <span class="texhtml mvar" style="font-style:italic;">i</span>.</li></ul> <p>Under <i>plane stress</i> conditions, <span class="texhtml"><i>σ<sub>zz</sub></i> = <i>σ<sub>zx</sub></i> = <i>σ<sub>yz</sub></i> = 0</span>, Hooke's law for an orthotropic material takes the form <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}\varepsilon _{xx}\\\varepsilon _{yy}\\2\varepsilon _{xy}\end{bmatrix}}\,=\,{\begin{bmatrix}{\frac {1}{E_{x}}}&-{\frac {\nu _{yx}}{E_{y}}}&0\\-{\frac {\nu _{xy}}{E_{x}}}&{\frac {1}{E_{y}}}&0\\0&0&{\frac {1}{G_{xy}}}\end{bmatrix}}{\begin{bmatrix}\sigma _{xx}\\\sigma _{yy}\\\sigma _{xy}\end{bmatrix}}\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>ε<!-- ε --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>x</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>ε<!-- ε --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> <mi>y</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mn>2</mn> <msub> <mi>ε<!-- ε --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>y</mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mspace width="thinmathspace" /> <mo>=</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mfrac> </mrow> </mtd> <mtd> <mo>−<!-- − --></mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>ν<!-- ν --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> <mi>x</mi> </mrow> </msub> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mfrac> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mo>−<!-- − --></mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>ν<!-- ν --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>y</mi> </mrow> </msub> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mfrac> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msub> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>y</mi> </mrow> </msub> </mfrac> </mrow> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>σ<!-- σ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>x</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>σ<!-- σ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> <mi>y</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>σ<!-- σ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>y</mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}\varepsilon _{xx}\\\varepsilon _{yy}\\2\varepsilon _{xy}\end{bmatrix}}\,=\,{\begin{bmatrix}{\frac {1}{E_{x}}}&-{\frac {\nu _{yx}}{E_{y}}}&0\\-{\frac {\nu _{xy}}{E_{x}}}&{\frac {1}{E_{y}}}&0\\0&0&{\frac {1}{G_{xy}}}\end{bmatrix}}{\begin{bmatrix}\sigma _{xx}\\\sigma _{yy}\\\sigma _{xy}\end{bmatrix}}\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48afddb72fa158772f533e563ef557aa2ae2598f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.171ex; width:42.597ex; height:13.509ex;" alt="{\displaystyle {\begin{bmatrix}\varepsilon _{xx}\\\varepsilon _{yy}\\2\varepsilon _{xy}\end{bmatrix}}\,=\,{\begin{bmatrix}{\frac {1}{E_{x}}}&-{\frac {\nu _{yx}}{E_{y}}}&0\\-{\frac {\nu _{xy}}{E_{x}}}&{\frac {1}{E_{y}}}&0\\0&0&{\frac {1}{G_{xy}}}\end{bmatrix}}{\begin{bmatrix}\sigma _{xx}\\\sigma _{yy}\\\sigma _{xy}\end{bmatrix}}\,.}"></span> The inverse relation is <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}\sigma _{xx}\\\sigma _{yy}\\\sigma _{xy}\end{bmatrix}}\,=\,{\frac {1}{1-\nu _{xy}\nu _{yx}}}{\begin{bmatrix}E_{x}&\nu _{yx}E_{x}&0\\\nu _{xy}E_{y}&E_{y}&0\\0&0&G_{xy}(1-\nu _{xy}\nu _{yx})\end{bmatrix}}{\begin{bmatrix}\varepsilon _{xx}\\\varepsilon _{yy}\\2\varepsilon _{xy}\end{bmatrix}}\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>σ<!-- σ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>x</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>σ<!-- σ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> <mi>y</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>σ<!-- σ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>y</mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mspace width="thinmathspace" /> <mo>=</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>1</mn> <mo>−<!-- − --></mo> <msub> <mi>ν<!-- ν --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>y</mi> </mrow> </msub> <msub> <mi>ν<!-- ν --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> <mi>x</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>ν<!-- ν --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> <mi>x</mi> </mrow> </msub> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <msub> <mi>ν<!-- ν --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>y</mi> </mrow> </msub> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <msub> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>y</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−<!-- − --></mo> <msub> <mi>ν<!-- ν --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>y</mi> </mrow> </msub> <msub> <mi>ν<!-- ν --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> <mi>x</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>ε<!-- ε --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>x</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>ε<!-- ε --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> <mi>y</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mn>2</mn> <msub> <mi>ε<!-- ε --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>y</mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}\sigma _{xx}\\\sigma _{yy}\\\sigma _{xy}\end{bmatrix}}\,=\,{\frac {1}{1-\nu _{xy}\nu _{yx}}}{\begin{bmatrix}E_{x}&\nu _{yx}E_{x}&0\\\nu _{xy}E_{y}&E_{y}&0\\0&0&G_{xy}(1-\nu _{xy}\nu _{yx})\end{bmatrix}}{\begin{bmatrix}\varepsilon _{xx}\\\varepsilon _{yy}\\2\varepsilon _{xy}\end{bmatrix}}\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d8ab0d83874095f926d648808090a7dc1b8ca05e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.338ex; width:67.732ex; height:9.843ex;" alt="{\displaystyle {\begin{bmatrix}\sigma _{xx}\\\sigma _{yy}\\\sigma _{xy}\end{bmatrix}}\,=\,{\frac {1}{1-\nu _{xy}\nu _{yx}}}{\begin{bmatrix}E_{x}&\nu _{yx}E_{x}&0\\\nu _{xy}E_{y}&E_{y}&0\\0&0&G_{xy}(1-\nu _{xy}\nu _{yx})\end{bmatrix}}{\begin{bmatrix}\varepsilon _{xx}\\\varepsilon _{yy}\\2\varepsilon _{xy}\end{bmatrix}}\,.}"></span> The transposed form of the above stiffness matrix is also often used. </p> <div class="mw-heading mw-heading4"><h4 id="Transversely_isotropic_materials">Transversely isotropic materials</h4></div> <p>A <a href="/wiki/Transversely_isotropic" class="mw-redirect" title="Transversely isotropic">transversely isotropic</a> material is symmetric with respect to a rotation about an <a href="/wiki/Axis_of_symmetry" class="mw-redirect" title="Axis of symmetry">axis of symmetry</a>. For such a material, if <span class="texhtml"><b>e</b><sub>3</sub></span> is the axis of symmetry, Hooke's law can be expressed as <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}\sigma _{1}\\\sigma _{2}\\\sigma _{3}\\\sigma _{4}\\\sigma _{5}\\\sigma _{6}\end{bmatrix}}\,=\,{\begin{bmatrix}C_{11}&C_{12}&C_{13}&0&0&0\\C_{12}&C_{11}&C_{13}&0&0&0\\C_{13}&C_{13}&C_{33}&0&0&0\\0&0&0&C_{44}&0&0\\0&0&0&0&C_{44}&0\\0&0&0&0&0&{\frac {C_{11}-C_{12}}{2}}\end{bmatrix}}{\begin{bmatrix}\varepsilon _{1}\\\varepsilon _{2}\\\varepsilon _{3}\\\varepsilon _{4}\\\varepsilon _{5}\\\varepsilon _{6}\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>σ<!-- σ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>σ<!-- σ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>σ<!-- σ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>σ<!-- σ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>σ<!-- σ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>σ<!-- σ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mspace width="thinmathspace" /> <mo>=</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>11</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>13</mn> </mrow> </msub> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>11</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>13</mn> </mrow> </msub> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>13</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>13</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>33</mn> </mrow> </msub> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>44</mn> </mrow> </msub> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>44</mn> </mrow> </msub> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>11</mn> </mrow> </msub> <mo>−<!-- − --></mo> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> </mrow> <mn>2</mn> </mfrac> </mrow> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>ε<!-- ε --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>ε<!-- ε --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>ε<!-- ε --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>ε<!-- ε --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>ε<!-- ε --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>ε<!-- ε --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}\sigma _{1}\\\sigma _{2}\\\sigma _{3}\\\sigma _{4}\\\sigma _{5}\\\sigma _{6}\end{bmatrix}}\,=\,{\begin{bmatrix}C_{11}&C_{12}&C_{13}&0&0&0\\C_{12}&C_{11}&C_{13}&0&0&0\\C_{13}&C_{13}&C_{33}&0&0&0\\0&0&0&C_{44}&0&0\\0&0&0&0&C_{44}&0\\0&0&0&0&0&{\frac {C_{11}-C_{12}}{2}}\end{bmatrix}}{\begin{bmatrix}\varepsilon _{1}\\\varepsilon _{2}\\\varepsilon _{3}\\\varepsilon _{4}\\\varepsilon _{5}\\\varepsilon _{6}\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a8d55554cdf1abd0e62360faf969edca00c6f28b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -9.505ex; width:56.714ex; height:20.176ex;" alt="{\displaystyle {\begin{bmatrix}\sigma _{1}\\\sigma _{2}\\\sigma _{3}\\\sigma _{4}\\\sigma _{5}\\\sigma _{6}\end{bmatrix}}\,=\,{\begin{bmatrix}C_{11}&C_{12}&C_{13}&0&0&0\\C_{12}&C_{11}&C_{13}&0&0&0\\C_{13}&C_{13}&C_{33}&0&0&0\\0&0&0&C_{44}&0&0\\0&0&0&0&C_{44}&0\\0&0&0&0&0&{\frac {C_{11}-C_{12}}{2}}\end{bmatrix}}{\begin{bmatrix}\varepsilon _{1}\\\varepsilon _{2}\\\varepsilon _{3}\\\varepsilon _{4}\\\varepsilon _{5}\\\varepsilon _{6}\end{bmatrix}}}"></span> </p><p>More frequently, the <span class="texhtml"><i>x</i> ≡ <b>e</b><sub>1</sub></span> axis is taken to be the axis of symmetry and the inverse Hooke's law is written as <sup id="cite_ref-Tan_15-0" class="reference"><a href="#cite_note-Tan-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}\varepsilon _{xx}\\\varepsilon _{yy}\\\varepsilon _{zz}\\2\varepsilon _{yz}\\2\varepsilon _{zx}\\2\varepsilon _{xy}\end{bmatrix}}\,=\,{\begin{bmatrix}{\frac {1}{E_{x}}}&-{\frac {\nu _{yx}}{E_{y}}}&-{\frac {\nu _{zx}}{E_{z}}}&0&0&0\\-{\frac {\nu _{xy}}{E_{x}}}&{\frac {1}{E_{y}}}&-{\frac {\nu _{zy}}{E_{z}}}&0&0&0\\-{\frac {\nu _{xz}}{E_{x}}}&-{\frac {\nu _{yz}}{E_{y}}}&{\frac {1}{E_{z}}}&0&0&0\\0&0&0&{\frac {1}{G_{yz}}}&0&0\\0&0&0&0&{\frac {1}{G_{xz}}}&0\\0&0&0&0&0&{\frac {1}{G_{xy}}}\\\end{bmatrix}}{\begin{bmatrix}\sigma _{xx}\\\sigma _{yy}\\\sigma _{zz}\\\sigma _{yz}\\\sigma _{zx}\\\sigma _{xy}\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>ε<!-- ε --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>x</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>ε<!-- ε --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> <mi>y</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>ε<!-- ε --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> <mi>z</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mn>2</mn> <msub> <mi>ε<!-- ε --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> <mi>z</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mn>2</mn> <msub> <mi>ε<!-- ε --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> <mi>x</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mn>2</mn> <msub> <mi>ε<!-- ε --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>y</mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mspace width="thinmathspace" /> <mo>=</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mfrac> </mrow> </mtd> <mtd> <mo>−<!-- − --></mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>ν<!-- ν --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> <mi>x</mi> </mrow> </msub> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mfrac> </mrow> </mtd> <mtd> <mo>−<!-- − --></mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>ν<!-- ν --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> <mi>x</mi> </mrow> </msub> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mfrac> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mo>−<!-- − --></mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>ν<!-- ν --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>y</mi> </mrow> </msub> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mfrac> </mrow> </mtd> <mtd> <mo>−<!-- − --></mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>ν<!-- ν --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> <mi>y</mi> </mrow> </msub> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mfrac> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mo>−<!-- − --></mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>ν<!-- ν --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>z</mi> </mrow> </msub> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mfrac> </mrow> </mtd> <mtd> <mo>−<!-- − --></mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>ν<!-- ν --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> <mi>z</mi> </mrow> </msub> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mfrac> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msub> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> <mi>z</mi> </mrow> </msub> </mfrac> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msub> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>z</mi> </mrow> </msub> </mfrac> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msub> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>y</mi> </mrow> </msub> </mfrac> </mrow> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>σ<!-- σ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>x</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>σ<!-- σ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> <mi>y</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>σ<!-- σ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> <mi>z</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>σ<!-- σ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> <mi>z</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>σ<!-- σ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> <mi>x</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>σ<!-- σ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>y</mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}\varepsilon _{xx}\\\varepsilon _{yy}\\\varepsilon _{zz}\\2\varepsilon _{yz}\\2\varepsilon _{zx}\\2\varepsilon _{xy}\end{bmatrix}}\,=\,{\begin{bmatrix}{\frac {1}{E_{x}}}&-{\frac {\nu _{yx}}{E_{y}}}&-{\frac {\nu _{zx}}{E_{z}}}&0&0&0\\-{\frac {\nu _{xy}}{E_{x}}}&{\frac {1}{E_{y}}}&-{\frac {\nu _{zy}}{E_{z}}}&0&0&0\\-{\frac {\nu _{xz}}{E_{x}}}&-{\frac {\nu _{yz}}{E_{y}}}&{\frac {1}{E_{z}}}&0&0&0\\0&0&0&{\frac {1}{G_{yz}}}&0&0\\0&0&0&0&{\frac {1}{G_{xz}}}&0\\0&0&0&0&0&{\frac {1}{G_{xy}}}\\\end{bmatrix}}{\begin{bmatrix}\sigma _{xx}\\\sigma _{yy}\\\sigma _{zz}\\\sigma _{yz}\\\sigma _{zx}\\\sigma _{xy}\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cbcd77de6d545d6c86ad5c224ba3a6b71ab9ecb6" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -13.005ex; width:60.8ex; height:27.176ex;" alt="{\displaystyle {\begin{bmatrix}\varepsilon _{xx}\\\varepsilon _{yy}\\\varepsilon _{zz}\\2\varepsilon _{yz}\\2\varepsilon _{zx}\\2\varepsilon _{xy}\end{bmatrix}}\,=\,{\begin{bmatrix}{\frac {1}{E_{x}}}&-{\frac {\nu _{yx}}{E_{y}}}&-{\frac {\nu _{zx}}{E_{z}}}&0&0&0\\-{\frac {\nu _{xy}}{E_{x}}}&{\frac {1}{E_{y}}}&-{\frac {\nu _{zy}}{E_{z}}}&0&0&0\\-{\frac {\nu _{xz}}{E_{x}}}&-{\frac {\nu _{yz}}{E_{y}}}&{\frac {1}{E_{z}}}&0&0&0\\0&0&0&{\frac {1}{G_{yz}}}&0&0\\0&0&0&0&{\frac {1}{G_{xz}}}&0\\0&0&0&0&0&{\frac {1}{G_{xy}}}\\\end{bmatrix}}{\begin{bmatrix}\sigma _{xx}\\\sigma _{yy}\\\sigma _{zz}\\\sigma _{yz}\\\sigma _{zx}\\\sigma _{xy}\end{bmatrix}}}"></span> </p> <dl><dd></dd></dl> <div class="mw-heading mw-heading4"><h4 id="Universal_elastic_anisotropy_index">Universal elastic anisotropy index</h4></div> <p>To grasp the degree of anisotropy of any class, a <b>universal elastic anisotropy index</b> (AU)<sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> was formulated. It replaces the <a href="/wiki/Zener_ratio" title="Zener ratio">Zener ratio</a>, which is suited for <a href="/wiki/Cubic_crystal_system" title="Cubic crystal system">cubic crystals</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Thermodynamic_basis">Thermodynamic basis</h2></div> <p>Linear deformations of elastic materials can be approximated as <a href="/wiki/Adiabatic" class="mw-redirect" title="Adiabatic">adiabatic</a>. Under these conditions and for quasistatic processes the <a href="/wiki/First_law_of_thermodynamics" title="First law of thermodynamics">first law of thermodynamics</a> for a deformed body can be expressed as <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta W=\delta U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ<!-- δ --></mi> <mi>W</mi> <mo>=</mo> <mi>δ<!-- δ --></mi> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta W=\delta U}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e0d6067e0a73913089b728cfc5b6ce14b53f4a1" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.414ex; height:2.343ex;" alt="{\displaystyle \delta W=\delta U}"></span> where <span class="texhtml mvar" style="font-style:italic;">δU</span> is the increase in <a href="/wiki/Internal_energy" title="Internal energy">internal energy</a> and <span class="texhtml mvar" style="font-style:italic;">δW</span> is the <a href="/wiki/Work_(physics)" title="Work (physics)">work</a> done by external forces. The work can be split into two terms <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta W=\delta W_{\mathrm {s} }+\delta W_{\mathrm {b} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ<!-- δ --></mi> <mi>W</mi> <mo>=</mo> <mi>δ<!-- δ --></mi> <msub> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">s</mi> </mrow> </mrow> </msub> <mo>+</mo> <mi>δ<!-- δ --></mi> <msub> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">b</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta W=\delta W_{\mathrm {s} }+\delta W_{\mathrm {b} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cb2c3072eb58b6cf4089381c76869283c2d85856" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:17.934ex; height:2.676ex;" alt="{\displaystyle \delta W=\delta W_{\mathrm {s} }+\delta W_{\mathrm {b} }}"></span> where <span class="texhtml"><i>δW</i><sub>s</sub></span> is the work done by <a href="/wiki/Surface_force" title="Surface force">surface forces</a> while <span class="texhtml"><i>δW</i><sub>b</sub></span> is the work done by <a href="/wiki/Body_force" title="Body force">body forces</a>. If <span class="texhtml"><i>δ</i><b>u</b></span> is a <a href="/wiki/Calculus_of_variations" title="Calculus of variations">variation</a> of the displacement field <span class="texhtml"><b>u</b></span> in the body, then the two external work terms can be expressed as <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta W_{\mathrm {s} }=\int _{\partial \Omega }\mathbf {t} \cdot \delta \mathbf {u} \,dS\,;\qquad \delta W_{\mathrm {b} }=\int _{\Omega }\mathbf {b} \cdot \delta \mathbf {u} \,dV}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ<!-- δ --></mi> <msub> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">s</mi> </mrow> </mrow> </msub> <mo>=</mo> <msub> <mo>∫<!-- ∫ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∂<!-- ∂ --></mi> <mi mathvariant="normal">Ω<!-- Ω --></mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">t</mi> </mrow> <mo>⋅<!-- ⋅ --></mo> <mi>δ<!-- δ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mi>S</mi> <mspace width="thinmathspace" /> <mo>;</mo> <mspace width="2em" /> <mi>δ<!-- δ --></mi> <msub> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">b</mi> </mrow> </mrow> </msub> <mo>=</mo> <msub> <mo>∫<!-- ∫ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Ω<!-- Ω --></mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mo>⋅<!-- ⋅ --></mo> <mi>δ<!-- δ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta W_{\mathrm {s} }=\int _{\partial \Omega }\mathbf {t} \cdot \delta \mathbf {u} \,dS\,;\qquad \delta W_{\mathrm {b} }=\int _{\Omega }\mathbf {b} \cdot \delta \mathbf {u} \,dV}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/886cdfe57770bfbd2ebbbfa0cdcec62702532bb6" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:45.346ex; height:5.676ex;" alt="{\displaystyle \delta W_{\mathrm {s} }=\int _{\partial \Omega }\mathbf {t} \cdot \delta \mathbf {u} \,dS\,;\qquad \delta W_{\mathrm {b} }=\int _{\Omega }\mathbf {b} \cdot \delta \mathbf {u} \,dV}"></span> where <span class="texhtml"><b>t</b></span> is the surface <a href="/wiki/Stress_(mechanics)" title="Stress (mechanics)">traction</a> vector, <span class="texhtml"><b>b</b></span> is the body force vector, <span class="texhtml mvar" style="font-style:italic;">Ω</span> represents the body and <span class="texhtml">∂<i>Ω</i></span> represents its surface. Using the relation between the <a href="/wiki/Stress_(mechanics)" title="Stress (mechanics)">Cauchy stress</a> and the surface traction, <span class="texhtml"><b>t</b> = <b>n</b> · <b>σ</b></span> (where <span class="texhtml"><b>n</b></span> is the unit outward normal to <span class="texhtml">∂<i>Ω</i></span>), we have <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta W=\delta U=\int _{\partial \Omega }(\mathbf {n} \cdot {\boldsymbol {\sigma }})\cdot \delta \mathbf {u} \,dS+\int _{\Omega }\mathbf {b} \cdot \delta \mathbf {u} \,dV\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ<!-- δ --></mi> <mi>W</mi> <mo>=</mo> <mi>δ<!-- δ --></mi> <mi>U</mi> <mo>=</mo> <msub> <mo>∫<!-- ∫ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∂<!-- ∂ --></mi> <mi mathvariant="normal">Ω<!-- Ω --></mi> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">n</mi> </mrow> <mo>⋅<!-- ⋅ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">σ<!-- σ --></mi> </mrow> <mo stretchy="false">)</mo> <mo>⋅<!-- ⋅ --></mo> <mi>δ<!-- δ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mi>S</mi> <mo>+</mo> <msub> <mo>∫<!-- ∫ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Ω<!-- Ω --></mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mo>⋅<!-- ⋅ --></mo> <mi>δ<!-- δ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mi>V</mi> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta W=\delta U=\int _{\partial \Omega }(\mathbf {n} \cdot {\boldsymbol {\sigma }})\cdot \delta \mathbf {u} \,dS+\int _{\Omega }\mathbf {b} \cdot \delta \mathbf {u} \,dV\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bd85e0fb67b1d440dc161c7dd7c2a7c5ef93f638" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:46.1ex; height:5.676ex;" alt="{\displaystyle \delta W=\delta U=\int _{\partial \Omega }(\mathbf {n} \cdot {\boldsymbol {\sigma }})\cdot \delta \mathbf {u} \,dS+\int _{\Omega }\mathbf {b} \cdot \delta \mathbf {u} \,dV\,.}"></span> Converting the <a href="/wiki/Surface_integral" title="Surface integral">surface integral</a> into a <a href="/wiki/Volume_integral" title="Volume integral">volume integral</a> via the <a href="/wiki/Divergence_theorem" title="Divergence theorem">divergence theorem</a> gives <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta U=\int _{\Omega }{\big (}\nabla \cdot ({\boldsymbol {\sigma }}\cdot \delta \mathbf {u} )+\mathbf {b} \cdot \delta \mathbf {u} {\big )}\,dV\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ<!-- δ --></mi> <mi>U</mi> <mo>=</mo> <msub> <mo>∫<!-- ∫ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Ω<!-- Ω --></mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mi mathvariant="normal">∇<!-- ∇ --></mi> <mo>⋅<!-- ⋅ --></mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">σ<!-- σ --></mi> </mrow> <mo>⋅<!-- ⋅ --></mo> <mi>δ<!-- δ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo stretchy="false">)</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mo>⋅<!-- ⋅ --></mo> <mi>δ<!-- δ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mi>V</mi> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta U=\int _{\Omega }{\big (}\nabla \cdot ({\boldsymbol {\sigma }}\cdot \delta \mathbf {u} )+\mathbf {b} \cdot \delta \mathbf {u} {\big )}\,dV\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a6fcc7b1e52a282fb9cff067b0aa9d681b20ac78" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:35.353ex; height:5.676ex;" alt="{\displaystyle \delta U=\int _{\Omega }{\big (}\nabla \cdot ({\boldsymbol {\sigma }}\cdot \delta \mathbf {u} )+\mathbf {b} \cdot \delta \mathbf {u} {\big )}\,dV\,.}"></span> Using the symmetry of the Cauchy stress and the identity <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nabla \cdot (\mathbf {a} \cdot \mathbf {b} )=(\nabla \cdot \mathbf {a} )\cdot \mathbf {b} +{\tfrac {1}{2}}\left(\mathbf {a} ^{\mathsf {T}}:\nabla \mathbf {b} +\mathbf {a} :(\nabla \mathbf {b} )^{\mathsf {T}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∇<!-- ∇ --></mi> <mo>⋅<!-- ⋅ --></mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>⋅<!-- ⋅ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi mathvariant="normal">∇<!-- ∇ --></mi> <mo>⋅<!-- ⋅ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo stretchy="false">)</mo> <mo>⋅<!-- ⋅ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mrow> <mo>(</mo> <mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">T</mi> </mrow> </mrow> </msup> <mo>:</mo> <mi mathvariant="normal">∇<!-- ∇ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>:</mo> <mo stretchy="false">(</mo> <mi mathvariant="normal">∇<!-- ∇ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">T</mi> </mrow> </mrow> </msup> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nabla \cdot (\mathbf {a} \cdot \mathbf {b} )=(\nabla \cdot \mathbf {a} )\cdot \mathbf {b} +{\tfrac {1}{2}}\left(\mathbf {a} ^{\mathsf {T}}:\nabla \mathbf {b} +\mathbf {a} :(\nabla \mathbf {b} )^{\mathsf {T}}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/14649ac9347a092aca4283f6fd50d581f15d4f17" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:50.559ex; height:3.509ex;" alt="{\displaystyle \nabla \cdot (\mathbf {a} \cdot \mathbf {b} )=(\nabla \cdot \mathbf {a} )\cdot \mathbf {b} +{\tfrac {1}{2}}\left(\mathbf {a} ^{\mathsf {T}}:\nabla \mathbf {b} +\mathbf {a} :(\nabla \mathbf {b} )^{\mathsf {T}}\right)}"></span> we have the following </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta U=\int _{\Omega }\left({\boldsymbol {\sigma }}:{\tfrac {1}{2}}\left(\nabla \delta \mathbf {u} +(\nabla \delta \mathbf {u} )^{\mathsf {T}}\right)+\left(\nabla \cdot {\boldsymbol {\sigma }}+\mathbf {b} \right)\cdot \delta \mathbf {u} \right)\,dV\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ<!-- δ --></mi> <mi>U</mi> <mo>=</mo> <msub> <mo>∫<!-- ∫ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Ω<!-- Ω --></mi> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">σ<!-- σ --></mi> </mrow> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mrow> <mo>(</mo> <mrow> <mi mathvariant="normal">∇<!-- ∇ --></mi> <mi>δ<!-- δ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo>+</mo> <mo stretchy="false">(</mo> <mi mathvariant="normal">∇<!-- ∇ --></mi> <mi>δ<!-- δ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">T</mi> </mrow> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mrow> <mo>(</mo> <mrow> <mi mathvariant="normal">∇<!-- ∇ --></mi> <mo>⋅<!-- ⋅ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">σ<!-- σ --></mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> </mrow> <mo>)</mo> </mrow> <mo>⋅<!-- ⋅ --></mo> <mi>δ<!-- δ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> </mrow> <mo>)</mo> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mi>V</mi> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta U=\int _{\Omega }\left({\boldsymbol {\sigma }}:{\tfrac {1}{2}}\left(\nabla \delta \mathbf {u} +(\nabla \delta \mathbf {u} )^{\mathsf {T}}\right)+\left(\nabla \cdot {\boldsymbol {\sigma }}+\mathbf {b} \right)\cdot \delta \mathbf {u} \right)\,dV\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/666f86c6fd71dcbac526537e9c954fbc367e1c91" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:57.015ex; height:5.676ex;" alt="{\displaystyle \delta U=\int _{\Omega }\left({\boldsymbol {\sigma }}:{\tfrac {1}{2}}\left(\nabla \delta \mathbf {u} +(\nabla \delta \mathbf {u} )^{\mathsf {T}}\right)+\left(\nabla \cdot {\boldsymbol {\sigma }}+\mathbf {b} \right)\cdot \delta \mathbf {u} \right)\,dV\,.}"></span> From the definition of <a href="/wiki/Infinitesimal_strain_theory" title="Infinitesimal strain theory">strain</a> and from the equations of <a href="/wiki/Linear_elasticity" title="Linear elasticity">equilibrium</a> we have <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta {\boldsymbol {\varepsilon }}={\tfrac {1}{2}}\left(\nabla \delta \mathbf {u} +(\nabla \delta \mathbf {u} )^{\mathsf {T}}\right)\,;\qquad \nabla \cdot {\boldsymbol {\sigma }}+\mathbf {b} =\mathbf {0} \,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ<!-- δ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ε<!-- ε --></mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mrow> <mo>(</mo> <mrow> <mi mathvariant="normal">∇<!-- ∇ --></mi> <mi>δ<!-- δ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo>+</mo> <mo stretchy="false">(</mo> <mi mathvariant="normal">∇<!-- ∇ --></mi> <mi>δ<!-- δ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">T</mi> </mrow> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mspace width="thinmathspace" /> <mo>;</mo> <mspace width="2em" /> <mi mathvariant="normal">∇<!-- ∇ --></mi> <mo>⋅<!-- ⋅ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">σ<!-- σ --></mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">0</mn> </mrow> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta {\boldsymbol {\varepsilon }}={\tfrac {1}{2}}\left(\nabla \delta \mathbf {u} +(\nabla \delta \mathbf {u} )^{\mathsf {T}}\right)\,;\qquad \nabla \cdot {\boldsymbol {\sigma }}+\mathbf {b} =\mathbf {0} \,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/25d5306a319e75bb8037f12111ce3a1fe3e613b2" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:45.95ex; height:3.509ex;" alt="{\displaystyle \delta {\boldsymbol {\varepsilon }}={\tfrac {1}{2}}\left(\nabla \delta \mathbf {u} +(\nabla \delta \mathbf {u} )^{\mathsf {T}}\right)\,;\qquad \nabla \cdot {\boldsymbol {\sigma }}+\mathbf {b} =\mathbf {0} \,.}"></span> Hence we can write <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta U=\int _{\Omega }{\boldsymbol {\sigma }}:\delta {\boldsymbol {\varepsilon }}\,dV}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ<!-- δ --></mi> <mi>U</mi> <mo>=</mo> <msub> <mo>∫<!-- ∫ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Ω<!-- Ω --></mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">σ<!-- σ --></mi> </mrow> <mo>:</mo> <mi>δ<!-- δ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ε<!-- ε --></mi> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta U=\int _{\Omega }{\boldsymbol {\sigma }}:\delta {\boldsymbol {\varepsilon }}\,dV}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/39dfee64e6c4b7c7e7a79d90d077101c771fa6ee" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:18.228ex; height:5.676ex;" alt="{\displaystyle \delta U=\int _{\Omega }{\boldsymbol {\sigma }}:\delta {\boldsymbol {\varepsilon }}\,dV}"></span> and therefore the variation in the <a href="/wiki/Internal_energy" title="Internal energy">internal energy</a> density is given by <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta U_{0}={\boldsymbol {\sigma }}:\delta {\boldsymbol {\varepsilon }}\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ<!-- δ --></mi> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">σ<!-- σ --></mi> </mrow> <mo>:</mo> <mi>δ<!-- δ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ε<!-- ε --></mi> </mrow> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta U_{0}={\boldsymbol {\sigma }}:\delta {\boldsymbol {\varepsilon }}\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e4c61baf65d92be418989874a3ac3ef801bf011" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.633ex; height:2.676ex;" alt="{\displaystyle \delta U_{0}={\boldsymbol {\sigma }}:\delta {\boldsymbol {\varepsilon }}\,.}"></span> An <a href="/wiki/Elasticity_(physics)" title="Elasticity (physics)">elastic</a> material is defined as one in which the total internal energy is equal to the <a href="/wiki/Potential_energy" title="Potential energy">potential energy</a> of the internal forces (also called the <b>elastic strain energy</b>). Therefore, the internal energy density is a function of the strains, <span class="texhtml"><i>U</i><sub>0</sub> = <i>U</i><sub>0</sub>(<b>ε</b>)</span> and the variation of the internal energy can be expressed as <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta U_{0}={\frac {\partial U_{0}}{\partial {\boldsymbol {\varepsilon }}}}:\delta {\boldsymbol {\varepsilon }}\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ<!-- δ --></mi> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂<!-- ∂ --></mi> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂<!-- ∂ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ε<!-- ε --></mi> </mrow> </mrow> </mfrac> </mrow> <mo>:</mo> <mi>δ<!-- δ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ε<!-- ε --></mi> </mrow> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta U_{0}={\frac {\partial U_{0}}{\partial {\boldsymbol {\varepsilon }}}}:\delta {\boldsymbol {\varepsilon }}\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bcb5556d24da4fe1e92c2170638fb91adf768ef6" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:16.834ex; height:5.509ex;" alt="{\displaystyle \delta U_{0}={\frac {\partial U_{0}}{\partial {\boldsymbol {\varepsilon }}}}:\delta {\boldsymbol {\varepsilon }}\,.}"></span> Since the variation of strain is arbitrary, the stress–strain relation of an elastic material is given by <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {\sigma }}={\frac {\partial U_{0}}{\partial {\boldsymbol {\varepsilon }}}}\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">σ<!-- σ --></mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂<!-- ∂ --></mi> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂<!-- ∂ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ε<!-- ε --></mi> </mrow> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {\sigma }}={\frac {\partial U_{0}}{\partial {\boldsymbol {\varepsilon }}}}\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf28aa9f1aef51b9ec3740dcf8edbb81a3f69bc2" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:10.523ex; height:5.509ex;" alt="{\displaystyle {\boldsymbol {\sigma }}={\frac {\partial U_{0}}{\partial {\boldsymbol {\varepsilon }}}}\,.}"></span> For a linear elastic material, the quantity <span class="texhtml"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">∂<i>U</i><sub>0</sub></span><span class="sr-only">/</span><span class="den">∂<b>ε</b></span></span>⁠</span></span> is a linear function of <span class="texhtml"><b>ε</b></span>, and can therefore be expressed as <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {\sigma }}={\mathsf {c}}:{\boldsymbol {\varepsilon }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">σ<!-- σ --></mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">c</mi> </mrow> </mrow> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ε<!-- ε --></mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {\sigma }}={\mathsf {c}}:{\boldsymbol {\varepsilon }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/90f83135a1824c537537e06e4ac1d65bee1cfd48" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.892ex; height:1.676ex;" alt="{\displaystyle {\boldsymbol {\sigma }}={\mathsf {c}}:{\boldsymbol {\varepsilon }}}"></span> where <b>c</b> is a fourth-rank tensor of material constants, also called the <b>stiffness tensor</b>. We can see why <b>c</b> must be a fourth-rank tensor by noting that, for a linear elastic material, <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial }{\partial {\boldsymbol {\varepsilon }}}}{\boldsymbol {\sigma }}({\boldsymbol {\varepsilon }})={\text{constant}}={\mathsf {c}}\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂<!-- ∂ --></mi> <mrow> <mi mathvariant="normal">∂<!-- ∂ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ε<!-- ε --></mi> </mrow> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">σ<!-- σ --></mi> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ε<!-- ε --></mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>constant</mtext> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">c</mi> </mrow> </mrow> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial }{\partial {\boldsymbol {\varepsilon }}}}{\boldsymbol {\sigma }}({\boldsymbol {\varepsilon }})={\text{constant}}={\mathsf {c}}\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32c70d526c7e316d5278c40be42819500be3d8c8" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:24.949ex; height:5.509ex;" alt="{\displaystyle {\frac {\partial }{\partial {\boldsymbol {\varepsilon }}}}{\boldsymbol {\sigma }}({\boldsymbol {\varepsilon }})={\text{constant}}={\mathsf {c}}\,.}"></span> In index notation <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial \sigma _{ij}}{\partial \varepsilon _{kl}}}={\text{constant}}=c_{ijkl}\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂<!-- ∂ --></mi> <msub> <mi>σ<!-- σ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂<!-- ∂ --></mi> <msub> <mi>ε<!-- ε --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mi>l</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>constant</mtext> </mrow> <mo>=</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> <mi>k</mi> <mi>l</mi> </mrow> </msub> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial \sigma _{ij}}{\partial \varepsilon _{kl}}}={\text{constant}}=c_{ijkl}\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e4f9fe7b08bdfefd9fd9b4dc7ff2c217b1650817" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:24.688ex; height:6.176ex;" alt="{\displaystyle {\frac {\partial \sigma _{ij}}{\partial \varepsilon _{kl}}}={\text{constant}}=c_{ijkl}\,.}"></span> </p><p>The right-hand side constant requires four indices and is a fourth-rank quantity. We can also see that this quantity must be a tensor because it is a linear transformation that takes the strain tensor to the stress tensor. We can also show that the constant obeys the tensor transformation rules for fourth-rank tensors. </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2></div> <ul><li><a href="/wiki/Acoustoelastic_effect" title="Acoustoelastic effect">Acoustoelastic effect</a></li> <li><a href="/wiki/Elastic_potential_energy" class="mw-redirect" title="Elastic potential energy">Elastic potential energy</a></li> <li><a href="/wiki/Laws_of_science" class="mw-redirect" title="Laws of science">Laws of science</a></li> <li><a href="/wiki/List_of_scientific_laws_named_after_people" title="List of scientific laws named after people">List of scientific laws named after people</a></li> <li><a href="/wiki/Quadratic_form" title="Quadratic form">Quadratic form</a></li> <li><a href="/wiki/Series_and_parallel_springs" title="Series and parallel springs">Series and parallel springs</a></li> <li><a href="/wiki/Spring_system" title="Spring system">Spring system</a></li> <li><a href="/wiki/Simple_harmonic_motion#Mass_on_a_spring" title="Simple harmonic motion">Simple harmonic motion of a mass on a spring</a></li> <li><a href="/wiki/Sine_wave" title="Sine wave">Sine wave</a></li> <li><a href="/wiki/Solid_mechanics" title="Solid mechanics">Solid mechanics</a></li> <li><a href="/wiki/Spring_pendulum" class="mw-redirect" title="Spring pendulum">Spring pendulum</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist reflist-columns references-column-width" style="column-width: 30em;"> <ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text">The anagram was given in alphabetical order, <i>ceiiinosssttuv</i>, representing <span title="Latin-language text"><i lang="la">Ut tensio, sic vis</i></span> – "As the extension, so the force": <style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFPetroski1996" class="citation book cs1"><a href="/wiki/Henry_Petroski" title="Henry Petroski">Petroski, Henry</a> (1996). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/inventionbydesig00petr"><i>Invention by Design: How Engineers Get from Thought to Thing</i></a></span>. Cambridge, MA: Harvard University Press. p. <a rel="nofollow" class="external text" href="https://archive.org/details/inventionbydesig00petr/page/11">11</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0674463684" title="Special:BookSources/978-0674463684"><bdi>978-0674463684</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Invention+by+Design%3A+How+Engineers+Get+from+Thought+to+Thing&rft.place=Cambridge%2C+MA&rft.pages=11&rft.pub=Harvard+University+Press&rft.date=1996&rft.isbn=978-0674463684&rft.aulast=Petroski&rft.aufirst=Henry&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Finventionbydesig00petr&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHooke%27s+law" class="Z3988"></span></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text">See <a rel="nofollow" class="external free" href="http://civil.lindahall.org/design.shtml">http://civil.lindahall.org/design.shtml</a>, where one can find also an anagram for <a href="/wiki/Catenary" title="Catenary">catenary</a>.</span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><a href="/wiki/Robert_Hooke" title="Robert Hooke">Robert Hooke</a>, <i>De Potentia Restitutiva, or of Spring. Explaining the Power of Springing Bodies</i>, London, 1678.</span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFYoungFreedmanFord2016" class="citation book cs1">Young, Hugh D.; Freedman, Roger A.; Ford, A. Lewis (2016). <i>Sears and Zemansky's University Physics: With Modern Physics</i> (14th ed.). Pearson. p. 209.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Sears+and+Zemansky%27s+University+Physics%3A+With+Modern+Physics&rft.pages=209&rft.edition=14th&rft.pub=Pearson&rft.date=2016&rft.aulast=Young&rft.aufirst=Hugh+D.&rft.au=Freedman%2C+Roger+A.&rft.au=Ford%2C+A.+Lewis&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHooke%27s+law" class="Z3988"></span></span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFUshibaMasuiTaguchiHamano2015" class="citation journal cs1">Ushiba, Shota; Masui, Kyoko; Taguchi, Natsuo; Hamano, Tomoki; Kawata, Satoshi; Shoji, Satoru (2015). <a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4661696">"Size dependent nanomechanics of coil spring shaped polymer nanowires"</a>. <i>Scientific Reports</i>. <b>5</b>: 17152. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2015NatSR...517152U">2015NatSR...517152U</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1038%2Fsrep17152">10.1038/srep17152</a>. <a href="/wiki/PMC_(identifier)" class="mw-redirect" title="PMC (identifier)">PMC</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4661696">4661696</a></span>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/26612544">26612544</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Scientific+Reports&rft.atitle=Size+dependent+nanomechanics+of+coil+spring+shaped+polymer+nanowires&rft.volume=5&rft.pages=17152&rft.date=2015&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC4661696%23id-name%3DPMC&rft_id=info%3Apmid%2F26612544&rft_id=info%3Adoi%2F10.1038%2Fsrep17152&rft_id=info%3Abibcode%2F2015NatSR...517152U&rft.aulast=Ushiba&rft.aufirst=Shota&rft.au=Masui%2C+Kyoko&rft.au=Taguchi%2C+Natsuo&rft.au=Hamano%2C+Tomoki&rft.au=Kawata%2C+Satoshi&rft.au=Shoji%2C+Satoru&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC4661696&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHooke%27s+law" class="Z3988"></span></span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBelen'kiiSalaev1988" class="citation journal cs1">Belen'kii; Salaev (1988). <a rel="nofollow" class="external text" href="https://doi.org/10.3367%2FUFNr.0155.198805c.0089">"Deformation effects in layer crystals"</a>. <i>Uspekhi Fizicheskikh Nauk</i>. <b>155</b> (5): 89. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.3367%2FUFNr.0155.198805c.0089">10.3367/UFNr.0155.198805c.0089</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Uspekhi+Fizicheskikh+Nauk&rft.atitle=Deformation+effects+in+layer+crystals&rft.volume=155&rft.issue=5&rft.pages=89&rft.date=1988&rft_id=info%3Adoi%2F10.3367%2FUFNr.0155.198805c.0089&rft.au=Belen%27kii&rft.au=Salaev&rft_id=https%3A%2F%2Fdoi.org%2F10.3367%252FUFNr.0155.198805c.0089&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHooke%27s+law" class="Z3988"></span></span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMouhatCoudert2014" class="citation journal cs1">Mouhat, Félix; Coudert, François-Xavier (5 December 2014). <a rel="nofollow" class="external text" href="https://link.aps.org/doi/10.1103/PhysRevB.90.224104">"Necessary and sufficient elastic stability conditions in various crystal systems"</a>. <i>Physical Review B</i>. <b>90</b> (22): 224104. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1410.0065">1410.0065</a></span>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2014PhRvB..90v4104M">2014PhRvB..90v4104M</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRevB.90.224104">10.1103/PhysRevB.90.224104</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/1098-0121">1098-0121</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:54058316">54058316</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Physical+Review+B&rft.atitle=Necessary+and+sufficient+elastic+stability+conditions+in+various+crystal+systems&rft.volume=90&rft.issue=22&rft.pages=224104&rft.date=2014-12-05&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A54058316%23id-name%3DS2CID&rft_id=info%3Abibcode%2F2014PhRvB..90v4104M&rft_id=info%3Aarxiv%2F1410.0065&rft.issn=1098-0121&rft_id=info%3Adoi%2F10.1103%2FPhysRevB.90.224104&rft.aulast=Mouhat&rft.aufirst=F%C3%A9lix&rft.au=Coudert%2C+Fran%C3%A7ois-Xavier&rft_id=https%3A%2F%2Flink.aps.org%2Fdoi%2F10.1103%2FPhysRevB.90.224104&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHooke%27s+law" class="Z3988"></span></span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFVijay_MadhavManogaran2009" class="citation journal cs1">Vijay Madhav, M.; Manogaran, S. (2009). "A relook at the compliance constants in redundant internal coordinates and some new insights". <i>J. Chem. Phys</i>. <b>131</b> (17): <span class="nowrap">174112–</span>174116. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2009JChPh.131q4112V">2009JChPh.131q4112V</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1063%2F1.3259834">10.1063/1.3259834</a>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/19895003">19895003</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=J.+Chem.+Phys.&rft.atitle=A+relook+at+the+compliance+constants+in+redundant+internal+coordinates+and+some+new+insights&rft.volume=131&rft.issue=17&rft.pages=%3Cspan+class%3D%22nowrap%22%3E174112-%3C%2Fspan%3E174116&rft.date=2009&rft_id=info%3Apmid%2F19895003&rft_id=info%3Adoi%2F10.1063%2F1.3259834&rft_id=info%3Abibcode%2F2009JChPh.131q4112V&rft.aulast=Vijay+Madhav&rft.aufirst=M.&rft.au=Manogaran%2C+S.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHooke%27s+law" class="Z3988"></span></span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPonomarevaYurenkoZhurakivskyVan_Mourik2012" class="citation journal cs1">Ponomareva, Alla; Yurenko, Yevgen; Zhurakivsky, Roman; Van Mourik, Tanja; Hovorun, Dmytro (2012). "Complete conformational space of the potential HIV-1 reverse transcriptase inhibitors d4U and d4C. A quantum chemical study". <i>Phys. Chem. Chem. Phys</i>. <b>14</b> (19): <span class="nowrap">6787–</span>6795. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2012PCCP...14.6787P">2012PCCP...14.6787P</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1039%2FC2CP40290D">10.1039/C2CP40290D</a>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/22461011">22461011</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Phys.+Chem.+Chem.+Phys.&rft.atitle=Complete+conformational+space+of+the+potential+HIV-1+reverse+transcriptase+inhibitors+d4U+and+d4C.+A+quantum+chemical+study&rft.volume=14&rft.issue=19&rft.pages=%3Cspan+class%3D%22nowrap%22%3E6787-%3C%2Fspan%3E6795&rft.date=2012&rft_id=info%3Apmid%2F22461011&rft_id=info%3Adoi%2F10.1039%2FC2CP40290D&rft_id=info%3Abibcode%2F2012PCCP...14.6787P&rft.aulast=Ponomareva&rft.aufirst=Alla&rft.au=Yurenko%2C+Yevgen&rft.au=Zhurakivsky%2C+Roman&rft.au=Van+Mourik%2C+Tanja&rft.au=Hovorun%2C+Dmytro&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHooke%27s+law" class="Z3988"></span></span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSymon1971" class="citation book cs1">Symon, Keith R. (1971). "Chapter 10". <i>Mechanics</i>. Reading, Massachusetts: Addison-Wesley. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780201073928" title="Special:BookSources/9780201073928"><bdi>9780201073928</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Chapter+10&rft.btitle=Mechanics&rft.place=Reading%2C+Massachusetts&rft.pub=Addison-Wesley&rft.date=1971&rft.isbn=9780201073928&rft.aulast=Symon&rft.aufirst=Keith+R.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHooke%27s+law" class="Z3988"></span></span> </li> <li id="cite_note-Simo98-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-Simo98_11-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSimoHughes1998" class="citation book cs1">Simo, J. C.; Hughes, T. J. R. (1998). <i>Computational Inelasticity</i>. Springer. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780387975207" title="Special:BookSources/9780387975207"><bdi>9780387975207</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Computational+Inelasticity&rft.pub=Springer&rft.date=1998&rft.isbn=9780387975207&rft.aulast=Simo&rft.aufirst=J.+C.&rft.au=Hughes%2C+T.+J.+R.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHooke%27s+law" class="Z3988"></span></span> </li> <li id="cite_note-Milton02-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-Milton02_12-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMilton2002" class="citation book cs1">Milton, Graeme W. (2002). <i>The Theory of Composites</i>. Cambridge Monographs on Applied and Computational Mathematics. Cambridge University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780521781251" title="Special:BookSources/9780521781251"><bdi>9780521781251</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Theory+of+Composites&rft.series=Cambridge+Monographs+on+Applied+and+Computational+Mathematics&rft.pub=Cambridge+University+Press&rft.date=2002&rft.isbn=9780521781251&rft.aulast=Milton&rft.aufirst=Graeme+W.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHooke%27s+law" class="Z3988"></span></span> </li> <li id="cite_note-Slaughter-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-Slaughter_13-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSlaughter2001" class="citation book cs1">Slaughter, William S. (2001). <i>The Linearized Theory of Elasticity</i>. Birkhäuser. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0817641177" title="Special:BookSources/978-0817641177"><bdi>978-0817641177</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Linearized+Theory+of+Elasticity&rft.pub=Birkh%C3%A4user&rft.date=2001&rft.isbn=978-0817641177&rft.aulast=Slaughter&rft.aufirst=William+S.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHooke%27s+law" class="Z3988"></span></span> </li> <li id="cite_note-Boresi-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-Boresi_14-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBoresiSchmidtSidebottom1993" class="citation book cs1">Boresi, A. P.; Schmidt, R. J.; Sidebottom, O. M. (1993). <i>Advanced Mechanics of Materials</i> (5th ed.). Wiley. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780471600091" title="Special:BookSources/9780471600091"><bdi>9780471600091</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Advanced+Mechanics+of+Materials&rft.edition=5th&rft.pub=Wiley&rft.date=1993&rft.isbn=9780471600091&rft.aulast=Boresi&rft.aufirst=A.+P.&rft.au=Schmidt%2C+R.+J.&rft.au=Sidebottom%2C+O.+M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHooke%27s+law" class="Z3988"></span></span> </li> <li id="cite_note-Tan-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-Tan_15-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTan1994" class="citation book cs1">Tan, S. C. (1994). <i>Stress Concentrations in Laminated Composites</i>. Lancaster, PA: Technomic Publishing Company. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9781566760775" title="Special:BookSources/9781566760775"><bdi>9781566760775</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Stress+Concentrations+in+Laminated+Composites&rft.place=Lancaster%2C+PA&rft.pub=Technomic+Publishing+Company&rft.date=1994&rft.isbn=9781566760775&rft.aulast=Tan&rft.aufirst=S.+C.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHooke%27s+law" class="Z3988"></span></span> </li> <li id="cite_note-16"><span class="mw-cite-backlink"><b><a href="#cite_ref-16">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRanganathanOstoja-Starzewski2008" class="citation journal cs1">Ranganathan, S.I.; <a href="/wiki/Martin_Ostoja-Starzewski" title="Martin Ostoja-Starzewski">Ostoja-Starzewski, M.</a> (2008). "Universal Elastic Anisotropy Index". <i>Physical Review Letters</i>. <b>101</b> (5): 055504–1–4. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2008PhRvL.101e5504R">2008PhRvL.101e5504R</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRevLett.101.055504">10.1103/PhysRevLett.101.055504</a>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/18764407">18764407</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:6668703">6668703</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Physical+Review+Letters&rft.atitle=Universal+Elastic+Anisotropy+Index&rft.volume=101&rft.issue=5&rft.pages=055504-1-4&rft.date=2008&rft_id=info%3Adoi%2F10.1103%2FPhysRevLett.101.055504&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A6668703%23id-name%3DS2CID&rft_id=info%3Apmid%2F18764407&rft_id=info%3Abibcode%2F2008PhRvL.101e5504R&rft.aulast=Ranganathan&rft.aufirst=S.I.&rft.au=Ostoja-Starzewski%2C+M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHooke%27s+law" class="Z3988"></span></span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2></div> <ul><li><a rel="nofollow" class="external text" href="https://feynmanlectures.caltech.edu/II_38.html#Ch38-S1">Hooke's law - The Feynman Lectures on Physics</a></li> <li><a rel="nofollow" class="external text" href="https://ocw.mit.edu/courses/8-01sc-classical-mechanics-fall-2016/pages/week-2-newtons-laws/7-4-hookes-law/">Hooke's Law - Classical Mechanics - Physics - MIT OpenCourseWare</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div> <ul><li><a rel="nofollow" class="external text" href="https://www.compadre.org/Physlets/mechanics/illustration5_4.cfm">JavaScript Applet demonstrating Springs and Hooke's law</a></li> <li><a rel="nofollow" class="external text" href="https://www.compadre.org/Physlets/mechanics/ex5_3.cfm">JavaScript Applet demonstrating Spring Force</a></li></ul> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1236075235">.mw-parser-output .navbox{box-sizing:border-box;border:1px solid #a2a9b1;width:100%;clear:both;font-size:88%;text-align:center;padding:1px;margin:1em auto 0}.mw-parser-output .navbox .navbox{margin-top:0}.mw-parser-output .navbox+.navbox,.mw-parser-output .navbox+.navbox-styles+.navbox{margin-top:-1px}.mw-parser-output .navbox-inner,.mw-parser-output .navbox-subgroup{width:100%}.mw-parser-output .navbox-group,.mw-parser-output .navbox-title,.mw-parser-output .navbox-abovebelow{padding:0.25em 1em;line-height:1.5em;text-align:center}.mw-parser-output .navbox-group{white-space:nowrap;text-align:right}.mw-parser-output .navbox,.mw-parser-output .navbox-subgroup{background-color:#fdfdfd}.mw-parser-output .navbox-list{line-height:1.5em;border-color:#fdfdfd}.mw-parser-output .navbox-list-with-group{text-align:left;border-left-width:2px;border-left-style:solid}.mw-parser-output tr+tr>.navbox-abovebelow,.mw-parser-output tr+tr>.navbox-group,.mw-parser-output tr+tr>.navbox-image,.mw-parser-output tr+tr>.navbox-list{border-top:2px solid #fdfdfd}.mw-parser-output .navbox-title{background-color:#ccf}.mw-parser-output .navbox-abovebelow,.mw-parser-output .navbox-group,.mw-parser-output .navbox-subgroup .navbox-title{background-color:#ddf}.mw-parser-output .navbox-subgroup .navbox-group,.mw-parser-output .navbox-subgroup .navbox-abovebelow{background-color:#e6e6ff}.mw-parser-output .navbox-even{background-color:#f7f7f7}.mw-parser-output .navbox-odd{background-color:transparent}.mw-parser-output .navbox .hlist td dl,.mw-parser-output .navbox .hlist td ol,.mw-parser-output .navbox .hlist td ul,.mw-parser-output .navbox td.hlist dl,.mw-parser-output .navbox td.hlist ol,.mw-parser-output .navbox td.hlist ul{padding:0.125em 0}.mw-parser-output .navbox .navbar{display:block;font-size:100%}.mw-parser-output .navbox-title .navbar{float:left;text-align:left;margin-right:0.5em}body.skin--responsive .mw-parser-output .navbox-image img{max-width:none!important}@media print{body.ns-0 .mw-parser-output .navbox{display:none!important}}</style></div><div role="navigation" class="navbox" aria-labelledby="Elastic_moduli_for_homogeneous_isotropic_materials74" style="padding:3px"><table class="nowraplinks mw-collapsible show navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Elastic_moduli" title="Template:Elastic moduli"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Elastic_moduli" title="Template talk:Elastic moduli"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Elastic_moduli" title="Special:EditPage/Template:Elastic moduli"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Elastic_moduli_for_homogeneous_isotropic_materials74" style="font-size:114%;margin:0 4em"><a href="/wiki/Elastic_modulus" title="Elastic modulus">Elastic moduli</a> for homogeneous <a href="/wiki/Isotropic" class="mw-redirect" title="Isotropic">isotropic</a> materials</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Bulk_modulus" title="Bulk modulus">Bulk modulus</a> (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}"></span>)</li> <li><a href="/wiki/Young%27s_modulus" title="Young's modulus">Young's modulus</a> (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}"></span>)</li> <li><a href="/wiki/Lam%C3%A9_parameters" title="Lamé parameters">Lamé's first parameter</a> (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ<!-- λ --></mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b43d0ea3c9c025af1be9128e62a18fa74bedda2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }"></span>)</li> <li><a href="/wiki/Shear_modulus" title="Shear modulus">Shear modulus</a> (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G,\mu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mo>,</mo> <mi>μ<!-- μ --></mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G,\mu }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ed0d290f164079e9704807191c18f9415457bea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.262ex; height:2.676ex;" alt="{\displaystyle G,\mu }"></span>)</li> <li><a href="/wiki/Poisson%27s_ratio" title="Poisson's ratio">Poisson's ratio</a> (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ν<!-- ν --></mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nu }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c15bbbb971240cf328aba572178f091684585468" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.232ex; height:1.676ex;" alt="{\displaystyle \nu }"></span>)</li> <li><a href="/wiki/P-wave_modulus" title="P-wave modulus">P-wave modulus</a> (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}"></span>)</li></ul> </div></td></tr></tbody></table></div> <table class="wikitable mw-collapsible" width="100%" style="font-size:smaller; background:white" align="center"> <tbody><tr> <th colspan="8">Conversion formulae </th></tr> <tr> <td colspan="8">Homogeneous isotropic linear elastic materials have their elastic properties uniquely determined by any two moduli among these; thus, given any two, any other of the elastic moduli can be calculated according to these formulas, provided both for 3D materials (first part of the table) and for 2D materials (second part). </td></tr> <tr> <td style="background:#F0F0FF;"><b>3D formulae</b> </td> <td style="text-align:center; background:#F0F0FF;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K=\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo>=</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K=\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d12e227970b986c1ff219badec7438f7383bfdb6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.906ex; height:2.176ex;" alt="{\displaystyle K=\,}"></span> </td> <td style="text-align:center; background:#F0F0FF;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E=\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo>=</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E=\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b75f04c79769676b88c45d4fef023a01294b670a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.616ex; height:2.176ex;" alt="{\displaystyle E=\,}"></span> </td> <td style="text-align:center; background:#F0F0FF;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda =\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ<!-- λ --></mi> <mo>=</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda =\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fdf13c116950ce78ddbc7a016c3cd804e23ea86d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.196ex; height:2.176ex;" alt="{\displaystyle \lambda =\,}"></span> </td> <td style="text-align:center; background:#F0F0FF;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G=\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mo>=</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G=\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fa8a892590600356bf9957268977269f386821c6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.667ex; height:2.176ex;" alt="{\displaystyle G=\,}"></span> </td> <td style="text-align:center; background:#F0F0FF;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu =\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ν<!-- ν --></mi> <mo>=</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nu =\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b632d33a8bb2255e8a086b63953c599d3b6a718" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.073ex; height:1.676ex;" alt="{\displaystyle \nu =\,}"></span> </td> <td style="text-align:center; background:#F0F0FF;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M=\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> <mo>=</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M=\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/843dc939c547010c55bd7753f7c94e6996934ca4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.283ex; height:2.176ex;" alt="{\displaystyle M=\,}"></span> </td> <td style="text-align:center; background:#F0F0FF;">Notes </td></tr> <tr> <td style="text-align:center;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (K,\,E)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>K</mi> <mo>,</mo> <mspace width="thinmathspace" /> <mi>E</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (K,\,E)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/996a015d617843aa00212a26839fbae210c834a4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.072ex; height:2.843ex;" alt="{\displaystyle (K,\,E)}"></span> </td> <td style="text-align:center;"> </td> <td style="text-align:center;"> </td> <td style="text-align:center;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {3K(3K-E)}{9K-E}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mn>3</mn> <mi>K</mi> <mo stretchy="false">(</mo> <mn>3</mn> <mi>K</mi> <mo>−<!-- − --></mo> <mi>E</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mn>9</mn> <mi>K</mi> <mo>−<!-- − --></mo> <mi>E</mi> </mrow> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {3K(3K-E)}{9K-E}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0cad8348883df915b9b03fb2820ec6b096640f4f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:9.215ex; height:4.343ex;" alt="{\displaystyle {\tfrac {3K(3K-E)}{9K-E}}}"></span> </td> <td style="text-align:center;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {3KE}{9K-E}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mn>3</mn> <mi>K</mi> <mi>E</mi> </mrow> <mrow> <mn>9</mn> <mi>K</mi> <mo>−<!-- − --></mo> <mi>E</mi> </mrow> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {3KE}{9K-E}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cdd679253fa436a6900d1cf949d909d3b09cf3c0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:5.653ex; height:3.843ex;" alt="{\displaystyle {\tfrac {3KE}{9K-E}}}"></span> </td> <td style="text-align:center;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {3K-E}{6K}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mn>3</mn> <mi>K</mi> <mo>−<!-- − --></mo> <mi>E</mi> </mrow> <mrow> <mn>6</mn> <mi>K</mi> </mrow> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {3K-E}{6K}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/52e22f238cec78b55c3b21bb03bd0f6bc122006b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:5.653ex; height:3.843ex;" alt="{\displaystyle {\tfrac {3K-E}{6K}}}"></span> </td> <td style="text-align:center;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {3K(3K+E)}{9K-E}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mn>3</mn> <mi>K</mi> <mo stretchy="false">(</mo> <mn>3</mn> <mi>K</mi> <mo>+</mo> <mi>E</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mn>9</mn> <mi>K</mi> <mo>−<!-- − --></mo> <mi>E</mi> </mrow> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {3K(3K+E)}{9K-E}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c924e463a20a0f63bd2f876283e3fb1a965db68" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:9.215ex; height:4.343ex;" alt="{\displaystyle {\tfrac {3K(3K+E)}{9K-E}}}"></span> </td> <td> </td></tr> <tr> <td style="text-align:center;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (K,\,\lambda )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>K</mi> <mo>,</mo> <mspace width="thinmathspace" /> <mi>λ<!-- λ --></mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (K,\,\lambda )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/423e5f4d1c8ee39ba6de7bc5e48df6837fff232a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.652ex; height:2.843ex;" alt="{\displaystyle (K,\,\lambda )}"></span> </td> <td style="text-align:center;"> </td> <td style="text-align:center;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {9K(K-\lambda )}{3K-\lambda }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mn>9</mn> <mi>K</mi> <mo stretchy="false">(</mo> <mi>K</mi> <mo>−<!-- − --></mo> <mi>λ<!-- λ --></mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mn>3</mn> <mi>K</mi> <mo>−<!-- − --></mo> <mi>λ<!-- λ --></mi> </mrow> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {9K(K-\lambda )}{3K-\lambda }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3233ebb13df867ba5551fefa7e092733fea80c42" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:8.096ex; height:4.343ex;" alt="{\displaystyle {\tfrac {9K(K-\lambda )}{3K-\lambda }}}"></span> </td> <td style="text-align:center;"> </td> <td style="text-align:center;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {3(K-\lambda )}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mn>3</mn> <mo stretchy="false">(</mo> <mi>K</mi> <mo>−<!-- − --></mo> <mi>λ<!-- λ --></mi> <mo stretchy="false">)</mo> </mrow> <mn>2</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {3(K-\lambda )}{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e4d4ffacd8fb2de88a240a663a13067a7c3e921f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:6.635ex; height:4.176ex;" alt="{\displaystyle {\tfrac {3(K-\lambda )}{2}}}"></span> </td> <td style="text-align:center;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {\lambda }{3K-\lambda }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>λ<!-- λ --></mi> <mrow> <mn>3</mn> <mi>K</mi> <mo>−<!-- − --></mo> <mi>λ<!-- λ --></mi> </mrow> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {\lambda }{3K-\lambda }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f1c70a68ad578ac446b83ae2af368a85554e2f4d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:5.356ex; height:3.843ex;" alt="{\displaystyle {\tfrac {\lambda }{3K-\lambda }}}"></span> </td> <td style="text-align:center;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 3K-2\lambda \,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>3</mn> <mi>K</mi> <mo>−<!-- − --></mo> <mn>2</mn> <mi>λ<!-- λ --></mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3K-2\lambda \,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dccc234a5195e4cc32a106e7280ce2e082a9da43" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:8.974ex; height:2.343ex;" alt="{\displaystyle 3K-2\lambda \,}"></span> </td> <td> </td></tr> <tr> <td style="text-align:center;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (K,\,G)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>K</mi> <mo>,</mo> <mspace width="thinmathspace" /> <mi>G</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (K,\,G)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bae48568e7a818eb72621f7cd602d0effc7e3648" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.123ex; height:2.843ex;" alt="{\displaystyle (K,\,G)}"></span> </td> <td style="text-align:center;"> </td> <td style="text-align:center;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {9KG}{3K+G}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mn>9</mn> <mi>K</mi> <mi>G</mi> </mrow> <mrow> <mn>3</mn> <mi>K</mi> <mo>+</mo> <mi>G</mi> </mrow> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {9KG}{3K+G}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5440c8c124cded6c55e978cbf920e76e4883a51b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:5.689ex; height:4.009ex;" alt="{\displaystyle {\tfrac {9KG}{3K+G}}}"></span> </td> <td style="text-align:center;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K-{\tfrac {2G}{3}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo>−<!-- − --></mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mn>2</mn> <mi>G</mi> </mrow> <mn>3</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K-{\tfrac {2G}{3}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a5272b70cff4a6a17225acf399ea27a1948b4f59" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:7.856ex; height:3.843ex;" alt="{\displaystyle K-{\tfrac {2G}{3}}}"></span> </td> <td style="text-align:center;"> </td> <td style="text-align:center;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {3K-2G}{2(3K+G)}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mn>3</mn> <mi>K</mi> <mo>−<!-- − --></mo> <mn>2</mn> <mi>G</mi> </mrow> <mrow> <mn>2</mn> <mo stretchy="false">(</mo> <mn>3</mn> <mi>K</mi> <mo>+</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {3K-2G}{2(3K+G)}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b11dec1ab89c26fc5ad5719fd2932e1dc68f4cb3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:7.791ex; height:4.343ex;" alt="{\displaystyle {\tfrac {3K-2G}{2(3K+G)}}}"></span> </td> <td style="text-align:center;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K+{\tfrac {4G}{3}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mn>4</mn> <mi>G</mi> </mrow> <mn>3</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K+{\tfrac {4G}{3}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a71d02abd602def118ed9f9228f0f686a06448d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:7.856ex; height:3.843ex;" alt="{\displaystyle K+{\tfrac {4G}{3}}}"></span> </td> <td> </td></tr> <tr> <td style="text-align:center;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (K,\,\nu )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>K</mi> <mo>,</mo> <mspace width="thinmathspace" /> <mi>ν<!-- ν --></mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (K,\,\nu )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b2f6df6fb9936ea5a0e15358e66802cafe9b747d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.528ex; height:2.843ex;" alt="{\displaystyle (K,\,\nu )}"></span> </td> <td style="text-align:center;"> </td> <td style="text-align:center;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 3K(1-2\nu )\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>3</mn> <mi>K</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−<!-- − --></mo> <mn>2</mn> <mi>ν<!-- ν --></mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3K(1-2\nu )\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dd567543f973005ad7031789e6219f93463e6d87" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.822ex; height:2.843ex;" alt="{\displaystyle 3K(1-2\nu )\,}"></span> </td> <td style="text-align:center;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {3K\nu }{1+\nu }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mn>3</mn> <mi>K</mi> <mi>ν<!-- ν --></mi> </mrow> <mrow> <mn>1</mn> <mo>+</mo> <mi>ν<!-- ν --></mi> </mrow> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {3K\nu }{1+\nu }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1194366850f7d5421ab50b6c2706e58498f7ea96" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:3.99ex; height:3.843ex;" alt="{\displaystyle {\tfrac {3K\nu }{1+\nu }}}"></span> </td> <td style="text-align:center;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {3K(1-2\nu )}{2(1+\nu )}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mn>3</mn> <mi>K</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−<!-- − --></mo> <mn>2</mn> <mi>ν<!-- ν --></mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mn>2</mn> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>ν<!-- ν --></mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {3K(1-2\nu )}{2(1+\nu )}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/31c8588d1514e15c703af716b1d98c94ef2ad4c9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:8.192ex; height:4.843ex;" alt="{\displaystyle {\tfrac {3K(1-2\nu )}{2(1+\nu )}}}"></span> </td> <td style="text-align:center;"> </td> <td style="text-align:center;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {3K(1-\nu )}{1+\nu }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mn>3</mn> <mi>K</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−<!-- − --></mo> <mi>ν<!-- ν --></mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mn>1</mn> <mo>+</mo> <mi>ν<!-- ν --></mi> </mrow> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {3K(1-\nu )}{1+\nu }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c09e8d1337a766b61ba2b9948fa4af1503d2ede0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:7.37ex; height:4.343ex;" alt="{\displaystyle {\tfrac {3K(1-\nu )}{1+\nu }}}"></span> </td> <td> </td></tr> <tr> <td style="text-align:center;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (K,\,M)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>K</mi> <mo>,</mo> <mspace width="thinmathspace" /> <mi>M</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (K,\,M)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1bb74f97b79a7d923d7036a9006af1437a58615e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.738ex; height:2.843ex;" alt="{\displaystyle (K,\,M)}"></span> </td> <td style="text-align:center;"> </td> <td style="text-align:center;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {9K(M-K)}{3K+M}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mn>9</mn> <mi>K</mi> <mo stretchy="false">(</mo> <mi>M</mi> <mo>−<!-- − --></mo> <mi>K</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mn>3</mn> <mi>K</mi> <mo>+</mo> <mi>M</mi> </mrow> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {9K(M-K)}{3K+M}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f0a754879220d62e4a49b2d4c5a9a5ac9ef939a8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:8.865ex; height:4.343ex;" alt="{\displaystyle {\tfrac {9K(M-K)}{3K+M}}}"></span> </td> <td style="text-align:center;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {3K-M}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mn>3</mn> <mi>K</mi> <mo>−<!-- − --></mo> <mi>M</mi> </mrow> <mn>2</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {3K-M}{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7da3dd9c26c427aaeb80a6f841626124c66dc665" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:6.124ex; height:3.676ex;" alt="{\displaystyle {\tfrac {3K-M}{2}}}"></span> </td> <td style="text-align:center;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {3(M-K)}{4}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mn>3</mn> <mo stretchy="false">(</mo> <mi>M</mi> <mo>−<!-- − --></mo> <mi>K</mi> <mo stretchy="false">)</mo> </mrow> <mn>4</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {3(M-K)}{4}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/322bd48e7d8cf39df6f267e9d4fb836db4a73702" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:7.404ex; height:4.176ex;" alt="{\displaystyle {\tfrac {3(M-K)}{4}}}"></span> </td> <td style="text-align:center;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {3K-M}{3K+M}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mn>3</mn> <mi>K</mi> <mo>−<!-- − --></mo> <mi>M</mi> </mrow> <mrow> <mn>3</mn> <mi>K</mi> <mo>+</mo> <mi>M</mi> </mrow> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {3K-M}{3K+M}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/73b822c78e943da6485611dfbdc74be5f78b15b6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:6.124ex; height:3.843ex;" alt="{\displaystyle {\tfrac {3K-M}{3K+M}}}"></span> </td> <td style="text-align:center;"> </td> <td> </td></tr> <tr> <td style="text-align:center;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (E,\,\lambda )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>E</mi> <mo>,</mo> <mspace width="thinmathspace" /> <mi>λ<!-- λ --></mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (E,\,\lambda )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b5c948c9a642e3318131878edf4a3d45bfb90f8d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.361ex; height:2.843ex;" alt="{\displaystyle (E,\,\lambda )}"></span> </td> <td style="text-align:center;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {E+3\lambda +R}{6}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mi>E</mi> <mo>+</mo> <mn>3</mn> <mi>λ<!-- λ --></mi> <mo>+</mo> <mi>R</mi> </mrow> <mn>6</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {E+3\lambda +R}{6}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a38418330bce4527d86325b426550cd9e160c575" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:7.676ex; height:3.843ex;" alt="{\displaystyle {\tfrac {E+3\lambda +R}{6}}}"></span> </td> <td style="text-align:center;"> </td> <td style="text-align:center;"> </td> <td style="text-align:center;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {E-3\lambda +R}{4}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mi>E</mi> <mo>−<!-- − --></mo> <mn>3</mn> <mi>λ<!-- λ --></mi> <mo>+</mo> <mi>R</mi> </mrow> <mn>4</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {E-3\lambda +R}{4}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/afb261c4ee6f07fd89951fa28248a90f7b9d3025" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:7.676ex; height:3.676ex;" alt="{\displaystyle {\tfrac {E-3\lambda +R}{4}}}"></span> </td> <td style="text-align:center;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {2\lambda }{E+\lambda +R}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mn>2</mn> <mi>λ<!-- λ --></mi> </mrow> <mrow> <mi>E</mi> <mo>+</mo> <mi>λ<!-- λ --></mi> <mo>+</mo> <mi>R</mi> </mrow> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {2\lambda }{E+\lambda +R}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c2a996d0bd87fb28da723345b090392d12e01118" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:6.854ex; height:3.843ex;" alt="{\displaystyle {\tfrac {2\lambda }{E+\lambda +R}}}"></span> </td> <td style="text-align:center;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {E-\lambda +R}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mi>E</mi> <mo>−<!-- − --></mo> <mi>λ<!-- λ --></mi> <mo>+</mo> <mi>R</mi> </mrow> <mn>2</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {E-\lambda +R}{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/372ceea38591e2c69d5087588cab00087c749abb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:6.854ex; height:3.676ex;" alt="{\displaystyle {\tfrac {E-\lambda +R}{2}}}"></span> </td> <td style="text-align:center;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R={\sqrt {E^{2}+9\lambda ^{2}+2E\lambda }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>9</mn> <msup> <mi>λ<!-- λ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>2</mn> <mi>E</mi> <mi>λ<!-- λ --></mi> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R={\sqrt {E^{2}+9\lambda ^{2}+2E\lambda }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8fedc22d764b6af93b68ad27937582d75919d76b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.58ex; height:3.509ex;" alt="{\displaystyle R={\sqrt {E^{2}+9\lambda ^{2}+2E\lambda }}}"></span> </td></tr> <tr> <td style="text-align:center;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (E,\,G)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>E</mi> <mo>,</mo> <mspace width="thinmathspace" /> <mi>G</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (E,\,G)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee52d776f2cb07c88fbac6567029158dff718e01" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.833ex; height:2.843ex;" alt="{\displaystyle (E,\,G)}"></span> </td> <td style="text-align:center;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {EG}{3(3G-E)}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mi>E</mi> <mi>G</mi> </mrow> <mrow> <mn>3</mn> <mo stretchy="false">(</mo> <mn>3</mn> <mi>G</mi> <mo>−<!-- − --></mo> <mi>E</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {EG}{3(3G-E)}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/94fef5491daefd3b9667bfd32eb6e4b3f31a67aa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:7.585ex; height:4.343ex;" alt="{\displaystyle {\tfrac {EG}{3(3G-E)}}}"></span> </td> <td style="text-align:center;"> </td> <td style="text-align:center;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {G(E-2G)}{3G-E}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mi>G</mi> <mo stretchy="false">(</mo> <mi>E</mi> <mo>−<!-- − --></mo> <mn>2</mn> <mi>G</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mn>3</mn> <mi>G</mi> <mo>−<!-- − --></mo> <mi>E</mi> </mrow> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {G(E-2G)}{3G-E}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b592ec0480b2ca3a99ea29d9cbd0b631d777d4d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:8.055ex; height:4.509ex;" alt="{\displaystyle {\tfrac {G(E-2G)}{3G-E}}}"></span> </td> <td style="text-align:center;"> </td> <td style="text-align:center;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {E}{2G}}-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>E</mi> <mrow> <mn>2</mn> <mi>G</mi> </mrow> </mfrac> </mstyle> </mrow> <mo>−<!-- − --></mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {E}{2G}}-1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b9fb70b8d93a1714d230f745d4bb0a3aa81031a5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:6.953ex; height:3.676ex;" alt="{\displaystyle {\tfrac {E}{2G}}-1}"></span> </td> <td style="text-align:center;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {G(4G-E)}{3G-E}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mi>G</mi> <mo stretchy="false">(</mo> <mn>4</mn> <mi>G</mi> <mo>−<!-- − --></mo> <mi>E</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mn>3</mn> <mi>G</mi> <mo>−<!-- − --></mo> <mi>E</mi> </mrow> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {G(4G-E)}{3G-E}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06fef6489910b32a86944f5fe2f5f0c1473e87c1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:8.055ex; height:4.509ex;" alt="{\displaystyle {\tfrac {G(4G-E)}{3G-E}}}"></span> </td> <td> </td></tr> <tr> <td style="text-align:center;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (E,\,\nu )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>E</mi> <mo>,</mo> <mspace width="thinmathspace" /> <mi>ν<!-- ν --></mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (E,\,\nu )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7d858acfba62fc12da4ae6fa0759b9df87c51d96" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.238ex; height:2.843ex;" alt="{\displaystyle (E,\,\nu )}"></span> </td> <td style="text-align:center;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {E}{3(1-2\nu )}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>E</mi> <mrow> <mn>3</mn> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−<!-- − --></mo> <mn>2</mn> <mi>ν<!-- ν --></mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {E}{3(1-2\nu )}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/49073e20128a9ecf62b06e95a6e611808b4a1015" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:6.731ex; height:4.176ex;" alt="{\displaystyle {\tfrac {E}{3(1-2\nu )}}}"></span> </td> <td style="text-align:center;"> </td> <td style="text-align:center;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {E\nu }{(1+\nu )(1-2\nu )}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mi>E</mi> <mi>ν<!-- ν --></mi> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>ν<!-- ν --></mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−<!-- − --></mo> <mn>2</mn> <mi>ν<!-- ν --></mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {E\nu }{(1+\nu )(1-2\nu )}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7aa443ecbbd2c92a028d26ee25863bbc1592b4b5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:10.16ex; height:4.176ex;" alt="{\displaystyle {\tfrac {E\nu }{(1+\nu )(1-2\nu )}}}"></span> </td> <td style="text-align:center;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {E}{2(1+\nu )}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>E</mi> <mrow> <mn>2</mn> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>ν<!-- ν --></mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {E}{2(1+\nu )}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ea91afdc4bb2364d6deb22ee5653d33451c8d7b2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:5.909ex; height:4.176ex;" alt="{\displaystyle {\tfrac {E}{2(1+\nu )}}}"></span> </td> <td style="text-align:center;"> </td> <td style="text-align:center;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {E(1-\nu )}{(1+\nu )(1-2\nu )}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mi>E</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−<!-- − --></mo> <mi>ν<!-- ν --></mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>ν<!-- ν --></mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−<!-- − --></mo> <mn>2</mn> <mi>ν<!-- ν --></mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {E(1-\nu )}{(1+\nu )(1-2\nu )}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7abf15539e84aef12ea7f6aed6ac61b939e0897a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:10.16ex; height:4.843ex;" alt="{\displaystyle {\tfrac {E(1-\nu )}{(1+\nu )(1-2\nu )}}}"></span> </td> <td> </td></tr> <tr> <td style="text-align:center;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (E,\,M)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>E</mi> <mo>,</mo> <mspace width="thinmathspace" /> <mi>M</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (E,\,M)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/029a8efce5680f60845e7b4029cd8d714726474c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.448ex; height:2.843ex;" alt="{\displaystyle (E,\,M)}"></span> </td> <td style="text-align:center;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {3M-E+S}{6}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mn>3</mn> <mi>M</mi> <mo>−<!-- − --></mo> <mi>E</mi> <mo>+</mo> <mi>S</mi> </mrow> <mn>6</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {3M-E+S}{6}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6df880ae07d8878db46e66ec91cde699a1172bbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:8.258ex; height:3.843ex;" alt="{\displaystyle {\tfrac {3M-E+S}{6}}}"></span> </td> <td style="text-align:center;"> </td> <td style="text-align:center;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {M-E+S}{4}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mi>M</mi> <mo>−<!-- − --></mo> <mi>E</mi> <mo>+</mo> <mi>S</mi> </mrow> <mn>4</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {M-E+S}{4}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cd8aa8a8f133559b802bf384ff144d2bfec3280c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:7.436ex; height:3.676ex;" alt="{\displaystyle {\tfrac {M-E+S}{4}}}"></span> </td> <td style="text-align:center;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {3M+E-S}{8}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mn>3</mn> <mi>M</mi> <mo>+</mo> <mi>E</mi> <mo>−<!-- − --></mo> <mi>S</mi> </mrow> <mn>8</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {3M+E-S}{8}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/53cddf1576995522248b1a396e8a790309691870" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:8.258ex; height:3.843ex;" alt="{\displaystyle {\tfrac {3M+E-S}{8}}}"></span> </td> <td style="text-align:center;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {E-M+S}{4M}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mi>E</mi> <mo>−<!-- − --></mo> <mi>M</mi> <mo>+</mo> <mi>S</mi> </mrow> <mrow> <mn>4</mn> <mi>M</mi> </mrow> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {E-M+S}{4M}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ba7b57d39482e416644c8d99a1ac18029b84711" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:7.436ex; height:3.676ex;" alt="{\displaystyle {\tfrac {E-M+S}{4M}}}"></span> </td> <td style="text-align:center;"> </td> <td style="text-align:center;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S=\pm {\sqrt {E^{2}+9M^{2}-10EM}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>=</mo> <mo>±<!-- ± --></mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>9</mn> <msup> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−<!-- − --></mo> <mn>10</mn> <mi>E</mi> <mi>M</mi> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S=\pm {\sqrt {E^{2}+9M^{2}-10EM}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/91fb683e4525bf2f8fef0c060113d446025ec2a5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.517ex; height:3.509ex;" alt="{\displaystyle S=\pm {\sqrt {E^{2}+9M^{2}-10EM}}}"></span><br /> <p>There are two valid solutions.<br /> The plus sign leads to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu \geq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ν<!-- ν --></mi> <mo>≥<!-- ≥ --></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nu \geq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dec80f16f256f3990d91eb9966a90bbe0baee4b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.493ex; height:2.343ex;" alt="{\displaystyle \nu \geq 0}"></span>.<br /> </p> The minus sign leads to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu \leq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ν<!-- ν --></mi> <mo>≤<!-- ≤ --></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nu \leq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/030f3e40f7ced9ec8cc73d667aa15f61f0a377e9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.493ex; height:2.343ex;" alt="{\displaystyle \nu \leq 0}"></span>.<br /><p class="mw-empty-elt"></p> </td></tr> <tr> <td style="text-align:center;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\lambda ,\,G)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>λ<!-- λ --></mi> <mo>,</mo> <mspace width="thinmathspace" /> <mi>G</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\lambda ,\,G)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7bffd6d22cc67fd1a35a585b13d7b32a74ce6b1f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.412ex; height:2.843ex;" alt="{\displaystyle (\lambda ,\,G)}"></span> </td> <td style="text-align:center;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda +{\tfrac {2G}{3}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ<!-- λ --></mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mn>2</mn> <mi>G</mi> </mrow> <mn>3</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda +{\tfrac {2G}{3}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bfcd78f0e042c88176c6ba5c0f8dcebf89fe223a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:7.145ex; height:3.843ex;" alt="{\displaystyle \lambda +{\tfrac {2G}{3}}}"></span> </td> <td style="text-align:center;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {G(3\lambda +2G)}{\lambda +G}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mi>G</mi> <mo stretchy="false">(</mo> <mn>3</mn> <mi>λ<!-- λ --></mi> <mo>+</mo> <mn>2</mn> <mi>G</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>λ<!-- λ --></mi> <mo>+</mo> <mi>G</mi> </mrow> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {G(3\lambda +2G)}{\lambda +G}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d0994ccfa74411f35a8f0762379eddc3660bfec4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:8.58ex; height:4.509ex;" alt="{\displaystyle {\tfrac {G(3\lambda +2G)}{\lambda +G}}}"></span> </td> <td style="text-align:center;"> </td> <td style="text-align:center;"> </td> <td style="text-align:center;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {\lambda }{2(\lambda +G)}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>λ<!-- λ --></mi> <mrow> <mn>2</mn> <mo stretchy="false">(</mo> <mi>λ<!-- λ --></mi> <mo>+</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {\lambda }{2(\lambda +G)}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/569977b3b51798797b8ed7d287e828ef07cdb54a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:6.466ex; height:4.343ex;" alt="{\displaystyle {\tfrac {\lambda }{2(\lambda +G)}}}"></span> </td> <td style="text-align:center;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda +2G\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ<!-- λ --></mi> <mo>+</mo> <mn>2</mn> <mi>G</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda +2G\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f04bd3c70df3389cd672961e8ba5636aa8d5fb48" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.572ex; height:2.343ex;" alt="{\displaystyle \lambda +2G\,}"></span> </td> <td> </td></tr> <tr> <td style="text-align:center;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\lambda ,\,\nu )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>λ<!-- λ --></mi> <mo>,</mo> <mspace width="thinmathspace" /> <mi>ν<!-- ν --></mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\lambda ,\,\nu )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cbafda646cd2c8251ec73dbe5efd428aee76544e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.818ex; height:2.843ex;" alt="{\displaystyle (\lambda ,\,\nu )}"></span> </td> <td style="text-align:center;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {\lambda (1+\nu )}{3\nu }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mi>λ<!-- λ --></mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>ν<!-- ν --></mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mn>3</mn> <mi>ν<!-- ν --></mi> </mrow> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {\lambda (1+\nu )}{3\nu }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f8668d7ae1d4ba0bea79bc24b22969de2e0020b1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:6.046ex; height:4.343ex;" alt="{\displaystyle {\tfrac {\lambda (1+\nu )}{3\nu }}}"></span> </td> <td style="text-align:center;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {\lambda (1+\nu )(1-2\nu )}{\nu }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mi>λ<!-- λ --></mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>ν<!-- ν --></mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−<!-- − --></mo> <mn>2</mn> <mi>ν<!-- ν --></mi> <mo stretchy="false">)</mo> </mrow> <mi>ν<!-- ν --></mi> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {\lambda (1+\nu )(1-2\nu )}{\nu }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/16ba478690ce90bc61e86d5b21ef89e8d97c0223" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.119ex; height:4.009ex;" alt="{\displaystyle {\tfrac {\lambda (1+\nu )(1-2\nu )}{\nu }}}"></span> </td> <td style="text-align:center;"> </td> <td style="text-align:center;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {\lambda (1-2\nu )}{2\nu }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mi>λ<!-- λ --></mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−<!-- − --></mo> <mn>2</mn> <mi>ν<!-- ν --></mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mn>2</mn> <mi>ν<!-- ν --></mi> </mrow> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {\lambda (1-2\nu )}{2\nu }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c0ae2f9d9801984c3a421791f68ed6d206906ce4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:6.868ex; height:4.176ex;" alt="{\displaystyle {\tfrac {\lambda (1-2\nu )}{2\nu }}}"></span> </td> <td style="text-align:center;"> </td> <td style="text-align:center;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {\lambda (1-\nu )}{\nu }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mi>λ<!-- λ --></mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−<!-- − --></mo> <mi>ν<!-- ν --></mi> <mo stretchy="false">)</mo> </mrow> <mi>ν<!-- ν --></mi> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {\lambda (1-\nu )}{\nu }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a4b0e21d0522f123eb9043c4e4352bbfd0944365" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.046ex; height:4.009ex;" alt="{\displaystyle {\tfrac {\lambda (1-\nu )}{\nu }}}"></span> </td> <td style="text-align:center;">Cannot be used when <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu =0\Leftrightarrow \lambda =0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ν<!-- ν --></mi> <mo>=</mo> <mn>0</mn> <mo stretchy="false">⇔<!-- ⇔ --></mo> <mi>λ<!-- λ --></mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nu =0\Leftrightarrow \lambda =0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d7e6655bc4e2949eee61d990716a2b8af684651d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:14.723ex; height:2.176ex;" alt="{\displaystyle \nu =0\Leftrightarrow \lambda =0}"></span> </td></tr> <tr> <td style="text-align:center;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\lambda ,\,M)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>λ<!-- λ --></mi> <mo>,</mo> <mspace width="thinmathspace" /> <mi>M</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\lambda ,\,M)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3242e9c8067dcfac72d5c96bcec80ccda592d34e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.028ex; height:2.843ex;" alt="{\displaystyle (\lambda ,\,M)}"></span> </td> <td style="text-align:center;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {M+2\lambda }{3}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mi>M</mi> <mo>+</mo> <mn>2</mn> <mi>λ<!-- λ --></mi> </mrow> <mn>3</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {M+2\lambda }{3}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1df1550f26280e13b63e387c950d8e73a06d1820" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:5.622ex; height:3.843ex;" alt="{\displaystyle {\tfrac {M+2\lambda }{3}}}"></span> </td> <td style="text-align:center;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {(M-\lambda )(M+2\lambda )}{M+\lambda }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mi>M</mi> <mo>−<!-- − --></mo> <mi>λ<!-- λ --></mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>M</mi> <mo>+</mo> <mn>2</mn> <mi>λ<!-- λ --></mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>M</mi> <mo>+</mo> <mi>λ<!-- λ --></mi> </mrow> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {(M-\lambda )(M+2\lambda )}{M+\lambda }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/94c594e157229dfa81918f2d6de968ffe4cea963" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:12.144ex; height:4.343ex;" alt="{\displaystyle {\tfrac {(M-\lambda )(M+2\lambda )}{M+\lambda }}}"></span> </td> <td style="text-align:center;"> </td> <td style="text-align:center;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {M-\lambda }{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mi>M</mi> <mo>−<!-- − --></mo> <mi>λ<!-- λ --></mi> </mrow> <mn>2</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {M-\lambda }{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba0c175995ebb2020f1b31ad369bf52374d87816" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:4.8ex; height:3.676ex;" alt="{\displaystyle {\tfrac {M-\lambda }{2}}}"></span> </td> <td style="text-align:center;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {\lambda }{M+\lambda }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>λ<!-- λ --></mi> <mrow> <mi>M</mi> <mo>+</mo> <mi>λ<!-- λ --></mi> </mrow> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {\lambda }{M+\lambda }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6926e4029cdea679bccd9d770206e1d6ba9cc097" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:4.8ex; height:3.843ex;" alt="{\displaystyle {\tfrac {\lambda }{M+\lambda }}}"></span> </td> <td style="text-align:center;"> </td> <td> </td></tr> <tr> <td style="text-align:center;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (G,\,\nu )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>G</mi> <mo>,</mo> <mspace width="thinmathspace" /> <mi>ν<!-- ν --></mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (G,\,\nu )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/03de031e73561f10f24638feaf69d90abe32aea5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.289ex; height:2.843ex;" alt="{\displaystyle (G,\,\nu )}"></span> </td> <td style="text-align:center;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {2G(1+\nu )}{3(1-2\nu )}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mn>2</mn> <mi>G</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>ν<!-- ν --></mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mn>3</mn> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−<!-- − --></mo> <mn>2</mn> <mi>ν<!-- ν --></mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {2G(1+\nu )}{3(1-2\nu )}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e7f8f5046df3cc6a7152f9fa56845d5dbce78033" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:7.201ex; height:4.843ex;" alt="{\displaystyle {\tfrac {2G(1+\nu )}{3(1-2\nu )}}}"></span> </td> <td style="text-align:center;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2G(1+\nu )\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>G</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>ν<!-- ν --></mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2G(1+\nu )\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ecc39d8499aa0848351a6510b116bd2dfea06d8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.421ex; height:2.843ex;" alt="{\displaystyle 2G(1+\nu )\,}"></span> </td> <td style="text-align:center;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {2G\nu }{1-2\nu }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mn>2</mn> <mi>G</mi> <mi>ν<!-- ν --></mi> </mrow> <mrow> <mn>1</mn> <mo>−<!-- − --></mo> <mn>2</mn> <mi>ν<!-- ν --></mi> </mrow> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {2G\nu }{1-2\nu }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d011b18df127233873f4af03b50318509583006e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:4.63ex; height:3.843ex;" alt="{\displaystyle {\tfrac {2G\nu }{1-2\nu }}}"></span> </td> <td style="text-align:center;"> </td> <td style="text-align:center;"> </td> <td style="text-align:center;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {2G(1-\nu )}{1-2\nu }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mn>2</mn> <mi>G</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−<!-- − --></mo> <mi>ν<!-- ν --></mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mn>1</mn> <mo>−<!-- − --></mo> <mn>2</mn> <mi>ν<!-- ν --></mi> </mrow> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {2G(1-\nu )}{1-2\nu }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/036c5b70861f3145642c2d797f6cbf666ea26dbc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:7.201ex; height:4.343ex;" alt="{\displaystyle {\tfrac {2G(1-\nu )}{1-2\nu }}}"></span> </td> <td> </td></tr> <tr> <td style="text-align:center;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (G,\,M)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>G</mi> <mo>,</mo> <mspace width="thinmathspace" /> <mi>M</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (G,\,M)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/40060aa6964ead07257199a60095ec47599c9455" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.499ex; height:2.843ex;" alt="{\displaystyle (G,\,M)}"></span> </td> <td style="text-align:center;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M-{\tfrac {4G}{3}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> <mo>−<!-- − --></mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mn>4</mn> <mi>G</mi> </mrow> <mn>3</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M-{\tfrac {4G}{3}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf22c9741623f0209e7585845196a3fa1ed6282a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:8.232ex; height:3.843ex;" alt="{\displaystyle M-{\tfrac {4G}{3}}}"></span> </td> <td style="text-align:center;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {G(3M-4G)}{M-G}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mi>G</mi> <mo stretchy="false">(</mo> <mn>3</mn> <mi>M</mi> <mo>−<!-- − --></mo> <mn>4</mn> <mi>G</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>M</mi> <mo>−<!-- − --></mo> <mi>G</mi> </mrow> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {G(3M-4G)}{M-G}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0b53b71a131246c16879e071ac283f0cf16bc255" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:9.348ex; height:4.509ex;" alt="{\displaystyle {\tfrac {G(3M-4G)}{M-G}}}"></span> </td> <td style="text-align:center;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M-2G\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> <mo>−<!-- − --></mo> <mn>2</mn> <mi>G</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M-2G\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4777994467bbd835a560ddf3d9b7a9bb88f6be15" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:8.659ex; height:2.343ex;" alt="{\displaystyle M-2G\,}"></span> </td> <td style="text-align:center;"> </td> <td style="text-align:center;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {M-2G}{2M-2G}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mi>M</mi> <mo>−<!-- − --></mo> <mn>2</mn> <mi>G</mi> </mrow> <mrow> <mn>2</mn> <mi>M</mi> <mo>−<!-- − --></mo> <mn>2</mn> <mi>G</mi> </mrow> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {M-2G}{2M-2G}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d95ab5446ac8be62e025bbc4461d19ad03ddb0ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:6.777ex; height:4.009ex;" alt="{\displaystyle {\tfrac {M-2G}{2M-2G}}}"></span> </td> <td style="text-align:center;"> </td> <td> </td></tr> <tr> <td style="text-align:center;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\nu ,\,M)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>ν<!-- ν --></mi> <mo>,</mo> <mspace width="thinmathspace" /> <mi>M</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\nu ,\,M)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1fb7e1e592dc27b6b7dd908dc0f1eaae129495bb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.905ex; height:2.843ex;" alt="{\displaystyle (\nu ,\,M)}"></span> </td> <td style="text-align:center;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {M(1+\nu )}{3(1-\nu )}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mi>M</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>ν<!-- ν --></mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mn>3</mn> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−<!-- − --></mo> <mi>ν<!-- ν --></mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {M(1+\nu )}{3(1-\nu )}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6b555e35103ae080362cc5ea5e71c0a5fb01dfb3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:6.814ex; height:4.843ex;" alt="{\displaystyle {\tfrac {M(1+\nu )}{3(1-\nu )}}}"></span> </td> <td style="text-align:center;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {M(1+\nu )(1-2\nu )}{1-\nu }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mi>M</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>ν<!-- ν --></mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−<!-- − --></mo> <mn>2</mn> <mi>ν<!-- ν --></mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mn>1</mn> <mo>−<!-- − --></mo> <mi>ν<!-- ν --></mi> </mrow> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {M(1+\nu )(1-2\nu )}{1-\nu }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c18235e12644cbb022f23fb36c11fca9164c19ee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:11.887ex; height:4.343ex;" alt="{\displaystyle {\tfrac {M(1+\nu )(1-2\nu )}{1-\nu }}}"></span> </td> <td style="text-align:center;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {M\nu }{1-\nu }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mi>M</mi> <mi>ν<!-- ν --></mi> </mrow> <mrow> <mn>1</mn> <mo>−<!-- − --></mo> <mi>ν<!-- ν --></mi> </mrow> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {M\nu }{1-\nu }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a015af678fe0800a1b50744d82084099f15f462a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:3.808ex; height:3.676ex;" alt="{\displaystyle {\tfrac {M\nu }{1-\nu }}}"></span> </td> <td style="text-align:center;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {M(1-2\nu )}{2(1-\nu )}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mi>M</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−<!-- − --></mo> <mn>2</mn> <mi>ν<!-- ν --></mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mn>2</mn> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−<!-- − --></mo> <mi>ν<!-- ν --></mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {M(1-2\nu )}{2(1-\nu )}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a076de87f5e08e5bb0733e7008687fec9f88396d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:7.636ex; height:4.843ex;" alt="{\displaystyle {\tfrac {M(1-2\nu )}{2(1-\nu )}}}"></span> </td> <td style="text-align:center;"> </td> <td style="text-align:center;"> </td> <td> </td></tr> <tr> <td style="background:#F0F0FF;"><b>2D formulae</b> </td> <td style="text-align:center; background:#F0F0FF;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K_{\mathrm {2D} }=\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <mo>=</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K_{\mathrm {2D} }=\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5a6c0547931c4c1e3cc4ac771a535617a442d98d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.123ex; height:2.509ex;" alt="{\displaystyle K_{\mathrm {2D} }=\,}"></span> </td> <td style="text-align:center; background:#F0F0FF;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{\mathrm {2D} }=\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <mo>=</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{\mathrm {2D} }=\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/145ff1eddd10383e2d0bf4f6eff8b5eeedb448f1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.865ex; height:2.509ex;" alt="{\displaystyle E_{\mathrm {2D} }=\,}"></span> </td> <td style="text-align:center; background:#F0F0FF;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda _{\mathrm {2D} }=\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>λ<!-- λ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <mo>=</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda _{\mathrm {2D} }=\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/70b691041358c0200dc4c170da6b8b62706e44a0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.505ex; height:2.509ex;" alt="{\displaystyle \lambda _{\mathrm {2D} }=\,}"></span> </td> <td style="text-align:center; background:#F0F0FF;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G_{\mathrm {2D} }=\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <mo>=</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G_{\mathrm {2D} }=\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b43b00c53cda00e94658abbcd45ed156af4c1d4a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.977ex; height:2.509ex;" alt="{\displaystyle G_{\mathrm {2D} }=\,}"></span> </td> <td style="text-align:center; background:#F0F0FF;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu _{\mathrm {2D} }=\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ν<!-- ν --></mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <mo>=</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nu _{\mathrm {2D} }=\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2eb28b6f1e836172c154325538936f09b55de993" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.299ex; height:2.009ex;" alt="{\displaystyle \nu _{\mathrm {2D} }=\,}"></span> </td> <td style="text-align:center; background:#F0F0FF;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{\mathrm {2D} }=\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <mo>=</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M_{\mathrm {2D} }=\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ebd093c0d6e6d5c2b5f88ffbd7b23c3f0164cb20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.404ex; height:2.509ex;" alt="{\displaystyle M_{\mathrm {2D} }=\,}"></span> </td> <td style="text-align:center; background:#F0F0FF;">Notes </td></tr> <tr> <td style="text-align:center;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (K_{\mathrm {2D} },\,E_{\mathrm {2D} })}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <mo>,</mo> <mspace width="thinmathspace" /> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (K_{\mathrm {2D} },\,E_{\mathrm {2D} })}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e6610d0252ab47240ee324d52f781214686e42b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.538ex; height:2.843ex;" alt="{\displaystyle (K_{\mathrm {2D} },\,E_{\mathrm {2D} })}"></span> </td> <td style="text-align:center;"> </td> <td style="text-align:center;"> </td> <td style="text-align:center;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {2K_{\mathrm {2D} }(2K_{\mathrm {2D} }-E_{\mathrm {2D} })}{4K_{\mathrm {2D} }-E_{\mathrm {2D} }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mn>2</mn> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mn>2</mn> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <mo>−<!-- − --></mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <mn>4</mn> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <mo>−<!-- − --></mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> </mrow> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {2K_{\mathrm {2D} }(2K_{\mathrm {2D} }-E_{\mathrm {2D} })}{4K_{\mathrm {2D} }-E_{\mathrm {2D} }}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/012be07cda111d8d43c481b55fba1c8cd9aa81da" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:14.594ex; height:4.676ex;" alt="{\displaystyle {\tfrac {2K_{\mathrm {2D} }(2K_{\mathrm {2D} }-E_{\mathrm {2D} })}{4K_{\mathrm {2D} }-E_{\mathrm {2D} }}}}"></span> </td> <td style="text-align:center;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {K_{\mathrm {2D} }E_{\mathrm {2D} }}{4K_{\mathrm {2D} }-E_{\mathrm {2D} }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> </mrow> <mrow> <mn>4</mn> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <mo>−<!-- − --></mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> </mrow> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {K_{\mathrm {2D} }E_{\mathrm {2D} }}{4K_{\mathrm {2D} }-E_{\mathrm {2D} }}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d225deea4bae144ad3163a592c84cba353a18b4e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:9.246ex; height:4.509ex;" alt="{\displaystyle {\tfrac {K_{\mathrm {2D} }E_{\mathrm {2D} }}{4K_{\mathrm {2D} }-E_{\mathrm {2D} }}}}"></span> </td> <td style="text-align:center;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {2K_{\mathrm {2D} }-E_{\mathrm {2D} }}{2K_{\mathrm {2D} }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mn>2</mn> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <mo>−<!-- − --></mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> </mrow> <mrow> <mn>2</mn> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> </mrow> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {2K_{\mathrm {2D} }-E_{\mathrm {2D} }}{2K_{\mathrm {2D} }}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/886783b8c6eba83c0af80170a833041a2bc91ff3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:9.246ex; height:4.509ex;" alt="{\displaystyle {\tfrac {2K_{\mathrm {2D} }-E_{\mathrm {2D} }}{2K_{\mathrm {2D} }}}}"></span> </td> <td style="text-align:center;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {4K_{\mathrm {2D} }^{2}}{4K_{\mathrm {2D} }-E_{\mathrm {2D} }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mn>4</mn> <msubsup> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mrow> <mrow> <mn>4</mn> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <mo>−<!-- − --></mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> </mrow> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {4K_{\mathrm {2D} }^{2}}{4K_{\mathrm {2D} }-E_{\mathrm {2D} }}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ce32fb675b775c5bdfca3f4b6659163a4b55c7b2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:9.246ex; height:5.009ex;" alt="{\displaystyle {\tfrac {4K_{\mathrm {2D} }^{2}}{4K_{\mathrm {2D} }-E_{\mathrm {2D} }}}}"></span> </td> <td> </td></tr> <tr> <td style="text-align:center;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (K_{\mathrm {2D} },\,\lambda _{\mathrm {2D} })}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <mo>,</mo> <mspace width="thinmathspace" /> <msub> <mi>λ<!-- λ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (K_{\mathrm {2D} },\,\lambda _{\mathrm {2D} })}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8e41320590a7de1bdd74d77ec4fb8c5de98c735" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.178ex; height:2.843ex;" alt="{\displaystyle (K_{\mathrm {2D} },\,\lambda _{\mathrm {2D} })}"></span> </td> <td style="text-align:center;"> </td> <td style="text-align:center;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {4K_{\mathrm {2D} }(K_{\mathrm {2D} }-\lambda _{\mathrm {2D} })}{2K_{\mathrm {2D} }-\lambda _{\mathrm {2D} }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mn>4</mn> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <mo>−<!-- − --></mo> <msub> <mi>λ<!-- λ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <mn>2</mn> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <mo>−<!-- − --></mo> <msub> <mi>λ<!-- λ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> </mrow> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {4K_{\mathrm {2D} }(K_{\mathrm {2D} }-\lambda _{\mathrm {2D} })}{2K_{\mathrm {2D} }-\lambda _{\mathrm {2D} }}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ff62a0404edd2ddaa137a027c26dcc9f5d54aa35" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:13.517ex; height:4.676ex;" alt="{\displaystyle {\tfrac {4K_{\mathrm {2D} }(K_{\mathrm {2D} }-\lambda _{\mathrm {2D} })}{2K_{\mathrm {2D} }-\lambda _{\mathrm {2D} }}}}"></span> </td> <td style="text-align:center;"> </td> <td style="text-align:center;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K_{\mathrm {2D} }-\lambda _{\mathrm {2D} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <mo>−<!-- − --></mo> <msub> <mi>λ<!-- λ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K_{\mathrm {2D} }-\lambda _{\mathrm {2D} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e0553633b1cd67e18d005fa1233921bba612bc7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.788ex; height:2.509ex;" alt="{\displaystyle K_{\mathrm {2D} }-\lambda _{\mathrm {2D} }}"></span> </td> <td style="text-align:center;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {\lambda _{\mathrm {2D} }}{2K_{\mathrm {2D} }-\lambda _{\mathrm {2D} }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <msub> <mi>λ<!-- λ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <mrow> <mn>2</mn> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <mo>−<!-- − --></mo> <msub> <mi>λ<!-- λ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> </mrow> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {\lambda _{\mathrm {2D} }}{2K_{\mathrm {2D} }-\lambda _{\mathrm {2D} }}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c03fbcd4f8880f2e465c96ab0105a73d82e4d4b6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:8.992ex; height:4.509ex;" alt="{\displaystyle {\tfrac {\lambda _{\mathrm {2D} }}{2K_{\mathrm {2D} }-\lambda _{\mathrm {2D} }}}}"></span> </td> <td style="text-align:center;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2K_{\mathrm {2D} }-\lambda _{\mathrm {2D} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <mo>−<!-- − --></mo> <msub> <mi>λ<!-- λ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2K_{\mathrm {2D} }-\lambda _{\mathrm {2D} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dadcbb465024372d667d981060d1e22f78032843" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.951ex; height:2.509ex;" alt="{\displaystyle 2K_{\mathrm {2D} }-\lambda _{\mathrm {2D} }}"></span> </td> <td> </td></tr> <tr> <td style="text-align:center;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (K_{\mathrm {2D} },\,G_{\mathrm {2D} })}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <mo>,</mo> <mspace width="thinmathspace" /> <msub> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (K_{\mathrm {2D} },\,G_{\mathrm {2D} })}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c0d212f7396611e34fe1d734027cedaef6af70a4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.65ex; height:2.843ex;" alt="{\displaystyle (K_{\mathrm {2D} },\,G_{\mathrm {2D} })}"></span> </td> <td style="text-align:center;"> </td> <td style="text-align:center;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {4K_{\mathrm {2D} }G_{\mathrm {2D} }}{K_{\mathrm {2D} }+G_{\mathrm {2D} }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mn>4</mn> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <msub> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> </mrow> <mrow> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <mo>+</mo> <msub> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> </mrow> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {4K_{\mathrm {2D} }G_{\mathrm {2D} }}{K_{\mathrm {2D} }+G_{\mathrm {2D} }}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5a1d3ae4367d4e17dfd09606481b88a4ed978fb6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:8.503ex; height:4.509ex;" alt="{\displaystyle {\tfrac {4K_{\mathrm {2D} }G_{\mathrm {2D} }}{K_{\mathrm {2D} }+G_{\mathrm {2D} }}}}"></span> </td> <td style="text-align:center;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K_{\mathrm {2D} }-G_{\mathrm {2D} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <mo>−<!-- − --></mo> <msub> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K_{\mathrm {2D} }-G_{\mathrm {2D} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b0c746a534f84ea0488149bfc6547318c71d73b5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.26ex; height:2.509ex;" alt="{\displaystyle K_{\mathrm {2D} }-G_{\mathrm {2D} }}"></span> </td> <td style="text-align:center;"> </td> <td style="text-align:center;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {K_{\mathrm {2D} }-G_{\mathrm {2D} }}{K_{\mathrm {2D} }+G_{\mathrm {2D} }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <mo>−<!-- − --></mo> <msub> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> </mrow> <mrow> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <mo>+</mo> <msub> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> </mrow> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {K_{\mathrm {2D} }-G_{\mathrm {2D} }}{K_{\mathrm {2D} }+G_{\mathrm {2D} }}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/de713b3e1c47a75f90ab0fbdd3f076a6b602edd6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:8.503ex; height:4.509ex;" alt="{\displaystyle {\tfrac {K_{\mathrm {2D} }-G_{\mathrm {2D} }}{K_{\mathrm {2D} }+G_{\mathrm {2D} }}}}"></span> </td> <td style="text-align:center;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K_{\mathrm {2D} }+G_{\mathrm {2D} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <mo>+</mo> <msub> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K_{\mathrm {2D} }+G_{\mathrm {2D} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fbb4c1d4cc074eaf126bacc62266e79366205b58" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.26ex; height:2.509ex;" alt="{\displaystyle K_{\mathrm {2D} }+G_{\mathrm {2D} }}"></span> </td> <td> </td></tr> <tr> <td style="text-align:center;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (K_{\mathrm {2D} },\,\nu _{\mathrm {2D} })}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <mo>,</mo> <mspace width="thinmathspace" /> <msub> <mi>ν<!-- ν --></mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (K_{\mathrm {2D} },\,\nu _{\mathrm {2D} })}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e283c72d13d40002703e71729b759b3e50c537f9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.971ex; height:2.843ex;" alt="{\displaystyle (K_{\mathrm {2D} },\,\nu _{\mathrm {2D} })}"></span> </td> <td style="text-align:center;"> </td> <td style="text-align:center;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2K_{\mathrm {2D} }(1-\nu _{\mathrm {2D} })\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−<!-- − --></mo> <msub> <mi>ν<!-- ν --></mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2K_{\mathrm {2D} }(1-\nu _{\mathrm {2D} })\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3ceee5a40e74656050ebc02c44486365b4eff639" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.103ex; height:2.843ex;" alt="{\displaystyle 2K_{\mathrm {2D} }(1-\nu _{\mathrm {2D} })\,}"></span> </td> <td style="text-align:center;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {2K_{\mathrm {2D} }\nu _{\mathrm {2D} }}{1+\nu _{\mathrm {2D} }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mn>2</mn> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <msub> <mi>ν<!-- ν --></mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> </mrow> <mrow> <mn>1</mn> <mo>+</mo> <msub> <mi>ν<!-- ν --></mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> </mrow> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {2K_{\mathrm {2D} }\nu _{\mathrm {2D} }}{1+\nu _{\mathrm {2D} }}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf645b1d01ce663dbe446ca262fe7d8982a5987f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:7.567ex; height:4.343ex;" alt="{\displaystyle {\tfrac {2K_{\mathrm {2D} }\nu _{\mathrm {2D} }}{1+\nu _{\mathrm {2D} }}}}"></span> </td> <td style="text-align:center;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {K_{\mathrm {2D} }(1-\nu _{\mathrm {2D} })}{1+\nu _{\mathrm {2D} }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−<!-- − --></mo> <msub> <mi>ν<!-- ν --></mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <mn>1</mn> <mo>+</mo> <msub> <mi>ν<!-- ν --></mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> </mrow> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {K_{\mathrm {2D} }(1-\nu _{\mathrm {2D} })}{1+\nu _{\mathrm {2D} }}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7feea6eb18fa6e18abbb7f5156a077c877e44a0c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:10.125ex; height:4.509ex;" alt="{\displaystyle {\tfrac {K_{\mathrm {2D} }(1-\nu _{\mathrm {2D} })}{1+\nu _{\mathrm {2D} }}}}"></span> </td> <td style="text-align:center;"> </td> <td style="text-align:center;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {2K_{\mathrm {2D} }}{1+\nu _{\mathrm {2D} }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mn>2</mn> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> </mrow> <mrow> <mn>1</mn> <mo>+</mo> <msub> <mi>ν<!-- ν --></mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> </mrow> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {2K_{\mathrm {2D} }}{1+\nu _{\mathrm {2D} }}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/478b33d574f890091f0231524ec05f9f8b8ee8f1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:5.6ex; height:4.343ex;" alt="{\displaystyle {\tfrac {2K_{\mathrm {2D} }}{1+\nu _{\mathrm {2D} }}}}"></span> </td> <td> </td></tr> <tr> <td style="text-align:center;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (E_{\mathrm {2D} },\,G_{\mathrm {2D} })}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <mo>,</mo> <mspace width="thinmathspace" /> <msub> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (E_{\mathrm {2D} },\,G_{\mathrm {2D} })}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/82f4b920c8f2b88eeddad760d3092c3fe6ade926" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.392ex; height:2.843ex;" alt="{\displaystyle (E_{\mathrm {2D} },\,G_{\mathrm {2D} })}"></span> </td> <td style="text-align:center;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {E_{\mathrm {2D} }G_{\mathrm {2D} }}{4G_{\mathrm {2D} }-E_{\mathrm {2D} }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <msub> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> </mrow> <mrow> <mn>4</mn> <msub> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <mo>−<!-- − --></mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> </mrow> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {E_{\mathrm {2D} }G_{\mathrm {2D} }}{4G_{\mathrm {2D} }-E_{\mathrm {2D} }}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3d08312a061b4190727b8cf4b3374b3a487ee0cd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:9.143ex; height:4.509ex;" alt="{\displaystyle {\tfrac {E_{\mathrm {2D} }G_{\mathrm {2D} }}{4G_{\mathrm {2D} }-E_{\mathrm {2D} }}}}"></span> </td> <td style="text-align:center;"> </td> <td style="text-align:center;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {2G_{\mathrm {2D} }(E_{\mathrm {2D} }-2G_{\mathrm {2D} })}{4G_{\mathrm {2D} }-E_{\mathrm {2D} }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mn>2</mn> <msub> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <mo>−<!-- − --></mo> <mn>2</mn> <msub> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <mn>4</mn> <msub> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <mo>−<!-- − --></mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> </mrow> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {2G_{\mathrm {2D} }(E_{\mathrm {2D} }-2G_{\mathrm {2D} })}{4G_{\mathrm {2D} }-E_{\mathrm {2D} }}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ba3f089d5a578367ccfdde36e561d651b70db82" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:14.387ex; height:4.676ex;" alt="{\displaystyle {\tfrac {2G_{\mathrm {2D} }(E_{\mathrm {2D} }-2G_{\mathrm {2D} })}{4G_{\mathrm {2D} }-E_{\mathrm {2D} }}}}"></span> </td> <td style="text-align:center;"> </td> <td style="text-align:center;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {E_{\mathrm {2D} }}{2G_{\mathrm {2D} }}}-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <mrow> <mn>2</mn> <msub> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> </mrow> </mfrac> </mstyle> </mrow> <mo>−<!-- − --></mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {E_{\mathrm {2D} }}{2G_{\mathrm {2D} }}}-1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1bc0b53a7580b8e0c920704ed43ffc27b9120799" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:8.804ex; height:4.343ex;" alt="{\displaystyle {\tfrac {E_{\mathrm {2D} }}{2G_{\mathrm {2D} }}}-1}"></span> </td> <td style="text-align:center;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {4G_{\mathrm {2D} }^{2}}{4G_{\mathrm {2D} }-E_{\mathrm {2D} }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mn>4</mn> <msubsup> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mrow> <mrow> <mn>4</mn> <msub> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <mo>−<!-- − --></mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> </mrow> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {4G_{\mathrm {2D} }^{2}}{4G_{\mathrm {2D} }-E_{\mathrm {2D} }}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2235f7699f8a7c484ebe6ce70240ed6f08da4c9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:9.143ex; height:5.009ex;" alt="{\displaystyle {\tfrac {4G_{\mathrm {2D} }^{2}}{4G_{\mathrm {2D} }-E_{\mathrm {2D} }}}}"></span> </td> <td> </td></tr> <tr> <td style="text-align:center;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (E_{\mathrm {2D} },\,\nu _{\mathrm {2D} })}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <mo>,</mo> <mspace width="thinmathspace" /> <msub> <mi>ν<!-- ν --></mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (E_{\mathrm {2D} },\,\nu _{\mathrm {2D} })}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b77009ceec3987a55112023183a7ee301731b675" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.714ex; height:2.843ex;" alt="{\displaystyle (E_{\mathrm {2D} },\,\nu _{\mathrm {2D} })}"></span> </td> <td style="text-align:center;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {E_{\mathrm {2D} }}{2(1-\nu _{\mathrm {2D} })}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <mrow> <mn>2</mn> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−<!-- − --></mo> <msub> <mi>ν<!-- ν --></mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {E_{\mathrm {2D} }}{2(1-\nu _{\mathrm {2D} })}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c7a7b40ff685661adb0b9dfee6bd644d70b2e914" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:7.701ex; height:4.509ex;" alt="{\displaystyle {\tfrac {E_{\mathrm {2D} }}{2(1-\nu _{\mathrm {2D} })}}}"></span> </td> <td style="text-align:center;"> </td> <td style="text-align:center;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {E_{\mathrm {2D} }\nu _{\mathrm {2D} }}{(1+\nu _{\mathrm {2D} })(1-\nu _{\mathrm {2D} })}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <msub> <mi>ν<!-- ν --></mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <msub> <mi>ν<!-- ν --></mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−<!-- − --></mo> <msub> <mi>ν<!-- ν --></mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {E_{\mathrm {2D} }\nu _{\mathrm {2D} }}{(1+\nu _{\mathrm {2D} })(1-\nu _{\mathrm {2D} })}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/099970a35cbfe96ff5d719bf331a6d26b1a70d47" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:12.922ex; height:4.509ex;" alt="{\displaystyle {\tfrac {E_{\mathrm {2D} }\nu _{\mathrm {2D} }}{(1+\nu _{\mathrm {2D} })(1-\nu _{\mathrm {2D} })}}}"></span> </td> <td style="text-align:center;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {E_{\mathrm {2D} }}{2(1+\nu _{\mathrm {2D} })}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <mrow> <mn>2</mn> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <msub> <mi>ν<!-- ν --></mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {E_{\mathrm {2D} }}{2(1+\nu _{\mathrm {2D} })}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d2caca86f5b8824866bbbb6db17c6ce009ea598e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:7.701ex; height:4.509ex;" alt="{\displaystyle {\tfrac {E_{\mathrm {2D} }}{2(1+\nu _{\mathrm {2D} })}}}"></span> </td> <td style="text-align:center;"> </td> <td style="text-align:center;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {E_{\mathrm {2D} }}{(1+\nu _{\mathrm {2D} })(1-\nu _{\mathrm {2D} })}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <msub> <mi>ν<!-- ν --></mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−<!-- − --></mo> <msub> <mi>ν<!-- ν --></mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {E_{\mathrm {2D} }}{(1+\nu _{\mathrm {2D} })(1-\nu _{\mathrm {2D} })}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b2e9fe243a025d14cc19c4feb002fa281be09200" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:12.922ex; height:4.509ex;" alt="{\displaystyle {\tfrac {E_{\mathrm {2D} }}{(1+\nu _{\mathrm {2D} })(1-\nu _{\mathrm {2D} })}}}"></span> </td> <td> </td></tr> <tr> <td style="text-align:center;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\lambda _{\mathrm {2D} },\,G_{\mathrm {2D} })}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>λ<!-- λ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <mo>,</mo> <mspace width="thinmathspace" /> <msub> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\lambda _{\mathrm {2D} },\,G_{\mathrm {2D} })}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e19fd95c2596c2d28dba16de09f7c918a4fc9262" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.032ex; height:2.843ex;" alt="{\displaystyle (\lambda _{\mathrm {2D} },\,G_{\mathrm {2D} })}"></span> </td> <td style="text-align:center;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda _{\mathrm {2D} }+G_{\mathrm {2D} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>λ<!-- λ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <mo>+</mo> <msub> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda _{\mathrm {2D} }+G_{\mathrm {2D} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2793b8b4c3ec84a326439a25264e5b988b625799" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.642ex; height:2.509ex;" alt="{\displaystyle \lambda _{\mathrm {2D} }+G_{\mathrm {2D} }}"></span> </td> <td style="text-align:center;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {4G_{\mathrm {2D} }(\lambda _{\mathrm {2D} }+G_{\mathrm {2D} })}{\lambda _{\mathrm {2D} }+2G_{\mathrm {2D} }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mn>4</mn> <msub> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>λ<!-- λ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <mo>+</mo> <msub> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <msub> <mi>λ<!-- λ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <mo>+</mo> <mn>2</mn> <msub> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> </mrow> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {4G_{\mathrm {2D} }(\lambda _{\mathrm {2D} }+G_{\mathrm {2D} })}{\lambda _{\mathrm {2D} }+2G_{\mathrm {2D} }}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fadfb075fa8e027e9917dcd1ba9bc67175d7c735" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:13.31ex; height:4.676ex;" alt="{\displaystyle {\tfrac {4G_{\mathrm {2D} }(\lambda _{\mathrm {2D} }+G_{\mathrm {2D} })}{\lambda _{\mathrm {2D} }+2G_{\mathrm {2D} }}}}"></span> </td> <td style="text-align:center;"> </td> <td style="text-align:center;"> </td> <td style="text-align:center;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {\lambda _{\mathrm {2D} }}{\lambda _{\mathrm {2D} }+2G_{\mathrm {2D} }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <msub> <mi>λ<!-- λ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <mrow> <msub> <mi>λ<!-- λ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <mo>+</mo> <mn>2</mn> <msub> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> </mrow> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {\lambda _{\mathrm {2D} }}{\lambda _{\mathrm {2D} }+2G_{\mathrm {2D} }}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eab1dc2f5aab249f9dc05a1ee174b19186396346" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:8.888ex; height:4.509ex;" alt="{\displaystyle {\tfrac {\lambda _{\mathrm {2D} }}{\lambda _{\mathrm {2D} }+2G_{\mathrm {2D} }}}}"></span> </td> <td style="text-align:center;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda _{\mathrm {2D} }+2G_{\mathrm {2D} }\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>λ<!-- λ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <mo>+</mo> <mn>2</mn> <msub> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda _{\mathrm {2D} }+2G_{\mathrm {2D} }\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5308206440fd0cc39d9baa35004ee0675fec397" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.191ex; height:2.509ex;" alt="{\displaystyle \lambda _{\mathrm {2D} }+2G_{\mathrm {2D} }\,}"></span> </td> <td> </td></tr> <tr> <td style="text-align:center;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\lambda _{\mathrm {2D} },\,\nu _{\mathrm {2D} })}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>λ<!-- λ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <mo>,</mo> <mspace width="thinmathspace" /> <msub> <mi>ν<!-- ν --></mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\lambda _{\mathrm {2D} },\,\nu _{\mathrm {2D} })}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5761b070cd3f5e66294b5ff948d0d5f2ce9fd73b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.354ex; height:2.843ex;" alt="{\displaystyle (\lambda _{\mathrm {2D} },\,\nu _{\mathrm {2D} })}"></span> </td> <td style="text-align:center;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {\lambda _{\mathrm {2D} }(1+\nu _{\mathrm {2D} })}{2\nu _{\mathrm {2D} }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <msub> <mi>λ<!-- λ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <msub> <mi>ν<!-- ν --></mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <mn>2</mn> <msub> <mi>ν<!-- ν --></mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> </mrow> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {\lambda _{\mathrm {2D} }(1+\nu _{\mathrm {2D} })}{2\nu _{\mathrm {2D} }}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/372ee40f2f39a0530c89865308f0cfe92d46a88a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:9.688ex; height:4.509ex;" alt="{\displaystyle {\tfrac {\lambda _{\mathrm {2D} }(1+\nu _{\mathrm {2D} })}{2\nu _{\mathrm {2D} }}}}"></span> </td> <td style="text-align:center;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {\lambda _{\mathrm {2D} }(1+\nu _{\mathrm {2D} })(1-\nu _{\mathrm {2D} })}{\nu _{\mathrm {2D} }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <msub> <mi>λ<!-- λ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <msub> <mi>ν<!-- ν --></mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−<!-- − --></mo> <msub> <mi>ν<!-- ν --></mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <msub> <mi>ν<!-- ν --></mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {\lambda _{\mathrm {2D} }(1+\nu _{\mathrm {2D} })(1-\nu _{\mathrm {2D} })}{\nu _{\mathrm {2D} }}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0fe813ef63bd6a0a2158043a1f270e7f2736bf49" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:15.731ex; height:4.343ex;" alt="{\displaystyle {\tfrac {\lambda _{\mathrm {2D} }(1+\nu _{\mathrm {2D} })(1-\nu _{\mathrm {2D} })}{\nu _{\mathrm {2D} }}}}"></span> </td> <td style="text-align:center;"> </td> <td style="text-align:center;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {\lambda _{\mathrm {2D} }(1-\nu _{\mathrm {2D} })}{2\nu _{\mathrm {2D} }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <msub> <mi>λ<!-- λ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−<!-- − --></mo> <msub> <mi>ν<!-- ν --></mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <mn>2</mn> <msub> <mi>ν<!-- ν --></mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> </mrow> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {\lambda _{\mathrm {2D} }(1-\nu _{\mathrm {2D} })}{2\nu _{\mathrm {2D} }}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e496450f903c59f63dc9ff43185abfc8771b7cf9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:9.688ex; height:4.509ex;" alt="{\displaystyle {\tfrac {\lambda _{\mathrm {2D} }(1-\nu _{\mathrm {2D} })}{2\nu _{\mathrm {2D} }}}}"></span> </td> <td style="text-align:center;"> </td> <td style="text-align:center;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {\lambda _{\mathrm {2D} }}{\nu _{\mathrm {2D} }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <msub> <mi>λ<!-- λ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <msub> <mi>ν<!-- ν --></mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {\lambda _{\mathrm {2D} }}{\nu _{\mathrm {2D} }}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/775d9920e7cedfb6290becbb23ccae4407dc2131" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:3.645ex; height:4.176ex;" alt="{\displaystyle {\tfrac {\lambda _{\mathrm {2D} }}{\nu _{\mathrm {2D} }}}}"></span> </td> <td style="text-align:center;">Cannot be used when <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu _{\mathrm {2D} }=0\Leftrightarrow \lambda _{\mathrm {2D} }=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ν<!-- ν --></mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <mo>=</mo> <mn>0</mn> <mo stretchy="false">⇔<!-- ⇔ --></mo> <msub> <mi>λ<!-- λ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nu _{\mathrm {2D} }=0\Leftrightarrow \lambda _{\mathrm {2D} }=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/abda36be11717593314a048856e008148eab22aa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:19.259ex; height:2.509ex;" alt="{\displaystyle \nu _{\mathrm {2D} }=0\Leftrightarrow \lambda _{\mathrm {2D} }=0}"></span> </td></tr> <tr> <td style="text-align:center;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (G_{\mathrm {2D} },\,\nu _{\mathrm {2D} })}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <mo>,</mo> <mspace width="thinmathspace" /> <msub> <mi>ν<!-- ν --></mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (G_{\mathrm {2D} },\,\nu _{\mathrm {2D} })}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7a15ad10fa657886cd82c74e9cc0cd8faabb72b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.825ex; height:2.843ex;" alt="{\displaystyle (G_{\mathrm {2D} },\,\nu _{\mathrm {2D} })}"></span> </td> <td style="text-align:center;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {G_{\mathrm {2D} }(1+\nu _{\mathrm {2D} })}{1-\nu _{\mathrm {2D} }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <msub> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <msub> <mi>ν<!-- ν --></mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <mn>1</mn> <mo>−<!-- − --></mo> <msub> <mi>ν<!-- ν --></mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> </mrow> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {G_{\mathrm {2D} }(1+\nu _{\mathrm {2D} })}{1-\nu _{\mathrm {2D} }}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/82c30e9987de68d9409550f95a806033d6bf05b8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:10.022ex; height:4.509ex;" alt="{\displaystyle {\tfrac {G_{\mathrm {2D} }(1+\nu _{\mathrm {2D} })}{1-\nu _{\mathrm {2D} }}}}"></span> </td> <td style="text-align:center;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2G_{\mathrm {2D} }(1+\nu _{\mathrm {2D} })\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <msub> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <msub> <mi>ν<!-- ν --></mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2G_{\mathrm {2D} }(1+\nu _{\mathrm {2D} })\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/20b78ca05d547b5b77f79626a3b1735716468790" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.957ex; height:2.843ex;" alt="{\displaystyle 2G_{\mathrm {2D} }(1+\nu _{\mathrm {2D} })\,}"></span> </td> <td style="text-align:center;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {2G_{\mathrm {2D} }\nu _{\mathrm {2D} }}{1-\nu _{\mathrm {2D} }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mn>2</mn> <msub> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <msub> <mi>ν<!-- ν --></mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> </mrow> <mrow> <mn>1</mn> <mo>−<!-- − --></mo> <msub> <mi>ν<!-- ν --></mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> </mrow> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {2G_{\mathrm {2D} }\nu _{\mathrm {2D} }}{1-\nu _{\mathrm {2D} }}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f7b74885d6aa702ca6d72909e74a0172966fb8d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:7.464ex; height:4.343ex;" alt="{\displaystyle {\tfrac {2G_{\mathrm {2D} }\nu _{\mathrm {2D} }}{1-\nu _{\mathrm {2D} }}}}"></span> </td> <td style="text-align:center;"> </td> <td style="text-align:center;"> </td> <td style="text-align:center;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {2G_{\mathrm {2D} }}{1-\nu _{\mathrm {2D} }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mn>2</mn> <msub> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> </mrow> <mrow> <mn>1</mn> <mo>−<!-- − --></mo> <msub> <mi>ν<!-- ν --></mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> </mrow> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {2G_{\mathrm {2D} }}{1-\nu _{\mathrm {2D} }}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b6109dc44b8f000901eb7cd762c273aca1e2571" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:5.6ex; height:4.343ex;" alt="{\displaystyle {\tfrac {2G_{\mathrm {2D} }}{1-\nu _{\mathrm {2D} }}}}"></span> </td> <td> </td></tr> <tr> <td style="text-align:center;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (G_{\mathrm {2D} },\,M_{\mathrm {2D} })}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <mo>,</mo> <mspace width="thinmathspace" /> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (G_{\mathrm {2D} },\,M_{\mathrm {2D} })}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6b59af291aea1f6bf2193881aa8272fabfd9d7d4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.931ex; height:2.843ex;" alt="{\displaystyle (G_{\mathrm {2D} },\,M_{\mathrm {2D} })}"></span> </td> <td style="text-align:center;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{\mathrm {2D} }-G_{\mathrm {2D} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <mo>−<!-- − --></mo> <msub> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M_{\mathrm {2D} }-G_{\mathrm {2D} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b720400365f3d72ef91064bcf17b97f5e6564cc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.541ex; height:2.509ex;" alt="{\displaystyle M_{\mathrm {2D} }-G_{\mathrm {2D} }}"></span> </td> <td style="text-align:center;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {4G_{\mathrm {2D} }(M_{\mathrm {2D} }-G_{\mathrm {2D} })}{M_{\mathrm {2D} }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mn>4</mn> <msub> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <mo>−<!-- − --></mo> <msub> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {4G_{\mathrm {2D} }(M_{\mathrm {2D} }-G_{\mathrm {2D} })}{M_{\mathrm {2D} }}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2e5e0a815db624389832f101dfb77a9a3d2c92b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:13.946ex; height:4.676ex;" alt="{\displaystyle {\tfrac {4G_{\mathrm {2D} }(M_{\mathrm {2D} }-G_{\mathrm {2D} })}{M_{\mathrm {2D} }}}}"></span> </td> <td style="text-align:center;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{\mathrm {2D} }-2G_{\mathrm {2D} }\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <mo>−<!-- − --></mo> <mn>2</mn> <msub> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M_{\mathrm {2D} }-2G_{\mathrm {2D} }\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6a8e1e8ddab785855909fc6450ee784327f49fff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.09ex; height:2.509ex;" alt="{\displaystyle M_{\mathrm {2D} }-2G_{\mathrm {2D} }\,}"></span> </td> <td style="text-align:center;"> </td> <td style="text-align:center;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {M_{\mathrm {2D} }-2G_{\mathrm {2D} }}{M_{\mathrm {2D} }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <mo>−<!-- − --></mo> <mn>2</mn> <msub> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> </mrow> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {M_{\mathrm {2D} }-2G_{\mathrm {2D} }}{M_{\mathrm {2D} }}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e1048dd64851e8dd14301a6894c023470ff055dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:9.524ex; height:4.509ex;" alt="{\displaystyle {\tfrac {M_{\mathrm {2D} }-2G_{\mathrm {2D} }}{M_{\mathrm {2D} }}}}"></span> </td> <td style="text-align:center;"> </td> <td> <p><br /> </p><p><br /> </p> </td></tr></tbody></table> <!-- NewPP limit report Parsed by mw‐web.codfw.main‐84749c7844‐dbnxn Cached time: 20250210044853 Cache expiry: 2592000 Reduced expiry: false Complications: [vary‐revision‐sha1, show‐toc] CPU time usage: 1.254 seconds Real time usage: 1.642 seconds Preprocessor visited node count: 7340/1000000 Post‐expand include size: 103935/2097152 bytes Template argument size: 13348/2097152 bytes Highest expansion depth: 17/100 Expensive parser function count: 11/500 Unstrip recursion depth: 1/20 Unstrip post‐expand size: 134447/5000000 bytes Lua time usage: 0.626/10.000 seconds Lua memory usage: 27192089/52428800 bytes Number of Wikibase entities loaded: 0/400 --> <!-- Transclusion expansion time report (%,ms,calls,template) 100.00% 1067.554 1 -total 19.01% 202.961 1 Template:Reflist 14.61% 155.918 1 Template:Short_description 13.90% 148.375 2 Template:Lang 12.70% 135.572 1 Template:Continuum_mechanics 10.89% 116.237 8 Template:Cite_book 10.02% 107.006 83 Template:Math 8.44% 90.089 2 Template:Pagetype 8.06% 86.047 3 Template:Sidebar 6.42% 68.564 89 Template:Main_other --> <!-- Saved in parser cache with key enwiki:pcache:229553:|#|:idhash:canonical and timestamp 20250210044853 and revision id 1261498479. Rendering was triggered because: page-view --> </div><!--esi <esi:include src="/esitest-fa8a495983347898/content" /> --><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?useformat=desktop&type=1x1&usesul3=0" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Hooke%27s_law&oldid=1261498479">https://en.wikipedia.org/w/index.php?title=Hooke%27s_law&oldid=1261498479</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Categories</a>: <ul><li><a href="/wiki/Category:1676_in_science" title="Category:1676 in science">1676 in science</a></li><li><a href="/wiki/Category:Elasticity_(physics)" title="Category:Elasticity (physics)">Elasticity (physics)</a></li><li><a href="/wiki/Category:Solid_mechanics" title="Category:Solid mechanics">Solid mechanics</a></li><li><a href="/wiki/Category:Springs_(mechanical)" title="Category:Springs (mechanical)">Springs (mechanical)</a></li><li><a href="/wiki/Category:Structural_analysis" title="Category:Structural analysis">Structural analysis</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Hidden categories: <ul><li><a href="/wiki/Category:Articles_containing_Latin-language_text" title="Category:Articles containing Latin-language text">Articles containing Latin-language text</a></li><li><a href="/wiki/Category:Articles_with_short_description" title="Category:Articles with short description">Articles with short description</a></li><li><a href="/wiki/Category:Short_description_is_different_from_Wikidata" title="Category:Short description is different from Wikidata">Short description is different from Wikidata</a></li><li><a href="/wiki/Category:Wikipedia_indefinitely_semi-protected_pages" title="Category:Wikipedia indefinitely semi-protected pages">Wikipedia indefinitely semi-protected pages</a></li><li><a href="/wiki/Category:Articles_lacking_in-text_citations_from_July_2017" title="Category:Articles lacking in-text citations from July 2017">Articles lacking in-text citations from July 2017</a></li><li><a href="/wiki/Category:All_articles_lacking_in-text_citations" title="Category:All articles lacking in-text citations">All articles lacking in-text citations</a></li><li><a href="/wiki/Category:Use_dmy_dates_from_October_2017" title="Category:Use dmy dates from October 2017">Use dmy dates from October 2017</a></li><li><a href="/wiki/Category:Pages_using_sidebar_with_the_child_parameter" title="Category:Pages using sidebar with the child parameter">Pages using sidebar with the child parameter</a></li><li><a href="/wiki/Category:Wikipedia_articles_needing_page_number_citations_from_November_2016" title="Category:Wikipedia articles needing page number citations from November 2016">Wikipedia articles needing page number citations from November 2016</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> This page was last edited on 6 December 2024, at 12:23<span class="anonymous-show"> (UTC)</span>.</li> <li id="footer-info-copyright">Text is available under the <a href="/wiki/Wikipedia:Text_of_the_Creative_Commons_Attribution-ShareAlike_4.0_International_License" title="Wikipedia:Text of the Creative Commons Attribution-ShareAlike 4.0 International License">Creative Commons Attribution-ShareAlike 4.0 License</a>; additional terms may apply. By using this site, you agree to the <a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Terms_of_Use" class="extiw" title="foundation:Special:MyLanguage/Policy:Terms of Use">Terms of Use</a> and <a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Privacy_policy" class="extiw" title="foundation:Special:MyLanguage/Policy:Privacy policy">Privacy Policy</a>. Wikipedia® is a registered trademark of the <a rel="nofollow" class="external text" href="https://wikimediafoundation.org/">Wikimedia Foundation, Inc.</a>, a non-profit organization.</li> </ul> <ul id="footer-places"> <li id="footer-places-privacy"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Privacy_policy">Privacy policy</a></li> <li id="footer-places-about"><a href="/wiki/Wikipedia:About">About Wikipedia</a></li> <li id="footer-places-disclaimers"><a href="/wiki/Wikipedia:General_disclaimer">Disclaimers</a></li> <li id="footer-places-contact"><a href="//en.wikipedia.org/wiki/Wikipedia:Contact_us">Contact Wikipedia</a></li> <li id="footer-places-wm-codeofconduct"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Universal_Code_of_Conduct">Code of Conduct</a></li> <li id="footer-places-developers"><a href="https://developer.wikimedia.org">Developers</a></li> <li id="footer-places-statslink"><a href="https://stats.wikimedia.org/#/en.wikipedia.org">Statistics</a></li> <li id="footer-places-cookiestatement"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Cookie_statement">Cookie statement</a></li> <li id="footer-places-mobileview"><a href="//en.m.wikipedia.org/w/index.php?title=Hooke%27s_law&mobileaction=toggle_view_mobile" class="noprint stopMobileRedirectToggle">Mobile view</a></li> </ul> <ul id="footer-icons" class="noprint"> <li id="footer-copyrightico"><a href="https://wikimediafoundation.org/" class="cdx-button cdx-button--fake-button cdx-button--size-large cdx-button--fake-button--enabled"><img src="/static/images/footer/wikimedia-button.svg" width="84" height="29" alt="Wikimedia Foundation" lang="en" loading="lazy"></a></li> <li id="footer-poweredbyico"><a href="https://www.mediawiki.org/" class="cdx-button cdx-button--fake-button cdx-button--size-large cdx-button--fake-button--enabled"><img src="/w/resources/assets/poweredby_mediawiki.svg" alt="Powered by MediaWiki" width="88" height="31" loading="lazy"></a></li> </ul> </footer> </div> </div> </div> <div class="vector-header-container vector-sticky-header-container"> <div id="vector-sticky-header" class="vector-sticky-header"> <div class="vector-sticky-header-start"> <div class="vector-sticky-header-icon-start vector-button-flush-left vector-button-flush-right" aria-hidden="true"> <button class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-sticky-header-search-toggle" tabindex="-1" data-event-name="ui.vector-sticky-search-form.icon"><span class="vector-icon mw-ui-icon-search mw-ui-icon-wikimedia-search"></span> <span>Search</span> </button> </div> <div role="search" class="vector-search-box-vue vector-search-box-show-thumbnail vector-search-box"> <div class="vector-typeahead-search-container"> <div class="cdx-typeahead-search cdx-typeahead-search--show-thumbnail"> <form action="/w/index.php" id="vector-sticky-search-form" class="cdx-search-input cdx-search-input--has-end-button"> <div class="cdx-search-input__input-wrapper" data-search-loc="header-moved"> <div class="cdx-text-input cdx-text-input--has-start-icon"> <input class="cdx-text-input__input" type="search" name="search" placeholder="Search Wikipedia"> <span class="cdx-text-input__icon cdx-text-input__start-icon"></span> </div> <input type="hidden" name="title" value="Special:Search"> </div> <button class="cdx-button cdx-search-input__end-button">Search</button> </form> </div> </div> </div> <div class="vector-sticky-header-context-bar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-sticky-header-toc" class="vector-dropdown mw-portlet mw-portlet-sticky-header-toc vector-sticky-header-toc vector-button-flush-left" > <input type="checkbox" id="vector-sticky-header-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-sticky-header-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-sticky-header-toc-label" for="vector-sticky-header-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Toggle the table of contents</span> </label> <div class="vector-dropdown-content"> <div id="vector-sticky-header-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <div class="vector-sticky-header-context-bar-primary" aria-hidden="true" ><span class="mw-page-title-main">Hooke's law</span></div> </div> </div> <div class="vector-sticky-header-end" aria-hidden="true"> <div class="vector-sticky-header-icons"> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only" id="ca-talk-sticky-header" tabindex="-1" data-event-name="talk-sticky-header"><span class="vector-icon mw-ui-icon-speechBubbles mw-ui-icon-wikimedia-speechBubbles"></span> <span></span> </a> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only" id="ca-subject-sticky-header" tabindex="-1" data-event-name="subject-sticky-header"><span class="vector-icon mw-ui-icon-article mw-ui-icon-wikimedia-article"></span> <span></span> </a> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only" id="ca-history-sticky-header" tabindex="-1" data-event-name="history-sticky-header"><span class="vector-icon mw-ui-icon-wikimedia-history mw-ui-icon-wikimedia-wikimedia-history"></span> <span></span> </a> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only mw-watchlink" id="ca-watchstar-sticky-header" tabindex="-1" data-event-name="watch-sticky-header"><span class="vector-icon mw-ui-icon-wikimedia-star mw-ui-icon-wikimedia-wikimedia-star"></span> <span></span> </a> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only" id="ca-edit-sticky-header" tabindex="-1" data-event-name="wikitext-edit-sticky-header"><span class="vector-icon mw-ui-icon-wikimedia-wikiText mw-ui-icon-wikimedia-wikimedia-wikiText"></span> <span></span> </a> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only" id="ca-ve-edit-sticky-header" tabindex="-1" data-event-name="ve-edit-sticky-header"><span class="vector-icon mw-ui-icon-wikimedia-edit mw-ui-icon-wikimedia-wikimedia-edit"></span> <span></span> </a> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only" id="ca-viewsource-sticky-header" tabindex="-1" data-event-name="ve-edit-protected-sticky-header"><span class="vector-icon mw-ui-icon-wikimedia-editLock mw-ui-icon-wikimedia-wikimedia-editLock"></span> <span></span> </a> </div> <div class="vector-sticky-header-buttons"> <button class="cdx-button cdx-button--weight-quiet mw-interlanguage-selector" id="p-lang-btn-sticky-header" tabindex="-1" data-event-name="ui.dropdown-p-lang-btn-sticky-header"><span class="vector-icon mw-ui-icon-wikimedia-language mw-ui-icon-wikimedia-wikimedia-language"></span> <span>69 languages</span> </button> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive" id="ca-addsection-sticky-header" tabindex="-1" data-event-name="addsection-sticky-header"><span class="vector-icon mw-ui-icon-speechBubbleAdd-progressive mw-ui-icon-wikimedia-speechBubbleAdd-progressive"></span> <span>Add topic</span> </a> </div> <div class="vector-sticky-header-icon-end"> <div class="vector-user-links"> </div> </div> </div> </div> </div> <div class="vector-settings" id="p-dock-bottom"> <ul></ul> </div><script>(RLQ=window.RLQ||[]).push(function(){mw.config.set({"wgHostname":"mw-web.codfw.main-654b9d4bd7-gqf4l","wgBackendResponseTime":115,"wgPageParseReport":{"limitreport":{"cputime":"1.254","walltime":"1.642","ppvisitednodes":{"value":7340,"limit":1000000},"postexpandincludesize":{"value":103935,"limit":2097152},"templateargumentsize":{"value":13348,"limit":2097152},"expansiondepth":{"value":17,"limit":100},"expensivefunctioncount":{"value":11,"limit":500},"unstrip-depth":{"value":1,"limit":20},"unstrip-size":{"value":134447,"limit":5000000},"entityaccesscount":{"value":0,"limit":400},"timingprofile":["100.00% 1067.554 1 -total"," 19.01% 202.961 1 Template:Reflist"," 14.61% 155.918 1 Template:Short_description"," 13.90% 148.375 2 Template:Lang"," 12.70% 135.572 1 Template:Continuum_mechanics"," 10.89% 116.237 8 Template:Cite_book"," 10.02% 107.006 83 Template:Math"," 8.44% 90.089 2 Template:Pagetype"," 8.06% 86.047 3 Template:Sidebar"," 6.42% 68.564 89 Template:Main_other"]},"scribunto":{"limitreport-timeusage":{"value":"0.626","limit":"10.000"},"limitreport-memusage":{"value":27192089,"limit":52428800}},"cachereport":{"origin":"mw-web.codfw.main-84749c7844-dbnxn","timestamp":"20250210044853","ttl":2592000,"transientcontent":false}}});});</script> <script type="application/ld+json">{"@context":"https:\/\/schema.org","@type":"Article","name":"Hooke's law","url":"https:\/\/en.wikipedia.org\/wiki\/Hooke%27s_law","sameAs":"http:\/\/www.wikidata.org\/entity\/Q170282","mainEntity":"http:\/\/www.wikidata.org\/entity\/Q170282","author":{"@type":"Organization","name":"Contributors to Wikimedia projects"},"publisher":{"@type":"Organization","name":"Wikimedia Foundation, Inc.","logo":{"@type":"ImageObject","url":"https:\/\/www.wikimedia.org\/static\/images\/wmf-hor-googpub.png"}},"datePublished":"2003-05-19T01:20:49Z","dateModified":"2024-12-06T12:23:01Z","image":"https:\/\/upload.wikimedia.org\/wikipedia\/commons\/f\/fc\/Hookes-law-springs.png","headline":"empirical physical law of mechanics that the force on a spring is proportional to its displacement"}</script> </body> </html>