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<span>Units</span> </div> </a> <button aria-controls="toc-Units-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 Units subsection</span> </button> <ul id="toc-Units-sublist" class="vector-toc-list"> <li id="toc-International_System" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#International_System"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>International System</span> </div> </a> <ul id="toc-International_System-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Linear_elasticity" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Linear_elasticity"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Linear elasticity</span> </div> </a> <ul id="toc-Linear_elasticity-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Finite_elasticity" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Finite_elasticity"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Finite elasticity</span> </div> </a> <button aria-controls="toc-Finite_elasticity-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 Finite elasticity subsection</span> </button> <ul id="toc-Finite_elasticity-sublist" class="vector-toc-list"> <li id="toc-Cauchy_elastic_materials" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Cauchy_elastic_materials"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Cauchy elastic materials</span> </div> </a> <ul id="toc-Cauchy_elastic_materials-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Hypoelastic_materials" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Hypoelastic_materials"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2</span> <span>Hypoelastic materials</span> </div> </a> <ul id="toc-Hypoelastic_materials-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Hyperelastic_materials" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Hyperelastic_materials"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.3</span> <span>Hyperelastic materials</span> </div> </a> <ul id="toc-Hyperelastic_materials-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Applications" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Applications"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Applications</span> </div> </a> <ul id="toc-Applications-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Factors_affecting_elasticity" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Factors_affecting_elasticity"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Factors affecting elasticity</span> </div> </a> <ul id="toc-Factors_affecting_elasticity-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</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 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</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 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</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 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</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" > <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">Elasticity (physics)</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 66 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-66" 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">66 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%85%D8%B1%D9%88%D9%86%D8%A9_(%D9%81%D9%8A%D8%B2%D9%8A%D8%A7%D8%A1)" title="مرونة (فيزياء) – Arabic" lang="ar" hreflang="ar" data-title="مرونة (فيزياء)" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/Elastiklik" title="Elastiklik – Azerbaijani" lang="az" hreflang="az" data-title="Elastiklik" 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%B8%E0%A7%8D%E0%A6%A5%E0%A6%BF%E0%A6%A4%E0%A6%BF%E0%A6%B8%E0%A7%8D%E0%A6%A5%E0%A6%BE%E0%A6%AA%E0%A6%95%E0%A6%A4%E0%A6%BE_(%E0%A6%AA%E0%A6%A6%E0%A6%BE%E0%A6%B0%E0%A7%8D%E0%A6%A5%E0%A6%AC%E0%A6%BF%E0%A6%9C%E0%A7%8D%E0%A6%9E%E0%A6%BE%E0%A6%A8)" title="স্থিতিস্থাপকতা (পদার্থবিজ্ঞান) – Bangla" lang="bn" hreflang="bn" data-title="স্থিতিস্থাপকতা (পদার্থবিজ্ঞান)" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-zh-min-nan mw-list-item"><a href="https://zh-min-nan.wikipedia.org/wiki/T%C3%A2n-s%C3%A8ng" title="Tân-sèng – Minnan" lang="nan" hreflang="nan" data-title="Tân-sèng" data-language-autonym="閩南語 / Bân-lâm-gú" data-language-local-name="Minnan" class="interlanguage-link-target"><span>閩南語 / Bân-lâm-gú</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%9F%D1%80%D1%83%D0%B3%D0%BA%D0%B0%D1%81%D1%86%D1%8C" title="Пругкасць – Belarusian" lang="be" hreflang="be" data-title="Пругкасць" 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%95%D0%BB%D0%B0%D1%81%D1%82%D0%B8%D1%87%D0%BD%D0%B0_%D0%B4%D0%B5%D1%84%D0%BE%D1%80%D0%BC%D0%B0%D1%86%D0%B8%D1%8F" title="Еластична деформация – Bulgarian" lang="bg" hreflang="bg" data-title="Еластична деформация" 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/Elasti%C4%8Dnost_(fizika)" title="Elastičnost (fizika) – Bosnian" lang="bs" hreflang="bs" data-title="Elastičnost (fizika)" 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/Elasticitat" title="Elasticitat – Catalan" lang="ca" hreflang="ca" data-title="Elasticitat" 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%9F%D0%B8%C3%A7%C4%95%D0%BB%C4%95%D1%85" title="Пиçĕлĕх – Chuvash" lang="cv" hreflang="cv" data-title="Пиçĕлĕх" 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/Pru%C5%BEnost" title="Pružnost – Czech" lang="cs" hreflang="cs" data-title="Pružnost" 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/Elasticitet_(fysik)" title="Elasticitet (fysik) – Danish" lang="da" hreflang="da" data-title="Elasticitet (fysik)" 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/Elastizit%C3%A4t_(Physik)" title="Elastizität (Physik) – German" lang="de" hreflang="de" data-title="Elastizität (Physik)" 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/Elastsus" title="Elastsus – Estonian" lang="et" hreflang="et" data-title="Elastsus" 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%95%CE%BB%CE%B1%CF%83%CF%84%CE%B9%CE%BA%CF%8C%CF%84%CE%B7%CF%84%CE%B1" title="Ελαστικότητα – Greek" lang="el" hreflang="el" data-title="Ελαστικότητα" 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/Elasticidad_(mec%C3%A1nica_de_s%C3%B3lidos)" title="Elasticidad (mecánica de sólidos) – Spanish" lang="es" hreflang="es" data-title="Elasticidad (mecánica de sólidos)" 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/Elasteco_(me%C4%A5aniko)" title="Elasteco (meĥaniko) – Esperanto" lang="eo" hreflang="eo" data-title="Elasteco (meĥaniko)" 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/Elastikotasun_(fisika)" title="Elastikotasun (fisika) – Basque" lang="eu" hreflang="eu" data-title="Elastikotasun (fisika)" 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/%DA%A9%D8%B4%D8%B3%D8%A7%D9%86%DB%8C" title="کشسانی – Persian" lang="fa" hreflang="fa" data-title="کشسانی" 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/D%C3%A9formation_%C3%A9lastique" title="Déformation élastique – French" lang="fr" hreflang="fr" data-title="Déformation élastique" 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/Leaisteachas" title="Leaisteachas – Irish" lang="ga" hreflang="ga" data-title="Leaisteachas" 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/Elasticidade_(mec%C3%A1nica)" title="Elasticidade (mecánica) – Galician" lang="gl" hreflang="gl" data-title="Elasticidade (mecánica)" 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%83%84%EC%84%B1" title="탄성 – Korean" lang="ko" hreflang="ko" data-title="탄성" 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/%D4%B1%D5%BC%D5%A1%D5%B1%D5%A3%D5%A1%D5%AF%D5%A1%D5%B6%D5%B8%D6%82%D5%A9%D5%B5%D5%B8%D6%82%D5%B6" title="Առաձգականություն – Armenian" lang="hy" hreflang="hy" data-title="Առաձգականություն" 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%AA%E0%A5%8D%E0%A4%B0%E0%A4%A4%E0%A5%8D%E0%A4%AF%E0%A4%BE%E0%A4%B8%E0%A5%8D%E0%A4%A5%E0%A4%A4%E0%A4%BE" title="प्रत्यास्थता – Hindi" lang="hi" hreflang="hi" data-title="प्रत्यास्थता" 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/Elasti%C4%8Dnost" title="Elastičnost – Croatian" lang="hr" hreflang="hr" data-title="Elastičnost" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-io mw-list-item"><a href="https://io.wikipedia.org/wiki/Elastikeso" title="Elastikeso – Ido" lang="io" hreflang="io" data-title="Elastikeso" data-language-autonym="Ido" data-language-local-name="Ido" class="interlanguage-link-target"><span>Ido</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Elastisitas_(fisika)" title="Elastisitas (fisika) – Indonesian" lang="id" hreflang="id" data-title="Elastisitas (fisika)" 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/Elasticit%C3%A0" title="Elasticità – Italian" lang="it" hreflang="it" data-title="Elasticità" 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%90%D7%9C%D7%A1%D7%98%D7%99%D7%95%D7%AA" title="אלסטיות – Hebrew" lang="he" hreflang="he" data-title="אלסטיות" 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%B8%E0%B3%8D%E0%B2%A5%E0%B2%BF%E0%B2%A4%E0%B2%BF%E0%B2%B8%E0%B3%8D%E0%B2%A5%E0%B2%BE%E0%B2%AA%E0%B2%95%E0%B2%A4%E0%B3%8D%E0%B2%B5" title="ಸ್ಥಿತಿಸ್ಥಾಪಕತ್ವ – Kannada" lang="kn" hreflang="kn" data-title="ಸ್ಥಿತಿಸ್ಥಾಪಕತ್ವ" data-language-autonym="ಕನ್ನಡ" data-language-local-name="Kannada" class="interlanguage-link-target"><span>ಕನ್ನಡ</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%A1%D0%B5%D1%80%D0%BF%D1%96%D0%BC%D0%B4%D1%96%D0%BB%D1%96%D0%BA" title="Серпімділік – Kazakh" lang="kk" hreflang="kk" data-title="Серпімділік" data-language-autonym="Қазақша" data-language-local-name="Kazakh" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-lo mw-list-item"><a href="https://lo.wikipedia.org/wiki/%E0%BA%84%E0%BA%A7%E0%BA%B2%E0%BA%A1%E0%BA%8D%E0%BA%B7%E0%BA%94%E0%BA%AB%E0%BA%8D%E0%BA%B8%E0%BB%88%E0%BA%99_(%E0%BA%9F%E0%BA%B5%E0%BA%8A%E0%BA%B4%E0%BA%81)" title="ຄວາມຍືດຫຍຸ່ນ (ຟີຊິກ) – Lao" lang="lo" hreflang="lo" data-title="ຄວາມຍືດຫຍຸ່ນ (ຟີຊິກ)" data-language-autonym="ລາວ" data-language-local-name="Lao" class="interlanguage-link-target"><span>ລາວ</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Rugalmass%C3%A1g_(fizika)" title="Rugalmasság (fizika) – Hungarian" lang="hu" hreflang="hu" data-title="Rugalmasság (fizika)" 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%A0%D0%B0%D1%81%D1%82%D0%B5%D0%B3%D0%BB%D0%B8%D0%B2%D0%BE%D1%81%D1%82" title="Растегливост – Macedonian" lang="mk" hreflang="mk" data-title="Растегливост" data-language-autonym="Македонски" data-language-local-name="Macedonian" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%87%E0%B4%B2%E0%B4%BE%E0%B4%B8%E0%B5%8D%E0%B4%A4%E0%B4%BF%E0%B4%95%E0%B4%A4" title="ഇലാസ്തികത – Malayalam" lang="ml" hreflang="ml" data-title="ഇലാസ്തികത" data-language-autonym="മലയാളം" data-language-local-name="Malayalam" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Keanjalan" title="Keanjalan – Malay" lang="ms" hreflang="ms" data-title="Keanjalan" 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%A3%D1%8F%D0%BD_%D0%BD%D0%B0%D0%BB%D0%B0%D1%80%D1%85%D0%B0%D0%B9" title="Уян налархай – Mongolian" lang="mn" hreflang="mn" data-title="Уян налархай" 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/Elasticiteit_(materiaalkunde)" title="Elasticiteit (materiaalkunde) – Dutch" lang="nl" hreflang="nl" data-title="Elasticiteit (materiaalkunde)" 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/%E5%BC%BE%E6%80%A7" title="弾性 – Japanese" lang="ja" hreflang="ja" data-title="弾性" 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/Elastisitet" title="Elastisitet – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Elastisitet" 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/Elastisitet" title="Elastisitet – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Elastisitet" 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-om mw-list-item"><a href="https://om.wikipedia.org/wiki/Dhifamummaa" title="Dhifamummaa – Oromo" lang="om" hreflang="om" data-title="Dhifamummaa" data-language-autonym="Oromoo" data-language-local-name="Oromo" class="interlanguage-link-target"><span>Oromoo</span></a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%B2%E0%A8%9A%E0%A8%95_(%E0%A8%AD%E0%A9%8C%E0%A8%A4%E0%A8%BF%E0%A8%95_%E0%A8%B5%E0%A8%BF%E0%A8%97%E0%A8%BF%E0%A8%86%E0%A8%A8)" title="ਲਚਕ (ਭੌਤਿਕ ਵਿਗਿਆਨ) – Punjabi" lang="pa" hreflang="pa" data-title="ਲਚਕ (ਭੌਤਿਕ ਵਿਗਿਆਨ)" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="Punjabi" class="interlanguage-link-target"><span>ਪੰਜਾਬੀ</span></a></li><li class="interlanguage-link interwiki-pms mw-list-item"><a href="https://pms.wikipedia.org/wiki/Elasticit%C3%A0" title="Elasticità – Piedmontese" lang="pms" hreflang="pms" data-title="Elasticità" 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/Spr%C4%99%C5%BCysto%C5%9B%C4%87" title="Sprężystość – Polish" lang="pl" hreflang="pl" data-title="Sprężystość" 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/Elasticidade" title="Elasticidade – Portuguese" lang="pt" hreflang="pt" data-title="Elasticidade" 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/Elasticitate" title="Elasticitate – Romanian" lang="ro" hreflang="ro" data-title="Elasticitate" 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%A3%D0%BF%D1%80%D1%83%D0%B3%D0%BE%D1%81%D1%82%D1%8C" title="Упругость – Russian" lang="ru" hreflang="ru" data-title="Упругость" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Elasticiteti_(fizik%C3%AB)" title="Elasticiteti (fizikë) – Albanian" lang="sq" hreflang="sq" data-title="Elasticiteti (fizikë)" data-language-autonym="Shqip" data-language-local-name="Albanian" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-si mw-list-item"><a href="https://si.wikipedia.org/wiki/%E0%B6%B4%E0%B7%8A%E2%80%8D%E0%B6%BB%E0%B6%AD%E0%B7%8A%E2%80%8D%E0%B6%BA%E0%B7%8F%E0%B7%83%E0%B7%8A%E0%B6%AE%E0%B6%AD%E0%B7%8F%E0%B7%80" title="ප්‍රත්‍යාස්ථතාව – Sinhala" lang="si" hreflang="si" data-title="ප්‍රත්‍යාස්ථතාව" 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/Elasticity_(physics)" title="Elasticity (physics) – Simple English" lang="en-simple" hreflang="en-simple" data-title="Elasticity (physics)" 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/Te%C3%B3ria_pru%C5%BEnosti" title="Teória pružnosti – Slovak" lang="sk" hreflang="sk" data-title="Teória pružnosti" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%D9%84%D8%A7%D8%B3%D8%AA%DB%8C%DA%A9%DB%8C" title="لاستیکی – Central Kurdish" lang="ckb" hreflang="ckb" data-title="لاستیکی" data-language-autonym="کوردی" data-language-local-name="Central Kurdish" class="interlanguage-link-target"><span>کوردی</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/Elasti%C4%8Dnost_(fizika)" title="Elastičnost (fizika) – Serbian" lang="sr" hreflang="sr" data-title="Elastičnost (fizika)" 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/Elasti%C4%8Dnost" title="Elastičnost – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Elastičnost" 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/Elastisuus" title="Elastisuus – Finnish" lang="fi" hreflang="fi" data-title="Elastisuus" 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/Elasticitet" title="Elasticitet – Swedish" lang="sv" hreflang="sv" data-title="Elasticitet" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Elastisidad" title="Elastisidad – Tagalog" lang="tl" hreflang="tl" data-title="Elastisidad" data-language-autonym="Tagalog" data-language-local-name="Tagalog" class="interlanguage-link-target"><span>Tagalog</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a 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<div class="mw-indicators"> </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 property when materials or objects return to original shape after deformation</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">"Elasticity theory" redirects here. For the economics measurement, see <a href="/wiki/Elasticity_(economics)" title="Elasticity (economics)">Elasticity (economics)</a>. 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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</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: 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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> and <a href="/wiki/Materials_science" title="Materials science">materials science</a>, <b>elasticity</b> is the ability of a <a href="/wiki/Physical_object" title="Physical object">body</a> to resist a distorting influence and to return to its original size and shape when that influence or <a href="/wiki/Force" title="Force">force</a> is removed. Solid objects will <a href="/wiki/Deformation_(engineering)" title="Deformation (engineering)">deform</a> when adequate <a href="/wiki/Structural_load" title="Structural load">loads</a> are applied to them; if the material is elastic, the object will return to its initial shape and size after removal. This is in contrast to <a href="/wiki/Plasticity_(physics)" title="Plasticity (physics)"><i>plasticity</i></a>, in which the object fails to do so and instead remains in its deformed state. </p><p>The physical reasons for elastic behavior can be quite different for different materials. In <a href="/wiki/Metal" title="Metal">metals</a>, the <a href="/wiki/Crystal_structure" title="Crystal structure">atomic lattice</a> changes size and shape when forces are applied (energy is added to the system). When forces are removed, the lattice goes back to the original lower energy state. For <a href="/wiki/Rubber_elasticity" title="Rubber elasticity">rubbers</a> and other <a href="/wiki/Polymer" title="Polymer">polymers</a>, elasticity is caused by the stretching of polymer chains when forces are applied. </p><p><a href="/wiki/Hooke%27s_law" title="Hooke's law">Hooke's law</a> states that the force required to deform elastic objects should be <a href="/wiki/Proportionality_(mathematics)" title="Proportionality (mathematics)">directly proportional</a> to the distance of deformation, regardless of how large that distance becomes. This is known as <i>perfect elasticity</i>, in which a given object will return to its original shape no matter how strongly it is deformed. This is an <a href="/wiki/Idealization_(science_philosophy)" class="mw-redirect" title="Idealization (science philosophy)">ideal concept</a> only; most materials which possess elasticity in practice remain purely elastic only up to very small deformations, after which plastic (permanent) deformation occurs. </p><p>In <a href="/wiki/Engineering" title="Engineering">engineering</a>, the elasticity of a material is quantified by the <a href="/wiki/Elastic_modulus" title="Elastic modulus">elastic modulus</a> such as the <a href="/wiki/Young%27s_modulus" title="Young's modulus">Young's modulus</a>, <a href="/wiki/Bulk_modulus" title="Bulk modulus">bulk modulus</a> or <a href="/wiki/Shear_modulus" title="Shear modulus">shear modulus</a> which measure the amount of <a href="/wiki/Stress_(mechanics)" title="Stress (mechanics)">stress</a> needed to achieve a unit of <a href="/wiki/Deformation_(engineering)" title="Deformation (engineering)">strain</a>; a higher modulus indicates that the material is harder to deform. The <a href="/wiki/International_System_of_Units" title="International System of Units">SI unit</a> of this modulus is the <a href="/wiki/Pascal_(unit)" title="Pascal (unit)">pascal</a> (Pa). The material's <i>elastic limit</i> or <a href="/wiki/Yield_strength" class="mw-redirect" title="Yield strength">yield strength</a> is the maximum <a href="/wiki/Stress_(mechanics)" title="Stress (mechanics)">stress</a> that can arise before the onset of plastic deformation. Its SI unit is also the pascal (Pa). </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Overview">Overview</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Elasticity_(physics)&action=edit&section=1" title="Edit section: Overview"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>When an elastic material is deformed due to an external force, it experiences internal resistance to the deformation and restores it to its original state if the external force is no longer applied. There are various <a href="/wiki/Elastic_modulus" title="Elastic modulus">elastic moduli</a>, such as <a href="/wiki/Young%27s_modulus" title="Young's modulus">Young's modulus</a>, the <a href="/wiki/Shear_modulus" title="Shear modulus">shear modulus</a>, and the <a href="/wiki/Bulk_modulus" title="Bulk modulus">bulk modulus</a>, all of which are measures of the inherent elastic properties of a material as a resistance to deformation under an applied load. The various moduli apply to different kinds of deformation. For instance, Young's modulus applies to extension/compression of a body, whereas the shear modulus applies to its <a href="/wiki/Shearing_(physics)" title="Shearing (physics)">shear</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> Young's modulus and shear modulus are only for solids, whereas the <a href="/wiki/Bulk_modulus" title="Bulk modulus">bulk modulus</a> is for solids, liquids, and gases. </p><p>The elasticity of materials is described by a <a href="/wiki/Stress%E2%80%93strain_curve" title="Stress–strain curve">stress–strain curve</a>, which shows the relation between <a href="/wiki/Stress_(mechanics)" title="Stress (mechanics)">stress</a> (the average restorative internal <a href="/wiki/Force" title="Force">force</a> per unit area) and <a href="/wiki/Strain_(engineering)" class="mw-redirect" title="Strain (engineering)">strain</a> (the relative deformation).<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> The curve is generally nonlinear, but it can (by use of a <a href="/wiki/Taylor_series" title="Taylor series">Taylor series</a>) be approximated as linear for sufficiently small deformations (in which higher-order terms are negligible). If the material is <a href="/wiki/Isotropic" class="mw-redirect" title="Isotropic">isotropic</a>, the linearized stress–strain relationship is called <a href="/wiki/Hooke%27s_law" title="Hooke's law">Hooke's law</a>, which is often presumed to apply up to the elastic limit for most metals or crystalline materials whereas <a href="/wiki/Nonlinear_elasticity" class="mw-redirect" title="Nonlinear elasticity">nonlinear elasticity</a> is generally required to model large deformations of rubbery materials even in the elastic range. For even higher stresses, materials exhibit <a href="/wiki/Plasticity_(physics)" title="Plasticity (physics)">plastic behavior</a>, that is, they deform irreversibly and do not return to their original shape after stress is no longer applied.<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> For rubber-like materials such as <a href="/wiki/Elastomer" title="Elastomer">elastomers</a>, the slope of the stress–strain curve increases with stress, meaning that rubbers progressively become more difficult to stretch, while for most metals, the gradient decreases at very high stresses, meaning that they progressively become easier to stretch.<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> Elasticity is not exhibited only by solids; <a href="/wiki/Non-Newtonian_fluid" title="Non-Newtonian fluid">non-Newtonian fluids</a>, such as <a href="/wiki/Viscoelasticity" title="Viscoelasticity">viscoelastic fluids</a>, will also exhibit elasticity in certain conditions quantified by the <a href="/wiki/Deborah_number" title="Deborah number">Deborah number</a>. In response to a small, rapidly applied and removed strain, these fluids may deform and then return to their original shape. Under larger strains, or strains applied for longer periods of time, these fluids may start to flow like a <a href="/wiki/Viscosity" title="Viscosity">viscous</a> liquid. </p><p>Because the elasticity of a material is described in terms of a stress–strain relation, it is essential that the terms <i>stress</i> and <i>strain</i> be defined without ambiguity. Typically, two types of relation are considered. The first type deals with materials that are elastic only for small strains. The second deals with materials that are not limited to small strains. Clearly, the second type of relation is more general in the sense that it must include the first type as a special case. </p><p>For small strains, the measure of stress that is used is the <a href="/wiki/Cauchy_stress_tensor" title="Cauchy stress tensor">Cauchy stress</a> while the measure of strain that is used is the <a href="/wiki/Infinitesimal_strain_theory" title="Infinitesimal strain theory">infinitesimal strain tensor</a>; the resulting (predicted) material behavior is termed <a href="/wiki/Linear_elasticity" title="Linear elasticity">linear elasticity</a>, which (for <a href="/wiki/Isotropic" class="mw-redirect" title="Isotropic">isotropic</a> media) is called the generalized <a href="/wiki/Hooke%27s_law" title="Hooke's law">Hooke's law</a>. <a href="/wiki/Cauchy_elastic_material" title="Cauchy elastic material">Cauchy elastic materials</a> and <a href="/wiki/Hypoelastic_material" title="Hypoelastic material">hypoelastic materials</a> are models that extend Hooke's law to allow for the possibility of large rotations, large distortions, and intrinsic or induced <a href="/wiki/Anisotropy" title="Anisotropy">anisotropy</a>. </p><p>For more general situations, any of a number of <a href="/wiki/Stress_measures" class="mw-redirect" title="Stress measures">stress measures</a> can be used, and it is generally desired (but not required) that the elastic stress–strain relation be phrased in terms of a <a href="/wiki/Finite_strain_theory" title="Finite strain theory">finite strain</a> measure that is <a href="/w/index.php?title=Work_conjugate&action=edit&redlink=1" class="new" title="Work conjugate (page does not exist)">work conjugate</a> to the selected stress measure, i.e., the time integral of the inner product of the stress measure with the rate of the strain measure should be equal to the change in internal energy for any <a href="/wiki/Adiabatic_process" title="Adiabatic process">adiabatic process</a> that remains below the elastic limit. </p> <div class="mw-heading mw-heading2"><h2 id="Units">Units</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Elasticity_(physics)&action=edit&section=2" title="Edit section: Units"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="International_System">International System</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Elasticity_(physics)&action=edit&section=3" title="Edit section: International System"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The SI unit for elasticity and the elastic modulus is the <a href="/wiki/Pascal_(unit)" title="Pascal (unit)">pascal</a> (Pa). This unit is defined as force per unit area, generally a measurement of <a href="/wiki/Pressure" title="Pressure">pressure</a>, which in mechanics corresponds to <a href="/wiki/Stress_(mechanics)" title="Stress (mechanics)">stress</a>. The pascal and therefore elasticity have the <a href="/wiki/Dimensional_analysis" title="Dimensional analysis">dimension</a> L<sup>−1</sup>⋅M⋅T<sup>−2</sup>. </p><p>For most commonly used engineering materials, the elastic modulus is on the scale of gigapascals (GPa, 10<sup>9</sup> Pa). </p> <div class="mw-heading mw-heading2"><h2 id="Linear_elasticity">Linear elasticity</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Elasticity_(physics)&action=edit&section=4" title="Edit section: Linear elasticity"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></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/Linear_elasticity" title="Linear elasticity">Linear elasticity</a></div> <p>As noted above, for small deformations, most elastic materials such as <a href="/wiki/Spring_(device)" title="Spring (device)">springs</a> exhibit linear elasticity and can be described by a linear relation between the stress and strain. This relationship is known as <a href="/wiki/Hooke%27s_law" title="Hooke's law">Hooke's law</a>. A geometry-dependent version of the idea<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>a<span class="cite-bracket">]</span></a></sup> was first formulated by <a href="/wiki/Robert_Hooke" title="Robert Hooke">Robert Hooke</a> in 1675 as a Latin <a href="/wiki/Anagram" title="Anagram">anagram</a>, "ceiiinosssttuv". He published the answer in 1678: "<i>Ut tensio, sic vis</i>" meaning "<i>As the extension, so the force</i>",<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> a linear relationship commonly referred to as <a href="/wiki/Hooke%27s_law" title="Hooke's law">Hooke's law</a>. This law can be stated as a relationship between tensile <a href="/wiki/Force" title="Force">force</a> <span class="texhtml"><i>F</i></span> and corresponding extension <a href="/wiki/Displacement_(vector)" class="mw-redirect" title="Displacement (vector)">displacement</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 x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span>, </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 F=kx,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo>=</mo> <mi>k</mi> <mi>x</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F=kx,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f74a73805ff00848046471a073c2de934338616b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.027ex; height:2.509ex;" alt="{\displaystyle F=kx,}"></span></dd></dl> <p>where <span class="texhtml"><i>k</i></span> is a constant known as the <i>rate</i> or <i>spring constant</i>. It can also be stated as a relationship between <a href="/wiki/Tensile_stress" class="mw-redirect" title="Tensile stress">stress</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 \sigma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>σ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59f59b7c3e6fdb1d0365a494b81fb9a696138c36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle \sigma }"></span> and <a href="/wiki/Strain_(materials_science)" class="mw-redirect" title="Strain (materials science)">strain</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 \varepsilon }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ε</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a30c89172e5b88edbd45d3e2772c7f5e562e5173" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.083ex; height:1.676ex;" alt="{\displaystyle \varepsilon }"></span>: </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 \sigma =E\varepsilon ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>σ</mi> <mo>=</mo> <mi>E</mi> <mi>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/f5d1bcaec7375695bb02d441903fab7501b59f98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.934ex; height:2.509ex;" alt="{\displaystyle \sigma =E\varepsilon ,}"></span></dd></dl> <p>where <span class="texhtml"><i>E</i></span> is known as the <a href="/wiki/Young%27s_modulus" title="Young's modulus">Young's modulus</a>.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </p><p>Although the general proportionality constant between stress and strain in three dimensions is a 4th-order <a href="/wiki/Tensor" title="Tensor">tensor</a> called <a href="/wiki/Stiffness" title="Stiffness">stiffness</a>, systems that exhibit <a href="/wiki/Symmetry" title="Symmetry">symmetry</a>, such as a one-dimensional rod, can often be reduced to applications of Hooke's law. </p> <div class="mw-heading mw-heading2"><h2 id="Finite_elasticity">Finite elasticity</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Elasticity_(physics)&action=edit&section=5" title="Edit section: Finite elasticity"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The elastic behavior of objects that undergo finite deformations has been described using a number of models, such as <a href="/wiki/Cauchy_elastic_material" title="Cauchy elastic material">Cauchy elastic material</a> models, <a href="/wiki/Hypoelastic_material" title="Hypoelastic material">Hypoelastic material</a> models, and <a href="/wiki/Hyperelastic_material" title="Hyperelastic material">Hyperelastic material</a> models. The <a href="/wiki/Deformation_gradient" class="mw-redirect" title="Deformation gradient">deformation gradient</a> (<i><b>F</b></i>) is the primary deformation measure used in <a href="/wiki/Finite_strain_theory" title="Finite strain theory">finite strain theory</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Cauchy_elastic_materials">Cauchy elastic materials</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Elasticity_(physics)&action=edit&section=6" title="Edit section: Cauchy elastic materials"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></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/Cauchy_elastic_material" title="Cauchy elastic material">Cauchy elastic material</a></div> <p>A material is said to be Cauchy-elastic if the <a href="/wiki/Cauchy_stress_tensor" title="Cauchy stress tensor">Cauchy stress tensor</a> <i><b>σ</b></i> is a function of the <a href="/wiki/Deformation_gradient" class="mw-redirect" title="Deformation gradient">deformation gradient</a> <i><b>F</b></i> alone: </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 \ {\boldsymbol {\sigma }}={\mathcal {G}}({\boldsymbol {F}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">σ</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">G</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">F</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ {\boldsymbol {\sigma }}={\mathcal {G}}({\boldsymbol {F}})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92f688c7239a089ba5fe489052bde61a2acb9726" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.355ex; height:2.843ex;" alt="{\displaystyle \ {\boldsymbol {\sigma }}={\mathcal {G}}({\boldsymbol {F}})}"></span></dd></dl> <p>It is generally incorrect to state that Cauchy stress is a function of merely a <a href="/wiki/Strain_tensor" class="mw-redirect" title="Strain tensor">strain tensor</a>, as such a model lacks crucial information about material rotation needed to produce correct results for an anisotropic medium subjected to vertical extension in comparison to the same extension applied horizontally and then subjected to a 90-degree rotation; both these deformations have the same spatial strain tensors yet must produce different values of the Cauchy stress tensor. </p><p>Even though the stress in a Cauchy-elastic material depends only on the state of deformation, the work done by stresses might depend on the path of deformation. Therefore, Cauchy elasticity includes non-conservative "non-hyperelastic" models (in which work of deformation is path dependent) as well as conservative "<a href="/wiki/Hyperelastic_material" title="Hyperelastic material">hyperelastic material</a>" models (for which stress can be derived from a scalar "elastic potential" function). </p> <div class="mw-heading mw-heading3"><h3 id="Hypoelastic_materials">Hypoelastic materials</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Elasticity_(physics)&action=edit&section=7" title="Edit section: Hypoelastic materials"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></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/Hypoelastic_material" title="Hypoelastic material">Hypoelastic material</a></div> <p>A hypoelastic material can be rigorously defined as one that is modeled using a <a href="/wiki/Constitutive_equation" title="Constitutive equation">constitutive equation</a> satisfying the following two criteria:<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> </p> <ol><li>The Cauchy stress <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 }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">σ</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {\sigma }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e45fe1b9d8dcbc3103fc7805d69798bfe5ca5b16" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.594ex; height:1.676ex;" alt="{\displaystyle {\boldsymbol {\sigma }}}"></span> at time <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 t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}"></span> depends only on the order in which the body has occupied its past configurations, but not on the time rate at which these past configurations were traversed. As a special case, this criterion includes a <a href="/wiki/Cauchy_elastic_material" title="Cauchy elastic material">Cauchy elastic material</a>, for which the current stress depends only on the current configuration rather than the history of past configurations.</li> <li>There is a tensor-valued function <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> </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/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}"></span> such that <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 {\dot {\boldsymbol {\sigma }}}=G({\boldsymbol {\sigma }},{\boldsymbol {L}})\,,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold-italic">σ</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mo>=</mo> <mi>G</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">σ</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">L</mi> </mrow> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\dot {\boldsymbol {\sigma }}}=G({\boldsymbol {\sigma }},{\boldsymbol {L}})\,,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f3d91b6d2bd793458770043fad3d2612419dd882" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.748ex; height:2.843ex;" alt="{\displaystyle {\dot {\boldsymbol {\sigma }}}=G({\boldsymbol {\sigma }},{\boldsymbol {L}})\,,}"></span> in which <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 {\dot {\boldsymbol {\sigma }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold-italic">σ</mi> <mo>˙</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\dot {\boldsymbol {\sigma }}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c126b5187517f81caf7772ac4d05aad833923146" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.594ex; height:2.176ex;" alt="{\displaystyle {\dot {\boldsymbol {\sigma }}}}"></span> is the material rate of the Cauchy stress tensor, and <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 {L}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">L</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {L}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/514b1d6d678587b0411952c87010c9c8beed0529" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.757ex; height:2.176ex;" alt="{\displaystyle {\boldsymbol {L}}}"></span> is the spatial <a href="/wiki/Velocity_gradient" class="mw-redirect" title="Velocity gradient">velocity gradient</a> tensor.</li></ol> <p>If only these two original criteria are used to define hypoelasticity, then <a href="/wiki/Hyperelasticity" class="mw-redirect" title="Hyperelasticity">hyperelasticity</a> would be included as a special case, which prompts some constitutive modelers to append a third criterion that specifically requires a hypoelastic model to <i>not</i> be hyperelastic (i.e., hypoelasticity implies that stress is not derivable from an energy potential). If this third criterion is adopted, it follows that a hypoelastic material might admit nonconservative adiabatic loading paths that start and end with the same <a href="/wiki/Deformation_gradient" class="mw-redirect" title="Deformation gradient">deformation gradient</a> but do <i>not</i> start and end at the same internal energy. </p><p>Note that the second criterion requires only that the function <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> </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/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}"></span> <i>exists</i>. As detailed in the main <a href="/wiki/Hypoelastic_material" title="Hypoelastic material">hypoelastic material</a> article, specific formulations of hypoelastic models typically employ so-called objective rates so that the <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> </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/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}"></span> function exists only implicitly and is typically needed explicitly only for numerical stress updates performed via direct integration of the actual (not objective) stress rate. </p> <div class="mw-heading mw-heading3"><h3 id="Hyperelastic_materials">Hyperelastic materials</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Elasticity_(physics)&action=edit&section=8" title="Edit section: Hyperelastic materials"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></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/Hyperelastic_material" title="Hyperelastic material">Hyperelastic material</a></div> <p>Hyperelastic materials (also called Green elastic materials) are conservative models that are derived from a <a href="/wiki/Strain_energy_density_function" title="Strain energy density function">strain energy density function</a> (<i>W</i>). A model is hyperelastic if and only if it is possible to express the <a href="/wiki/Cauchy_stress_tensor" title="Cauchy stress tensor">Cauchy stress tensor</a> as a function of the <a href="/wiki/Deformation_gradient" class="mw-redirect" title="Deformation gradient">deformation gradient</a> via a relationship of the form </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 {\boldsymbol {\sigma }}={\cfrac {1}{J}}~{\cfrac {\partial W}{\partial {\boldsymbol {F}}}}{\boldsymbol {F}}^{\textsf {T}}\quad {\text{where}}\quad J:=\det {\boldsymbol {F}}\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">σ</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi>J</mi> </mrow> </mstyle> </mrow> </mfrac> </mrow> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∂</mi> <mi>W</mi> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">F</mi> </mrow> </mrow> </mstyle> </mrow> </mfrac> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">F</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext mathvariant="sans-serif">T</mtext> </mrow> </mrow> </msup> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>where</mtext> </mrow> <mspace width="1em" /> <mi>J</mi> <mo>:=</mo> <mo movablelimits="true" form="prefix">det</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">F</mi> </mrow> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {\sigma }}={\cfrac {1}{J}}~{\cfrac {\partial W}{\partial {\boldsymbol {F}}}}{\boldsymbol {F}}^{\textsf {T}}\quad {\text{where}}\quad J:=\det {\boldsymbol {F}}\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e11fba319131e9401e13da13faff9ef49f94e65" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:37.825ex; height:7.176ex;" alt="{\displaystyle {\boldsymbol {\sigma }}={\cfrac {1}{J}}~{\cfrac {\partial W}{\partial {\boldsymbol {F}}}}{\boldsymbol {F}}^{\textsf {T}}\quad {\text{where}}\quad J:=\det {\boldsymbol {F}}\,.}"></span></dd></dl> <p>This formulation takes the energy potential (<i>W</i>) as a function of the <a href="/wiki/Deformation_gradient" class="mw-redirect" title="Deformation gradient">deformation gradient</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 {\boldsymbol {F}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">F</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {F}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a728145135614a4f2c53b9332be4ea077ed10156" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.88ex; height:2.176ex;" alt="{\displaystyle {\boldsymbol {F}}}"></span>). By also requiring satisfaction of <a href="/wiki/Material_objectivity" class="mw-redirect" title="Material objectivity">material objectivity</a>, the energy potential may be alternatively regarded as a function of the <a href="/wiki/Cauchy-Green_deformation_tensor" class="mw-redirect" title="Cauchy-Green deformation tensor">Cauchy-Green deformation tensor</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 {\boldsymbol {C}}:={\boldsymbol {F}}^{\textsf {T}}{\boldsymbol {F}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">C</mi> </mrow> <mo>:=</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">F</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext mathvariant="sans-serif">T</mtext> </mrow> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">F</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {C}}:={\boldsymbol {F}}^{\textsf {T}}{\boldsymbol {F}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7c0ff7af91d5063c4e2a646765fc29cb3cabd1fc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.928ex; height:2.676ex;" alt="{\displaystyle {\boldsymbol {C}}:={\boldsymbol {F}}^{\textsf {T}}{\boldsymbol {F}}}"></span>), in which case the hyperelastic model may be written alternatively 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 {\boldsymbol {\sigma }}={\cfrac {2}{J}}~{\boldsymbol {F}}{\cfrac {\partial W}{\partial {\boldsymbol {C}}}}{\boldsymbol {F}}^{\textsf {T}}\quad {\text{where}}\quad J:=\det {\boldsymbol {F}}\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">σ</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi>J</mi> </mrow> </mstyle> </mrow> </mfrac> </mrow> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">F</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∂</mi> <mi>W</mi> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">C</mi> </mrow> </mrow> </mstyle> </mrow> </mfrac> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">F</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext mathvariant="sans-serif">T</mtext> </mrow> </mrow> </msup> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>where</mtext> </mrow> <mspace width="1em" /> <mi>J</mi> <mo>:=</mo> <mo movablelimits="true" form="prefix">det</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">F</mi> </mrow> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {\sigma }}={\cfrac {2}{J}}~{\boldsymbol {F}}{\cfrac {\partial W}{\partial {\boldsymbol {C}}}}{\boldsymbol {F}}^{\textsf {T}}\quad {\text{where}}\quad J:=\det {\boldsymbol {F}}\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3db3109a93ecdf36ed0a1b4aa2bf502a8dcd2e76" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:39.706ex; height:7.176ex;" alt="{\displaystyle {\boldsymbol {\sigma }}={\cfrac {2}{J}}~{\boldsymbol {F}}{\cfrac {\partial W}{\partial {\boldsymbol {C}}}}{\boldsymbol {F}}^{\textsf {T}}\quad {\text{where}}\quad J:=\det {\boldsymbol {F}}\,.}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Applications">Applications</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Elasticity_(physics)&action=edit&section=9" title="Edit section: Applications"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Linear elasticity is used widely in the design and analysis of structures such as <a href="/wiki/Beam_bending" class="mw-redirect" title="Beam bending">beams</a>, <a href="/wiki/Plate_theory" title="Plate theory">plates and shells</a>, and <a href="/wiki/Sandwich_theory" title="Sandwich theory">sandwich composites</a>. This theory is also the basis of much of <a href="/wiki/Fracture_mechanics" title="Fracture mechanics">fracture mechanics</a>. </p><p>Hyperelasticity is primarily used to determine the response of <a href="/wiki/Elastomer" title="Elastomer">elastomer</a>-based objects such as <a href="/wiki/Gasket" title="Gasket">gaskets</a> and of biological materials such as <a href="/wiki/Soft_tissue" title="Soft tissue">soft tissues</a> and <a href="/wiki/Elasticity_of_cell_membranes" title="Elasticity of cell membranes">cell membranes</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Factors_affecting_elasticity">Factors affecting elasticity</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Elasticity_(physics)&action=edit&section=10" title="Edit section: Factors affecting elasticity"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In a given <a href="/wiki/Isotropic_material" class="mw-redirect" title="Isotropic material"> isotropic solid</a>, with known theoretical elasticity for the bulk material in terms of Young's modulus,the effective elasticity will be governed by <a href="/wiki/Porosity" title="Porosity">porosity</a>. Generally a more porous material will exhibit lower stiffness. More specifically, the fraction of pores, their distribution at different sizes and the nature of the fluid with which they are filled give rise to different elastic behaviours in solids.<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> </p><p>For <a href="/wiki/Isotropic_material" class="mw-redirect" title="Isotropic material">isotropic materials</a> containing cracks, the presence of fractures affects the Young and the shear moduli perpendicular to the planes of the cracks, which decrease (Young's modulus faster than the shear modulus) as the fracture <a href="/wiki/Number_density" title="Number density">density</a> increases,<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> indicating that the presence of cracks makes bodies brittler. <a href="/wiki/Microscopic" class="mw-redirect" title="Microscopic">Microscopically</a>, the stress–strain relationship of materials is in general governed by the <a href="/wiki/Helmholtz_free_energy" title="Helmholtz free energy">Helmholtz free energy</a>, a <a href="/wiki/Functions_of_state" class="mw-redirect" title="Functions of state">thermodynamic quantity</a>. <a href="/wiki/Molecule" title="Molecule">Molecules</a> settle in the configuration which minimizes the free energy, subject to constraints derived from their structure, and, depending on whether the energy or the <a href="/wiki/Entropy" title="Entropy">entropy</a> term dominates the free energy, materials can broadly be classified as <i>energy-elastic</i> and <i>entropy-elastic</i>. As such, microscopic factors affecting the free energy, such as the <a href="/wiki/Mechanical_equilibrium" title="Mechanical equilibrium">equilibrium</a> distance between molecules, can affect the elasticity of materials: for instance, in <a href="/wiki/Inorganic" class="mw-redirect" title="Inorganic">inorganic</a> materials, as the equilibrium distance between molecules at <a href="/wiki/Absolute_zero" title="Absolute zero">0 K</a> increases, the <a href="/wiki/Bulk_modulus" title="Bulk modulus">bulk modulus</a> decreases.<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> The effect of temperature on elasticity is difficult to isolate, because there are numerous factors affecting it. For instance, the bulk modulus of a material is dependent on the form of its <a href="/wiki/Crystal_structure" title="Crystal structure">lattice</a>, its behavior under <a href="/wiki/Thermal_expansion" title="Thermal expansion">expansion</a>, as well as the <a href="/wiki/Vibrations" class="mw-redirect" title="Vibrations">vibrations</a> of the molecules, all of which are dependent on temperature.<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Elasticity_(physics)&action=edit&section=11" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1184024115">.mw-parser-output .div-col{margin-top:0.3em;column-width:30em}.mw-parser-output .div-col-small{font-size:90%}.mw-parser-output .div-col-rules{column-rule:1px solid #aaa}.mw-parser-output .div-col dl,.mw-parser-output .div-col ol,.mw-parser-output .div-col ul{margin-top:0}.mw-parser-output .div-col li,.mw-parser-output .div-col dd{page-break-inside:avoid;break-inside:avoid-column}</style><div class="div-col" style="column-width: 20em;"> <ul><li><a href="/wiki/Elasticity_tensor" title="Elasticity tensor">Elasticity tensor</a></li> <li><a href="/wiki/Elastography" title="Elastography">Elastography</a></li> <li><a href="/wiki/Tactile_imaging" class="mw-redirect" title="Tactile imaging">Tactile imaging</a></li> <li><a href="/wiki/Elastic_modulus" title="Elastic modulus">Elastic modulus</a></li> <li><a href="/wiki/Linear_elasticity" title="Linear elasticity">Linear elasticity</a></li> <li><a href="/wiki/Pseudoelasticity" title="Pseudoelasticity">Pseudoelasticity</a></li> <li><a href="/wiki/Resilience_(materials_science)" title="Resilience (materials science)">Resilience</a></li> <li><a href="/wiki/Rubber_elasticity" title="Rubber elasticity">Rubber elasticity</a></li> <li><a href="/wiki/Stiffness" title="Stiffness">Stiffness</a></li> <li><a href="/wiki/Ductility" title="Ductility">Ductility</a></li></ul></div> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Elasticity_(physics)&action=edit&section=12" title="Edit section: Notes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></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-lower-alpha"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text">Descriptions of material behavior should be independent of the geometry and shape of the object made of the material under consideration. The original version of Hooke's law involves a stiffness constant that depends on the initial size and shape of the object. The stiffness constant is therefore not strictly a material property.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (October 2023)">citation needed</span></a></i>]</sup></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Elasticity_(physics)&action=edit&section=13" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239543626"><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><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">Landau LD, Lipshitz EM. Theory of Elasticity, 3rd Edition, 1970: 1–172.</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"><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="CITEREFTreloar1975" class="citation book cs1">Treloar, L. R. G. (1975). <span class="id-lock-limited" title="Free access subject to limited trial, subscription normally required"><a rel="nofollow" class="external text" href="https://archive.org/details/physicsrubberela00trel_285"><i>The Physics of Rubber Elasticity</i></a></span>. Oxford: Clarendon Press. p. <a rel="nofollow" class="external text" href="https://archive.org/details/physicsrubberela00trel_285/page/n14">2</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-1985-1355-1" title="Special:BookSources/978-0-1985-1355-1"><bdi>978-0-1985-1355-1</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+Physics+of+Rubber+Elasticity&rft.place=Oxford&rft.pages=2&rft.pub=Clarendon+Press&rft.date=1975&rft.isbn=978-0-1985-1355-1&rft.aulast=Treloar&rft.aufirst=L.+R.+G.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fphysicsrubberela00trel_285&rfr_id=info%3Asid%2Fen.wikipedia.org%3AElasticity+%28physics%29" class="Z3988"></span></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"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSadd2005" class="citation book cs1">Sadd, Martin H. (2005). <span class="id-lock-limited" title="Free access subject to limited trial, subscription normally required"><a rel="nofollow" class="external text" href="https://archive.org/details/elasticitytheory00sadd"><i>Elasticity: Theory, Applications, and Numerics</i></a></span>. Oxford: Elsevier. p. <a rel="nofollow" class="external text" href="https://archive.org/details/elasticitytheory00sadd/page/n74">70</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-1237-4446-3" title="Special:BookSources/978-0-1237-4446-3"><bdi>978-0-1237-4446-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Elasticity%3A+Theory%2C+Applications%2C+and+Numerics&rft.place=Oxford&rft.pages=70&rft.pub=Elsevier&rft.date=2005&rft.isbn=978-0-1237-4446-3&rft.aulast=Sadd&rft.aufirst=Martin+H.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Felasticitytheory00sadd&rfr_id=info%3Asid%2Fen.wikipedia.org%3AElasticity+%28physics%29" class="Z3988"></span></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="CITEREFde_With2006" class="citation book cs1">de With, Gijsbertus (2006). <i>Structure, Deformation, and Integrity of Materials, Volume I: Fundamentals and Elasticity</i>. Weinheim: Wiley VCH. p. 32. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-527-31426-3" title="Special:BookSources/978-3-527-31426-3"><bdi>978-3-527-31426-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Structure%2C+Deformation%2C+and+Integrity+of+Materials%2C+Volume+I%3A+Fundamentals+and+Elasticity&rft.place=Weinheim&rft.pages=32&rft.pub=Wiley+VCH&rft.date=2006&rft.isbn=978-3-527-31426-3&rft.aulast=de+With&rft.aufirst=Gijsbertus&rfr_id=info%3Asid%2Fen.wikipedia.org%3AElasticity+%28physics%29" 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="CITEREFAtanackovicGuran2000" class="citation book cs1">Atanackovic, Teodor M.; Guran, Ardéshir (2000). "Hooke's law". <span class="id-lock-limited" title="Free access subject to limited trial, subscription normally required"><a rel="nofollow" class="external text" href="https://archive.org/details/theoryelasticity00atan"><i>Theory of elasticity for scientists and engineers</i></a></span>. Boston, Mass.: Birkhäuser. p. <a rel="nofollow" class="external text" href="https://archive.org/details/theoryelasticity00atan/page/n94">85</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8176-4072-9" title="Special:BookSources/978-0-8176-4072-9"><bdi>978-0-8176-4072-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Hooke%27s+law&rft.btitle=Theory+of+elasticity+for+scientists+and+engineers&rft.place=Boston%2C+Mass.&rft.pages=85&rft.pub=Birkh%C3%A4user&rft.date=2000&rft.isbn=978-0-8176-4072-9&rft.aulast=Atanackovic&rft.aufirst=Teodor+M.&rft.au=Guran%2C+Ard%C3%A9shir&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Ftheoryelasticity00atan&rfr_id=info%3Asid%2Fen.wikipedia.org%3AElasticity+%28physics%29" 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 class="citation web cs1"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20101113210736/http://www.lindahall.org/events_exhib/exhibit/exhibits/civil/design.shtml">"Strength and Design"</a>. <i>Centuries of Civil Engineering: A Rare Book Exhibition Celebrating the Heritage of Civil Engineering</i>. Linda Hall Library of Science, Engineering & Technology. Archived from <a rel="nofollow" class="external text" href="http://www.lindahall.org/events_exhib/exhibit/exhibits/civil/design.shtml">the original</a> on 13 November 2010.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Centuries+of+Civil+Engineering%3A+A+Rare+Book+Exhibition+Celebrating+the+Heritage+of+Civil+Engineering&rft.atitle=Strength+and+Design&rft_id=http%3A%2F%2Fwww.lindahall.org%2Fevents_exhib%2Fexhibit%2Fexhibits%2Fcivil%2Fdesign.shtml&rfr_id=info%3Asid%2Fen.wikipedia.org%3AElasticity+%28physics%29" class="Z3988"></span><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 2012)">page needed</span></a></i>]</sup></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="CITEREFIbrahimbegovic2009" class="citation book cs1">Ibrahimbegovic, Adnan (2 June 2009). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=z8oI0hCMt40C&dq=%22hooke%27&pg=PA20"><i>Nonlinear Solid Mechanics: Theoretical Formulations and Finite Element Solution Methods</i></a>. Springer Science & Business Media. pp. 20–26. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-90-481-2330-8" title="Special:BookSources/978-90-481-2330-8"><bdi>978-90-481-2330-8</bdi></a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20240528030534/https://books.google.com/books?id=z8oI0hCMt40C&dq=%22hooke%27&pg=PA20#v=onepage&q=%22hooke'&f=false">Archived</a> from the original on 28 May 2024<span class="reference-accessdate">. Retrieved <span class="nowrap">9 July</span> 2023</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Nonlinear+Solid+Mechanics%3A+Theoretical+Formulations+and+Finite+Element+Solution+Methods&rft.pages=20-26&rft.pub=Springer+Science+%26+Business+Media&rft.date=2009-06-02&rft.isbn=978-90-481-2330-8&rft.aulast=Ibrahimbegovic&rft.aufirst=Adnan&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3Dz8oI0hCMt40C%26dq%3D%2522hooke%2527%26pg%3DPA20&rfr_id=info%3Asid%2Fen.wikipedia.org%3AElasticity+%28physics%29" 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="CITEREFTruesdellNoll2004" class="citation book cs1">Truesdell, Clifford; Noll, Walter (2004). <i>The Non-linear Field Theories of Mechanics</i> (3rd ed.). Berlin Heidelberg New York: Springer-Verlag. p. 401. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-540-02779-9" title="Special:BookSources/978-3-540-02779-9"><bdi>978-3-540-02779-9</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+Non-linear+Field+Theories+of+Mechanics&rft.place=Berlin+Heidelberg+New+York&rft.pages=401&rft.edition=3rd&rft.pub=Springer-Verlag&rft.date=2004&rft.isbn=978-3-540-02779-9&rft.aulast=Truesdell&rft.aufirst=Clifford&rft.au=Noll%2C+Walter&rfr_id=info%3Asid%2Fen.wikipedia.org%3AElasticity+%28physics%29" 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="CITEREFLiuWuGanHanaor2019" class="citation journal cs1">Liu, Mingchao; Wu, Jian; Gan, Yixiang; Hanaor, Dorian AH; Chen, C.Q. (1 May 2019). <a rel="nofollow" class="external text" href="https://unsworks.unsw.edu.au/bitstreams/1dd2eacb-33d5-41d6-9ff2-401e255cee3a/download">"Multiscale modeling of the effective elastic properties of fluid-filled porous materials"</a>. <i>International Journal of Solids and Structures</i>. <b>162</b>: 36–44. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.ijsolstr.2018.11.028">10.1016/j.ijsolstr.2018.11.028</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=International+Journal+of+Solids+and+Structures&rft.atitle=Multiscale+modeling+of+the+effective+elastic+properties+of+fluid-filled+porous+materials&rft.volume=162&rft.pages=36-44&rft.date=2019-05-01&rft_id=info%3Adoi%2F10.1016%2Fj.ijsolstr.2018.11.028&rft.aulast=Liu&rft.aufirst=Mingchao&rft.au=Wu%2C+Jian&rft.au=Gan%2C+Yixiang&rft.au=Hanaor%2C+Dorian+AH&rft.au=Chen%2C+C.Q.&rft_id=https%3A%2F%2Funsworks.unsw.edu.au%2Fbitstreams%2F1dd2eacb-33d5-41d6-9ff2-401e255cee3a%2Fdownload&rfr_id=info%3Asid%2Fen.wikipedia.org%3AElasticity+%28physics%29" class="Z3988"></span></span> </li> <li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSadd2005" class="citation book cs1">Sadd, Martin H. (2005). <span class="id-lock-limited" title="Free access subject to limited trial, subscription normally required"><a rel="nofollow" class="external text" href="https://archive.org/details/elasticitytheory00sadd"><i>Elasticity: Theory, Applications, and Numerics</i></a></span>. Oxford: Elsevier. p. <a rel="nofollow" class="external text" href="https://archive.org/details/elasticitytheory00sadd/page/n386">387</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-1237-4446-3" title="Special:BookSources/978-0-1237-4446-3"><bdi>978-0-1237-4446-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Elasticity%3A+Theory%2C+Applications%2C+and+Numerics&rft.place=Oxford&rft.pages=387&rft.pub=Elsevier&rft.date=2005&rft.isbn=978-0-1237-4446-3&rft.aulast=Sadd&rft.aufirst=Martin+H.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Felasticitytheory00sadd&rfr_id=info%3Asid%2Fen.wikipedia.org%3AElasticity+%28physics%29" class="Z3988"></span></span> </li> <li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSadd2005" class="citation book cs1">Sadd, Martin H. (2005). <span class="id-lock-limited" title="Free access subject to limited trial, subscription normally required"><a rel="nofollow" class="external text" href="https://archive.org/details/elasticitytheory00sadd"><i>Elasticity: Theory, Applications, and Numerics</i></a></span>. Oxford: Elsevier. p. <a rel="nofollow" class="external text" href="https://archive.org/details/elasticitytheory00sadd/page/n344">344</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-1237-4446-3" title="Special:BookSources/978-0-1237-4446-3"><bdi>978-0-1237-4446-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Elasticity%3A+Theory%2C+Applications%2C+and+Numerics&rft.place=Oxford&rft.pages=344&rft.pub=Elsevier&rft.date=2005&rft.isbn=978-0-1237-4446-3&rft.aulast=Sadd&rft.aufirst=Martin+H.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Felasticitytheory00sadd&rfr_id=info%3Asid%2Fen.wikipedia.org%3AElasticity+%28physics%29" class="Z3988"></span></span> </li> <li id="cite_note-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-13">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSadd2005" class="citation book cs1">Sadd, Martin H. (2005). <span class="id-lock-limited" title="Free access subject to limited trial, subscription normally required"><a rel="nofollow" class="external text" href="https://archive.org/details/elasticitytheory00sadd"><i>Elasticity: Theory, Applications, and Numerics</i></a></span>. Oxford: Elsevier. p. <a rel="nofollow" class="external text" href="https://archive.org/details/elasticitytheory00sadd/page/n364">365</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-1237-4446-3" title="Special:BookSources/978-0-1237-4446-3"><bdi>978-0-1237-4446-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Elasticity%3A+Theory%2C+Applications%2C+and+Numerics&rft.place=Oxford&rft.pages=365&rft.pub=Elsevier&rft.date=2005&rft.isbn=978-0-1237-4446-3&rft.aulast=Sadd&rft.aufirst=Martin+H.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Felasticitytheory00sadd&rfr_id=info%3Asid%2Fen.wikipedia.org%3AElasticity+%28physics%29" class="Z3988"></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Elasticity_(physics)&action=edit&section=14" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a rel="nofollow" class="external text" href="https://feynmanlectures.caltech.edu/II_38.html">The Feynman Lectures on Physics Vol. II Ch. 38: Elasticity</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 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navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Solid_mechanics" title="Solid mechanics">Solid mechanics</a></li> <li><a href="/wiki/Fluid_mechanics" title="Fluid mechanics">Fluid mechanics</a></li> <li><a href="/wiki/Acoustics" title="Acoustics">Acoustics</a></li> <li><a href="/wiki/Vibrations" class="mw-redirect" title="Vibrations">Vibrations</a></li> <li><a href="/wiki/Rigid_body_dynamics" title="Rigid body dynamics">Rigid body dynamics</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Laws and Definitions</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <dl><dt>Laws</dt></dl> <ul><li><a href="/wiki/Conservation_of_mass" title="Conservation of mass">Conservation of mass</a></li> <li><a href="/wiki/Momentum" title="Momentum">Conservation of momentum</a> <ul><li><a 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measures</a></li></ul></li> <li><a href="/wiki/Deformation_(mechanics)" class="mw-redirect" title="Deformation (mechanics)">Deformation</a></li> <li><a href="/wiki/Infinitesimal_strain_theory" title="Infinitesimal strain theory">Small strain</a> <ul><li><a href="/wiki/Antiplane_shear" title="Antiplane shear">Antiplane shear</a></li></ul></li> <li><a href="/wiki/Finite_strain_theory" title="Finite strain theory">Large strain</a></li> <li><a href="/wiki/Compatibility_(mechanics)" title="Compatibility (mechanics)">Compatibility</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Solid_mechanics" title="Solid mechanics">Solid</a> and <a href="/wiki/Structural_mechanics" title="Structural mechanics">Structural mechanics</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <dl><dt><a href="/wiki/Solid" title="Solid">Solids</a></dt></dl> <ul><li><a class="mw-selflink selflink">Elasticity</a> <ul><li><a href="/wiki/Linear_elasticity" title="Linear elasticity">linear</a></li> <li><a href="/wiki/Hooke%27s_law" title="Hooke's law">Hooke's law</a></li> <li><a href="/wiki/Transverse_isotropy" title="Transverse isotropy">Transverse isotropy</a></li> <li><a href="/wiki/Orthotropic_material" title="Orthotropic material">Orthotropy</a></li> <li><a href="/wiki/Hyperelastic_material" title="Hyperelastic material">hyperelasticity</a> <ul><li><a href="/wiki/Elasticity_of_cell_membranes" title="Elasticity of cell membranes">Membrane elasticity</a></li></ul></li> <li><a href="/wiki/Equation_of_state" title="Equation of state">Equation of state</a> <ul><li><a href="/wiki/Hugoniot_elastic_limit" class="mw-redirect" title="Hugoniot elastic limit">Hugoniot</a></li> <li><a href="/wiki/Equation_of_state#Jones–Wilkins–Lee_equation_of_state_for_explosives_(JWL_equation)" title="Equation of state">JWL</a></li></ul></li> <li><a href="/wiki/Hypoelastic_material" title="Hypoelastic material">hypoelasticity</a></li> <li><a href="/wiki/Cauchy_elastic_material" title="Cauchy elastic material">Cauchy elasticity</a></li> <li><a href="/wiki/Viscoelasticity" title="Viscoelasticity">Viscoelasticity</a> <ul><li><a href="/wiki/Creep_(deformation)" title="Creep (deformation)">Creep</a></li> <li><a href="/wiki/Creep_and_shrinkage_of_concrete" title="Creep and shrinkage of concrete">Concrete creep</a></li></ul></li></ul></li> <li><a href="/wiki/Plasticity_(physics)" title="Plasticity (physics)">Plasticity</a> <ul><li><a href="/wiki/Rock_mass_plasticity" title="Rock mass plasticity">Rock mass plasticity</a></li> <li><a href="/wiki/Viscoplasticity" title="Viscoplasticity">Viscoplasticity</a></li> <li><a href="/wiki/Yield_surface" title="Yield surface">Yield criterion</a> <ul><li><a href="/wiki/Bresler%E2%80%93Pister_yield_criterion" title="Bresler–Pister yield criterion">Bresler-Pister</a></li></ul></li></ul></li> <li><a href="/wiki/Contact_mechanics" title="Contact mechanics">Contact mechanics</a> <ul><li><a href="/wiki/Contact_mechanics" title="Contact mechanics">Frictionless</a></li> <li><a href="/wiki/Frictional_contact_mechanics" title="Frictional contact mechanics">Frictional</a></li></ul></li></ul> <dl><dt><a href="/wiki/Failure" title="Failure">Material failure theory</a></dt></dl> <ul><li><a href="/wiki/Drucker_stability" title="Drucker stability">Drucker stability</a></li> <li><a href="/wiki/Material_failure_theory" title="Material failure theory">Material failure theory</a></li> <li><a href="/wiki/Fatigue_(material)" title="Fatigue (material)">Fatigue</a></li> <li><a href="/wiki/Fracture_mechanics" title="Fracture mechanics">Fracture mechanics</a> <ul><li><a href="/wiki/J-integral" title="J-integral">J-integral</a></li> <li><a href="/wiki/Compact_tension_specimen" title="Compact tension specimen">Compact tension specimen</a></li></ul></li> <li><a href="/wiki/Damage_mechanics" title="Damage mechanics">Damage mechanics</a> <ul><li><a href="/wiki/Johnson-Holmquist_damage_model" class="mw-redirect" title="Johnson-Holmquist damage model">Johnson-Holmquist</a></li></ul></li></ul> <dl><dt><a href="/wiki/Structural_mechanics" title="Structural mechanics">Structures</a></dt></dl> <ul><li><a href="/wiki/Bending" title="Bending">Bending</a> <ul><li><a href="/wiki/Bending_moment" title="Bending moment">Bending moment</a></li> <li><a href="/wiki/Bending_of_plates" title="Bending of plates">Bending of plates</a></li> <li><a href="/wiki/Sandwich_theory" title="Sandwich theory">Sandwich theory</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Fluid_mechanics" title="Fluid mechanics">Fluid mechanics</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <dl><dt><a href="/wiki/Fluid" title="Fluid">Fluids</a></dt></dl> <ul><li><a href="/wiki/Fluid_statics" class="mw-redirect" title="Fluid statics">Fluid statics</a></li> <li><a href="/wiki/Fluid_dynamics" title="Fluid dynamics">Fluid dynamics</a></li> <li><a href="/wiki/Navier%E2%80%93Stokes_equations" title="Navier–Stokes equations">Navier–Stokes equations</a></li> <li><a href="/wiki/Bernoulli%27s_principle" title="Bernoulli's principle">Bernoulli's principle</a></li> <li><a href="/wiki/Poiseuille_equation" class="mw-redirect" title="Poiseuille equation">Poiseuille equation</a></li> <li><a href="/wiki/Buoyancy" title="Buoyancy">Buoyancy</a></li> <li><a href="/wiki/Viscosity" title="Viscosity">Viscosity</a> <ul><li><a href="/wiki/Newtonian_fluid" title="Newtonian fluid">Newtonian</a></li> <li><a href="/wiki/Non-Newtonian_fluid" title="Non-Newtonian fluid">Non-Newtonian</a></li></ul></li> <li><a href="/wiki/Archimedes%27_principle" title="Archimedes' principle">Archimedes' principle</a></li> <li><a href="/wiki/Pascal%27s_law" title="Pascal's law">Pascal's law</a></li> <li><a href="/wiki/Pressure" title="Pressure">Pressure</a></li> <li><a href="/wiki/Liquid" title="Liquid">Liquids</a></li> <li><a href="/wiki/Surface_tension" title="Surface tension">Surface tension</a></li> <li><a href="/wiki/Capillary_action" title="Capillary action">Capillary action</a></li></ul> <dl><dt><a href="/wiki/Gas" title="Gas">Gases</a></dt></dl> <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/Gay-Lussac%27s_law" title="Gay-Lussac's law">Gay-Lussac's law</a></li> <li><a href="/wiki/Combined_gas_law" class="mw-redirect" title="Combined gas law">Combined gas law</a></li></ul> <dl><dt><a href="/wiki/Plasma_(physics)" title="Plasma (physics)">Plasma</a></dt></dl> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Acoustics" title="Acoustics">Acoustics</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Acoustic_theory" title="Acoustic theory">Acoustic theory</a></li> <li><a href="/wiki/Aeroacoustics" title="Aeroacoustics">Aeroacoustics</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Rheology" title="Rheology">Rheology</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Viscoelasticity" title="Viscoelasticity">Viscoelasticity</a></li> <li><a href="/wiki/Smart_fluid" title="Smart fluid">Smart fluids</a> <ul><li><a href="/wiki/Magnetorheological_fluid" title="Magnetorheological fluid">Magnetorheological</a></li> <li><a href="/wiki/Electrorheological_fluid" title="Electrorheological fluid">Electrorheological</a></li> <li><a href="/wiki/Ferrofluid" title="Ferrofluid">Ferrofluids</a></li></ul></li> <li><a href="/wiki/Rheometry" title="Rheometry">Rheometry</a></li> <li><a href="/wiki/Rheometer" title="Rheometer">Rheometer</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Scientist" title="Scientist">Scientists</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <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/Joseph_Louis_Gay-Lussac" title="Joseph Louis Gay-Lussac">Gay-Lussac</a></li> <li><a href="/wiki/Robert_Hooke" title="Robert 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