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vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>See also</span> </div> </a> <ul id="toc-See_also-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">6</span> <span>References</span> </div> </a> <button aria-controls="toc-References-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 References subsection</span> </button> <ul id="toc-References-sublist" class="vector-toc-list"> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.1</span> <span>Notes</span> </div> </a> <ul id="toc-Notes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Bibliography" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Bibliography"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.2</span> <span>Bibliography</span> </div> </a> <ul id="toc-Bibliography-sublist" class="vector-toc-list"> </ul> </li> </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 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Available in 38 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-38" 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">38 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%D9%8A%D9%83%D8%A7%D9%86%D9%8A%D9%83%D8%A7_%D8%A7%D9%84%D9%85%D9%88%D8%A7%D8%AF_%D8%A7%D9%84%D8%B5%D9%84%D8%A8%D8%A9" title="ميكانيكا المواد الصلبة – Arabic" lang="ar" hreflang="ar" data-title="ميكانيكا المواد الصلبة" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Mec%C3%A1nica_de_s%C3%B3lidos_deformables" title="Mecánica de sólidos deformables – Asturian" lang="ast" hreflang="ast" data-title="Mecánica de sólidos deformables" data-language-autonym="Asturianu" data-language-local-name="Asturian" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%95%E0%A6%A0%E0%A6%BF%E0%A6%A8_%E0%A6%AC%E0%A6%B2%E0%A6%AC%E0%A6%BF%E0%A6%9C%E0%A7%8D%E0%A6%9E%E0%A6%BE%E0%A6%A8" title="কঠিন বলবিজ্ঞান – Bangla" lang="bn" hreflang="bn" data-title="কঠিন বলবিজ্ঞান" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%9C%D0%B5%D1%85%D0%B0%D0%BD%D1%96%D0%BA%D0%B0_%D1%86%D0%B2%D1%91%D1%80%D0%B4%D0%B0%D0%B3%D0%B0_%D0%B4%D1%8D%D1%84%D0%B0%D1%80%D0%BC%D0%B0%D0%B2%D0%B0%D0%BD%D0%B0%D0%B3%D0%B0_%D1%86%D0%B5%D0%BB%D0%B0" title="Механіка цвёрдага дэфармаванага цела – Belarusian" lang="be" hreflang="be" data-title="Механіка цвёрдага дэфармаванага цела" data-language-autonym="Беларуская" data-language-local-name="Belarusian" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Mehanika_%C4%8Dvrstih_tijela" title="Mehanika čvrstih tijela – Bosnian" lang="bs" hreflang="bs" data-title="Mehanika čvrstih tijela" 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/Mec%C3%A0nica_del_s%C3%B2lid_deformable" title="Mecànica del sòlid deformable – Catalan" lang="ca" hreflang="ca" data-title="Mecànica del sòlid deformable" 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%94%D0%B5%D1%84%D0%BE%D1%80%D0%BC%D0%B0%D1%86%D0%B8%D0%BB%D0%B5%D0%BD%D0%B5%D0%BA%D0%B5%D0%BD_%D1%85%D1%8B%D1%82%C4%83_%C4%95%D1%81%D0%BA%D0%B5%D1%80_%D0%BC%D0%B5%D1%85%D0%B0%D0%BD%D0%B8%D0%BA%D0%B8" title="Деформациленекен хытă ĕскер механики – Chuvash" lang="cv" hreflang="cv" data-title="Деформациленекен хытă ĕскер механики" 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/Mechanika_tuh%C3%A9ho_t%C4%9Blesa" title="Mechanika tuhého tělesa – Czech" lang="cs" hreflang="cs" data-title="Mechanika tuhého tělesa" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Mechanik_fester_K%C3%B6rper" title="Mechanik fester Körper – German" lang="de" hreflang="de" data-title="Mechanik fester Körper" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Mec%C3%A1nica_de_s%C3%B3lidos_deformables" title="Mecánica de sólidos deformables – Spanish" lang="es" hreflang="es" data-title="Mecánica de sólidos deformables" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Solido_deformagarrien_mekanika" title="Solido deformagarrien mekanika – Basque" lang="eu" hreflang="eu" data-title="Solido deformagarrien mekanika" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%85%DA%A9%D8%A7%D9%86%DB%8C%DA%A9_%D8%AC%D8%A7%D9%85%D8%AF%D8%A7%D8%AA" title="مکانیک جامدات – Persian" lang="fa" hreflang="fa" data-title="مکانیک جامدات" 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/M%C3%A9canique_des_solides_d%C3%A9formables" title="Mécanique des solides déformables – French" lang="fr" hreflang="fr" data-title="Mécanique des solides déformables" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EA%B3%A0%EC%B2%B4%EC%97%AD%ED%95%99" title="고체역학 – Korean" lang="ko" hreflang="ko" data-title="고체역학" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D5%80%D5%B8%D5%AE_%D5%B4%D5%AB%D5%BB%D5%A1%D5%BE%D5%A1%D5%B5%D6%80%D5%AB_%D5%B4%D5%A5%D5%AD%D5%A1%D5%B6%D5%AB%D5%AF%D5%A1" title="Հոծ միջավայրի մեխանիկա – Armenian" lang="hy" hreflang="hy" data-title="Հոծ միջավայրի մեխանիկա" 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%A0%E0%A5%8B%E0%A4%B8_%E0%A4%AF%E0%A4%BE%E0%A4%82%E0%A4%A4%E0%A5%8D%E0%A4%B0%E0%A4%BF%E0%A4%95%E0%A5%80" title="ठोस यांत्रिकी – Hindi" lang="hi" hreflang="hi" data-title="ठोस यांत्रिकी" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-io mw-list-item"><a href="https://io.wikipedia.org/wiki/Solida_mekaniko" title="Solida mekaniko – Ido" lang="io" hreflang="io" data-title="Solida mekaniko" 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/Mekanika_zat_padat" title="Mekanika zat padat – Indonesian" lang="id" hreflang="id" data-title="Mekanika zat padat" 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/Meccanica_dei_solidi" title="Meccanica dei solidi – Italian" lang="it" hreflang="it" data-title="Meccanica dei solidi" 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%9E%D7%9B%D7%A0%D7%99%D7%A7%D7%AA_%D7%94%D7%9E%D7%95%D7%A6%D7%A7" title="מכניקת המוצק – Hebrew" lang="he" hreflang="he" data-title="מכניקת המוצק" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%9C%D0%B5%D1%85%D0%B0%D0%BD%D0%B8%D0%BA%D0%B0_%D0%BD%D0%B0_%D1%86%D0%B2%D1%80%D1%81%D1%82%D0%B8%D1%82%D0%B5_%D1%82%D0%B5%D0%BB%D0%B0" title="Механика на цврстите тела – Macedonian" lang="mk" hreflang="mk" data-title="Механика на цврстите тела" data-language-autonym="Македонски" data-language-local-name="Macedonian" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-mr mw-list-item"><a href="https://mr.wikipedia.org/wiki/%E0%A4%98%E0%A4%A8%E0%A4%AF%E0%A4%BE%E0%A4%AE%E0%A4%BF%E0%A4%95%E0%A5%80" title="घनयामिकी – Marathi" lang="mr" hreflang="mr" data-title="घनयामिकी" data-language-autonym="मराठी" data-language-local-name="Marathi" class="interlanguage-link-target"><span>मराठी</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Mekanik_pepejal" title="Mekanik pepejal – Malay" lang="ms" hreflang="ms" data-title="Mekanik pepejal" 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-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E5%9B%BA%E4%BD%93%E5%8A%9B%E5%AD%A6" title="固体力学 – Japanese" lang="ja" hreflang="ja" data-title="固体力学" 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-km mw-list-item"><a href="https://km.wikipedia.org/wiki/%E1%9E%98%E1%9F%81%E1%9E%80%E1%9E%B6%E1%9E%93%E1%9E%B7%E1%9E%85%E1%9E%9F%E1%9E%BC%E1%9E%9B%E1%9E%B8%E1%9E%8A" title="មេកានិចសូលីដ – Khmer" lang="km" hreflang="km" data-title="មេកានិចសូលីដ" data-language-autonym="ភាសាខ្មែរ" data-language-local-name="Khmer" class="interlanguage-link-target"><span>ភាសាខ្មែរ</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Mec%C3%A2nica_dos_s%C3%B3lidos" title="Mecânica dos sólidos – Portuguese" lang="pt" hreflang="pt" data-title="Mecânica dos sólidos" 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/Mecanica_solidului" title="Mecanica solidului – Romanian" lang="ro" hreflang="ro" data-title="Mecanica solidului" 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%9C%D0%B5%D1%85%D0%B0%D0%BD%D0%B8%D0%BA%D0%B0_%D1%82%D0%B2%D1%91%D1%80%D0%B4%D0%BE%D0%B3%D0%BE_%D0%B4%D0%B5%D1%84%D0%BE%D1%80%D0%BC%D0%B8%D1%80%D1%83%D0%B5%D0%BC%D0%BE%D0%B3%D0%BE_%D1%82%D0%B5%D0%BB%D0%B0" title="Механика твёрдого деформируемого тела – Russian" lang="ru" hreflang="ru" data-title="Механика твёрдого деформируемого тела" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-si mw-list-item"><a href="https://si.wikipedia.org/wiki/%E0%B6%9D%E0%B6%B1_%E0%B6%AF%E0%B7%8A%E2%80%8D%E0%B6%BB%E0%B7%80%E0%B7%8A%E2%80%8D%E0%B6%BA_%E0%B6%B4%E0%B7%92%E0%B7%85%E0%B7%92%E0%B6%B6%E0%B6%B3_%E0%B6%BA%E0%B7%8F%E0%B6%B1%E0%B7%8A%E0%B6%AD%E0%B7%8A%E2%80%8D%E0%B6%BB_%E0%B7%80%E0%B7%92%E0%B6%AF%E0%B7%8A%E2%80%8D%E0%B6%BA%E0%B7%8F%E0%B7%80" title="ඝන ද්‍රව්‍ය පිළිබඳ යාන්ත්‍ර විද්‍යාව – Sinhala" lang="si" hreflang="si" data-title="ඝන ද්‍රව්‍ය පිළිබඳ යාන්ත්‍ර විද්‍යාව" 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/Solid_mechanics" title="Solid mechanics – Simple English" lang="en-simple" hreflang="en-simple" data-title="Solid mechanics" 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-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/Mehanika_%C4%8Dvrstih_tela" title="Mehanika čvrstih tela – Serbian" lang="sr" hreflang="sr" data-title="Mehanika čvrstih tela" 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-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/H%C3%A5llfasthetsl%C3%A4ra" title="Hållfasthetslära – Swedish" lang="sv" hreflang="sv" data-title="Hållfasthetslära" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Kat%C4%B1_mekani%C4%9Fi" title="Katı mekaniği – Turkish" lang="tr" hreflang="tr" data-title="Katı mekaniği" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%9C%D0%B5%D1%85%D0%B0%D0%BD%D1%96%D0%BA%D0%B0_%D0%B4%D0%B5%D1%84%D0%BE%D1%80%D0%BC%D1%96%D0%B2%D0%BD%D0%BE%D0%B3%D0%BE_%D1%82%D0%B2%D0%B5%D1%80%D0%B4%D0%BE%D0%B3%D0%BE_%D1%82%D1%96%D0%BB%D0%B0" title="Механіка деформівного твердого тіла – Ukrainian" lang="uk" hreflang="uk" data-title="Механіка деформівного твердого тіла" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/C%C6%A1_h%E1%BB%8Dc_v%E1%BA%ADt_r%E1%BA%AFn" title="Cơ học vật rắn – Vietnamese" lang="vi" hreflang="vi" data-title="Cơ học vật rắn" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E5%9B%BA%E4%BD%93%E5%8A%9B%E5%AD%A6" title="固体力学 – Wu" lang="wuu" hreflang="wuu" data-title="固体力学" data-language-autonym="吴语" data-language-local-name="Wu" class="interlanguage-link-target"><span>吴语</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E5%9B%BA%E9%AB%94%E5%8A%9B%E5%AD%B8" title="固體力學 – Cantonese" lang="yue" hreflang="yue" data-title="固體力學" data-language-autonym="粵語" data-language-local-name="Cantonese" class="interlanguage-link-target"><span>粵語</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E5%9B%BA%E4%BD%93%E5%8A%9B%E5%AD%A6" 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rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1246091330"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><table class="sidebar sidebar-collapse nomobile nowraplinks plainlist"><tbody><tr><td class="sidebar-pretitle">Part of a series on</td></tr><tr><th class="sidebar-title-with-pretitle"><a href="/wiki/Continuum_mechanics" title="Continuum mechanics">Continuum mechanics</a></th></tr><tr><td class="sidebar-image"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J=-D{\frac {d\varphi }{dx}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>J</mi> <mo>=</mo> <mo>−</mo> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>φ</mi> </mrow> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle J=-D{\frac {d\varphi }{dx}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1856f88def2056f28ed27c7d31180a6240820ea6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:11.874ex; height:5.509ex;" alt="{\displaystyle J=-D{\frac {d\varphi }{dx}}}"></span><div class="sidebar-caption"><a href="/wiki/Fick%27s_laws_of_diffusion" title="Fick's laws of diffusion">Fick's laws of diffusion</a></div></td></tr><tr><td class="sidebar-content-with-subgroup"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:transparent;border-top:1px solid #aaa;text-align:center;;color: var(--color-base)">Laws</div><div class="sidebar-list-content mw-collapsible-content"><table class="sidebar-subgroup"><tbody><tr><th class="sidebar-heading" style="font-style:italic;font-weight:normal;"> Conservations</th></tr><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Conservation_of_mass" title="Conservation of mass">Mass</a></li> <li><a href="/wiki/Conservation_of_momentum" class="mw-redirect" title="Conservation of momentum">Momentum</a></li> <li><a href="/wiki/Conservation_of_energy" title="Conservation of energy">Energy</a></li></ul></td> </tr><tr><th class="sidebar-heading" style="font-style:italic;font-weight:normal;"> Inequalities</th></tr><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Clausius%E2%80%93Duhem_inequality" title="Clausius–Duhem inequality">Clausius–Duhem (entropy)</a></li></ul></td> </tr></tbody></table></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible"><div class="sidebar-list-title" style="background:transparent;border-top:1px solid #aaa;text-align:center;;color: var(--color-base)"><a class="mw-selflink selflink">Solid mechanics</a></div><div class="sidebar-list-content mw-collapsible-content"><div class="hlist"> <ul><li><a href="/wiki/Deformation_(physics)" title="Deformation (physics)">Deformation</a></li> <li><a href="/wiki/Elasticity_(physics)" title="Elasticity (physics)">Elasticity</a> <ul><li><a href="/wiki/Linear_elasticity" title="Linear elasticity">linear</a></li></ul></li> <li><a href="/wiki/Plasticity_(physics)" title="Plasticity (physics)">Plasticity</a></li> <li><a href="/wiki/Hooke%27s_law" title="Hooke's law">Hooke's law</a></li> <li><a href="/wiki/Stress_(mechanics)" title="Stress (mechanics)">Stress</a></li> <li><a href="/wiki/Strain_(mechanics)" title="Strain (mechanics)">Strain</a> <ul><li><a href="/wiki/Finite_strain_theory" title="Finite strain theory">Finite strain</a></li> <li><a href="/wiki/Infinitesimal_strain_theory" title="Infinitesimal strain theory">Infinitesimal strain</a></li></ul></li> <li><a href="/wiki/Compatibility_(mechanics)" title="Compatibility (mechanics)">Compatibility</a></li> <li><a href="/wiki/Bending" title="Bending">Bending</a></li> <li><a href="/wiki/Contact_mechanics" title="Contact mechanics">Contact mechanics</a> <ul><li><a href="/wiki/Frictional_contact_mechanics" title="Frictional contact mechanics">frictional</a></li></ul></li> <li><a href="/wiki/Material_failure_theory" title="Material failure theory">Material failure theory</a></li> <li><a href="/wiki/Fracture_mechanics" title="Fracture mechanics">Fracture mechanics</a></li></ul> </div></div></div></td> </tr><tr><td class="sidebar-content-with-subgroup"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:transparent;border-top:1px solid #aaa;text-align:center;;color: var(--color-base)"><a href="/wiki/Fluid_mechanics" title="Fluid mechanics">Fluid mechanics</a></div><div class="sidebar-list-content mw-collapsible-content"><table class="sidebar-subgroup"><tbody><tr><th class="sidebar-heading" style="font-style:italic;"> <a href="/wiki/Fluid" title="Fluid">Fluids</a></th></tr><tr><td class="sidebar-content"> <div class="wraplinks"> <ul><li><a href="/wiki/Hydrostatics" title="Hydrostatics">Statics</a> <b>·</b> <a href="/wiki/Fluid_dynamics" title="Fluid dynamics">Dynamics</a></li> <li><a href="/wiki/Archimedes%27_principle" title="Archimedes' principle">Archimedes' principle</a> <b>·</b> <a href="/wiki/Bernoulli%27s_principle" title="Bernoulli's principle">Bernoulli's principle</a></li> <li><a href="/wiki/Navier%E2%80%93Stokes_equations" title="Navier–Stokes equations">Navier–Stokes equations</a></li> <li><a href="/wiki/Hagen%E2%80%93Poiseuille_equation" title="Hagen–Poiseuille equation">Poiseuille equation</a> <b>·</b> <a href="/wiki/Pascal%27s_law" title="Pascal's law">Pascal's law</a></li> <li><a href="/wiki/Viscosity" title="Viscosity">Viscosity</a> <ul><li>(<a href="/wiki/Newtonian_fluid" title="Newtonian fluid">Newtonian</a> <b>·</b> <a href="/wiki/Non-Newtonian_fluid" title="Non-Newtonian fluid">non-Newtonian</a>)</li></ul></li> <li><a href="/wiki/Buoyancy" title="Buoyancy">Buoyancy</a> <b>·</b> <a href="/wiki/Mixing_(process_engineering)" title="Mixing (process engineering)">Mixing</a> <b>·</b> <a href="/wiki/Pressure" title="Pressure">Pressure</a></li></ul> </div></td> </tr><tr><th class="sidebar-heading" style="font-style:italic;"> <a href="/wiki/Liquid" title="Liquid">Liquids</a></th></tr><tr><td class="sidebar-content"> <div class="hlist"> <ul><li><a href="/wiki/Adhesion" title="Adhesion">Adhesion</a></li> <li><a href="/wiki/Capillary_action" title="Capillary action">Capillary action</a></li> <li><a href="/wiki/Chromatography" title="Chromatography">Chromatography</a></li> <li><a href="/wiki/Cohesion_(chemistry)" title="Cohesion (chemistry)">Cohesion (chemistry)</a></li> <li><a href="/wiki/Surface_tension" title="Surface tension">Surface tension</a></li></ul> </div></td> </tr><tr><th class="sidebar-heading" style="font-style:italic;"> <a href="/wiki/Gas" title="Gas">Gases</a></th></tr><tr><td class="sidebar-content"> <div class="hlist"> <ul><li><a href="/wiki/Atmosphere" title="Atmosphere">Atmosphere</a></li> <li><a href="/wiki/Boyle%27s_law" title="Boyle's law">Boyle's law</a></li> <li><a href="/wiki/Charles%27s_law" title="Charles's law">Charles's law</a></li> <li><a href="/wiki/Combined_gas_law" class="mw-redirect" title="Combined gas law">Combined gas law</a></li> <li><a href="/wiki/Fick%27s_law" class="mw-redirect" title="Fick's law">Fick's law</a></li> <li><a href="/wiki/Gay-Lussac%27s_law" title="Gay-Lussac's law">Gay-Lussac's law</a></li> <li><a href="/wiki/Graham%27s_law" title="Graham's law">Graham's law</a></li></ul> </div></td> </tr><tr><th class="sidebar-heading" style="font-style:italic;"> <a href="/wiki/Plasma_(physics)" title="Plasma (physics)">Plasma</a></th></tr></tbody></table></div></div></td> </tr><tr><td class="sidebar-content-with-subgroup"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:transparent;border-top:1px solid #aaa;text-align:center;;color: var(--color-base)"><a href="/wiki/Rheology" title="Rheology">Rheology</a></div><div class="sidebar-list-content mw-collapsible-content"><table class="sidebar-subgroup"><tbody><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Viscoelasticity" title="Viscoelasticity">Viscoelasticity</a></li> <li><a href="/wiki/Rheometry" title="Rheometry">Rheometry</a></li> <li><a href="/wiki/Rheometer" title="Rheometer">Rheometer</a></li></ul></td> </tr><tr><th class="sidebar-heading" style="font-style:italic;"> <a href="/wiki/Smart_fluid" title="Smart fluid">Smart fluids</a></th></tr><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Electrorheological_fluid" title="Electrorheological fluid">Electrorheological</a></li> <li><a href="/wiki/Magnetorheological_fluid" title="Magnetorheological fluid">Magnetorheological</a></li> <li><a href="/wiki/Ferrofluid" title="Ferrofluid">Ferrofluids</a></li></ul></td> </tr></tbody></table></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:transparent;border-top:1px solid #aaa;text-align:center;;color: var(--color-base)">Scientists</div><div class="sidebar-list-content mw-collapsible-content"><div class="hlist"> <ul><li><a href="/wiki/Daniel_Bernoulli" title="Daniel Bernoulli">Bernoulli</a></li> <li><a href="/wiki/Robert_Boyle" title="Robert Boyle">Boyle</a></li> <li><a href="/wiki/Augustin-Louis_Cauchy" title="Augustin-Louis Cauchy">Cauchy</a></li> <li><a href="/wiki/Jacques_Charles" title="Jacques Charles">Charles</a></li> <li><a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Euler</a></li> <li><a href="/wiki/Adolf_Eugen_Fick" title="Adolf Eugen Fick">Fick</a></li> <li><a href="/wiki/Joseph_Louis_Gay-Lussac" title="Joseph Louis Gay-Lussac">Gay-Lussac</a></li> <li><a href="/wiki/Thomas_Graham_(chemist)" title="Thomas Graham (chemist)">Graham</a></li> <li><a href="/wiki/Robert_Hooke" title="Robert Hooke">Hooke</a></li> <li><a href="/wiki/Isaac_Newton" title="Isaac Newton">Newton</a></li> <li><a href="/wiki/Claude-Louis_Navier" title="Claude-Louis Navier">Navier</a></li> <li><a href="/wiki/Walter_Noll" title="Walter Noll">Noll</a></li> <li><a href="/wiki/Blaise_Pascal" title="Blaise Pascal">Pascal</a></li> <li><a href="/wiki/Sir_George_Stokes,_1st_Baronet" title="Sir George Stokes, 1st Baronet">Stokes</a></li> <li><a href="/wiki/Clifford_Truesdell" title="Clifford Truesdell">Truesdell</a></li></ul> </div></div></div></td> </tr><tr><td class="sidebar-navbar"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Continuum_mechanics" title="Template:Continuum mechanics"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Continuum_mechanics" title="Template talk:Continuum mechanics"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Continuum_mechanics" title="Special:EditPage/Template:Continuum mechanics"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> <p><b>Solid mechanics</b> (also known as <b>mechanics of solids</b>) is the branch of <a href="/wiki/Continuum_mechanics" title="Continuum mechanics">continuum mechanics</a> that studies the behavior of <a href="/wiki/Solid" title="Solid">solid</a> materials, especially their motion and <a href="/wiki/Deformation_(mechanics)" class="mw-redirect" title="Deformation (mechanics)">deformation</a> under the action of <a href="/wiki/Force" title="Force">forces</a>, <a href="/wiki/Temperature" title="Temperature">temperature</a> changes, <a href="/wiki/Phase_(chemistry)" class="mw-redirect" title="Phase (chemistry)">phase</a> changes, and other external or internal agents. </p><p>Solid mechanics is fundamental for <a href="/wiki/Civil_engineering" title="Civil engineering">civil</a>, <a href="/wiki/Aerospace_engineering" title="Aerospace engineering">aerospace</a>, <a href="/wiki/Nuclear_engineering" title="Nuclear engineering">nuclear</a>, <a href="/wiki/Biomedical_engineering" title="Biomedical engineering">biomedical</a> and <a href="/wiki/Mechanical_engineering" title="Mechanical engineering">mechanical engineering</a>, for <a href="/wiki/Geology" title="Geology">geology</a>, and for many branches of <a href="/wiki/Physics" title="Physics">physics</a> and <a href="/wiki/Chemistry" title="Chemistry">chemistry</a> such as <a href="/wiki/Materials_science" title="Materials science">materials science</a>.<sup id="cite_ref-applied_mechanics_1-0" class="reference"><a href="#cite_note-applied_mechanics-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> It has specific applications in many other areas, such as understanding the <a href="/wiki/Anatomy" title="Anatomy">anatomy</a> of living beings, and the design of <a href="/wiki/Dental_prosthesis" title="Dental prosthesis">dental prostheses</a> and <a href="/wiki/Surgical_implant" class="mw-redirect" title="Surgical implant">surgical implants</a>. One of the most common practical applications of solid mechanics is the <a href="/wiki/Euler%E2%80%93Bernoulli_beam_theory" title="Euler–Bernoulli beam theory">Euler–Bernoulli beam equation</a>. Solid mechanics extensively uses <a href="/wiki/Tensor" title="Tensor">tensors</a> to describe stresses, strains, and the relationship between them. </p><p>Solid mechanics is a vast subject because of the wide range of solid materials available, such as steel, wood, concrete, biological materials, textiles, geological materials, and plastics. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Fundamental_aspects">Fundamental aspects</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Solid_mechanics&action=edit&section=1" title="Edit section: Fundamental aspects"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A <i>solid</i> is a material that can support a substantial amount of <a href="/wiki/Shear_force" title="Shear force">shearing force</a> over a given time scale during a natural or industrial process or action. This is what distinguishes solids from <a href="/wiki/Fluid" title="Fluid">fluids</a>, because fluids also support <i>normal forces</i> which are those forces that are directed perpendicular to the material plane across from which they act and <i>normal stress</i> is the <a href="/wiki/Normal_force" title="Normal force">normal force</a> per unit area of that material plane. <i>Shearing forces</i> in contrast with <i>normal forces</i>, act parallel rather than perpendicular to the material plane and the shearing force per unit area is called <i>shear stress</i>. </p><p>Therefore, solid mechanics examines the shear stress, deformation and the failure of solid materials and structures. </p><p>The most common topics covered in solid mechanics include: </p> <ol><li><b>stability of structures</b> - examining whether structures can return to a given equilibrium after disturbance or partial/complete failure, <i>see <a href="/wiki/Structure_mechanics" class="mw-redirect" title="Structure mechanics">Structure mechanics</a></i></li> <li><b>dynamical systems and chaos</b> - dealing with mechanical systems highly sensitive to their given initial position</li> <li><b>thermomechanics</b> - analyzing materials with models derived from principles of <a href="/wiki/Thermodynamics" title="Thermodynamics">thermodynamics</a></li> <li><b><a href="/wiki/Biomechanics" title="Biomechanics">biomechanics</a></b> - solid mechanics applied to biological materials e.g. bones, heart tissue</li> <li><b>geomechanics</b> - solid mechanics applied to geological materials e.g. ice, soil, rock</li> <li><b>vibrations of solids and structures</b> - examining vibration and wave propagation from vibrating particles and structures i.e. vital in mechanical, civil, mining, aeronautical, maritime/marine, aerospace engineering</li> <li><b>fracture and damage mechanics</b> - dealing with crack-growth mechanics in solid materials</li> <li><b>composite materials</b> - solid mechanics applied to materials made up of more than one compound e.g. <a href="/wiki/Fibre-reinforced_plastic" title="Fibre-reinforced plastic">reinforced plastics</a>, <a href="/wiki/Reinforced_concrete" title="Reinforced concrete">reinforced concrete</a>, <a href="/wiki/Fiber_glass" class="mw-redirect" title="Fiber glass">fiber glass</a></li> <li><b>variational formulations and computational mechanics</b> - numerical solutions to mathematical equations arising from various branches of solid mechanics e.g. <a href="/wiki/Finite_element_method" title="Finite element method">finite element method (FEM)</a></li> <li><b>experimental mechanics</b> - design and analysis of experimental methods to examine the behavior of solid materials and structures</li></ol> <div class="mw-heading mw-heading2"><h2 id="Relationship_to_continuum_mechanics">Relationship to continuum mechanics</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Solid_mechanics&action=edit&section=2" title="Edit section: Relationship to continuum mechanics"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>As shown in the following table, solid mechanics inhabits a central place within continuum mechanics. The field of <a href="/wiki/Rheology" title="Rheology">rheology</a> presents an overlap between solid and <a href="/wiki/Fluid_mechanics" title="Fluid mechanics">fluid mechanics</a>. </p> <table class="wikitable"> <tbody><tr> <td rowspan="4"><a href="/wiki/Continuum_mechanics" title="Continuum mechanics">Continuum mechanics</a><br /><small>The study of the physics of continuous materials</small> </td> <td rowspan="2"><b>Solid mechanics</b><br /><small>The study of the physics of continuous materials with a defined rest shape.</small> </td> <td colspan="2"><a href="/wiki/Elasticity_(physics)" title="Elasticity (physics)">Elasticity</a><br /><small>Describes materials that return to their rest shape after applied <a href="/wiki/Stress_(physics)" class="mw-redirect" title="Stress (physics)">stresses</a> are removed.</small> </td></tr> <tr> <td><a href="/wiki/Plasticity_(physics)" title="Plasticity (physics)">Plasticity</a><br /><small>Describes materials that permanently deform after a sufficient applied stress.</small> </td> <td rowspan="2"><a href="/wiki/Rheology" title="Rheology">Rheology</a><br /><small>The study of materials with both solid and fluid characteristics.</small> </td></tr> <tr> <td rowspan="2"><a href="/wiki/Fluid_mechanics" title="Fluid mechanics">Fluid mechanics</a><br /><small>The study of the physics of continuous materials which deform when subjected to a force.</small> </td> <td><a href="/wiki/Non-Newtonian_fluid" title="Non-Newtonian fluid">Non-Newtonian fluid</a><br /><small>Do not undergo strain rates proportional to the applied shear stress.</small> </td></tr> <tr> <td colspan="2"><a href="/wiki/Newtonian_fluid" title="Newtonian fluid">Newtonian fluids</a> undergo strain rates proportional to the applied shear stress. </td></tr></tbody></table> <div class="mw-heading mw-heading2"><h2 id="Response_models">Response models</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Solid_mechanics&action=edit&section=3" title="Edit section: Response models"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A material has a rest shape and its shape departs away from the rest shape due to stress. The amount of departure from rest shape is called <a href="/wiki/Deformation_(mechanics)" class="mw-redirect" title="Deformation (mechanics)">deformation</a>, the proportion of deformation to original size is called strain. If the applied stress is sufficiently low (or the imposed strain is small enough), almost all solid materials behave in such a way that the strain is directly proportional to the stress; the coefficient of the proportion is called the <a href="/wiki/Modulus_of_elasticity" class="mw-redirect" title="Modulus of elasticity">modulus of elasticity</a>. This region of deformation is known as the linearly elastic region. </p><p>It is most common for analysts in solid mechanics to use <a href="/wiki/Linear" class="mw-redirect" title="Linear">linear</a> material models, due to ease of computation. However, real materials often exhibit <a href="/wiki/Non-linear" class="mw-redirect" title="Non-linear">non-linear</a> behavior. As new materials are used and old ones are pushed to their limits, non-linear material models are becoming more common. </p><p>These are basic models that describe how a solid responds to an applied stress: </p> <ol><li><a href="/wiki/Elasticity_(physics)" title="Elasticity (physics)">Elasticity</a> – When an applied stress is removed, the material returns to its undeformed state. Linearly elastic materials, those that deform proportionally to the applied load, can be described by the <a href="/wiki/Linear_elasticity" title="Linear elasticity">linear elasticity</a> equations such as <a href="/wiki/Hooke%27s_law" title="Hooke's law">Hooke's law</a>.</li> <li><a href="/wiki/Viscoelasticity" title="Viscoelasticity">Viscoelasticity</a> – These are materials that behave elastically, but also have <a href="/wiki/Friction" title="Friction">damping</a>: when the stress is applied and removed, work has to be done against the damping effects and is converted in heat within the material resulting in a <a href="/wiki/Hysteresis_loop" class="mw-redirect" title="Hysteresis loop">hysteresis loop</a> in the stress–strain curve. This implies that the material response has time-dependence.</li> <li><a href="/wiki/Plasticity_(physics)" title="Plasticity (physics)">Plasticity</a> – Materials that behave elastically generally do so when the applied stress is less than a yield value. When the stress is greater than the yield stress, the material behaves plastically and does not return to its previous state. That is, deformation that occurs after yield is permanent.</li> <li><a href="/wiki/Viscoplasticity" title="Viscoplasticity">Viscoplasticity</a> - Combines theories of viscoelasticity and plasticity and applies to materials like <a href="/wiki/Gel" title="Gel">gels</a> and <a href="/wiki/Mud" title="Mud">mud</a>.</li> <li>Thermoelasticity - There is coupling of mechanical with thermal responses. In general, thermoelasticity is concerned with elastic solids under conditions that are neither isothermal nor adiabatic. The simplest theory involves the <a href="/wiki/Fourier%27s_law" class="mw-redirect" title="Fourier's law">Fourier's law</a> of heat conduction, as opposed to advanced theories with physically more realistic models.</li></ol> <div class="mw-heading mw-heading2"><h2 id="Timeline">Timeline</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Solid_mechanics&action=edit&section=4" title="Edit section: Timeline"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>1452–1519 <a href="/wiki/Leonardo_da_Vinci" title="Leonardo da Vinci">Leonardo da Vinci</a> made many contributions<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></li> <li>1638: <a href="/wiki/Galileo_Galilei" title="Galileo Galilei">Galileo Galilei</a> published the book "<a href="/wiki/Two_New_Sciences" title="Two New Sciences">Two New Sciences</a>" in which he examined the failure of simple structures</li></ul> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Galileo_Galilei_by_Ottavio_Leoni_Marucelliana_(cropped).jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/dd/Galileo_Galilei_by_Ottavio_Leoni_Marucelliana_%28cropped%29.jpg/170px-Galileo_Galilei_by_Ottavio_Leoni_Marucelliana_%28cropped%29.jpg" decoding="async" width="170" height="194" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/dd/Galileo_Galilei_by_Ottavio_Leoni_Marucelliana_%28cropped%29.jpg/255px-Galileo_Galilei_by_Ottavio_Leoni_Marucelliana_%28cropped%29.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/dd/Galileo_Galilei_by_Ottavio_Leoni_Marucelliana_%28cropped%29.jpg/340px-Galileo_Galilei_by_Ottavio_Leoni_Marucelliana_%28cropped%29.jpg 2x" data-file-width="483" data-file-height="552" /></a><figcaption> <a href="/wiki/Galileo_Galilei" title="Galileo Galilei">Galileo Galilei</a> published the book "<a href="/wiki/Two_New_Sciences" title="Two New Sciences">Two New Sciences</a>" in which he examined the failure of simple structures</figcaption></figure> <ul><li>1660: <a href="/wiki/Hooke%27s_law" title="Hooke's law">Hooke's law</a> by <a href="/wiki/Robert_Hooke" title="Robert Hooke">Robert Hooke</a></li> <li>1687: <a href="/wiki/Isaac_Newton" title="Isaac Newton">Isaac Newton</a> published "<a href="/wiki/Philosophiae_Naturalis_Principia_Mathematica" class="mw-redirect" title="Philosophiae Naturalis Principia Mathematica">Philosophiae Naturalis Principia Mathematica</a>" which contains <a href="/wiki/Newton%27s_laws_of_motion" title="Newton's laws of motion">Newton's laws of motion</a></li></ul> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Sir_Isaac_Newton_by_Sir_Godfrey_Kneller,_Bt.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/50/Sir_Isaac_Newton_by_Sir_Godfrey_Kneller%2C_Bt.jpg/170px-Sir_Isaac_Newton_by_Sir_Godfrey_Kneller%2C_Bt.jpg" decoding="async" width="170" height="206" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/50/Sir_Isaac_Newton_by_Sir_Godfrey_Kneller%2C_Bt.jpg/255px-Sir_Isaac_Newton_by_Sir_Godfrey_Kneller%2C_Bt.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/50/Sir_Isaac_Newton_by_Sir_Godfrey_Kneller%2C_Bt.jpg/340px-Sir_Isaac_Newton_by_Sir_Godfrey_Kneller%2C_Bt.jpg 2x" data-file-width="2400" data-file-height="2912" /></a><figcaption> <a href="/wiki/Isaac_Newton" title="Isaac Newton">Isaac Newton</a> published "<a href="/wiki/Philosophiae_Naturalis_Principia_Mathematica" class="mw-redirect" title="Philosophiae Naturalis Principia Mathematica">Philosophiae Naturalis Principia Mathematica</a>" which contains the <a href="/wiki/Newton%27s_laws_of_motion" title="Newton's laws of motion">Newton's laws of motion</a></figcaption></figure> <ul><li>1750: <a href="/wiki/Euler%E2%80%93Bernoulli_beam_equation" class="mw-redirect" title="Euler–Bernoulli beam equation">Euler–Bernoulli beam equation</a></li> <li>1700–1782: <a href="/wiki/Daniel_Bernoulli" title="Daniel Bernoulli">Daniel Bernoulli</a> introduced the principle of <a href="/wiki/Virtual_work" title="Virtual work">virtual work</a></li> <li>1707–1783: <a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Leonhard Euler</a> developed the theory of <a href="/wiki/Buckling" title="Buckling">buckling</a> of columns</li></ul> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Leonhard_Euler_2.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/60/Leonhard_Euler_2.jpg/170px-Leonhard_Euler_2.jpg" decoding="async" width="170" height="212" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/60/Leonhard_Euler_2.jpg/255px-Leonhard_Euler_2.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/60/Leonhard_Euler_2.jpg/340px-Leonhard_Euler_2.jpg 2x" data-file-width="614" data-file-height="767" /></a><figcaption> <a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Leonhard Euler</a> developed the theory of <a href="/wiki/Buckling" title="Buckling">buckling</a> of columns</figcaption></figure> <ul><li>1826: <a href="/wiki/Claude-Louis_Navier" title="Claude-Louis Navier">Claude-Louis Navier</a> published a treatise on the elastic behaviors of structures</li> <li>1873: <a href="/wiki/Carlo_Alberto_Castigliano" title="Carlo Alberto Castigliano">Carlo Alberto Castigliano</a> presented his dissertation "Intorno ai sistemi elastici", which contains <a href="/wiki/Castigliano%27s_method" title="Castigliano's method">his theorem</a> for computing displacement as partial derivative of the strain energy. This theorem includes the method of <i>least work</i> as a special case</li> <li>1874: <a href="/wiki/Otto_Mohr" class="mw-redirect" title="Otto Mohr">Otto Mohr</a> formalized the idea of a statically indeterminate structure.</li> <li>1922: <a href="/wiki/Stephen_Timoshenko" title="Stephen Timoshenko">Timoshenko</a> corrects the <a href="/wiki/Euler%E2%80%93Bernoulli_beam_theory" title="Euler–Bernoulli beam theory">Euler–Bernoulli beam equation</a></li> <li>1936: <a href="/wiki/Hardy_Cross" title="Hardy Cross">Hardy Cross</a>' publication of the moment distribution method, an important innovation in the design of continuous frames.</li> <li>1941: <a href="/wiki/Alexander_Hrennikoff" title="Alexander Hrennikoff">Alexander Hrennikoff</a> solved the discretization of plane elasticity problems using a lattice framework</li> <li>1942: <a href="/wiki/Richard_Courant" title="Richard Courant">R. Courant</a> divided a domain into finite subregions</li> <li>1956: J. Turner, R. W. Clough, H. C. Martin, and L. J. Topp's paper on the "Stiffness and Deflection of Complex Structures" introduces the name "finite-element method" and is widely recognized as the first comprehensive treatment of the method as it is known today</li></ul> <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=Solid_mechanics&action=edit&section=5" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1235681985">.mw-parser-output .side-box{margin:4px 0;box-sizing:border-box;border:1px solid #aaa;font-size:88%;line-height:1.25em;background-color:var(--background-color-interactive-subtle,#f8f9fa);display:flow-root}.mw-parser-output .side-box-abovebelow,.mw-parser-output .side-box-text{padding:0.25em 0.9em}.mw-parser-output .side-box-image{padding:2px 0 2px 0.9em;text-align:center}.mw-parser-output .side-box-imageright{padding:2px 0.9em 2px 0;text-align:center}@media(min-width:500px){.mw-parser-output .side-box-flex{display:flex;align-items:center}.mw-parser-output .side-box-text{flex:1;min-width:0}}@media(min-width:720px){.mw-parser-output .side-box{width:238px}.mw-parser-output .side-box-right{clear:right;float:right;margin-left:1em}.mw-parser-output .side-box-left{margin-right:1em}}</style><style data-mw-deduplicate="TemplateStyles:r1237033735">@media print{body.ns-0 .mw-parser-output .sistersitebox{display:none!important}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}</style><div class="side-box side-box-right plainlinks sistersitebox"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409"> <div class="side-box-flex"> <div class="side-box-image"><span class="noviewer" typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Wikiversity_logo_2017.svg/40px-Wikiversity_logo_2017.svg.png" decoding="async" width="40" height="33" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Wikiversity_logo_2017.svg/60px-Wikiversity_logo_2017.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Wikiversity_logo_2017.svg/80px-Wikiversity_logo_2017.svg.png 2x" data-file-width="626" data-file-height="512" /></span></span></div> <div class="side-box-text plainlist">Wikiversity has learning resources about <i><b><a href="https://en.wikiversity.org/wiki/Solid_mechanics" class="extiw" title="v:Solid mechanics">Solid mechanics</a></b></i></div></div> </div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1235681985"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1237033735"><div class="side-box side-box-right plainlinks sistersitebox"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409"> <div class="side-box-flex"> <div class="side-box-image"><span class="noviewer" typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/d/df/Wikibooks-logo-en-noslogan.svg/40px-Wikibooks-logo-en-noslogan.svg.png" decoding="async" width="40" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/df/Wikibooks-logo-en-noslogan.svg/60px-Wikibooks-logo-en-noslogan.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/df/Wikibooks-logo-en-noslogan.svg/80px-Wikibooks-logo-en-noslogan.svg.png 2x" data-file-width="400" data-file-height="400" /></span></span></div> <div class="side-box-text plainlist">Wikibooks has a book on the topic of: <i><b><a href="https://en.wikibooks.org/wiki/Solid_mechanics" class="extiw" title="wikibooks:Solid mechanics">Solid mechanics</a></b></i></div></div> </div> <ul><li><a href="/wiki/Strength_of_materials" title="Strength of materials">Strength of materials</a> - Specific definitions and the relationships between stress and strain.</li> <li><a href="/wiki/Applied_mechanics" title="Applied mechanics">Applied mechanics</a></li> <li><a href="/wiki/Materials_science" title="Materials science">Materials science</a></li> <li><a href="/wiki/Continuum_mechanics" title="Continuum mechanics">Continuum mechanics</a></li> <li><a href="/wiki/Fracture_mechanics" title="Fracture mechanics">Fracture mechanics</a></li> <li><a href="/wiki/Impact_(mechanics)" title="Impact (mechanics)">Impact (mechanics)</a></li></ul> <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=Solid_mechanics&action=edit&section=6" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Notes">Notes</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Solid_mechanics&action=edit&section=7" 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"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-applied_mechanics-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-applied_mechanics_1-0">^</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="CITEREFAllan_Bower2009" class="citation book cs1">Allan Bower (2009). <a rel="nofollow" class="external text" href="http://solidmechanics.org"><i>Applied mechanics of solids</i></a>. CRC press<span class="reference-accessdate">. Retrieved <span class="nowrap">March 5,</span> 2017</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Applied+mechanics+of+solids&rft.pub=CRC+press&rft.date=2009&rft.au=Allan+Bower&rft_id=http%3A%2F%2Fsolidmechanics.org&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASolid+mechanics" class="Z3988"></span></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://qeprize.org/news/leonardo-da-vinci">"Leonardo da Vinci: The Mechanics of Genius"</a>. <i>Queen Elizabeth Prize for Engineering</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2024-05-27</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Queen+Elizabeth+Prize+for+Engineering&rft.atitle=Leonardo+da+Vinci%3A+The+Mechanics+of+Genius&rft_id=https%3A%2F%2Fqeprize.org%2Fnews%2Fleonardo-da-vinci&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASolid+mechanics" class="Z3988"></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading3"><h3 id="Bibliography">Bibliography</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Solid_mechanics&action=edit&section=8" title="Edit section: Bibliography"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Lev_Davidovich_Landau" class="mw-redirect" title="Lev Davidovich Landau">L.D. Landau</a>, <a href="/wiki/Evgeny_Mikhailovich_Lifshitz" class="mw-redirect" title="Evgeny Mikhailovich Lifshitz">E.M. Lifshitz</a>, <i><a href="/wiki/Course_of_Theoretical_Physics" title="Course of Theoretical Physics">Course of Theoretical Physics</a>: Theory of Elasticity</i> Butterworth-Heinemann, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-7506-2633-X" title="Special:BookSources/0-7506-2633-X">0-7506-2633-X</a></li> <li>J.E. Marsden, T.J. Hughes, <i>Mathematical Foundations of Elasticity</i>, Dover, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-486-67865-2" title="Special:BookSources/0-486-67865-2">0-486-67865-2</a></li> <li>P.C. Chou, N. J. Pagano, <i>Elasticity: Tensor, Dyadic, and Engineering Approaches</i>, Dover, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-486-66958-0" title="Special:BookSources/0-486-66958-0">0-486-66958-0</a></li> <li>R.W. Ogden, <i>Non-linear Elastic Deformation</i>, Dover, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-486-69648-0" title="Special:BookSources/0-486-69648-0">0-486-69648-0</a></li> <li><a href="/wiki/Stephen_Timoshenko" title="Stephen Timoshenko">S. Timoshenko</a> and J.N. Goodier," Theory of elasticity", 3d ed., New York, McGraw-Hill, 1970.</li> <li><a href="/wiki/Gerhard_A._Holzapfel" title="Gerhard A. Holzapfel">G.A. Holzapfel</a>, <i>Nonlinear Solid Mechanics: A Continuum Approach for Engineering</i>, Wiley, 2000</li> <li>A.I. Lurie, <i>Theory of Elasticity</i>, Springer, 1999.</li> <li>L.B. Freund, <i>Dynamic Fracture Mechanics</i>, Cambridge University Press, 1990.</li> <li>R. Hill, <i>The Mathematical Theory of Plasticity</i>, Oxford University, 1950.</li> <li>J. Lubliner, <i>Plasticity Theory</i>, Macmillan Publishing Company, 1990.</li> <li>J. Ignaczak, <a href="/wiki/M._Ostoja-Starzewski" class="mw-redirect" title="M. Ostoja-Starzewski">M. Ostoja-Starzewski</a>, <i>Thermoelasticity with Finite Wave Speeds</i>, Oxford University Press, 2010.</li> <li>D. Bigoni, <i>Nonlinear Solid Mechanics: Bifurcation Theory and Material Instability</i>, Cambridge University Press, 2012.</li> <li>Y. C. Fung, Pin Tong and Xiaohong Chen, <i>Classical and Computational Solid Mechanics</i>, 2nd Edition, World Scientific Publishing, 2017, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-981-4713-64-1" title="Special:BookSources/978-981-4713-64-1">978-981-4713-64-1</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 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