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<span>Definition</span> </div> </a> <button aria-controls="toc-Definition-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Definition subsection</span> </button> <ul id="toc-Definition-sublist" class="vector-toc-list"> <li id="toc-Normal_and_shear" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Normal_and_shear"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Normal and shear</span> </div> </a> <ul id="toc-Normal_and_shear-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Units" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Units"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Units</span> </div> </a> <ul id="toc-Units-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Causes_and_effects" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Causes_and_effects"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Causes and effects</span> </div> </a> <ul id="toc-Causes_and_effects-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Simple_types" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Simple_types"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Simple types</span> </div> </a> <button aria-controls="toc-Simple_types-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 Simple types subsection</span> </button> <ul id="toc-Simple_types-sublist" class="vector-toc-list"> <li id="toc-Uniaxial_normal" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Uniaxial_normal"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1</span> <span>Uniaxial normal</span> </div> </a> <ul id="toc-Uniaxial_normal-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Shear" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Shear"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.2</span> <span>Shear</span> </div> </a> <ul id="toc-Shear-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Isotropic" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Isotropic"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.3</span> <span>Isotropic</span> </div> </a> <ul id="toc-Isotropic-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Cylinder" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Cylinder"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.4</span> <span>Cylinder</span> </div> </a> <ul id="toc-Cylinder-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-General_types" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#General_types"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>General types</span> </div> </a> <ul id="toc-General_types-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Cauchy_tensor" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Cauchy_tensor"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Cauchy tensor</span> </div> </a> <button aria-controls="toc-Cauchy_tensor-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 Cauchy tensor subsection</span> </button> <ul id="toc-Cauchy_tensor-sublist" class="vector-toc-list"> <li id="toc-Change_of_coordinates" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Change_of_coordinates"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.1</span> <span>Change of coordinates</span> </div> </a> <ul id="toc-Change_of_coordinates-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Tensor_field" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Tensor_field"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.2</span> <span>Tensor field</span> </div> </a> <ul id="toc-Tensor_field-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Thin_plates" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Thin_plates"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.3</span> <span>Thin plates</span> </div> </a> <ul id="toc-Thin_plates-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Thin_beams" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Thin_beams"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.4</span> <span>Thin beams</span> </div> </a> <ul id="toc-Thin_beams-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Analysis" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Analysis"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Analysis</span> </div> </a> <button aria-controls="toc-Analysis-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 Analysis subsection</span> </button> <ul id="toc-Analysis-sublist" class="vector-toc-list"> <li id="toc-Goals_and_assumptions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Goals_and_assumptions"> <div class="vector-toc-text"> <span class="vector-toc-numb">8.1</span> <span>Goals and assumptions</span> </div> </a> <ul id="toc-Goals_and_assumptions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Methods" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Methods"> <div class="vector-toc-text"> <span class="vector-toc-numb">8.2</span> <span>Methods</span> </div> </a> <ul id="toc-Methods-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Measures" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Measures"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>Measures</span> </div> </a> <ul id="toc-Measures-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">10</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">11</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Further_reading" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Further_reading"> <div class="vector-toc-text"> <span class="vector-toc-numb">12</span> <span>Further reading</span> </div> </a> <ul id="toc-Further_reading-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">Stress (mechanics)</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 63 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-63" 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">63 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/%D8%A5%D8%AC%D9%87%D8%A7%D8%AF_(%D9%85%D9%8A%D9%83%D8%A7%D9%86%D9%8A%D9%83%D8%A7)" title="إجهاد (ميكانيكا) – Arabic" lang="ar" hreflang="ar" data-title="إجهاد (ميكانيكا)" 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/Tensi%C3%B3n_(mec%C3%A1nica)" title="Tensión (mecánica) – Asturian" lang="ast" hreflang="ast" data-title="Tensión (mecánica)" data-language-autonym="Asturianu" data-language-local-name="Asturian" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/G%C9%99rginlik_(mexanika)" title="Gərginlik (mexanika) – Azerbaijani" lang="az" hreflang="az" data-title="Gərginlik (mexanika)" 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%AA%E0%A7%80%E0%A6%A1%E0%A6%BC%E0%A6%A8" title="পীড়ন – Bangla" lang="bn" hreflang="bn" data-title="পীড়ন" 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%D1%87%D0%BD%D0%B0%D0%B5_%D0%BD%D0%B0%D0%BF%D1%80%D1%83%D0%B6%D0%B0%D0%BD%D0%BD%D0%B5" title="Механічнае напружанне – Belarusian" lang="be" hreflang="be" data-title="Механічнае напружанне" data-language-autonym="Беларуская" data-language-local-name="Belarusian" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-be-x-old mw-list-item"><a href="https://be-tarask.wikipedia.org/wiki/%D0%9C%D1%8D%D1%85%D0%B0%D0%BD%D1%96%D1%87%D0%BD%D0%B0%D0%B5_%D0%BD%D0%B0%D0%BF%D1%80%D1%83%D0%B6%D0%B0%D0%BD%D1%8C%D0%BD%D0%B5" title="Мэханічнае напружаньне – Belarusian (Taraškievica orthography)" lang="be-tarask" hreflang="be-tarask" data-title="Мэханічнае напружаньне" data-language-autonym="Беларуская (тарашкевіца)" data-language-local-name="Belarusian (Taraškievica orthography)" class="interlanguage-link-target"><span>Беларуская (тарашкевіца)</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%9D%D0%B0%D0%BF%D1%80%D0%B5%D0%B6%D0%B5%D0%BD%D0%B8%D0%B5_(%D0%BC%D0%B5%D1%85%D0%B0%D0%BD%D0%B8%D0%BA%D0%B0)" title="Напрежение (механика) – Bulgarian" lang="bg" hreflang="bg" data-title="Напрежение (механика)" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bar mw-list-item"><a href="https://bar.wikipedia.org/wiki/Spaunnung" title="Spaunnung – Bavarian" lang="bar" hreflang="bar" data-title="Spaunnung" data-language-autonym="Boarisch" data-language-local-name="Bavarian" class="interlanguage-link-target"><span>Boarisch</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Mehani%C4%8Dki_napon" title="Mehanički napon – Bosnian" lang="bs" hreflang="bs" data-title="Mehanički napon" 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/Tensi%C3%B3_(mec%C3%A0nica)" title="Tensió (mecànica) – Catalan" lang="ca" hreflang="ca" data-title="Tensió (mecànica)" 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%9C%D0%B5%D1%85%D0%B0%D0%BD%D0%B8%D0%BA%C4%83%D0%BB%D0%BB%D0%B0_%D1%85%D0%B8%D0%B2%D1%80%D0%B5%D0%BB%C4%95%D1%85" title="Механикăлла хиврелĕх – Chuvash" lang="cv" hreflang="cv" data-title="Механикăлла хиврелĕх" 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/Mechanick%C3%A9_nap%C4%9Bt%C3%AD" title="Mechanické napětí – Czech" lang="cs" hreflang="cs" data-title="Mechanické napětí" 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/Sp%C3%A6nding_(mekanik)" title="Spænding (mekanik) – Danish" lang="da" hreflang="da" data-title="Spænding (mekanik)" 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/Mechanische_Spannung" title="Mechanische Spannung – German" lang="de" hreflang="de" data-title="Mechanische Spannung" 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/Pinge_(mehaanika)" title="Pinge (mehaanika) – Estonian" lang="et" hreflang="et" data-title="Pinge (mehaanika)" 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%A4%CE%AC%CF%83%CE%B7_(%CE%BC%CE%B7%CF%87%CE%B1%CE%BD%CE%B9%CE%BA%CE%AE)" title="Τάση (μηχανική) – Greek" lang="el" hreflang="el" data-title="Τάση (μηχανική)" 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/Tensi%C3%B3n_mec%C3%A1nica" title="Tensión mecánica – Spanish" lang="es" hreflang="es" data-title="Tensión mecánica" 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/Stre%C4%89o_(mekaniko)" title="Streĉo (mekaniko) – Esperanto" lang="eo" hreflang="eo" data-title="Streĉo (mekaniko)" 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/Tentsio_mekaniko" title="Tentsio mekaniko – Basque" lang="eu" hreflang="eu" data-title="Tentsio mekaniko" 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/%D8%AA%D9%86%D8%B4_(%D9%85%DA%A9%D8%A7%D9%86%DB%8C%DA%A9)" title="تنش (مکانیک) – Persian" lang="fa" hreflang="fa" data-title="تنش (مکانیک)" 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/Contrainte_(m%C3%A9canique)" title="Contrainte (mécanique) – French" lang="fr" hreflang="fr" data-title="Contrainte (mécanique)" 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/Strus_(fisic)" title="Strus (fisic) – Irish" lang="ga" hreflang="ga" data-title="Strus (fisic)" 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/Tensi%C3%B3n_mec%C3%A1nica" title="Tensión mecánica – Galician" lang="gl" hreflang="gl" data-title="Tensión 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/%EB%B3%80%ED%98%95%EB%A0%A5" title="변형력 – Korean" lang="ko" hreflang="ko" data-title="변형력" 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%84%D5%A5%D5%AD%D5%A1%D5%B6%D5%AB%D5%AF%D5%A1%D5%AF%D5%A1%D5%B6_%D5%AC%D5%A1%D6%80%D5%B8%D6%82%D5%B4" title="Մեխանիկական լարում – Armenian" lang="hy" hreflang="hy" data-title="Մեխանիկական լարում" 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%A4%BF%E0%A4%AC%E0%A4%B2" title="प्रतिबल – Hindi" lang="hi" hreflang="hi" data-title="प्रतिबल" 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/Naprezanje" title="Naprezanje – Croatian" lang="hr" hreflang="hr" data-title="Naprezanje" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Tegangan_(mekanika)" title="Tegangan (mekanika) – Indonesian" lang="id" hreflang="id" data-title="Tegangan (mekanika)" 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/Tensione_interna" title="Tensione interna – Italian" lang="it" hreflang="it" data-title="Tensione interna" 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%90%D7%9E%D7%A5_(%D7%94%D7%A0%D7%93%D7%A1%D7%94)" title="מאמץ (הנדסה) – Hebrew" lang="he" hreflang="he" data-title="מאמץ (הנדסה)" 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%AA%E0%B3%80%E0%B2%A1%E0%B2%A8" title="ಪೀಡನ – Kannada" lang="kn" hreflang="kn" data-title="ಪೀಡನ" data-language-autonym="ಕನ್ನಡ" data-language-local-name="Kannada" class="interlanguage-link-target"><span>ಕನ್ನಡ</span></a></li><li class="interlanguage-link interwiki-ht mw-list-item"><a href="https://ht.wikipedia.org/wiki/Estr%C3%A8s" title="Estrès – Haitian Creole" lang="ht" hreflang="ht" data-title="Estrès" data-language-autonym="Kreyòl ayisyen" data-language-local-name="Haitian Creole" class="interlanguage-link-target"><span>Kreyòl ayisyen</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Meh%C4%81niskais_spriegums" title="Mehāniskais spriegums – Latvian" lang="lv" hreflang="lv" data-title="Mehāniskais spriegums" data-language-autonym="Latviešu" data-language-local-name="Latvian" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/%C4%AEtempis" title="Įtempis – Lithuanian" lang="lt" hreflang="lt" data-title="Įtempis" data-language-autonym="Lietuvių" data-language-local-name="Lithuanian" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Mechanikai_fesz%C3%BClts%C3%A9g" title="Mechanikai feszültség – Hungarian" lang="hu" hreflang="hu" data-title="Mechanikai feszültség" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%86%E0%B4%AF%E0%B4%BE%E0%B4%B8%E0%B4%82" title="ആയാസം – Malayalam" lang="ml" hreflang="ml" data-title="ആയാസം" 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/Tegasan_(mekanik)" title="Tegasan (mekanik) – Malay" lang="ms" hreflang="ms" data-title="Tegasan (mekanik)" 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-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Mechanische_spanning" title="Mechanische spanning – Dutch" lang="nl" hreflang="nl" data-title="Mechanische spanning" 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%BF%9C%E5%8A%9B" title="応力 – Japanese" lang="ja" hreflang="ja" data-title="応力" 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/Spenning_(mekanikk)" title="Spenning (mekanikk) – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Spenning (mekanikk)" 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/Mekanisk_spenning" title="Mekanisk spenning – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Mekanisk spenning" 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-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Mexanik_kuchlanish" title="Mexanik kuchlanish – Uzbek" lang="uz" hreflang="uz" data-title="Mexanik kuchlanish" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="Uzbek" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Napr%C4%99%C5%BCenie" title="Naprężenie – Polish" lang="pl" hreflang="pl" data-title="Naprężenie" 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/Tens%C3%A3o_(mec%C3%A2nica)" title="Tensão (mecânica) – Portuguese" lang="pt" hreflang="pt" data-title="Tensão (mecânica)" 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/Tensiune_(mecanic%C4%83)" title="Tensiune (mecanică) – Romanian" lang="ro" hreflang="ro" data-title="Tensiune (mecanică)" 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%D1%87%D0%B5%D1%81%D0%BA%D0%BE%D0%B5_%D0%BD%D0%B0%D0%BF%D1%80%D1%8F%D0%B6%D0%B5%D0%BD%D0%B8%D0%B5" title="Механическое напряжение – Russian" lang="ru" hreflang="ru" data-title="Механическое напряжение" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sco mw-list-item"><a href="https://sco.wikipedia.org/wiki/Stress_(mechanics)" title="Stress (mechanics) – Scots" lang="sco" hreflang="sco" data-title="Stress (mechanics)" data-language-autonym="Scots" data-language-local-name="Scots" class="interlanguage-link-target"><span>Scots</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Stress_(mechanics)" title="Stress (mechanics) – Simple English" lang="en-simple" hreflang="en-simple" data-title="Stress (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-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Mechanick%C3%A9_nap%C3%A4tie" title="Mechanické napätie – Slovak" lang="sk" hreflang="sk" data-title="Mechanické napätie" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Mehanska_napetost" title="Mehanska napetost – Slovenian" lang="sl" hreflang="sl" data-title="Mehanska napetost" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%9D%D0%B0%D0%BF%D0%BE%D0%BD_(%D0%BC%D0%B5%D1%85%D0%B0%D0%BD%D0%B8%D0%BA%D0%B0)" title="Напон (механика) – Serbian" lang="sr" hreflang="sr" data-title="Напон (механика)" 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/Napor_(mehanika)" title="Napor (mehanika) – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Napor (mehanika)" 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/J%C3%A4nnitys" title="Jännitys – Finnish" lang="fi" hreflang="fi" data-title="Jännitys" 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/Dragsp%C3%A4nning" title="Dragspänning – Swedish" lang="sv" hreflang="sv" data-title="Dragspänning" 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/Stress_(mekanika)" title="Stress (mekanika) – Tagalog" lang="tl" hreflang="tl" data-title="Stress (mekanika)" 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 href="https://ta.wikipedia.org/wiki/%E0%AE%A4%E0%AE%95%E0%AF%88%E0%AE%B5%E0%AF%81" title="தகைவு – Tamil" lang="ta" hreflang="ta" data-title="தகைவு" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%84%E0%B8%A7%E0%B8%B2%E0%B8%A1%E0%B9%80%E0%B8%84%E0%B9%89%E0%B8%99" title="ความเค้น – Thai" lang="th" hreflang="th" data-title="ความเค้น" data-language-autonym="ไทย" data-language-local-name="Thai" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Gerilme" title="Gerilme – Turkish" lang="tr" hreflang="tr" data-title="Gerilme" 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 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Unsourced material may be challenged and removed.<br /><small><span class="plainlinks"><i>Find sources:</i> <a rel="nofollow" class="external text" href="https://www.google.com/search?as_eq=wikipedia&q=%22Stress%22+mechanics">"Stress" mechanics</a> – <a rel="nofollow" class="external text" href="https://www.google.com/search?tbm=nws&q=%22Stress%22+mechanics+-wikipedia&tbs=ar:1">news</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.google.com/search?&q=%22Stress%22+mechanics&tbs=bkt:s&tbm=bks">newspapers</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.google.com/search?tbs=bks:1&q=%22Stress%22+mechanics+-wikipedia">books</a> <b>·</b> <a rel="nofollow" class="external text" href="https://scholar.google.com/scholar?q=%22Stress%22+mechanics">scholar</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.jstor.org/action/doBasicSearch?Query=%22Stress%22+mechanics&acc=on&wc=on">JSTOR</a></span></small></span> <span class="date-container"><i>(<span class="date">August 2021</span>)</i></span><span class="hide-when-compact"><i> (<small><a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span></div></td></tr></tbody></table> <style data-mw-deduplicate="TemplateStyles:r1257001546">.mw-parser-output .infobox-subbox{padding:0;border:none;margin:-3px;width:auto;min-width:100%;font-size:100%;clear:none;float:none;background-color:transparent}.mw-parser-output .infobox-3cols-child{margin:auto}.mw-parser-output .infobox .navbar{font-size:100%}@media screen{html.skin-theme-clientpref-night .mw-parser-output .infobox-full-data:not(.notheme)>div:not(.notheme)[style]{background:#1f1f23!important;color:#f8f9fa}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .infobox-full-data:not(.notheme) div:not(.notheme){background:#1f1f23!important;color:#f8f9fa}}@media(min-width:640px){body.skin--responsive .mw-parser-output .infobox-table{display:table!important}body.skin--responsive .mw-parser-output .infobox-table>caption{display:table-caption!important}body.skin--responsive .mw-parser-output .infobox-table>tbody{display:table-row-group}body.skin--responsive .mw-parser-output .infobox-table tr{display:table-row!important}body.skin--responsive .mw-parser-output .infobox-table th,body.skin--responsive .mw-parser-output .infobox-table td{padding-left:inherit;padding-right:inherit}}</style><table class="infobox"><tbody><tr><th colspan="2" class="infobox-above">Stress</th></tr><tr><td colspan="2" class="infobox-image"><span class="mw-default-size" typeof="mw:File/Frameless"><a href="/wiki/File:Plastic_Protractor_Polarized_05375.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e4/Plastic_Protractor_Polarized_05375.jpg/220px-Plastic_Protractor_Polarized_05375.jpg" decoding="async" width="220" height="165" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e4/Plastic_Protractor_Polarized_05375.jpg/330px-Plastic_Protractor_Polarized_05375.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e4/Plastic_Protractor_Polarized_05375.jpg/440px-Plastic_Protractor_Polarized_05375.jpg 2x" data-file-width="2048" data-file-height="1536" /></a></span><div class="infobox-caption"><a href="/wiki/Residual_stress" title="Residual stress">Residual stresses</a> inside a plastic protractor are revealed by <a href="/wiki/Photoelasticity" title="Photoelasticity">polarized light</a>.</div></td></tr><tr><th scope="row" class="infobox-label"><div style="display: inline-block; line-height: 1.2em; padding: .1em 0;">Common symbols</div></th><td class="infobox-data"><i>σ</i></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/SI_unit" class="mw-redirect" title="SI unit">SI unit</a></th><td class="infobox-data"><a href="/wiki/Pascal_(unit)" title="Pascal (unit)">pascal</a></td></tr><tr><th scope="row" class="infobox-label"><div style="display: inline-block; line-height: 1.2em; padding: .1em 0;">Other units</div></th><td class="infobox-data"><a href="/wiki/Pound_per_square_inch" title="Pound per square inch">psi</a>, <a href="/wiki/Bar_(unit)" title="Bar (unit)">bar</a></td></tr><tr><th scope="row" class="infobox-label">In <a href="/wiki/SI_base_unit" title="SI base unit"><span class="wrap">SI base units</span></a></th><td class="infobox-data">Pa = <a href="/wiki/Kilogram" title="Kilogram">kg</a>⋅<a href="/wiki/Metre" title="Metre">m</a><sup>−1</sup>⋅<a href="/wiki/Second" title="Second">s</a><sup>−2</sup></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Dimensional_analysis#Formulation" title="Dimensional analysis">Dimension</a></th><td class="infobox-data"><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 {\mathsf {L}}^{-1}{\mathsf {M}}{\mathsf {T}}^{-2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">M</mi> </mrow> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">T</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {L}}^{-1}{\mathsf {M}}{\mathsf {T}}^{-2}}</annotation> </semantics> 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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: -2.005ex; width:11.874ex; height:5.509ex;" alt="{\displaystyle J=-D{\frac {d\varphi }{dx}}}"></span><div class="sidebar-caption"><a href="/wiki/Fick%27s_laws_of_diffusion" title="Fick's laws of diffusion">Fick's laws of diffusion</a></div></td></tr><tr><td class="sidebar-content-with-subgroup"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:transparent;border-top:1px solid #aaa;text-align:center;;color: var(--color-base)">Laws</div><div class="sidebar-list-content mw-collapsible-content"><table class="sidebar-subgroup"><tbody><tr><th class="sidebar-heading" style="font-style:italic;font-weight:normal;"> Conservations</th></tr><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Conservation_of_mass" title="Conservation of mass">Mass</a></li> <li><a href="/wiki/Conservation_of_momentum" class="mw-redirect" title="Conservation of momentum">Momentum</a></li> <li><a href="/wiki/Conservation_of_energy" title="Conservation of energy">Energy</a></li></ul></td> </tr><tr><th class="sidebar-heading" style="font-style:italic;font-weight:normal;"> Inequalities</th></tr><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Clausius%E2%80%93Duhem_inequality" title="Clausius–Duhem inequality">Clausius–Duhem (entropy)</a></li></ul></td> </tr></tbody></table></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible"><div class="sidebar-list-title" style="background:transparent;border-top:1px solid #aaa;text-align:center;;color: var(--color-base)"><a href="/wiki/Solid_mechanics" title="Solid mechanics">Solid mechanics</a></div><div class="sidebar-list-content mw-collapsible-content"><div class="hlist"> <ul><li><a href="/wiki/Deformation_(physics)" title="Deformation (physics)">Deformation</a></li> <li><a href="/wiki/Elasticity_(physics)" title="Elasticity (physics)">Elasticity</a> <ul><li><a href="/wiki/Linear_elasticity" title="Linear elasticity">linear</a></li></ul></li> <li><a href="/wiki/Plasticity_(physics)" title="Plasticity (physics)">Plasticity</a></li> <li><a href="/wiki/Hooke%27s_law" title="Hooke's law">Hooke's law</a></li> <li><a class="mw-selflink selflink">Stress</a></li> <li><a href="/wiki/Strain_(mechanics)" title="Strain (mechanics)">Strain</a> <ul><li><a href="/wiki/Finite_strain_theory" title="Finite strain theory">Finite strain</a></li> <li><a href="/wiki/Infinitesimal_strain_theory" title="Infinitesimal strain theory">Infinitesimal strain</a></li></ul></li> <li><a href="/wiki/Compatibility_(mechanics)" title="Compatibility (mechanics)">Compatibility</a></li> <li><a href="/wiki/Bending" title="Bending">Bending</a></li> <li><a href="/wiki/Contact_mechanics" title="Contact mechanics">Contact mechanics</a> <ul><li><a href="/wiki/Frictional_contact_mechanics" title="Frictional contact mechanics">frictional</a></li></ul></li> <li><a href="/wiki/Material_failure_theory" title="Material failure theory">Material failure theory</a></li> <li><a href="/wiki/Fracture_mechanics" title="Fracture mechanics">Fracture mechanics</a></li></ul> </div></div></div></td> </tr><tr><td class="sidebar-content-with-subgroup"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:transparent;border-top:1px solid #aaa;text-align:center;;color: var(--color-base)"><a href="/wiki/Fluid_mechanics" title="Fluid mechanics">Fluid mechanics</a></div><div class="sidebar-list-content mw-collapsible-content"><table class="sidebar-subgroup"><tbody><tr><th class="sidebar-heading" style="font-style:italic;"> <a href="/wiki/Fluid" title="Fluid">Fluids</a></th></tr><tr><td class="sidebar-content"> <div class="wraplinks"> <ul><li><a href="/wiki/Hydrostatics" title="Hydrostatics">Statics</a> <b>·</b> <a href="/wiki/Fluid_dynamics" title="Fluid dynamics">Dynamics</a></li> <li><a href="/wiki/Archimedes%27_principle" title="Archimedes' principle">Archimedes' principle</a> <b>·</b> <a href="/wiki/Bernoulli%27s_principle" title="Bernoulli's principle">Bernoulli's principle</a></li> <li><a href="/wiki/Navier%E2%80%93Stokes_equations" title="Navier–Stokes equations">Navier–Stokes equations</a></li> <li><a href="/wiki/Hagen%E2%80%93Poiseuille_equation" title="Hagen–Poiseuille equation">Poiseuille equation</a> <b>·</b> <a href="/wiki/Pascal%27s_law" title="Pascal's law">Pascal's law</a></li> <li><a href="/wiki/Viscosity" title="Viscosity">Viscosity</a> <ul><li>(<a href="/wiki/Newtonian_fluid" title="Newtonian fluid">Newtonian</a> <b>·</b> <a href="/wiki/Non-Newtonian_fluid" title="Non-Newtonian fluid">non-Newtonian</a>)</li></ul></li> <li><a href="/wiki/Buoyancy" title="Buoyancy">Buoyancy</a> <b>·</b> <a href="/wiki/Mixing_(process_engineering)" title="Mixing (process engineering)">Mixing</a> <b>·</b> <a href="/wiki/Pressure" title="Pressure">Pressure</a></li></ul> </div></td> </tr><tr><th class="sidebar-heading" style="font-style:italic;"> <a href="/wiki/Liquid" title="Liquid">Liquids</a></th></tr><tr><td class="sidebar-content"> <div class="hlist"> <ul><li><a href="/wiki/Adhesion" title="Adhesion">Adhesion</a></li> <li><a href="/wiki/Capillary_action" title="Capillary action">Capillary action</a></li> <li><a href="/wiki/Chromatography" title="Chromatography">Chromatography</a></li> <li><a href="/wiki/Cohesion_(chemistry)" title="Cohesion (chemistry)">Cohesion (chemistry)</a></li> <li><a href="/wiki/Surface_tension" title="Surface tension">Surface tension</a></li></ul> </div></td> </tr><tr><th class="sidebar-heading" style="font-style:italic;"> <a href="/wiki/Gas" title="Gas">Gases</a></th></tr><tr><td class="sidebar-content"> <div class="hlist"> <ul><li><a href="/wiki/Atmosphere" title="Atmosphere">Atmosphere</a></li> <li><a href="/wiki/Boyle%27s_law" title="Boyle's law">Boyle's law</a></li> <li><a href="/wiki/Charles%27s_law" title="Charles's law">Charles's law</a></li> <li><a href="/wiki/Combined_gas_law" class="mw-redirect" title="Combined gas law">Combined gas law</a></li> <li><a href="/wiki/Fick%27s_law" class="mw-redirect" title="Fick's law">Fick's law</a></li> <li><a href="/wiki/Gay-Lussac%27s_law" title="Gay-Lussac's law">Gay-Lussac's law</a></li> <li><a href="/wiki/Graham%27s_law" title="Graham's law">Graham's law</a></li></ul> </div></td> </tr><tr><th class="sidebar-heading" style="font-style:italic;"> <a href="/wiki/Plasma_(physics)" title="Plasma (physics)">Plasma</a></th></tr></tbody></table></div></div></td> </tr><tr><td class="sidebar-content-with-subgroup"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:transparent;border-top:1px solid #aaa;text-align:center;;color: var(--color-base)"><a href="/wiki/Rheology" title="Rheology">Rheology</a></div><div class="sidebar-list-content mw-collapsible-content"><table class="sidebar-subgroup"><tbody><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Viscoelasticity" title="Viscoelasticity">Viscoelasticity</a></li> <li><a href="/wiki/Rheometry" title="Rheometry">Rheometry</a></li> <li><a href="/wiki/Rheometer" title="Rheometer">Rheometer</a></li></ul></td> </tr><tr><th class="sidebar-heading" style="font-style:italic;"> <a href="/wiki/Smart_fluid" title="Smart fluid">Smart fluids</a></th></tr><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Electrorheological_fluid" title="Electrorheological fluid">Electrorheological</a></li> <li><a href="/wiki/Magnetorheological_fluid" title="Magnetorheological fluid">Magnetorheological</a></li> <li><a href="/wiki/Ferrofluid" title="Ferrofluid">Ferrofluids</a></li></ul></td> </tr></tbody></table></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:transparent;border-top:1px solid #aaa;text-align:center;;color: var(--color-base)">Scientists</div><div class="sidebar-list-content mw-collapsible-content"><div class="hlist"> <ul><li><a href="/wiki/Daniel_Bernoulli" title="Daniel Bernoulli">Bernoulli</a></li> <li><a href="/wiki/Robert_Boyle" title="Robert Boyle">Boyle</a></li> <li><a href="/wiki/Augustin-Louis_Cauchy" title="Augustin-Louis Cauchy">Cauchy</a></li> <li><a href="/wiki/Jacques_Charles" title="Jacques Charles">Charles</a></li> <li><a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Euler</a></li> <li><a href="/wiki/Adolf_Eugen_Fick" title="Adolf Eugen Fick">Fick</a></li> <li><a href="/wiki/Joseph_Louis_Gay-Lussac" title="Joseph Louis Gay-Lussac">Gay-Lussac</a></li> <li><a href="/wiki/Thomas_Graham_(chemist)" title="Thomas Graham (chemist)">Graham</a></li> <li><a href="/wiki/Robert_Hooke" title="Robert Hooke">Hooke</a></li> <li><a href="/wiki/Isaac_Newton" title="Isaac Newton">Newton</a></li> <li><a href="/wiki/Claude-Louis_Navier" title="Claude-Louis Navier">Navier</a></li> <li><a href="/wiki/Walter_Noll" title="Walter Noll">Noll</a></li> <li><a href="/wiki/Blaise_Pascal" title="Blaise Pascal">Pascal</a></li> <li><a href="/wiki/Sir_George_Stokes,_1st_Baronet" title="Sir George Stokes, 1st Baronet">Stokes</a></li> <li><a href="/wiki/Clifford_Truesdell" title="Clifford Truesdell">Truesdell</a></li></ul> </div></div></div></td> </tr><tr><td class="sidebar-navbar"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Continuum_mechanics" title="Template:Continuum mechanics"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Continuum_mechanics" title="Template talk:Continuum mechanics"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Continuum_mechanics" title="Special:EditPage/Template:Continuum mechanics"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> <p>In <a href="/wiki/Continuum_mechanics" title="Continuum mechanics">continuum mechanics</a>, <b>stress</b> is a <a href="/wiki/Physical_quantity" title="Physical quantity">physical quantity</a> that describes <a href="/wiki/Force" title="Force">forces</a> present during <a href="/wiki/Deformation_(physics)" title="Deformation (physics)">deformation</a>. For example, an object being pulled apart, such as a stretched elastic band, is subject to <a href="/wiki/Tension_(physics)" title="Tension (physics)"><i>tensile</i></a> stress and may undergo <a href="/wiki/Elongation_(materials_science)" class="mw-redirect" title="Elongation (materials science)">elongation</a>. An object being pushed together, such as a crumpled sponge, is subject to <a href="/wiki/Compression_(physics)" title="Compression (physics)"><i>compressive</i></a> stress and may undergo shortening.<sup id="cite_ref-:0_1-0" class="reference"><a href="#cite_note-:0-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> The greater the force and the smaller the cross-sectional area of the body on which it acts, the greater the stress. Stress has <a href="/wiki/Dimension_(physics)" class="mw-redirect" title="Dimension (physics)">dimension</a> of force per area, with <a href="/wiki/SI_Units" class="mw-redirect" title="SI Units">SI units</a> of newtons per square meter (N/m<sup>2</sup>) or <a href="/wiki/Pascal_(unit)" title="Pascal (unit)">pascal</a> (Pa).<sup id="cite_ref-:0_1-1" class="reference"><a href="#cite_note-:0-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p><p>Stress expresses the internal forces that neighbouring <a href="/wiki/Particle" title="Particle">particles</a> of a continuous material exert on each other, while <a href="/wiki/Strain_(mechanics)" title="Strain (mechanics)"><i>strain</i></a> is the measure of the relative <a href="/wiki/Deformation_(mechanics)" class="mw-redirect" title="Deformation (mechanics)">deformation</a> of the material.<sup id="cite_ref-:1_3-0" class="reference"><a href="#cite_note-:1-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> For example, when a <a href="/wiki/Solid" title="Solid">solid</a> vertical bar is supporting an overhead <a href="/wiki/Weight" title="Weight">weight</a>, each particle in the bar pushes on the particles immediately below it. When a <a href="/wiki/Liquid" title="Liquid">liquid</a> is in a closed container under <a href="/wiki/Pressure" title="Pressure">pressure</a>, each particle gets pushed against by all the surrounding particles. The container walls and the <a href="/wiki/Pressure" title="Pressure">pressure</a>-inducing surface (such as a piston) push against them in (Newtonian) <a href="/wiki/Reaction_force" class="mw-redirect" title="Reaction force">reaction</a>. These macroscopic forces are actually the net result of a very large number of <a href="/wiki/Intermolecular_force" title="Intermolecular force">intermolecular forces</a> and <a href="/wiki/Statistical_mechanics" title="Statistical mechanics">collisions</a> between the particles in those <a href="/wiki/Molecule" title="Molecule">molecules</a>. Stress is frequently represented by a lowercase Greek letter sigma (<i>σ</i>).<sup id="cite_ref-:1_3-1" class="reference"><a href="#cite_note-:1-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p><p>Strain inside a material may arise by various mechanisms, such as <i>stress</i> as applied by external forces to the bulk material (like <a href="/wiki/Gravity" title="Gravity">gravity</a>) or to its surface (like <a href="/wiki/Contact_force" title="Contact force">contact forces</a>, external pressure, or <a href="/wiki/Friction" title="Friction">friction</a>). Any <a href="/wiki/Deformation_(mechanics)" class="mw-redirect" title="Deformation (mechanics)">strain (deformation)</a> of a solid material generates an internal <i>elastic stress</i>, analogous to the reaction force of a <a href="/wiki/Spring_(device)" title="Spring (device)">spring</a>, that tends to restore the material to its original non-deformed state. In liquids and <a href="/wiki/Gas" title="Gas">gases</a>, only deformations that change the volume generate persistent elastic stress. If the deformation changes gradually with time, even in fluids there will usually be some <i>viscous stress</i>, opposing that change. Elastic and viscous stresses are usually combined under the name <i>mechanical stress</i>. </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Different-types-of-mechanical-stress_EN.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/2a/Different-types-of-mechanical-stress_EN.svg/220px-Different-types-of-mechanical-stress_EN.svg.png" decoding="async" width="220" height="121" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/2a/Different-types-of-mechanical-stress_EN.svg/330px-Different-types-of-mechanical-stress_EN.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/2a/Different-types-of-mechanical-stress_EN.svg/440px-Different-types-of-mechanical-stress_EN.svg.png 2x" data-file-width="612" data-file-height="337" /></a><figcaption>Mechanical stress</figcaption></figure> <p>Significant stress may exist even when deformation is negligible or non-existent (a common assumption when modeling the flow of water). Stress may exist in the absence of external forces; such <i>built-in stress</i> is important, for example, in <a href="/wiki/Prestressed_concrete" title="Prestressed concrete">prestressed concrete</a> and <a href="/wiki/Tempered_glass" title="Tempered glass">tempered glass</a>. Stress may also be imposed on a material without the application of <a href="/wiki/Newton%27s_laws_of_motion#Newton's_second_law" title="Newton's laws of motion">net forces</a>, for example by <a href="/wiki/Thermal_expansion" title="Thermal expansion">changes in temperature</a> or <a href="/wiki/Chemistry" title="Chemistry">chemical</a> composition, or by external <a href="/wiki/Electromagnetic_field" title="Electromagnetic field">electromagnetic fields</a> (as in <a href="/wiki/Piezoelectricity" title="Piezoelectricity">piezoelectric</a> and <a href="/wiki/Magnetostriction" title="Magnetostriction">magnetostrictive</a> materials). </p><p>The relation between mechanical stress, strain, and the <a href="/wiki/Strain_rate" title="Strain rate">strain rate</a> can be quite complicated, although a <a href="/wiki/Linear_elasticity" title="Linear elasticity">linear approximation</a> may be adequate in practice if the quantities are sufficiently small. Stress that exceeds certain <a href="/wiki/Strength_of_materials" title="Strength of materials">strength limits</a> of the material will result in permanent deformation (such as <a href="/wiki/Plasticity_(physics)" title="Plasticity (physics)">plastic flow</a>, <a href="/wiki/Fracture" title="Fracture">fracture</a>, <a href="/wiki/Cavitation" title="Cavitation">cavitation</a>) or even change its <a href="/wiki/Crystal_structure" title="Crystal structure">crystal structure</a> and <a href="/wiki/Chemistry" title="Chemistry">chemical composition</a>. </p> <style data-mw-deduplicate="TemplateStyles:r886046785">.mw-parser-output .toclimit-2 .toclevel-1 ul,.mw-parser-output .toclimit-3 .toclevel-2 ul,.mw-parser-output .toclimit-4 .toclevel-3 ul,.mw-parser-output .toclimit-5 .toclevel-4 ul,.mw-parser-output .toclimit-6 .toclevel-5 ul,.mw-parser-output .toclimit-7 .toclevel-6 ul{display:none}</style><div class="toclimit-2"><meta property="mw:PageProp/toc" /></div> <div class="mw-heading mw-heading2"><h2 id="History">History</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Stress_(mechanics)&action=edit&section=1" title="Edit section: History"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1237032888/mw-parser-output/.tmulti">.mw-parser-output .tmulti .multiimageinner{display:flex;flex-direction:column}.mw-parser-output .tmulti .trow{display:flex;flex-direction:row;clear:left;flex-wrap:wrap;width:100%;box-sizing:border-box}.mw-parser-output .tmulti .tsingle{margin:1px;float:left}.mw-parser-output .tmulti .theader{clear:both;font-weight:bold;text-align:center;align-self:center;background-color:transparent;width:100%}.mw-parser-output .tmulti .thumbcaption{background-color:transparent}.mw-parser-output .tmulti .text-align-left{text-align:left}.mw-parser-output .tmulti .text-align-right{text-align:right}.mw-parser-output .tmulti .text-align-center{text-align:center}@media all and (max-width:720px){.mw-parser-output .tmulti .thumbinner{width:100%!important;box-sizing:border-box;max-width:none!important;align-items:center}.mw-parser-output .tmulti .trow{justify-content:center}.mw-parser-output .tmulti .tsingle{float:none!important;max-width:100%!important;box-sizing:border-box;text-align:center}.mw-parser-output .tmulti .tsingle .thumbcaption{text-align:left}.mw-parser-output .tmulti .trow>.thumbcaption{text-align:center}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .tmulti .multiimageinner img{background-color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .tmulti .multiimageinner img{background-color:white}}</style><div class="thumb tmulti tright"><div class="thumbinner multiimageinner" style="width:292px;max-width:292px"><div class="trow"><div class="tsingle" style="width:290px;max-width:290px"><div class="thumbimage" style="height:115px;overflow:hidden"><span typeof="mw:File"><a href="/wiki/File:Roman_era_stone_arch_bridge,_Ticino,_Switzerland_cropped.JPG" class="mw-file-description"><img alt="Stone arch bridge spanning a river." src="//upload.wikimedia.org/wikipedia/commons/thumb/9/9c/Roman_era_stone_arch_bridge%2C_Ticino%2C_Switzerland_cropped.JPG/288px-Roman_era_stone_arch_bridge%2C_Ticino%2C_Switzerland_cropped.JPG" decoding="async" width="288" height="115" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/9c/Roman_era_stone_arch_bridge%2C_Ticino%2C_Switzerland_cropped.JPG/432px-Roman_era_stone_arch_bridge%2C_Ticino%2C_Switzerland_cropped.JPG 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/9c/Roman_era_stone_arch_bridge%2C_Ticino%2C_Switzerland_cropped.JPG/576px-Roman_era_stone_arch_bridge%2C_Ticino%2C_Switzerland_cropped.JPG 2x" data-file-width="4000" data-file-height="1600" /></a></span></div><div class="thumbcaption"><a href="/wiki/Roman_Empire" title="Roman Empire">Roman</a>-era bridge in <a href="/wiki/Switzerland" title="Switzerland">Switzerland</a>. The stone arches in the bridge are subject to <i>compressive</i> stresses.</div></div></div><div class="trow"><div class="tsingle" style="width:290px;max-width:290px"><div class="thumbimage" style="height:188px;overflow:hidden"><span typeof="mw:File"><a href="/wiki/File:Inca_bridge.jpg" class="mw-file-description"><img alt="Rope bridge spanning a deep river valley." src="//upload.wikimedia.org/wikipedia/commons/thumb/8/87/Inca_bridge.jpg/288px-Inca_bridge.jpg" decoding="async" width="288" height="188" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/8/87/Inca_bridge.jpg 1.5x" data-file-width="431" data-file-height="282" /></a></span></div><div class="thumbcaption"><a href="/wiki/Inca" class="mw-redirect" title="Inca">Inca</a> bridge on the <a href="/wiki/Apurimac_River" class="mw-redirect" title="Apurimac River">Apurimac River</a>. The rope in the bridge is subject to <i>tensile</i> stresses.</div></div></div></div></div> <p>Humans have known about stress inside materials since ancient times. Until the 17th century, this understanding was largely intuitive and empirical, though this did not prevent the development of relatively advanced technologies like the <a href="/wiki/Composite_bow" title="Composite bow">composite bow</a> and <a href="/wiki/Glass_blowing" class="mw-redirect" title="Glass blowing">glass blowing</a>.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p><p>Over several millennia, architects and builders in particular, learned how to put together carefully shaped wood beams and stone blocks to withstand, transmit, and distribute stress in the most effective manner, with ingenious devices such as the <a href="/wiki/Capital_(architecture)" title="Capital (architecture)">capitals</a>, <a href="/wiki/Arch" title="Arch">arches</a>, <a href="/wiki/Cupola" title="Cupola">cupolas</a>, <a href="/wiki/Truss" title="Truss">trusses</a> and the <a href="/wiki/Flying_buttress" title="Flying buttress">flying buttresses</a> of <a href="/wiki/Gothic_architecture" title="Gothic architecture">Gothic cathedrals</a>. </p><p>Ancient and medieval architects did develop some geometrical methods and simple formulas to compute the proper sizes of pillars and beams, but the scientific understanding of stress became possible only after the necessary tools were invented in the 17th and 18th centuries: <a href="/wiki/Galileo_Galilei" title="Galileo Galilei">Galileo Galilei</a>'s rigorous <a href="/wiki/Experimental_method" class="mw-redirect" title="Experimental method">experimental method</a>, <a href="/wiki/Ren%C3%A9_Descartes" title="René Descartes">René Descartes</a>'s <a href="/wiki/Cartesian_coordinates" class="mw-redirect" title="Cartesian coordinates">coordinates</a> and <a href="/wiki/Analytic_geometry" title="Analytic geometry">analytic geometry</a>, and <a href="/wiki/Isaac_Newton" title="Isaac Newton">Newton</a>'s <a href="/wiki/Newton%27s_laws" class="mw-redirect" title="Newton's laws">laws of motion and equilibrium</a> and <a href="/wiki/Calculus" title="Calculus">calculus of infinitesimals</a>.<sup id="cite_ref-Lubliner_5-0" class="reference"><a href="#cite_note-Lubliner-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> With those tools, <a href="/wiki/Augustin-Louis_Cauchy" title="Augustin-Louis Cauchy">Augustin-Louis Cauchy</a> was able to give the first rigorous and general mathematical model of a deformed elastic body by introducing the notions of stress and strain.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> Cauchy observed that the force across an imaginary surface was a linear function of its normal vector; and, moreover, that it must be a symmetric function (with zero total momentum). The understanding of stress in liquids started with Newton, who provided a differential formula for friction forces (shear stress) in parallel <a href="/wiki/Laminar_flow" title="Laminar flow">laminar flow</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Stress_(mechanics)&action=edit&section=2" title="Edit section: Definition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Stress is defined as the force across a small boundary per unit area of that boundary, for all orientations of the boundary.<sup id="cite_ref-Chen_7-0" class="reference"><a href="#cite_note-Chen-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> Derived from a fundamental physical quantity (force) and a purely geometrical quantity (area), stress is also a fundamental quantity, like velocity, <a href="/wiki/Torque" title="Torque">torque</a> or <a href="/wiki/Energy" title="Energy">energy</a>, that can be quantified and analyzed without explicit consideration of the nature of the material or of its physical causes. </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Cmec_stress_defn_f02_t6.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Cmec_stress_defn_f02_t6.png/220px-Cmec_stress_defn_f02_t6.png" decoding="async" width="220" height="293" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Cmec_stress_defn_f02_t6.png/330px-Cmec_stress_defn_f02_t6.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Cmec_stress_defn_f02_t6.png/440px-Cmec_stress_defn_f02_t6.png 2x" data-file-width="600" data-file-height="800" /></a><figcaption>The stress across a surface element (yellow disk) is the force that the material on one side (top ball) exerts on the material on the other side (bottom ball), divided by the area of the surface.</figcaption></figure> <p>Following the basic premises of continuum mechanics, stress is a <a href="/wiki/Macroscopic" class="mw-redirect" title="Macroscopic">macroscopic</a> concept. Namely, the particles considered in its definition and analysis should be just small enough to be treated as homogeneous in composition and state, but still large enough to ignore <a href="/wiki/Quantum_mechanics" title="Quantum mechanics">quantum</a> effects and the detailed motions of molecules. Thus, the force between two particles is actually the average of a very large number of atomic forces between their molecules; and physical quantities like mass, velocity, and forces that act through the bulk of three-dimensional bodies, like gravity, are assumed to be smoothly distributed over them.<sup id="cite_ref-Chadwick_8-0" class="reference"><a href="#cite_note-Chadwick-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Pages: 90–106">: 90–106 </span></sup> Depending on the context, one may also assume that the particles are large enough to allow the averaging out of other microscopic features, like the grains of a <a href="/wiki/Metal" title="Metal">metal</a> rod or the <a href="/wiki/Fiber" title="Fiber">fibers</a> of a piece of <a href="/wiki/Wood" title="Wood">wood</a>. </p><p>Quantitatively, the stress is expressed by the <i>Cauchy traction vector</i> <i>T</i> defined as the <a href="/wiki/Traction_(mechanics)" title="Traction (mechanics)">traction force</a> <i>F</i> between adjacent parts of the material across an imaginary separating surface <i>S</i>, divided by the area of <i>S</i>.<sup id="cite_ref-Liu_9-0" class="reference"><a href="#cite_note-Liu-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Pages: 41–50">: 41–50 </span></sup> In a <a href="/wiki/Fluid" title="Fluid">fluid</a> at rest the force is perpendicular to the surface, and is the familiar <a href="/wiki/Hydrostatic_pressure" class="mw-redirect" title="Hydrostatic pressure">pressure</a>. In a <a href="/wiki/Solid" title="Solid">solid</a>, or in a <a href="/wiki/Fluid_dynamics" title="Fluid dynamics">flow</a> of viscous <a href="/wiki/Liquid" title="Liquid">liquid</a>, the force <i>F</i> may not be perpendicular to <i>S</i>; hence the stress across a surface must be regarded a vector quantity, not a scalar. Moreover, the direction and magnitude generally depend on the orientation of <i>S</i>. Thus the stress state of the material must be described by a <a href="/wiki/Tensor" title="Tensor">tensor</a>, called the <a href="/wiki/Cauchy_stress_tensor" title="Cauchy stress tensor">(Cauchy) stress tensor</a>; which is a <a href="/wiki/Linear_map" title="Linear map">linear function</a> that relates the <a href="/wiki/Surface_normal" class="mw-redirect" title="Surface normal">normal vector</a> <i>n</i> of a surface <i>S</i> to the traction vector <i>T</i> across <i>S</i>. With respect to any chosen <a href="/wiki/Cartesian_coordinates" class="mw-redirect" title="Cartesian coordinates">coordinate system</a>, the Cauchy stress tensor can be represented as a <a href="/wiki/Symmetric_matrix" title="Symmetric matrix">symmetric matrix</a> of 3×3 real numbers. Even within a <a href="/wiki/Homogeneous" class="mw-redirect" title="Homogeneous">homogeneous</a> body, the stress tensor may vary from place to place, and may change over time; therefore, the stress within a material is, in general, a time-varying <a href="/wiki/Tensor_field" title="Tensor field">tensor field</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Normal_and_shear">Normal and shear</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Stress_(mechanics)&action=edit&section=3" title="Edit section: Normal and shear"><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">Further information: <a href="/wiki/Compression_(physical)" class="mw-redirect" title="Compression (physical)">Compression (physical)</a> and <a href="/wiki/Shear_stress" title="Shear stress">Shear stress</a></div> <p>In general, the stress <i>T</i> that a particle <i>P</i> applies on another particle <i>Q</i> across a surface <i>S</i> can have any direction relative to <i>S</i>. The vector <i>T</i> may be regarded as the sum of two components: the <i><a href="/wiki/Normal_(geometry)" title="Normal (geometry)">normal</a> stress</i> (<a href="/wiki/Compression_(physical)" class="mw-redirect" title="Compression (physical)">compression</a> or <a href="/wiki/Tension_(physics)" title="Tension (physics)">tension</a>) perpendicular to the surface, and the <i><a href="/wiki/Shear_stress" title="Shear stress">shear stress</a></i> that is parallel to the surface. </p><p>If the normal unit vector <i>n</i> of the surface (pointing from <i>Q</i> towards <i>P</i>) is assumed fixed, the normal component can be expressed by a single number, the <a href="/wiki/Inner_product" class="mw-redirect" title="Inner product">dot product</a> <span class="nowrap"><i>T</i> · <i>n</i></span>. This number will be positive if <i>P</i> is "pulling" on <i>Q</i> (tensile stress), and negative if <i>P</i> is "pushing" against <i>Q</i> (compressive stress). The shear component is then the vector <span class="nowrap"><i>T</i> − (<i>T</i> · <i>n</i>)<i>n</i></span>. </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=Stress_(mechanics)&action=edit&section=4" title="Edit section: Units"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The dimension of stress is that of <a href="/wiki/Pressure" title="Pressure">pressure</a>, and therefore its coordinates are measured in the same units as pressure: namely, <a href="/wiki/Pascal_(unit)" title="Pascal (unit)">pascals</a> (Pa, that is, <a href="/wiki/Newton_(force)" class="mw-redirect" title="Newton (force)">newtons</a> per <a href="/wiki/Square_metre" title="Square metre">square metre</a>) in the <a href="/wiki/International_System_of_Units" title="International System of Units">International System</a>, or <a href="/wiki/Pound-force" class="mw-redirect" title="Pound-force">pounds</a> per <a href="/wiki/Square_inch" title="Square inch">square inch</a> (psi) in the <a href="/wiki/Imperial_units" title="Imperial units">Imperial system</a>. Because mechanical stresses easily exceed a million Pascals, MPa, which stands for megapascal, is a common unit of stress. </p> <div class="mw-heading mw-heading2"><h2 id="Causes_and_effects">Causes and effects</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Stress_(mechanics)&action=edit&section=5" title="Edit section: Causes and effects"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Vase-craquele-Emile-Galle-vers-1880-decor-mante-religieuse-cigale-1301.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/22/Vase-craquele-Emile-Galle-vers-1880-decor-mante-religieuse-cigale-1301.jpg/170px-Vase-craquele-Emile-Galle-vers-1880-decor-mante-religieuse-cigale-1301.jpg" decoding="async" width="170" height="259" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/22/Vase-craquele-Emile-Galle-vers-1880-decor-mante-religieuse-cigale-1301.jpg/255px-Vase-craquele-Emile-Galle-vers-1880-decor-mante-religieuse-cigale-1301.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/22/Vase-craquele-Emile-Galle-vers-1880-decor-mante-religieuse-cigale-1301.jpg/340px-Vase-craquele-Emile-Galle-vers-1880-decor-mante-religieuse-cigale-1301.jpg 2x" data-file-width="1972" data-file-height="3000" /></a><figcaption>Glass vase with the <i><a href="/wiki/Craquel%C3%A9" class="mw-redirect" title="Craquelé">craquelé</a></i> effect. The cracks are the result of brief but intense stress created when the semi-molten piece is briefly dipped in water.<sup id="cite_ref-lamglass_10-0" class="reference"><a href="#cite_note-lamglass-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup></figcaption></figure> <p>Stress in a material body may be due to multiple physical causes, including external influences and internal physical processes. Some of these agents (like gravity, changes in <a href="/wiki/Temperature" title="Temperature">temperature</a> and <a href="/wiki/Phase_(chemistry)" class="mw-redirect" title="Phase (chemistry)">phase</a>, and electromagnetic fields) act on the bulk of the material, varying continuously with position and time. Other agents (like external loads and friction, ambient pressure, and contact forces) may create stresses and forces that are concentrated on certain surfaces, lines or points; and possibly also on very short time intervals (as in the <a href="/wiki/Impulse_(physics)" title="Impulse (physics)">impulses</a> due to collisions). In <a href="/wiki/Active_matter" title="Active matter">active matter</a>, self-propulsion of microscopic particles generates macroscopic stress profiles.<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> In general, the stress distribution in a body is expressed as a <a href="/wiki/Piecewise" class="mw-redirect" title="Piecewise">piecewise</a> <a href="/wiki/Continuous_function" title="Continuous function">continuous function</a> of space and time. </p><p>Conversely, stress is usually correlated with various effects on the material, possibly including changes in physical properties like <a href="/wiki/Birefringence" title="Birefringence">birefringence</a>, <a href="/wiki/Polarizability" title="Polarizability">polarization</a>, and <a href="/wiki/Permeability_(earth_sciences)" class="mw-redirect" title="Permeability (earth sciences)">permeability</a>. The imposition of stress by an external agent usually creates some <a href="/wiki/Deformation_(mechanics)" class="mw-redirect" title="Deformation (mechanics)">strain (deformation)</a> in the material, even if it is too small to be detected. In a solid material, such strain will in turn generate an internal elastic stress, analogous to the reaction force of a stretched <a href="/wiki/Spring_(device)" title="Spring (device)">spring</a>, tending to restore the material to its original undeformed state. Fluid materials (liquids, <a href="/wiki/Gas" title="Gas">gases</a> and <a href="/wiki/Plasma_(physics)" title="Plasma (physics)">plasmas</a>) by definition can only oppose deformations that would change their volume. If the deformation changes with time, even in fluids there will usually be some viscous stress, opposing that change. Such stresses can be either shear or normal in nature. Molecular origin of shear stresses in fluids is given in the article on <a href="/wiki/Viscosity" title="Viscosity">viscosity</a>. The same for normal viscous stresses can be found in Sharma (2019).<sup id="cite_ref-sharma2019_12-0" class="reference"><a href="#cite_note-sharma2019-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> </p><p>The relation between stress and its effects and causes, including deformation and rate of change of deformation, can be quite complicated (although a <a href="/wiki/Linear_elasticity" title="Linear elasticity">linear approximation</a> may be adequate in practice if the quantities are small enough). Stress that exceeds certain <a href="/wiki/Strength_of_materials" title="Strength of materials">strength limits</a> of the material will result in permanent deformation (such as <a href="/wiki/Plasticity_(physics)" title="Plasticity (physics)">plastic flow</a>, <a href="/wiki/Fracture" title="Fracture">fracture</a>, <a href="/wiki/Cavitation" title="Cavitation">cavitation</a>) or even change its <a href="/wiki/Crystal_structure" title="Crystal structure">crystal structure</a> and <a href="/wiki/Chemistry" title="Chemistry">chemical composition</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Simple_types">Simple types</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Stress_(mechanics)&action=edit&section=6" title="Edit section: Simple types"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In some situations, the stress within a body may adequately be described by a single number, or by a single vector (a number and a direction). Three such <b>simple stress</b> situations, that are often encountered in engineering design, are the <i>uniaxial normal stress</i>, the <i>simple shear stress</i>, and the <i>isotropic normal stress</i>.<sup id="cite_ref-Huston_13-0" class="reference"><a href="#cite_note-Huston-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Uniaxial_normal">Uniaxial normal</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Stress_(mechanics)&action=edit&section=7" title="Edit section: Uniaxial normal"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Axial_stress_noavg.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0f/Axial_stress_noavg.svg/240px-Axial_stress_noavg.svg.png" decoding="async" width="240" height="197" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0f/Axial_stress_noavg.svg/360px-Axial_stress_noavg.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/0f/Axial_stress_noavg.svg/480px-Axial_stress_noavg.svg.png 2x" data-file-width="491" data-file-height="403" /></a><figcaption>Idealized stress in a straight bar with uniform cross-section.</figcaption></figure> <p>A common situation with a simple stress pattern is when a straight rod, with uniform material and cross section, is subjected to <a href="/wiki/Tension_(physics)" title="Tension (physics)">tension</a> by opposite forces of magnitude <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/545fd099af8541605f7ee55f08225526be88ce57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}"></span> along its axis. If the system is in <a href="/wiki/Mechanical_equilibrium" title="Mechanical equilibrium">equilibrium</a> and not changing with time, and the weight of the bar can be neglected, then through each transversal section of the bar the top part must pull on the bottom part with the same force, <i>F</i> with continuity through the full cross-sectional area<i>, A</i>. Therefore, the stress σ throughout the bar, across any horizontal surface, can be expressed simply by the single number σ, calculated simply with the magnitude of those forces, <i>F</i>, and cross sectional area, <i>A</i>.<span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma ={\frac {F}{A}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>σ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>F</mi> <mi>A</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma ={\frac {F}{A}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/31def95c6a21e9623f365a856d09d467dfba3983" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:7.007ex; height:5.343ex;" alt="{\displaystyle \sigma ={\frac {F}{A}}}"></span> On the other hand, if one imagines the bar being cut along its length, parallel to the axis, there will be no force (hence no stress) between the two halves across the cut. This type of stress may be called (simple) normal stress or uniaxial stress; specifically, (uniaxial, simple, etc.) tensile stress.<sup id="cite_ref-Huston_13-1" class="reference"><a href="#cite_note-Huston-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> If the load is <a href="/wiki/Compression_(physical)" class="mw-redirect" title="Compression (physical)">compression</a> on the bar, rather than stretching it, the analysis is the same except that the force <i>F</i> and the 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 \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> change sign, and the stress is called compressive stress. </p> <figure class="mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/File:Normal_stress.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4f/Normal_stress.svg/240px-Normal_stress.svg.png" decoding="async" width="240" height="222" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4f/Normal_stress.svg/360px-Normal_stress.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4f/Normal_stress.svg/480px-Normal_stress.svg.png 2x" data-file-width="477" data-file-height="441" /></a><figcaption>The ratio <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 =F/A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>σ</mi> <mo>=</mo> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma =F/A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f889dfaa80d2d7ebb35ee0dc03353460d196c57f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.074ex; height:2.843ex;" alt="{\displaystyle \sigma =F/A}"></span> may be only an average stress. The stress may be unevenly distributed over the cross section (<i>m</i>–<i>m</i>), especially near the attachment points (<i>n</i>–<i>n</i>).</figcaption></figure> <p>This analysis assumes the stress is evenly distributed over the entire cross-section. In practice, depending on how the bar is attached at the ends and how it was manufactured, this assumption may not be valid. In that case, the value <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> = <i>F</i>/<i>A</i> will be only the average stress, called <i>engineering stress</i> or <i>nominal stress</i>. If the bar's length <i>L</i> is many times its diameter <i>D</i>, and it has no gross defects or <a href="/w/index.php?title=Built-in_stress&action=edit&redlink=1" class="new" title="Built-in stress (page does not exist)">built-in stress</a>, then the stress can be assumed to be uniformly distributed over any cross-section that is more than a few times <i>D</i> from both ends. (This observation is known as the <a href="/wiki/Saint-Venant%27s_principle" title="Saint-Venant's principle">Saint-Venant's principle</a>). </p><p>Normal stress occurs in many other situations besides axial tension and compression. If an elastic bar with uniform and symmetric cross-section is bent in one of its planes of symmetry, the resulting <i>bending stress</i> will still be normal (perpendicular to the cross-section), but will vary over the cross section: the outer part will be under tensile stress, while the inner part will be compressed. Another variant of normal stress is the <i>hoop stress</i> that occurs on the walls of a cylindrical <a href="/wiki/Pipe_(fluid_conveyance)" title="Pipe (fluid conveyance)">pipe</a> or <a href="/wiki/Pressure_vessel" title="Pressure vessel">vessel</a> filled with pressurized fluid. </p> <div class="mw-heading mw-heading3"><h3 id="Shear">Shear</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Stress_(mechanics)&action=edit&section=8" title="Edit section: Shear"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Shear_stress.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/df/Shear_stress.svg/240px-Shear_stress.svg.png" decoding="async" width="240" height="106" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/df/Shear_stress.svg/360px-Shear_stress.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/df/Shear_stress.svg/480px-Shear_stress.svg.png 2x" data-file-width="583" data-file-height="258" /></a><figcaption>Shear stress in a horizontal bar loaded by two offset blocks.</figcaption></figure> <p>Another simple type of stress occurs when a uniformly thick layer of elastic material like glue or rubber is firmly attached to two stiff bodies that are pulled in opposite directions by forces parallel to the layer; or a section of a soft metal bar that is being cut by the jaws of a <a href="/wiki/Snips" title="Snips">scissors-like tool</a>. Let <i>F</i> be the magnitude of those forces, and <i>M</i> be the midplane of that layer. Just as in the normal stress case, the part of the layer on one side of <i>M</i> must pull the other part with the same force <i>F</i>. Assuming that the direction of the forces is known, the stress across <i>M</i> can be expressed simply by the single number <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>τ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tau }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38a7dcde9730ef0853809fefc18d88771f95206c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.202ex; height:1.676ex;" alt="{\displaystyle \tau }"></span>, calculated simply with the magnitude of those forces, <i>F</i> and the cross sectional area, <i>A</i>.<span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau ={\frac {F}{A}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>τ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>F</mi> <mi>A</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tau ={\frac {F}{A}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/44c5ea1cd4f9cd7c880fc64744787b9bed2ce32a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:6.88ex; height:5.343ex;" alt="{\displaystyle \tau ={\frac {F}{A}}}"></span>Unlike normal stress, this <i>simple shear stress</i> is directed parallel to the cross-section considered, rather than perpendicular to it.<sup id="cite_ref-Huston_13-2" class="reference"><a href="#cite_note-Huston-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> For any plane <i>S</i> that is perpendicular to the layer, the net internal force across <i>S</i>, and hence the stress, will be zero. </p><p>As in the case of an axially loaded bar, in practice the shear stress may not be uniformly distributed over the layer; so, as before, the ratio <i>F</i>/<i>A</i> will only be an average ("nominal", "engineering") stress. That average is often sufficient for practical purposes.<sup id="cite_ref-Pilkey_14-0" class="reference"><a href="#cite_note-Pilkey-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page: 292">: 292 </span></sup> Shear stress is observed also when a cylindrical bar such as a <a href="/wiki/Axle" title="Axle">shaft</a> is subjected to opposite torques at its ends. In that case, the shear stress on each cross-section is parallel to the cross-section, but oriented tangentially relative to the axis, and increases with distance from the axis. Significant shear stress occurs in the middle plate (the "web") of <a href="/wiki/I-beam" title="I-beam">I-beams</a> under bending loads, due to the web constraining the end plates ("flanges"). </p> <div class="mw-heading mw-heading3"><h3 id="Isotropic">Isotropic</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Stress_(mechanics)&action=edit&section=9" title="Edit section: Isotropic"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Isotropic_stress_noavg.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4d/Isotropic_stress_noavg.svg/240px-Isotropic_stress_noavg.svg.png" decoding="async" width="240" height="240" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4d/Isotropic_stress_noavg.svg/360px-Isotropic_stress_noavg.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4d/Isotropic_stress_noavg.svg/480px-Isotropic_stress_noavg.svg.png 2x" data-file-width="480" data-file-height="480" /></a><figcaption>Isotropic tensile stress. Top left: Each face of a cube of homogeneous material is pulled by a force with magnitude <i>F</i>, applied evenly over the entire face whose area is <i>A</i>. The force across any section <i>S</i> of the cube must balance the forces applied below the section. In the three sections shown, the forces are <i>F</i> (top right), <i>F</i><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 {\sqrt {2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4afc1e27d418021bf10898eb44a7f5f315735ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.098ex; height:3.009ex;" alt="{\displaystyle {\sqrt {2}}}"></span> (bottom left), and <i>F</i><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 {\sqrt {3}}/2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>3</mn> </msqrt> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {3}}/2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3245e1141ec36a954dd702c886bba16d8c6cb057" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.423ex; height:3.009ex;" alt="{\displaystyle {\sqrt {3}}/2}"></span> (bottom right); and the area of <i>S</i> is <i>A</i>, <i>A</i><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 {\sqrt {2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4afc1e27d418021bf10898eb44a7f5f315735ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.098ex; height:3.009ex;" alt="{\displaystyle {\sqrt {2}}}"></span> and <i>A</i><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 {\sqrt {3}}/2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>3</mn> </msqrt> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {3}}/2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3245e1141ec36a954dd702c886bba16d8c6cb057" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.423ex; height:3.009ex;" alt="{\displaystyle {\sqrt {3}}/2}"></span>, respectively. So the stress across <i>S</i> is <i>F</i>/<i>A</i> in all three cases.</figcaption></figure> <p>Another simple type of stress occurs when the material body is under equal compression or tension in all directions. This is the case, for example, in a portion of liquid or gas at rest, whether enclosed in some container or as part of a larger mass of fluid; or inside a cube of elastic material that is being pressed or pulled on all six faces by equal perpendicular forces — provided, in both cases, that the material is homogeneous, without built-in stress, and that the effect of gravity and other external forces can be neglected. </p><p>In these situations, the stress across any imaginary internal surface turns out to be equal in magnitude and always directed perpendicularly to the surface independently of the surface's orientation. This type of stress may be called <i>isotropic normal</i> or just <i>isotropic</i>; if it is compressive, it is called <i>hydrostatic pressure</i> or just <i>pressure</i>. Gases by definition cannot withstand tensile stresses, but some liquids may withstand very large amounts of isotropic tensile stress under some circumstances. see <a href="/wiki/Z-tube" title="Z-tube">Z-tube</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Cylinder">Cylinder</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Stress_(mechanics)&action=edit&section=10" title="Edit section: Cylinder"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Parts with <a href="/wiki/Rotational_symmetry" title="Rotational symmetry">rotational symmetry</a>, such as wheels, axles, pipes, and pillars, are very common in engineering. Often the stress patterns that occur in such parts have rotational or even <a href="/wiki/Cylindrical_symmetry" class="mw-redirect" title="Cylindrical symmetry">cylindrical symmetry</a>. The analysis of such <a href="/wiki/Cylinder_stress" title="Cylinder stress">cylinder stresses</a> can take advantage of the symmetry to reduce the dimension of the domain and/or of the stress tensor. </p> <div class="mw-heading mw-heading2"><h2 id="General_types">General types</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Stress_(mechanics)&action=edit&section=11" title="Edit section: General types"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Often, mechanical bodies experience more than one type of stress at the same time; this is called <i>combined stress</i>. In normal and shear stress, the magnitude of the stress is maximum for surfaces that are perpendicular to a certain direction <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 d}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e85ff03cbe0c7341af6b982e47e9f90d235c66ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.216ex; height:2.176ex;" alt="{\displaystyle d}"></span>, and zero across any surfaces that are parallel to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e85ff03cbe0c7341af6b982e47e9f90d235c66ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.216ex; height:2.176ex;" alt="{\displaystyle d}"></span>. When the shear stress is zero only across surfaces that are perpendicular to one particular direction, the stress is called <i>biaxial</i>, and can be viewed as the sum of two normal or shear stresses. In the most general case, called <i>triaxial stress</i>, the stress is nonzero across every surface element. </p> <div class="mw-heading mw-heading2"><h2 id="Cauchy_tensor">Cauchy tensor</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Stress_(mechanics)&action=edit&section=12" title="Edit section: Cauchy tensor"><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_stress_tensor" title="Cauchy stress tensor">Cauchy stress tensor</a></div> <figure class="mw-default-size mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/File:Components_stress_tensor_cartesian.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b3/Components_stress_tensor_cartesian.svg/220px-Components_stress_tensor_cartesian.svg.png" decoding="async" width="220" height="185" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b3/Components_stress_tensor_cartesian.svg/330px-Components_stress_tensor_cartesian.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b3/Components_stress_tensor_cartesian.svg/440px-Components_stress_tensor_cartesian.svg.png 2x" data-file-width="505" data-file-height="425" /></a><figcaption>Components of stress in three dimensions</figcaption></figure> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Cmec_stress_ball_f02_t6.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Cmec_stress_ball_f02_t6.png/220px-Cmec_stress_ball_f02_t6.png" decoding="async" width="220" height="181" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Cmec_stress_ball_f02_t6.png/330px-Cmec_stress_ball_f02_t6.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Cmec_stress_ball_f02_t6.png/440px-Cmec_stress_ball_f02_t6.png 2x" data-file-width="510" data-file-height="420" /></a><figcaption>Illustration of typical stresses (arrows) across various surface elements on the boundary of a particle (sphere), in a homogeneous material under uniform (but not isotropic) triaxial stress. The normal stresses on the principal axes are +5, +2, and −3 units.</figcaption></figure> <p>Combined stresses cannot be described by a single vector. Even if the material is stressed in the same way throughout the volume of the body, the stress across any imaginary surface will depend on the orientation of that surface, in a non-trivial way. </p><p>Cauchy observed that the stress vector <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/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}"></span> across a surface will always be a <a href="/wiki/Linear_function" title="Linear function">linear function</a> of the surface's <a href="/wiki/Surface_normal" class="mw-redirect" title="Surface normal">normal vector</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 n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span>, the unit-length vector that is perpendicular to it. That is, <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={\boldsymbol {\sigma }}(n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">σ</mi> </mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T={\boldsymbol {\sigma }}(n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cb5195cd0cec03b77b89578cd0bbdb1ababb0356" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.533ex; height:2.843ex;" alt="{\displaystyle T={\boldsymbol {\sigma }}(n)}"></span>, where 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 {\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> satisfies <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {\sigma }}(\alpha u+\beta v)=\alpha {\boldsymbol {\sigma }}(u)+\beta {\boldsymbol {\sigma }}(v)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">σ</mi> </mrow> <mo stretchy="false">(</mo> <mi>α</mi> <mi>u</mi> <mo>+</mo> <mi>β</mi> <mi>v</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">σ</mi> </mrow> <mo stretchy="false">(</mo> <mi>u</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">σ</mi> </mrow> <mo stretchy="false">(</mo> <mi>v</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {\sigma }}(\alpha u+\beta v)=\alpha {\boldsymbol {\sigma }}(u)+\beta {\boldsymbol {\sigma }}(v)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b237b2108952c93505349a4c64d00b52646d3253" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.544ex; height:2.843ex;" alt="{\displaystyle {\boldsymbol {\sigma }}(\alpha u+\beta v)=\alpha {\boldsymbol {\sigma }}(u)+\beta {\boldsymbol {\sigma }}(v)}"></span> for any vectors <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 u,v}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> <mo>,</mo> <mi>v</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u,v}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e66f4b32a0181923cc1337a5634f38241e5c697" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.491ex; height:2.009ex;" alt="{\displaystyle u,v}"></span> and any real numbers <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 \alpha ,\beta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>,</mo> <mi>β</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha ,\beta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e4b46b57cfa0011b643037751809904d915c1b48" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.854ex; height:2.509ex;" alt="{\displaystyle \alpha ,\beta }"></span>. 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 {\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>, now called the <a href="/wiki/Cauchy_stress_tensor" title="Cauchy stress tensor">(Cauchy) stress tensor</a>, completely describes the stress state of a uniformly stressed body. (Today, any linear connection between two physical vector quantities is called a <a href="/wiki/Tensor" title="Tensor">tensor</a>, reflecting Cauchy's original use to describe the "tensions" (stresses) in a material.) In <a href="/wiki/Tensor_calculus" class="mw-redirect" title="Tensor calculus">tensor calculus</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 {\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> is classified as a second-order tensor of <a href="/wiki/Type_of_a_tensor" class="mw-redirect" title="Type of a tensor">type</a> (0,2) or (1,1) depending on convention. </p><p>Like any linear map between vectors, the stress tensor can be represented in any chosen <a href="/wiki/Cartesian_coordinate_system" title="Cartesian coordinate system">Cartesian coordinate system</a> by a 3×3 matrix of real numbers. Depending on whether the coordinates are numbered <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_{1},x_{2},x_{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{1},x_{2},x_{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d4102ba3aa87d8bd353467896b23eae57f4fb06" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.22ex; height:2.009ex;" alt="{\displaystyle x_{1},x_{2},x_{3}}"></span> or named <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,y,z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x,y,z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bbeca34b28f569a407ef74a955d041df9f360268" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.641ex; height:2.009ex;" alt="{\displaystyle x,y,z}"></span>, the matrix may be written as <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}\sigma _{11}&\sigma _{12}&\sigma _{13}\\\sigma _{21}&\sigma _{22}&\sigma _{23}\\\sigma _{31}&\sigma _{32}&\sigma _{33}\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>11</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>13</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>21</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>22</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>23</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>31</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>32</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>33</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}\sigma _{11}&\sigma _{12}&\sigma _{13}\\\sigma _{21}&\sigma _{22}&\sigma _{23}\\\sigma _{31}&\sigma _{32}&\sigma _{33}\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6b61f5ca0cb2060e56f0ecf9ae6aa8941b1c82de" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:18.108ex; height:9.176ex;" alt="{\displaystyle {\begin{bmatrix}\sigma _{11}&\sigma _{12}&\sigma _{13}\\\sigma _{21}&\sigma _{22}&\sigma _{23}\\\sigma _{31}&\sigma _{32}&\sigma _{33}\end{bmatrix}}}"></span> or <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}\sigma _{xx}&\sigma _{xy}&\sigma _{xz}\\\sigma _{yx}&\sigma _{yy}&\sigma _{yz}\\\sigma _{zx}&\sigma _{zy}&\sigma _{zz}\\\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>x</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>y</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>z</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> <mi>x</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> <mi>y</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> <mi>z</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> <mi>x</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> <mi>y</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> <mi>z</mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}\sigma _{xx}&\sigma _{xy}&\sigma _{xz}\\\sigma _{yx}&\sigma _{yy}&\sigma _{yz}\\\sigma _{zx}&\sigma _{zy}&\sigma _{zz}\\\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/589f35de026b0ecac396bacf0887f7ff5b476ae3" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.338ex; width:18.524ex; height:9.843ex;" alt="{\displaystyle {\begin{bmatrix}\sigma _{xx}&\sigma _{xy}&\sigma _{xz}\\\sigma _{yx}&\sigma _{yy}&\sigma _{yz}\\\sigma _{zx}&\sigma _{zy}&\sigma _{zz}\\\end{bmatrix}}}"></span> The stress vector <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={\boldsymbol {\sigma }}(n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">σ</mi> </mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T={\boldsymbol {\sigma }}(n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cb5195cd0cec03b77b89578cd0bbdb1ababb0356" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.533ex; height:2.843ex;" alt="{\displaystyle T={\boldsymbol {\sigma }}(n)}"></span> across a surface with <a href="/wiki/Normal_vector" class="mw-redirect" title="Normal vector">normal vector</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 n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> (which is <a href="/wiki/Covariance_and_contravariance_of_vectors#Covariant_transformation" title="Covariance and contravariance of vectors">covariant</a> - <a href="/wiki/Row_and_column_vectors" title="Row and column vectors">"row; horizontal"</a> - vector) with coordinates <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 n_{1},n_{2},n_{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n_{1},n_{2},n_{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b5ad7be042695833cfb38db88f2293c75cfcccd3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.415ex; height:2.009ex;" alt="{\displaystyle n_{1},n_{2},n_{3}}"></span> is then a matrix product <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=n\cdot {\boldsymbol {\sigma }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo>=</mo> <mi>n</mi> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">σ</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T=n\cdot {\boldsymbol {\sigma }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5f7ea9f96f6cea34a82a15527d441d09af7a8679" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.403ex; height:2.176ex;" alt="{\displaystyle T=n\cdot {\boldsymbol {\sigma }}}"></span> (where T in upper index is <a href="/wiki/Transpose" title="Transpose">transposition</a>, and as a result we get <a href="/wiki/Covariance_and_contravariance_of_vectors#Covariant_transformation" title="Covariance and contravariance of vectors">covariant</a> (row) vector) (look on <a href="/wiki/Cauchy_stress_tensor" title="Cauchy stress tensor">Cauchy stress tensor</a>), that is <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}T_{1}&T_{2}&T_{3}\end{bmatrix}}={\begin{bmatrix}n_{1}&n_{2}&n_{3}\end{bmatrix}}\cdot {\begin{bmatrix}\sigma _{11}&\sigma _{21}&\sigma _{31}\\\sigma _{12}&\sigma _{22}&\sigma _{32}\\\sigma _{13}&\sigma _{23}&\sigma _{33}\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>11</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>21</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>31</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>22</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>32</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>13</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>23</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>33</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}T_{1}&T_{2}&T_{3}\end{bmatrix}}={\begin{bmatrix}n_{1}&n_{2}&n_{3}\end{bmatrix}}\cdot {\begin{bmatrix}\sigma _{11}&\sigma _{21}&\sigma _{31}\\\sigma _{12}&\sigma _{22}&\sigma _{32}\\\sigma _{13}&\sigma _{23}&\sigma _{33}\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e3df4858c33a33e90f0d2b04e0fac5c40ddf6ddc" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:50.849ex; height:9.176ex;" alt="{\displaystyle {\begin{bmatrix}T_{1}&T_{2}&T_{3}\end{bmatrix}}={\begin{bmatrix}n_{1}&n_{2}&n_{3}\end{bmatrix}}\cdot {\begin{bmatrix}\sigma _{11}&\sigma _{21}&\sigma _{31}\\\sigma _{12}&\sigma _{22}&\sigma _{32}\\\sigma _{13}&\sigma _{23}&\sigma _{33}\end{bmatrix}}}"></span> </p><p>The linear relation between <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/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}"></span> 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 n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> follows from the fundamental laws of <a href="/wiki/Conservation_of_linear_momentum" class="mw-redirect" title="Conservation of linear momentum">conservation of linear momentum</a> and <a href="/wiki/Static_equilibrium" class="mw-redirect" title="Static equilibrium">static equilibrium</a> of forces, and is therefore mathematically exact, for any material and any stress situation. The components of the Cauchy stress tensor at every point in a material satisfy the equilibrium equations (<a href="/wiki/Cauchy_momentum_equation" title="Cauchy momentum equation">Cauchy's equations of motion</a> for zero acceleration). Moreover, the principle of <a href="/wiki/Conservation_of_angular_momentum" class="mw-redirect" title="Conservation of angular momentum">conservation of angular momentum</a> implies that the stress tensor is <a href="/wiki/Symmetric_matrix" title="Symmetric matrix">symmetric</a>, that is <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 _{12}=\sigma _{21}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>21</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{12}=\sigma _{21}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/67a18cec116597945dace5009f601b570edb9c84" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.506ex; height:2.009ex;" alt="{\displaystyle \sigma _{12}=\sigma _{21}}"></span>, <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 _{13}=\sigma _{31}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>13</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>31</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{13}=\sigma _{31}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/90610519cc6a5be456f989fa1a95b444345e14bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.506ex; height:2.009ex;" alt="{\displaystyle \sigma _{13}=\sigma _{31}}"></span>, 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 \sigma _{23}=\sigma _{32}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>23</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>32</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{23}=\sigma _{32}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6ade8f57282d2f8ff2b7790352ba85d9c86ca16c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.506ex; height:2.009ex;" alt="{\displaystyle \sigma _{23}=\sigma _{32}}"></span>. Therefore, the stress state of the medium at any point and instant can be specified by only six independent parameters, rather than nine. These may be written <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}\sigma _{x}&\tau _{xy}&\tau _{xz}\\\tau _{xy}&\sigma _{y}&\tau _{yz}\\\tau _{xz}&\tau _{yz}&\sigma _{z}\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>y</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>z</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>y</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> <mi>z</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>z</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> <mi>z</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}\sigma _{x}&\tau _{xy}&\tau _{xz}\\\tau _{xy}&\sigma _{y}&\tau _{yz}\\\tau _{xz}&\tau _{yz}&\sigma _{z}\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2cd8311ce8d01c50e0744deceb2b704659463113" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.338ex; width:17.467ex; height:9.843ex;" alt="{\displaystyle {\begin{bmatrix}\sigma _{x}&\tau _{xy}&\tau _{xz}\\\tau _{xy}&\sigma _{y}&\tau _{yz}\\\tau _{xz}&\tau _{yz}&\sigma _{z}\end{bmatrix}}}"></span> where the elements <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 _{x},\sigma _{y},\sigma _{z}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{x},\sigma _{y},\sigma _{z}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4dd4e3c33f292dac57e8aefab903caace2eeebd4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.273ex; height:2.343ex;" alt="{\displaystyle \sigma _{x},\sigma _{y},\sigma _{z}}"></span> are called the <i>orthogonal normal stresses</i> (relative to the chosen coordinate system), 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 \tau _{xy},\tau _{xz},\tau _{yz}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>y</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>z</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> <mi>z</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tau _{xy},\tau _{xz},\tau _{yz}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3de24e5031f23a643987baec9cfea87494896537" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.866ex; height:2.343ex;" alt="{\displaystyle \tau _{xy},\tau _{xz},\tau _{yz}}"></span> the <i>orthogonal shear stresses</i>.<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 2021)">citation needed</span></a></i>]</sup> </p> <div class="mw-heading mw-heading3"><h3 id="Change_of_coordinates">Change of coordinates</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Stress_(mechanics)&action=edit&section=13" title="Edit section: Change of coordinates"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The Cauchy stress tensor obeys the tensor transformation law under a change in the system of coordinates. A graphical representation of this transformation law is the <a href="/wiki/Mohr%27s_circle" title="Mohr's circle">Mohr's circle</a> of stress distribution. </p><p>As a symmetric 3×3 real matrix, the stress tensor <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> has three mutually orthogonal unit-length <a href="/wiki/Eigenvalues_and_eigenvectors" title="Eigenvalues and eigenvectors">eigenvectors</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e_{1},e_{2},e_{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e_{1},e_{2},e_{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e35821963b9761d80f5b4079f1ba179ac101da80" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.481ex; height:2.009ex;" alt="{\displaystyle e_{1},e_{2},e_{3}}"></span> and three real <a href="/wiki/Eigenvalues_and_eigenvectors" title="Eigenvalues and eigenvectors">eigenvalues</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda _{1},\lambda _{2},\lambda _{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda _{1},\lambda _{2},\lambda _{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f2b8c04bd1b2d2ea63bf44bf3ef1712baa2e7248" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.296ex; height:2.509ex;" alt="{\displaystyle \lambda _{1},\lambda _{2},\lambda _{3}}"></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 {\boldsymbol {\sigma }}e_{i}=\lambda _{i}e_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">σ</mi> </mrow> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {\sigma }}e_{i}=\lambda _{i}e_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2a3b4adc882ba1b2c24887d32d2b449fe198419c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.614ex; height:2.509ex;" alt="{\displaystyle {\boldsymbol {\sigma }}e_{i}=\lambda _{i}e_{i}}"></span>. Therefore, in a coordinate system with axes <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e_{1},e_{2},e_{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e_{1},e_{2},e_{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e35821963b9761d80f5b4079f1ba179ac101da80" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.481ex; height:2.009ex;" alt="{\displaystyle e_{1},e_{2},e_{3}}"></span>, the stress tensor is a diagonal matrix, and has only the three normal components <span class="mwe-math-element" data-qid="Q112270193"><a href="/w/index.php?title=Special:MathWikibase&qid=Q112270193" style="color:inherit;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda _{1},\lambda _{2},\lambda _{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda _{1},\lambda _{2},\lambda _{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f2b8c04bd1b2d2ea63bf44bf3ef1712baa2e7248" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.296ex; height:2.509ex;" alt="{\displaystyle \lambda _{1},\lambda _{2},\lambda _{3}}"></a></span> the <a href="/wiki/Principal_stresses" class="mw-redirect" title="Principal stresses">principal stresses</a>. If the three eigenvalues are equal, the stress is an <a href="/wiki/Isotropic" class="mw-redirect" title="Isotropic">isotropic</a> compression or tension, always perpendicular to any surface, there is no shear stress, and the tensor is a diagonal matrix in any coordinate frame. </p> <div class="mw-heading mw-heading3"><h3 id="Tensor_field">Tensor field</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Stress_(mechanics)&action=edit&section=14" title="Edit section: Tensor field"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In general, stress is not uniformly distributed over a material body, and may vary with time. Therefore, the stress tensor must be defined for each point and each moment, by considering an <a href="/wiki/Infinitesimal" title="Infinitesimal">infinitesimal</a> particle of the medium surrounding that point, and taking the average stresses in that particle as being the stresses at the point. </p> <div class="mw-heading mw-heading3"><h3 id="Thin_plates">Thin plates</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Stress_(mechanics)&action=edit&section=15" title="Edit section: Thin plates"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:W39504_stat_Nbk2007.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/6d/W39504_stat_Nbk2007.jpg/220px-W39504_stat_Nbk2007.jpg" decoding="async" width="220" height="149" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/6d/W39504_stat_Nbk2007.jpg/330px-W39504_stat_Nbk2007.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/6d/W39504_stat_Nbk2007.jpg/440px-W39504_stat_Nbk2007.jpg 2x" data-file-width="1500" data-file-height="1013" /></a><figcaption>A <a href="/wiki/Tank_car" title="Tank car">tank car</a> made from bent and welded steel plates.</figcaption></figure> <p>Human-made objects are often made from stock plates of various materials by operations that do not change their essentially two-dimensional character, like cutting, drilling, gentle bending and welding along the edges. The description of stress in such bodies can be simplified by modeling those parts as two-dimensional surfaces rather than three-dimensional bodies. </p><p>In that view, one redefines a "particle" as being an infinitesimal patch of the plate's surface, so that the boundary between adjacent particles becomes an infinitesimal line element; both are implicitly extended in the third dimension, normal to (straight through) the plate. "Stress" is then redefined as being a measure of the internal forces between two adjacent "particles" across their common line element, divided by the length of that line. Some components of the stress tensor can be ignored, but since particles are not infinitesimal in the third dimension one can no longer ignore the torque that a particle applies on its neighbors. That torque is modeled as a <i>bending stress</i> that tends to change the <a href="/wiki/Curvature" title="Curvature">curvature</a> of the plate. These simplifications may not hold at welds, at sharp bends and creases (where the <a href="/wiki/Radius_of_curvature_(mathematics)" class="mw-redirect" title="Radius of curvature (mathematics)">radius of curvature</a> is comparable to the thickness of the plate). </p> <div class="mw-heading mw-heading3"><h3 id="Thin_beams">Thin beams</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Stress_(mechanics)&action=edit&section=16" title="Edit section: Thin beams"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Sandy_Hook_NJ_beach_fisherman%27s_pole.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/02/Sandy_Hook_NJ_beach_fisherman%27s_pole.jpg/140px-Sandy_Hook_NJ_beach_fisherman%27s_pole.jpg" decoding="async" width="140" height="290" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/02/Sandy_Hook_NJ_beach_fisherman%27s_pole.jpg/210px-Sandy_Hook_NJ_beach_fisherman%27s_pole.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/02/Sandy_Hook_NJ_beach_fisherman%27s_pole.jpg/280px-Sandy_Hook_NJ_beach_fisherman%27s_pole.jpg 2x" data-file-width="640" data-file-height="1327" /></a><figcaption>For stress modeling, a <a href="/wiki/Fishing_pole" class="mw-redirect" title="Fishing pole">fishing pole</a> may be considered one-dimensional.</figcaption></figure> <p>The analysis of stress can be considerably simplified also for thin bars, beams or wires of uniform (or smoothly varying) composition and cross-section that are subjected to moderate bending and twisting. For those bodies, one may consider only cross-sections that are perpendicular to the bar's axis, and redefine a "particle" as being a piece of wire with infinitesimal length between two such cross sections. The ordinary stress is then reduced to a scalar (tension or compression of the bar), but one must take into account also a <i>bending stress</i> (that tries to change the bar's curvature, in some direction perpendicular to the axis) and a <i>torsional stress</i> (that tries to twist or un-twist it about its axis). </p> <div class="mw-heading mw-heading2"><h2 id="Analysis">Analysis</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Stress_(mechanics)&action=edit&section=17" title="Edit section: Analysis"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Stress_analysis" class="mw-redirect" title="Stress analysis">Stress analysis</a> is a branch of <a href="/wiki/Applied_physics" title="Applied physics">applied physics</a> that covers the determination of the internal distribution of internal forces in solid objects. It is an essential tool in engineering for the study and design of structures such as tunnels, dams, mechanical parts, and structural frames, under prescribed or expected loads. It is also important in many other disciplines; for example, in geology, to study phenomena like <a href="/wiki/Plate_tectonics" title="Plate tectonics">plate tectonics</a>, vulcanism and <a href="/wiki/Avalanche" title="Avalanche">avalanches</a>; and in biology, to understand the anatomy of living beings. </p> <div class="mw-heading mw-heading3"><h3 id="Goals_and_assumptions">Goals and assumptions</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Stress_(mechanics)&action=edit&section=18" title="Edit section: Goals and assumptions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Stress analysis is generally concerned with objects and structures that can be assumed to be in macroscopic <a href="/wiki/Static_equilibrium" class="mw-redirect" title="Static equilibrium">static equilibrium</a>. By <a href="/wiki/Newton%27s_laws_of_motion" title="Newton's laws of motion">Newton's laws of motion</a>, any external forces being applied to such a system must be balanced by internal reaction forces,<sup id="cite_ref-Smith_15-0" class="reference"><a href="#cite_note-Smith-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page: 97">: 97 </span></sup> which are almost always surface contact forces between adjacent particles — that is, as stress.<sup id="cite_ref-Liu_9-1" class="reference"><a href="#cite_note-Liu-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> Since every particle needs to be in equilibrium, this reaction stress will generally propagate from particle to particle, creating a stress distribution throughout the body. The typical problem in stress analysis is to determine these internal stresses, given the external forces that are acting on the system. The latter may be <a href="/wiki/Body_force" title="Body force">body forces</a> (such as gravity or magnetic attraction), that act throughout the volume of a material;<sup id="cite_ref-Irgens_16-0" class="reference"><a href="#cite_note-Irgens-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Pages: 42–81">: 42–81 </span></sup> or concentrated loads (such as friction between an axle and a <a href="/wiki/Bearing_(mechanical)" title="Bearing (mechanical)">bearing</a>, or the weight of a train wheel on a rail), that are imagined to act over a two-dimensional area, or along a line, or at single point. </p><p>In stress analysis one normally disregards the physical causes of the forces or the precise nature of the materials. Instead, one assumes that the stresses are related to deformation (and, in non-static problems, to the rate of deformation) of the material by known <a href="/wiki/Constitutive_equations" class="mw-redirect" title="Constitutive equations">constitutive equations</a>.<sup id="cite_ref-Slaughter_17-0" class="reference"><a href="#cite_note-Slaughter-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Methods">Methods</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Stress_(mechanics)&action=edit&section=19" title="Edit section: Methods"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Stress analysis may be carried out experimentally, by applying loads to the actual artifact or to scale model, and measuring the resulting stresses, by any of several available methods. This approach is often used for safety certification and monitoring. Most stress is analysed by mathematical methods, especially during design. The basic stress analysis problem can be formulated by <a href="/wiki/Euler%27s_laws" class="mw-redirect" title="Euler's laws">Euler's equations of motion</a> for continuous bodies (which are consequences of <a href="/wiki/Newton%27s_laws_of_motion" title="Newton's laws of motion">Newton's laws</a> for conservation of <a href="/wiki/Linear_momentum" class="mw-redirect" title="Linear momentum">linear momentum</a> and <a href="/wiki/Angular_momentum" title="Angular momentum">angular momentum</a>) and the <a href="/wiki/Euler-Cauchy_stress_principle" class="mw-redirect" title="Euler-Cauchy stress principle">Euler-Cauchy stress principle</a>, together with the appropriate constitutive equations. Thus one obtains a system of <a href="/wiki/Partial_differential_equations" class="mw-redirect" title="Partial differential equations">partial differential equations</a> involving the stress tensor field and the <a href="/wiki/Strain_tensor" class="mw-redirect" title="Strain tensor">strain tensor</a> field, as unknown functions to be determined. The external body forces appear as the independent ("right-hand side") term in the differential equations, while the concentrated forces appear as boundary conditions. The basic stress analysis problem is therefore a <a href="/wiki/Boundary-value_problem" class="mw-redirect" title="Boundary-value problem">boundary-value problem</a>. </p><p>Stress analysis for <a href="/wiki/Elasticity_(physics)" title="Elasticity (physics)">elastic</a> structures is based on the <a href="/wiki/Theory_of_elasticity" class="mw-redirect" title="Theory of elasticity">theory of elasticity</a> and <a href="/wiki/Infinitesimal_strain_theory" title="Infinitesimal strain theory">infinitesimal strain theory</a>. When the applied loads cause permanent deformation, one must use more complicated constitutive equations, that can account for the physical processes involved (<a href="/wiki/Plasticity_(physics)" title="Plasticity (physics)">plastic flow</a>, <a href="/wiki/Fracture" title="Fracture">fracture</a>, <a href="/wiki/Phase_transition" title="Phase transition">phase change</a>, etc.). Engineered structures are usually designed so the maximum expected stresses are well within the range of <a href="/wiki/Linear_elasticity" title="Linear elasticity">linear elasticity</a> (the generalization of <a href="/wiki/Hooke%E2%80%99s_law" class="mw-redirect" title="Hooke’s law">Hooke's law</a> for continuous media); that is, the deformations caused by internal stresses are linearly related to them. In this case the differential equations that define the stress tensor are linear, and the problem becomes much easier. For one thing, the stress at any point will be a linear function of the loads, too. For small enough stresses, even non-linear systems can usually be assumed to be linear. </p> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Loaded_truss.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b3/Loaded_truss.svg/240px-Loaded_truss.svg.png" decoding="async" width="240" height="144" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b3/Loaded_truss.svg/360px-Loaded_truss.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b3/Loaded_truss.svg/480px-Loaded_truss.svg.png 2x" data-file-width="500" data-file-height="300" /></a><figcaption>Simplified model of a truss for stress analysis, assuming unidimensional elements under uniform axial tension or compression.</figcaption></figure> <p>Stress analysis is simplified when the physical dimensions and the distribution of loads allow the structure to be treated as one- or two-dimensional. In the analysis of trusses, for example, the stress field may be assumed to be uniform and uniaxial over each member. Then the differential equations reduce to a finite set of equations (usually linear) with finitely many unknowns. In other contexts one may be able to reduce the three-dimensional problem to a two-dimensional one, and/or replace the general stress and strain tensors by simpler models like uniaxial tension/compression, simple shear, etc. </p><p>Still, for two- or three-dimensional cases one must solve a partial differential equation problem. Analytical or closed-form solutions to the differential equations can be obtained when the geometry, constitutive relations, and boundary conditions are simple enough. Otherwise one must generally resort to numerical approximations such as the <a href="/wiki/Finite_element_method" title="Finite element method">finite element method</a>, the <a href="/wiki/Finite_difference_method" title="Finite difference method">finite difference method</a>, and the <a href="/wiki/Boundary_element_method" title="Boundary element method">boundary element method</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Measures">Measures</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Stress_(mechanics)&action=edit&section=20" title="Edit section: Measures"><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/Stress_measures" class="mw-redirect" title="Stress measures">Stress measures</a></div> <p>Other useful stress measures include the first and second <a href="/wiki/Piola%E2%80%93Kirchhoff_stress_tensor" class="mw-redirect" title="Piola–Kirchhoff stress tensor">Piola–Kirchhoff stress tensors</a>, the <a href="/wiki/Biot_stress_tensor" class="mw-redirect" title="Biot stress tensor">Biot stress tensor</a>, and the <a href="/wiki/Kirchhoff_stress_tensor" class="mw-redirect" title="Kirchhoff stress tensor">Kirchhoff stress tensor</a>. </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=Stress_(mechanics)&action=edit&section=21" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1246091330"><table class="sidebar nomobile nowraplinks wraplinks" style="width:18em;"><tbody><tr><td class="sidebar-above" style="background:lavender; font-size:120%;"> <a href="/wiki/Conjugate_variables_(thermodynamics)" title="Conjugate variables (thermodynamics)">Conjugate variables<br />of thermodynamics</a></td></tr><tr><td class="sidebar-content" style="padding:2px;"> <table class="wikitable" style="width:100%;border-collapse:collapse;border-spacing:0px 0px;border:none;margin:1px 0 0 0;"><tbody><tr style="vertical-align:top"><td> <a href="/wiki/Pressure" title="Pressure">Pressure</a></td><td> <a href="/wiki/Volume_(thermodynamics)" title="Volume (thermodynamics)">Volume</a></td></tr><tr style="vertical-align:top"><td> (<a class="mw-selflink selflink">Stress</a>)</td><td> (<a href="/wiki/Deformation_(physics)#Strain" title="Deformation (physics)">Strain</a>)</td></tr><tr style="vertical-align:top"><td> <a href="/wiki/Temperature" title="Temperature">Temperature</a></td><td> <a href="/wiki/Entropy" title="Entropy">Entropy</a></td></tr><tr style="vertical-align:top"><td> <a href="/wiki/Chemical_potential" title="Chemical potential">Chemical potential</a></td><td> <a href="/wiki/Particle_number" title="Particle number">Particle number</a></td></tr></tbody></table></td> </tr></tbody></table><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/Bending" title="Bending">Bending</a></li> <li><a href="/wiki/Compressive_strength" title="Compressive strength">Compressive strength</a></li> <li><a href="/wiki/Critical_plane_analysis" title="Critical plane analysis">Critical plane analysis</a></li> <li><a href="/wiki/Kelvin_probe_force_microscope" title="Kelvin probe force microscope">Kelvin probe force microscope</a></li> <li><a href="/wiki/Mohr%27s_circle" title="Mohr's circle">Mohr's circle</a></li> <li><a href="/wiki/Lam%C3%A9%27s_stress_ellipsoid" title="Lamé's stress ellipsoid">Lamé's stress ellipsoid</a></li> <li><a href="/wiki/Reinforced_solid" title="Reinforced solid">Reinforced solid</a></li> <li><a href="/wiki/Residual_stress" title="Residual stress">Residual stress</a></li> <li><a href="/wiki/Shear_strength" title="Shear strength">Shear strength</a></li> <li><a href="/wiki/Shot_peening" title="Shot peening">Shot peening</a></li> <li><a href="/wiki/Strain_(materials_science)" class="mw-redirect" title="Strain (materials science)">Strain</a></li> <li><a href="/wiki/Strain_tensor" class="mw-redirect" title="Strain tensor">Strain tensor</a></li> <li><a href="/wiki/Strain_rate_tensor" class="mw-redirect" title="Strain rate tensor">Strain rate tensor</a></li> <li><a href="/wiki/Stress%E2%80%93energy_tensor" title="Stress–energy tensor">Stress–energy tensor</a></li> <li><a href="/wiki/Stress%E2%80%93strain_curve" title="Stress–strain curve">Stress–strain curve</a></li> <li><a href="/wiki/Stress_concentration" title="Stress concentration">Stress concentration</a></li> <li><a href="/wiki/Transient_friction_loading" title="Transient friction loading">Transient friction loading</a></li> <li><a href="/wiki/Tensile_strength" class="mw-redirect" title="Tensile strength">Tensile strength</a></li> <li><a href="/wiki/Thermal_stress" title="Thermal stress">Thermal stress</a></li> <li><a href="/wiki/Virial_stress" title="Virial stress">Virial stress</a></li> <li><a href="/wiki/Yield_(engineering)" title="Yield (engineering)">Yield (engineering)</a></li> <li><a href="/wiki/Yield_surface" title="Yield surface">Yield surface</a></li> <li><a href="/wiki/Virial_theorem" title="Virial theorem">Virial theorem</a></li></ul> </div> <div style="clear:both;" class=""></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=Stress_(mechanics)&action=edit&section=22" title="Edit section: References"><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-columns references-column-width" style="column-width: 30em;"> <ol class="references"> <li id="cite_note-:0-1"><span class="mw-cite-backlink">^ <a href="#cite_ref-:0_1-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-:0_1-1"><sup><i><b>b</b></i></sup></a></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 class="citation web cs1"><a rel="nofollow" class="external text" href="https://openstax.org/books/university-physics-volume-1/pages/12-3-stress-strain-and-elastic-modulus">"12.3 Stress, Strain, and Elastic Modulus 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Accessed on 2013-02-08.</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="CITEREFMarchettiJoannyRamaswamyLiverpool2013" class="citation journal cs1">Marchetti, M. C.; Joanny, J. F.; Ramaswamy, S.; Liverpool, T. B.; Prost, J.; Rao, Madan; Simha, R. Aditi (2013). "Hydrodynamics of soft active matter". <i>Reviews of Modern Physics</i>. <b>85</b> (3): 1143–1189. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2013RvMP...85.1143M">2013RvMP...85.1143M</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FRevModPhys.85.1143">10.1103/RevModPhys.85.1143</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Reviews+of+Modern+Physics&rft.atitle=Hydrodynamics+of+soft+active+matter&rft.volume=85&rft.issue=3&rft.pages=1143-1189&rft.date=2013&rft_id=info%3Adoi%2F10.1103%2FRevModPhys.85.1143&rft_id=info%3Abibcode%2F2013RvMP...85.1143M&rft.aulast=Marchetti&rft.aufirst=M.+C.&rft.au=Joanny%2C+J.+F.&rft.au=Ramaswamy%2C+S.&rft.au=Liverpool%2C+T.+B.&rft.au=Prost%2C+J.&rft.au=Rao%2C+Madan&rft.au=Simha%2C+R.+Aditi&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStress+%28mechanics%29" class="Z3988"></span></span> </li> <li id="cite_note-sharma2019-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-sharma2019_12-0">^</a></b></span> <span class="reference-text">Sharma, B and Kumar, R "Estimation of bulk viscosity of dilute gases using a nonequilibrium molecular dynamics approach.", <i>Physical Review E</i>,100, 013309 (2019)</span> </li> <li id="cite_note-Huston-13"><span class="mw-cite-backlink">^ <a href="#cite_ref-Huston_13-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Huston_13-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-Huston_13-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text">Ronald L. Huston and Harold Josephs (2009), "Practical Stress Analysis in Engineering Design". 3rd edition, CRC Press, 634 pages. <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/9781574447132" title="Special:BookSources/9781574447132">9781574447132</a></span> </li> <li id="cite_note-Pilkey-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-Pilkey_14-0">^</a></b></span> <span class="reference-text">Walter D. Pilkey, Orrin H. Pilkey (1974), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=d7I8AAAAIAAJ">"Mechanics of solids"</a> (book)</span> </li> <li id="cite_note-Smith-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-Smith_15-0">^</a></b></span> <span class="reference-text">Donald Ray Smith and Clifford Truesdell (1993) <a rel="nofollow" class="external text" href="https://books.google.com/books?id=ZcWC7YVdb4wC&pg=PA97">"An Introduction to Continuum Mechanics after Truesdell and Noll". Springer.</a> <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/0-7923-2454-4" title="Special:BookSources/0-7923-2454-4">0-7923-2454-4</a></span> </li> <li id="cite_note-Irgens-16"><span class="mw-cite-backlink"><b><a href="#cite_ref-Irgens_16-0">^</a></b></span> <span class="reference-text">Fridtjov Irgens (2008), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=q5dB7Gf4bIoC&pg=PA46">"Continuum Mechanics"</a>. Springer. <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/3-540-74297-2" title="Special:BookSources/3-540-74297-2">3-540-74297-2</a></span> </li> <li id="cite_note-Slaughter-17"><span class="mw-cite-backlink"><b><a href="#cite_ref-Slaughter_17-0">^</a></b></span> <span class="reference-text">William S. Slaughter (2012), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=TxvnBwAAQBAJ&pg=PA193">"The Linearized Theory of Elasticity"</a>. Birkhäuser Basel <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/978-0-8176-4117-7" title="Special:BookSources/978-0-8176-4117-7">978-0-8176-4117-7</a></span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Stress_(mechanics)&action=edit&section=23" title="Edit section: Further reading"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239549316">.mw-parser-output .refbegin{margin-bottom:0.5em}.mw-parser-output .refbegin-hanging-indents>ul{margin-left:0}.mw-parser-output .refbegin-hanging-indents>ul>li{margin-left:0;padding-left:3.2em;text-indent:-3.2em}.mw-parser-output .refbegin-hanging-indents ul,.mw-parser-output .refbegin-hanging-indents ul li{list-style:none}@media(max-width:720px){.mw-parser-output .refbegin-hanging-indents>ul>li{padding-left:1.6em;text-indent:-1.6em}}.mw-parser-output .refbegin-columns{margin-top:0.3em}.mw-parser-output .refbegin-columns ul{margin-top:0}.mw-parser-output .refbegin-columns li{page-break-inside:avoid;break-inside:avoid-column}@media screen{.mw-parser-output .refbegin{font-size:90%}}</style><div class="refbegin" style=""> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFChakrabarty2006" class="citation book cs1">Chakrabarty, J. (2006). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=9CZsqgsfwEAC&q=related%3AISBN0486435946&pg=PA17"><i>Theory of plasticity</i></a> (3 ed.). Butterworth-Heinemann. pp. 17–32. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-7506-6638-2" title="Special:BookSources/0-7506-6638-2"><bdi>0-7506-6638-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Theory+of+plasticity&rft.pages=17-32&rft.edition=3&rft.pub=Butterworth-Heinemann&rft.date=2006&rft.isbn=0-7506-6638-2&rft.aulast=Chakrabarty&rft.aufirst=J.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D9CZsqgsfwEAC%26q%3Drelated%253AISBN0486435946%26pg%3DPA17&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStress+%28mechanics%29" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBeerElwood_Russell_JohnstonJohn_T._DeWolf1992" class="citation book cs1">Beer, Ferdinand Pierre; Elwood Russell Johnston; John T. DeWolf (1992). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/mechanicsofmater00ferd_1"><i>Mechanics of Materials</i></a></span>. McGraw-Hill Professional. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-07-112939-1" title="Special:BookSources/0-07-112939-1"><bdi>0-07-112939-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mechanics+of+Materials&rft.pub=McGraw-Hill+Professional&rft.date=1992&rft.isbn=0-07-112939-1&rft.aulast=Beer&rft.aufirst=Ferdinand+Pierre&rft.au=Elwood+Russell+Johnston&rft.au=John+T.+DeWolf&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fmechanicsofmater00ferd_1&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStress+%28mechanics%29" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBradyE.T._Brown1993" class="citation book cs1">Brady, B.H.G.; E.T. Brown (1993). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=s0BaKxL11KsC&pg=PA18"><i>Rock Mechanics For Underground Mining</i></a> (Third ed.). Kluwer Academic Publisher. pp. 17–29. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-412-47550-2" title="Special:BookSources/0-412-47550-2"><bdi>0-412-47550-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Rock+Mechanics+For+Underground+Mining&rft.pages=17-29&rft.edition=Third&rft.pub=Kluwer+Academic+Publisher&rft.date=1993&rft.isbn=0-412-47550-2&rft.aulast=Brady&rft.aufirst=B.H.G.&rft.au=E.T.+Brown&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3Ds0BaKxL11KsC%26pg%3DPA18&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStress+%28mechanics%29" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFChenBaladi,_G.Y.1985" class="citation book cs1">Chen, Wai-Fah; Baladi, G.Y. (1985). <i>Soil Plasticity, Theory and Implementation</i>. Elsevier Science. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-444-42455-5" title="Special:BookSources/0-444-42455-5"><bdi>0-444-42455-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Soil+Plasticity%2C+Theory+and+Implementation&rft.pub=Elsevier+Science&rft.date=1985&rft.isbn=0-444-42455-5&rft.aulast=Chen&rft.aufirst=Wai-Fah&rft.au=Baladi%2C+G.Y.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStress+%28mechanics%29" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFChouPagano,_N.J.1992" class="citation book cs1">Chou, Pei Chi; Pagano, N.J. (1992). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=9-pJ7Kg5XmAC&pg=PA1"><i>Elasticity: tensor, dyadic, and engineering approaches</i></a>. Dover books on engineering. Dover Publications. pp. 1–33. <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"><bdi>0-486-66958-0</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+tensor%2C+dyadic%2C+and+engineering+approaches&rft.series=Dover+books+on+engineering&rft.pages=1-33&rft.pub=Dover+Publications&rft.date=1992&rft.isbn=0-486-66958-0&rft.aulast=Chou&rft.aufirst=Pei+Chi&rft.au=Pagano%2C+N.J.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D9-pJ7Kg5XmAC%26pg%3DPA1&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStress+%28mechanics%29" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDavisSelvadurai._A._P._S.1996" class="citation book cs1">Davis, R. O.; Selvadurai. A. P. S. (1996). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=4Z11rZaUn1UC&pg=PA16"><i>Elasticity and geomechanics</i></a>. Cambridge University Press. pp. 16–26. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-521-49827-9" title="Special:BookSources/0-521-49827-9"><bdi>0-521-49827-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Elasticity+and+geomechanics&rft.pages=16-26&rft.pub=Cambridge+University+Press&rft.date=1996&rft.isbn=0-521-49827-9&rft.aulast=Davis&rft.aufirst=R.+O.&rft.au=Selvadurai.+A.+P.+S.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D4Z11rZaUn1UC%26pg%3DPA16&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStress+%28mechanics%29" class="Z3988"></span></li> <li>Dieter, G. E. (3 ed.). (1989). <i>Mechanical Metallurgy</i>. New York: McGraw-Hill. <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/0-07-100406-8" title="Special:BookSources/0-07-100406-8">0-07-100406-8</a>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHoltzKovacs,_William_D.1981" class="citation book cs1">Holtz, Robert D.; Kovacs, William D. (1981). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=yYkYAQAAIAAJ"><i>An introduction to geotechnical engineering</i></a>. Prentice-Hall civil engineering and engineering mechanics series. Prentice-Hall. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-13-484394-0" title="Special:BookSources/0-13-484394-0"><bdi>0-13-484394-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=An+introduction+to+geotechnical+engineering&rft.series=Prentice-Hall+civil+engineering+and+engineering+mechanics+series&rft.pub=Prentice-Hall&rft.date=1981&rft.isbn=0-13-484394-0&rft.aulast=Holtz&rft.aufirst=Robert+D.&rft.au=Kovacs%2C+William+D.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DyYkYAQAAIAAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStress+%28mechanics%29" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJones2008" class="citation book cs1">Jones, Robert Millard (2008). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=kiCVc3AJhVwC&pg=PA95"><i>Deformation Theory of Plasticity</i></a>. Bull Ridge Corporation. pp. 95–112. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-9787223-1-9" title="Special:BookSources/978-0-9787223-1-9"><bdi>978-0-9787223-1-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Deformation+Theory+of+Plasticity&rft.pages=95-112&rft.pub=Bull+Ridge+Corporation&rft.date=2008&rft.isbn=978-0-9787223-1-9&rft.aulast=Jones&rft.aufirst=Robert+Millard&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DkiCVc3AJhVwC%26pg%3DPA95&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStress+%28mechanics%29" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJumikis1969" class="citation book cs1">Jumikis, Alfreds R. (1969). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=NPZRAAAAMAAJ"><i>Theoretical soil mechanics: with practical applications to soil mechanics and foundation engineering</i></a>. Van Nostrand Reinhold Co. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-442-04199-3" title="Special:BookSources/0-442-04199-3"><bdi>0-442-04199-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Theoretical+soil+mechanics%3A+with+practical+applications+to+soil+mechanics+and+foundation+engineering&rft.pub=Van+Nostrand+Reinhold+Co.&rft.date=1969&rft.isbn=0-442-04199-3&rft.aulast=Jumikis&rft.aufirst=Alfreds+R.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DNPZRAAAAMAAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStress+%28mechanics%29" class="Z3988"></span></li> <li>Landau, L.D. and E.M.Lifshitz. (1959). <i>Theory of Elasticity</i>.</li> <li>Love, A. E. H. (4 ed.). (1944). <i>Treatise on the Mathematical Theory of Elasticity</i>. New York: Dover Publications. <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/0-486-60174-9" title="Special:BookSources/0-486-60174-9">0-486-60174-9</a>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMarsdenHughes,_T._J._R.1994" class="citation book cs1">Marsden, J. E.; Hughes, T. J. R. (1994). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/mathematicalfoun00mars"><i>Mathematical Foundations of Elasticity</i></a></span>. Dover Publications. pp. <a rel="nofollow" class="external text" href="https://archive.org/details/mathematicalfoun00mars/page/132">132</a>–142. <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"><bdi>0-486-67865-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mathematical+Foundations+of+Elasticity&rft.pages=132-142&rft.pub=Dover+Publications&rft.date=1994&rft.isbn=0-486-67865-2&rft.aulast=Marsden&rft.aufirst=J.+E.&rft.au=Hughes%2C+T.+J.+R.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fmathematicalfoun00mars&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStress+%28mechanics%29" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFParry2004" class="citation book cs1">Parry, Richard Hawley Grey (2004). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=u_rec9uQnLcC&q=mohr%20circles%2C%20sterss%20paths%20and%20geotechnics&pg=PA1"><i>Mohr circles, stress paths and geotechnics</i></a> (2 ed.). Taylor & Francis. pp. 1–30. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-415-27297-1" title="Special:BookSources/0-415-27297-1"><bdi>0-415-27297-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mohr+circles%2C+stress+paths+and+geotechnics&rft.pages=1-30&rft.edition=2&rft.pub=Taylor+%26+Francis&rft.date=2004&rft.isbn=0-415-27297-1&rft.aulast=Parry&rft.aufirst=Richard+Hawley+Grey&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3Du_rec9uQnLcC%26q%3Dmohr%2520circles%252C%2520sterss%2520paths%2520and%2520geotechnics%26pg%3DPA1&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStress+%28mechanics%29" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRees2006" class="citation book cs1">Rees, David (2006). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=4KWbmn_1hcYC&pg=PA1"><i>Basic Engineering Plasticity – An Introduction with Engineering and Manufacturing Applications</i></a>. Butterworth-Heinemann. pp. 1–32. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-7506-8025-3" title="Special:BookSources/0-7506-8025-3"><bdi>0-7506-8025-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Basic+Engineering+Plasticity+%E2%80%93+An+Introduction+with+Engineering+and+Manufacturing+Applications&rft.pages=1-32&rft.pub=Butterworth-Heinemann&rft.date=2006&rft.isbn=0-7506-8025-3&rft.aulast=Rees&rft.aufirst=David&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D4KWbmn_1hcYC%26pg%3DPA1&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStress+%28mechanics%29" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTimoshenkoJames_Norman_Goodier1970" class="citation book cs1"><a href="/wiki/Stephen_Timoshenko" title="Stephen Timoshenko">Timoshenko, Stephen P.</a>; James Norman Goodier (1970). <i>Theory of Elasticity</i> (Third ed.). McGraw-Hill International Editions. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-07-085805-5" title="Special:BookSources/0-07-085805-5"><bdi>0-07-085805-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Theory+of+Elasticity&rft.edition=Third&rft.pub=McGraw-Hill+International+Editions&rft.date=1970&rft.isbn=0-07-085805-5&rft.aulast=Timoshenko&rft.aufirst=Stephen+P.&rft.au=James+Norman+Goodier&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStress+%28mechanics%29" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTimoshenko1983" class="citation book cs1">Timoshenko, Stephen P. (1983). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=tkScQmyhsb8C&q=stress"><i>History of strength of materials: with a brief account of the history of theory of elasticity and theory of structures</i></a>. Dover Books on Physics. Dover Publications. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-486-61187-6" title="Special:BookSources/0-486-61187-6"><bdi>0-486-61187-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=History+of+strength+of+materials%3A+with+a+brief+account+of+the+history+of+theory+of+elasticity+and+theory+of+structures&rft.series=Dover+Books+on+Physics&rft.pub=Dover+Publications&rft.date=1983&rft.isbn=0-486-61187-6&rft.aulast=Timoshenko&rft.aufirst=Stephen+P.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DtkScQmyhsb8C%26q%3Dstress&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStress+%28mechanics%29" class="Z3988"></span></li></ul> </div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1236075235">.mw-parser-output .navbox{box-sizing:border-box;border:1px solid #a2a9b1;width:100%;clear:both;font-size:88%;text-align:center;padding:1px;margin:1em auto 0}.mw-parser-output .navbox .navbox{margin-top:0}.mw-parser-output .navbox+.navbox,.mw-parser-output .navbox+.navbox-styles+.navbox{margin-top:-1px}.mw-parser-output .navbox-inner,.mw-parser-output .navbox-subgroup{width:100%}.mw-parser-output .navbox-group,.mw-parser-output .navbox-title,.mw-parser-output .navbox-abovebelow{padding:0.25em 1em;line-height:1.5em;text-align:center}.mw-parser-output .navbox-group{white-space:nowrap;text-align:right}.mw-parser-output .navbox,.mw-parser-output .navbox-subgroup{background-color:#fdfdfd}.mw-parser-output .navbox-list{line-height:1.5em;border-color:#fdfdfd}.mw-parser-output .navbox-list-with-group{text-align:left;border-left-width:2px;border-left-style:solid}.mw-parser-output tr+tr>.navbox-abovebelow,.mw-parser-output tr+tr>.navbox-group,.mw-parser-output tr+tr>.navbox-image,.mw-parser-output tr+tr>.navbox-list{border-top:2px solid #fdfdfd}.mw-parser-output .navbox-title{background-color:#ccf}.mw-parser-output .navbox-abovebelow,.mw-parser-output .navbox-group,.mw-parser-output .navbox-subgroup .navbox-title{background-color:#ddf}.mw-parser-output .navbox-subgroup .navbox-group,.mw-parser-output .navbox-subgroup .navbox-abovebelow{background-color:#e6e6ff}.mw-parser-output .navbox-even{background-color:#f7f7f7}.mw-parser-output .navbox-odd{background-color:transparent}.mw-parser-output .navbox .hlist td dl,.mw-parser-output .navbox .hlist td ol,.mw-parser-output .navbox .hlist td ul,.mw-parser-output .navbox td.hlist dl,.mw-parser-output .navbox td.hlist ol,.mw-parser-output .navbox td.hlist ul{padding:0.125em 0}.mw-parser-output .navbox .navbar{display:block;font-size:100%}.mw-parser-output .navbox-title .navbar{float:left;text-align:left;margin-right:0.5em}body.skin--responsive .mw-parser-output .navbox-image img{max-width:none!important}@media print{body.ns-0 .mw-parser-output .navbox{display:none!important}}</style></div><div role="navigation" class="navbox authority-control" aria-label="Navbox" style="padding:3px"><table class="nowraplinks hlist navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Help:Authority_control" title="Help:Authority control">Authority control databases</a>: National <span class="mw-valign-text-top noprint" typeof="mw:File/Frameless"><a href="https://www.wikidata.org/wiki/Q206175#identifiers" title="Edit this at Wikidata"><img alt="Edit this at Wikidata" src="//upload.wikimedia.org/wikipedia/en/thumb/8/8a/OOjs_UI_icon_edit-ltr-progressive.svg/10px-OOjs_UI_icon_edit-ltr-progressive.svg.png" decoding="async" width="10" height="10" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/8/8a/OOjs_UI_icon_edit-ltr-progressive.svg/15px-OOjs_UI_icon_edit-ltr-progressive.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/8/8a/OOjs_UI_icon_edit-ltr-progressive.svg/20px-OOjs_UI_icon_edit-ltr-progressive.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"><ul><li><span class="uid"><a rel="nofollow" class="external text" href="https://d-nb.info/gnd/4134428-5">Germany</a></span></li><li><span class="uid"><a rel="nofollow" class="external text" href="https://id.ndl.go.jp/auth/ndlna/00568941">Japan</a></span></li></ul></div></td></tr></tbody></table></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; 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