Strain (mechanics)

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class="vector-toc-numb">2</span> <span>Strain measures</span> </div> </a> <button aria-controls="toc-Strain_measures-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 Strain measures subsection</span> </button> <ul id="toc-Strain_measures-sublist" class="vector-toc-list"> <li id="toc-Engineering_strain" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Engineering_strain"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Engineering strain</span> </div> </a> <ul id="toc-Engineering_strain-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Stretch_ratio" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Stretch_ratio"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Stretch ratio</span> </div> </a> <ul id="toc-Stretch_ratio-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Logarithmic_strain" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Logarithmic_strain"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>Logarithmic strain</span> </div> </a> <ul id="toc-Logarithmic_strain-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Green_strain" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Green_strain"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.4</span> <span>Green strain</span> </div> </a> <ul id="toc-Green_strain-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Almansi_strain" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Almansi_strain"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.5</span> <span>Almansi strain</span> </div> </a> <ul id="toc-Almansi_strain-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Strain_tensor" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Strain_tensor"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Strain tensor</span> </div> </a> <button aria-controls="toc-Strain_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 Strain tensor subsection</span> </button> <ul id="toc-Strain_tensor-sublist" class="vector-toc-list"> <li id="toc-Geometric_setting" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Geometric_setting"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Geometric setting</span> </div> </a> <ul id="toc-Geometric_setting-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Normal_strain" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Normal_strain"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Normal strain</span> </div> </a> <ul id="toc-Normal_strain-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Shear_strain" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Shear_strain"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3</span> <span>Shear strain</span> </div> </a> <ul id="toc-Shear_strain-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Volume_strain" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Volume_strain"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.4</span> <span>Volume strain</span> </div> </a> <ul id="toc-Volume_strain-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Metric_tensor" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Metric_tensor"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Metric tensor</span> </div> </a> <ul id="toc-Metric_tensor-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">5</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>References</span> </div> </a> <ul id="toc-References-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 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class="firstHeading mw-first-heading"><span class="mw-page-title-main">Strain (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. 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href="https://bar.wikipedia.org/wiki/Dehnung" title="Dehnung – Bavarian" lang="bar" hreflang="bar" data-title="Dehnung" data-language-autonym="Boarisch" data-language-local-name="Bavarian" class="interlanguage-link-target"><span>Boarisch</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/T%C3%B8jning" title="Tøjning – Danish" lang="da" hreflang="da" data-title="Tøjning" 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/Dehnung" title="Dehnung – German" lang="de" hreflang="de" data-title="Dehnung" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%DA%A9%D8%B1%D9%86%D8%B4" title="کرنش – Persian" lang="fa" hreflang="fa" data-title="کرنش" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/Straidhn" title="Straidhn – Irish" lang="ga" hreflang="ga" data-title="Straidhn" data-language-autonym="Gaeilge" data-language-local-name="Irish" class="interlanguage-link-target"><span>Gaeilge</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_(%EC%97%AD%ED%95%99)" title="변형 (역학) – Korean" lang="ko" hreflang="ko" data-title="변형 (역학)" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Alakv%C3%A1ltoz%C3%A1s" title="Alakváltozás – Hungarian" lang="hu" hreflang="hu" data-title="Alakváltozás" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Rek_(mechanica)" title="Rek (mechanica) – Dutch" lang="nl" hreflang="nl" data-title="Rek (mechanica)" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E3%81%B2%E3%81%9A%E3%81%BF" title="ひずみ – Japanese" lang="ja" hreflang="ja" data-title="ひずみ" 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-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Relativni_raztezek" title="Relativni raztezek – Slovenian" lang="sl" hreflang="sl" data-title="Relativni 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title="Shear strain">Shear strain</a>)</span></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Relative deformation of a physical body</div> <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">Strain</th></tr><tr><th scope="row" class="infobox-label"><div style="display: inline-block; line-height: 1.2em; padding: .1em 0;">Other names</div></th><td class="infobox-data">Strain tensor</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">1</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">%</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">m/m</td></tr><tr><th scope="row" class="infobox-label"><div style="display: inline-block; line-height: 1.2em; padding: .1em 0;">Behaviour under<br /><span class="nowrap"><a href="/wiki/Coordinate_transformation" class="mw-redirect" title="Coordinate transformation">coord transformation</a></span></div></th><td class="infobox-data">tensor</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: 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rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><table class="sidebar sidebar-collapse nomobile nowraplinks plainlist"><tbody><tr><td class="sidebar-pretitle">Part of a series on</td></tr><tr><th class="sidebar-title-with-pretitle"><a href="/wiki/Continuum_mechanics" title="Continuum mechanics">Continuum mechanics</a></th></tr><tr><td class="sidebar-image"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J=-D{\frac {d\varphi }{dx}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>J</mi> <mo>=</mo> <mo>−</mo> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>φ</mi> </mrow> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle J=-D{\frac {d\varphi }{dx}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1856f88def2056f28ed27c7d31180a6240820ea6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:11.874ex; height:5.509ex;" alt="{\displaystyle J=-D{\frac {d\varphi }{dx}}}"></span><div class="sidebar-caption"><a href="/wiki/Fick%27s_laws_of_diffusion" title="Fick's laws of diffusion">Fick's laws of diffusion</a></div></td></tr><tr><td class="sidebar-content-with-subgroup"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:transparent;border-top:1px solid #aaa;text-align:center;;color: var(--color-base)">Laws</div><div class="sidebar-list-content mw-collapsible-content"><table class="sidebar-subgroup"><tbody><tr><th class="sidebar-heading" style="font-style:italic;font-weight:normal;"> Conservations</th></tr><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Conservation_of_mass" title="Conservation of mass">Mass</a></li> <li><a href="/wiki/Conservation_of_momentum" class="mw-redirect" title="Conservation of momentum">Momentum</a></li> <li><a href="/wiki/Conservation_of_energy" title="Conservation of energy">Energy</a></li></ul></td> </tr><tr><th class="sidebar-heading" style="font-style:italic;font-weight:normal;"> Inequalities</th></tr><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Clausius%E2%80%93Duhem_inequality" title="Clausius–Duhem inequality">Clausius–Duhem (entropy)</a></li></ul></td> </tr></tbody></table></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible"><div class="sidebar-list-title" style="background:transparent;border-top:1px solid #aaa;text-align:center;;color: var(--color-base)"><a 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 href="/wiki/Stress_(mechanics)" title="Stress (mechanics)">Stress</a></li> <li><a class="mw-selflink selflink">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/Mechanics" title="Mechanics">mechanics</a>, <b>strain</b> is defined as relative <a href="/wiki/Deformation_(mechanics)" class="mw-redirect" title="Deformation (mechanics)">deformation</a>, compared to a <em>reference</em> <a href="/wiki/Position_(geometry)" title="Position (geometry)">position</a> configuration. Different equivalent choices may be made for the expression of a strain field depending on whether it is defined with respect to the initial or the final configuration of the body and on whether the <a href="/wiki/Metric_tensor" title="Metric tensor">metric tensor</a> or its dual is considered. </p><p>Strain has <a href="/wiki/Dimension_(physics)" class="mw-redirect" title="Dimension (physics)">dimension</a> of a <a href="/wiki/Length" title="Length">length</a> <a href="/wiki/Ratio" title="Ratio">ratio</a>, with <a href="/wiki/SI_base_units" class="mw-redirect" title="SI base units">SI base units</a> of meter per meter (m/m). Hence strains are <a href="/wiki/Dimensionless" class="mw-redirect" title="Dimensionless">dimensionless</a> and are usually expressed as a <a href="/wiki/Decimal" title="Decimal">decimal fraction</a> or a <a href="/wiki/Percentage" title="Percentage">percentage</a>. <a href="/wiki/Parts-per_notation" title="Parts-per notation">Parts-per notation</a> is also used, e.g., <a href="/wiki/Parts_per_million" class="mw-redirect" title="Parts per million">parts per million</a> or <a href="/wiki/Parts_per_billion" class="mw-redirect" title="Parts per billion">parts per billion</a> (sometimes called "microstrains" and "nanostrains", respectively), corresponding to <a href="/wiki/Micrometre" title="Micrometre">μm</a>/m and <a href="/wiki/Nanometre" title="Nanometre">nm</a>/m. </p><p>Strain can be formulated as the <a href="/wiki/Spatial_derivative" class="mw-redirect" title="Spatial derivative">spatial derivative</a> of <a href="/wiki/Displacement_(physics)" class="mw-redirect" title="Displacement (physics)">displacement</a>: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {\varepsilon }}\doteq {\cfrac {\partial }{\partial \mathbf {X} }}\left(\mathbf {x} -\mathbf {X} \right)={\boldsymbol {F}}'-{\boldsymbol {I}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ε</mi> </mrow> <mo>≐</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∂</mi> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> </mrow> </mstyle> </mrow> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">F</mi> </mrow> <mo>′</mo> </msup> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">I</mi> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {\varepsilon }}\doteq {\cfrac {\partial }{\partial \mathbf {X} }}\left(\mathbf {x} -\mathbf {X} \right)={\boldsymbol {F}}'-{\boldsymbol {I}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/77b3bbb9d1485775abe0dc7069df39c807ba6988" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:27.536ex; height:7.176ex;" alt="{\displaystyle {\boldsymbol {\varepsilon }}\doteq {\cfrac {\partial }{\partial \mathbf {X} }}\left(\mathbf {x} -\mathbf {X} \right)={\boldsymbol {F}}'-{\boldsymbol {I}},}"></span> where <span class="texhtml mvar" style="font-style:italic;"><b>I</b></span> is the <a href="/wiki/Identity_matrix" title="Identity matrix">identity tensor</a>. The displacement of a body may be expressed in the form <span class="texhtml"><b>x</b> = <i><b>F</b></i>(<b>X</b>)</span>, where <span class="texhtml"><b>X</b></span> is the reference position of material points of the body; displacement has units of length and does not distinguish between rigid body motions (translations and rotations) and deformations (changes in shape and size) of the body. The spatial derivative of a uniform translation is zero, thus strains measure how much a given displacement differs locally from a rigid-body motion.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p><p>A strain is in general a <a href="/wiki/Tensor" title="Tensor">tensor</a> quantity. Physical insight into strains can be gained by observing that a given strain can be decomposed into normal and shear components. The amount of stretch or compression along material line elements or fibers is the <i>normal strain</i>, and the amount of distortion associated with the sliding of plane layers over each other is the <i>shear strain</i>, within a deforming body.<sup id="cite_ref-rees_2-0" class="reference"><a href="#cite_note-rees-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> This could be applied by elongation, shortening, or volume changes, or angular distortion.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p><p>The state of strain at a <a href="/wiki/Continuum_mechanics" title="Continuum mechanics">material point</a> of a continuum body is defined as the totality of all the changes in length of material lines or fibers, the <i>normal strain</i>, which pass through that point and also the totality of all the changes in the angle between pairs of lines initially perpendicular to each other, the <i>shear strain</i>, radiating from this point. However, it is sufficient to know the normal and shear components of strain on a set of three mutually perpendicular directions. </p><p>If there is an increase in length of the material line, the normal strain is called <i>tensile strain</i>; otherwise, if there is reduction or compression in the length of the material line, it is called <i>compressive strain</i>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Strain_regimes">Strain regimes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Strain_(mechanics)&action=edit&section=1" title="Edit section: Strain regimes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Depending on the amount of strain, or local deformation, the analysis of deformation is subdivided into three deformation theories: </p> <ul><li><a href="/wiki/Finite_strain_theory" title="Finite strain theory">Finite strain theory</a>, also called <i>large strain theory</i>, <i>large deformation theory</i>, deals with deformations in which both rotations and strains are arbitrarily large. In this case, the undeformed and deformed configurations of the <a href="/wiki/Continuum_mechanics" title="Continuum mechanics">continuum</a> are significantly different and a clear distinction has to be made between them. This is commonly the case with <a href="/wiki/Elastomer" title="Elastomer">elastomers</a>, <a href="/wiki/Plasticity_(physics)" title="Plasticity (physics)">plastically-deforming</a> materials and other <a href="/wiki/Fluid" title="Fluid">fluids</a> and biological <a href="/wiki/Soft_tissue" title="Soft tissue">soft tissue</a>.</li> <li><a href="/wiki/Infinitesimal_strain_theory" title="Infinitesimal strain theory">Infinitesimal strain theory</a>, also called <i>small strain theory</i>, <i>small deformation theory</i>, <i>small displacement theory</i>, or <i>small displacement-gradient theory</i> where strains and rotations are both small. In this case, the undeformed and deformed configurations of the body can be assumed identical. The infinitesimal strain theory is used in the analysis of deformations of materials exhibiting <a href="/wiki/Deformation_(engineering)#Elastic_deformation" title="Deformation (engineering)">elastic</a> behavior, such as materials found in mechanical and civil engineering applications, e.g. concrete and steel.</li> <li><i>Large-displacement</i> or <i>large-rotation theory</i>, which assumes small strains but large rotations and displacements.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Strain_measures">Strain measures</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Strain_(mechanics)&action=edit&section=2" title="Edit section: Strain measures"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In each of these theories the strain is then defined differently. The <i>engineering strain</i> is the most common definition applied to materials used in mechanical and structural engineering, which are subjected to very small deformations. On the other hand, for some materials, e.g., <a href="/wiki/Elastomers" class="mw-redirect" title="Elastomers">elastomers</a> and polymers, subjected to large deformations, the engineering definition of strain is not applicable, e.g. typical engineering strains greater than 1%;<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> thus other more complex definitions of strain are required, such as <i>stretch</i>, <i>logarithmic strain</i>, <i>Green strain</i>, and <i>Almansi strain</i>. </p> <div class="mw-heading mw-heading3"><h3 id="Engineering_strain">Engineering strain</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Strain_(mechanics)&action=edit&section=3" title="Edit section: Engineering strain"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><b>Engineering strain</b>, also known as <b>Cauchy strain</b>, is expressed as the ratio of total deformation to the initial dimension of the material body on which forces are applied. In the case of a material line element or fiber axially loaded, its <a href="/wiki/Elongation_(mechanics)" class="mw-redirect" title="Elongation (mechanics)">elongation</a> gives rise to an <i>engineering normal strain</i> or <i>engineering extensional strain</i> <span class="texhtml mvar" style="font-style:italic;">e</span>, which equals the <i>relative elongation</i> or the change in length <span class="texhtml">Δ<i>L</i></span> per unit of the original length <span class="texhtml mvar" style="font-style:italic;">L</span> of the line element or fibers (in meters per meter). The normal strain is positive if the material fibers are stretched and negative if they are compressed. Thus, we have <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e={\frac {\Delta L}{L}}={\frac {l-L}{L}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>e</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">Δ</mi> <mi>L</mi> </mrow> <mi>L</mi> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>l</mi> <mo>−</mo> <mi>L</mi> </mrow> <mi>L</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e={\frac {\Delta L}{L}}={\frac {l-L}{L}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/13013ba7d723c7fb2971828c1c6c7574516db856" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:17.588ex; height:5.343ex;" alt="{\displaystyle e={\frac {\Delta L}{L}}={\frac {l-L}{L}}}"></span>, where <span class="texhtml mvar" style="font-style:italic;">e</span> is the <i>engineering normal strain</i>, <span class="texhtml mvar" style="font-style:italic;">L</span> is the original length of the fiber and <span class="texhtml mvar" style="font-style:italic;">l</span> is the final length of the fiber. </p><p>The <i>true shear strain</i> is defined as the change in the angle (in radians) between two material line elements initially perpendicular to each other in the undeformed or initial configuration. The <i>engineering shear strain</i> is defined as the tangent of that angle, and is equal to the length of deformation at its maximum divided by the perpendicular length in the plane of force application, which sometimes makes it easier to calculate. </p> <div class="mw-heading mw-heading3"><h3 id="Stretch_ratio">Stretch ratio</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Strain_(mechanics)&action=edit&section=4" title="Edit section: Stretch ratio"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <b>stretch ratio</b> or <b>extension ratio</b> (symbol λ) is an alternative measure related to the extensional or normal strain of an axially loaded differential line element. It is defined as the ratio between the final length <span class="texhtml mvar" style="font-style:italic;">l</span> and the initial length <span class="texhtml mvar" style="font-style:italic;">L</span> of the material line. <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 \lambda ={\frac {l}{L}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>l</mi> <mi>L</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda ={\frac {l}{L}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cb212ca383139404977d5ebd04350a2c0f463ee7" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:6.873ex; height:5.343ex;" alt="{\displaystyle \lambda ={\frac {l}{L}}}"></span> </p><p>The extension ratio λ is related to the engineering strain <i>e</i> by <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e=\lambda -1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>e</mi> <mo>=</mo> <mi>λ</mi> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e=\lambda -1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f022d7954fa935280d25c79e1436be6c96573e21" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.54ex; height:2.343ex;" alt="{\displaystyle e=\lambda -1}"></span> This equation implies that when the normal strain is zero, so that there is no deformation, the stretch ratio is equal to unity. </p><p>The stretch ratio is used in the analysis of materials that exhibit large deformations, such as <a href="/wiki/Elastomer" title="Elastomer">elastomers</a>, which can sustain stretch ratios of 3 or 4 before they fail. On the other hand, traditional engineering materials, such as concrete or steel, fail at much lower stretch ratios. </p> <div class="mw-heading mw-heading3"><h3 id="Logarithmic_strain">Logarithmic strain<span class="anchor" id="True_strain"></span></h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Strain_(mechanics)&action=edit&section=5" title="Edit section: Logarithmic strain"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <b>logarithmic strain</b> <span class="texhtml mvar" style="font-style:italic;">ε</span>, also called, <i>true strain</i> or <i>Hencky strain</i>.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> Considering an incremental strain (Ludwik) <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta \varepsilon ={\frac {\delta l}{l}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <mi>ε</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>δ</mi> <mi>l</mi> </mrow> <mi>l</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta \varepsilon ={\frac {\delta l}{l}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ad2d3f3f4c48b10e6ff26c05c00d42a6b06f564b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:7.809ex; height:5.509ex;" alt="{\displaystyle \delta \varepsilon ={\frac {\delta l}{l}}}"></span> the logarithmic strain is obtained by integrating this incremental strain: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\int \delta \varepsilon &=\int _{L}^{l}{\frac {\delta l}{l}}\\\varepsilon &=\ln \left({\frac {l}{L}}\right)=\ln(\lambda )\\&=\ln(1+e)\\&=e-{\frac {e^{2}}{2}}+{\frac {e^{3}}{3}}-\cdots \end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mo>∫</mo> <mi>δ</mi> <mi>ε</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>δ</mi> <mi>l</mi> </mrow> <mi>l</mi> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>ε</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>ln</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>l</mi> <mi>L</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>λ</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>e</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mi>e</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mn>3</mn> </mfrac> </mrow> <mo>−</mo> <mo>⋯</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\int \delta \varepsilon &=\int _{L}^{l}{\frac {\delta l}{l}}\\\varepsilon &=\ln \left({\frac {l}{L}}\right)=\ln(\lambda )\\&=\ln(1+e)\\&=e-{\frac {e^{2}}{2}}+{\frac {e^{3}}{3}}-\cdots \end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/77f2eb0f9ffc8475d745799d5a9ba44a37c3b5e7" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -10.171ex; width:26.838ex; height:21.509ex;" alt="{\displaystyle {\begin{aligned}\int \delta \varepsilon &=\int _{L}^{l}{\frac {\delta l}{l}}\\\varepsilon &=\ln \left({\frac {l}{L}}\right)=\ln(\lambda )\\&=\ln(1+e)\\&=e-{\frac {e^{2}}{2}}+{\frac {e^{3}}{3}}-\cdots \end{aligned}}}"></span> where <span class="texhtml mvar" style="font-style:italic;">e</span> is the engineering strain. The logarithmic strain provides the correct measure of the final strain when deformation takes place in a series of increments, taking into account the influence of the strain path.<sup id="cite_ref-rees_2-1" class="reference"><a href="#cite_note-rees-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Green_strain">Green strain</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Strain_(mechanics)&action=edit&section=6" title="Edit section: Green strain"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Finite_strain_theory#Finite_strain_tensors" title="Finite strain theory">Finite strain theory</a></div> <p>The Green strain is defined 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 \varepsilon _{G}={\tfrac {1}{2}}\left({\frac {l^{2}-L^{2}}{L^{2}}}\right)={\tfrac {1}{2}}(\lambda ^{2}-1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>G</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo stretchy="false">(</mo> <msup> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon _{G}={\tfrac {1}{2}}\left({\frac {l^{2}-L^{2}}{L^{2}}}\right)={\tfrac {1}{2}}(\lambda ^{2}-1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/be052ec0fc0f835a9e1d480bcbede6c3b83835e2" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:32.212ex; height:6.343ex;" alt="{\displaystyle \varepsilon _{G}={\tfrac {1}{2}}\left({\frac {l^{2}-L^{2}}{L^{2}}}\right)={\tfrac {1}{2}}(\lambda ^{2}-1)}"></span> </p> <div class="mw-heading mw-heading3"><h3 id="Almansi_strain">Almansi strain</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Strain_(mechanics)&action=edit&section=7" title="Edit section: Almansi strain"><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/Finite_strain_theory#Finite_strain_tensors" title="Finite strain theory">Finite strain theory</a></div> <p>The Euler-Almansi strain is defined 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 \varepsilon _{E}={\tfrac {1}{2}}\left({\frac {l^{2}-L^{2}}{l^{2}}}\right)={\tfrac {1}{2}}\left(1-{\frac {1}{\lambda ^{2}}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>E</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <msup> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon _{E}={\tfrac {1}{2}}\left({\frac {l^{2}-L^{2}}{l^{2}}}\right)={\tfrac {1}{2}}\left(1-{\frac {1}{\lambda ^{2}}}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a4bd1a2d4affc6ed95a5bf7608b3d49bc4a351bd" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:35.011ex; height:6.343ex;" alt="{\displaystyle \varepsilon _{E}={\tfrac {1}{2}}\left({\frac {l^{2}-L^{2}}{l^{2}}}\right)={\tfrac {1}{2}}\left(1-{\frac {1}{\lambda ^{2}}}\right)}"></span> </p> <div class="mw-heading mw-heading2"><h2 id="Strain_tensor">Strain tensor<span class="anchor" id="Tensor"></span></h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Strain_(mechanics)&action=edit&section=8" title="Edit section: Strain 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">Further information: <a href="/wiki/Infinitesimal_strain_theory#Infinitesimal_strain_tensor" title="Infinitesimal strain theory">Infinitesimal strain theory § Infinitesimal strain tensor</a></div> <p>The (infinitesimal) <b>strain tensor</b> (symbol <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 {\varepsilon }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ε</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {\varepsilon }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8445af5ff7da70714382bc35e78bedcacf68e825" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle {\boldsymbol {\varepsilon }}}"></span>) is defined in the <a href="/wiki/International_System_of_Quantities" title="International System of Quantities">International System of Quantities</a> (ISQ), more specifically in <a href="/wiki/ISO_80000-4" class="mw-redirect" title="ISO 80000-4">ISO 80000-4</a> (Mechanics), as a "tensor quantity representing the deformation of matter caused by stress. Strain tensor is symmetric and has three linear strain and three shear strain (Cartesian) components."<sup id="cite_ref-iso_6-0" class="reference"><a href="#cite_note-iso-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> ISO 80000-4 further defines <b>linear strain</b> as the "quotient of change in length of an object and its length" and <b>shear strain</b> as the "quotient of parallel displacement of two surfaces of a layer and the thickness of the layer".<sup id="cite_ref-iso_6-1" class="reference"><a href="#cite_note-iso-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> Thus, strains are classified as either <i>normal</i> or <i>shear</i>. A <i>normal strain</i> is perpendicular to the face of an element, and a <i>shear strain</i> is parallel to it. These definitions are consistent with those of <a href="/wiki/Normal_stress" class="mw-redirect" title="Normal stress">normal stress</a> and <a href="/wiki/Shear_stress" title="Shear stress">shear stress</a>. </p><p>The strain tensor can then be expressed in terms of normal and shear components 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 {\underline {\underline {\boldsymbol {\varepsilon }}}}={\begin{bmatrix}\varepsilon _{xx}&\varepsilon _{xy}&\varepsilon _{xz}\\\varepsilon _{yx}&\varepsilon _{yy}&\varepsilon _{yz}\\\varepsilon _{zx}&\varepsilon _{zy}&\varepsilon _{zz}\\\end{bmatrix}}={\begin{bmatrix}\varepsilon _{xx}&{\tfrac {1}{2}}\gamma _{xy}&{\tfrac {1}{2}}\gamma _{xz}\\{\tfrac {1}{2}}\gamma _{yx}&\varepsilon _{yy}&{\tfrac {1}{2}}\gamma _{yz}\\{\tfrac {1}{2}}\gamma _{zx}&{\tfrac {1}{2}}\gamma _{zy}&\varepsilon _{zz}\\\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <munder> <munder> <mi mathvariant="bold-italic">ε</mi> <mo>_</mo> </munder> <mo>_</mo> </munder> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>x</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>y</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>z</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> <mi>x</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> <mi>y</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> <mi>z</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> <mi>x</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> <mi>y</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> <mi>z</mi> </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>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>x</mi> </mrow> </msub> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>y</mi> </mrow> </msub> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>z</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> <mi>x</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> <mi>y</mi> </mrow> </msub> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> <mi>z</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> <mi>x</mi> </mrow> </msub> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> <mi>y</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>ε</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 {\underline {\underline {\boldsymbol {\varepsilon }}}}={\begin{bmatrix}\varepsilon _{xx}&\varepsilon _{xy}&\varepsilon _{xz}\\\varepsilon _{yx}&\varepsilon _{yy}&\varepsilon _{yz}\\\varepsilon _{zx}&\varepsilon _{zy}&\varepsilon _{zz}\\\end{bmatrix}}={\begin{bmatrix}\varepsilon _{xx}&{\tfrac {1}{2}}\gamma _{xy}&{\tfrac {1}{2}}\gamma _{xz}\\{\tfrac {1}{2}}\gamma _{yx}&\varepsilon _{yy}&{\tfrac {1}{2}}\gamma _{yz}\\{\tfrac {1}{2}}\gamma _{zx}&{\tfrac {1}{2}}\gamma _{zy}&\varepsilon _{zz}\\\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4473b4826d40e4ad7fbd15fb902635f3dbd528fe" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.171ex; width:48.229ex; height:11.509ex;" alt="{\displaystyle {\underline {\underline {\boldsymbol {\varepsilon }}}}={\begin{bmatrix}\varepsilon _{xx}&\varepsilon _{xy}&\varepsilon _{xz}\\\varepsilon _{yx}&\varepsilon _{yy}&\varepsilon _{yz}\\\varepsilon _{zx}&\varepsilon _{zy}&\varepsilon _{zz}\\\end{bmatrix}}={\begin{bmatrix}\varepsilon _{xx}&{\tfrac {1}{2}}\gamma _{xy}&{\tfrac {1}{2}}\gamma _{xz}\\{\tfrac {1}{2}}\gamma _{yx}&\varepsilon _{yy}&{\tfrac {1}{2}}\gamma _{yz}\\{\tfrac {1}{2}}\gamma _{zx}&{\tfrac {1}{2}}\gamma _{zy}&\varepsilon _{zz}\\\end{bmatrix}}}"></span> </p> <div class="mw-heading mw-heading3"><h3 id="Geometric_setting">Geometric setting</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Strain_(mechanics)&action=edit&section=9" title="Edit section: Geometric setting"><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:2D_geometric_strain.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/23/2D_geometric_strain.svg/350px-2D_geometric_strain.svg.png" decoding="async" width="350" height="272" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/23/2D_geometric_strain.svg/525px-2D_geometric_strain.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/23/2D_geometric_strain.svg/700px-2D_geometric_strain.svg.png 2x" data-file-width="900" data-file-height="700" /></a><figcaption>Two-dimensional geometric deformation of an infinitesimal material element</figcaption></figure> <p>Consider a two-dimensional, infinitesimal, rectangular material element with dimensions <span class="texhtml"><i>dx</i> × <i>dy</i></span>, which, after deformation, takes the form of a <a href="/wiki/Rhombus" title="Rhombus">rhombus</a>. The deformation is described by the <a href="/wiki/Displacement_field_(mechanics)" title="Displacement field (mechanics)">displacement field</a> <span class="texhtml"><b>u</b></span>. From the geometry of the adjacent figure we have <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {length} (AB)=dx}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">l</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">g</mi> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">h</mi> </mrow> <mo stretchy="false">(</mo> <mi>A</mi> <mi>B</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>d</mi> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {length} (AB)=dx}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4d5ee25e319882f4a2527a6fa9d7a4457865030b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.292ex; height:2.843ex;" alt="{\displaystyle \mathrm {length} (AB)=dx}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\mathrm {length} (ab)&={\sqrt {\left(dx+{\frac {\partial u_{x}}{\partial x}}dx\right)^{2}+\left({\frac {\partial u_{y}}{\partial x}}dx\right)^{2}}}\\&={\sqrt {dx^{2}\left(1+{\frac {\partial u_{x}}{\partial x}}\right)^{2}+dx^{2}\left({\frac {\partial u_{y}}{\partial x}}\right)^{2}}}\\&=dx~{\sqrt {\left(1+{\frac {\partial u_{x}}{\partial x}}\right)^{2}+\left({\frac {\partial u_{y}}{\partial x}}\right)^{2}}}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">l</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">g</mi> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">h</mi> </mrow> <mo stretchy="false">(</mo> <mi>a</mi> <mi>b</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mrow> <mo>(</mo> <mrow> <mi>d</mi> <mi>x</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mi>d</mi> <mi>x</mi> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mi>d</mi> <mi>x</mi> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>d</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>d</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mi>d</mi> <mi>x</mi> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\mathrm {length} (ab)&={\sqrt {\left(dx+{\frac {\partial u_{x}}{\partial x}}dx\right)^{2}+\left({\frac {\partial u_{y}}{\partial x}}dx\right)^{2}}}\\&={\sqrt {dx^{2}\left(1+{\frac {\partial u_{x}}{\partial x}}\right)^{2}+dx^{2}\left({\frac {\partial u_{y}}{\partial x}}\right)^{2}}}\\&=dx~{\sqrt {\left(1+{\frac {\partial u_{x}}{\partial x}}\right)^{2}+\left({\frac {\partial u_{y}}{\partial x}}\right)^{2}}}\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e351cc522ede215c58c778bf6eced035fe0e29d8" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -11.005ex; width:48.725ex; height:23.176ex;" alt="{\displaystyle {\begin{aligned}\mathrm {length} (ab)&={\sqrt {\left(dx+{\frac {\partial u_{x}}{\partial x}}dx\right)^{2}+\left({\frac {\partial u_{y}}{\partial x}}dx\right)^{2}}}\\&={\sqrt {dx^{2}\left(1+{\frac {\partial u_{x}}{\partial x}}\right)^{2}+dx^{2}\left({\frac {\partial u_{y}}{\partial x}}\right)^{2}}}\\&=dx~{\sqrt {\left(1+{\frac {\partial u_{x}}{\partial x}}\right)^{2}+\left({\frac {\partial u_{y}}{\partial x}}\right)^{2}}}\end{aligned}}}"></span> For very small displacement gradients the squares of the derivative of <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_{y}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u_{y}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ad3979cb7bef75dc4cfb429817450ae8bc370d00" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.379ex; height:2.343ex;" alt="{\displaystyle u_{y}}"></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 u_{x}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u_{x}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/045896a773b67c720bdff8d660cfcc906f5f6b20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.502ex; height:2.009ex;" alt="{\displaystyle u_{x}}"></span> are negligible and we have <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {length} (ab)\approx dx\left(1+{\frac {\partial u_{x}}{\partial x}}\right)=dx+{\frac {\partial u_{x}}{\partial x}}dx}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">l</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">g</mi> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">h</mi> </mrow> <mo stretchy="false">(</mo> <mi>a</mi> <mi>b</mi> <mo stretchy="false">)</mo> <mo>≈</mo> <mi>d</mi> <mi>x</mi> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mi>d</mi> <mi>x</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mi>d</mi> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {length} (ab)\approx dx\left(1+{\frac {\partial u_{x}}{\partial x}}\right)=dx+{\frac {\partial u_{x}}{\partial x}}dx}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ccdc04a61d5b6bfe0b0df626e516d66e500a3c7" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:44.166ex; height:6.176ex;" alt="{\displaystyle \mathrm {length} (ab)\approx dx\left(1+{\frac {\partial u_{x}}{\partial x}}\right)=dx+{\frac {\partial u_{x}}{\partial x}}dx}"></span> </p> <div class="mw-heading mw-heading3"><h3 id="Normal_strain">Normal strain</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Strain_(mechanics)&action=edit&section=10" title="Edit section: Normal strain"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>For an <a href="/wiki/Isotropic" class="mw-redirect" title="Isotropic">isotropic</a> material that obeys <a href="/wiki/Hooke%27s_law" title="Hooke's law">Hooke's law</a>, a <a href="/wiki/Normal_stress" class="mw-redirect" title="Normal stress">normal stress</a> will cause a normal strain. Normal strains produce <i>dilations</i>. </p><p>The normal strain in the <span class="texhtml mvar" style="font-style:italic;">x</span>-direction of the rectangular element is defined by <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon _{x}={\frac {\text{extension}}{\text{original length}}}={\frac {\mathrm {length} (ab)-\mathrm {length} (AB)}{\mathrm {length} (AB)}}={\frac {\partial u_{x}}{\partial x}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mtext>extension</mtext> <mtext>original length</mtext> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">l</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">g</mi> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">h</mi> </mrow> <mo stretchy="false">(</mo> <mi>a</mi> <mi>b</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">l</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">g</mi> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">h</mi> </mrow> <mo stretchy="false">(</mo> <mi>A</mi> <mi>B</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">l</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">g</mi> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">h</mi> </mrow> <mo stretchy="false">(</mo> <mi>A</mi> <mi>B</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon _{x}={\frac {\text{extension}}{\text{original length}}}={\frac {\mathrm {length} (ab)-\mathrm {length} (AB)}{\mathrm {length} (AB)}}={\frac {\partial u_{x}}{\partial x}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3a334659cddda62f131dbbd4b06d79821527c3a4" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:57.28ex; height:6.509ex;" alt="{\displaystyle \varepsilon _{x}={\frac {\text{extension}}{\text{original length}}}={\frac {\mathrm {length} (ab)-\mathrm {length} (AB)}{\mathrm {length} (AB)}}={\frac {\partial u_{x}}{\partial x}}}"></span> Similarly, the normal strain in the <span class="texhtml mvar" style="font-style:italic;">y</span>- and <span class="texhtml mvar" style="font-style:italic;">z</span>-directions becomes <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon _{y}={\frac {\partial u_{y}}{\partial y}}\quad ,\qquad \varepsilon _{z}={\frac {\partial u_{z}}{\partial z}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> </mfrac> </mrow> <mspace width="1em" /> <mo>,</mo> <mspace width="2em" /> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>z</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon _{y}={\frac {\partial u_{y}}{\partial y}}\quad ,\qquad \varepsilon _{z}={\frac {\partial u_{z}}{\partial z}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/070af43d769b55f7666be6bc1a673eb98186ac73" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:27.435ex; height:6.343ex;" alt="{\displaystyle \varepsilon _{y}={\frac {\partial u_{y}}{\partial y}}\quad ,\qquad \varepsilon _{z}={\frac {\partial u_{z}}{\partial z}}}"></span> </p> <div class="mw-heading mw-heading3"><h3 id="Shear_strain">Shear strain</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Strain_(mechanics)&action=edit&section=11" title="Edit section: Shear strain"><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">See also: <a href="/wiki/Shear_stress" title="Shear stress">Shear stress</a></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1257001546"><table class="infobox"><tbody><tr><th colspan="2" class="infobox-above">Shear strain</th></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"><span class="texhtml mvar" style="font-style:italic;"><a href="/wiki/Gamma" title="Gamma">γ</a></span> or <span class="texhtml mvar" style="font-style:italic;"><a href="/wiki/Epsilon" title="Epsilon">ε</a></span></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/Dimensionless" class="mw-redirect" title="Dimensionless">1</a>, or <a href="/wiki/Radian" title="Radian">radian</a></td></tr><tr><th scope="row" class="infobox-label"><div style="display: inline-block; line-height: 1.2em; padding: .1em 0;">Derivations from<br />other quantities</div></th><td class="infobox-data"><span class="texhtml"><i>γ</i> = <style data-mw-deduplicate="TemplateStyles:r1214402035">.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}</style><span class="sfrac">⁠<span class="tion"><span class="num"><a href="/wiki/Shear_stress" title="Shear stress"><i>τ</i></a></span><span class="sr-only">/</span><span class="den"><a href="/wiki/Shear_modulus" title="Shear modulus"><i>G</i></a></span></span>⁠</span></span></td></tr></tbody></table> <p>The engineering shear strain (<span class="texhtml"><i>γ<sub>xy</sub></i></span>) is defined as the change in angle between lines <span style="text-decoration:overline;"><i>AC</i></span> and <span style="text-decoration:overline;"><i>AB</i></span>. Therefore, <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 \gamma _{xy}=\alpha +\beta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>y</mi> </mrow> </msub> <mo>=</mo> <mi>α</mi> <mo>+</mo> <mi>β</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma _{xy}=\alpha +\beta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6de672680fe1c7e78d5d3187d432d814311f4214" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.952ex; height:2.843ex;" alt="{\displaystyle \gamma _{xy}=\alpha +\beta }"></span> </p><p>From the geometry of the figure, we have <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\tan \alpha &={\frac {{\tfrac {\partial u_{y}}{\partial x}}dx}{dx+{\tfrac {\partial u_{x}}{\partial x}}dx}}={\frac {\tfrac {\partial u_{y}}{\partial x}}{1+{\tfrac {\partial u_{x}}{\partial x}}}}\\\tan \beta &={\frac {{\tfrac {\partial u_{x}}{\partial y}}dy}{dy+{\tfrac {\partial u_{y}}{\partial y}}dy}}={\frac {\tfrac {\partial u_{x}}{\partial y}}{1+{\tfrac {\partial u_{y}}{\partial y}}}}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>tan</mi> <mo>⁡</mo> <mi>α</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mstyle> </mrow> <mi>d</mi> <mi>x</mi> </mrow> <mrow> <mi>d</mi> <mi>x</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mstyle> </mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mstyle> <mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mstyle> </mrow> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>tan</mi> <mo>⁡</mo> <mi>β</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> </mfrac> </mstyle> </mrow> <mi>d</mi> <mi>y</mi> </mrow> <mrow> <mi>d</mi> <mi>y</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> </mfrac> </mstyle> </mrow> <mi>d</mi> <mi>y</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> </mfrac> </mstyle> <mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> </mfrac> </mstyle> </mrow> </mrow> </mfrac> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\tan \alpha &={\frac {{\tfrac {\partial u_{y}}{\partial x}}dx}{dx+{\tfrac {\partial u_{x}}{\partial x}}dx}}={\frac {\tfrac {\partial u_{y}}{\partial x}}{1+{\tfrac {\partial u_{x}}{\partial x}}}}\\\tan \beta &={\frac {{\tfrac {\partial u_{x}}{\partial y}}dy}{dy+{\tfrac {\partial u_{y}}{\partial y}}dy}}={\frac {\tfrac {\partial u_{x}}{\partial y}}{1+{\tfrac {\partial u_{y}}{\partial y}}}}\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ce730390cc6019e0ec20a5120d4dcaf301e297a9" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -9.005ex; width:33.061ex; height:19.176ex;" alt="{\displaystyle {\begin{aligned}\tan \alpha &={\frac {{\tfrac {\partial u_{y}}{\partial x}}dx}{dx+{\tfrac {\partial u_{x}}{\partial x}}dx}}={\frac {\tfrac {\partial u_{y}}{\partial x}}{1+{\tfrac {\partial u_{x}}{\partial x}}}}\\\tan \beta &={\frac {{\tfrac {\partial u_{x}}{\partial y}}dy}{dy+{\tfrac {\partial u_{y}}{\partial y}}dy}}={\frac {\tfrac {\partial u_{x}}{\partial y}}{1+{\tfrac {\partial u_{y}}{\partial y}}}}\end{aligned}}}"></span> For small displacement gradients we have <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial u_{x}}{\partial x}}\ll 1~;~~{\frac {\partial u_{y}}{\partial y}}\ll 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>≪</mo> <mn>1</mn> <mtext> </mtext> <mo>;</mo> <mtext> </mtext> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> </mfrac> </mrow> <mo>≪</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial u_{x}}{\partial x}}\ll 1~;~~{\frac {\partial u_{y}}{\partial y}}\ll 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/85ffb4c958cd83e85a00fe5ea9094b6588a8e6f8" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:21.519ex; height:6.343ex;" alt="{\displaystyle {\frac {\partial u_{x}}{\partial x}}\ll 1~;~~{\frac {\partial u_{y}}{\partial y}}\ll 1}"></span> For small rotations, i.e. <span class="texhtml mvar" style="font-style:italic;">α</span> and <span class="texhtml mvar" style="font-style:italic;">β</span> are ≪ 1 we have <span class="texhtml">tan <i>α</i> ≈ <i>α</i></span>, <span class="texhtml">tan <i>β</i> ≈ <i>β</i></span>. Therefore, <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 \alpha \approx {\frac {\partial u_{y}}{\partial x}}~;~~\beta \approx {\frac {\partial u_{x}}{\partial y}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>≈</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mtext> </mtext> <mo>;</mo> <mtext> </mtext> <mtext> </mtext> <mi>β</mi> <mo>≈</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha \approx {\frac {\partial u_{y}}{\partial x}}~;~~\beta \approx {\frac {\partial u_{x}}{\partial y}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b9c1cc9987dd7b6e168fbdf95f060fa07d8c5e8d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:20.982ex; height:6.343ex;" alt="{\displaystyle \alpha \approx {\frac {\partial u_{y}}{\partial x}}~;~~\beta \approx {\frac {\partial u_{x}}{\partial y}}}"></span> thus <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 \gamma _{xy}=\alpha +\beta ={\frac {\partial u_{y}}{\partial x}}+{\frac {\partial u_{x}}{\partial y}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>y</mi> </mrow> </msub> <mo>=</mo> <mi>α</mi> <mo>+</mo> <mi>β</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma _{xy}=\alpha +\beta ={\frac {\partial u_{y}}{\partial x}}+{\frac {\partial u_{x}}{\partial y}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/62f89d140b5e017430a1de77c5554e1af080df36" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:27.081ex; height:6.343ex;" alt="{\displaystyle \gamma _{xy}=\alpha +\beta ={\frac {\partial u_{y}}{\partial x}}+{\frac {\partial u_{x}}{\partial y}}}"></span> By interchanging <span class="texhtml mvar" style="font-style:italic;">x</span> and <span class="texhtml mvar" style="font-style:italic;">y</span> and <span class="texhtml"><i>u<sub>x</sub></i></span> and <span class="texhtml"><i>u<sub>y</sub></i></span>, it can be shown that <span class="texhtml"><i>γ<sub>xy</sub></i> = <i>γ<sub>yx</sub></i></span>. </p><p>Similarly, for the <span class="texhtml mvar" style="font-style:italic;">yz</span>- and <span class="texhtml mvar" style="font-style:italic;">xz</span>-planes, we have <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma _{yz}=\gamma _{zy}={\frac {\partial u_{y}}{\partial z}}+{\frac {\partial u_{z}}{\partial y}}\quad ,\qquad \gamma _{zx}=\gamma _{xz}={\frac {\partial u_{z}}{\partial x}}+{\frac {\partial u_{x}}{\partial z}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> <mi>z</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> <mi>y</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>z</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> </mfrac> </mrow> <mspace width="1em" /> <mo>,</mo> <mspace width="2em" /> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> <mi>x</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>z</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>z</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma _{yz}=\gamma _{zy}={\frac {\partial u_{y}}{\partial z}}+{\frac {\partial u_{z}}{\partial y}}\quad ,\qquad \gamma _{zx}=\gamma _{xz}={\frac {\partial u_{z}}{\partial x}}+{\frac {\partial u_{x}}{\partial z}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e47a1b5d92e97d82c32d925c8066de22a9fe6fd4" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:56.575ex; height:6.343ex;" alt="{\displaystyle \gamma _{yz}=\gamma _{zy}={\frac {\partial u_{y}}{\partial z}}+{\frac {\partial u_{z}}{\partial y}}\quad ,\qquad \gamma _{zx}=\gamma _{xz}={\frac {\partial u_{z}}{\partial x}}+{\frac {\partial u_{x}}{\partial z}}}"></span> </p> <div class="mw-heading mw-heading3"><h3 id="Volume_strain">Volume strain</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Strain_(mechanics)&action=edit&section=12" title="Edit section: Volume strain"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="excerpt-block"><style data-mw-deduplicate="TemplateStyles:r1066933788">.mw-parser-output .excerpt-hat .mw-editsection-like{font-style:normal}</style><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable dablink excerpt-hat selfref">This section is an excerpt from <a href="/wiki/Infinitesimal_strain_theory#Volumetric_strain" title="Infinitesimal strain theory">Infinitesimal strain theory § Volumetric strain</a>.<span class="mw-editsection-like plainlinks"><span class="mw-editsection-bracket">[</span><a class="external text" href="https://en.wikipedia.org/w/index.php?title=Infinitesimal_strain_theory&action=edit#Volumetric_strain">edit</a><span class="mw-editsection-bracket">]</span></span></div><div class="excerpt"> <p>The volumetric strain, also called bulk strain, is the relative variation of the volume, as arising from <i><a href="/wiki/Dilation_(physics)" class="mw-redirect" title="Dilation (physics)">dilation</a></i> or <i>compression</i>; it is the <a href="#Strain_invariants">first strain invariant</a> or <a href="/wiki/Trace_(matrix)" class="mw-redirect" title="Trace (matrix)">trace</a> of the tensor: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta ={\frac {\Delta V}{V_{0}}}=I_{1}=\varepsilon _{11}+\varepsilon _{22}+\varepsilon _{33}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">Δ</mi> <mi>V</mi> </mrow> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mfrac> </mrow> <mo>=</mo> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>11</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>22</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>33</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta ={\frac {\Delta V}{V_{0}}}=I_{1}=\varepsilon _{11}+\varepsilon _{22}+\varepsilon _{33}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48a4bcd3b38d879b3a2d424679c0c293febf8303" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:31.541ex; height:5.843ex;" alt="{\displaystyle \delta ={\frac {\Delta V}{V_{0}}}=I_{1}=\varepsilon _{11}+\varepsilon _{22}+\varepsilon _{33}}"></span> Actually, if we consider a cube with an edge length <i>a</i>, it is a quasi-cube after the deformation (the variations of the angles do not change the volume) with the dimensions <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 a\cdot (1+\varepsilon _{11})\times a\cdot (1+\varepsilon _{22})\times a\cdot (1+\varepsilon _{33})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>⋅</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>11</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>×</mo> <mi>a</mi> <mo>⋅</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>22</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>×</mo> <mi>a</mi> <mo>⋅</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>33</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\cdot (1+\varepsilon _{11})\times a\cdot (1+\varepsilon _{22})\times a\cdot (1+\varepsilon _{33})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6ae36ca9b95a617c02e6a3a4b5e083a5284aac85" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:40.723ex; height:2.843ex;" alt="{\displaystyle a\cdot (1+\varepsilon _{11})\times a\cdot (1+\varepsilon _{22})\times a\cdot (1+\varepsilon _{33})}"></span> and <i>V</i><sub>0</sub> = <i>a</i><sup>3</sup>, thus <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\Delta V}{V_{0}}}={\frac {\left(1+\varepsilon _{11}+\varepsilon _{22}+\varepsilon _{33}+\varepsilon _{11}\cdot \varepsilon _{22}+\varepsilon _{11}\cdot \varepsilon _{33}+\varepsilon _{22}\cdot \varepsilon _{33}+\varepsilon _{11}\cdot \varepsilon _{22}\cdot \varepsilon _{33}\right)\cdot a^{3}-a^{3}}{a^{3}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">Δ</mi> <mi>V</mi> </mrow> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>11</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>22</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>33</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>11</mn> </mrow> </msub> <mo>⋅</mo> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>22</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>11</mn> </mrow> </msub> <mo>⋅</mo> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>33</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>22</mn> </mrow> </msub> <mo>⋅</mo> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>33</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>11</mn> </mrow> </msub> <mo>⋅</mo> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>22</mn> </mrow> </msub> <mo>⋅</mo> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>33</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>⋅</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\Delta V}{V_{0}}}={\frac {\left(1+\varepsilon _{11}+\varepsilon _{22}+\varepsilon _{33}+\varepsilon _{11}\cdot \varepsilon _{22}+\varepsilon _{11}\cdot \varepsilon _{33}+\varepsilon _{22}\cdot \varepsilon _{33}+\varepsilon _{11}\cdot \varepsilon _{22}\cdot \varepsilon _{33}\right)\cdot a^{3}-a^{3}}{a^{3}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3f1826bd3c96a6f777db8c6917829359e3f7afe3" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:84.348ex; height:6.343ex;" alt="{\displaystyle {\frac {\Delta V}{V_{0}}}={\frac {\left(1+\varepsilon _{11}+\varepsilon _{22}+\varepsilon _{33}+\varepsilon _{11}\cdot \varepsilon _{22}+\varepsilon _{11}\cdot \varepsilon _{33}+\varepsilon _{22}\cdot \varepsilon _{33}+\varepsilon _{11}\cdot \varepsilon _{22}\cdot \varepsilon _{33}\right)\cdot a^{3}-a^{3}}{a^{3}}}}"></span> as we consider small deformations, <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 1\gg \varepsilon _{ii}\gg \varepsilon _{ii}\cdot \varepsilon _{jj}\gg \varepsilon _{11}\cdot \varepsilon _{22}\cdot \varepsilon _{33}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>≫</mo> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>i</mi> </mrow> </msub> <mo>≫</mo> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>i</mi> </mrow> </msub> <mo>⋅</mo> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mi>j</mi> </mrow> </msub> <mo>≫</mo> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>11</mn> </mrow> </msub> <mo>⋅</mo> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>22</mn> </mrow> </msub> <mo>⋅</mo> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>33</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1\gg \varepsilon _{ii}\gg \varepsilon _{ii}\cdot \varepsilon _{jj}\gg \varepsilon _{11}\cdot \varepsilon _{22}\cdot \varepsilon _{33}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b5c1ef9daeba20511fa574620a6f7830e09435cc" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:33.493ex; height:2.843ex;" alt="{\displaystyle 1\gg \varepsilon _{ii}\gg \varepsilon _{ii}\cdot \varepsilon _{jj}\gg \varepsilon _{11}\cdot \varepsilon _{22}\cdot \varepsilon _{33}}"></span> therefore the formula. </p><p><span typeof="mw:File"><a href="/wiki/File:Approximation_volume_deformation.png" class="mw-file-description" title="Real variation of volume (top) and the approximated one (bottom): the green drawing shows the estimated volume and the orange drawing the neglected volume"><img alt="Real variation of volume (top) and the approximated one (bottom): the green drawing shows the estimated volume and the orange drawing the neglected volume" src="//upload.wikimedia.org/wikipedia/commons/thumb/8/81/Approximation_volume_deformation.png/400px-Approximation_volume_deformation.png" decoding="async" width="400" height="429" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/8/81/Approximation_volume_deformation.png 1.5x" data-file-width="455" data-file-height="488" /></a></span> </p> In case of pure shear, we can see that there is no change of the volume.</div></div> <div class="mw-heading mw-heading2"><h2 id="Metric_tensor">Metric tensor</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Strain_(mechanics)&action=edit&section=13" title="Edit section: Metric 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/Finite_strain_theory#Deformation_tensors_in_curvilinear_coordinates" title="Finite strain theory">Finite strain theory § Deformation tensors in curvilinear coordinates</a></div> <p>A strain field associated with a displacement is defined, at any point, by the change in length of the <a href="/wiki/Tangent_vector" title="Tangent vector">tangent vectors</a> representing the speeds of arbitrarily <a href="/wiki/Parametrization_(geometry)" title="Parametrization (geometry)">parametrized</a> curves passing through that point. A basic geometric result, due to <a href="/wiki/Maurice_Fr%C3%A9chet" class="mw-redirect" title="Maurice Fréchet">Fréchet</a>, <a href="/wiki/John_von_Neumann" title="John von Neumann">von Neumann</a> and <a href="/wiki/Pascual_Jordan" title="Pascual Jordan">Jordan</a>, states that, if the lengths of the tangent vectors fulfil the axioms of a <a href="/wiki/Norm_(mathematics)" title="Norm (mathematics)">norm</a> and the <a href="/wiki/Parallelogram_law" title="Parallelogram law">parallelogram law</a>, then the length of a vector is the square root of the value of the <a href="/wiki/Quadratic_form" title="Quadratic form">quadratic form</a> associated, by the <a href="/wiki/Polarization_formula" class="mw-redirect" title="Polarization formula">polarization formula</a>, with a <a href="/wiki/Positive_definite" class="mw-redirect" title="Positive definite">positive definite</a> <a href="/wiki/Bilinear_map" title="Bilinear map">bilinear map</a> called the <a href="/wiki/Metric_tensor" title="Metric tensor">metric 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=Strain_(mechanics)&action=edit&section=14" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Stress_measures" class="mw-redirect" title="Stress measures">Stress measures</a></li> <li><a href="/wiki/Strain_rate" title="Strain rate">Strain rate</a></li> <li><a href="/wiki/Strain_tensor" class="mw-redirect" title="Strain tensor">Strain tensor</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Strain_(mechanics)&action=edit&section=15" 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"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFLubliner2008" class="citation book cs1">Lubliner, Jacob (2008). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20100331022415/http://www.ce.berkeley.edu/~coby/plas/pdf/book.pdf"><i>Plasticity Theory</i></a> <span class="cs1-format">(PDF)</span> (Revised ed.). Dover Publications. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-46290-5" title="Special:BookSources/978-0-486-46290-5"><bdi>978-0-486-46290-5</bdi></a>. Archived from <a rel="nofollow" class="external text" href="http://www.ce.berkeley.edu/~coby/plas/pdf/book.pdf">the original</a> <span class="cs1-format">(PDF)</span> on 2010-03-31.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Plasticity+Theory&rft.edition=Revised&rft.pub=Dover+Publications&rft.date=2008&rft.isbn=978-0-486-46290-5&rft.aulast=Lubliner&rft.aufirst=Jacob&rft_id=http%3A%2F%2Fwww.ce.berkeley.edu%2F~coby%2Fplas%2Fpdf%2Fbook.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStrain+%28mechanics%29" class="Z3988"></span></span> </li> <li id="cite_note-rees-2"><span class="mw-cite-backlink">^ <a href="#cite_ref-rees_2-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-rees_2-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><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"><i>Basic Engineering Plasticity: An Introduction with Engineering and Manufacturing Applications</i></a>. Butterworth-Heinemann. <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>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20171222205706/https://books.google.com/books?id=4KWbmn_1hcYC">Archived</a> from the original on 2017-12-22.</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%3A+An+Introduction+with+Engineering+and+Manufacturing+Applications&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&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStrain+%28mechanics%29" class="Z3988"></span></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text">"Earth."Encyclopædia Britannica from <a href="/wiki/Encyclop%C3%A6dia_Britannica_2006_Ultimate_Reference_Suite_DVD" class="mw-redirect" title="Encyclopædia Britannica 2006 Ultimate Reference Suite DVD">Encyclopædia Britannica 2006 Ultimate Reference Suite DVD</a> .[2009].</span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRees2006" class="citation book cs1">Rees, David (2006). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=4KWbmn_1hcYC"><i>Basic Engineering Plasticity: An Introduction with Engineering and Manufacturing Applications</i></a>. Butterworth-Heinemann. p. 41. <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>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20171222205706/https://books.google.com/books?id=4KWbmn_1hcYC">Archived</a> from the original on 2017-12-22.</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%3A+An+Introduction+with+Engineering+and+Manufacturing+Applications&rft.pages=41&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&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStrain+%28mechanics%29" class="Z3988"></span></span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHencky1928" class="citation journal cs1">Hencky, H. 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Retrieved <span class="nowrap">2023-08-28</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=ISO&rft.atitle=ISO+80000-4%3A2019&rft.date=2013-08-20&rft_id=https%3A%2F%2Fwww.iso.org%2Fobp%2Fui%2Fen%2F%23iso%3Astd%3Aiso%3A80000%3A-4%3Aed-2%3Av1%3Aen&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStrain+%28mechanics%29" class="Z3988"></span></span> </li> </ol></div></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><style data-mw-deduplicate="TemplateStyles:r1038841319">.mw-parser-output .tooltip-dotted{border-bottom:1px dotted;cursor:help}</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/Q1990546#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"><span class="rt-commentedText tooltip tooltip-dotted" title="Dehnung"><a rel="nofollow" class="external text" href="https://d-nb.info/gnd/4123045-0">Germany</a></span></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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