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class="vector-toc-numb">2</span> <span>Physical mechanisms</span> </div> </a> <button aria-controls="toc-Physical_mechanisms-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 Physical mechanisms subsection</span> </button> <ul id="toc-Physical_mechanisms-sublist" class="vector-toc-list"> <li id="toc-In_metals" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#In_metals"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>In metals</span> </div> </a> <ul id="toc-In_metals-sublist" class="vector-toc-list"> <li id="toc-Slip_systems" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Slip_systems"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1.1</span> <span>Slip systems</span> </div> </a> <ul id="toc-Slip_systems-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Reversible_plasticity" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Reversible_plasticity"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1.2</span> <span>Reversible plasticity</span> </div> </a> <ul id="toc-Reversible_plasticity-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Shear_banding" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Shear_banding"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1.3</span> <span>Shear banding</span> </div> </a> <ul id="toc-Shear_banding-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Microplasticity" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Microplasticity"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1.4</span> <span>Microplasticity</span> </div> </a> <ul id="toc-Microplasticity-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Amorphous_materials" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Amorphous_materials"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Amorphous materials</span> </div> </a> <ul id="toc-Amorphous_materials-sublist" class="vector-toc-list"> <li id="toc-Crazing" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Crazing"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2.1</span> <span>Crazing</span> </div> </a> <ul id="toc-Crazing-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Cellular_materials" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Cellular_materials"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>Cellular materials</span> </div> </a> <ul id="toc-Cellular_materials-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Soils_and_sand" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Soils_and_sand"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.4</span> <span>Soils and sand</span> </div> </a> <ul id="toc-Soils_and_sand-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Rocks_and_concrete" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Rocks_and_concrete"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.5</span> <span>Rocks and concrete</span> </div> </a> <ul id="toc-Rocks_and_concrete-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Time-independent_yielding_and_plastic_flow_in_crystalline_materials" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Time-independent_yielding_and_plastic_flow_in_crystalline_materials"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Time-independent yielding and plastic flow in crystalline materials</span> </div> </a> <button aria-controls="toc-Time-independent_yielding_and_plastic_flow_in_crystalline_materials-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 Time-independent yielding and plastic flow in crystalline materials subsection</span> </button> <ul id="toc-Time-independent_yielding_and_plastic_flow_in_crystalline_materials-sublist" class="vector-toc-list"> <li id="toc-Time-independent_yielding_and_plastic_flow_in_single_crystals" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Time-independent_yielding_and_plastic_flow_in_single_crystals"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Time-independent yielding and plastic flow in single crystals</span> </div> </a> <ul id="toc-Time-independent_yielding_and_plastic_flow_in_single_crystals-sublist" class="vector-toc-list"> <li id="toc-Critical_resolved_shear_stress_dependence_on_temperature,_strain_rate,_and_point_defects" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Critical_resolved_shear_stress_dependence_on_temperature,_strain_rate,_and_point_defects"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1.1</span> <span>Critical resolved shear stress dependence on temperature, strain rate, and point defects</span> </div> </a> <ul id="toc-Critical_resolved_shear_stress_dependence_on_temperature,_strain_rate,_and_point_defects-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Stages_of_time-independent_plastic_flow,_post_yielding" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Stages_of_time-independent_plastic_flow,_post_yielding"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1.2</span> <span>Stages of time-independent plastic flow, post yielding</span> </div> </a> <ul id="toc-Stages_of_time-independent_plastic_flow,_post_yielding-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Time-independent_yielding_and_plastic_flow_in_polycrystals" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Time-independent_yielding_and_plastic_flow_in_polycrystals"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Time-independent yielding and plastic flow in polycrystals</span> </div> </a> <ul id="toc-Time-independent_yielding_and_plastic_flow_in_polycrystals-sublist" class="vector-toc-list"> <li id="toc-Grain_boundary_constraint_in_polycrystals" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Grain_boundary_constraint_in_polycrystals"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2.1</span> <span>Grain boundary constraint in polycrystals</span> </div> </a> <ul id="toc-Grain_boundary_constraint_in_polycrystals-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Implications_of_the_grain_boundary_constraint_in_polycrystals" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Implications_of_the_grain_boundary_constraint_in_polycrystals"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2.2</span> <span>Implications of the grain boundary constraint in polycrystals</span> </div> </a> <ul id="toc-Implications_of_the_grain_boundary_constraint_in_polycrystals-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </li> <li id="toc-Mathematical_descriptions" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Mathematical_descriptions"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Mathematical descriptions</span> </div> </a> <button aria-controls="toc-Mathematical_descriptions-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 Mathematical descriptions subsection</span> </button> <ul id="toc-Mathematical_descriptions-sublist" class="vector-toc-list"> <li id="toc-Deformation_theory" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Deformation_theory"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Deformation theory</span> </div> </a> <ul id="toc-Deformation_theory-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Flow_plasticity_theory" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Flow_plasticity_theory"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2</span> <span>Flow plasticity theory</span> </div> </a> <ul id="toc-Flow_plasticity_theory-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Yield_criteria" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Yield_criteria"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Yield criteria</span> </div> </a> <button aria-controls="toc-Yield_criteria-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 Yield criteria subsection</span> </button> <ul id="toc-Yield_criteria-sublist" class="vector-toc-list"> <li id="toc-Tresca_criterion" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Tresca_criterion"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1</span> <span>Tresca criterion</span> </div> </a> <ul id="toc-Tresca_criterion-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Huber–von_Mises_criterion" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Huber–von_Mises_criterion"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.2</span> <span>Huber–von Mises criterion</span> </div> </a> <ul id="toc-Huber–von_Mises_criterion-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</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"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Further_reading" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Further_reading"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Further reading</span> </div> </a> <ul id="toc-Further_reading-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span 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Available in 48 languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-48" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">48 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%84%D8%AF%D9%88%D9%86%D8%A9" title="لدونة – Arabic" lang="ar" hreflang="ar" data-title="لدونة" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/Plastiklik_(fizika)" title="Plastiklik (fizika) – Azerbaijani" lang="az" hreflang="az" data-title="Plastiklik (fizika)" data-language-autonym="Azərbaycanca" data-language-local-name="Azerbaijani" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%85%E0%A6%B8%E0%A7%8D%E0%A6%A5%E0%A6%BF%E0%A6%A4%E0%A6%BF%E0%A6%B8%E0%A7%8D%E0%A6%A5%E0%A6%BE%E0%A6%AA%E0%A6%95%E0%A6%A4%E0%A6%BE_(%E0%A6%A8%E0%A6%AE%E0%A6%A8%E0%A7%80%E0%A6%AF%E0%A6%BC_%E0%A6%AA%E0%A6%A6%E0%A6%BE%E0%A6%B0%E0%A7%8D%E0%A6%A5)" title="অস্থিতিস্থাপকতা (নমনীয় পদার্থ) – Bangla" lang="bn" hreflang="bn" data-title="অস্থিতিস্থাপকতা (নমনীয় পদার্থ)" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%9F%D0%BB%D0%B0%D1%81%D1%82%D1%8B%D1%87%D0%BD%D0%B0%D1%8F_%D0%B4%D1%8D%D1%84%D0%B0%D1%80%D0%BC%D0%B0%D1%86%D1%8B%D1%8F" title="Пластычная дэфармацыя – Belarusian" lang="be" hreflang="be" data-title="Пластычная дэфармацыя" data-language-autonym="Беларуская" data-language-local-name="Belarusian" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-be-x-old mw-list-item"><a href="https://be-tarask.wikipedia.org/wiki/%D0%9F%D0%BB%D1%8F%D1%81%D1%82%D1%8B%D1%87%D0%BD%D0%B0%D1%8F_%D0%B4%D1%8D%D1%84%D0%B0%D1%80%D0%BC%D0%B0%D1%86%D1%8B%D1%8F" title="Плястычная дэфармацыя – Belarusian (Taraškievica orthography)" lang="be-tarask" hreflang="be-tarask" data-title="Плястычная дэфармацыя" data-language-autonym="Беларуская (тарашкевіца)" data-language-local-name="Belarusian (Taraškievica orthography)" class="interlanguage-link-target"><span>Беларуская (тарашкевіца)</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%9F%D0%BB%D0%B0%D1%81%D1%82%D0%B8%D1%87%D0%BD%D0%B0_%D0%B4%D0%B5%D1%84%D0%BE%D1%80%D0%BC%D0%B0%D1%86%D0%B8%D1%8F" title="Пластична деформация – Bulgarian" lang="bg" hreflang="bg" data-title="Пластична деформация" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Plasticitat" title="Plasticitat – Catalan" lang="ca" hreflang="ca" data-title="Plasticitat" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%A2%D3%B3%D1%81%C4%95%D0%BC%D0%BB%C4%95%D1%85_(%D1%84%D0%B8%D0%B7%D0%B8%D0%BA%D0%B0)" title="Тӳсĕмлĕх (физика) – Chuvash" lang="cv" hreflang="cv" data-title="Тӳсĕмлĕх (физика)" data-language-autonym="Чӑвашла" data-language-local-name="Chuvash" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Plastizit%C3%A4t_(Physik)" title="Plastizität (Physik) – German" lang="de" hreflang="de" data-title="Plastizität (Physik)" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Plastsus" title="Plastsus – Estonian" lang="et" hreflang="et" data-title="Plastsus" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Plasticidad_(mec%C3%A1nica_de_s%C3%B3lidos)" title="Plasticidad (mecánica de sólidos) – Spanish" lang="es" hreflang="es" data-title="Plasticidad (mecánica de sólidos)" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Plasta_aliformi%C4%9Do" title="Plasta aliformiĝo – Esperanto" lang="eo" hreflang="eo" data-title="Plasta aliformiĝo" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Plastikotasun" title="Plastikotasun – Basque" lang="eu" hreflang="eu" data-title="Plastikotasun" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%85%D9%88%D9%85%D8%B3%D8%A7%D9%86%DB%8C" title="مومسانی – Persian" lang="fa" hreflang="fa" data-title="مومسانی" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/D%C3%A9formation_plastique" title="Déformation plastique – French" lang="fr" hreflang="fr" data-title="Déformation plastique" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/D%C3%ADchumadh_plaisteach" title="Díchumadh plaisteach – Irish" lang="ga" hreflang="ga" data-title="Díchumadh plaisteach" 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/%EC%86%8C%EC%84%B1_(%EB%AC%BC%EB%A6%AC%ED%95%99)" title="소성 (물리학) – Korean" lang="ko" hreflang="ko" data-title="소성 (물리학)" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D5%8A%D5%AC%D5%A1%D5%BD%D5%BF%D5%AB%D5%AF%D5%B8%D6%82%D5%A9%D5%B5%D5%B8%D6%82%D5%B6_(%D5%B4%D5%A5%D5%AD%D5%A1%D5%B6%D5%AB%D5%AF%D5%A1)" title="Պլաստիկություն (մեխանիկա) – Armenian" lang="hy" hreflang="hy" data-title="Պլաստիկություն (մեխանիկա)" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%B8%E0%A5%81%E0%A4%98%E0%A4%9F%E0%A5%8D%E0%A4%AF%E0%A4%A4%E0%A4%BE" title="सुघट्यता – Hindi" lang="hi" hreflang="hi" data-title="सुघट्यता" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Plasti%C4%8Dnost" title="Plastičnost – Croatian" lang="hr" hreflang="hr" data-title="Plastičnost" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Plastisitas_(fisika)" title="Plastisitas (fisika) – Indonesian" lang="id" hreflang="id" data-title="Plastisitas (fisika)" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-zu mw-list-item"><a href="https://zu.wikipedia.org/wiki/Ubucwazi" title="Ubucwazi – Zulu" lang="zu" hreflang="zu" data-title="Ubucwazi" data-language-autonym="IsiZulu" data-language-local-name="Zulu" class="interlanguage-link-target"><span>IsiZulu</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Plasticit%C3%A0_(fisica)" title="Plasticità (fisica) – Italian" lang="it" hreflang="it" data-title="Plasticità (fisica)" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%A4%D7%9C%D7%A1%D7%98%D7%99%D7%95%D7%AA" title="פלסטיות – Hebrew" lang="he" hreflang="he" data-title="פלסטיות" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%9E%E1%83%9A%E1%83%90%E1%83%A1%E1%83%A2%E1%83%98%E1%83%99%E1%83%A3%E1%83%A0%E1%83%9D%E1%83%91%E1%83%90" title="პლასტიკურობა – Georgian" lang="ka" hreflang="ka" data-title="პლასტიკურობა" data-language-autonym="ქართული" data-language-local-name="Georgian" class="interlanguage-link-target"><span>ქართული</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%98%D0%BB%D0%B5%D0%BC%D0%B4%D1%96%D0%BB%D1%96%D0%BA_(%D1%84%D0%B8%D0%B7%D0%B8%D0%BA%D0%B0)" title="Илемділік (физика) – Kazakh" lang="kk" hreflang="kk" data-title="Илемділік (физика)" data-language-autonym="Қазақша" data-language-local-name="Kazakh" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Plasticitas" title="Plasticitas – Latin" lang="la" hreflang="la" data-title="Plasticitas" data-language-autonym="Latina" data-language-local-name="Latin" class="interlanguage-link-target"><span>Latina</span></a></li><li class="interlanguage-link interwiki-hu badge-Q17437796 badge-featuredarticle mw-list-item" title="featured article badge"><a href="https://hu.wikipedia.org/wiki/K%C3%A9pl%C3%A9kenys%C3%A9gtan" title="Képlékenységtan – Hungarian" lang="hu" hreflang="hu" data-title="Képlékenységtan" 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/Plasticiteit_(materiaalkunde)" title="Plasticiteit (materiaalkunde) – Dutch" lang="nl" hreflang="nl" data-title="Plasticiteit (materiaalkunde)" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-new mw-list-item"><a href="https://new.wikipedia.org/wiki/%E0%A4%86%E0%A4%B2%E0%A5%81%E0%A4%AA%E0%A4%B9%E0%A4%83" title="आलुपहः – Newari" lang="new" hreflang="new" data-title="आलुपहः" data-language-autonym="नेपाल भाषा" data-language-local-name="Newari" class="interlanguage-link-target"><span>नेपाल भाषा</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E5%A1%91%E6%80%A7" title="塑性 – Japanese" lang="ja" hreflang="ja" data-title="塑性" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Plastisitet" title="Plastisitet – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Plastisitet" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Plastiklik" title="Plastiklik – Uzbek" lang="uz" hreflang="uz" data-title="Plastiklik" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="Uzbek" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%A2%E0%A8%BF%E0%A8%B2%E0%A8%95_(%E0%A8%AD%E0%A9%8C%E0%A8%A4%E0%A8%BF%E0%A8%95_%E0%A8%B5%E0%A8%BF%E0%A8%97%E0%A8%BF%E0%A8%86%E0%A8%A8)" title="ਢਿਲਕ (ਭੌਤਿਕ ਵਿਗਿਆਨ) – Punjabi" lang="pa" hreflang="pa" data-title="ਢਿਲਕ (ਭੌਤਿਕ ਵਿਗਿਆਨ)" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="Punjabi" class="interlanguage-link-target"><span>ਪੰਜਾਬੀ</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Plastyczno%C5%9B%C4%87" title="Plastyczność – Polish" lang="pl" hreflang="pl" data-title="Plastyczność" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Plasticidade" title="Plasticidade – Portuguese" lang="pt" hreflang="pt" data-title="Plasticidade" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9F%D0%BB%D0%B0%D1%81%D1%82%D0%B8%D1%87%D0%BD%D0%BE%D1%81%D1%82%D1%8C_(%D1%84%D0%B8%D0%B7%D0%B8%D0%BA%D0%B0)" title="Пластичность (физика) – Russian" lang="ru" hreflang="ru" data-title="Пластичность (физика)" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Plasticity_(physics)" title="Plasticity (physics) – Simple English" lang="en-simple" hreflang="en-simple" data-title="Plasticity (physics)" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Plastick%C3%A1_deform%C3%A1cia" title="Plastická deformácia – Slovak" lang="sk" hreflang="sk" data-title="Plastická deformácia" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/Plasti%C4%8Dnost_(fizika)" title="Plastičnost (fizika) – Serbian" lang="sr" hreflang="sr" data-title="Plastičnost (fizika)" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Plasti%C4%8Dnost" title="Plastičnost – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Plastičnost" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="Serbo-Croatian" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Plastisuus" title="Plastisuus – Finnish" lang="fi" hreflang="fi" data-title="Plastisuus" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Plasticitet_(fysik)" title="Plasticitet (fysik) – Swedish" lang="sv" hreflang="sv" data-title="Plasticitet (fysik)" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%A8%E0%AF%86%E0%AE%95%E0%AE%BF%E0%AE%B4%E0%AF%8D%E0%AE%B5%E0%AF%81_%E0%AE%A4%E0%AE%A9%E0%AF%8D%E0%AE%AE%E0%AF%88" title="நெகிழ்வு தன்மை – Tamil" lang="ta" hreflang="ta" data-title="நெகிழ்வு தன்மை" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%9F%D0%BB%D0%B0%D1%81%D1%82%D0%B8%D1%87%D0%BD%D1%96%D1%81%D1%82%D1%8C_%D0%BC%D0%B0%D1%82%D0%B5%D1%80%D1%96%D0%B0%D0%BB%D1%83" title="Пластичність матеріалу – Ukrainian" lang="uk" hreflang="uk" data-title="Пластичність матеріалу" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-ur mw-list-item"><a href="https://ur.wikipedia.org/wiki/%D9%84%D8%AF%D9%88%D9%86%D8%AA_(%D8%B7%D8%A8%DB%8C%D8%B9%DB%8C%D8%A7%D8%AA)" title="لدونت (طبیعیات) – Urdu" lang="ur" hreflang="ur" data-title="لدونت (طبیعیات)" data-language-autonym="اردو" data-language-local-name="Urdu" class="interlanguage-link-target"><span>اردو</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/Bi%E1%BA%BFn_d%E1%BA%A1ng_d%E1%BA%BBo" title="Biến dạng dẻo – Vietnamese" lang="vi" hreflang="vi" data-title="Biến dạng dẻo" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E5%A1%91%E6%80%A7%E8%AE%8A%E5%BD%A2" title="塑性變形 – Chinese" lang="zh" hreflang="zh" data-title="塑性變形" data-language-autonym="中文" data-language-local-name="Chinese" 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<div class="mw-indicators"> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Non-reversible deformation of a solid material in response to applied forces</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">"Plastic material" redirects here. For the material used in manufacturing, see <a href="/wiki/Plastic" title="Plastic">Plastic</a>.</div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Metal_yield.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/8e/Metal_yield.svg/290px-Metal_yield.svg.png" decoding="async" width="290" height="217" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/8e/Metal_yield.svg/435px-Metal_yield.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/8e/Metal_yield.svg/580px-Metal_yield.svg.png 2x" data-file-width="331" data-file-height="248" /></a><figcaption> <a href="/wiki/Stress%E2%80%93strain_curve" title="Stress–strain curve">Stress–strain curve</a> showing typical <a href="/wiki/Yield_(engineering)" title="Yield (engineering)">yield</a> behavior for <a href="/wiki/Non-ferrous_metal" title="Non-ferrous metal">nonferrous alloys</a> (<a href="/wiki/Stress_(mechanics)" title="Stress (mechanics)">stress</a>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>σ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59f59b7c3e6fdb1d0365a494b81fb9a696138c36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle \sigma }"></span>, shown as a function of <a href="/wiki/Strain_(mechanics)" title="Strain (mechanics)">strain</a>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \epsilon }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϵ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \epsilon }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3837cad72483d97bcdde49c85d3b7b859fb3fd2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.944ex; height:1.676ex;" alt="{\displaystyle \epsilon }"></span>): <style data-mw-deduplicate="TemplateStyles:r1184024115">.mw-parser-output .div-col{margin-top:0.3em;column-width:30em}.mw-parser-output .div-col-small{font-size:90%}.mw-parser-output .div-col-rules{column-rule:1px solid #aaa}.mw-parser-output .div-col dl,.mw-parser-output .div-col ol,.mw-parser-output .div-col ul{margin-top:0}.mw-parser-output .div-col li,.mw-parser-output .div-col dd{page-break-inside:avoid;break-inside:avoid-column}</style><div class="div-col" style="column-width: 10em;"> <div><ol style="margin:0; list-style-position:inside;"><li style="white-space:nowrap;"><a href="/wiki/True_elastic_limit" class="mw-redirect" title="True elastic limit">True elastic limit</a></li><li style="white-space:nowrap;"><a href="/wiki/Proportionality_limit" class="mw-redirect" title="Proportionality limit">Proportionality limit</a></li><li style="white-space:nowrap;"><a href="/wiki/Elastic_limit" class="mw-redirect" title="Elastic limit">Elastic limit</a></li><li style="white-space:nowrap;">Offset <a href="/wiki/Yield_strength" class="mw-redirect" title="Yield strength">yield strength</a></li></ol></div> </div> </figcaption></figure> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Stress_v_strain_A36_2.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f1/Stress_v_strain_A36_2.svg/290px-Stress_v_strain_A36_2.svg.png" decoding="async" width="290" height="300" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f1/Stress_v_strain_A36_2.svg/435px-Stress_v_strain_A36_2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f1/Stress_v_strain_A36_2.svg/580px-Stress_v_strain_A36_2.svg.png 2x" data-file-width="777" data-file-height="804" /></a><figcaption> A <a href="/wiki/Stress%E2%80%93strain_curve" title="Stress–strain curve">stress–strain curve</a> typical of <a href="/wiki/Structural_steel" title="Structural steel">structural steel</a>:<hr /> <div><ol style="margin-left:0;"><li style="list-style-position:inside; white-space:nowrap;"><a href="/wiki/Ultimate_tensile_strength" title="Ultimate tensile strength">Ultimate strength</a></li><li style="list-style-position:inside; white-space:nowrap;"><a href="/wiki/Yield_(engineering)" title="Yield (engineering)">Yield strength (yield point)</a></li><li style="list-style-position:inside; white-space:nowrap;">Rupture</li><li style="list-style-position:inside; white-space:nowrap;"><a href="/wiki/Strain_hardening" class="mw-redirect" title="Strain hardening">Strain hardening</a> region</li><li style="list-style-position:inside; white-space:nowrap;"><a href="/wiki/Necking_(engineering)" title="Necking (engineering)">Necking</a> region</li></ol></div> <div><ol style="margin-left:0; margin-top:0; list-style-type:upper-alpha;"><li style="list-style-position:inside; white-space:nowrap;">Apparent stress (<i>F</i>/<i>A</i><sub>0</sub>)</li><li style="list-style-position:inside; white-space:nowrap;">Actual stress (<i>F</i>/<i>A</i>)</li></ol></div> </figcaption></figure> <style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output 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href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1246091330"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><table class="sidebar sidebar-collapse nomobile nowraplinks plainlist"><tbody><tr><td class="sidebar-pretitle">Part of a series on</td></tr><tr><th class="sidebar-title-with-pretitle"><a href="/wiki/Continuum_mechanics" title="Continuum mechanics">Continuum mechanics</a></th></tr><tr><td class="sidebar-image"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J=-D{\frac {d\varphi }{dx}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>J</mi> <mo>=</mo> <mo>−</mo> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>φ</mi> </mrow> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle J=-D{\frac {d\varphi }{dx}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1856f88def2056f28ed27c7d31180a6240820ea6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:11.874ex; height:5.509ex;" alt="{\displaystyle J=-D{\frac {d\varphi }{dx}}}"></span><div class="sidebar-caption"><a href="/wiki/Fick%27s_laws_of_diffusion" title="Fick's laws of diffusion">Fick's laws of diffusion</a></div></td></tr><tr><td class="sidebar-content-with-subgroup"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:transparent;border-top:1px solid #aaa;text-align:center;;color: var(--color-base)">Laws</div><div class="sidebar-list-content mw-collapsible-content"><table class="sidebar-subgroup"><tbody><tr><th class="sidebar-heading" style="font-style:italic;font-weight:normal;"> Conservations</th></tr><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Conservation_of_mass" title="Conservation of mass">Mass</a></li> <li><a href="/wiki/Conservation_of_momentum" class="mw-redirect" title="Conservation of momentum">Momentum</a></li> <li><a href="/wiki/Conservation_of_energy" title="Conservation of energy">Energy</a></li></ul></td> </tr><tr><th class="sidebar-heading" style="font-style:italic;font-weight:normal;"> Inequalities</th></tr><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Clausius%E2%80%93Duhem_inequality" title="Clausius–Duhem inequality">Clausius–Duhem (entropy)</a></li></ul></td> </tr></tbody></table></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible"><div class="sidebar-list-title" style="background:transparent;border-top:1px solid #aaa;text-align:center;;color: var(--color-base)"><a 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 class="mw-selflink selflink">Plasticity</a></li> <li><a href="/wiki/Hooke%27s_law" title="Hooke's law">Hooke's law</a></li> <li><a href="/wiki/Stress_(mechanics)" title="Stress (mechanics)">Stress</a></li> <li><a href="/wiki/Strain_(mechanics)" title="Strain (mechanics)">Strain</a> <ul><li><a href="/wiki/Finite_strain_theory" title="Finite strain theory">Finite strain</a></li> <li><a href="/wiki/Infinitesimal_strain_theory" title="Infinitesimal strain theory">Infinitesimal strain</a></li></ul></li> <li><a href="/wiki/Compatibility_(mechanics)" title="Compatibility (mechanics)">Compatibility</a></li> <li><a href="/wiki/Bending" title="Bending">Bending</a></li> <li><a href="/wiki/Contact_mechanics" title="Contact mechanics">Contact mechanics</a> <ul><li><a href="/wiki/Frictional_contact_mechanics" title="Frictional contact mechanics">frictional</a></li></ul></li> <li><a href="/wiki/Material_failure_theory" title="Material failure theory">Material failure theory</a></li> <li><a href="/wiki/Fracture_mechanics" title="Fracture mechanics">Fracture mechanics</a></li></ul> </div></div></div></td> </tr><tr><td class="sidebar-content-with-subgroup"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:transparent;border-top:1px solid #aaa;text-align:center;;color: var(--color-base)"><a href="/wiki/Fluid_mechanics" title="Fluid mechanics">Fluid mechanics</a></div><div class="sidebar-list-content mw-collapsible-content"><table class="sidebar-subgroup"><tbody><tr><th class="sidebar-heading" style="font-style:italic;"> <a href="/wiki/Fluid" title="Fluid">Fluids</a></th></tr><tr><td class="sidebar-content"> <div class="wraplinks"> <ul><li><a href="/wiki/Hydrostatics" title="Hydrostatics">Statics</a> <b>·</b> <a href="/wiki/Fluid_dynamics" title="Fluid dynamics">Dynamics</a></li> <li><a href="/wiki/Archimedes%27_principle" title="Archimedes' principle">Archimedes' principle</a> <b>·</b> <a href="/wiki/Bernoulli%27s_principle" title="Bernoulli's principle">Bernoulli's principle</a></li> <li><a href="/wiki/Navier%E2%80%93Stokes_equations" title="Navier–Stokes equations">Navier–Stokes equations</a></li> <li><a href="/wiki/Hagen%E2%80%93Poiseuille_equation" title="Hagen–Poiseuille equation">Poiseuille equation</a> <b>·</b> <a href="/wiki/Pascal%27s_law" title="Pascal's law">Pascal's law</a></li> <li><a href="/wiki/Viscosity" title="Viscosity">Viscosity</a> <ul><li>(<a href="/wiki/Newtonian_fluid" title="Newtonian fluid">Newtonian</a> <b>·</b> <a href="/wiki/Non-Newtonian_fluid" title="Non-Newtonian fluid">non-Newtonian</a>)</li></ul></li> <li><a href="/wiki/Buoyancy" title="Buoyancy">Buoyancy</a> <b>·</b> <a href="/wiki/Mixing_(process_engineering)" title="Mixing (process engineering)">Mixing</a> <b>·</b> <a href="/wiki/Pressure" title="Pressure">Pressure</a></li></ul> </div></td> </tr><tr><th class="sidebar-heading" style="font-style:italic;"> <a href="/wiki/Liquid" title="Liquid">Liquids</a></th></tr><tr><td class="sidebar-content"> <div class="hlist"> <ul><li><a href="/wiki/Adhesion" title="Adhesion">Adhesion</a></li> <li><a href="/wiki/Capillary_action" title="Capillary action">Capillary action</a></li> <li><a href="/wiki/Chromatography" title="Chromatography">Chromatography</a></li> <li><a href="/wiki/Cohesion_(chemistry)" title="Cohesion (chemistry)">Cohesion (chemistry)</a></li> <li><a href="/wiki/Surface_tension" title="Surface tension">Surface tension</a></li></ul> </div></td> </tr><tr><th class="sidebar-heading" style="font-style:italic;"> <a href="/wiki/Gas" title="Gas">Gases</a></th></tr><tr><td class="sidebar-content"> <div class="hlist"> <ul><li><a href="/wiki/Atmosphere" title="Atmosphere">Atmosphere</a></li> <li><a href="/wiki/Boyle%27s_law" title="Boyle's law">Boyle's law</a></li> <li><a href="/wiki/Charles%27s_law" title="Charles's law">Charles's law</a></li> <li><a href="/wiki/Combined_gas_law" class="mw-redirect" title="Combined gas law">Combined gas law</a></li> <li><a href="/wiki/Fick%27s_law" class="mw-redirect" title="Fick's law">Fick's law</a></li> <li><a href="/wiki/Gay-Lussac%27s_law" title="Gay-Lussac's law">Gay-Lussac's law</a></li> <li><a href="/wiki/Graham%27s_law" title="Graham's law">Graham's law</a></li></ul> </div></td> </tr><tr><th class="sidebar-heading" style="font-style:italic;"> <a href="/wiki/Plasma_(physics)" title="Plasma (physics)">Plasma</a></th></tr></tbody></table></div></div></td> </tr><tr><td class="sidebar-content-with-subgroup"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:transparent;border-top:1px solid #aaa;text-align:center;;color: var(--color-base)"><a href="/wiki/Rheology" title="Rheology">Rheology</a></div><div class="sidebar-list-content mw-collapsible-content"><table class="sidebar-subgroup"><tbody><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Viscoelasticity" title="Viscoelasticity">Viscoelasticity</a></li> <li><a href="/wiki/Rheometry" title="Rheometry">Rheometry</a></li> <li><a href="/wiki/Rheometer" title="Rheometer">Rheometer</a></li></ul></td> </tr><tr><th class="sidebar-heading" style="font-style:italic;"> <a href="/wiki/Smart_fluid" title="Smart fluid">Smart fluids</a></th></tr><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Electrorheological_fluid" title="Electrorheological fluid">Electrorheological</a></li> <li><a href="/wiki/Magnetorheological_fluid" title="Magnetorheological fluid">Magnetorheological</a></li> <li><a href="/wiki/Ferrofluid" title="Ferrofluid">Ferrofluids</a></li></ul></td> </tr></tbody></table></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:transparent;border-top:1px solid #aaa;text-align:center;;color: var(--color-base)">Scientists</div><div class="sidebar-list-content mw-collapsible-content"><div class="hlist"> <ul><li><a href="/wiki/Daniel_Bernoulli" title="Daniel Bernoulli">Bernoulli</a></li> <li><a href="/wiki/Robert_Boyle" title="Robert Boyle">Boyle</a></li> <li><a href="/wiki/Augustin-Louis_Cauchy" title="Augustin-Louis Cauchy">Cauchy</a></li> <li><a href="/wiki/Jacques_Charles" title="Jacques Charles">Charles</a></li> <li><a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Euler</a></li> <li><a href="/wiki/Adolf_Eugen_Fick" title="Adolf Eugen Fick">Fick</a></li> <li><a href="/wiki/Joseph_Louis_Gay-Lussac" title="Joseph Louis Gay-Lussac">Gay-Lussac</a></li> <li><a href="/wiki/Thomas_Graham_(chemist)" title="Thomas Graham (chemist)">Graham</a></li> <li><a href="/wiki/Robert_Hooke" title="Robert Hooke">Hooke</a></li> <li><a href="/wiki/Isaac_Newton" title="Isaac Newton">Newton</a></li> <li><a href="/wiki/Claude-Louis_Navier" title="Claude-Louis Navier">Navier</a></li> <li><a href="/wiki/Walter_Noll" title="Walter Noll">Noll</a></li> <li><a href="/wiki/Blaise_Pascal" title="Blaise Pascal">Pascal</a></li> <li><a href="/wiki/Sir_George_Stokes,_1st_Baronet" title="Sir George Stokes, 1st Baronet">Stokes</a></li> <li><a href="/wiki/Clifford_Truesdell" title="Clifford Truesdell">Truesdell</a></li></ul> </div></div></div></td> </tr><tr><td class="sidebar-navbar"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Continuum_mechanics" title="Template:Continuum mechanics"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Continuum_mechanics" title="Template talk:Continuum mechanics"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Continuum_mechanics" title="Special:EditPage/Template:Continuum mechanics"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> <p>In <a href="/wiki/Physics" title="Physics">physics</a> and <a href="/wiki/Materials_science" title="Materials science">materials science</a>, <b>plasticity</b> (also known as <b>plastic deformation</b>) is the ability of a <a href="/wiki/Solid" title="Solid">solid</a> <a href="/wiki/Material" title="Material">material</a> to undergo permanent <a href="/wiki/Deformation_(engineering)" title="Deformation (engineering)">deformation</a>, a non-reversible change of shape in response to applied forces.<sup id="cite_ref-Lubliner_1-0" class="reference"><a href="#cite_note-Lubliner-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> For example, a solid piece of metal being bent or pounded into a new shape displays plasticity as permanent changes occur within the material itself. In engineering, the transition from <a href="/wiki/Elasticity_(physics)" title="Elasticity (physics)">elastic</a> behavior to plastic behavior is known as <a href="/wiki/Yield_(engineering)" title="Yield (engineering)">yielding</a>. </p><p>Plastic deformation is observed in most materials, particularly <a href="/wiki/Metal" title="Metal">metals</a>, <a href="/wiki/Soil" title="Soil">soils</a>, <a href="/wiki/Rock_(geology)" title="Rock (geology)">rocks</a>, <a href="/wiki/Concrete" title="Concrete">concrete</a>, and <a href="/wiki/Foam" title="Foam">foams</a>.<sup id="cite_ref-Jirasek_3-0" class="reference"><a href="#cite_note-Jirasek-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Chen_4-0" class="reference"><a href="#cite_note-Chen-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Yu_5-0" class="reference"><a href="#cite_note-Yu-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Chen1_6-0" class="reference"><a href="#cite_note-Chen1-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> However, the physical mechanisms that cause plastic deformation can vary widely. At a <a href="/wiki/Crystalline" class="mw-redirect" title="Crystalline">crystalline</a> scale, plasticity in metals is usually a consequence of <a href="/wiki/Dislocation" title="Dislocation">dislocations</a>. Such defects are relatively rare in most crystalline materials, but are numerous in some and part of their crystal structure; in such cases, <a href="/wiki/Plastic_crystallinity" class="mw-redirect" title="Plastic crystallinity">plastic crystallinity</a> can result. In <a href="/wiki/Brittleness" title="Brittleness">brittle</a> materials such as rock, concrete and bone, plasticity is caused predominantly by <a href="/wiki/Slip_(materials_science)" title="Slip (materials science)">slip</a> at <a href="/wiki/Microcrack" class="mw-redirect" title="Microcrack">microcracks</a>. In cellular materials such as liquid <a href="/wiki/Foams" class="mw-redirect" title="Foams">foams</a> or <a href="/wiki/Tissue_(biology)" title="Tissue (biology)">biological tissues</a>, plasticity is mainly a consequence of bubble or cell rearrangements, notably <a href="/wiki/T1_process" title="T1 process">T1 processes</a>. </p><p>For many <a href="/wiki/Ductile" class="mw-redirect" title="Ductile">ductile</a> metals, <a href="/wiki/Tensile_loading" class="mw-redirect" title="Tensile loading">tensile loading</a> applied to a sample will cause it to behave in an elastic manner. Each increment of load is accompanied by a proportional increment in extension. When the load is removed, the piece returns to its original size. However, once the load exceeds a threshold – the yield strength – the extension increases more rapidly than in the elastic region; now when the load is removed, some degree of extension will remain. </p><p><a href="/wiki/Elastic_deformation" class="mw-redirect" title="Elastic deformation">Elastic deformation</a>, however, is an approximation and its quality depends on the time frame considered and loading speed. If, as indicated in the graph opposite, the deformation includes elastic deformation, it is also often referred to as "elasto-plastic deformation" or "elastic-plastic deformation". </p><p>Perfect plasticity is a property of materials to undergo irreversible deformation without any increase in stresses or loads. Plastic materials that have been <a href="/wiki/Work_hardening" title="Work hardening">hardened</a> by prior deformation, such as <a href="/wiki/Cold_forming" class="mw-redirect" title="Cold forming">cold forming</a>, may need increasingly higher stresses to deform further. Generally, plastic deformation is also dependent on the deformation speed, i.e. higher stresses usually have to be applied to increase the rate of deformation. Such materials are said to deform <a href="/wiki/Viscoplasticity" title="Viscoplasticity">visco-plastically</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Contributing_properties">Contributing properties</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Plasticity_(physics)&action=edit&section=1" title="Edit section: Contributing properties"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The plasticity of a material is directly proportional to the <a href="/wiki/Ductility" title="Ductility">ductility</a> and <a href="/wiki/Malleability" class="mw-redirect" title="Malleability">malleability</a> of the material. </p> <div class="mw-heading mw-heading2"><h2 id="Physical_mechanisms">Physical mechanisms</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Plasticity_(physics)&action=edit&section=2" title="Edit section: Physical mechanisms"><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:PlasticityIn111Copper.jpg" class="mw-file-description"><img alt="A large sphere on a flat plane of very small spheres with multiple sets of very small spheres contiguously extending below the plane (all with a black background)" src="//upload.wikimedia.org/wikipedia/commons/thumb/7/7f/PlasticityIn111Copper.jpg/220px-PlasticityIn111Copper.jpg" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/7f/PlasticityIn111Copper.jpg/330px-PlasticityIn111Copper.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/7f/PlasticityIn111Copper.jpg/440px-PlasticityIn111Copper.jpg 2x" data-file-width="4000" data-file-height="4000" /></a><figcaption>Plasticity under a spherical nanoindenter in (111) copper. All particles in ideal lattice positions are omitted and the color code refers to the von Mises stress field.</figcaption></figure> <div class="mw-heading mw-heading3"><h3 id="In_metals">In metals</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Plasticity_(physics)&action=edit&section=3" title="Edit section: In metals"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Plasticity in a crystal of pure metal is primarily caused by two modes of deformation in the crystal lattice: slip and twinning. Slip is a shear deformation which moves the atoms through many interatomic distances relative to their initial positions. Twinning is the plastic deformation which takes place along two planes due to a set of forces applied to a given metal piece. </p><p>Most metals show more plasticity when hot than when cold. Lead shows sufficient plasticity at room temperature, while cast iron does not possess sufficient plasticity for any forging operation even when hot. This property is of importance in forming, shaping and extruding operations on metals. Most metals are rendered plastic by heating and hence shaped hot. </p> <div class="mw-heading mw-heading4"><h4 id="Slip_systems">Slip systems</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Plasticity_(physics)&action=edit&section=4" title="Edit section: Slip systems"><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/Slip_(materials_science)#Slip_systems" title="Slip (materials science)">Slip (materials science) § Slip systems</a></div> <p>Crystalline materials contain uniform planes of atoms organized with long-range order. Planes may slip past each other along their close-packed directions, as is shown on the slip systems page. The result is a permanent change of shape within the crystal and plastic deformation. The presence of dislocations increases the likelihood of planes. </p> <div class="mw-heading mw-heading4"><h4 id="Reversible_plasticity">Reversible plasticity</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Plasticity_(physics)&action=edit&section=5" title="Edit section: Reversible plasticity"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>On the nanoscale the primary plastic deformation in simple <a href="/wiki/Face-centered_cubic" class="mw-redirect" title="Face-centered cubic">face-centered cubic</a> metals is reversible, as long as there is no material transport in form of <a href="/wiki/Cross_Slip" class="mw-redirect" title="Cross Slip">cross-slip</a>.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Shape-memory_alloy" title="Shape-memory alloy">Shape-memory alloys</a> such as Nitinol wire also exhibit a reversible form of plasticity which is more properly called <a href="/wiki/Pseudoelasticity" title="Pseudoelasticity">pseudoelasticity</a>. </p> <div class="mw-heading mw-heading4"><h4 id="Shear_banding">Shear banding</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Plasticity_(physics)&action=edit&section=6" title="Edit section: Shear banding"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The presence of other defects within a crystal may entangle dislocations or otherwise prevent them from gliding. When this happens, plasticity is localized to particular regions in the material. For crystals, these regions of localized plasticity are called <a href="/wiki/Shear_band" title="Shear band">shear bands</a>. </p> <div class="mw-heading mw-heading4"><h4 id="Microplasticity">Microplasticity</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Plasticity_(physics)&action=edit&section=7" title="Edit section: Microplasticity"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Microplasticity is a local phenomenon in metals. It occurs for <a href="/wiki/Stress_(physics)" class="mw-redirect" title="Stress (physics)">stress</a> values where the metal is globally in the <a href="/wiki/Elasticity_(physics)" title="Elasticity (physics)">elastic</a> domain while some local areas are in the plastic domain.<sup id="cite_ref-Maaß2018_8-0" class="reference"><a href="#cite_note-Maaß2018-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Amorphous_materials">Amorphous materials</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Plasticity_(physics)&action=edit&section=8" title="Edit section: Amorphous materials"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading4"><h4 id="Crazing">Crazing</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Plasticity_(physics)&action=edit&section=9" title="Edit section: Crazing"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In <a href="/wiki/Amorphous" class="mw-redirect" title="Amorphous">amorphous</a> materials, the discussion of "dislocations" is inapplicable, since the entire material lacks long range order. These materials can still undergo plastic deformation. Since amorphous materials, like polymers, are not well-ordered, they contain a large amount of free volume, or wasted space. Pulling these materials in tension opens up these regions and can give materials a hazy appearance. This haziness is the result of <i><a href="/wiki/Crazing" title="Crazing">crazing</a></i>, where <a href="/wiki/Fibrils" class="mw-redirect" title="Fibrils">fibrils</a> are formed within the material in regions of high <a href="/wiki/Hydrostatic_stress" title="Hydrostatic stress">hydrostatic stress</a>. The material may go from an ordered appearance to a "crazy" pattern of strain and stretch marks. </p> <div class="mw-heading mw-heading3"><h3 id="Cellular_materials">Cellular materials</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Plasticity_(physics)&action=edit&section=10" title="Edit section: Cellular materials"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>These materials plastically deform when the bending moment exceeds the fully plastic moment. This applies to open cell foams where the bending moment is exerted on the cell walls. The foams can be made of any material with a plastic yield point which includes rigid polymers and metals. This method of modeling the foam as beams is only valid if the ratio of the density of the foam to the density of the matter is less than 0.3. This is because beams yield axially instead of bending. In closed cell foams, the yield strength is increased if the material is under tension because of the membrane that spans the face of the cells. </p> <div class="mw-heading mw-heading3"><h3 id="Soils_and_sand">Soils and sand</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Plasticity_(physics)&action=edit&section=11" title="Edit section: Soils and sand"><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/Critical_state_soil_mechanics" title="Critical state soil mechanics">critical state soil mechanics</a></div> <p>Soils, particularly clays, display a significant amount of inelasticity under load. The causes of plasticity in soils can be quite complex and are strongly dependent on the <a href="/wiki/Microstructure" title="Microstructure">microstructure</a>, chemical composition, and water content. Plastic behavior in soils is caused primarily by the rearrangement of clusters of adjacent grains. </p> <div class="mw-heading mw-heading3"><h3 id="Rocks_and_concrete">Rocks and concrete</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Plasticity_(physics)&action=edit&section=12" title="Edit section: Rocks and concrete"><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/Rock_mass_plasticity" title="Rock mass plasticity">rock mass plasticity</a></div> <p>Inelastic deformations of rocks and concrete are primarily caused by the formation of microcracks and sliding motions relative to these cracks. At high temperatures and pressures, plastic behavior can also be affected by the motion of dislocations in individual grains in the microstructure. </p> <div class="mw-heading mw-heading2"><h2 id="Time-independent_yielding_and_plastic_flow_in_crystalline_materials">Time-independent yielding and plastic flow in crystalline materials</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Plasticity_(physics)&action=edit&section=13" title="Edit section: Time-independent yielding and plastic flow in crystalline materials"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> </p><p>Time-independent plastic flow in both single crystals and polycrystals is defined by a critical/maximum resolved <a href="/wiki/Shear_stress" title="Shear stress">shear stress</a> (<i>τ</i><sub>CRSS</sub>), initiating dislocation migration along parallel slip planes of a single slip system, thereby defining the transition from elastic to plastic deformation behavior in crystalline materials. </p> <div class="mw-heading mw-heading3"><h3 id="Time-independent_yielding_and_plastic_flow_in_single_crystals">Time-independent yielding and plastic flow in single crystals</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Plasticity_(physics)&action=edit&section=14" title="Edit section: Time-independent yielding and plastic flow in single crystals"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The critical resolved shear stress for single crystals is defined by Schmid’s law <i>τ</i><sub>CRSS</sub>=σ<sub>y</sub>/m, where σ<sub>y</sub> is the yield strength of the single crystal and <i>m</i> is the Schmid factor. The Schmid factor comprises two variables λ and φ, defining the angle between the slip plane direction and the tensile force applied, and the angle between the slip plane normal and the tensile force applied, respectively. Notably, because <i>m</i> > 1, <i>σ</i><sub><i>y</i></sub> > <i>τ</i><sub>CRSS</sub>. </p> <div class="mw-heading mw-heading4"><h4 id="Critical_resolved_shear_stress_dependence_on_temperature,_strain_rate,_and_point_defects"><span id="Critical_resolved_shear_stress_dependence_on_temperature.2C_strain_rate.2C_and_point_defects"></span>Critical resolved shear stress dependence on temperature, strain rate, and point defects</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Plasticity_(physics)&action=edit&section=15" title="Edit section: Critical resolved shear stress dependence on temperature, strain rate, and point defects"><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:Critical_Resolved_Shear_Stress_Versus_Temperature.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b3/Critical_Resolved_Shear_Stress_Versus_Temperature.png/220px-Critical_Resolved_Shear_Stress_Versus_Temperature.png" decoding="async" width="220" height="180" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b3/Critical_Resolved_Shear_Stress_Versus_Temperature.png/330px-Critical_Resolved_Shear_Stress_Versus_Temperature.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b3/Critical_Resolved_Shear_Stress_Versus_Temperature.png/440px-Critical_Resolved_Shear_Stress_Versus_Temperature.png 2x" data-file-width="627" data-file-height="512" /></a><figcaption>The three characteristic regions of the critical resolved shear stress as a function of temperature</figcaption></figure><p>There are three characteristic regions of the critical resolved shear stress as a function of temperature. In the low temperature region 1 (<i>T</i> ≤ 0.25<i>T</i><sub>m</sub>), the <a href="/wiki/Strain_rate" title="Strain rate">strain rate</a> must be high to achieve high <i>τ</i><sub>CRSS</sub> which is required to initiate dislocation glide and equivalently plastic flow. In region 1, the critical resolved shear stress has two components: athermal (<i>τ</i><sub><i>a</i></sub>) and thermal (<i>τ</i>*) shear stresses, arising from the stress required to move dislocations in the presence of other dislocations, and the resistance of point defect obstacles to dislocation migration, respectively. At <i>T</i> = <i>T</i>*, the moderate temperature region 2 (0.25<i>T</i><sub>m</sub> < <i>T</i> < 0.7<i>T</i><sub>m</sub>) is defined, where the thermal shear stress component <i>τ</i>* → 0, representing the elimination of point defect impedance to dislocation migration. Thus the temperature-independent critical resolved shear stress τ<sub>CRSS</sub> = τ<sub>a</sub> remains so until region 3 is defined. Notably, in region 2 moderate temperature time-dependent plastic deformation (creep) mechanisms such as solute-drag should be considered. Furthermore, in the high temperature region 3 (<i>T</i> ≥ 0.7<i>T</i><sub>m</sub>) έ can be low, contributing to low τ<sub>CRSS</sub>, however plastic flow will still occur due to thermally activated high temperature time-dependent plastic deformation mechanisms such as Nabarro–Herring (NH) and Coble diffusional flow through the lattice and along the single crystal surfaces, respectively, as well as dislocation climb-glide creep. </p><div class="mw-heading mw-heading4"><h4 id="Stages_of_time-independent_plastic_flow,_post_yielding"><span id="Stages_of_time-independent_plastic_flow.2C_post_yielding"></span>Stages of time-independent plastic flow, post yielding</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Plasticity_(physics)&action=edit&section=16" title="Edit section: Stages of time-independent plastic flow, post yielding"><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:Plastic_Stress_Versus_Strain.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a6/Plastic_Stress_Versus_Strain.png/220px-Plastic_Stress_Versus_Strain.png" decoding="async" width="220" height="171" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a6/Plastic_Stress_Versus_Strain.png/330px-Plastic_Stress_Versus_Strain.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a6/Plastic_Stress_Versus_Strain.png/440px-Plastic_Stress_Versus_Strain.png 2x" data-file-width="621" data-file-height="483" /></a><figcaption>The three stages of time-independent plastic deformation of single crystals</figcaption></figure><p>During the easy glide stage 1, the work hardening rate, defined by the change in shear stress with respect to shear strain (<i>dτ</i>/<i>dγ</i>) is low, representative of a small amount of applied shear stress necessary to induce a large amount of shear strain. Facile dislocation glide and corresponding flow is attributed to dislocation migration along parallel slip planes only (i.e. one slip system). Moderate impedance to dislocation migration along parallel slip planes is exhibited according to the weak stress field interactions between these dislocations, which heightens with smaller interplanar spacing. Overall, these migrating dislocations within a single slip system act as weak obstacles to flow, and a modest rise in stress is observed in comparison to the yield stress. During the linear hardening stage 2 of flow, the work hardening rate becomes high as considerable stress is required to overcome the stress field interactions of dislocations migrating on non-parallel slip planes (i.e. multiple slip systems), acting as strong obstacles to flow. Much stress is required to drive continual dislocation migration for small strains. The shear flow stress is directly proportional to the square root of the dislocation density (τ<sub>flow</sub> ~<i>ρ</i><sup>½</sup>), irrespective of the evolution of dislocation configurations, displaying the reliance of hardening on the number of dislocations present. Regarding this evolution of dislocation configurations, at small strains the dislocation arrangement is a random 3D array of intersecting lines. Moderate strains correspond to cellular dislocation structures of heterogeneous dislocation distribution with large dislocation density at the cell boundaries, and small dislocation density within the cell interior. At even larger strains the cellular dislocation structure reduces in size until a minimum size is achieved. Finally, the work hardening rate becomes low again in the exhaustion/saturation of hardening stage 3 of plastic flow, as small shear stresses produce large shear strains. Notably, instances when multiple slip systems are oriented favorably with respect to the applied stress, the τ<sub>CRSS</sub> for these systems may be similar and yielding may occur according to dislocation migration along multiple slip systems with non-parallel slip planes, displaying a stage 1 work-hardening rate typically characteristic of stage 2. Lastly, distinction between time-independent plastic deformation in body-centered cubic transition metals and face centered cubic metals is summarized below. </p><table class="wikitable"> <caption>Comparison between the time-independent plastic deformation of body centered cubic transition metals and face centered cubic metals, highlighting the critical resolved shear stress, work hardening rate, and necking strain during tensile testing. </caption> <tbody><tr> <th>Body-centered cubic transition metals</th> <th>Face-centered cubic metals </th></tr> <tr> <td>Critical resolved shear stress = high (relatively) & strongly temperature-dependent</td> <td>Critical resolved shear stress = low (relatively) & weakly temperature-dependent </td></tr> <tr> <td>Work hardening rate = temperature-independent</td> <td>Work hardening rate = temperature-dependent </td></tr> <tr> <td>Necking strain increases with temperature</td> <td>Necking strain decreases with temperature </td></tr></tbody></table> <div class="mw-heading mw-heading3"><h3 id="Time-independent_yielding_and_plastic_flow_in_polycrystals">Time-independent yielding and plastic flow in polycrystals</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Plasticity_(physics)&action=edit&section=17" title="Edit section: Time-independent yielding and plastic flow in polycrystals"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Plasticity in polycrystals differs substantially from that in single crystals due to the presence of grain boundary (GB) planar defects, which act as very strong obstacles to plastic flow by impeding dislocation migration along the entire length of the activated slip plane(s). Hence, dislocations cannot pass from one grain to another across the grain boundary. The following sections explore specific GB requirements for extensive plastic deformation of polycrystals prior to fracture, as well as the influence of microscopic yielding within individual crystallites on macroscopic yielding of the polycrystal. The critical resolved shear stress for polycrystals is defined by Schmid’s law as well (τ<sub>CRSS</sub>=σ<sub>y</sub>/ṁ), where σ<sub>y</sub> is the yield strength of the polycrystal and <i>ṁ</i> is the weighted Schmid factor. The weighted Schmid factor reflects the least favorably oriented slip system among the most favorably oriented slip systems of the grains constituting the GB. </p> <div class="mw-heading mw-heading4"><h4 id="Grain_boundary_constraint_in_polycrystals">Grain boundary constraint in polycrystals</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Plasticity_(physics)&action=edit&section=18" title="Edit section: Grain boundary constraint in polycrystals"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The GB constraint for polycrystals can be explained by considering a grain boundary in the xz plane between two single crystals A and B of identical composition, structure, and slip systems, but misoriented with respect to each other. To ensure that voids do not form between individually deforming grains, the GB constraint for the bicrystal is as follows: ε<sub>xx</sub><sup>A</sup> = ε<sub>xx</sub><sup>B</sup> (the x-axial strain at the GB must be equivalent for A and B), ε<sub>zz</sub><sup>A</sup> = ε<sub>zz</sub><sup>B</sup> (the z-axial strain at the GB must be equivalent for A and B), and ε<sub>xz</sub><sup>A</sup> = ε<sub>xz</sub><sup>B</sup> (the xz shear strain along the xz-GB plane must be equivalent for A and B). In addition, this GB constraint requires that five independent slip systems be activated per crystallite constituting the GB. Notably, because independent slip systems are defined as slip planes on which dislocation migrations cannot be reproduced by any combination of dislocation migrations along other slip system’s planes, the number of geometrical slip systems for a given crystal system - which by definition can be constructed by slip system combinations - is typically greater than that of independent slip systems. Significantly, there is a maximum of five independent slip systems for each of the seven crystal systems, however, not all seven crystal systems acquire this upper limit. In fact, even within a given crystal system, the composition and Bravais lattice diversifies the number of independent slip systems (see the table below). In cases for which crystallites of a polycrystal do not obtain five independent slip systems, the GB condition cannot be met, and thus the time-independent deformation of individual crystallites results in cracks and voids at the GBs of the polycrystal, and soon fracture is realized. Hence, for a given composition and structure, a single crystal with less than five independent slip systems is stronger (exhibiting a greater extent of plasticity) than its polycrystalline form. </p> <table class="wikitable"> <caption>The number of independent slip systems for a given composition (primary material class) and structure (Bravais lattice).<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> </caption> <tbody><tr> <th>Bravais lattice</th> <th>Primary material class: # Independent slip systems </th></tr> <tr> <td>Face centered cubic</td> <td>Metal: 5, ceramic (covalent): 5, ceramic (ionic): 2 </td></tr> <tr> <td>Body centered cubic</td> <td>Metal: 5 </td></tr> <tr> <td>Simple cubic</td> <td>Ceramic (ionic): 3 </td></tr> <tr> <td>Hexagonal</td> <td>Metal: 2, ceramic (mixed): 2 </td></tr></tbody></table> <div class="mw-heading mw-heading4"><h4 id="Implications_of_the_grain_boundary_constraint_in_polycrystals">Implications of the grain boundary constraint in polycrystals</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Plasticity_(physics)&action=edit&section=19" title="Edit section: Implications of the grain boundary constraint in polycrystals"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Although the two crystallites A and B discussed in the above section have identical slip systems, they are misoriented with respect to each other, and therefore misoriented with respect to the applied force. Thus, microscopic yielding within a crystallite interior may occur according to the rules governing single crystal time-independent yielding. Eventually, the activated slip planes within the grain interiors will permit dislocation migration to the GB where many dislocations then pile up as geometrically necessary dislocations. This pile up corresponds to strain gradients across individual grains as the dislocation density near the GB is greater than that in the grain interior, imposing a stress on the adjacent grain in contact. When considering the AB bicrystal as a whole, the most favorably oriented slip system in A will not be the that in B, and hence τ<sup>A</sup><sub>CRSS</sub> ≠ τ<sup>B</sup><sub>CRSS</sub>. Paramount is the fact that macroscopic yielding of the bicrystal is prolonged until the higher value of τ<sub>CRSS</sub> between grains A and B is achieved, according to the GB constraint. Thus, for a given composition and structure, a polycrystal with five independent slip systems is stronger (greater extent of plasticity) than its single crystalline form. Correspondingly, the work hardening rate will be higher for the polycrystal than the single crystal, as more stress is required in the polycrystal to produce strains. Importantly, just as with single crystal flow stress, τ<sub>flow</sub> ~ρ<sup>½</sup>, but is also inversely proportional to the square root of average grain diameter (τ<sub>flow</sub> ~d<sup>-½</sup> ). Therefore, the flow stress of a polycrystal, and hence the polycrystal’s strength, increases with small grain size. The reason for this is that smaller grains have a relatively smaller number of slip planes to be activated, corresponding to a fewer number of dislocations migrating to the GBs, and therefore less stress induced on adjacent grains due to dislocation pile up. In addition, for a given volume of polycrystal, smaller grains present more strong obstacle grain boundaries. These two factors provide an understanding as to why the onset of macroscopic flow in fine-grained polycrystals occurs at larger applied stresses than in coarse-grained polycrystals. </p> <div class="mw-heading mw-heading2"><h2 id="Mathematical_descriptions">Mathematical descriptions</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Plasticity_(physics)&action=edit&section=20" title="Edit section: Mathematical descriptions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Deformation_theory">Deformation theory</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Plasticity_(physics)&action=edit&section=21" title="Edit section: Deformation theory"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Stress-strain1.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/5a/Stress-strain1.svg/220px-Stress-strain1.svg.png" decoding="async" width="220" height="143" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/5a/Stress-strain1.svg/330px-Stress-strain1.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/5a/Stress-strain1.svg/440px-Stress-strain1.svg.png 2x" data-file-width="515" data-file-height="335" /></a><figcaption>An idealized uniaxial <a href="/wiki/Stress-strain_curve" class="mw-redirect" title="Stress-strain curve">stress-strain curve</a> showing elastic and plastic deformation regimes for the deformation theory of plasticity</figcaption></figure> <p>There are several mathematical descriptions of plasticity.<sup id="cite_ref-Hill_12-0" class="reference"><a href="#cite_note-Hill-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> One is deformation theory (see e.g. <a href="/wiki/Hooke%27s_law" title="Hooke's law">Hooke's law</a>) where the <a href="/wiki/Cauchy_stress_tensor" title="Cauchy stress tensor">Cauchy stress tensor</a> (of order d-1 in d dimensions) is a function of the strain tensor. Although this description is accurate when a small part of matter is subjected to increasing loading (such as strain loading), this theory cannot account for irreversibility. </p><p>Ductile materials can sustain large plastic deformations without <a href="/wiki/Fracture" title="Fracture">fracture</a>. However, even ductile metals will fracture when the <a href="/wiki/Strain_(materials_science)" class="mw-redirect" title="Strain (materials science)">strain</a> becomes large enough—this is as a result of <a href="/wiki/Work_hardening" title="Work hardening">work hardening</a> of the material, which causes it to become <a href="/wiki/Brittle" class="mw-redirect" title="Brittle">brittle</a>. <a href="/wiki/Heat_treatment" class="mw-redirect" title="Heat treatment">Heat treatment</a> such as <a href="/wiki/Annealing_(metallurgy)" class="mw-redirect" title="Annealing (metallurgy)">annealing</a> can restore the <a href="/wiki/Ductility" title="Ductility">ductility</a> of a worked piece, so that shaping can continue. </p> <div class="mw-heading mw-heading3"><h3 id="Flow_plasticity_theory">Flow plasticity theory</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Plasticity_(physics)&action=edit&section=22" title="Edit section: Flow plasticity theory"><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/Flow_plasticity_theory" title="Flow plasticity theory">Flow plasticity theory</a></div> <p>In 1934, <a href="/wiki/Egon_Orowan" title="Egon Orowan">Egon Orowan</a>, <a href="/wiki/Michael_Polanyi" title="Michael Polanyi">Michael Polanyi</a> and <a href="/wiki/Geoffrey_Ingram_Taylor" class="mw-redirect" title="Geoffrey Ingram Taylor">Geoffrey Ingram Taylor</a>, roughly simultaneously, realized that the plastic deformation of ductile materials could be explained in terms of the theory of <a href="/wiki/Dislocations" class="mw-redirect" title="Dislocations">dislocations</a>. The mathematical theory of plasticity, <a href="/wiki/Flow_plasticity_theory" title="Flow plasticity theory">flow plasticity theory</a>, uses a set of non-linear, non-integrable equations to describe the set of changes on strain and stress with respect to a previous state and a small increase of deformation. </p> <div class="mw-heading mw-heading2"><h2 id="Yield_criteria">Yield criteria</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Plasticity_(physics)&action=edit&section=23" title="Edit section: Yield criteria"><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:Critere_tresca_von_mises.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f2/Critere_tresca_von_mises.svg/220px-Critere_tresca_von_mises.svg.png" decoding="async" width="220" height="228" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f2/Critere_tresca_von_mises.svg/330px-Critere_tresca_von_mises.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f2/Critere_tresca_von_mises.svg/440px-Critere_tresca_von_mises.svg.png 2x" data-file-width="346" data-file-height="358" /></a><figcaption>Comparison of Tresca criterion to Von Mises criterion</figcaption></figure> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Yield_(engineering)" title="Yield (engineering)">Yield (engineering)</a></div> <p>If the stress exceeds a critical value, as was mentioned above, the material will undergo plastic, or irreversible, deformation. This critical stress can be tensile or compressive. The Tresca and the <a href="/wiki/Von_Mises_yield_criterion" title="Von Mises yield criterion">von Mises</a> criteria are commonly used to determine whether a material has yielded. However, these criteria have proved inadequate for a large range of materials and several other yield criteria are also in widespread use. </p> <div class="mw-heading mw-heading3"><h3 id="Tresca_criterion">Tresca criterion</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Plasticity_(physics)&action=edit&section=24" title="Edit section: Tresca criterion"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The Tresca criterion is based on the notion that when a material fails, it does so in shear, which is a relatively good assumption when considering metals. Given the principal stress state, we can use <a href="/wiki/Mohr%27s_circle" title="Mohr's circle">Mohr's circle</a> to solve for the maximum shear stresses our material will experience and conclude that the material will fail if </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{1}-\sigma _{3}\geq \sigma _{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>≥</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{1}-\sigma _{3}\geq \sigma _{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/57121e427c45a24ee8a0c1ffe1a7d6cdb34e51ac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.084ex; height:2.343ex;" alt="{\displaystyle \sigma _{1}-\sigma _{3}\geq \sigma _{0}}"></span></dd></dl> <p>where <i>σ</i><sub>1</sub> is the maximum normal stress, <i>σ</i><sub>3</sub> is the minimum normal stress, and <i>σ</i><sub>0</sub> is the stress under which the material fails in uniaxial loading. A <a href="/wiki/Yield_surface" title="Yield surface">yield surface</a> may be constructed, which provides a visual representation of this concept. Inside of the yield surface, deformation is elastic. On the surface, deformation is plastic. It is impossible for a material to have stress states outside its yield surface. </p> <div class="mw-heading mw-heading3"><h3 id="Huber–von_Mises_criterion"><span id="Huber.E2.80.93von_Mises_criterion"></span>Huber–von Mises criterion</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Plasticity_(physics)&action=edit&section=25" title="Edit section: Huber–von Mises criterion"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Yield_surfaces.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/cc/Yield_surfaces.svg/220px-Yield_surfaces.svg.png" decoding="async" width="220" height="179" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/cc/Yield_surfaces.svg/330px-Yield_surfaces.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/cc/Yield_surfaces.svg/440px-Yield_surfaces.svg.png 2x" data-file-width="512" data-file-height="417" /></a><figcaption>The von Mises yield surfaces in principal stress coordinates circumscribes a cylinder around the hydrostatic axis. Also shown is <a href="/wiki/Henri_Tresca" title="Henri Tresca">Tresca</a>'s hexagonal yield surface.</figcaption></figure> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Von_Mises_yield_criterion" title="Von Mises yield criterion">Von Mises yield criterion</a></div> <p>The Huber–von Mises criterion<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> is based on the Tresca criterion but takes into account the assumption that hydrostatic stresses do not contribute to material failure. <a href="/wiki/Tytus_Maksymilian_Huber" title="Tytus Maksymilian Huber">M. T. Huber</a> was the first who proposed the criterion of shear energy.<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> Von Mises solves for an <a href="/wiki/Effective_stress" title="Effective stress">effective stress</a> under uniaxial loading, subtracting out hydrostatic stresses, and states that all effective stresses greater than that which causes material failure in uniaxial loading will result in plastic deformation. </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{v}^{2}={\tfrac {1}{2}}[(\sigma _{11}-\sigma _{22})^{2}+(\sigma _{22}-\sigma _{33})^{2}+(\sigma _{11}-\sigma _{33})^{2}+6(\sigma _{23}^{2}+\sigma _{31}^{2}+\sigma _{12}^{2})]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <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> <mo stretchy="false">(</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>11</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>22</mn> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mo stretchy="false">(</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>22</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>33</mn> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mo stretchy="false">(</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>11</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>33</mn> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>6</mn> <mo stretchy="false">(</mo> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>23</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>31</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{v}^{2}={\tfrac {1}{2}}[(\sigma _{11}-\sigma _{22})^{2}+(\sigma _{22}-\sigma _{33})^{2}+(\sigma _{11}-\sigma _{33})^{2}+6(\sigma _{23}^{2}+\sigma _{31}^{2}+\sigma _{12}^{2})]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/918b083c9c0c4f9cb190d778ecfdc3fd9eb6e4f9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:71.553ex; height:3.509ex;" alt="{\displaystyle \sigma _{v}^{2}={\tfrac {1}{2}}[(\sigma _{11}-\sigma _{22})^{2}+(\sigma _{22}-\sigma _{33})^{2}+(\sigma _{11}-\sigma _{33})^{2}+6(\sigma _{23}^{2}+\sigma _{31}^{2}+\sigma _{12}^{2})]}"></span></dd></dl> <p>Again, a visual representation of the yield surface may be constructed using the above equation, which takes the shape of an ellipse. Inside the surface, materials undergo elastic deformation. Reaching the surface means the material undergoes plastic deformations. </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=Plasticity_(physics)&action=edit&section=26" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Yield_(engineering)" title="Yield (engineering)">Yield (engineering)</a></li> <li><a href="/wiki/Atterberg_limits" title="Atterberg limits">Atterberg limits</a></li> <li><a href="/wiki/Deformation_(mechanics)" class="mw-redirect" title="Deformation (mechanics)">Deformation (mechanics)</a></li> <li><a href="/wiki/Deformation_(engineering)" title="Deformation (engineering)">Deformation (engineering)</a></li> <li><a href="/w/index.php?title=Plastometer&action=edit&redlink=1" class="new" title="Plastometer (page does not exist)">Plastometer</a></li> <li><a href="/wiki/Poisson%27s_ratio" title="Poisson's ratio">Poisson's ratio</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=Plasticity_(physics)&action=edit&section=27" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist reflist-columns references-column-width" style="column-width: 35em;"> <ol class="references"> <li id="cite_note-Lubliner-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-Lubliner_1-0">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFLubliner2008" class="citation book cs1">Lubliner, Jacob (2008). <i>Plasticity theory</i>. Dover. <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>.</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.pub=Dover&rft.date=2008&rft.isbn=978-0-486-46290-5&rft.aulast=Lubliner&rft.aufirst=Jacob&rfr_id=info%3Asid%2Fen.wikipedia.org%3APlasticity+%28physics%29" class="Z3988"></span></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBigoni2012" class="citation book cs1">Bigoni, Davide (2012). <i>Nonlinear Solid Mechanics: Bifurcation Theory and Material Instability</i>. Cambridge University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-107-02541-7" title="Special:BookSources/978-1-107-02541-7"><bdi>978-1-107-02541-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Nonlinear+Solid+Mechanics%3A+Bifurcation+Theory+and+Material+Instability&rft.pub=Cambridge+University+Press&rft.date=2012&rft.isbn=978-1-107-02541-7&rft.aulast=Bigoni&rft.aufirst=Davide&rfr_id=info%3Asid%2Fen.wikipedia.org%3APlasticity+%28physics%29" class="Z3988"></span></span> </li> <li id="cite_note-Jirasek-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-Jirasek_3-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJirásekBažant2002" class="citation book cs1">Jirásek, Milan; <a href="/wiki/Zden%C4%9Bk_P._Ba%C5%BEant" title="Zdeněk P. Bažant">Bažant, Zdeněk P.</a> (2002). <i>Inelastic analysis of structures</i>. John Wiley and Sons. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-471-98716-6" title="Special:BookSources/0-471-98716-6"><bdi>0-471-98716-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Inelastic+analysis+of+structures&rft.pub=John+Wiley+and+Sons&rft.date=2002&rft.isbn=0-471-98716-6&rft.aulast=Jir%C3%A1sek&rft.aufirst=Milan&rft.au=Ba%C5%BEant%2C+Zden%C4%9Bk+P.&rfr_id=info%3Asid%2Fen.wikipedia.org%3APlasticity+%28physics%29" class="Z3988"></span></span> </li> <li id="cite_note-Chen-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-Chen_4-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFChen2008" class="citation book cs1">Chen, Wai-Fah (2008). <i>Limit Analysis and Soil Plasticity</i>. J. Ross Publishing. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-932159-73-8" title="Special:BookSources/978-1-932159-73-8"><bdi>978-1-932159-73-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Limit+Analysis+and+Soil+Plasticity&rft.pub=J.+Ross+Publishing&rft.date=2008&rft.isbn=978-1-932159-73-8&rft.aulast=Chen&rft.aufirst=Wai-Fah&rfr_id=info%3Asid%2Fen.wikipedia.org%3APlasticity+%28physics%29" class="Z3988"></span></span> </li> <li id="cite_note-Yu-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-Yu_5-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFYuMaQiangZhang2006" class="citation book cs1"><a href="/wiki/Yu_Mao-Hong" title="Yu Mao-Hong">Yu, Mao-Hong</a>; Ma, Guo-Wei; Qiang, Hong-Fu; Zhang, Yong-Qiang (2006). <i>Generalized Plasticity</i>. Springer. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/3-540-25127-8" title="Special:BookSources/3-540-25127-8"><bdi>3-540-25127-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Generalized+Plasticity&rft.pub=Springer&rft.date=2006&rft.isbn=3-540-25127-8&rft.aulast=Yu&rft.aufirst=Mao-Hong&rft.au=Ma%2C+Guo-Wei&rft.au=Qiang%2C+Hong-Fu&rft.au=Zhang%2C+Yong-Qiang&rfr_id=info%3Asid%2Fen.wikipedia.org%3APlasticity+%28physics%29" class="Z3988"></span></span> </li> <li id="cite_note-Chen1-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-Chen1_6-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFChen2007" class="citation book cs1">Chen, Wai-Fah (2007). <i>Plasticity in Reinforced Concrete</i>. J. Ross Publishing. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-932159-74-5" title="Special:BookSources/978-1-932159-74-5"><bdi>978-1-932159-74-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Plasticity+in+Reinforced+Concrete&rft.pub=J.+Ross+Publishing&rft.date=2007&rft.isbn=978-1-932159-74-5&rft.aulast=Chen&rft.aufirst=Wai-Fah&rfr_id=info%3Asid%2Fen.wikipedia.org%3APlasticity+%28physics%29" class="Z3988"></span></span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text">Ziegenhain, Gerolf; and Urbassek, Herbert M.; "Reversible Plasticity in fcc metals" in <i>Philosophical Magazine Letters</i>, 89(11):717-723, 2009 <a rel="nofollow" class="external text" href="https://dx.doi.org/10.1080/09500830903272900">DOI 10.1080/09500830903272900</a></span> </li> <li id="cite_note-Maaß2018-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-Maaß2018_8-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMaaßDerlet2018" class="citation journal cs1">Maaß, Robert; Derlet, Peter M. (January 2018). "Micro-plasticity and recent insights from intermittent and small-scale plasticity". <i>Acta Materialia</i>. <b>143</b>: 338–363. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1704.07297">1704.07297</a></span>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2018AcMat.143..338M">2018AcMat.143..338M</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.actamat.2017.06.023">10.1016/j.actamat.2017.06.023</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:119387816">119387816</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Acta+Materialia&rft.atitle=Micro-plasticity+and+recent+insights+from+intermittent+and+small-scale+plasticity&rft.volume=143&rft.pages=338-363&rft.date=2018-01&rft_id=info%3Aarxiv%2F1704.07297&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A119387816%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1016%2Fj.actamat.2017.06.023&rft_id=info%3Abibcode%2F2018AcMat.143..338M&rft.aulast=Maa%C3%9F&rft.aufirst=Robert&rft.au=Derlet%2C+Peter+M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3APlasticity+%28physics%29" class="Z3988"></span></span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCourtney2005" class="citation book cs1">Courtney, Thomas (2005). <i>Mechanical Behavior of Materials</i> (Second ed.). Long Grove, Illinois: Waveland Press, Inc. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-57766-425-3" title="Special:BookSources/978-1-57766-425-3"><bdi>978-1-57766-425-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mechanical+Behavior+of+Materials&rft.place=Long+Grove%2C+Illinois&rft.edition=Second&rft.pub=Waveland+Press%2C+Inc&rft.date=2005&rft.isbn=978-1-57766-425-3&rft.aulast=Courtney&rft.aufirst=Thomas&rfr_id=info%3Asid%2Fen.wikipedia.org%3APlasticity+%28physics%29" class="Z3988"></span></span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPartridge1969" class="citation book cs1">Partridge, Peter (1969). <i>Deformation and Fatigue of Hexagonal Close Packed Metals</i>. University of Surrey.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Deformation+and+Fatigue+of+Hexagonal+Close+Packed+Metals&rft.place=University+of+Surrey&rft.date=1969&rft.aulast=Partridge&rft.aufirst=Peter&rfr_id=info%3Asid%2Fen.wikipedia.org%3APlasticity+%28physics%29" class="Z3988"></span><span class="cs1-maint citation-comment"><code class="cs1-code">{{<a href="/wiki/Template:Cite_book" title="Template:Cite book">cite book</a>}}</code>: CS1 maint: location missing publisher (<a href="/wiki/Category:CS1_maint:_location_missing_publisher" title="Category:CS1 maint: location missing publisher">link</a>)</span></span> </li> <li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGrovesKelly1963" class="citation journal cs1">Groves, Geoffrey W.; Kelly, Anthony (1963). "Independent Slip Systems in Crystals". <i>Philosophical Magazine</i>. <b>8</b> (89): 877–887. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1963PMag....8..877G">1963PMag....8..877G</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1080%2F14786436308213843">10.1080/14786436308213843</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Philosophical+Magazine&rft.atitle=Independent+Slip+Systems+in+Crystals&rft.volume=8&rft.issue=89&rft.pages=877-887&rft.date=1963&rft_id=info%3Adoi%2F10.1080%2F14786436308213843&rft_id=info%3Abibcode%2F1963PMag....8..877G&rft.aulast=Groves&rft.aufirst=Geoffrey+W.&rft.au=Kelly%2C+Anthony&rfr_id=info%3Asid%2Fen.wikipedia.org%3APlasticity+%28physics%29" class="Z3988"></span></span> </li> <li id="cite_note-Hill-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-Hill_12-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHill1998" class="citation book cs1"><a href="/wiki/Rodney_Hill" title="Rodney Hill">Hill, Rodney</a> (1998). <i>The Mathematical Theory of Plasticity</i>. Oxford University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-19-850367-9" title="Special:BookSources/0-19-850367-9"><bdi>0-19-850367-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Mathematical+Theory+of+Plasticity&rft.pub=Oxford+University+Press&rft.date=1998&rft.isbn=0-19-850367-9&rft.aulast=Hill&rft.aufirst=Rodney&rfr_id=info%3Asid%2Fen.wikipedia.org%3APlasticity+%28physics%29" class="Z3988"></span></span> </li> <li id="cite_note-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-13">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFvon_Mises1913" class="citation journal cs1"><a href="/wiki/Richard_von_Mises" title="Richard von Mises">von Mises, Richard</a> (1913). <a rel="nofollow" class="external text" href="http://www.digizeitschriften.de/dms/resolveppn/?PID=GDZPPN002503697">"Mechanik der festen Körper im plastisch-deformablen Zustand"</a>. <i>Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen</i>. Mathematisch-Physikalische Klasse. <b>1913</b> (1): 582–592.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Nachrichten+von+der+Gesellschaft+der+Wissenschaften+zu+G%C3%B6ttingen&rft.atitle=Mechanik+der+festen+K%C3%B6rper+im+plastisch-deformablen+Zustand&rft.volume=1913&rft.issue=1&rft.pages=582-592&rft.date=1913&rft.aulast=von+Mises&rft.aufirst=Richard&rft_id=http%3A%2F%2Fwww.digizeitschriften.de%2Fdms%2Fresolveppn%2F%3FPID%3DGDZPPN002503697&rfr_id=info%3Asid%2Fen.wikipedia.org%3APlasticity+%28physics%29" class="Z3988"></span></span> </li> <li id="cite_note-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-14">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHuber1904" class="citation journal cs1"><a href="/wiki/Tytus_Maksymilian_Huber" title="Tytus Maksymilian Huber">Huber, Maksymilian Tytus</a> (1904). "Właściwa praca odkształcenia jako miara wytezenia materiału". <i>Czasopismo Techniczne</i>. <b>22</b>. Lwów.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Czasopismo+Techniczne&rft.atitle=W%C5%82a%C5%9Bciwa+praca+odkszta%C5%82cenia+jako+miara+wytezenia+materia%C5%82u&rft.volume=22&rft.date=1904&rft.aulast=Huber&rft.aufirst=Maksymilian+Tytus&rfr_id=info%3Asid%2Fen.wikipedia.org%3APlasticity+%28physics%29" class="Z3988"></span> Translated as <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation journal cs1"><a rel="nofollow" class="external text" href="http://am.ippt.pan.pl/am/article/viewFile/v56p173/pdf">"Specific Work of Strain as a Measure of Material Effort"</a>. <i>Archives of Mechanics</i>. <b>56</b>: 173–190. 2004.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Archives+of+Mechanics&rft.atitle=Specific+Work+of+Strain+as+a+Measure+of+Material+Effort&rft.volume=56&rft.pages=173-190&rft.date=2004&rft_id=http%3A%2F%2Fam.ippt.pan.pl%2Fam%2Farticle%2FviewFile%2Fv56p173%2Fpdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3APlasticity+%28physics%29" class="Z3988"></span></span> </li> <li id="cite_note-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-15">^</a></b></span> <span class="reference-text">See <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTimoshenko1953" class="citation book cs1"><a href="/wiki/Stephen_Timoshenko" title="Stephen Timoshenko">Timoshenko, Stephen P.</a> (1953). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=tkScQmyhsb8C&pg=PA369"><i>History of Strength of Materials</i></a>. New York: McGraw-Hill. p. 369. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780486611877" title="Special:BookSources/9780486611877"><bdi>9780486611877</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=History+of+Strength+of+Materials&rft.place=New+York&rft.pages=369&rft.pub=McGraw-Hill&rft.date=1953&rft.isbn=9780486611877&rft.aulast=Timoshenko&rft.aufirst=Stephen+P.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DtkScQmyhsb8C%26pg%3DPA369&rfr_id=info%3Asid%2Fen.wikipedia.org%3APlasticity+%28physics%29" class="Z3988"></span></span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Plasticity_(physics)&action=edit&section=28" title="Edit section: Further reading"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAshby2001" class="citation book cs1"><a href="/wiki/Michael_F._Ashby" title="Michael F. Ashby">Ashby, Michael F.</a> (2001). "Plastic Deformation of Cellular Materials". <i>Encyclopedia of Materials: Science and Technology</i>. Vol. 7. Oxford: Elsevier. pp. 7068–7071. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-08-043152-6" title="Special:BookSources/0-08-043152-6"><bdi>0-08-043152-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Plastic+Deformation+of+Cellular+Materials&rft.btitle=Encyclopedia+of+Materials%3A+Science+and+Technology&rft.place=Oxford&rft.pages=7068-7071&rft.pub=Elsevier&rft.date=2001&rft.isbn=0-08-043152-6&rft.aulast=Ashby&rft.aufirst=Michael+F.&rfr_id=info%3Asid%2Fen.wikipedia.org%3APlasticity+%28physics%29" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHanReddy2013" class="citation book cs1">Han, Weimin; Reddy, B. Daya (2013). <i>Plasticity: Mathematical Theory and Numerical Analysis</i> (2nd ed.). New York: Springer. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4614-5939-2" title="Special:BookSources/978-1-4614-5939-2"><bdi>978-1-4614-5939-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Plasticity%3A+Mathematical+Theory+and+Numerical+Analysis&rft.place=New+York&rft.edition=2nd&rft.pub=Springer&rft.date=2013&rft.isbn=978-1-4614-5939-2&rft.aulast=Han&rft.aufirst=Weimin&rft.au=Reddy%2C+B.+Daya&rfr_id=info%3Asid%2Fen.wikipedia.org%3APlasticity+%28physics%29" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKachanov2004" class="citation book cs1">Kachanov, Lazar' Markovich (2004). <i>Fundamentals of the Theory of Plasticity</i>. Dover Books. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-486-43583-0" title="Special:BookSources/0-486-43583-0"><bdi>0-486-43583-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Fundamentals+of+the+Theory+of+Plasticity&rft.pub=Dover+Books&rft.date=2004&rft.isbn=0-486-43583-0&rft.aulast=Kachanov&rft.aufirst=Lazar%27+Markovich&rfr_id=info%3Asid%2Fen.wikipedia.org%3APlasticity+%28physics%29" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKhanHuang1995" class="citation book cs1">Khan, Akhtar S.; Huang, Sujian (1995). <i>Continuum Theory of Plasticity</i>. Wiley. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-471-31043-3" title="Special:BookSources/0-471-31043-3"><bdi>0-471-31043-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Continuum+Theory+of+Plasticity&rft.pub=Wiley&rft.date=1995&rft.isbn=0-471-31043-3&rft.aulast=Khan&rft.aufirst=Akhtar+S.&rft.au=Huang%2C+Sujian&rfr_id=info%3Asid%2Fen.wikipedia.org%3APlasticity+%28physics%29" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSimoHughes1998" class="citation book cs1"><a href="/w/index.php?title=Juan_C._Simo&action=edit&redlink=1" class="new" title="Juan C. Simo (page does not exist)">Simo, Juan C.</a>; <a href="/wiki/Thomas_J._R._Hughes" class="mw-redirect" title="Thomas J. R. Hughes">Hughes, Thomas J. R.</a> (1998). <i>Computational Inelasticity</i>. Springer. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-387-97520-9" title="Special:BookSources/0-387-97520-9"><bdi>0-387-97520-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Computational+Inelasticity&rft.pub=Springer&rft.date=1998&rft.isbn=0-387-97520-9&rft.aulast=Simo&rft.aufirst=Juan+C.&rft.au=Hughes%2C+Thomas+J.+R.&rfr_id=info%3Asid%2Fen.wikipedia.org%3APlasticity+%28physics%29" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFVan_Vliet2006" class="citation web cs1"><a href="/wiki/Krystyn_Van_Vliet" title="Krystyn Van Vliet">Van Vliet, Krystyn J.</a> (2006). <a rel="nofollow" class="external text" href="https://ocw.mit.edu/courses/materials-science-and-engineering/3-032-mechanical-behavior-of-materials-fall-2007/">"Mechanical Behavior of Materials"</a>. <i>MIT Course Number 3.032</i>. Massachusetts Institute of Technology.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=MIT+Course+Number+3.032&rft.atitle=Mechanical+Behavior+of+Materials&rft.date=2006&rft.aulast=Van+Vliet&rft.aufirst=Krystyn+J.&rft_id=https%3A%2F%2Focw.mit.edu%2Fcourses%2Fmaterials-science-and-engineering%2F3-032-mechanical-behavior-of-materials-fall-2007%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3APlasticity+%28physics%29" class="Z3988"></span></li></ul> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1236075235">.mw-parser-output .navbox{box-sizing:border-box;border:1px solid #a2a9b1;width:100%;clear:both;font-size:88%;text-align:center;padding:1px;margin:1em auto 0}.mw-parser-output .navbox .navbox{margin-top:0}.mw-parser-output .navbox+.navbox,.mw-parser-output .navbox+.navbox-styles+.navbox{margin-top:-1px}.mw-parser-output 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