Deformation (engineering)

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class="vector-toc-numb">2</span> <span>Types of deformation</span> </div> </a> <button aria-controls="toc-Types_of_deformation-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 Types of deformation subsection</span> </button> <ul id="toc-Types_of_deformation-sublist" class="vector-toc-list"> <li id="toc-Elastic_deformation" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Elastic_deformation"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Elastic deformation</span> </div> </a> <ul id="toc-Elastic_deformation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Plastic_deformation" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Plastic_deformation"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Plastic deformation</span> </div> </a> <ul id="toc-Plastic_deformation-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Failure" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Failure"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Failure</span> </div> </a> <button aria-controls="toc-Failure-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 Failure subsection</span> </button> <ul id="toc-Failure-sublist" class="vector-toc-list"> <li id="toc-Compressive_failure" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Compressive_failure"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Compressive failure</span> </div> </a> <ul id="toc-Compressive_failure-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Fracture" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Fracture"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Fracture</span> </div> </a> <ul id="toc-Fracture-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Types_of_stress_and_strain" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Types_of_stress_and_strain"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Types of stress and strain</span> </div> </a> <button aria-controls="toc-Types_of_stress_and_strain-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 Types of stress and strain subsection</span> </button> <ul id="toc-Types_of_stress_and_strain-sublist" class="vector-toc-list"> <li id="toc-Engineering_stress_and_strain" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Engineering_stress_and_strain"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Engineering stress and strain</span> </div> </a> <ul id="toc-Engineering_stress_and_strain-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-True_stress_and_strain" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#True_stress_and_strain"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2</span> <span>True stress and strain</span> </div> </a> <ul id="toc-True_stress_and_strain-sublist" class="vector-toc-list"> <li id="toc-Discussion" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Discussion"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2.1</span> <span>Discussion</span> </div> </a> <ul id="toc-Discussion-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </li> <li 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class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Deformation (engineering)</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. Available in 16 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-16" 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">16 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%AA%D8%B4%D9%88%D9%87_(%D9%87%D9%86%D8%AF%D8%B3%D8%A9)" title="تشوه (هندسة) – Arabic" lang="ar" hreflang="ar" data-title="تشوه (هندسة)" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%B0%E0%A7%82%E0%A6%AA%E0%A6%AC%E0%A6%BF%E0%A6%95%E0%A6%BE%E0%A6%B0_(%E0%A6%AA%E0%A7%8D%E0%A6%B0%E0%A6%95%E0%A7%8C%E0%A6%B6%E0%A6%B2)" title="রূপবিকার (প্রকৌশল) – Bangla" lang="bn" hreflang="bn" data-title="রূপবিকার (প্রকৌশল)" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Verformung" title="Verformung – German" lang="de" hreflang="de" data-title="Verformung" 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/Deformatsioon" title="Deformatsioon – Estonian" lang="et" hreflang="et" data-title="Deformatsioon" 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/Deformaci%C3%B3n_(ingenier%C3%ADa)" title="Deformación (ingeniería) – Spanish" lang="es" hreflang="es" data-title="Deformación (ingeniería)" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%AA%D8%BA%DB%8C%DB%8C%D8%B1_%D8%B4%DA%A9%D9%84_(%D9%85%D9%87%D9%86%D8%AF%D8%B3%DB%8C)" title="تغییر شکل (مهندسی) – Persian" lang="fa" hreflang="fa" data-title="تغییر شکل (مهندسی)" 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_(ing%C3%A9nierie)" title="Déformation (ingénierie) – French" lang="fr" hreflang="fr" data-title="Déformation (ingénierie)" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%9D%BC%EA%B7%B8%EB%9F%AC%EC%A7%90_(%EA%B3%B5%ED%95%99)" title="일그러짐 (공학) – Korean" lang="ko" hreflang="ko" data-title="일그러짐 (공학)" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Deformasi_(teknik)" title="Deformasi (teknik) – Indonesian" lang="id" hreflang="id" data-title="Deformasi (teknik)" 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-kn mw-list-item"><a href="https://kn.wikipedia.org/wiki/%E0%B2%B5%E0%B2%BF%E0%B2%B0%E0%B3%82%E0%B2%AA%E0%B2%A4%E0%B3%86" title="ವಿರೂಪತೆ – Kannada" lang="kn" hreflang="kn" data-title="ವಿರೂಪತೆ" data-language-autonym="ಕನ್ನಡ" data-language-local-name="Kannada" class="interlanguage-link-target"><span>ಕನ್ನಡ</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%94%D0%B5%D1%84%D0%BE%D1%80%D0%BC%D0%B0%D1%86%D0%B8%D1%8F%D0%BB%D0%B0%D1%80" title="Деформациялар – Kazakh" lang="kk" hreflang="kk" data-title="Деформациялар" data-language-autonym="Қазақша" data-language-local-name="Kazakh" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%98%D0%B7%D0%BE%D0%B1%D0%BB%D0%B8%D1%87%D1%83%D0%B2%D0%B0%D1%9A%D0%B5" title="Изобличување – Macedonian" lang="mk" hreflang="mk" data-title="Изобличување" data-language-autonym="Македонски" data-language-local-name="Macedonian" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Deformatie_(materiaalkunde)" title="Deformatie (materiaalkunde) – Dutch" lang="nl" hreflang="nl" data-title="Deformatie (materiaalkunde)" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-si mw-list-item"><a href="https://si.wikipedia.org/wiki/%E0%B7%80%E0%B7%92%E0%B6%BB%E0%B7%96%E0%B6%B4%E0%B6%AB%E0%B6%BA_(%E0%B6%89%E0%B6%82%E0%B6%A2%E0%B7%92%E0%B6%B1%E0%B7%9A%E0%B6%BB%E0%B7%94_%E0%B7%80%E0%B7%92%E0%B6%AF%E0%B7%8A%E2%80%8D%E0%B6%BA%E0%B7%8F%E0%B7%80)" title="විරූපණය (ඉංජිනේරු විද්‍යාව) – Sinhala" lang="si" hreflang="si" data-title="විරූපණය (ඉංජිනේරු විද්‍යාව)" data-language-autonym="සිංහල" data-language-local-name="Sinhala" class="interlanguage-link-target"><span>සිංහල</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E8%AE%8A%E5%BD%A2" title="變形 – Cantonese" lang="yue" hreflang="yue" data-title="變形" data-language-autonym="粵語" data-language-local-name="Cantonese" class="interlanguage-link-target"><span>粵語</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E5%BD%A2%E8%AE%8A" title="形變 – Chinese" lang="zh" hreflang="zh" data-title="形變" data-language-autonym="中文" data-language-local-name="Chinese" class="interlanguage-link-target"><span>中文</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q2672013#sitelinks-wikipedia" title="Edit interlanguage links" class="wbc-editpage">Edit links</a></span></div> </div> </div> </div> </header> <div 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dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Change in the shape or size of an object</div> <style data-mw-deduplicate="TemplateStyles:r1251242444">.mw-parser-output .ambox{border:1px solid #a2a9b1;border-left:10px solid #36c;background-color:#fbfbfb;box-sizing:border-box}.mw-parser-output .ambox+link+.ambox,.mw-parser-output .ambox+link+style+.ambox,.mw-parser-output .ambox+link+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+style+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+link+.ambox{margin-top:-1px}html body.mediawiki .mw-parser-output .ambox.mbox-small-left{margin:4px 1em 4px 0;overflow:hidden;width:238px;border-collapse:collapse;font-size:88%;line-height:1.25em}.mw-parser-output .ambox-speedy{border-left:10px solid #b32424;background-color:#fee7e6}.mw-parser-output .ambox-delete{border-left:10px solid #b32424}.mw-parser-output .ambox-content{border-left:10px solid #f28500}.mw-parser-output .ambox-style{border-left:10px solid #fc3}.mw-parser-output .ambox-move{border-left:10px solid #9932cc}.mw-parser-output .ambox-protection{border-left:10px solid #a2a9b1}.mw-parser-output .ambox .mbox-text{border:none;padding:0.25em 0.5em;width:100%}.mw-parser-output .ambox .mbox-image{border:none;padding:2px 0 2px 0.5em;text-align:center}.mw-parser-output .ambox .mbox-imageright{border:none;padding:2px 0.5em 2px 0;text-align:center}.mw-parser-output .ambox .mbox-empty-cell{border:none;padding:0;width:1px}.mw-parser-output .ambox .mbox-image-div{width:52px}@media(min-width:720px){.mw-parser-output .ambox{margin:0 10%}}@media print{body.ns-0 .mw-parser-output .ambox{display:none!important}}</style><table class="box-More_citations_needed plainlinks metadata ambox ambox-content ambox-Refimprove" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span typeof="mw:File"><a href="/wiki/File:Question_book-new.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/50px-Question_book-new.svg.png" decoding="async" width="50" height="39" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/75px-Question_book-new.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/100px-Question_book-new.svg.png 2x" data-file-width="512" data-file-height="399" /></a></span></div></td><td class="mbox-text"><div class="mbox-text-span">This article <b>needs additional citations for <a href="/wiki/Wikipedia:Verifiability" title="Wikipedia:Verifiability">verification</a></b>.<span class="hide-when-compact"> Please help <a href="/wiki/Special:EditPage/Deformation_(engineering)" title="Special:EditPage/Deformation (engineering)">improve this article</a> by <a href="/wiki/Help:Referencing_for_beginners" title="Help:Referencing for beginners">adding citations to reliable sources</a>. Unsourced material may be challenged and removed.<br /><small><span class="plainlinks"><i>Find sources:</i> <a rel="nofollow" class="external text" href="https://www.google.com/search?as_eq=wikipedia&q=%22Deformation%22+engineering">"Deformation" engineering</a> – <a rel="nofollow" class="external text" href="https://www.google.com/search?tbm=nws&q=%22Deformation%22+engineering+-wikipedia&tbs=ar:1">news</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.google.com/search?&q=%22Deformation%22+engineering&tbs=bkt:s&tbm=bks">newspapers</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.google.com/search?tbs=bks:1&q=%22Deformation%22+engineering+-wikipedia">books</a> <b>·</b> <a rel="nofollow" class="external text" href="https://scholar.google.com/scholar?q=%22Deformation%22+engineering">scholar</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.jstor.org/action/doBasicSearch?Query=%22Deformation%22+engineering&acc=on&wc=on">JSTOR</a></span></small></span> <span class="date-container"><i>(<span class="date">September 2008</span>)</i></span><span class="hide-when-compact"><i> (<small><a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span></div></td></tr></tbody></table> <style data-mw-deduplicate="TemplateStyles: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">For usage in physics, see <a href="/wiki/Deformation_(physics)" title="Deformation (physics)">Deformation (physics)</a>.</div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Deformation_due_to_compression.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/99/Deformation_due_to_compression.svg/100px-Deformation_due_to_compression.svg.png" decoding="async" width="100" height="158" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/99/Deformation_due_to_compression.svg/150px-Deformation_due_to_compression.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/99/Deformation_due_to_compression.svg/200px-Deformation_due_to_compression.svg.png 2x" data-file-width="331" data-file-height="524" /></a><figcaption>Compressive stress results in deformation which shortens the object but also expands it outwards.</figcaption></figure> <p>In <a href="/wiki/Engineering" title="Engineering">engineering</a>, <b>deformation</b> (the change in size or shape of an object) may be <i>elastic</i> or <i>plastic</i>. If the deformation is negligible, the object is said to be <a href="/wiki/Rigid_body" title="Rigid body"><i>rigid</i></a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Main_concepts">Main concepts</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Deformation_(engineering)&action=edit&section=1" title="Edit section: Main concepts"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Occurrence of deformation in engineering applications is based on the following background concepts: </p> <ul><li><a href="/wiki/Displacement_(mechanics)" class="mw-redirect" title="Displacement (mechanics)"><i>Displacements</i></a> are any change in position of a point on the object, including whole-body translations and rotations (<a href="/wiki/Rigid_transformation" title="Rigid transformation">rigid transformations</a>).</li> <li><a href="/wiki/Deformation_(mechanics)" class="mw-redirect" title="Deformation (mechanics)"><i>Deformation</i></a> are changes in the relative position between internals points on the object, excluding rigid transformations, causing the body to change shape or size.</li> <li><a href="/wiki/Strain_(mechanics)" title="Strain (mechanics)"><i>Strain</i></a> is the <i>relative</i> <i>internal</i> deformation, the <a href="/wiki/Dimensionless" class="mw-redirect" title="Dimensionless">dimensionless</a> change in shape of an infinitesimal cube of material relative to a reference configuration. Mechanical strains are caused by <a href="/wiki/Stress_(mechanics)" title="Stress (mechanics)">mechanical stress</a>, <i>see <a href="/wiki/Stress-strain_curve" class="mw-redirect" title="Stress-strain curve">stress-strain curve</a></i>.</li></ul> <p>The relationship between stress and strain is generally linear and reversible up until the <a href="/wiki/Yield_point" class="mw-redirect" title="Yield point">yield point</a> and the deformation is <a href="/wiki/Elasticity_(physics)" title="Elasticity (physics)">elastic</a>. Elasticity in materials occurs when applied stress does not surpass the energy required to break molecular bonds, allowing the material to deform reversibly and return to its original shape once the stress is removed. The linear relationship for a material is known as <a href="/wiki/Young%27s_modulus" title="Young's modulus">Young's modulus</a>. Above the yield point, some degree of permanent distortion remains after unloading and is termed <a href="/wiki/Plasticity_(physics)" title="Plasticity (physics)">plastic deformation</a>. The determination of the stress and strain throughout a solid object is given by the field of <a href="/wiki/Strength_of_materials" title="Strength of materials">strength of materials</a> and for a structure by <a href="/wiki/Structural_analysis" title="Structural analysis">structural analysis</a>. </p><p>In the above figure, it can be seen that the compressive loading (indicated by the arrow) has caused deformation in the <a href="/wiki/Cylinder_(geometry)" class="mw-redirect" title="Cylinder (geometry)">cylinder</a> so that the original shape (dashed lines) has changed (deformed) into one with bulging sides. The sides bulge because the material, although strong enough to not crack or otherwise fail, is not strong enough to support the load without change. As a result, the material is forced out laterally. Internal forces (in this case at right angles to the deformation) resist the applied load. </p> <div class="mw-heading mw-heading2"><h2 id="Types_of_deformation">Types of deformation</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Deformation_(engineering)&action=edit&section=2" title="Edit section: Types of deformation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Depending on the type of material, size and geometry of the object, and the forces applied, various types of deformation may result. The image to the right shows the engineering stress vs. strain diagram for a typical ductile material such as steel. Different deformation modes may occur under different conditions, as can be depicted using a <a href="/wiki/Deformation_mechanism_map" class="mw-redirect" title="Deformation mechanism map">deformation mechanism map</a>. </p><p>Permanent deformation is irreversible; the deformation stays even after removal of the applied forces, while the temporary deformation is recoverable as it disappears after the removal of applied forces. Temporary deformation is also called <b>elastic</b> deformation, while the permanent deformation is called <b>plastic</b> deformation. </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Stress_Strain_Ductile_Material.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/84/Stress_Strain_Ductile_Material.png/330px-Stress_Strain_Ductile_Material.png" decoding="async" width="330" height="216" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/84/Stress_Strain_Ductile_Material.png/495px-Stress_Strain_Ductile_Material.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/84/Stress_Strain_Ductile_Material.png/660px-Stress_Strain_Ductile_Material.png 2x" data-file-width="885" data-file-height="578" /></a><figcaption>Typical stress vs. strain diagram indicating the various stages of deformation.</figcaption></figure> <div class="mw-heading mw-heading3"><h3 id="Elastic_deformation">Elastic deformation</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Deformation_(engineering)&action=edit&section=3" title="Edit section: Elastic deformation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Further information: <a href="/wiki/Elasticity_(physics)" title="Elasticity (physics)">Elasticity (physics)</a></div> <p>The study of temporary or elastic deformation in the case of <a href="/wiki/Deformation_(mechanics)#Engineering_strain" class="mw-redirect" title="Deformation (mechanics)">engineering strain</a> is applied to materials used in mechanical and structural engineering, such as <a href="/wiki/Concrete" title="Concrete">concrete</a> and <a href="/wiki/Steel" title="Steel">steel</a>, which are subjected to very small deformations. Engineering strain is modeled by <a href="/wiki/Infinitesimal_strain_theory" title="Infinitesimal strain theory">infinitesimal strain theory</a>, also called <i>small strain theory</i>, <i>small deformation theory</i>, <i>small displacement theory</i>, or <i>small displacement-gradient theory</i> where strains and rotations are both small. </p><p>For some materials, e.g. <a href="/wiki/Elastomers" class="mw-redirect" title="Elastomers">elastomers</a> and polymers, subjected to large deformations, the engineering definition of strain is not applicable, e.g. typical engineering strains greater than 1%,<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> thus other more complex definitions of strain are required, such as <i>stretch</i>, <i>logarithmic strain</i>, <i>Green strain</i>, and <i>Almansi strain</i>. <a href="/wiki/Elastomer" title="Elastomer">Elastomers</a> and <a href="/wiki/Shape_memory" class="mw-redirect" title="Shape memory">shape memory</a> metals such as <a href="/wiki/Nitinol" class="mw-redirect" title="Nitinol">Nitinol</a> exhibit large elastic deformation ranges, as does <a href="/wiki/Rubber" class="mw-redirect" title="Rubber">rubber</a>. However, elasticity is nonlinear in these materials. </p><p>Normal metals, ceramics and most crystals show linear elasticity and a smaller elastic range. </p><p><b>Linear elastic deformation</b> is governed by <a href="/wiki/Hooke%27s_law" title="Hooke's law">Hooke's law</a>, which states: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma =E\varepsilon }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>σ</mi> <mo>=</mo> <mi>E</mi> <mi>ε</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma =E\varepsilon }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c429bd4b14acaf624ac17c9d7e59e729b9d08513" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.287ex; height:2.176ex;" alt="{\displaystyle \sigma =E\varepsilon }"></span></dd></dl> <p>where </p> <ul><li><span class="texhtml mvar" style="font-style:italic;">σ</span> is the applied <a href="/wiki/Stress_(physics)" class="mw-redirect" title="Stress (physics)">stress</a>;</li> <li><span class="texhtml mvar" style="font-style:italic;">E</span> is a material constant called <a href="/wiki/Young%27s_modulus" title="Young's modulus">Young's modulus</a> or <a href="/wiki/Elastic_modulus" title="Elastic modulus">elastic modulus</a>;</li> <li><span class="texhtml mvar" style="font-style:italic;">ε</span> is the resulting <a href="/wiki/Strain_(materials_science)" class="mw-redirect" title="Strain (materials science)">strain</a>.</li></ul> <p>This relationship only applies in the elastic range and indicates that the slope of the stress vs. strain curve can be used to find Young's modulus (<span class="texhtml mvar" style="font-style:italic;">E</span>). Engineers often use this calculation in tensile tests. The area under this elastic region is known as resilience. </p><p>Note that not all elastic materials undergo linear elastic deformation; some, such as <a href="/wiki/Concrete" title="Concrete">concrete</a>, <a href="/wiki/Gray_cast_iron" class="mw-redirect" title="Gray cast iron">gray cast iron</a>, and many polymers, respond in a nonlinear fashion. For these materials Hooke's law is inapplicable.<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> </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Stress_strain_comparison.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/2/24/Stress_strain_comparison.gif" decoding="async" width="328" height="226" class="mw-file-element" data-file-width="328" data-file-height="226" /></a><figcaption>Difference in true and engineering <a href="/wiki/Stress-strain_curve" class="mw-redirect" title="Stress-strain curve">stress-strain curves</a></figcaption></figure> <div class="mw-heading mw-heading3"><h3 id="Plastic_deformation">Plastic deformation</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Deformation_(engineering)&action=edit&section=4" title="Edit section: Plastic deformation"><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:Pannzerung_plastische_deformation.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/67/Pannzerung_plastische_deformation.png/220px-Pannzerung_plastische_deformation.png" decoding="async" width="220" height="250" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/67/Pannzerung_plastische_deformation.png/330px-Pannzerung_plastische_deformation.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/67/Pannzerung_plastische_deformation.png/440px-Pannzerung_plastische_deformation.png 2x" data-file-width="5963" data-file-height="6782" /></a><figcaption>Swebor-brand high-strength low alloy steel plate, showing both sides, after plastic deformation from bringing to rest <a href="/wiki/Projectile" title="Projectile">projectiles</a> in <a href="/wiki/Ballistics" title="Ballistics">ballistics</a> testing.</figcaption></figure> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Plasticity_(physics)" title="Plasticity (physics)">Plasticity (physics)</a></div> <p>This type of deformation is not undone simply by removing the applied force. An object in the plastic deformation range, however, will first have undergone elastic deformation, which is undone simply be removing the applied force, so the object will return part way to its original shape. Soft <a href="/wiki/Thermoplastics" class="mw-redirect" title="Thermoplastics">thermoplastics</a> have a rather large plastic deformation range as do ductile metals such as <a href="/wiki/Copper" title="Copper">copper</a>, <a href="/wiki/Silver" title="Silver">silver</a>, and <a href="/wiki/Gold" title="Gold">gold</a>. <a href="/wiki/Steel" title="Steel">Steel</a> does, too, but not <a href="/wiki/Cast_iron" title="Cast iron">cast iron</a>. Hard thermosetting plastics, rubber, crystals, and ceramics have minimal plastic deformation ranges. An example of a material with a large plastic deformation range is wet <a href="/wiki/Chewing_gum" title="Chewing gum">chewing gum</a>, which can be stretched to dozens of times its original length. </p><p>Under tensile stress, plastic deformation is characterized by a <a href="/wiki/Strain_hardening" class="mw-redirect" title="Strain hardening">strain hardening</a> region and a <a href="/wiki/Necking_(engineering)" title="Necking (engineering)">necking</a> region and finally, fracture (also called rupture). During strain hardening the material becomes stronger through the movement of <a href="/wiki/Dislocation" title="Dislocation">atomic dislocations</a>. The necking phase is indicated by a reduction in cross-sectional area of the specimen. Necking begins after the ultimate strength is reached. During necking, the material can no longer withstand the maximum stress and the strain in the specimen rapidly increases. Plastic deformation ends with the fracture of the material. </p> <figure class="mw-default-size" 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/290px-Stress-strain1.svg.png" decoding="async" width="290" height="189" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/5a/Stress-strain1.svg/435px-Stress-strain1.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/5a/Stress-strain1.svg/580px-Stress-strain1.svg.png 2x" data-file-width="515" data-file-height="335" /></a><figcaption>Diagram of a <a href="/wiki/Stress%E2%80%93strain_curve" title="Stress–strain curve">stress–strain curve</a>, showing the relationship between stress (force applied) and strain (deformation) of a ductile metal.</figcaption></figure> <div class="mw-heading mw-heading2"><h2 id="Failure">Failure</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Deformation_(engineering)&action=edit&section=5" title="Edit section: Failure"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Compressive_failure">Compressive failure</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Deformation_(engineering)&action=edit&section=6" title="Edit section: Compressive failure"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Usually, compressive stress applied to bars, <a href="/wiki/Column" title="Column">columns</a>, etc. leads to shortening. </p><p>Loading a structural element or specimen will increase the compressive stress until it reaches its <a href="/wiki/Compressive_strength" title="Compressive strength">compressive strength</a>. According to the properties of the material, failure modes are <a href="/wiki/Yield_(engineering)" title="Yield (engineering)">yielding</a> for materials with <a href="/wiki/Ductile" class="mw-redirect" title="Ductile">ductile</a> behavior (most <a href="/wiki/Metal" title="Metal">metals</a>, some <a href="/wiki/Soil" title="Soil">soils</a> and <a href="/wiki/Plastic" title="Plastic">plastics</a>) or rupturing for brittle behavior (geomaterials, <a href="/wiki/Cast_iron" title="Cast iron">cast iron</a>, <a href="/wiki/Glass" title="Glass">glass</a>, etc.). </p><p>In long, slender structural elements — such as columns or <a href="/wiki/Truss" title="Truss">truss</a> bars — an increase of compressive force <i>F</i> leads to <a href="/wiki/Structural_failure" class="mw-redirect" title="Structural failure">structural failure</a> due to <a href="/wiki/Buckling" title="Buckling">buckling</a> at lower stress than the compressive strength. </p> <div class="mw-heading mw-heading3"><h3 id="Fracture">Fracture</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Deformation_(engineering)&action=edit&section=7" title="Edit section: Fracture"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Concrete_fracture_analysis" class="mw-redirect" title="Concrete fracture analysis">Concrete fracture analysis</a> and <a href="/wiki/Fracture_mechanics" title="Fracture mechanics">Fracture mechanics</a></div> <p>A break occurs after the material has reached the end of the elastic, and then plastic, deformation ranges. At this point forces accumulate until they are sufficient to cause a fracture. All materials will eventually fracture, if sufficient forces are applied. </p> <div class="mw-heading mw-heading2"><h2 id="Types_of_stress_and_strain">Types of stress and strain</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Deformation_(engineering)&action=edit&section=8" title="Edit section: Types of stress and strain"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><b>Engineering stress</b> and <b>engineering strain</b> are approximations to the internal state that may be determined from the external forces and deformations of an object, provided that there is no significant change in size. When there is a significant change in size, the <b>true stress</b> and <b>true strain</b> can be derived from the instantaneous size of the object. </p> <div class="mw-heading mw-heading3"><h3 id="Engineering_stress_and_strain">Engineering stress and strain</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Deformation_(engineering)&action=edit&section=9" title="Edit section: Engineering stress and strain"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Consider a bar of original <a href="/wiki/Cross_section_(geometry)" title="Cross section (geometry)">cross sectional area</a> <span class="texhtml"><i>A</i><sub>0</sub></span> being subjected to equal and opposite forces <span class="texhtml mvar" style="font-style:italic;">F</span> pulling at the ends so the bar is under tension. The material is experiencing a stress defined to be the ratio of the force to the cross sectional area of the bar, as well as an axial elongation: </p> <table class="wikitable"> <caption>Eng. stress & strain equations </caption> <tbody><tr> <th>Stress </th> <th>Strain </th></tr> <tr> <td><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 ={\frac {F}{A_{0}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>σ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>F</mi> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma ={\frac {F}{A_{0}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d905f300468eab871934639bc5f784690b254928" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:8.062ex; height:5.676ex;" alt="{\displaystyle \sigma ={\frac {F}{A_{0}}}}"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon ={\frac {L-L_{0}}{L_{0}}}={\frac {\Delta L}{L_{0}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ε</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>L</mi> <mo>−</mo> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">Δ</mi> <mi>L</mi> </mrow> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon ={\frac {L-L_{0}}{L_{0}}}={\frac {\Delta L}{L_{0}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d961244bf47f65518d600c64154f00ea3810b3fe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:19.532ex; height:5.843ex;" alt="{\displaystyle \varepsilon ={\frac {L-L_{0}}{L_{0}}}={\frac {\Delta L}{L_{0}}}}"></span> </td></tr></tbody></table> <p>Subscript 0 denotes the original dimensions of the sample. The <a href="/wiki/SI_derived_unit" title="SI derived unit">SI derived unit</a> for stress is <a href="/wiki/Newton_(unit)" title="Newton (unit)">newtons</a> per square metre, or <a href="/wiki/Pascal_(unit)" title="Pascal (unit)">pascals</a> (1 pascal = 1 Pa = 1 N/m<sup>2</sup>), and strain is <a href="/wiki/Dimensionless_quantity" title="Dimensionless quantity">unitless</a>. The stress–strain curve for this material is plotted by elongating the sample and recording the stress variation with strain until the sample <a href="/wiki/Fracture" title="Fracture">fractures</a>. By convention, the strain is set to the horizontal axis and stress is set to vertical axis. Note that for engineering purposes we often assume the cross-section area of the material does not change during the whole deformation process. This is not true since the actual area will decrease while deforming due to elastic and plastic deformation. The curve based on the original cross-section and gauge length is called the <i>engineering stress–strain curve</i>, while the curve based on the instantaneous cross-section area and length is called the <i>true stress–strain curve</i>. Unless stated otherwise, engineering stress–strain is generally used. </p> <div class="mw-heading mw-heading3"><h3 id="True_stress_and_strain">True stress and strain</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Deformation_(engineering)&action=edit&section=10" title="Edit section: True stress and strain"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Stress_strain_comparison.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a0/Stress_strain_comparison.svg/328px-Stress_strain_comparison.svg.png" decoding="async" width="328" height="205" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a0/Stress_strain_comparison.svg/492px-Stress_strain_comparison.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a0/Stress_strain_comparison.svg/656px-Stress_strain_comparison.svg.png 2x" data-file-width="400" data-file-height="250" /></a><figcaption>The difference between true stress–strain curve and engineering stress–strain curve</figcaption></figure> <p>In the above definitions of engineering stress and strain, two behaviors of materials in tensile tests are ignored: </p> <ul><li>the shrinking of section area</li> <li>compounding development of elongation</li></ul> <p><i>True stress</i> and <i>true strain</i> are defined differently than engineering stress and strain to account for these behaviors. They are given as </p> <table class="wikitable"> <caption>True stress & strain equations </caption> <tbody><tr> <th>Stress </th> <th>Strain </th></tr> <tr> <td><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 _{\mathrm {t} }={\frac {F}{A}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">t</mi> </mrow> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>F</mi> <mi>A</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{\mathrm {t} }={\frac {F}{A}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2174bc49fb18e46ed29a789971abbd546790eeaa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:7.877ex; height:5.343ex;" alt="{\displaystyle \sigma _{\mathrm {t} }={\frac {F}{A}}}"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon _{\mathrm {t} }=\int {\frac {\delta L}{L}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">t</mi> </mrow> </mrow> </msub> <mo>=</mo> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>δ</mi> <mi>L</mi> </mrow> <mi>L</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon _{\mathrm {t} }=\int {\frac {\delta L}{L}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f15ee5537239509a069240beb6fd094443d3f120" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:11.102ex; height:5.843ex;" alt="{\displaystyle \varepsilon _{\mathrm {t} }=\int {\frac {\delta L}{L}}}"></span> </td></tr></tbody></table> <dl><dd></dd></dl> <p>Here the dimensions are instantaneous values. Assuming volume of the sample conserves and deformation happens uniformly, </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 A_{0}L_{0}=AL}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mi>A</mi> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{0}L_{0}=AL}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/15b7f9e8cb8e0bc541d7ba4185e40504503e923e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.859ex; height:2.509ex;" alt="{\displaystyle A_{0}L_{0}=AL}"></span></dd></dl> <p>The true stress and strain can be expressed by engineering stress and strain. For true stress, </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 _{\mathrm {t} }={\frac {F}{A}}={\frac {F}{A_{0}}}{\frac {A_{0}}{A}}={\frac {F}{A_{0}}}{\frac {L}{L_{0}}}=\sigma (1+\varepsilon )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">t</mi> </mrow> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>F</mi> <mi>A</mi> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>F</mi> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mi>A</mi> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>F</mi> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>L</mi> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mfrac> </mrow> <mo>=</mo> <mi>σ</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>ε</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{\mathrm {t} }={\frac {F}{A}}={\frac {F}{A_{0}}}{\frac {A_{0}}{A}}={\frac {F}{A_{0}}}{\frac {L}{L_{0}}}=\sigma (1+\varepsilon )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2768aed6ec245046f0d7cda61ccf9714b605dbdc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:39.771ex; height:5.843ex;" alt="{\displaystyle \sigma _{\mathrm {t} }={\frac {F}{A}}={\frac {F}{A_{0}}}{\frac {A_{0}}{A}}={\frac {F}{A_{0}}}{\frac {L}{L_{0}}}=\sigma (1+\varepsilon )}"></span></dd></dl> <p>For the strain, </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 \delta \varepsilon _{\mathrm {t} }={\frac {\delta L}{L}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">t</mi> </mrow> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>δ</mi> <mi>L</mi> </mrow> <mi>L</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta \varepsilon _{\mathrm {t} }={\frac {\delta L}{L}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bcd9b1b5a95842904982669246633a85870f4902" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:9.57ex; height:5.343ex;" alt="{\displaystyle \delta \varepsilon _{\mathrm {t} }={\frac {\delta L}{L}}}"></span></dd></dl> <p>Integrate both sides and apply the boundary condition, </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 \varepsilon _{\mathrm {t} }=\ln \left({\frac {L}{L_{0}}}\right)=\ln(1+\varepsilon )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">t</mi> </mrow> </mrow> </msub> <mo>=</mo> <mi>ln</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>L</mi> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>ε</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon _{\mathrm {t} }=\ln \left({\frac {L}{L_{0}}}\right)=\ln(1+\varepsilon )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7c022b96d0ff9dbc1de425f11e4738c57ad65a3a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:25.821ex; height:6.176ex;" alt="{\displaystyle \varepsilon _{\mathrm {t} }=\ln \left({\frac {L}{L_{0}}}\right)=\ln(1+\varepsilon )}"></span></dd></dl> <p>So in a <a href="/wiki/Tensile_testing" title="Tensile testing">tension test</a>, true stress is larger than engineering stress and true strain is less than engineering strain. Thus, a point defining true stress–strain curve is displaced upwards and to the left to define the equivalent engineering stress–strain curve. The difference between the true and engineering stresses and strains will increase with <a href="/wiki/Plasticity_(physics)" title="Plasticity (physics)">plastic</a> deformation. At low strains (such as <a href="/wiki/Elasticity_(physics)" title="Elasticity (physics)">elastic</a> deformation), the differences between the two is negligible. As for the tensile strength point, it is the maximal point in engineering stress–strain curve but is not a special point in true stress–strain curve. Because engineering stress is proportional to the force applied along the sample, the criterion for <a href="/wiki/Necking_(engineering)" title="Necking (engineering)">necking</a> formation can be set as <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 \delta F=0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <mi>F</mi> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta F=0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/db3ca96d224c925fd6d264c94e03d7564702cce8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.697ex; height:2.343ex;" alt="{\displaystyle \delta F=0.}"></span> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}&\delta F=\sigma _{\text{t}}\,\delta A+A\,\delta \sigma _{\text{t}}=0\\&-{\frac {\delta A}{A}}={\frac {\delta \sigma _{\mathrm {t} }}{\sigma _{\mathrm {t} }}}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd /> <mtd> <mi>δ</mi> <mi>F</mi> <mo>=</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>t</mtext> </mrow> </msub> <mspace width="thinmathspace" /> <mi>δ</mi> <mi>A</mi> <mo>+</mo> <mi>A</mi> <mspace width="thinmathspace" /> <mi>δ</mi> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>t</mtext> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>δ</mi> <mi>A</mi> </mrow> <mi>A</mi> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>δ</mi> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">t</mi> </mrow> </mrow> </msub> </mrow> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">t</mi> </mrow> </mrow> </msub> </mfrac> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}&\delta F=\sigma _{\text{t}}\,\delta A+A\,\delta \sigma _{\text{t}}=0\\&-{\frac {\delta A}{A}}={\frac {\delta \sigma _{\mathrm {t} }}{\sigma _{\mathrm {t} }}}\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe7e60a441e91fcfe5b5a44844cee4ed544bc428" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.838ex; width:24.497ex; height:8.843ex;" alt="{\displaystyle {\begin{aligned}&\delta F=\sigma _{\text{t}}\,\delta A+A\,\delta \sigma _{\text{t}}=0\\&-{\frac {\delta A}{A}}={\frac {\delta \sigma _{\mathrm {t} }}{\sigma _{\mathrm {t} }}}\end{aligned}}}"></span></dd></dl> <p>This analysis suggests nature of the <a href="/wiki/Ultimate_tensile_strength" title="Ultimate tensile strength">ultimate tensile strength</a> (UTS) point. The <a href="/wiki/Work_hardening" title="Work hardening">work strengthening</a> effect is exactly balanced by the shrinking of section area at UTS point. </p><p>After the formation of necking, the sample undergoes heterogeneous deformation, so equations above are not valid. The stress and strain at the necking can be expressed as: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\sigma _{\mathrm {t} }&={\frac {F}{A_{\mathrm {neck} }}}\\\varepsilon _{\mathrm {t} }&=\ln \left({\frac {A_{0}}{A_{\mathrm {neck} }}}\right)\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">t</mi> </mrow> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>F</mi> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">k</mi> </mrow> </mrow> </msub> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">t</mi> </mrow> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>ln</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">k</mi> </mrow> </mrow> </msub> </mfrac> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\sigma _{\mathrm {t} }&={\frac {F}{A_{\mathrm {neck} }}}\\\varepsilon _{\mathrm {t} }&=\ln \left({\frac {A_{0}}{A_{\mathrm {neck} }}}\right)\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e94d96a4b3393db71d77db747296b1ea282a325e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.173ex; margin-bottom: -0.332ex; width:17.463ex; height:12.176ex;" alt="{\displaystyle {\begin{aligned}\sigma _{\mathrm {t} }&={\frac {F}{A_{\mathrm {neck} }}}\\\varepsilon _{\mathrm {t} }&=\ln \left({\frac {A_{0}}{A_{\mathrm {neck} }}}\right)\end{aligned}}}"></span></dd></dl> <p>An <a href="/wiki/Empirical_equation" class="mw-redirect" title="Empirical equation">empirical equation</a> is commonly used to describe the relationship between true stress and true strain. </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 _{\mathrm {t} }=K(\varepsilon _{\mathrm {t} })^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">t</mi> </mrow> </mrow> </msub> <mo>=</mo> <mi>K</mi> <mo stretchy="false">(</mo> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">t</mi> </mrow> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{\mathrm {t} }=K(\varepsilon _{\mathrm {t} })^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee6ab26a0e573d05f0e1dff06696c0396b2b9a26" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.347ex; height:2.843ex;" alt="{\displaystyle \sigma _{\mathrm {t} }=K(\varepsilon _{\mathrm {t} })^{n}}"></span></dd></dl> <p>Here, <span class="texhtml mvar" style="font-style:italic;">n</span> is the strain-hardening exponent and <span class="texhtml mvar" style="font-style:italic;">K</span> is the strength coefficient. <span class="texhtml mvar" style="font-style:italic;">n</span> is a measure of a material's work hardening behavior. Materials with a higher <span class="texhtml mvar" style="font-style:italic;">n</span> have a greater resistance to necking. Typically, metals at room temperature have <span class="texhtml mvar" style="font-style:italic;">n</span> ranging from 0.02 to 0.5.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading4"><h4 id="Discussion">Discussion</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Deformation_(engineering)&action=edit&section=11" title="Edit section: Discussion"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Since we disregard the change of area during deformation above, the true stress and strain curve should be re-derived. For deriving the stress strain curve, we can assume that the volume change is 0 even if we deformed the materials. We can assume that: </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 A_{i}\times \varepsilon _{i}=A_{f}\times \varepsilon _{f}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>×</mo> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> <mo>×</mo> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{i}\times \varepsilon _{i}=A_{f}\times \varepsilon _{f}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6be202515833815a4cb1fa8ab7ec54986e698f25" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:18.305ex; height:2.843ex;" alt="{\displaystyle A_{i}\times \varepsilon _{i}=A_{f}\times \varepsilon _{f}}"></span></dd></dl> <p>Then, the true stress can be expressed as below: </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 {\begin{aligned}\sigma _{T}={\frac {F}{A_{f}}}&={\frac {F}{A_{i}}}\times {\frac {A_{i}}{A_{f}}}\\&=\sigma _{e}\times {\frac {l_{f}}{l_{i}}}\\[2pt]&=\sigma _{E}\times {\frac {l_{i}+\delta l}{l_{i}}}\\[2pt]&=\sigma _{E}(1+\varepsilon _{E})\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt 0.5em 0.5em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>F</mi> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> </mfrac> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>F</mi> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mfrac> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msub> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> <msub> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>E</mi> </mrow> </msub> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>+</mo> <mi>δ</mi> <mi>l</mi> </mrow> <msub> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>E</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>E</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\sigma _{T}={\frac {F}{A_{f}}}&={\frac {F}{A_{i}}}\times {\frac {A_{i}}{A_{f}}}\\&=\sigma _{e}\times {\frac {l_{f}}{l_{i}}}\\[2pt]&=\sigma _{E}\times {\frac {l_{i}+\delta l}{l_{i}}}\\[2pt]&=\sigma _{E}(1+\varepsilon _{E})\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c9910460a3a7c87f08f619708a3813bc7e0808a8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -10.671ex; width:25.948ex; height:22.509ex;" alt="{\displaystyle {\begin{aligned}\sigma _{T}={\frac {F}{A_{f}}}&={\frac {F}{A_{i}}}\times {\frac {A_{i}}{A_{f}}}\\&=\sigma _{e}\times {\frac {l_{f}}{l_{i}}}\\[2pt]&=\sigma _{E}\times {\frac {l_{i}+\delta l}{l_{i}}}\\[2pt]&=\sigma _{E}(1+\varepsilon _{E})\end{aligned}}}"></span></dd></dl> <p>Additionally, the true strain <span class="texhtml mvar" style="font-style:italic;">ε<sub>T</sub></span> can be expressed as below: </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 \varepsilon _{T}={\frac {dl}{l_{0}}}+{\frac {dl}{l_{1}}}+{\frac {dl}{l_{2}}}+\cdots =\sum _{i}{\frac {dl}{l_{i}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>l</mi> </mrow> <msub> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>l</mi> </mrow> <msub> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>l</mi> </mrow> <msub> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mfrac> </mrow> <mo>+</mo> <mo>⋯</mo> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>l</mi> </mrow> <msub> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon _{T}={\frac {dl}{l_{0}}}+{\frac {dl}{l_{1}}}+{\frac {dl}{l_{2}}}+\cdots =\sum _{i}{\frac {dl}{l_{i}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/95ab25d10e758a9af70d3029a536cc1d7ea8cf00" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:34.637ex; height:6.509ex;" alt="{\displaystyle \varepsilon _{T}={\frac {dl}{l_{0}}}+{\frac {dl}{l_{1}}}+{\frac {dl}{l_{2}}}+\cdots =\sum _{i}{\frac {dl}{l_{i}}}}"></span></dd></dl> <p>Then, we can express the value as </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{l_{0}}^{l_{i}}{\frac {dl}{l}}\,dx=\ln \left({\frac {l_{i}}{l_{0}}}\right)=\ln(1+\varepsilon _{E})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>l</mi> </mrow> <mi>l</mi> </mfrac> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <mo>=</mo> <mi>ln</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>E</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{l_{0}}^{l_{i}}{\frac {dl}{l}}\,dx=\ln \left({\frac {l_{i}}{l_{0}}}\right)=\ln(1+\varepsilon _{E})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/13253a85898088f221462ed1797a424bc10643bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:34.341ex; height:6.676ex;" alt="{\displaystyle \int _{l_{0}}^{l_{i}}{\frac {dl}{l}}\,dx=\ln \left({\frac {l_{i}}{l_{0}}}\right)=\ln(1+\varepsilon _{E})}"></span></dd></dl> <p>Thus, we can induce the plot in terms of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{T}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{T}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/da1facbbe5ddcf256e6d88dbaf0ee200fdc6632f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.717ex; height:2.009ex;" alt="{\displaystyle \sigma _{T}}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon _{E}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>E</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon _{E}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48d05bd7a19d82b387f70f5f468f317d47d55582" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.571ex; height:2.009ex;" alt="{\displaystyle \varepsilon _{E}}"></span> as right figure. </p><p>Additionally, based on the true stress-strain curve, we can estimate the region where necking starts to happen. Since necking starts to appear after ultimate tensile stress where the maximum force applied, we can express this situation as below: </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 dF=0=\sigma _{T}dA_{i}+A_{i}d\sigma _{T}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mi>F</mi> <mo>=</mo> <mn>0</mn> <mo>=</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msub> <mi>d</mi> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mi>d</mi> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle dF=0=\sigma _{T}dA_{i}+A_{i}d\sigma _{T}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/11e0678914dd34a73acf0a4dea2596df5cc53140" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:26.107ex; height:2.509ex;" alt="{\displaystyle dF=0=\sigma _{T}dA_{i}+A_{i}d\sigma _{T}}"></span></dd></dl> <p>so this form can be expressed as below: </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 {\frac {d\sigma _{T}}{\sigma _{T}}}=-{\frac {dA_{i}}{A_{i}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msub> </mrow> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msub> </mfrac> </mrow> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {d\sigma _{T}}{\sigma _{T}}}=-{\frac {dA_{i}}{A_{i}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f437606173a600ef4029263b24245024c1ba187" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:14.27ex; height:5.843ex;" alt="{\displaystyle {\frac {d\sigma _{T}}{\sigma _{T}}}=-{\frac {dA_{i}}{A_{i}}}}"></span></dd></dl> <p>It indicates that the necking starts to appear where reduction of area becomes much significant compared to the stress change. Then the stress will be localized to specific area where the necking appears. </p><p>Additionally, we can induce various relation based on true stress-strain curve. </p><p>1) True strain and stress curve can be expressed by the approximate linear relationship by taking a log on true stress and strain. The relation can be expressed as below: </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 _{T}=K\times (\varepsilon _{T})^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msub> <mo>=</mo> <mi>K</mi> <mo>×</mo> <mo stretchy="false">(</mo> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{T}=K\times (\varepsilon _{T})^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6ae139ae08f7145fd67757503bbf38961b343590" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.222ex; height:2.843ex;" alt="{\displaystyle \sigma _{T}=K\times (\varepsilon _{T})^{n}}"></span></dd></dl> <p>Where <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 K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}"></span> is stress coefficient and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> is strain-hardening coefficient. Usually, the value of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> has range around 0.02 to 0.5 at room temperature. If <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> is 1, we can express this material as perfect elastic material.<sup id="cite_ref-:0_4-0" class="reference"><a href="#cite_note-:0-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p><p>2) In reality, stress is also highly dependent on the rate of strain variation. Thus, we can induce the empirical equation based on the strain rate variation. </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 _{T}=K'\times ({\dot {\varepsilon _{T}}})^{m}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msub> <mo>=</mo> <msup> <mi>K</mi> <mo>′</mo> </msup> <mo>×</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msub> <mo>˙</mo> </mover> </mrow> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{T}=K'\times ({\dot {\varepsilon _{T}}})^{m}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cfece0348d0c65c2be79d1d4def9793dbe5cbb9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.391ex; height:3.009ex;" alt="{\displaystyle \sigma _{T}=K'\times ({\dot {\varepsilon _{T}}})^{m}}"></span></dd></dl> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:The_way_to_calculate_the_true_strain_where_necking_starts.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/d/d1/The_way_to_calculate_the_true_strain_where_necking_starts.png" decoding="async" width="463" height="340" class="mw-file-element" data-file-width="463" data-file-height="340" /></a><figcaption>True stress-strain curve of FCC metal and its derivative form<sup id="cite_ref-:0_4-1" class="reference"><a href="#cite_note-:0-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup></figcaption></figure> <p>Where <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 K'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>K</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K'}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/63399d2b53fc472cfd349fd6167c21585d0a37f9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.779ex; height:2.509ex;" alt="{\displaystyle K'}"></span> is constant related to the material flow stress. <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\dot {\varepsilon _{T}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msub> <mo>˙</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\dot {\varepsilon _{T}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a35e7596bd126689f473373196b28e49f49ccf9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.473ex; height:2.509ex;" alt="{\displaystyle {\dot {\varepsilon _{T}}}}"></span> indicates the derivative of strain by the time, which is also known as strain rate. <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 m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}"></span> is the strain-rate sensitivity. Moreover, value of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}"></span> is related to the resistance toward the necking. Usually, the value of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}"></span> is at the range of 0-0.1 at room temperature and as high as 0.8 when the temperature is increased. </p><p>By combining the 1) and 2), we can create the ultimate relation as below: </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 _{T}=K''\times (\varepsilon _{T})^{n}({\dot {\varepsilon _{T}}})^{m}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msub> <mo>=</mo> <msup> <mi>K</mi> <mo>″</mo> </msup> <mo>×</mo> <mo stretchy="false">(</mo> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msub> <mo>˙</mo> </mover> </mrow> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{T}=K''\times (\varepsilon _{T})^{n}({\dot {\varepsilon _{T}}})^{m}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/37963967419e66fbd0b05fac17296eb25fcb1ecc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.344ex; height:3.009ex;" alt="{\displaystyle \sigma _{T}=K''\times (\varepsilon _{T})^{n}({\dot {\varepsilon _{T}}})^{m}}"></span></dd></dl> <p>Where <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 K''}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>K</mi> <mo>″</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K''}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5a926f6f13facbbf843fbcb17f56ec0193dc99a1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.231ex; height:2.509ex;" alt="{\displaystyle K''}"></span> is the global constant for relating strain, strain rate and stress. </p><p>3) Based on the true stress-strain curve and its derivative form, we can estimate the strain necessary to start necking. This can be calculated based on the intersection between true stress-strain curve as shown in right. </p><p>This figure also shows the dependency of the necking strain at different temperature. In case of FCC metals, both of the stress-strain curve at its derivative are highly dependent on temperature. Therefore, at higher temperature, necking starts to appear even under lower strain value. </p><p>All of these properties indicate the importance of calculating the true stress-strain curve for further analyzing the behavior of materials in sudden environment. </p><p>4) A graphical method, so-called "Considere construction", can help determine the behavior of stress-strain curve whether necking or drawing happens on the sample. By setting <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda =L/L_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> <mo>=</mo> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda =L/L_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7cfde329c0f8b7b5ef7915097e308b5064280eba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.836ex; height:2.843ex;" alt="{\displaystyle \lambda =L/L_{0}}"></span>as determinant, the true stress and strain can be expressed with engineering stress and strain as below: </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 _{T}=\sigma _{e}\times \lambda ,\qquad \varepsilon _{T}=\ln \lambda .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msub> <mo>×</mo> <mi>λ</mi> <mo>,</mo> <mspace width="2em" /> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msub> <mo>=</mo> <mi>ln</mi> <mo>⁡</mo> <mi>λ</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{T}=\sigma _{e}\times \lambda ,\qquad \varepsilon _{T}=\ln \lambda .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/12e53c5a0cca16ce7ccc85bc0d83ce24c6670eee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:27.915ex; height:2.509ex;" alt="{\displaystyle \sigma _{T}=\sigma _{e}\times \lambda ,\qquad \varepsilon _{T}=\ln \lambda .}"></span></dd></dl> <p>Therefore, the value of engineering stress can be expressed by the secant line from made by true stress and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b43d0ea3c9c025af1be9128e62a18fa74bedda2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }"></span> value where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda =0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda =0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/00c4bba30544017fe76932de5a4e25adb5512d95" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.616ex; height:2.176ex;" alt="{\displaystyle \lambda =0}"></span> to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda =1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda =1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/543b4490416437b7c80ea473bbcac0e4ab7a7f11" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.616ex; height:2.176ex;" alt="{\displaystyle \lambda =1}"></span>. By analyzing the shape of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{T}-\lambda }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msub> <mo>−</mo> <mi>λ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{T}-\lambda }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca44b97e4392f9ec2738e4600788ae9b38731315" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.912ex; height:2.509ex;" alt="{\displaystyle \sigma _{T}-\lambda }"></span> diagram and secant line, we can determine whether the materials show drawing or necking. </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Figureforneckdraw.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/cd/Figureforneckdraw.jpg/590px-Figureforneckdraw.jpg" decoding="async" width="590" height="142" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/cd/Figureforneckdraw.jpg/885px-Figureforneckdraw.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/cd/Figureforneckdraw.jpg/1180px-Figureforneckdraw.jpg 2x" data-file-width="1608" data-file-height="388" /></a><figcaption>Considere Plot. (a) True stress-strain curve without tangents. There is neither necking nor drawing. (b) With one tangent. There is only necking. (c) With two tangents. There are both necking and drawing.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup></figcaption></figure> <p>On the figure (a), there is only concave upward Considere plot. It indicates that there is no yield drop so the material will be suffered from fracture before it yields. On the figure (b), there is specific point where the tangent matches with secant line at point where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda =\lambda _{Y}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> <mo>=</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>Y</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda =\lambda _{Y}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b3e8a0b77cdaacdefbdbc4280314886af18de360" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.295ex; height:2.509ex;" alt="{\displaystyle \lambda =\lambda _{Y}}"></span>. After this value, the slope becomes smaller than the secant line where necking starts to appear. On the figure (c), there is point where yielding starts to appear but when <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda =\lambda _{d}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> <mo>=</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda =\lambda _{d}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4858a4501cd869834c38b55f8da2987fe58136f8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.901ex; height:2.509ex;" alt="{\displaystyle \lambda =\lambda _{d}}"></span>, the drawing happens. After drawing, all the material will stretch and eventually show fracture. Between <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda _{Y}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>Y</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda _{Y}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c1054d5f0729584d371d2198e6e09d228a77eed9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.841ex; height:2.509ex;" alt="{\displaystyle \lambda _{Y}}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda _{d}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda _{d}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b0e0e5ea8d81402f5112f5bab9944f2d34c0086" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.447ex; height:2.509ex;" alt="{\displaystyle \lambda _{d}}"></span>, the material itself does not stretch but rather, only the neck starts to stretch out. </p> <div class="mw-heading mw-heading2"><h2 id="Misconceptions">Misconceptions</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Deformation_(engineering)&action=edit&section=12" title="Edit section: Misconceptions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A popular misconception is that all materials that bend are "weak" and those that do not are "strong". In reality, many materials that undergo large elastic and plastic deformations, such as steel, are able to absorb stresses that would cause brittle materials, such as glass, with minimal plastic deformation ranges, to break.<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> </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=Deformation_(engineering)&action=edit&section=13" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1184024115">.mw-parser-output .div-col{margin-top:0.3em;column-width:30em}.mw-parser-output .div-col-small{font-size:90%}.mw-parser-output .div-col-rules{column-rule:1px solid #aaa}.mw-parser-output .div-col dl,.mw-parser-output .div-col ol,.mw-parser-output .div-col ul{margin-top:0}.mw-parser-output .div-col li,.mw-parser-output .div-col dd{page-break-inside:avoid;break-inside:avoid-column}</style><div class="div-col" style="column-width: 18em;"> <ul><li><a href="/wiki/Artificial_cranial_deformation" title="Artificial cranial deformation">Artificial cranial deformation</a></li> <li><a href="/wiki/Buff_strength" title="Buff strength">Buff strength</a></li> <li><a href="/wiki/Creep_(deformation)" title="Creep (deformation)">Creep (deformation)</a></li> <li><a href="/wiki/Deflection_(engineering)" title="Deflection (engineering)">Deflection (engineering)</a></li> <li><a href="/wiki/Deformation_(mechanics)" class="mw-redirect" title="Deformation (mechanics)">Deformation (mechanics)</a></li> <li><a href="/wiki/Deformation_mechanism_maps" class="mw-redirect" title="Deformation mechanism maps">Deformation mechanism maps</a></li> <li><a href="/wiki/Deformation_monitoring" title="Deformation monitoring">Deformation monitoring</a></li> <li><a href="/wiki/Deformation_retract" class="mw-redirect" title="Deformation retract">Deformation retract</a></li> <li><a href="/wiki/Deformation_theory" class="mw-redirect" title="Deformation theory">Deformation theory</a></li> <li><a href="/wiki/Elastic_(solid_mechanics)" class="mw-redirect" title="Elastic (solid mechanics)">Elasticity</a></li> <li><a href="/wiki/Malleability" class="mw-redirect" title="Malleability">Malleability</a></li> <li><a href="/wiki/Planar_deformation_features" title="Planar deformation features">Planar deformation features</a></li> <li><a href="/wiki/Plasticity_(physics)" title="Plasticity (physics)">Plasticity (physics)</a></li> <li><a href="/wiki/Poisson%27s_ratio" title="Poisson's ratio">Poisson's ratio</a></li> <li><a href="/wiki/Strain_tensor" class="mw-redirect" title="Strain tensor">Strain tensor</a></li> <li><a href="/wiki/Strength_of_materials" title="Strength of materials">Strength of materials</a></li> <li><a href="/wiki/Wood_warping" title="Wood warping">Wood warping</a></li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Deformation_(engineering)&action=edit&section=14" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFRees2006" class="citation book cs1">Rees, David (2006). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=4KWbmn_1hcYC"><i>Basic Engineering Plasticity: An Introduction with Engineering and Manufacturing Applications</i></a>. Butterworth-Heinemann. p. 41. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-7506-8025-3" title="Special:BookSources/0-7506-8025-3"><bdi>0-7506-8025-3</bdi></a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20171222205706/https://books.google.com/books?id=4KWbmn_1hcYC">Archived</a> from the original on 2017-12-22.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Basic+Engineering+Plasticity%3A+An+Introduction+with+Engineering+and+Manufacturing+Applications&rft.pages=41&rft.pub=Butterworth-Heinemann&rft.date=2006&rft.isbn=0-7506-8025-3&rft.aulast=Rees&rft.aufirst=David&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D4KWbmn_1hcYC&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADeformation+%28engineering%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">Callister, William D. (2004) <i>Fundamentals of Materials Science and Engineering</i>, John Wiley and Sons, 2nd ed. p. 184. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-471-66081-7" title="Special:BookSources/0-471-66081-7">0-471-66081-7</a>.</span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCourtney2005" class="citation book cs1">Courtney, Thomas (2005). <i>Mechanical behavior of materials</i>. Waveland Press, Inc. pp. 6–13.</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.pages=6-13&rft.pub=Waveland+Press%2C+Inc&rft.date=2005&rft.aulast=Courtney&rft.aufirst=Thomas&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADeformation+%28engineering%29" class="Z3988"></span></span> </li> <li id="cite_note-:0-4"><span class="mw-cite-backlink">^ <a href="#cite_ref-:0_4-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-:0_4-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCourtney2000" class="citation book cs1">Courtney, Thomas (2000). <i>Mechanical Behavior of Materials</i>. Illinois: Waveland Press. p. 165. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780073228242" title="Special:BookSources/9780073228242"><bdi>9780073228242</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=Illinois&rft.pages=165&rft.pub=Waveland+Press&rft.date=2000&rft.isbn=9780073228242&rft.aulast=Courtney&rft.aufirst=Thomas&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADeformation+%28engineering%29" class="Z3988"></span></span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20180127221742/http://web.adanabtu.edu.tr/Files/iyilmaz/Duyuru/dosya/ME%20207%20%E2%80%93%20Chapter%203_P3.pdf">"True Stress and Strain"</a> <span class="cs1-format">(PDF)</span>. Archived from <a rel="nofollow" class="external text" href="https://web.adanabtu.edu.tr/Files/iyilmaz/Duyuru/dosya/ME%20207%20%E2%80%93%20Chapter%203_P3.pdf">the original</a> <span class="cs1-format">(PDF)</span> on 2018-01-27<span class="reference-accessdate">. Retrieved <span class="nowrap">2018-05-15</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=True+Stress+and+Strain&rft_id=https%3A%2F%2Fweb.adanabtu.edu.tr%2FFiles%2Fiyilmaz%2FDuyuru%2Fdosya%2FME%2520207%2520%25E2%2580%2593%2520Chapter%25203_P3.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADeformation+%28engineering%29" class="Z3988"></span></span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRoland" class="citation web cs1">Roland, David. <a rel="nofollow" class="external text" href="http://web.mit.edu/course/3/3.11/www/modules/ss.pdf">"STRESS-STRAIN CURVES"</a> <span class="cs1-format">(PDF)</span>. <i>MIT</i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=MIT&rft.atitle=STRESS-STRAIN+CURVES&rft.aulast=Roland&rft.aufirst=David&rft_id=http%3A%2F%2Fweb.mit.edu%2Fcourse%2F3%2F3.11%2Fwww%2Fmodules%2Fss.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADeformation+%28engineering%29" class="Z3988"></span></span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRice,_Peter_and_Dutton,_Hugh1995" class="citation book cs1">Rice, Peter and Dutton, Hugh (1995). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=7t9wgJEUWHYC&pg=PA33"><i>Structural glass</i></a>. Taylor & Francis. p. 33. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-419-19940-3" title="Special:BookSources/0-419-19940-3"><bdi>0-419-19940-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Structural+glass&rft.pages=33&rft.pub=Taylor+%26+Francis&rft.date=1995&rft.isbn=0-419-19940-3&rft.au=Rice%2C+Peter+and+Dutton%2C+Hugh&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D7t9wgJEUWHYC%26pg%3DPA33&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADeformation+%28engineering%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: multiple names: authors list (<a href="/wiki/Category:CS1_maint:_multiple_names:_authors_list" title="Category:CS1 maint: multiple names: authors list">link</a>)</span></span> </li> </ol></div></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Deformation_(engineering)&oldid=1257785337">https://en.wikipedia.org/w/index.php?title=Deformation_(engineering)&oldid=1257785337</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Categories</a>: <ul><li><a href="/wiki/Category:Solid_mechanics" title="Category:Solid mechanics">Solid mechanics</a></li><li><a href="/wiki/Category:Deformation_(mechanics)" title="Category:Deformation (mechanics)">Deformation (mechanics)</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Hidden categories: <ul><li><a href="/wiki/Category:CS1_maint:_multiple_names:_authors_list" title="Category:CS1 maint: multiple names: authors list">CS1 maint: multiple names: authors list</a></li><li><a href="/wiki/Category:Articles_with_short_description" title="Category:Articles with short description">Articles with short description</a></li><li><a href="/wiki/Category:Short_description_is_different_from_Wikidata" title="Category:Short description is different from Wikidata">Short description is different from Wikidata</a></li><li><a href="/wiki/Category:Articles_needing_additional_references_from_September_2008" title="Category:Articles needing additional references from September 2008">Articles needing additional references from September 2008</a></li><li><a href="/wiki/Category:All_articles_needing_additional_references" title="Category:All articles needing additional references">All articles needing additional references</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> This page was last edited on 16 November 2024, at 17:01<span class="anonymous-show"> (UTC)</span>.</li> <li id="footer-info-copyright">Text is available under the <a href="/wiki/Wikipedia:Text_of_the_Creative_Commons_Attribution-ShareAlike_4.0_International_License" title="Wikipedia:Text of the Creative Commons Attribution-ShareAlike 4.0 International License">Creative Commons Attribution-ShareAlike 4.0 License</a>; additional terms may apply. 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Deformation (engineering) - Wikipedia