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class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Contact_with_friction"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Contact with friction</span> </div> </a> <ul id="toc-Contact_with_friction-sublist" class="vector-toc-list"> <li id="toc-Correction_formulas" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Correction_formulas"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2.1</span> <span>Correction formulas</span> </div> </a> <ul id="toc-Correction_formulas-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Geometric_extrapolation" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Geometric_extrapolation"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2.2</span> <span>Geometric extrapolation</span> </div> </a> <ul id="toc-Geometric_extrapolation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Finite_element_analysis" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Finite_element_analysis"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2.3</span> <span>Finite element analysis</span> </div> </a> <ul id="toc-Finite_element_analysis-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </li> <li id="toc-Comparison_of_compressive_and_tensile_strengths" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Comparison_of_compressive_and_tensile_strengths"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Comparison of compressive and tensile strengths</span> </div> </a> <ul id="toc-Comparison_of_compressive_and_tensile_strengths-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Compressive_failure_modes" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Compressive_failure_modes"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Compressive failure modes</span> </div> </a> <button aria-controls="toc-Compressive_failure_modes-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 Compressive failure modes subsection</span> </button> <ul id="toc-Compressive_failure_modes-sublist" class="vector-toc-list"> <li id="toc-Microcracking" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Microcracking"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Microcracking</span> </div> </a> <ul id="toc-Microcracking-sublist" class="vector-toc-list"> <li id="toc-Shear_bands" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Shear_bands"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1.1</span> <span>Shear bands</span> </div> </a> <ul id="toc-Shear_bands-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </li> <li id="toc-Typical_values" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Typical_values"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Typical values</span> </div> </a> <ul id="toc-Typical_values-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Compressive_strength_of_concrete" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Compressive_strength_of_concrete"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Compressive strength of concrete</span> </div> </a> <ul id="toc-Compressive_strength_of_concrete-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" 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interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Esfor%C3%A7_de_compressi%C3%B3" title="Esforç de compressió – Catalan" lang="ca" hreflang="ca" data-title="Esforç de compressió" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Pevnost_v_tlaku" title="Pevnost v tlaku – Czech" lang="cs" hreflang="cs" data-title="Pevnost v tlaku" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Trykstyrke" title="Trykstyrke – Danish" lang="da" hreflang="da" data-title="Trykstyrke" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Druckfestigkeit" title="Druckfestigkeit – German" lang="de" hreflang="de" data-title="Druckfestigkeit" 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/Survetugevus" title="Survetugevus – Estonian" lang="et" hreflang="et" data-title="Survetugevus" 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/Esfuerzo_de_compresi%C3%B3n" title="Esfuerzo de compresión – Spanish" lang="es" hreflang="es" data-title="Esfuerzo de compresión" 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/%D9%85%D9%82%D8%A7%D9%88%D9%85%D8%AA_%D9%81%D8%B4%D8%A7%D8%B1%DB%8C" title="مقاومت فشاری – Persian" lang="fa" hreflang="fa" data-title="مقاومت فشاری" 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/Essai_de_compression" title="Essai de compression – French" lang="fr" hreflang="fr" data-title="Essai de compression" 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%95%95%EC%B6%95%EA%B0%95%EB%8F%84" title="압축강도 – Korean" lang="ko" hreflang="ko" data-title="압축강도" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%B8%E0%A4%82%E0%A4%AA%E0%A5%80%E0%A4%A1%E0%A4%A8_%E0%A4%AA%E0%A5%81%E0%A4%B7%E0%A5%8D%E0%A4%9F%E0%A4%BF" title="संपीडन पुष्टि – Hindi" lang="hi" hreflang="hi" data-title="संपीडन पुष्टि" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Kekuatan_tekan" title="Kekuatan tekan – Indonesian" lang="id" hreflang="id" data-title="Kekuatan tekan" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Resistenza_a_compressione" title="Resistenza a compressione – Italian" lang="it" hreflang="it" data-title="Resistenza a compressione" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Spiedes_stipr%C4%ABba" title="Spiedes stiprība – Latvian" lang="lv" hreflang="lv" data-title="Spiedes stiprība" data-language-autonym="Latviešu" data-language-local-name="Latvian" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Druksterkte" title="Druksterkte – Dutch" lang="nl" hreflang="nl" data-title="Druksterkte" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Trykkstyrke" title="Trykkstyrke – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Trykkstyrke" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Trykkfastleik" title="Trykkfastleik – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Trykkfastleik" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/%C5%9Aciskanie" title="Ściskanie – Polish" lang="pl" hreflang="pl" data-title="Ściskanie" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Esfor%C3%A7o_de_compress%C3%A3o" 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id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Capacity of a material or structure to withstand loads tending to reduce size</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 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.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/Compressive_strength" title="Special:EditPage/Compressive strength">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=%22Compressive+strength%22">"Compressive strength"</a> – <a rel="nofollow" class="external text" href="https://www.google.com/search?tbm=nws&q=%22Compressive+strength%22+-wikipedia&tbs=ar:1">news</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.google.com/search?&q=%22Compressive+strength%22&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=%22Compressive+strength%22+-wikipedia">books</a> <b>·</b> <a rel="nofollow" class="external text" href="https://scholar.google.com/scholar?q=%22Compressive+strength%22">scholar</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.jstor.org/action/doBasicSearch?Query=%22Compressive+strength%22&acc=on&wc=on">JSTOR</a></span></small></span> <span class="date-container"><i>(<span class="date">April 2014</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> <figure typeof="mw:File/Thumb"><a href="/wiki/File:US_military_drum_compression_test.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/86/US_military_drum_compression_test.jpg/200px-US_military_drum_compression_test.jpg" decoding="async" width="200" height="211" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/8/86/US_military_drum_compression_test.jpg 1.5x" data-file-width="300" data-file-height="317" /></a><figcaption>Measuring the compressive strength of a steel <a href="/wiki/Drum_(container)" title="Drum (container)">drum</a></figcaption></figure> <p>In <a href="/wiki/Mechanics" title="Mechanics">mechanics</a>, <b>compressive strength</b> (or <b>compression strength</b>) is the capacity of a material or <a href="/wiki/Structural_system" title="Structural system">structure</a> to withstand <a href="/wiki/Structural_load" title="Structural load">loads</a> tending to reduce size (<a href="/wiki/Compression_(physics)" title="Compression (physics)">compression</a>). It is opposed to <i><a href="/wiki/Tensile_strength" class="mw-redirect" title="Tensile strength">tensile strength</a></i> which withstands loads tending to elongate, resisting <a href="/wiki/Tension_(physics)" title="Tension (physics)">tension</a> (being pulled apart). In the study of <a href="/wiki/Strength_of_materials" title="Strength of materials">strength of materials</a>, compressive strength, tensile strength, and <a href="/wiki/Shear_strength" title="Shear strength">shear strength</a> can be analyzed independently. </p><p>Some materials <a href="/wiki/Fracture" title="Fracture">fracture</a> at their compressive strength limit; others <a href="/wiki/Plasticity_(physics)" title="Plasticity (physics)">deform irreversibly</a>, so a given amount of <a href="/wiki/Deformation_(engineering)" title="Deformation (engineering)">deformation</a> may be considered as the limit for compressive load. Compressive strength is a key value for <a href="/wiki/Structural_engineering" title="Structural engineering">design of structures</a>. </p><p>Compressive strength is often measured on a <a href="/wiki/Universal_testing_machine" title="Universal testing machine">universal testing machine</a>. Measurements of compressive strength are affected by the specific <a href="/wiki/Test_method" title="Test method">test method</a> and conditions of measurement. Compressive strengths are usually reported in relationship to a specific <a href="/wiki/Technical_standard" title="Technical standard">technical standard</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Introduction">Introduction</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Compressive_strength&action=edit&section=1" title="Edit section: Introduction"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1237032888/mw-parser-output/.tmulti">.mw-parser-output .tmulti .multiimageinner{display:flex;flex-direction:column}.mw-parser-output .tmulti .trow{display:flex;flex-direction:row;clear:left;flex-wrap:wrap;width:100%;box-sizing:border-box}.mw-parser-output .tmulti .tsingle{margin:1px;float:left}.mw-parser-output .tmulti .theader{clear:both;font-weight:bold;text-align:center;align-self:center;background-color:transparent;width:100%}.mw-parser-output .tmulti .thumbcaption{background-color:transparent}.mw-parser-output .tmulti .text-align-left{text-align:left}.mw-parser-output .tmulti .text-align-right{text-align:right}.mw-parser-output .tmulti .text-align-center{text-align:center}@media all and (max-width:720px){.mw-parser-output .tmulti .thumbinner{width:100%!important;box-sizing:border-box;max-width:none!important;align-items:center}.mw-parser-output .tmulti .trow{justify-content:center}.mw-parser-output .tmulti .tsingle{float:none!important;max-width:100%!important;box-sizing:border-box;text-align:center}.mw-parser-output .tmulti .tsingle .thumbcaption{text-align:left}.mw-parser-output .tmulti .trow>.thumbcaption{text-align:center}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .tmulti .multiimageinner img{background-color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .tmulti .multiimageinner img{background-color:white}}</style><div class="thumb tmulti tright"><div class="thumbinner multiimageinner" style="width:142px;max-width:142px"><div class="trow"><div class="tsingle" style="width:70px;max-width:70px"><div class="thumbimage" style="height:566px;overflow:hidden"><span typeof="mw:File"><a href="/wiki/File:Tension_applied.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d1/Tension_applied.svg/68px-Tension_applied.svg.png" decoding="async" width="68" height="570" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d1/Tension_applied.svg/102px-Tension_applied.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d1/Tension_applied.svg/136px-Tension_applied.svg.png 2x" data-file-width="21" data-file-height="176" /></a></span></div><div class="thumbcaption"><a href="/wiki/Tension_(physics)" title="Tension (physics)">Tension</a></div></div><div class="tsingle" style="width:68px;max-width:68px"><div class="thumbimage" style="height:566px;overflow:hidden"><span typeof="mw:File"><a href="/wiki/File:Compression_applied.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/5/53/Compression_applied.svg/66px-Compression_applied.svg.png" decoding="async" width="66" height="563" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/53/Compression_applied.svg/99px-Compression_applied.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/53/Compression_applied.svg/132px-Compression_applied.svg.png 2x" data-file-width="21" data-file-height="179" /></a></span></div><div class="thumbcaption"><a href="/wiki/Compression_(physics)" title="Compression (physics)">Compression</a></div></div></div></div></div> <p>When a specimen of material is loaded in such a way that it extends it is said to be in <i>tension</i>. On the other hand, if the material <a href="/wiki/Compression_(physical)" class="mw-redirect" title="Compression (physical)">compresses</a> and shortens it is said to be in <i>compression</i>. </p><p>On an atomic level, the molecules or <a href="/wiki/Atoms" class="mw-redirect" title="Atoms">atoms</a> are forced apart when in tension whereas in compression they are forced together. Since atoms in solids always try to find an equilibrium position, and distance between other atoms, forces arise throughout the entire material which oppose both tension or compression. The phenomena prevailing on an atomic level are therefore similar. </p><p>The "strain" is the relative change in length under applied stress; positive strain characterizes an object under tension load which tends to lengthen it, and a compressive stress that shortens an object gives negative strain. Tension tends to pull small sideways deflections back into alignment, while compression tends to amplify such deflection into <a href="/wiki/Buckling" title="Buckling">buckling</a>. </p><p>Compressive strength is measured on materials, components,<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> and structures.<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><p> The ultimate compressive strength of a material is the maximum uniaxial <a href="/wiki/Compressive_stress" title="Compressive stress">compressive stress</a> that it can withstand before complete failure. This value is typically determined through a compressive test conducted using a <a href="/wiki/Universal_testing_machine" title="Universal testing machine">universal testing machine</a>. During the test, a steadily increasing uniaxial compressive load is applied to the test specimen until it fails.The specimen, often cylindrical in shape, experiences both axial shortening and <a href="/wiki/Geometric_terms_of_location" title="Geometric terms of location">lateral</a> expansion under the load. As the load increases, the machine records the corresponding deformation, plotting a <a href="/wiki/Stress%E2%80%93strain_curve" title="Stress–strain curve">stress-strain curve</a> that would look similar to the following:</p><figure class="mw-default-size mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/File:Engineering_stress_strain.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d0/Engineering_stress_strain.svg/220px-Engineering_stress_strain.svg.png" decoding="async" width="220" height="201" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d0/Engineering_stress_strain.svg/330px-Engineering_stress_strain.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d0/Engineering_stress_strain.svg/440px-Engineering_stress_strain.svg.png 2x" data-file-width="217" data-file-height="198" /></a><figcaption>True stress-strain curve for a typical specimen</figcaption></figure> <p>The compressive strength of the material corresponds to the stress at the red point shown on the curve. In a compression test, there is a linear region where the material follows <a href="/wiki/Hooke%27s_law" title="Hooke's law">Hooke's law</a>. Hence, for this region, <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> <mo>,</mo> </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/f5d1bcaec7375695bb02d441903fab7501b59f98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.934ex; height:2.509ex;" alt="{\displaystyle \sigma =E\varepsilon ,}"></span> where, this time, <span class="texhtml mvar" style="font-style:italic;">E</span> refers to the Young's modulus for compression. In this region, the material deforms elastically and returns to its original length when the stress is removed. </p><p>This linear region terminates at what is known as the <a href="/wiki/Yield_point" class="mw-redirect" title="Yield point">yield point</a>. Above this point the material behaves <a href="/wiki/Plasticity_(physics)" title="Plasticity (physics)">plastically</a> and will not return to its original length once the load is removed. </p><p>There is a difference between the engineering stress and the true stress. By its basic definition the uniaxial stress is given by: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\acute {\sigma }}={\frac {F}{A}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>σ</mi> <mo>´</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>F</mi> <mi>A</mi> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\acute {\sigma }}={\frac {F}{A}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4498b7ddd377dfbca23dc60cbf97fbe58c245f2" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:7.654ex; height:5.343ex;" alt="{\displaystyle {\acute {\sigma }}={\frac {F}{A}},}"></span>where <span class="texhtml mvar" style="font-style:italic;">F</span> is load applied [N] and <span class="texhtml mvar" style="font-style:italic;">A</span> is area [m<sup>2</sup>]. </p><p>As stated, the area of the specimen varies on compression. In reality therefore the area is some function of the applied load i.e. <span class="texhtml"><i>A</i> = <i>f</i> (<i>F</i>)</span>. Indeed, stress is defined as the force divided by the area at the start of the experiment. This is known as the engineering stress, and is defined by<span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{e}={\frac {F}{A_{0}}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msub> <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> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{e}={\frac {F}{A_{0}}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/61d0055b61c3194428787ffedf260a353d1fab36" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:9.705ex; height:5.676ex;" alt="{\displaystyle \sigma _{e}={\frac {F}{A_{0}}},}"></span>where <span class="texhtml"><i>A</i><sub>0</sub></span> is the original specimen area [m<sup>2</sup>]. </p><p>Correspondingly, the engineering <a href="/wiki/Strain_(materials_science)" class="mw-redirect" title="Strain (materials science)">strain</a> is defined by<span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon _{e}={\frac {l-l_{0}}{l_{0}}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon _{e}={\frac {l-l_{0}}{l_{0}}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d148a560528c784ea60c67c0292d435436d64394" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:11.945ex; height:5.843ex;" alt="{\displaystyle \varepsilon _{e}={\frac {l-l_{0}}{l_{0}}},}"></span>where <span class="texhtml mvar" style="font-style:italic;">l</span> is the current specimen length [m] and <span class="texhtml"><i>l</i><sub>0</sub></span> is the original specimen length [m]. True strain, also known as logarithmic strain or natural strain, provides a more accurate measure of large deformations, such as in materials like ductile metals<sup id="cite_ref-:0_3-0" class="reference"><a href="#cite_note-:0-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\acute {\epsilon }}=\ln(l/l_{o})=ln(1+\epsilon _{e})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ϵ</mi> <mo>´</mo> </mover> </mrow> </mrow> <mo>=</mo> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msub> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>o</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mi>l</mi> <mi>n</mi> <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 {\acute {\epsilon }}=\ln(l/l_{o})=ln(1+\epsilon _{e})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/097a375360785626146c7b7193bf59c20513b6dd" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.659ex; height:2.843ex;" alt="{\displaystyle {\acute {\epsilon }}=\ln(l/l_{o})=ln(1+\epsilon _{e})}"></span>The compressive strength therefore corresponds to the point on the engineering stress–strain curve <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 \left(\varepsilon _{e}^{*},\sigma _{e}^{*}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mrow> <msubsup> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <mo>,</mo> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(\varepsilon _{e}^{*},\sigma _{e}^{*}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3543980ce9928edb67f25f3983cf5aa42c673c3d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.366ex; height:2.843ex;" alt="{\displaystyle \left(\varepsilon _{e}^{*},\sigma _{e}^{*}\right)}"></span> defined by<span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{e}^{*}={\frac {F^{*}}{A_{0}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <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 _{e}^{*}={\frac {F^{*}}{A_{0}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e3a5f65bdebe18f6a14c7f2626c8686667a6ff72" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:9.188ex; height:5.843ex;" alt="{\displaystyle \sigma _{e}^{*}={\frac {F^{*}}{A_{0}}}}"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon _{e}^{*}={\frac {l^{*}-l_{0}}{l_{0}}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon _{e}^{*}={\frac {l^{*}-l_{0}}{l_{0}}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/efbb6ec407a9350d07454f253348edf8e7edb0d4" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:13.055ex; height:5.843ex;" alt="{\displaystyle \varepsilon _{e}^{*}={\frac {l^{*}-l_{0}}{l_{0}}},}"></span> </p><p>where <span class="texhtml"><i>F</i><sup>*</sup></span> is the load applied just before crushing and <span class="texhtml"><i>l</i><sup>*</sup></span> is the specimen length just before crushing. </p> <div class="mw-heading mw-heading2"><h2 id="Deviation_of_engineering_stress_from_true_stress">Deviation of engineering stress from true stress</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Compressive_strength&action=edit&section=2" title="Edit section: Deviation of engineering stress from true stress"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Barelling.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/eb/Barelling.svg/75px-Barelling.svg.png" decoding="async" width="75" height="221" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/eb/Barelling.svg/113px-Barelling.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/eb/Barelling.svg/150px-Barelling.svg.png 2x" data-file-width="41" data-file-height="121" /></a><figcaption>Barrelling</figcaption></figure> <p>When a uniaxial compressive load is applied to an object it will become shorter and spread laterally so its original cross sectional area (<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="{\textstyle A_{o}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>o</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle A_{o}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/41627a0fe3a06811b0bd2ea08b19ab934c97c024" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.773ex; height:2.509ex;" alt="{\textstyle A_{o}}"></span>) increases to the loaded area (<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="{\textstyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a118c6ad00742b3f5dccd2f0e74b5e369df6fd31" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\textstyle A}"></span>).<sup id="cite_ref-:0_3-1" class="reference"><a href="#cite_note-:0-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> Thus the true 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 {\acute {\sigma }}=F/A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>σ</mi> <mo>´</mo> </mover> </mrow> </mrow> <mo>=</mo> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\acute {\sigma }}=F/A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a55ccf01fc6ab2eaf5800d447e802de17437e11e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.074ex; height:2.843ex;" alt="{\displaystyle {\acute {\sigma }}=F/A}"></span>) deviates from engineering stress (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{e}=F/A_{o}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msub> <mo>=</mo> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>o</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{e}=F/A_{o}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1e57e709f5066db34f78cc3a60cc894afe56d873" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.1ex; height:2.843ex;" alt="{\displaystyle \sigma _{e}=F/A_{o}}"></span>). Tests that measure the engineering stress at the point of failure in a material are often sufficient for many routine applications, such as quality control in concrete production. However, determining the true stress in materials under compressive loads is important for research focused on the properties on new materials and their processing. </p><p>The geometry of test specimens and friction can significantly influence the results of compressive stress tests.<sup id="cite_ref-:0_3-2" class="reference"><a href="#cite_note-:0-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-:1_4-0" class="reference"><a href="#cite_note-:1-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> Friction at the contact points between the testing machine and the specimen can restrict the lateral expansion at its ends (also known as 'barreling') leading to non-uniform stress distribution. This is discussed in section on <a href="#Contact_with_friction">contact with friction</a>. <span class="anchor" id="Frictionless_contact"></span> </p> <div class="mw-heading mw-heading3"><h3 id="Frictionless_contact">Frictionless contact</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Compressive_strength&action=edit&section=3" title="Edit section: Frictionless contact"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>With a compressive load on a test specimen it will become shorter and spread laterally so its cross sectional area increases and the true compressive stress is<span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\acute {\sigma }}=F/A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>σ</mi> <mo>´</mo> </mover> </mrow> </mrow> <mo>=</mo> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\acute {\sigma }}=F/A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a55ccf01fc6ab2eaf5800d447e802de17437e11e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.074ex; height:2.843ex;" alt="{\displaystyle {\acute {\sigma }}=F/A}"></span>and the engineering stress is<span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sigma _{e}}=F/A_{o}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msub> </mrow> <mo>=</mo> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>o</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sigma _{e}}=F/A_{o}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ff4ca5faea3588ffbafaf3d8235b24ecd73d7271" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.1ex; height:2.843ex;" alt="{\displaystyle {\sigma _{e}}=F/A_{o}}"></span>The cross sectional area (<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="{\textstyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a118c6ad00742b3f5dccd2f0e74b5e369df6fd31" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\textstyle A}"></span>) and consequently the stress ( <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\acute {\sigma }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>σ</mi> <mo>´</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {\acute {\sigma }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5810bb3743bc43d4287c317a8699e49b4e5a788c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:2.343ex;" alt="{\textstyle {\acute {\sigma }}}"></span>) are uniform along the length of the specimen because there are no external lateral constraints. This condition represents an ideal test condition. For all practical purposes the volume of a high <a href="/wiki/Bulk_modulus" title="Bulk modulus">bulk modulus</a> material (e.g. solid metals) is not changed by uniaxial compression.<sup id="cite_ref-:0_3-3" class="reference"><a href="#cite_note-:0-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> So<span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Al=A_{o}l_{o}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mi>l</mi> <mo>=</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>o</mi> </mrow> </msub> <msub> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>o</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Al=A_{o}l_{o}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d8531b36a8913cd730428c4c0921a177660be1e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.03ex; height:2.509ex;" alt="{\displaystyle Al=A_{o}l_{o}}"></span>Using the strain equation from above<sup id="cite_ref-:0_3-4" class="reference"><a href="#cite_note-:0-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A=A_{o}/(1+\epsilon _{e})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>o</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <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 A=A_{o}/(1+\epsilon _{e})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4a58e08836139963b97bc63a7e8eac04146e51a3" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.531ex; height:2.843ex;" alt="{\displaystyle A=A_{o}/(1+\epsilon _{e})}"></span>and<span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\acute {\sigma }}=\sigma _{e}(1+\epsilon _{e})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>σ</mi> <mo>´</mo> </mover> </mrow> </mrow> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\acute {\sigma }}=\sigma _{e}(1+\epsilon _{e})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/89a6cab3ce907d82834181b9e91e7ca5843da707" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.509ex; height:2.843ex;" alt="{\displaystyle {\acute {\sigma }}=\sigma _{e}(1+\epsilon _{e})}"></span>Note that compressive strain is negative, so the true 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 {\acute {\sigma }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>σ</mi> <mo>´</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\acute {\sigma }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aa55fce19611b160e40c1553dd22ccbd04549b51" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:2.343ex;" alt="{\displaystyle {\acute {\sigma }}}"></span> ) is less than the engineering 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="{\textstyle \sigma _{e}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \sigma _{e}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b7172ebd080a367cb9c1c795ff1ec9b34bc627ef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.326ex; height:2.009ex;" alt="{\textstyle \sigma _{e}}"></span>). The true strain (<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 {\acute {\epsilon }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ϵ</mi> <mo>´</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\acute {\epsilon }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d84f866baf77fea7712425e2306bbf63a6c47279" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.292ex; height:2.343ex;" alt="{\displaystyle {\acute {\epsilon }}}"></span>) can be used in these formulas instead of engineering strain (<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="{\textstyle \epsilon _{e}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msub> <mi>ϵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \epsilon _{e}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/37889e30a14fc161379cc689d0f593b28cb5e100" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.943ex; height:2.009ex;" alt="{\textstyle \epsilon _{e}}"></span>) when the deformation is large.<span class="anchor" id="Contact_with_friction"></span> </p> <div class="mw-heading mw-heading3"><h3 id="Contact_with_friction">Contact with friction</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Compressive_strength&action=edit&section=4" title="Edit section: Contact with friction"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>As the load is applied, friction at the interface between the specimen and the test machine restricts the lateral expansion at its ends. This has two effects: </p> <ul><li>It can cause non-uniform stress distribution across the specimen, with higher stress at the centre and lower stress at the edges, which affects the accuracy of the result.</li></ul> <ul><li>It causes a barreling effect (bulging at the centre) in ductile materials. This changes the specimen’s geometry and affects its load-bearing capacity, leading to a higher apparent compressive strength.</li></ul> <p>Various methods can be used to reduce the friction according to the application: </p> <ul><li>Applying a suitable lubricant, such as <a href="/wiki/MoS2" class="mw-redirect" title="MoS2">MoS2</a>, oil or grease; however, care must be taken not to affect the material properties with the lubricant used.</li></ul> <ul><li>Use of <a href="/wiki/PTFE" class="mw-redirect" title="PTFE">PTFE</a> or other low-friction sheets between the test machine and specimen.</li> <li>A spherical or self-aligning test fixture, which can minimize friction by applying the load more evenly across the specimen's surface.</li></ul> <p>Three methods can be used to compensate for the effects of friction on the test result: </p> <ol><li><a href="#Correction_formula">Correction formulas</a></li> <li><a href="#Geometric_extrapolation">Geometric extrapolation</a></li> <li><a href="#Finite_element_analysis">Finite element analysis</a></li></ol> <p><span class="anchor" id="Correction_formula"></span> </p> <div class="mw-heading mw-heading4"><h4 id="Correction_formulas">Correction formulas</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Compressive_strength&action=edit&section=5" title="Edit section: Correction formulas"><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:Compression_Test_Specimen.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/9c/Compression_Test_Specimen.jpg/220px-Compression_Test_Specimen.jpg" decoding="async" width="220" height="224" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/9c/Compression_Test_Specimen.jpg/330px-Compression_Test_Specimen.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/9c/Compression_Test_Specimen.jpg/440px-Compression_Test_Specimen.jpg 2x" data-file-width="587" data-file-height="599" /></a><figcaption></figcaption></figure> <p>Round test specimens made from ductile materials with a high bulk modulus, such as metals, tend to form a barrel shape under axial compressive loading due to frictional contact at the ends. For this case the equivalent true compressive stress for this condition can be calculated using<sup id="cite_ref-:1_4-1" class="reference"><a href="#cite_note-:1-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\acute {\sigma }}=C\sigma _{a}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>σ</mi> <mo>´</mo> </mover> </mrow> </mrow> <mo>=</mo> <mi>C</mi> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\acute {\sigma }}=C\sigma _{a}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ab28612acb6eb55cad5ff0be7a4992e6712297a8" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.624ex; height:2.676ex;" alt="{\displaystyle {\acute {\sigma }}=C\sigma _{a}}"></span>where </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 C={(1-2R/d_{2})\ln(1-d_{2})/(2R))}^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mo>=</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mn>2</mn> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>R</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C={(1-2R/d_{2})\ln(1-d_{2})/(2R))}^{-1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d48a01630b68dd34852dd261e13e0b6161cace88" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:36.566ex; height:3.343ex;" alt="{\displaystyle C={(1-2R/d_{2})\ln(1-d_{2})/(2R))}^{-1}}"></span></dd> <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 R=(l^{2}+(d_{2}-d_{1})^{2})/(4(d_{2}-d_{1}))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo>=</mo> <mo stretchy="false">(</mo> <msup> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mo stretchy="false">(</mo> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">(</mo> <mn>4</mn> <mo stretchy="false">(</mo> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R=(l^{2}+(d_{2}-d_{1})^{2})/(4(d_{2}-d_{1}))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d41f8764665d1838df610e4f9efadf8e67625f5d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:34.8ex; height:3.176ex;" alt="{\displaystyle R=(l^{2}+(d_{2}-d_{1})^{2})/(4(d_{2}-d_{1}))}"></span></dd> <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 _{a}=4F/(\pi d_{2}^{2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mo>=</mo> <mn>4</mn> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">(</mo> <mi>π</mi> <msubsup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{a}=4F/(\pi d_{2}^{2})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/61180da724b075b2109fc8ba5039d3f3460f5095" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:15.007ex; height:3.176ex;" alt="{\displaystyle \sigma _{a}=4F/(\pi d_{2}^{2})}"></span></dd> <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 l}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>l</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle l}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/829091f745070b9eb97a80244129025440a1cfac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.693ex; height:2.176ex;" alt="{\displaystyle l}"></span> is the loaded length of the test specimen,</dd> <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 d_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4cccb5a6a2f1acab4ca255e0be86c224ed82282a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.263ex; height:2.509ex;" alt="{\displaystyle d_{1}}"></span>is the loaded diameter of the test specimen at its ends, and</dd> <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 d_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9276f8f68c5c23329de74ad76e69f6801358fb1f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.263ex; height:2.509ex;" alt="{\displaystyle d_{2}}"></span>is the maximum loaded diameter of the test specimen.</dd></dl> <p>Note that if there is frictionless contact between the ends of the specimen and the test machine, the bulge radius becomes infinite (<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="{\textstyle R=\infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>R</mi> <mo>=</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle R=\infty }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4f494211fa70e5831ab45d85957cebd920ba56cf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.186ex; height:2.176ex;" alt="{\textstyle R=\infty }"></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="{\textstyle C=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>C</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle C=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/337497b436909438d6dda65193f8c5b4f1ac50fb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.027ex; height:2.176ex;" alt="{\textstyle C=1}"></span>.<sup id="cite_ref-:1_4-2" class="reference"><a href="#cite_note-:1-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> In this case, the formulas yield the same result 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="{\textstyle {\acute {\sigma }}=\sigma _{e}(1+\epsilon _{e})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>σ</mi> <mo>´</mo> </mover> </mrow> </mrow> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {\acute {\sigma }}=\sigma _{e}(1+\epsilon _{e})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d4f62b8fc442da9a9c22441491c5c98d88635f96" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.509ex; height:2.843ex;" alt="{\textstyle {\acute {\sigma }}=\sigma _{e}(1+\epsilon _{e})}"></span> because <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="{\textstyle \sigma _{a}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \sigma _{a}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fc15793a5e7906379c85975c9ad7b4f56ae9b624" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.429ex; height:2.009ex;" alt="{\textstyle \sigma _{a}}"></span> changes according to the ratio <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (d_{o}/d_{2})^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>o</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (d_{o}/d_{2})^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/347ce54b9a63ed6bb84dd48a40b6b7a9a01b00c9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.528ex; height:3.176ex;" alt="{\displaystyle (d_{o}/d_{2})^{2}}"></span>. </p><p>The parameters (<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="{\textstyle F,d_{1},d_{2},l}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>F</mi> <mo>,</mo> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mi>l</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle F,d_{1},d_{2},l}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b90b22b5742e593103be572808a48ccbdf7e2c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.062ex; height:2.509ex;" alt="{\textstyle F,d_{1},d_{2},l}"></span>) obtained from a test result can be used with these formulas to calculate the equivalent true 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="{\textstyle {\acute {\sigma }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>σ</mi> <mo>´</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {\acute {\sigma }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5810bb3743bc43d4287c317a8699e49b4e5a788c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:2.343ex;" alt="{\textstyle {\acute {\sigma }}}"></span> at failure. </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Compression_Test_Specimen_Shape_Effect.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/35/Compression_Test_Specimen_Shape_Effect.jpg/220px-Compression_Test_Specimen_Shape_Effect.jpg" decoding="async" width="220" height="244" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/35/Compression_Test_Specimen_Shape_Effect.jpg/330px-Compression_Test_Specimen_Shape_Effect.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/35/Compression_Test_Specimen_Shape_Effect.jpg/440px-Compression_Test_Specimen_Shape_Effect.jpg 2x" data-file-width="489" data-file-height="542" /></a><figcaption>Specimen shape effect</figcaption></figure> <p>The graph of <a href="#Compression_Test_Specimen_Shape_Effect.jpg">specimen shape effect</a> shows how the ratio of true stress to engineering stress (σ´/σ<sub>e</sub>) varies with the aspect ratio of the test specimen (<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="{\textstyle d_{o}/l_{o}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>o</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msub> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>o</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle d_{o}/l_{o}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ef5d28bfb9e2b8f5662292b46c46493c4b588cd4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.124ex; height:2.843ex;" alt="{\textstyle d_{o}/l_{o}}"></span>). The curves were calculated using the formulas provided above, based on the specific values presented in the table for <a href="#Specimen_shape_effect_calculations">specimen shape effect calculations</a>. For the curves where end restraint is applied to the specimens, they are assumed to be fully laterally restrained, meaning that the coefficient of friction at the contact points between the specimen and the testing machine is greater than or equal to one (μ ⩾ 1). As shown in the graph, as the relative length of the specimen increases (<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="{\textstyle d_{o}/l_{o}\rightarrow 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>o</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msub> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>o</mi> </mrow> </msub> <mo stretchy="false">→</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle d_{o}/l_{o}\rightarrow 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eed8152cb0c6ea8256103d59d72c40565f022f68" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.9ex; height:2.843ex;" alt="{\textstyle d_{o}/l_{o}\rightarrow 0}"></span>), the ratio of true to engineering 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 {\acute {\sigma }}/\sigma _{e}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>σ</mi> <mo>´</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\acute {\sigma }}/\sigma _{e}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9538e8470b90a976b633b83de930c526d4a11c71" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.818ex; height:2.843ex;" alt="{\displaystyle {\acute {\sigma }}/\sigma _{e}}"></span>) approaches the value corresponding to <a href="#Frictionless_contact">frictionless contact</a> between the specimen and the machine, which is the ideal test condition. </p> <table class="wikitable"> <caption>Specimen shape effect calculations </caption> <tbody><tr> <th> </th> <th>Frictionless </th> <th>Laterally Constrained </th></tr> <tr> <td>Constant volume </td> <td colspan="2"><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 \pi l_{o}d_{o}^{2}/4=\pi l(d_{2}^{2}+d_{1}^{2})/12}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> <msub> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>o</mi> </mrow> </msub> <msubsup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>o</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>4</mn> <mo>=</mo> <mi>π</mi> <mi>l</mi> <mo stretchy="false">(</mo> <msubsup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>12</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi l_{o}d_{o}^{2}/4=\pi l(d_{2}^{2}+d_{1}^{2})/12}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/31805c5029b51e93fcab24c760edf2e8bb19efb9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:25.458ex; height:3.176ex;" alt="{\displaystyle \pi l_{o}d_{o}^{2}/4=\pi l(d_{2}^{2}+d_{1}^{2})/12}"></span> </td></tr> <tr> <td>Equal diameters </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 d_{1}=d_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d_{1}=d_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/70d6765423ac7ea2f8f6d2b02c44869adde879f2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.625ex; height:2.509ex;" alt="{\displaystyle d_{1}=d_{2}}"></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 d_{o}=d_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>o</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d_{o}=d_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/007b9b6306c28669bb0b113ff5faa1cc83ea5f37" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.6ex; height:2.509ex;" alt="{\displaystyle d_{o}=d_{1}}"></span> </td></tr> <tr> <td rowspan="2">Solve for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9276f8f68c5c23329de74ad76e69f6801358fb1f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.263ex; height:2.509ex;" alt="{\displaystyle d_{2}}"></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 \pi l_{o}d_{o}^{2}/4=\pi l(d_{2}^{2}+d_{2}^{2})/12}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> <msub> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>o</mi> </mrow> </msub> <msubsup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>o</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>4</mn> <mo>=</mo> <mi>π</mi> <mi>l</mi> <mo stretchy="false">(</mo> <msubsup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>12</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi l_{o}d_{o}^{2}/4=\pi l(d_{2}^{2}+d_{2}^{2})/12}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72f9136f90dcf91ddbb2e3feb2d83de8f2290056" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:25.458ex; height:3.176ex;" alt="{\displaystyle \pi l_{o}d_{o}^{2}/4=\pi l(d_{2}^{2}+d_{2}^{2})/12}"></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 \pi l_{o}d_{o}^{2}/4=\pi l(d_{2}^{2}+d_{o}^{2})/12}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> <msub> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>o</mi> </mrow> </msub> <msubsup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>o</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>4</mn> <mo>=</mo> <mi>π</mi> <mi>l</mi> <mo stretchy="false">(</mo> <msubsup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>o</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>12</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi l_{o}d_{o}^{2}/4=\pi l(d_{2}^{2}+d_{o}^{2})/12}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5e9512f7a4d3265207fd265f671ad8e8011bc8fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:25.458ex; height:3.176ex;" alt="{\displaystyle \pi l_{o}d_{o}^{2}/4=\pi l(d_{2}^{2}+d_{o}^{2})/12}"></span> </td></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 d_{2}=d_{o}{\sqrt {l_{o}/l}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>o</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msub> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>o</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>l</mi> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d_{2}=d_{o}{\sqrt {l_{o}/l}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5103ab5bd9b987176640a0af6606cf3dd31d2daf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:13.503ex; height:4.843ex;" alt="{\displaystyle d_{2}=d_{o}{\sqrt {l_{o}/l}}}"></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 d_{2}=3d_{o}{\sqrt {(3l_{o}/l-1)/18}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mn>3</mn> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>o</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo stretchy="false">(</mo> <mn>3</mn> <msub> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>o</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>l</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>18</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d_{2}=3d_{o}{\sqrt {(3l_{o}/l-1)/18}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6c57798443153c45bf69774e08ea092d9cf1a35" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:25.127ex; height:4.843ex;" alt="{\displaystyle d_{2}=3d_{o}{\sqrt {(3l_{o}/l-1)/18}}}"></span> </td></tr> <tr> <td>Equivalent stress ratio </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 {\acute {\sigma }}/\sigma _{a}=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>σ</mi> <mo>´</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\acute {\sigma }}/\sigma _{a}=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b7e0e33fc290650f3cd1062bea7f57579ae7e9c2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.182ex; height:2.843ex;" alt="{\displaystyle {\acute {\sigma }}/\sigma _{a}=1}"></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 {\acute {\sigma }}/\sigma _{a}=C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>σ</mi> <mo>´</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mo>=</mo> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\acute {\sigma }}/\sigma _{a}=C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4f9f029ab402ddf252491d1bdb12105b5035c04f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.786ex; height:2.843ex;" alt="{\displaystyle {\acute {\sigma }}/\sigma _{a}=C}"></span> </td></tr> <tr> <td>Engineering stress </td> <td colspan="2"><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}=4F/\pi d_{o}^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msub> <mo>=</mo> <mn>4</mn> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>π</mi> <msubsup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>o</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{e}=4F/\pi d_{o}^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d762bfdd84eb0dbd1770ee7dce4cbb1c3c09abb0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.094ex; height:3.009ex;" alt="{\displaystyle \sigma _{e}=4F/\pi d_{o}^{2}}"></span> </td></tr> <tr> <td>Average stress </td> <td colspan="2"><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 _{a}=4F/\pi d_{2}^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mo>=</mo> <mn>4</mn> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>π</mi> <msubsup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{a}=4F/\pi d_{2}^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c74903ddb8eb7b77ebac8db27ce92a0ed892d825" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:13.198ex; height:3.176ex;" alt="{\displaystyle \sigma _{a}=4F/\pi d_{2}^{2}}"></span> </td></tr> <tr> <td>Average stress ratio </td> <td colspan="2"><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 _{a}/\sigma _{e}=(d_{o}/d_{2})^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msub> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>o</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{a}/\sigma _{e}=(d_{o}/d_{2})^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e9a869ebd9be1e99aa7aba72c80836f50904959a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.544ex; height:3.176ex;" alt="{\displaystyle \sigma _{a}/\sigma _{e}=(d_{o}/d_{2})^{2}}"></span> </td></tr> <tr> <td>True strain </td> <td colspan="2"><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 {\acute {\epsilon }}=\ln(l/l_{o})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ϵ</mi> <mo>´</mo> </mover> </mrow> </mrow> <mo>=</mo> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msub> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>o</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\acute {\epsilon }}=\ln(l/l_{o})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/73470211e586e56eb014b72e4c1fbf337756eedb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.717ex; height:2.843ex;" alt="{\displaystyle {\acute {\epsilon }}=\ln(l/l_{o})}"></span> </td></tr></tbody></table> <p><span class="anchor" id="Geometric_extrapolation"></span> </p> <div class="mw-heading mw-heading4"><h4 id="Geometric_extrapolation">Geometric extrapolation</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Compressive_strength&action=edit&section=6" title="Edit section: Geometric extrapolation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Illustration_of_Extrapolation.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b6/Illustration_of_Extrapolation.jpg/300px-Illustration_of_Extrapolation.jpg" decoding="async" width="300" height="303" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/b/b6/Illustration_of_Extrapolation.jpg 1.5x" data-file-width="445" data-file-height="450" /></a><figcaption></figcaption></figure><p>As shown in the section on <a href="#Correction_formulas">correction formulas</a>, as the length of test specimens is increased and their aspect ratio approaches zero (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d_{o}/l_{o}\longrightarrow 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>o</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msub> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>o</mi> </mrow> </msub> <mo stretchy="false">⟶</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d_{o}/l_{o}\longrightarrow 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/84dd8b667eabdc82bba6dfc07ef0441dfcf84b4a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.382ex; height:2.843ex;" alt="{\displaystyle d_{o}/l_{o}\longrightarrow 0}"></span>), the compressive stresses (σ) approach the true value (σ′). However, conducting tests with excessively long specimens is impractical, as they would fail by <a href="/wiki/Buckling" title="Buckling">buckling</a> before reaching the material's true compressive strength. To overcome this, a series of tests can be conducted using specimens with varying aspect ratios, and the true compressive strength can then be determined through extrapolation.<sup id="cite_ref-:0_3-5" class="reference"><a href="#cite_note-:0-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p><p><br /> </p><p><br /> <span class="anchor" id="Finite_element_analysis"></span> </p> <div class="mw-heading mw-heading4"><h4 id="Finite_element_analysis">Finite element analysis</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Compressive_strength&action=edit&section=7" title="Edit section: Finite element analysis"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1251242444"><table class="box-Expand_section plainlinks metadata ambox mbox-small-left ambox-content" role="presentation"><tbody><tr><td class="mbox-image"><span typeof="mw:File"><a href="/wiki/File:Wiki_letter_w_cropped.svg" class="mw-file-description"><img alt="[icon]" src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1c/Wiki_letter_w_cropped.svg/20px-Wiki_letter_w_cropped.svg.png" decoding="async" width="20" height="14" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/1c/Wiki_letter_w_cropped.svg/30px-Wiki_letter_w_cropped.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/1c/Wiki_letter_w_cropped.svg/40px-Wiki_letter_w_cropped.svg.png 2x" data-file-width="44" data-file-height="31" /></a></span></td><td class="mbox-text"><div class="mbox-text-span">This section <b>needs expansion</b>. You can help by <a class="external text" href="https://en.wikipedia.org/w/index.php?title=Compressive_strength&action=edit&section=">adding to it</a>. <span class="date-container"><i>(<span class="date">September 2024</span>)</i></span></div></td></tr></tbody></table> <div class="mw-heading mw-heading2"><h2 id="Comparison_of_compressive_and_tensile_strengths">Comparison of compressive and tensile strengths</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Compressive_strength&action=edit&section=8" title="Edit section: Comparison of compressive and tensile strengths"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Concrete and ceramics typically have much higher compressive strengths than tensile strengths. Composite materials, such as glass fiber epoxy matrix composite, tend to have higher tensile strengths than compressive strengths. Metals are difficult to test to failure in tension vs compression. In compression metals fail from buckling/crumbling/45° shear which is much different (though higher stresses) than tension which fails from defects or necking down. </p> <div class="mw-heading mw-heading2"><h2 id="Compressive_failure_modes">Compressive failure modes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Compressive_strength&action=edit&section=9" title="Edit section: Compressive failure modes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Universal_Testing_Machine.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/61/Universal_Testing_Machine.jpg/220px-Universal_Testing_Machine.jpg" decoding="async" width="220" height="293" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/61/Universal_Testing_Machine.jpg/330px-Universal_Testing_Machine.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/61/Universal_Testing_Machine.jpg/440px-Universal_Testing_Machine.jpg 2x" data-file-width="3024" data-file-height="4032" /></a><figcaption>A cylinder being crushed under a UTM</figcaption></figure> <p>If the ratio of the length to the effective radius of the material loaded in compression (<a href="/wiki/Slenderness_ratio" title="Slenderness ratio">Slenderness ratio</a>) is too high, it is likely that the material will fail under <a href="/wiki/Buckling" title="Buckling">buckling</a>. Otherwise, if the material is ductile yielding usually occurs which displaying the barreling effect discussed above. A brittle material in compression typically will fail by axial splitting, shear fracture, or ductile failure depending on the level of constraint in the direction perpendicular to the direction of loading. If there is no constraint (also called confining pressure), the brittle material is likely to fail by axial splitting. Moderate confining pressure often results in shear fracture, while high confining pressure often leads to ductile failure, even in brittle materials.<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>Axial Splitting relieves elastic energy in brittle material by releasing strain energy in the directions perpendicular to the applied compressive stress. As defined by a materials <a href="/wiki/Poisson%27s_ratio" title="Poisson's ratio">Poisson ratio</a> a material compressed elastically in one direction will strain in the other two directions. During axial splitting a crack may release that tensile strain by forming a new surface parallel to the applied load. The material then proceeds to separate in two or more pieces. Hence the axial splitting occurs most often when there is no confining pressure, i.e. a lesser compressive load on axis perpendicular to the main applied load.<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> The material now split into micro columns will feel different frictional forces either due to inhomogeneity of interfaces on the free end or stress shielding. In the case of <a href="/wiki/Stress_shielding" title="Stress shielding">stress shielding</a>, inhomogeneity in the materials can lead to different <a href="/wiki/Young%27s_modulus" title="Young's modulus">Young's modulus</a>. This will in turn cause the stress to be disproportionately distributed, leading to a difference in frictional forces. In either case this will cause the material sections to begin bending and lead to ultimate failure.<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-heading3"><h3 id="Microcracking">Microcracking</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Compressive_strength&action=edit&section=10" title="Edit section: Microcracking"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1251242444"><table class="box-More_citations_needed_section 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 section <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/Compressive_strength" title="Special:EditPage/Compressive strength">improve this article</a> by <a href="/wiki/Help:Referencing_for_beginners" title="Help:Referencing for beginners">adding citations to reliable sources</a> in this section. Unsourced material may be challenged and removed.</span> <span class="date-container"><i>(<span class="date">December 2021</span>)</i></span><span class="hide-when-compact"><i> (<small><a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span></div></td></tr></tbody></table> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Customhw406figure.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/ac/Customhw406figure.jpg/220px-Customhw406figure.jpg" decoding="async" width="220" height="124" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/ac/Customhw406figure.jpg/330px-Customhw406figure.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/ac/Customhw406figure.jpg/440px-Customhw406figure.jpg 2x" data-file-width="2304" data-file-height="1296" /></a><figcaption>Figure 1: microcrack nucleation and propagation</figcaption></figure><p>Microcracks are a leading cause of failure under compression for <a href="/wiki/Brittle" class="mw-redirect" title="Brittle">brittle</a> and quasi-brittle materials. Sliding along crack tips leads to tensile forces along the tip of the crack. Microcracks tend to form around any pre-existing crack tips. In all cases it is the overall global compressive stress interacting with local microstructural anomalies to create local areas of tension.  Microcracks can stem from a few factors. </p><ol><li>Porosity is the controlling factor for compressive strength in many materials. Microcracks can form around pores, until about they reach approximately the same size as their parent pores. (a)</li> <li>Stiff inclusions within a material such as a precipitate can cause localized areas of tension. (b) When inclusions are grouped up or larger, this effect can be amplified.</li> <li>Even without pores or stiff inclusions, a material can develop microcracks between weak inclined (relative to applied stress) interfaces. These interfaces can slip and create a secondary crack. These secondary cracks can continue opening, as the slip of the original interfaces keeps opening the secondary crack (c). The slipping of interfaces alone is not solely responsible for secondary crack growth as inhomogeneities in the material's <a href="/wiki/Young%27s_modulus" title="Young's modulus">Young's modulus</a> can lead to an increase in effective misfit strain. Cracks that grow this way are known as wingtip microcracks.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup></li></ol> <p>The growth of microcracks is not the growth of the original crack or imperfection. The cracks that nucleate do so perpendicular to the original crack and are known as secondary cracks.<sup id="cite_ref-auto_9-0" class="reference"><a href="#cite_note-auto-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> The figure below emphasizes this point for wingtip cracks. </p><p> These secondary cracks can grow to as long as 10-15 times the length of the original cracks in simple (uniaxial) compression. However, if a transverse compressive load is applied. The growth is limited to a few integer multiples of the original crack's length.<sup id="cite_ref-auto_9-1" class="reference"><a href="#cite_note-auto-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup></p><figure class="mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/File:Wingtipmicrocrack.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/5b/Wingtipmicrocrack.jpg/183px-Wingtipmicrocrack.jpg" decoding="async" width="183" height="147" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/5b/Wingtipmicrocrack.jpg/275px-Wingtipmicrocrack.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/5b/Wingtipmicrocrack.jpg/366px-Wingtipmicrocrack.jpg 2x" data-file-width="963" data-file-height="775" /></a><figcaption>A secondary crack growing from the tip of a preexisting crack</figcaption></figure><figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Shear_band.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/6b/Shear_band.jpg/220px-Shear_band.jpg" decoding="async" width="220" height="192" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/6b/Shear_band.jpg/330px-Shear_band.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/6b/Shear_band.jpg/440px-Shear_band.jpg 2x" data-file-width="646" data-file-height="563" /></a><figcaption>shear band formation</figcaption></figure> <div class="mw-heading mw-heading4"><h4 id="Shear_bands">Shear bands</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Compressive_strength&action=edit&section=11" title="Edit section: Shear bands"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>If the sample size is large enough such that the worse defect's secondary cracks cannot grow large enough to break the sample, other defects within the sample will begin to grow secondary cracks as well. This will occur homogeneously over the entire sample. These micro-cracks form an echelon that can form an “intrinsic” fracture behavior, the nucleus of a shear fault instability. Shown right: </p><p>Eventually this leads the material deforming non-homogeneously. That is the strain caused by the material will no longer vary linearly with the load. Creating localized <a href="/wiki/Shear_band" title="Shear band">shear bands</a> on which the material will fail according to deformation theory. “The onset of localized banding does not necessarily constitute final failure of a material element, but it presumably is at least the beginning of the primary failure process under compressive loading.”<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Typical_values">Typical values</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Compressive_strength&action=edit&section=12" title="Edit section: Typical values"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <table class="wikitable"> <tbody><tr> <th>Material </th> <th>R<sub>s</sub> (<a href="/wiki/Pascal_(unit)" title="Pascal (unit)">MPa</a>) </th></tr> <tr> <td><a href="/wiki/Steel" title="Steel">Steel</a> </td> <td>250-1,500 </td></tr> <tr> <td><a href="/wiki/Porcelain" title="Porcelain">Porcelain</a></td> <td>20-1,000<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> </td></tr> <tr> <td><a href="/wiki/Bone" title="Bone">Bone</a></td> <td>106-131<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> </td></tr> <tr> <td><a href="/wiki/Concrete" title="Concrete">Concrete</a></td> <td>17-70<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> </td></tr> <tr> <td><a href="/wiki/Ice" title="Ice">Ice</a> (−5 to −20 °C) </td> <td>5–25<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> </td></tr> <tr> <td><a href="/wiki/Ice" title="Ice">Ice</a> (0 °C)</td> <td>3<sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> </td></tr> <tr> <td><a href="/wiki/Styrofoam" title="Styrofoam">Styrofoam</a></td> <td>~1 </td></tr></tbody></table> <div class="mw-heading mw-heading2"><h2 id="Compressive_strength_of_concrete">Compressive strength of concrete</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Compressive_strength&action=edit&section=13" title="Edit section: Compressive strength of concrete"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Compressive_strength_test.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/de/Compressive_strength_test.gif/200px-Compressive_strength_test.gif" decoding="async" width="200" height="200" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/de/Compressive_strength_test.gif/300px-Compressive_strength_test.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/de/Compressive_strength_test.gif/400px-Compressive_strength_test.gif 2x" data-file-width="500" data-file-height="500" /></a><figcaption>Compressive strength test of concrete in UTM</figcaption></figure> <p>For designers, compressive strength is one of the most important engineering properties of <a href="/wiki/Concrete" title="Concrete">concrete</a>. It is standard industrial practice that the compressive strength of a given concrete mix is classified by grade. Cubic or cylindrical samples of concrete are tested under a compression testing machine to measure this value. Test requirements vary by country based on their differing design codes. Use of a <a href="/wiki/Compressometer" title="Compressometer">Compressometer</a> is common. As per Indian codes, compressive strength of concrete is defined as: </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Concrete_cube_mold.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/74/Concrete_cube_mold.jpg/220px-Concrete_cube_mold.jpg" decoding="async" width="220" height="138" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/74/Concrete_cube_mold.jpg/330px-Concrete_cube_mold.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/74/Concrete_cube_mold.jpg/440px-Concrete_cube_mold.jpg 2x" data-file-width="2842" data-file-height="1778" /></a><figcaption>Field cured concrete in cubic steel molds (Greece)</figcaption></figure> <p><i>The </i>compressive strength of concrete<i> is given in terms of the </i>characteristic compressive strength<i> of 150 mm size cubes tested after 28 days (fck). In field, compressive strength tests are also conducted at interim duration i.e. after 7 days to verify the anticipated compressive strength expected after 28 days. The same is done to be forewarned of an event of failure and take necessary precautions. The <b>characteristic strength</b> is defined as the <b>strength</b> of the <b>concrete</b> below which not more than 5% of the test results are expected to fall.</i><sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> </p><p>For design purposes, this compressive strength value is restricted by dividing with a factor of safety, whose value depends on the design philosophy used. </p><p>The construction industry is often involved in a wide array of testing. In addition to simple compression testing, testing standards such as ASTM C39, ASTM C109, ASTM C469, ASTM C1609 are among the test methods that can be followed to measure the mechanical properties of concrete. When measuring the compressive strength and other material properties of concrete, testing equipment that can be manually controlled or servo-controlled may be selected depending on the procedure followed. Certain test methods specify or limit the loading rate to a certain value or a range, whereas other methods request data based on test procedures run at very low rates.<sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> </p><p>Ultra-high performance concrete (UHPC) is defined as having a compressive strength over 150 MPa.<sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>18<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=Compressive_strength&action=edit&section=14" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Buff_strength" title="Buff strength">Buff strength</a></li> <li><a href="/wiki/Container_compression_test" title="Container compression test">Container compression test</a></li> <li><a href="/wiki/Crashworthiness" title="Crashworthiness">Crashworthiness</a></li> <li><a href="/wiki/Deformation_(engineering)" title="Deformation (engineering)">Deformation (engineering)</a></li> <li><a href="/wiki/Schmidt_hammer" title="Schmidt hammer">Schmidt hammer</a>, for measuring compressive strength of materials</li> <li><a href="/wiki/Plane_strain_compression_test" title="Plane strain compression test">Plane strain compression test</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Compressive_strength&action=edit&section=15" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><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 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.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="CITEREFUrbanekLeeJohnson" class="citation journal cs1">Urbanek, T.; Lee, S.; Johnson, C. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20140514075837/http://www.fpl.fs.fed.us/documnts/pdf2006/fpl_2006_urbanik001.pdf">"Column Compression Strength of Tubular Packaging Forms Made of Paper"</a> <span class="cs1-format">(PDF)</span>. <i>Journal of Testing and Evaluation</i>. <b>34</b> (6): 31–40. 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Retrieved <span class="nowrap">2022-09-15</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=icme.hpc.msstate.edu&rft.atitle=Multiscale+structure-property+relationships+of+ultra-high+performance+concrete+-+EVOCD&rft_id=https%3A%2F%2Ficme.hpc.msstate.edu%2Fmediawiki%2Findex.php%2FMultiscale_structure-property_relationships_of_ultra-high_performance_concrete.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACompressive+strength" class="Z3988"></span></span> </li> </ol></div></div> <ul><li>Mikell P. Groover, <i>Fundamentals of Modern Manufacturing</i>, John Wiley & Sons, 2002 U.S.A, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-471-40051-3" title="Special:BookSources/0-471-40051-3">0-471-40051-3</a></li> <li>Callister W.D. Jr., <i>Materials Science & Engineering an Introduction</i>, John Wiley & Sons, 2003 U.S.A, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-471-22471-5" title="Special:BookSources/0-471-22471-5">0-471-22471-5</a></li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol 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