Composite material

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class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Composite_materials"> <div class="vector-toc-text"> 2.1 Composite materials </div> </a> <ul id="toc-Composite_materials-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Products" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Products"> <div class="vector-toc-text"> 2.2 Products </div> </a> <ul id="toc-Products-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Overview" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Overview"> <div class="vector-toc-text"> 3 Overview </div> </a> <ul id="toc-Overview-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Cores_in_composites" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Cores_in_composites"> <div class="vector-toc-text"> 4 Cores in composites </div> </a> <button aria-controls="toc-Cores_in_composites-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Cores in composites subsection </button> <ul id="toc-Cores_in_composites-sublist" class="vector-toc-list"> <li id="toc-Semi-crystalline_polymers" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Semi-crystalline_polymers"> <div class="vector-toc-text"> 4.1 Semi-crystalline polymers </div> </a> <ul id="toc-Semi-crystalline_polymers-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Methods_of_fabrication" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Methods_of_fabrication"> <div class="vector-toc-text"> 5 Methods of fabrication </div> </a> <button aria-controls="toc-Methods_of_fabrication-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Methods of fabrication subsection </button> <ul id="toc-Methods_of_fabrication-sublist" class="vector-toc-list"> <li id="toc-Overview_of_mould" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Overview_of_mould"> <div class="vector-toc-text"> 5.1 Overview of mould </div> </a> <ul id="toc-Overview_of_mould-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Other_fabrication_methods" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Other_fabrication_methods"> <div class="vector-toc-text"> 5.2 Other fabrication methods </div> </a> <ul id="toc-Other_fabrication_methods-sublist" class="vector-toc-list"> <li id="toc-Finishing_methods" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Finishing_methods"> <div class="vector-toc-text"> 5.2.1 Finishing methods </div> </a> <ul id="toc-Finishing_methods-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Tooling" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Tooling"> <div class="vector-toc-text"> 5.3 Tooling </div> </a> <ul id="toc-Tooling-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Physical_properties" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Physical_properties"> <div class="vector-toc-text"> 6 Physical properties </div> </a> <button aria-controls="toc-Physical_properties-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Physical properties subsection </button> <ul id="toc-Physical_properties-sublist" class="vector-toc-list"> <li id="toc-Isostrain_rule_of_mixtures" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Isostrain_rule_of_mixtures"> <div class="vector-toc-text"> 6.1 Isostrain rule of mixtures </div> </a> <ul id="toc-Isostrain_rule_of_mixtures-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Isostress_rule_of_mixtures" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Isostress_rule_of_mixtures"> <div class="vector-toc-text"> 6.2 Isostress rule of mixtures </div> </a> <ul id="toc-Isostress_rule_of_mixtures-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Mechanical_properties_of_composites" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Mechanical_properties_of_composites"> <div class="vector-toc-text"> 7 Mechanical properties of composites </div> </a> <button aria-controls="toc-Mechanical_properties_of_composites-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Mechanical properties of composites subsection </button> <ul id="toc-Mechanical_properties_of_composites-sublist" class="vector-toc-list"> <li id="toc-Particle_reinforcement" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Particle_reinforcement"> <div class="vector-toc-text"> 7.1 Particle reinforcement </div> </a> <ul id="toc-Particle_reinforcement-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Continuous_fiber_reinforcement" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Continuous_fiber_reinforcement"> <div class="vector-toc-text"> 7.2 Continuous fiber reinforcement </div> </a> <ul id="toc-Continuous_fiber_reinforcement-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-The_effect_of_fiber_orientation" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#The_effect_of_fiber_orientation"> <div class="vector-toc-text"> 7.3 The effect of fiber orientation </div> </a> <ul id="toc-The_effect_of_fiber_orientation-sublist" class="vector-toc-list"> <li id="toc-Aligned_fibers" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Aligned_fibers"> <div class="vector-toc-text"> 7.3.1 Aligned fibers </div> </a> <ul id="toc-Aligned_fibers-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Randomly_oriented_fibers" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Randomly_oriented_fibers"> <div class="vector-toc-text"> 7.3.2 Randomly oriented fibers </div> </a> <ul id="toc-Randomly_oriented_fibers-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Stiffness_and_Compliance_Elasticity" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Stiffness_and_Compliance_Elasticity"> <div class="vector-toc-text"> 7.4 Stiffness and Compliance Elasticity </div> </a> <ul id="toc-Stiffness_and_Compliance_Elasticity-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Types_of_fibers_and_mechanical_properties" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Types_of_fibers_and_mechanical_properties"> <div class="vector-toc-text"> 7.5 Types of fibers and mechanical properties </div> </a> <ul id="toc-Types_of_fibers_and_mechanical_properties-sublist" class="vector-toc-list"> <li id="toc-Mechanical_properties_of_fiber_composite_materials" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Mechanical_properties_of_fiber_composite_materials"> <div class="vector-toc-text"> 7.5.1 Mechanical properties of fiber composite materials </div> </a> <ul id="toc-Mechanical_properties_of_fiber_composite_materials-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Carbon_fiber_&_fiberglass_composites_vs._aluminum_alloy_and_steel" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Carbon_fiber_&_fiberglass_composites_vs._aluminum_alloy_and_steel"> <div class="vector-toc-text"> 7.5.2 Carbon fiber & fiberglass composites vs. aluminum alloy and steel </div> </a> <ul id="toc-Carbon_fiber_&_fiberglass_composites_vs._aluminum_alloy_and_steel-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Failure" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Failure"> <div class="vector-toc-text"> 7.6 Failure </div> </a> <ul id="toc-Failure-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Testing" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Testing"> <div class="vector-toc-text"> 7.7 Testing </div> </a> <ul id="toc-Testing-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> 8 See also </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> 9 References </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Further_reading" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Further_reading"> <div class="vector-toc-text"> 10 Further reading </div> </a> <ul id="toc-Further_reading-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> 11 External links </div> </a> <ul id="toc-External_links-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" title="Table of Contents" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" > Toggle the table of contents </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading">Composite material</h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. Available in 52 languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-52" aria-hidden="true" > 52 languages </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%85%D8%A7%D8%AF%D8%A9_%D9%85%D8%A4%D9%84%D9%81%D8%A9" title="مادة مؤلفة – Arabic" lang="ar" hreflang="ar" data-title="مادة مؤلفة" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target">العربية</a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/Kompozit_material" title="Kompozit material – Azerbaijani" lang="az" hreflang="az" data-title="Kompozit material" data-language-autonym="Azərbaycanca" data-language-local-name="Azerbaijani" class="interlanguage-link-target">Azərbaycanca</a></li><li class="interlanguage-link interwiki-zh-min-nan mw-list-item"><a href="https://zh-min-nan.wikipedia.org/wiki/Ho%CC%8Dk-ha%CC%8Dp_ch%C3%A2i-li%C4%81u" title="Ho̍k-ha̍p châi-liāu – Minnan" lang="nan" hreflang="nan" data-title="Ho̍k-ha̍p châi-liāu" data-language-autonym="閩南語 / Bân-lâm-gú" data-language-local-name="Minnan" class="interlanguage-link-target">閩南語 / Bân-lâm-gú</a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%9A%D0%BE%D0%BC%D0%BF%D0%BE%D0%B7%D0%B8%D1%82%D0%B5%D0%BD_%D0%BC%D0%B0%D1%82%D0%B5%D1%80%D0%B8%D0%B0%D0%BB" title="Композитен материал – Bulgarian" lang="bg" hreflang="bg" data-title="Композитен материал" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target">Български</a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Comp%C3%B2sit" title="Compòsit – Catalan" lang="ca" hreflang="ca" data-title="Compòsit" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target">Català</a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Kompozitn%C3%AD_materi%C3%A1l" title="Kompozitní materiál – Czech" lang="cs" hreflang="cs" data-title="Kompozitní materiál" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target">Čeština</a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Defnydd_cyfansawdd" title="Defnydd cyfansawdd – Welsh" lang="cy" hreflang="cy" data-title="Defnydd cyfansawdd" data-language-autonym="Cymraeg" data-language-local-name="Welsh" class="interlanguage-link-target">Cymraeg</a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Komposit" title="Komposit – Danish" lang="da" hreflang="da" data-title="Komposit" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target">Dansk</a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Verbundwerkstoff" title="Verbundwerkstoff – German" lang="de" hreflang="de" data-title="Verbundwerkstoff" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target">Deutsch</a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Komposiitmaterjal" title="Komposiitmaterjal – Estonian" lang="et" hreflang="et" data-title="Komposiitmaterjal" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target">Eesti</a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Material_compuesto" title="Material compuesto – Spanish" lang="es" hreflang="es" data-title="Material compuesto" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target">Español</a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Kompozita_materialo" title="Kompozita materialo – Esperanto" lang="eo" hreflang="eo" data-title="Kompozita materialo" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target">Esperanto</a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Material_konposatu" title="Material konposatu – Basque" lang="eu" hreflang="eu" data-title="Material konposatu" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target">Euskara</a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%85%D8%A7%D8%AF%D9%87_%DA%A9%D8%A7%D9%85%D9%BE%D9%88%D8%B2%DB%8C%D8%AA" title="ماده کامپوزیت – Persian" lang="fa" hreflang="fa" data-title="ماده کامپوزیت" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target">فارسی</a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Mat%C3%A9riau_composite" title="Matériau composite – French" lang="fr" hreflang="fr" data-title="Matériau composite" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target">Français</a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EB%B3%B5%ED%95%A9_%EC%9E%AC%EB%A3%8C" title="복합 재료 – Korean" lang="ko" hreflang="ko" data-title="복합 재료" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target">한국어</a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%95%E0%A4%AE%E0%A5%8D%E0%A4%AA%E0%A5%8B%E0%A4%9C%E0%A4%BF%E0%A4%9F_%E0%A4%AA%E0%A4%A6%E0%A4%BE%E0%A4%B0%E0%A5%8D%E0%A4%A5" title="कम्पोजिट पदार्थ – Hindi" lang="hi" hreflang="hi" data-title="कम्पोजिट पदार्थ" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target">हिन्दी</a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Kompozit" title="Kompozit – Croatian" lang="hr" hreflang="hr" data-title="Kompozit" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target">Hrvatski</a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Material_komposit" title="Material komposit – Indonesian" lang="id" hreflang="id" data-title="Material komposit" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target">Bahasa Indonesia</a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Materiale_composito" title="Materiale composito – Italian" lang="it" hreflang="it" data-title="Materiale composito" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target">Italiano</a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%97%D7%95%D7%9E%D7%A8_%D7%9E%D7%A8%D7%95%D7%9B%D7%91" title="חומר מרוכב – Hebrew" lang="he" hreflang="he" data-title="חומר מרוכב" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target">עברית</a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%99%E1%83%9D%E1%83%9B%E1%83%9E%E1%83%9D%E1%83%96%E1%83%98%E1%83%A2%E1%83%A3%E1%83%A0%E1%83%98_%E1%83%9B%E1%83%90%E1%83%A1%E1%83%90%E1%83%9A%E1%83%90" title="კომპოზიტური მასალა – Georgian" lang="ka" hreflang="ka" data-title="კომპოზიტური მასალა" data-language-autonym="ქართული" data-language-local-name="Georgian" class="interlanguage-link-target">ქართული</a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%9A%D0%BE%D0%BC%D0%BF%D0%BE%D0%B7%D0%B8%D1%82%D1%82%D0%B5%D1%80" title="Композиттер – Kazakh" lang="kk" hreflang="kk" data-title="Композиттер" data-language-autonym="Қазақша" data-language-local-name="Kazakh" class="interlanguage-link-target">Қазақша</a></li><li class="interlanguage-link interwiki-ht mw-list-item"><a href="https://ht.wikipedia.org/wiki/Konpozit" title="Konpozit – Haitian Creole" lang="ht" hreflang="ht" data-title="Konpozit" data-language-autonym="Kreyòl ayisyen" data-language-local-name="Haitian Creole" class="interlanguage-link-target">Kreyòl ayisyen</a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Kompoz%C4%ABtmateri%C4%81li" title="Kompozītmateriāli – Latvian" lang="lv" hreflang="lv" data-title="Kompozītmateriāli" data-language-autonym="Latviešu" data-language-local-name="Latvian" class="interlanguage-link-target">Latviešu</a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Kompozitanyagok" title="Kompozitanyagok – Hungarian" lang="hu" hreflang="hu" data-title="Kompozitanyagok" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target">Magyar</a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%9A%D0%BE%D0%BC%D0%BF%D0%BE%D0%B7%D0%B8%D1%82%D0%B5%D0%BD_%D0%BC%D0%B0%D1%82%D0%B5%D1%80%D0%B8%D1%98%D0%B0%D0%BB" title="Композитен материјал – Macedonian" lang="mk" hreflang="mk" data-title="Композитен материјал" data-language-autonym="Македонски" data-language-local-name="Macedonian" class="interlanguage-link-target">Македонски</a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Bahan_komposit" title="Bahan komposit – Malay" lang="ms" hreflang="ms" data-title="Bahan komposit" data-language-autonym="Bahasa Melayu" data-language-local-name="Malay" class="interlanguage-link-target">Bahasa Melayu</a></li><li class="interlanguage-link interwiki-mn mw-list-item"><a href="https://mn.wikipedia.org/wiki/%D0%9D%D0%B8%D0%B9%D0%BB%D0%BC%D1%8D%D0%BB_%D0%BC%D0%B0%D1%82%D0%B5%D1%80%D0%B8%D0%B0%D0%BB" title="Нийлмэл материал – Mongolian" lang="mn" hreflang="mn" data-title="Нийлмэл материал" data-language-autonym="Монгол" data-language-local-name="Mongolian" class="interlanguage-link-target">Монгол</a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Composiet_(materiaal)" title="Composiet (materiaal) – Dutch" lang="nl" hreflang="nl" data-title="Composiet (materiaal)" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target">Nederlands</a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E8%A4%87%E5%90%88%E6%9D%90%E6%96%99" title="複合材料 – Japanese" lang="ja" hreflang="ja" data-title="複合材料" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target">日本語</a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Komposittmateriale" title="Komposittmateriale – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Komposittmateriale" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target">Norsk bokmål</a></li><li class="interlanguage-link interwiki-pms mw-list-item"><a href="https://pms.wikipedia.org/wiki/Material_%C3%ACbrid" title="Material ìbrid – Piedmontese" lang="pms" hreflang="pms" data-title="Material ìbrid" data-language-autonym="Piemontèis" data-language-local-name="Piedmontese" class="interlanguage-link-target">Piemontèis</a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Materia%C5%82_kompozytowy" title="Materiał kompozytowy – Polish" lang="pl" hreflang="pl" data-title="Materiał kompozytowy" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target">Polski</a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Comp%C3%B3sito" title="Compósito – Portuguese" lang="pt" hreflang="pt" data-title="Compósito" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target">Português</a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Material_compozit" title="Material compozit – Romanian" lang="ro" hreflang="ro" data-title="Material compozit" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target">Română</a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9A%D0%BE%D0%BC%D0%BF%D0%BE%D0%B7%D0%B8%D1%82%D0%BD%D1%8B%D0%B9_%D0%BC%D0%B0%D1%82%D0%B5%D1%80%D0%B8%D0%B0%D0%BB" title="Композитный материал – Russian" lang="ru" hreflang="ru" data-title="Композитный материал" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target">Русский</a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Material_kompozit" title="Material kompozit – Albanian" lang="sq" hreflang="sq" data-title="Material kompozit" data-language-autonym="Shqip" data-language-local-name="Albanian" class="interlanguage-link-target">Shqip</a></li><li class="interlanguage-link interwiki-si mw-list-item"><a href="https://si.wikipedia.org/wiki/%E0%B7%83%E0%B6%82%E0%B6%BA%E0%B7%94%E0%B6%9A%E0%B7%8A%E0%B6%AD_%E0%B6%AF%E0%B7%8A%E2%80%8D%E0%B6%BB%E0%B7%80%E0%B7%8A%E2%80%8D%E0%B6%BA%E0%B6%BA" title="සංයුක්ත ද්‍රව්‍යය – Sinhala" lang="si" hreflang="si" data-title="සංයුක්ත ද්‍රව්‍යය" data-language-autonym="සිංහල" data-language-local-name="Sinhala" class="interlanguage-link-target">සිංහල</a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Composite_material" title="Composite material – Simple English" lang="en-simple" hreflang="en-simple" data-title="Composite material" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target">Simple English</a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Kompozitn%C3%BD_materi%C3%A1l" title="Kompozitný materiál – Slovak" lang="sk" hreflang="sk" data-title="Kompozitný materiál" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target">Slovenčina</a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Kompozit" title="Kompozit – Slovenian" lang="sl" hreflang="sl" data-title="Kompozit" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target">Slovenščina</a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Komposiitti" title="Komposiitti – Finnish" lang="fi" hreflang="fi" data-title="Komposiitti" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target">Suomi</a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Komposit" title="Komposit – Swedish" lang="sv" hreflang="sv" data-title="Komposit" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target">Svenska</a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%95%E0%AF%82%E0%AE%9F%E0%AF%8D%E0%AE%9F%E0%AF%81%E0%AE%AA%E0%AF%8D_%E0%AE%AA%E0%AF%8A%E0%AE%B0%E0%AF%81%E0%AE%B3%E0%AF%8D" title="கூட்டுப் பொருள் – Tamil" lang="ta" hreflang="ta" data-title="கூட்டுப் பொருள்" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target">தமிழ்</a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%A7%E0%B8%B1%E0%B8%AA%E0%B8%94%E0%B8%B8%E0%B8%9C%E0%B8%AA%E0%B8%A1" title="วัสดุผสม – Thai" lang="th" hreflang="th" data-title="วัสดุผสม" data-language-autonym="ไทย" data-language-local-name="Thai" class="interlanguage-link-target">ไทย</a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Kompozit_malzemeler" title="Kompozit malzemeler – Turkish" lang="tr" hreflang="tr" data-title="Kompozit malzemeler" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target">Türkçe</a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%9A%D0%BE%D0%BC%D0%BF%D0%BE%D0%B7%D0%B8%D1%82%D0%B8" title="Композити – Ukrainian" lang="uk" hreflang="uk" data-title="Композити" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target">Українська</a></li><li class="interlanguage-link interwiki-ug mw-list-item"><a href="https://ug.wikipedia.org/wiki/%D8%A8%D9%89%D8%B1%D9%89%D9%83%D9%85%DB%95_%D9%85%D8%A7%D8%AA%DB%90%D8%B1%D9%89%D9%8A%D8%A7%D9%84" title="بىرىكمە ماتېرىيال – Uyghur" lang="ug" hreflang="ug" data-title="بىرىكمە ماتېرىيال" data-language-autonym="ئۇيغۇرچە / Uyghurche" data-language-local-name="Uyghur" class="interlanguage-link-target">ئۇيغۇرچە / Uyghurche</a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/V%E1%BA%ADt_li%E1%BB%87u_composite" title="Vật liệu composite – Vietnamese" lang="vi" hreflang="vi" data-title="Vật liệu composite" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" class="interlanguage-link-target">Tiếng Việt</a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E8%A4%87%E5%90%88%E6%9D%90%E6%96%99" title="複合材料 – Cantonese" lang="yue" hreflang="yue" data-title="複合材料" data-language-autonym="粵語" data-language-local-name="Cantonese" class="interlanguage-link-target">粵語</a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E5%A4%8D%E5%90%88%E6%9D%90%E6%96%99" title="复合材料 – Chinese" lang="zh" hreflang="zh" data-title="复合材料" data-language-autonym="中文" data-language-local-name="Chinese" class="interlanguage-link-target">中文</a></li> </ul> <div class="after-portlet after-portlet-lang"><a 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page hosted by Wikidata [g]" accesskey="g">Wikidata item</a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> </div> </div> </div> <div class="vector-column-end"> <div class="vector-sticky-pinned-container"> <nav class="vector-page-tools-landmark" aria-label="Page tools"> <div id="vector-page-tools-pinned-container" class="vector-pinned-container"> </div> </nav> <nav class="vector-appearance-landmark" aria-label="Appearance"> <div id="vector-appearance-pinned-container" class="vector-pinned-container"> <div id="vector-appearance" class="vector-appearance vector-pinnable-element"> <div class="vector-pinnable-header vector-appearance-pinnable-header vector-pinnable-header-pinned" data-feature-name="appearance-pinned" data-pinnable-element-id="vector-appearance" data-pinned-container-id="vector-appearance-pinned-container" data-unpinned-container-id="vector-appearance-unpinned-container" > <div class="vector-pinnable-header-label">Appearance</div> <button 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typeof="mw:File/Thumb"><a href="/wiki/File:Concrete_aggregate_grinding.JPG" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/25/Concrete_aggregate_grinding.JPG/220px-Concrete_aggregate_grinding.JPG" decoding="async" width="220" height="165" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/25/Concrete_aggregate_grinding.JPG/330px-Concrete_aggregate_grinding.JPG 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/25/Concrete_aggregate_grinding.JPG/440px-Concrete_aggregate_grinding.JPG 2x" data-file-width="1600" data-file-height="1200" /></a><figcaption>Concrete is a mixture of adhesive and aggregate, giving a robust, strong material that is very widely used.</figcaption></figure> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Spruce_plywood.JPG" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/Spruce_plywood.JPG/220px-Spruce_plywood.JPG" decoding="async" width="220" height="166" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/Spruce_plywood.JPG/330px-Spruce_plywood.JPG 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/Spruce_plywood.JPG/440px-Spruce_plywood.JPG 2x" data-file-width="935" data-file-height="705" /></a><figcaption>Plywood is used widely in construction</figcaption></figure> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Glare_honeycomb.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/84/Glare_honeycomb.jpg/220px-Glare_honeycomb.jpg" decoding="async" width="220" height="146" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/84/Glare_honeycomb.jpg/330px-Glare_honeycomb.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/84/Glare_honeycomb.jpg/440px-Glare_honeycomb.jpg 2x" data-file-width="2576" data-file-height="1713" /></a><figcaption>Composite sandwich structure panel used for testing at NASA</figcaption></figure> <style data-mw-deduplicate="TemplateStyles:r1251242444">.mw-parser-output .ambox{border:1px solid #a2a9b1;border-left:10px solid #36c;background-color:#fbfbfb;box-sizing:border-box}.mw-parser-output .ambox+link+.ambox,.mw-parser-output .ambox+link+style+.ambox,.mw-parser-output .ambox+link+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+style+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+link+.ambox{margin-top:-1px}html body.mediawiki .mw-parser-output .ambox.mbox-small-left{margin:4px 1em 4px 0;overflow:hidden;width:238px;border-collapse:collapse;font-size:88%;line-height:1.25em}.mw-parser-output .ambox-speedy{border-left:10px solid #b32424;background-color:#fee7e6}.mw-parser-output .ambox-delete{border-left:10px solid #b32424}.mw-parser-output .ambox-content{border-left:10px solid #f28500}.mw-parser-output .ambox-style{border-left:10px solid #fc3}.mw-parser-output .ambox-move{border-left:10px solid #9932cc}.mw-parser-output .ambox-protection{border-left:10px solid #a2a9b1}.mw-parser-output .ambox .mbox-text{border:none;padding:0.25em 0.5em;width:100%}.mw-parser-output .ambox .mbox-image{border:none;padding:2px 0 2px 0.5em;text-align:center}.mw-parser-output .ambox .mbox-imageright{border:none;padding:2px 0.5em 2px 0;text-align:center}.mw-parser-output .ambox .mbox-empty-cell{border:none;padding:0;width:1px}.mw-parser-output .ambox .mbox-image-div{width:52px}@media(min-width:720px){.mw-parser-output .ambox{margin:0 10%}}@media print{body.ns-0 .mw-parser-output .ambox{display:none!important}}</style><table class="box-Split plainlinks metadata ambox ambox-move" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a7/Split-arrows.svg/50px-Split-arrows.svg.png" decoding="async" width="50" height="17" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a7/Split-arrows.svg/75px-Split-arrows.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a7/Split-arrows.svg/100px-Split-arrows.svg.png 2x" data-file-width="60" data-file-height="20" /></div></td><td class="mbox-text"><div class="mbox-text-span">It has been suggested that this article be <a href="/wiki/Wikipedia:Splitting" title="Wikipedia:Splitting">split</a> into articles titled <a href="/wiki/Autoclave_moulding" title="Autoclave moulding">Autoclave moulding</a>, <a href="/wiki/Resin_transfer_moulding" title="Resin transfer moulding">Resin transfer moulding</a>, <a href="/wiki/Pressure_bag_moulding" title="Pressure bag moulding">Pressure bag moulding</a> and <a href="/wiki/Light_resin_transfer_moulding" title="Light resin transfer moulding">Light resin transfer moulding</a>. (<a href="/wiki/Talk:Composite_material" title="Talk:Composite material">Discuss</a>) (November 2020)</div></td></tr></tbody></table> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Cfaser_haarrp.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/71/Cfaser_haarrp.jpg/220px-Cfaser_haarrp.jpg" decoding="async" width="220" height="124" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/71/Cfaser_haarrp.jpg/330px-Cfaser_haarrp.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/71/Cfaser_haarrp.jpg/440px-Cfaser_haarrp.jpg 2x" data-file-width="849" data-file-height="478" /></a><figcaption>A black <a href="/wiki/Carbon_fibre" class="mw-redirect" title="Carbon fibre">carbon fibre</a> (used as a reinforcement component) compared to a <a href="/wiki/Human_hair" class="mw-redirect" title="Human hair">human hair</a></figcaption></figure><figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Composite_3d.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/13/Composite_3d.png/220px-Composite_3d.png" decoding="async" width="220" height="165" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/13/Composite_3d.png/330px-Composite_3d.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/13/Composite_3d.png/440px-Composite_3d.png 2x" data-file-width="800" data-file-height="600" /></a><figcaption>Composites are formed by combining materials together to form an overall structure with properties that differ from that of the individual components</figcaption></figure> A composite or composite material (also composition material) is a <a href="/wiki/Material" title="Material">material</a> which is produced from two or more constituent materials.<a href="#cite_note-1">[1]</a> These constituent materials have notably dissimilar chemical or physical properties and are merged to create a material with properties unlike the individual elements. Within the finished structure, the individual elements remain separate and distinct, distinguishing composites from <a href="/wiki/Mixture" title="Mixture">mixtures</a> and <a href="/wiki/Solid_solution" title="Solid solution">solid solutions</a>. Composite materials with more than one distinct layer are called <a href="/wiki/Composite_laminate" title="Composite laminate">composite laminates</a>. Typical engineered composite <a href="/wiki/Materials" class="mw-redirect" title="Materials">materials</a> are made up of a <a href="/wiki/Binding_agent" class="mw-redirect" title="Binding agent">binding agent</a> forming the matrix and a <a href="/wiki/Filler_(materials)" title="Filler (materials)">filler material</a> (<a href="/wiki/Particulate" class="mw-redirect" title="Particulate">particulates</a> or <a href="/wiki/Fibre" class="mw-redirect" title="Fibre">fibres</a>) giving substance, e.g.: <ul><li><a href="/wiki/Concrete" title="Concrete">Concrete</a>, <a href="/wiki/Reinforced_concrete" title="Reinforced concrete">reinforced concrete</a> and <a href="/wiki/Masonry" title="Masonry">masonry</a> with cement, lime or <a href="/wiki/Mortar_(masonry)" title="Mortar (masonry)">mortar</a> (which is itself a composite material) as a binder</li> <li><a href="/wiki/Composite_wood" class="mw-redirect" title="Composite wood">Composite wood</a> such as <a href="/wiki/Glulam" class="mw-redirect" title="Glulam">glulam</a> and <a href="/wiki/Plywood" title="Plywood">plywood</a> with <a href="/wiki/Wood_glue" title="Wood glue">wood glue</a> as a binder</li> <li><a href="/wiki/Reinforced_plastic" class="mw-redirect" title="Reinforced plastic">Reinforced plastics</a>, such as <a href="/wiki/Fiberglass" title="Fiberglass">fiberglass</a> and <a href="/wiki/Fibre-reinforced_polymer" class="mw-redirect" title="Fibre-reinforced polymer">fibre-reinforced polymer</a> with <a href="/wiki/Resin" title="Resin">resin</a> or <a href="/wiki/Thermoplastic" title="Thermoplastic">thermoplastics</a> as a binder</li> <li><a href="/wiki/Ceramic_matrix_composite" title="Ceramic matrix composite">Ceramic matrix composites</a> (<a href="/wiki/Composite_armor" class="mw-redirect" title="Composite armor">composite ceramic and metal matrices</a>)</li> <li><a href="/wiki/Metal_matrix_composite" title="Metal matrix composite">Metal matrix composites</a><a href="#cite_note-2">[2]</a></li> <li><a href="/wiki/Advanced_composite_materials_(engineering)" title="Advanced composite materials (engineering)">advanced composite materials</a>, often first developed for <a href="/wiki/Spacecraft" title="Spacecraft">spacecraft</a> and <a href="/wiki/Aircraft" title="Aircraft">aircraft</a> applications.</li></ul> Composite materials can be less expensive, lighter, stronger or more durable than common materials. Some are inspired by biological structures found in plants and animals.<a href="#cite_note-3">[3]</a> <a href="/wiki/Robotic_materials" title="Robotic materials">Robotic materials</a> are composites that include sensing, actuation, computation, and communication components.<a href="#cite_note-4">[4]</a><a href="#cite_note-5">[5]</a> Composite materials are used for <a href="/wiki/Construction" title="Construction">construction</a> and technical <a href="/wiki/Structure" title="Structure">structures</a> such as <a href="/wiki/Boat_hulls" class="mw-redirect" title="Boat hulls">boat hulls</a>, <a href="/wiki/Swimming_pool" title="Swimming pool">swimming pool</a> panels, <a href="/wiki/Racing_car" class="mw-redirect" title="Racing car">racing car</a> bodies, <a href="/wiki/Shower" title="Shower">shower</a> stalls, <a href="/wiki/Bathtub" title="Bathtub">bathtubs</a>, <a href="/wiki/Storage_tank" title="Storage tank">storage tanks</a>, <a href="/wiki/Imitation" title="Imitation">imitation</a> <a href="/wiki/Granite" title="Granite">granite</a>, and <a href="/wiki/Cultured_marble" class="mw-redirect" title="Cultured marble">cultured marble</a> <a href="/wiki/Sink" title="Sink">sinks</a> and countertops.<a href="#cite_note-6">[6]</a><a href="#cite_note-7">[7]</a> They are also being increasingly used in general automotive applications.<a href="#cite_note-8">[8]</a> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="History">History</h2>[<a href="/w/index.php?title=Composite_material&action=edit&section=1" title="Edit section: History">edit</a>]</div> The earliest composite materials were made from <a href="/wiki/Straw" title="Straw">straw</a> and <a href="/wiki/Mud" title="Mud">mud</a> combined to form <a href="/wiki/Brick" title="Brick">bricks</a> for <a href="/wiki/Building" title="Building">building</a> <a href="/wiki/Construction" title="Construction">construction</a>. Ancient <a href="/wiki/Brick#Mud_bricks" title="Brick">brick-making</a> was documented by <a href="/wiki/Art_of_Ancient_Egypt#Painting" class="mw-redirect" title="Art of Ancient Egypt">Egyptian tomb paintings</a>.<a href="#cite_note-Haka-9">[9]</a> <a href="/wiki/Wattle_and_daub" title="Wattle and daub">Wattle and daub</a> might be the oldest composite materials, at over 6000 years old.<a href="#cite_note-Shaffer-10">[10]</a> <ul><li>Woody <a href="/wiki/Plant" title="Plant">plants</a>, both true <a href="/wiki/Wood" title="Wood">wood</a> from <a href="/wiki/Trees" class="mw-redirect" title="Trees">trees</a> and such plants as <a href="/wiki/Palm_(plant)" class="mw-redirect" title="Palm (plant)">palms</a> and <a href="/wiki/Bamboo" title="Bamboo">bamboo</a>, yield natural composites that were used prehistorically by humankind and are still used widely in construction and scaffolding.</li> <li><a href="/wiki/Plywood" title="Plywood">Plywood</a>, 3400 BC,<a href="#cite_note-auto-11">[11]</a> by the Ancient Mesopotamians; gluing wood at different angles gives better properties than natural wood.</li> <li><a href="/wiki/Cartonnage" title="Cartonnage">Cartonnage</a>, layers of <a href="/wiki/Linen" title="Linen">linen</a> or <a href="/wiki/Papyrus" title="Papyrus">papyrus</a> soaked in plaster dates to the <a href="/wiki/First_Intermediate_Period_of_Egypt" title="First Intermediate Period of Egypt">First Intermediate Period of Egypt</a> c. 2181–2055 BC<a href="#cite_note-auto-11">[11]</a> and was used for <a href="/wiki/Death_mask" title="Death mask">death masks</a>.</li> <li><a href="/wiki/Cob_(material)" title="Cob (material)">Cob</a> mud bricks, or mud walls, (using mud (clay) with straw or gravel as a binder) have been used for thousands of years.<a href="#cite_note-12">[12]</a></li> <li><a href="/wiki/Concrete" title="Concrete">Concrete</a> was described by <a href="/wiki/Vitruvius" title="Vitruvius">Vitruvius</a>, writing around 25 BC in his <a href="/wiki/De_architectura" title="De architectura">Ten Books on Architecture</a>, distinguished types of aggregate appropriate for the preparation of <a href="/wiki/Lime_mortar" title="Lime mortar">lime mortars</a>. For structural mortars, he recommended <a href="/wiki/Pozzolana" title="Pozzolana">pozzolana</a>, which were volcanic sands from the sandlike beds of <a href="/wiki/Pozzuoli" title="Pozzuoli">Pozzuoli</a> brownish-yellow-gray in colour near <a href="/wiki/Naples" title="Naples">Naples</a> and reddish-brown at <a href="/wiki/Rome" title="Rome">Rome</a>. Vitruvius specifies a ratio of 1 part lime to 3 parts pozzolana for cements used in buildings and a 1:2 ratio of lime to pulvis Puteolanus for underwater work, essentially the same ratio mixed today for concrete used at sea.<a href="#cite_note-13">[13]</a> <a href="/wiki/Cement" title="Cement">Natural cement-stones</a>, after burning, produced cements used in concretes from post-Roman times into the 20th century, with some properties superior to manufactured <a href="/wiki/Portland_cement" title="Portland cement">Portland cement</a>.</li> <li><a href="/wiki/Papier-m%C3%A2ch%C3%A9" title="Papier-mâché">Papier-mâché</a>, a composite of paper and glue, has been used for hundreds of years.<a href="#cite_note-14">[14]</a></li> <li>The first artificial <a href="/wiki/Fibre_reinforced_plastic" class="mw-redirect" title="Fibre reinforced plastic">fibre reinforced plastic</a> was a combination of fiber glass and <a href="/wiki/Bakelite" title="Bakelite">bakelite</a>, performed in 1935 by Al Simison and Arthur D Little in Owens Corning Company<a href="#cite_note-15">[15]</a></li> <li>One of the most common and familiar composite is <a href="/wiki/Fibreglass" class="mw-redirect" title="Fibreglass">fibreglass</a>, in which small glass fibre are embedded within a polymeric material (normally an epoxy or polyester). The glass fibre is relatively strong and stiff (but also brittle), whereas the polymer is ductile (but also weak and flexible). Thus the resulting fibreglass is relatively stiff, strong, flexible, and ductile.<a href="#cite_note-16">[16]</a></li> <li><a href="/wiki/Composite_bow" title="Composite bow">Composite bow</a></li> <li><a href="/wiki/Leather_cannon" title="Leather cannon">Leather cannon</a>, <a href="/wiki/Wooden_cannon" title="Wooden cannon">wooden cannon</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Examples">Examples</h2>[<a href="/w/index.php?title=Composite_material&action=edit&section=2" title="Edit section: Examples">edit</a>]</div> <div class="mw-heading mw-heading3"><h3 id="Composite_materials">Composite materials</h3>[<a href="/w/index.php?title=Composite_material&action=edit&section=3" title="Edit section: Composite materials">edit</a>]</div> <a href="/wiki/Concrete" title="Concrete">Concrete</a> is the most common artificial composite material of all. As of 2009<a class="external text" href="https://en.wikipedia.org/w/index.php?title=Composite_material&action=edit">[update]</a>, about 7.5 billion cubic metres of concrete are made each year.<a href="#cite_note-17">[17]</a> Concrete typically consists of loose stones (<a href="/wiki/Construction_aggregate" title="Construction aggregate">construction aggregate</a>) held with a matrix of <a href="/wiki/Cement" title="Cement">cement</a>. Concrete is an inexpensive material resisting large compressive forces,<a href="#cite_note-18">[18]</a> however, susceptible to <a href="/wiki/Tensile_load" class="mw-redirect" title="Tensile load">tensile loading</a>.<a href="#cite_note-19">[19]</a> To give concrete the ability to resist being stretched, steel bars, which can resist high stretching (tensile) forces, are often added to concrete to form <a href="/wiki/Reinforced_concrete" title="Reinforced concrete">reinforced concrete</a>.<a href="#cite_note-20">[20]</a> <a href="/wiki/Fibre-reinforced_plastic" title="Fibre-reinforced plastic">Fibre-reinforced polymers</a> include <a href="/wiki/Carbon-fiber-reinforced_polymers" class="mw-redirect" title="Carbon-fiber-reinforced polymers">carbon-fiber-reinforced polymers</a> and <a href="/wiki/Glass-reinforced_plastic" class="mw-redirect" title="Glass-reinforced plastic">glass-reinforced plastic</a>. If classified by matrix then there are <a href="/wiki/Thermoplastic_composites" class="mw-redirect" title="Thermoplastic composites">thermoplastic composites</a>, <a href="/wiki/Short_fiber_thermoplastics" title="Short fiber thermoplastics">short fibre thermoplastics</a>, <a href="/wiki/Long_fibre_thermoplastic" class="mw-redirect" title="Long fibre thermoplastic">long fibre thermoplastics</a> or <a href="/wiki/Long-fiber-reinforced_thermoplastic" title="Long-fiber-reinforced thermoplastic">long-fiber-reinforced thermoplastics</a>. There are numerous <a href="/wiki/Thermoset" class="mw-redirect" title="Thermoset">thermoset</a> composites, including <a href="/wiki/Paper_composite_panels" title="Paper composite panels">paper composite panels</a>. Many advanced <a href="/wiki/Thermoset_polymer_matrix" title="Thermoset polymer matrix">thermoset polymer matrix</a> systems usually incorporate <a href="/wiki/Aramid" title="Aramid">aramid</a> <a href="/wiki/Fibre" class="mw-redirect" title="Fibre">fibre</a> and <a href="/wiki/Carbon_fibre" class="mw-redirect" title="Carbon fibre">carbon fibre</a> in an <a href="/wiki/Epoxy_resin" class="mw-redirect" title="Epoxy resin">epoxy resin</a> matrix.<a href="#cite_note-21">[21]</a><a href="#cite_note-22">[22]</a> <a href="/wiki/Shape-memory_polymer" title="Shape-memory polymer">Shape-memory polymer</a> composites are high-performance composites, formulated using fibre or fabric reinforcements and shape-memory polymer resin as the matrix. Since a shape-memory polymer resin is used as the matrix, these composites have the ability to be easily manipulated into various configurations when they are heated above their activation temperatures and will exhibit high strength and stiffness at lower temperatures. They can also be reheated and reshaped repeatedly without losing their material properties. These composites are ideal for applications such as lightweight, rigid, deployable structures; rapid manufacturing; and dynamic reinforcement.<a href="#cite_note-23">[23]</a><a href="#cite_note-24">[24]</a> <a href="/wiki/High_strain_composite_structure" title="High strain composite structure">High strain composites</a> are another type of high-performance composites that are designed to perform in a high deformation setting and are often used in deployable systems where structural flexing is advantageous.[<a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed">citation needed</a>] Although high strain composites exhibit many similarities to shape-memory polymers, their performance is generally dependent on the fibre layout as opposed to the resin content of the matrix.<a href="#cite_note-25">[25]</a> Composites can also use metal fibres reinforcing other metals, as in <a href="/wiki/Metal_matrix_composite" title="Metal matrix composite">metal matrix composites</a> (MMC)<a href="#cite_note-26">[26]</a> or <a href="/wiki/Ceramic_matrix_composite" title="Ceramic matrix composite">ceramic matrix composites</a> (CMC),<a href="#cite_note-27">[27]</a> which includes <a href="/wiki/Bone_mineral" title="Bone mineral">bone</a> (<a href="/wiki/Hydroxyapatite" title="Hydroxyapatite">hydroxyapatite</a> reinforced with <a href="/wiki/Collagen" title="Collagen">collagen</a> fibres), <a href="/wiki/Cermet" title="Cermet">cermet</a> (ceramic and metal), and <a href="/wiki/Concrete" title="Concrete">concrete</a>. Ceramic matrix composites are built primarily for <a href="/wiki/Fracture_toughness" title="Fracture toughness">fracture toughness</a>, not for strength. Another class of composite materials involve woven fabric composite consisting of longitudinal and transverse laced yarns. Woven fabric composites are flexible as they are in form of fabric. Organic matrix/ceramic aggregate composites include <a href="/wiki/Asphalt_concrete" title="Asphalt concrete">asphalt concrete</a>, <a href="/wiki/Polymer_concrete" title="Polymer concrete">polymer concrete</a>, <a href="/wiki/Mastic_asphalt" class="mw-redirect" title="Mastic asphalt">mastic asphalt</a>, mastic roller hybrid, <a href="/wiki/Dental_composite" title="Dental composite">dental composite</a>, <a href="/wiki/Syntactic_foam" title="Syntactic foam">syntactic foam</a>, and <a href="/wiki/Nacre" title="Nacre">mother of pearl</a>.<a href="#cite_note-28">[28]</a> <a href="/wiki/Chobham_armour" title="Chobham armour">Chobham armour</a> is a special type of <a href="/wiki/Composite_armour" title="Composite armour">composite armour</a> used in military applications.[<a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed">citation needed</a>] Additionally, <a href="/wiki/Thermoplastic" title="Thermoplastic">thermoplastic</a> composite materials can be formulated with specific metal powders resulting in materials with a density range from 2 g/cm3 to 11 g/cm3 (same density as lead). The most common name for this type of material is "high gravity compound" (HGC), although "lead replacement" is also used. These materials can be used in place of traditional materials such as aluminium, stainless steel, brass, bronze, copper, lead, and even tungsten in weighting, balancing (for example, modifying the centre of gravity of a tennis <a href="/wiki/Racquet" class="mw-redirect" title="Racquet">racquet</a>), vibration damping, and radiation shielding applications. High density composites are an economically viable option when certain materials are deemed hazardous and are banned (such as lead) or when secondary operations costs (such as machining, finishing, or coating) are a factor.<a href="#cite_note-29">[29]</a> There have been several studies indicating that interleaving stiff and brittle epoxy-based <a href="/wiki/Carbon-fiber-reinforced_polymers" class="mw-redirect" title="Carbon-fiber-reinforced polymers">carbon-fiber-reinforced polymer</a> laminates with flexible thermoplastic laminates can help to make highly toughened composites that show improved impact resistance.<a href="#cite_note-30">[30]</a> Another interesting aspect of such interleaved composites is that they are able to have shape memory behaviour without needing any <a href="/wiki/Shape-memory_polymer" title="Shape-memory polymer">shape-memory polymers</a> or <a href="/wiki/Shape-memory_alloy" title="Shape-memory alloy">shape-memory alloys</a> e.g. balsa plies interleaved with hot glue,<a href="#cite_note-31">[31]</a> aluminium plies interleaved with <a href="/wiki/Acrylate_polymer" title="Acrylate polymer">acrylic polymers</a> or <a href="/wiki/Polyvinyl_chloride" title="Polyvinyl chloride">PVC</a><a href="#cite_note-32">[32]</a> and <a href="/wiki/Carbon-fiber-reinforced_polymers" class="mw-redirect" title="Carbon-fiber-reinforced polymers">carbon-fiber-reinforced polymer</a> laminates interleaved with <a href="/wiki/Polystyrene" title="Polystyrene">polystyrene</a>.<a href="#cite_note-33">[33]</a> A <a href="/wiki/Sandwich-structured_composite" title="Sandwich-structured composite">sandwich-structured composite</a> is a special class of composite material that is fabricated by attaching two thin but stiff skins to a lightweight but thick core. The core material is normally low strength material, but its higher thickness provides the sandwich composite with high <a href="/wiki/Bending" title="Bending">bending</a> <a href="/wiki/Stiffness" title="Stiffness">stiffness</a> with overall low <a href="/wiki/Density" title="Density">density</a>.<a href="#cite_note-34">[34]</a><a href="#cite_note-35">[35]</a> Wood is a naturally occurring composite comprising cellulose fibres in a <a href="/wiki/Lignin" title="Lignin">lignin</a> and <a href="/wiki/Hemicellulose" title="Hemicellulose">hemicellulose</a> matrix.<a href="#cite_note-36">[36]</a> <a href="/wiki/Engineered_wood" title="Engineered wood">Engineered wood</a> includes a wide variety of different products such as wood fibre board, <a href="/wiki/Plywood" title="Plywood">plywood</a>, <a href="/wiki/Oriented_strand_board" title="Oriented strand board">oriented strand board</a>, <a href="/wiki/Wood_plastic_composite" class="mw-redirect" title="Wood plastic composite">wood plastic composite</a> (recycled wood fibre in polyethylene matrix), <a href="/wiki/Pykrete" title="Pykrete">Pykrete</a> (sawdust in ice matrix), plastic-impregnated or <a href="/wiki/Plastic-coated_paper" class="mw-redirect" title="Plastic-coated paper">laminated paper</a> or textiles, <a href="/wiki/Arborite" title="Arborite">Arborite</a>, <a href="/wiki/Formica_(plastic)" title="Formica (plastic)">Formica (plastic)</a>, and <a href="/wiki/Micarta" title="Micarta">Micarta</a>. Other engineered laminate composites, such as <a href="/wiki/Mallite" title="Mallite">Mallite</a>, use a central core of end grain <a href="/wiki/Balsa_wood" class="mw-redirect" title="Balsa wood">balsa wood</a>, bonded to surface skins of light <a href="/wiki/Alloy" title="Alloy">alloy</a> or GRP. These generate low-weight, high rigidity materials.<a href="#cite_note-37">[37]</a> Particulate composites have particle as filler material dispersed in matrix, which may be nonmetal, such as glass, epoxy. Automobile tire is an example of particulate composite.<a href="#cite_note-38">[38]</a> Advanced diamond-like carbon (DLC) coated polymer composites have been reported<a href="#cite_note-ZiaShah2015-39">[39]</a> where the coating increases the surface hydrophobicity, hardness and wear resistance. Ferromagnetic composites, including those with a polymer matrix consisting, for example, of nanocrystalline filler of Fe-based powders and polymers matrix. Amorphous and nanocrystalline powders obtained, for example, from metallic glasses can be used. Their use makes it possible to obtain ferromagnetic nanocomposites with controlled magnetic properties.<a href="#cite_note-40">[40]</a> <div class="mw-heading mw-heading3"><h3 id="Products">Products</h3>[<a href="/w/index.php?title=Composite_material&action=edit&section=4" title="Edit section: Products">edit</a>]</div> Fibre-reinforced composite materials have gained popularity (despite their generally high cost) in high-performance products that need to be lightweight, yet strong enough to take harsh loading conditions such as <a href="/wiki/Aerospace" title="Aerospace">aerospace</a> components (<a href="/wiki/Empennage" title="Empennage">tails</a>, <a href="/wiki/Wing" title="Wing">wings</a>, <a href="/wiki/Fuselage" title="Fuselage">fuselages</a>, <a href="/wiki/Propeller_(aircraft)" class="mw-redirect" title="Propeller (aircraft)">propellers</a>), boat and <a href="/wiki/Scull" class="mw-redirect" title="Scull">scull</a> hulls, <a href="/wiki/Bicycle" title="Bicycle">bicycle</a> frames, and <a href="/wiki/Racing_car" class="mw-redirect" title="Racing car">racing car</a> bodies. Other uses include <a href="/wiki/Fishing_rod" title="Fishing rod">fishing rods</a>, <a href="/wiki/Storage_tank" title="Storage tank">storage tanks</a>, swimming pool panels, and <a href="/wiki/Composite_baseball_bat" title="Composite baseball bat">baseball bats</a>. The <a href="/wiki/Boeing_787" class="mw-redirect" title="Boeing 787">Boeing 787</a> and <a href="/wiki/Airbus_A350" title="Airbus A350">Airbus A350</a> structures including the wings and fuselage are composed largely of composites.<a href="#cite_note-41">[41]</a> Composite materials are also becoming more common in the realm of <a href="/wiki/Orthopedic_surgery" title="Orthopedic surgery">orthopedic surgery</a>,<a href="#cite_note-42">[42]</a> and it is the most common hockey stick material. Carbon composite is a key material in today's launch vehicles and <a href="/wiki/Heat_shield" title="Heat shield">heat shields</a> for the <a href="/wiki/Re-entry" class="mw-redirect" title="Re-entry">re-entry</a> phase of <a href="/wiki/Spacecraft" title="Spacecraft">spacecraft</a>. It is widely used in solar panel substrates, antenna reflectors and yokes of spacecraft. It is also used in payload adapters, inter-stage structures and heat shields of <a href="/wiki/Launch_vehicle" title="Launch vehicle">launch vehicles</a>. Furthermore, <a href="/wiki/Disk_brake" class="mw-redirect" title="Disk brake">disk brake</a> systems of <a href="/wiki/Airplane" title="Airplane">airplanes</a> and racing cars are using <a href="/wiki/Carbon/carbon" class="mw-redirect" title="Carbon/carbon">carbon/carbon</a> material, and the <a href="/wiki/Ceramic_matrix_composite" title="Ceramic matrix composite">composite material</a> with <a href="/wiki/Carbon_fibre" class="mw-redirect" title="Carbon fibre">carbon fibres</a> and <a href="/wiki/Silicon_carbide" title="Silicon carbide">silicon carbide</a> matrix has been introduced in <a href="/wiki/Luxury_vehicle" class="mw-redirect" title="Luxury vehicle">luxury vehicles</a> and <a href="/wiki/Sports_car" title="Sports car">sports cars</a>. In 2006, a fibre-reinforced composite pool panel was introduced for in-ground swimming pools, residential as well as commercial, as a non-corrosive alternative to galvanized steel. In 2007, an all-composite military <a href="/wiki/Humvee" title="Humvee">Humvee</a> was introduced by TPI Composites Inc and Armor Holdings Inc, the first all-composite <a href="/wiki/Military_vehicle" title="Military vehicle">military vehicle</a>. By using composites the vehicle is lighter, allowing higher payloads.<a href="#cite_note-43">[43]</a> In 2008, carbon fibre and <a href="/wiki/DuPont" title="DuPont">DuPont</a> Kevlar (five times stronger than steel) were combined with enhanced thermoset resins to make military transit cases by ECS Composites creating 30-percent lighter cases with high strength. Pipes and fittings for various purpose like transportation of potable water, fire-fighting, irrigation, seawater, desalinated water, chemical and industrial waste, and sewage are now manufactured in glass reinforced plastics. Composite materials used in tensile structures for facade application provides the advantage of being translucent. The woven base cloth combined with the appropriate coating allows better light transmission. This provides a very comfortable level of illumination compared to the full brightness of outside.<a href="#cite_note-44">[44]</a> The wings of wind turbines, in growing sizes in the order of 50 m length are fabricated in composites since several years.<a href="#cite_note-45">[45]</a> Two-lower-leg-amputees run on carbon-composite spring-like artificial feet as quick as non-amputee athletes.<a href="#cite_note-46">[46]</a> High-pressure gas cylinders typically about 7–9 litre volume x 300 bar pressure for firemen are nowadays constructed from carbon composite. <a href="/w/index.php?title=Type-4_gas_cylinder&action=edit&redlink=1" class="new" title="Type-4 gas cylinder (page does not exist)">Type-4-cylinders</a> include metal only as boss that carries the thread to screw in the valve. On 5 September 2019, <a href="/wiki/HMD_Global" title="HMD Global">HMD Global</a> unveiled the <a href="/wiki/Nokia_6.2" title="Nokia 6.2">Nokia 6.2</a> and <a href="/wiki/Nokia_7.2" title="Nokia 7.2">Nokia 7.2</a> which are claimed to be using polymer composite for the frames.<a href="#cite_note-47">[47]</a> <div class="mw-heading mw-heading2"><h2 id="Overview">Overview</h2>[<a href="/w/index.php?title=Composite_material&action=edit&section=5" title="Edit section: Overview">edit</a>]</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Cfk_heli_slw.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ee/Cfk_heli_slw.jpg/170px-Cfk_heli_slw.jpg" decoding="async" width="170" height="440" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ee/Cfk_heli_slw.jpg/255px-Cfk_heli_slw.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ee/Cfk_heli_slw.jpg/340px-Cfk_heli_slw.jpg 2x" data-file-width="536" data-file-height="1388" /></a><figcaption>Carbon fibre composite part.</figcaption></figure> Composite materials are created from individual materials. These individual materials are known as constituent materials, and there are two main categories of it. One is the <a href="/wiki/Matrix_(composite)" title="Matrix (composite)">matrix</a> (<a href="/wiki/Binder_(material)" title="Binder (material)">binder</a>) and the other <a href="/wiki/Reinforcement_(composite)" title="Reinforcement (composite)">reinforcement</a>.<a href="#cite_note-48">[48]</a> A portion of each kind is needed at least. The reinforcement receives support from the matrix as the matrix surrounds the reinforcement and maintains its relative positions. The properties of the matrix are improved as the reinforcements impart their exceptional physical and mechanical properties. The mechanical properties become unavailable from the individual constituent materials by synergism. At the same time, the designer of the product or structure receives options to choose an optimum combination from the variety of matrix and strengthening materials. To shape the engineered composites, it must be formed. The reinforcement is placed onto the mould surface or into the <a href="/wiki/Molding_(process)" title="Molding (process)">mould</a> cavity. Before or after this, the matrix can be introduced to the reinforcement. The matrix undergoes a melding event which sets the part shape necessarily. This melding event can happen in several ways, depending upon the matrix nature, such as solidification from the melted state for a thermoplastic polymer matrix composite or chemical <a href="/wiki/Polymerization" title="Polymerization">polymerization</a> for a <a href="/wiki/Thermoset_polymer_matrix" title="Thermoset polymer matrix">thermoset polymer matrix</a>. According to the requirements of end-item design, various methods of moulding can be used. The natures of the chosen matrix and reinforcement are the key factors influencing the methodology. The gross quantity of material to be made is another main factor. To support high capital investments for rapid and automated manufacturing technology, vast quantities can be used. Cheaper capital investments but higher labour and tooling expenses at a correspondingly slower rate assists the small production quantities. Many commercially produced composites use a <a href="/wiki/Polymer" title="Polymer">polymer</a> matrix material often called a resin solution. There are many different polymers available depending upon the starting raw ingredients. There are several broad categories, each with numerous variations. The most common are known as <a href="/wiki/Polyester" title="Polyester">polyester</a>, <a href="/wiki/Vinyl_ester_resin" title="Vinyl ester resin">vinyl ester</a>, <a href="/wiki/Epoxy" title="Epoxy">epoxy</a>, <a href="/wiki/Phenolic_resin" class="mw-redirect" title="Phenolic resin">phenolic</a>, <a href="/wiki/Polyimide" title="Polyimide">polyimide</a>, <a href="/wiki/Polyamide" title="Polyamide">polyamide</a>, <a href="/wiki/Polypropylene" title="Polypropylene">polypropylene</a>, <a href="/wiki/PEEK" class="mw-redirect" title="PEEK">PEEK</a>, and others. The reinforcement materials are often fibres but also commonly ground minerals. The various methods described below have been developed to reduce the resin content of the final product, or the fibre content is increased. As a rule of thumb, lay up results in a product containing 60% resin and 40% fibre, whereas vacuum infusion gives a final product with 40% resin and 60% fibre content. The strength of the product is greatly dependent on this ratio. Martin Hubbe and Lucian A Lucia consider <a href="/wiki/Wood" title="Wood">wood</a> to be a natural composite of <a href="/wiki/Cellulose_fibre" class="mw-redirect" title="Cellulose fibre">cellulose fibres</a> in a <a href="/wiki/Matrix_(biology)" title="Matrix (biology)">matrix</a> of <a href="/wiki/Lignin" title="Lignin">lignin</a>.<a href="#cite_note-49">[49]</a><a href="#cite_note-50">[50]</a> <div class="mw-heading mw-heading2"><h2 id="Cores_in_composites">Cores in composites</h2>[<a href="/w/index.php?title=Composite_material&action=edit&section=6" title="Edit section: Cores in composites">edit</a>]</div> Several layup designs of composite also involve a co-curing or post-curing of the prepreg with many other media, such as foam or honeycomb. Generally, this is known as a <a href="/wiki/Sandwich_structured_composite" class="mw-redirect" title="Sandwich structured composite">sandwich structure</a>. This is a more general layup for the production of cowlings, doors, radomes or non-structural parts. Open- and closed-cell-structured <a href="/wiki/Foam" title="Foam">foams</a> like <a href="/wiki/Polyvinyl_chloride" title="Polyvinyl chloride">polyvinyl chloride</a>, <a href="/wiki/Polyurethane" title="Polyurethane">polyurethane</a>, <a href="/wiki/Polyethylene" title="Polyethylene">polyethylene</a>, or <a href="/wiki/Polystyrene" title="Polystyrene">polystyrene</a> foams, <a href="/wiki/Balsa" class="mw-redirect" title="Balsa">balsa wood</a>, <a href="/wiki/Syntactic_foam" title="Syntactic foam">syntactic foams</a>, and <a href="/wiki/Composite_honeycomb" class="mw-redirect" title="Composite honeycomb">honeycombs</a> are generally utilized core materials. Open- and closed-cell <a href="/wiki/Metal_foam" title="Metal foam">metal foam</a> can also be utilized as core materials. Recently, 3D graphene structures ( also called graphene foam) have also been employed as core structures. A recent review by Khurram and Xu et al., have provided the summary of the state-of-the-art techniques for fabrication of the 3D structure of graphene, and the examples of the use of these foam like structures as a core for their respective polymer composites.<a href="#cite_note-51">[51]</a> <div class="mw-heading mw-heading3"><h3 id="Semi-crystalline_polymers">Semi-crystalline polymers</h3>[<a href="/w/index.php?title=Composite_material&action=edit&section=7" title="Edit section: Semi-crystalline polymers">edit</a>]</div> Although the two phases are chemically equivalent, semi-crystalline polymers can be described both quantitatively and qualitatively as composite materials. The crystalline portion has a higher elastic modulus and provides reinforcement for the less stiff, amorphous phase. Polymeric materials can range from 0% to 100%<a href="#cite_note-52">[52]</a> crystallinity aka volume fraction depending on molecular structure and thermal history. Different processing techniques can be employed to vary the percent crystallinity in these materials and thus the mechanical properties of these materials as described in the physical properties section. This effect is seen in a variety of places from industrial plastics like polyethylene shopping bags to spiders which can produce silks with different mechanical properties.<a href="#cite_note-53">[53]</a> In many cases these materials act like particle composites with randomly dispersed crystals known as spherulites. However they can also be engineered to be anisotropic and act more like fiber reinforced composites.<a href="#cite_note-54">[54]</a> In the case of spider silk, the properties of the material can even be dependent on the size of the crystals, independent of the volume fraction.<a href="#cite_note-55">[55]</a> Ironically, single component polymeric materials are some of the most easily tunable composite materials known. <div class="mw-heading mw-heading2"><h2 id="Methods_of_fabrication">Methods of fabrication</h2>[<a href="/w/index.php?title=Composite_material&action=edit&section=8" title="Edit section: Methods of fabrication">edit</a>]</div> Normally, the fabrication of composite includes wetting, mixing or saturating the reinforcement with the matrix. The matrix is then induced to bind together (with heat or a chemical reaction) into a rigid structure. Usually, the operation is done in an open or closed forming mould. However, the order and ways of introducing the constituents alters considerably. Composites fabrication is achieved by a wide variety of methods, including <a href="/wiki/Advanced_fiber_placement" class="mw-redirect" title="Advanced fiber placement">advanced fibre placement</a> (automated fibre placement),<a href="#cite_note-56">[56]</a> <a href="/wiki/Fiberglass_spray_lay-up_process" title="Fiberglass spray lay-up process">fibreglass spray lay-up process</a>,<a href="#cite_note-57">[57]</a> <a href="/wiki/Filament_winding" title="Filament winding">filament winding</a>,<a href="#cite_note-58">[58]</a> <a href="/wiki/Lanxide_process" title="Lanxide process">lanxide process</a>,<a href="#cite_note-59">[59]</a> <a href="/wiki/Tailored_fiber_placement" title="Tailored fiber placement">tailored fibre placement</a>,<a href="#cite_note-60">[60]</a> <a href="/wiki/Tufting_(composites)" title="Tufting (composites)">tufting</a>,<a href="#cite_note-61">[61]</a> and <a href="/wiki/Z-pinning" title="Z-pinning">z-pinning</a>.<a href="#cite_note-62">[62]</a> <div class="mw-heading mw-heading3"><h3 id="Overview_of_mould">Overview of mould</h3>[<a href="/w/index.php?title=Composite_material&action=edit&section=9" title="Edit section: Overview of mould">edit</a>]</div> The reinforcing and matrix materials are merged, compacted, and cured (processed) within a mould to undergo a melding event. The part shape is fundamentally set after the melding event. However, under particular process conditions, it can deform. The melding event for a <a href="/wiki/Thermoset_polymer_matrix" title="Thermoset polymer matrix">thermoset polymer matrix</a> material is a curing reaction that is caused by the possibility of extra heat or chemical reactivity such as an organic peroxide. The melding event for a thermoplastic polymeric matrix material is a solidification from the melted state. The melding event for a metal matrix material such as titanium foil is a fusing at high pressure and a temperature near the melting point. It is suitable for many moulding methods to refer to one mould piece as a "lower" mould and another mould piece as an "upper" mould. Lower and upper does not refer to the mould's configuration in space, but the different faces of the moulded panel. There is always a lower mould, and sometimes an upper mould in this convention. Part construction commences by applying materials to the lower mould. Lower mould and upper mould are more generalized descriptors than more common and specific terms such as male side, female side, a-side, b-side, tool side, bowl, hat, mandrel, etc. Continuous manufacturing utilizes a different nomenclature. Usually, the moulded product is referred to as a panel. It can be referred to as casting for certain geometries and material combinations. It can be referred to as a profile for certain continuous processes. Some of the processes are <a href="/wiki/Autoclave_moulding" title="Autoclave moulding">autoclave moulding</a>,<a href="#cite_note-63">[63]</a> <a href="/wiki/Vacuum_bag_moulding" title="Vacuum bag moulding">vacuum bag moulding</a>,<a href="#cite_note-64">[64]</a> <a href="/wiki/Pressure_bag_moulding" title="Pressure bag moulding">pressure bag moulding</a>,<a href="#cite_note-65">[65]</a> <a href="/wiki/Resin_transfer_moulding" title="Resin transfer moulding">resin transfer moulding</a>,<a href="#cite_note-66">[66]</a> and <a href="/wiki/Light_resin_transfer_moulding" title="Light resin transfer moulding">light resin transfer moulding</a>.<a href="#cite_note-67">[67]</a> <div class="mw-heading mw-heading3"><h3 id="Other_fabrication_methods">Other fabrication methods</h3>[<a href="/w/index.php?title=Composite_material&action=edit&section=10" title="Edit section: Other fabrication methods">edit</a>]</div> Other types of fabrication include <a href="/wiki/Casting" title="Casting">casting</a>,<a href="#cite_note-68">[68]</a> centrifugal casting,<a href="#cite_note-69">[69]</a> <a href="/wiki/Braiding_machine" title="Braiding machine">braiding</a> (onto a <a href="/wiki/Former" title="Former">former</a>), <a href="/wiki/Continuous_casting" title="Continuous casting">continuous casting</a>,<a href="#cite_note-70">[70]</a> <a href="/wiki/Filament_winding" title="Filament winding">filament winding</a>,<a href="#cite_note-71">[71]</a> press moulding,<a href="#cite_note-72">[72]</a> <a href="/wiki/Transfer_moulding" class="mw-redirect" title="Transfer moulding">transfer moulding</a>, <a href="/wiki/Pultrusion" title="Pultrusion">pultrusion</a> moulding,<a href="#cite_note-73">[73]</a> and <a href="/wiki/Slip_forming" title="Slip forming">slip forming</a>.<a href="#cite_note-74">[74]</a> There are also forming capabilities including <a href="/wiki/Numerical_control" title="Numerical control">CNC</a> filament winding, vacuum infusion, wet lay-up, <a href="/wiki/Compression_moulding" class="mw-redirect" title="Compression moulding">compression moulding</a>, and <a href="/wiki/Thermoplastic" title="Thermoplastic">thermoplastic</a> moulding, to name a few. The practice of curing ovens and paint booths is also required for some projects. <div class="mw-heading mw-heading4"><h4 id="Finishing_methods">Finishing methods</h4>[<a href="/w/index.php?title=Composite_material&action=edit&section=11" title="Edit section: Finishing methods">edit</a>]</div> The composite parts finishing is also crucial in the final design. Many of these finishes will involve rain-erosion coatings or polyurethane coatings. <div class="mw-heading mw-heading3"><h3 id="Tooling">Tooling</h3>[<a href="/w/index.php?title=Composite_material&action=edit&section=12" title="Edit section: Tooling">edit</a>]</div> The mould and mould inserts are referred to as "tooling". The mould/tooling can be built from different materials. Tooling materials include <a href="/wiki/Aluminium" title="Aluminium">aluminium</a>, <a href="/wiki/Carbon_fibre" class="mw-redirect" title="Carbon fibre">carbon fibre</a>, <a href="/wiki/Invar" title="Invar">invar</a>, <a href="/wiki/Nickel" title="Nickel">nickel</a>, reinforced <a href="/wiki/Silicone_rubber" title="Silicone rubber">silicone rubber</a> and steel. The tooling material selection is normally based on, but not limited to, the <a href="/wiki/Coefficient_of_thermal_expansion" class="mw-redirect" title="Coefficient of thermal expansion">coefficient of thermal expansion</a>, expected number of cycles, end item tolerance, desired or expected surface condition, cure method, <a href="/wiki/Glass_transition_temperature" class="mw-redirect" title="Glass transition temperature">glass transition temperature</a> of the material being moulded, moulding method, matrix, cost, and other various considerations. <div class="mw-heading mw-heading2"><h2 id="Physical_properties">Physical properties</h2>[<a href="/w/index.php?title=Composite_material&action=edit&section=13" title="Edit section: Physical properties">edit</a>]</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Composite_elastic_modulus.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/74/Composite_elastic_modulus.svg/220px-Composite_elastic_modulus.svg.png" decoding="async" width="220" height="176" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/74/Composite_elastic_modulus.svg/330px-Composite_elastic_modulus.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/74/Composite_elastic_modulus.svg/440px-Composite_elastic_modulus.svg.png 2x" data-file-width="600" data-file-height="480" /></a><figcaption>Plot of the overall strength of a composite material as a function of fiber volume fraction limited by the upper bound (isostrain) and lower bound (isostress) conditions.</figcaption></figure> Usually, the composite's physical properties are not <a href="/wiki/Isotropic" class="mw-redirect" title="Isotropic">isotropic</a> (independent of the direction of applied force) in nature. But they are typically <a href="/wiki/Anisotropy" title="Anisotropy">anisotropic</a> (different depending on the direction of the applied force or load). For instance, the composite panel's stiffness will usually depend upon the orientation of the applied forces and/or moments. The composite's strength is bounded by two loading conditions, as shown in the plot to the right. <div class="mw-heading mw-heading3"><h3 id="Isostrain_rule_of_mixtures">Isostrain rule of mixtures</h3>[<a href="/w/index.php?title=Composite_material&action=edit&section=14" title="Edit section: Isostrain rule of mixtures">edit</a>]</div> If both the fibres and matrix are aligned parallel to the loading direction, the deformation of both phases will be the same (assuming there is no delamination at the fibre-matrix interface). This isostrain condition provides the upper bound for composite strength, and is determined by the <a href="/wiki/Rule_of_mixtures" title="Rule of mixtures">rule of mixtures</a>: <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Isostress_and_isostrain_conditions_for_composite_materials.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a3/Isostress_and_isostrain_conditions_for_composite_materials.gif/220px-Isostress_and_isostrain_conditions_for_composite_materials.gif" decoding="async" width="220" height="171" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/a/a3/Isostress_and_isostrain_conditions_for_composite_materials.gif 1.5x" data-file-width="308" data-file-height="240" /></a><figcaption>Figure a) shows the isostress condition where the composite materials are perpendicular to the applied force and b) is the isostrain condition that has the layers parallel to the force.<a href="#cite_note-75">[75]</a></figcaption></figure> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{C}=\sum _{i=1}V_{i}E_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msub> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> </munder> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{C}=\sum _{i=1}V_{i}E_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01d12a891bceaffdf90b3d03d16993da034db3a3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:14.707ex; height:5.509ex;" alt="{\displaystyle E_{C}=\sum _{i=1}V_{i}E_{i}}"></dd></dl> where EC is the effective composite <a href="/wiki/Young%27s_modulus" title="Young's modulus">Young's modulus</a>, and Vi and Ei are the volume fraction and Young's moduli, respectively, of the composite phases. For example, a composite material made up of α and β phases as shown in the figure to the right under isostrain, the Young's modulus would be as follows:<math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{C}=V_{\alpha }E_{\alpha }+V_{\beta }E_{\beta }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msub> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{C}=V_{\alpha }E_{\alpha }+V_{\beta }E_{\beta }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b0a313957f339d9036a388d476806c1c282c1bf2" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:20.193ex; height:2.843ex;" alt="{\displaystyle E_{C}=V_{\alpha }E_{\alpha }+V_{\beta }E_{\beta }}">where Vα and Vβ are the respective volume fractions of each phase. This can be derived by considering that in the isostrain case,<math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \epsilon _{C}=\epsilon _{\alpha }=\epsilon _{\beta }=\epsilon }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ϵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>ϵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>ϵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msub> <mo>=</mo> <mi>ϵ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \epsilon _{C}=\epsilon _{\alpha }=\epsilon _{\beta }=\epsilon }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2608f42d2864e79b21ac7447f90adfc5e5f54a53" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:17.011ex; height:2.343ex;" alt="{\displaystyle \epsilon _{C}=\epsilon _{\alpha }=\epsilon _{\beta }=\epsilon }"> Assuming that the composite has a uniform cross section, the stress on the composite is a weighted average between the two phases,<math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{C}=\sigma _{\alpha }V_{\alpha }+\sigma _{\beta }V_{\beta }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msub> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{C}=\sigma _{\alpha }V_{\alpha }+\sigma _{\beta }V_{\beta }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/61af88897f359afee206258b216a1dfc6b09e7d0" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:19.029ex; height:2.843ex;" alt="{\displaystyle \sigma _{C}=\sigma _{\alpha }V_{\alpha }+\sigma _{\beta }V_{\beta }}"> The stresses in the individual phases are given by Hooke's Law,<math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{\beta }=E_{\beta }\epsilon }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msub> <mi>ϵ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{\beta }=E_{\beta }\epsilon }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b35ab865b75931777042f1f6467b942237bf1ab0" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.433ex; height:2.843ex;" alt="{\displaystyle \sigma _{\beta }=E_{\beta }\epsilon }"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{\alpha }=E_{\alpha }\epsilon }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> <mi>ϵ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{\alpha }=E_{\alpha }\epsilon }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d69bab1592422b162fd0d1b3517f211a8adac108" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.654ex; height:2.509ex;" alt="{\displaystyle \sigma _{\alpha }=E_{\alpha }\epsilon }"> Combining these equations gives that the overall stress in the composite is<math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{C}=E_{\alpha }V_{\alpha }\epsilon +E_{\beta }V_{\beta }\epsilon =(E_{\alpha }V_{\alpha }+E_{\beta }V_{\beta })\epsilon }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> <mi>ϵ</mi> <mo>+</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msub> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msub> <mi>ϵ</mi> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msub> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mi>ϵ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{C}=E_{\alpha }V_{\alpha }\epsilon +E_{\beta }V_{\beta }\epsilon =(E_{\alpha }V_{\alpha }+E_{\beta }V_{\beta })\epsilon }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0b0dddf6546a7de4b7fee94e0319984ee5116ed3" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:41.443ex; height:3.009ex;" alt="{\displaystyle \sigma _{C}=E_{\alpha }V_{\alpha }\epsilon +E_{\beta }V_{\beta }\epsilon =(E_{\alpha }V_{\alpha }+E_{\beta }V_{\beta })\epsilon }"> Then it can be shown that<math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{C}=(E_{\alpha }V_{\alpha }+E_{\beta }V_{\beta })}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msub> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msub> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{C}=(E_{\alpha }V_{\alpha }+E_{\beta }V_{\beta })}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/df14a08eea108dffff162c15fda039a26a510919" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:22.002ex; height:3.009ex;" alt="{\displaystyle E_{C}=(E_{\alpha }V_{\alpha }+E_{\beta }V_{\beta })}"> <div class="mw-heading mw-heading3"><h3 id="Isostress_rule_of_mixtures">Isostress rule of mixtures</h3>[<a href="/w/index.php?title=Composite_material&action=edit&section=15" title="Edit section: Isostress rule of mixtures">edit</a>]</div> The lower bound is dictated by the isostress condition, in which the fibres and matrix are oriented perpendicularly to the loading direction:<math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{C}=\sigma _{\alpha }=\sigma _{\beta }=\sigma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msub> <mo>=</mo> <mi>σ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{C}=\sigma _{\alpha }=\sigma _{\beta }=\sigma }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8fec430f7510eeb3f969b71e4a518ba53fd46a36" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:18.547ex; height:2.343ex;" alt="{\displaystyle \sigma _{C}=\sigma _{\alpha }=\sigma _{\beta }=\sigma }">and now the strains become a weighted average<math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \epsilon _{C}=\epsilon _{\alpha }V_{\alpha }+\epsilon _{\beta }V_{\beta }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ϵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>ϵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>ϵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msub> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \epsilon _{C}=\epsilon _{\alpha }V_{\alpha }+\epsilon _{\beta }V_{\beta }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9e36e889dfd348a82eaded319397a831054cbb33" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:17.88ex; height:2.843ex;" alt="{\displaystyle \epsilon _{C}=\epsilon _{\alpha }V_{\alpha }+\epsilon _{\beta }V_{\beta }}">Rewriting Hooke's Law for the individual phases<math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \epsilon _{\beta }={\frac {\sigma }{E_{\beta }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ϵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>σ</mi> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msub> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \epsilon _{\beta }={\frac {\sigma }{E_{\beta }}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e2264a4704911a375e731b135e81573c1b9c66f2" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:8.942ex; height:5.343ex;" alt="{\displaystyle \epsilon _{\beta }={\frac {\sigma }{E_{\beta }}}}"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \epsilon _{\alpha }={\frac {\sigma }{E_{\alpha }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ϵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>σ</mi> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \epsilon _{\alpha }={\frac {\sigma }{E_{\alpha }}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3733f84767d61ec2d7112dafb8e3af5e5211d4df" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:9.162ex; height:5.009ex;" alt="{\displaystyle \epsilon _{\alpha }={\frac {\sigma }{E_{\alpha }}}}"> This leads to<math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \epsilon _{c}=V_{\beta }{\frac {\sigma }{\epsilon _{\beta }}}+V_{\alpha }{\frac {\sigma }{\epsilon _{\alpha }}}=({\frac {V_{\alpha }}{\epsilon _{\alpha }}}+{\frac {V_{\beta }}{\epsilon _{\beta }}})\sigma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ϵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>σ</mi> <msub> <mi>ϵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msub> </mfrac> </mrow> <mo>+</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>σ</mi> <msub> <mi>ϵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> </mfrac> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> <msub> <mi>ϵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msub> <msub> <mi>ϵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msub> </mfrac> </mrow> <mo stretchy="false">)</mo> <mi>σ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \epsilon _{c}=V_{\beta }{\frac {\sigma }{\epsilon _{\beta }}}+V_{\alpha }{\frac {\sigma }{\epsilon _{\alpha }}}=({\frac {V_{\alpha }}{\epsilon _{\alpha }}}+{\frac {V_{\beta }}{\epsilon _{\beta }}})\sigma }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b0c5de3a79031e4aca536517aac395a01cfcb0ad" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:34.934ex; height:6.176ex;" alt="{\displaystyle \epsilon _{c}=V_{\beta }{\frac {\sigma }{\epsilon _{\beta }}}+V_{\alpha }{\frac {\sigma }{\epsilon _{\alpha }}}=({\frac {V_{\alpha }}{\epsilon _{\alpha }}}+{\frac {V_{\beta }}{\epsilon _{\beta }}})\sigma }">From the definition of Hooke's Law<math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{E_{C}}}={\frac {V_{\alpha }}{E_{\alpha }}}+{\frac {V_{\beta }}{E_{\beta }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msub> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msub> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msub> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{E_{C}}}={\frac {V_{\alpha }}{E_{\alpha }}}+{\frac {V_{\beta }}{E_{\beta }}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a72302c1085006222940ac8231c95b67c1e89fdc" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:17.532ex; height:6.176ex;" alt="{\displaystyle {\frac {1}{E_{C}}}={\frac {V_{\alpha }}{E_{\alpha }}}+{\frac {V_{\beta }}{E_{\beta }}}}">and, in general, <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{E_{C}}}=\sum _{i=1}{\frac {V_{i}}{E_{i}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msub> </mfrac> </mrow> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{E_{C}}}=\sum _{i=1}{\frac {V_{i}}{E_{i}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0678d6339af50b4c4521f6d8e885122b60093d4c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:14.224ex; height:6.343ex;" alt="{\displaystyle {\frac {1}{E_{C}}}=\sum _{i=1}{\frac {V_{i}}{E_{i}}}}"></dd></dl> Following the example above, if one had a composite material made up of α and β phases under isostress conditions as shown in the figure to the right, the composition Young's modulus would be:<math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{C}=(E_{\alpha }E_{\beta })/(V_{\alpha }E_{\beta }+V_{\beta }E_{\alpha })}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msub> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">(</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msub> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{C}=(E_{\alpha }E_{\beta })/(V_{\alpha }E_{\beta }+V_{\beta }E_{\alpha })}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cd1e2b6b6805242a982d388307a5ac88a34b3917" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:30.863ex; height:3.009ex;" alt="{\displaystyle E_{C}=(E_{\alpha }E_{\beta })/(V_{\alpha }E_{\beta }+V_{\beta }E_{\alpha })}"> The isostrain condition implies that under an applied load, both phases experience the same strain but will feel different stress. Comparatively, under isostress conditions both phases will feel the same stress but the strains will differ between each phase. A generalized equation for any loading condition between isostrain and isostress can be written as:<a href="#cite_note-76">[76]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (X_{c})^{n}=V_{m}(X_{m})^{n}+V_{r}(X_{r})^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>=</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>+</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (X_{c})^{n}=V_{m}(X_{m})^{n}+V_{r}(X_{r})^{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1e263a2977dc318bc27cdd861ac1540a3402b900" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.747ex; height:2.843ex;" alt="{\displaystyle (X_{c})^{n}=V_{m}(X_{m})^{n}+V_{r}(X_{r})^{n}}"></dd></dl> where X is a material property such as modulus or stress, c, m, and r stand for the properties of the composite, matrix, and reinforcement materials respectively, and n is a value between 1 and −1. The above equation can be further generalized beyond a two phase composite to an m-component system: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (X_{c})^{n}=\sum _{i=1}^{m}V_{i}(X_{i})^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </munderover> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (X_{c})^{n}=\sum _{i=1}^{m}V_{i}(X_{i})^{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/510b64453aa49926a8f715502f582f0fa9484bae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:20.643ex; height:6.843ex;" alt="{\displaystyle (X_{c})^{n}=\sum _{i=1}^{m}V_{i}(X_{i})^{n}}"></dd></dl> Though composite stiffness is maximized when fibres are aligned with the loading direction, so is the possibility of fibre tensile fracture, assuming the tensile strength exceeds that of the matrix. When a fibre has some angle of misorientation θ, several fracture modes are possible. For small values of θ the stress required to initiate fracture is increased by a factor of (cos θ)−2 due to the increased cross-sectional area (A cos θ) of the fibre and reduced force (F/cos θ) experienced by the fibre, leading to a composite tensile strength of σparallel /cos2 θ where σparallel is the tensile strength of the composite with fibres aligned parallel with the applied force. Intermediate angles of misorientation θ lead to matrix shear failure. Again the cross sectional area is modified but since <a href="/wiki/Shear_stress" title="Shear stress">shear stress</a> is now the driving force for failure the area of the matrix parallel to the fibres is of interest, increasing by a factor of 1/sin θ. Similarly, the force parallel to this area again decreases (F/cos θ) leading to a total tensile strength of τmy /sin θ cos θ where τmy is the matrix shear strength. Finally, for large values of θ (near π/2) transverse matrix failure is the most likely to occur, since the fibres no longer carry the majority of the load. Still, the tensile strength will be greater than for the purely perpendicular orientation, since the force perpendicular to the fibres will decrease by a factor of 1/sin θ and the area decreases by a factor of 1/sin θ producing a composite tensile strength of σperp /sin2θ where σperp is the tensile strength of the composite with fibres align perpendicular to the applied force.<a href="#cite_note-77">[77]</a> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Composite_Strength_as_a_Function_of_Fiber_Misalignment.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/7a/Composite_Strength_as_a_Function_of_Fiber_Misalignment.png/220px-Composite_Strength_as_a_Function_of_Fiber_Misalignment.png" decoding="async" width="220" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/7a/Composite_Strength_as_a_Function_of_Fiber_Misalignment.png/330px-Composite_Strength_as_a_Function_of_Fiber_Misalignment.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/7a/Composite_Strength_as_a_Function_of_Fiber_Misalignment.png/440px-Composite_Strength_as_a_Function_of_Fiber_Misalignment.png 2x" data-file-width="840" data-file-height="458" /></a><figcaption>The graph depicts the three fracture modes a composite material may experience depending on the angle of misorientation relative to aligning fibres parallel to the applied stress.</figcaption></figure> The majority of commercial composites are formed with random dispersion and orientation of the strengthening fibres, in which case the composite Young's modulus will fall between the isostrain and isostress bounds. However, in applications where the strength-to-weight ratio is engineered to be as high as possible (such as in the aerospace industry), fibre alignment may be tightly controlled. Panel stiffness is also dependent on the design of the panel. For instance, the fibre reinforcement and matrix used, the method of panel build, thermoset versus thermoplastic, and type of weave. In contrast to composites, isotropic materials (for example, aluminium or steel), in standard wrought forms, possess the same stiffness typically despite the directional orientation of the applied forces and/or moments. The relationship between forces/moments and strains/curvatures for an isotropic material can be described with the following material properties: Young's Modulus, the <a href="/wiki/Shear_modulus" title="Shear modulus">shear modulus</a>, and the <a href="/wiki/Poisson%27s_ratio" title="Poisson's ratio">Poisson's ratio</a>, in relatively simple mathematical relationships. For the anisotropic material, it needs the mathematics of a second-order tensor and up to 21 material property constants. For the special case of orthogonal isotropy, there are three distinct material property constants for each of Young's Modulus, Shear Modulus and Poisson's ratio—a total of 9 constants to express the relationship between forces/moments and strains/curvatures. Techniques that take benefit of the materials' anisotropic properties involve <a href="/wiki/Mortise_and_tenon" title="Mortise and tenon">mortise and tenon</a> joints (in natural composites such as wood) and <a href="/w/index.php?title=Pi_joint&action=edit&redlink=1" class="new" title="Pi joint (page does not exist)">pi joints</a> in synthetic composites. <div class="mw-heading mw-heading2"><h2 id="Mechanical_properties_of_composites">Mechanical properties of composites</h2>[<a href="/w/index.php?title=Composite_material&action=edit&section=16" title="Edit section: Mechanical properties of composites">edit</a>]</div> <div class="mw-heading mw-heading3"><h3 id="Particle_reinforcement">Particle reinforcement</h3>[<a href="/w/index.php?title=Composite_material&action=edit&section=17" title="Edit section: Particle reinforcement">edit</a>]</div> In general, particle reinforcement is <a href="/wiki/Strengthening_mechanisms_of_materials" title="Strengthening mechanisms of materials">strengthening</a> the composites less than <a href="/wiki/Fiber" title="Fiber">fiber</a> reinforcement. It is used to enhance the <a href="/wiki/Stiffness" title="Stiffness">stiffness</a> of the composites while increasing the <a href="/wiki/Yield_(engineering)" title="Yield (engineering)">strength</a> and the <a href="/wiki/Toughness" title="Toughness">toughness</a>. Because of their <a href="/wiki/Mechanical_properties" class="mw-redirect" title="Mechanical properties">mechanical properties</a>, they are used in applications in which <a href="/wiki/Wear" title="Wear">wear</a> resistance is required. For example, hardness of <a href="/wiki/Engineered_cementitious_composite" title="Engineered cementitious composite">cement</a> can be increased by reinforcing gravel particles, drastically. Particle reinforcement a highly advantageous method of tuning mechanical properties of materials since it is very easy implement while being low cost.<a href="#cite_note-78">[78]</a><a href="#cite_note-79">[79]</a><a href="#cite_note-80">[80]</a><a href="#cite_note-81">[81]</a> The <a href="/wiki/Elastic_modulus" title="Elastic modulus">elastic modulus</a> of particle-reinforced composites can be expressed as, <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{c}=V_{m}E_{m}+K_{c}V_{p}E_{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{c}=V_{m}E_{m}+K_{c}V_{p}E_{p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d7eaf9974aa941129df359b9eeed77a00e21e586" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:23.125ex; height:2.843ex;" alt="{\displaystyle E_{c}=V_{m}E_{m}+K_{c}V_{p}E_{p}}"></dd></dl> where E is the <a href="/wiki/Elastic_modulus" title="Elastic modulus">elastic modulus</a>, V is the <a href="/wiki/Volume_fraction" title="Volume fraction">volume fraction</a>. The subscripts c, p and m are indicating composite, particle and matrix, respectively. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K_{c}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K_{c}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86e37b50e18c03bf957b06bf74cf6ed0c1a45939" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.917ex; height:2.509ex;" alt="{\displaystyle K_{c}}"> is a constant can be found empirically. Similarly, tensile strength of particle-reinforced composites can be expressed as, <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (T.S.)_{c}=V_{m}(T.S.)_{m}+K_{s}V_{p}(T.S.)_{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>T</mi> <mo>.</mo> <mi>S</mi> <mo>.</mo> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>T</mi> <mo>.</mo> <mi>S</mi> <mo>.</mo> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>T</mi> <mo>.</mo> <mi>S</mi> <mo>.</mo> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (T.S.)_{c}=V_{m}(T.S.)_{m}+K_{s}V_{p}(T.S.)_{p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e17264133286cc31d6c57fb344b4e039e6246eb4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:39.076ex; height:3.009ex;" alt="{\displaystyle (T.S.)_{c}=V_{m}(T.S.)_{m}+K_{s}V_{p}(T.S.)_{p}}"></dd></dl> where T.S. is the <a href="/wiki/Ultimate_tensile_strength" title="Ultimate tensile strength">tensile strength</a>, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K_{s}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K_{s}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a44d82e5e2fdd046a55645a8b3f967cfa7fd0a8d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.976ex; height:2.509ex;" alt="{\displaystyle K_{s}}"> is a constant (not equal to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K_{c}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K_{c}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86e37b50e18c03bf957b06bf74cf6ed0c1a45939" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.917ex; height:2.509ex;" alt="{\displaystyle K_{c}}">) that can be found empirically. <div class="mw-heading mw-heading3"><h3 id="Continuous_fiber_reinforcement">Continuous fiber reinforcement</h3>[<a href="/w/index.php?title=Composite_material&action=edit&section=18" title="Edit section: Continuous fiber reinforcement">edit</a>]</div> In general, continuous <a href="/wiki/Fiber" title="Fiber">fiber</a> reinforcement is implemented by incorporating a <a href="/wiki/Fiber" title="Fiber">fiber</a> as the strong phase into a weak phase, matrix. The reason for the popularity of fiber usage is materials with extraordinary strength can be obtained in their fiber form. Non-metallic fibers are usually showing a very high strength to density ratio compared to metal fibers because of the <a href="/wiki/Covalent_bond" title="Covalent bond">covalent</a> nature of their <a href="/wiki/Chemical_bond" title="Chemical bond">bonds</a>. The most famous example of this is <a href="/wiki/Carbon_fibers" title="Carbon fibers">carbon fibers</a> that have many applications extending from <a href="/wiki/Sports_gear" class="mw-redirect" title="Sports gear">sports gear</a> to <a href="/wiki/Protective_gear_in_sports" title="Protective gear in sports">protective equipment</a> to <a href="/wiki/SpaceX" title="SpaceX">space industries</a>.<a href="#cite_note-:1-82">[82]</a><a href="#cite_note-83">[83]</a> The stress on the composite can be expressed in terms of the <a href="/wiki/Volume_fraction" title="Volume fraction">volume fraction</a> of the fiber and the matrix. <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{c}=V_{f}\sigma _{f}+V_{m}\sigma _{m}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{c}=V_{f}\sigma _{f}+V_{m}\sigma _{m}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2edf381f8c41e5fda1c60a65939c3f197d047cd9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:19.198ex; height:2.843ex;" alt="{\displaystyle \sigma _{c}=V_{f}\sigma _{f}+V_{m}\sigma _{m}}"></dd></dl> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>σ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59f59b7c3e6fdb1d0365a494b81fb9a696138c36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle \sigma }"> is the stress, V is the <a href="/wiki/Volume_fraction" title="Volume fraction">volume fraction</a>. The subscripts c, f and m are indicating composite, fiber and matrix, respectively. Although the <a href="/wiki/Stress%E2%80%93strain_analysis" title="Stress–strain analysis">stress–strain</a> behavior of fiber composites can only be determined by testing, there is an expected trend, three stages of the <a href="/wiki/Stress%E2%80%93strain_curve" title="Stress–strain curve">stress–strain curve</a>. The first stage is the region of the stress–strain curve where both fiber and the matrix are <a href="/wiki/Elastic_deformation" class="mw-redirect" title="Elastic deformation">elastically deformed</a>. This linearly elastic region can be expressed in the following form.<a href="#cite_note-:1-82">[82]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{c}-E_{c}\epsilon _{c}=\epsilon _{c}(V_{f}E_{f}+V_{m}E_{m})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> <msub> <mi>ϵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>ϵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{c}-E_{c}\epsilon _{c}=\epsilon _{c}(V_{f}E_{f}+V_{m}E_{m})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/93a267999b70164a516cead8de3b128a8661252b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:31.06ex; height:3.009ex;" alt="{\displaystyle \sigma _{c}-E_{c}\epsilon _{c}=\epsilon _{c}(V_{f}E_{f}+V_{m}E_{m})}"></dd></dl> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>σ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59f59b7c3e6fdb1d0365a494b81fb9a696138c36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle \sigma }"> is the stress, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \epsilon }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϵ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \epsilon }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3837cad72483d97bcdde49c85d3b7b859fb3fd2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.944ex; height:1.676ex;" alt="{\displaystyle \epsilon }"> is the strain, E is the <a href="/wiki/Elastic_modulus" title="Elastic modulus">elastic modulus</a>, and V is the <a href="/wiki/Volume_fraction" title="Volume fraction">volume fraction</a>. The subscripts c, f, and m are indicating composite, fiber, and matrix, respectively. After passing the elastic region for both fiber and the matrix, the second region of the stress–strain curve can be observed. In the second region, the fiber is still elastically deformed while the matrix is plastically deformed since the matrix is the weak phase. The instantaneous <a href="/wiki/Elastic_modulus" title="Elastic modulus">modulus</a> can be determined using the slope of the stress–strain curve in the second region. The relationship between <a href="/wiki/Stress_(mechanics)" title="Stress (mechanics)">stress</a> and strain can be expressed as, <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{c}=V_{f}E_{f}\epsilon _{c}+V_{m}\sigma _{m}(\epsilon _{c})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> <msub> <mi>ϵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>ϵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{c}=V_{f}E_{f}\epsilon _{c}+V_{m}\sigma _{m}(\epsilon _{c})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/083ef8840cbff7191b8f56c746c3d1bd3955f20c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:25.172ex; height:3.009ex;" alt="{\displaystyle \sigma _{c}=V_{f}E_{f}\epsilon _{c}+V_{m}\sigma _{m}(\epsilon _{c})}"></dd></dl> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>σ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59f59b7c3e6fdb1d0365a494b81fb9a696138c36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle \sigma }"> is the stress, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \epsilon }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϵ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \epsilon }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3837cad72483d97bcdde49c85d3b7b859fb3fd2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.944ex; height:1.676ex;" alt="{\displaystyle \epsilon }"> is the strain, E is the <a href="/wiki/Elastic_modulus" title="Elastic modulus">elastic modulus</a>, and V is the <a href="/wiki/Volume_fraction" title="Volume fraction">volume fraction</a>. The subscripts c, f, and m are indicating composite, fiber, and matrix, respectively. To find the modulus in the second region derivative of this equation can be used since the <a href="/wiki/Slope" title="Slope">slope of the curve</a> is equal to the modulus. <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{c}'={\frac {d\sigma _{c}}{d\epsilon _{c}}}=V_{f}E_{f}+V_{m}\left({\frac {d\sigma _{c}}{d\epsilon _{c}}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> <mo>′</mo> </msubsup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> </mrow> <mrow> <mi>d</mi> <msub> <mi>ϵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>=</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> </mrow> <mrow> <mi>d</mi> <msub> <mi>ϵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{c}'={\frac {d\sigma _{c}}{d\epsilon _{c}}}=V_{f}E_{f}+V_{m}\left({\frac {d\sigma _{c}}{d\epsilon _{c}}}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af085fe2777cf9e8364d1b65db430787f05d9017" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:32.526ex; height:6.176ex;" alt="{\displaystyle E_{c}'={\frac {d\sigma _{c}}{d\epsilon _{c}}}=V_{f}E_{f}+V_{m}\left({\frac {d\sigma _{c}}{d\epsilon _{c}}}\right)}"></dd></dl> In most cases it can be assumed<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{c}'=V_{f}E_{f}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> <mo>′</mo> </msubsup> <mo>=</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{c}'=V_{f}E_{f}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/019320ddb8ebcbd6d3169118505993a5a61a3bf0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.101ex; height:2.843ex;" alt="{\displaystyle E_{c}'=V_{f}E_{f}}"> since the second term is much less than the first one.<a href="#cite_note-:1-82">[82]</a> In reality, the <a href="/wiki/Derivative" title="Derivative">derivative</a> of stress with respect to strain is not always returning the modulus because of the <a href="/wiki/Chemical_bond" title="Chemical bond">binding interaction</a> between the fiber and matrix. The strength of the interaction between these two phases can result in changes in the <a href="/wiki/List_of_materials_properties" title="List of materials properties">mechanical properties</a> of the composite. The compatibility of the fiber and matrix is a measure of <a href="/wiki/Stress_(mechanics)" title="Stress (mechanics)">internal stress</a>.<a href="#cite_note-:1-82">[82]</a> The <a href="/wiki/Covalent_bond" title="Covalent bond">covalently bonded</a> high strength fibers (e.g. <a href="/wiki/Carbon_fibers" title="Carbon fibers">carbon fibers</a>) experience mostly <a href="/wiki/Deformation_(engineering)" title="Deformation (engineering)">elastic deformation</a> before the fracture since the <a href="/wiki/Deformation_(engineering)" title="Deformation (engineering)">plastic deformation</a> can happen due to <a href="/wiki/Dislocation" title="Dislocation">dislocation motion</a>. Whereas, <a href="/wiki/Metallic_fiber" title="Metallic fiber">metallic fibers</a> have more space to plastically deform, so their composites exhibit a third stage where both fiber and the matrix are plastically deforming. <a href="/wiki/Metallic_fiber" title="Metallic fiber">Metallic fibers</a> have <a href="/wiki/Cryogenic_hardening" title="Cryogenic hardening">many applications</a> to work at <a href="/wiki/Cryogenics" title="Cryogenics">cryogenic temperatures</a> that is one of the advantages of composites with <a href="/wiki/Steel_fibre-reinforced_shotcrete" title="Steel fibre-reinforced shotcrete">metal fibers</a> over nonmetallic. The stress in this region of the <a href="/wiki/Stress%E2%80%93strain_curve" title="Stress–strain curve">stress–strain curve</a> can be expressed as, <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{c}(\epsilon _{c})=V_{f}\sigma _{f}\epsilon _{c}+V_{m}\sigma _{m}(\epsilon _{c})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>ϵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> <msub> <mi>ϵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>ϵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{c}(\epsilon _{c})=V_{f}\sigma _{f}\epsilon _{c}+V_{m}\sigma _{m}(\epsilon _{c})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/03da7099f1381b5fcaade0688456ec2fa3ccdb73" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:28.482ex; height:3.009ex;" alt="{\displaystyle \sigma _{c}(\epsilon _{c})=V_{f}\sigma _{f}\epsilon _{c}+V_{m}\sigma _{m}(\epsilon _{c})}"></dd></dl> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>σ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59f59b7c3e6fdb1d0365a494b81fb9a696138c36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle \sigma }"> is the stress, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \epsilon }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϵ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \epsilon }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3837cad72483d97bcdde49c85d3b7b859fb3fd2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.944ex; height:1.676ex;" alt="{\displaystyle \epsilon }"> is the strain, E is the <a href="/wiki/Elastic_modulus" title="Elastic modulus">elastic modulus</a>, and V is the <a href="/wiki/Volume_fraction" title="Volume fraction">volume fraction</a>. The subscripts c, f, and m are indicating composite, fiber, and matrix, respectively. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{f}(\epsilon _{c})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>ϵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{f}(\epsilon _{c})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/701d7da44b2261602b0cf563daa12dab5a0207f2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.161ex; height:3.009ex;" alt="{\displaystyle \sigma _{f}(\epsilon _{c})}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{m}(\epsilon _{c})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>ϵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{m}(\epsilon _{c})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c25551f9fac0f320b05aaab41ddb1c7543b6b3f8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.7ex; height:2.843ex;" alt="{\displaystyle \sigma _{m}(\epsilon _{c})}"> are for fiber and matrix flow stresses respectively. Just after the third region the composite exhibit <a href="/wiki/Necking_(engineering)" title="Necking (engineering)">necking</a>. The necking strain of composite is happened to be between the necking strain of the fiber and the matrix just like other mechanical properties of the composites. The necking strain of the weak phase is delayed by the strong phase. The amount of the delay depends upon the volume fraction of the strong phase.<a href="#cite_note-:1-82">[82]</a> Thus, the <a href="/wiki/Ultimate_tensile_strength" title="Ultimate tensile strength">tensile strength</a> of the composite can be expressed in terms of the <a href="/wiki/Volume_fraction" title="Volume fraction">volume fraction</a>.<a href="#cite_note-:1-82">[82]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (T.S.)_{c}=V_{f}(T.S.)_{f}+V_{m}\sigma _{m}(\epsilon _{m})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>T</mi> <mo>.</mo> <mi>S</mi> <mo>.</mo> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>T</mi> <mo>.</mo> <mi>S</mi> <mo>.</mo> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>ϵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (T.S.)_{c}=V_{f}(T.S.)_{f}+V_{m}\sigma _{m}(\epsilon _{m})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/08218cafe93294f697f4f9de465c677ee389ee91" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:34.997ex; height:3.009ex;" alt="{\displaystyle (T.S.)_{c}=V_{f}(T.S.)_{f}+V_{m}\sigma _{m}(\epsilon _{m})}"></dd></dl> where T.S. is the <a href="/wiki/Ultimate_tensile_strength" title="Ultimate tensile strength">tensile strength</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>σ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59f59b7c3e6fdb1d0365a494b81fb9a696138c36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle \sigma }"> is the stress, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \epsilon }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϵ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \epsilon }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3837cad72483d97bcdde49c85d3b7b859fb3fd2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.944ex; height:1.676ex;" alt="{\displaystyle \epsilon }"> is the strain, E is the <a href="/wiki/Elastic_modulus" title="Elastic modulus">elastic modulus</a>, and V is the <a href="/wiki/Volume_fraction" title="Volume fraction">volume fraction</a>. The subscripts c, f, and m are indicating composite, fiber, and matrix, respectively. The composite tensile strength can be expressed as <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (T.S.)_{c}=V_{m}(T.S.)_{m}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>T</mi> <mo>.</mo> <mi>S</mi> <mo>.</mo> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>T</mi> <mo>.</mo> <mi>S</mi> <mo>.</mo> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (T.S.)_{c}=V_{m}(T.S.)_{m}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a168d209530330532c03dfadf10bbbda36faeb53" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.773ex; height:2.843ex;" alt="{\displaystyle (T.S.)_{c}=V_{m}(T.S.)_{m}}"> for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{f}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{f}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cdc0d1604bbf75069df35d14afa9fac3e883be3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.492ex; height:2.843ex;" alt="{\displaystyle V_{f}}"> is less than or equal to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{c}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{c}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/338b595db7a169754c5e088d9881010c2225f597" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.299ex; height:2.509ex;" alt="{\displaystyle V_{c}}"> (arbitrary critical value of volume fraction)</dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (T.S.)_{c}=V_{f}(T.S.)_{f}+V_{m}(\sigma _{m})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>T</mi> <mo>.</mo> <mi>S</mi> <mo>.</mo> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>T</mi> <mo>.</mo> <mi>S</mi> <mo>.</mo> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (T.S.)_{c}=V_{f}(T.S.)_{f}+V_{m}(\sigma _{m})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7fa45e6f37073f1dff3d387cfd113b772d0432a6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:32.378ex; height:3.009ex;" alt="{\displaystyle (T.S.)_{c}=V_{f}(T.S.)_{f}+V_{m}(\sigma _{m})}"> for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{f}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{f}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cdc0d1604bbf75069df35d14afa9fac3e883be3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.492ex; height:2.843ex;" alt="{\displaystyle V_{f}}"> is greater than or equal to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{c}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{c}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/338b595db7a169754c5e088d9881010c2225f597" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.299ex; height:2.509ex;" alt="{\displaystyle V_{c}}"></dd></dl> The critical value of <a href="/wiki/Volume_fraction" title="Volume fraction">volume fraction</a> can be expressed as, <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{c}={\frac {[(T.S.)_{m}-\sigma _{m}(\epsilon _{f})]}{[(T.S.)_{f}+(T.S.)_{m}-\sigma _{m}(\epsilon _{f})]}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">[</mo> <mo stretchy="false">(</mo> <mi>T</mi> <mo>.</mo> <mi>S</mi> <mo>.</mo> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>ϵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> </mrow> <mrow> <mo stretchy="false">[</mo> <mo stretchy="false">(</mo> <mi>T</mi> <mo>.</mo> <mi>S</mi> <mo>.</mo> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> <mo>+</mo> <mo stretchy="false">(</mo> <mi>T</mi> <mo>.</mo> <mi>S</mi> <mo>.</mo> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>ϵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{c}={\frac {[(T.S.)_{m}-\sigma _{m}(\epsilon _{f})]}{[(T.S.)_{f}+(T.S.)_{m}-\sigma _{m}(\epsilon _{f})]}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9a0cb48f61c071d3675fabb7c708af485af0ade4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:36.937ex; height:6.509ex;" alt="{\displaystyle V_{c}={\frac {[(T.S.)_{m}-\sigma _{m}(\epsilon _{f})]}{[(T.S.)_{f}+(T.S.)_{m}-\sigma _{m}(\epsilon _{f})]}}}"></dd></dl> Evidently, the composite <a href="/wiki/Ultimate_tensile_strength" title="Ultimate tensile strength">tensile strength</a> can be higher than the matrix if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (T.S.)_{c}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>T</mi> <mo>.</mo> <mi>S</mi> <mo>.</mo> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (T.S.)_{c}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f786959ecb139ba03f714665f1ede32bb5f7e92" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.957ex; height:2.843ex;" alt="{\displaystyle (T.S.)_{c}}"> is greater than <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (T.S.)_{m}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>T</mi> <mo>.</mo> <mi>S</mi> <mo>.</mo> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (T.S.)_{m}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a5a1f9408b4ff36412a80d8eb0ce36ebe064c796" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.688ex; height:2.843ex;" alt="{\displaystyle (T.S.)_{m}}">. Thus, the minimum volume fraction of the fiber can be expressed as, <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{c}={\frac {[(T.S.)_{m}-\sigma _{m}(\epsilon _{f})]}{[(T.S.)_{f}-\sigma _{m}(\epsilon _{f})]}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">[</mo> <mo stretchy="false">(</mo> <mi>T</mi> <mo>.</mo> <mi>S</mi> <mo>.</mo> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>ϵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> </mrow> <mrow> <mo stretchy="false">[</mo> <mo stretchy="false">(</mo> <mi>T</mi> <mo>.</mo> <mi>S</mi> <mo>.</mo> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>ϵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{c}={\frac {[(T.S.)_{m}-\sigma _{m}(\epsilon _{f})]}{[(T.S.)_{f}-\sigma _{m}(\epsilon _{f})]}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b4d04ff78d317dd93bc5657b9443fbe953067a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:25.948ex; height:6.509ex;" alt="{\displaystyle V_{c}={\frac {[(T.S.)_{m}-\sigma _{m}(\epsilon _{f})]}{[(T.S.)_{f}-\sigma _{m}(\epsilon _{f})]}}}"></dd></dl> Although this minimum value is very low in practice, it is very important to know since the reason for the incorporation of continuous fibers is to improve the mechanical properties of the materials/composites, and this value of volume fraction is the threshold of this improvement.<a href="#cite_note-:1-82">[82]</a> <div class="mw-heading mw-heading3"><h3 id="The_effect_of_fiber_orientation">The effect of fiber orientation</h3>[<a href="/w/index.php?title=Composite_material&action=edit&section=19" title="Edit section: The effect of fiber orientation">edit</a>]</div> <div class="mw-heading mw-heading4"><h4 id="Aligned_fibers">Aligned fibers</h4>[<a href="/w/index.php?title=Composite_material&action=edit&section=20" title="Edit section: Aligned fibers">edit</a>]</div> A change in the angle between the applied stress and fiber orientation will affect the mechanical properties of fiber-reinforced composites, especially the tensile strength. This angle, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }">, can be used predict the dominant tensile fracture mechanism. At small angles, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta \approx 0^{\circ }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> <mo>≈</mo> <msup> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta \approx 0^{\circ }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3accb9a6d2eb284e30c9ddb3ca731c69a12d525b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.406ex; height:2.343ex;" alt="{\displaystyle \theta \approx 0^{\circ }}">, the dominant fracture mechanism is the same as with load-fiber alignment, tensile fracture. The resolved force acting upon the length of the fibers is reduced by a factor of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos \theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/611e5c70de1d1cf4ebc3b70d2b5467f45d17a483" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.589ex; height:2.176ex;" alt="{\displaystyle \cos \theta }"> from rotation. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{\mbox{res}}=F\cos \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>res</mtext> </mstyle> </mrow> </msub> <mo>=</mo> <mi>F</mi> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{\mbox{res}}=F\cos \theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1e18710669fcafdf177dfcfb4123d0dff41a6031" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.402ex; height:2.509ex;" alt="{\displaystyle F_{\mbox{res}}=F\cos \theta }">. The resolved area on which the fiber experiences the force is increased by a factor of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos \theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/611e5c70de1d1cf4ebc3b70d2b5467f45d17a483" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.589ex; height:2.176ex;" alt="{\displaystyle \cos \theta }"> from rotation. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{\mbox{res}}=A_{0}/\cos \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>res</mtext> </mstyle> </mrow> </msub> <mo>=</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{\mbox{res}}=A_{0}/\cos \theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/11d00fdd60eb3f27fc29d1fd27a415481b4703db" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.87ex; height:2.843ex;" alt="{\displaystyle A_{\mbox{res}}=A_{0}/\cos \theta }">. Taking the effective <a href="/wiki/Ultimate_tensile_strength" title="Ultimate tensile strength">tensile strength</a> to be <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ({\mbox{T.S.}})_{\mbox{c}}=F_{\mbox{res}}/A_{\mbox{res}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>T.S.</mtext> </mstyle> </mrow> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>c</mtext> </mstyle> </mrow> </msub> <mo>=</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>res</mtext> </mstyle> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>res</mtext> </mstyle> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ({\mbox{T.S.}})_{\mbox{c}}=F_{\mbox{res}}/A_{\mbox{res}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f94fc2265fa3885d06598f1bb83e4de9be5589a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.022ex; height:2.843ex;" alt="{\displaystyle ({\mbox{T.S.}})_{\mbox{c}}=F_{\mbox{res}}/A_{\mbox{res}}}"> and the aligned <a href="/wiki/Ultimate_tensile_strength" title="Ultimate tensile strength">tensile strength</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{\parallel }^{*}=F/A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∥</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <mo>=</mo> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{\parallel }^{*}=F/A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3fbb7e035a324df52fb294d3e7455d95c8288f22" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:10.13ex; height:3.509ex;" alt="{\displaystyle \sigma _{\parallel }^{*}=F/A}">.<a href="#cite_note-:1-82">[82]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ({\mbox{T.S.}})_{\mbox{c}}\;({\mbox{longitudinal fracture}})={\frac {\sigma _{\parallel }^{*}}{\cos ^{2}\theta }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>T.S.</mtext> </mstyle> </mrow> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>c</mtext> </mstyle> </mrow> </msub> <mspace width="thickmathspace" /> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>longitudinal fracture</mtext> </mstyle> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∥</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <mrow> <msup> <mi>cos</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ({\mbox{T.S.}})_{\mbox{c}}\;({\mbox{longitudinal fracture}})={\frac {\sigma _{\parallel }^{*}}{\cos ^{2}\theta }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0708c92b4cc434643468a6aef77e4f38c6cc31bf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:40.06ex; height:6.343ex;" alt="{\displaystyle ({\mbox{T.S.}})_{\mbox{c}}\;({\mbox{longitudinal fracture}})={\frac {\sigma _{\parallel }^{*}}{\cos ^{2}\theta }}}"></dd></dl> At moderate angles, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta \approx 45^{\circ }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> <mo>≈</mo> <msup> <mn>45</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta \approx 45^{\circ }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/21552faa0ba6e8479fdb5f84cfebf2bdfff654ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.568ex; height:2.343ex;" alt="{\displaystyle \theta \approx 45^{\circ }}">, the material experiences shear failure. The effective force direction is reduced with respect to the aligned direction. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{\mbox{res}}=F\cos \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>res</mtext> </mstyle> </mrow> </msub> <mo>=</mo> <mi>F</mi> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{\mbox{res}}=F\cos \theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1e18710669fcafdf177dfcfb4123d0dff41a6031" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.402ex; height:2.509ex;" alt="{\displaystyle F_{\mbox{res}}=F\cos \theta }">. The resolved area on which the force acts is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{\mbox{res}}=A_{m}/\sin \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>res</mtext> </mstyle> </mrow> </msub> <mo>=</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{\mbox{res}}=A_{m}/\sin \theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/77c9216b30988e84f205b3dce941177f7f294b7c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.235ex; height:2.843ex;" alt="{\displaystyle A_{\mbox{res}}=A_{m}/\sin \theta }">. The resulting <a href="/wiki/Ultimate_tensile_strength" title="Ultimate tensile strength">tensile strength</a> depends on the <a href="/wiki/Shear_strength" title="Shear strength">shear strength</a> of the matrix, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau _{m}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tau _{m}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/253e09c325b5968be08898a0a9baf724bde446c5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.691ex; height:2.009ex;" alt="{\displaystyle \tau _{m}}">.<a href="#cite_note-:1-82">[82]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ({\mbox{T.S.}})_{\mbox{c}}\;({\mbox{shear failure}})={\frac {\tau _{m}}{\sin {\theta }\cos {\theta }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>T.S.</mtext> </mstyle> </mrow> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>c</mtext> </mstyle> </mrow> </msub> <mspace width="thickmathspace" /> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>shear failure</mtext> </mstyle> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mrow> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>θ</mi> </mrow> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>θ</mi> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ({\mbox{T.S.}})_{\mbox{c}}\;({\mbox{shear failure}})={\frac {\tau _{m}}{\sin {\theta }\cos {\theta }}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f7b6a2acea5dc06e312db54ae8b6aac54c0239ca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:35.337ex; height:4.843ex;" alt="{\displaystyle ({\mbox{T.S.}})_{\mbox{c}}\;({\mbox{shear failure}})={\frac {\tau _{m}}{\sin {\theta }\cos {\theta }}}}"></dd></dl> At extreme angles, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta \approx 90^{\circ }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> <mo>≈</mo> <msup> <mn>90</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta \approx 90^{\circ }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3993030fbbb5a9643818eab977fa0152df2e61f2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.568ex; height:2.343ex;" alt="{\displaystyle \theta \approx 90^{\circ }}">, the dominant mode of failure is tensile fracture in the matrix in the perpendicular direction. As in the <a href="#Isostress_rule_of_mixtures">isostress case</a> of layered composite materials, the strength in this direction is lower than in the aligned direction. The effective areas and forces act perpendicular to the aligned direction so they both scale by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin \theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/efa733f6703578b0c3af870a3170b4ab0dd99c00" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.333ex; height:2.176ex;" alt="{\displaystyle \sin \theta }">. The resolved tensile strength is proportional to the transverse strength, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{\perp }^{*}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>⊥</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{\perp }^{*}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/225dde12e75f54ffd5f3db9e8ec2d97ddd5b8f2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.838ex; height:2.843ex;" alt="{\displaystyle \sigma _{\perp }^{*}}">.<a href="#cite_note-:1-82">[82]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ({\mbox{T.S.}})_{\mbox{c}}\;({\mbox{transverse fracture}})={\frac {\sigma _{\perp }^{*}}{\sin ^{2}\theta }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>T.S.</mtext> </mstyle> </mrow> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>c</mtext> </mstyle> </mrow> </msub> <mspace width="thickmathspace" /> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>transverse fracture</mtext> </mstyle> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>⊥</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <mrow> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ({\mbox{T.S.}})_{\mbox{c}}\;({\mbox{transverse fracture}})={\frac {\sigma _{\perp }^{*}}{\sin ^{2}\theta }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dc91e049c6a8c390deb9babbf260448eb00edf92" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:37.963ex; height:6.009ex;" alt="{\displaystyle ({\mbox{T.S.}})_{\mbox{c}}\;({\mbox{transverse fracture}})={\frac {\sigma _{\perp }^{*}}{\sin ^{2}\theta }}}"></dd></dl> The critical angles from which the dominant fracture mechanism changes can be calculated as, <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta _{c_{1}}=\tan ^{-1}\left({\frac {\tau _{m}}{\sigma _{\parallel }^{*}}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msub> <mo>=</mo> <msup> <mi>tan</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∥</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> </mfrac> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta _{c_{1}}=\tan ^{-1}\left({\frac {\tau _{m}}{\sigma _{\parallel }^{*}}}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fea646d749e4257eab398c40830139726034050d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:18.866ex; height:7.509ex;" alt="{\displaystyle \theta _{c_{1}}=\tan ^{-1}\left({\frac {\tau _{m}}{\sigma _{\parallel }^{*}}}\right)}"></dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta _{c_{2}}=\tan ^{-1}\left({\frac {\sigma _{\perp }^{*}}{\tau _{m}}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </msub> <mo>=</mo> <msup> <mi>tan</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>⊥</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <msub> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mfrac> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta _{c_{2}}=\tan ^{-1}\left({\frac {\sigma _{\perp }^{*}}{\tau _{m}}}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/de60166ffa5d7281fee83dbd36245aa0b6bde101" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:18.753ex; height:6.176ex;" alt="{\displaystyle \theta _{c_{2}}=\tan ^{-1}\left({\frac {\sigma _{\perp }^{*}}{\tau _{m}}}\right)}"></dd></dl> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta _{c_{1}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta _{c_{1}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/292e5335ccd47820abbad7a877bc42d15566376b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.866ex; height:2.843ex;" alt="{\displaystyle \theta _{c_{1}}}"> is the critical angle between longitudinal fracture and shear failure, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta _{c_{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta _{c_{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9aa637d06e982d136ab0724880259820491120b5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.866ex; height:2.843ex;" alt="{\displaystyle \theta _{c_{2}}}"> is the critical angle between shear failure and transverse fracture.<a href="#cite_note-:1-82">[82]</a> By ignoring length effects, this model is most accurate for continuous fibers and does not effectively capture the strength-orientation relationship for short fiber reinforced composites. Furthermore, most realistic systems do not experience the <a href="/wiki/Maxima_and_minima" class="mw-redirect" title="Maxima and minima">local maxima</a> predicted at the critical angles.<a href="#cite_note-84">[84]</a><a href="#cite_note-85">[85]</a><a href="#cite_note-86">[86]</a><a href="#cite_note-87">[87]</a> The <a href="/wiki/Tsai-Hill_failure_criterion" title="Tsai-Hill failure criterion">Tsai-Hill criterion</a> provides a more complete description of fiber composite tensile strength as a function of orientation angle by coupling the contributing yield stresses: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{\parallel }^{*}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∥</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{\parallel }^{*}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4760dbb6ba4f5bbe92d15b574a046b7dbc62e6f8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:2.385ex; height:3.343ex;" alt="{\displaystyle \sigma _{\parallel }^{*}}">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{\perp }^{*}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>⊥</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{\perp }^{*}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/225dde12e75f54ffd5f3db9e8ec2d97ddd5b8f2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.838ex; height:2.843ex;" alt="{\displaystyle \sigma _{\perp }^{*}}">, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau _{m}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tau _{m}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/253e09c325b5968be08898a0a9baf724bde446c5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.691ex; height:2.009ex;" alt="{\displaystyle \tau _{m}}">.<a href="#cite_note-88">[88]</a><a href="#cite_note-:1-82">[82]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ({\mbox{T.S.}})_{\mbox{c}}\;({\mbox{Tsai-Hill}})={\bigg [}{\frac {\cos ^{4}\theta }{({\sigma _{\parallel }^{*}})^{2}}}+\cos ^{2}\theta \sin ^{2}\theta \left({\frac {1}{({\tau _{m}})^{2}}}-{\frac {1}{({\sigma _{\parallel }^{*}})^{2}}}\right)+{\frac {\sin ^{4}\theta }{({\sigma _{\perp }^{*}})^{2}}}{\bigg ]}^{-1/2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>T.S.</mtext> </mstyle> </mrow> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>c</mtext> </mstyle> </mrow> </msub> <mspace width="thickmathspace" /> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>Tsai-Hill</mtext> </mstyle> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">[</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>cos</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> </mrow> <mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∥</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>+</mo> <msup> <mi>cos</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∥</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> </mrow> <mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>⊥</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">]</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ({\mbox{T.S.}})_{\mbox{c}}\;({\mbox{Tsai-Hill}})={\bigg [}{\frac {\cos ^{4}\theta }{({\sigma _{\parallel }^{*}})^{2}}}+\cos ^{2}\theta \sin ^{2}\theta \left({\frac {1}{({\tau _{m}})^{2}}}-{\frac {1}{({\sigma _{\parallel }^{*}})^{2}}}\right)+{\frac {\sin ^{4}\theta }{({\sigma _{\perp }^{*}})^{2}}}{\bigg ]}^{-1/2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/970bbbef62730b15a7123400d6ff5504eb672a7b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:77.684ex; height:7.676ex;" alt="{\displaystyle ({\mbox{T.S.}})_{\mbox{c}}\;({\mbox{Tsai-Hill}})={\bigg [}{\frac {\cos ^{4}\theta }{({\sigma _{\parallel }^{*}})^{2}}}+\cos ^{2}\theta \sin ^{2}\theta \left({\frac {1}{({\tau _{m}})^{2}}}-{\frac {1}{({\sigma _{\parallel }^{*}})^{2}}}\right)+{\frac {\sin ^{4}\theta }{({\sigma _{\perp }^{*}})^{2}}}{\bigg ]}^{-1/2}}"></dd></dl> <div class="mw-heading mw-heading4"><h4 id="Randomly_oriented_fibers">Randomly oriented fibers</h4>[<a href="/w/index.php?title=Composite_material&action=edit&section=21" title="Edit section: Randomly oriented fibers">edit</a>]</div> Anisotropy in the tensile strength of fiber reinforced composites can be removed by randomly orienting the fiber directions within the material. It sacrifices the ultimate strength in the aligned direction for an overall, isotropically strengthened material. <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{c}=KV_{f}E_{f}+V_{m}E_{m}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> <mo>=</mo> <mi>K</mi> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{c}=KV_{f}E_{f}+V_{m}E_{m}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0df280cf648c71b612dce1ac734313befa9bd0b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:22.428ex; height:2.843ex;" alt="{\displaystyle E_{c}=KV_{f}E_{f}+V_{m}E_{m}}"></dd></dl> Where K is an empirically determined reinforcement factor; similar to the <a href="#Particle_Reinforcement">particle reinforcement</a> equation. For fibers with randomly distributed orientations in a plane, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K\approx 0.38}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo>≈</mo> <mn>0.38</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K\approx 0.38}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/684055bfb4690653c7e882f4cebefb4abdaefc8a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.299ex; height:2.176ex;" alt="{\displaystyle K\approx 0.38}">, and for a random distribution in 3D, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K\approx 0.20}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo>≈</mo> <mn>0.20</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K\approx 0.20}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54092980c4abcea9f1db3f501a26a148da25aaef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.299ex; height:2.176ex;" alt="{\displaystyle K\approx 0.20}">.<a href="#cite_note-:1-82">[82]</a> <div class="mw-heading mw-heading3"><h3 id="Stiffness_and_Compliance_Elasticity">Stiffness and Compliance Elasticity</h3>[<a href="/w/index.php?title=Composite_material&action=edit&section=22" title="Edit section: Stiffness and Compliance Elasticity">edit</a>]</div> For real application, most composite is <a href="/wiki/Anisotropy" title="Anisotropy">anisotropic material</a> or <a href="/wiki/Orthotropic_material" title="Orthotropic material">orthotropic material</a>. The three-dimension stress tensor is required for stress and strain analysis. The stiffness and compliance can be written as follows<a href="#cite_note-89">[89]</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}\sigma _{1}\\\sigma _{2}\\\sigma _{3}\\\sigma _{4}\\\sigma _{5}\\\sigma _{6}\end{bmatrix}}={\begin{bmatrix}C_{11}&C_{12}&C_{13}&C_{14}&C_{15}&C_{16}\\C_{12}&C_{22}&C_{23}&C_{24}&C_{25}&C_{26}\\C_{13}&C_{23}&C_{33}&C_{34}&C_{35}&C_{36}\\C_{14}&C_{24}&C_{34}&C_{44}&C_{45}&C_{46}\\C_{15}&C_{25}&C_{35}&C_{45}&C_{55}&C_{56}\\C_{16}&C_{26}&C_{36}&C_{46}&C_{56}&C_{66}\end{bmatrix}}{\begin{bmatrix}\varepsilon _{1}\\\varepsilon _{2}\\\varepsilon _{3}\\\varepsilon _{4}\\\varepsilon _{5}\\\varepsilon _{6}\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>11</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>13</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>14</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>15</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>16</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>22</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>23</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>24</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>25</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>26</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>13</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>23</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>33</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>34</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>35</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>36</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>14</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>24</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>34</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>44</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>45</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>46</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>15</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>25</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>35</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>45</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>55</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>56</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>16</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>26</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>36</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>46</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>56</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>66</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}\sigma _{1}\\\sigma _{2}\\\sigma _{3}\\\sigma _{4}\\\sigma _{5}\\\sigma _{6}\end{bmatrix}}={\begin{bmatrix}C_{11}&C_{12}&C_{13}&C_{14}&C_{15}&C_{16}\\C_{12}&C_{22}&C_{23}&C_{24}&C_{25}&C_{26}\\C_{13}&C_{23}&C_{33}&C_{34}&C_{35}&C_{36}\\C_{14}&C_{24}&C_{34}&C_{44}&C_{45}&C_{46}\\C_{15}&C_{25}&C_{35}&C_{45}&C_{55}&C_{56}\\C_{16}&C_{26}&C_{36}&C_{46}&C_{56}&C_{66}\end{bmatrix}}{\begin{bmatrix}\varepsilon _{1}\\\varepsilon _{2}\\\varepsilon _{3}\\\varepsilon _{4}\\\varepsilon _{5}\\\varepsilon _{6}\end{bmatrix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8192540547d26ebe176bd80dc6f5cf577b9c24b2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -9.005ex; width:52.015ex; height:19.176ex;" alt="{\displaystyle {\begin{bmatrix}\sigma _{1}\\\sigma _{2}\\\sigma _{3}\\\sigma _{4}\\\sigma _{5}\\\sigma _{6}\end{bmatrix}}={\begin{bmatrix}C_{11}&C_{12}&C_{13}&C_{14}&C_{15}&C_{16}\\C_{12}&C_{22}&C_{23}&C_{24}&C_{25}&C_{26}\\C_{13}&C_{23}&C_{33}&C_{34}&C_{35}&C_{36}\\C_{14}&C_{24}&C_{34}&C_{44}&C_{45}&C_{46}\\C_{15}&C_{25}&C_{35}&C_{45}&C_{55}&C_{56}\\C_{16}&C_{26}&C_{36}&C_{46}&C_{56}&C_{66}\end{bmatrix}}{\begin{bmatrix}\varepsilon _{1}\\\varepsilon _{2}\\\varepsilon _{3}\\\varepsilon _{4}\\\varepsilon _{5}\\\varepsilon _{6}\end{bmatrix}}}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}\varepsilon _{1}\\\varepsilon _{2}\\\varepsilon _{3}\\\varepsilon _{4}\\\varepsilon _{5}\\\varepsilon _{6}\end{bmatrix}}={\begin{bmatrix}S_{11}&S_{12}&S_{13}&S_{14}&S_{15}&S_{16}\\S_{12}&S_{22}&S_{23}&S_{24}&S_{25}&S_{26}\\S_{13}&S_{23}&S_{33}&S_{34}&S_{35}&S_{36}\\S_{14}&S_{24}&S_{34}&S_{44}&S_{45}&S_{46}\\S_{15}&S_{25}&S_{35}&S_{45}&S_{55}&S_{56}\\S_{16}&S_{26}&S_{36}&S_{46}&S_{56}&S_{66}\end{bmatrix}}{\begin{bmatrix}\sigma _{1}\\\sigma _{2}\\\sigma _{3}\\\sigma _{4}\\\sigma _{5}\\\sigma _{6}\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>11</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>13</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>14</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>15</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>16</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>22</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>23</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>24</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>25</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>26</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>13</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>23</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>33</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>34</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>35</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>36</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>14</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>24</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>34</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>44</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>45</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>46</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>15</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>25</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>35</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>45</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>55</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>56</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>16</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>26</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>36</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>46</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>56</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>66</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}\varepsilon _{1}\\\varepsilon _{2}\\\varepsilon _{3}\\\varepsilon _{4}\\\varepsilon _{5}\\\varepsilon _{6}\end{bmatrix}}={\begin{bmatrix}S_{11}&S_{12}&S_{13}&S_{14}&S_{15}&S_{16}\\S_{12}&S_{22}&S_{23}&S_{24}&S_{25}&S_{26}\\S_{13}&S_{23}&S_{33}&S_{34}&S_{35}&S_{36}\\S_{14}&S_{24}&S_{34}&S_{44}&S_{45}&S_{46}\\S_{15}&S_{25}&S_{35}&S_{45}&S_{55}&S_{56}\\S_{16}&S_{26}&S_{36}&S_{46}&S_{56}&S_{66}\end{bmatrix}}{\begin{bmatrix}\sigma _{1}\\\sigma _{2}\\\sigma _{3}\\\sigma _{4}\\\sigma _{5}\\\sigma _{6}\end{bmatrix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c980b5712b53a589f53d4de8c520061d17ad0b41" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -9.005ex; width:50.594ex; height:19.176ex;" alt="{\displaystyle {\begin{bmatrix}\varepsilon _{1}\\\varepsilon _{2}\\\varepsilon _{3}\\\varepsilon _{4}\\\varepsilon _{5}\\\varepsilon _{6}\end{bmatrix}}={\begin{bmatrix}S_{11}&S_{12}&S_{13}&S_{14}&S_{15}&S_{16}\\S_{12}&S_{22}&S_{23}&S_{24}&S_{25}&S_{26}\\S_{13}&S_{23}&S_{33}&S_{34}&S_{35}&S_{36}\\S_{14}&S_{24}&S_{34}&S_{44}&S_{45}&S_{46}\\S_{15}&S_{25}&S_{35}&S_{45}&S_{55}&S_{56}\\S_{16}&S_{26}&S_{36}&S_{46}&S_{56}&S_{66}\end{bmatrix}}{\begin{bmatrix}\sigma _{1}\\\sigma _{2}\\\sigma _{3}\\\sigma _{4}\\\sigma _{5}\\\sigma _{6}\end{bmatrix}}}"> In order to simplify the 3D stress direction, the plane stress assumption is apply that the out–of–plane stress and out–of–plane strain are insignificant or zero. That is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{3}=\sigma _{4}=\sigma _{5}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{3}=\sigma _{4}=\sigma _{5}=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fec5c739865716a5d5550f964057553b3d722343" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:17.603ex; height:2.509ex;" alt="{\displaystyle \sigma _{3}=\sigma _{4}=\sigma _{5}=0}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon _{4}=\varepsilon _{5}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon _{4}=\varepsilon _{5}=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5751aa34619e871967341a46cc737cbf85660e44" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.635ex; height:2.509ex;" alt="{\displaystyle \varepsilon _{4}=\varepsilon _{5}=0}">.<a href="#cite_note-:0-90">[90]</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}\varepsilon _{1}\\\varepsilon _{2}\\\varepsilon _{3}\\\varepsilon _{4}\\\varepsilon _{5}\\\varepsilon _{6}\end{bmatrix}}={\begin{bmatrix}{\tfrac {1}{E_{\rm {1}}}}&-{\tfrac {\nu _{\rm {21}}}{E_{\rm {2}}}}&-{\tfrac {\nu _{\rm {31}}}{E_{\rm {3}}}}&0&0&0\\-{\tfrac {\nu _{\rm {12}}}{E_{\rm {1}}}}&{\tfrac {1}{E_{\rm {2}}}}&-{\tfrac {\nu _{\rm {32}}}{E_{\rm {3}}}}&0&0&0\\-{\tfrac {\nu _{\rm {13}}}{E_{\rm {1}}}}&-{\tfrac {\nu _{\rm {23}}}{E_{\rm {2}}}}&{\tfrac {1}{E_{\rm {3}}}}&0&0&0\\0&0&0&{\tfrac {1}{G_{\rm {23}}}}&0&0\\0&0&0&0&{\tfrac {1}{G_{\rm {31}}}}&0\\0&0&0&0&0&{\tfrac {1}{G_{\rm {12}}}}\\\end{bmatrix}}{\begin{bmatrix}\sigma _{1}\\\sigma _{2}\\\sigma _{3}\\\sigma _{4}\\\sigma _{5}\\\sigma _{6}\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </mrow> </msub> </mfrac> </mstyle> </mrow> </mtd> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <msub> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>21</mn> </mrow> </mrow> </msub> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </mrow> </msub> </mfrac> </mstyle> </mrow> </mtd> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <msub> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>31</mn> </mrow> </mrow> </msub> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </mrow> </msub> </mfrac> </mstyle> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <msub> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </mrow> </msub> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </mrow> </msub> </mfrac> </mstyle> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </mrow> </msub> </mfrac> </mstyle> </mrow> </mtd> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <msub> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>32</mn> </mrow> </mrow> </msub> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </mrow> </msub> </mfrac> </mstyle> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <msub> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>13</mn> </mrow> </mrow> </msub> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </mrow> </msub> </mfrac> </mstyle> </mrow> </mtd> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <msub> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>23</mn> </mrow> </mrow> </msub> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </mrow> </msub> </mfrac> </mstyle> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </mrow> </msub> </mfrac> </mstyle> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <msub> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>23</mn> </mrow> </mrow> </msub> </mfrac> </mstyle> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <msub> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>31</mn> </mrow> </mrow> </msub> </mfrac> </mstyle> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <msub> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </mrow> </msub> </mfrac> </mstyle> </mrow> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}\varepsilon _{1}\\\varepsilon _{2}\\\varepsilon _{3}\\\varepsilon _{4}\\\varepsilon _{5}\\\varepsilon _{6}\end{bmatrix}}={\begin{bmatrix}{\tfrac {1}{E_{\rm {1}}}}&-{\tfrac {\nu _{\rm {21}}}{E_{\rm {2}}}}&-{\tfrac {\nu _{\rm {31}}}{E_{\rm {3}}}}&0&0&0\\-{\tfrac {\nu _{\rm {12}}}{E_{\rm {1}}}}&{\tfrac {1}{E_{\rm {2}}}}&-{\tfrac {\nu _{\rm {32}}}{E_{\rm {3}}}}&0&0&0\\-{\tfrac {\nu _{\rm {13}}}{E_{\rm {1}}}}&-{\tfrac {\nu _{\rm {23}}}{E_{\rm {2}}}}&{\tfrac {1}{E_{\rm {3}}}}&0&0&0\\0&0&0&{\tfrac {1}{G_{\rm {23}}}}&0&0\\0&0&0&0&{\tfrac {1}{G_{\rm {31}}}}&0\\0&0&0&0&0&{\tfrac {1}{G_{\rm {12}}}}\\\end{bmatrix}}{\begin{bmatrix}\sigma _{1}\\\sigma _{2}\\\sigma _{3}\\\sigma _{4}\\\sigma _{5}\\\sigma _{6}\end{bmatrix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/330206559f52e4e116cdd6c664d8c201030dfb2b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -12.505ex; width:56.533ex; height:26.176ex;" alt="{\displaystyle {\begin{bmatrix}\varepsilon _{1}\\\varepsilon _{2}\\\varepsilon _{3}\\\varepsilon _{4}\\\varepsilon _{5}\\\varepsilon _{6}\end{bmatrix}}={\begin{bmatrix}{\tfrac {1}{E_{\rm {1}}}}&-{\tfrac {\nu _{\rm {21}}}{E_{\rm {2}}}}&-{\tfrac {\nu _{\rm {31}}}{E_{\rm {3}}}}&0&0&0\\-{\tfrac {\nu _{\rm {12}}}{E_{\rm {1}}}}&{\tfrac {1}{E_{\rm {2}}}}&-{\tfrac {\nu _{\rm {32}}}{E_{\rm {3}}}}&0&0&0\\-{\tfrac {\nu _{\rm {13}}}{E_{\rm {1}}}}&-{\tfrac {\nu _{\rm {23}}}{E_{\rm {2}}}}&{\tfrac {1}{E_{\rm {3}}}}&0&0&0\\0&0&0&{\tfrac {1}{G_{\rm {23}}}}&0&0\\0&0&0&0&{\tfrac {1}{G_{\rm {31}}}}&0\\0&0&0&0&0&{\tfrac {1}{G_{\rm {12}}}}\\\end{bmatrix}}{\begin{bmatrix}\sigma _{1}\\\sigma _{2}\\\sigma _{3}\\\sigma _{4}\\\sigma _{5}\\\sigma _{6}\end{bmatrix}}}"> The stiffness matrix and compliance matrix can be reduced to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}\sigma _{1}\\\sigma _{2}\\\sigma _{6}\end{bmatrix}}={\begin{bmatrix}{\tfrac {E_{\rm {1}}}{1-{\nu _{\rm {12}}}{\nu _{\rm {21}}}}}&{\tfrac {E_{\rm {2}}{\nu _{\rm {12}}}}{1-{\nu _{\rm {12}}}{\nu _{\rm {21}}}}}&0\\{\tfrac {E_{\rm {2}}{\nu _{\rm {12}}}}{1-{\nu _{\rm {12}}}{\nu _{\rm {21}}}}}&{\tfrac {E_{\rm {2}}}{1-{\nu _{\rm {12}}}{\nu _{\rm {21}}}}}&0\\0&0&G_{\rm {12}}\\\end{bmatrix}}{\begin{bmatrix}\varepsilon _{1}\\\varepsilon _{2}\\\varepsilon _{6}\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </mrow> </msub> <mrow> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>21</mn> </mrow> </mrow> </msub> </mrow> </mrow> </mfrac> </mstyle> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </mrow> </msub> </mrow> </mrow> <mrow> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>21</mn> </mrow> </mrow> </msub> </mrow> </mrow> </mfrac> </mstyle> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </mrow> </msub> </mrow> </mrow> <mrow> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>21</mn> </mrow> </mrow> </msub> </mrow> </mrow> </mfrac> </mstyle> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </mrow> </msub> <mrow> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>21</mn> </mrow> </mrow> </msub> </mrow> </mrow> </mfrac> </mstyle> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <msub> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}\sigma _{1}\\\sigma _{2}\\\sigma _{6}\end{bmatrix}}={\begin{bmatrix}{\tfrac {E_{\rm {1}}}{1-{\nu _{\rm {12}}}{\nu _{\rm {21}}}}}&{\tfrac {E_{\rm {2}}{\nu _{\rm {12}}}}{1-{\nu _{\rm {12}}}{\nu _{\rm {21}}}}}&0\\{\tfrac {E_{\rm {2}}{\nu _{\rm {12}}}}{1-{\nu _{\rm {12}}}{\nu _{\rm {21}}}}}&{\tfrac {E_{\rm {2}}}{1-{\nu _{\rm {12}}}{\nu _{\rm {21}}}}}&0\\0&0&G_{\rm {12}}\\\end{bmatrix}}{\begin{bmatrix}\varepsilon _{1}\\\varepsilon _{2}\\\varepsilon _{6}\end{bmatrix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9783cbb8e73e64f69c2dbc4bba7d8c66d6865fc7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.505ex; width:42.64ex; height:12.176ex;" alt="{\displaystyle {\begin{bmatrix}\sigma _{1}\\\sigma _{2}\\\sigma _{6}\end{bmatrix}}={\begin{bmatrix}{\tfrac {E_{\rm {1}}}{1-{\nu _{\rm {12}}}{\nu _{\rm {21}}}}}&{\tfrac {E_{\rm {2}}{\nu _{\rm {12}}}}{1-{\nu _{\rm {12}}}{\nu _{\rm {21}}}}}&0\\{\tfrac {E_{\rm {2}}{\nu _{\rm {12}}}}{1-{\nu _{\rm {12}}}{\nu _{\rm {21}}}}}&{\tfrac {E_{\rm {2}}}{1-{\nu _{\rm {12}}}{\nu _{\rm {21}}}}}&0\\0&0&G_{\rm {12}}\\\end{bmatrix}}{\begin{bmatrix}\varepsilon _{1}\\\varepsilon _{2}\\\varepsilon _{6}\end{bmatrix}}}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}\varepsilon _{1}\\\varepsilon _{2}\\\varepsilon _{6}\end{bmatrix}}={\begin{bmatrix}{\tfrac {1}{E_{\rm {1}}}}&-{\tfrac {\nu _{\rm {21}}}{E_{\rm {2}}}}&0\\-{\tfrac {\nu _{\rm {12}}}{E_{\rm {1}}}}&{\tfrac {1}{E_{\rm {2}}}}&0\\0&0&{\tfrac {1}{G_{\rm {12}}}}\\\end{bmatrix}}{\begin{bmatrix}\sigma _{1}\\\sigma _{2}\\\sigma _{6}\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </mrow> </msub> </mfrac> </mstyle> </mrow> </mtd> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <msub> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>21</mn> </mrow> </mrow> </msub> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </mrow> </msub> </mfrac> </mstyle> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <msub> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </mrow> </msub> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </mrow> </msub> </mfrac> </mstyle> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </mrow> </msub> </mfrac> </mstyle> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <msub> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </mrow> </msub> </mfrac> </mstyle> </mrow> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}\varepsilon _{1}\\\varepsilon _{2}\\\varepsilon _{6}\end{bmatrix}}={\begin{bmatrix}{\tfrac {1}{E_{\rm {1}}}}&-{\tfrac {\nu _{\rm {21}}}{E_{\rm {2}}}}&0\\-{\tfrac {\nu _{\rm {12}}}{E_{\rm {1}}}}&{\tfrac {1}{E_{\rm {2}}}}&0\\0&0&{\tfrac {1}{G_{\rm {12}}}}\\\end{bmatrix}}{\begin{bmatrix}\sigma _{1}\\\sigma _{2}\\\sigma _{6}\end{bmatrix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/702192918ae7993fc8b48c9407a7b2b5cdc08dcb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.535ex; margin-bottom: -0.303ex; width:37.357ex; height:12.843ex;" alt="{\displaystyle {\begin{bmatrix}\varepsilon _{1}\\\varepsilon _{2}\\\varepsilon _{6}\end{bmatrix}}={\begin{bmatrix}{\tfrac {1}{E_{\rm {1}}}}&-{\tfrac {\nu _{\rm {21}}}{E_{\rm {2}}}}&0\\-{\tfrac {\nu _{\rm {12}}}{E_{\rm {1}}}}&{\tfrac {1}{E_{\rm {2}}}}&0\\0&0&{\tfrac {1}{G_{\rm {12}}}}\\\end{bmatrix}}{\begin{bmatrix}\sigma _{1}\\\sigma _{2}\\\sigma _{6}\end{bmatrix}}}"> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Transform_coordinate_system.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/40/Transform_coordinate_system.png/331px-Transform_coordinate_system.png" decoding="async" width="331" height="210" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/40/Transform_coordinate_system.png/497px-Transform_coordinate_system.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/40/Transform_coordinate_system.png/662px-Transform_coordinate_system.png 2x" data-file-width="1923" data-file-height="1222" /></a><figcaption>Two different coordinate systems of material. The structure has a (1-2) coordinate system. The material has a (x-y) principal coordinate system.</figcaption></figure> For fiber-reinforced composite, the fiber orientation in material affect anisotropic properties of the structure. From characterizing technique i.e. tensile testing, the material properties were measured based on sample (1-2) coordinate system. The tensors above express stress-strain relationship in (1-2) coordinate system. While the known material properties is in the principal coordinate system (x-y) of material. Transforming the tensor between two coordinate system help identify the material properties of the tested sample. The <a href="/wiki/Transformation_matrix" title="Transformation matrix">transformation matrix</a> with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }"> degree rotation is <a href="#cite_note-:0-90">[90]</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T(\theta )_{\epsilon }={\begin{bmatrix}\cos ^{2}\theta &\sin ^{2}\theta &\cos \theta \sin \theta \\sin^{2}\theta &\cos ^{2}\theta &-\cos \theta \sin \theta \\-2\cos \theta \sin \theta &2\cos \theta \sin \theta &\cos ^{2}\theta -\sin ^{2}\theta \end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo stretchy="false">(</mo> <mi>θ</mi> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>ϵ</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msup> <mi>cos</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> </mtd> <mtd> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> </mtd> <mtd> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> </mtd> </mtr> <mtr> <mtd> <mi>s</mi> <mi>i</mi> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>θ</mi> </mtd> <mtd> <msup> <mi>cos</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> </mtd> <mtd> <mo>−</mo> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mn>2</mn> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> </mtd> <mtd> <mn>2</mn> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> </mtd> <mtd> <msup> <mi>cos</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> <mo>−</mo> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T(\theta )_{\epsilon }={\begin{bmatrix}\cos ^{2}\theta &\sin ^{2}\theta &\cos \theta \sin \theta \\sin^{2}\theta &\cos ^{2}\theta &-\cos \theta \sin \theta \\-2\cos \theta \sin \theta &2\cos \theta \sin \theta &\cos ^{2}\theta -\sin ^{2}\theta \end{bmatrix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ec510c56fbfd6bbb217830f9a265ea5ab4f6cb2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.338ex; width:54.427ex; height:9.843ex;" alt="{\displaystyle T(\theta )_{\epsilon }={\begin{bmatrix}\cos ^{2}\theta &\sin ^{2}\theta &\cos \theta \sin \theta \\sin^{2}\theta &\cos ^{2}\theta &-\cos \theta \sin \theta \\-2\cos \theta \sin \theta &2\cos \theta \sin \theta &\cos ^{2}\theta -\sin ^{2}\theta \end{bmatrix}}}"> for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}{\acute {\epsilon }}\end{bmatrix}}=T(\theta )_{\epsilon }{\begin{bmatrix}\epsilon \end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ϵ</mi> <mo>´</mo> </mover> </mrow> </mrow> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>=</mo> <mi>T</mi> <mo stretchy="false">(</mo> <mi>θ</mi> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>ϵ</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>ϵ</mi> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}{\acute {\epsilon }}\end{bmatrix}}=T(\theta )_{\epsilon }{\begin{bmatrix}\epsilon \end{bmatrix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9b32dcefdacd36f8c91bf2da2c4b5ccc1e6d0c81" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.86ex; height:2.843ex;" alt="{\displaystyle {\begin{bmatrix}{\acute {\epsilon }}\end{bmatrix}}=T(\theta )_{\epsilon }{\begin{bmatrix}\epsilon \end{bmatrix}}}"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T(\theta )_{\sigma }={\begin{bmatrix}\cos ^{2}\theta &\sin ^{2}\theta &2\cos \theta \sin \theta \\sin^{2}\theta &\cos ^{2}\theta &-2\cos \theta \sin \theta \\-\cos \theta \sin \theta &\cos \theta \sin \theta &\cos ^{2}\theta -\sin ^{2}\theta \end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo stretchy="false">(</mo> <mi>θ</mi> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>σ</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msup> <mi>cos</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> </mtd> <mtd> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> </mtd> <mtd> <mn>2</mn> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> </mtd> </mtr> <mtr> <mtd> <mi>s</mi> <mi>i</mi> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>θ</mi> </mtd> <mtd> <msup> <mi>cos</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> </mtd> <mtd> <mo>−</mo> <mn>2</mn> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> </mtd> <mtd> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> </mtd> <mtd> <msup> <mi>cos</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> <mo>−</mo> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T(\theta )_{\sigma }={\begin{bmatrix}\cos ^{2}\theta &\sin ^{2}\theta &2\cos \theta \sin \theta \\sin^{2}\theta &\cos ^{2}\theta &-2\cos \theta \sin \theta \\-\cos \theta \sin \theta &\cos \theta \sin \theta &\cos ^{2}\theta -\sin ^{2}\theta \end{bmatrix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/699c42c7af7956bd2f7cfee585cfc2989ba61144" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.338ex; width:51.988ex; height:9.843ex;" alt="{\displaystyle T(\theta )_{\sigma }={\begin{bmatrix}\cos ^{2}\theta &\sin ^{2}\theta &2\cos \theta \sin \theta \\sin^{2}\theta &\cos ^{2}\theta &-2\cos \theta \sin \theta \\-\cos \theta \sin \theta &\cos \theta \sin \theta &\cos ^{2}\theta -\sin ^{2}\theta \end{bmatrix}}}"> for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}{\acute {\sigma }}\end{bmatrix}}=T(\theta )_{\sigma }{\begin{bmatrix}\sigma \end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>σ</mi> <mo>´</mo> </mover> </mrow> </mrow> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>=</mo> <mi>T</mi> <mo stretchy="false">(</mo> <mi>θ</mi> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>σ</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>σ</mi> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}{\acute {\sigma }}\end{bmatrix}}=T(\theta )_{\sigma }{\begin{bmatrix}\sigma \end{bmatrix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/40682743299cc80da928f799c800d3c7931c7269" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.557ex; height:2.843ex;" alt="{\displaystyle {\begin{bmatrix}{\acute {\sigma }}\end{bmatrix}}=T(\theta )_{\sigma }{\begin{bmatrix}\sigma \end{bmatrix}}}"> <div class="mw-heading mw-heading3"><h3 id="Types_of_fibers_and_mechanical_properties">Types of fibers and mechanical properties</h3>[<a href="/w/index.php?title=Composite_material&action=edit&section=23" title="Edit section: Types of fibers and mechanical properties">edit</a>]</div> The most common types of fibers used in industry are <a href="/wiki/Glass_fiber" title="Glass fiber">glass fibers</a>, <a href="/wiki/Carbon_fibers" title="Carbon fibers">carbon fibers</a>, and <a href="/wiki/Kevlar" title="Kevlar">kevlar</a> due to their ease of production and availability. Their mechanical properties are very important to know, therefore the table of their mechanical properties is given below to compare them with S97 <a href="/wiki/Steel" title="Steel">steel</a>.<a href="#cite_note-91">[91]</a><a href="#cite_note-92">[92]</a><a href="#cite_note-93">[93]</a><a href="#cite_note-94">[94]</a> The angle of fiber orientation is very important because of the anisotropy of fiber composites (please see the section "<a href="#Physical_properties">Physical properties</a>" for a more detailed explanation). The mechanical properties of the composites can be tested using standard <a href="/wiki/Mechanical_testing" title="Mechanical testing">mechanical testing</a> methods by positioning the samples at various angles (the standard angles are 0°, 45°, and 90°) with respect to the orientation of fibers within the composites. In general, 0° axial alignment makes composites resistant to longitudinal bending and axial tension/compression, 90° hoop alignment is used to obtain resistance to internal/external pressure, and ± 45° is the ideal choice to obtain resistance against pure torsion.<a href="#cite_note-95">[95]</a> <div class="mw-heading mw-heading4"><h4 id="Mechanical_properties_of_fiber_composite_materials">Mechanical properties of fiber composite materials</h4>[<a href="/w/index.php?title=Composite_material&action=edit&section=24" title="Edit section: Mechanical properties of fiber composite materials">edit</a>]</div> <table class="wikitable"> <caption>Fibres @ 0° (UD), 0/90° (fabric) to loading axis, Dry, Room Temperature, Vf = 60% (UD), 50% (fabric) Fibre / Epoxy Resin (cured at 120 °C)<a href="#cite_note-:2-96">[96]</a> </caption> <tbody><tr> <td> </td> <th>Symbol </th> <th>Units </th> <th>Standard Carbon Fiber Fabric </th> <th>High Modulus Carbon Fiber Fabric </th> <th>E-Glass Fibre Glass Fabric </th> <th>Kevlar Fabric </th> <th>Standard Unidirectional Carbon Fiber Fabric </th> <th>High Modulus Unidirectional Carbon Fiber Fabric </th> <th>E-Glass Unidirectional Fiber Glass Fabric </th> <th>Kevlar Unidirectional Fabric </th> <th>Steel S97 </th></tr> <tr> <th>Young's Modulus 0° </th> <td>E1 </td> <td>GPa </td> <td>70 </td> <td>85 </td> <td>25 </td> <td>30 </td> <td>135 </td> <td>175 </td> <td>40 </td> <td>75 </td> <td>207 </td></tr> <tr> <th>Young's Modulus 90° </th> <td>E2 </td> <td>GPa </td> <td>70 </td> <td>85 </td> <td>25 </td> <td>30 </td> <td>10 </td> <td>8 </td> <td>8 </td> <td>6 </td> <td>207 </td></tr> <tr> <th>In-plane Shear Modulus </th> <td>G12 </td> <td>GPa </td> <td>5 </td> <td>5 </td> <td>4 </td> <td>5 </td> <td>5 </td> <td>5 </td> <td>4 </td> <td>2 </td> <td>80 </td></tr> <tr> <th>Major Poisson's Ratio </th> <td>v12 </td> <td> </td> <td>0.10 </td> <td>0.10 </td> <td>0.20 </td> <td>0.20 </td> <td>0.30 </td> <td>0.30 </td> <td>0.25 </td> <td>0.34 </td> <td>– </td></tr> <tr> <th>Ult. Tensile Strength 0° </th> <td>Xt </td> <td>MPa </td> <td>600 </td> <td>350 </td> <td>440 </td> <td>480 </td> <td>1500 </td> <td>1000 </td> <td>1000 </td> <td>1300 </td> <td>990 </td></tr> <tr> <th>Ult. Comp. Strength 0° </th> <td>Xc </td> <td>MPa </td> <td>570 </td> <td>150 </td> <td>425 </td> <td>190 </td> <td>1200 </td> <td>850 </td> <td>600 </td> <td>280 </td> <td>– </td></tr> <tr> <th>Ult. Tensile Strength 90° </th> <td>Yt </td> <td>MPa </td> <td>600 </td> <td>350 </td> <td>440 </td> <td>480 </td> <td>50 </td> <td>40 </td> <td>30 </td> <td>30 </td> <td>– </td></tr> <tr> <th>Ult. Comp. Strength 90° </th> <td>Yc </td> <td>MPa </td> <td>570 </td> <td>150 </td> <td>425 </td> <td>190 </td> <td>250 </td> <td>200 </td> <td>110 </td> <td>140 </td> <td>– </td></tr> <tr> <th>Ult. In-plane Shear Stren. </th> <td>S </td> <td>MPa </td> <td>90 </td> <td>35 </td> <td>40 </td> <td>50 </td> <td>70 </td> <td>60 </td> <td>40 </td> <td>60 </td> <td>– </td></tr> <tr> <th>Ult. Tensile Strain 0° </th> <td>ext </td> <td>% </td> <td>0.85 </td> <td>0.40 </td> <td>1.75 </td> <td>1.60 </td> <td>1.05 </td> <td>0.55 </td> <td>2.50 </td> <td>1.70 </td> <td>– </td></tr> <tr> <th>Ult. Comp. Strain 0° </th> <td>exc </td> <td>% </td> <td>0.80 </td> <td>0.15 </td> <td>1.70 </td> <td>0.60 </td> <td>0.85 </td> <td>0.45 </td> <td>1.50 </td> <td>0.35 </td> <td>– </td></tr> <tr> <th>Ult. Tensile Strain 90° </th> <td>eyt </td> <td>% </td> <td>0.85 </td> <td>0.40 </td> <td>1.75 </td> <td>1.60 </td> <td>0.50 </td> <td>0.50 </td> <td>0.35 </td> <td>0.50 </td> <td>– </td></tr> <tr> <th>Ult. Comp. Strain 90° </th> <td>eyc </td> <td>% </td> <td>0.80 </td> <td>0.15 </td> <td>1.70 </td> <td>0.60 </td> <td>2.50 </td> <td>2.50 </td> <td>1.35 </td> <td>2.30 </td> <td>– </td></tr> <tr> <th>Ult. In-plane shear strain </th> <td>es </td> <td>% </td> <td>1.80 </td> <td>0.70 </td> <td>1.00 </td> <td>1.00 </td> <td>1.40 </td> <td>1.20 </td> <td>1.00 </td> <td>3.00 </td> <td>– </td></tr> <tr> <th>Density </th> <td> </td> <td>g/cc </td> <td>1.60 </td> <td>1.60 </td> <td>1.90 </td> <td>1.40 </td> <td>1.60 </td> <td>1.60 </td> <td>1.90 </td> <td>1.40 </td> <td>– </td></tr></tbody></table> <table class="wikitable"> <caption>Fibres @ ±45 Deg. to loading axis, Dry, Room Temperature, Vf = 60% (UD), 50% (fabric)<a href="#cite_note-:2-96">[96]</a> </caption> <tbody><tr> <th> </th> <th>Symbol </th> <th>Units </th> <th>Standard Carbon Fiber </th> <th>High Modulus Carbon Fiber </th> <th>E-Glass Fiber Glass </th> <th>Standard Carbon Fibers Fabric </th> <th>E-Glass Fiber Glass Fabric </th> <th>Steel </th> <th>Al </th></tr> <tr> <th>Longitudinal Modulus </th> <td>E1 </td> <td>GPa </td> <td>17 </td> <td>17 </td> <td>12.3 </td> <td>19.1 </td> <td>12.2 </td> <td>207 </td> <td>72 </td></tr> <tr> <th>Transverse Modulus </th> <td>E2 </td> <td>GPa </td> <td>17 </td> <td>17 </td> <td>12.3 </td> <td>19.1 </td> <td>12.2 </td> <td>207 </td> <td>72 </td></tr> <tr> <th>In Plane Shear Modulus </th> <td>G12 </td> <td>GPa </td> <td>33 </td> <td>47 </td> <td>11 </td> <td>30 </td> <td>8 </td> <td>80 </td> <td>25 </td></tr> <tr> <th>Poisson's Ratio </th> <td>v12 </td> <td> </td> <td>.77 </td> <td>.83 </td> <td>.53 </td> <td>.74 </td> <td>.53 </td> <td> </td> <td> </td></tr> <tr> <th>Tensile Strength </th> <td>Xt </td> <td>MPa </td> <td>110 </td> <td>110 </td> <td>90 </td> <td>120 </td> <td>120 </td> <td>990 </td> <td>460 </td></tr> <tr> <th>Compressive Strength </th> <td>Xc </td> <td>MPa </td> <td>110 </td> <td>110 </td> <td>90 </td> <td>120 </td> <td>120 </td> <td>990 </td> <td>460 </td></tr> <tr> <th>In Plane Shear Strength </th> <td>S </td> <td>MPa </td> <td>260 </td> <td>210 </td> <td>100 </td> <td>310 </td> <td>150 </td> <td> </td> <td> </td></tr> <tr> <th>Thermal Expansion Co-ef </th> <td>Alpha1 </td> <td>Strain/K </td> <td>2.15 E-6 </td> <td>0.9 E-6 </td> <td>12 E-6 </td> <td>4.9 E-6 </td> <td>10 E-6 </td> <td>11 E-6 </td> <td>23 E-6 </td></tr> <tr> <th>Moisture Co-ef </th> <td>Beta1 </td> <td>Strain/K </td> <td>3.22 E-4 </td> <td>2.49 E-4 </td> <td>6.9 E-4 </td> <td> </td> <td> </td> <td> </td> <td> </td></tr></tbody></table> <div class="mw-heading mw-heading4"><h4 id="Carbon_fiber_&_fiberglass_composites_vs._aluminum_alloy_and_steel">Carbon fiber & fiberglass composites vs. aluminum alloy and steel</h4>[<a href="/w/index.php?title=Composite_material&action=edit&section=25" title="Edit section: Carbon fiber & fiberglass composites vs. aluminum alloy and steel">edit</a>]</div> Although strength and stiffness of <a href="/wiki/Steel" title="Steel">steel</a> and <a href="/wiki/Aluminium_alloy" title="Aluminium alloy">aluminum alloys</a> are comparable to fiber composites, <a href="/wiki/Specific_strength" title="Specific strength">specific strength</a> and <a href="/wiki/Specific_modulus" title="Specific modulus">stiffness</a> of composites (i.e. in relation to their weight) are significantly higher. <table class="wikitable"> <caption>Comparison of Cost, Specific Strength, and Specific Stiffness<a href="#cite_note-97">[97]</a> </caption> <tbody><tr> <td> </td> <td>Carbon Fiber Composite (aerospace grade) </td> <td>Carbon Fiber Composite (commercial grade) </td> <td>Fiberglass Composite </td> <td>Aluminum 6061 T-6 </td> <td>Steel, Mild </td></tr> <tr> <td>Cost $/LB </td> <td>$20 – $250+ </td> <td>$5 – $20 </td> <td>$1.50 – $3.00 </td> <td>$3 </td> <td>$0.30 </td></tr> <tr> <td>Strength (psi) </td> <td>90,000 – 200,000 </td> <td>50,000 – 90,000 </td> <td>20,000 – 35,000 </td> <td>35,000 </td> <td>60,000 </td></tr> <tr> <td>Stiffness (psi) </td> <td>10 x 106– 50 x 106 </td> <td>8 x 106 – 10 x 106 </td> <td>1 x 106 – 1.5 x 106 </td> <td>10 x 106 </td> <td>30 x 106 </td></tr> <tr> <td>Density (lb/in3) </td> <td>0.050 </td> <td>0.050 </td> <td>0.055 </td> <td>0.10 </td> <td>0.30 </td></tr> <tr> <td>Specific Strength </td> <td>1.8 x 106 – 4 x 106 </td> <td>1 x 106 – 1.8 x 106 </td> <td>363,640–636,360 </td> <td>350,000 </td> <td>200,000 </td></tr> <tr> <td>Specific Stiffness </td> <td>200 x 106 – 1,000 x 106 </td> <td>160 x 106 – 200 x 106 </td> <td>18 x 106 – 27 x 106 </td> <td>100 x 106 </td> <td>100 x 106 </td></tr></tbody></table> <div class="mw-heading mw-heading3"><h3 id="Failure">Failure</h3>[<a href="/w/index.php?title=Composite_material&action=edit&section=26" title="Edit section: Failure">edit</a>]</div> Shock, impact of varying speed, or repeated cyclic stresses can provoke the laminate to separate at the interface between two layers, a condition known as <a href="/wiki/Delamination" title="Delamination">delamination</a>.<a href="#cite_note-98">[98]</a><a href="#cite_note-99">[99]</a> Individual fibres can separate from the matrix, for example, <a href="/wiki/Fiber_pull-out" title="Fiber pull-out">fibre pull-out</a>. Composites can fail on the <a href="/wiki/Macroscopic" class="mw-redirect" title="Macroscopic">macroscopic</a> or <a href="/wiki/Microscopic" class="mw-redirect" title="Microscopic">microscopic</a> scale. Compression failures can happen at both the macro scale or at each individual reinforcing fibre in compression buckling. Tension failures can be net section failures of the part or degradation of the composite at a microscopic scale where one or more of the layers in the composite fail in tension of the matrix or failure of the bond between the matrix and fibres. Some composites are brittle and possess little reserve strength beyond the initial onset of failure while others may have large deformations and have reserve energy absorbing capacity past the onset of damage. The distinctions in fibres and matrices that are available and the <a href="/wiki/Mixture" title="Mixture">mixtures</a> that can be made with blends leave a very broad range of properties that can be designed into a composite structure. The most famous failure of a brittle ceramic matrix composite occurred when the carbon-carbon composite tile on the leading edge of the wing of the <a href="/wiki/Space_Shuttle_Columbia" title="Space Shuttle Columbia">Space Shuttle Columbia</a> fractured when impacted during take-off. It directed to the catastrophic break-up of the vehicle when it re-entered the Earth's atmosphere on 1 February 2003. Composites have relatively poor bearing strength compared to metals. <div class="mw-heading mw-heading3"><h3 id="Testing">Testing</h3>[<a href="/w/index.php?title=Composite_material&action=edit&section=27" title="Edit section: Testing">edit</a>]</div> Composites are tested before and after construction to assist in predicting and preventing failures. Pre-construction testing may adopt finite element analysis (FEA) for ply-by-ply analysis of curved surfaces and predicting wrinkling, crimping and dimpling of composites.<a href="#cite_note-Waterman-100">[100]</a><a href="#cite_note-101">[101]</a><a href="#cite_note-102">[102]</a><a href="#cite_note-103">[103]</a> Materials may be tested during manufacturing and after construction by various non-destructive methods including ultrasonic, thermography, shearography and X-ray radiography,<a href="#cite_note-104">[104]</a> and laser bond inspection for NDT of relative bond strength integrity in a localized area. <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2>[<a href="/w/index.php?title=Composite_material&action=edit&section=28" title="Edit section: See also">edit</a>]</div> <ul><li><a href="/wiki/3D_composites" title="3D composites">3D composites</a></li> <li><a href="/wiki/Aluminium_composite_panel" class="mw-redirect" title="Aluminium composite panel">Aluminium composite panel</a></li> <li><a href="/wiki/American_Composites_Manufacturers_Association" title="American Composites Manufacturers Association">American Composites Manufacturers Association</a></li> <li><a href="/wiki/Chemical_vapour_infiltration" class="mw-redirect" title="Chemical vapour infiltration">Chemical vapour infiltration</a></li> <li><a href="/wiki/Composite_laminate" title="Composite laminate">Composite laminate</a></li> <li><a href="/wiki/Discontinuous_aligned_composite" title="Discontinuous aligned composite">Discontinuous aligned composite</a></li> <li><a href="/wiki/Epoxy_granite" title="Epoxy granite">Epoxy granite</a></li> <li><a href="/wiki/Hybrid_material" title="Hybrid material">Hybrid material</a></li> <li><a href="/wiki/Lay-up_process" title="Lay-up process">Lay-up process</a></li> <li><a href="/wiki/Nanocomposite" title="Nanocomposite">Nanocomposite</a></li> <li><a href="/wiki/Pykrete" title="Pykrete">Pykrete</a></li> <li><a href="/wiki/Rule_of_mixtures" title="Rule of mixtures">Rule of mixtures</a></li> <li><a href="/wiki/Scaled_Composites" title="Scaled Composites">Scaled Composites</a></li> <li><a href="/wiki/Smart_material" title="Smart material">Smart material</a></li> <li><a href="/wiki/Smart_Materials_and_Structures" title="Smart Materials and Structures">Smart Materials and Structures</a></li> <li><a href="/wiki/Void_(composites)" title="Void (composites)">Void (composites)</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2>[<a href="/w/index.php?title=Composite_material&action=edit&section=29" title="Edit section: References">edit</a>]</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 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Retrieved 2020-12-20.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=www.sicomin.com&rft.atitle=Composite+Casting+Processes&rft_id=http%3A%2F%2Fwww.sicomin.com%2Fprocesses%2Fcasting&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComposite+material" class="Z3988"> </li> <li id="cite_note-69"><a href="#cite_ref-69">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://compositeslab.com/composites-manufacturing-processes/closed-molding/centrifugal-casting/">"Centrifugal Casting - Closed Molding"</a>. CompositesLab. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20150926120921/http://compositeslab.com/composites-manufacturing-processes/closed-molding/centrifugal-casting/">Archived</a> from the original on 2015-09-26. Retrieved 2020-12-20.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=CompositesLab&rft.atitle=Centrifugal+Casting+-+Closed+Molding&rft_id=http%3A%2F%2Fcompositeslab.com%2Fcomposites-manufacturing-processes%2Fclosed-molding%2Fcentrifugal-casting%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComposite+material" class="Z3988"> </li> <li id="cite_note-70"><a href="#cite_ref-70">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKwaśniewskiKiesiewicz2014" class="citation journal cs1">Kwaśniewski, Paweł; Kiesiewicz, Grzegorz (2014). "Studies on Obtaining Cu-CNT Composites by Continuous Casting Method". Metallurgy and Foundry Engineering. 40 (2): 83. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.7494%2Fmafe.2014.40.2.83">10.7494/mafe.2014.40.2.83</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Metallurgy+and+Foundry+Engineering&rft.atitle=Studies+on+Obtaining+Cu-CNT+Composites+by+Continuous+Casting+Method&rft.volume=40&rft.issue=2&rft.pages=83&rft.date=2014&rft_id=info%3Adoi%2F10.7494%2Fmafe.2014.40.2.83&rft.aulast=Kwa%C5%9Bniewski&rft.aufirst=Pawe%C5%82&rft.au=Kiesiewicz%2C+Grzegorz&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComposite+material" class="Z3988"> </li> <li id="cite_note-71"><a href="#cite_ref-71">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1 cs1-prop-unfit"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20210523194853/https://netcomposites.com/guide/manufacturing/filament-winding/">"Filament Winding"</a>. NetComposites. Archived from the original on 2021-05-23. Retrieved 2020-12-20.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=NetComposites&rft.atitle=Filament+Winding&rft_id=https%3A%2F%2Fnetcomposites.com%2Fguide%2Fmanufacturing%2Ffilament-winding%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComposite+material" class="Z3988"> </li> <li id="cite_note-72"><a href="#cite_ref-72">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://www.shape-group.com/press-moulding-of-automotive-composites">"PRESS MOULDING OF AUTOMOTIVE COMPOSITES – Shape Group"</a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20200920183451/http://www.shape-group.com/press-moulding-of-automotive-composites">Archived</a> from the original on 2020-09-20. Retrieved 2020-12-20.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=PRESS+MOULDING+OF+AUTOMOTIVE+COMPOSITES+%E2%80%93+Shape+Group&rft_id=http%3A%2F%2Fwww.shape-group.com%2Fpress-moulding-of-automotive-composites&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComposite+material" class="Z3988"> </li> <li id="cite_note-73"><a href="#cite_ref-73">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFShrivastava2018" class="citation book cs1">Shrivastava, Anshuman (2018). "Plastics Processing". Introduction to Plastics Engineering. pp. 143–177. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2FB978-0-323-39500-7.00005-8">10.1016/B978-0-323-39500-7.00005-8</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-323-39500-7" title="Special:BookSources/978-0-323-39500-7"><bdi>978-0-323-39500-7</bdi></a>. <q>The term 'pultrusion' combines the word 'pull' and 'extrusion.' It is a continuous manufacturing process to produce products with constant cross sections such as profiles and sheets. Fig. 5.25 is a schematic illustration of general pultrusion setup. As shown in the figure, continuous fiber reinforcements are saturated (wet out) with desired resin matrix either in a resin bath or in resin injection chamber. The coated fibers then pass through heating and forming dies where curing of the resin and forming of the shape occur. After the die the composite is allowed to postcure while being pulled to the saw which cuts it into stock length. Different resin–fiber combinations are used to achieve the final desired properties</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Plastics+Processing&rft.btitle=Introduction+to+Plastics+Engineering&rft.pages=%3Cspan+class%3D%22nowrap%22%3E143-%3C%2Fspan%3E177&rft.date=2018&rft_id=info%3Adoi%2F10.1016%2FB978-0-323-39500-7.00005-8&rft.isbn=978-0-323-39500-7&rft.aulast=Shrivastava&rft.aufirst=Anshuman&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComposite+material" class="Z3988"> </li> <li id="cite_note-74"><a href="#cite_ref-74">^</a> <style data-mw-deduplicate="TemplateStyles:r1041539562">.mw-parser-output .citation{word-wrap:break-word}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}</style><a rel="nofollow" class="external autonumber" href="https://patents.google.com/patent/US9435085B1/en">[1]</a>, "System and method for slip forming monolithic reinforced composite concrete structures having multiple functionally discrete components", issued 2015-05-24<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Apatent&rft.number=&rft.cc=&rft.title=System+and+method+for+slip+forming+monolithic+reinforced+composite+concrete+structures+having+multiple+functionally+discrete+components&rft.date=2015-05-24">  <a rel="nofollow" class="external text" href="https://web.archive.org/web/20210608191250/https://patents.google.com/patent/US9435085B1/en">Archived</a> 2021-06-08 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a> </li> <li id="cite_note-75"><a href="#cite_ref-75">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKim2000" class="citation journal cs1">Kim, Hyoung Seop (September 2000). 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Materials Science and Engineering: A. 289 (1–2): 30–33. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2FS0921-5093%2800%2900909-6">10.1016/S0921-5093(00)00909-6</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Materials+Science+and+Engineering%3A+A&rft.atitle=On+the+rule+of+mixtures+for+the+hardness+of+particle+reinforced+composites&rft.volume=289&rft.issue=%3Cspan+class%3D%22nowrap%22%3E1%E2%80%93%3C%2Fspan%3E2&rft.pages=%3Cspan+class%3D%22nowrap%22%3E30-%3C%2Fspan%3E33&rft.date=2000-09&rft_id=info%3Adoi%2F10.1016%2FS0921-5093%2800%2900909-6&rft.aulast=Kim&rft.aufirst=Hyoung+Seop&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComposite+material" class="Z3988"> </li> <li id="cite_note-76"><a href="#cite_ref-76">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSoboyejo2003" class="citation book cs1">Soboyejo, W. O. (2003). "9.3.1 Constant-Strain and Constant-Stress Rules of Mixtures". Mechanical properties of engineered materials. Marcel Dekker. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-8247-8900-8" title="Special:BookSources/0-8247-8900-8"><bdi>0-8247-8900-8</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/300921090">300921090</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=9.3.1+Constant-Strain+and+Constant-Stress+Rules+of+Mixtures&rft.btitle=Mechanical+properties+of+engineered+materials&rft.pub=Marcel+Dekker&rft.date=2003&rft_id=info%3Aoclcnum%2F300921090&rft.isbn=0-8247-8900-8&rft.aulast=Soboyejo&rft.aufirst=W.+O.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComposite+material" class="Z3988"> </li> <li id="cite_note-77"><a href="#cite_ref-77">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCourtney2000" class="citation book cs1">Courtney, Thomas H. 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Materials and Structures. 46 (3): 405–420. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1617%2Fs11527-012-9909-z">10.1617/s11527-012-9909-z</a>. <a href="/wiki/Hdl_(identifier)" class="mw-redirect" title="Hdl (identifier)">hdl</a>:<a rel="nofollow" class="external text" href="https://hdl.handle.net/2027.42%2F94214">2027.42/94214</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Materials+and+Structures&rft.atitle=Rheology%2C+fiber+dispersion%2C+and+robust+properties+of+Engineered+Cementitious+Composites&rft.volume=46&rft.issue=3&rft.pages=%3Cspan+class%3D%22nowrap%22%3E405-%3C%2Fspan%3E420&rft.date=2013-03&rft_id=info%3Ahdl%2F2027.42%2F94214&rft_id=info%3Adoi%2F10.1617%2Fs11527-012-9909-z&rft.aulast=Li&rft.aufirst=Mo&rft.au=Li%2C+Victor+C.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComposite+material" class="Z3988"> </li> <li id="cite_note-80"><a href="#cite_ref-80">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation journal cs1">"Large-Scale Processing of Engineered Cementitious Composites". 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Retrieved 2020-05-22.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=www.performance-composites.com&rft.atitle=Mechanical+Properties+of+Carbon+Fibre+Composite+Materials&rft_id=http%3A%2F%2Fwww.performance-composites.com%2Fcarbonfibre%2Fmechanicalproperties_2.asp&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComposite+material" class="Z3988"> </li> <li id="cite_note-97"><a href="#cite_ref-97">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.performancecomposites.com/about-composites-technical-info/124-designing-with-carbon-fiber.pdf">"Carbon Fiber Composite Design Guide"</a> (PDF). www.performancecomposites.com. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20201030130724/https://www.performancecomposites.com/about-composites-technical-info/124-designing-with-carbon-fiber.pdf">Archived</a> (PDF) from the original on 2020-10-30. Retrieved 2020-05-22.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=www.performancecomposites.com&rft.atitle=Carbon+Fiber+Composite+Design+Guide&rft_id=https%3A%2F%2Fwww.performancecomposites.com%2Fabout-composites-technical-info%2F124-designing-with-carbon-fiber.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComposite+material" class="Z3988"> </li> <li id="cite_note-98"><a href="#cite_ref-98">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMaCaoFengSong2024" class="citation journal cs1">Ma, Binlin; Cao, Xiaofei; Feng, Yu; Song, Yujian; Yang, Fei; Li, Ying; Zhang, Deyue; Wang, Yipeng; He, Yuting (February 2024). "A comparative study on the low velocity impact behavior of UD, woven, and hybrid UD/woven FRP composite laminates". Composites Part B: Engineering. 271: 111133. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.compositesb.2023.111133">10.1016/j.compositesb.2023.111133</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Composites+Part+B%3A+Engineering&rft.atitle=A+comparative+study+on+the+low+velocity+impact+behavior+of+UD%2C+woven%2C+and+hybrid+UD%2Fwoven+FRP+composite+laminates&rft.volume=271&rft.pages=111133&rft.date=2024-02&rft_id=info%3Adoi%2F10.1016%2Fj.compositesb.2023.111133&rft.aulast=Ma&rft.aufirst=Binlin&rft.au=Cao%2C+Xiaofei&rft.au=Feng%2C+Yu&rft.au=Song%2C+Yujian&rft.au=Yang%2C+Fei&rft.au=Li%2C+Ying&rft.au=Zhang%2C+Deyue&rft.au=Wang%2C+Yipeng&rft.au=He%2C+Yuting&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComposite+material" class="Z3988"> </li> <li id="cite_note-99"><a href="#cite_ref-99">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSanchez-SaezBarberoZaeraNavarro2005" class="citation journal cs1">Sanchez-Saez, S.; Barbero, E.; Zaera, R.; Navarro, C. (October 2005). "Compression after impact of thin composite laminates". Composites Science and Technology. 65 (13): 1911–1919. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.compscitech.2005.04.009">10.1016/j.compscitech.2005.04.009</a>. <a href="/wiki/Hdl_(identifier)" class="mw-redirect" title="Hdl (identifier)">hdl</a>:<a rel="nofollow" class="external text" href="https://hdl.handle.net/10016%2F7498">10016/7498</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Composites+Science+and+Technology&rft.atitle=Compression+after+impact+of+thin+composite+laminates&rft.volume=65&rft.issue=13&rft.pages=%3Cspan+class%3D%22nowrap%22%3E1911-%3C%2Fspan%3E1919&rft.date=2005-10&rft_id=info%3Ahdl%2F10016%2F7498&rft_id=info%3Adoi%2F10.1016%2Fj.compscitech.2005.04.009&rft.aulast=Sanchez-Saez&rft.aufirst=S.&rft.au=Barbero%2C+E.&rft.au=Zaera%2C+R.&rft.au=Navarro%2C+C.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComposite+material" class="Z3988"> </li> <li id="cite_note-Waterman-100"><a href="#cite_ref-Waterman_100-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWaterman2007" class="citation news cs1">Waterman, Pamela (1 May 2007). <a rel="nofollow" class="external text" href="https://www.digitalengineering247.com/article/the-life-of-composite-materials/">"The Life of Composite Materials"</a>. Digital Engineering.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Digital+Engineering&rft.atitle=The+Life+of+Composite+Materials&rft.date=2007-05-01&rft.aulast=Waterman&rft.aufirst=Pamela&rft_id=https%3A%2F%2Fwww.digitalengineering247.com%2Farticle%2Fthe-life-of-composite-materials%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComposite+material" class="Z3988"> </li> <li id="cite_note-101"><a href="#cite_ref-101">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAghdamMorsali2013" class="citation journal cs1">Aghdam, M.M.; Morsali, S.R. (November 2013). "Damage initiation and collapse behavior of unidirectional metal matrix composites at elevated temperatures". Computational Materials Science. 79: 402–407. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.commatsci.2013.06.024">10.1016/j.commatsci.2013.06.024</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Computational+Materials+Science&rft.atitle=Damage+initiation+and+collapse+behavior+of+unidirectional+metal+matrix+composites+at+elevated+temperatures&rft.volume=79&rft.pages=%3Cspan+class%3D%22nowrap%22%3E402-%3C%2Fspan%3E407&rft.date=2013-11&rft_id=info%3Adoi%2F10.1016%2Fj.commatsci.2013.06.024&rft.aulast=Aghdam&rft.aufirst=M.M.&rft.au=Morsali%2C+S.R.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComposite+material" class="Z3988"> </li> <li id="cite_note-102"><a href="#cite_ref-102">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDebnathSingh2017" class="citation book cs1">Debnath, Kishore; Singh, Inderdeep, eds. (2017). Primary and Secondary Manufacturing of Polymer Matrix Composites. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1201%2F9781351228466">10.1201/9781351228466</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-351-22846-6" title="Special:BookSources/978-1-351-22846-6"><bdi>978-1-351-22846-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Primary+and+Secondary+Manufacturing+of+Polymer+Matrix+Composites&rft.date=2017&rft_id=info%3Adoi%2F10.1201%2F9781351228466&rft.isbn=978-1-351-22846-6&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComposite+material" class="Z3988">[<a href="/wiki/Wikipedia:Citing_sources" title="Wikipedia:Citing sources">page needed</a>] </li> <li id="cite_note-103"><a href="#cite_ref-103">^</a> <a rel="nofollow" class="external text" href="https://coventivecomposites.com/explainers/what-is-finite-element-analysis/">What is Finite Element Analysis?</a>[<a href="/wiki/Wikipedia:Link_rot" title="Wikipedia:Link rot">permanent dead link</a>‍] </li> <li id="cite_note-104"><a href="#cite_ref-104">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMatzkaninYolken" class="citation journal cs1">Matzkanin, George A.; Yolken, H. Thomas. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20081217033116/http://ammtiac.alionscience.com/pdf/AQV2N4.pdf">"Techniques for the Nondestructive Evaluation of Polymer Matrix Composites"</a> (PDF). AMMTIAC Quarterly. 2 (4). Archived from <a rel="nofollow" class="external text" href="http://ammtiac.alionscience.com/pdf/AQV2N4.pdf">the original</a> (PDF) on 2008-12-17.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=AMMTIAC+Quarterly&rft.atitle=Techniques+for+the+Nondestructive+Evaluation+of+Polymer+Matrix+Composites&rft.volume=2&rft.issue=4&rft.aulast=Matzkanin&rft.aufirst=George+A.&rft.au=Yolken%2C+H.+Thomas&rft_id=http%3A%2F%2Fammtiac.alionscience.com%2Fpdf%2FAQV2N4.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComposite+material" class="Z3988"> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2>[<a href="/w/index.php?title=Composite_material&action=edit&section=30" title="Edit section: Further reading">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1239549316">.mw-parser-output .refbegin{margin-bottom:0.5em}.mw-parser-output .refbegin-hanging-indents>ul{margin-left:0}.mw-parser-output .refbegin-hanging-indents>ul>li{margin-left:0;padding-left:3.2em;text-indent:-3.2em}.mw-parser-output .refbegin-hanging-indents ul,.mw-parser-output .refbegin-hanging-indents ul li{list-style:none}@media(max-width:720px){.mw-parser-output .refbegin-hanging-indents>ul>li{padding-left:1.6em;text-indent:-1.6em}}.mw-parser-output .refbegin-columns{margin-top:0.3em}.mw-parser-output .refbegin-columns ul{margin-top:0}.mw-parser-output .refbegin-columns li{page-break-inside:avoid;break-inside:avoid-column}@media screen{.mw-parser-output .refbegin{font-size:90%}}</style><div class="refbegin" style=""> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJones2018" class="citation book cs1">Jones, Robert M. (2018). Mechanics of Composite Materials. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1201%2F9781498711067">10.1201/9781498711067</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-315-27298-6" title="Special:BookSources/978-1-315-27298-6"><bdi>978-1-315-27298-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mechanics+of+Composite+Materials&rft.date=2018&rft_id=info%3Adoi%2F10.1201%2F9781498711067&rft.isbn=978-1-315-27298-6&rft.aulast=Jones&rft.aufirst=Robert+M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComposite+material" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAboudiCederbaumElishakoffLibrescu1992" class="citation book cs1">Aboudi, Jacob; Cederbaum, Gabriel; Elishakoff, Isaac; Librescu, Liviu (1992). Random Vibration and Reliability of Composite Structures. CRC Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-87762-865-1" title="Special:BookSources/978-0-87762-865-1"><bdi>978-0-87762-865-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Random+Vibration+and+Reliability+of+Composite+Structures&rft.pub=CRC+Press&rft.date=1992&rft.isbn=978-0-87762-865-1&rft.aulast=Aboudi&rft.aufirst=Jacob&rft.au=Cederbaum%2C+Gabriel&rft.au=Elishakoff%2C+Isaac&rft.au=Librescu%2C+Liviu&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComposite+material" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLibrescuSong2006" class="citation book cs1">Librescu, Liviu; Song, Ohseop (2006). Thin-Walled Composite Beams. Solid Mechanics and Its Applications. Vol. 131. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F1-4020-4203-5">10.1007/1-4020-4203-5</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4020-3457-2" title="Special:BookSources/978-1-4020-3457-2"><bdi>978-1-4020-3457-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Thin-Walled+Composite+Beams&rft.series=Solid+Mechanics+and+Its+Applications&rft.date=2006&rft_id=info%3Adoi%2F10.1007%2F1-4020-4203-5&rft.isbn=978-1-4020-3457-2&rft.aulast=Librescu&rft.aufirst=Liviu&rft.au=Song%2C+Ohseop&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComposite+material" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPalsule2016" class="citation book cs1">Palsule, Sanjay, ed. (2016). Polymers and Polymeric Composites: A Reference Series. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-3-642-37179-0">10.1007/978-3-642-37179-0</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-642-37179-0" title="Special:BookSources/978-3-642-37179-0"><bdi>978-3-642-37179-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Polymers+and+Polymeric+Composites%3A+A+Reference+Series&rft.date=2016&rft_id=info%3Adoi%2F10.1007%2F978-3-642-37179-0&rft.isbn=978-3-642-37179-0&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComposite+material" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKaw2005" class="citation book cs1">Kaw, Autar K. (2005). Mechanics of Composite Materials. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1201%2F9781420058291">10.1201/9781420058291</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-429-12539-3" title="Special:BookSources/978-0-429-12539-3"><bdi>978-0-429-12539-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mechanics+of+Composite+Materials&rft.date=2005&rft_id=info%3Adoi%2F10.1201%2F9781420058291&rft.isbn=978-0-429-12539-3&rft.aulast=Kaw&rft.aufirst=Autar+K.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComposite+material" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHollaway1994" class="citation book cs1">Hollaway, L. C. (1994). Handbook of Polymer Composites for Engineers. Woodhead Publishing. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-85573-129-5" title="Special:BookSources/978-1-85573-129-5"><bdi>978-1-85573-129-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Handbook+of+Polymer+Composites+for+Engineers&rft.pub=Woodhead+Publishing&rft.date=1994&rft.isbn=978-1-85573-129-5&rft.aulast=Hollaway&rft.aufirst=L.+C.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComposite+material" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMadboulyZhangKessler2015" class="citation book cs1">Madbouly, Samy; Zhang, Chaoqun; Kessler, Michael R. (2015). Bio-Based Plant Oil Polymers and Composites. William Andrew. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-323-37128-5" title="Special:BookSources/978-0-323-37128-5"><bdi>978-0-323-37128-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Bio-Based+Plant+Oil+Polymers+and+Composites&rft.pub=William+Andrew&rft.date=2015&rft.isbn=978-0-323-37128-5&rft.aulast=Madbouly&rft.aufirst=Samy&rft.au=Zhang%2C+Chaoqun&rft.au=Kessler%2C+Michael+R.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComposite+material" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMatthewsRawlings1999" class="citation book cs1">Matthews, F. L.; Rawlings, Rees D. (1999). Composite Materials: Engineering and Science. Woodhead Publishing. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8493-0621-1" title="Special:BookSources/978-0-8493-0621-1"><bdi>978-0-8493-0621-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Composite+Materials%3A+Engineering+and+Science&rft.pub=Woodhead+Publishing&rft.date=1999&rft.isbn=978-0-8493-0621-1&rft.aulast=Matthews&rft.aufirst=F.+L.&rft.au=Rawlings%2C+Rees+D.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComposite+material" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHaka2023" class="citation book cs1">Haka, Andreas T. (2023). Engineered Stability. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-3-658-41408-5">10.1007/978-3-658-41408-5</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-658-41407-8" title="Special:BookSources/978-3-658-41407-8"><bdi>978-3-658-41407-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Engineered+Stability&rft.date=2023&rft_id=info%3Adoi%2F10.1007%2F978-3-658-41408-5&rft.isbn=978-3-658-41407-8&rft.aulast=Haka&rft.aufirst=Andreas+T.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComposite+material" class="Z3988"></li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2>[<a href="/w/index.php?title=Composite_material&action=edit&section=31" title="Edit section: External links">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1235681985">.mw-parser-output .side-box{margin:4px 0;box-sizing:border-box;border:1px solid #aaa;font-size:88%;line-height:1.25em;background-color:var(--background-color-interactive-subtle,#f8f9fa);display:flow-root}.mw-parser-output .side-box-abovebelow,.mw-parser-output .side-box-text{padding:0.25em 0.9em}.mw-parser-output .side-box-image{padding:2px 0 2px 0.9em;text-align:center}.mw-parser-output .side-box-imageright{padding:2px 0.9em 2px 0;text-align:center}@media(min-width:500px){.mw-parser-output .side-box-flex{display:flex;align-items:center}.mw-parser-output .side-box-text{flex:1;min-width:0}}@media(min-width:720px){.mw-parser-output .side-box{width:238px}.mw-parser-output .side-box-right{clear:right;float:right;margin-left:1em}.mw-parser-output .side-box-left{margin-right:1em}}</style><style data-mw-deduplicate="TemplateStyles:r1237033735">@media print{body.ns-0 .mw-parser-output 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Composite material - Wikipedia