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class="vector-main-menu vector-pinnable-element"> <div class="vector-pinnable-header vector-main-menu-pinnable-header vector-pinnable-header-unpinned" data-feature-name="main-menu-pinned" data-pinnable-element-id="vector-main-menu" data-pinned-container-id="vector-main-menu-pinned-container" data-unpinned-container-id="vector-main-menu-unpinned-container" > <div class="vector-pinnable-header-label">Menu nagusia</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-main-menu.pin">mugitu alboko barrara</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-main-menu.unpin">ezkutatu</button> </div> <div id="p-navigation" class="vector-menu mw-portlet mw-portlet-navigation" > <div class="vector-menu-heading"> Nabigazioa </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="n-mainpage" class="mw-list-item"><a href="/wiki/Azala" title="Azala bisitatu [z]" accesskey="z"><span>Azala</span></a></li><li id="n-Txikipedia" class="mw-list-item"><a href="/wiki/Txikipedia:Azala"><span>Txikipedia</span></a></li><li id="n-Ikusgela" class="mw-list-item"><a href="/wiki/Atari:Hezkuntza/Ikusgela"><span>Ikusgela</span></a></li><li id="n-portal" class="mw-list-item"><a href="/wiki/Wikipedia:Txokoa" title="Proiektuaren inguruan, zer egin dezakezu, non aurkitu nahi duzuna"><span>Txokoa</span></a></li><li id="n-recentchanges" class="mw-list-item"><a href="/wiki/Berezi:AzkenAldaketak" title="Wikiko azken aldaketen zerrenda. [r]" accesskey="r"><span>Aldaketa berriak</span></a></li><li id="n-randompage" class="mw-list-item"><a href="/wiki/Berezi:Ausazkoa" title="Ausazko orrialde bat kargatu [x]" accesskey="x"><span>Ausazko orria</span></a></li><li id="n-help" class="mw-list-item"><a href="/wiki/Laguntza:Sarrera" title="Aurkitzeko lekua."><span>Laguntza</span></a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> <a href="/wiki/Azala" class="mw-logo"> <img class="mw-logo-icon" src="/static/images/icons/wikipedia.png" alt="" aria-hidden="true" height="50" width="50"> <span class="mw-logo-container skin-invert"> <img class="mw-logo-wordmark" alt="Wikipedia" src="/static/images/mobile/copyright/wikipedia-wordmark-en.svg" style="width: 7.5em; height: 1.125em;"> <img class="mw-logo-tagline" alt="Entziklopedia askea" src="/static/images/mobile/copyright/wikipedia-tagline-eu.svg" width="120" height="13" style="width: 7.5em; height: 0.8125em;"> </span> </a> </div> <div class="vector-header-end"> <div id="p-search" role="search" class="vector-search-box-vue vector-search-box-collapses vector-search-box-show-thumbnail vector-search-box-auto-expand-width vector-search-box"> <a href="/wiki/Berezi:Bilatu" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only search-toggle" title="Wikipedia(e)n bilatu [f]" accesskey="f"><span class="vector-icon mw-ui-icon-search mw-ui-icon-wikimedia-search"></span> <span>Bilatu</span> </a> <div class="vector-typeahead-search-container"> <div class="cdx-typeahead-search cdx-typeahead-search--show-thumbnail cdx-typeahead-search--auto-expand-width"> <form action="/w/index.php" id="searchform" class="cdx-search-input cdx-search-input--has-end-button"> <div id="simpleSearch" class="cdx-search-input__input-wrapper" data-search-loc="header-moved"> <div class="cdx-text-input cdx-text-input--has-start-icon"> <input class="cdx-text-input__input" type="search" name="search" placeholder="Wikipedia wikian bilatu" aria-label="Wikipedia wikian bilatu" autocapitalize="sentences" title="Wikipedia(e)n bilatu [f]" accesskey="f" id="searchInput" > <span class="cdx-text-input__icon cdx-text-input__start-icon"></span> </div> <input type="hidden" name="title" value="Berezi:Bilatu"> </div> <button class="cdx-button cdx-search-input__end-button">Bilatu</button> </form> </div> </div> </div> <nav class="vector-user-links vector-user-links-wide" aria-label="Tresna pertsonalak"> <div class="vector-user-links-main"> <div id="p-vector-user-menu-preferences" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <div id="p-vector-user-menu-userpage" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <nav class="vector-appearance-landmark" aria-label="Itxura"> <div id="vector-appearance-dropdown" class="vector-dropdown " title="Change the appearance of the page's font size, width, and color" > <input type="checkbox" id="vector-appearance-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-appearance-dropdown" class="vector-dropdown-checkbox " aria-label="Itxura" > <label id="vector-appearance-dropdown-label" for="vector-appearance-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-appearance mw-ui-icon-wikimedia-appearance"></span> <span class="vector-dropdown-label-text">Itxura</span> </label> <div class="vector-dropdown-content"> <div id="vector-appearance-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <div id="p-vector-user-menu-notifications" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <div id="p-vector-user-menu-overflow" class="vector-menu mw-portlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-sitesupport-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="//donate.wikimedia.org/wiki/Special:FundraiserRedirector?utm_source=donate&utm_medium=sidebar&utm_campaign=C13_eu.wikipedia.org&uselang=eu" class=""><span>Dohaintza egin</span></a> </li> <li id="pt-createaccount-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=Berezi:KontuaSortu&returnto=Plastikotasun" title="Kontu bat sortu eta horrekin saioa hastea eskatu nahi genizuke; ez da ezinbestekoa, ordea." class=""><span>Sortu kontua</span></a> </li> <li id="pt-login-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=Berezi:SaioaHasi&returnto=Plastikotasun" title="Izen ematera gonbidatzen zaitugu. [o]" accesskey="o" class=""><span>Hasi saioa</span></a> </li> </ul> </div> </div> </div> <div id="vector-user-links-dropdown" class="vector-dropdown vector-user-menu vector-button-flush-right vector-user-menu-logged-out" title="Aukera gehiago" > <input type="checkbox" id="vector-user-links-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-user-links-dropdown" class="vector-dropdown-checkbox " aria-label="Tresna pertsonalak" > <label id="vector-user-links-dropdown-label" for="vector-user-links-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-ellipsis mw-ui-icon-wikimedia-ellipsis"></span> <span class="vector-dropdown-label-text">Tresna pertsonalak</span> </label> <div class="vector-dropdown-content"> <div id="p-personal" class="vector-menu mw-portlet mw-portlet-personal user-links-collapsible-item" title="User menu" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-sitesupport" class="user-links-collapsible-item mw-list-item"><a href="//donate.wikimedia.org/wiki/Special:FundraiserRedirector?utm_source=donate&utm_medium=sidebar&utm_campaign=C13_eu.wikipedia.org&uselang=eu"><span>Dohaintza egin</span></a></li><li id="pt-createaccount" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Berezi:KontuaSortu&returnto=Plastikotasun" title="Kontu bat sortu eta horrekin saioa hastea eskatu nahi genizuke; ez da ezinbestekoa, ordea."><span class="vector-icon mw-ui-icon-userAdd mw-ui-icon-wikimedia-userAdd"></span> <span>Sortu kontua</span></a></li><li id="pt-login" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Berezi:SaioaHasi&returnto=Plastikotasun" title="Izen ematera gonbidatzen zaitugu. [o]" accesskey="o"><span class="vector-icon mw-ui-icon-logIn mw-ui-icon-wikimedia-logIn"></span> <span>Hasi saioa</span></a></li> </ul> </div> </div> <div id="p-user-menu-anon-editor" class="vector-menu mw-portlet mw-portlet-user-menu-anon-editor" > <div class="vector-menu-heading"> Izena eman gabeko erabiltzaileentzako orrialdeak <a href="/wiki/Laguntza:Sarrera" aria-label="Artikuluak aldatzeari buruz gehiago ikasi"><span>gehiago ikasi</span></a> </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-anoncontribs" class="mw-list-item"><a href="/wiki/Berezi:NireEkarpenak" title="IP helbide honetatik egindako aldaketen zerrenda [y]" accesskey="y"><span>Ekarpenak</span></a></li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/Berezi:NireEztabaida" title="Zure IParen eztabaida [n]" accesskey="n"><span>Eztabaida</span></a></li> </ul> </div> </div> </div> </div> </nav> </div> </header> </div> <div class="mw-page-container"> <div class="mw-page-container-inner"> <div class="vector-sitenotice-container"> <div id="siteNotice"></div> </div> <div class="vector-column-start"> <div class="vector-main-menu-container"> <div id="mw-navigation"> <nav id="mw-panel" class="vector-main-menu-landmark" aria-label="Gunea"> <div id="vector-main-menu-pinned-container" class="vector-pinned-container"> </div> </nav> </div> </div> <div class="vector-sticky-pinned-container"> <nav id="mw-panel-toc" aria-label="Edukiak" data-event-name="ui.sidebar-toc" class="mw-table-of-contents-container vector-toc-landmark"> <div id="vector-toc-pinned-container" class="vector-pinned-container"> <div id="vector-toc" class="vector-toc vector-pinnable-element"> <div class="vector-pinnable-header vector-toc-pinnable-header vector-pinnable-header-pinned" data-feature-name="toc-pinned" data-pinnable-element-id="vector-toc" > <h2 class="vector-pinnable-header-label">Edukiak</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">mugitu alboko barrara</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">ezkutatu</button> </div> <ul class="vector-toc-contents" id="mw-panel-toc-list"> <li id="toc-mw-content-text" class="vector-toc-list-item vector-toc-level-1"> <a href="#" class="vector-toc-link"> <div class="vector-toc-text">⇑ Gora</div> </a> </li> <li id="toc-Sarrera" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Sarrera"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Sarrera</span> </div> </a> <button aria-controls="toc-Sarrera-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Erakutsi/ezkutatu Sarrera azpiatal</span> </button> <ul id="toc-Sarrera-sublist" class="vector-toc-list"> <li id="toc-Plastikotasunaren_historia" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Plastikotasunaren_historia"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>Plastikotasunaren historia</span> </div> </a> <ul id="toc-Plastikotasunaren_historia-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Plastikotasun_ereduak" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Plastikotasun_ereduak"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.2</span> <span>Plastikotasun ereduak</span> </div> </a> <ul id="toc-Plastikotasun_ereduak-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Tentsio-deskonposizioa" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Tentsio-deskonposizioa"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.3</span> <span>Tentsio-deskonposizioa</span> </div> </a> <ul id="toc-Tentsio-deskonposizioa-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Erreferentziak" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Erreferentziak"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Erreferentziak</span> </div> </a> <ul id="toc-Erreferentziak-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Ikus,_gainera" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Ikus,_gainera"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Ikus, gainera</span> </div> </a> <ul id="toc-Ikus,_gainera-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="Edukiak" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Eduki taularen ikusgarritasuna aldatu" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Eduki taularen ikusgarritasuna aldatu</span> </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"><span class="mw-page-title-main">Plastikotasun</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Joan beste hizkuntza batean idatzitako artikulu batera. 48 hizkuntzatan eskuragarri." > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-48" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">48 hizkuntza</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%84%D8%AF%D9%88%D9%86%D8%A9" title="لدونة – arabiera" lang="ar" hreflang="ar" data-title="لدونة" data-language-autonym="العربية" data-language-local-name="arabiera" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/Plastiklik_(fizika)" title="Plastiklik (fizika) – azerbaijanera" lang="az" hreflang="az" data-title="Plastiklik (fizika)" data-language-autonym="Azərbaycanca" data-language-local-name="azerbaijanera" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%9F%D0%BB%D0%B0%D1%81%D1%82%D1%8B%D1%87%D0%BD%D0%B0%D1%8F_%D0%B4%D1%8D%D1%84%D0%B0%D1%80%D0%BC%D0%B0%D1%86%D1%8B%D1%8F" title="Пластычная дэфармацыя – bielorrusiera" lang="be" hreflang="be" data-title="Пластычная дэфармацыя" data-language-autonym="Беларуская" data-language-local-name="bielorrusiera" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-be-x-old mw-list-item"><a href="https://be-tarask.wikipedia.org/wiki/%D0%9F%D0%BB%D1%8F%D1%81%D1%82%D1%8B%D1%87%D0%BD%D0%B0%D1%8F_%D0%B4%D1%8D%D1%84%D0%B0%D1%80%D0%BC%D0%B0%D1%86%D1%8B%D1%8F" title="Плястычная дэфармацыя – Belarusian (Taraškievica orthography)" lang="be-tarask" hreflang="be-tarask" data-title="Плястычная дэфармацыя" data-language-autonym="Беларуская (тарашкевіца)" data-language-local-name="Belarusian (Taraškievica orthography)" class="interlanguage-link-target"><span>Беларуская (тарашкевіца)</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%9F%D0%BB%D0%B0%D1%81%D1%82%D0%B8%D1%87%D0%BD%D0%B0_%D0%B4%D0%B5%D1%84%D0%BE%D1%80%D0%BC%D0%B0%D1%86%D0%B8%D1%8F" title="Пластична деформация – bulgariera" lang="bg" hreflang="bg" data-title="Пластична деформация" data-language-autonym="Български" data-language-local-name="bulgariera" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%85%E0%A6%B8%E0%A7%8D%E0%A6%A5%E0%A6%BF%E0%A6%A4%E0%A6%BF%E0%A6%B8%E0%A7%8D%E0%A6%A5%E0%A6%BE%E0%A6%AA%E0%A6%95%E0%A6%A4%E0%A6%BE_(%E0%A6%A8%E0%A6%AE%E0%A6%A8%E0%A7%80%E0%A6%AF%E0%A6%BC_%E0%A6%AA%E0%A6%A6%E0%A6%BE%E0%A6%B0%E0%A7%8D%E0%A6%A5)" title="অস্থিতিস্থাপকতা (নমনীয় পদার্থ) – bengalera" lang="bn" hreflang="bn" data-title="অস্থিতিস্থাপকতা (নমনীয় পদার্থ)" data-language-autonym="বাংলা" data-language-local-name="bengalera" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Plasticitat" title="Plasticitat – katalana" lang="ca" hreflang="ca" data-title="Plasticitat" data-language-autonym="Català" data-language-local-name="katalana" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%A2%D3%B3%D1%81%C4%95%D0%BC%D0%BB%C4%95%D1%85_(%D1%84%D0%B8%D0%B7%D0%B8%D0%BA%D0%B0)" title="Тӳсĕмлĕх (физика) – txuvaxera" lang="cv" hreflang="cv" data-title="Тӳсĕмлĕх (физика)" data-language-autonym="Чӑвашла" data-language-local-name="txuvaxera" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Plastizit%C3%A4t_(Physik)" title="Plastizität (Physik) – alemana" lang="de" hreflang="de" data-title="Plastizität (Physik)" data-language-autonym="Deutsch" data-language-local-name="alemana" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Plasticity_(physics)" title="Plasticity (physics) – ingelesa" lang="en" hreflang="en" data-title="Plasticity (physics)" data-language-autonym="English" data-language-local-name="ingelesa" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Plasta_aliformi%C4%9Do" title="Plasta aliformiĝo – esperantoa" lang="eo" hreflang="eo" data-title="Plasta aliformiĝo" data-language-autonym="Esperanto" data-language-local-name="esperantoa" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Plasticidad_(mec%C3%A1nica_de_s%C3%B3lidos)" title="Plasticidad (mecánica de sólidos) – gaztelania" lang="es" hreflang="es" data-title="Plasticidad (mecánica de sólidos)" data-language-autonym="Español" data-language-local-name="gaztelania" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Plastsus" title="Plastsus – estoniera" lang="et" hreflang="et" data-title="Plastsus" data-language-autonym="Eesti" data-language-local-name="estoniera" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%85%D9%88%D9%85%D8%B3%D8%A7%D9%86%DB%8C" title="مومسانی – persiera" lang="fa" hreflang="fa" data-title="مومسانی" data-language-autonym="فارسی" data-language-local-name="persiera" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Plastisuus" title="Plastisuus – finlandiera" lang="fi" hreflang="fi" data-title="Plastisuus" data-language-autonym="Suomi" data-language-local-name="finlandiera" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/D%C3%A9formation_plastique" title="Déformation plastique – frantsesa" lang="fr" hreflang="fr" data-title="Déformation plastique" data-language-autonym="Français" data-language-local-name="frantsesa" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/D%C3%ADchumadh_plaisteach" title="Díchumadh plaisteach – irlandera" lang="ga" hreflang="ga" data-title="Díchumadh plaisteach" data-language-autonym="Gaeilge" data-language-local-name="irlandera" class="interlanguage-link-target"><span>Gaeilge</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%A4%D7%9C%D7%A1%D7%98%D7%99%D7%95%D7%AA" title="פלסטיות – hebreera" lang="he" hreflang="he" data-title="פלסטיות" data-language-autonym="עברית" data-language-local-name="hebreera" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%B8%E0%A5%81%E0%A4%98%E0%A4%9F%E0%A5%8D%E0%A4%AF%E0%A4%A4%E0%A4%BE" title="सुघट्यता – hindia" lang="hi" hreflang="hi" data-title="सुघट्यता" data-language-autonym="हिन्दी" data-language-local-name="hindia" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Plasti%C4%8Dnost" title="Plastičnost – kroaziera" lang="hr" hreflang="hr" data-title="Plastičnost" data-language-autonym="Hrvatski" data-language-local-name="kroaziera" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-hu badge-Q17437796 badge-featuredarticle mw-list-item" title="artikulu nabarmenduak"><a href="https://hu.wikipedia.org/wiki/K%C3%A9pl%C3%A9kenys%C3%A9gtan" title="Képlékenységtan – hungariera" lang="hu" hreflang="hu" data-title="Képlékenységtan" data-language-autonym="Magyar" data-language-local-name="hungariera" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D5%8A%D5%AC%D5%A1%D5%BD%D5%BF%D5%AB%D5%AF%D5%B8%D6%82%D5%A9%D5%B5%D5%B8%D6%82%D5%B6_(%D5%B4%D5%A5%D5%AD%D5%A1%D5%B6%D5%AB%D5%AF%D5%A1)" title="Պլաստիկություն (մեխանիկա) – armeniera" lang="hy" hreflang="hy" data-title="Պլաստիկություն (մեխանիկա)" data-language-autonym="Հայերեն" data-language-local-name="armeniera" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Plastisitas_(fisika)" title="Plastisitas (fisika) – indonesiera" lang="id" hreflang="id" data-title="Plastisitas (fisika)" data-language-autonym="Bahasa Indonesia" data-language-local-name="indonesiera" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Plasticit%C3%A0_(fisica)" title="Plasticità (fisica) – italiera" lang="it" hreflang="it" data-title="Plasticità (fisica)" data-language-autonym="Italiano" data-language-local-name="italiera" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E5%A1%91%E6%80%A7" title="塑性 – japoniera" lang="ja" hreflang="ja" data-title="塑性" data-language-autonym="日本語" data-language-local-name="japoniera" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%9E%E1%83%9A%E1%83%90%E1%83%A1%E1%83%A2%E1%83%98%E1%83%99%E1%83%A3%E1%83%A0%E1%83%9D%E1%83%91%E1%83%90" title="პლასტიკურობა – georgiera" lang="ka" hreflang="ka" data-title="პლასტიკურობა" data-language-autonym="ქართული" data-language-local-name="georgiera" class="interlanguage-link-target"><span>ქართული</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%98%D0%BB%D0%B5%D0%BC%D0%B4%D1%96%D0%BB%D1%96%D0%BA_(%D1%84%D0%B8%D0%B7%D0%B8%D0%BA%D0%B0)" title="Илемділік (физика) – kazakhera" lang="kk" hreflang="kk" data-title="Илемділік (физика)" data-language-autonym="Қазақша" data-language-local-name="kazakhera" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%86%8C%EC%84%B1_(%EB%AC%BC%EB%A6%AC%ED%95%99)" title="소성 (물리학) – koreera" lang="ko" hreflang="ko" data-title="소성 (물리학)" data-language-autonym="한국어" data-language-local-name="koreera" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Plasticitas" title="Plasticitas – latina" lang="la" hreflang="la" data-title="Plasticitas" data-language-autonym="Latina" data-language-local-name="latina" class="interlanguage-link-target"><span>Latina</span></a></li><li class="interlanguage-link interwiki-new mw-list-item"><a href="https://new.wikipedia.org/wiki/%E0%A4%86%E0%A4%B2%E0%A5%81%E0%A4%AA%E0%A4%B9%E0%A4%83" title="आलुपहः – newarera" lang="new" hreflang="new" data-title="आलुपहः" data-language-autonym="नेपाल भाषा" data-language-local-name="newarera" class="interlanguage-link-target"><span>नेपाल भाषा</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Plasticiteit_(materiaalkunde)" title="Plasticiteit (materiaalkunde) – nederlandera" lang="nl" hreflang="nl" data-title="Plasticiteit (materiaalkunde)" data-language-autonym="Nederlands" data-language-local-name="nederlandera" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Plastisitet" title="Plastisitet – bokmål (norvegiera)" lang="nb" hreflang="nb" data-title="Plastisitet" data-language-autonym="Norsk bokmål" data-language-local-name="bokmål (norvegiera)" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%A2%E0%A8%BF%E0%A8%B2%E0%A8%95_(%E0%A8%AD%E0%A9%8C%E0%A8%A4%E0%A8%BF%E0%A8%95_%E0%A8%B5%E0%A8%BF%E0%A8%97%E0%A8%BF%E0%A8%86%E0%A8%A8)" title="ਢਿਲਕ (ਭੌਤਿਕ ਵਿਗਿਆਨ) – punjabera" lang="pa" hreflang="pa" data-title="ਢਿਲਕ (ਭੌਤਿਕ ਵਿਗਿਆਨ)" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="punjabera" class="interlanguage-link-target"><span>ਪੰਜਾਬੀ</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Plastyczno%C5%9B%C4%87" title="Plastyczność – poloniera" lang="pl" hreflang="pl" data-title="Plastyczność" data-language-autonym="Polski" data-language-local-name="poloniera" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Plasticidade" title="Plasticidade – portugesa" lang="pt" hreflang="pt" data-title="Plasticidade" data-language-autonym="Português" data-language-local-name="portugesa" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9F%D0%BB%D0%B0%D1%81%D1%82%D0%B8%D1%87%D0%BD%D0%BE%D1%81%D1%82%D1%8C_(%D1%84%D0%B8%D0%B7%D0%B8%D0%BA%D0%B0)" title="Пластичность (физика) – errusiera" lang="ru" hreflang="ru" data-title="Пластичность (физика)" data-language-autonym="Русский" data-language-local-name="errusiera" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Plasti%C4%8Dnost" title="Plastičnost – serbokroaziera" lang="sh" hreflang="sh" data-title="Plastičnost" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="serbokroaziera" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Plasticity_(physics)" title="Plasticity (physics) – Simple English" lang="en-simple" hreflang="en-simple" data-title="Plasticity (physics)" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Plastick%C3%A1_deform%C3%A1cia" title="Plastická deformácia – eslovakiera" lang="sk" hreflang="sk" data-title="Plastická deformácia" data-language-autonym="Slovenčina" data-language-local-name="eslovakiera" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/Plasti%C4%8Dnost_(fizika)" title="Plastičnost (fizika) – serbiera" lang="sr" hreflang="sr" data-title="Plastičnost (fizika)" data-language-autonym="Српски / srpski" data-language-local-name="serbiera" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Plasticitet_(fysik)" title="Plasticitet (fysik) – suediera" lang="sv" hreflang="sv" data-title="Plasticitet (fysik)" data-language-autonym="Svenska" data-language-local-name="suediera" 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<div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="eu" dir="ltr"><p><b>Plastikotasuna</b> iraunkorki eta itzulezin <a href="/wiki/Deformazio" title="Deformazio">deformatzeko</a> gaitasuna duten material anelastiko batzuen portaera mekaniko bereizgarria da beren tarte <a href="/wiki/Elastikotasun_(fisika)" title="Elastikotasun (fisika)">elastikotik</a> gorako <a href="/wiki/Tentsio_mekaniko" title="Tentsio mekaniko">tentsioak</a> jasaten dituztenean, hau da, beren muga elastikotik gora. </p><p>Metaletan, plastikotasuna dislokazioen desplazamendu itzulezinei dagokienez azaltzen da. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Sarrera">Sarrera</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Plastikotasun&veaction=edit&section=1" title="Aldatu atal hau: «Sarrera»" class="mw-editsection-visualeditor"><span>aldatu</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Plastikotasun&action=edit&section=1" title="Edit section's source code: Sarrera"><span>aldatu iturburu kodea</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure typeof="mw:File/Thumb"><a href="/wiki/Fitxategi:Stress-strain1-es.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d7/Stress-strain1-es.svg/350px-Stress-strain1-es.svg.png" decoding="async" width="350" height="228" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d7/Stress-strain1-es.svg/525px-Stress-strain1-es.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d7/Stress-strain1-es.svg/700px-Stress-strain1-es.svg.png 2x" data-file-width="515" data-file-height="335" /></a><figcaption> Tentsio-tentsio uniaxial baterako tentsio-deformazio-kurbaren adibide tipikoa, portaera elastoplastikoa duen metal harikor batean: portaera elastiko lineala da deformazio txikietarako (sekzio zuzena urdinez) eta plastikotasuna aurkezten du muga jakin batetik.</figcaption></figure> <p>Material elastikoetan, bereziki metal harikor askotan, <a href="/wiki/Trakzio" title="Trakzio">trakzio</a>-esfortzu ardatz-bakar txiki batek portaera elastikoa dakar. Horrek esan nahi du trakzio-esfortzuaren igoera txikiek deformazioaren gehikuntza txikiak ekartzen dituztela, karga berriro zero bihurtzen bada gorputzak bere jatorrizko forma zehatza berreskuratzen du; hau da, deformazioa guztiz itzulgarria da. Hala ere, esperimentalki egiaztatu da muga bat dagoela, muga elastikoa deritzona; tentsioen funtzio homogeneo jakin batek muga hori gainditzen badu, deformazio hondarrak geratzen dira karga desagertzen denean, eta gorputza ez da zehatz-mehatz bere formara itzultzen. Hau da, deformazio itzulezinak agertzen dira. </p><p>Goian deskribatutako <b>portaera elasto-plastiko</b> mota hori metal ezagun gehienetan aurkitzen da, baita beste material askotan ere. <b>Jokaera guztiz plastikoa</b> ez da hain ohikoa, eta deformazio itzulezinak agertzea dakar tentsioa txikia izan arren; modelatzeko buztina eta <a href="/wiki/Plastilina" title="Plastilina">plastilina</a> portaera guztiz plastikora hurbiltzen da. Beste material batzuek, gainera, <b>gogortze</b> <b>plastikotasuna</b> erakusten dute, eta gero eta tentsio handiagoak behar dituzte beren deformazio plastiko osoa handitzeko. Aurreko jokabideak efektu likatsuekin batera etor daitezke, tentsio-tasa handien kasuan tentsioak handiagoak izatea eragiten dutenak; portaera horri <b>bisko-plastikotasuna deritzo</b>. </p><p>Materialen plastikotasuna material horien aldaketa itzulezinekin lotuta dago. <a href="/wiki/Termodinamika" title="Termodinamika">Termodinamikoki</a> itzulgarria den <a href="/wiki/Elastikotasun_(fisika)" title="Elastikotasun (fisika)">portaera elastikoa</a> ez bezala, plastikoki deformatzen den gorputz batek entropia-aldaketak jasaten ditu, hala nola dislokazio-aldaketak. <b>Portaera plastikoan,</b> <a href="/wiki/Energia_mekaniko" title="Energia mekaniko">energia mekanikoaren</a> zati bat barrutik xahutzen da <a href="/wiki/Energia_potentzial" title="Energia potentzial">energia potentzial elastiko</a> bihurtu beharrean. </p><p><a href="/wiki/Mikroskopio" title="Mikroskopio">Mikroskopikoki</a>, metalen kristal-sarearen eskalan, plastikotasuna dislokazio izeneko sarean, zenbait inperfekzio egotearen ondorioa da. <a href="/wiki/1934" title="1934">1934an</a>, Egon Orowan-ek, Michael Polanyi-k eta Geoffrey Ingram Taylor-ek, gutxi gorabehera aldi berean, ondorioztatu zuten material harikorren deformazio plastikoa dislokazioaren teoriaren arabera azal zitekeela. Plastikotasuna deskribatzeko, deformazio-tentsore eta tentsio-tentsore osagaien aldaketak deskribatzen dituen ekuazio diferentzial ez-lineal eta denboraren mendeko multzo bat erabili ohi da, aurreko deformazio- eta tentsio-egoerarekiko eta une bakoitzeko deformazio-gehikuntza. </p> <div class="mw-heading mw-heading3"><h3 id="Plastikotasunaren_historia">Plastikotasunaren historia</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Plastikotasun&veaction=edit&section=2" title="Aldatu atal hau: «Plastikotasunaren historia»" class="mw-editsection-visualeditor"><span>aldatu</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Plastikotasun&action=edit&section=2" title="Edit section's source code: Plastikotasunaren historia"><span>aldatu iturburu kodea</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/XIX._mendea" title="XIX. mendea">XIX. mendean</a> Tresca, Saint-Venant, Lévy eta Bauschinger-en lanekin finkatu zen plastikotasunaren teoria modernoaren oinarria. <a href="/wiki/XX._mendea" title="XX. mendea">XX. mendearen</a> hasieran, zenbait aurrerapen eman zituzten <a href="/wiki/Ludwig_Prandtl" title="Ludwig Prandtl">Prandtl</a>, <a href="/wiki/Richard_von_Mises" title="Richard von Mises">Von Mises</a> eta A. Reuss-ek fenomenoaren ulermenean. Lehen fase horretan, deformazio itzulezina, <a href="/wiki/Hutsegite-teoriak" class="mw-redirect" title="Hutsegite-teoriak">hutsegite irizpideak</a>, gogortzea eta plastikotasun perfektuaren kontzeptua sartu ziren, baita deformazio plastikoaren ekuazio eratzaileen gehikuntza forma ere. </p><p><a href="/wiki/Bigarren_Mundu_Gerra" title="Bigarren Mundu Gerra">Bigarren Mundu Gerraren ostean</a>, Prager, Drucker eta Hill-en lanak agertu ziren, non formulazioaren argitasun handiagoa lortu eta azalaren isurpen <a href="/wiki/Multzo_konbexu" title="Multzo konbexu">ganbiltasuna</a> ezarri zen. Handik gutxira, <a href="/wiki/1960" title="1960">1960ko hamarkadan</a> hasita, ekuazio diferentzial partzialen eta aldaera-desberdintasun teoriaren zenbait aurrerapen matematiko eman ziren, plastikotasunaren teoriarako bereziki lagungarriak izango zirenak. Aurrerapen horiek frogatu zuten solido elastoplastikoen hasierako balio-arazoak ebazteko marko naturala desberdintasun aldaerak zirela. Solidoen mekanika eta matematika alorreko zenbait aurrerapen bat egiteak garapen teoriko berriak sortu zituen, eta horien adibide dira Moreauren artikuluak, Duvauten monografiak eta J.L Lions eta Temam. </p> <div class="mw-heading mw-heading3"><h3 id="Plastikotasun_ereduak">Plastikotasun ereduak</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Plastikotasun&veaction=edit&section=3" title="Aldatu atal hau: «Plastikotasun ereduak»" class="mw-editsection-visualeditor"><span>aldatu</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Plastikotasun&action=edit&section=3" title="Edit section's source code: Plastikotasun ereduak"><span>aldatu iturburu kodea</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Oro har, plastikotasun-eredu batek hainbat elementu definitzea eskatzen du: </p> <ul><li>Lehenik eta behin, tentsio nagusien espazioan, tentsio <b>onargarrien eremua</b> deritzona definitu behar da, tentsio-espazio horren multzo itxia (eta, agian, trinkoa) izango dena. Multzo horren mugari, <b>etekin-azalera</b> deitu ohi zaio.</li> <li>Esfortzu nagusiak, tentsio onargarrien eskualdean dauden solidoaren puntuetarako, portaera elastikoa da. Dena den, errendimendu-azaleko puntuetarako, beharrezkoa da <b>fluxuaren araua</b> definitzea, zeinak deformazio plastikoa nola handituko den (tentsioaren gehikuntza-tasa eta beste barne-parametro batzuen arabera) bere isurpen puntura iritsi den material baten tentsioa gehitzen bada.</li> <li>Plastikotasun inperfektuaren ereduek eskatuko dute tentsio-eremu onargarria denboran zehar gogortzea eta desplazatzea kontuan hartzen duten barne-aldagaien multzoa definitzea gainerako aldagaien gehikuntza-tasen arabera.</li></ul><p> Barne-aldagaiak egoteak —esaterako, plastifikazio-maila (deformazio plastikoa), gogortzea eta beste— tentsioen eta deformazioen arteko erlazioa kasu elastikoan baino konplexuagoa bihurtzen du; bereziki, barne-aldagaiak nola aldatu diren jakin ezean, deformazio-esfortzu elastikoen maila kontuan hartuta ezin dira tentsioak jakin. Barne-aldagaiak nola aldatzen diren kontuan hartzeak esan nahi du problema elastoplastiko bat, orokorrean, problema dinamiko gisa unibokoki ebatz daitekeela sistema honen ekuazioak aldi berean ebatziz:</p><blockquote style="padding: 5px 10px;background-color: white; color:black; text-align:left; margin-left:30px; margin-bottom:0.4em; margin-top:0.2em; min-width:50%;"> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{cases}{\boldsymbol {\sigma }}(\mathbf {x,t} )=T({\boldsymbol {\varepsilon }}(\mathbf {x,t} ),{\boldsymbol {\xi }}(\mathbf {x,t} ),\mathbf {x} )\\{\boldsymbol {\nabla }}{\boldsymbol {\sigma }}=\mathbf {f} (\mathbf {x,t} )\\{\dot {\boldsymbol {\xi }}}={\boldsymbol {\Phi }}({\boldsymbol {\varepsilon }}(\mathbf {x,t} ),{\boldsymbol {\xi }}(\mathbf {x,t} ),{\dot {\boldsymbol {\varepsilon }}}(\mathbf {x,t} ),\mathbf {x} ),\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">σ</mi> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> <mo mathvariant="bold">,</mo> <mi mathvariant="bold">t</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mi>T</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ε</mi> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> <mo mathvariant="bold">,</mo> <mi mathvariant="bold">t</mi> </mrow> <mo stretchy="false">)</mo> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ξ</mi> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> <mo mathvariant="bold">,</mo> <mi mathvariant="bold">t</mi> </mrow> <mo stretchy="false">)</mo> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">∇</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">σ</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">f</mi> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> <mo mathvariant="bold">,</mo> <mi mathvariant="bold">t</mi> </mrow> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold-italic">ξ</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Φ</mi> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ε</mi> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> <mo mathvariant="bold">,</mo> <mi mathvariant="bold">t</mi> </mrow> <mo stretchy="false">)</mo> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ξ</mi> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> <mo mathvariant="bold">,</mo> <mi mathvariant="bold">t</mi> </mrow> <mo stretchy="false">)</mo> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold-italic">ε</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> <mo mathvariant="bold">,</mo> <mi mathvariant="bold">t</mi> </mrow> <mo stretchy="false">)</mo> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo stretchy="false">)</mo> <mo>,</mo> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{cases}{\boldsymbol {\sigma }}(\mathbf {x,t} )=T({\boldsymbol {\varepsilon }}(\mathbf {x,t} ),{\boldsymbol {\xi }}(\mathbf {x,t} ),\mathbf {x} )\\{\boldsymbol {\nabla }}{\boldsymbol {\sigma }}=\mathbf {f} (\mathbf {x,t} )\\{\dot {\boldsymbol {\xi }}}={\boldsymbol {\Phi }}({\boldsymbol {\varepsilon }}(\mathbf {x,t} ),{\boldsymbol {\xi }}(\mathbf {x,t} ),{\dot {\boldsymbol {\varepsilon }}}(\mathbf {x,t} ),\mathbf {x} ),\end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ecec6621380cb0d6859b9af102dc6ee391678202" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:36.281ex; height:9.176ex;" alt="{\displaystyle {\begin{cases}{\boldsymbol {\sigma }}(\mathbf {x,t} )=T({\boldsymbol {\varepsilon }}(\mathbf {x,t} ),{\boldsymbol {\xi }}(\mathbf {x,t} ),\mathbf {x} )\\{\boldsymbol {\nabla }}{\boldsymbol {\sigma }}=\mathbf {f} (\mathbf {x,t} )\\{\dot {\boldsymbol {\xi }}}={\boldsymbol {\Phi }}({\boldsymbol {\varepsilon }}(\mathbf {x,t} ),{\boldsymbol {\xi }}(\mathbf {x,t} ),{\dot {\boldsymbol {\varepsilon }}}(\mathbf {x,t} ),\mathbf {x} ),\end{cases}}}"></span> </p> </blockquote><p>non lehen erlazioak <a href="/wiki/Tentsio_mekaniko" title="Tentsio mekaniko">tentsio mekanikoaren</a> arteko ekuazio eratzailea adierazten duen (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \scriptstyle {\boldsymbol {\sigma }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">σ</mi> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle {\boldsymbol {\sigma }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ad1a3bc637b28d156d8be0481b1a2810ba12d64e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.127ex; height:1.343ex;" alt="{\displaystyle \scriptstyle {\boldsymbol {\sigma }}}"></span>), <a href="/wiki/Deformazio" title="Deformazio">deformazioak</a> (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \scriptstyle {\boldsymbol {\varepsilon }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ε</mi> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle {\boldsymbol {\varepsilon }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/797f52e8f0949a2dbaa846159953ef4b4c08f1ee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.87ex; height:1.343ex;" alt="{\displaystyle \scriptstyle {\boldsymbol {\varepsilon }}}"></span>), barne aldagaiak (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \scriptstyle {\boldsymbol {\xi }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ξ</mi> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle {\boldsymbol {\xi }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1324169cf1f0354e260071947844ef8cb8f1053f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:0.835ex; height:2.009ex;" alt="{\displaystyle \scriptstyle {\boldsymbol {\xi }}}"></span>), solidoaren puntu bakoitzeko. Bigarren erlazioa barneko esfortzuen eta aplikatutako indarren arteko indar oreka biltzen duen <a href="/wiki/Ekuazio_diferentzial_partzial" title="Ekuazio diferentzial partzial">deribatu partzialetako ekuazioa da</a> (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \scriptstyle \mathbf {f} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">f</mi> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle \mathbf {f} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e20a54124abeb15c9d99a4fad6645ba171bf9b0c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.745ex; height:1.676ex;" alt="{\displaystyle \scriptstyle \mathbf {f} }"></span>), eta azkena ekuazio diferentzial arrunta da, barne-aldagaiak (bereziki deformazio plastikoa) denborarekin nola handitzen diren adierazten duen fluxu-araua ematen duena <a href="/wiki/Ekuazio_diferentzial_arrunt" title="Ekuazio diferentzial arrunt">ekuazio diferentzial arrunta</a> materiala tentsio-egoerara iristen denean, non isurpena agertzen den. </p><div class="mw-heading mw-heading3"><h3 id="Tentsio-deskonposizioa">Tentsio-deskonposizioa</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Plastikotasun&veaction=edit&section=4" title="Aldatu atal hau: «Tentsio-deskonposizioa»" class="mw-editsection-visualeditor"><span>aldatu</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Plastikotasun&action=edit&section=4" title="Edit section's source code: Tentsio-deskonposizioa"><span>aldatu iturburu kodea</span></a><span class="mw-editsection-bracket">]</span></span></div><p> Material plastiko baten deskribapenak deformazio osoa deskribatzen duten aldagaiak eta barne aldagaiak behar ditu. <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \xi _{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \xi _{k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d15cfc3ac865cb9b3fefec7096afaedc65eb81bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.107ex; height:2.509ex;" alt="{\displaystyle \xi _{k}}"></span> materialaren barruan gertatzen diren aldaketa atzeraezinak deskribatzen dituztenak. Aldagai horiek materialaren xahutze-erlazioetan ere esku hartzen dute. Kontsiderazio termodinamikoek daramate <a href="/wiki/Gibbs-en_energia" class="mw-redirect" title="Gibbs-en energia">Gibbsen energia askea</a> <i>g</i> (bolumen unitateko) Helmholtz energia askearekin <i>f</i>, , tentsioak deformazioak erlazionatuta egotera erlazio honen bidez:</p><blockquote style="padding: 5px 10px;background-color: white; color:black; text-align:left; margin-left:30px; margin-bottom:0.4em; margin-top:0.2em; min-width:50%;"> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(\sigma _{ij},\xi _{k})=\left(\sum _{i,j}\sigma _{ij}\varepsilon _{ij}\right)-f(\sigma _{ij},\xi _{k}),\qquad \varepsilon _{ij}={\frac {\partial g}{\partial \sigma _{ij}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo stretchy="false">(</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>,</mo> <mi>j</mi> </mrow> </munder> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>,</mo> <mspace width="2em" /> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>g</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g(\sigma _{ij},\xi _{k})=\left(\sum _{i,j}\sigma _{ij}\varepsilon _{ij}\right)-f(\sigma _{ij},\xi _{k}),\qquad \varepsilon _{ij}={\frac {\partial g}{\partial \sigma _{ij}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bb602410a22f65fb55092797f19f722be8c208af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:52.929ex; height:7.676ex;" alt="{\displaystyle g(\sigma _{ij},\xi _{k})=\left(\sum _{i,j}\sigma _{ij}\varepsilon _{ij}\right)-f(\sigma _{ij},\xi _{k}),\qquad \varepsilon _{ij}={\frac {\partial g}{\partial \sigma _{ij}}}}"></span> </p> </blockquote><p>Non: </p><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g,f\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo>,</mo> <mi>f</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g,f\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b22c8eca2ec3edb1a56be13d3dd98bc03655d737" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.816ex; height:2.509ex;" alt="{\displaystyle g,f\,}"></span> Gibbs energia askea eta Helmhotz energia askea bolumen unitateko,</dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{ij}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{ij}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a9993319cd516732d5f4a593c23c19c2dedbc501" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.192ex; height:2.343ex;" alt="{\displaystyle \sigma _{ij}\,}"></span> tentsio-tentsorearen osagaiak dira;</dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon _{ij}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon _{ij}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/871f80731933e658f992d04fc42d4cd0ae4e388b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.948ex; height:2.343ex;" alt="{\displaystyle \varepsilon _{ij}\,}"></span> deformazio-tentsorearen osagaiak dira, eta</dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \xi _{k}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \xi _{k}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1849f8c9ebf56cb426964064697847950c5f37f3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.494ex; height:2.509ex;" alt="{\displaystyle \xi _{k}\,}"></span> materialaren aldaketa itzulezinekin lotutako barne-aldagaien multzoa dira</dd></dl><p> Aurreko erlazioak honakoa dakar:</p><blockquote style="padding: 5px 10px;background-color: white; color:black; text-align:left; margin-left:30px; margin-bottom:0.4em; margin-top:0.2em; min-width:50%;"> <p><span style="float:right; width:10%; text-align:right;">(<cite id="Equation_*" style="font-style: normal;"><a href="#Eqnref_*">*</a></cite>)</span> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\dot {\varepsilon _{ij}}}={\frac {\partial \varepsilon _{ij}}{\partial \sigma _{mn}}}{\dot {\sigma }}_{mn}+{\frac {\partial \varepsilon _{ij}}{\partial \xi _{k}}}{\dot {\xi }}_{k}=\mathbf {A} :{\boldsymbol {\dot {\sigma }}}+\mathbf {B} \cdot {\boldsymbol {\dot {\xi }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>˙</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mi>n</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>σ</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mi>n</mi> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ξ</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold-italic">σ</mi> <mo mathvariant="bold">˙</mo> </mover> </mrow> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold-italic">ξ</mi> <mo mathvariant="bold">˙</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\dot {\varepsilon _{ij}}}={\frac {\partial \varepsilon _{ij}}{\partial \sigma _{mn}}}{\dot {\sigma }}_{mn}+{\frac {\partial \varepsilon _{ij}}{\partial \xi _{k}}}{\dot {\xi }}_{k}=\mathbf {A} :{\boldsymbol {\dot {\sigma }}}+\mathbf {B} \cdot {\boldsymbol {\dot {\xi }}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ea8cf516f8eae7d9f32bd016727e1a0931d4d0a2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:42.569ex; height:6.343ex;" alt="{\displaystyle {\dot {\varepsilon _{ij}}}={\frac {\partial \varepsilon _{ij}}{\partial \sigma _{mn}}}{\dot {\sigma }}_{mn}+{\frac {\partial \varepsilon _{ij}}{\partial \xi _{k}}}{\dot {\xi }}_{k}=\mathbf {A} :{\boldsymbol {\dot {\sigma }}}+\mathbf {B} \cdot {\boldsymbol {\dot {\xi }}}}"></span> </p> </blockquote><p>Esperimentalki jakina da <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {A} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {A} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0795cc96c75d81520a120482662b90f024c9a1a1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.019ex; height:2.176ex;" alt="{\displaystyle \mathbf {A} }"></span> betetze tentsorea ez dirudiela deformazio plastikoaren prozesu itzulezinek eragiten duenik, eta horrek esan nahi du:</p><blockquote style="padding: 5px 10px;background-color: white; color:black; text-align:left; margin-left:30px; margin-bottom:0.4em; margin-top:0.2em; min-width:50%;"> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0={\frac {\partial }{\partial \xi _{k}}}{\frac {\partial \varepsilon _{ij}}{\partial \sigma _{mn}}}={\frac {\partial }{\partial \sigma _{mn}}}{\frac {\partial \varepsilon _{ij}}{\partial \xi _{k}}}={\frac {\partial \mathbf {B} }{\partial {\boldsymbol {\sigma }}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mi>n</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mi>n</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">σ</mi> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0={\frac {\partial }{\partial \xi _{k}}}{\frac {\partial \varepsilon _{ij}}{\partial \sigma _{mn}}}={\frac {\partial }{\partial \sigma _{mn}}}{\frac {\partial \varepsilon _{ij}}{\partial \xi _{k}}}={\frac {\partial \mathbf {B} }{\partial {\boldsymbol {\sigma }}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b88496c35d6199a61cc0d52ad61cd54e2097bb16" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:35.775ex; height:6.343ex;" alt="{\displaystyle 0={\frac {\partial }{\partial \xi _{k}}}{\frac {\partial \varepsilon _{ij}}{\partial \sigma _{mn}}}={\frac {\partial }{\partial \sigma _{mn}}}{\frac {\partial \varepsilon _{ij}}{\partial \xi _{k}}}={\frac {\partial \mathbf {B} }{\partial {\boldsymbol {\sigma }}}}}"></span> </p> </blockquote><p>Eta kasu horretan, tentsioaren deskonposizio gehigarria gertatzen da, <a href="/wiki/Deformazio" title="Deformazio">tentsio</a> elastikoan eta tentsio plastikoan, zeren <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {A} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {A} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0795cc96c75d81520a120482662b90f024c9a1a1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.019ex; height:2.176ex;" alt="{\displaystyle \mathbf {A} }"></span>-ren deformazio plastikoaren independentziaren hipotesiaren arabera ( forma honetan integra daiteke:</p><blockquote style="padding: 5px 10px;background-color: white; color:black; text-align:left; margin-left:30px; margin-bottom:0.4em; margin-top:0.2em; min-width:50%;"> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon _{ij}=\varepsilon _{ij}^{e}({\boldsymbol {\sigma }})+\varepsilon _{ij}^{p}({\boldsymbol {\xi }})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>=</mo> <msubsup> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msubsup> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">σ</mi> </mrow> <mo stretchy="false">)</mo> <mo>+</mo> <msubsup> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msubsup> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ξ</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon _{ij}=\varepsilon _{ij}^{e}({\boldsymbol {\sigma }})+\varepsilon _{ij}^{p}({\boldsymbol {\xi }})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3d3ca5ed0132fe238542c690d72a61eae817b321" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:20.015ex; height:3.509ex;" alt="{\displaystyle \varepsilon _{ij}=\varepsilon _{ij}^{e}({\boldsymbol {\sigma }})+\varepsilon _{ij}^{p}({\boldsymbol {\xi }})}"></span> </p> </blockquote><p>Bestalde, <b>fluxu-legea</b> energiaren xahupen plastikoarekin lotutako desberdintasun batek mugatzen du. Desberdintasun hau <a href="/wiki/Termodinamikaren_bigarren_legea" title="Termodinamikaren bigarren legea">termodinamikaren bigarren legetik</a> dator Clausius-Duhem forman:</p><blockquote style="padding: 5px 10px;background-color: white; color:black; text-align:left; margin-left:30px; margin-bottom:0.4em; margin-top:0.2em; min-width:50%;"> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\dot {f}}+s{\dot {T}}-{\boldsymbol {\sigma }}:{\dot {\boldsymbol {\varepsilon }}}+{\frac {\mathbf {q} }{T}}\cdot {\boldsymbol {\nabla }}T\leq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mo>+</mo> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>T</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">σ</mi> </mrow> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold-italic">ε</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> <mi>T</mi> </mfrac> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">∇</mi> </mrow> <mi>T</mi> <mo>≤</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\dot {f}}+s{\dot {T}}-{\boldsymbol {\sigma }}:{\dot {\boldsymbol {\varepsilon }}}+{\frac {\mathbf {q} }{T}}\cdot {\boldsymbol {\nabla }}T\leq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4f9acd42593bfef688ce3e2adedf0f75bc0ac0bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:30.31ex; height:4.843ex;" alt="{\displaystyle {\dot {f}}+s{\dot {T}}-{\boldsymbol {\sigma }}:{\dot {\boldsymbol {\varepsilon }}}+{\frac {\mathbf {q} }{T}}\cdot {\boldsymbol {\nabla }}T\leq 0}"></span> </p> </blockquote><p>Non: </p><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f,s\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>,</mo> <mi>s</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f,s\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8f419f41a8e2a5f90cc027fc2e657283e4225f3d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.79ex; height:2.509ex;" alt="{\displaystyle f,s\,}"></span> Helmholtz energia librea eta bolumen unitateko <a href="/wiki/Entropia" title="Entropia">entropia</a> dira.</dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T,\mathbf {q} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T,\mathbf {q} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a06fedb50dd3c600eba8f410674eb06eb0e088dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.086ex; height:2.509ex;" alt="{\displaystyle T,\mathbf {q} }"></span> gainazaleko tenperatura eta <a href="/wiki/Beroaren_eroapen" title="Beroaren eroapen">bero-fluxua</a> dira.</dd></dl> <div class="mw-heading mw-heading2"><h2 id="Erreferentziak">Erreferentziak</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Plastikotasun&veaction=edit&section=5" title="Aldatu atal hau: «Erreferentziak»" class="mw-editsection-visualeditor"><span>aldatu</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Plastikotasun&action=edit&section=5" title="Edit section's source code: Erreferentziak"><span>aldatu iturburu kodea</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="reflist columns references-column-width" style="-moz-column-width: 30em; -webkit-column-width: 30em; column-width: 30em; list-style-type: decimal;"> </div> <div class="mw-heading mw-heading2"><h2 id="Ikus,_gainera"><span id="Ikus.2C_gainera"></span>Ikus, gainera</h2><span class="mw-editsection"><span 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