ความเครียด (กลศาสตร์)

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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="หน้าตา"> <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="หน้าตา" > <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">หน้าตา</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_th.wikipedia.org&uselang=th" class=""><span>บริจาคให้วิกิพีเดีย</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=%E0%B8%9E%E0%B8%B4%E0%B9%80%E0%B8%A8%E0%B8%A9:%E0%B8%AA%E0%B8%A3%E0%B9%89%E0%B8%B2%E0%B8%87%E0%B8%9A%E0%B8%B1%E0%B8%8D%E0%B8%8A%E0%B8%B5%E0%B8%9C%E0%B8%B9%E0%B9%89%E0%B9%83%E0%B8%8A%E0%B9%89%E0%B9%83%E0%B8%AB%E0%B8%A1%E0%B9%88&returnto=%E0%B8%84%E0%B8%A7%E0%B8%B2%E0%B8%A1%E0%B9%80%E0%B8%84%E0%B8%A3%E0%B8%B5%E0%B8%A2%E0%B8%94+%28%E0%B8%81%E0%B8%A5%E0%B8%A8%E0%B8%B2%E0%B8%AA%E0%B8%95%E0%B8%A3%E0%B9%8C%29" title="แนะนำให้คุณสร้างบัญชีและเข้าสู่ระบบ แต่ไม่บังคับ" class=""><span>สร้างบัญชี</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=%E0%B8%9E%E0%B8%B4%E0%B9%80%E0%B8%A8%E0%B8%A9:%E0%B8%A5%E0%B9%87%E0%B8%AD%E0%B8%81%E0%B8%AD%E0%B8%B4%E0%B8%99&returnto=%E0%B8%84%E0%B8%A7%E0%B8%B2%E0%B8%A1%E0%B9%80%E0%B8%84%E0%B8%A3%E0%B8%B5%E0%B8%A2%E0%B8%94+%28%E0%B8%81%E0%B8%A5%E0%B8%A8%E0%B8%B2%E0%B8%AA%E0%B8%95%E0%B8%A3%E0%B9%8C%29" title="แนะนำให้คุณเข้าสู่ระบบ แต่ไม่บังคับ [o]" accesskey="o" class=""><span>เข้าสู่ระบบ</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="ตัวเลือกเพิ่มเติม" > <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="เครื่องมือส่วนตัว" > <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">เครื่องมือส่วนตัว</span> </label> <div class="vector-dropdown-content"> <div id="p-personal" class="vector-menu mw-portlet mw-portlet-personal user-links-collapsible-item" title="เมนูผู้ใช้" > <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_th.wikipedia.org&uselang=th"><span>บริจาคให้วิกิพีเดีย</span></a></li><li id="pt-createaccount" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=%E0%B8%9E%E0%B8%B4%E0%B9%80%E0%B8%A8%E0%B8%A9:%E0%B8%AA%E0%B8%A3%E0%B9%89%E0%B8%B2%E0%B8%87%E0%B8%9A%E0%B8%B1%E0%B8%8D%E0%B8%8A%E0%B8%B5%E0%B8%9C%E0%B8%B9%E0%B9%89%E0%B9%83%E0%B8%8A%E0%B9%89%E0%B9%83%E0%B8%AB%E0%B8%A1%E0%B9%88&returnto=%E0%B8%84%E0%B8%A7%E0%B8%B2%E0%B8%A1%E0%B9%80%E0%B8%84%E0%B8%A3%E0%B8%B5%E0%B8%A2%E0%B8%94+%28%E0%B8%81%E0%B8%A5%E0%B8%A8%E0%B8%B2%E0%B8%AA%E0%B8%95%E0%B8%A3%E0%B9%8C%29" title="แนะนำให้คุณสร้างบัญชีและเข้าสู่ระบบ แต่ไม่บังคับ"><span class="vector-icon mw-ui-icon-userAdd mw-ui-icon-wikimedia-userAdd"></span> <span>สร้างบัญชี</span></a></li><li id="pt-login" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=%E0%B8%9E%E0%B8%B4%E0%B9%80%E0%B8%A8%E0%B8%A9:%E0%B8%A5%E0%B9%87%E0%B8%AD%E0%B8%81%E0%B8%AD%E0%B8%B4%E0%B8%99&returnto=%E0%B8%84%E0%B8%A7%E0%B8%B2%E0%B8%A1%E0%B9%80%E0%B8%84%E0%B8%A3%E0%B8%B5%E0%B8%A2%E0%B8%94+%28%E0%B8%81%E0%B8%A5%E0%B8%A8%E0%B8%B2%E0%B8%AA%E0%B8%95%E0%B8%A3%E0%B9%8C%29" title="แนะนำให้คุณเข้าสู่ระบบ แต่ไม่บังคับ [o]" accesskey="o"><span class="vector-icon mw-ui-icon-logIn mw-ui-icon-wikimedia-logIn"></span> <span>เข้าสู่ระบบ</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"> หน้าสำหรับผู้แก้ไขที่ออกจากระบบ <a href="/wiki/%E0%B8%A7%E0%B8%B4%E0%B8%98%E0%B8%B5%E0%B9%83%E0%B8%8A%E0%B9%89:%E0%B8%9A%E0%B8%97%E0%B8%99%E0%B8%B3" aria-label="เรียนรู้เพิ่มเกี่ยวกับการแก้ไข"><span>เรียนรู้เพิ่มเติม</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/%E0%B8%9E%E0%B8%B4%E0%B9%80%E0%B8%A8%E0%B8%A9:%E0%B9%80%E0%B8%A3%E0%B8%B7%E0%B9%88%E0%B8%AD%E0%B8%87%E0%B8%97%E0%B8%B5%E0%B9%88%E0%B8%89%E0%B8%B1%E0%B8%99%E0%B9%80%E0%B8%82%E0%B8%B5%E0%B8%A2%E0%B8%99" title="รายการการแก้ไขจากเลขที่อยู่ไอพีนี้ [y]" accesskey="y"><span>ส่วนร่วม</span></a></li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/%E0%B8%9E%E0%B8%B4%E0%B9%80%E0%B8%A8%E0%B8%A9:%E0%B8%AB%E0%B8%99%E0%B9%89%E0%B8%B2%E0%B8%9E%E0%B8%B9%E0%B8%94%E0%B8%84%E0%B8%B8%E0%B8%A2%E0%B8%82%E0%B8%AD%E0%B8%87%E0%B8%89%E0%B8%B1%E0%B8%99" title="อภิปรายเกี่ยวกับการแก้ไขจากเลขที่อยู่ไอพีนี้ [n]" accesskey="n"><span>คุย</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="ไซต์"> <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="สารบัญ" 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">สารบัญ</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">ย้ายเมนูไปที่แถบด้านข้าง</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">ซ่อน</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">บทนำ</div> </a> </li> <li id="toc-นิยามความเครียด" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#นิยามความเครียด"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>นิยามความเครียด</span> </div> </a> <button aria-controls="toc-นิยามความเครียด-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle นิยามความเครียด subsection</span> </button> <ul id="toc-นิยามความเครียด-sublist" class="vector-toc-list"> <li id="toc-ความเครียดทางวิศวกรรม" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#ความเครียดทางวิศวกรรม"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>ความเครียดทางวิศวกรรม</span> </div> </a> <ul id="toc-ความเครียดทางวิศวกรรม-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-อัตราส่วนความยืด" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#อัตราส่วนความยืด"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.2</span> <span>อัตราส่วนความยืด</span> </div> </a> <ul id="toc-อัตราส่วนความยืด-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-ความเครียดจริง" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#ความเครียดจริง"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.3</span> <span>ความเครียดจริง</span> </div> </a> <ul id="toc-ความเครียดจริง-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-ความเครียดตั้งฉากและความเครียดเฉือน" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#ความเครียดตั้งฉากและความเครียดเฉือน"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>ความเครียดตั้งฉากและความเครียดเฉือน</span> </div> </a> <button aria-controls="toc-ความเครียดตั้งฉากและความเครียดเฉือน-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle ความเครียดตั้งฉากและความเครียดเฉือน subsection</span> </button> <ul id="toc-ความเครียดตั้งฉากและความเครียดเฉือน-sublist" class="vector-toc-list"> <li id="toc-ความเครียดตั้งฉาก" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#ความเครียดตั้งฉาก"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>ความเครียดตั้งฉาก</span> </div> </a> <ul id="toc-ความเครียดตั้งฉาก-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-ความเครียดเฉือน" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#ความเครียดเฉือน"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>ความเครียดเฉือน</span> </div> </a> <ul id="toc-ความเครียดเฉือน-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-อ้างอิง" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#อ้างอิง"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>อ้างอิง</span> </div> </a> <ul id="toc-อ้างอิง-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="สารบัญ" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Toggle the table of contents</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">ความเครียด (กลศาสตร์)</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="ไปที่บทความในภาษาอื่น ซึ่งมีใน 55 ภาษา" > <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-55" 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">55 ภาษา</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-af mw-list-item"><a href="https://af.wikipedia.org/wiki/Vervorming" title="Vervorming – แอฟริกานส์" lang="af" hreflang="af" data-title="Vervorming" data-language-autonym="Afrikaans" data-language-local-name="แอฟริกานส์" class="interlanguage-link-target"><span>Afrikaans</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%A7%D9%86%D9%81%D8%B9%D8%A7%D9%84_(%D8%B9%D9%84%D9%85_%D8%A7%D9%84%D9%85%D9%88%D8%A7%D8%AF)" title="انفعال (علم المواد) – อาหรับ" lang="ar" hreflang="ar" data-title="انفعال (علم المواد)" data-language-autonym="العربية" data-language-local-name="อาหรับ" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Deformaci%C3%B3n" title="Deformación – อัสตูเรียส" lang="ast" hreflang="ast" data-title="Deformación" data-language-autonym="Asturianu" data-language-local-name="อัสตูเรียส" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/Deformasiya" title="Deformasiya – อาเซอร์ไบจาน" lang="az" hreflang="az" data-title="Deformasiya" data-language-autonym="Azərbaycanca" data-language-local-name="อาเซอร์ไบจาน" 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%94%D1%8D%D1%84%D0%B0%D1%80%D0%BC%D0%B0%D1%86%D1%8B%D1%8F" title="Дэфармацыя – เบลารุส" lang="be" hreflang="be" data-title="Дэфармацыя" data-language-autonym="Беларуская" data-language-local-name="เบลารุส" 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%94%D1%8D%D1%84%D0%B0%D1%80%D0%BC%D0%B0%D1%86%D1%8B%D1%8F" title="Дэфармацыя – Belarusian (Taraškievica orthography)" lang="be-tarask" hreflang="be-tarask" data-title="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%94%D0%B5%D1%84%D0%BE%D1%80%D0%BC%D0%B0%D1%86%D0%B8%D1%8F" title="Деформация – บัลแกเรีย" lang="bg" hreflang="bg" data-title="Деформация" data-language-autonym="Български" data-language-local-name="บัลแกเรีย" 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%B0%E0%A7%82%E0%A6%AA%E0%A6%AC%E0%A6%BF%E0%A6%95%E0%A6%BE%E0%A6%B0_(%E0%A6%AA%E0%A6%A6%E0%A6%BE%E0%A6%B0%E0%A7%8D%E0%A6%A5%E0%A6%AC%E0%A6%BF%E0%A6%9C%E0%A7%8D%E0%A6%9E%E0%A6%BE%E0%A6%A8)" title="রূপবিকার (পদার্থবিজ্ঞান) – บังกลา" lang="bn" hreflang="bn" data-title="রূপবিকার (পদার্থবিজ্ঞান)" data-language-autonym="বাংলা" data-language-local-name="บังกลา" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Deformacija_(mehanika)" title="Deformacija (mehanika) – บอสเนีย" lang="bs" hreflang="bs" data-title="Deformacija (mehanika)" data-language-autonym="Bosanski" data-language-local-name="บอสเนีย" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Deformaci%C3%B3" title="Deformació – คาตาลัน" lang="ca" hreflang="ca" data-title="Deformació" data-language-autonym="Català" data-language-local-name="คาตาลัน" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Deformace" title="Deformace – เช็ก" lang="cs" hreflang="cs" data-title="Deformace" data-language-autonym="Čeština" data-language-local-name="เช็ก" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%94%D0%B5%D1%84%D0%BE%D1%80%D0%BC%D0%B0%D1%86%D0%B8" title="Деформаци – ชูวัช" lang="cv" hreflang="cv" data-title="Деформаци" data-language-autonym="Чӑвашла" data-language-local-name="ชูวัช" 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/Verformung" title="Verformung – เยอรมัน" lang="de" hreflang="de" data-title="Verformung" data-language-autonym="Deutsch" data-language-local-name="เยอรมัน" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%A0%CE%B1%CF%81%CE%B1%CE%BC%CF%8C%CF%81%CF%86%CF%89%CF%83%CE%B7_(%CE%BC%CE%B7%CF%87%CE%B1%CE%BD%CE%B9%CE%BA%CE%AE)" title="Παραμόρφωση (μηχανική) – กรีก" lang="el" hreflang="el" data-title="Παραμόρφωση (μηχανική)" data-language-autonym="Ελληνικά" data-language-local-name="กรีก" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Deformation_(physics)" title="Deformation (physics) – อังกฤษ" lang="en" hreflang="en" data-title="Deformation (physics)" data-language-autonym="English" data-language-local-name="อังกฤษ" 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/Deformi%C4%9Do" title="Deformiĝo – เอสเปรันโต" lang="eo" hreflang="eo" data-title="Deformiĝo" data-language-autonym="Esperanto" data-language-local-name="เอสเปรันโต" 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/Deformaci%C3%B3n" title="Deformación – สเปน" lang="es" hreflang="es" data-title="Deformación" data-language-autonym="Español" data-language-local-name="สเปน" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Deformazio" title="Deformazio – บาสก์" lang="eu" hreflang="eu" data-title="Deformazio" data-language-autonym="Euskara" data-language-local-name="บาสก์" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%AA%D8%BA%DB%8C%DB%8C%D8%B1_%D8%B4%DA%A9%D9%84_(%D9%81%DB%8C%D8%B2%DB%8C%DA%A9)" title="تغییر شکل (فیزیک) – เปอร์เซีย" lang="fa" hreflang="fa" data-title="تغییر شکل (فیزیک)" data-language-autonym="فارسی" data-language-local-name="เปอร์เซีย" 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/Venym%C3%A4" title="Venymä – ฟินแลนด์" lang="fi" hreflang="fi" data-title="Venymä" data-language-autonym="Suomi" data-language-local-name="ฟินแลนด์" 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_d%27un_mat%C3%A9riau" title="Déformation d'un matériau – ฝรั่งเศส" lang="fr" hreflang="fr" data-title="Déformation d'un matériau" data-language-autonym="Français" data-language-local-name="ฝรั่งเศส" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Deformaci%C3%B3n" title="Deformación – กาลิเซีย" lang="gl" hreflang="gl" data-title="Deformación" data-language-autonym="Galego" data-language-local-name="กาลิเซีย" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%9E%D7%A2%D7%95%D7%95%D7%AA" title="מעוות – ฮิบรู" lang="he" hreflang="he" data-title="מעוות" data-language-autonym="עברית" data-language-local-name="ฮิบรู" 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%B5%E0%A4%BF%E0%A4%B0%E0%A5%82%E0%A4%AA%E0%A4%A3_(%E0%A4%AF%E0%A4%BE%E0%A4%82%E0%A4%A4%E0%A5%8D%E0%A4%B0%E0%A4%BF%E0%A4%95%E0%A5%80)" title="विरूपण (यांत्रिकी) – ฮินดี" lang="hi" hreflang="hi" data-title="विरूपण (यांत्रिकी)" data-language-autonym="हिन्दी" data-language-local-name="ฮินดี" 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/Deformacija" title="Deformacija – โครเอเชีย" lang="hr" hreflang="hr" data-title="Deformacija" data-language-autonym="Hrvatski" data-language-local-name="โครเอเชีย" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D4%B4%D5%A5%D6%86%D5%B8%D6%80%D5%B4%D5%A1%D6%81%D5%AB%D5%A1" title="Դեֆորմացիա – อาร์เมเนีย" lang="hy" hreflang="hy" data-title="Դեֆորմացիա" data-language-autonym="Հայերեն" data-language-local-name="อาร์เมเนีย" 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/Deformasi_(fisika)" title="Deformasi (fisika) – อินโดนีเซีย" lang="id" hreflang="id" data-title="Deformasi (fisika)" data-language-autonym="Bahasa Indonesia" data-language-local-name="อินโดนีเซีย" 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/Deformazione" title="Deformazione – อิตาลี" lang="it" hreflang="it" data-title="Deformazione" data-language-autonym="Italiano" data-language-local-name="อิตาลี" 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%A4%89%E5%BD%A2" title="変形 – ญี่ปุ่น" lang="ja" hreflang="ja" data-title="変形" data-language-autonym="日本語" data-language-local-name="ญี่ปุ่น" 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%93%E1%83%94%E1%83%A4%E1%83%9D%E1%83%A0%E1%83%9B%E1%83%90%E1%83%AA%E1%83%98%E1%83%90" title="დეფორმაცია – จอร์เจีย" lang="ka" hreflang="ka" data-title="დეფორმაცია" data-language-autonym="ქართული" data-language-local-name="จอร์เจีย" 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%94%D0%B5%D1%84%D0%BE%D1%80%D0%BC%D0%B0%D1%86%D0%B8%D1%8F" title="Деформация – คาซัค" lang="kk" hreflang="kk" data-title="Деформация" data-language-autonym="Қазақша" data-language-local-name="คาซัค" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-kn mw-list-item"><a href="https://kn.wikipedia.org/wiki/%E0%B2%B5%E0%B2%BF%E0%B2%95%E0%B3%83%E0%B2%A4%E0%B2%BF" title="ವಿಕೃತಿ – กันนาดา" lang="kn" hreflang="kn" data-title="ವಿಕೃತಿ" data-language-autonym="ಕನ್ನಡ" data-language-local-name="กันนาดา" class="interlanguage-link-target"><span>ಕನ್ನಡ</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%9D%BC%EA%B7%B8%EB%9F%AC%EC%A7%90_(%EB%AC%BC%EB%A6%AC%ED%95%99)" title="일그러짐 (물리학) – เกาหลี" lang="ko" hreflang="ko" data-title="일그러짐 (물리학)" data-language-autonym="한국어" data-language-local-name="เกาหลี" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-ky mw-list-item"><a href="https://ky.wikipedia.org/wiki/%D0%94%D0%B5%D1%84%D0%BE%D1%80%D0%BC%D0%B0%D1%86%D0%B8%D1%8F_(%D1%84%D0%B8%D0%B7%D0%B8%D0%BA%D0%B0%D0%B4%D0%B0)" title="Деформация (физикада) – คีร์กีซ" lang="ky" hreflang="ky" data-title="Деформация (физикада)" data-language-autonym="Кыргызча" data-language-local-name="คีร์กีซ" class="interlanguage-link-target"><span>Кыргызча</span></a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Deformatio" title="Deformatio – ละติน" lang="la" hreflang="la" data-title="Deformatio" data-language-autonym="Latina" data-language-local-name="ละติน" class="interlanguage-link-target"><span>Latina</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/K%C5%ABno_deformacija" title="Kūno deformacija – ลิทัวเนีย" lang="lt" hreflang="lt" data-title="Kūno deformacija" data-language-autonym="Lietuvių" data-language-local-name="ลิทัวเนีย" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Deform%C4%81cija" title="Deformācija – ลัตเวีย" lang="lv" hreflang="lv" data-title="Deformācija" data-language-autonym="Latviešu" data-language-local-name="ลัตเวีย" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-mn mw-list-item"><a href="https://mn.wikipedia.org/wiki/%D0%94%D0%B5%D1%84%D0%BE%D1%80%D0%BC%D0%B0%D1%86" title="Деформац – มองโกเลีย" lang="mn" hreflang="mn" data-title="Деформац" data-language-autonym="Монгол" data-language-local-name="มองโกเลีย" class="interlanguage-link-target"><span>Монгол</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Deformatie_(natuurkunde)" title="Deformatie (natuurkunde) – ดัตช์" lang="nl" hreflang="nl" data-title="Deformatie (natuurkunde)" data-language-autonym="Nederlands" data-language-local-name="ดัตช์" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Deformasjon" title="Deformasjon – นอร์เวย์นีนอสก์" lang="nn" hreflang="nn" data-title="Deformasjon" data-language-autonym="Norsk nynorsk" data-language-local-name="นอร์เวย์นีนอสก์" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Deformasjon" title="Deformasjon – นอร์เวย์บุคมอล" lang="nb" hreflang="nb" data-title="Deformasjon" data-language-autonym="Norsk bokmål" data-language-local-name="นอร์เวย์บุคมอล" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Odkszta%C5%82cenie" title="Odkształcenie – โปแลนด์" lang="pl" hreflang="pl" data-title="Odkształcenie" data-language-autonym="Polski" data-language-local-name="โปแลนด์" 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/Deforma%C3%A7%C3%A3o" title="Deformação – โปรตุเกส" lang="pt" hreflang="pt" data-title="Deformação" data-language-autonym="Português" data-language-local-name="โปรตุเกส" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Deformare" title="Deformare – โรมาเนีย" lang="ro" hreflang="ro" data-title="Deformare" data-language-autonym="Română" data-language-local-name="โรมาเนีย" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%94%D0%B5%D1%84%D0%BE%D1%80%D0%BC%D0%B0%D1%86%D0%B8%D1%8F" title="Деформация – รัสเซีย" lang="ru" hreflang="ru" data-title="Деформация" data-language-autonym="Русский" data-language-local-name="รัสเซีย" 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/Deformacija_(mehanika)" title="Deformacija (mehanika) – เซอร์โบ-โครเอเชีย" lang="sh" hreflang="sh" data-title="Deformacija (mehanika)" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="เซอร์โบ-โครเอเชีย" 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/Deformation" title="Deformation – Simple English" lang="en-simple" hreflang="en-simple" data-title="Deformation" 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/Deform%C3%A1cia_(mechanika)" title="Deformácia (mechanika) – สโลวัก" lang="sk" hreflang="sk" data-title="Deformácia (mechanika)" data-language-autonym="Slovenčina" data-language-local-name="สโลวัก" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Deformimi_(fizik%C3%AB)" title="Deformimi (fizikë) – แอลเบเนีย" lang="sq" hreflang="sq" data-title="Deformimi (fizikë)" data-language-autonym="Shqip" data-language-local-name="แอลเบเนีย" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%94%D0%B5%D1%84%D0%BE%D1%80%D0%BC%D0%B0%D1%86%D0%B8%D1%98%D0%B0" title="Деформација – เซอร์เบีย" lang="sr" hreflang="sr" data-title="Деформација" data-language-autonym="Српски / srpski" data-language-local-name="เซอร์เบีย" 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/Deformation" title="Deformation – สวีเดน" lang="sv" hreflang="sv" data-title="Deformation" data-language-autonym="Svenska" data-language-local-name="สวีเดน" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/%C5%9Eekil_de%C4%9Fi%C5%9Ftirme" title="Şekil değiştirme – ตุรกี" lang="tr" hreflang="tr" data-title="Şekil değiştirme" data-language-autonym="Türkçe" data-language-local-name="ตุรกี" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%94%D0%B5%D1%84%D0%BE%D1%80%D0%BC%D0%B0%D1%86%D1%96%D1%8F" title="Деформація – ยูเครน" lang="uk" hreflang="uk" data-title="Деформація" data-language-autonym="Українська" data-language-local-name="ยูเครน" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Deformatsiya" title="Deformatsiya – อุซเบก" lang="uz" hreflang="uz" data-title="Deformatsiya" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="อุซเบก" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-vec mw-list-item"><a href="https://vec.wikipedia.org/wiki/Deformasion" title="Deformasion – เวเนโต้" lang="vec" hreflang="vec" data-title="Deformasion" data-language-autonym="Vèneto" data-language-local-name="เวเนโต้" class="interlanguage-link-target"><span>Vèneto</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q193514#sitelinks-wikipedia" title="แก้ไขลิงก์ข้ามภาษา" class="wbc-editpage">แก้ไขลิงก์</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="เนมสเปซ"> <div id="p-associated-pages" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-associated-pages" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-nstab-main" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/%E0%B8%84%E0%B8%A7%E0%B8%B2%E0%B8%A1%E0%B9%80%E0%B8%84%E0%B8%A3%E0%B8%B5%E0%B8%A2%E0%B8%94_(%E0%B8%81%E0%B8%A5%E0%B8%A8%E0%B8%B2%E0%B8%AA%E0%B8%95%E0%B8%A3%E0%B9%8C)" title="ดูหน้าเนื้อหา [c]" accesskey="c"><span>บทความ</span></a></li><li id="ca-talk" class="new vector-tab-noicon mw-list-item"><a href="/w/index.php?title=%E0%B8%9E%E0%B8%B9%E0%B8%94%E0%B8%84%E0%B8%B8%E0%B8%A2:%E0%B8%84%E0%B8%A7%E0%B8%B2%E0%B8%A1%E0%B9%80%E0%B8%84%E0%B8%A3%E0%B8%B5%E0%B8%A2%E0%B8%94_(%E0%B8%81%E0%B8%A5%E0%B8%A8%E0%B8%B2%E0%B8%AA%E0%B8%95%E0%B8%A3%E0%B9%8C)&action=edit&redlink=1" rel="discussion" class="new" title="อภิปรายเกี่ยวกับหน้าเนื้อหา (ไม่มีหน้านี้) [t]" accesskey="t"><span>อภิปราย</span></a></li> </ul> </div> </div> <div id="vector-variants-dropdown" class="vector-dropdown emptyPortlet" > <input type="checkbox" id="vector-variants-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-variants-dropdown" class="vector-dropdown-checkbox " aria-label="เปลี่ยนรูปแบบภาษา" > <label id="vector-variants-dropdown-label" for="vector-variants-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span class="vector-dropdown-label-text">ไทย</span> </label> <div class="vector-dropdown-content"> <div id="p-variants" class="vector-menu mw-portlet mw-portlet-variants emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> </div> </div> </nav> </div> <div id="right-navigation" class="vector-collapsible"> <nav aria-label="ดู"> <div id="p-views" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-views" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-view" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/%E0%B8%84%E0%B8%A7%E0%B8%B2%E0%B8%A1%E0%B9%80%E0%B8%84%E0%B8%A3%E0%B8%B5%E0%B8%A2%E0%B8%94_(%E0%B8%81%E0%B8%A5%E0%B8%A8%E0%B8%B2%E0%B8%AA%E0%B8%95%E0%B8%A3%E0%B9%8C)"><span>อ่าน</span></a></li><li id="ca-edit" class="vector-tab-noicon mw-list-item"><a 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href="/w/index.php?title=%E0%B8%84%E0%B8%A7%E0%B8%B2%E0%B8%A1%E0%B9%80%E0%B8%84%E0%B8%A3%E0%B8%B5%E0%B8%A2%E0%B8%94_(%E0%B8%81%E0%B8%A5%E0%B8%A8%E0%B8%B2%E0%B8%AA%E0%B8%95%E0%B8%A3%E0%B9%8C)&action=edit" title="แก้ไขรหัสต้นฉบับของหน้านี้ [e]" accesskey="e"><span>แก้ไข</span></a></li><li id="ca-more-history" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=%E0%B8%84%E0%B8%A7%E0%B8%B2%E0%B8%A1%E0%B9%80%E0%B8%84%E0%B8%A3%E0%B8%B5%E0%B8%A2%E0%B8%94_(%E0%B8%81%E0%B8%A5%E0%B8%A8%E0%B8%B2%E0%B8%AA%E0%B8%95%E0%B8%A3%E0%B9%8C)&action=history"><span>ดูประวัติ</span></a></li> </ul> </div> </div> <div id="p-tb" class="vector-menu mw-portlet mw-portlet-tb" > <div class="vector-menu-heading"> ทั่วไป </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-whatlinkshere" class="mw-list-item"><a href="/wiki/%E0%B8%9E%E0%B8%B4%E0%B9%80%E0%B8%A8%E0%B8%A9:%E0%B8%9A%E0%B8%97%E0%B8%84%E0%B8%A7%E0%B8%B2%E0%B8%A1%E0%B8%97%E0%B8%B5%E0%B9%88%E0%B9%82%E0%B8%A2%E0%B8%87%E0%B8%A1%E0%B8%B2/%E0%B8%84%E0%B8%A7%E0%B8%B2%E0%B8%A1%E0%B9%80%E0%B8%84%E0%B8%A3%E0%B8%B5%E0%B8%A2%E0%B8%94_(%E0%B8%81%E0%B8%A5%E0%B8%A8%E0%B8%B2%E0%B8%AA%E0%B8%95%E0%B8%A3%E0%B9%8C)" title="รายการหน้าวิกิทุกหน้าที่ลิงก์มาที่นี่ [j]" accesskey="j"><span>หน้าที่ลิงก์มา</span></a></li><li id="t-recentchangeslinked" class="mw-list-item"><a href="/wiki/%E0%B8%9E%E0%B8%B4%E0%B9%80%E0%B8%A8%E0%B8%A9:%E0%B8%81%E0%B8%B2%E0%B8%A3%E0%B8%9B%E0%B8%A3%E0%B8%B1%E0%B8%9A%E0%B8%9B%E0%B8%A3%E0%B8%B8%E0%B8%87%E0%B8%97%E0%B8%B5%E0%B9%88%E0%B9%82%E0%B8%A2%E0%B8%87%E0%B8%A1%E0%B8%B2/%E0%B8%84%E0%B8%A7%E0%B8%B2%E0%B8%A1%E0%B9%80%E0%B8%84%E0%B8%A3%E0%B8%B5%E0%B8%A2%E0%B8%94_(%E0%B8%81%E0%B8%A5%E0%B8%A8%E0%B8%B2%E0%B8%AA%E0%B8%95%E0%B8%A3%E0%B9%8C)" rel="nofollow" title="รายการเปลี่ยนแปลงล่าสุดในหน้าที่ลิงก์จากหน้านี้ [k]" 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class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">ย้ายเมนูไปที่แถบด้านข้าง</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">ซ่อน</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">จากวิกิพีเดีย สารานุกรมเสรี</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="th" dir="ltr"><p><b>ความเครียด</b> (<a href="/wiki/%E0%B8%A0%E0%B8%B2%E0%B8%A9%E0%B8%B2%E0%B8%AD%E0%B8%B1%E0%B8%87%E0%B8%81%E0%B8%A4%E0%B8%A9" title="ภาษาอังกฤษ">อังกฤษ</a>: <b>strain</b>) คือปริมาณการเปลี่ยนแปลงขนาดของวัตถุเทียบกับขนาดตั้งต้น </p><p>การเปลี่ยนแปลงรูปร่างของวัตถุสามารถเขียนได้ด้วยการเขียนตำแหน่งสุดท้ายของวัตถุ <span class="texhtml"><b>x</b> = <i><b>F</b></i>(<b>X</b>)</span> โดยที่ <span class="texhtml"><b>X</b></span> คือตำแหน่งตั้งต้นของวัตถุ การเขียนสมการแบบนี้จะไม่แยกระหว่างการเคลื่อนที่ของวัตถุ (การเปลี่ยนตำแหน่งและหมุน) และการเปลี่ยนรูปร่างและขนาดของวัตถุ </p><p>เราสามารถนิยามความเครียดได้จาก </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 {\boldsymbol {\varepsilon }}\doteq {\cfrac {\partial }{\partial \mathbf {X} }}\left(\mathbf {x} -\mathbf {X} \right)={\boldsymbol {F}}'-{\boldsymbol {I}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ε</mi> </mrow> <mo>≐</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∂</mi> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> </mrow> </mstyle> </mrow> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">F</mi> </mrow> <mo>′</mo> </msup> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">I</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {\varepsilon }}\doteq {\cfrac {\partial }{\partial \mathbf {X} }}\left(\mathbf {x} -\mathbf {X} \right)={\boldsymbol {F}}'-{\boldsymbol {I}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b02fc07ffc15363ae1397c107d6b9306e54c5c89" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:26.889ex; height:7.176ex;" alt="{\displaystyle {\boldsymbol {\varepsilon }}\doteq {\cfrac {\partial }{\partial \mathbf {X} }}\left(\mathbf {x} -\mathbf {X} \right)={\boldsymbol {F}}'-{\boldsymbol {I}}}"></span></dd></dl> <p>โดยที่ <span class="texhtml mvar" style="font-style:italic;"><b>I</b></span> คือ <a href="/wiki/%E0%B9%80%E0%B8%A1%E0%B8%97%E0%B8%A3%E0%B8%B4%E0%B8%81%E0%B8%8B%E0%B9%8C%E0%B9%80%E0%B8%AD%E0%B8%81%E0%B8%A5%E0%B8%B1%E0%B8%81%E0%B8%A9%E0%B8%93%E0%B9%8C" title="เมทริกซ์เอกลักษณ์">เมทริกซ์เอกลักษณ์</a> ดังนั้นความเครียดจึงไม่มีหน่วย และมักจะนิยมเขียนเป็นจุดทศนิยม เปอร์เซ็นต์ หรือ ร้อยละ ความเครียดบอกถึงการเปลี่ยนรูปร่างที่ตำแหน่งใดๆในวัตถุจากการเปลี่ยนแปลงทั้งหมด </p><p>ความเครียดเป็นปริมาณเทนเซอร์ ความเครียดสามารถแบ่งองค์ประกอบเป็นความ<i>เครียดตั้งฉาก</i> (normal strain) และ<i>ความเครียดเฉือน</i> (shear strain) เมื่อวัตถุถูกเปลี่ยนรูปร่าง ความเครียดตั้งฉาก บอกถึงอัตราส่วนการเปลี่ยนขนาดหรือความยาวของวัตถุ ในขณะที่ความเครียดเฉือนบอกถึงมุมที่วัตถุถูกเบือนจากทิศทางตั้งต้น โดยความเครียดทั้งสอบแบบนี้บอกถึงการเปลี่ยนรูปร่างในทิศตั้งฉากกัน <sup id="cite_ref-rees_1-0" class="reference"><a href="#cite_note-rees-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p><p>ถ้าความยาวของวัตถุเพิ่มขึ้น ความเครียดเฉือนตั้งฉากจะเรียกว่า <i>ความเครียดดึง</i> (tensile strain) ในทางกลับกันถ้าความยาวลดลง เราจะเรียกว่า <i>ความเครียดอัด</i> (compressive strain) </p><p>นอกเหนือจากนิยามที่กล่าวมายังมีการนิยามความเครียดหลายแบบ อาทิเช่น <i>ความเครียดทางวิศวกรรม</i> (engineering strain)ซึ่งมักจะใช้กับวัสดุที่ใช้ในเคลื่องกลและโครงสร้างทางวิศวกรรมซึ่งจะเปลี่ยนรูปร่างได้เพียงเล็กน้อย ในขณะที่วัสดุบางประเภท อาทิ อีลาสโตเมอร์และพอลิเมอร์สามารถเปลี่ยนรูปร่างได้เยอะ การใช้นิยามความเครียดทางวิศวกรรมนั้นอาจจะไม่เหมาะสมเมื่อวัตถุขยายขนาดมากกว่า 1% <sup id="cite_ref-rees_1-1" class="reference"><a href="#cite_note-rees-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> จึงต้องใช้นิยามแบบอื่น เช่น อัตราส่วนการยืด หรือ ความเครียดจริง </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="นิยามความเครียด"><span id=".E0.B8.99.E0.B8.B4.E0.B8.A2.E0.B8.B2.E0.B8.A1.E0.B8.84.E0.B8.A7.E0.B8.B2.E0.B8.A1.E0.B9.80.E0.B8.84.E0.B8.A3.E0.B8.B5.E0.B8.A2.E0.B8.94"></span>นิยามความเครียด</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E0%B8%84%E0%B8%A7%E0%B8%B2%E0%B8%A1%E0%B9%80%E0%B8%84%E0%B8%A3%E0%B8%B5%E0%B8%A2%E0%B8%94_(%E0%B8%81%E0%B8%A5%E0%B8%A8%E0%B8%B2%E0%B8%AA%E0%B8%95%E0%B8%A3%E0%B9%8C)&action=edit&section=1" title="แก้ไขส่วน: นิยามความเครียด"><span>แก้</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading4"><h4 id="ความเครียดทางวิศวกรรม"><span id=".E0.B8.84.E0.B8.A7.E0.B8.B2.E0.B8.A1.E0.B9.80.E0.B8.84.E0.B8.A3.E0.B8.B5.E0.B8.A2.E0.B8.94.E0.B8.97.E0.B8.B2.E0.B8.87.E0.B8.A7.E0.B8.B4.E0.B8.A8.E0.B8.A7.E0.B8.81.E0.B8.A3.E0.B8.A3.E0.B8.A1"></span>ความเครียดทางวิศวกรรม</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E0%B8%84%E0%B8%A7%E0%B8%B2%E0%B8%A1%E0%B9%80%E0%B8%84%E0%B8%A3%E0%B8%B5%E0%B8%A2%E0%B8%94_(%E0%B8%81%E0%B8%A5%E0%B8%A8%E0%B8%B2%E0%B8%AA%E0%B8%95%E0%B8%A3%E0%B9%8C)&action=edit&section=2" title="แก้ไขส่วน: ความเครียดทางวิศวกรรม"><span>แก้</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><b>ความเครียดทางวิศวกรรม</b> (engineering strain หรือ Cauchy strain) คืออัตราส่วนระหว่างขนาดที่เปลี่ยนไปต่อขนาดตั้งต้น สำหรับวัตถุขนาดยาวตั้งต้น <span class="texhtml"><i>L</i></span> ที่เปลี่ยนความยาว <span class="texhtml">Δ<i>L</i></span> สามารถเขียนได้ว่า </p><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 \ e={\frac {\Delta L}{L}}={\frac {l-L}{L}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>e</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">Δ</mi> <mi>L</mi> </mrow> <mi>L</mi> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>l</mi> <mo>−</mo> <mi>L</mi> </mrow> <mi>L</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ e={\frac {\Delta L}{L}}={\frac {l-L}{L}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/08d50292b716c1e2b6aaf9ae78cbedf128beae78" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:18.169ex; height:5.343ex;" alt="{\displaystyle \ e={\frac {\Delta L}{L}}={\frac {l-L}{L}}}"></span> </p><p>โดย <span class="texhtml mvar" style="font-style:italic;">e</span> คือความเครียดตั้งฉากทางวิศวกรรมและ <span class="texhtml mvar" style="font-style:italic;">l</span> คือความยาวสุดท้าย ถ้าความเครียดนี้เป็นบวกหมายความว่าความยาวเพิ่มขึ้น แต่ถ้าเป็นลบแสดงว่าความยาวลดลง </p><p>ส่วนความเครียดเฉือนทางวิศวกรรมนิยามจากค่า tan ของมุมเฉือน ซึ่งเท่ากับอัตราส่วนระหว่างความยาวที่เปลี่ยนไปในทิศของแรงเฉือนต่อความยาวตั้งฉากกับแรงเฉือน </p><p>การคำนวณความเครียดทางวิศวกรรมนั้นคำนวณได้ง่าย แต่บอกได้ถึงการเปลี่ยนแปลงขนาดโดยรวมเท่านั้น ไม่สามารถบอกถึงว่าวัตถุถูกเปลี่ยนแปลงรุปร่างผ่านขั้นตอนอะไร </p> <div class="mw-heading mw-heading4"><h4 id="อัตราส่วนความยืด"><span id=".E0.B8.AD.E0.B8.B1.E0.B8.95.E0.B8.A3.E0.B8.B2.E0.B8.AA.E0.B9.88.E0.B8.A7.E0.B8.99.E0.B8.84.E0.B8.A7.E0.B8.B2.E0.B8.A1.E0.B8.A2.E0.B8.B7.E0.B8.94"></span>อัตราส่วนความยืด</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E0%B8%84%E0%B8%A7%E0%B8%B2%E0%B8%A1%E0%B9%80%E0%B8%84%E0%B8%A3%E0%B8%B5%E0%B8%A2%E0%B8%94_(%E0%B8%81%E0%B8%A5%E0%B8%A8%E0%B8%B2%E0%B8%AA%E0%B8%95%E0%B8%A3%E0%B9%8C)&action=edit&section=3" title="แก้ไขส่วน: อัตราส่วนความยืด"><span>แก้</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><b>อัตราส่วนความยืด</b> (stretch ratio หรือ extension ratio ) คือปริมาณที่บอกว่าวัตถุเปลี่ยนความยาวไปแค่ไหน ซึ่งนิยามจจากอัตราส่วนระหว่างความสุดท้าย <span class="texhtml mvar" style="font-style:italic;">l</span> ต่อความยาวตั้งต้น <span class="texhtml"><i>L</i></span> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ \lambda ={\frac {l}{L}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>λ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>l</mi> <mi>L</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ \lambda ={\frac {l}{L}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1b5403ed7a2db9b8a8175e530b5cb6432e46150d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:7.453ex; height:5.343ex;" alt="{\displaystyle \ \lambda ={\frac {l}{L}}}"></span></dd></dl> <p>ซึ่งสัมพันธ์กับความเครียดทางวิศวกรรมโดย </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 \ e={\frac {l-L}{L}}=\lambda -1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>e</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>l</mi> <mo>−</mo> <mi>L</mi> </mrow> <mi>L</mi> </mfrac> </mrow> <mo>=</mo> <mi>λ</mi> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ e={\frac {l-L}{L}}=\lambda -1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75c8c95a779e7f64c7bce248e5274d61f6344276" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:19.172ex; height:5.343ex;" alt="{\displaystyle \ e={\frac {l-L}{L}}=\lambda -1}"></span></dd></dl> <p>ถ้าความเครียดเป็นศูนย์ซึ่งแปลวาสวัตถุไม่เปลี่ยนความยาว อัตราการยืดจะเท่ากับ 1 </p><p>อัตราส่วนความยืดนั้งมักถูกใช้ในการวิเคราะห์วัสดุที่เปลี่ยนรูปร่างขนาดใหญ่ อาทิเช่น <a href="/wiki/%E0%B8%AD%E0%B8%B5%E0%B8%A5%E0%B8%B2%E0%B8%AA%E0%B9%82%E0%B8%95%E0%B9%80%E0%B8%A1%E0%B8%AD%E0%B8%A3%E0%B9%8C" title="อีลาสโตเมอร์">อีลาสโตเมอร์</a>หรือ<a href="/wiki/%E0%B8%A2%E0%B8%B2%E0%B8%87" title="ยาง">ยาง</a> ซึ่งสามารถเปลี่ยนขนาดได้ที่อัตราส่วน 3 หรือ 4 เท่า ก่อนที่มันจะเสียหาย ในทางตรงกันข้ามวัสดุทางวิศกรรมเช่น คอนกรีต หรือ เหล็กกล้า จะเสียหายที่อัตราส่วนที่ต่ำกว่ามาก </p> <div class="mw-heading mw-heading4"><h4 id="ความเครียดจริง"><span id=".E0.B8.84.E0.B8.A7.E0.B8.B2.E0.B8.A1.E0.B9.80.E0.B8.84.E0.B8.A3.E0.B8.B5.E0.B8.A2.E0.B8.94.E0.B8.88.E0.B8.A3.E0.B8.B4.E0.B8.87"></span>ความเครียดจริง</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E0%B8%84%E0%B8%A7%E0%B8%B2%E0%B8%A1%E0%B9%80%E0%B8%84%E0%B8%A3%E0%B8%B5%E0%B8%A2%E0%B8%94_(%E0%B8%81%E0%B8%A5%E0%B8%A8%E0%B8%B2%E0%B8%AA%E0%B8%95%E0%B8%A3%E0%B9%8C)&action=edit&section=4" title="แก้ไขส่วน: ความเครียดจริง"><span>แก้</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><b>ความเครียดจริง</b> (true strain or logarithmic strain) คำนวณจากการพิจารณาการเปลี่ยนแปลงความยาวเล็กน้อย <span class="texhtml">𝛿<i>l</i></span> จากความยาว <span class="texhtml"><i>l</i></span> ซึ่งทำให้เกิดความเครียดตั้งฉาก </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ \delta \varepsilon ={\frac {\delta l}{l}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>δ</mi> <mi>ε</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>δ</mi> <mi>l</mi> </mrow> <mi>l</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ \delta \varepsilon ={\frac {\delta l}{l}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cac3346649aec80f881e3c298012eddfb6835ff3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:8.389ex; height:5.509ex;" alt="{\displaystyle \ \delta \varepsilon ={\frac {\delta l}{l}}}"></span></dd></dl> <p>เนื่องจากความยาวนั้นเปลี่ยนเรื่อยๆ ความเครียดโดยรวมจึงหาได้จาก </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ {\begin{aligned}\int \delta \varepsilon &=\int _{L}^{l}{\frac {\delta l}{l}}\\\varepsilon &=\ln \left({\frac {l}{L}}\right)=\ln(\lambda )\\&=\ln(1+e)\\&=e-{\frac {e^{2}}{2}}+{\frac {e^{3}}{3}}-\cdots \\\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mo>∫</mo> <mi>δ</mi> <mi>ε</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>δ</mi> <mi>l</mi> </mrow> <mi>l</mi> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>ε</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>ln</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>l</mi> <mi>L</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>λ</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>e</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mi>e</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mn>3</mn> </mfrac> </mrow> <mo>−</mo> <mo>⋯</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ {\begin{aligned}\int \delta \varepsilon &=\int _{L}^{l}{\frac {\delta l}{l}}\\\varepsilon &=\ln \left({\frac {l}{L}}\right)=\ln(\lambda )\\&=\ln(1+e)\\&=e-{\frac {e^{2}}{2}}+{\frac {e^{3}}{3}}-\cdots \\\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/edc249c6aad5556e87cc3b80c00354a3d328614a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -10.171ex; width:27.419ex; height:21.509ex;" alt="{\displaystyle \ {\begin{aligned}\int \delta \varepsilon &=\int _{L}^{l}{\frac {\delta l}{l}}\\\varepsilon &=\ln \left({\frac {l}{L}}\right)=\ln(\lambda )\\&=\ln(1+e)\\&=e-{\frac {e^{2}}{2}}+{\frac {e^{3}}{3}}-\cdots \\\end{aligned}}}"></span></dd></dl> <p>โดย <span class="texhtml mvar" style="font-style:italic;">e</span> คือความเครียดทางวิศวกรรม ความเครียดจริงบอกถึงการเปลี่ยนแปลงรูปร่างของวัตถุโดยนำขั้นตอนการเปลี่ยนขนาดมาพิจารณาด้วย </p><p>ส่วนความเครียดเฉือนจริงนั้นคำนวณได้จากมุม(ในหน่วยเรเดียน)ของวัตถุที่เปลี่ยนไปหลังจากการเปลี่ยนรูปร่าง </p> <div class="mw-heading mw-heading2"><h2 id="ความเครียดตั้งฉากและความเครียดเฉือน"><span id=".E0.B8.84.E0.B8.A7.E0.B8.B2.E0.B8.A1.E0.B9.80.E0.B8.84.E0.B8.A3.E0.B8.B5.E0.B8.A2.E0.B8.94.E0.B8.95.E0.B8.B1.E0.B9.89.E0.B8.87.E0.B8.89.E0.B8.B2.E0.B8.81.E0.B9.81.E0.B8.A5.E0.B8.B0.E0.B8.84.E0.B8.A7.E0.B8.B2.E0.B8.A1.E0.B9.80.E0.B8.84.E0.B8.A3.E0.B8.B5.E0.B8.A2.E0.B8.94.E0.B9.80.E0.B8.89.E0.B8.B7.E0.B8.AD.E0.B8.99"></span>ความเครียดตั้งฉากและความเครียดเฉือน</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E0%B8%84%E0%B8%A7%E0%B8%B2%E0%B8%A1%E0%B9%80%E0%B8%84%E0%B8%A3%E0%B8%B5%E0%B8%A2%E0%B8%94_(%E0%B8%81%E0%B8%A5%E0%B8%A8%E0%B8%B2%E0%B8%AA%E0%B8%95%E0%B8%A3%E0%B9%8C)&action=edit&section=5" title="แก้ไขส่วน: ความเครียดตั้งฉากและความเครียดเฉือน"><span>แก้</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>อย่างที่กล่าวมาว่าความเครียดแบ่งเป็นสองแบบคือความเครียดตั้งฉากซึ้งตั้งฉากกับหน้าตัดของวัตถุและความเครียดเฉือนที่ขนานกับหน้าตัดของวัตถุ โดยนิยามเหล่านี้สอดคล้องกับ<a href="/wiki/%E0%B8%84%E0%B8%A7%E0%B8%B2%E0%B8%A1%E0%B9%80%E0%B8%84%E0%B9%89%E0%B8%99" title="ความเค้น">ความเค้น</a>ตั้งฉาก(normal stress)และความเค้นเฉือน(shear stress) </p> <div class="mw-heading mw-heading4"><h4 id="ความเครียดตั้งฉาก"><span id=".E0.B8.84.E0.B8.A7.E0.B8.B2.E0.B8.A1.E0.B9.80.E0.B8.84.E0.B8.A3.E0.B8.B5.E0.B8.A2.E0.B8.94.E0.B8.95.E0.B8.B1.E0.B9.89.E0.B8.87.E0.B8.89.E0.B8.B2.E0.B8.81"></span>ความเครียดตั้งฉาก</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E0%B8%84%E0%B8%A7%E0%B8%B2%E0%B8%A1%E0%B9%80%E0%B8%84%E0%B8%A3%E0%B8%B5%E0%B8%A2%E0%B8%94_(%E0%B8%81%E0%B8%A5%E0%B8%A8%E0%B8%B2%E0%B8%AA%E0%B8%95%E0%B8%A3%E0%B9%8C)&action=edit&section=6" title="แก้ไขส่วน: ความเครียดตั้งฉาก"><span>แก้</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>สำหรับวัสดุที่สม่ำเสมอและมีพฤติกรรมตามกฎของฮุค ความเค้นตั้งฉากจะทำให้เกิดความเครียดตั้งฉาก </p><p><span typeof="mw:File"><a href="/wiki/%E0%B9%84%E0%B8%9F%E0%B8%A5%E0%B9%8C:2D_geometric_strain.svg" class="mw-file-description" title="2D geometric strain"><img alt="2D geometric strain" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/23/2D_geometric_strain.svg/485px-2D_geometric_strain.svg.png" decoding="async" width="485" height="377" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/23/2D_geometric_strain.svg/728px-2D_geometric_strain.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/23/2D_geometric_strain.svg/970px-2D_geometric_strain.svg.png 2x" data-file-width="900" data-file-height="700" /></a></span> </p><p>พิจารณา element สี่เหลี่ยมในสองมิติมีขนาด <span class="texhtml"><i>dx</i> × <i>dy</i></span> ซึ่งเมื่อถูกเปลี่ยนรูปร่างเป็น<a href="/wiki/%E0%B8%A3%E0%B8%B9%E0%B8%9B%E0%B8%AA%E0%B8%B5%E0%B9%88%E0%B9%80%E0%B8%AB%E0%B8%A5%E0%B8%B5%E0%B9%88%E0%B8%A2%E0%B8%A1%E0%B8%82%E0%B8%99%E0%B8%A1%E0%B9%80%E0%B8%9B%E0%B8%B5%E0%B8%A2%E0%B8%81%E0%B8%9B%E0%B8%B9%E0%B8%99" title="รูปสี่เหลี่ยมขนมเปียกปูน">สี่เหลี่ยมขนมเปียกปูน</a> การเปลี่ยนรูปร่างสามารถบรรยายโดยใช้สนามกระจัด (displacement field) จากรูปเราสามารถเขียนความยาวด้าน </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 \mathrm {length} (AB)=dx\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">l</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">g</mi> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">h</mi> </mrow> <mo stretchy="false">(</mo> <mi>A</mi> <mi>B</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>d</mi> <mi>x</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {length} (AB)=dx\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/19246765185d36a8776abd4f196305000820e2f7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.679ex; height:2.843ex;" alt="{\displaystyle \mathrm {length} (AB)=dx\,}"></span></dd></dl> <p>และ </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\mathrm {length} (ab)&={\sqrt {\left(dx+{\frac {\partial u_{x}}{\partial x}}dx\right)^{2}+\left({\frac {\partial u_{y}}{\partial x}}dx\right)^{2}}}\\&=dx~{\sqrt {1+2{\frac {\partial u_{x}}{\partial x}}+\left({\frac {\partial u_{x}}{\partial x}}\right)^{2}+\left({\frac {\partial u_{y}}{\partial x}}\right)^{2}}}\\\end{aligned}}\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">l</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">g</mi> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">h</mi> </mrow> <mo stretchy="false">(</mo> <mi>a</mi> <mi>b</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mrow> <mo>(</mo> <mrow> <mi>d</mi> <mi>x</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mi>d</mi> <mi>x</mi> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mi>d</mi> <mi>x</mi> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mi>d</mi> <mi>x</mi> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>+</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mtd> </mtr> </mtable> </mrow> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\mathrm {length} (ab)&={\sqrt {\left(dx+{\frac {\partial u_{x}}{\partial x}}dx\right)^{2}+\left({\frac {\partial u_{y}}{\partial x}}dx\right)^{2}}}\\&=dx~{\sqrt {1+2{\frac {\partial u_{x}}{\partial x}}+\left({\frac {\partial u_{x}}{\partial x}}\right)^{2}+\left({\frac {\partial u_{y}}{\partial x}}\right)^{2}}}\\\end{aligned}}\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/17ac8d56c061ed8bf71fb0b80427bc71878aca10" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.171ex; margin-right: -0.387ex; width:53.698ex; height:15.509ex;" alt="{\displaystyle {\begin{aligned}\mathrm {length} (ab)&={\sqrt {\left(dx+{\frac {\partial u_{x}}{\partial x}}dx\right)^{2}+\left({\frac {\partial u_{y}}{\partial x}}dx\right)^{2}}}\\&=dx~{\sqrt {1+2{\frac {\partial u_{x}}{\partial x}}+\left({\frac {\partial u_{x}}{\partial x}}\right)^{2}+\left({\frac {\partial u_{y}}{\partial x}}\right)^{2}}}\\\end{aligned}}\,\!}"></span></dd></dl> <p>สำหรับการเปลี่ยนแปลงขนาดเล็ก พจน์ยกกำลังสองนั้นมีขนาดเล็กและสามารถละทิ้งได้ ดังนั้น </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 \mathrm {length} (ab)\approx dx+{\frac {\partial u_{x}}{\partial x}}dx}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">l</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">g</mi> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">h</mi> </mrow> <mo stretchy="false">(</mo> <mi>a</mi> <mi>b</mi> <mo stretchy="false">)</mo> <mo>≈</mo> <mi>d</mi> <mi>x</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mi>d</mi> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {length} (ab)\approx dx+{\frac {\partial u_{x}}{\partial x}}dx}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b17d530d5f039f27a8b66a40cc02cb039bb0fa35" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:26.054ex; height:5.509ex;" alt="{\displaystyle \mathrm {length} (ab)\approx dx+{\frac {\partial u_{x}}{\partial x}}dx}"></span></dd></dl> <p>ความเครียดตั้งฉากในทิศ x ของสี่เหลี่ยมนั้นนิยามว่า </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon _{x}={\frac {\text{extension}}{\text{original length}}}={\frac {\mathrm {length} (ab)-\mathrm {length} (AB)}{\mathrm {length} (AB)}}={\frac {\partial u_{x}}{\partial x}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mtext>extension</mtext> <mtext>original length</mtext> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">l</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">g</mi> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">h</mi> </mrow> <mo stretchy="false">(</mo> <mi>a</mi> <mi>b</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">l</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">g</mi> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">h</mi> </mrow> <mo stretchy="false">(</mo> <mi>A</mi> <mi>B</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">l</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">g</mi> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">h</mi> </mrow> <mo stretchy="false">(</mo> <mi>A</mi> <mi>B</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon _{x}={\frac {\text{extension}}{\text{original length}}}={\frac {\mathrm {length} (ab)-\mathrm {length} (AB)}{\mathrm {length} (AB)}}={\frac {\partial u_{x}}{\partial x}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3a334659cddda62f131dbbd4b06d79821527c3a4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:57.28ex; height:6.509ex;" alt="{\displaystyle \varepsilon _{x}={\frac {\text{extension}}{\text{original length}}}={\frac {\mathrm {length} (ab)-\mathrm {length} (AB)}{\mathrm {length} (AB)}}={\frac {\partial u_{x}}{\partial x}}}"></span></dd></dl> <p>ในทิศ y และ ทิศ z ก็เขียนได้ในลักษณะเดียวกัน </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon _{y}={\frac {\partial u_{y}}{\partial y}}\quad ,\qquad \varepsilon _{z}={\frac {\partial u_{z}}{\partial z}}\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> </mfrac> </mrow> <mspace width="1em" /> <mo>,</mo> <mspace width="2em" /> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>z</mi> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon _{y}={\frac {\partial u_{y}}{\partial y}}\quad ,\qquad \varepsilon _{z}={\frac {\partial u_{z}}{\partial z}}\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/856f69ebff996d88361e4a1f0c4520a213530bfe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; margin-right: -0.387ex; width:27.822ex; height:6.343ex;" alt="{\displaystyle \varepsilon _{y}={\frac {\partial u_{y}}{\partial y}}\quad ,\qquad \varepsilon _{z}={\frac {\partial u_{z}}{\partial z}}\,\!}"></span></dd></dl> <div class="mw-heading mw-heading4"><h4 id="ความเครียดเฉือน"><span id=".E0.B8.84.E0.B8.A7.E0.B8.B2.E0.B8.A1.E0.B9.80.E0.B8.84.E0.B8.A3.E0.B8.B5.E0.B8.A2.E0.B8.94.E0.B9.80.E0.B8.89.E0.B8.B7.E0.B8.AD.E0.B8.99"></span>ความเครียดเฉือน</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E0%B8%84%E0%B8%A7%E0%B8%B2%E0%B8%A1%E0%B9%80%E0%B8%84%E0%B8%A3%E0%B8%B5%E0%B8%A2%E0%B8%94_(%E0%B8%81%E0%B8%A5%E0%B8%A8%E0%B8%B2%E0%B8%AA%E0%B8%95%E0%B8%A3%E0%B9%8C)&action=edit&section=7" title="แก้ไขส่วน: ความเครียดเฉือน"><span>แก้</span></a><span class="mw-editsection-bracket">]</span></span></div> <table class="infobox" style="width: 24em; text-align: left; font-size: 90%" cellspacing="2"> <tbody><tr> <td style="font-size: 110%;" colspan="2"><center><i><b>ความเครียดเฉือน</b></i></center> </td></tr> <tr> <td><b>หน่วยการวัด (<a href="/wiki/%E0%B8%AB%E0%B8%99%E0%B9%88%E0%B8%A7%E0%B8%A2%E0%B9%80%E0%B8%AD%E0%B8%AA%E0%B9%84%E0%B8%AD" class="mw-redirect" title="หน่วยเอสไอ">SI</a>):</b></td> <td><a href="/w/index.php?title=Dimensionless&action=edit&redlink=1" class="new" title="Dimensionless (ไม่มีหน้านี้)">1</a>, or <a href="/wiki/%E0%B9%80%E0%B8%A3%E0%B9%80%E0%B8%94%E0%B8%B5%E0%B8%A2%E0%B8%99" title="เรเดียน">เรเดียน</a> </td></tr> <tr> <td><b>สัญลักษณ์ที่มักจะใช้:</b></td> <td><a href="/wiki/Gamma" class="mw-redirect" title="Gamma"><span class="texhtml mvar" style="font-style:italic;">γ</span></a> or <a href="/wiki/Epsilon" class="mw-redirect" title="Epsilon"><span class="texhtml mvar" style="font-style:italic;">ε</span></a> </td></tr> </tbody></table><p>ความเครียดเฉือนทางวิศวกรรม(<span class="texhtml"><i>γ<sub>xy</sub></i></span>) นิยามจากมุมที่เปลี่ยนไประหว่างส่วนของเส้นตรง <span style="text-decoration:overline;"><i>AC</i></span> และ <span style="text-decoration:overline;"><i>AB</i></span> ดังนั้น </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 \gamma _{xy}=\alpha +\beta \,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>y</mi> </mrow> </msub> <mo>=</mo> <mi>α</mi> <mo>+</mo> <mi>β</mi> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma _{xy}=\alpha +\beta \,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6444ae19e350186648683ea1d88af5a19bb81aeb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; margin-right: -0.387ex; width:12.339ex; height:2.843ex;" alt="{\displaystyle \gamma _{xy}=\alpha +\beta \,\!}"></span></dd></dl> <p>จากรูปเราสามารถคำนวณมุมได้ </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\tan \alpha &={\frac {{\tfrac {\partial u_{y}}{\partial x}}dx}{dx+{\tfrac {\partial u_{x}}{\partial x}}dx}}={\frac {\tfrac {\partial u_{y}}{\partial x}}{1+{\tfrac {\partial u_{x}}{\partial x}}}}\\\tan \beta &={\frac {{\tfrac {\partial u_{x}}{\partial y}}dy}{dy+{\tfrac {\partial u_{y}}{\partial y}}dy}}={\frac {\tfrac {\partial u_{x}}{\partial y}}{1+{\tfrac {\partial u_{y}}{\partial y}}}}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>tan</mi> <mo>⁡</mo> <mi>α</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mstyle> </mrow> <mi>d</mi> <mi>x</mi> </mrow> <mrow> <mi>d</mi> <mi>x</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mstyle> </mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mstyle> <mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mstyle> </mrow> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>tan</mi> <mo>⁡</mo> <mi>β</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> </mfrac> </mstyle> </mrow> <mi>d</mi> <mi>y</mi> </mrow> <mrow> <mi>d</mi> <mi>y</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> </mfrac> </mstyle> </mrow> <mi>d</mi> <mi>y</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> </mfrac> </mstyle> <mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> </mfrac> </mstyle> </mrow> </mrow> </mfrac> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\tan \alpha &={\frac {{\tfrac {\partial u_{y}}{\partial x}}dx}{dx+{\tfrac {\partial u_{x}}{\partial x}}dx}}={\frac {\tfrac {\partial u_{y}}{\partial x}}{1+{\tfrac {\partial u_{x}}{\partial x}}}}\\\tan \beta &={\frac {{\tfrac {\partial u_{x}}{\partial y}}dy}{dy+{\tfrac {\partial u_{y}}{\partial y}}dy}}={\frac {\tfrac {\partial u_{x}}{\partial y}}{1+{\tfrac {\partial u_{y}}{\partial y}}}}\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ce730390cc6019e0ec20a5120d4dcaf301e297a9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -9.005ex; width:33.061ex; height:19.176ex;" alt="{\displaystyle {\begin{aligned}\tan \alpha &={\frac {{\tfrac {\partial u_{y}}{\partial x}}dx}{dx+{\tfrac {\partial u_{x}}{\partial x}}dx}}={\frac {\tfrac {\partial u_{y}}{\partial x}}{1+{\tfrac {\partial u_{x}}{\partial x}}}}\\\tan \beta &={\frac {{\tfrac {\partial u_{x}}{\partial y}}dy}{dy+{\tfrac {\partial u_{y}}{\partial y}}dy}}={\frac {\tfrac {\partial u_{x}}{\partial y}}{1+{\tfrac {\partial u_{y}}{\partial y}}}}\end{aligned}}}"></span></dd></dl> <p>สำหรับการเปลี่ยนแปลงขนาดเล็ก </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 {\cfrac {\partial u_{x}}{\partial x}}\ll 1~;~~{\cfrac {\partial u_{y}}{\partial y}}\ll 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∂</mi> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mstyle> </mrow> </mfrac> </mrow> <mo>≪</mo> <mn>1</mn> <mtext> </mtext> <mo>;</mo> <mtext> </mtext> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∂</mi> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> </mstyle> </mrow> </mfrac> </mrow> <mo>≪</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\cfrac {\partial u_{x}}{\partial x}}\ll 1~;~~{\cfrac {\partial u_{y}}{\partial y}}\ll 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c05bb7ae32ef894cfd3c55cd07a37ede00254fd9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:21.519ex; height:7.176ex;" alt="{\displaystyle {\cfrac {\partial u_{x}}{\partial x}}\ll 1~;~~{\cfrac {\partial u_{y}}{\partial y}}\ll 1}"></span></dd></dl> <p>สำหรับการเปลี่ยนมุมขนาดเล็ก (<span class="texhtml mvar" style="font-style:italic;">α</span> และ <span class="texhtml mvar" style="font-style:italic;">β</span> ≪ 1) เราสามารถประมาณ <span class="texhtml">tan <i>α</i> ≈ <i>α</i></span>, <span class="texhtml">tan <i>β</i> ≈ <i>β</i></span> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha \approx {\cfrac {\partial u_{y}}{\partial x}}~;~~\beta \approx {\cfrac {\partial u_{x}}{\partial y}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>≈</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∂</mi> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mstyle> </mrow> </mfrac> </mrow> <mtext> </mtext> <mo>;</mo> <mtext> </mtext> <mtext> </mtext> <mi>β</mi> <mo>≈</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∂</mi> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha \approx {\cfrac {\partial u_{y}}{\partial x}}~;~~\beta \approx {\cfrac {\partial u_{x}}{\partial y}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee43d0fe4bb8f9b5fdf6f034902c97e9219ce89b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:20.982ex; height:7.176ex;" alt="{\displaystyle \alpha \approx {\cfrac {\partial u_{y}}{\partial x}}~;~~\beta \approx {\cfrac {\partial u_{x}}{\partial y}}}"></span></dd></dl> <p>ดังนั้น </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 \gamma _{xy}=\alpha +\beta ={\frac {\partial u_{y}}{\partial x}}+{\frac {\partial u_{x}}{\partial y}}\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>y</mi> </mrow> </msub> <mo>=</mo> <mi>α</mi> <mo>+</mo> <mi>β</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma _{xy}=\alpha +\beta ={\frac {\partial u_{y}}{\partial x}}+{\frac {\partial u_{x}}{\partial y}}\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e29692f39a443381ec3e29b6797dc828a8c91501" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; margin-right: -0.387ex; width:27.468ex; height:6.343ex;" alt="{\displaystyle \gamma _{xy}=\alpha +\beta ={\frac {\partial u_{y}}{\partial x}}+{\frac {\partial u_{x}}{\partial y}}\,\!}"></span></dd></dl> <p>โดยการสลับสัญลักษณ์ <span class="texhtml mvar" style="font-style:italic;">x</span> กับ <span class="texhtml mvar" style="font-style:italic;">y</span> และ <span class="texhtml"><i>u<sub>x</sub></i></span> กับ <span class="texhtml"><i>u<sub>y</sub></i></span> ดังนั้นจึงเขียนได้อีกว่า <span class="texhtml"><i>γ<sub>xy</sub></i> = <i>γ<sub>yx</sub></i></span>. </p><p>ในทางเดียวกัน สามารถคำนวณความเครียดเฉือนในระนาบ <span class="texhtml mvar" style="font-style:italic;">yz</span> และ <span class="texhtml mvar" style="font-style:italic;">xz</span> ได้ </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 \gamma _{yz}=\gamma _{zy}={\frac {\partial u_{y}}{\partial z}}+{\frac {\partial u_{z}}{\partial y}}\quad ,\qquad \gamma _{zx}=\gamma _{xz}={\frac {\partial u_{z}}{\partial x}}+{\frac {\partial u_{x}}{\partial z}}\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> <mi>z</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> <mi>y</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>z</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> </mfrac> </mrow> <mspace width="1em" /> <mo>,</mo> <mspace width="2em" /> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> <mi>x</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>z</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>z</mi> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma _{yz}=\gamma _{zy}={\frac {\partial u_{y}}{\partial z}}+{\frac {\partial u_{z}}{\partial y}}\quad ,\qquad \gamma _{zx}=\gamma _{xz}={\frac {\partial u_{z}}{\partial x}}+{\frac {\partial u_{x}}{\partial z}}\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/94f7a41cfb6f2a4f056f61c83af37f08d772902a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; margin-right: -0.387ex; width:56.963ex; height:6.343ex;" alt="{\displaystyle \gamma _{yz}=\gamma _{zy}={\frac {\partial u_{y}}{\partial z}}+{\frac {\partial u_{z}}{\partial y}}\quad ,\qquad \gamma _{zx}=\gamma _{xz}={\frac {\partial u_{z}}{\partial x}}+{\frac {\partial u_{x}}{\partial z}}\,\!}"></span></dd></dl> <p>เราสามารถเขียนความเครียดในรูปแบบเทนเซอร์ซึ่งจะรวมทั้งความเค้นตั้งฉากและเฉือนเข้าด้วยกัน </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 {\underline {\underline {\boldsymbol {\varepsilon }}}}=\left[{\begin{matrix}\varepsilon _{xx}&\varepsilon _{xy}&\varepsilon _{xz}\\\varepsilon _{yx}&\varepsilon _{yy}&\varepsilon _{yz}\\\varepsilon _{zx}&\varepsilon _{zy}&\varepsilon _{zz}\\\end{matrix}}\right]=\left[{\begin{matrix}\varepsilon _{xx}&{\tfrac {1}{2}}\gamma _{xy}&{\tfrac {1}{2}}\gamma _{xz}\\{\tfrac {1}{2}}\gamma _{yx}&\varepsilon _{yy}&{\tfrac {1}{2}}\gamma _{yz}\\{\tfrac {1}{2}}\gamma _{zx}&{\tfrac {1}{2}}\gamma _{zy}&\varepsilon _{zz}\\\end{matrix}}\right]\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <munder> <munder> <mi mathvariant="bold-italic">ε</mi> <mo>_</mo> </munder> <mo>_</mo> </munder> </mrow> <mo>=</mo> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>x</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>y</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>z</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> <mi>x</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> <mi>y</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> <mi>z</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> <mi>x</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> <mi>y</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> <mi>z</mi> </mrow> </msub> </mtd> </mtr> </mtable> </mrow> <mo>]</mo> </mrow> <mo>=</mo> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>x</mi> </mrow> </msub> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>y</mi> </mrow> </msub> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>z</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> <mi>x</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> <mi>y</mi> </mrow> </msub> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> <mi>z</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> <mi>x</mi> </mrow> </msub> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> <mi>y</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> <mi>z</mi> </mrow> </msub> </mtd> </mtr> </mtable> </mrow> <mo>]</mo> </mrow> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\underline {\underline {\boldsymbol {\varepsilon }}}}=\left[{\begin{matrix}\varepsilon _{xx}&\varepsilon _{xy}&\varepsilon _{xz}\\\varepsilon _{yx}&\varepsilon _{yy}&\varepsilon _{yz}\\\varepsilon _{zx}&\varepsilon _{zy}&\varepsilon _{zz}\\\end{matrix}}\right]=\left[{\begin{matrix}\varepsilon _{xx}&{\tfrac {1}{2}}\gamma _{xy}&{\tfrac {1}{2}}\gamma _{xz}\\{\tfrac {1}{2}}\gamma _{yx}&\varepsilon _{yy}&{\tfrac {1}{2}}\gamma _{yz}\\{\tfrac {1}{2}}\gamma _{zx}&{\tfrac {1}{2}}\gamma _{zy}&\varepsilon _{zz}\\\end{matrix}}\right]\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/85bc0a67fe8c2fe990429bb95c096bb0e5f49dbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.171ex; margin-right: -0.387ex; width:48.616ex; height:11.509ex;" alt="{\displaystyle {\underline {\underline {\boldsymbol {\varepsilon }}}}=\left[{\begin{matrix}\varepsilon _{xx}&\varepsilon _{xy}&\varepsilon _{xz}\\\varepsilon _{yx}&\varepsilon _{yy}&\varepsilon _{yz}\\\varepsilon _{zx}&\varepsilon _{zy}&\varepsilon _{zz}\\\end{matrix}}\right]=\left[{\begin{matrix}\varepsilon _{xx}&{\tfrac {1}{2}}\gamma _{xy}&{\tfrac {1}{2}}\gamma _{xz}\\{\tfrac {1}{2}}\gamma _{yx}&\varepsilon _{yy}&{\tfrac {1}{2}}\gamma _{yz}\\{\tfrac {1}{2}}\gamma _{zx}&{\tfrac {1}{2}}\gamma _{zy}&\varepsilon _{zz}\\\end{matrix}}\right]\,\!}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="อ้างอิง"><span id=".E0.B8.AD.E0.B9.89.E0.B8.B2.E0.B8.87.E0.B8.AD.E0.B8.B4.E0.B8.87"></span>อ้างอิง</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E0%B8%84%E0%B8%A7%E0%B8%B2%E0%B8%A1%E0%B9%80%E0%B8%84%E0%B8%A3%E0%B8%B5%E0%B8%A2%E0%B8%94_(%E0%B8%81%E0%B8%A5%E0%B8%A8%E0%B8%B2%E0%B8%AA%E0%B8%95%E0%B8%A3%E0%B9%8C)&action=edit&section=8" title="แก้ไขส่วน: อ้างอิง"><span>แก้</span></a><span class="mw-editsection-bracket">]</span></span></div> <ol class="references"> <li id="cite_note-rees-1"><span class="mw-cite-backlink">↑ <sup><a href="#cite_ref-rees_1-0">1.0</a></sup> <sup><a href="#cite_ref-rees_1-1">1.1</a></sup></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r10205087">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free a,.mw-parser-output .citation .cs1-lock-free a{background:linear-gradient(transparent,transparent),url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited a,.mw-parser-output .id-lock-registration a,.mw-parser-output .citation .cs1-lock-limited a,.mw-parser-output .citation .cs1-lock-registration a{background:linear-gradient(transparent,transparent),url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription a,.mw-parser-output .citation .cs1-lock-subscription a{background:linear-gradient(transparent,transparent),url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:linear-gradient(transparent,transparent),url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:#d33}.mw-parser-output .cs1-visible-error{color:#d33}.mw-parser-output .cs1-maint{display:none;color:#3a3;margin-left:0.3em}.mw-parser-output .cs1-format{font-size:95%}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}</style><cite id="CITEREFRees2006" class="citation book cs1">Rees, David (2006). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=4KWbmn_1hcYC"><i>Basic Engineering Plasticity: An Introduction with Engineering and Manufacturing Applications</i></a>. Butterworth-Heinemann. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/%E0%B8%9E%E0%B8%B4%E0%B9%80%E0%B8%A8%E0%B8%A9:%E0%B9%81%E0%B8%AB%E0%B8%A5%E0%B9%88%E0%B8%87%E0%B8%AB%E0%B8%99%E0%B8%B1%E0%B8%87%E0%B8%AA%E0%B8%B7%E0%B8%AD/0-7506-8025-3" title="พิเศษ:แหล่งหนังสือ/0-7506-8025-3"><bdi>0-7506-8025-3</bdi></a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20171222205706/https://books.google.com/books?id=4KWbmn_1hcYC">เก็บ</a>จากแหล่งเดิมเมื่อ 2017-12-22.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Basic+Engineering+Plasticity%3A+An+Introduction+with+Engineering+and+Manufacturing+Applications&rft.pub=Butterworth-Heinemann&rft.date=2006&rft.isbn=0-7506-8025-3&rft.aulast=Rees&rft.aufirst=David&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D4KWbmn_1hcYC&rfr_id=info%3Asid%2Fth.wikipedia.org%3A%E0%B8%84%E0%B8%A7%E0%B8%B2%E0%B8%A1%E0%B9%80%E0%B8%84%E0%B8%A3%E0%B8%B5%E0%B8%A2%E0%B8%94+%28%E0%B8%81%E0%B8%A5%E0%B8%A8%E0%B8%B2%E0%B8%AA%E0%B8%95%E0%B8%A3%E0%B9%8C%29" class="Z3988"></span></span> </li> </ol>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">เข้าถึงจาก "<a dir="ltr" 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