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Micron Document

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<title>Verschiebungsgradient</title>

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<h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Verschiebungsgradient</span></h1>

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<p>Der <b>Verschiebungsgradient</b> (Formelzeichen: <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {H} }">

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {H} }</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f017b876ed763037d8818ec5dfbbdc6703e0f683.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.091ex; height:2.176ex;" alt="{\displaystyle \mathbf {H} }" loading="lazy"></span>) ist in der <a href="Kontinuumsmechanik" title="Kontinuumsmechanik">Kontinuumsmechanik</a> ein einheitenfreier <a href="Tensor" title="Tensor">Tensor</a> zweiter Stufe, der die lokale Verformung in einem materiellen Punkt eines Körpers beschreibt. Tensoren zweiter Stufe werden hier als <a href="Lineare_Abbildung" title="Lineare Abbildung">lineare Abbildungen</a> von geometrischen Vektoren auf geometrische Vektoren benutzt, die im Allgemeinen dabei gedreht und gestreckt werden, siehe Abbildung rechts.

</p><p>Die Verschiebung des Partikels eines Körpers ist die Strecke zwischen seiner aktuellen Lage und seiner Position in der (undeformierten) Ausgangslage. Der Verschiebungsgradient beschreibt nun, wie sich die Verschiebung ändert, wenn die Position in der Ausgangslage variiert. Mathematisch ist er der <a href="Gradient_(Mathematik)" title="Gradient (Mathematik)">Gradient</a> der den Verschiebungen zugeordneten <a href="Vektor" title="Vektor">Vektoren</a>, daher der Name. Im allgemeinen Fall ist der Verschiebungsgradient sowohl vom Ort als auch von der Zeit abhängig. Die Komponenten des Verschiebungsgradienten berechnen sich wie eine <a href="Jacobimatrix" class="mw-redirect" title="Jacobimatrix">Jacobimatrix</a> und können auch in einer <a href="Matrix_(Mathematik)" title="Matrix (Mathematik)">Matrix</a> notiert werden.

</p><p>Der Verschiebungsgradient unterscheidet sich vom <a href="Deformationsgradient" title="Deformationsgradient">Deformationsgradient</a> nur durch den konstanten <a href="Einheitstensor" title="Einheitstensor">Einheitstensor</a>, wird aber vor allem im Fall <i>kleiner</i> Verschiebungen benutzt. Kleine Verschiebungen liegen vor, wenn die größten, im Körper auftretenden Verschiebungen immer noch wesentlich kleiner sind als eine charakteristische Abmessung des Körpers. Bei kleinen Verschiebungen ist der Verschiebungsgradient eine grundlegende Größe mit der lokale Drehungen, Streckungen und Dehnungen quantifiziert werden. Sein <a href="Symmetrische_Matrix" title="Symmetrische Matrix">symmetrischer Anteil</a> entspricht beispielsweise der <a href="Verzerrungstensor" title="Verzerrungstensor">Ingenieursdehnung</a>.

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<div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2></div>

<p>Der materielle Körper wird mit <a href="Konfiguration_(Mechanik)" title="Konfiguration (Mechanik)">Konfigurationen</a> in einen <a href="Pr%C3%A4hilbertraum" title="Prähilbertraum">euklidischen Vektorraum</a> abgebildet. In ihm wird die Bewegung eines materiellen Punktes mit der <i>Bewegungsfunktion</i>

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<annotation encoding="application/x-tex">{\displaystyle {\vec {x}}=x_{1}{\vec {e}}_{1}+x_{2}{\vec {e}}_{2}+x_{3}{\vec {e}}_{3}={\vec {\chi }}({\vec {X}},t)}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/dbdbed2899a6b04c3187dcb91c043cc359b61972.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:34.309ex; height:3.343ex;" alt="{\displaystyle {\vec {x}}=x_{1}{\vec {e}}_{1}+x_{2}{\vec {e}}_{2}+x_{3}{\vec {e}}_{3}={\vec {\chi }}({\vec {X}},t)}" loading="lazy"></span></dd></dl>

<p>beschrieben. Der Vektor <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {x}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\vec {x}}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/db2dc6ced9cc3bc7e8b9f2707cbec033f6d3759c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:2.343ex;" alt="{\displaystyle {\vec {x}}}" loading="lazy"></span> ist die aktuelle Position des materiellen Punktes <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {X}}}">

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5dc35b5a0226cf11a2c3f2d2dbbac6ab5ade6036.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.843ex;" alt="{\displaystyle {\vec {X}}}" loading="lazy"></span> zur Zeit <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}">

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<annotation encoding="application/x-tex">{\displaystyle t}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/65658b7b223af9e1acc877d848888ecdb4466560.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}" loading="lazy"></span> in der Momentankonfiguration. Die Komponenten <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{1,2,3}}">

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b67eda9b9b24758489f6004e13d51444f494e207.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.943ex; height:2.343ex;" alt="{\displaystyle x_{1,2,3}}" loading="lazy"></span> sind die <i>räumlichen</i> Koordinaten des Punktes bezüglich der <a href="Standardbasis" title="Standardbasis">Standardbasis</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {e}}_{1,2,3}}">

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4e8e19dca26cce96d3f35ced4c72a9875faf1a99.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.836ex; height:3.009ex;" alt="{\displaystyle {\vec {e}}_{1,2,3}}" loading="lazy"></span>. Der Vektor

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<annotation encoding="application/x-tex">{\displaystyle {\vec {X}}=X{\vec {e}}_{1}+Y{\vec {e}}_{2}+Z{\vec {e}}_{3}=X_{1}{\vec {e}}_{1}+X_{2}{\vec {e}}_{2}+X_{3}{\vec {e}}_{3}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/daaab93b3763d1c5de8833a060386a5270148fff.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:47.572ex; height:3.176ex;" alt="{\displaystyle {\vec {X}}=X{\vec {e}}_{1}+Y{\vec {e}}_{2}+Z{\vec {e}}_{3}=X_{1}{\vec {e}}_{1}+X_{2}{\vec {e}}_{2}+X_{3}{\vec {e}}_{3}}" loading="lazy"></span></dd></dl>

<p>ist genauer die Position des betrachteten materiellen Punktes im undeformierten Körper in der Ausgangs- oder Referenzkonfiguration. Die Komponenten <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{1,2,3}}">

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<annotation encoding="application/x-tex">{\displaystyle X_{1,2,3}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/96012b6106782ee51bb4c56f86300dd37421d2d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.537ex; height:2.843ex;" alt="{\displaystyle X_{1,2,3}}" loading="lazy"></span> sind die <i>materiellen</i> Koordinaten des betrachteten Punktes.

</p><p>Bei festgehaltenem materiellen Punkt <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {X}}}">

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5dc35b5a0226cf11a2c3f2d2dbbac6ab5ade6036.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.843ex;" alt="{\displaystyle {\vec {X}}}" loading="lazy"></span> beschreibt die Bewegungsfunktion dessen <a href="Bahnlinie" title="Bahnlinie">Bahnlinie</a> durch den Raum. Die Verschiebung ist nun der Differenzvektor zwischen der aktuellen Lage des Punktes im deformierten Körper und seiner ursprünglichen Lage im undeformierten Körper:

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<annotation encoding="application/x-tex">{\displaystyle {\vec {u}}({\vec {X}},t)=u{\vec {e}}_{1}+v{\vec {e}}_{2}+w{\vec {e}}_{3}=u_{1}{\vec {e}}_{1}+u_{2}{\vec {e}}_{2}+u_{3}{\vec {e}}_{3}={\vec {\chi }}({\vec {X}},t)-{\vec {X}}={\vec {x}}-{\vec {X}}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/99af5961ca5f983261ef5df8b52bcc662ebb127e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:73.774ex; height:3.343ex;" alt="{\displaystyle {\vec {u}}({\vec {X}},t)=u{\vec {e}}_{1}+v{\vec {e}}_{2}+w{\vec {e}}_{3}=u_{1}{\vec {e}}_{1}+u_{2}{\vec {e}}_{2}+u_{3}{\vec {e}}_{3}={\vec {\chi }}({\vec {X}},t)-{\vec {X}}={\vec {x}}-{\vec {X}}}" loading="lazy"></span>.</dd></dl>

<p>Um zu untersuchen wie sich die Verschiebung ändert, wenn die Position in der undeformierten Ausgangslage variiert wird, wird die Ableitung gebildet:

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<annotation encoding="application/x-tex">{\displaystyle H_{kl}={\frac {\mathrm {d} u_{k}({\vec {X}},t)}{\mathrm {d} X_{l}}}\quad {\mathsf {mit}}\quad k,l=1,2,3}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d0ca79c326edfd955363172b10f4dac7a9949f4d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:36.297ex; height:6.676ex;" alt="{\displaystyle H_{kl}={\frac {\mathrm {d} u_{k}({\vec {X}},t)}{\mathrm {d} X_{l}}}\quad {\mathsf {mit}}\quad k,l=1,2,3}" loading="lazy"></span>.</dd></dl>

<p>Darin sind <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H_{kl}}">

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<annotation encoding="application/x-tex">{\displaystyle H_{kl}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e9c577e8f9b51a07eb121b829fef159a6699d0dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.51ex; height:2.509ex;" alt="{\displaystyle H_{kl}}" loading="lazy"></span> die Komponenten des Verschiebungsgradienten bezüglich des Basissystems <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {e}}_{1,2,3}}">

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</p><p>Um zu einer koordinatenfreien Darstellung zu gelangen, wird das <a href="Dyadisches_Produkt" title="Dyadisches Produkt">dyadische Produkt</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \otimes }">

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/de29098f5a34ee296a505681a0d5e875070f2aea.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \otimes }" loading="lazy"></span> benutzt:

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {H} =\sum _{k,l=1}^{3}H_{kl}{\vec {e}}_{k}\otimes {\vec {e}}_{l}:=\operatorname {GRAD} \,{\vec {u}}({\vec {X}},t)={\begin{pmatrix}{\frac {\mathrm {d} u}{\mathrm {d} X}}&{\frac {\mathrm {d} u}{\mathrm {d} Y}}&{\frac {\mathrm {d} u}{\mathrm {d} Z}}\\{\frac {\mathrm {d} v}{\mathrm {d} X}}&{\frac {\mathrm {d} v}{\mathrm {d} Y}}&{\frac {\mathrm {d} v}{\mathrm {d} Z}}\\{\frac {\mathrm {d} w}{\mathrm {d} X}}&{\frac {\mathrm {d} w}{\mathrm {d} Y}}&{\frac {\mathrm {d} w}{\mathrm {d} Z}}\end{pmatrix}}_{{\vec {e}}_{k}\otimes {\vec {e}}_{l}}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/896b749cbf635eae858cab1d7964f55b323828c8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.838ex; width:65.048ex; height:12.509ex;" alt="{\displaystyle \mathbf {H} =\sum _{k,l=1}^{3}H_{kl}{\vec {e}}_{k}\otimes {\vec {e}}_{l}:=\operatorname {GRAD} \,{\vec {u}}({\vec {X}},t)={\begin{pmatrix}{\frac {\mathrm {d} u}{\mathrm {d} X}}&{\frac {\mathrm {d} u}{\mathrm {d} Y}}&{\frac {\mathrm {d} u}{\mathrm {d} Z}}\\{\frac {\mathrm {d} v}{\mathrm {d} X}}&{\frac {\mathrm {d} v}{\mathrm {d} Y}}&{\frac {\mathrm {d} v}{\mathrm {d} Z}}\\{\frac {\mathrm {d} w}{\mathrm {d} X}}&{\frac {\mathrm {d} w}{\mathrm {d} Y}}&{\frac {\mathrm {d} w}{\mathrm {d} Z}}\end{pmatrix}}_{{\vec {e}}_{k}\otimes {\vec {e}}_{l}}}" loading="lazy"></span>.</dd></dl>

<p>Der Tensor <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {H} }">

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {H} }</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f017b876ed763037d8818ec5dfbbdc6703e0f683.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.091ex; height:2.176ex;" alt="{\displaystyle \mathbf {H} }" loading="lazy"></span> ist der Verschiebungsgradient und <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {GRAD} }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {GRAD} }</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1a38ad2f23f69c80b41d45cbc4531a9cd269ceab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.054ex; height:2.176ex;" alt="{\displaystyle \operatorname {GRAD} }" loading="lazy"></span> ist das Symbol für den materiellen <a href="Gradient_(Mathematik)" title="Gradient (Mathematik)">Gradienten</a>, denn es wird nach den materiellen Koordinaten <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{1,2,3}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X_{1,2,3}}</annotation>

</semantics>

</p>

<div class="mw-heading mw-heading2"><h2 id="Geometrische_Linearisierung">Geometrische Linearisierung</h2></div>

<p>In der <a href="Festk%C3%B6rpermechanik" class="mw-redirect" title="Festkörpermechanik">Festkörpermechanik</a> treten in vielen, vor allem in technischen Anwendungsbereichen, nur kleine Verschiebungen auf. In diesem Fall erfahren die Gleichungen der Kontinuumsmechanik eine erhebliche Vereinfachung durch <i><a href="Geometrische_Linearisierung" title="Geometrische Linearisierung">geometrische Linearisierung</a></i>. Wenn <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle L}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}" loading="lazy"></span> eine charakteristische Abmessung des Körpers ist, dann wird bei kleinen Verschiebungen sowohl <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |{\vec {u}}|\ll L}">

</mrow>

<mover>

</mover>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle |{\vec {u}}|\ll L}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0a006f6e8aaf8b6e44b3ccbbfcfaf598d3e6e621.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.82ex; height:2.843ex;" alt="{\displaystyle |{\vec {u}}|\ll L}" loading="lazy"></span> als auch <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \parallel \mathbf {H} \parallel \ll 1}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \parallel \mathbf {H} \parallel \ll 1}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c45a0b7b63ce5444f06e147e2346853b369bf5c9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.193ex; height:2.843ex;" alt="{\displaystyle \parallel \mathbf {H} \parallel \ll 1}" loading="lazy"></span> gefordert, so dass alle Terme, die höhere Potenzen von <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {u}}}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\vec {u}}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/89c41e9cf70c5e5b56e2128a136985a75f90ba43.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:2.343ex;" alt="{\displaystyle {\vec {u}}}" loading="lazy"></span> oder <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {H} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {H} }</annotation>

</semantics>

</p>

</mrow>

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

<mover>

</mover>

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</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {F} =\operatorname {GRAD} {\vec {\chi }}=\operatorname {GRAD} {\vec {X}}+\operatorname {GRAD} {\vec {u}}=\mathbf {I} +\mathbf {H} }</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/85a41f3ed4c613906aa3195f1e5e07b655c424d4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:46.851ex; height:3.176ex;" alt="{\displaystyle \mathbf {F} =\operatorname {GRAD} {\vec {\chi }}=\operatorname {GRAD} {\vec {X}}+\operatorname {GRAD} {\vec {u}}=\mathbf {I} +\mathbf {H} }" loading="lazy"></span>,</dd></dl>

<p>den symmetrischen-

</p>

<msup>

</mrow>

</mrow>

</msup>

<mfrac>

</mfrac>

</mrow>

</mrow>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {H} ^{\mathrm {S} }={\frac {1}{2}}(\mathbf {H} +\mathbf {H} ^{\mathrm {T} })}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e691b7b806203a2ab93e8c9d37552392af954151.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:18.586ex; height:5.176ex;" alt="{\displaystyle \mathbf {H} ^{\mathrm {S} }={\frac {1}{2}}(\mathbf {H} +\mathbf {H} ^{\mathrm {T} })}" loading="lazy"></span></dd></dl>

<p>und schiefsymmetrischen Anteil

</p>

<msup>

</mrow>

</mrow>

</msup>

<mfrac>

</mfrac>

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<msup>

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</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {H} ^{\mathrm {A} }={\frac {1}{2}}(\mathbf {H} -\mathbf {H} ^{\mathrm {T} })}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/14f61cf3e2770b2bac2a8603ae505bcb0ac56e3e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:18.905ex; height:5.176ex;" alt="{\displaystyle \mathbf {H} ^{\mathrm {A} }={\frac {1}{2}}(\mathbf {H} -\mathbf {H} ^{\mathrm {T} })}" loading="lazy"></span></dd></dl>

<p>des Verschiebungsgradienten werden im Folgenden benutzt. Der linearisierte Verzerrungstensor

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {\varepsilon }}=\mathbf {H} ^{\mathrm {S} }={\frac {1}{2}}\left(\mathbf {H} +\mathbf {H} ^{\mathrm {T} }\right)={\begin{pmatrix}{\frac {\mathrm {d} u}{\mathrm {d} x}}&{\frac {1}{2}}\left({\frac {\mathrm {d} u}{\mathrm {d} y}}+{\frac {\mathrm {d} v}{\mathrm {d} x}}\right)&{\frac {1}{2}}\left({\frac {\mathrm {d} u}{\mathrm {d} z}}+{\frac {\mathrm {d} w}{\mathrm {d} x}}\right)\\{\frac {1}{2}}\left({\frac {\mathrm {d} v}{\mathrm {d} x}}+{\frac {\mathrm {d} u}{\mathrm {d} y}}\right)&{\frac {\mathrm {d} v}{\mathrm {d} y}}&{\frac {1}{2}}\left({\frac {\mathrm {d} v}{\mathrm {d} z}}+{\frac {\mathrm {d} w}{\mathrm {d} y}}\right)\\{\frac {1}{2}}\left({\frac {\mathrm {d} w}{\mathrm {d} x}}+{\frac {\mathrm {d} u}{\mathrm {d} z}}\right)&{\frac {1}{2}}\left({\frac {\mathrm {d} w}{\mathrm {d} y}}+{\frac {\mathrm {d} v}{\mathrm {d} z}}\right)&{\frac {\mathrm {d} w}{\mathrm {d} z}}\end{pmatrix}}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/175d2d4fca060fde021275cdacd07d9e3316446a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.838ex; width:75.876ex; height:14.843ex;" alt="{\displaystyle {\boldsymbol {\varepsilon }}=\mathbf {H} ^{\mathrm {S} }={\frac {1}{2}}\left(\mathbf {H} +\mathbf {H} ^{\mathrm {T} }\right)={\begin{pmatrix}{\frac {\mathrm {d} u}{\mathrm {d} x}}&{\frac {1}{2}}\left({\frac {\mathrm {d} u}{\mathrm {d} y}}+{\frac {\mathrm {d} v}{\mathrm {d} x}}\right)&{\frac {1}{2}}\left({\frac {\mathrm {d} u}{\mathrm {d} z}}+{\frac {\mathrm {d} w}{\mathrm {d} x}}\right)\\{\frac {1}{2}}\left({\frac {\mathrm {d} v}{\mathrm {d} x}}+{\frac {\mathrm {d} u}{\mathrm {d} y}}\right)&{\frac {\mathrm {d} v}{\mathrm {d} y}}&{\frac {1}{2}}\left({\frac {\mathrm {d} v}{\mathrm {d} z}}+{\frac {\mathrm {d} w}{\mathrm {d} y}}\right)\\{\frac {1}{2}}\left({\frac {\mathrm {d} w}{\mathrm {d} x}}+{\frac {\mathrm {d} u}{\mathrm {d} z}}\right)&{\frac {1}{2}}\left({\frac {\mathrm {d} w}{\mathrm {d} y}}+{\frac {\mathrm {d} v}{\mathrm {d} z}}\right)&{\frac {\mathrm {d} w}{\mathrm {d} z}}\end{pmatrix}}}" loading="lazy"></span></dd></dl>

<p>ist in der <a href="Technische_Mechanik" title="Technische Mechanik">technischen Mechanik</a> wohlbekannt und wird auch Ingenieursdehnung genannt.

</p>

<div class="mw-heading mw-heading3"><h3 id="Deformationsgradient_und_seine_Polarzerlegung">Deformationsgradient und seine Polarzerlegung</h3></div>

<div class="hauptartikel" role="navigation"><span class="hauptartikel-pfeil" title="siehe" aria-hidden="true" role="presentation">→ </span><i><span class="hauptartikel-text">Hauptartikel</span>: <a href="Deformationsgradient" title="Deformationsgradient">Deformationsgradient</a></i></div>

<p>Bei kleinen Verschiebungen sind die Invarianten des Deformationsgradienten Funktionen der <a href="Spur_(Mathematik)" title="Spur (Mathematik)">Spur</a> des Verschiebungsgradienten:

</p>

<th>Operator</th>

<th>Allgemeine Definition</th>

<th>Form bei kleinen Verschiebungen

</th></tr>

<tr>

<td>Spur

</td>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {Sp} (\mathbf {F} )=3+\operatorname {Sp} (\mathbf {H} )}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/869e8d2c8d1b4a694c384d0e742ea3cf47a29699.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.664ex; height:2.843ex;" alt="{\displaystyle \operatorname {Sp} (\mathbf {F} )=3+\operatorname {Sp} (\mathbf {H} )}" loading="lazy"></span>

</td>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {Sp} (\mathbf {F} )=3+\operatorname {Sp} (\mathbf {H} )}</annotation>

</semantics>

</td></tr>

<tr>

<td>Zweite <a href="Hauptinvariante" title="Hauptinvariante">Hauptinvariante</a>

</td>

<msub>

</mrow>

</msub>

</mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

<msup>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {I} _{2}(\mathbf {F} )={\frac {1}{2}}(\operatorname {Sp} (\mathbf {F} )^{2}-\operatorname {Sp} (\mathbf {F\cdot F} ))}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a4b0225b6399aaf538d326e3cdab064dbc2b2114.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:31.703ex; height:5.176ex;" alt="{\displaystyle \operatorname {I} _{2}(\mathbf {F} )={\frac {1}{2}}(\operatorname {Sp} (\mathbf {F} )^{2}-\operatorname {Sp} (\mathbf {F\cdot F} ))}" loading="lazy"></span>

</td>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {I} _{2}(\mathbf {F} )\approx 3+2\operatorname {Sp} (\mathbf {H} )}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3f06da562d3e5e60bf5e5e044089bcf37ad82287.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.523ex; height:2.843ex;" alt="{\displaystyle \operatorname {I} _{2}(\mathbf {F} )\approx 3+2\operatorname {Sp} (\mathbf {H} )}" loading="lazy"></span>

</td></tr>

<tr>

<td><a href="Determinante" title="Determinante">Determinante</a>

</td>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {det} (\mathbf {F} )=\operatorname {det} (\mathbf {I} +\mathbf {H} )}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d409e53e707d7caf61ddc4b527fdd8f57cac00bf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.805ex; height:2.843ex;" alt="{\displaystyle \operatorname {det} (\mathbf {F} )=\operatorname {det} (\mathbf {I} +\mathbf {H} )}" loading="lazy"></span>

</td>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {det} (\mathbf {F} )\approx 1+\operatorname {Sp} (\mathbf {H} )}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b5c499ec2c9b82cc45eb8cd8379c5ad6c3137993.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.309ex; height:2.843ex;" alt="{\displaystyle \operatorname {det} (\mathbf {F} )\approx 1+\operatorname {Sp} (\mathbf {H} )}" loading="lazy"></span>

</td></tr>

<tr>

<td><a href="Frobeniusnorm" title="Frobeniusnorm">Frobeniusnorm</a>

</td>

</mrow>

<msqrt>

<msup>

</mrow>

</msup>

</mrow>

</msqrt>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \parallel \mathbf {F} \parallel ={\sqrt {\operatorname {Sp} (\mathbf {F^{\mathrm {T} }\cdot F} )}}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4742e792e4663e60463ea8de57ac8c78cd873930.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:20.288ex; height:4.843ex;" alt="{\displaystyle \parallel \mathbf {F} \parallel ={\sqrt {\operatorname {Sp} (\mathbf {F^{\mathrm {T} }\cdot F} )}}}" loading="lazy"></span>

</td>

</mrow>

<msqrt>

</msqrt>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \parallel \mathbf {F} \parallel \approx {\sqrt {3}}+\operatorname {Sp} (\mathbf {H} )}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/02c87181e4a58314ae450fdb9fce762bce1aeadc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.531ex; height:3.009ex;" alt="{\displaystyle \parallel \mathbf {F} \parallel \approx {\sqrt {3}}+\operatorname {Sp} (\mathbf {H} )}" loading="lazy"></span>

</td></tr></tbody></table>

<p>Der Deformationsgradient <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {F} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {F} }</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/da18bef8c979f3548bb0d8976f5844012d7b8256.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.683ex; height:2.176ex;" alt="{\displaystyle \mathbf {F} }" loading="lazy"></span> lässt sich eindeutig „polar“ in eine Rotation und eine reine Streckung zerlegen. Durch Anwendung der <a href="Polarzerlegung" title="Polarzerlegung">Polarzerlegung</a> resultiert die Darstellung

</p>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {F} =\mathbf {R\cdot U} =\mathbf {v\cdot R} }</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7b814ff10dfebace620e1fe9a351eb8f071ca3ae.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:18.712ex; height:2.176ex;" alt="{\displaystyle \mathbf {F} =\mathbf {R\cdot U} =\mathbf {v\cdot R} }" loading="lazy"></span>.</dd></dl>

<p>Der Rotationstensor <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {R} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {R} }</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5de85fcc2a00d8ba14aae84aeef812d7fef4b3d5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.003ex; height:2.176ex;" alt="{\displaystyle \mathbf {R} }" loading="lazy"></span> ist ein <a href="Orthogonaler_Tensor" title="Orthogonaler Tensor"> „eigentlich orthogonaler Tensor“</a>. Der materielle Rechte <a href="Strecktensor" title="Strecktensor">Strecktensor</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {U} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {U} }</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b2141bec2344e3dc5241ff50b0fd366755e00223.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.057ex; height:2.176ex;" alt="{\displaystyle \mathbf {U} }" loading="lazy"></span> und der räumliche Linke Strecktensor <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {v} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {v} }</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/35c1866e359fbfd2e0f606c725ba5cc37a5195d6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.411ex; height:1.676ex;" alt="{\displaystyle \mathbf {v} }" loading="lazy"></span> sind <a href="Symmetrische_Matrix" title="Symmetrische Matrix">symmetrisch</a> und <a href="Positiv_definit" class="mw-redirect" title="Positiv definit">positiv definit</a>. Bei kleinen Verschiebungen sind sie identisch und linear in den linearisierten Dehnungen, wie die folgende Tabelle zeigt:

</p>

<th>Allgemeine Definition</th>

<th>Form bei kleinen Verschiebungen

</th></tr>

<tr>

<td>Rechter Strecktensor

</td>

</mrow>

<msqrt>

<msup>

</mrow>

</msup>

</mrow>

</msqrt>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {U} ={\sqrt {\mathbf {F^{\mathrm {T} }\cdot F} }}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1e5337274df6e77d84f39b1694c91017f5ed0f90.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.942ex; height:3.509ex;" alt="{\displaystyle \mathbf {U} ={\sqrt {\mathbf {F^{\mathrm {T} }\cdot F} }}}" loading="lazy"></span> <sup id="cite_ref-funktion_1-0" class="reference"><a href="#cite_note-funktion-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>

</td>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {U} \approx \mathbf {I} +{\boldsymbol {\varepsilon }}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/10b63319fbdf91bad09358e97e9fa9ee938f3a3e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.239ex; height:2.343ex;" alt="{\displaystyle \mathbf {U} \approx \mathbf {I} +{\boldsymbol {\varepsilon }}}" loading="lazy"></span>

</td></tr>

<tr>

<td>Linker Strecktensor

</td>

</mrow>

<msqrt>

<msup>

</mrow>

</mrow>

</msup>

</msqrt>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {v} ={\sqrt {\mathbf {F\cdot F} ^{\mathrm {T} }}}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bac5debdf9442e94c7535155480a36f3b4a45511.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.297ex; height:3.509ex;" alt="{\displaystyle \mathbf {v} ={\sqrt {\mathbf {F\cdot F} ^{\mathrm {T} }}}}" loading="lazy"></span><sup id="cite_ref-funktion_1-1" class="reference"><a href="#cite_note-funktion-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>

</td>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {v} \approx \mathbf {I} +{\boldsymbol {\varepsilon }}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68bca7a727735ff960fe39ed121f1c171ea51af6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.593ex; height:2.343ex;" alt="{\displaystyle \mathbf {v} \approx \mathbf {I} +{\boldsymbol {\varepsilon }}}" loading="lazy"></span>

</td></tr>

<tr>

<td>Rotationstensor

</td>

</mrow>

<msup>

</mrow>

</mrow>

</msup>

<msup>

</mrow>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {R} =\mathbf {F\cdot U} ^{-1}=\mathbf {v} ^{-1}\cdot \mathbf {F} }</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a1e1294dfe50a351d120e985a4a3671672b94a4b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:23.057ex; height:2.676ex;" alt="{\displaystyle \mathbf {R} =\mathbf {F\cdot U} ^{-1}=\mathbf {v} ^{-1}\cdot \mathbf {F} }" loading="lazy"></span>

</td>

</mrow>

</mrow>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {R} \approx \mathbf {I} +\mathbf {H} ^{\mathrm {A} }}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c1e67d46bd5cea4553e5a51fc2be35bedc9c328f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:12.512ex; height:2.843ex;" alt="{\displaystyle \mathbf {R} \approx \mathbf {I} +\mathbf {H} ^{\mathrm {A} }}" loading="lazy"></span>

</td></tr></tbody></table>

<p>Die Identitäten

</p>

<mtr>

<mtd>

</mrow>

</mtd>

<mtd>

</mtd>

<mtd>

</mrow>

</mtd>

<mtd>

</mtd>

<mtd>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

</mrow>

</mtd>

<mtd>

</mtd>

<mtd>

<msub>

</mrow>

</mrow>

</msub>

</mtd>

<mtd>

</mtd>

<mtd>

</mrow>

</mtd>

<mtd>

</mtd>

<mtd>

<msup>

</mrow>

</mrow>

</msup>

<msup>

</mrow>

</mrow>

</msup>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{array}{lclclll}\mathbf {F} &=&\mathbf {R} &\cdot &\mathbf {U} \\\mathbf {H} &=&\mathbf {R} _{L}&+&{\boldsymbol {\varepsilon }}&=&\mathbf {H} ^{\mathrm {A} }+\mathbf {H} ^{\mathrm {S} }\end{array}}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d871c77ae91975d88da832ed8835cc92957d6161.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:37.249ex; height:6.509ex;" alt="{\displaystyle {\begin{array}{lclclll}\mathbf {F} &=&\mathbf {R} &\cdot &\mathbf {U} \\\mathbf {H} &=&\mathbf {R} _{L}&+&{\boldsymbol {\varepsilon }}&=&\mathbf {H} ^{\mathrm {A} }+\mathbf {H} ^{\mathrm {S} }\end{array}}}" loading="lazy"></span></dd></dl>

<p>zeigen, dass bei kleinen Verzerrungen die Polarzerlegung des Deformationsgradienten in die additive Zerlegung des Verschiebungsgradienten in seinen schiefsymmetrischen und symmetrischen Anteil übergeht. Der Anteil

</p>

<msub>

</mrow>

</mrow>

</msub>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {R} _{L}:=\mathbf {H} ^{\mathrm {A} }}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d63d940f7a36c9f4e50de1fb815cde46dc6a2f87.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.656ex; height:3.009ex;" alt="{\displaystyle \mathbf {R} _{L}:=\mathbf {H} ^{\mathrm {A} }}" loading="lazy"></span></dd></dl>

<p>wird linearisierter Rotationstensor und der symmetrische Anteil

</p>

</mrow>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {\varepsilon }}:=\mathbf {H} ^{\mathrm {S} }}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f4d8ed8c25fa44b8f0cb0305afbe72dcef07e613.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.213ex; height:2.676ex;" alt="{\displaystyle {\boldsymbol {\varepsilon }}:=\mathbf {H} ^{\mathrm {S} }}" loading="lazy"></span>,</dd></dl>

<p>wird, wie oben erwähnt, linearisierter Verzerrungstensor oder Ingenieursdehnung genannt.

</p><p>Bei den Inversen der Tensoren in der Tabelle dreht sich bei geometrischer Linearisierung das Vorzeichen des Anteils des Verschiebungsgradienten um:

</p>

<mtr>

<mtd>

<msup>

</mrow>

</mrow>

</msup>

</mtd>

<mtd>

</mtd>

<mtd>

<msup>

</mrow>

</mrow>

</msup>

</mrow>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

<msup>

</mrow>

</mrow>

</msup>

</mtd>

<mtd>

</mtd>

<mtd>

<msup>

</mrow>

</mrow>

</msup>

</mrow>

<msup>

</mrow>

</mrow>

</msup>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{array}{rcl}\mathbf {U} ^{-1}&=&\mathbf {v} ^{-1}\approx \mathbf {I} -{\boldsymbol {\varepsilon }}\\\mathbf {R} ^{-1}&=&\mathbf {R} ^{\mathrm {T} }\approx \mathbf {I} -\mathbf {H} ^{A}\end{array}}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7e3026fd315820dda3b05aa813de2787be0de139.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:25.525ex; height:6.509ex;" alt="{\displaystyle {\begin{array}{rcl}\mathbf {U} ^{-1}&=&\mathbf {v} ^{-1}\approx \mathbf {I} -{\boldsymbol {\varepsilon }}\\\mathbf {R} ^{-1}&=&\mathbf {R} ^{\mathrm {T} }\approx \mathbf {I} -\mathbf {H} ^{A}\end{array}}}" loading="lazy"></span></dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Strecktensoren">Strecktensoren</h3></div>

<div class="hauptartikel" role="navigation"><span class="hauptartikel-pfeil" title="siehe" aria-hidden="true" role="presentation">→ </span><i><span class="hauptartikel-text">Hauptartikel</span>: <a href="Strecktensor" title="Strecktensor">Strecktensor</a></i></div>

<p>Der rechte und linke Cauchy-Green Tensor sind bei kleinen Verschiebungen identisch und linear in den linearisierten Dehnungen:

</p>

<th>Allgemeine Definition</th>

<th>Form bei kleinen Verschiebungen

</th></tr>

<tr>

<td>Rechter Cauchy-Green Tensor

</td>

</mrow>

<msup>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {C} =\mathbf {F^{\mathrm {T} }\cdot F} }</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8dc3100426bc2c9d95df885ff1c5afb7605ba477.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.493ex; height:2.676ex;" alt="{\displaystyle \mathbf {C} =\mathbf {F^{\mathrm {T} }\cdot F} }" loading="lazy"></span>

</td>

</mrow>

</mrow>

</mrow>

<msup>

</mrow>

</mrow>

</msup>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {C} \approx \mathbf {I} +\mathbf {H} +\mathbf {H} ^{\mathrm {T} }=\mathbf {I} +2{\boldsymbol {\varepsilon }}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5106f9d00a33199ff341293abf9eb945397b0a2f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:26.671ex; height:2.843ex;" alt="{\displaystyle \mathbf {C} \approx \mathbf {I} +\mathbf {H} +\mathbf {H} ^{\mathrm {T} }=\mathbf {I} +2{\boldsymbol {\varepsilon }}}" loading="lazy"></span>

</td></tr>

<tr>

<td>Linker Cauchy-Green Tensor

</td>

</mrow>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {b} =\mathbf {F\cdot F} ^{\mathrm {T} }}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3999466b376e0a1fe034cd11005f2fb6e0b5476f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.047ex; height:2.676ex;" alt="{\displaystyle \mathbf {b} =\mathbf {F\cdot F} ^{\mathrm {T} }}" loading="lazy"></span>

</td>

</mrow>

</mrow>

</mrow>

<msup>

</mrow>

</mrow>

</msup>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {b} \approx \mathbf {I} +\mathbf {H} +\mathbf {H} ^{\mathrm {T} }=\mathbf {I} +2{\boldsymbol {\varepsilon }}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9d9566658e006f7bc6b29089bbc454f7e1090405.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:26.225ex; height:2.843ex;" alt="{\displaystyle \mathbf {b} \approx \mathbf {I} +\mathbf {H} +\mathbf {H} ^{\mathrm {T} }=\mathbf {I} +2{\boldsymbol {\varepsilon }}}" loading="lazy"></span>

</td></tr></tbody></table>

<p>Auch hier dreht sich bei Invertierung im geometrisch linearen Fall das Vorzeichen von <span class="mwe-math-element mwe-math-element-inline"><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 }}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {\varepsilon }}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8445af5ff7da70714382bc35e78bedcacf68e825.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle {\boldsymbol {\varepsilon }}}" loading="lazy"></span> um:

</p>

<msup>

</mrow>

</mrow>

</msup>

<msup>

</mrow>

</mrow>

</msup>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {C} ^{-1}\approx \mathbf {b} ^{-1}\approx \mathbf {I} -2{\boldsymbol {\varepsilon }}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7c1c2067b23885e5e6dbfeccb1eff4428e8cab60.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:20.525ex; height:2.843ex;" alt="{\displaystyle \mathbf {C} ^{-1}\approx \mathbf {b} ^{-1}\approx \mathbf {I} -2{\boldsymbol {\varepsilon }}}" loading="lazy"></span>.</dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Verzerrungstensoren">Verzerrungstensoren</h3></div>

<div class="hauptartikel" role="navigation"><span class="hauptartikel-pfeil" title="siehe" aria-hidden="true" role="presentation">→ </span><i><span class="hauptartikel-text">Hauptartikel</span>: <a href="Verzerrungstensor" title="Verzerrungstensor">Verzerrungstensor</a></i></div>

<p>Mit den obigen Ergebnissen für die Strecktensoren kann sofort bestätigt werden, dass die Verzerrungstensoren bei kleinen Verschiebungen in den linearisierten Verzerrungstensor <span class="mwe-math-element mwe-math-element-inline"><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 }}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {\varepsilon }}}</annotation>

</semantics>

</p>

<th>Allgemeine Definition</th>

<th>Form bei kleinen Verschiebungen

</th></tr>

<tr>

<td>Green-Lagrange Verzerrungstensor

</td>

</mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {E} ={\frac {1}{2}}(\mathbf {C} -\mathbf {I} )}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b3f93c2454c1bcdeb693b5490fc3dca453e5f10a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:14.449ex; height:5.176ex;" alt="{\displaystyle \mathbf {E} ={\frac {1}{2}}(\mathbf {C} -\mathbf {I} )}" loading="lazy"></span>

</td>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {E} \approx {\boldsymbol {\varepsilon }}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6bf0fa9942f12afa7eb06bf08003d44b21cd86f0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.085ex; height:2.176ex;" alt="{\displaystyle \mathbf {E} \approx {\boldsymbol {\varepsilon }}}" loading="lazy"></span>

</td></tr>

<tr>

<td>Biot-Verzerrungstensor

</td>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {E} _{N}=\mathbf {U} -\mathbf {I} }</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/84c0faa1f6f778a3e4c36eb52d5e4dea66db62cc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.458ex; height:2.509ex;" alt="{\displaystyle \mathbf {E} _{N}=\mathbf {U} -\mathbf {I} }" loading="lazy"></span>

</td>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {E} _{N}\approx {\boldsymbol {\varepsilon }}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/09001b88bf11a2b9d5292a15308b5f2ea183d91f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.777ex; height:2.509ex;" alt="{\displaystyle \mathbf {E} _{N}\approx {\boldsymbol {\varepsilon }}}" loading="lazy"></span>

</td></tr>

<tr>

<td>Hencky Dehnungen

</td>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {E} _{H}=\ln(\mathbf {U} )}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5854d17e776cf534405634069d9c48d14cdb3052.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.352ex; height:2.843ex;" alt="{\displaystyle \mathbf {E} _{H}=\ln(\mathbf {U} )}" loading="lazy"></span> <sup id="cite_ref-funktion_1-2" class="reference"><a href="#cite_note-funktion-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>

</td>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {E} _{H}\approx \ln(\mathbf {I} +{\boldsymbol {\varepsilon }})\approx {\boldsymbol {\varepsilon }}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1b04b4bd75cfba637658ebdd640efb211ad0e362.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.708ex; height:2.843ex;" alt="{\displaystyle \mathbf {E} _{H}\approx \ln(\mathbf {I} +{\boldsymbol {\varepsilon }})\approx {\boldsymbol {\varepsilon }}}" loading="lazy"></span> <sup id="cite_ref-funktion_1-3" class="reference"><a href="#cite_note-funktion-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>

</td></tr>

<tr>

<td>Piola-Verzerrungstensor

</td>

<msub>

</mrow>

</mrow>

</msub>

<mfrac>

</mfrac>

</mrow>

<msup>

</mrow>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {E} _{P}={\frac {1}{2}}(\mathbf {C} ^{-1}-\mathbf {I} )}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7ab8c735fa7f716e3d4d65360316bee8cb81d219.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:18.248ex; height:5.176ex;" alt="{\displaystyle \mathbf {E} _{P}={\frac {1}{2}}(\mathbf {C} ^{-1}-\mathbf {I} )}" loading="lazy"></span>

</td>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {E} _{P}\approx -{\boldsymbol {\varepsilon }}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9189e0fa89e9b505f62bb3dd50498d3a657d1c0d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.36ex; height:2.509ex;" alt="{\displaystyle \mathbf {E} _{P}\approx -{\boldsymbol {\varepsilon }}}" loading="lazy"></span>

</td></tr>

<tr>

<td>Euler-Almansi Verzerrungstensor

</td>

</mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {e} ={\frac {1}{2}}(\mathbf {I} -\mathbf {b} ^{-1})}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bf7d98edff7d9d223759a99316a9549e5ff359ac.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:15.804ex; height:5.176ex;" alt="{\displaystyle \mathbf {e} ={\frac {1}{2}}(\mathbf {I} -\mathbf {b} ^{-1})}" loading="lazy"></span>

</td>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {e} \approx {\boldsymbol {\varepsilon }}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d660c6f5839151d32e47dd3e674b8592f4e830c1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.553ex; height:1.676ex;" alt="{\displaystyle \mathbf {e} \approx {\boldsymbol {\varepsilon }}}" loading="lazy"></span>

</td></tr>

<tr>

<td>Finger-Tensor

</td>

<msub>

</mrow>

</mrow>

</msub>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {e} _{F}={\frac {1}{2}}(\mathbf {I} -\mathbf {b} )}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0b981d0124bf1e6d1a34dcd3931f57b84daa6f40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:14.934ex; height:5.176ex;" alt="{\displaystyle \mathbf {e} _{F}={\frac {1}{2}}(\mathbf {I} -\mathbf {b} )}" loading="lazy"></span>

</td>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {e} _{F}\approx -{\boldsymbol {\varepsilon }}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/13833ddb933ab9cb5b1dc8e7fd86608fa0a83859.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.825ex; height:2.343ex;" alt="{\displaystyle \mathbf {e} _{F}\approx -{\boldsymbol {\varepsilon }}}" loading="lazy"></span>

</td></tr>

<tr>

<td>Swainger-Verzerrungstensor

</td>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {e} _{S}=\mathbf {I} -\mathbf {v} ^{-1}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e873d80d708b803850facd9e0f19bcf30f509899.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.214ex; height:3.009ex;" alt="{\displaystyle \mathbf {e} _{S}=\mathbf {I} -\mathbf {v} ^{-1}}" loading="lazy"></span>

</td>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {e} _{S}\approx {\boldsymbol {\varepsilon }}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6ff0f94c38ab96657abc5e9eb5244f25588472f6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.846ex; height:2.009ex;" alt="{\displaystyle \mathbf {e} _{S}\approx {\boldsymbol {\varepsilon }}}" loading="lazy"></span>.

</td></tr></tbody></table>

<div class="mw-heading mw-heading2"><h2 id="Siehe_auch">Siehe auch</h2></div>

<ul><li><a href="Linearisierung" title="Linearisierung">Linearisierung</a></li>

<li><a href="Formelsammlung_Tensoralgebra" title="Formelsammlung Tensoralgebra">Formelsammlung Tensoralgebra</a></li>

<li><a href="Formelsammlung_Tensoranalysis" title="Formelsammlung Tensoranalysis">Formelsammlung Tensoranalysis</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="Fußnoten"><span id="Fu.C3.9Fnoten"></span>Fußnoten</h2></div>

<li id="cite_note-funktion-1"><span class="mw-cite-backlink">↑ <sup><a href="#cite_ref-funktion_1-0">a</a></sup> <sup><a href="#cite_ref-funktion_1-1">b</a></sup> <sup><a href="#cite_ref-funktion_1-2">c</a></sup> <sup><a href="#cite_ref-funktion_1-3">d</a></sup></span> <span class="reference-text">Der Funktionswert eines symmetrischen, positiv definiten Tensors zweiter Stufe berechnet sich mittels seiner <a href="Hauptachsentransformation" title="Hauptachsentransformation">Hauptachsentransformation</a>, Bildung des Funktionswertes der Diagonalelemente und Rücktransformation.</span>

</li>

</ol>

<div class="mw-heading mw-heading2"><h2 id="Literatur">Literatur</h2></div>

<ul><li>H. Altenbach: <cite style="font-style:italic">Kontinuumsmechanik</cite>. Springer, 2012, ISBN 978-3-642-24118-5.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Verschiebungsgradient&rft.au=H.+Altenbach&rft.btitle=Kontinuumsmechanik&rft.date=2012&rft.genre=book&rft.isbn=9783642241185&rft.pub=Springer" style="display:none"> </span></li>

<li>P. Haupt: <cite style="font-style:italic">Continuum Mechanics and Theory of Materials</cite>. Springer, 2000, ISBN 3-540-66114-X.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Verschiebungsgradient&rft.au=P.+Haupt&rft.btitle=Continuum+Mechanics+and+Theory+of+Materials&rft.date=2000&rft.genre=book&rft.isbn=354066114X&rft.pub=Springer" style="display:none"> </span></li></ul></div><div><div class="zim-footer">

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