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<title>Verzerrungstensor</title>

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<h1 id="firstHeading" class="firstHeading mw-first-heading">Verzerrungstensor</h1>

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<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="de" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="de" dir="ltr">Verzerrungstensoren sind <a href="Dimension_(Gr%C3%B6%C3%9Fensystem)" title="Dimension (Größensystem)">dimensionslose</a> <a href="Tensor" title="Tensor">Tensoren</a> zweiter Stufe, die das Verhältnis von <a href="Konfiguration_(Mechanik)" title="Konfiguration (Mechanik)">Momentankonfiguration</a> zur Ausgangskonfiguration bei der Deformation von kontinuierlichen Körpern und damit Veränderung der gegenseitigen Lagebeziehungen der Materieelemente beschreiben. Diese Änderung (Deformation) der inneren Anordnung korrespondiert mit einer Änderung der äußeren Gestalt des <a href="Festk%C3%B6rper" title="Festkörper">Festkörpers</a> und wird damit beispielsweise als <a href="Dehnung" title="Dehnung">Dehnung</a>, <a href="Stauchung" title="Stauchung">Stauchung</a>, <a href="Scherung_(Mechanik)" title="Scherung (Mechanik)">Scherung</a> usw. sichtbar. Die Verzerrungstensoren sind eine wesentliche Größe in der Beschreibung der <a href="Kinematik" title="Kinematik">Kinematik</a> der Deformation. In der <a href="Kontinuumsmechanik" title="Kontinuumsmechanik">Kontinuumsmechanik</a> werden eine Reihe von verschiedenen Verzerrungstensoren definiert, deren Benennung nicht einheitlich ist.

Die Verzerrungstensoren werden vor allem für die Formulierung von Materialmodellen, z. B. der <a href="Hyperelastizit%C3%A4t" title="Hyperelastizität">Hyperelastizität</a>, verwendet, die eine Relation zwischen den <a href="Spannungstensor" title="Spannungstensor">Spannungen</a> im Material und seinen Deformationen herstellen. Solche Materialmodelle werden dazu benutzt, Verformungen von Körpern zu berechnen.

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

In der Literatur ist eine Vielzahl von Verzerrungstensoren bekannt, die aus dem <a href="Deformationsgradient" title="Deformationsgradient">Deformationsgradienten</a> gebildet werden. Für deren Definition werden die Verschiebungen

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<annotation encoding="application/x-tex">{\displaystyle {\vec {u}}({\vec {X}},t)={\vec {\chi }}({\vec {X}},t)-{\vec {X}}=\sum _{i=1}^{3}u_{i}{\vec {e}}_{i}={\begin{pmatrix}u({\vec {X}},t)\\v({\vec {X}},t)\\w({\vec {X}},t)\end{pmatrix}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b0b4d532ab95b3abaec2a0d03180b5a31c6f18db.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.005ex; width:48.266ex; height:11.176ex;" alt="{\displaystyle {\vec {u}}({\vec {X}},t)={\vec {\chi }}({\vec {X}},t)-{\vec {X}}=\sum _{i=1}^{3}u_{i}{\vec {e}}_{i}={\begin{pmatrix}u({\vec {X}},t)\\v({\vec {X}},t)\\w({\vec {X}},t)\end{pmatrix}}}" loading="lazy"></dd></dl>

als Differenzvektor zwischen der momentanen Lage <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {x}}={\vec {\chi }}({\vec {X}},t)}">

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<annotation encoding="application/x-tex">{\displaystyle {\vec {x}}={\vec {\chi }}({\vec {X}},t)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9e92bff89d59a995104a9f1d246741c880d1b2b5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.546ex; height:3.343ex;" alt="{\displaystyle {\vec {x}}={\vec {\chi }}({\vec {X}},t)}" loading="lazy"> eines Partikels und seiner Ausgangslage <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {X}}=\sum _{i=1}^{3}X_{i}{\vec {e}}_{i}}">

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<annotation encoding="application/x-tex">{\displaystyle {\vec {X}}=\sum _{i=1}^{3}X_{i}{\vec {e}}_{i}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/35d921e005ca874fe342618e253215957e896d2d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:13.567ex; height:7.176ex;" alt="{\displaystyle {\vec {X}}=\sum _{i=1}^{3}X_{i}{\vec {e}}_{i}}" loading="lazy"> eingeführt, mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{i}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/af4a0955af42beb5f85aa05fb8c07abedc13990d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.724ex; height:2.509ex;" alt="{\displaystyle X_{i}}" loading="lazy"> als den materiellen Koordinaten des Partikels bezüglich der <a href="Standardbasis" title="Standardbasis">Standardbasis</a>. Der <a href="Verschiebungsgradient" title="Verschiebungsgradient">Verschiebungsgradient</a><a href="#cite_note-Frechet-1">[1]</a>

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {H} :=\operatorname {GRAD} ({\vec {u}}):=\sum _{i,j=1}^{3}{\frac {\mathrm {d} u_{i}}{\mathrm {d} X_{j}}}{\vec {e}}_{i}\otimes {\vec {e}}_{j}={\frac {\mathrm {d} {\vec {u}}}{\mathrm {d} {\vec {X}}}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ee34372ae234a64bb738258a3debef944f3ec7b5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:43.13ex; height:7.509ex;" alt="{\displaystyle \mathbf {H} :=\operatorname {GRAD} ({\vec {u}}):=\sum _{i,j=1}^{3}{\frac {\mathrm {d} u_{i}}{\mathrm {d} X_{j}}}{\vec {e}}_{i}\otimes {\vec {e}}_{j}={\frac {\mathrm {d} {\vec {u}}}{\mathrm {d} {\vec {X}}}}}" loading="lazy"></dd></dl>

ist dann die Ableitung des Verschiebungsvektors <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {u}}}">

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</math><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"> nach den materiellen Koordinaten <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {X}}}">

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</math><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"> und enthält die Ableitungen der Verschiebungen ui nach den Koordinaten Xj. Damit bekommt der Deformationsgradient die Form

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {F} :={\frac {\mathrm {d} {\vec {\chi }}}{\mathrm {d} {\vec {X}}}}={\frac {\mathrm {d} }{\mathrm {d} {\vec {X}}}}({\vec {u}}+{\vec {X}})=\mathbf {H} +\mathbf {1} }</annotation>

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worin 1 der <a href="Einheitstensor" title="Einheitstensor">Einheitstensor</a> ist. Zunächst lassen sich damit der rechte Cauchy-Green-Tensor

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d4697e83ddb26138705a98f59188bc3447557283.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:12.232ex; height:2.676ex;" alt="{\displaystyle \mathbf {C} :=\mathbf {F^{\top }\cdot F} }" loading="lazy"></dd></dl>

bzgl. der Ausgangskonfiguration und der linke Cauchy-Green-Tensor

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/89a82aef8ed4b36830508e3d1fe5619db2dffa50.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.786ex; height:2.676ex;" alt="{\displaystyle \mathbf {b} :=\mathbf {F\cdot F^{\top }} }" loading="lazy"></dd></dl>

bzgl. der Momentankonfiguration bilden. Diese beiden <a href="Strecktensor" title="Strecktensor">Strecktensoren</a> sind symmetrisch und im Fall einer Nicht-Deformation gleich dem Einheitstensor.

Für ingenieurtechnische Anwendungen werden gewöhnlich allerdings Größen gewünscht, die bei Nicht-Deformation eine Null darstellen.

Dies führt auf Definitionen des Green-Lagrange-Verzerrungstensors

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {E} :={\frac {1}{2}}(\mathbf {F^{\top }\cdot F-1} )}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/024e7c0ccce1182813e3633b99b70b55beabe10a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:20.312ex; height:5.176ex;" alt="{\displaystyle \mathbf {E} :={\frac {1}{2}}(\mathbf {F^{\top }\cdot F-1} )}" loading="lazy"></dd></dl>

oder des Euler-Almansi-Verzerrungstensors

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {e} :={\frac {1}{2}}(\mathbf {1-(F\cdot F^{\top })} ^{-1})\,.}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2e0ddb0e5a56b011901d99b9bef50abac08f63be.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:25.226ex; height:5.176ex;" alt="{\displaystyle \mathbf {e} :={\frac {1}{2}}(\mathbf {1-(F\cdot F^{\top })} ^{-1})\,.}" loading="lazy"></dd></dl>

Daneben existiert aber noch eine Vielzahl weiterer ähnlicher Definitionen, die jeweils ihre Berechtigung und Vorteile in verschiedenen Theorien besitzen, siehe unten. Dort erklärt sich auch der oben auftretende Faktor ½.

<div class="mw-heading mw-heading2"><h2 id="Linearisierter_Verzerrungstensor">Linearisierter Verzerrungstensor</h2></div>

Zur Beschreibung <a href="Geometrische_Linearisierung" title="Geometrische Linearisierung">kleiner Verzerrungen</a> wird in der <a href="Technische_Mechanik" title="Technische Mechanik">technischen Mechanik</a> üblicherweise der linearisierte Verzerrungstensor <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {\varepsilon }}}">

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</math><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"> verwendet. Dieser Verzerrungstensor wird auch Ingenieursdehnung genannt, denn bei vielen Anwendungen im technischen Bereich liegen kleine Dehnungen vor oder sie müssen aus sicherheitstechnischen Gründen klein gehalten werden. Der linearisierte Verzerrungstensor entsteht durch Linearisierung der Größen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {E} }">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0d7f22b39d51f780fc02859059c1757c606b9de2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.757ex; height:2.176ex;" alt="{\displaystyle \mathbf {E} }" loading="lazy"> oder <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {e} \,.}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8186d4b888839751048ac3d980c81113e762b4e7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.259ex; height:1.676ex;" alt="{\displaystyle \mathbf {e} \,.}" loading="lazy">

Hierzu wird die Definition des Deformationsgradienten in den Verzerrungstensor eingesetzt:

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {E} ={\frac {1}{2}}{\Bigl [}\mathbf {F^{\top }\cdot F} -\mathbf {1} {\Bigr ]}={\frac {1}{2}}{\Bigl [}{\Bigl (}\mathbf {H+1} {\Bigr )}^{\top }\cdot {\Bigl (}\mathbf {H} +\mathbf {1} {\Bigr )}-\mathbf {1} {\Bigr ]}={\frac {1}{2}}{\Bigl [}\mathbf {H+H^{\top }} +\mathbf {H^{\top }\cdot H} {\Bigr ]}\,.}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/172c0700b8f6b6d1a3112f41224cca27de48d35b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:80.14ex; height:5.176ex;" alt="{\displaystyle \mathbf {E} ={\frac {1}{2}}{\Bigl [}\mathbf {F^{\top }\cdot F} -\mathbf {1} {\Bigr ]}={\frac {1}{2}}{\Bigl [}{\Bigl (}\mathbf {H+1} {\Bigr )}^{\top }\cdot {\Bigl (}\mathbf {H} +\mathbf {1} {\Bigr )}-\mathbf {1} {\Bigr ]}={\frac {1}{2}}{\Bigl [}\mathbf {H+H^{\top }} +\mathbf {H^{\top }\cdot H} {\Bigr ]}\,.}" loading="lazy"></dd></dl>

Bei kleinen Verzerrungen kann der letzte Term vernachlässigt werden und so entsteht der linearisierte Verzerrungstensor

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<annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {\varepsilon }}:={\frac {1}{2}}{\Bigl [}\mathbf {H+H^{\top }} {\Bigr ]}={\begin{pmatrix}\varepsilon _{xx}&\varepsilon _{xy}&\varepsilon _{xz}\\\varepsilon _{yx}&\varepsilon _{yy}&\varepsilon _{yz}\\\varepsilon _{zx}&\varepsilon _{zy}&\varepsilon _{zz}\end{pmatrix}},}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9f3b88b1a99c0c867f20ce565155d3ac2116ba8c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.338ex; width:40.476ex; height:9.843ex;" alt="{\displaystyle {\boldsymbol {\varepsilon }}:={\frac {1}{2}}{\Bigl [}\mathbf {H+H^{\top }} {\Bigr ]}={\begin{pmatrix}\varepsilon _{xx}&\varepsilon _{xy}&\varepsilon _{xz}\\\varepsilon _{yx}&\varepsilon _{yy}&\varepsilon _{yz}\\\varepsilon _{zx}&\varepsilon _{zy}&\varepsilon _{zz}\end{pmatrix}},}" loading="lazy"></dd></dl>

mit den Komponenten

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<annotation encoding="application/x-tex">{\displaystyle {\begin{matrix}&\varepsilon _{xx}={\frac {\partial u_{x}}{\partial X_{x}}},\;\varepsilon _{yy}={\frac {\partial u_{y}}{\partial X_{y}}},\;\varepsilon _{zz}={\frac {\partial u_{z}}{\partial X_{z}}},\;\varepsilon _{xy}=\varepsilon _{yx}={\frac {1}{2}}\left({\frac {\partial u_{x}}{\partial X_{y}}}+{\frac {\partial u_{y}}{\partial X_{x}}}\right),\\&\varepsilon _{yz}=\varepsilon _{zy}={\frac {1}{2}}\left({\frac {\partial u_{y}}{\partial X_{z}}}+{\frac {\partial u_{z}}{\partial X_{y}}}\right),\;\varepsilon _{zx}=\varepsilon _{xz}={\frac {1}{2}}\left({\frac {\partial u_{z}}{\partial X_{x}}}+{\frac {\partial u_{x}}{\partial X_{z}}}\right).\end{matrix}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/741869834995dc5e8514581661279bff35c6df7c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.505ex; width:67.389ex; height:10.176ex;" alt="{\displaystyle {\begin{matrix}&\varepsilon _{xx}={\frac {\partial u_{x}}{\partial X_{x}}},\;\varepsilon _{yy}={\frac {\partial u_{y}}{\partial X_{y}}},\;\varepsilon _{zz}={\frac {\partial u_{z}}{\partial X_{z}}},\;\varepsilon _{xy}=\varepsilon _{yx}={\frac {1}{2}}\left({\frac {\partial u_{x}}{\partial X_{y}}}+{\frac {\partial u_{y}}{\partial X_{x}}}\right),\\&\varepsilon _{yz}=\varepsilon _{zy}={\frac {1}{2}}\left({\frac {\partial u_{y}}{\partial X_{z}}}+{\frac {\partial u_{z}}{\partial X_{y}}}\right),\;\varepsilon _{zx}=\varepsilon _{xz}={\frac {1}{2}}\left({\frac {\partial u_{z}}{\partial X_{x}}}+{\frac {\partial u_{x}}{\partial X_{z}}}\right).\end{matrix}}}" loading="lazy"></dd></dl>

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

In <a href="Zylinderkoordinaten" class="mw-redirect" title="Zylinderkoordinaten">Zylinderkoordinaten</a> mit radialer Koordinate <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho }">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1f7d439671d1289b6a816e6af7a304be40608d64.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.202ex; height:2.176ex;" alt="{\displaystyle \rho }" loading="lazy">, <a href="Azimut" title="Azimut">Azimut</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi }">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }" loading="lazy"> und <a href="H%C3%B6he" title="Höhe">Höhe</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bf368e72c009decd9b6686ee84a375632e11de98.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}" loading="lazy"> über der <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho \varphi }">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/192a517e18f5f6495fd91ad3eed563422ba0b3cb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.722ex; height:2.176ex;" alt="{\displaystyle \rho \varphi }" loading="lazy">-Ebene lauten der Verzerrungstensor<a href="#cite_note-sadd-2">[2]</a><a href="#cite_note-Landau-4-3">[3]</a>

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\varepsilon _{\rho \rho }=&{\frac {\partial u_{\rho }}{\partial \rho }}\qquad \varepsilon _{\varphi \varphi }={\frac {1}{\rho }}\left(u_{\rho }+{\frac {\partial u_{\varphi }}{\partial \varphi }}\right)\qquad \varepsilon _{zz}={\frac {\partial u_{z}}{\partial z}}\\\varepsilon _{\rho \varphi }=&{\frac {1}{2}}\left({\frac {1}{\rho }}{\frac {\partial u_{\rho }}{\partial \varphi }}+{\frac {\partial u_{\varphi }}{\partial \rho }}-{\frac {u_{\varphi }}{\rho }}\right)\\\varepsilon _{\varphi z}=&{\frac {1}{2}}\left({\frac {\partial u_{\varphi }}{\partial z}}+{\frac {1}{\rho }}{\frac {\partial u_{z}}{\partial \varphi }}\right)\\\varepsilon _{\rho z}=&{\frac {1}{2}}\left({\frac {\partial u_{\rho }}{\partial z}}+{\frac {\partial u_{z}}{\partial \rho }}\right)\end{aligned}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f3274f2187902965697fa34935275f9a6915ae9e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -12.049ex; margin-bottom: -0.289ex; width:53.581ex; height:25.843ex;" alt="{\displaystyle {\begin{aligned}\varepsilon _{\rho \rho }=&{\frac {\partial u_{\rho }}{\partial \rho }}\qquad \varepsilon _{\varphi \varphi }={\frac {1}{\rho }}\left(u_{\rho }+{\frac {\partial u_{\varphi }}{\partial \varphi }}\right)\qquad \varepsilon _{zz}={\frac {\partial u_{z}}{\partial z}}\\\varepsilon _{\rho \varphi }=&{\frac {1}{2}}\left({\frac {1}{\rho }}{\frac {\partial u_{\rho }}{\partial \varphi }}+{\frac {\partial u_{\varphi }}{\partial \rho }}-{\frac {u_{\varphi }}{\rho }}\right)\\\varepsilon _{\varphi z}=&{\frac {1}{2}}\left({\frac {\partial u_{\varphi }}{\partial z}}+{\frac {1}{\rho }}{\frac {\partial u_{z}}{\partial \varphi }}\right)\\\varepsilon _{\rho z}=&{\frac {1}{2}}\left({\frac {\partial u_{\rho }}{\partial z}}+{\frac {\partial u_{z}}{\partial \rho }}\right)\end{aligned}}}" loading="lazy"></dd></dl>

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

In <a href="Kugelkoordinaten" title="Kugelkoordinaten">Kugelkoordinaten</a> mit Abstand <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0d1ecb613aa2984f0576f70f86650b7c2a132538.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}" loading="lazy"> vom Ursprung, <a href="Zenitwinkel" class="mw-redirect" title="Zenitwinkel">Zenitwinkel</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \vartheta }">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d00eaf197c35bbfa391b9477490a4af955416837.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.374ex; height:2.176ex;" alt="{\displaystyle \vartheta }" loading="lazy"> und <a href="Azimut" title="Azimut">Azimut</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi }">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }" loading="lazy"> lauten der Verzerrungstensor<a href="#cite_note-sadd-2">[2]</a><a href="#cite_note-Landau-4-3">[3]</a>

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\varepsilon _{rr}=&{\frac {\partial u_{r}}{\partial r}}\qquad \varepsilon _{\vartheta \vartheta }={\frac {1}{r}}\left(u_{r}+{\frac {\partial u_{\vartheta }}{\partial \vartheta }}\right)\\\varepsilon _{\varphi \varphi }=&{\frac {1}{r\sin \vartheta }}\left({\frac {\partial u_{\varphi }}{\partial \varphi }}+\sin \vartheta \,u_{r}+\cos \vartheta \,u_{\vartheta }\right)={\frac {1}{r\sin \vartheta }}{\frac {\partial u_{\varphi }}{\partial \varphi }}+{\frac {u_{\vartheta }}{r}}\cot \vartheta +{\frac {u_{r}}{r}}\\\varepsilon _{r\vartheta }=&{\frac {1}{2}}\left({\frac {1}{r}}{\frac {\partial u_{r}}{\partial \vartheta }}+{\frac {\partial u_{\vartheta }}{\partial r}}-{\frac {u_{\vartheta }}{r}}\right)\\\varepsilon _{\vartheta \varphi }=&{\frac {1}{2r}}\left({\frac {1}{\sin \vartheta }}{\frac {\partial u_{\vartheta }}{\partial \varphi }}+{\frac {\partial u_{\varphi }}{\partial \vartheta }}-\cot \vartheta \;u_{\varphi }\right)\\\varepsilon _{r\varphi }=&{\frac {1}{2}}\left({\frac {1}{r\sin \vartheta }}{\frac {\partial u_{r}}{\partial \varphi }}+{\frac {\partial u_{\varphi }}{\partial r}}-{\frac {u_{\varphi }}{r}}\right)\end{aligned}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/aab5fc4b0dcb25be1d77d1a150b3cf6422e132ad.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -15.081ex; margin-bottom: -0.257ex; width:75.154ex; height:31.843ex;" alt="{\displaystyle {\begin{aligned}\varepsilon _{rr}=&{\frac {\partial u_{r}}{\partial r}}\qquad \varepsilon _{\vartheta \vartheta }={\frac {1}{r}}\left(u_{r}+{\frac {\partial u_{\vartheta }}{\partial \vartheta }}\right)\\\varepsilon _{\varphi \varphi }=&{\frac {1}{r\sin \vartheta }}\left({\frac {\partial u_{\varphi }}{\partial \varphi }}+\sin \vartheta \,u_{r}+\cos \vartheta \,u_{\vartheta }\right)={\frac {1}{r\sin \vartheta }}{\frac {\partial u_{\varphi }}{\partial \varphi }}+{\frac {u_{\vartheta }}{r}}\cot \vartheta +{\frac {u_{r}}{r}}\\\varepsilon _{r\vartheta }=&{\frac {1}{2}}\left({\frac {1}{r}}{\frac {\partial u_{r}}{\partial \vartheta }}+{\frac {\partial u_{\vartheta }}{\partial r}}-{\frac {u_{\vartheta }}{r}}\right)\\\varepsilon _{\vartheta \varphi }=&{\frac {1}{2r}}\left({\frac {1}{\sin \vartheta }}{\frac {\partial u_{\vartheta }}{\partial \varphi }}+{\frac {\partial u_{\varphi }}{\partial \vartheta }}-\cot \vartheta \;u_{\varphi }\right)\\\varepsilon _{r\varphi }=&{\frac {1}{2}}\left({\frac {1}{r\sin \vartheta }}{\frac {\partial u_{r}}{\partial \varphi }}+{\frac {\partial u_{\varphi }}{\partial r}}-{\frac {u_{\varphi }}{r}}\right)\end{aligned}}}" loading="lazy"></dd></dl>

mit den Winkelfunktionen <a href="Sinus_und_Cosinus" class="mw-redirect" title="Sinus und Cosinus">Sinus sin</a> und <a href="Tangens_und_Kotangens" title="Tangens und Kotangens">Kotangens cot</a>.

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

Ein Tensor E ist ein geeignetes Verzerrungsmaß, wenn er drei Forderungen genügt<a href="#cite_note-4">[4]</a>:

<ol><li>E verschwindet bei Starrkörperbewegungen (Verschiebung und/oder Drehung ohne Formänderung)</li>

<li>E ist eine monotone, stetige und stetig differenzierbare Funktion des Verschiebungsgradienten H und</li>

<li>E geht bei kleinen Verzerrungen in den linearisierten Verzerrungstensor ε über.</li></ol>

Die <a href="Polarzerlegung" title="Polarzerlegung">Polarzerlegung</a> des Deformationsgradienten F = R · U = v · R spaltet die Verformung lokal in eine reine Drehung, vermittelt durch den <a href="Orthogonaler_Tensor" title="Orthogonaler Tensor">orthogonalen</a> Rotationstensor R (mit R · RT und der <a href="Determinante" title="Determinante">Determinante</a> det(R) = 1), und eine reine Streckung, vermittelt durch die <a href="Symmetrische_Matrix" title="Symmetrische Matrix">symmetrischen</a> <a href="Definitheit" title="Definitheit">positiv definiten</a> rechten bzw. linken Strecktensoren U bzw. v. Letztere dienen der Definition einer Vielzahl von Verzerrungstensoren.

In seiner natürlichen Darstellung in <a href="Konvektive_Koordinaten" title="Konvektive Koordinaten">konvektiven Koordinaten</a> ist der rechte Strecktensor U kovariant und der linke Strecktensor v kontravariant. Diese Eigenschaft überträgt sich auf die mit ihnen gebildeten Verzerrungstensoren. Durch Invertierung werden kovariante Tensoren kontravariant und umgekehrt.

<div class="mw-heading mw-heading3"><h3 id="Seth-Hill-Familie_von_Verzerrungstensoren">Seth-Hill-Familie von Verzerrungstensoren</h3></div>

Die Verzerrungstensoren

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cd02201edb750a79fbf2364dd1381a2991ab8949.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:39.119ex; height:5.176ex;" alt="{\displaystyle \mathbf {E} _{(m)}={\frac {1}{2m}}(\mathbf {U} ^{2m}-\mathbf {1} )={\frac {1}{2m}}(\mathbf {C} ^{m}-\mathbf {1} )}" loading="lazy"></dd></dl>

und

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/039039c740630ae13f2abd8cf96e6ab4124cd01a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:38.529ex; height:5.176ex;" alt="{\displaystyle \mathbf {e} _{(m)}={\frac {1}{2m}}(\mathbf {v} ^{2m}-\mathbf {1} )={\frac {1}{2m}}(\mathbf {b} ^{m}-\mathbf {1} )\,,}" loading="lazy"></dd></dl>

die sich für verschiedene Werte des Parameters <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0a07d98bb302f3856cbabc47b2b9016692e3f7bc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}" loading="lazy"> ergeben, genügen den Bedingungen der allgemeinen Definition<a href="#cite_note-5">[5]</a>. Die einigen gebräuchlichen Werten von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}">

</mstyle>

</mrow>

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

</semantics>

<th>Verzerrungstensor</th>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/953b15a08bf405d8293ebe6185454b0e8efcc54f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.872ex; height:2.176ex;" alt="{\displaystyle \nabla \Delta }" loading="lazy"></th>

<th>Namen<a href="#cite_note-Bertram-6">[6]</a><a href="#cite_note-einheit-7">[7]</a><a href="#cite_note-altenbach-8">[8]</a>

</th></tr>

<tr>

<msub>

</mrow>

</mrow>

</msub>

<mfrac>

</mfrac>

</mrow>

<msup>

</mrow>

</mrow>

</msup>

</mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/83964e104f452fd6c117b737440d63de897b9b6d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:31.569ex; height:5.176ex;" alt="{\displaystyle \mathbf {E} _{(1)}={\frac {1}{2}}(\mathbf {U} ^{2}-\mathbf {1} )={\frac {1}{2}}(\mathbf {C-1} )}" loading="lazy">

</td>

</mstyle>

</mrow>

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

</semantics>

</td>

<td>Green-Lagrange-Verzerrungstensor, Green- oder St.-Venant-Dehnungen

</td></tr>

<tr>

<td>½

</td>

<msub>

</mrow>

</mrow>

</mrow>

</msub>

</mrow>

</mrow>

<msqrt>

</mrow>

</msqrt>

</mrow>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fb21738fdf3f23efa18795e58d4971c9c603de22.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:26.209ex; height:3.509ex;" alt="{\displaystyle \mathbf {E} _{(1/2)}=\mathbf {U} -\mathbf {1} ={\sqrt {\mathbf {C} }}-\mathbf {1} }" loading="lazy">

</td>

</mstyle>

</mrow>

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

</semantics>

</td>

<td>Biot-Verzerrungstensor, Materieller Biot-, Cauchy- oder Swainger-Verzerrungstensor

</td></tr>

<tr>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {E} _{(0)}=\ln(\mathbf {U} )={\frac {1}{2}}\ln(\mathbf {C} )}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b125930cb8607f1f3b1a7a8643d80f2452d85765.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:24.158ex; height:5.176ex;" alt="{\displaystyle \mathbf {E} _{(0)}=\ln(\mathbf {U} )={\frac {1}{2}}\ln(\mathbf {C} )}" loading="lazy">

</td>

</mstyle>

</mrow>

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

</semantics>

</td>

<td><a href="Dehnung#Logarithmische_Dehnung" title="Dehnung">Hencky-Dehnungen</a>, materielle logarithmische Dehnungen

</td></tr>

<tr>

<msub>

</mrow>

</mrow>

</msub>

<mfrac>

</mfrac>

</mrow>

</mrow>

<msup>

</mrow>

</mrow>

</msup>

<mfrac>

</mfrac>

</mrow>

</mrow>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/971a289144cb608ac54a27a951a36b547c421705.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:36.189ex; height:5.176ex;" alt="{\displaystyle \mathbf {E} _{(-1)}={\frac {1}{2}}(\mathbf {1} -\mathbf {U} ^{-2})={\frac {1}{2}}(\mathbf {1} -\mathbf {C} ^{-1})}" loading="lazy">

</td>

</mstyle>

</mrow>

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

</semantics>

</td>

<td>negativer Piola-Verzerrungstensor, Lagrange-Karni-Reiner-Verzerrungstensor

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

Die hier benutzten Namen stehen jeweils kursiv hervorgehoben an erster Stelle. In der räumlichen Beschreibung ergeben sich die Entsprechungen:

<th>Verzerrungstensor</th>

</mstyle>

</mrow>

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

</semantics>

<th>Namen<a href="#cite_note-Bertram-6">[6]</a><a href="#cite_note-einheit-7">[7]</a><a href="#cite_note-altenbach-8">[8]</a>

</th></tr>

<tr>

<msub>

</mrow>

</mrow>

</msub>

<mfrac>

</mfrac>

</mrow>

<msup>

</mrow>

</mrow>

</msup>

</mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/67456ca80fc711d7b01ef121c2ccb65c1ebcc6be.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:29.676ex; height:5.176ex;" alt="{\displaystyle \mathbf {e} _{(1)}={\frac {1}{2}}(\mathbf {v} ^{2}-\mathbf {1} )={\frac {1}{2}}(\mathbf {b} -\mathbf {1} )}" loading="lazy">

</td>

</mstyle>

</mrow>

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

</semantics>

</td>

<td>negativer Finger-Tensor, Euler-Karni-Reiner-Verzerrungstensor

</td></tr>

<tr>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {e} _{(0)}=\ln(\mathbf {v} )={\frac {1}{2}}\ln(\mathbf {b} )}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/51d8e53aa6335d7973f2f1b3b4fc0a9c8e4e28de.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:22.535ex; height:5.176ex;" alt="{\displaystyle \mathbf {e} _{(0)}=\ln(\mathbf {v} )={\frac {1}{2}}\ln(\mathbf {b} )}" loading="lazy">

</td>

</mstyle>

</mrow>

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

</semantics>

</td>

<td>Räumliche Hencky-Dehnungen, räumliche logarithmische Dehnungen

</td></tr>

<tr>

<msub>

</mrow>

</mrow>

</mrow>

</msub>

<msup>

</mrow>

</mrow>

</msup>

</mrow>

<msup>

<msqrt>

</mrow>

</msqrt>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {e} _{(-1/2)}=\mathbf {1-v} ^{-1}=\mathbf {1} -{\sqrt {\mathbf {b} }}^{\,-1}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/62bfef33919fd248f536a3b29183d2c95d4fba39.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:31.187ex; height:4.009ex;" alt="{\displaystyle \mathbf {e} _{(-1/2)}=\mathbf {1-v} ^{-1}=\mathbf {1} -{\sqrt {\mathbf {b} }}^{\,-1}}" loading="lazy">

</td>

</mstyle>

</mrow>

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

</semantics>

</td>

<td>Swainger-Verzerrungstensor, räumlicher Biot-Verzerrungstensor

</td></tr>

<tr>

<msub>

</mrow>

</mrow>

</msub>

<mfrac>

</mfrac>

</mrow>

<msup>

</mrow>

</mrow>

</msup>

<mfrac>

</mfrac>

</mrow>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f07bf55c4bee838abce41679c677510d27977f73.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:35.105ex; height:5.176ex;" alt="{\displaystyle \mathbf {e} _{(-1)}={\frac {1}{2}}(\mathbf {1-v} ^{-2})={\frac {1}{2}}(\mathbf {1-b} ^{-1})}" loading="lazy">

</td>

</mstyle>

</mrow>

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

</semantics>

</td>

<td>Euler-Almansi-Verzerrungstensor, Almansis- oder Hamels-Verzerrungstensor

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

In den Tabellen bedeutet „<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/32769037c408874e1890f77554c65f39c523ebe2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.936ex; height:2.176ex;" alt="{\displaystyle \Delta }" loading="lazy">“ Kovarianz und „<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nabla }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a3d0e93b78c50237f9ea83d027e4ebbdaef354b2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.936ex; height:2.176ex;" alt="{\displaystyle \nabla }" loading="lazy">“ Kontravarianz. Der Funktionswert eines Tensors (z. B. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {\mathbf {C} }}\,,\;\ln(\mathbf {C} )}">

<msqrt>

</mrow>

</msqrt>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\sqrt {\mathbf {C} }}\,,\;\ln(\mathbf {C} )}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b38ec47f22ac38f89ad791f4e9ba85218816f5c1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.613ex; height:3.176ex;" alt="{\displaystyle {\sqrt {\mathbf {C} }}\,,\;\ln(\mathbf {C} )}" loading="lazy">) berechnet sich durch <a href="Hauptachsentransformation" title="Hauptachsentransformation">Hauptachsentransformation</a>, Bildung der Funktionswerte der Diagonalelemente und Rücktransformation.

<div class="mw-heading mw-heading2"><h2 id="Beschreibung_einiger_Verzerrungstensoren">Beschreibung einiger Verzerrungstensoren</h2></div>

Weil die Verzerrungstensoren der Seth-Hill-Familie bei kleinen Verzerrungen in den linearisierten Verzerrungstensor übergehen, trifft das hier gesagte bei kleinen Verzerrungen auch auf den linearisierten Verzerrungstensor zu.

<div class="mw-heading mw-heading3"><h3 id="Der_Green-Lagrange-Verzerrungstensor">Der Green-Lagrange-Verzerrungstensor</h3></div>

Der Green-Lagrange-Verzerrungstensor ist aus dem Vergleich zweier materieller Linienelemente <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {d} {\vec {X}}}">

</mrow>

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1e78464ce193ac66c8337573f8f0117d64ea0cdc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.273ex; height:2.843ex;" alt="{\displaystyle \mathrm {d} {\vec {X}}}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {d} {\vec {Y}}}">

</mrow>

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5199ff63b04de5e8f2f0d8178dd43df2156d1405.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.066ex; height:2.843ex;" alt="{\displaystyle \mathrm {d} {\vec {Y}}}" loading="lazy"> im Punkt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {X}}}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</mrow>

<mover>

</mover>

</mrow>

</mrow>

<mover>

</mover>

</mrow>

</mrow>

<mover>

</mover>

</mrow>

</mrow>

<mover>

</mover>

</mrow>

</mrow>

</mrow>

<mover>

</mover>

</mrow>

</mrow>

</mrow>

<mover>

</mover>

</mrow>

</mrow>

<mover>

</mover>

</mrow>

</mrow>

<mover>

</mover>

</mrow>

</mrow>

<mover>

</mover>

</mrow>

</mrow>

</mrow>

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {d} {\vec {x}}\cdot \mathrm {d} {\vec {y}}-\mathrm {d} {\vec {X}}\cdot \mathrm {d} {\vec {Y}}=(\mathbf {F} \cdot \mathrm {d} {\vec {X}})\cdot (\mathbf {F} \cdot \mathrm {d} {\vec {Y}})-\mathrm {d} {\vec {X}}\cdot \mathrm {d} {\vec {Y}}=2\mathrm {d} {\vec {X}}\cdot \mathbf {E} \cdot \mathrm {d} {\vec {Y}}\,.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a7a698dd87dcf3efb5b83004ef051c1e0373b950.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:66.781ex; height:3.343ex;" alt="{\displaystyle \mathrm {d} {\vec {x}}\cdot \mathrm {d} {\vec {y}}-\mathrm {d} {\vec {X}}\cdot \mathrm {d} {\vec {Y}}=(\mathbf {F} \cdot \mathrm {d} {\vec {X}})\cdot (\mathbf {F} \cdot \mathrm {d} {\vec {Y}})-\mathrm {d} {\vec {X}}\cdot \mathrm {d} {\vec {Y}}=2\mathrm {d} {\vec {X}}\cdot \mathbf {E} \cdot \mathrm {d} {\vec {Y}}\,.}" loading="lazy"></dd></dl>

In einer Richtung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {e}}_{1}={\tfrac {\mathrm {d} {\vec {X}}}{|\mathrm {d} {\vec {X}}|}}}">

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<mfrac>

<mrow>

</mrow>

<mover>

</mover>

</mrow>

<mrow>

</mrow>

</mrow>

<mover>

</mover>

</mrow>

</mrow>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\vec {e}}_{1}={\tfrac {\mathrm {d} {\vec {X}}}{|\mathrm {d} {\vec {X}}|}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/14f2f7a8271a1624c02d2016d1f4f28e0a321e28.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:9.441ex; height:5.176ex;" alt="{\displaystyle {\vec {e}}_{1}={\tfrac {\mathrm {d} {\vec {X}}}{|\mathrm {d} {\vec {X}}|}}}" loading="lazy"> ergibt sich über

<mtr>

<mtd>

</mrow>

<mover>

</mover>

</mrow>

</mrow>

<mover>

</mover>

</mrow>

</mtd>

<mtd>

</mrow>

<mover>

</mover>

</mrow>

</mrow>

</mrow>

<mover>

</mover>

</mrow>

</mrow>

<mover>

</mover>

</mrow>

</mrow>

<mover>

</mover>

</mrow>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

</mrow>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

</mrow>

<mover>

</mover>

</mrow>

</mrow>

<mover>

</mover>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

</mrow>

</mrow>

<mover>

</mover>

</mrow>

</mrow>

</mtd>

<mtd>

<msqrt>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

</mrow>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

</msqrt>

</mrow>

</mrow>

</mrow>

<mover>

</mover>

</mrow>

</mrow>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\mathrm {d} {\vec {x}}\cdot \mathrm {d} {\vec {x}}=&2\mathrm {d} {\vec {X}}\cdot \mathbf {E} \cdot \mathrm {d} {\vec {X}}+\mathrm {d} {\vec {X}}\cdot \mathrm {d} {\vec {X}}=(2{\vec {e}}_{1}\cdot \mathbf {E} \cdot {\vec {e}}_{1}+1)(\mathrm {d} {\vec {X}}\cdot \mathrm {d} {\vec {X}})\\\rightarrow \quad |\mathrm {d} {\vec {x}}|=&{\sqrt {2{\vec {e}}_{1}\cdot \mathbf {E} \cdot {\vec {e}}_{1}+1}}\;|\mathrm {d} {\vec {X}}|\end{aligned}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5cad63705a01270f92e05bc880f50ae74b924e97.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:66.076ex; height:7.176ex;" alt="{\displaystyle {\begin{aligned}\mathrm {d} {\vec {x}}\cdot \mathrm {d} {\vec {x}}=&2\mathrm {d} {\vec {X}}\cdot \mathbf {E} \cdot \mathrm {d} {\vec {X}}+\mathrm {d} {\vec {X}}\cdot \mathrm {d} {\vec {X}}=(2{\vec {e}}_{1}\cdot \mathbf {E} \cdot {\vec {e}}_{1}+1)(\mathrm {d} {\vec {X}}\cdot \mathrm {d} {\vec {X}})\\\rightarrow \quad |\mathrm {d} {\vec {x}}|=&{\sqrt {2{\vec {e}}_{1}\cdot \mathbf {E} \cdot {\vec {e}}_{1}+1}}\;|\mathrm {d} {\vec {X}}|\end{aligned}}}" loading="lazy"></dd></dl>

die <a href="Dehnung" title="Dehnung">Dehnung</a>:

<mfrac>

<mrow>

</mrow>

</mrow>

<mover>

</mover>

</mrow>

</mrow>

</mrow>

</mrow>

<mover>

</mover>

</mrow>

</mrow>

<mrow>

</mrow>

</mrow>

<mover>

</mover>

</mrow>

</mrow>

</mfrac>

</mrow>

<msqrt>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

</mrow>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

</msqrt>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varepsilon :={\frac {|\mathrm {d} {\vec {x}}|-|\mathrm {d} {\vec {X}}|}{|\mathrm {d} {\vec {X}}|}}={\sqrt {1+2{\vec {e}}_{1}\cdot \mathbf {E} \cdot {\vec {e}}_{1}}}-1}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/170d662265561ef2dda7f884064b2cb4b2a8a580.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:41.248ex; height:7.509ex;" alt="{\displaystyle \varepsilon :={\frac {|\mathrm {d} {\vec {x}}|-|\mathrm {d} {\vec {X}}|}{|\mathrm {d} {\vec {X}}|}}={\sqrt {1+2{\vec {e}}_{1}\cdot \mathbf {E} \cdot {\vec {e}}_{1}}}-1}" loading="lazy"></dd></dl>

Wenn in der Ausgangskonfiguration <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {d} {\vec {X}}\cdot \mathrm {d} {\vec {Y}}=0}">

</mrow>

<mover>

</mover>

</mrow>

</mrow>

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {d} {\vec {X}}\cdot \mathrm {d} {\vec {Y}}=0}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/44f4f70e3cacbe287d086530e6afbf0f69d5886a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:12.278ex; height:2.843ex;" alt="{\displaystyle \mathrm {d} {\vec {X}}\cdot \mathrm {d} {\vec {Y}}=0}" loading="lazy"> ist, berechnet sich:

<mtr>

<mtd>

</mrow>

<mover>

</mover>

</mrow>

</mrow>

<mover>

</mover>

</mrow>

</mtd>

<mtd>

</mrow>

<mover>

</mover>

</mrow>

</mrow>

</mrow>

<mover>

</mover>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

<mfrac>

<mrow>

</mrow>

<mover>

</mover>

</mrow>

</mrow>

<mover>

</mover>

</mrow>

<mrow>

</mrow>

</mrow>

<mover>

</mover>

</mrow>

</mrow>

</mrow>

</mrow>

<mover>

</mover>

</mrow>

</mrow>

</mfrac>

</mrow>

</mtd>

<mtd>

<mfrac>

<mrow>

</mrow>

<mover>

</mover>

</mrow>

</mrow>

</mrow>

<mover>

</mover>

</mrow>

<mrow>

</mrow>

</mrow>

<mover>

</mover>

</mrow>

</mrow>

</mrow>

</mrow>

<mover>

</mover>

</mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

<mover>

</mover>

</mrow>

</mrow>

</mrow>

<mover>

</mover>

</mrow>

<mrow>

<msqrt>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

</mrow>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

</msqrt>

</mrow>

</mrow>

</mrow>

<mover>

</mover>

</mrow>

</mrow>

<msqrt>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

</mrow>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

</msqrt>

</mrow>

</mrow>

</mrow>

<mover>

</mover>

</mrow>

</mrow>

</mfrac>

</mrow>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\mathrm {d} {\vec {x}}\cdot \mathrm {d} {\vec {y}}=&2\mathrm {d} {\vec {X}}\cdot \mathbf {E} \cdot \mathrm {d} {\vec {Y}}\\{\frac {\mathrm {d} {\vec {x}}\cdot \mathrm {d} {\vec {y}}}{|\mathrm {d} {\vec {x}}||\mathrm {d} {\vec {y}}|}}=&{\frac {2\mathrm {d} {\vec {X}}\cdot \mathbf {E} \cdot \mathrm {d} {\vec {Y}}}{|\mathrm {d} {\vec {x}}||\mathrm {d} {\vec {y}}|}}={\frac {2\mathrm {d} {\vec {X}}\cdot \mathbf {E} \cdot \mathrm {d} {\vec {Y}}}{{\sqrt {2{\vec {e}}_{1}\cdot \mathbf {E} \cdot {\vec {e}}_{1}+1}}\;|\mathrm {d} {\vec {X}}|\;{\sqrt {2{\vec {e}}_{2}\cdot \mathbf {E} \cdot {\vec {e}}_{2}+1}}\;|\mathrm {d} {\vec {Y}}|}}\end{aligned}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/345948ee1cde387187614f2f0f8a7d896d0f31d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.838ex; width:74.373ex; height:10.843ex;" alt="{\displaystyle {\begin{aligned}\mathrm {d} {\vec {x}}\cdot \mathrm {d} {\vec {y}}=&2\mathrm {d} {\vec {X}}\cdot \mathbf {E} \cdot \mathrm {d} {\vec {Y}}\\{\frac {\mathrm {d} {\vec {x}}\cdot \mathrm {d} {\vec {y}}}{|\mathrm {d} {\vec {x}}||\mathrm {d} {\vec {y}}|}}=&{\frac {2\mathrm {d} {\vec {X}}\cdot \mathbf {E} \cdot \mathrm {d} {\vec {Y}}}{|\mathrm {d} {\vec {x}}||\mathrm {d} {\vec {y}}|}}={\frac {2\mathrm {d} {\vec {X}}\cdot \mathbf {E} \cdot \mathrm {d} {\vec {Y}}}{{\sqrt {2{\vec {e}}_{1}\cdot \mathbf {E} \cdot {\vec {e}}_{1}+1}}\;|\mathrm {d} {\vec {X}}|\;{\sqrt {2{\vec {e}}_{2}\cdot \mathbf {E} \cdot {\vec {e}}_{2}+1}}\;|\mathrm {d} {\vec {Y}}|}}\end{aligned}}}" loading="lazy"></dd></dl>

Mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {e}}_{2}={\tfrac {\mathrm {d} {\vec {Y}}}{|\mathrm {d} {\vec {Y}}|}}}">

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<mfrac>

<mrow>

</mrow>

<mover>

</mover>

</mrow>

<mrow>

</mrow>

</mrow>

<mover>

</mover>

</mrow>

</mrow>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\vec {e}}_{2}={\tfrac {\mathrm {d} {\vec {Y}}}{|\mathrm {d} {\vec {Y}}|}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9b280a71556702de9e425e0d4c97416c6eb8d414.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:9.295ex; height:5.176ex;" alt="{\displaystyle {\vec {e}}_{2}={\tfrac {\mathrm {d} {\vec {Y}}}{|\mathrm {d} {\vec {Y}}|}}}" loading="lazy"> resultiert für die <a href="Scherung_(Mechanik)" title="Scherung (Mechanik)">Scherung</a> γ dann:

<mfrac>

<mrow>

</mrow>

<mover>

</mover>

</mrow>

</mrow>

<mover>

</mover>

</mrow>

<mrow>

</mrow>

</mrow>

<mover>

</mover>

</mrow>

</mrow>

</mrow>

</mrow>

<mover>

</mover>

</mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

</mrow>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

</mrow>

<mrow>

<msqrt>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

</mrow>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

</msqrt>

</mrow>

<msqrt>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

</mrow>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

</msqrt>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sin(\gamma ):={\frac {\mathrm {d} {\vec {x}}\cdot \mathrm {d} {\vec {y}}}{|\mathrm {d} {\vec {x}}||\mathrm {d} {\vec {y}}|}}={\frac {2{\vec {e}}_{1}\cdot \mathbf {E} \cdot {\vec {e}}_{2}}{{\sqrt {1+2{\vec {e}}_{1}\cdot \mathbf {E} \cdot {\vec {e}}_{1}}}{\sqrt {1+2{\vec {e}}_{2}\cdot \mathbf {E} \cdot {\vec {e}}_{2}}}}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/731f5596d06a63a199b242e1c8cd821ef71786de.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:56.528ex; height:6.843ex;" alt="{\displaystyle \sin(\gamma ):={\frac {\mathrm {d} {\vec {x}}\cdot \mathrm {d} {\vec {y}}}{|\mathrm {d} {\vec {x}}||\mathrm {d} {\vec {y}}|}}={\frac {2{\vec {e}}_{1}\cdot \mathbf {E} \cdot {\vec {e}}_{2}}{{\sqrt {1+2{\vec {e}}_{1}\cdot \mathbf {E} \cdot {\vec {e}}_{1}}}{\sqrt {1+2{\vec {e}}_{2}\cdot \mathbf {E} \cdot {\vec {e}}_{2}}}}}}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Der_Euler-Almansi-Verzerrungstensor">Der Euler-Almansi-Verzerrungstensor</h3></div>

Der Euler-Almansi-Verzerrungstensor

</mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

<msup>

</mrow>

</mrow>

</msup>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a6d38d217de80e87773317eb2847737485ce3649.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:23.297ex; height:5.176ex;" alt="{\displaystyle \mathbf {e} ={\frac {1}{2}}(\mathbf {1} -\mathbf {F} ^{\top -1}\cdot \mathbf {F} ^{-1})}" loading="lazy"></dd></dl>

kann analog zum Green-Lagrange-Verzerrungstensor aus dem Vergleich zweier materieller Linienelemente <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {d} {\vec {x}}}">

</mrow>

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/da40a7b7c1a44346cdd7e57c5aeddbef55dcacdb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.622ex; height:2.343ex;" alt="{\displaystyle \mathrm {d} {\vec {x}}}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {d} {\vec {y}}}">

</mrow>

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a0c7179cd5a46e2ee0ce8b7e74ebeebb180e2f8a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.557ex; height:2.676ex;" alt="{\displaystyle \mathrm {d} {\vec {y}}}" loading="lazy"> im Punkt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {x}}}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><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"> motiviert werden:

</mrow>

<mover>

</mover>

</mrow>

</mrow>

<mover>

</mover>

</mrow>

</mrow>

<mover>

</mover>

</mrow>

</mrow>

<mover>

</mover>

</mrow>

</mrow>

<mover>

</mover>

</mrow>

</mrow>

<mover>

</mover>

</mrow>

<msup>

</mrow>

</mrow>

</msup>

</mrow>

<mover>

</mover>

</mrow>

<msup>

</mrow>

</mrow>

</msup>

</mrow>

<mover>

</mover>

</mrow>

</mrow>

<mover>

</mover>

</mrow>

</mrow>

</mrow>

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {d} {\vec {x}}\cdot \mathrm {d} {\vec {y}}-\mathrm {d} {\vec {X}}\cdot \mathrm {d} {\vec {Y}}=\mathrm {d} {\vec {x}}\cdot \mathrm {d} {\vec {y}}-(\mathbf {F} ^{-1}\cdot \mathrm {d} {\vec {x}})\cdot (\mathbf {F} ^{-1}\cdot \mathrm {d} {\vec {y}})=2\mathrm {d} {\vec {x}}\cdot \mathbf {e} \cdot \mathrm {d} {\vec {y}}\,.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/99d98f355381b90b0614e82ada15dee9461ffb69.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:67.437ex; height:3.343ex;" alt="{\displaystyle \mathrm {d} {\vec {x}}\cdot \mathrm {d} {\vec {y}}-\mathrm {d} {\vec {X}}\cdot \mathrm {d} {\vec {Y}}=\mathrm {d} {\vec {x}}\cdot \mathrm {d} {\vec {y}}-(\mathbf {F} ^{-1}\cdot \mathrm {d} {\vec {x}})\cdot (\mathbf {F} ^{-1}\cdot \mathrm {d} {\vec {y}})=2\mathrm {d} {\vec {x}}\cdot \mathbf {e} \cdot \mathrm {d} {\vec {y}}\,.}" loading="lazy"></dd></dl>

Für die Dehnung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a30c89172e5b88edbd45d3e2772c7f5e562e5173.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.083ex; height:1.676ex;" alt="{\displaystyle \varepsilon }" loading="lazy"> in eine Richtung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {e}}_{1}}">

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\vec {e}}_{1}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bad1143107d7fbfc245bd9fcdcc422d341ec852e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.277ex; height:2.676ex;" alt="{\displaystyle {\vec {e}}_{1}}" loading="lazy"> ergibt sich dann:

<mfrac>

<msqrt>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

</mrow>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

</msqrt>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varepsilon ={\frac {1}{\sqrt {1-2{\vec {e}}_{1}\cdot \mathbf {e} \cdot {\vec {e}}_{1}}}}-1}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4452d7bcb7df6d3bdbd2e588624d5e9a5a68c327.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:25.648ex; height:6.509ex;" alt="{\displaystyle \varepsilon ={\frac {1}{\sqrt {1-2{\vec {e}}_{1}\cdot \mathbf {e} \cdot {\vec {e}}_{1}}}}-1}" loading="lazy"> mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {e}}_{1}={\tfrac {\mathrm {d} {\vec {x}}}{|\mathrm {d} {\vec {x}}|}}\,.}">

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<mfrac>

<mrow>

</mrow>

<mover>

</mover>

</mrow>

<mrow>

</mrow>

</mrow>

<mover>

</mover>

</mrow>

</mrow>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\vec {e}}_{1}={\tfrac {\mathrm {d} {\vec {x}}}{|\mathrm {d} {\vec {x}}|}}\,.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5760539c2813d9a61ee05c49b2a4d35ed8b0a57e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:10.015ex; height:5.009ex;" alt="{\displaystyle {\vec {e}}_{1}={\tfrac {\mathrm {d} {\vec {x}}}{|\mathrm {d} {\vec {x}}|}}\,.}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Der_Hencky-Verzerrungstensor">Der Hencky-Verzerrungstensor</h3></div>

Der Hencky-Verzerrungstensor wird über die Hauptachsentransformation des rechten Strecktensors <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>

</mrow>

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {U} =\sum _{i=1}^{3}\lambda _{i}{\hat {v}}_{i}\otimes {\hat {v}}_{i}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/01bcdd9c9630828f982536f93c2e174a8133ffd5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:17.946ex; height:7.176ex;" alt="{\displaystyle \mathbf {U} =\sum _{i=1}^{3}\lambda _{i}{\hat {v}}_{i}\otimes {\hat {v}}_{i}}" loading="lazy"></dd></dl>

wobei λi die sämtlich positiven <a href="Eigenwertproblem" class="mw-redirect" title="Eigenwertproblem">Eigenwerte</a> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {v}}_{i}}">

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2fda848e6a1d7255581b5853435e8319267a42bd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.027ex; height:2.509ex;" alt="{\displaystyle {\hat {v}}_{i}}" loading="lazy"> die auf eins normierten und paarweise orthogonalen Eigenvektoren von <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>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {E} _{H}:=\ln(\mathbf {U} ):=\sum _{i=1}^{3}\ln(\lambda _{i}){\hat {v}}_{i}\otimes {\hat {v}}_{i}\,.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c8754aef5984fc4bb7c081ca6a7131f78fd1917f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:34.318ex; height:7.176ex;" alt="{\displaystyle \mathbf {E} _{H}:=\ln(\mathbf {U} ):=\sum _{i=1}^{3}\ln(\lambda _{i}){\hat {v}}_{i}\otimes {\hat {v}}_{i}\,.}" loading="lazy"></dd></dl>

Seine <a href="Hauptinvariante" title="Hauptinvariante">Spur</a> ist wegen

<msub>

</mrow>

</mrow>

</msub>

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {Sp} (\mathbf {E} _{H})=\sum _{i=1}^{3}\ln(\lambda _{i}){\hat {v}}_{i}\cdot {\hat {v}}_{i}=\ln(\lambda _{1}\lambda _{2}\lambda _{3})=\ln(\operatorname {det} (\mathbf {U} ))=\ln(\operatorname {det} (\mathbf {F} ))}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c3a54bb93aab7519530d137be79902d171e4ebe4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:67.906ex; height:7.176ex;" alt="{\displaystyle \operatorname {Sp} (\mathbf {E} _{H})=\sum _{i=1}^{3}\ln(\lambda _{i}){\hat {v}}_{i}\cdot {\hat {v}}_{i}=\ln(\lambda _{1}\lambda _{2}\lambda _{3})=\ln(\operatorname {det} (\mathbf {U} ))=\ln(\operatorname {det} (\mathbf {F} ))}" loading="lazy"></dd></dl>

ein Maß für die Kompression am Ort. Bei <a href="Geometrische_Linearisierung" title="Geometrische Linearisierung">kleinen Verzerrungen</a> ist

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d4aa1074811eb81928bef480c18be49952da9458.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.777ex; height:2.843ex;" alt="{\displaystyle \ln(\operatorname {det} (\mathbf {F} ))\approx \operatorname {Sp} (\mathbf {H} )=\operatorname {Sp} ({\boldsymbol {\varepsilon }})}" loading="lazy"></dd></dl>

weswegen dann die Spur des Verschiebungsgradienten oder des linearisierten Verzerrungstensors diese Rolle übernimmt.

<div class="mw-heading mw-heading3"><h3 id="Der_Piola-_und_Finger-Verzerrungstensor">Der Piola- und Finger-Verzerrungstensor</h3></div>

Der Piola-Verzerrungstensor <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {E} _{P}=-\mathbf {E} _{(-1)}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/46c5b3987090d18635e9b0950233d62096df70af.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:13.499ex; height:3.009ex;" alt="{\displaystyle \mathbf {E} _{P}=-\mathbf {E} _{(-1)}}" loading="lazy"> ist aus dem Vergleich der <a href="Normalenvektor" title="Normalenvektor">Normalenvektoren</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {N}}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/13e01cc1cec4201098af497311170ea68412c6b7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.843ex;" alt="{\displaystyle {\vec {N}}}" loading="lazy"> an materielle Flächen motiviert. Eine Familie von <a href="Fl%C3%A4che_(Mathematik)" title="Fläche (Mathematik)">Flächen</a> kann durch eine skalare Funktion

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<annotation encoding="application/x-tex">{\displaystyle \Phi ({\vec {X}},t)=C}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cbf612b11d8c56df620676bdcd37d7e8b6897098.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.206ex; height:3.343ex;" alt="{\displaystyle \Phi ({\vec {X}},t)=C}" loading="lazy"></dd></dl>

und einen Flächenparameter <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4fc55753007cd3c18576f7933f6f089196732029.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}" loading="lazy"> definiert werden. Die Normalenvektoren an diese Flächen sind die Gradienten

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<annotation encoding="application/x-tex">{\displaystyle {\vec {N}}:=\operatorname {GRAD} (\Phi )=\sum _{i=1}^{3}{\frac {\mathrm {d} \Phi }{\mathrm {d} X_{i}}}{\vec {e}}_{i}\,.}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d007d4572dacbd27601de8ea2963f6d36ead4c2b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:31.1ex; height:7.176ex;" alt="{\displaystyle {\vec {N}}:=\operatorname {GRAD} (\Phi )=\sum _{i=1}^{3}{\frac {\mathrm {d} \Phi }{\mathrm {d} X_{i}}}{\vec {e}}_{i}\,.}" loading="lazy"></dd></dl>

Im Zuge einer Deformation wird daraus

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\vec {n}}=&\operatorname {grad} (\Phi )=\sum _{i=1}^{3}{\frac {\mathrm {d} \Phi }{\mathrm {d} x_{i}}}{\vec {e}}_{i}=\sum _{i,j=1}^{3}{\frac {\mathrm {d} \Phi }{\mathrm {d} X_{j}}}{\frac {\mathrm {d} X_{j}}{\mathrm {d} x_{i}}}{\vec {e}}_{i}\\=&\sum _{i,j=1}^{3}{\frac {\mathrm {d} X_{j}}{\mathrm {d} x_{i}}}{\vec {e}}_{i}\otimes {\vec {e}}_{j}\cdot {\frac {\mathrm {d} \Phi }{\mathrm {d} X_{k}}}{\vec {e}}_{k}=\mathbf {F} ^{\top -1}\cdot {\vec {N}}.\end{aligned}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/85860dd95b3a207e46e982ed9bce4436576b2f34.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.171ex; width:44.974ex; height:15.509ex;" alt="{\displaystyle {\begin{aligned}{\vec {n}}=&\operatorname {grad} (\Phi )=\sum _{i=1}^{3}{\frac {\mathrm {d} \Phi }{\mathrm {d} x_{i}}}{\vec {e}}_{i}=\sum _{i,j=1}^{3}{\frac {\mathrm {d} \Phi }{\mathrm {d} X_{j}}}{\frac {\mathrm {d} X_{j}}{\mathrm {d} x_{i}}}{\vec {e}}_{i}\\=&\sum _{i,j=1}^{3}{\frac {\mathrm {d} X_{j}}{\mathrm {d} x_{i}}}{\vec {e}}_{i}\otimes {\vec {e}}_{j}\cdot {\frac {\mathrm {d} \Phi }{\mathrm {d} X_{k}}}{\vec {e}}_{k}=\mathbf {F} ^{\top -1}\cdot {\vec {N}}.\end{aligned}}}" loading="lazy"></dd></dl>

Mit einer anderen skalaren Funktion <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Psi ({\vec {X}},t)}">

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<annotation encoding="application/x-tex">{\displaystyle \Psi ({\vec {X}},t)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6b4cacab43539ce18aa7a708e9cd7a151159ecbe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.471ex; height:3.343ex;" alt="{\displaystyle \Psi ({\vec {X}},t)}" loading="lazy"> kann eine andere Familie von Flächen definiert werden, deren Normalenvektoren <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {M}}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b903a0e453efca49cfda6ba33edb465de21a6b1d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.843ex;" alt="{\displaystyle {\vec {M}}}" loading="lazy"> bzw. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {m}}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/24e08685bde682d6b67c3a22b026d16dbbf8dfcc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:2.343ex;" alt="{\displaystyle {\vec {m}}}" loading="lazy"> über

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<annotation encoding="application/x-tex">{\displaystyle {\vec {m}}=\mathbf {F} ^{\top -1}\cdot {\vec {M}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2e0e3d6439f97b8fa0f6fec3f8ee9baa07aab3e5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:14.554ex; height:2.843ex;" alt="{\displaystyle {\vec {m}}=\mathbf {F} ^{\top -1}\cdot {\vec {M}}}" loading="lazy"></dd></dl>

in Beziehung stehen. Der Vergleich der Skalarprodukte der Normalenvektoren in der deformierten und undeformierten Lage in einem materiellen Punkt <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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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\vec {m}}\cdot {\vec {n}}-{\vec {M}}\cdot {\vec {N}}=&(\mathbf {F} ^{\top -1}\cdot {\vec {M}})\cdot (\mathbf {F} ^{\top -1}\cdot {\vec {N}})-{\vec {M}}\cdot {\vec {N}}\\=&2{\vec {M}}\cdot {\frac {1}{2}}(\mathbf {F} ^{-1}\cdot \mathbf {F} ^{\top -1}-\mathbf {1} )\cdot {\vec {N}}=2{\vec {M}}\cdot \mathbf {E} _{P}\cdot {\vec {N}}\,,\end{aligned}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f2902841ce439df537ef47f9dc51a5035f213503.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.57ex; margin-bottom: -0.268ex; width:61.727ex; height:8.843ex;" alt="{\displaystyle {\begin{aligned}{\vec {m}}\cdot {\vec {n}}-{\vec {M}}\cdot {\vec {N}}=&(\mathbf {F} ^{\top -1}\cdot {\vec {M}})\cdot (\mathbf {F} ^{\top -1}\cdot {\vec {N}})-{\vec {M}}\cdot {\vec {N}}\\=&2{\vec {M}}\cdot {\frac {1}{2}}(\mathbf {F} ^{-1}\cdot \mathbf {F} ^{\top -1}-\mathbf {1} )\cdot {\vec {N}}=2{\vec {M}}\cdot \mathbf {E} _{P}\cdot {\vec {N}}\,,\end{aligned}}}" loading="lazy"></dd></dl>

der also ein Maß für die Deformationen der materiellen Flächen ist. Der Piola-Verzerrungstensor operiert in der Ausgangskonfiguration.

Sein Gegenstück in der Momentankonfiguration ist der Finger-Tensor<a href="#cite_note-einheit-7">[7]</a>

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d0a54bf86d4967b03abaf8e6cdf479c772e93573.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:33.543ex; height:5.176ex;" alt="{\displaystyle \mathbf {e} _{F}:={\frac {1}{2}}(\mathbf {1} -\mathbf {F\cdot F} ^{\top })={\frac {1}{2}}(\mathbf {1} -\mathbf {b} )}" loading="lazy"></dd></dl>

für den

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\vec {m}}\cdot {\vec {n}}-{\vec {M}}\cdot {\vec {N}}=&{\vec {m}}\cdot {\vec {n}}-(\mathbf {F} ^{\top }\cdot {\vec {m}})\cdot (\mathbf {F} ^{\top }\cdot {\vec {n}})\\=&2{\vec {m}}\cdot {\frac {1}{2}}(\mathbf {1-F\cdot F} ^{\top })\cdot {\vec {n}}=2{\vec {m}}\cdot \mathbf {e} _{F}\cdot {\vec {n}}\,.\end{aligned}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/db1fabfcb2c8cf7a9ac5a315583d4d5426b1c044.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.57ex; margin-bottom: -0.268ex; width:54.886ex; height:8.843ex;" alt="{\displaystyle {\begin{aligned}{\vec {m}}\cdot {\vec {n}}-{\vec {M}}\cdot {\vec {N}}=&{\vec {m}}\cdot {\vec {n}}-(\mathbf {F} ^{\top }\cdot {\vec {m}})\cdot (\mathbf {F} ^{\top }\cdot {\vec {n}})\\=&2{\vec {m}}\cdot {\frac {1}{2}}(\mathbf {1-F\cdot F} ^{\top })\cdot {\vec {n}}=2{\vec {m}}\cdot \mathbf {e} _{F}\cdot {\vec {n}}\,.\end{aligned}}}" loading="lazy"></dd></dl>

abgeleitet werden kann.

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

<div class="hauptartikel" role="navigation">→ Hauptartikel: <a href="Geschwindigkeitsgradient" title="Geschwindigkeitsgradient">Geschwindigkeitsgradient</a></div>

Alle realen Materialien sind mehr oder weniger ratenabhängig, das heißt ihr Widerstand gegen eine Deformation hängt davon ab, mit welcher Geschwindigkeit diese Deformation herbeigeführt wird. Für die Beschreibung eines solchen Zusammenhangs werden Verzerrungsgeschwindigkeiten benutzt. Das Materialverhalten ist <a href="Euklidische_Transformation" title="Euklidische Transformation">beobachterinvariant</a>, die meisten Zeitableitungen der Verzerrungen jedoch nicht. Es wurden aber eine Reihe von Verzerrungsgeschwindigkeiten definiert, die beobachterinvariant sind.

Der rechte Strecktensor <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {U} }">

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

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</math><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"> ist körperbezogen objektiv, was bedeutet, dass er von einem Wechsel des Bezugssystems unbeeinflusst ist. Gleiches gilt auch für seine <a href="Substantielle_Ableitung" title="Substantielle Ableitung">materielle Zeitableitung</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\dot {\mathbf {U} }}\,.}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/34e2404f71103c0fecb983d5b5e75224d1bd2359.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.091ex; height:2.676ex;" alt="{\displaystyle {\dot {\mathbf {U} }}\,.}" loading="lazy"> Dementsprechend sind auch alle aus dieser Zeitableitung gebildeten Verzerrungsgeschwindigkeiten, z. B.

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<annotation encoding="application/x-tex">{\displaystyle {\dot {\mathbf {E} }}={\frac {1}{2}}({\dot {\mathbf {U} }}\cdot \mathbf {U} +\mathbf {U} \cdot {\dot {\mathbf {U} }})\,,}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bf8f3600a8ff120f81be51b327dd7429252ccf95.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:24.123ex; height:5.176ex;" alt="{\displaystyle {\dot {\mathbf {E} }}={\frac {1}{2}}({\dot {\mathbf {U} }}\cdot \mathbf {U} +\mathbf {U} \cdot {\dot {\mathbf {U} }})\,,}" loading="lazy"></dd></dl>

körperbezogen objektiv.

In der räumlichen Beschreibung kann nachgewiesen werden, dass der linke Strecktensor <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {v} }">

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

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</math><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"> objektiv ist, seine Rate <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\dot {\mathbf {v} }}}">

<mover>

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/151ab5eaf7ec6c13d04ed456f49c608586953e84.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.411ex; height:2.176ex;" alt="{\displaystyle {\dot {\mathbf {v} }}}" loading="lazy"> jedoch nicht. Für die Formulierung objektiver Raten der räumlichen Verzerrungstensoren wird der räumliche Geschwindigkeitsgradient

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {l} ={\dot {\mathbf {F} }}\cdot \mathbf {F} ^{-1}=\mathbf {d+w} }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d7a57834273aded996b7e7f0206f1f267f32be39.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:20.843ex; height:2.843ex;" alt="{\displaystyle \mathbf {l} ={\dot {\mathbf {F} }}\cdot \mathbf {F} ^{-1}=\mathbf {d+w} }" loading="lazy"></dd></dl>

definiert, dessen symmetrischer Anteil

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {d} :={\frac {1}{2}}(\mathbf {l+l} ^{\top })}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/85039384fa25562e39f037168f2de226626a944b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:15.143ex; height:5.176ex;" alt="{\displaystyle \mathbf {d} :={\frac {1}{2}}(\mathbf {l+l} ^{\top })}" loading="lazy"></dd></dl>

räumlicher Verzerrungsgeschwindigkeitstensor und dessen unsymmetrischer Anteil

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {w} :={\frac {1}{2}}(\mathbf {l-l} ^{\top })}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b599b3a4a8e54adac71db4e90021f7b9fc2ea4fb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:15.589ex; height:5.176ex;" alt="{\displaystyle \mathbf {w} :={\frac {1}{2}}(\mathbf {l-l} ^{\top })}" loading="lazy"></dd></dl>

Spin- oder Wirbeltensor heißt. Dann lautet die (objektive) kovariante Oldroyd-Ableitung eines Tensors <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {a} }">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1a957216653a9ee0d0133dcefd13fb75e36b8b9d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.299ex; height:1.676ex;" alt="{\displaystyle \mathbf {a} }" loading="lazy">:

<mover>

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</mrow>

</mover>

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<mover>

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<msup>

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</msup>

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<annotation encoding="application/x-tex">{\displaystyle {\stackrel {\Delta }{\mathbf {a} }}:={\dot {\mathbf {a} }}+\mathbf {a\cdot l+l^{\top }\cdot a} \,.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/70c1e29f4e2119a374b58bd2ffa479987da52bfa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:22.35ex; height:3.676ex;" alt="{\displaystyle {\stackrel {\Delta }{\mathbf {a} }}:={\dot {\mathbf {a} }}+\mathbf {a\cdot l+l^{\top }\cdot a} \,.}" loading="lazy"></dd></dl>

Für den Euler-Almansi-Tensor e gilt insbesondere

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<annotation encoding="application/x-tex">{\displaystyle {\stackrel {\Delta }{\mathbf {e} }}=\mathbf {d} =\mathbf {F} ^{\top -1}\cdot {\dot {\mathbf {E} }}\cdot \mathbf {F} ^{-1}\,.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a6197e0249641fb7810e7c12f4bae21070c20449.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:24.51ex; height:3.509ex;" alt="{\displaystyle {\stackrel {\Delta }{\mathbf {e} }}=\mathbf {d} =\mathbf {F} ^{\top -1}\cdot {\dot {\mathbf {E} }}\cdot \mathbf {F} ^{-1}\,.}" loading="lazy"></dd></dl>

Die kontravariante Oldroyd-Ableitung eines Tensors <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {a} }">

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

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<mover>

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<annotation encoding="application/x-tex">{\displaystyle {\stackrel {\nabla }{\mathbf {a} }}:={\dot {\mathbf {a} }}-\mathbf {l\cdot a-a\cdot l} ^{\top }\,.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d5fcf13949d07d07c82c6de4307928264db3e20d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:22.35ex; height:3.676ex;" alt="{\displaystyle {\stackrel {\nabla }{\mathbf {a} }}:={\dot {\mathbf {a} }}-\mathbf {l\cdot a-a\cdot l} ^{\top }\,.}" loading="lazy"></dd></dl>

Die Raten der kovarianten Tensoren werden üblicherweise mit der kovarianten Oldroyd-Ableitung gebildet und die der kontravarianten Tensoren mit der kontravarianten Oldroyd-Ableitung. Die Zaremba-Jaumann-Rate eines Tensors <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {a} }">

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

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<annotation encoding="application/x-tex">{\displaystyle {\stackrel {\circ }{\mathbf {a} }}:={\dot {\mathbf {a} }}+\mathbf {a\cdot w-w\cdot a} \,.}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4f70e372a974498a52fcf3b09ecf4909dc24458c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:23.148ex; height:3.176ex;" alt="{\displaystyle {\stackrel {\circ }{\mathbf {a} }}:={\dot {\mathbf {a} }}+\mathbf {a\cdot w-w\cdot a} \,.}" loading="lazy"></dd></dl>

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

<ul><li><a href="Mohrscher_Spannungskreis" title="Mohrscher Spannungskreis">Mohrscher Spannungskreis</a></li>

<li><a href="Dyadisches_Produkt" title="Dyadisches Produkt">Dyadisches Produkt</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="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:Verzerrungstensor&rft.au=H.+Altenbach&rft.btitle=Kontinuumsmechanik&rft.date=2012&rft.genre=book&rft.isbn=9783642241185&rft.pub=Springer" style="display:none"> </li>

<li>G. Holzapfel: <cite style="font-style:italic">Nonlinear Solid Mechanics: A Continuum Approach for Engineering</cite>. Wiley, 2000, ISBN 978-0-471-82319-3.<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:Verzerrungstensor&rft.au=G.+Holzapfel&rft.btitle=Nonlinear+Solid+Mechanics%3A+A+Continuum+Approach+for+Engineering&rft.date=2000&rft.genre=book&rft.isbn=9780471823193&rft.pub=Wiley" style="display:none"> </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:Verzerrungstensor&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"> </li>

<li>A. Bertram: <cite style="font-style:italic">Elasticity and Plasticity of Large Deformations: An Introduction</cite>. Springer, 2012, ISBN 978-3-642-24614-2.<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:Verzerrungstensor&rft.au=A.+Bertram&rft.btitle=Elasticity+and+Plasticity+of+Large+Deformations%3A+An+Introduction&rft.date=2012&rft.genre=book&rft.isbn=9783642246142&rft.pub=Springer" style="display:none"> </li></ul>

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

<li id="cite_note-Frechet-1"><a href="#cite_ref-Frechet_1-0">↑</a> Die <a href="Fr%C3%A9chet-Ableitung" title="Fréchet-Ableitung">Fréchet-Ableitung</a> einer Funktion <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> nach <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

</mrow>

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

</semantics>

ist der beschränkte lineare Operator <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {A}}}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/280ae03440942ab348c2ca9b8db6b56ffa9618f8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.903ex; height:2.343ex;" alt="{\displaystyle {\mathcal {A}}}" loading="lazy"> der – sofern er existiert – in alle Richtungen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h}">

</mstyle>

</mrow>

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b26be3e694314bc90c3215047e4a2010c6ee184a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:2.176ex;" alt="{\displaystyle h}" loading="lazy"> dem <a href="G%C3%A2teaux-Differential" title="Gâteaux-Differential">Gâteaux-Differential</a> entspricht, also

</mrow>

<msub>

<mrow>

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</munder>

<mfrac>

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {A}}(h)=\left.{\frac {\mathrm {d} }{\mathrm {d} s}}f(x+sh)\right|_{s=0}=\lim _{s\rightarrow 0}{\frac {f(x+sh)-f(x)}{s}}\quad \forall \;h}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f4b6858094685423f4c5c8fbc9378b5ed9dec452.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:54.91ex; height:6.176ex;" alt="{\displaystyle {\mathcal {A}}(h)=\left.{\frac {\mathrm {d} }{\mathrm {d} s}}f(x+sh)\right|_{s=0}=\lim _{s\rightarrow 0}{\frac {f(x+sh)-f(x)}{s}}\quad \forall \;h}" loading="lazy"></dd></dl>

gilt. Darin ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s\in \mathbb {R} \,,f,x\,{\textsf {und}}\,h}">

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<annotation encoding="application/x-tex">{\displaystyle s\in \mathbb {R} \,,f,x\,{\textsf {und}}\,h}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a9dbea5741432858342457ca6073c0098dfaf2ca.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.391ex; height:2.509ex;" alt="{\displaystyle s\in \mathbb {R} \,,f,x\,{\textsf {und}}\,h}" loading="lazy"> skalar-, vektor- oder tensorwertig aber <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h}">

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

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {A}}={\frac {\partial f}{\partial x}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a5d851fc35445648bf933225a1d553ef67b9458f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:8.486ex; height:5.676ex;" alt="{\displaystyle {\mathcal {A}}={\frac {\partial f}{\partial x}}}" loading="lazy"></dd></dl>

geschrieben.

</li>

<li id="cite_note-sadd-2">↑ <a href="#cite_ref-sadd_2-0">a</a> <a href="#cite_ref-sadd_2-1">b</a> Martin H. Sadd: <cite style="font-style:italic">Elasticity – Theory, applications and numerics</cite>. Elsevier Butterworth-Heinemann, 2005, ISBN 0-12-605811-3, S. 47 f.<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:Verzerrungstensor&rft.au=Martin+H.+Sadd&rft.btitle=Elasticity+-+Theory%2C+applications+and+numerics&rft.date=2005&rft.genre=book&rft.isbn=0126058113&rft.pages=47f.&rft.pub=Elsevier+Butterworth-Heinemann" style="display:none">

</li>

<li id="cite_note-Landau-4-3">↑ <a href="#cite_ref-Landau-4_3-0">a</a> <a href="#cite_ref-Landau-4_3-1">b</a> L. D. Landau, E. M. Lifschitz: <cite style="font-style:italic">Lehrbuch der theoretischen Physik, Band 7, Elastizitätstheorie -</cite>. 4. Auflage. Akademie Verlag, Berlin 1975, S. 4.<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:Verzerrungstensor&rft.au=L.+D.+Landau%2C+E.+M.+Lifschitz&rft.btitle=Lehrbuch+der+theoretischen+Physik%2C+Band+7%2C+Elastizit%C3%A4tstheorie+-&rft.date=1975&rft.edition=4.&rft.genre=book&rft.pages=4&rft.place=Berlin&rft.pub=Akademie+Verlag" style="display:none">

</li>

<li id="cite_note-4"><a href="#cite_ref-4">↑</a> Z. P. Bazant, L. Cedolin: Stability of Structures. Elastic, Inelastic, Fracture and Damage Theories. Oxford Univ. Press, 2003, ISBN 0-486-42568-1.

</li>

<li id="cite_note-5"><a href="#cite_ref-5">↑</a> B. R. Seth vom Indian Institute of Technology in Kharagpur war der erste der gezeigt hat, dass der Green-Lagrange- und der Euler-Almansi-Verzerrungstensor Spezialfälle dieses allgemeineren Verzerrungsmaßes sind [a][b]. Die Idee wurde von <a href="Rodney_Hill" title="Rodney Hill">Rodney Hill</a> in [c] weiterentwickelt.

<dl><dd>[a] B. R. Seth: Generalized strain measure with applications to physical problems. MRC Technical Summary Report #248 des Mathematics Research Center, United States Army, University of Wisconsin, 1961, S. 1–18, <a rel="nofollow" class="external text" href="http://www.dtic.mil/cgi-bin/GetTRDoc?AD=AD0266913">AD0266913.pdf</a></dd>

<dd>[b] B. R. Seth: Generalized strain measure with applications to physical problems. IUTAM Symposium on Second Order Effects in Elasticity, Plasticity and Fluid Mechanics, Haifa 1962.</dd>

<dd>[c] R. Hill: On constitutive inequalities for simple materials-I. In: Journal of the Mechanics and Physics of Solids. 16, Nr. 4, 1968, S. 229–242. </dd></dl>

</li>

<li id="cite_note-Bertram-6">↑ <a href="#cite_ref-Bertram_6-0">a</a> <a href="#cite_ref-Bertram_6-1">b</a> Bertram (2012)

</li>

<li id="cite_note-einheit-7">↑ <a href="#cite_ref-einheit_7-0">a</a> <a href="#cite_ref-einheit_7-1">b</a> <a href="#cite_ref-einheit_7-2">c</a> Haupt (2000)

</li>

<li id="cite_note-altenbach-8">↑ <a href="#cite_ref-altenbach_8-0">a</a> <a href="#cite_ref-altenbach_8-1">b</a> Altenbach (2012)

</li>

</ol>

<div>

Normdaten (Sachbegriff): <a href="Gemeinsame_Normdatei" title="Gemeinsame Normdatei">GND</a>: <a rel="nofollow" class="external text" href="https://d-nb.info/gnd/4316421-3">4316421-3</a> </div>

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