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<title>Strecktensor</title>

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<h1 id="firstHeading" class="firstHeading mw-first-heading">Strecktensor</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">Strecktensoren oder Deformationstensoren sind einheitenfreie <a href="Tensor" title="Tensor">Tensoren</a> (bestimmte mathematische Objekte der <a href="Lineare_Algebra" title="Lineare Algebra">linearen Algebra</a>) zweiter Stufe, die lokale Distanzänderungen von Materieelementen bei einer Deformation eines Körpers bemessen. Distanzänderungen von Materieelementen entsprechen der Streckung bzw. Stauchung der materiellen Linien, die die betrachteten Materieelemente verbinden. Diese Änderungen der inneren Anordnung korrespondieren mit einer Änderung der äußeren Gestalt des Festkörpers und werden beispielsweise als <a href="Dehnung" title="Dehnung">Dehnung</a> oder <a href="Stauchung" title="Stauchung">Stauchung</a> sichtbar.

Die Strecktensoren sind eine wesentliche Größe in der Beschreibung der <a href="Kinematik" title="Kinematik">Kinematik</a> der Deformation und in der <a href="Kontinuumsmechanik" title="Kontinuumsmechanik">Kontinuumsmechanik</a> werden eine Reihe von verschiedenen Strecktensoren definiert, die ihrerseits der Definition von <a href="Verzerrungstensor" title="Verzerrungstensor">Verzerrungstensoren</a> dienen. In einigen Materialmodellen der <a href="Hyperelastizit%C3%A4t" title="Hyperelastizität">Hyperelastizität</a> werden Strecktensoren direkt eingesetzt.

<div class="mw-heading mw-heading2"><h2 id="Streckung_von_Linienelementen">Streckung von Linienelementen</h2></div>

Bei der quantitativen Beurteilung einer Deformation eines Körpers werden materielle Linien des Körpers vor und nach Deformation miteinander verglichen. In der Praxis können dazu <a href="Dehnungsmessstreifen" title="Dehnungsmessstreifen">Dehnungsmessstreifen</a> (DMS) auf dem Körper aufgeklebt werden. Die Richtung des DMS wird mathematisch mit einem materiellen Linienelement <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {d} {\vec {X}}}">

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

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</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"> in der undeformierten Ausgangskonfiguration und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {d} {\vec {x}}}">

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

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</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"> in der deformierten Momentankonfiguration beschrieben, siehe Abbildung rechts. Diese Linienelemente stehen in linearer Näherung über den <a href="Deformationsgradient" title="Deformationsgradient">Deformationsgradient</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {F} }">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/da18bef8c979f3548bb0d8976f5844012d7b8256.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.683ex; height:2.176ex;" alt="{\displaystyle \mathbf {F} }" loading="lazy"> in Beziehung:

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/33f27871f1552fdda21249e92e9e05262a40f083.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:12.355ex; height:2.843ex;" alt="{\displaystyle \mathrm {d} {\vec {x}}=\mathbf {F} \cdot \mathrm {d} {\vec {X}}}" loading="lazy"></dd></dl>

Die Streckung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda }">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b43d0ea3c9c025af1be9128e62a18fa74bedda2a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }" loading="lazy"> eines Linienelementes in der Richtung

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ca4452f40f12615ce19591fc2e16a319a423f48e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:9.724ex; height:7.343ex;" alt="{\displaystyle {\vec {e}}={\frac {\mathrm {d} {\vec {X}}}{|\mathrm {d} {\vec {X}}|}}}" loading="lazy"></dd></dl>

ist das Verhältnis

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<annotation encoding="application/x-tex">{\displaystyle \lambda ={\frac {|\mathrm {d} {\vec {x}}|}{|\mathrm {d} {\vec {X}}|}}={\sqrt {\frac {\mathrm {d} {\vec {x}}\cdot \mathrm {d} {\vec {x}}}{\mathrm {d} {\vec {X}}\cdot \mathrm {d} {\vec {X}}}}}={\sqrt {\frac {(\mathbf {F} \cdot \mathrm {d} {\vec {X}})\cdot (\mathbf {F} \cdot \mathrm {d} {\vec {X}})}{\mathrm {d} {\vec {X}}\cdot \mathrm {d} {\vec {X}}}}}={\sqrt {{\vec {e}}\cdot \mathbf {C} \cdot {\vec {e}}}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a1da31ffbbf60d55d081ae86726ee54096ab4c94.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:62.063ex; height:8.343ex;" alt="{\displaystyle \lambda ={\frac {|\mathrm {d} {\vec {x}}|}{|\mathrm {d} {\vec {X}}|}}={\sqrt {\frac {\mathrm {d} {\vec {x}}\cdot \mathrm {d} {\vec {x}}}{\mathrm {d} {\vec {X}}\cdot \mathrm {d} {\vec {X}}}}}={\sqrt {\frac {(\mathbf {F} \cdot \mathrm {d} {\vec {X}})\cdot (\mathbf {F} \cdot \mathrm {d} {\vec {X}})}{\mathrm {d} {\vec {X}}\cdot \mathrm {d} {\vec {X}}}}}={\sqrt {{\vec {e}}\cdot \mathbf {C} \cdot {\vec {e}}}}}" loading="lazy"></dd></dl>

Der Strecktensor

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a6e3763f7cb41ac67a3da1e36a2f860442e74b41.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.585ex; height:2.676ex;" alt="{\displaystyle \mathbf {C} =\mathbf {F} ^{\top }\cdot \mathbf {F} }" loading="lazy"></dd></dl>

heißt rechter Cauchy-Green Tensor und ist demnach ein Maß für die Streckung von Linienelementen. Das Superskript „<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \top }">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cf12e436fef2365e76fcb1034a51179d8328bb33.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \top }" loading="lazy">“ steht für die <a href="Transponierte_Matrix" title="Transponierte Matrix">Transposition</a>. In Richtung der <a href="Eigenwertproblem" class="mw-redirect" title="Eigenwertproblem">Eigenvektoren</a> von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {C} }">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/11de80478fce9090e43eed19100b37cc841661e8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.931ex; height:2.176ex;" alt="{\displaystyle \mathbf {C} }" loading="lazy"> sind die Streckungen extremal. In der deformierten Lage berechnet sich die Streckung aus

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<annotation encoding="application/x-tex">{\displaystyle \lambda ={\sqrt {\frac {\mathrm {d} {\vec {x}}\cdot \mathrm {d} {\vec {x}}}{(\mathbf {F} ^{-1}\cdot \mathrm {d} {\vec {x}})\cdot (\mathbf {F} ^{-1}\cdot \mathrm {d} {\vec {x}})}}}=\left({\frac {\mathrm {d} {\vec {x}}}{|\mathrm {d} {\vec {x}}|}}\cdot \mathbf {c} \cdot {\frac {\mathrm {d} {\vec {x}}}{|\mathrm {d} {\vec {x}}|}}\right)^{-{\frac {1}{2}}}\,.}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/554d6a30873d92f41083ad6458712d396af23a49.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:54.163ex; height:8.176ex;" alt="{\displaystyle \lambda ={\sqrt {\frac {\mathrm {d} {\vec {x}}\cdot \mathrm {d} {\vec {x}}}{(\mathbf {F} ^{-1}\cdot \mathrm {d} {\vec {x}})\cdot (\mathbf {F} ^{-1}\cdot \mathrm {d} {\vec {x}})}}}=\left({\frac {\mathrm {d} {\vec {x}}}{|\mathrm {d} {\vec {x}}|}}\cdot \mathbf {c} \cdot {\frac {\mathrm {d} {\vec {x}}}{|\mathrm {d} {\vec {x}}|}}\right)^{-{\frac {1}{2}}}\,.}" loading="lazy"></dd></dl>

Der Cauchysche Strecktensor

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7bc15fda94110a4460bdd4fae22df32a2cfede96.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:15.922ex; height:2.676ex;" alt="{\displaystyle \mathbf {c} :=\mathbf {F} ^{\top -1}\cdot \mathbf {F} ^{-1}}" loading="lazy"></dd></dl>

ist also ein räumliches Maß für die Streckung von Linienelementen.

<div class="mw-heading mw-heading2"><h2 id="Streckung_von_Normalvektoren">Streckung von Normalvektoren</h2></div>

Mit Strecktensoren kann auch die Streckung von <a href="Normalvektor" class="mw-redirect" title="Normalvektor">Normalvektoren</a> ermittelt werden. 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/ed4d58a2dd0a083336756ad0fb8affc9a5ddeb1b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.555ex; height:2.843ex;" 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_/eb734a37dd21ce173a46342d1cc64c92/7c097ad2573c505f337f982c83d567d556366092.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:27.312ex; 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>

Diese hängen mit der Normale in der Referenzkonfiguration wie folgt zusammen:

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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 _{j,k=1}^{3}{\frac {\mathrm {d} \Phi }{\mathrm {d} x_{k}}}{\frac {\mathrm {d} x_{k}}{\mathrm {d} X_{j}}}{\vec {e}}_{j}\\=&\sum _{i,j,k=1}^{3}{\frac {\mathrm {d} \Phi }{\mathrm {d} x_{k}}}{\vec {e}}_{k}\cdot {\frac {\mathrm {d} x_{i}}{\mathrm {d} X_{j}}}{\vec {e}}_{i}\otimes {\vec {e}}_{j}={\vec {n}}\cdot \mathbf {F} =\mathbf {F} ^{\top }\cdot {\vec {n}}\\\rightarrow {\vec {n}}=&\mathbf {F} ^{\top -1}\cdot {\vec {N}}\end{aligned}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ba3d6a30cbca4eb266f579ab269d6fb02a6932a2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -8.838ex; width:51.177ex; height:18.843ex;" 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 _{j,k=1}^{3}{\frac {\mathrm {d} \Phi }{\mathrm {d} x_{k}}}{\frac {\mathrm {d} x_{k}}{\mathrm {d} X_{j}}}{\vec {e}}_{j}\\=&\sum _{i,j,k=1}^{3}{\frac {\mathrm {d} \Phi }{\mathrm {d} x_{k}}}{\vec {e}}_{k}\cdot {\frac {\mathrm {d} x_{i}}{\mathrm {d} X_{j}}}{\vec {e}}_{i}\otimes {\vec {e}}_{j}={\vec {n}}\cdot \mathbf {F} =\mathbf {F} ^{\top }\cdot {\vec {n}}\\\rightarrow {\vec {n}}=&\mathbf {F} ^{\top -1}\cdot {\vec {N}}\end{aligned}}}" loading="lazy"></dd></dl>

Das Rechenzeichen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \otimes }">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/de29098f5a34ee296a505681a0d5e875070f2aea.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \otimes }" loading="lazy"> bezeichnet das <a href="Dyadisches_Produkt" title="Dyadisches Produkt">dyadische Produkt</a>. Die Streckung der Normalvektoren 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 {\begin{array}{rcl}\lambda &=&\displaystyle {\frac {|{\vec {n}}|}{|{\vec {N}}|}}={\sqrt {\frac {{\vec {n}}\cdot {\vec {n}}}{{\vec {N}}\cdot {\vec {N}}}}}={\sqrt {\frac {(\mathbf {F} ^{\top -1}\cdot {\vec {N}})\cdot (\mathbf {F} ^{\top -1}\cdot {\vec {N}})}{{\vec {N}}\cdot {\vec {N}}}}}={\sqrt {{\frac {\vec {N}}{|{\vec {N}}|}}\cdot \mathbf {f} \cdot {\frac {\vec {N}}{|{\vec {N}}|}}}}\\\rightarrow \mathbf {f} &=&\mathbf {F} ^{-1}\cdot \mathbf {F} ^{\top -1}=\mathbf {C} ^{-1}\,,\end{array}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/01fb31aee343a34d7ae96bfe11e1be2137543c7f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.171ex; width:75.465ex; height:11.509ex;" alt="{\displaystyle {\begin{array}{rcl}\lambda &=&\displaystyle {\frac {|{\vec {n}}|}{|{\vec {N}}|}}={\sqrt {\frac {{\vec {n}}\cdot {\vec {n}}}{{\vec {N}}\cdot {\vec {N}}}}}={\sqrt {\frac {(\mathbf {F} ^{\top -1}\cdot {\vec {N}})\cdot (\mathbf {F} ^{\top -1}\cdot {\vec {N}})}{{\vec {N}}\cdot {\vec {N}}}}}={\sqrt {{\frac {\vec {N}}{|{\vec {N}}|}}\cdot \mathbf {f} \cdot {\frac {\vec {N}}{|{\vec {N}}|}}}}\\\rightarrow \mathbf {f} &=&\mathbf {F} ^{-1}\cdot \mathbf {F} ^{\top -1}=\mathbf {C} ^{-1}\,,\end{array}}}" loading="lazy"></dd></dl>

der also ein Maß für die Streckung der materiellen Flächennormalen ist. Der Finger-Tensor operiert in der Ausgangskonfiguration.

Sein Gegenstück in der Momentankonfiguration ist der linke Cauchy-Green Tensor

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/536cefd7637b61572caaffa21628b7697e13e415.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:17.758ex; height:2.676ex;" alt="{\displaystyle \mathbf {b} =\mathbf {F} \cdot \mathbf {F} ^{\top }=\mathbf {c} ^{-1}}" loading="lazy"></dd></dl>

für den

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<annotation encoding="application/x-tex">{\displaystyle \lambda ={\sqrt {\frac {{\vec {n}}\cdot {\vec {n}}}{(\mathbf {F} ^{\top }\cdot {\vec {n}})\cdot (\mathbf {F} ^{\top }\cdot {\vec {n}})}}}=\left({\frac {\vec {n}}{|{\vec {n}}|}}\cdot \mathbf {b} \cdot {\frac {\vec {n}}{|{\vec {n}}|}}\right)^{-{\frac {1}{2}}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/09e24b79ab5d9486f11435ea3bfd2d8f8fa73a15.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:46.872ex; height:8.176ex;" alt="{\displaystyle \lambda ={\sqrt {\frac {{\vec {n}}\cdot {\vec {n}}}{(\mathbf {F} ^{\top }\cdot {\vec {n}})\cdot (\mathbf {F} ^{\top }\cdot {\vec {n}})}}}=\left({\frac {\vec {n}}{|{\vec {n}}|}}\cdot \mathbf {b} \cdot {\frac {\vec {n}}{|{\vec {n}}|}}\right)^{-{\frac {1}{2}}}}" loading="lazy"></dd></dl>

abgeleitet werden kann.

<div class="mw-heading mw-heading2"><h2 id="Hauptinvarianten_des_rechten_Cauchy-Green_Tensors">Hauptinvarianten des rechten Cauchy-Green Tensors</h2></div>

Bei einer Deformation werden die materiellen <a href="Deformationsgradient#Linien-,_Flächen-_und_Volumenelemente" title="Deformationsgradient">Linien-, Flächen- und Volumenelemente</a> mit dem Deformationsgradient von der Ausgangskonfiguration in die Momentankonfiguration transformiert

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<annotation encoding="application/x-tex">{\displaystyle {\begin{array}{rccl}\mathrm {d} {\vec {x}}&=&\mathbf {F} &\cdot \;\mathrm {d} {\vec {X}}\\{\vec {n}}\,\mathrm {d} a&=&\operatorname {cof} (\mathbf {F} )&\cdot \;{\vec {N}}\,\mathrm {d} A\\\mathrm {d} v&=&\operatorname {det} (\mathbf {F} )&\mathrm {d} V\,.\end{array}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/abf4ef1f1320cc2bb3b8fad60e5188c0f33d7232.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.671ex; width:27.331ex; height:10.509ex;" alt="{\displaystyle {\begin{array}{rccl}\mathrm {d} {\vec {x}}&=&\mathbf {F} &\cdot \;\mathrm {d} {\vec {X}}\\{\vec {n}}\,\mathrm {d} a&=&\operatorname {cof} (\mathbf {F} )&\cdot \;{\vec {N}}\,\mathrm {d} A\\\mathrm {d} v&=&\operatorname {det} (\mathbf {F} )&\mathrm {d} V\,.\end{array}}}" loading="lazy"></dd></dl>

Der <a href="Minor_(Mathematik)#Kofaktoren" class="mw-redirect" title="Minor (Mathematik)">Kofaktor</a> des Deformationsgradienten ist seine transponierte <a href="Adjunkte" title="Adjunkte">Adjunkte</a>:

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {cof} (\mathbf {F} ):=\operatorname {det} (\mathbf {F} )\mathbf {F} ^{\top -1}\,.}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/def8442f5f64bf99a1edbc41723bbbe64c14f4d4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.194ex; height:3.176ex;" alt="{\displaystyle \operatorname {cof} (\mathbf {F} ):=\operatorname {det} (\mathbf {F} )\mathbf {F} ^{\top -1}\,.}" loading="lazy"></dd></dl>

Es zeigt sich, dass die <a href="Hauptinvariante" title="Hauptinvariante">Hauptinvarianten</a> des rechten Cauchy-Green Tensors Maße für die Änderung der Linien-, Flächen- und Volumenelemente sind:

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<annotation encoding="application/x-tex">{\displaystyle {\begin{array}{rclclcc}\operatorname {I} _{1}(\mathbf {C} )&=&\operatorname {Sp} (\mathbf {C} )&=&\operatorname {Sp} (\mathbf {F^{\top }\cdot F} )=\mathbf {F} :\mathbf {F} &=&\|\mathbf {F} \|^{2}\\\operatorname {I} _{2}(\mathbf {C} )&=&\operatorname {Sp(cof} (\mathbf {C} ))&=&\operatorname {Sp(cof} (\mathbf {F} )^{\top }\cdot \operatorname {cof} (\mathbf {F} ))&=&\|\operatorname {cof} (\mathbf {F} )\|^{2}\\\operatorname {I} _{3}(\mathbf {C} )&=&\operatorname {det} (\mathbf {C} )&=&\operatorname {det} (\mathbf {F^{\top }\cdot F} )&=&\operatorname {det} (\mathbf {F} )^{2}\end{array}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b489be241222eb8ecd296e48f718849a79e1699c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.338ex; width:67.641ex; height:9.843ex;" alt="{\displaystyle {\begin{array}{rclclcc}\operatorname {I} _{1}(\mathbf {C} )&=&\operatorname {Sp} (\mathbf {C} )&=&\operatorname {Sp} (\mathbf {F^{\top }\cdot F} )=\mathbf {F} :\mathbf {F} &=&\|\mathbf {F} \|^{2}\\\operatorname {I} _{2}(\mathbf {C} )&=&\operatorname {Sp(cof} (\mathbf {C} ))&=&\operatorname {Sp(cof} (\mathbf {F} )^{\top }\cdot \operatorname {cof} (\mathbf {F} ))&=&\|\operatorname {cof} (\mathbf {F} )\|^{2}\\\operatorname {I} _{3}(\mathbf {C} )&=&\operatorname {det} (\mathbf {C} )&=&\operatorname {det} (\mathbf {F^{\top }\cdot F} )&=&\operatorname {det} (\mathbf {F} )^{2}\end{array}}}" loading="lazy"></dd></dl>

Die <a href="Frobeniusnorm" title="Frobeniusnorm">Frobeniusnorm</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|(\cdot )\|}">

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<annotation encoding="application/x-tex">{\displaystyle \|(\cdot )\|}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e8365efc2a147b774655bd8fb519aa73a77b98dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.781ex; height:2.843ex;" alt="{\displaystyle \|(\cdot )\|}" loading="lazy"> wird mit dem <a href="Frobenius-Skalarprodukt" title="Frobenius-Skalarprodukt">Frobenius-Skalarprodukt</a> „:“ von Tensoren definiert:

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {A} :\mathbf {B} :=\operatorname {Sp} (\mathbf {A^{\top }\cdot B} )\quad {\text{und}}\quad \|\mathbf {A} \|:={\sqrt {\mathbf {A} :\mathbf {A} }}\,.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/17239c216d9c643f943c8b695ae84af8e4479595.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:46.666ex; height:3.176ex;" alt="{\displaystyle \mathbf {A} :\mathbf {B} :=\operatorname {Sp} (\mathbf {A^{\top }\cdot B} )\quad {\text{und}}\quad \|\mathbf {A} \|:={\sqrt {\mathbf {A} :\mathbf {A} }}\,.}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Physikalische_Interpretation">Physikalische Interpretation</h2></div>

Der Zusammenhang zwischen dem rechten Cauchy-Green Tensor und der Änderung der Linien-, Flächen- und Volumenelemente macht sich makroskopisch bemerkbar.

Sei

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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"></dd></dl>

die Bewegungsfunktion der Partikel eines materiellen Körpers. Die 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"> nehmen die Partikel zu einer bestimmten Zeit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t_{0}}">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/02d3006c4190b1939b04d9b9bb21006fb4e6fa4a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.894ex; height:2.343ex;" alt="{\displaystyle t_{0}}" loading="lazy"> ein, zu der der Körper in seiner undeformierten Ausgangslage vorliegt. Der zeitabhängige Vektor <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {x}}}">

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</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"> bezeichnet die räumlichen Koordinaten, die die Partikel bei ihrer Bewegung – inklusive Deformation – zur Zeit t einnehmen.

<div class="mw-heading mw-heading3"><h3 id="Längen_von_Linien">Längen von Linien</h3></div>

Wenn im undeformierten Ausgangszustand eine materielle Linie <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {X}}(s)}">

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<annotation encoding="application/x-tex">{\displaystyle L=\int _{0}^{1}\left|{\frac {\mathrm {d} {\vec {X}}}{\mathrm {d} s}}\right|\mathrm {d} s=\int _{0}^{1}{\sqrt {{\frac {\mathrm {d} {\vec {X}}}{\mathrm {d} s}}\cdot {\frac {\mathrm {d} {\vec {X}}}{\mathrm {d} s}}}}\mathrm {d} s=\int _{0}^{1}{\sqrt {{\frac {\mathrm {d} {\vec {X}}}{\mathrm {d} s}}\cdot \mathbf {I} \cdot {\frac {\mathrm {d} {\vec {X}}}{\mathrm {d} s}}}}\mathrm {d} s\,.}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/38384f7df2e4d8b1bf5e1ba7be37cbe6a0215286.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:63.7ex; height:7.843ex;" alt="{\displaystyle L=\int _{0}^{1}\left|{\frac {\mathrm {d} {\vec {X}}}{\mathrm {d} s}}\right|\mathrm {d} s=\int _{0}^{1}{\sqrt {{\frac {\mathrm {d} {\vec {X}}}{\mathrm {d} s}}\cdot {\frac {\mathrm {d} {\vec {X}}}{\mathrm {d} s}}}}\mathrm {d} s=\int _{0}^{1}{\sqrt {{\frac {\mathrm {d} {\vec {X}}}{\mathrm {d} s}}\cdot \mathbf {I} \cdot {\frac {\mathrm {d} {\vec {X}}}{\mathrm {d} s}}}}\mathrm {d} s\,.}" loading="lazy"></dd></dl>

Darin ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {I} }">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8a458c8aeb096ce732abf346ae8edf3e4f53a126.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.014ex; height:2.176ex;" alt="{\displaystyle \mathbf {I} }" loading="lazy"> der <a href="Einheitstensor" title="Einheitstensor">Einheitstensor</a>. In der deformierten Lage verändert sich diese Länge zu

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<mfrac>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle l=\int _{0}^{1}\left|{\frac {\mathrm {d} {\vec {\chi }}({\vec {X}}(s),t)}{\mathrm {d} s}}\right|\mathrm {d} s=\int _{0}^{1}{\sqrt {{\frac {\mathrm {d} {\vec {x}}}{\mathrm {d} s}}\cdot {\frac {\mathrm {d} {\vec {x}}}{\mathrm {d} s}}}}\mathrm {d} s=\int _{0}^{1}{\sqrt {\left(\mathbf {F} \cdot {\frac {\mathrm {d} {\vec {X}}}{\mathrm {d} s}}\right)\cdot \left(\mathbf {F} \cdot {\frac {\mathrm {d} {\vec {X}}}{\mathrm {d} s}}\right)}}\mathrm {d} s=\int _{0}^{1}{\sqrt {{\frac {\mathrm {d} {\vec {X}}}{\mathrm {d} s}}\cdot \mathbf {C} \cdot {\frac {\mathrm {d} {\vec {X}}}{\mathrm {d} s}}}}\mathrm {d} s\,.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/369b761b8a09bbb0ee4829285f085e2323563960.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:106.289ex; height:8.176ex;" alt="{\displaystyle l=\int _{0}^{1}\left|{\frac {\mathrm {d} {\vec {\chi }}({\vec {X}}(s),t)}{\mathrm {d} s}}\right|\mathrm {d} s=\int _{0}^{1}{\sqrt {{\frac {\mathrm {d} {\vec {x}}}{\mathrm {d} s}}\cdot {\frac {\mathrm {d} {\vec {x}}}{\mathrm {d} s}}}}\mathrm {d} s=\int _{0}^{1}{\sqrt {\left(\mathbf {F} \cdot {\frac {\mathrm {d} {\vec {X}}}{\mathrm {d} s}}\right)\cdot \left(\mathbf {F} \cdot {\frac {\mathrm {d} {\vec {X}}}{\mathrm {d} s}}\right)}}\mathrm {d} s=\int _{0}^{1}{\sqrt {{\frac {\mathrm {d} {\vec {X}}}{\mathrm {d} s}}\cdot \mathbf {C} \cdot {\frac {\mathrm {d} {\vec {X}}}{\mathrm {d} s}}}}\mathrm {d} s\,.}" loading="lazy"></dd></dl>

Die Änderung der Länge der markierten Linie wird also vom Strecktensor <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {C} }">

</mrow>

</mstyle>

</mrow>

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

</semantics>

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

Wenn in gleicher Weise im undeformierten Ausgangszustand eine materielle Fläche <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {X}}(u,v)}">

<mover>

</mover>

</mrow>

</mstyle>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/38ee211f4e83a15d83c2615e148cbe569acb362b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.281ex; height:3.343ex;" alt="{\displaystyle {\vec {X}}(u,v)}" loading="lazy"> mit den Flächenparametern <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (u,v)\in [0,1]^{2}}">

<msup>

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</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (u,v)\in [0,1]^{2}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bfb3a3f028eed9d8d5ac42fd0b842c732ba13c6d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.848ex; height:3.176ex;" alt="{\displaystyle (u,v)\in [0,1]^{2}}" loading="lazy"> bezeichnet wird, ergibt sich der Inhalt der Fläche zu

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle F=\int _{0}^{1}\int _{0}^{1}{\biggl |}\overbrace {{\frac {\mathrm {d} {\vec {X}}}{\mathrm {d} u}}\times {\frac {\mathrm {d} {\vec {X}}}{\mathrm {d} v}}} ^{A{\vec {N}}}{\biggr |}\,\mathrm {d} u\mathrm {d} v=\int _{0}^{1}\int _{0}^{1}|{\vec {N}}|\,\overbrace {A\mathrm {d} u\mathrm {d} v} ^{\mathrm {d} A}=\int _{0}^{1}\int _{0}^{1}{\sqrt {{\vec {N}}\cdot \mathbf {I} \cdot {\vec {N}}}}\,\mathrm {d} A\,.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/887b51269e03bd6e7d234e4b3dc272add3372d80.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:78.059ex; height:10.343ex;" alt="{\displaystyle F=\int _{0}^{1}\int _{0}^{1}{\biggl |}\overbrace {{\frac {\mathrm {d} {\vec {X}}}{\mathrm {d} u}}\times {\frac {\mathrm {d} {\vec {X}}}{\mathrm {d} v}}} ^{A{\vec {N}}}{\biggr |}\,\mathrm {d} u\mathrm {d} v=\int _{0}^{1}\int _{0}^{1}|{\vec {N}}|\,\overbrace {A\mathrm {d} u\mathrm {d} v} ^{\mathrm {d} A}=\int _{0}^{1}\int _{0}^{1}{\sqrt {{\vec {N}}\cdot \mathbf {I} \cdot {\vec {N}}}}\,\mathrm {d} A\,.}" loading="lazy"></dd></dl>

In der deformierten Lage verändert sich diese Fläche zu

<mtr>

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<mover>

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<msup>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{array}{rcl}f&=&\displaystyle \int _{0}^{1}\int _{0}^{1}{\biggl |}\overbrace {{\frac {\mathrm {d} {\vec {x}}}{\mathrm {d} u}}\times {\frac {\mathrm {d} {\vec {x}}}{\mathrm {d} v}}} ^{a{\vec {n}}}{\biggr |}\,\mathrm {d} u\mathrm {d} v=\int _{0}^{1}\int _{0}^{1}|{\vec {n}}|\,\overbrace {a\mathrm {d} u\mathrm {d} v} ^{\mathrm {d} a}=\int _{0}^{1}\int _{0}^{1}|\operatorname {cof} (\mathbf {F} )\cdot {\vec {N}}|\,\mathrm {d} A\\&=&\displaystyle \int _{0}^{1}\int _{0}^{1}{\sqrt {{\vec {N}}\cdot \operatorname {cof} (\mathbf {F} )^{\top }\cdot \operatorname {cof} (\mathbf {F} )\cdot {\vec {N}}}}\,\mathrm {d} A=\int _{0}^{1}\int _{0}^{1}{\sqrt {{\vec {N}}\cdot \operatorname {cof} (\mathbf {C} )\cdot {\vec {N}}}}\,\mathrm {d} A\end{array}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6511a57a57b62a7b787ce9358e8ab1077d0379a9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.671ex; width:79.186ex; height:16.509ex;" alt="{\displaystyle {\begin{array}{rcl}f&=&\displaystyle \int _{0}^{1}\int _{0}^{1}{\biggl |}\overbrace {{\frac {\mathrm {d} {\vec {x}}}{\mathrm {d} u}}\times {\frac {\mathrm {d} {\vec {x}}}{\mathrm {d} v}}} ^{a{\vec {n}}}{\biggr |}\,\mathrm {d} u\mathrm {d} v=\int _{0}^{1}\int _{0}^{1}|{\vec {n}}|\,\overbrace {a\mathrm {d} u\mathrm {d} v} ^{\mathrm {d} a}=\int _{0}^{1}\int _{0}^{1}|\operatorname {cof} (\mathbf {F} )\cdot {\vec {N}}|\,\mathrm {d} A\\&=&\displaystyle \int _{0}^{1}\int _{0}^{1}{\sqrt {{\vec {N}}\cdot \operatorname {cof} (\mathbf {F} )^{\top }\cdot \operatorname {cof} (\mathbf {F} )\cdot {\vec {N}}}}\,\mathrm {d} A=\int _{0}^{1}\int _{0}^{1}{\sqrt {{\vec {N}}\cdot \operatorname {cof} (\mathbf {C} )\cdot {\vec {N}}}}\,\mathrm {d} A\end{array}}}" loading="lazy"></dd></dl>

Die Inhaltsänderung der markierten Fläche wird also vom Kofaktor des Strecktensors <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {C} }">

</mrow>

</mstyle>

</mrow>

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

</semantics>

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

Im undeformierten Ausgangszustand wird ein materielles Volumen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {X}}(\xi ,\eta ,\zeta )}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d673130526da18e934882571e0324c07c68704cb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.152ex; height:3.343ex;" alt="{\displaystyle {\vec {X}}(\xi ,\eta ,\zeta )}" loading="lazy"> mit den Ortsparametern <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\xi ,\eta ,\zeta )\in [0,1]^{3}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (\xi ,\eta ,\zeta )\in [0,1]^{3}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f46e221710afb9d70ec9113d8f012b78a0836998.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.719ex; height:3.176ex;" alt="{\displaystyle (\xi ,\eta ,\zeta )\in [0,1]^{3}}" loading="lazy"> markiert. Das Volumen berechnet sich dann zu

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<msqrt>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle V=\int _{0}^{1}\int _{0}^{1}\int _{0}^{1}\overbrace {\operatorname {det} {\begin{pmatrix}{\frac {\mathrm {d} {\vec {X}}}{\mathrm {d} \xi }}&{\frac {\mathrm {d} {\vec {X}}}{\mathrm {d} \eta }}&{\frac {\mathrm {d} {\vec {X}}}{\mathrm {d} \zeta }}\end{pmatrix}}\,\mathrm {d} \xi \mathrm {d} \eta \mathrm {d} \zeta } ^{\mathrm {d} V}=\int _{0}^{1}\int _{0}^{1}\int _{0}^{1}{\sqrt {\operatorname {det} (\mathbf {I} )}}\,\mathrm {d} V}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cb6ae216896db98faeafe2065b03a93ba59f26d9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:72.081ex; height:8.676ex;" alt="{\displaystyle V=\int _{0}^{1}\int _{0}^{1}\int _{0}^{1}\overbrace {\operatorname {det} {\begin{pmatrix}{\frac {\mathrm {d} {\vec {X}}}{\mathrm {d} \xi }}&{\frac {\mathrm {d} {\vec {X}}}{\mathrm {d} \eta }}&{\frac {\mathrm {d} {\vec {X}}}{\mathrm {d} \zeta }}\end{pmatrix}}\,\mathrm {d} \xi \mathrm {d} \eta \mathrm {d} \zeta } ^{\mathrm {d} V}=\int _{0}^{1}\int _{0}^{1}\int _{0}^{1}{\sqrt {\operatorname {det} (\mathbf {I} )}}\,\mathrm {d} V}" loading="lazy"></dd></dl>

In der deformierten Lage verändert sich dieses Volumen zu

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{array}{rcl}v&=&\displaystyle \int _{0}^{1}\int _{0}^{1}\int _{0}^{1}\operatorname {det} {\begin{pmatrix}{\frac {\mathrm {d} {\vec {x}}}{\mathrm {d} \xi }}&{\frac {\mathrm {d} {\vec {x}}}{\mathrm {d} \eta }}&{\frac {\mathrm {d} {\vec {x}}}{\mathrm {d} \zeta }}\end{pmatrix}}\,\mathrm {d} \xi \mathrm {d} \eta \mathrm {d} \zeta =\int _{0}^{1}\int _{0}^{1}\int _{0}^{1}\operatorname {det} {\begin{pmatrix}\mathbf {F} \cdot {\frac {\mathrm {d} {\vec {X}}}{\mathrm {d} \xi }}&\mathbf {F} \cdot {\frac {\mathrm {d} {\vec {X}}}{\mathrm {d} \eta }}&\mathbf {F} \cdot {\frac {\mathrm {d} {\vec {X}}}{\mathrm {d} \zeta }}\end{pmatrix}}\,\mathrm {d} \xi \mathrm {d} \eta \mathrm {d} \zeta \\&=&\displaystyle \int _{0}^{1}\int _{0}^{1}\int _{0}^{1}\operatorname {det} \left[\mathbf {F} \cdot {\begin{pmatrix}{\frac {\mathrm {d} {\vec {X}}}{\mathrm {d} \xi }}&{\frac {\mathrm {d} {\vec {X}}}{\mathrm {d} \eta }}&{\frac {\mathrm {d} {\vec {X}}}{\mathrm {d} \zeta }}\end{pmatrix}}\right]\,\mathrm {d} \xi \mathrm {d} \eta \mathrm {d} \zeta =\int _{0}^{1}\int _{0}^{1}\int _{0}^{1}\operatorname {det} (\mathbf {F} )\,\mathrm {d} V=\int _{0}^{1}\int _{0}^{1}\int _{0}^{1}{\sqrt {\operatorname {det} (\mathbf {C} )}}\,\mathrm {d} V\,,\end{array}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ac1a64ad6cf3dc584c7e09196acc80404056779c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.838ex; width:107.63ex; height:12.843ex;" alt="{\displaystyle {\begin{array}{rcl}v&=&\displaystyle \int _{0}^{1}\int _{0}^{1}\int _{0}^{1}\operatorname {det} {\begin{pmatrix}{\frac {\mathrm {d} {\vec {x}}}{\mathrm {d} \xi }}&{\frac {\mathrm {d} {\vec {x}}}{\mathrm {d} \eta }}&{\frac {\mathrm {d} {\vec {x}}}{\mathrm {d} \zeta }}\end{pmatrix}}\,\mathrm {d} \xi \mathrm {d} \eta \mathrm {d} \zeta =\int _{0}^{1}\int _{0}^{1}\int _{0}^{1}\operatorname {det} {\begin{pmatrix}\mathbf {F} \cdot {\frac {\mathrm {d} {\vec {X}}}{\mathrm {d} \xi }}&\mathbf {F} \cdot {\frac {\mathrm {d} {\vec {X}}}{\mathrm {d} \eta }}&\mathbf {F} \cdot {\frac {\mathrm {d} {\vec {X}}}{\mathrm {d} \zeta }}\end{pmatrix}}\,\mathrm {d} \xi \mathrm {d} \eta \mathrm {d} \zeta \\&=&\displaystyle \int _{0}^{1}\int _{0}^{1}\int _{0}^{1}\operatorname {det} \left[\mathbf {F} \cdot {\begin{pmatrix}{\frac {\mathrm {d} {\vec {X}}}{\mathrm {d} \xi }}&{\frac {\mathrm {d} {\vec {X}}}{\mathrm {d} \eta }}&{\frac {\mathrm {d} {\vec {X}}}{\mathrm {d} \zeta }}\end{pmatrix}}\right]\,\mathrm {d} \xi \mathrm {d} \eta \mathrm {d} \zeta =\int _{0}^{1}\int _{0}^{1}\int _{0}^{1}\operatorname {det} (\mathbf {F} )\,\mathrm {d} V=\int _{0}^{1}\int _{0}^{1}\int _{0}^{1}{\sqrt {\operatorname {det} (\mathbf {C} )}}\,\mathrm {d} V\,,\end{array}}}" loading="lazy"></dd></dl>

worin der <a href="Determinante#Determinantenproduktsatz" title="Determinante">Determinantenproduktsatz</a> ausgenutzt wurde. Die Volumenänderung kann also wie bei den materiellen Linien und Flächen mit dem Strecktensor ausgedrückt werden.

<div class="mw-heading mw-heading2"><h2 id="Linker_und_rechter_Strecktensor_und_Drehungen">Linker und rechter Strecktensor und Drehungen</h2></div>

Bei Nicht-Deformation sind die Strecktensoren gleich dem <a href="Einheitstensor" title="Einheitstensor">Einheitstensor</a> und das unabhängig von eventuell auftretenden Drehungen des Körpers. Der Grund hierfür liegt in der <a href="Polarzerlegung" title="Polarzerlegung">Polarzerlegung</a> des Deformationsgradienten

</mrow>

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</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e3d723f8ec523d924fb6991b2fb2eccdd8793e59.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:19.746ex; height:2.509ex;" alt="{\displaystyle \mathbf {F} =\mathbf {R\cdot U} =\mathbf {v\cdot R} \,,}" loading="lazy"></dd></dl>

die die Deformation lokal in eine Drehung, vermittelt durch den <a href="Orthogonaler_Tensor" title="Orthogonaler Tensor">orthogonalen</a> Rotationstensor <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {R} }">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5de85fcc2a00d8ba14aae84aeef812d7fef4b3d5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.003ex; height:2.176ex;" alt="{\displaystyle \mathbf {R} }" loading="lazy"> (mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {R} ^{\top }=\mathbf {R} ^{-1}}">

<msup>

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</mrow>

</msup>

<msup>

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {R} ^{\top }=\mathbf {R} ^{-1}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ba00cb88ca319e1c90cb8e80a21971989d917544.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.949ex; height:2.676ex;" alt="{\displaystyle \mathbf {R} ^{\top }=\mathbf {R} ^{-1}}" loading="lazy"> und der <a href="Determinante" title="Determinante">Determinante</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {det} (\mathbf {R} )=+1}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0304f94cf60ad7c1e52a1beac3eec16939de368b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.111ex; height:2.843ex;" alt="{\displaystyle \operatorname {det} (\mathbf {R} )=+1}" loading="lazy">), 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 <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>

</semantics>

</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"> bzw. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {v} }">

</mrow>

</mstyle>

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

</semantics>

</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">, aufspaltet. Durch die Multiplikation des Deformationsgradienten mit seiner <a href="Transponierte_Matrix" title="Transponierte Matrix">transponierten</a> heben sich die Drehungen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {R} }">

</mrow>

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5de85fcc2a00d8ba14aae84aeef812d7fef4b3d5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.003ex; height:2.176ex;" alt="{\displaystyle \mathbf {R} }" loading="lazy"> und „Rückdrehungen“ <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {R} ^{\top }=\mathbf {R} ^{-1}}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {R} ^{\top }=\mathbf {R} ^{-1}}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle {\begin{array}{ccccccc}\mathbf {F} ^{\top }\cdot \mathbf {F} &=&\mathbf {U\cdot R} ^{\top }\cdot \mathbf {R\cdot U} &=&\mathbf {U\cdot U} &=&\mathbf {C} \\\mathbf {F\cdot F} ^{\top }&=&\mathbf {v\cdot R\cdot R} ^{\top }\cdot \mathbf {v} &=&\mathbf {v\cdot v} &=&\mathbf {b} \end{array}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b6fa68e0a4d294d1fdfc8981e88847b8cf0e6ec7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:49.058ex; height:6.509ex;" alt="{\displaystyle {\begin{array}{ccccccc}\mathbf {F} ^{\top }\cdot \mathbf {F} &=&\mathbf {U\cdot R} ^{\top }\cdot \mathbf {R\cdot U} &=&\mathbf {U\cdot U} &=&\mathbf {C} \\\mathbf {F\cdot F} ^{\top }&=&\mathbf {v\cdot R\cdot R} ^{\top }\cdot \mathbf {v} &=&\mathbf {v\cdot v} &=&\mathbf {b} \end{array}}}" loading="lazy"></dd></dl>

was natürlich auch für die Inversen des rechten und linken Cauchy-Green-Tensors zutrifft. Der rechte und linke Cauchy-Green-Tensor und ihre Inversen sind daher von Drehungen des Körpers unbeeinflusst.

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

Der rechte und linke Strecktensor ebenso wie der rechte und linke Cauchy-Green-Tensor sind also <a href="%C3%84hnlichkeit_(Matrix)" title="Ähnlichkeit (Matrix)">ähnlich</a>, weswegen sie dieselben <a href="Eigenwertproblem" class="mw-redirect" title="Eigenwertproblem">Eigenwerte</a> und daher auch dieselben <a href="Hauptinvariante" title="Hauptinvariante">Hauptinvarianten</a> besitzen. Die Eigenwerte der Strecktensoren werden Hauptstreckungen genannt. Sämtliche Strecktensoren sind symmetrisch positiv definit und daher sind alle drei Eigenwerte positiv und die drei Eigenvektoren sind paarweise zueinander senkrecht (oder orthogonalisierbar), so dass sie eine <a href="Orthonormalbasis" title="Orthonormalbasis">Orthonormalbasis</a> bilden. Seien <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {N}}_{1,2,3}}">

<msub>

<mover>

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9e26ed3dd9ec0e797300d3771574965fc7a82672.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.677ex; height:3.509ex;" alt="{\displaystyle {\vec {N}}_{1,2,3}}" loading="lazy"> die Eigenvektoren von <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">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {n}}_{1,2,3}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fff5a029bbad184e6f818dab98ca9364b2ba6387.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.008ex; height:3.009ex;" alt="{\displaystyle {\vec {n}}_{1,2,3}}" loading="lazy"> die Eigenvektoren von <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"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda _{1,2,3}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6a778b0288d979c9ac3db5a2191d3d4d014d64f2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.968ex; height:2.843ex;" alt="{\displaystyle \lambda _{1,2,3}}" loading="lazy"> dessen Eigenwerte. Dann lauten die Hauptachsentransformationen:

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {U} =\sum _{i=1}^{3}\lambda _{i}{\vec {N}}_{i}\otimes {\vec {N}}_{i}\qquad {\textsf {und}}\qquad \mathbf {v} =\sum _{i=1}^{3}\lambda _{i}{\vec {n}}_{i}\otimes {\vec {n}}_{i}\,.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/56a171fb6bb959297213a548983fd3e10c9a294a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:51.185ex; height:7.176ex;" alt="{\displaystyle \mathbf {U} =\sum _{i=1}^{3}\lambda _{i}{\vec {N}}_{i}\otimes {\vec {N}}_{i}\qquad {\textsf {und}}\qquad \mathbf {v} =\sum _{i=1}^{3}\lambda _{i}{\vec {n}}_{i}\otimes {\vec {n}}_{i}\,.}" loading="lazy"></dd></dl>

Aus <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {v} =\mathbf {R} \cdot \mathbf {U} \cdot \mathbf {R} ^{\top }}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e5bfe485c1f2bfba9475979d6de1289cbb19a615.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:15.442ex; height:2.676ex;" alt="{\displaystyle \mathbf {v} =\mathbf {R} \cdot \mathbf {U} \cdot \mathbf {R} ^{\top }}" loading="lazy"> folgt:

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {R} \cdot {\vec {N}}_{i}={\vec {n}}_{i}\qquad {\text{also}}\qquad \mathbf {R} =\sum _{i=1}^{3}{\vec {n}}_{i}\otimes {\vec {N}}_{i}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1402468d7d8597ceeb4cde648edb575072d7200e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:41.759ex; height:7.176ex;" alt="{\displaystyle \mathbf {R} \cdot {\vec {N}}_{i}={\vec {n}}_{i}\qquad {\text{also}}\qquad \mathbf {R} =\sum _{i=1}^{3}{\vec {n}}_{i}\otimes {\vec {N}}_{i}}" loading="lazy"></dd></dl>

und weiter:

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {F} =\mathbf {R\cdot U} =\mathbf {R} \cdot \sum _{i=1}^{3}\lambda _{i}{\vec {N}}_{i}\otimes {\vec {N}}_{i}=\sum _{i=1}^{3}\lambda _{i}{\vec {n}}_{i}\otimes {\vec {N}}_{i}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/675c7febd2d702d1e7fdbe646548b370fe1183a3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:48.658ex; height:7.176ex;" alt="{\displaystyle \mathbf {F} =\mathbf {R\cdot U} =\mathbf {R} \cdot \sum _{i=1}^{3}\lambda _{i}{\vec {N}}_{i}\otimes {\vec {N}}_{i}=\sum _{i=1}^{3}\lambda _{i}{\vec {n}}_{i}\otimes {\vec {N}}_{i}}" loading="lazy"></dd></dl>

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {C} =\mathbf {U\cdot U} =\sum _{i=1}^{3}\lambda _{i}^{2}{\vec {N}}_{i}\otimes {\vec {N}}_{i}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/450402fbb487d59c7e6a366ef7af08fd82aeb1f9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:28.639ex; height:7.176ex;" alt="{\displaystyle \mathbf {C} =\mathbf {U\cdot U} =\sum _{i=1}^{3}\lambda _{i}^{2}{\vec {N}}_{i}\otimes {\vec {N}}_{i}}" loading="lazy"></dd></dl>

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {f} =\mathbf {C} ^{-1}=\sum _{i=1}^{3}\lambda _{i}^{-2}{\vec {N}}_{i}\otimes {\vec {N}}_{i}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b81e038461b09af840ded0b042e14510f90530d7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:27.511ex; height:7.176ex;" alt="{\displaystyle \mathbf {f} =\mathbf {C} ^{-1}=\sum _{i=1}^{3}\lambda _{i}^{-2}{\vec {N}}_{i}\otimes {\vec {N}}_{i}}" loading="lazy"></dd></dl>

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {b} =\mathbf {v\cdot v} =\sum _{i=1}^{3}\lambda _{i}^{2}{\vec {n}}_{i}\otimes {\vec {n}}_{i}}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {c} =\mathbf {b} ^{-1}=\sum _{i=1}^{3}\lambda _{i}^{-2}{\vec {n}}_{i}\otimes {\vec {n}}_{i}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/090ff2b5b324952e4d2a0c99c999c10a078a190f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:25.862ex; height:7.176ex;" alt="{\displaystyle \mathbf {c} =\mathbf {b} ^{-1}=\sum _{i=1}^{3}\lambda _{i}^{-2}{\vec {n}}_{i}\otimes {\vec {n}}_{i}}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Ableitung_der_Streckungen">Ableitung der Streckungen</h2></div>

Manche Materialmodelle der <a href="Hyperelastizit%C3%A4t" title="Hyperelastizität">Hyperelastizität</a> beinhalten Funktionen der Eigenwerte <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda _{i}}">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/72fde940918edf84caf3d406cc7d31949166820f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.155ex; height:2.509ex;" alt="{\displaystyle \lambda _{i}}" loading="lazy"> des linken Strecktensors <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {v} }">

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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"> und die Spannungen ergeben sich aus der Ableitung dieser Funktionen nach dem linken Cauchy-Green-Tensor <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {b} }">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/13ebf4628a1adf07133a6009e4a78bdd990c6eb9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.485ex; height:2.176ex;" alt="{\displaystyle \mathbf {b} }" loading="lazy">. Deshalb lohnt es sich, die Ableitung der Eigenwerte <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda _{i}}">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/72fde940918edf84caf3d406cc7d31949166820f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.155ex; height:2.509ex;" alt="{\displaystyle \lambda _{i}}" loading="lazy"> nach dem Strecktensor <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {b} }">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/13ebf4628a1adf07133a6009e4a78bdd990c6eb9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.485ex; height:2.176ex;" alt="{\displaystyle \mathbf {b} }" loading="lazy"> bereitzustellen<a href="#cite_note-Frechet-1">[1]</a>. Es ergibt sich

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<annotation encoding="application/x-tex">{\displaystyle {\frac {\mathrm {d} \lambda _{i}}{\mathrm {d} \mathbf {b} }}={\frac {1}{2\lambda _{i}}}{\hat {n}}_{i}\otimes {\hat {n}}_{i}\,.}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1bdc546c5147e6577c0950a14c3ec9ba01f86bbd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:19.799ex; height:5.843ex;" alt="{\displaystyle {\frac {\mathrm {d} \lambda _{i}}{\mathrm {d} \mathbf {b} }}={\frac {1}{2\lambda _{i}}}{\hat {n}}_{i}\otimes {\hat {n}}_{i}\,.}" loading="lazy"></dd></dl>

Entsprechend berechnet sich

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<annotation encoding="application/x-tex">{\displaystyle {\frac {\mathrm {d} \lambda _{i}}{\mathrm {d} \mathbf {C} }}={\frac {1}{2\lambda _{i}}}{\hat {N}}_{i}\otimes {\hat {N}}_{i}\,.}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1a406439f1ee4e21fbd150e363b258c96213c522.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:21.136ex; height:5.843ex;" alt="{\displaystyle {\frac {\mathrm {d} \lambda _{i}}{\mathrm {d} \mathbf {C} }}={\frac {1}{2\lambda _{i}}}{\hat {N}}_{i}\otimes {\hat {N}}_{i}\,.}" loading="lazy"></dd></dl>

<td>Beweis

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<td>Betrachtet werden zunächst die Eigenwerte <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \eta _{i}=\lambda _{i}^{2}}">

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<annotation encoding="application/x-tex">{\displaystyle \eta _{i}=\lambda _{i}^{2}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/54c3de51ef16e12ae8015a22cac9a4eadf446294.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.463ex; height:3.176ex;" alt="{\displaystyle \eta _{i}=\lambda _{i}^{2}}" loading="lazy"> des linken Cauchy-Green-Tensors. Die Eigenwerte <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \eta _{i}}">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b336ed51c970728280e38f5a131ac52f69833c67.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.955ex; height:2.176ex;" alt="{\displaystyle \eta _{i}}" loading="lazy"> lösen das charakteristische Polynom des linken Cauchy-Green-Tensors:

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {det} (\mathbf {b} -\eta _{i}\operatorname {I} )=-\eta _{i}^{3}+\operatorname {I} _{1}\eta _{i}^{2}-\operatorname {I} _{2}\eta _{i}+\operatorname {I} _{3}=0\,.}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e00611cb82fccd6c81de477f827a960c787c2624.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:44.136ex; height:3.176ex;" alt="{\displaystyle \operatorname {det} (\mathbf {b} -\eta _{i}\operatorname {I} )=-\eta _{i}^{3}+\operatorname {I} _{1}\eta _{i}^{2}-\operatorname {I} _{2}\eta _{i}+\operatorname {I} _{3}=0\,.}" loading="lazy">

Die Koeffizienten dieses Polynoms sind die drei <a href="Hauptinvariante" title="Hauptinvariante">Hauptinvarianten</a> des linken Cauchy-Green-Tensors. <a href="Implizite_Differentiation" title="Implizite Differentiation">Implizite</a> Differentiation des charakteristischen Polynoms unter Benutzung der <a href="Kettenregel" title="Kettenregel">Kettenregel</a> und der <a href="Hauptinvariante#Ableitungen_der_Hauptinvarianten" title="Hauptinvariante">Ableitungen der Hauptinvarianten</a> bei <a href="Symmetrische_Matrix" title="Symmetrische Matrix">symmetrischen</a> Tensoren

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<annotation encoding="application/x-tex">{\displaystyle {\frac {\mathrm {d} \operatorname {I} _{1}(\mathbf {b} )}{\mathrm {d} \mathbf {b} }}=\mathbf {I} ,\quad {\frac {\mathrm {d} \operatorname {I} _{2}(\mathbf {b} )}{\mathrm {d} \mathbf {b} }}=\operatorname {I} _{1}\mathbf {I} -\mathbf {b} \quad {\text{und}}\quad {\frac {\mathrm {d} \operatorname {I} _{3}(\mathbf {b} )}{\mathrm {d} \mathbf {b} }}=\operatorname {I} _{3}\mathbf {b} ^{-1}=\mathbf {b\cdot b} -\operatorname {I} _{1}\mathbf {b} +\operatorname {I} _{2}\mathbf {I} }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4a8d1b30447ee84f94f1b6fbd29f9538e9190592.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:79.511ex; height:5.843ex;" alt="{\displaystyle {\frac {\mathrm {d} \operatorname {I} _{1}(\mathbf {b} )}{\mathrm {d} \mathbf {b} }}=\mathbf {I} ,\quad {\frac {\mathrm {d} \operatorname {I} _{2}(\mathbf {b} )}{\mathrm {d} \mathbf {b} }}=\operatorname {I} _{1}\mathbf {I} -\mathbf {b} \quad {\text{und}}\quad {\frac {\mathrm {d} \operatorname {I} _{3}(\mathbf {b} )}{\mathrm {d} \mathbf {b} }}=\operatorname {I} _{3}\mathbf {b} ^{-1}=\mathbf {b\cdot b} -\operatorname {I} _{1}\mathbf {b} +\operatorname {I} _{2}\mathbf {I} }" loading="lazy">

liefert

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}0=&-3\eta _{i}^{2}{\frac {\mathrm {d} \eta _{i}}{\mathrm {d} \mathbf {b} }}+\eta _{i}^{2}\mathbf {I} +2\operatorname {I} _{1}\eta _{i}{\frac {\mathrm {d} \eta _{i}}{\mathrm {d} \mathbf {b} }}-(\operatorname {I} _{1}\mathbf {I} -\mathbf {b} )\eta _{i}-\operatorname {I} _{2}{\frac {\mathrm {d} \eta _{i}}{\mathrm {d} \mathbf {b} }}+\mathbf {b\cdot b} -\operatorname {I} _{1}\mathbf {b} +\operatorname {I} _{2}\mathbf {I} \\=&-(3\eta _{i}^{2}-2\operatorname {I} _{1}\eta _{i}+\operatorname {I} _{2}){\frac {\mathrm {d} \eta _{i}}{\mathrm {d} \mathbf {b} }}+(\eta _{i}^{2}-\eta _{i}\operatorname {I} _{1}+\operatorname {I} _{2})\mathbf {I} +(\eta _{i}-\operatorname {I} _{1})\mathbf {b} +\mathbf {b\cdot b} \\\rightarrow {\frac {\mathrm {d} \eta _{i}}{\mathrm {d} \mathbf {b} }}=&{\frac {(\eta _{i}^{2}-\eta _{i}\operatorname {I} _{1}+\operatorname {I} _{2})\mathbf {I} +(\eta _{i}-\operatorname {I} _{1})\mathbf {b} +\mathbf {b\cdot b} }{3\eta _{i}^{2}-2\operatorname {I} _{1}\eta _{i}+\operatorname {I} _{2}}}\end{aligned}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/daabee8037df1278fa8f5362b968adfcb54cc4fa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -8.671ex; width:83.027ex; height:18.509ex;" alt="{\displaystyle {\begin{aligned}0=&-3\eta _{i}^{2}{\frac {\mathrm {d} \eta _{i}}{\mathrm {d} \mathbf {b} }}+\eta _{i}^{2}\mathbf {I} +2\operatorname {I} _{1}\eta _{i}{\frac {\mathrm {d} \eta _{i}}{\mathrm {d} \mathbf {b} }}-(\operatorname {I} _{1}\mathbf {I} -\mathbf {b} )\eta _{i}-\operatorname {I} _{2}{\frac {\mathrm {d} \eta _{i}}{\mathrm {d} \mathbf {b} }}+\mathbf {b\cdot b} -\operatorname {I} _{1}\mathbf {b} +\operatorname {I} _{2}\mathbf {I} \\=&-(3\eta _{i}^{2}-2\operatorname {I} _{1}\eta _{i}+\operatorname {I} _{2}){\frac {\mathrm {d} \eta _{i}}{\mathrm {d} \mathbf {b} }}+(\eta _{i}^{2}-\eta _{i}\operatorname {I} _{1}+\operatorname {I} _{2})\mathbf {I} +(\eta _{i}-\operatorname {I} _{1})\mathbf {b} +\mathbf {b\cdot b} \\\rightarrow {\frac {\mathrm {d} \eta _{i}}{\mathrm {d} \mathbf {b} }}=&{\frac {(\eta _{i}^{2}-\eta _{i}\operatorname {I} _{1}+\operatorname {I} _{2})\mathbf {I} +(\eta _{i}-\operatorname {I} _{1})\mathbf {b} +\mathbf {b\cdot b} }{3\eta _{i}^{2}-2\operatorname {I} _{1}\eta _{i}+\operatorname {I} _{2}}}\end{aligned}}}" loading="lazy">

Einsetzen der Formeln von Vieta

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {I} _{1}=\eta _{1}+\eta _{2}+\eta _{3}\quad {\text{und}}\quad \operatorname {I} _{2}=\eta _{1}\eta _{2}+\eta _{2}\eta _{3}+\eta _{3}\eta _{1}\,,}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e76e98c39f55577d475bd844e5a1bc476490b8fe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:51.178ex; height:2.676ex;" alt="{\displaystyle \operatorname {I} _{1}=\eta _{1}+\eta _{2}+\eta _{3}\quad {\text{und}}\quad \operatorname {I} _{2}=\eta _{1}\eta _{2}+\eta _{2}\eta _{3}+\eta _{3}\eta _{1}\,,}" loading="lazy">

der Hauptachsentransformation des linken Cauchy-Green-Tensors

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {b} =\eta _{1}{\hat {n}}_{1}\otimes {\hat {n}}_{1}+\eta _{2}{\hat {n}}_{2}\otimes {\hat {n}}_{2}+\eta _{3}{\hat {n}}_{3}\otimes {\hat {n}}_{3}}</annotation>

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mit seinen auf Betrag eins normierten, paarweise orthogonalen Eigenvektoren <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {n}}_{1,2,3}\,,}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/aaa8e893d7896cd8a22c02765398f91c75ef3628.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.042ex; height:2.843ex;" alt="{\displaystyle {\hat {n}}_{1,2,3}\,,}" loading="lazy"> die mit den Eigenvektoren des linken Strecktensors übereinstimmen, und der Form <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {I} ={\hat {n}}_{1}\otimes {\hat {n}}_{1}+{\hat {n}}_{2}\otimes {\hat {n}}_{2}+{\hat {n}}_{3}\otimes {\hat {n}}_{3}}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {I} ={\hat {n}}_{1}\otimes {\hat {n}}_{1}+{\hat {n}}_{2}\otimes {\hat {n}}_{2}+{\hat {n}}_{3}\otimes {\hat {n}}_{3}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/aaa85a3b3ab90da99b627913f5c08cdc45abad3c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:33.008ex; height:2.509ex;" alt="{\displaystyle \mathbf {I} ={\hat {n}}_{1}\otimes {\hat {n}}_{1}+{\hat {n}}_{2}\otimes {\hat {n}}_{2}+{\hat {n}}_{3}\otimes {\hat {n}}_{3}}" loading="lazy"> des Einheitstensors ergibt beispielsweise für i=1:

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\frac {\mathrm {d} \eta _{1}}{\mathrm {d} \mathbf {b} }}=&{\frac {\eta _{1}^{2}-\eta _{1}(\eta _{1}+\eta _{2}+\eta _{3})+\eta _{1}\eta _{2}+\eta _{2}\eta _{3}+\eta _{3}\eta _{1}}{3\eta _{1}^{2}-2(\eta _{1}+\eta _{2}+\eta _{3})\eta _{1}+\eta _{1}\eta _{2}+\eta _{2}\eta _{3}+\eta _{3}\eta _{1}}}({\hat {n}}_{1}\otimes {\hat {n}}_{1}+{\hat {n}}_{2}\otimes {\hat {n}}_{2}+{\hat {n}}_{3}\otimes {\hat {n}}_{3})\\&+{\frac {[\eta _{1}-(\eta _{1}+\eta _{2}+\eta _{3})](\eta _{1}{\hat {n}}_{1}\otimes {\hat {n}}_{1}+\eta _{2}{\hat {n}}_{2}\otimes {\hat {n}}_{2}+\eta _{3}{\hat {n}}_{3}\otimes {\hat {n}}_{3})}{3\eta _{1}^{2}-2(\eta _{1}+\eta _{2}+\eta _{3})\eta _{1}+\eta _{1}\eta _{2}+\eta _{2}\eta _{3}+\eta _{3}\eta _{1}}}\\&+{\frac {\eta _{1}^{2}{\hat {n}}_{1}\otimes {\hat {n}}_{1}+\eta _{2}^{2}{\hat {n}}_{2}\otimes {\hat {n}}_{2}+\eta _{3}^{2}{\hat {n}}_{3}\otimes {\hat {n}}_{3}}{3\eta _{1}^{2}-2(\eta _{1}+\eta _{2}+\eta _{3})\eta _{1}+\eta _{1}\eta _{2}+\eta _{2}\eta _{3}+\eta _{3}\eta _{1}}}\\=&{\hat {n}}_{1}\otimes {\hat {n}}_{1}\end{aligned}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/96ed6332daab42eb41a9e836c86a23759cf6a17e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -11.505ex; width:84.586ex; height:24.176ex;" alt="{\displaystyle {\begin{aligned}{\frac {\mathrm {d} \eta _{1}}{\mathrm {d} \mathbf {b} }}=&{\frac {\eta _{1}^{2}-\eta _{1}(\eta _{1}+\eta _{2}+\eta _{3})+\eta _{1}\eta _{2}+\eta _{2}\eta _{3}+\eta _{3}\eta _{1}}{3\eta _{1}^{2}-2(\eta _{1}+\eta _{2}+\eta _{3})\eta _{1}+\eta _{1}\eta _{2}+\eta _{2}\eta _{3}+\eta _{3}\eta _{1}}}({\hat {n}}_{1}\otimes {\hat {n}}_{1}+{\hat {n}}_{2}\otimes {\hat {n}}_{2}+{\hat {n}}_{3}\otimes {\hat {n}}_{3})\\&+{\frac {[\eta _{1}-(\eta _{1}+\eta _{2}+\eta _{3})](\eta _{1}{\hat {n}}_{1}\otimes {\hat {n}}_{1}+\eta _{2}{\hat {n}}_{2}\otimes {\hat {n}}_{2}+\eta _{3}{\hat {n}}_{3}\otimes {\hat {n}}_{3})}{3\eta _{1}^{2}-2(\eta _{1}+\eta _{2}+\eta _{3})\eta _{1}+\eta _{1}\eta _{2}+\eta _{2}\eta _{3}+\eta _{3}\eta _{1}}}\\&+{\frac {\eta _{1}^{2}{\hat {n}}_{1}\otimes {\hat {n}}_{1}+\eta _{2}^{2}{\hat {n}}_{2}\otimes {\hat {n}}_{2}+\eta _{3}^{2}{\hat {n}}_{3}\otimes {\hat {n}}_{3}}{3\eta _{1}^{2}-2(\eta _{1}+\eta _{2}+\eta _{3})\eta _{1}+\eta _{1}\eta _{2}+\eta _{2}\eta _{3}+\eta _{3}\eta _{1}}}\\=&{\hat {n}}_{1}\otimes {\hat {n}}_{1}\end{aligned}}}" loading="lazy">

Für i=2 und i=3 berechnet sich entsprechendes, womit

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<annotation encoding="application/x-tex">{\displaystyle {\frac {\mathrm {d} \eta _{i}}{\mathrm {d} \mathbf {b} }}={\hat {n}}_{i}\otimes {\hat {n}}_{i}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/43be6224777ee491202e0739ec510a54cffafed0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:14.411ex; height:5.509ex;" alt="{\displaystyle {\frac {\mathrm {d} \eta _{i}}{\mathrm {d} \mathbf {b} }}={\hat {n}}_{i}\otimes {\hat {n}}_{i}}" loading="lazy">

feststeht. Die gesuchte Ableitung der Eigenwerte des linken Strecktensors nach dem linken-Cauchy-Green Tensor ermittelt sich schließlich aus

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<annotation encoding="application/x-tex">{\displaystyle {\frac {\mathrm {d} \eta _{i}}{\mathrm {d} \mathbf {b} }}={\frac {\mathrm {d} \lambda _{i}^{2}}{\mathrm {d} \mathbf {b} }}=2\lambda _{i}{\frac {\mathrm {d} \lambda _{i}}{\mathrm {d} \mathbf {b} }}={\hat {n}}_{i}\otimes {\hat {n}}_{i}\quad \rightarrow \quad {\frac {\mathrm {d} \lambda _{i}}{\mathrm {d} \mathbf {b} }}={\frac {1}{2\lambda _{i}}}{\hat {n}}_{i}\otimes {\hat {n}}_{i}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a1a38a46057e1c67644f61f2c203c26dc31e2eb0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:59.771ex; height:6.343ex;" alt="{\displaystyle {\frac {\mathrm {d} \eta _{i}}{\mathrm {d} \mathbf {b} }}={\frac {\mathrm {d} \lambda _{i}^{2}}{\mathrm {d} \mathbf {b} }}=2\lambda _{i}{\frac {\mathrm {d} \lambda _{i}}{\mathrm {d} \mathbf {b} }}={\hat {n}}_{i}\otimes {\hat {n}}_{i}\quad \rightarrow \quad {\frac {\mathrm {d} \lambda _{i}}{\mathrm {d} \mathbf {b} }}={\frac {1}{2\lambda _{i}}}{\hat {n}}_{i}\otimes {\hat {n}}_{i}}" loading="lazy">

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

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

Ein Quadrat der Seitenlänge eins wird zu einem Rechteck mit Breite <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f11423fbb2e967f986e36804a8ae4271734917c3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}" loading="lazy"> und Höhe <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h}">

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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"> gestreckt und um einen Winkel <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b79333175c8b3f0840bfb4ec41b8072c83ea88d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }" loading="lazy"> verdreht, siehe die Abbildung rechts. Das Quadrat sei im Ursprung eines <a href="Kartesisches_Koordinatensystem" title="Kartesisches Koordinatensystem">kartesischen Koordinatensystems</a> positioniert, so dass für die Punkte des Quadrates

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<annotation encoding="application/x-tex">{\displaystyle {\vec {X}}=\left({\begin{array}{c}X\\Y\end{array}}\right)\in {[{0,1}]}^{2}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8b011e853b60af540637e22e98c8274c026e2884.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:19.779ex; height:6.176ex;" alt="{\displaystyle {\vec {X}}=\left({\begin{array}{c}X\\Y\end{array}}\right)\in {[{0,1}]}^{2}}" loading="lazy"></dd></dl>

gilt. Im deformierten Zustand ist dann

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<annotation encoding="application/x-tex">{\displaystyle {\begin{array}{rcl}{\vec {x}}&=&\left({\begin{array}{c}x\\y\end{array}}\right)=\left({\begin{array}{cc}\cos \alpha &-\sin \alpha \\\sin \alpha &\cos \alpha \end{array}}\right)\left({\begin{array}{c}bX\\hY\end{array}}\right)\\&=&\left({\begin{array}{c}bX\cos \alpha -hY\sin \alpha \\bX\sin \alpha +hY\cos \alpha \end{array}}\right)\end{array}}\,.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/783304639bd21a0839a0a1b860d9f62abdb7a1ab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.671ex; width:44.248ex; height:12.509ex;" alt="{\displaystyle {\begin{array}{rcl}{\vec {x}}&=&\left({\begin{array}{c}x\\y\end{array}}\right)=\left({\begin{array}{cc}\cos \alpha &-\sin \alpha \\\sin \alpha &\cos \alpha \end{array}}\right)\left({\begin{array}{c}bX\\hY\end{array}}\right)\\&=&\left({\begin{array}{c}bX\cos \alpha -hY\sin \alpha \\bX\sin \alpha +hY\cos \alpha \end{array}}\right)\end{array}}\,.}" loading="lazy"></dd></dl>

Damit berechnen sich der Deformationsgradient und die Strecktensoren zu

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<annotation encoding="application/x-tex">{\displaystyle {\begin{array}{rcl}\mathbf {F} &=&\left({\begin{array}{cc}b\cos \alpha &-h\sin \alpha \\b\sin \alpha &h\cos \alpha \end{array}}\right)=\left({\begin{array}{cc}\cos \alpha &-\sin \alpha \\\sin \alpha &\cos \alpha \end{array}}\right)\left({\begin{array}{cc}b&0\\0&h\end{array}}\right)\\\rightarrow \mathbf {R} &=&\left({\begin{array}{cc}\cos \alpha &-\sin \alpha \\\sin \alpha &\cos \alpha \end{array}}\right)\\\rightarrow \mathbf {U} &=&\left({\begin{array}{cc}b&0\\0&h\end{array}}\right),\lambda _{1}=b,{\vec {N}}_{1}=\left({\begin{array}{c}1\\0\end{array}}\right),\lambda _{2}=h,{\vec {N}}_{2}=\left({\begin{array}{c}0\\1\end{array}}\right)\\\rightarrow \mathbf {C} &=&\mathbf {U\cdot U} =\left({\begin{array}{cc}b&0\\0&h\end{array}}\right)\left({\begin{array}{cc}b&0\\0&h\end{array}}\right)=\left({\begin{array}{cc}b^{2}&0\\0&h^{2}\end{array}}\right)=\mathbf {F} ^{\top }\cdot \mathbf {F} \\\rightarrow \mathbf {v} &=&b\mathbf {R} \cdot {\vec {N}}_{1}\otimes \mathbf {R} \cdot {\vec {N}}_{1}+h\mathbf {R} \cdot {\vec {N}}_{2}\otimes \mathbf {R} \cdot {\vec {N}}_{2}\\&=&b\left({\begin{array}{c}\cos \alpha \\\sin \alpha \end{array}}\right)\otimes \left({\begin{array}{c}\cos \alpha \\\sin \alpha \end{array}}\right)+h\left({\begin{array}{c}-\sin \alpha \\\cos \alpha \end{array}}\right)\otimes \left({\begin{array}{c}-\sin \alpha \\\cos \alpha \end{array}}\right)\\&=&\left({\begin{array}{cc}b{\cos }^{2}\alpha +h{\sin }^{2}\alpha &\left(b-h\right)\cos \alpha \sin \alpha \\\left(b-h\right)\sin \alpha \cos \alpha &b{\sin }^{2}\alpha +h{\cos }^{2}\alpha \end{array}}\right)\\&&\rightarrow \lambda _{1}=b,{\vec {n}}_{1}=\left({\begin{array}{c}\cos \alpha \\\sin \alpha \end{array}}\right),\lambda _{2}=h,{\vec {n}}_{2}=\left({\begin{array}{c}-\sin \alpha \\\cos \alpha \end{array}}\right)\\\rightarrow \mathbf {b} &=&\mathbf {v\cdot v} =\left({\begin{array}{cc}b^{2}{\cos }^{2}\alpha +h^{2}{\sin }^{2}\alpha &\left(b^{2}-h^{2}\right)\sin \alpha \cos \alpha \\\left(b^{2}-h^{2}\right)\sin \alpha \cos \alpha &b^{2}{\sin }^{2}\alpha +h^{2}{\cos }^{2}\alpha \end{array}}\right)=\mathbf {F\cdot F} ^{\top }\end{array}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/670d22828b33d76b820455dd17f5d7e86ce6f93b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -28.171ex; width:76.05ex; height:57.509ex;" alt="{\displaystyle {\begin{array}{rcl}\mathbf {F} &=&\left({\begin{array}{cc}b\cos \alpha &-h\sin \alpha \\b\sin \alpha &h\cos \alpha \end{array}}\right)=\left({\begin{array}{cc}\cos \alpha &-\sin \alpha \\\sin \alpha &\cos \alpha \end{array}}\right)\left({\begin{array}{cc}b&0\\0&h\end{array}}\right)\\\rightarrow \mathbf {R} &=&\left({\begin{array}{cc}\cos \alpha &-\sin \alpha \\\sin \alpha &\cos \alpha \end{array}}\right)\\\rightarrow \mathbf {U} &=&\left({\begin{array}{cc}b&0\\0&h\end{array}}\right),\lambda _{1}=b,{\vec {N}}_{1}=\left({\begin{array}{c}1\\0\end{array}}\right),\lambda _{2}=h,{\vec {N}}_{2}=\left({\begin{array}{c}0\\1\end{array}}\right)\\\rightarrow \mathbf {C} &=&\mathbf {U\cdot U} =\left({\begin{array}{cc}b&0\\0&h\end{array}}\right)\left({\begin{array}{cc}b&0\\0&h\end{array}}\right)=\left({\begin{array}{cc}b^{2}&0\\0&h^{2}\end{array}}\right)=\mathbf {F} ^{\top }\cdot \mathbf {F} \\\rightarrow \mathbf {v} &=&b\mathbf {R} \cdot {\vec {N}}_{1}\otimes \mathbf {R} \cdot {\vec {N}}_{1}+h\mathbf {R} \cdot {\vec {N}}_{2}\otimes \mathbf {R} \cdot {\vec {N}}_{2}\\&=&b\left({\begin{array}{c}\cos \alpha \\\sin \alpha \end{array}}\right)\otimes \left({\begin{array}{c}\cos \alpha \\\sin \alpha \end{array}}\right)+h\left({\begin{array}{c}-\sin \alpha \\\cos \alpha \end{array}}\right)\otimes \left({\begin{array}{c}-\sin \alpha \\\cos \alpha \end{array}}\right)\\&=&\left({\begin{array}{cc}b{\cos }^{2}\alpha +h{\sin }^{2}\alpha &\left(b-h\right)\cos \alpha \sin \alpha \\\left(b-h\right)\sin \alpha \cos \alpha &b{\sin }^{2}\alpha +h{\cos }^{2}\alpha \end{array}}\right)\\&&\rightarrow \lambda _{1}=b,{\vec {n}}_{1}=\left({\begin{array}{c}\cos \alpha \\\sin \alpha \end{array}}\right),\lambda _{2}=h,{\vec {n}}_{2}=\left({\begin{array}{c}-\sin \alpha \\\cos \alpha \end{array}}\right)\\\rightarrow \mathbf {b} &=&\mathbf {v\cdot v} =\left({\begin{array}{cc}b^{2}{\cos }^{2}\alpha +h^{2}{\sin }^{2}\alpha &\left(b^{2}-h^{2}\right)\sin \alpha \cos \alpha \\\left(b^{2}-h^{2}\right)\sin \alpha \cos \alpha &b^{2}{\sin }^{2}\alpha +h^{2}{\cos }^{2}\alpha \end{array}}\right)=\mathbf {F\cdot F} ^{\top }\end{array}}}" loading="lazy"></dd></dl>

In der Mitte des Quadrates wird eine gerade Linie der Länge ½ in einem Winkel <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta }">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7ed48a5e36207156fb792fa79d29925d2f7901e8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.332ex; height:2.509ex;" alt="{\displaystyle \beta }" loading="lazy"> zur x-Achse markiert, siehe Abbildung. Die Punkte auf der Linie haben in der Ausgangslage dann für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s\in [{0,1}]}">

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<annotation encoding="application/x-tex">{\displaystyle s\in [{0,1}]}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ca70db75212a08e3912c54b2e74735f4f8dd6074.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.584ex; height:2.843ex;" alt="{\displaystyle s\in [{0,1}]}" loading="lazy"> die Koordinaten:

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<annotation encoding="application/x-tex">{\displaystyle {\begin{array}{rcl}{\vec {X}}\left(s\right)&=&{\dfrac {1}{2}}\left({\begin{array}{c}1+s\,\cos \beta \\1+s\,\sin \beta \end{array}}\right)\\\rightarrow {\dfrac {\mathrm {d} {\vec {X}}(s)}{\mathrm {d} s}}&=&{\dfrac {1}{2}}\left({\begin{array}{c}\cos \beta \\\sin \beta \end{array}}\right)\rightarrow \left|{\dfrac {\mathrm {d} {\vec {X}}\left(s\right)}{\mathrm {d} s}}\right|={\dfrac {1}{2}}\end{array}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c87ac5a4cae9fc8e2c90d41435f69442c19e1f91.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.505ex; width:45.971ex; height:14.176ex;" alt="{\displaystyle {\begin{array}{rcl}{\vec {X}}\left(s\right)&=&{\dfrac {1}{2}}\left({\begin{array}{c}1+s\,\cos \beta \\1+s\,\sin \beta \end{array}}\right)\\\rightarrow {\dfrac {\mathrm {d} {\vec {X}}(s)}{\mathrm {d} s}}&=&{\dfrac {1}{2}}\left({\begin{array}{c}\cos \beta \\\sin \beta \end{array}}\right)\rightarrow \left|{\dfrac {\mathrm {d} {\vec {X}}\left(s\right)}{\mathrm {d} s}}\right|={\dfrac {1}{2}}\end{array}}}" loading="lazy"></dd></dl>

Die Länge der Linie ist definitionsgemäß unabhängig von deren Richtung:

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<annotation encoding="application/x-tex">{\displaystyle L(\beta )=\int _{0}^{1}\left|{\dfrac {\mathrm {d} {\vec {X}}}{\mathrm {d} s}}\right|\mathrm {d} s=\int _{0}^{1}{\dfrac {1}{2}}\mathrm {d} s={\dfrac {1}{2}}\,.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c6d0a629f8d09e7c7cdc38a02e87c5e43405a36c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:37.417ex; height:7.176ex;" alt="{\displaystyle L(\beta )=\int _{0}^{1}\left|{\dfrac {\mathrm {d} {\vec {X}}}{\mathrm {d} s}}\right|\mathrm {d} s=\int _{0}^{1}{\dfrac {1}{2}}\mathrm {d} s={\dfrac {1}{2}}\,.}" loading="lazy"></dd></dl>

In der deformierten Lage haben die Punkte die Koordinaten

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<annotation encoding="application/x-tex">{\displaystyle {\begin{array}{rcl}{\vec {x}}(s)&=&\left({\begin{array}{c}x\\y\end{array}}\right)={\dfrac {1}{2}}\left({\begin{array}{c}b(1+s\cos \beta )\cos \alpha -h(1+s\,\sin \beta )\sin \alpha \\b(1+s\,\cos \beta )\sin \alpha +h(1+s\,\sin \beta )\cos \alpha \end{array}}\right)\\\rightarrow {\dfrac {\mathrm {d} {\vec {x}}}{\mathrm {d} s}}&=&{\dfrac {1}{2}}\left({\begin{array}{c}b\cos \alpha \cos \beta -h\sin \alpha \sin \beta \\b\sin \alpha \cos \beta +h\cos \alpha \sin \beta \end{array}}\right)=\left({\begin{array}{cc}b\cos \alpha &-h\sin \alpha \\b\sin \alpha &h\cos \alpha \end{array}}\right){\dfrac {1}{2}}\left({\begin{array}{c}\cos \beta \\\sin \beta \end{array}}\right)=\mathbf {F} \cdot {\dfrac {\mathrm {d} {\vec {X}}}{\mathrm {d} s}}\\\rightarrow \left|{\dfrac {\mathrm {d} {\vec {x}}}{\mathrm {d} s}}\right|&=&{\dfrac {1}{2}}{\sqrt {b^{2}{\cos }^{2}\beta +h^{2}{\sin }^{2}\beta }}\\&=&{\sqrt {{\dfrac {1}{2}}\left({\begin{array}{c}\cos \beta \\\sin \beta \end{array}}\right)\cdot \left({\begin{array}{cc}b^{2}&0\\0&h^{2}\end{array}}\right){\dfrac {1}{2}}\left({\begin{array}{c}\cos \beta \\\sin \beta \end{array}}\right)}}={\sqrt {{\dfrac {\mathrm {d} {\vec {X}}}{\mathrm {d} s}}\cdot \mathbf {C} \cdot {\dfrac {\mathrm {d} {\vec {X}}}{\mathrm {d} s}}}}\end{array}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e80a5e8254837a71c8fb05cd7844a7ec4840c134.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -13.305ex; margin-bottom: -0.199ex; width:93.904ex; height:28.176ex;" alt="{\displaystyle {\begin{array}{rcl}{\vec {x}}(s)&=&\left({\begin{array}{c}x\\y\end{array}}\right)={\dfrac {1}{2}}\left({\begin{array}{c}b(1+s\cos \beta )\cos \alpha -h(1+s\,\sin \beta )\sin \alpha \\b(1+s\,\cos \beta )\sin \alpha +h(1+s\,\sin \beta )\cos \alpha \end{array}}\right)\\\rightarrow {\dfrac {\mathrm {d} {\vec {x}}}{\mathrm {d} s}}&=&{\dfrac {1}{2}}\left({\begin{array}{c}b\cos \alpha \cos \beta -h\sin \alpha \sin \beta \\b\sin \alpha \cos \beta +h\cos \alpha \sin \beta \end{array}}\right)=\left({\begin{array}{cc}b\cos \alpha &-h\sin \alpha \\b\sin \alpha &h\cos \alpha \end{array}}\right){\dfrac {1}{2}}\left({\begin{array}{c}\cos \beta \\\sin \beta \end{array}}\right)=\mathbf {F} \cdot {\dfrac {\mathrm {d} {\vec {X}}}{\mathrm {d} s}}\\\rightarrow \left|{\dfrac {\mathrm {d} {\vec {x}}}{\mathrm {d} s}}\right|&=&{\dfrac {1}{2}}{\sqrt {b^{2}{\cos }^{2}\beta +h^{2}{\sin }^{2}\beta }}\\&=&{\sqrt {{\dfrac {1}{2}}\left({\begin{array}{c}\cos \beta \\\sin \beta \end{array}}\right)\cdot \left({\begin{array}{cc}b^{2}&0\\0&h^{2}\end{array}}\right){\dfrac {1}{2}}\left({\begin{array}{c}\cos \beta \\\sin \beta \end{array}}\right)}}={\sqrt {{\dfrac {\mathrm {d} {\vec {X}}}{\mathrm {d} s}}\cdot \mathbf {C} \cdot {\dfrac {\mathrm {d} {\vec {X}}}{\mathrm {d} s}}}}\end{array}}}" loading="lazy"></dd></dl>

weswegen sich die Länge der Linie zu

<mtr>

<mtd>

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<mtd>

</mtd>

<mtd>

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</msubsup>

<mrow>

<mfrac>

<mrow>

</mrow>

<mover>

</mover>

</mrow>

<mrow>

</mrow>

</mrow>

</mfrac>

</mstyle>

</mrow>

</mrow>

</mrow>

</mrow>

</mrow>

</msubsup>

<mfrac>

</mfrac>

</mstyle>

</mrow>

<msqrt>

<msup>

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</msup>

<msup>

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</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

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</msqrt>

</mrow>

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</mstyle>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

<mfrac>

</mfrac>

</mstyle>

</mrow>

<msqrt>

<msup>

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</msup>

<msup>

</mrow>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</mrow>

</msup>

</msqrt>

</mrow>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{array}{rcl}l(\beta )&=&\displaystyle \int _{0}^{1}\left|{\dfrac {\mathrm {d} {\vec {x}}}{\mathrm {d} s}}\right|\mathrm {d} s=\int _{0}^{1}{\dfrac {1}{2}}{\sqrt {b^{2}{\cos }^{2}\beta +h^{2}{\sin }^{2}\beta }}\mathrm {d} s\\&=&{\dfrac {1}{2}}{\sqrt {b^{2}{\cos }^{2}\beta +h^{2}{\sin }^{2}\beta }}\end{array}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8f45a73cd79c869aeded2d0fdc4d304220905ed4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.338ex; width:54.201ex; height:11.843ex;" alt="{\displaystyle {\begin{array}{rcl}l(\beta )&=&\displaystyle \int _{0}^{1}\left|{\dfrac {\mathrm {d} {\vec {x}}}{\mathrm {d} s}}\right|\mathrm {d} s=\int _{0}^{1}{\dfrac {1}{2}}{\sqrt {b^{2}{\cos }^{2}\beta +h^{2}{\sin }^{2}\beta }}\mathrm {d} s\\&=&{\dfrac {1}{2}}{\sqrt {b^{2}{\cos }^{2}\beta +h^{2}{\sin }^{2}\beta }}\end{array}}}" loading="lazy"></dd></dl>

verändert. Das Ergebnis ist wiederum unabhängig vom Drehwinkel <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }">

</mstyle>

</mrow>

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

</semantics>

<mfrac>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle b\cdot h=1\rightarrow h={\dfrac {1}{b}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0645242ec34f660bb1256456d4b623d9decf773a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:18.327ex; height:5.343ex;" alt="{\displaystyle b\cdot h=1\rightarrow h={\dfrac {1}{b}}}" loading="lazy"></dd></dl>

und die Längen der deformierten Linie bilden in einem Polardiagramm eine Kurve wie in der Abbildung rechts. Dort ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b=1,5}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle b=1,5}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/01290f79ac07201734d9ef64d392120e9e296c2f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.455ex; height:2.509ex;" alt="{\displaystyle b=1,5}" loading="lazy">.

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

Mechanik:

<ul><li><a href="Konvektive_Koordinaten" title="Konvektive Koordinaten">Konvektive Koordinaten</a></li>

<li><a href="Konfiguration_(Mechanik)" title="Konfiguration (Mechanik)">Konfiguration (Mechanik)</a></li></ul>

Mathematik:

<ul><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="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 skalaren Funktion <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(\mathbf {T} )}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a23c54679dd35f3d85a0c49ebe9e80c45ab2d87d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.947ex; height:2.843ex;" alt="{\displaystyle f(\mathbf {T} )}" loading="lazy"> nach einem Tensor <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {T} }">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9593e3b995a1b57c078873a5ea186c7012e1a5ee.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.859ex; height:2.176ex;" alt="{\displaystyle \mathbf {T} }" loading="lazy">

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

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0795cc96c75d81520a120482662b90f024c9a1a1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.019ex; height:2.176ex;" alt="{\displaystyle \mathbf {A} }" loading="lazy"> für den - sofern er existiert - gilt:

</mrow>

</mrow>

<msub>

<mrow>

<mrow>

<mfrac>

</mrow>

<mrow>

</mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

</mrow>

</mrow>

</mrow>

</msub>

</mrow>

</munder>

<mfrac>

<mrow>

</mrow>

</mrow>

</mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/dc67d762a9992a7e0a390acfd075cdf4e08802ee.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:59.752ex; height:6.176ex;" alt="{\displaystyle \mathbf {A} :\mathbf {H} =\left.{\frac {\mathrm {d} }{\mathrm {d} s}}f(\mathbf {T} +s\mathbf {H} )\right|_{s=0}=\lim _{s\rightarrow 0}{\frac {f(\mathbf {T} +s\mathbf {H} )-f(\mathbf {T} )}{s}}\quad \forall \;\mathbf {H} }" loading="lazy"></dd></dl>

Darin ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s\in \mathbb {R} }">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/36efff902c6854b1196e79dec095b31e0c6a8ee9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.609ex; height:2.176ex;" alt="{\displaystyle s\in \mathbb {R} }" loading="lazy"> und ":" das <a href="Frobenius-Skalarprodukt" title="Frobenius-Skalarprodukt">Frobenius-Skalarprodukt</a>. Dann wird auch

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/710684048f01795e32460d2156e6272786d3f07d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:9.131ex; height:5.676ex;" alt="{\displaystyle {\frac {\partial f}{\partial \mathbf {T} }}=\mathbf {A} }" loading="lazy"></dd></dl>

geschrieben.

</li>

</ol>

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

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

<li>P. Haupt: <cite style="font-style:italic">Continuum Mechanics and Theory of Materials</cite>. Springer, 2010, ISBN 978-3-642-07718-0.<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:Strecktensor&rft.au=P.+Haupt&rft.btitle=Continuum+Mechanics+and+Theory+of+Materials&rft.date=2010&rft.genre=book&rft.isbn=9783642077180&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:Strecktensor&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>

<li>Richard P. Feynman, Robert B. Leigthon, Matthew Sands: <cite class="lang" lang="en" dir="auto" style="font-style:italic"><a href="Feynman-Vorlesungen_%C3%BCber_Physik" title="Feynman-Vorlesungen über Physik">The Feynman Lectures on Physics</a></cite>. Band 2. Addison-Wesley, Reading, Massachusetts 1964, 39-1 The tensor of strain (englisch, <a rel="nofollow" class="external text" href="https://www.feynmanlectures.caltech.edu/II_39.html">caltech.edu</a> – anschauliche Beschreibung).<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abookitem&rfr_id=info:sid/de.wikipedia.org:Strecktensor&rft.atitle=39-1+The+tensor+of+strain&rft.au=Richard+P.+Feynman%2C+Robert+B.+Leigthon%2C+Matthew+Sands&rft.btitle=The+Feynman+Lectures+on+Physics&rft.date=1964&rft.genre=bookitem&rft.place=Reading%2C+Massachusetts&rft.pub=Addison-Wesley&rft.volume=2" style="display:none"> </li></ul></div><div><div class="zim-footer">

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