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<h1 id="firstHeading" class="firstHeading mw-first-heading">Geschwindigkeitsgradient</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">Der (räumliche) Geschwindigkeitsgradient (Formelzeichen l oder L, <a href="Dimension_(Gr%C3%B6%C3%9Fensystem)" title="Dimension (Größensystem)">Dimension</a> T -1) ist in der <a href="Kontinuumsmechanik" title="Kontinuumsmechanik">Kontinuumsmechanik</a> ein Mittel zur Beschreibung der lokalen Verformungsgeschwindigkeit eines Körpers. Der Körper mag fest, flüssig oder gasförmig sein und der Begriff der Verformung wird hier so weit gefasst, dass auch das Fließen einer Flüssigkeit und das Strömen eines Gases darunter fallen. Als <a href="Gradient_(Mathematik)" title="Gradient (Mathematik)">Gradient</a> bemisst der Geschwindigkeitsgradient die örtlichen Änderungen des Geschwindigkeitsfeldes. In <a href="Kartesisches_Koordinatensystem" title="Kartesisches Koordinatensystem">kartesischen Koordinaten</a> hat er die Form:

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {l} :=\operatorname {grad} {\vec {v}}:={\begin{pmatrix}{\frac {\partial v_{x}}{\partial x}}&{\frac {\partial v_{x}}{\partial y}}&{\frac {\partial v_{x}}{\partial z}}\\{\frac {\partial v_{y}}{\partial x}}&{\frac {\partial v_{y}}{\partial y}}&{\frac {\partial v_{y}}{\partial z}}\\{\frac {\partial v_{z}}{\partial x}}&{\frac {\partial v_{z}}{\partial y}}&{\frac {\partial v_{z}}{\partial z}}\end{pmatrix}}\,.}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e77e37d3b859574a1fb029d8b72b02657f2aa98a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.262ex; margin-bottom: -0.243ex; width:35.301ex; height:14.176ex;" alt="{\displaystyle \mathbf {l} :=\operatorname {grad} {\vec {v}}:={\begin{pmatrix}{\frac {\partial v_{x}}{\partial x}}&{\frac {\partial v_{x}}{\partial y}}&{\frac {\partial v_{x}}{\partial z}}\\{\frac {\partial v_{y}}{\partial x}}&{\frac {\partial v_{y}}{\partial y}}&{\frac {\partial v_{y}}{\partial z}}\\{\frac {\partial v_{z}}{\partial x}}&{\frac {\partial v_{z}}{\partial y}}&{\frac {\partial v_{z}}{\partial z}}\end{pmatrix}}\,.}" loading="lazy"></dd></dl>

Die Komponenten <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{x,y,z}}">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a4d24c0aa6f3f7c43075efa22bebba68bd257fe6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.801ex; height:2.343ex;" alt="{\displaystyle v_{x,y,z}}" loading="lazy"> sind die Geschwindigkeitsanteile in x-, y- bzw. z-Richtung. Der räumliche Geschwindigkeitsgradient enthält alle Informationen über die bezugssysteminvarianten <a href="Schergeschwindigkeit" title="Schergeschwindigkeit">Schergeschwindigkeiten</a>, die <a href="Divergenz_eines_Vektorfeldes" title="Divergenz eines Vektorfeldes">Divergenz</a> und die <a href="Winkelgeschwindigkeit" title="Winkelgeschwindigkeit">Winkelgeschwindigkeit</a> oder <a href="Wirbelst%C3%A4rke" title="Wirbelstärke">Wirbelstärke</a> des Geschwindigkeitsfeldes.

Der Geschwindigkeitsgradient wird bei der mathematischen Formulierung von <a href="Physikalisches_Gesetz" title="Physikalisches Gesetz">physikalischen Gesetzen</a> und <a href="Materialmodell" title="Materialmodell">Materialmodellen</a> benutzt und ist – vergleichbar zum <a href="Deformationsgradient" title="Deformationsgradient">Deformationsgradienten</a> bezüglich der Deformation von Festkörpern – in der <a href="Str%C3%B6mungsmechanik" title="Strömungsmechanik">Strömungsmechanik</a> von zentraler Bedeutung.

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

Das <a href="Geschwindigkeitsfeld" title="Geschwindigkeitsfeld">Geschwindigkeitsfeld</a> eines Körpers gibt an, wie schnell sich die einzelnen Partikel (Fluidelemente) des Körpers bewegen, siehe Abb. 1. Wenn sich der Körper gleichförmig bewegt, dann sind die Geschwindigkeiten benachbarter Partikel gleich und der Geschwindigkeitsgradient verschwindet, denn als <a href="Gradient_(Mathematik)" title="Gradient (Mathematik)">Gradient</a> bemisst er die örtlichen Änderungen, siehe den oberen Bildteil. Wenn sich aber die Geschwindigkeiten zweier benachbarter Partikel unterscheiden, dann liegt lokal entweder eine Drehung oder eine Deformation vor und der Geschwindigkeitsgradient ist von null verschieden wie im unteren Bildteil.

Das Geschwindigkeitsfeld kann für die sich bewegenden Partikel eines Körpers oder an den Raumpunkten innerhalb des Körpers aufgestellt werden. Ersteres ist die <a href="Lagrangesche_Betrachtungsweise" title="Lagrangesche Betrachtungsweise">materielle</a> letzteres die <a href="Eulersche_Betrachtungsweise" title="Eulersche Betrachtungsweise">räumliche Formulierung</a>. Weil das Geschwindigkeitsfeld üblicherweise räumlich begriffen wird, bezieht sich der Begriff „Geschwindigkeitsgradient“ zumeist auf den räumlichen Geschwindigkeitsgradient und dieser wird hier vorrangig behandelt.

Der räumliche Geschwindigkeitsgradient taucht in den lokalen, räumlichen Formulierungen der Massen-, Impuls- und Energiebilanzen auf und ist für die kinematische Nichtlinearität der <a href="Kontinuumsmechanik#Impulsbilanz" title="Kontinuumsmechanik">Impulsbilanz</a> in dieser Formulierung verantwortlich.

Der Bewegungszustand eines Beobachters beeinflusst seine Einschätzung der Geschwindigkeit der Partikel des Körpers und damit auch den von ihm beobachteten Geschwindigkeitsgradient. Weil also unterschiedlich bewegte Beobachter verschiedene Geschwindigkeitsgradienten wahrnehmen, ist dieser keine objektive Größe. Mit dem räumlichen Geschwindigkeitsgradient werden objektive Zeitableitungen von Vektoren und Tensoren definiert, die für die Formulierung bezugssysteminvarianter Materialgleichungen benötigt werden. Mehr zu dem Thema ist unter <a href="Euklidische_Transformation" title="Euklidische Transformation">Euklidische Transformation</a> zu finden.

Mathematisch ist der Geschwindigkeitsgradient ein Tensor zweiter Stufe, mit dem Vektoren linear auf andere Vektoren abgebildet werden, siehe Abb. 2. Ein solcher Tensor kann wie eine 3×3-<a href="Matrix_(Mathematik)" title="Matrix (Mathematik)">Matrix</a> betrachtet werden, deren Komponenten auf <a href="Dyadisches_Produkt" title="Dyadisches Produkt">Dyaden</a> referenzieren so wie die Komponenten eines <a href="Vektor" title="Vektor">Vektors</a> auf <a href="Basisvektor" class="mw-redirect" title="Basisvektor">Basisvektoren</a> referenzieren.

Die Summe der Diagonalelemente, die <a href="Spur_(Mathematik)" title="Spur (Mathematik)">Spur</a>, ist die <a href="Divergenz_eines_Vektorfeldes" title="Divergenz eines Vektorfeldes">Divergenz</a> des Geschwindigkeitsfeldes und ein Maß für die Ausdehnungsgeschwindigkeit eines (infinitesimal) kleinen <a href="Volumenelement" class="mw-redirect" title="Volumenelement">Volumenelementes</a> des Körpers.

Der <a href="Symmetrische_Matrix" title="Symmetrische Matrix">symmetrische</a> Anteil des räumlichen Geschwindigkeitsgradienten, der räumliche Verzerrungs-, Streck- oder Deformationsgeschwindigkeitstensor (Formelzeichen d oder D) verschwindet bei <a href="Starrer_K%C3%B6rper#Allgemeine_Bewegungen_starrer_Körper" title="Starrer Körper">Starrkörperbewegungen</a> inklusive Drehungen, tritt also nur bei „echten“ Verformungen auf und ist objektiv. Der Verzerrungsgeschwindigkeitstensor wird in Materialmodellen geschwindigkeitsabhängiger Materialien eingesetzt, z. B. beim <a href="Newtonsches_Fluid" title="Newtonsches Fluid">linear viskosen Fluid</a>, dessen Geschwindigkeitsfeld den Navier-Stokes-Gleichungen gehorcht, die Fluidströmungen wirklichkeitsnah abbilden.

Der <a href="Schiefsymmetrische_Matrix" title="Schiefsymmetrische Matrix">schiefsymmetrische</a> Anteil des räumlichen Geschwindigkeitsgradienten, der Wirbel-, Spin- oder Drehgeschwindigkeitstensor (Formelzeichen w oder W) besitzt einen dualen Vektor <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {\omega }}\,,}">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fc3ef1f8e087140b58812c885facac43535004be.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.48ex; height:2.676ex;" alt="{\displaystyle {\vec {\omega }}\,,}" loading="lazy"> die <a href="Winkelgeschwindigkeit" title="Winkelgeschwindigkeit">Winkelgeschwindigkeit</a>, die proportional zum Wirbelvektor oder der <a href="Wirbelst%C3%A4rke" title="Wirbelstärke">Wirbelstärke</a> ist, die in Flüssigkeits- und Gasströmungen eine wichtige Rolle spielt.

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

<div class="mw-heading mw-heading3"><h3 id="Materielle_und_räumliche_Koordinaten_und_das_Geschwindigkeitsfeld">Materielle und räumliche Koordinaten und das Geschwindigkeitsfeld</h3></div>

Die Bewegung eines materiellen Punktes (Fluidelementes) wird mathematisch mit der Bewegungsfunktion

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

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beschrieben. Der 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"> ist die aktuelle Position des materiellen Punktes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {X}}}">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/65658b7b223af9e1acc877d848888ecdb4466560.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}" loading="lazy"> in der <a href="Konfiguration_(Mechanik)" title="Konfiguration (Mechanik)">Momentankonfiguration</a> (Kleinbuchstaben). Genauer ist <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"> die Position des betrachteten materiellen Punktes in der Ausgangs- oder Referenzkonfiguration des Körpers zu einer vergangenen Zeit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t_{0}\leq t}">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/59054322d2e0183cf04f4e89c5ac3b0b74d1b287.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.832ex; height:2.343ex;" alt="{\displaystyle t_{0}\leq t}" loading="lazy"> (Großbuchstaben). Bei festgehaltenem materiellen Punkt <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"> gibt die Bewegungsfunktion dessen <a href="Bahnlinie" title="Bahnlinie">Bahnlinie</a> durch den Raum wieder und bei festgehaltenem räumlichen Punkt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {x}}}">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9766745636fc303b8f8b010f843adfdf09c40267.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.879ex; height:3.343ex;" alt="{\displaystyle {\vec {X}}={\vec {\chi }}^{-1}({\vec {x}},t)}" loading="lazy"> die <a href="Streichlinie" title="Streichlinie">Streichlinie</a> durch den betrachteten Punkt wieder. Im kartesischen Koordinatensystem mit der <a href="Standardbasis" title="Standardbasis">Standardbasis</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {e}}_{1,2,3}}">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/91ec2fd7cc46b4427cfd09bab244026d27d81519.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.905ex; height:2.843ex;" alt="{\displaystyle {\hat {e}}_{1,2,3}}" loading="lazy"> hat der Raumpunkt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {x}}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/94901db673b3083a8f91c76e3f8f8b691262b166.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:29.24ex; height:7.176ex;" alt="{\displaystyle {\vec {x}}=\sum _{i=1}^{3}x_{i}{\hat {e}}_{i}=\sum _{i=1}^{3}\chi _{i}({\vec {X}},t){\hat {e}}_{i}}" loading="lazy"></dd></dl>

und entsprechend gilt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {X}}=\sum _{i=1}^{3}{X}_{i}{\hat {e}}_{i}}">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3b32ffc2d99348aecd797beb506b5b3b870e6f10.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:13.636ex; height:7.176ex;" alt="{\displaystyle {\vec {X}}=\sum _{i=1}^{3}{X}_{i}{\hat {e}}_{i}}" loading="lazy">. Die Zahlen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{1,2,3}}">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b67eda9b9b24758489f6004e13d51444f494e207.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.943ex; height:2.343ex;" alt="{\displaystyle x_{1,2,3}}" loading="lazy"> werden räumliche Koordinaten genannt, weil diese einen Raumpunkt kennzeichnen, und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{1,2,3}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/96012b6106782ee51bb4c56f86300dd37421d2d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.537ex; height:2.843ex;" alt="{\displaystyle X_{1,2,3}}" loading="lazy"> werden materielle Koordinaten genannt, denn diese haften einem materiellen Punkt an. Die Bewegungsfunktion ist zu jeder Zeit an jedem Ort invertierbar

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<annotation encoding="application/x-tex">{\displaystyle {\vec {x}}={\vec {\chi }}({\vec {X}},t)\quad \leftrightarrow \quad {\vec {X}}={\vec {\chi }}^{-1}({\vec {x}},t)\,,}</annotation>

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weil sich an einem Punkt im Raum immer nur ein materieller Punkt aufhalten kann und ein materieller Punkt zu einer Zeit nur an einem Ort sein kann. Die Ableitung der Bewegungsfunktion nach der Zeit liefert das Geschwindigkeitsfeld:

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<annotation encoding="application/x-tex">{\displaystyle {\vec {x}}={\vec {\chi }}({\vec {X}},t)\quad \rightarrow \quad {\vec {v}}({\vec {x}},t)=\sum _{i=1}^{3}v_{i}({\vec {x}},t){\hat {e}}_{i}=\sum _{i=1}^{3}{\frac {\mathrm {D} \chi _{i}({\vec {X}},t)}{\mathrm {D} t}}{\hat {e}}_{i}=\sum _{i=1}^{3}{\dot {\chi }}_{i}({\vec {X}},t){\hat {e}}_{i}={\dot {\vec {\chi }}}({\vec {X}},t)\,.}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fdd80c8274c3f11fa118f6a2d17b301ef753b312.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:89.426ex; height:7.343ex;" alt="{\displaystyle {\vec {x}}={\vec {\chi }}({\vec {X}},t)\quad \rightarrow \quad {\vec {v}}({\vec {x}},t)=\sum _{i=1}^{3}v_{i}({\vec {x}},t){\hat {e}}_{i}=\sum _{i=1}^{3}{\frac {\mathrm {D} \chi _{i}({\vec {X}},t)}{\mathrm {D} t}}{\hat {e}}_{i}=\sum _{i=1}^{3}{\dot {\chi }}_{i}({\vec {X}},t){\hat {e}}_{i}={\dot {\vec {\chi }}}({\vec {X}},t)\,.}" loading="lazy"></dd></dl>

Die materiellen Koordinaten <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {X}}={\vec {\chi }}^{-1}({\vec {x}},t)}">

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<annotation encoding="application/x-tex">{\displaystyle {\vec {X}}={\vec {\chi }}^{-1}({\vec {x}},t)}</annotation>

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4bc75511fedd7635904ff72ac3c7481227b21622.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.404ex; height:3.343ex;" alt="{\displaystyle {\vec {v}}({\vec {x}},t)={\dot {\vec {\chi }}}({\vec {X}},t)}" loading="lazy"> ist. Das Geschwindigkeitsfeld wird üblicherweise räumlich begriffen, weshalb es hier nur in der räumlichen Darstellung mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {v}}}">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/85820588abd7333ef4d0c56539cb31c20e730753.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.175ex; height:2.343ex;" alt="{\displaystyle {\vec {v}}}" loading="lazy"> (für <a href="Englische_Sprache" title="Englische Sprache">englisch</a> velocity „Geschwindigkeit“) bezeichnet wird. Ganz rechts steht das materielle Geschwindigkeitsfeld, das mit der <a href="Substantielle_Ableitung" title="Substantielle Ableitung">substantiellen Zeitableitung</a> der Bewegungsfunktion berechnet wird. Die <a href="%C3%9Cberpunkt#Als_wissenschaftliches_Symbol" title="Überpunkt">Punktnotation</a> wird hier ausschließlich für die substantielle Zeitableitung verwendet.

<div class="mw-heading mw-heading3"><h3 id="Geschwindigkeitsgradient_und_Deformationsgradient">Geschwindigkeitsgradient und Deformationsgradient</h3></div>

Der <a href="Deformationsgradient" title="Deformationsgradient">Deformationsgradient</a> ist die Ableitung der Bewegung nach den materiellen Koordinaten<a href="#cite_note-Frechet-1">[F 1]</a>:

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5920b54e8ad394a87820db8014a78915a3aba0de.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:55.344ex; height:7.676ex;" alt="{\displaystyle \mathbf {F} :=\operatorname {GRAD} \left({\vec {\chi }}({\vec {X}},t)\right):=\sum _{i,j=1}^{3}{\frac {\mathrm {d} \chi _{i}({\vec {X}},t)}{\mathrm {d} X_{j}}}{\hat {e}}_{i}\otimes {\hat {e}}_{j}=:{\frac {\mathrm {d} {\vec {x}}}{\mathrm {d} {\vec {X}}}}}" 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">“ bildet das <a href="Dyadisches_Produkt" title="Dyadisches Produkt">dyadische Produkt</a> und „GRAD“ den materiellen Gradienten mit Ableitungen nach den materiellen Koordinaten. Durch die substantielle Zeitableitung des Deformationsgradienten entstehen die Geschwindigkeitsgradienten:

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\dot {\mathbf {F} }}=&{\frac {\mathrm {D} }{\mathrm {D} t}}\sum _{i,j=1}^{3}{\frac {\mathrm {d} \chi _{i}({\vec {X}},t)}{\mathrm {d} X_{j}}}{\hat {e}}_{i}\otimes {\hat {e}}_{j}=\sum _{i,j=1}^{3}{\frac {\mathrm {d} {\dot {\chi }}_{i}({\vec {X}},t)}{\mathrm {d} X_{j}}}{\hat {e}}_{i}\otimes {\hat {e}}_{j}\\=&\sum _{i,j=1}^{3}{\frac {\mathrm {d} v_{i}({\vec {x}},t)}{\mathrm {d} X_{j}}}{\hat {e}}_{i}\otimes {\hat {e}}_{j}=\sum _{i,j,k=1}^{3}{\frac {\mathrm {d} v_{i}({\vec {x}},t)}{\mathrm {d} x_{k}}}{\frac {\mathrm {d} \chi _{k}({\vec {X}},t)}{\mathrm {d} X_{j}}}{\hat {e}}_{i}\otimes {\hat {e}}_{j}=\underbrace {\sum _{i,k=1}^{3}{\frac {\mathrm {d} v_{i}({\vec {x}},t)}{\mathrm {d} x_{k}}}{\hat {e}}_{i}\otimes {\hat {e}}_{k}} _{\operatorname {grad} {\vec {v}}}\cdot \underbrace {\sum _{j,l=1}^{3}{\frac {\mathrm {d} \chi _{l}({\vec {X}},t)}{\mathrm {d} X_{j}}}{\hat {e}}_{l}\otimes {\hat {e}}_{j}} _{=\operatorname {GRAD} {\vec {\chi }}}\\=&\operatorname {grad} {\bigl (}{\vec {v}}({\vec {x}},t){\bigl )}\cdot \operatorname {GRAD} {\bigl (}{\vec {\chi }}({\vec {X}},t){\bigl )}=:\mathbf {l\cdot F} \,.\end{aligned}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a3d844f62def3093ec1bd61f7ca0410fcde99fb5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -11.338ex; width:107.469ex; height:23.676ex;" alt="{\displaystyle {\begin{aligned}{\dot {\mathbf {F} }}=&{\frac {\mathrm {D} }{\mathrm {D} t}}\sum _{i,j=1}^{3}{\frac {\mathrm {d} \chi _{i}({\vec {X}},t)}{\mathrm {d} X_{j}}}{\hat {e}}_{i}\otimes {\hat {e}}_{j}=\sum _{i,j=1}^{3}{\frac {\mathrm {d} {\dot {\chi }}_{i}({\vec {X}},t)}{\mathrm {d} X_{j}}}{\hat {e}}_{i}\otimes {\hat {e}}_{j}\\=&\sum _{i,j=1}^{3}{\frac {\mathrm {d} v_{i}({\vec {x}},t)}{\mathrm {d} X_{j}}}{\hat {e}}_{i}\otimes {\hat {e}}_{j}=\sum _{i,j,k=1}^{3}{\frac {\mathrm {d} v_{i}({\vec {x}},t)}{\mathrm {d} x_{k}}}{\frac {\mathrm {d} \chi _{k}({\vec {X}},t)}{\mathrm {d} X_{j}}}{\hat {e}}_{i}\otimes {\hat {e}}_{j}=\underbrace {\sum _{i,k=1}^{3}{\frac {\mathrm {d} v_{i}({\vec {x}},t)}{\mathrm {d} x_{k}}}{\hat {e}}_{i}\otimes {\hat {e}}_{k}} _{\operatorname {grad} {\vec {v}}}\cdot \underbrace {\sum _{j,l=1}^{3}{\frac {\mathrm {d} \chi _{l}({\vec {X}},t)}{\mathrm {d} X_{j}}}{\hat {e}}_{l}\otimes {\hat {e}}_{j}} _{=\operatorname {GRAD} {\vec {\chi }}}\\=&\operatorname {grad} {\bigl (}{\vec {v}}({\vec {x}},t){\bigl )}\cdot \operatorname {GRAD} {\bigl (}{\vec {\chi }}({\vec {X}},t){\bigl )}=:\mathbf {l\cdot F} \,.\end{aligned}}}" loading="lazy"></dd></dl>

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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">“ bildet das <a href="Dyadisches_Produkt" title="Dyadisches Produkt">dyadische Produkt</a>, „grad“ den räumlichen und „GRAD“ den materiellen Gradient mit Ableitungen nach den räumlichen bzw. den materiellen Koordinaten. Der materielle Geschwindigkeitsgradient <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\dot {\mathbf {F} }}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ecd10a2737311dc76a3e39b3c2b1bb2e5b5a14aa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.683ex; height:2.676ex;" alt="{\displaystyle {\dot {\mathbf {F} }}}" loading="lazy"> ist die Zeitableitung des Deformationsgradienten oder – weil die Reihenfolge der Ableitungen vertauscht werden darf – die materielle Ableitung der Geschwindigkeit nach den materiellen Koordinaten:

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<annotation encoding="application/x-tex">{\displaystyle {\dot {\mathbf {F} }}({\vec {X}},t):=\operatorname {GRAD} \left({\dot {\vec {\chi }}}({\vec {X}},t)\right)=\sum _{i,j=1}^{3}{\frac {\mathrm {d} {\dot {\chi }}_{i}}{\mathrm {d} X_{j}}}{\hat {e}}_{i}\otimes {\hat {e}}_{j}=:{\frac {\mathrm {d} {\dot {\vec {\chi }}}}{\mathrm {d} {\vec {X}}}}={\frac {\mathrm {d} {\vec {v}}}{\mathrm {d} {\vec {x}}}}\cdot {\frac {\mathrm {d} {\vec {x}}}{\mathrm {d} {\vec {X}}}}=\mathbf {l\cdot F} \,.}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5c5e64319fa5aa75158deefb3be69ca53164342d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:75.857ex; height:7.509ex;" alt="{\displaystyle {\dot {\mathbf {F} }}({\vec {X}},t):=\operatorname {GRAD} \left({\dot {\vec {\chi }}}({\vec {X}},t)\right)=\sum _{i,j=1}^{3}{\frac {\mathrm {d} {\dot {\chi }}_{i}}{\mathrm {d} X_{j}}}{\hat {e}}_{i}\otimes {\hat {e}}_{j}=:{\frac {\mathrm {d} {\dot {\vec {\chi }}}}{\mathrm {d} {\vec {X}}}}={\frac {\mathrm {d} {\vec {v}}}{\mathrm {d} {\vec {x}}}}\cdot {\frac {\mathrm {d} {\vec {x}}}{\mathrm {d} {\vec {X}}}}=\mathbf {l\cdot F} \,.}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Räumlicher_Geschwindigkeitsgradient">Räumlicher Geschwindigkeitsgradient</h3></div>

Der räumliche Geschwindigkeitsgradient <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {l} }">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/dca0b04733c4e44533df8a7eb12145d74cdbefef.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.742ex; height:2.176ex;" alt="{\displaystyle \mathbf {l} }" loading="lazy"> ist die räumliche Ableitung der Geschwindigkeit nach den räumlichen Koordinaten<a href="#cite_note-Frechet-1">[F 1]</a>:

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {l} ({\vec {x}},t):=\operatorname {grad} {\bigl (}{\vec {v}}({\vec {x}},t){\bigr )}:=\sum _{i,j=1}^{3}{\frac {\mathrm {d} v_{i}}{\mathrm {d} x_{j}}}{\hat {e}}_{i}\otimes {\hat {e}}_{j}=:{\frac {\mathrm {d} {\vec {v}}}{\mathrm {d} {\vec {x}}}}={\frac {\mathrm {d} {\dot {\vec {\chi }}}}{\mathrm {d} {\vec {X}}}}\cdot {\frac {\mathrm {d} {\vec {X}}}{\mathrm {d} {\vec {x}}}}={\dot {\mathbf {F} }}\cdot \mathbf {F} ^{-1}\,.}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1323f011b8e43b925be10fbcc353c1bd2234fd5b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:73.878ex; height:7.509ex;" alt="{\displaystyle \mathbf {l} ({\vec {x}},t):=\operatorname {grad} {\bigl (}{\vec {v}}({\vec {x}},t){\bigr )}:=\sum _{i,j=1}^{3}{\frac {\mathrm {d} v_{i}}{\mathrm {d} x_{j}}}{\hat {e}}_{i}\otimes {\hat {e}}_{j}=:{\frac {\mathrm {d} {\vec {v}}}{\mathrm {d} {\vec {x}}}}={\frac {\mathrm {d} {\dot {\vec {\chi }}}}{\mathrm {d} {\vec {X}}}}\cdot {\frac {\mathrm {d} {\vec {X}}}{\mathrm {d} {\vec {x}}}}={\dot {\mathbf {F} }}\cdot \mathbf {F} ^{-1}\,.}" loading="lazy"></dd></dl>

Das Geschwindigkeitsfeld wird meistens räumlich dargestellt, weshalb mit dem Begriff „Geschwindigkeitsgradient“ in der Regel der räumliche Geschwindigkeitsgradient gemeint ist. Materielle Größen werden in der Kontinuumsmechanik gemeinhin groß geschrieben und räumliche klein, weswegen hier auch die Kleinschreibung des räumlichen Geschwindigkeitsgradienten benutzt wird. Sein symmetrischer Anteil

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {d} :={\frac {1}{2}}(\mathbf {l+l} ^{\top })={\frac {1}{2}}{\begin{pmatrix}2{\frac {\partial v_{x}}{\partial x}}&{\frac {\partial v_{x}}{\partial y}}+{\frac {\partial v_{y}}{\partial x}}&{\frac {\partial v_{x}}{\partial z}}+{\frac {\partial v_{z}}{\partial x}}\\{\frac {\partial v_{y}}{\partial x}}+{\frac {\partial v_{x}}{\partial y}}&2{\frac {\partial v_{y}}{\partial y}}&{\frac {\partial v_{y}}{\partial z}}+{\frac {\partial v_{z}}{\partial y}}\\{\frac {\partial v_{z}}{\partial x}}+{\frac {\partial v_{x}}{\partial z}}&{\frac {\partial v_{z}}{\partial y}}+{\frac {\partial v_{y}}{\partial z}}&2{\frac {\partial v_{z}}{\partial z}}\end{pmatrix}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/20f895d65e9aee1d4b966db278930deb49a72299.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.521ex; margin-bottom: -0.317ex; width:58.845ex; height:14.843ex;" alt="{\displaystyle \mathbf {d} :={\frac {1}{2}}(\mathbf {l+l} ^{\top })={\frac {1}{2}}{\begin{pmatrix}2{\frac {\partial v_{x}}{\partial x}}&{\frac {\partial v_{x}}{\partial y}}+{\frac {\partial v_{y}}{\partial x}}&{\frac {\partial v_{x}}{\partial z}}+{\frac {\partial v_{z}}{\partial x}}\\{\frac {\partial v_{y}}{\partial x}}+{\frac {\partial v_{x}}{\partial y}}&2{\frac {\partial v_{y}}{\partial y}}&{\frac {\partial v_{y}}{\partial z}}+{\frac {\partial v_{z}}{\partial y}}\\{\frac {\partial v_{z}}{\partial x}}+{\frac {\partial v_{x}}{\partial z}}&{\frac {\partial v_{z}}{\partial y}}+{\frac {\partial v_{y}}{\partial z}}&2{\frac {\partial v_{z}}{\partial z}}\end{pmatrix}}}" loading="lazy"></dd></dl>

ist der (räumliche) Verzerrungsgeschwindigkeitstensor und sein schiefsymmetrischer Anteil

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {w} :={\frac {1}{2}}(\mathbf {l-l} ^{\top })={\frac {1}{2}}{\begin{pmatrix}0&{\frac {\partial v_{x}}{\partial y}}-{\frac {\partial v_{y}}{\partial x}}&{\frac {\partial v_{x}}{\partial z}}-{\frac {\partial v_{z}}{\partial x}}\\{\frac {\partial v_{y}}{\partial x}}-{\frac {\partial v_{x}}{\partial y}}&0&{\frac {\partial v_{y}}{\partial z}}-{\frac {\partial v_{z}}{\partial y}}\\{\frac {\partial v_{z}}{\partial x}}-{\frac {\partial v_{x}}{\partial z}}&{\frac {\partial v_{z}}{\partial y}}-{\frac {\partial v_{y}}{\partial z}}&0\end{pmatrix}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0159578f1a9047b80ca96ddee93a0670f26b5816.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.521ex; margin-bottom: -0.317ex; width:59.291ex; height:14.843ex;" alt="{\displaystyle \mathbf {w} :={\frac {1}{2}}(\mathbf {l-l} ^{\top })={\frac {1}{2}}{\begin{pmatrix}0&{\frac {\partial v_{x}}{\partial y}}-{\frac {\partial v_{y}}{\partial x}}&{\frac {\partial v_{x}}{\partial z}}-{\frac {\partial v_{z}}{\partial x}}\\{\frac {\partial v_{y}}{\partial x}}-{\frac {\partial v_{x}}{\partial y}}&0&{\frac {\partial v_{y}}{\partial z}}-{\frac {\partial v_{z}}{\partial y}}\\{\frac {\partial v_{z}}{\partial x}}-{\frac {\partial v_{x}}{\partial z}}&{\frac {\partial v_{z}}{\partial y}}-{\frac {\partial v_{y}}{\partial z}}&0\end{pmatrix}}}" loading="lazy"></dd></dl>

ist der (räumliche) Spin-, Wirbel- oder Drehgeschwindigkeitstensor. 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"> kennzeichnet die <a href="Transponierte_Matrix" title="Transponierte Matrix">Transposition</a>. In den Matrixdarstellungen beziehen sich die Geschwindigkeitsanteile <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{x,y,z}}">

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<div class="mw-heading mw-heading3"><h3 id="Die_Winkelgeschwindigkeit_oder_Wirbelstärke">Die Winkelgeschwindigkeit oder Wirbelstärke</h3></div>

Dem Wirbeltensor kann, weil er <a href="Schiefsymmetrische_Matrix#Kreuzprodukt" title="Schiefsymmetrische Matrix">schiefsymmetrisch</a> ist, ein <a href="Vektorinvariante#Dualer_Vektor_und_Kreuzproduktmatrix" title="Vektorinvariante">dualer Vektor</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {\omega }}\,,}">

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<annotation encoding="application/x-tex">{\displaystyle {\vec {\omega }}\times {\vec {u}}=\mathbf {w} \cdot {\vec {u}}\quad {\text{für alle}}\quad {\vec {u}}\quad \Leftrightarrow \quad \mathbf {w} ={\vec {\omega }}\times \mathbf {1} =\sum _{i=1}^{3}{\vec {\omega }}\times {\hat {e}}_{i}\otimes {\hat {e}}_{i}\,,}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/11f47dd0fe284926b7496b6538af3ea0cb203e68.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:64.834ex; height:7.176ex;" alt="{\displaystyle {\vec {\omega }}\times {\vec {u}}=\mathbf {w} \cdot {\vec {u}}\quad {\text{für alle}}\quad {\vec {u}}\quad \Leftrightarrow \quad \mathbf {w} ={\vec {\omega }}\times \mathbf {1} =\sum _{i=1}^{3}{\vec {\omega }}\times {\hat {e}}_{i}\otimes {\hat {e}}_{i}\,,}" loading="lazy"></dd></dl>

zugeordnet werden. Der Tensor 1 ist der <a href="Einheitstensor" title="Einheitstensor">Einheitstensor</a>, „<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \otimes }">

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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">“ das <a href="Dyadisches_Produkt" title="Dyadisches Produkt">dyadische</a> und „×“ das <a href="Kreuzprodukt" title="Kreuzprodukt">Kreuzprodukt</a>. Im Fall des Wirbeltensors ist der duale Vektor die <a href="Winkelgeschwindigkeit" title="Winkelgeschwindigkeit">Winkelgeschwindigkeit</a>, die der Drehgeschwindigkeitsvektor bei Starrkörperbewegungen ist, wie der gleichnamige Abschnitt unten ausführt. Die Winkelgeschwindigkeit berechnet sich mit dem <a href="Nabla-Operator" title="Nabla-Operator">Nabla-Operator</a>

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<annotation encoding="application/x-tex">{\displaystyle \nabla :=\sum _{k=1}^{3}{\hat {e}}_{k}{\frac {\partial }{\partial x_{k}}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/25898f6536c9e608cda5ae08e455205d64c57998.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:16.376ex; height:7.176ex;" alt="{\displaystyle \nabla :=\sum _{k=1}^{3}{\hat {e}}_{k}{\frac {\partial }{\partial x_{k}}}}" loading="lazy"></dd></dl>

nach der Vorschrift<a href="#cite_note-2">[L 1]</a>

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<annotation encoding="application/x-tex">{\displaystyle {\vec {\omega }}=-{\frac {1}{2}}\mathbf {1\cdot \!\!\times w} =-{\frac {1}{2}}\mathbf {1} \cdot \!\times {\frac {1}{2}}{\bigl [}\overbrace {(\nabla \otimes {\vec {v}})^{\top }} ^{=\operatorname {grad} {\vec {v}}=\mathbf {l} }-\overbrace {\nabla \otimes {\vec {v}}} ^{\mathbf {l} ^{\top }}{\bigr ]}:=-{\frac {1}{4}}(-\nabla \times {\vec {v}}-\nabla \times {\vec {v}})={\frac {1}{2}}\nabla \times {\vec {v}}={\frac {1}{2}}\operatorname {rot} ({\vec {v}})\,,}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7850b2c4c13b767e67f159da78e2c0922c594169.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:97.119ex; height:8.009ex;" alt="{\displaystyle {\vec {\omega }}=-{\frac {1}{2}}\mathbf {1\cdot \!\!\times w} =-{\frac {1}{2}}\mathbf {1} \cdot \!\times {\frac {1}{2}}{\bigl [}\overbrace {(\nabla \otimes {\vec {v}})^{\top }} ^{=\operatorname {grad} {\vec {v}}=\mathbf {l} }-\overbrace {\nabla \otimes {\vec {v}}} ^{\mathbf {l} ^{\top }}{\bigr ]}:=-{\frac {1}{4}}(-\nabla \times {\vec {v}}-\nabla \times {\vec {v}})={\frac {1}{2}}\nabla \times {\vec {v}}={\frac {1}{2}}\operatorname {rot} ({\vec {v}})\,,}" loading="lazy"></dd></dl>

denn das <a href="Formelsammlung_Tensoralgebra#Skalarkreuzprodukt_von_Tensoren" title="Formelsammlung Tensoralgebra">Skalarkreuzprodukt</a> „<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cdot \!\times }">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/40dde55084a6377f91e7469710fe13caa1650154.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:2.068ex; height:1.509ex;" alt="{\displaystyle \cdot \!\times }" loading="lazy">“ des Einheitstensors mit einer Dyade vertauscht das dyadische Produkt mit dem Kreuzprodukt. Der Differentialoperator „rot“ steht für die <a href="Rotation_eines_Vektorfeldes" title="Rotation eines Vektorfeldes">Rotation</a> des Geschwindigkeitsfeldes.

Die Winkelgeschwindigkeit ist proportional zur <a href="Wirbelst%C3%A4rke" title="Wirbelstärke">Wirbelstärke</a>, die eine besondere Bedeutung in Flüssigkeits- und Gasströmungen hat.

<div class="mw-heading mw-heading3"><h3 id="Darstellung_in_Zylinder-_und_Kugelkoordinaten">Darstellung in Zylinder- und Kugelkoordinaten</h3></div>

In <a href="Achsensymmetrie#Rotationskörper" title="Achsensymmetrie">achsensymmetrischen</a> Strömungen bietet es sich an, ein <a href="Zylinderkoordinaten" class="mw-redirect" title="Zylinderkoordinaten">Zylinder-</a> oder <a href="Kugelkoordinaten" title="Kugelkoordinaten">Kugelkoordinatensystem</a> zu benutzen. In Zylinderkoordinaten {ρ,φ,z} mit <a href="Vektorraumbasis" class="mw-redirect" title="Vektorraumbasis">Basisvektoren</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {e}}_{\rho ,\varphi ,z}}">

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<annotation encoding="application/x-tex">{\displaystyle {\hat {e}}_{\rho ,\varphi ,z}}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\operatorname {grad} {\vec {v}}=&{\hat {e}}_{\rho }\otimes (\operatorname {grad} v_{\rho })+{\frac {v_{\rho }}{\rho }}{\hat {e}}_{\varphi }\otimes {\hat {e}}_{\varphi }+{\hat {e}}_{\varphi }\otimes (\operatorname {grad} v_{\varphi })-{\frac {v_{\varphi }}{\rho }}{\hat {e}}_{\rho }\otimes {\hat {e}}_{\varphi }+{\hat {e}}_{z}\otimes (\operatorname {grad} v_{z})\\=&{\begin{pmatrix}{\frac {\partial v_{\rho }}{\partial \rho }}&{\frac {1}{\rho }}{\frac {\partial v_{\rho }}{\partial \varphi }}-{\frac {v_{\varphi }}{\rho }}&{\frac {\partial v_{\rho }}{\partial z}}\\{\frac {\partial v_{\varphi }}{\partial \rho }}&{\frac {v_{\rho }}{\rho }}+{\frac {1}{\rho }}{\frac {\partial v_{\varphi }}{\partial \varphi }}&{\frac {\partial v_{\varphi }}{\partial z}}\\{\frac {\partial v_{z}}{\partial \rho }}&{\frac {1}{\rho }}{\frac {\partial v_{z}}{\partial \varphi }}&{\frac {\partial v_{z}}{\partial z}}\end{pmatrix}}_{{\hat {e}}_{\rho ,\varphi ,z}\otimes {\hat {e}}_{\rho ,\varphi ,z}}\\{\text{mit}}\quad \operatorname {grad} f=&{\frac {\partial f}{\partial \rho }}{\hat {e}}_{\rho }+{\frac {1}{\rho }}{\frac {\partial f}{\partial \varphi }}{\hat {e}}_{\varphi }+{\frac {\partial f}{\partial z}}{\hat {e}}_{z}\,.\end{aligned}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4857a9aaf8770427934c107d230091828cd6ca8e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -13.005ex; width:91.863ex; height:27.176ex;" alt="{\displaystyle {\begin{aligned}\operatorname {grad} {\vec {v}}=&{\hat {e}}_{\rho }\otimes (\operatorname {grad} v_{\rho })+{\frac {v_{\rho }}{\rho }}{\hat {e}}_{\varphi }\otimes {\hat {e}}_{\varphi }+{\hat {e}}_{\varphi }\otimes (\operatorname {grad} v_{\varphi })-{\frac {v_{\varphi }}{\rho }}{\hat {e}}_{\rho }\otimes {\hat {e}}_{\varphi }+{\hat {e}}_{z}\otimes (\operatorname {grad} v_{z})\\=&{\begin{pmatrix}{\frac {\partial v_{\rho }}{\partial \rho }}&{\frac {1}{\rho }}{\frac {\partial v_{\rho }}{\partial \varphi }}-{\frac {v_{\varphi }}{\rho }}&{\frac {\partial v_{\rho }}{\partial z}}\\{\frac {\partial v_{\varphi }}{\partial \rho }}&{\frac {v_{\rho }}{\rho }}+{\frac {1}{\rho }}{\frac {\partial v_{\varphi }}{\partial \varphi }}&{\frac {\partial v_{\varphi }}{\partial z}}\\{\frac {\partial v_{z}}{\partial \rho }}&{\frac {1}{\rho }}{\frac {\partial v_{z}}{\partial \varphi }}&{\frac {\partial v_{z}}{\partial z}}\end{pmatrix}}_{{\hat {e}}_{\rho ,\varphi ,z}\otimes {\hat {e}}_{\rho ,\varphi ,z}}\\{\text{mit}}\quad \operatorname {grad} f=&{\frac {\partial f}{\partial \rho }}{\hat {e}}_{\rho }+{\frac {1}{\rho }}{\frac {\partial f}{\partial \varphi }}{\hat {e}}_{\varphi }+{\frac {\partial f}{\partial z}}{\hat {e}}_{z}\,.\end{aligned}}}" loading="lazy"></dd></dl>

In Kugelkoordinaten {r,θ,φ} mit Basisvektoren <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {e}}_{r,\theta ,\varphi }}">

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<annotation encoding="application/x-tex">{\displaystyle {\hat {e}}_{r,\theta ,\varphi }}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\operatorname {grad} {\vec {v}}=&{\hat {e}}_{r}\otimes (\operatorname {grad} v_{r})+{\frac {v_{r}}{r}}\mathbf {1} -{\frac {v_{r}}{r}}{\hat {e}}_{r}\otimes {\hat {e}}_{r}+{\hat {e}}_{\theta }\otimes (\operatorname {grad} v_{\theta })+{\frac {v_{\theta }}{r\tan \theta }}{\hat {e}}_{\varphi }\otimes {\hat {e}}_{\varphi }-{\frac {v_{\theta }}{r}}{\hat {e}}_{r}\otimes {\hat {e}}_{\theta }\\&+{\hat {e}}_{\varphi }\otimes (\operatorname {grad} v_{\varphi })-{\frac {v_{\varphi }}{r\tan \theta }}{\hat {e}}_{\theta }\otimes {\hat {e}}_{\varphi }-{\frac {v_{\varphi }}{r}}{\hat {e}}_{r}\otimes {\hat {e}}_{\varphi }\\=&{\begin{pmatrix}{\frac {\partial v_{r}}{\partial r}}&{\frac {1}{r}}{\frac {\partial v_{r}}{\partial \theta }}-{\frac {v_{\theta }}{r}}&{\frac {1}{r\sin \theta }}{\frac {\partial v_{r}}{\partial \varphi }}-{\frac {v_{\varphi }}{r}}\\{\frac {\partial v_{\theta }}{\partial r}}&{\frac {1}{r}}{\frac {\partial v_{\theta }}{\partial \theta }}+{\frac {v_{r}}{r}}&{\frac {1}{r\sin \theta }}{\frac {\partial v_{\theta }}{\partial \varphi }}-{\frac {v_{\varphi }}{r\tan \theta }}\\{\frac {\partial v_{\varphi }}{\partial r}}&{\frac {1}{r}}{\frac {\partial v_{\varphi }}{\partial \theta }}&{\frac {1}{r\sin \theta }}{\frac {\partial v_{\varphi }}{\partial \varphi }}+{\frac {v_{r}}{r}}+{\frac {v_{\theta }}{r\tan \theta }}\end{pmatrix}}_{{\hat {e}}_{r,\theta ,\varphi }\otimes {\hat {e}}_{r,\theta ,\varphi }}\\{\text{mit}}\quad \operatorname {grad} f=&{\frac {\partial f}{\partial r}}{\hat {e}}_{r}+{\frac {1}{r}}{\frac {\partial f}{\partial \theta }}{\hat {e}}_{\theta }+{\frac {1}{r\sin \theta }}{\frac {\partial f}{\partial \varphi }}{\hat {e}}_{\varphi }\,.\end{aligned}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c7849ee8f09bf01005398b67d7ed9c976cdba229.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -15.338ex; width:97.82ex; height:31.843ex;" alt="{\displaystyle {\begin{aligned}\operatorname {grad} {\vec {v}}=&{\hat {e}}_{r}\otimes (\operatorname {grad} v_{r})+{\frac {v_{r}}{r}}\mathbf {1} -{\frac {v_{r}}{r}}{\hat {e}}_{r}\otimes {\hat {e}}_{r}+{\hat {e}}_{\theta }\otimes (\operatorname {grad} v_{\theta })+{\frac {v_{\theta }}{r\tan \theta }}{\hat {e}}_{\varphi }\otimes {\hat {e}}_{\varphi }-{\frac {v_{\theta }}{r}}{\hat {e}}_{r}\otimes {\hat {e}}_{\theta }\\&+{\hat {e}}_{\varphi }\otimes (\operatorname {grad} v_{\varphi })-{\frac {v_{\varphi }}{r\tan \theta }}{\hat {e}}_{\theta }\otimes {\hat {e}}_{\varphi }-{\frac {v_{\varphi }}{r}}{\hat {e}}_{r}\otimes {\hat {e}}_{\varphi }\\=&{\begin{pmatrix}{\frac {\partial v_{r}}{\partial r}}&{\frac {1}{r}}{\frac {\partial v_{r}}{\partial \theta }}-{\frac {v_{\theta }}{r}}&{\frac {1}{r\sin \theta }}{\frac {\partial v_{r}}{\partial \varphi }}-{\frac {v_{\varphi }}{r}}\\{\frac {\partial v_{\theta }}{\partial r}}&{\frac {1}{r}}{\frac {\partial v_{\theta }}{\partial \theta }}+{\frac {v_{r}}{r}}&{\frac {1}{r\sin \theta }}{\frac {\partial v_{\theta }}{\partial \varphi }}-{\frac {v_{\varphi }}{r\tan \theta }}\\{\frac {\partial v_{\varphi }}{\partial r}}&{\frac {1}{r}}{\frac {\partial v_{\varphi }}{\partial \theta }}&{\frac {1}{r\sin \theta }}{\frac {\partial v_{\varphi }}{\partial \varphi }}+{\frac {v_{r}}{r}}+{\frac {v_{\theta }}{r\tan \theta }}\end{pmatrix}}_{{\hat {e}}_{r,\theta ,\varphi }\otimes {\hat {e}}_{r,\theta ,\varphi }}\\{\text{mit}}\quad \operatorname {grad} f=&{\frac {\partial f}{\partial r}}{\hat {e}}_{r}+{\frac {1}{r}}{\frac {\partial f}{\partial \theta }}{\hat {e}}_{\theta }+{\frac {1}{r\sin \theta }}{\frac {\partial f}{\partial \varphi }}{\hat {e}}_{\varphi }\,.\end{aligned}}}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Darstellung_in_konvektiven_Koordinaten">Darstellung in konvektiven Koordinaten</h3></div>

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

Konvektive Koordinaten sind <a href="Krummlinige_Koordinaten" title="Krummlinige Koordinaten">krummlinige Koordinatensysteme</a>, die an einen Körper gebunden sind und von allen Deformationen, die der Körper erfährt, mitgeführt werden, siehe Bild. Konvektive Koordinatensysteme werden in der Kinematik schlanker oder dünnwandiger Strukturen (z. B. <a href="Stab_(Statik)" title="Stab (Statik)">Stäbe</a> oder <a href="Schale_(Technische_Mechanik)" title="Schale (Technische Mechanik)">Schalen</a>) eingesetzt. Auch materielle Vorzugsrichtungen nicht isotroper Materialien, wie z. B. von Holz, können in konvektiven Koordinaten beschrieben werden. Die Geschwindigkeitsgradienten bekommen, in konvektiven Koordinaten ausgedrückt, besonders einfache Darstellungen.

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

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

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</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"> werden über eine <a href="Konfiguration_(Mechanik)#Referenzkonfiguration" title="Konfiguration (Mechanik)">Referenzkonfiguration</a> <a href="Bijektive_Funktion" title="Bijektive Funktion">eineindeutig</a> konvektive Koordinaten <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {\Theta }}=(\Theta _{1},\Theta _{2},\Theta _{3})}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/18cf39c4e6ef834f459142e3a0d99e105f6b1388.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.371ex; height:3.509ex;" alt="{\displaystyle {\vec {\Theta }}=(\Theta _{1},\Theta _{2},\Theta _{3})}" loading="lazy"> zugeordnet. Die Tangentenvektoren

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<annotation encoding="application/x-tex">{\displaystyle {\vec {G}}_{i}:={\dfrac {\mathrm {d} {\vec {X}}({\vec {\Theta }})}{\mathrm {d} \Theta _{i}}}\quad {\textsf {bzw.}}\quad {\vec {g}}_{i}:={\dfrac {\mathrm {d} {\vec {\chi }}\left({\vec {X}}({\vec {\Theta }}),t\right)}{\mathrm {d} \Theta _{i}}}={\dfrac {\mathrm {d} {\vec {\chi }}}{\mathrm {d} {\vec {X}}}}\cdot {\dfrac {\mathrm {d} {\vec {X}}}{\mathrm {d} \Theta _{i}}}=\mathbf {F} \cdot {\vec {G}}_{i}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e4c7b8cb586a253fed499cc59e6110e492eb6b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:65.837ex; height:8.343ex;" alt="{\displaystyle {\vec {G}}_{i}:={\dfrac {\mathrm {d} {\vec {X}}({\vec {\Theta }})}{\mathrm {d} \Theta _{i}}}\quad {\textsf {bzw.}}\quad {\vec {g}}_{i}:={\dfrac {\mathrm {d} {\vec {\chi }}\left({\vec {X}}({\vec {\Theta }}),t\right)}{\mathrm {d} \Theta _{i}}}={\dfrac {\mathrm {d} {\vec {\chi }}}{\mathrm {d} {\vec {X}}}}\cdot {\dfrac {\mathrm {d} {\vec {X}}}{\mathrm {d} \Theta _{i}}}=\mathbf {F} \cdot {\vec {G}}_{i}}" loading="lazy"></dd></dl>

bilden dann kovariante Basen im Punkt <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"> bzw. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {x}}={\vec {\chi }}({\vec {X}},t)}">

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<annotation encoding="application/x-tex">{\displaystyle {\vec {G}}^{i}:=\operatorname {GRAD} \Theta _{i}:=\sum _{j=1}^{3}{\dfrac {\mathrm {d} \Theta _{i}}{\mathrm {d} X_{j}}}{\hat {e}}_{j}=:{\frac {\mathrm {d} \Theta _{i}}{\mathrm {d} {\vec {X}}}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/005eb7ff7d7e37e85cafcbd0eb4328a2c6a62036.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:39.553ex; height:7.509ex;" alt="{\displaystyle {\vec {G}}^{i}:=\operatorname {GRAD} \Theta _{i}:=\sum _{j=1}^{3}{\dfrac {\mathrm {d} \Theta _{i}}{\mathrm {d} X_{j}}}{\hat {e}}_{j}=:{\frac {\mathrm {d} \Theta _{i}}{\mathrm {d} {\vec {X}}}}}" loading="lazy"></dd></dl>

bzw.

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<annotation encoding="application/x-tex">{\displaystyle {\vec {g}}^{i}:=\operatorname {grad} \Theta _{i}:=\sum _{j=1}^{3}{\frac {\mathrm {d} \Theta _{i}}{\mathrm {d} x_{j}}}{\hat {e}}_{j}=:{\frac {\mathrm {d} \Theta _{i}}{\mathrm {d} {\vec {x}}}}={\frac {\mathrm {d} \Theta _{i}}{\mathrm {d} {\vec {X}}}}\cdot {\frac {\mathrm {d} {\vec {X}}}{\mathrm {d} {\vec {x}}}}={\vec {G}}^{i}\cdot \mathbf {F} ^{-1}=\mathbf {F} ^{\top -1}\cdot {\vec {G}}^{i}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/631e4b1f8040e546bc0da730e0b8dcf5d2663d5f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:73.89ex; height:7.509ex;" alt="{\displaystyle {\vec {g}}^{i}:=\operatorname {grad} \Theta _{i}:=\sum _{j=1}^{3}{\frac {\mathrm {d} \Theta _{i}}{\mathrm {d} x_{j}}}{\hat {e}}_{j}=:{\frac {\mathrm {d} \Theta _{i}}{\mathrm {d} {\vec {x}}}}={\frac {\mathrm {d} \Theta _{i}}{\mathrm {d} {\vec {X}}}}\cdot {\frac {\mathrm {d} {\vec {X}}}{\mathrm {d} {\vec {x}}}}={\vec {G}}^{i}\cdot \mathbf {F} ^{-1}=\mathbf {F} ^{\top -1}\cdot {\vec {G}}^{i}}" loading="lazy"></dd></dl>

formen die kontravarianten Basen, die zu den kovarianten <a href="Duale_Basis" title="Duale Basis">dual</a> sind. In diesen Basissystemen ausgedrückt, bekommt der Deformationsgradient die besonders einfache Form

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {F} =\sum _{i=1}^{3}{\vec {g}}_{i}\otimes {\vec {G}}^{i}\quad \rightarrow \quad \mathbf {F} ^{-1}=\sum _{i=1}^{3}{\vec {G}}_{i}\otimes {\vec {g}}^{i}\,.}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/23c5270f273a2adbcb8ec859197f17788e574043.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:43.554ex; height:7.176ex;" alt="{\displaystyle \mathbf {F} =\sum _{i=1}^{3}{\vec {g}}_{i}\otimes {\vec {G}}^{i}\quad \rightarrow \quad \mathbf {F} ^{-1}=\sum _{i=1}^{3}{\vec {G}}_{i}\otimes {\vec {g}}^{i}\,.}" loading="lazy"></dd></dl>

Aus der Zeitableitung des Deformationsgradienten und der Zeitableitung der Inversen ergibt sich

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<annotation encoding="application/x-tex">{\displaystyle {\dot {\mathbf {F} }}=\sum _{i=1}^{3}{\dot {\vec {g}}}_{i}\otimes {\vec {G}}^{i}\,,\quad (\mathbf {F} ^{-1}){\dot {}}=\sum _{i=1}^{3}{\vec {G}}_{i}\otimes {\dot {\vec {g}}}^{i}\,,}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0f759500df611c9efb1c5d7d3155bd76975b70e5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:42.074ex; height:7.176ex;" alt="{\displaystyle {\dot {\mathbf {F} }}=\sum _{i=1}^{3}{\dot {\vec {g}}}_{i}\otimes {\vec {G}}^{i}\,,\quad (\mathbf {F} ^{-1}){\dot {}}=\sum _{i=1}^{3}{\vec {G}}_{i}\otimes {\dot {\vec {g}}}^{i}\,,}" loading="lazy"></dd></dl>

denn die Ausgangskonfiguration und die in ihr definierten Basisvektoren hängen nicht von der Zeit ab. Mit diesen Ergebnissen schreibt sich der räumliche Geschwindigkeitsgradient:

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\mathbf {l} =&{\dot {\mathbf {F} }}\cdot \mathbf {F} ^{-1}=\sum _{i=1}^{3}{\dot {\vec {g}}}_{i}\otimes {\vec {g}}^{i}\\=&\sum _{i=1}^{3}{\dot {\vec {g}}}_{i}\otimes {\vec {g}}^{i}-\underbrace {{\frac {\mathrm {D} }{\mathrm {D} t}}\left(\sum _{i=1}^{3}{\vec {g}}_{i}\otimes {\vec {g}}^{i}\right)} _{={\dot {\mathbf {1} }}=\mathbf {0} }=\sum _{i=1}^{3}({\dot {\vec {g}}}_{i}\otimes {\vec {g}}^{i}-{\dot {\vec {g}}}_{i}\otimes {\vec {g}}^{i}-{\vec {g}}_{i}\otimes {\dot {\vec {g}}}^{i})=-\sum _{i=1}^{3}{\vec {g}}_{i}\otimes {\dot {\vec {g}}}^{i}\,,\end{aligned}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fe2fda809c32bbf141123f9aaec35e0afad84d2a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -8.573ex; margin-bottom: -0.265ex; width:87.079ex; height:18.843ex;" alt="{\displaystyle {\begin{aligned}\mathbf {l} =&{\dot {\mathbf {F} }}\cdot \mathbf {F} ^{-1}=\sum _{i=1}^{3}{\dot {\vec {g}}}_{i}\otimes {\vec {g}}^{i}\\=&\sum _{i=1}^{3}{\dot {\vec {g}}}_{i}\otimes {\vec {g}}^{i}-\underbrace {{\frac {\mathrm {D} }{\mathrm {D} t}}\left(\sum _{i=1}^{3}{\vec {g}}_{i}\otimes {\vec {g}}^{i}\right)} _{={\dot {\mathbf {1} }}=\mathbf {0} }=\sum _{i=1}^{3}({\dot {\vec {g}}}_{i}\otimes {\vec {g}}^{i}-{\dot {\vec {g}}}_{i}\otimes {\vec {g}}^{i}-{\vec {g}}_{i}\otimes {\dot {\vec {g}}}^{i})=-\sum _{i=1}^{3}{\vec {g}}_{i}\otimes {\dot {\vec {g}}}^{i}\,,\end{aligned}}}" loading="lazy"></dd></dl>

worin das Verschwinden der Zeitableitung des Einheitstensors 1 ausgenutzt wurde. Die Geschwindigkeitsgradienten bilden die Basisvektoren auf ihre Raten ab:

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\dot {\vec {g}}}_{i}=&{\dot {\mathbf {F} }}\cdot {\vec {G}}_{i}\,,\quad {\dot {\vec {g}}}^{i}=(\mathbf {F} ^{\top -1}){\dot {}}\cdot {\vec {G}}^{i}\\{\dot {\vec {g}}}_{i}=&\mathbf {l} \cdot {\vec {g}}_{i}\,,\quad {\dot {\vec {g}}}^{i}=-\mathbf {l} ^{\top }\cdot {\vec {g}}^{i}.\end{aligned}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/81b28888678e84bed927eeb5ebed8c06751e4652.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:32.618ex; height:7.843ex;" alt="{\displaystyle {\begin{aligned}{\dot {\vec {g}}}_{i}=&{\dot {\mathbf {F} }}\cdot {\vec {G}}_{i}\,,\quad {\dot {\vec {g}}}^{i}=(\mathbf {F} ^{\top -1}){\dot {}}\cdot {\vec {G}}^{i}\\{\dot {\vec {g}}}_{i}=&\mathbf {l} \cdot {\vec {g}}_{i}\,,\quad {\dot {\vec {g}}}^{i}=-\mathbf {l} ^{\top }\cdot {\vec {g}}^{i}.\end{aligned}}}" loading="lazy"></dd></dl>

Der symmetrische Anteil des räumlichen Geschwindigkeitsgradienten ist der Verzerrungsgeschwindigkeitstensor:

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\mathbf {d} ={\dfrac {1}{2}}(\mathbf {l+l} ^{\top })=&{\frac {1}{2}}\sum _{j=1}^{3}{\dot {\vec {g}}}_{j}\otimes {\vec {g}}^{j}+{\frac {1}{2}}\sum _{i=1}^{3}{\vec {g}}^{i}\otimes {\dot {\vec {g}}}_{i}={\frac {1}{2}}\sum _{i,j=1}^{3}\left({\vec {g}}_{i}\cdot {\dot {\vec {g}}}_{j}+{\dot {\vec {g}}}_{i}\cdot {\vec {g}}_{j}\right){\vec {g}}^{i}\otimes {\vec {g}}^{j}\\=&-{\frac {1}{2}}\sum _{i=1}^{3}{\vec {g}}_{i}\otimes {\dot {\vec {g}}}^{i}-{\frac {1}{2}}\sum _{j=1}^{3}{\dot {\vec {g}}}^{j}\otimes {\vec {g}}_{j}=-{\frac {1}{2}}\sum _{i,j=1}^{3}\left({\dot {\vec {g}}}^{i}\cdot {\vec {g}}^{j}+{\vec {g}}^{i}\cdot {\dot {\vec {g}}}^{j}\right){\vec {g}}_{i}\otimes {\vec {g}}_{j}\,.\end{aligned}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2cc77807983ca2d541c58305e61cbc356de9cb9e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.171ex; width:86.452ex; height:15.509ex;" alt="{\displaystyle {\begin{aligned}\mathbf {d} ={\dfrac {1}{2}}(\mathbf {l+l} ^{\top })=&{\frac {1}{2}}\sum _{j=1}^{3}{\dot {\vec {g}}}_{j}\otimes {\vec {g}}^{j}+{\frac {1}{2}}\sum _{i=1}^{3}{\vec {g}}^{i}\otimes {\dot {\vec {g}}}_{i}={\frac {1}{2}}\sum _{i,j=1}^{3}\left({\vec {g}}_{i}\cdot {\dot {\vec {g}}}_{j}+{\dot {\vec {g}}}_{i}\cdot {\vec {g}}_{j}\right){\vec {g}}^{i}\otimes {\vec {g}}^{j}\\=&-{\frac {1}{2}}\sum _{i=1}^{3}{\vec {g}}_{i}\otimes {\dot {\vec {g}}}^{i}-{\frac {1}{2}}\sum _{j=1}^{3}{\dot {\vec {g}}}^{j}\otimes {\vec {g}}_{j}=-{\frac {1}{2}}\sum _{i,j=1}^{3}\left({\dot {\vec {g}}}^{i}\cdot {\vec {g}}^{j}+{\vec {g}}^{i}\cdot {\dot {\vec {g}}}^{j}\right){\vec {g}}_{i}\otimes {\vec {g}}_{j}\,.\end{aligned}}}" loading="lazy"></dd></dl>

Mit den <a href="Einheitstensor#Metrikkoeffizienten" title="Einheitstensor">Metrikkoeffizienten</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g_{ij}:={\vec {g}}_{i}\cdot {\vec {g}}_{j}}">

<msub>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g_{ij}:={\vec {g}}_{i}\cdot {\vec {g}}_{j}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d778b711d7bfc362ed4e7d785c08006cf5600e21.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:12.068ex; height:3.176ex;" alt="{\displaystyle g_{ij}:={\vec {g}}_{i}\cdot {\vec {g}}_{j}}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g^{ij}:={\vec {g}}^{i}\cdot {\vec {g}}^{j}}">

<msup>

</mrow>

</msup>

<msup>

<mover>

</mover>

</mrow>

</mrow>

</msup>

<msup>

<mover>

</mover>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g^{ij}:={\vec {g}}^{i}\cdot {\vec {g}}^{j}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bc38b283f59898d4271bf8e95cee23db94b16da0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.077ex; height:3.176ex;" alt="{\displaystyle g^{ij}:={\vec {g}}^{i}\cdot {\vec {g}}^{j}}" loading="lazy"> sowie der <a href="Produktregel" title="Produktregel">Produktregel</a> schreibt sich das:

</mrow>

<mfrac>

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</mrow>

</mrow>

</mrow>

</munderover>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<msup>

<mover>

</mover>

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</mrow>

</msup>

<msup>

<mover>

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</mrow>

</mrow>

</msup>

<mfrac>

</mfrac>

</mrow>

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</mrow>

</munderover>

<msup>

<mover>

</mover>

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</mrow>

</msup>

<msub>

<mover>

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<msub>

<mover>

</mover>

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</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {d} ={\frac {1}{2}}\sum _{i,j=1}^{3}{\dot {g}}_{ij}{\vec {g}}^{i}\otimes {\vec {g}}^{j}=-{\frac {1}{2}}\sum _{i,j=1}^{3}{\dot {g}}^{ij}{\vec {g}}_{i}\otimes {\vec {g}}_{j}\,.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/135de5c2ad26f40dc42e3d0a7b0ca2056fb2f2cb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:42.889ex; height:7.509ex;" alt="{\displaystyle \mathbf {d} ={\frac {1}{2}}\sum _{i,j=1}^{3}{\dot {g}}_{ij}{\vec {g}}^{i}\otimes {\vec {g}}^{j}=-{\frac {1}{2}}\sum _{i,j=1}^{3}{\dot {g}}^{ij}{\vec {g}}_{i}\otimes {\vec {g}}_{j}\,.}" loading="lazy"></dd></dl>

Die <a href="Frobenius-Skalarprodukt" title="Frobenius-Skalarprodukt">Frobenius-Skalarprodukte</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {g}}_{i}\cdot {\vec {g}}_{j}}">

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\vec {g}}_{i}\cdot {\vec {g}}_{j}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6970b31f4a287fbd7ba6c5cd8b788ea4bbf91f75.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:5.737ex; height:3.176ex;" alt="{\displaystyle {\vec {g}}_{i}\cdot {\vec {g}}_{j}}" loading="lazy"> bleiben bei einer Rotation oder Translation unverändert, weswegen der Verzerrungsgeschwindigkeitstensor genau dann verschwindet, nämlich bei Starrkörperbewegungen.

<div class="mw-heading mw-heading3"><h3 id="Geometrische_Linearisierung">Geometrische Linearisierung</h3></div>

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

In der Festkörpermechanik treten in vielen Anwendungsbereichen nur kleine Deformationen auf. In diesem Fall erfahren die Gleichungen der Kontinuumsmechanik eine erhebliche Vereinfachung durch geometrische Linearisierung. Dazu werden die Verschiebungen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {u}}({\vec {X}},t)}">

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4d50d4c60e9e2ddf8c4c9a2f46808c9cc5023f94.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.993ex; height:3.343ex;" alt="{\displaystyle {\vec {u}}({\vec {X}},t)}" loading="lazy"> betrachtet, die ein materieller Punkt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {X}}}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5dc35b5a0226cf11a2c3f2d2dbbac6ab5ade6036.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"> im Laufe seiner Bewegung erfährt. Weil <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {x}}={\vec {\chi }}({\vec {X}},t)}">

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</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"> die aktuelle Position des Punktes ist, der in der Ausgangskonfiguration die Position <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {X}}}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

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

</semantics>

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1d084f14bcf549563f2c61c568e0b467fc173d5f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.4ex; height:3.343ex;" alt="{\displaystyle {\vec {u}}={\vec {\chi }}({\vec {X}},t)-{\vec {X}}\,.}" loading="lazy"></dd></dl>

Der materielle Gradient der Verschiebungen ist der Tensor

</mrow>

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<mover>

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<mover>

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</mrow>

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {H} =\mathrm {GRAD} \,{\vec {u}}=\mathrm {GRAD} \,{\vec {\chi }}-\mathrm {GRAD} \,{\vec {X}}=\mathbf {F} -\mathbf {1} \,.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/028fa495fc66e318b30f50958073c7680d2a0002.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:48.208ex; height:3.176ex;" alt="{\displaystyle \mathbf {H} =\mathrm {GRAD} \,{\vec {u}}=\mathrm {GRAD} \,{\vec {\chi }}-\mathrm {GRAD} \,{\vec {X}}=\mathbf {F} -\mathbf {1} \,.}" loading="lazy"></dd></dl>

und wird <a href="Verschiebungsgradient" title="Verschiebungsgradient">Verschiebungsgradient</a> genannt. Er unterscheidet sich vom Deformationsgradient nur durch den Einheitstensor 1. Wenn <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L_{0}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle L_{0}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/db742b8c210fc611329a4c2dcc3af4b4e1a110cb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.637ex; height:2.509ex;" alt="{\displaystyle L_{0}}" loading="lazy"> eine charakteristische Abmessung des Körpers ist, dann wird bei kleinen Verschiebungen sowohl <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |{\vec {u}}|\ll L_{0}}">

</mrow>

<mover>

</mover>

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<msub>

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</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle |{\vec {u}}|\ll L_{0}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c5361f75e29b4aff61d87cd3850c551c73797878.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.875ex; height:2.843ex;" alt="{\displaystyle |{\vec {u}}|\ll L_{0}}" loading="lazy"> als auch <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \parallel \mathbf {H} \parallel \ll 1}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c45a0b7b63ce5444f06e147e2346853b369bf5c9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.193ex; height:2.843ex;" alt="{\displaystyle \parallel \mathbf {H} \parallel \ll 1}" loading="lazy"> und hier <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \parallel {\dot {\mathbf {H} }}\parallel \ll 1/s}">

<mover>

</mrow>

</mover>

</mrow>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d7d3da60386efe82b2fe9d57a8f036dfdbc1b63b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.446ex; height:3.176ex;" alt="{\displaystyle \parallel {\dot {\mathbf {H} }}\parallel \ll 1/s}" loading="lazy"> gefordert, so dass alle Terme, die höhere Potenzen von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {u}},\,\mathbf {H} }">

<mover>

</mover>

</mrow>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/770e16b6b9cf7af6e6ce15c223dc6a35441f34af.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.842ex; height:2.676ex;" alt="{\displaystyle {\vec {u}},\,\mathbf {H} }" loading="lazy"> oder <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\dot {\mathbf {H} }}}">

<mover>

</mrow>

</mover>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cf56717599edbea01447acf70febd8142bc54e43.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.091ex; height:2.676ex;" alt="{\displaystyle {\dot {\mathbf {H} }}}" loading="lazy"> beinhalten, vernachlässigt werden können. Dann gilt:

<mtr>

<mtd>

</mrow>

</mtd>

<mtd>

<mover>

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</mover>

</mrow>

<msup>

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</msup>

<mover>

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</mrow>

<mover>

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</mover>

</mrow>

<mover>

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</mover>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

</mrow>

</mtd>

<mtd>

<mfrac>

</mfrac>

</mrow>

<mover>

</mrow>

</mover>

</mrow>

<msup>

<mover>

</mrow>

</mover>

</mrow>

</mrow>

</msup>

<mover>

</mover>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

</mrow>

</mtd>

<mtd>

<mfrac>

</mfrac>

</mrow>

<mover>

</mrow>

</mover>

</mrow>

<msup>

<mover>

</mrow>

</mover>

</mrow>

</mrow>

</msup>

<msub>

<mover>

</mrow>

</mover>

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</mrow>

</msub>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\mathbf {l} =&{\dot {\mathbf {F} }}\cdot \mathbf {F} ^{-1}\approx {\dot {\mathbf {H} }}\cdot (\mathbf {1-H} )\approx {\dot {\mathbf {H} }}={\dot {\mathbf {F} }}\\\mathbf {d} \approx &{\frac {1}{2}}({\dot {\mathbf {H} }}+{\dot {\mathbf {H} }}^{\top })={\dot {\boldsymbol {\varepsilon }}}\\\mathbf {w} \approx &{\frac {1}{2}}({\dot {\mathbf {H} }}-{\dot {\mathbf {H} }}^{\top })={\dot {\mathbf {R} }}_{L}\,.\end{aligned}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/44f4e42c9eaf0e6d1e238c94405ed38a529d65db.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.189ex; margin-bottom: -0.316ex; width:37.701ex; height:14.176ex;" alt="{\displaystyle {\begin{aligned}\mathbf {l} =&{\dot {\mathbf {F} }}\cdot \mathbf {F} ^{-1}\approx {\dot {\mathbf {H} }}\cdot (\mathbf {1-H} )\approx {\dot {\mathbf {H} }}={\dot {\mathbf {F} }}\\\mathbf {d} \approx &{\frac {1}{2}}({\dot {\mathbf {H} }}+{\dot {\mathbf {H} }}^{\top })={\dot {\boldsymbol {\varepsilon }}}\\\mathbf {w} \approx &{\frac {1}{2}}({\dot {\mathbf {H} }}-{\dot {\mathbf {H} }}^{\top })={\dot {\mathbf {R} }}_{L}\,.\end{aligned}}}" loading="lazy"></dd></dl>

Der Tensor <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {\varepsilon }}}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8445af5ff7da70714382bc35e78bedcacf68e825.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle {\boldsymbol {\varepsilon }}}" loading="lazy"> ist der <a href="Verzerrungstensor#Linearisierter_Verzerrungstensor" title="Verzerrungstensor">linearisierte Verzerrungstensor</a> und RL ist der <a href="Verschiebungsgradient#Deformationsgradient_und_seine_Polarzerlegung" title="Verschiebungsgradient">linearisierte Rotationstensor</a>. Eine Unterscheidung des materiellen und räumlichen Geschwindigkeitsgradienten ist bei kleinen Deformationen demnach nicht nötig.

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

<div class="mw-heading mw-heading3"><h3 id="Linien-,_Flächen-_und_Volumenelemente">Linien-, Flächen- und Volumenelemente</h3></div>

Der räumliche Geschwindigkeitsgradient transformiert in der Momentankonfiguration die <a href="Linienelement" class="mw-redirect" title="Linienelement">Linien-</a>,

<a href="Oberfl%C3%A4chenelement" class="mw-redirect" title="Oberflächenelement">Flächen-</a> und <a href="Volumenelement" class="mw-redirect" title="Volumenelement">Volumenelemente</a> in ihre Raten:

<mtr>

<mtd>

</mrow>

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

</mtd>

<mtd>

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</mrow>

<mover>

</mover>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

</mrow>

<mover>

</mover>

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<mover>

</mover>

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<mtd>

</mrow>

</mrow>

<msup>

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</mrow>

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</mrow>

<mover>

</mover>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

</mrow>

<mover>

</mover>

</mrow>

</mtd>

<mtd>

</mrow>

</mrow>

<mover>

</mover>

</mrow>

</mrow>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}(\mathrm {d} {\vec {x}}){\dot {}}&=\mathbf {l} \cdot \mathrm {d} {\vec {x}}\\(\mathrm {d} {\vec {a}}){\dot {}}&=(\operatorname {Sp} (\mathbf {l} )\mathbf {1} -\mathbf {l} ^{\top })\cdot \mathrm {d} {\vec {a}}\\(\mathrm {d} v){\dot {}}&=\operatorname {Sp} (\mathbf {l} )\mathrm {d} v=\operatorname {div} ({\vec {v}})\,\mathrm {d} v\,.\end{aligned}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ffe86487f0fb2c06d1308aca70b33be7a056f9c6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.171ex; width:30.092ex; height:9.509ex;" alt="{\displaystyle {\begin{aligned}(\mathrm {d} {\vec {x}}){\dot {}}&=\mathbf {l} \cdot \mathrm {d} {\vec {x}}\\(\mathrm {d} {\vec {a}}){\dot {}}&=(\operatorname {Sp} (\mathbf {l} )\mathbf {1} -\mathbf {l} ^{\top })\cdot \mathrm {d} {\vec {a}}\\(\mathrm {d} v){\dot {}}&=\operatorname {Sp} (\mathbf {l} )\mathrm {d} v=\operatorname {div} ({\vec {v}})\,\mathrm {d} v\,.\end{aligned}}}" loading="lazy"></dd></dl>

Darin ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {d} {\vec {a}}}">

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<mover>

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/073ceacc07f2d7e5060a0f9d7603797968e2a860.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.522ex; height:2.343ex;" alt="{\displaystyle \mathrm {d} {\vec {a}}}" loading="lazy"> (für <a href="Englische_Sprache" title="Englische Sprache">englisch</a> area „Fläche“) das vektorielle Oberflächenelement und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {d} v}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c6c4081d342e25c7d393c46a7ce4fa8502288d40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.42ex; height:2.176ex;" alt="{\displaystyle \mathrm {d} v}" loading="lazy"> (für <a href="Englische_Sprache" title="Englische Sprache">englisch</a> volume „Volumen“) das Volumenelement. Der Operator <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Sp} }">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/37bbee2d752fc809d3a8f9f8977451fce98fef77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.585ex; height:2.509ex;" alt="{\displaystyle \operatorname {Sp} }" loading="lazy"> berechnet die <a href="Spur_(Mathematik)" title="Spur (Mathematik)">Spur</a> seines Argumentes, die im Fall des Geschwindigkeitsgradienten die <a href="Divergenz_eines_Vektorfeldes" title="Divergenz eines Vektorfeldes">Divergenz</a> des Geschwindigkeitsfeldes ist:

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {Sp} (\mathbf {l} )=\operatorname {Sp} (\mathbf {d} )=\operatorname {div} ({\vec {v}})\,.}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9d38fafafbac04323cc9d0e2fe711bdee1b4f33f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.398ex; height:2.843ex;" alt="{\displaystyle \operatorname {Sp} (\mathbf {l} )=\operatorname {Sp} (\mathbf {d} )=\operatorname {div} ({\vec {v}})\,.}" loading="lazy"></dd></dl>

<td>Beweis

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<td>Der <a href="Deformationsgradient" title="Deformationsgradient">Deformationsgradient</a> F transformiert die Linien-, Flächen- und Volumenelement von der Referenzkonfiguration in die Momentankonfiguration:

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\mathrm {d} {\vec {x}}=&\mathbf {F} \cdot \mathrm {d} {\vec {X}}\\\mathrm {d} {\vec {a}}=&\operatorname {det} (\mathbf {F} )\mathbf {F} ^{\top -1}\cdot \mathrm {d} {\vec {A}}\\\mathrm {d} v=&\operatorname {det} (\mathbf {F} )\;\mathrm {d} V\,.\end{aligned}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/424e1243888cedf9b490f8c35717123607d7a82a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.505ex; width:22.557ex; height:10.176ex;" alt="{\displaystyle {\begin{aligned}\mathrm {d} {\vec {x}}=&\mathbf {F} \cdot \mathrm {d} {\vec {X}}\\\mathrm {d} {\vec {a}}=&\operatorname {det} (\mathbf {F} )\mathbf {F} ^{\top -1}\cdot \mathrm {d} {\vec {A}}\\\mathrm {d} v=&\operatorname {det} (\mathbf {F} )\;\mathrm {d} V\,.\end{aligned}}}" loading="lazy">

Der Operator <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {det} (\cdot )}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a01a1992993c72c69453e12ed5266eb5e5204d02.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.686ex; height:2.843ex;" alt="{\displaystyle \operatorname {det} (\cdot )}" loading="lazy"> bildet die <a href="Determinante" title="Determinante">Determinante</a> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\cdot )^{\top -1}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7b8c1c9e1aca96d683e702498ed5506f48f43274.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.067ex; height:3.176ex;" alt="{\displaystyle (\cdot )^{\top -1}}" loading="lazy"> die <a href="Transponierte_Matrix" title="Transponierte Matrix">transponiert</a> <a href="Inverse_Matrix" title="Inverse Matrix">Inverse</a>. Die Oberfläche des Körpers in der Referenzkonfiguration hat das <a href="Oberfl%C3%A4chenelement" class="mw-redirect" title="Oberflächenelement">Oberflächenelement</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {d} {\vec {A}}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d9f9388557a681a60c4b6e29cd5e877b13d41622.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.036ex; height:3.009ex;" alt="{\displaystyle \mathrm {d} {\vec {A}}}" loading="lazy">, d. h. die mit dem Flächenstück <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {d} A}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2c176b538dffc74abcc1a74bbf51a65b18dc4ef5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.036ex; height:2.176ex;" alt="{\displaystyle \mathrm {d} A}" loading="lazy"> multiplizierte Normale <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {N}}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/13e01cc1cec4201098af497311170ea68412c6b7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.843ex;" alt="{\displaystyle {\vec {N}}}" loading="lazy"> des Flächenstücks, und Gleiches gilt für das räumliche Flächenelement <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {d} {\vec {a}}}">

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

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<annotation encoding="application/x-tex">{\displaystyle {\frac {\mathrm {D} }{\mathrm {D} t}}(\mathrm {d} {\vec {x}})={\frac {\mathrm {D} }{\mathrm {D} t}}(\mathbf {F} \cdot \mathrm {d} {\vec {X}})={\dot {\mathbf {F} }}\cdot \mathrm {d} {\vec {X}}={\dot {\mathbf {F} }}\cdot \mathbf {F} ^{-1}\cdot \mathrm {d} {\vec {x}}=\mathbf {l} \cdot \mathrm {d} {\vec {x}}\,.}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/97f5d3dd659264785c400618e8b0b526ec20dcbf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:56.562ex; height:5.176ex;" alt="{\displaystyle {\frac {\mathrm {D} }{\mathrm {D} t}}(\mathrm {d} {\vec {x}})={\frac {\mathrm {D} }{\mathrm {D} t}}(\mathbf {F} \cdot \mathrm {d} {\vec {X}})={\dot {\mathbf {F} }}\cdot \mathrm {d} {\vec {X}}={\dot {\mathbf {F} }}\cdot \mathbf {F} ^{-1}\cdot \mathrm {d} {\vec {x}}=\mathbf {l} \cdot \mathrm {d} {\vec {x}}\,.}" loading="lazy">

Die materielle Zeitableitung des Volumenelements ergibt sich mit der <a href="Hauptinvariante#Ableitungen_der_Hauptinvarianten" title="Hauptinvariante">Ableitung der Determinante</a> aus

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\frac {\mathrm {D} }{\mathrm {D} t}}(\mathrm {d} v)=&{\frac {\mathrm {D} }{\mathrm {D} t}}[\operatorname {det} (\mathbf {F} )\mathrm {d} V]=\operatorname {det} (\mathbf {F} )(\mathbf {F} ^{\top -1}:{\dot {\mathbf {F} }})\mathrm {d} V=\operatorname {Sp} (\mathbf {F} ^{-1}\cdot {\dot {\mathbf {F} }})\mathrm {d} v=\operatorname {Sp} ({\dot {\mathbf {F} }}\cdot \mathbf {F} ^{-1})\mathrm {d} v=\operatorname {Sp} (\mathbf {l} )\mathrm {d} v\,.\end{aligned}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c51bbddb4babc173a17a415bbf02e6a687777ed4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.83ex; margin-bottom: -0.175ex; width:95.324ex; height:5.176ex;" alt="{\displaystyle {\begin{aligned}{\frac {\mathrm {D} }{\mathrm {D} t}}(\mathrm {d} v)=&{\frac {\mathrm {D} }{\mathrm {D} t}}[\operatorname {det} (\mathbf {F} )\mathrm {d} V]=\operatorname {det} (\mathbf {F} )(\mathbf {F} ^{\top -1}:{\dot {\mathbf {F} }})\mathrm {d} V=\operatorname {Sp} (\mathbf {F} ^{-1}\cdot {\dot {\mathbf {F} }})\mathrm {d} v=\operatorname {Sp} ({\dot {\mathbf {F} }}\cdot \mathbf {F} ^{-1})\mathrm {d} v=\operatorname {Sp} (\mathbf {l} )\mathrm {d} v\,.\end{aligned}}}" loading="lazy">

Der Doppelpunkt „:“ steht für das <a href="Frobenius-Skalarprodukt" title="Frobenius-Skalarprodukt">Frobenius-Skalarprodukt</a> von Tensoren, das für zwei Tensoren A und B über <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {A:B} :=\operatorname {Sp} (\mathbf {A^{\top }\cdot B} )}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {A:B} :=\operatorname {Sp} (\mathbf {A^{\top }\cdot B} )}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e4e90ab4208740e2b5a21df9d9e7c7134a629e67.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.203ex; height:3.176ex;" alt="{\displaystyle \mathbf {A:B} :=\operatorname {Sp} (\mathbf {A^{\top }\cdot B} )}" loading="lazy"> definiert ist. Der Gradient eines Vektorfeldes ist mit dem <a href="Nabla-Operator" title="Nabla-Operator">Nabla-Operator</a> und dem <a href="Dyadisches_Produkt" title="Dyadisches Produkt">dyadischen Produkt</a> „<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">“ definiert: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {grad} {\vec {v}}:=(\nabla \otimes {\vec {v}})^{\top }\,.}">

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {grad} {\vec {v}}:=(\nabla \otimes {\vec {v}})^{\top }\,.}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d154a7e535e31d7a02f89c97fad3e797e92f688f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.142ex; height:3.176ex;" alt="{\displaystyle \operatorname {grad} {\vec {v}}:=(\nabla \otimes {\vec {v}})^{\top }\,.}" loading="lazy"> Die Spur eines dyadischen Produkts ist das Skalarprodukt ihrer Faktoren: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Sp(grad} {\vec {v}})=\nabla \cdot {\vec {v}}\,,}">

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {Sp(grad} {\vec {v}})=\nabla \cdot {\vec {v}}\,,}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d70a56e7a1de178b0e81d2916be66f298c2e7e61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.409ex; height:2.843ex;" alt="{\displaystyle \operatorname {Sp(grad} {\vec {v}})=\nabla \cdot {\vec {v}}\,,}" loading="lazy"> denn die Transposition hat keinen Einfluss auf die Spur. Das Skalarprodukt des Nabla-Operators mit dem Geschwindigkeitsfeld ist aber dessen Divergenz: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {div} {\vec {v}}:=\nabla \cdot {\vec {v}}\,.}">

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {div} {\vec {v}}:=\nabla \cdot {\vec {v}}\,.}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bbb9d76d2419ad9af463cf6bcc63fbe7b1444ec4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:14.299ex; height:2.343ex;" alt="{\displaystyle \operatorname {div} {\vec {v}}:=\nabla \cdot {\vec {v}}\,.}" loading="lazy"> Also ist die Spur des Geschwindigkeitsgradienten gleich der Divergenz des Geschwindigkeitsfeldes.

Abschließend berechnet sich noch die materielle Zeitableitung des Flächenelements mit der <a href="Produktregel" title="Produktregel">Produktregel</a> zu

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\frac {\mathrm {D} }{\mathrm {D} t}}(\mathrm {d} {\vec {a}})=&{\frac {\mathrm {D} }{\mathrm {D} t}}(\operatorname {det} (\mathbf {F} )\mathbf {F} ^{\top -1}\cdot \mathrm {d} {\vec {A}})=[\operatorname {det} (\mathbf {F} )(\mathbf {F} ^{\top -1}:{\dot {\mathbf {F} }})\mathbf {F} ^{\top -1}-\operatorname {det} (\mathbf {F} )\mathbf {F} ^{\top -1}\cdot {\dot {\mathbf {F} }}^{\top }\cdot \mathbf {F} ^{\top -1}]\cdot \mathrm {d} {\vec {A}}\\=&[\operatorname {Sp} (\mathbf {F} ^{-1}\cdot {\dot {\mathbf {F} }})\mathbf {1} -\mathbf {F} ^{\top -1}\cdot {\dot {\mathbf {F} }}^{\top }]\cdot \operatorname {det} (\mathbf {F} )\mathbf {F} ^{\top -1}\cdot \mathrm {d} {\vec {A}}=[\operatorname {Sp} ({\dot {\mathbf {F} }}\cdot \mathbf {F} ^{-1})\mathbf {1} -({\dot {\mathbf {F} }}\cdot \mathbf {F} ^{-1})^{\top }]\cdot \mathrm {d} {\vec {a}}\\=&[\operatorname {Sp} (\mathbf {l} )\mathbf {1} -\mathbf {l} ^{\top }]\cdot \mathrm {d} {\vec {a}}\,.\end{aligned}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5d6ad549f8fb7b24f06f85a7db769dbc24e64799.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.671ex; width:92.046ex; height:12.509ex;" alt="{\displaystyle {\begin{aligned}{\frac {\mathrm {D} }{\mathrm {D} t}}(\mathrm {d} {\vec {a}})=&{\frac {\mathrm {D} }{\mathrm {D} t}}(\operatorname {det} (\mathbf {F} )\mathbf {F} ^{\top -1}\cdot \mathrm {d} {\vec {A}})=[\operatorname {det} (\mathbf {F} )(\mathbf {F} ^{\top -1}:{\dot {\mathbf {F} }})\mathbf {F} ^{\top -1}-\operatorname {det} (\mathbf {F} )\mathbf {F} ^{\top -1}\cdot {\dot {\mathbf {F} }}^{\top }\cdot \mathbf {F} ^{\top -1}]\cdot \mathrm {d} {\vec {A}}\\=&[\operatorname {Sp} (\mathbf {F} ^{-1}\cdot {\dot {\mathbf {F} }})\mathbf {1} -\mathbf {F} ^{\top -1}\cdot {\dot {\mathbf {F} }}^{\top }]\cdot \operatorname {det} (\mathbf {F} )\mathbf {F} ^{\top -1}\cdot \mathrm {d} {\vec {A}}=[\operatorname {Sp} ({\dot {\mathbf {F} }}\cdot \mathbf {F} ^{-1})\mathbf {1} -({\dot {\mathbf {F} }}\cdot \mathbf {F} ^{-1})^{\top }]\cdot \mathrm {d} {\vec {a}}\\=&[\operatorname {Sp} (\mathbf {l} )\mathbf {1} -\mathbf {l} ^{\top }]\cdot \mathrm {d} {\vec {a}}\,.\end{aligned}}}" loading="lazy">

Hier wurde die Konstanz des <a href="Einheitstensor" title="Einheitstensor">Einheitstensors</a> benutzt:

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {0} ={\frac {\mathrm {D} }{\mathrm {D} t}}\mathbf {1} ={\frac {\mathrm {D} }{\mathrm {D} t}}(\mathbf {F} ^{\top -1}\cdot \mathbf {F} ^{\top })={\frac {\mathrm {D} }{\mathrm {D} t}}(\mathbf {F} ^{\top -1})\cdot \mathbf {F} ^{\top }+\mathbf {F} ^{\top -1}\cdot {\dot {\mathbf {F} }}^{\top }\quad \rightarrow \quad {\frac {\mathrm {D} }{\mathrm {D} t}}(\mathbf {F} ^{\top -1})=-\mathbf {F} ^{\top -1}\cdot {\dot {\mathbf {F} }}^{\top }\cdot \mathbf {F} ^{\top -1}\,.}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/571871ea12e53cbff4e90e69e507a27a1775a4a5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:101.176ex; height:5.176ex;" alt="{\displaystyle \mathbf {0} ={\frac {\mathrm {D} }{\mathrm {D} t}}\mathbf {1} ={\frac {\mathrm {D} }{\mathrm {D} t}}(\mathbf {F} ^{\top -1}\cdot \mathbf {F} ^{\top })={\frac {\mathrm {D} }{\mathrm {D} t}}(\mathbf {F} ^{\top -1})\cdot \mathbf {F} ^{\top }+\mathbf {F} ^{\top -1}\cdot {\dot {\mathbf {F} }}^{\top }\quad \rightarrow \quad {\frac {\mathrm {D} }{\mathrm {D} t}}(\mathbf {F} ^{\top -1})=-\mathbf {F} ^{\top -1}\cdot {\dot {\mathbf {F} }}^{\top }\cdot \mathbf {F} ^{\top -1}\,.}" loading="lazy">

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

Wenn die Spur des räumlichen Geschwindigkeitsgradienten l oder – gleichbedeutend – des räumlichen Verzerrungsgeschwindigkeitstensors d oder die Divergenz des Geschwindigkeitsfeldes verschwindet, dann ist die Bewegung lokal volumenerhaltend. Bei einer Starrkörperbewegung ist, wie unten nachgewiesen, Sp(l)=Sp(w)=0, was die Konstanz des Volumens bei einer solchen Bewegung bestätigt. Eine positive Divergenz bedeutet Expansion, was namensgebend für die Divergenz ist (<a href="Latein" title="Latein">lateinisch</a> divergere „auseinanderstreben“) und was in der Realität mit einer <a href="Kontinuumsmechanik#Massenbilanz" title="Kontinuumsmechanik">Abnahme der Dichte</a> einher geht.

<div class="mw-heading mw-heading3"><h3 id="Dehn-_und_Schergeschwindigkeiten">Dehn- und Schergeschwindigkeiten</h3></div>

Bei der Verformung eines Körpers ändern sich in den deformierten Stellen die Abstände seiner Partikel und/oder die Winkel zwischen Verbindungslinien seiner Partikel. Mathematisch werden die Tangentenvektoren an solche Verbindungslinien betrachtet, siehe Abbildung rechts. Ändern diese Tangentenvektoren ihre Längen oder die Winkel untereinander, was im gleichen Maß geschieht wie die Verbindungslinien gedehnt oder geschert werden, dann ändern sich ihre <a href="Skalarprodukt" title="Skalarprodukt">Skalarprodukte</a> und es liegen Deformationen vor. Die Änderungsrate dieser Skalarprodukte bemisst der räumliche Verzerrungsgeschwindigkeitstensor d:

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<annotation encoding="application/x-tex">{\displaystyle {\frac {\mathrm {D} }{\mathrm {D} t}}(\mathrm {d} {\vec {x}}\cdot \mathrm {d} {\vec {y}})=(\mathbf {l} \cdot \mathrm {d} {\vec {x}})\cdot \mathrm {d} {\vec {y}}+\mathrm {d} {\vec {x}}\cdot (\mathbf {l} \cdot \mathrm {d} {\vec {y}})=2\mathrm {d} {\vec {x}}\cdot \mathbf {d} \cdot \mathrm {d} {\vec {y}}.}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/51bc9cc6f79b049c05eca314126079fecc3375fa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:55.166ex; height:5.176ex;" alt="{\displaystyle {\frac {\mathrm {D} }{\mathrm {D} t}}(\mathrm {d} {\vec {x}}\cdot \mathrm {d} {\vec {y}})=(\mathbf {l} \cdot \mathrm {d} {\vec {x}})\cdot \mathrm {d} {\vec {y}}+\mathrm {d} {\vec {x}}\cdot (\mathbf {l} \cdot \mathrm {d} {\vec {y}})=2\mathrm {d} {\vec {x}}\cdot \mathbf {d} \cdot \mathrm {d} {\vec {y}}.}" loading="lazy"></dd></dl>

Die <a href="Dehnrate" title="Dehnrate">Dehnrate</a> in einer bestimmten Richtung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {e}}=\mathrm {d} {\vec {x}}/|\mathrm {d} {\vec {x}}|}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bca15ff627b5edc982fe9979a88f715bda8d9c8b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.091ex; height:2.843ex;" alt="{\displaystyle {\hat {e}}=\mathrm {d} {\vec {x}}/|\mathrm {d} {\vec {x}}|}" loading="lazy"> berechnet sich aus<a href="#cite_note-spin-3">[F 2]</a>:

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<annotation encoding="application/x-tex">{\displaystyle {\dot {\varepsilon }}={\frac {|\mathrm {d} {\vec {x}}|{\dot {}}}{|\mathrm {d} {\vec {x}}|}}={\frac {{\frac {\mathrm {d} {\vec {x}}}{|\mathrm {d} {\vec {x}}|}}\cdot (\mathrm {d} {\vec {x}}){\dot {}}}{|\mathrm {d} {\vec {x}}|}}={\frac {\mathrm {d} {\vec {x}}\cdot \mathbf {l} \cdot \mathrm {d} {\vec {x}}}{|\mathrm {d} {\vec {x}}|^{2}}}={\hat {e}}\cdot (\mathbf {d+w} )\cdot {\hat {e}}={\hat {e}}\cdot \mathbf {d} \cdot {\hat {e}}={\frac {\partial v}{\partial x}}\,,}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2f1bb807a5f66759617e148b5f3a2778a0f15dc7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:73.978ex; height:9.009ex;" alt="{\displaystyle {\dot {\varepsilon }}={\frac {|\mathrm {d} {\vec {x}}|{\dot {}}}{|\mathrm {d} {\vec {x}}|}}={\frac {{\frac {\mathrm {d} {\vec {x}}}{|\mathrm {d} {\vec {x}}|}}\cdot (\mathrm {d} {\vec {x}}){\dot {}}}{|\mathrm {d} {\vec {x}}|}}={\frac {\mathrm {d} {\vec {x}}\cdot \mathbf {l} \cdot \mathrm {d} {\vec {x}}}{|\mathrm {d} {\vec {x}}|^{2}}}={\hat {e}}\cdot (\mathbf {d+w} )\cdot {\hat {e}}={\hat {e}}\cdot \mathbf {d} \cdot {\hat {e}}={\frac {\partial v}{\partial x}}\,,}" loading="lazy"></dd></dl>

wo die Geschwindigkeit v und die Koordinate x in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {e}}}">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7ac09328845eecc01a117acbf303c1bc1decc4a1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.292ex; height:2.176ex;" alt="{\displaystyle {\hat {e}}}" loading="lazy">-Richtung zählen. Die <a href="Scherung_(Mechanik)" title="Scherung (Mechanik)">Schergeschwindigkeit</a> ergibt sich im Zustand <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma =0}">

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<annotation encoding="application/x-tex">{\displaystyle \gamma =0}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0a5e84cac32e896a80a89f8cd1917c2defcf4108.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.523ex; height:2.676ex;" alt="{\displaystyle \gamma =0}" loading="lazy"> aus

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\frac {\mathrm {D} }{\mathrm {D} t}}(\mathrm {d} {\vec {x}}\cdot \mathrm {d} {\vec {y}})=&{\frac {\mathrm {D} }{\mathrm {D} t}}(|\mathrm {d} {\vec {x}}|\,|\mathrm {d} {\vec {y}}|\sin(\gamma ))={\frac {\mathrm {D} }{\mathrm {D} t}}(|\mathrm {d} {\vec {x}}|\,|\mathrm {d} {\vec {y}}|)\sin(\gamma )+|\mathrm {d} {\vec {x}}|\,|\mathrm {d} {\vec {y}}|\cos(\gamma ){\dot {\gamma }}\\\gamma =0\rightarrow &{\frac {\mathrm {D} }{\mathrm {D} t}}(\mathrm {d} {\vec {x}}\cdot \mathrm {d} {\vec {y}})=2\mathrm {d} {\vec {x}}\cdot \mathbf {d} \cdot \mathrm {d} {\vec {y}}=|\mathrm {d} {\vec {x}}|\,|\mathrm {d} {\vec {y}}|\,{\dot {\gamma }}\\\rightarrow {\dot {\gamma }}=&2{\frac {\mathrm {d} {\vec {x}}}{|\mathrm {d} {\vec {x}}|}}\cdot \mathbf {d} \cdot {\frac {\mathrm {d} {\vec {y}}}{|\mathrm {d} {\vec {y}}|}}={\frac {\partial v_{x}}{\partial y}}+{\frac {\partial v_{y}}{\partial x}}\,.\end{aligned}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/41d11ffd3f38b65a8982b84aa2b5b9f64391cd55.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -8.171ex; width:76.706ex; height:17.509ex;" alt="{\displaystyle {\begin{aligned}{\frac {\mathrm {D} }{\mathrm {D} t}}(\mathrm {d} {\vec {x}}\cdot \mathrm {d} {\vec {y}})=&{\frac {\mathrm {D} }{\mathrm {D} t}}(|\mathrm {d} {\vec {x}}|\,|\mathrm {d} {\vec {y}}|\sin(\gamma ))={\frac {\mathrm {D} }{\mathrm {D} t}}(|\mathrm {d} {\vec {x}}|\,|\mathrm {d} {\vec {y}}|)\sin(\gamma )+|\mathrm {d} {\vec {x}}|\,|\mathrm {d} {\vec {y}}|\cos(\gamma ){\dot {\gamma }}\\\gamma =0\rightarrow &{\frac {\mathrm {D} }{\mathrm {D} t}}(\mathrm {d} {\vec {x}}\cdot \mathrm {d} {\vec {y}})=2\mathrm {d} {\vec {x}}\cdot \mathbf {d} \cdot \mathrm {d} {\vec {y}}=|\mathrm {d} {\vec {x}}|\,|\mathrm {d} {\vec {y}}|\,{\dot {\gamma }}\\\rightarrow {\dot {\gamma }}=&2{\frac {\mathrm {d} {\vec {x}}}{|\mathrm {d} {\vec {x}}|}}\cdot \mathbf {d} \cdot {\frac {\mathrm {d} {\vec {y}}}{|\mathrm {d} {\vec {y}}|}}={\frac {\partial v_{x}}{\partial y}}+{\frac {\partial v_{y}}{\partial x}}\,.\end{aligned}}}" loading="lazy"></dd></dl>

Hier zählen die Geschwindigkeit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{x}}">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/704b7ad1ece77840fde455daa6d2e51e64282b5e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.3ex; height:2.009ex;" alt="{\displaystyle v_{x}}" loading="lazy"> sowie die Koordinate x in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {d} {\vec {x}}}">

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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">-Richtung und die Geschwindigkeit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{y}}">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/922aa64dad09633a401e14be9b4389795835cd8c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.177ex; height:2.343ex;" alt="{\displaystyle v_{y}}" loading="lazy"> sowie die Koordinate y in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {d} {\vec {y}}}">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a0c7179cd5a46e2ee0ce8b7e74ebeebb180e2f8a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.557ex; height:2.676ex;" alt="{\displaystyle \mathrm {d} {\vec {y}}}" loading="lazy">-Richtung.

Der Verzerrungsgeschwindigkeitstensor d legt also die Dehn- und Scherraten in der Momentankonfiguration fest.

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

Sind die im vorigen Abschnitt betrachteten Tangentenvektoren Eigenvektoren des Geschwindigkeitsgradienten oder des Verzerrungsgeschwindigkeitstensors, dann hat das bemerkenswerte Konsequenzen. Für einen solchen Eigenvektor des Geschwindigkeitsgradienten gilt:

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/46e98d7e7e855d6152aadfeec77d5e459215c413.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.492ex; height:2.176ex;" alt="{\displaystyle \mathbf {l} \cdot {\hat {e}}=\lambda {\hat {e}}\,.}" loading="lazy"></dd></dl>

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

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</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7ac09328845eecc01a117acbf303c1bc1decc4a1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.292ex; height:2.176ex;" alt="{\displaystyle {\hat {e}}}" loading="lazy"> gehörende Eigenwert. Die <a href="Frobeniusnorm" title="Frobeniusnorm">Frobeniusnorm</a> der Eigenvektoren ist unbestimmt, weswegen ihr Betrag hier auf eins festgelegt wird, was im <a href="Zirkumflex" title="Zirkumflex">Hut</a> über dem e zum Ausdruck kommt. Die Zeitableitung eines Tangentenvektors <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {e}}=\mathrm {d} {\vec {x}}/|\mathrm {d} {\vec {x}}|}">

<mover>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bca15ff627b5edc982fe9979a88f715bda8d9c8b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.091ex; height:2.843ex;" alt="{\displaystyle {\hat {e}}=\mathrm {d} {\vec {x}}/|\mathrm {d} {\vec {x}}|}" loading="lazy"> der Länge eins in der Momentankonfiguration liefert

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<mover>

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<mover>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\dot {\hat {e}}}={\frac {(\mathrm {d} {\vec {x}}){\dot {}}}{|\mathrm {d} {\vec {x}}|}}-{\frac {\mathrm {d} {\vec {x}}}{|\mathrm {d} {\vec {x}}|^{2}}}{\frac {\mathrm {d} {\vec {x}}\cdot (\mathrm {d} {\vec {x}}){\dot {}}}{|\mathrm {d} {\vec {x}}|}}=\mathbf {l} \cdot {\frac {\mathrm {d} {\vec {x}}}{|\mathrm {d} {\vec {x}}|}}-\left({\frac {\mathrm {d} {\vec {x}}}{|\mathrm {d} {\vec {x}}|}}\cdot \mathbf {l} \cdot {\frac {\mathrm {d} {\vec {x}}}{|\mathrm {d} {\vec {x}}|}}\right){\frac {\mathrm {d} {\vec {x}}}{|\mathrm {d} {\vec {x}}|}}=\mathbf {l} \cdot {\hat {e}}-({\hat {e}}\cdot \mathbf {l} \cdot {\hat {e}}){\hat {e}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4e822a4a0cb1cd52b5d5e6c4dc8b8d515036529a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:84.911ex; height:6.843ex;" alt="{\displaystyle {\dot {\hat {e}}}={\frac {(\mathrm {d} {\vec {x}}){\dot {}}}{|\mathrm {d} {\vec {x}}|}}-{\frac {\mathrm {d} {\vec {x}}}{|\mathrm {d} {\vec {x}}|^{2}}}{\frac {\mathrm {d} {\vec {x}}\cdot (\mathrm {d} {\vec {x}}){\dot {}}}{|\mathrm {d} {\vec {x}}|}}=\mathbf {l} \cdot {\frac {\mathrm {d} {\vec {x}}}{|\mathrm {d} {\vec {x}}|}}-\left({\frac {\mathrm {d} {\vec {x}}}{|\mathrm {d} {\vec {x}}|}}\cdot \mathbf {l} \cdot {\frac {\mathrm {d} {\vec {x}}}{|\mathrm {d} {\vec {x}}|}}\right){\frac {\mathrm {d} {\vec {x}}}{|\mathrm {d} {\vec {x}}|}}=\mathbf {l} \cdot {\hat {e}}-({\hat {e}}\cdot \mathbf {l} \cdot {\hat {e}}){\hat {e}}}" loading="lazy"></dd></dl>

In Richtung der Eigenvektoren des räumlichen Geschwindigkeitsgradienten verschwindet diese Rate<a href="#cite_note-4">[F 3]</a>. Einsetzen des Verzerrungsgeschwindigkeitstensors und des Wirbeltensors ergibt weiterhin<a href="#cite_note-spin-3">[F 2]</a>:

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<mover>

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<mover>

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<mover>

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<mover>

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<mover>

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<mover>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\dot {\hat {e}}}=(\mathbf {d+w} )\cdot {\hat {e}}-({\hat {e}}\cdot (\mathbf {d+w} )\cdot {\hat {e}}){\hat {e}}=\mathbf {w} \cdot {\hat {e}}+\mathbf {d} \cdot {\hat {e}}-({\hat {e}}\cdot \mathbf {d} \cdot {\hat {e}}){\hat {e}}\,.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7ed319c0887c96682e2cf1025585351d67c50731.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:65.613ex; height:3.343ex;" alt="{\displaystyle {\dot {\hat {e}}}=(\mathbf {d+w} )\cdot {\hat {e}}-({\hat {e}}\cdot (\mathbf {d+w} )\cdot {\hat {e}}){\hat {e}}=\mathbf {w} \cdot {\hat {e}}+\mathbf {d} \cdot {\hat {e}}-({\hat {e}}\cdot \mathbf {d} \cdot {\hat {e}}){\hat {e}}\,.}" loading="lazy"></dd></dl>

Sei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {e}}}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7ac09328845eecc01a117acbf303c1bc1decc4a1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.292ex; height:2.176ex;" alt="{\displaystyle {\hat {e}}}" loading="lazy"> Eigenvektor von d. Dann ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {d} \cdot {\hat {e}}-({\hat {e}}\cdot \mathbf {d} \cdot {\hat {e}}){\hat {e}}=0}">

</mrow>

<mover>

</mover>

</mrow>

<mover>

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<mover>

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<mover>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {d} \cdot {\hat {e}}-({\hat {e}}\cdot \mathbf {d} \cdot {\hat {e}}){\hat {e}}=0}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/771042742fadc4d19222eb63a7d33d3bf8946afd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.085ex; height:2.843ex;" alt="{\displaystyle \mathbf {d} \cdot {\hat {e}}-({\hat {e}}\cdot \mathbf {d} \cdot {\hat {e}}){\hat {e}}=0}" loading="lazy"> und daher lautet die Zeitableitung

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<mover>

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<mover>

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<mover>

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<mover>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\dot {\hat {e}}}=\mathbf {w} \cdot {\hat {e}}={\vec {\omega }}\times {\hat {e}}\,.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e02d6d6eda2dcaa384ba41385ef12d65d13fa076.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:19.002ex; height:2.843ex;" alt="{\displaystyle {\dot {\hat {e}}}=\mathbf {w} \cdot {\hat {e}}={\vec {\omega }}\times {\hat {e}}\,.}" loading="lazy"></dd></dl>

In Kombination mit dem obigen Ergebnis

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<annotation encoding="application/x-tex">{\displaystyle {\dot {\varepsilon }}={\hat {e}}\cdot \mathbf {d} \cdot {\hat {e}}\quad \rightarrow \quad \mathbf {d} \cdot {\hat {e}}={\dot {\varepsilon }}{\hat {e}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a81ff765c4ade8c860c1de39903c84c026b53ba6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:30.342ex; height:2.176ex;" alt="{\displaystyle {\dot {\varepsilon }}={\hat {e}}\cdot \mathbf {d} \cdot {\hat {e}}\quad \rightarrow \quad \mathbf {d} \cdot {\hat {e}}={\dot {\varepsilon }}{\hat {e}}}" loading="lazy"></dd></dl>

zeigt sich für Eigenvektoren von d:

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {l} \cdot {\hat {e}}=(\mathbf {d+w} )\cdot {\hat {e}}={\dot {\varepsilon }}{\hat {e}}+{\vec {\omega }}\times {\hat {e}}\,.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8696aa559aac24d23dcc8799086d20db9fbdf9e0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.316ex; height:2.843ex;" alt="{\displaystyle \mathbf {l} \cdot {\hat {e}}=(\mathbf {d+w} )\cdot {\hat {e}}={\dot {\varepsilon }}{\hat {e}}+{\vec {\omega }}\times {\hat {e}}\,.}" loading="lazy"></dd></dl>

Die polare Zerlegung des Deformationsgradienten in eine Drehung und eine rotationsfreie Streckung entspricht beim räumlichen Geschwindigkeitsgradient der additiven Zerlegung in die Dehnrate und Drehgeschwindigkeit.

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

<div class="mw-heading mw-heading3"><h3 id="Substantielle_Beschleunigung">Substantielle Beschleunigung</h3></div>

Das zweite <a href="Newtonsche_Gesetze" title="Newtonsche Gesetze">Newton’sche Gesetz</a> besagt, dass eine <a href="Kraft" title="Kraft">Kraft</a> einen materiellen Körper in Richtung der Kraft <a href="Beschleunigung" title="Beschleunigung">beschleunigt</a>. Auf lokaler Ebene werden dann die materiellen Punkte von einem von außen aufgeprägten Beschleunigungsvektor <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {b}}}">

<mover>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3c9ef58be7103eb0b2bfcb460df23430f6a36216.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.094ex; height:2.843ex;" alt="{\displaystyle {\vec {b}}}" loading="lazy"> angetrieben:

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<mover>

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<annotation encoding="application/x-tex">{\displaystyle {\dot {\vec {v}}}({\vec {x}},t)={\vec {b}}({\vec {x}},t)\,.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/532ec730d2cdd0fa4ef5b4db16bdfd4672a119cf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.461ex; height:3.343ex;" alt="{\displaystyle {\dot {\vec {v}}}({\vec {x}},t)={\vec {b}}({\vec {x}},t)\,.}" loading="lazy"></dd></dl>

Weil aber in der klassischen Mechanik ein Raumpunkt nicht beschleunigt werden kann, sondern nur ein materieller Punkt, muss auf der linken Seite der Gleichung die <a href="Substantielle_Ableitung" title="Substantielle Ableitung">materielle Zeitableitung</a> der Geschwindigkeit gebildet werden, die – wie üblich – mit einem aufgesetzten Punkt notiert wird<a href="#cite_note-Frechet-1">[F 1]</a>:

<mover>

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<mfrac>

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<mfrac>

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<mfrac>

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<mfrac>

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<mfrac>

<mrow>

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<mover>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\dot {\vec {v}}}({\vec {x}},t):={\frac {\mathrm {D} }{\mathrm {D} t}}{\vec {v}}({\vec {\chi }}({\vec {X}},t),t)=\underbrace {\frac {\partial {\vec {v}}({\vec {x}},t)}{\partial {\vec {x}}}} _{=\operatorname {grad} {\vec {v}}=\mathbf {l} }\cdot \underbrace {\frac {\mathrm {D} {\vec {\chi }}({\vec {X}},t)}{\mathrm {D} t}} _{={\vec {v}}}+{\frac {\partial {\vec {v}}({\vec {x}},t)}{\partial t}}={\frac {\partial {\vec {v}}}{\partial t}}+(\operatorname {grad} {\vec {v}})\cdot {\vec {v}}={\frac {\partial {\vec {v}}}{\partial t}}+\mathbf {l} \cdot {\vec {v}}\,.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c8143e05b3a5f51b418d03469d08a31bd7d199f9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.338ex; width:93.349ex; height:10.676ex;" alt="{\displaystyle {\dot {\vec {v}}}({\vec {x}},t):={\frac {\mathrm {D} }{\mathrm {D} t}}{\vec {v}}({\vec {\chi }}({\vec {X}},t),t)=\underbrace {\frac {\partial {\vec {v}}({\vec {x}},t)}{\partial {\vec {x}}}} _{=\operatorname {grad} {\vec {v}}=\mathbf {l} }\cdot \underbrace {\frac {\mathrm {D} {\vec {\chi }}({\vec {X}},t)}{\mathrm {D} t}} _{={\vec {v}}}+{\frac {\partial {\vec {v}}({\vec {x}},t)}{\partial t}}={\frac {\partial {\vec {v}}}{\partial t}}+(\operatorname {grad} {\vec {v}})\cdot {\vec {v}}={\frac {\partial {\vec {v}}}{\partial t}}+\mathbf {l} \cdot {\vec {v}}\,.}" loading="lazy"></dd></dl>

Darin gehört der festgehaltene Vektor <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {X}}={\vec {\chi }}^{-1}({\vec {x}},t)}">

<mover>

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<msup>

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<annotation encoding="application/x-tex">{\displaystyle {\vec {X}}={\vec {\chi }}^{-1}({\vec {x}},t)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9766745636fc303b8f8b010f843adfdf09c40267.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.879ex; height:3.343ex;" alt="{\displaystyle {\vec {X}}={\vec {\chi }}^{-1}({\vec {x}},t)}" loading="lazy"> zu dem beschleunigten Partikel, das sich zur Zeit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/65658b7b223af9e1acc877d848888ecdb4466560.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}" loading="lazy"> am Ort <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {x}}}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/db2dc6ced9cc3bc7e8b9f2707cbec033f6d3759c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:2.343ex;" alt="{\displaystyle {\vec {x}}}" loading="lazy"> aufhält und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {v}}({\vec {x}},t)={\dot {\vec {\chi }}}({\vec {X}},t)}">

<mover>

</mover>

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<mover>

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<mover>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4bc75511fedd7635904ff72ac3c7481227b21622.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.404ex; height:3.343ex;" alt="{\displaystyle {\vec {v}}({\vec {x}},t)={\dot {\vec {\chi }}}({\vec {X}},t)}" loading="lazy"> ist dessen Geschwindigkeit zur Zeit t. Der letzte Term in obiger Gleichung ist ein <a href="Konvektion" title="Konvektion">konvektiver</a> Anteil, der die kinematische Nichtlinearität der Impulsbilanz in der <a href="Eulersche_Betrachtungsweise" title="Eulersche Betrachtungsweise">Euler’schen Betrachtungsweise</a> bewirkt.

Im geometrisch linearen Fall fällt der quadratische konvektive Anteil weg und es gilt:

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<annotation encoding="application/x-tex">{\displaystyle {\dot {\vec {v}}}({\vec {x}},t)\approx {\frac {\partial {\vec {v}}({\vec {x}},t)}{\partial t}}\,.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6cc68bfdd22ea2b4f09c4d1475c9d6394fc4223e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:18.697ex; height:5.843ex;" alt="{\displaystyle {\dot {\vec {v}}}({\vec {x}},t)\approx {\frac {\partial {\vec {v}}({\vec {x}},t)}{\partial t}}\,.}" loading="lazy"></dd></dl>

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

Jede <a href="Starrk%C3%B6rper#Allgemeine_Bewegungen_starrer_Körper" class="mw-redirect" title="Starrkörper">Starrkörperbewegung</a> lässt sich in eine Translation und eine Rotation zerlegen. Als Drehzentrum eignet sich jeder ruhende oder bewegte Punkt und auch der Schwerpunkt des Körpers, siehe Abbildung rechts. Sei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {r}}({\vec {X}})={\vec {X}}-{\vec {S}}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/093437c9b4c263faeada82267f3738ddd667c0d5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.47ex; height:3.509ex;" alt="{\displaystyle {\vec {r}}({\vec {X}})={\vec {X}}-{\vec {S}}}" loading="lazy"> der zeitlich fixierte Differenzvektor zwischen einem Partikel <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"> des starren Körpers und seinem Schwerpunkt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {S}}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c71a6b104c40975c738d5f0e22d445ebd509eb81.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.538ex; height:3.009ex;" alt="{\displaystyle {\vec {S}}}" loading="lazy"> zu einem Zeitpunkt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {t}_{0}}">

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<annotation encoding="application/x-tex">{\displaystyle {t}_{0}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5ba2b1dfa671eececd364f5f7b25fc12fed3a163.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">. Die Translation des Körpers kann dann mit seiner Schwerpunktsbewegung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {s}}(t)}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bd4ac913cab83bbf9f828d6ac7c2c0bf159160a6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.872ex; height:2.843ex;" alt="{\displaystyle {\vec {s}}(t)}" loading="lazy"> (mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {s}}(t_{0})={\vec {S}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\vec {s}}(t_{0})={\vec {S}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0fe46d66b85edfbbdfb8ac95f7d38915f2ed2ab4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.563ex; height:3.509ex;" alt="{\displaystyle {\vec {s}}(t_{0})={\vec {S}}}" loading="lazy">) und seine Drehung mit einem von der Zeit aber nicht vom Ort abhängigen <a href="Orthogonaler_Tensor" title="Orthogonaler Tensor">orthogonalen Tensor</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {Q} (t)}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/37671fb9281ca1cee6c0740768bf7800105a470b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.657ex; height:2.843ex;" alt="{\displaystyle \mathbf {Q} (t)}" loading="lazy"> (mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {Q} (t)\cdot \mathbf {Q} (t)^{\top }=\mathbf {1} \,,\;\operatorname {det} (\mathbf {Q} (t))=+1\,,\;\mathbf {Q} ({t}_{0})=\mathbf {1} }">

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {Q} (t)\cdot \mathbf {Q} (t)^{\top }=\mathbf {1} \,,\;\operatorname {det} (\mathbf {Q} (t))=+1\,,\;\mathbf {Q} ({t}_{0})=\mathbf {1} }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1468a49b6d04007bbf52cc8ba06d328604986837.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:46.982ex; height:3.176ex;" alt="{\displaystyle \mathbf {Q} (t)\cdot \mathbf {Q} (t)^{\top }=\mathbf {1} \,,\;\operatorname {det} (\mathbf {Q} (t))=+1\,,\;\mathbf {Q} ({t}_{0})=\mathbf {1} }" loading="lazy">) dargestellt werden. Translation und Rotation zusammengenommen definieren die Bewegungsfunktion und das materielle Geschwindigkeitsfeld:

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\vec {\chi }}({\vec {X}},t)=&{\vec {s}}(t)+\mathbf {Q} (t)\cdot ({\vec {X}}-{\vec {S}})={\vec {x}}\;\rightarrow \;\mathbf {F} (t)=\mathbf {Q} (t)\\\rightarrow {\dot {\vec {\chi }}}({\vec {X}},t)=&{\dot {\vec {s}}}(t)+{\dot {\mathbf {Q} }}(t)\cdot ({\vec {X}}-{\vec {S}})\;\rightarrow \;{\dot {\mathbf {F} }}(t)={\dot {\mathbf {Q} }}(t)\,.\end{aligned}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/16c35f64eb7b649c7f12620320e542b8cc9fda38.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:55.927ex; height:7.176ex;" alt="{\displaystyle {\begin{aligned}{\vec {\chi }}({\vec {X}},t)=&{\vec {s}}(t)+\mathbf {Q} (t)\cdot ({\vec {X}}-{\vec {S}})={\vec {x}}\;\rightarrow \;\mathbf {F} (t)=\mathbf {Q} (t)\\\rightarrow {\dot {\vec {\chi }}}({\vec {X}},t)=&{\dot {\vec {s}}}(t)+{\dot {\mathbf {Q} }}(t)\cdot ({\vec {X}}-{\vec {S}})\;\rightarrow \;{\dot {\mathbf {F} }}(t)={\dot {\mathbf {Q} }}(t)\,.\end{aligned}}}" loading="lazy"></dd></dl>

Im materiellen Geschwindigkeitsgradient taucht die gleichförmige Schwerpunktsgeschwindigkeit nicht mehr auf. Das räumliche Geschwindigkeitsfeld entsteht durch die Ersetzung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {X}}-{\vec {S}}=\mathbf {Q} ^{\top }(t)\cdot {\bigl (}{\vec {x}}-{\vec {s}}(t){\bigr )}}">

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<annotation encoding="application/x-tex">{\displaystyle {\vec {X}}-{\vec {S}}=\mathbf {Q} ^{\top }(t)\cdot {\bigl (}{\vec {x}}-{\vec {s}}(t){\bigr )}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0823d16955a246d1904add6f9baa3af0c3c62c86.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:27.476ex; height:3.676ex;" alt="{\displaystyle {\vec {X}}-{\vec {S}}=\mathbf {Q} ^{\top }(t)\cdot {\bigl (}{\vec {x}}-{\vec {s}}(t){\bigr )}}" loading="lazy"> im materiellen Geschwindigkeitsfeld:

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\dot {\vec {\chi }}}({\vec {X}},t)=&{\dot {\vec {s}}}(t)+{\dot {\mathbf {Q} }}(t)\cdot ({\vec {X}}-{\vec {S}})={\dot {\vec {s}}}(t)+{\dot {\mathbf {Q} }}(t)\cdot \mathbf {Q} ^{\top }(t)\cdot {\bigl (}{\vec {x}}-{\vec {s}}(t){\bigr )}\\\rightarrow {\vec {v}}({\vec {x}},t)=&{\dot {\vec {s}}}(t)+{\dot {\mathbf {Q} }}(t)\cdot \mathbf {Q} ^{\top }(t)\cdot {\bigl (}{\vec {x}}-{\vec {s}}(t){\bigr )}\,.\end{aligned}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ef60b32109c54527e9143c8754236ae185093aa1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:67.812ex; height:7.509ex;" alt="{\displaystyle {\begin{aligned}{\dot {\vec {\chi }}}({\vec {X}},t)=&{\dot {\vec {s}}}(t)+{\dot {\mathbf {Q} }}(t)\cdot ({\vec {X}}-{\vec {S}})={\dot {\vec {s}}}(t)+{\dot {\mathbf {Q} }}(t)\cdot \mathbf {Q} ^{\top }(t)\cdot {\bigl (}{\vec {x}}-{\vec {s}}(t){\bigr )}\\\rightarrow {\vec {v}}({\vec {x}},t)=&{\dot {\vec {s}}}(t)+{\dot {\mathbf {Q} }}(t)\cdot \mathbf {Q} ^{\top }(t)\cdot {\bigl (}{\vec {x}}-{\vec {s}}(t){\bigr )}\,.\end{aligned}}}" loading="lazy"></dd></dl>

woraus der ebenfalls vom Ort und der gleichförmigen Schwerpunktsgeschwindigkeit unabhängige räumliche Geschwindigkeitsgradient <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {l} (t)={\dot {\mathbf {Q} }}\cdot \mathbf {Q} ^{\top }}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {l} (t)={\dot {\mathbf {Q} }}\cdot \mathbf {Q} ^{\top }}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/81914e3ca09a025eeddc7e7ef63b0b4930f8922c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.695ex; height:3.176ex;" alt="{\displaystyle \mathbf {l} (t)={\dot {\mathbf {Q} }}\cdot \mathbf {Q} ^{\top }}" loading="lazy"> folgt. Der räumliche Geschwindigkeitsgradient ist hier schiefsymmetrisch

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {l} +\mathbf {l} ^{\top }={\dot {\mathbf {Q} }}\cdot \mathbf {Q} ^{\top }+\mathbf {Q} \cdot {\dot {\mathbf {Q} }}^{\top }={\frac {\text{d}}{{\text{d}}t}}(\mathbf {Q\cdot Q} ^{\top })={\dot {\mathbf {1} }}=\mathbf {0} \rightarrow \mathbf {l} ^{\top }=-\mathbf {l} }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c6bd8773f9c169689744d426fa2eb047950e7bac.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:61.653ex; height:5.509ex;" alt="{\displaystyle \mathbf {l} +\mathbf {l} ^{\top }={\dot {\mathbf {Q} }}\cdot \mathbf {Q} ^{\top }+\mathbf {Q} \cdot {\dot {\mathbf {Q} }}^{\top }={\frac {\text{d}}{{\text{d}}t}}(\mathbf {Q\cdot Q} ^{\top })={\dot {\mathbf {1} }}=\mathbf {0} \rightarrow \mathbf {l} ^{\top }=-\mathbf {l} }" loading="lazy"></dd></dl>

und daher identisch zu seinem Wirbeltensor (l=w) was bestätigt, dass der symmetrische Verzerrungsgeschwindigkeitstensor d bei Starrkörperbewegungen verschwindet. Der axiale duale Wirbelvektor des Wirbeltensors wird in das Geschwindigkeitsfeld eingesetzt

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<annotation encoding="application/x-tex">{\displaystyle {\vec {v}}({\vec {x}},t)={\dot {\vec {s}}}(t)+{\vec {\omega }}(t)\times {\bigl (}{\vec {x}}-{\vec {s}}(t){\bigr )}\,,}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6fad0df4a5d9ecfb835bbea682407476ae7d4cc7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:34.174ex; height:3.509ex;" alt="{\displaystyle {\vec {v}}({\vec {x}},t)={\dot {\vec {s}}}(t)+{\vec {\omega }}(t)\times {\bigl (}{\vec {x}}-{\vec {s}}(t){\bigr )}\,,}" loading="lazy"></dd></dl>

das nun keinen sichtbaren Tensor mehr enthält. Nur im Kreuzprodukt, das einer Tensortransformation entspricht, verbirgt sich noch ein Hinweis auf den Wirbeltensor.

Die Drehachse <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {e}}:={\vec {\omega }}/|{\vec {\omega }}|}">

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<annotation encoding="application/x-tex">{\displaystyle {\hat {e}}:={\vec {\omega }}/|{\vec {\omega }}|}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b1999068587c4fb48848d405d556f26a87ff4141.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.385ex; height:2.843ex;" alt="{\displaystyle {\hat {e}}:={\vec {\omega }}/|{\vec {\omega }}|}" loading="lazy"> ist ein Eigenvektor des Geschwindigkeitsgradienten (mit Eigenwert null), weswegen ihre Zeitableitung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\dot {\hat {e}}}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/04608a94549829ac1eb73dbb1c18f19d9f6d15d9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.292ex; height:2.843ex;" alt="{\displaystyle {\dot {\hat {e}}}}" loading="lazy"> zu jeder Zeit verschwindet (siehe oben), was sich auch dadurch bemerkbar macht, dass alle Punkte, deren Distanz sich in Vielfachen des Drehgeschwindigkeitsvektors bemisst, dieselbe Geschwindigkeit aufweisen: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {v}}({\vec {x}}+\lambda {\vec {\omega }},t)={\vec {v}}({\vec {x}},t)}">

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<annotation encoding="application/x-tex">{\displaystyle {\vec {v}}({\vec {x}}+\lambda {\vec {\omega }},t)={\vec {v}}({\vec {x}},t)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/becfc3f8318d527d6c055d708006869348f2c1b8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.116ex; height:2.843ex;" alt="{\displaystyle {\vec {v}}({\vec {x}}+\lambda {\vec {\omega }},t)={\vec {v}}({\vec {x}},t)}" loading="lazy"> für alle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {x}}\in \mathbb {R} ^{3},\,\lambda \in \mathbb {R} }">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2da4bd52d062167ecb98b107c0fd0f709f036723.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.198ex; height:3.009ex;" alt="{\displaystyle {\vec {x}}\in \mathbb {R} ^{3},\,\lambda \in \mathbb {R} }" loading="lazy">. Wäre die Drehachse mit diesen Partikeln verknüpft, dürfte sie sich höchstens parallel verschieben aber nicht neigen, so wie es <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\dot {\hat {e}}}={\vec {0}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\dot {\hat {e}}}={\vec {0}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68db72ea0289778ba1e20b4336164fe9992f8cab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.553ex; height:2.843ex;" alt="{\displaystyle {\dot {\hat {e}}}={\vec {0}}}" loading="lazy"> glauben macht. Als geometrisches Objekt ist der Parameter der Bewegung „Drehachse“, der sich aus dem vorgegebenen orthogonalen Tensor Q ableitet, aber an keine Partikel gebunden und kann ja sogar außerhalb des Starrkörpers liegen. An die <a href="Winkelbeschleunigung" title="Winkelbeschleunigung">Winkelbeschleunigung</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\dot {\vec {\omega }}}}">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/86ced775bc189194817dc8d335041da5cefed5f8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.446ex; height:2.843ex;" alt="{\displaystyle {\dot {\vec {\omega }}}}" loading="lazy"> resultiert an dieser Stelle mithin keinerlei Einschränkung.<a href="#cite_note-5">[F 4]</a>

Aus der lokalen Zeitableitung des Geschwindigkeitsfeldes (bei festgehaltenem Raumpunkt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {x}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\frac {\partial }{\partial t}}{\vec {v}}({\vec {x}},t)=&{\ddot {\vec {s}}}(t)+{\dot {\vec {\omega }}}(t)\times {\bigl (}{\vec {x}}-{\vec {s}}(t){\bigr )}-{\vec {\omega }}(t)\times {\dot {\vec {s}}}(t)\\[1ex]=&{\ddot {\vec {s}}}(t)+{\dot {\vec {\omega }}}(t)\times {\bigl (}{\vec {x}}-{\vec {s}}(t){\bigr )}-{\vec {\omega }}(t)\times \left[{\vec {v}}({\vec {x}},t)-{\vec {\omega }}(t)\times {\bigl (}{\vec {x}}-{\vec {s}}(t){\bigr )}\right]\,,\end{aligned}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f2f128a48848d7441ba52ae26ce31249d00c8a75.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.312ex; margin-bottom: -0.193ex; width:75.511ex; height:10.176ex;" alt="{\displaystyle {\begin{aligned}{\frac {\partial }{\partial t}}{\vec {v}}({\vec {x}},t)=&{\ddot {\vec {s}}}(t)+{\dot {\vec {\omega }}}(t)\times {\bigl (}{\vec {x}}-{\vec {s}}(t){\bigr )}-{\vec {\omega }}(t)\times {\dot {\vec {s}}}(t)\\[1ex]=&{\ddot {\vec {s}}}(t)+{\dot {\vec {\omega }}}(t)\times {\bigl (}{\vec {x}}-{\vec {s}}(t){\bigr )}-{\vec {\omega }}(t)\times \left[{\vec {v}}({\vec {x}},t)-{\vec {\omega }}(t)\times {\bigl (}{\vec {x}}-{\vec {s}}(t){\bigr )}\right]\,,\end{aligned}}}" loading="lazy"></dd></dl>

hervor was zusammen mit der materiellen Zeitableitung des Geschwindigkeitsfeldes

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<annotation encoding="application/x-tex">{\displaystyle {\dot {\vec {v}}}({\vec {x}},t)={\frac {\partial }{\partial t}}{\vec {v}}({\vec {x}},t)+\mathbf {l} \cdot {\vec {v}}({\vec {x}},t)={\frac {\partial }{\partial t}}{\vec {v}}({\vec {x}},t)+{\vec {\omega }}(t)\times {\vec {v}}({\vec {x}},t)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d28f1d21dc5e6e00b8063c6330692caa14a99d8b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:58.195ex; height:5.509ex;" alt="{\displaystyle {\dot {\vec {v}}}({\vec {x}},t)={\frac {\partial }{\partial t}}{\vec {v}}({\vec {x}},t)+\mathbf {l} \cdot {\vec {v}}({\vec {x}},t)={\frac {\partial }{\partial t}}{\vec {v}}({\vec {x}},t)+{\vec {\omega }}(t)\times {\vec {v}}({\vec {x}},t)}" loading="lazy"></dd></dl>

im Beschleunigungsfeld <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {a}}}">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/546e6615827e17295718741fd0b86f639a947f16.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:2.343ex;" alt="{\displaystyle {\vec {a}}}" loading="lazy"> (für <a href="Englische_Sprache" title="Englische Sprache">englisch</a> acceleration „Beschleunigung“) einer Starrkörperbewegung mündet:

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<annotation encoding="application/x-tex">{\displaystyle {\vec {a}}({\vec {x}},t):={\dot {\vec {v}}}({\vec {x}},t)={\ddot {\vec {s}}}(t)+{\dot {\vec {\omega }}}(t)\times {\bigl (}{\vec {x}}-{\vec {s}}(t){\bigr )}+{\vec {\omega }}(t)\times \left[{\vec {\omega }}(t)\times {\bigl (}{\vec {x}}-{\vec {s}}(t){\bigr )}\right]\,.}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/87f1349770bdc2e8a206dba00fbcbb21e55630dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:73.405ex; height:3.509ex;" alt="{\displaystyle {\vec {a}}({\vec {x}},t):={\dot {\vec {v}}}({\vec {x}},t)={\ddot {\vec {s}}}(t)+{\dot {\vec {\omega }}}(t)\times {\bigl (}{\vec {x}}-{\vec {s}}(t){\bigr )}+{\vec {\omega }}(t)\times \left[{\vec {\omega }}(t)\times {\bigl (}{\vec {x}}-{\vec {s}}(t){\bigr )}\right]\,.}" loading="lazy"></dd></dl>

Diese Herleitung beleuchtet die <a href="Kontinuumsmechanik#Lokale_und_materielle_Zeitableitung" title="Kontinuumsmechanik">lokale und materielle Zeitableitung</a> und ihre Ausprägung bei einer Starrkörperbewegung.

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

Der Potentialwirbel oder freie Wirbel ist ein klassisches Beispiel einer rotationsfreien <a href="Potentialstr%C3%B6mung" title="Potentialströmung">Potentialströmung</a>, siehe Bild rechts. Große Wirbel in Fluiden mit niedriger Viskosität werden mit diesem Modell gut beschrieben. Beispiele für einen Potentialwirbel sind der Badewannenablauf fern des Ausflusses, aber auch in guter Näherung ein <a href="Tornado" title="Tornado">Tornado</a>. Das Geschwindigkeitsfeld des <a href="Potentialstr%C3%B6mung#Potentialwirbel" title="Potentialströmung">Potentialwirbels</a> ist in <a href="Zylinderkoordinaten" class="mw-redirect" title="Zylinderkoordinaten">Zylinderkoordinaten</a> mit dem Abstand ρ vom Wirbelzentrum gegeben durch:

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<annotation encoding="application/x-tex">{\displaystyle {\vec {v}}=v_{\varphi }{\hat {e}}_{\varphi }\quad {\text{mit}}\quad v_{\varphi }:={\frac {\Gamma _{0}}{2\pi \rho }}\,.}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/06abf807e8a29a9ac33c0433c7357266a0a1a7df.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:29.186ex; height:5.843ex;" alt="{\displaystyle {\vec {v}}=v_{\varphi }{\hat {e}}_{\varphi }\quad {\text{mit}}\quad v_{\varphi }:={\frac {\Gamma _{0}}{2\pi \rho }}\,.}" loading="lazy"></dd></dl>

Der Parameter <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma _{0}}">

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<annotation encoding="application/x-tex">{\displaystyle \Gamma _{0}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/44f27c3fa0660b68ef8fa747442140655cff65cd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.507ex; height:2.509ex;" alt="{\displaystyle \Gamma _{0}}" loading="lazy"> kontrolliert die Strömungsgeschwindigkeit und es ergibt sich der Geschwindigkeitsgradient

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\operatorname {grad} {\vec {v}}=&{\hat {e}}_{\varphi }\otimes \left(-{\frac {\Gamma _{0}}{2\pi \rho ^{2}}}\right){\hat {e}}_{\rho }-{\frac {\Gamma _{0}}{2\pi \rho ^{2}}}{\hat {e}}_{\rho }\otimes {\hat {e}}_{\varphi }=-{\frac {\Gamma _{0}}{2\pi \rho ^{2}}}({\hat {e}}_{\varphi }\otimes {\hat {e}}_{\rho }+{\hat {e}}_{\rho }\otimes {\hat {e}}_{\varphi })=\mathbf {d} +\mathbf {w} \\\rightarrow \mathbf {d} =&-{\frac {\Gamma _{0}}{2\pi \rho ^{2}}}({\hat {e}}_{\varphi }\otimes {\hat {e}}_{\rho }+{\hat {e}}_{\rho }\otimes {\hat {e}}_{\varphi })\quad {\text{und}}\quad \mathbf {w} =\mathbf {0} \,.\end{aligned}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e9a3be582ae2447a4e34c2e2c7b5485072d33032.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.671ex; width:84.677ex; height:12.509ex;" alt="{\displaystyle {\begin{aligned}\operatorname {grad} {\vec {v}}=&{\hat {e}}_{\varphi }\otimes \left(-{\frac {\Gamma _{0}}{2\pi \rho ^{2}}}\right){\hat {e}}_{\rho }-{\frac {\Gamma _{0}}{2\pi \rho ^{2}}}{\hat {e}}_{\rho }\otimes {\hat {e}}_{\varphi }=-{\frac {\Gamma _{0}}{2\pi \rho ^{2}}}({\hat {e}}_{\varphi }\otimes {\hat {e}}_{\rho }+{\hat {e}}_{\rho }\otimes {\hat {e}}_{\varphi })=\mathbf {d} +\mathbf {w} \\\rightarrow \mathbf {d} =&-{\frac {\Gamma _{0}}{2\pi \rho ^{2}}}({\hat {e}}_{\varphi }\otimes {\hat {e}}_{\rho }+{\hat {e}}_{\rho }\otimes {\hat {e}}_{\varphi })\quad {\text{und}}\quad \mathbf {w} =\mathbf {0} \,.\end{aligned}}}" loading="lazy"></dd></dl>

Die Drehgeschwindigkeit der Fluidelemente um sich selbst verschwindet wegen w=0 und infolge von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {div} {\vec {v}}=\operatorname {Sp} (\mathbf {l} )=\operatorname {Sp} (\mathbf {d} )=0}">

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {div} {\vec {v}}=\operatorname {Sp} (\mathbf {l} )=\operatorname {Sp} (\mathbf {d} )=0}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c08ed8463ea40c0c6e33a9d8e96f855e61e35b2a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.203ex; height:2.843ex;" alt="{\displaystyle \operatorname {div} {\vec {v}}=\operatorname {Sp} (\mathbf {l} )=\operatorname {Sp} (\mathbf {d} )=0}" loading="lazy"> ist die Bewegung volumenerhaltend. Bei Annäherung an das Wirbelzentrum wächst die Schergeschwindigkeit aufgrund von

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<annotation encoding="application/x-tex">{\displaystyle {\dot {\gamma }}_{\rho \varphi }=2{\hat {e}}_{\rho }\cdot \mathbf {d} \cdot {\hat {e}}_{\varphi }=-{\frac {\Gamma _{0}}{\pi \rho ^{2}}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2422f0f5fc12e477f6c461b1739c8c4ea89eef19.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:26.827ex; height:6.009ex;" alt="{\displaystyle {\dot {\gamma }}_{\rho \varphi }=2{\hat {e}}_{\rho }\cdot \mathbf {d} \cdot {\hat {e}}_{\varphi }=-{\frac {\Gamma _{0}}{\pi \rho ^{2}}}}" loading="lazy"></dd></dl>

über alle Grenzen, was in realen Strömungen nicht auftreten kann, weil die immer vorhandene aber hier vernachlässigte <a href="Viskosit%C3%A4t" title="Viskosität">Viskosität</a> das wie im <a href="Hamel-Oseenscher-Wirbel" class="mw-redirect" title="Hamel-Oseenscher-Wirbel">Hamel-Oseen’schen Wirbel</a> verhindert.

<div class="mw-heading mw-heading3"><h3 id="Wechsel_des_Bezugssystems">Wechsel des Bezugssystems</h3></div>

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

Zwei Beobachter, die die Deformation eines Körpers analysieren, können sich über das Bewegungs- und Geschwindigkeitsfeld des Körpers austauschen. Beide Beobachter werden über den Deformationsgradient Einigkeit erzielen, denn er ist eine objektive Größe. Genauso wie der Insasse eines fahrenden Zuges die Geschwindigkeit eines vorbeifliegenden Vogels anders beurteilt wie ein in der Nähe befindlicher Fußgänger, werden verschieden bewegte Beobachter – wie eingangs erwähnt – unterschiedliche Geschwindigkeitsfelder und Geschwindigkeitsgradienten messen. Das Geschwindigkeitsfeld und der Geschwindigkeitsgradient sind nicht objektiv. Für den Nachweis der Objektivität – oder des Gegenteils – ist die Drehbewegung des Bezugssystems des Beobachters ausschlaggebend. Die Drehung des bewegten Beobachters relativ zum materiellen Körper wird mit einem <a href="Orthogonaler_Tensor" title="Orthogonaler Tensor">orthogonalen Tensor</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {Q} }">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/132d0144479d6f47c30ad82a65d458966ccbe928.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.008ex; height:2.509ex;" alt="{\displaystyle \mathbf {Q} }" loading="lazy"> aus der <a href="Spezielle_orthogonale_Gruppe" class="mw-redirect" title="Spezielle orthogonale Gruppe">speziellen orthogonalen Gruppe</a>

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {SO}}=\{\mathbf {Q} \in {\mathcal {L}}|\mathbf {Q} ^{-1}=\mathbf {Q} ^{\top }\;\wedge \;\det(\mathbf {Q} )=+1\}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f125f14124ef64227643aaf2c6b95a5281f993bb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:43.811ex; height:3.176ex;" alt="{\displaystyle {\mathcal {SO}}=\{\mathbf {Q} \in {\mathcal {L}}|\mathbf {Q} ^{-1}=\mathbf {Q} ^{\top }\;\wedge \;\det(\mathbf {Q} )=+1\}}" loading="lazy"></dd></dl>

beschrieben. Die Menge <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {L}}}">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9027196ecb178d598958555ea01c43157d83597c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.604ex; height:2.176ex;" alt="{\displaystyle {\mathcal {L}}}" loading="lazy"> enthält alle <a href="Tensor" title="Tensor">Tensoren</a> (zweiter Stufe), <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\cdot )^{\top }}">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ee3d42f886650bb46f2a5062fdaafa0a9a63399f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.967ex; height:3.176ex;" alt="{\displaystyle (\cdot )^{\top }}" loading="lazy"> bezeichnet die <a href="Transponierte_Matrix" title="Transponierte Matrix">Transposition</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\cdot )^{-1}}">

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Es gibt drei Arten objektiver Tensoren, die sich auf unterschiedliche Weise bei einer Euklidischen Transformation verhalten:

<td>Körperbezogen objektive, materielle, ein-Feld Tensoren

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {Q} \in {\mathcal {SO}}}</annotation>

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<td>Objektive, räumliche, ein-Feld Tensoren

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {T} '=\mathbf {Q\cdot T\cdot Q} ^{\top }}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ef5e0855c1a65dd445a5ada25cf84f1576a6ef58.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.386ex; height:3.009ex;" alt="{\displaystyle \mathbf {T} '=\mathbf {Q\cdot T\cdot Q} ^{\top }}" loading="lazy">

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<td>Objektive zwei-Feld Tensoren wie der Deformationsgradient

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

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Stellt der relativ zum Körper ruhende Beobachter in einem materiellen Punkt den Deformationsgradienten <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} '=\mathbf {Q\cdot F} \quad \rightarrow \quad {\dot {\mathbf {F} }}'={\dot {\mathbf {Q} }}\cdot \mathbf {F} +\mathbf {Q} \cdot {\dot {\mathbf {F} }}\neq \mathbf {Q} \cdot {\dot {\mathbf {F} }}\,.}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9cd0ea9c840946a95c85a680515b0a1dac632b60.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:47.643ex; height:3.509ex;" alt="{\displaystyle \mathbf {F} '=\mathbf {Q\cdot F} \quad \rightarrow \quad {\dot {\mathbf {F} }}'={\dot {\mathbf {Q} }}\cdot \mathbf {F} +\mathbf {Q} \cdot {\dot {\mathbf {F} }}\neq \mathbf {Q} \cdot {\dot {\mathbf {F} }}\,.}" loading="lazy"></dd></dl>

Der materielle Geschwindigkeitsgradient <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\dot {\mathbf {F} }}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ecd10a2737311dc76a3e39b3c2b1bb2e5b5a14aa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.683ex; height:2.676ex;" alt="{\displaystyle {\dot {\mathbf {F} }}}" loading="lazy"> ist also nicht objektiv. Es kann weiter der räumliche Geschwindigkeitsgradient des bewegten Beobachters berechnet werden

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\mathbf {l} '=&{\dot {\mathbf {F} }}'\cdot {\mathbf {F} '}^{-1}=({\dot {\mathbf {Q} }}\cdot \mathbf {F} +\mathbf {Q} \cdot {\dot {\mathbf {F} }})\cdot {\mathbf {F} }^{-1}\cdot \mathbf {Q} ^{-1}={\dot {\mathbf {Q} }}\cdot \mathbf {F} \cdot {\mathbf {F} }^{-1}\cdot \mathbf {Q} ^{\top }+\mathbf {Q} \cdot {\dot {\mathbf {F} }}\cdot {\mathbf {F} }^{-1}\cdot \mathbf {Q} ^{\top }\\\rightarrow \mathbf {l} '=&\mathbf {Q\cdot l\cdot Q} ^{\top }+{\dot {\mathbf {Q} }}\cdot \mathbf {Q} ^{\top }\,,\end{aligned}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/88df5d497569083a93b6f59691c42223834c365e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:85.012ex; height:6.843ex;" alt="{\displaystyle {\begin{aligned}\mathbf {l} '=&{\dot {\mathbf {F} }}'\cdot {\mathbf {F} '}^{-1}=({\dot {\mathbf {Q} }}\cdot \mathbf {F} +\mathbf {Q} \cdot {\dot {\mathbf {F} }})\cdot {\mathbf {F} }^{-1}\cdot \mathbf {Q} ^{-1}={\dot {\mathbf {Q} }}\cdot \mathbf {F} \cdot {\mathbf {F} }^{-1}\cdot \mathbf {Q} ^{\top }+\mathbf {Q} \cdot {\dot {\mathbf {F} }}\cdot {\mathbf {F} }^{-1}\cdot \mathbf {Q} ^{\top }\\\rightarrow \mathbf {l} '=&\mathbf {Q\cdot l\cdot Q} ^{\top }+{\dot {\mathbf {Q} }}\cdot \mathbf {Q} ^{\top }\,,\end{aligned}}}" loading="lazy"></dd></dl>

der somit ebenfalls nicht objektiv ist. Der letzte Term in der letzten Gleichung ist wegen

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<annotation encoding="application/x-tex">{\displaystyle {\dot {\mathbf {Q} }}\cdot \mathbf {Q} ^{\top }+({\dot {\mathbf {Q} }}\cdot \mathbf {Q} ^{\top })^{\top }={\dot {\mathbf {Q} }}\cdot \mathbf {Q} ^{\top }+\mathbf {Q} \cdot {\dot {\mathbf {Q} }}^{\top }=(\mathbf {Q\cdot Q} ^{\top }){\dot {}}={\dot {\mathbf {1} }}=\mathbf {0} }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e92433db287490b81c7b2dabbc906e6b8c6fc700.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:63.068ex; height:3.676ex;" alt="{\displaystyle {\dot {\mathbf {Q} }}\cdot \mathbf {Q} ^{\top }+({\dot {\mathbf {Q} }}\cdot \mathbf {Q} ^{\top })^{\top }={\dot {\mathbf {Q} }}\cdot \mathbf {Q} ^{\top }+\mathbf {Q} \cdot {\dot {\mathbf {Q} }}^{\top }=(\mathbf {Q\cdot Q} ^{\top }){\dot {}}={\dot {\mathbf {1} }}=\mathbf {0} }" loading="lazy"></dd></dl>

schiefsymmetrisch und hebt sich beim symmetrischen Verzerrungsgeschwindigkeitstensor auf:

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}2\mathbf {d} '=&\mathbf {l'+l'} ^{\top }=\mathbf {Q\cdot l\cdot Q} ^{\top }+{\dot {\mathbf {Q} }}\cdot \mathbf {Q} ^{\top }+\mathbf {Q\cdot l^{\top }\cdot Q} ^{\top }+\mathbf {Q} \cdot {\dot {\mathbf {Q} }}^{\top }=\mathbf {Q\cdot (l+l^{\top })\cdot Q} ^{\top }\\\rightarrow \mathbf {d} '=&\mathbf {Q\cdot d\cdot Q} ^{\top }\quad {\text{für alle}}\quad \mathbf {Q} \in {\mathcal {SO}}\,.\end{aligned}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b08098a5b808bc502748583247570ecd7cf8050.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:82.78ex; height:7.509ex;" alt="{\displaystyle {\begin{aligned}2\mathbf {d} '=&\mathbf {l'+l'} ^{\top }=\mathbf {Q\cdot l\cdot Q} ^{\top }+{\dot {\mathbf {Q} }}\cdot \mathbf {Q} ^{\top }+\mathbf {Q\cdot l^{\top }\cdot Q} ^{\top }+\mathbf {Q} \cdot {\dot {\mathbf {Q} }}^{\top }=\mathbf {Q\cdot (l+l^{\top })\cdot Q} ^{\top }\\\rightarrow \mathbf {d} '=&\mathbf {Q\cdot d\cdot Q} ^{\top }\quad {\text{für alle}}\quad \mathbf {Q} \in {\mathcal {SO}}\,.\end{aligned}}}" loading="lazy"></dd></dl>

Der Verzerrungsgeschwindigkeitstensor ist also objektiv, denn er transformiert sich wie ein objektiver, räumlicher, ein-Feld Tensor. Aus der Differenz <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {w=l-d} }">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a7c1a36945bce95d23d8f6643fedda9eeeb840fa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.636ex; height:2.176ex;" alt="{\displaystyle \mathbf {w=l-d} }" loading="lazy"> ergibt sich, dass der Wirbeltensor wieder nicht objektiv ist:

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\mathbf {w} '=&\mathbf {l'-d'} =\mathbf {Q\cdot l\cdot Q} ^{\top }+{\dot {\mathbf {Q} }}\cdot \mathbf {Q} ^{\top }-{\frac {1}{2}}\mathbf {Q\cdot (l+l^{\top })\cdot Q} ^{\top }={\frac {1}{2}}\mathbf {Q\cdot (l-l^{\top })\cdot Q} ^{\top }+{\dot {\mathbf {Q} }}\cdot \mathbf {Q} ^{\top }\\\rightarrow \mathbf {w} '=&\mathbf {Q\cdot w\cdot Q} ^{\top }+{\dot {\mathbf {Q} }}\cdot \mathbf {Q} ^{\top }\,.\end{aligned}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/066d391a21a35fd6cb9163fa6ca2cb3fabdf4ebc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.671ex; width:92.386ex; height:8.509ex;" alt="{\displaystyle {\begin{aligned}\mathbf {w} '=&\mathbf {l'-d'} =\mathbf {Q\cdot l\cdot Q} ^{\top }+{\dot {\mathbf {Q} }}\cdot \mathbf {Q} ^{\top }-{\frac {1}{2}}\mathbf {Q\cdot (l+l^{\top })\cdot Q} ^{\top }={\frac {1}{2}}\mathbf {Q\cdot (l-l^{\top })\cdot Q} ^{\top }+{\dot {\mathbf {Q} }}\cdot \mathbf {Q} ^{\top }\\\rightarrow \mathbf {w} '=&\mathbf {Q\cdot w\cdot Q} ^{\top }+{\dot {\mathbf {Q} }}\cdot \mathbf {Q} ^{\top }\,.\end{aligned}}}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Objektive_Zeitableitungen">Objektive Zeitableitungen</h3></div>

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

Für die Formulierung ratenabhängiger <a href="Materialmodell" title="Materialmodell">Materialmodelle</a> werden in der <a href="Eulersche_Betrachtungsweise" title="Eulersche Betrachtungsweise">räumlichen Betrachtungsweise</a> objektive Zeitableitungen für konstitutive Variablen benötigt, denn es entspricht nicht der Erfahrung, dass ein bewegter Beobachter ein anderes Materialverhalten misst als ein ruhender. Somit müssen die Materialmodelle mit objektiven Zeitableitungen formuliert werden. So wie die Geschwindigkeit und ihr Gradient nicht objektiv sind – siehe die <a href="#Beschreibung">#Beschreibung</a> oben – sind auch die Zeitableitungen anderer vom Fluid transponierter Größen nicht objektiv. Es existieren jedoch mehrere bezugssysteminvariante Raten, die für objektive Größen ebenfalls objektiv sind und mit Hilfe vom Geschwindigkeitsgradienten formuliert werden, unter anderem<a href="#cite_note-6">[F 5]</a>:

Zaremba-<a href="Gustav_Jaumann" title="Gustav Jaumann">Jaumann</a> Ableitung: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\stackrel {\circ }{\mathbf {T} }}:={\dot {\mathbf {T} }}+\mathbf {T\cdot w} -\mathbf {w\cdot T} }">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e4b70f8a28f2c1e70036c1d9434fdef02730a00b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:24.084ex; height:3.676ex;" alt="{\displaystyle {\stackrel {\circ }{\mathbf {T} }}:={\dot {\mathbf {T} }}+\mathbf {T\cdot w} -\mathbf {w\cdot T} }" loading="lazy">

<a href="Konvektive_Koordinaten#Objektive_Zeitableitungen" title="Konvektive Koordinaten">Kovariante Oldroyd</a><a href="#cite_note-7">[L 2]</a> Ableitung: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\stackrel {\triangle }{\mathbf {T} }}:={\dot {\mathbf {T} }}+\mathbf {T\cdot l} +\mathbf {l} ^{\top }\cdot \mathbf {T} ={\stackrel {\circ }{\mathbf {T} }}+\mathbf {T\cdot d} +\mathbf {d\cdot T} }">

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<annotation encoding="application/x-tex">{\displaystyle {\stackrel {\triangle }{\mathbf {T} }}:={\dot {\mathbf {T} }}+\mathbf {T\cdot l} +\mathbf {l} ^{\top }\cdot \mathbf {T} ={\stackrel {\circ }{\mathbf {T} }}+\mathbf {T\cdot d} +\mathbf {d\cdot T} }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5b3d3d68bfb54b83461c5471ab6ff92f3c81fcf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:43.902ex; height:4.176ex;" alt="{\displaystyle {\stackrel {\triangle }{\mathbf {T} }}:={\dot {\mathbf {T} }}+\mathbf {T\cdot l} +\mathbf {l} ^{\top }\cdot \mathbf {T} ={\stackrel {\circ }{\mathbf {T} }}+\mathbf {T\cdot d} +\mathbf {d\cdot T} }" loading="lazy">

<a href="Konvektive_Koordinaten#Objektive_Zeitableitungen" title="Konvektive Koordinaten">Kontravariante Oldroyd Ableitung</a>: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\stackrel {\nabla }{\mathbf {T} }}:={\dot {\mathbf {T} }}-\mathbf {l\cdot T} -\mathbf {T} \cdot \mathbf {l} ^{\top }={\stackrel {\circ }{\mathbf {T} }}-\mathbf {T\cdot d} -\mathbf {d\cdot T} }">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a8182073829366af7c01e8ded122d4ff5fa94eb3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:43.902ex; height:4.176ex;" alt="{\displaystyle {\stackrel {\nabla }{\mathbf {T} }}:={\dot {\mathbf {T} }}-\mathbf {l\cdot T} -\mathbf {T} \cdot \mathbf {l} ^{\top }={\stackrel {\circ }{\mathbf {T} }}-\mathbf {T\cdot d} -\mathbf {d\cdot T} }" loading="lazy">

Cauchy-Ableitung:<a href="#cite_note-8">[F 6]</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\stackrel {\diamond }{\mathbf {T} }}={\dot {\mathbf {T} }}+\operatorname {Sp} (\mathbf {l} )\mathbf {T} -\mathbf {l\cdot T} -\mathbf {T\cdot l} ^{\top }\,.}">

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<annotation encoding="application/x-tex">{\displaystyle {\stackrel {\diamond }{\mathbf {T} }}={\dot {\mathbf {T} }}+\operatorname {Sp} (\mathbf {l} )\mathbf {T} -\mathbf {l\cdot T} -\mathbf {T\cdot l} ^{\top }\,.}</annotation>

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Für einen objektiven Vektor <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {v}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\begin{array}{rclcl}{\stackrel {\circ }{\vec {v}}}&=&{\dot {\vec {v}}}-\mathbf {w} \cdot {\vec {v}}\\{\stackrel {\Delta }{\vec {v}}}&=&{\dot {\vec {v}}}+\mathbf {l} ^{\top }\cdot {\vec {v}}&=&{\stackrel {\circ }{\vec {v}}}+\mathbf {d} \cdot {\vec {v}}\\{\stackrel {\nabla }{\vec {v}}}&=&{\dot {\vec {v}}}-\mathbf {l} \cdot {\vec {v}}&=&{\stackrel {\circ }{\vec {v}}}-\mathbf {d} \cdot {\vec {v}}\end{array}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1c196ab71b98fc26cf031790813c215432c57fba.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.505ex; width:32.54ex; height:14.176ex;" alt="{\displaystyle {\begin{array}{rclcl}{\stackrel {\circ }{\vec {v}}}&=&{\dot {\vec {v}}}-\mathbf {w} \cdot {\vec {v}}\\{\stackrel {\Delta }{\vec {v}}}&=&{\dot {\vec {v}}}+\mathbf {l} ^{\top }\cdot {\vec {v}}&=&{\stackrel {\circ }{\vec {v}}}+\mathbf {d} \cdot {\vec {v}}\\{\stackrel {\nabla }{\vec {v}}}&=&{\dot {\vec {v}}}-\mathbf {l} \cdot {\vec {v}}&=&{\stackrel {\circ }{\vec {v}}}-\mathbf {d} \cdot {\vec {v}}\end{array}}}" loading="lazy"></dd></dl>

objektiv. Mehr dazu ist im Hauptartikel nachzuschlagen.

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

Ein Einheitsquadrat aus einer viskoelastischen Flüssigkeit wird mit konstanter Schergeschwindigkeit zu einem Parallelogramm verformt, siehe Abbildung rechts. Die Referenzkonfiguration ist das Einheitsquadrat

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<annotation encoding="application/x-tex">{\displaystyle {\begin{pmatrix}X\\Y\end{pmatrix}}\in [0,1]^{2}}</annotation>

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In der Momentankonfiguration haben die Punkte des Quadrates die räumlichen Koordinaten

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<annotation encoding="application/x-tex">{\displaystyle {\vec {\chi }}({\vec {X}},t)={\begin{pmatrix}x\\y\end{pmatrix}}={\begin{pmatrix}X+\gamma Y\\Y\end{pmatrix}}\quad \rightarrow \quad {\dot {\vec {\chi }}}({\vec {X}},t)={\begin{pmatrix}{\dot {x}}\\{\dot {y}}\end{pmatrix}}={\begin{pmatrix}{\dot {\gamma }}Y\\0\end{pmatrix}}={\begin{pmatrix}{\dot {\gamma }}y\\0\end{pmatrix}}={\vec {v}}({\vec {x}},t)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/93ced68189842ddbb4b390ebd42141e1b604e5a1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:84.106ex; height:6.176ex;" alt="{\displaystyle {\vec {\chi }}({\vec {X}},t)={\begin{pmatrix}x\\y\end{pmatrix}}={\begin{pmatrix}X+\gamma Y\\Y\end{pmatrix}}\quad \rightarrow \quad {\dot {\vec {\chi }}}({\vec {X}},t)={\begin{pmatrix}{\dot {x}}\\{\dot {y}}\end{pmatrix}}={\begin{pmatrix}{\dot {\gamma }}Y\\0\end{pmatrix}}={\begin{pmatrix}{\dot {\gamma }}y\\0\end{pmatrix}}={\vec {v}}({\vec {x}},t)}" loading="lazy"></dd></dl>

woraus sich der Deformations- und (räumliche) Geschwindigkeitsgradient berechnen:

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\mathbf {F} =&{\begin{pmatrix}1&\gamma \\0&1\end{pmatrix}}\;\rightarrow \quad {\dot {\mathbf {F} }}={\begin{pmatrix}0&{\dot {\gamma }}\\0&0\end{pmatrix}}=\operatorname {GRAD} {\dot {\vec {\chi }}}\,,\quad \mathbf {F} ^{-1}={\begin{pmatrix}1&-\gamma \\0&1\end{pmatrix}}\\\rightarrow \mathbf {l} =&\operatorname {grad} {\vec {v}}={\begin{pmatrix}0&{\dot {\gamma }}\\0&0\end{pmatrix}}={\dot {\mathbf {F} }}\cdot \mathbf {F} ^{-1}={\begin{pmatrix}0&{\dot {\gamma }}\\0&0\end{pmatrix}}\cdot {\begin{pmatrix}1&-\gamma \\0&1\end{pmatrix}}\\\rightarrow \mathbf {d} =&{\dfrac {\dot {\gamma }}{2}}{\begin{pmatrix}0&1\\1&0\end{pmatrix}}\;,\quad \mathbf {w} ={\dfrac {\dot {\gamma }}{2}}{\begin{pmatrix}0&1\\-1&0\end{pmatrix}}\,.\end{aligned}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6d8a177f4d74b5a3615a3c6999adb54793fd58d8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -8.838ex; width:70.442ex; height:18.843ex;" alt="{\displaystyle {\begin{aligned}\mathbf {F} =&{\begin{pmatrix}1&\gamma \\0&1\end{pmatrix}}\;\rightarrow \quad {\dot {\mathbf {F} }}={\begin{pmatrix}0&{\dot {\gamma }}\\0&0\end{pmatrix}}=\operatorname {GRAD} {\dot {\vec {\chi }}}\,,\quad \mathbf {F} ^{-1}={\begin{pmatrix}1&-\gamma \\0&1\end{pmatrix}}\\\rightarrow \mathbf {l} =&\operatorname {grad} {\vec {v}}={\begin{pmatrix}0&{\dot {\gamma }}\\0&0\end{pmatrix}}={\dot {\mathbf {F} }}\cdot \mathbf {F} ^{-1}={\begin{pmatrix}0&{\dot {\gamma }}\\0&0\end{pmatrix}}\cdot {\begin{pmatrix}1&-\gamma \\0&1\end{pmatrix}}\\\rightarrow \mathbf {d} =&{\dfrac {\dot {\gamma }}{2}}{\begin{pmatrix}0&1\\1&0\end{pmatrix}}\;,\quad \mathbf {w} ={\dfrac {\dot {\gamma }}{2}}{\begin{pmatrix}0&1\\-1&0\end{pmatrix}}\,.\end{aligned}}}" loading="lazy"></dd></dl>

Eine Verallgemeinerung des Materialgesetzes für eine <a href="Rheologisches_Modell#Viskoelastizität" title="Rheologisches Modell">viskoelastische Flüssigkeit</a> (Maxwell-Körper) <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda {\dot {\sigma }}+\sigma =\eta \,{\dot {\varepsilon }}}">

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<annotation encoding="application/x-tex">{\displaystyle \lambda {\dot {\sigma }}+\sigma =\eta \,{\dot {\varepsilon }}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b8007cfb31a5d796705d8ab8415270250709cf0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.866ex; height:2.676ex;" alt="{\displaystyle \lambda {\dot {\sigma }}+\sigma =\eta \,{\dot {\varepsilon }}}" loading="lazy"> mit Materialparametern <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda ,\eta }">

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<annotation encoding="application/x-tex">{\displaystyle \lambda {\stackrel {\circ }{\boldsymbol {\sigma }}}+{\boldsymbol {\sigma }}=\eta \,\mathbf {d} \,.}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5a834f1db9975d51dfb9f54f348d3f547abef7bd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.559ex; height:3.509ex;" alt="{\displaystyle \lambda {\stackrel {\circ }{\boldsymbol {\sigma }}}+{\boldsymbol {\sigma }}=\eta \,\mathbf {d} \,.}" loading="lazy"></dd></dl>

Der Cauchy’sche <a href="Spannungstensor" title="Spannungstensor">Spannungstensor</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {\sigma }}}">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e45fe1b9d8dcbc3103fc7805d69798bfe5ca5b16.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.594ex; height:1.676ex;" alt="{\displaystyle {\boldsymbol {\sigma }}}" loading="lazy"> ist hier <a href="Deviator" title="Deviator">deviatorisch</a> und besitzt daher die Form

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<annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {\sigma }}={\begin{pmatrix}\sigma &\tau \\\tau &-\sigma \end{pmatrix}}\,.}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/82b7e79c959d43366b1cf0697ce53cddf4f82434.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:16.69ex; height:6.176ex;" alt="{\displaystyle {\boldsymbol {\sigma }}={\begin{pmatrix}\sigma &\tau \\\tau &-\sigma \end{pmatrix}}\,.}" loading="lazy"></dd></dl>

So berechnet sich die Zaremba-Jaumann Ableitung zu:

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<annotation encoding="application/x-tex">{\displaystyle {\stackrel {\circ }{\boldsymbol {\sigma }}}={\begin{pmatrix}{\dot {\sigma }}&{\dot {\tau }}\\{\dot {\tau }}&-{\dot {\sigma }}\end{pmatrix}}+{\frac {\dot {\gamma }}{2}}{\begin{pmatrix}\sigma &\tau \\\tau &-\sigma \end{pmatrix}}\cdot {\begin{pmatrix}0&1\\-1&0\end{pmatrix}}-{\frac {\dot {\gamma }}{2}}{\begin{pmatrix}0&1\\-1&0\end{pmatrix}}\cdot {\begin{pmatrix}\sigma &\tau \\\tau &-\sigma \end{pmatrix}}={\begin{pmatrix}{\dot {\sigma }}-{\dot {\gamma }}\tau &{\dot {\tau }}+{\dot {\gamma }}\sigma \\{\dot {\tau }}+{\dot {\gamma }}\sigma &-{\dot {\sigma }}+{\dot {\gamma }}\tau \end{pmatrix}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/86ee90d6230bc02075ffbcda40dbec152d62513f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:96.948ex; height:6.176ex;" alt="{\displaystyle {\stackrel {\circ }{\boldsymbol {\sigma }}}={\begin{pmatrix}{\dot {\sigma }}&{\dot {\tau }}\\{\dot {\tau }}&-{\dot {\sigma }}\end{pmatrix}}+{\frac {\dot {\gamma }}{2}}{\begin{pmatrix}\sigma &\tau \\\tau &-\sigma \end{pmatrix}}\cdot {\begin{pmatrix}0&1\\-1&0\end{pmatrix}}-{\frac {\dot {\gamma }}{2}}{\begin{pmatrix}0&1\\-1&0\end{pmatrix}}\cdot {\begin{pmatrix}\sigma &\tau \\\tau &-\sigma \end{pmatrix}}={\begin{pmatrix}{\dot {\sigma }}-{\dot {\gamma }}\tau &{\dot {\tau }}+{\dot {\gamma }}\sigma \\{\dot {\tau }}+{\dot {\gamma }}\sigma &-{\dot {\sigma }}+{\dot {\gamma }}\tau \end{pmatrix}}}" loading="lazy"></dd></dl>

was über das Materialgesetz auf zwei Differentialgleichungen für die Spannungskomponenten führt:

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<annotation encoding="application/x-tex">{\displaystyle \lambda ({\dot {\sigma }}-{\dot {\gamma }}\tau )+\sigma =0\quad {\textsf {und}}\quad \lambda ({\dot {\tau }}+{\dot {\gamma }}\sigma )+\tau ={\frac {\eta }{2}}{\dot {\gamma }}\,.}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2edbb80435d7a96022da62d4776846e72b93fd10.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:47.888ex; height:4.843ex;" alt="{\displaystyle \lambda ({\dot {\sigma }}-{\dot {\gamma }}\tau )+\sigma =0\quad {\textsf {und}}\quad \lambda ({\dot {\tau }}+{\dot {\gamma }}\sigma )+\tau ={\frac {\eta }{2}}{\dot {\gamma }}\,.}" loading="lazy"></dd></dl>

Bei konstanter Schergeschwindigkeit kommt nach Eliminierung der Normalspannung die Differentialgleichung

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<annotation encoding="application/x-tex">{\displaystyle {\ddot {\tau }}+{\frac {2}{\lambda }}{\dot {\tau }}+(\lambda ^{-2}+{\dot {\gamma }}^{2})\tau ={\frac {\eta {\dot {\gamma }}}{2\lambda ^{2}}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bb5ebcd1c13273c20978b117e172753bc91cb621.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:29.97ex; height:5.676ex;" alt="{\displaystyle {\ddot {\tau }}+{\frac {2}{\lambda }}{\dot {\tau }}+(\lambda ^{-2}+{\dot {\gamma }}^{2})\tau ={\frac {\eta {\dot {\gamma }}}{2\lambda ^{2}}}}" loading="lazy"></dd></dl>

für die Schubspannung heraus, die als Lösung eine gedämpfte Schwingung besitzt. Dies ist ein bei Verwendung der Zaremba-Jaumann Rate bekanntes unphysikalisches Phänomen,<a href="#cite_note-9">[L 3]</a> siehe Abbildung rechts.

Verwendung der kontravarianten Oldroyd Ableitung liefert einen nicht-deviatorischen Spannungstensor:

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<annotation encoding="application/x-tex">{\displaystyle {\stackrel {\nabla }{\boldsymbol {\sigma }}}={\begin{pmatrix}{\dot {\sigma }}_{xx}&{\dot {\tau }}\\{\dot {\tau }}&{\dot {\sigma }}_{yy}\end{pmatrix}}-{\begin{pmatrix}0&{\dot {\gamma }}\\0&0\end{pmatrix}}\cdot {\begin{pmatrix}\sigma _{xx}&\tau \\\tau &\sigma _{yy}\end{pmatrix}}-{\begin{pmatrix}\sigma _{xx}&\tau \\\tau &\sigma _{yy}\end{pmatrix}}\cdot {\begin{pmatrix}0&0\\{\dot {\gamma }}&0\end{pmatrix}}={\begin{pmatrix}{\dot {\sigma }}_{xx}-2{\dot {\gamma }}\tau &{\dot {\tau }}-{\dot {\gamma }}\sigma _{yy}\\{\dot {\tau }}-{\dot {\gamma }}\sigma _{yy}&{\dot {\sigma }}_{yy}\end{pmatrix}}\,.}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/46db1d5d285dace58dbb6ab6b4a59ada14c7ec27.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:100.165ex; height:6.509ex;" alt="{\displaystyle {\stackrel {\nabla }{\boldsymbol {\sigma }}}={\begin{pmatrix}{\dot {\sigma }}_{xx}&{\dot {\tau }}\\{\dot {\tau }}&{\dot {\sigma }}_{yy}\end{pmatrix}}-{\begin{pmatrix}0&{\dot {\gamma }}\\0&0\end{pmatrix}}\cdot {\begin{pmatrix}\sigma _{xx}&\tau \\\tau &\sigma _{yy}\end{pmatrix}}-{\begin{pmatrix}\sigma _{xx}&\tau \\\tau &\sigma _{yy}\end{pmatrix}}\cdot {\begin{pmatrix}0&0\\{\dot {\gamma }}&0\end{pmatrix}}={\begin{pmatrix}{\dot {\sigma }}_{xx}-2{\dot {\gamma }}\tau &{\dot {\tau }}-{\dot {\gamma }}\sigma _{yy}\\{\dot {\tau }}-{\dot {\gamma }}\sigma _{yy}&{\dot {\sigma }}_{yy}\end{pmatrix}}\,.}" loading="lazy"></dd></dl>

Die Materialgleichung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda {\stackrel {\nabla }{\boldsymbol {\sigma }}}+{\boldsymbol {\sigma }}=\eta \,\mathbf {d} }">

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<annotation encoding="application/x-tex">{\displaystyle \lambda {\stackrel {\nabla }{\boldsymbol {\sigma }}}+{\boldsymbol {\sigma }}=\eta \,\mathbf {d} }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6a946eb40808a807f2b211035d191267c0ebc43b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.525ex; height:4.009ex;" alt="{\displaystyle \lambda {\stackrel {\nabla }{\boldsymbol {\sigma }}}+{\boldsymbol {\sigma }}=\eta \,\mathbf {d} }" loading="lazy"> ergibt:

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<annotation encoding="application/x-tex">{\displaystyle {\begin{pmatrix}\lambda ({\dot {\sigma }}_{xx}-2{\dot {\gamma }}\tau )+\sigma _{xx}&\lambda ({\dot {\tau }}-{\dot {\gamma }}\sigma _{yy})+\tau \\\lambda ({\dot {\tau }}-{\dot {\gamma }}\sigma _{yy})+\tau &\lambda {\dot {\sigma }}_{yy}+\sigma _{yy}\end{pmatrix}}={\frac {\eta {\dot {\gamma }}}{2}}{\begin{pmatrix}0&1\\1&0\end{pmatrix}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/25ab3fdbb7e57c3d983f7b37fe8d5c265e7ce6ab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:56.907ex; height:6.509ex;" alt="{\displaystyle {\begin{pmatrix}\lambda ({\dot {\sigma }}_{xx}-2{\dot {\gamma }}\tau )+\sigma _{xx}&\lambda ({\dot {\tau }}-{\dot {\gamma }}\sigma _{yy})+\tau \\\lambda ({\dot {\tau }}-{\dot {\gamma }}\sigma _{yy})+\tau &\lambda {\dot {\sigma }}_{yy}+\sigma _{yy}\end{pmatrix}}={\frac {\eta {\dot {\gamma }}}{2}}{\begin{pmatrix}0&1\\1&0\end{pmatrix}}}" loading="lazy"></dd></dl>

was sich bei anfänglich verschwindenden Spannungen und konstanter Scherrate geschlossen integrieren lässt:

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\sigma _{yy}=&0\\\tau =&{\frac {\eta {\dot {\gamma }}}{2}}(1-e^{-{\frac {t}{\lambda }}})\\\sigma _{xx}=&\eta {\dot {\gamma }}^{2}(\lambda -\lambda e^{-{\frac {t}{\lambda }}}-te^{-{\frac {t}{\lambda }}}).\end{aligned}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/650286b9326b5c618f8e53bada75f302a95fcaca.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.588ex; margin-bottom: -0.25ex; width:30.235ex; height:12.843ex;" alt="{\displaystyle {\begin{aligned}\sigma _{yy}=&0\\\tau =&{\frac {\eta {\dot {\gamma }}}{2}}(1-e^{-{\frac {t}{\lambda }}})\\\sigma _{xx}=&\eta {\dot {\gamma }}^{2}(\lambda -\lambda e^{-{\frac {t}{\lambda }}}-te^{-{\frac {t}{\lambda }}}).\end{aligned}}}" loading="lazy"></dd></dl>

Hier treten keine Schwingungen auf. Die Abbildung rechts zeigt die bei einer Scherrate <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\dot {\gamma }}=}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e350ada8d7a72f6891d3be0a21af182a6dd4b500.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.716ex; height:2.676ex;" alt="{\displaystyle {\dot {\gamma }}=}" loading="lazy">10/s mit der Zaremba-Jaumann und der kontravarianten Oldroyd Ableitung und den in der Tabelle angegebenen Materialparametern berechneten Spannungen.

<th>Parameter</th>

<th><a href="Relaxationszeit" class="mw-redirect" title="Relaxationszeit">Relaxationszeit</a></th>

<th>dynamische <a href="Viskosit%C3%A4t" title="Viskosität">Viskosität</a>

</th></tr>

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<th>Formelzeichen

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

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e4d701857cf5fbec133eebaf94deadf722537f64.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.169ex; height:2.176ex;" alt="{\displaystyle \eta }" loading="lazy">

</td></tr>

<tr>

<th>Einheit

</th>

<td>MPa s

</td></tr>

<tr>

<th>Zaremba-Jaumann Ableitung

</th>

<td>45,2

</td></tr>

<tr>

<th>Kontravariante Oldroyd Ableitung

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<td>0,2

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

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

<li id="cite_note-Frechet-1">↑ <a href="#cite_ref-Frechet_1-0">a</a> <a href="#cite_ref-Frechet_1-1">b</a> <a href="#cite_ref-Frechet_1-2">c</a> Die <a href="Fr%C3%A9chet-Ableitung" title="Fréchet-Ableitung">Fréchet-Ableitung</a> einer Funktion <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

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

</semantics>

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

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

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

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

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

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

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

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f92353beffbd1e9633a251d46d7e6ff2d15c69e0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:62.404ex; height:6.176ex;" alt="{\displaystyle {\mathcal {A}}(h)=\left.{\frac {\mathrm {d} }{\mathrm {d} s}}f(x+sh)\right|_{s=0}=\lim _{s\rightarrow 0}{\frac {f(x+sh)-f(x)}{s}}\quad {\text{für alle}}\quad h}" loading="lazy"></dd></dl>

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

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

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

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

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

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

geschrieben.

</li>

<li id="cite_note-spin-3">↑ <a href="#cite_ref-spin_3-0">a</a> <a href="#cite_ref-spin_3-1">b</a> Denn mit dem Wirbelvektor ergibt sich <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {e}}\cdot \mathbf {w} \cdot {\hat {e}}={\hat {e}}\cdot ({\vec {\omega }}\times {\hat {e}})=0}">

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<annotation encoding="application/x-tex">{\displaystyle {\hat {e}}\cdot \mathbf {w} \cdot {\hat {e}}={\hat {e}}\cdot ({\vec {\omega }}\times {\hat {e}})=0}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9f8216ddb1a53fb521330011d0ebb66a1e6f1b5d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.59ex; height:2.843ex;" alt="{\displaystyle {\hat {e}}\cdot \mathbf {w} \cdot {\hat {e}}={\hat {e}}\cdot ({\vec {\omega }}\times {\hat {e}})=0}" loading="lazy">

</li>

<li id="cite_note-4"><a href="#cite_ref-4">↑</a> Denn aus

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {l} \cdot {\hat {e}}=\lambda {\hat {e}}\quad {\text{und}}\quad {\hat {e}}\cdot {\hat {e}}=1}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ccb4be92b8b0a8eb7a9796f8ede44e86803488a9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:26.504ex; height:2.176ex;" alt="{\displaystyle \mathbf {l} \cdot {\hat {e}}=\lambda {\hat {e}}\quad {\text{und}}\quad {\hat {e}}\cdot {\hat {e}}=1}" loading="lazy">

folgt:

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\mathbf {l} \cdot {\hat {e}}-({\hat {e}}\cdot \mathbf {l} \cdot {\hat {e}}){\hat {e}}=\\=\lambda {\hat {e}}-({\hat {e}}\cdot \lambda {\hat {e}}){\hat {e}}=0\end{aligned}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/539e800538eaf99aaf443cddfbab8298dae54d1e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:21.671ex; height:6.176ex;" alt="{\displaystyle {\begin{aligned}\mathbf {l} \cdot {\hat {e}}-({\hat {e}}\cdot \mathbf {l} \cdot {\hat {e}}){\hat {e}}=\\=\lambda {\hat {e}}-({\hat {e}}\cdot \lambda {\hat {e}}){\hat {e}}=0\end{aligned}}}" loading="lazy">

</li>

<li id="cite_note-5"><a href="#cite_ref-5">↑</a> Dieses <a href="Paradoxon" title="Paradoxon">Paradoxon</a> tritt nur bei nicht materiellen Objekten wie der Drehachse hier oder dem <a href="Momentanpol" title="Momentanpol">Momentanpol</a> auf.

</li>

<li id="cite_note-6"><a href="#cite_ref-6">↑</a> Die Formelzeichen für die objektiven Raten variieren von Quelle zu Quelle. Die hier angegebenen folgen P. Haupt, S. 48ff. In H. Altenbach wird

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cca605c8eaa91f6fc3e272d6477459236557f248.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.46ex; height:2.676ex;" alt="{\displaystyle \mathbf {T} ^{\nabla }}" loading="lazy"> für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\stackrel {\circ }{\mathbf {T} }}}">

<mover>

</mrow>

</mrow>

</mover>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d9257c42c7f7bae7b51bcafb1e39056390df12df.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.859ex; height:3.509ex;" alt="{\displaystyle {\stackrel {\circ }{\mathbf {T} }}}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {T} ^{O}}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6156664e6daa8797c98b873ab1b13d7647993b32.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.345ex; height:2.676ex;" alt="{\displaystyle \mathbf {T} ^{O}}" loading="lazy"> für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\stackrel {\triangle }{\mathbf {T} }}}">

<mover>

</mrow>

</mrow>

</mover>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d372b89efc07720c96589564e58195b6e6e2f241.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.859ex; height:4.009ex;" alt="{\displaystyle {\stackrel {\triangle }{\mathbf {T} }}}" loading="lazy"> benutzt.

</li>

<li id="cite_note-8"><a href="#cite_ref-8">↑</a> Diese Ableitung kommt in der <a href="Cauchy-Elastizit%C3%A4t" title="Cauchy-Elastizität">Cauchy-Elastizität</a> vor und wird auch nach C. Truesdell benannt. Er selbst benannte die Ableitung aber nach Cauchy und schrieb 1963, dass diese Rate ohne erfindlichen Grund nach ihm benannt wurde („came to be named, for no good reason, after […] me“) siehe C. Truesdell: Remarks on Hypo-Elasticity, Journal of Research of the National Bureau of Standards - B. Mathematics and Mathematical Physics, Vol. 67B, No. 3, July-September 1963, S. 141.

</li>

</ol>

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

<ul><li><a href="Euler-Gleichungen_(Str%C3%B6mungsmechanik)" title="Euler-Gleichungen (Strömungsmechanik)">Euler-Gleichungen (Strömungsmechanik)</a></li>

<li><a href="Navier-Stokes-Gleichungen" title="Navier-Stokes-Gleichungen">Navier-Stokes-Gleichungen</a></li>

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

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

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

<ul><li>H. Altenbach: <cite style="font-style:italic">Kontinuumsmechanik</cite>. Springer, 2012, ISBN 978-3-642-24118-5.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Geschwindigkeitsgradient&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, 2000, ISBN 3-540-66114-X.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Geschwindigkeitsgradient&rft.au=P.+Haupt&rft.btitle=Continuum+Mechanics+and+Theory+of+Materials&rft.date=2000&rft.genre=book&rft.isbn=354066114X&rft.pub=Springer" style="display:none"> </li></ul>

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

<li id="cite_note-2"><a href="#cite_ref-2">↑</a> Altenbach (2012), S. 109 und 32.

</li>

<li id="cite_note-7"><a href="#cite_ref-7">↑</a> nach James G. Oldroyd (1921 - 1982), <a href="https://en.wikipedia.org/wiki/James_G._Oldroyd" class="extiw external" title="en:James G. Oldroyd">James G. Oldroyd in engl. Wikipedia</a> (engl.)

</li>

<li id="cite_note-9"><a href="#cite_ref-9">↑</a> P. Haupt (2000), S. 302ff

</li>

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