Cauchy-eulersche Bewegungsgesetze

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id="mw-content-text" class="mw-body-content"> <script>function mfTempOpenSection(id){var block=document.getElementById("mf-section-"+id);block.className+=" open-block";block.previousSibling.className+=" open-block";}</script> <div class="mw-content-ltr mw-parser-output" lang="de" dir="ltr"> <section class="mf-section-0" id="mf-section-0"> Die cauchy-eulerschen Bewegungsgesetze von <a href="https://de-m-wikipedia-org.translate.goog/wiki/Augustin-Louis_Cauchy?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Augustin-Louis Cauchy">Augustin-Louis Cauchy</a> und <a href="https://de-m-wikipedia-org.translate.goog/wiki/Leonhard_Euler?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Leonhard Euler">Leonhard Euler</a> sind die lokalen Formen des <a href="https://de-m-wikipedia-org.translate.goog/wiki/Kinetik_(Mechanik)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Schwerpunktsatz_oder_Impulssatz" title="Kinetik (Mechanik)">Impuls-</a> und <a href="https://de-m-wikipedia-org.translate.goog/wiki/Drallsatz?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Drallsatz">Drallsatzes</a> in der <a href="https://de-m-wikipedia-org.translate.goog/wiki/Kontinuumsmechanik?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Kontinuumsmechanik">Kontinuumsmechanik</a>. Es sind <a href="https://de-m-wikipedia-org.translate.goog/wiki/Bewegungsgleichung?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Bewegungsgleichung">Bewegungsgleichungen</a>, die, wenn sie lokal, d. h. in jedem Punkt eines <a href="https://de-m-wikipedia-org.translate.goog/wiki/K%C3%B6rper_(Physik)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Körper (Physik)">Körpers</a> erfüllt sind, sicherstellen, dass die Bewegung des Körpers als Ganzes – inklusive Verformungen – dem Impuls- bzw. Drallsatz gehorcht. Das erste cauchy-eulersche Bewegungsgesetz korrespondiert mit dem Impulssatz und lautet im <a href="https://de-m-wikipedia-org.translate.goog/wiki/Geometrische_Linearisierung?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Geometrische Linearisierung">geometrisch linearen</a> Fall an einem materiellen Punkt des Körpers: <dl> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho {\vec {a}}=\rho {\vec {k}}+\operatorname {div} \;{\boldsymbol {\sigma }}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> ρ </mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> a </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo> = </mo> <mi> ρ </mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> k </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo> + </mo> <mi> div </mi> <mspace width="thickmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> σ </mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \rho {\vec {a}}=\rho {\vec {k}}+\operatorname {div} \;{\boldsymbol {\sigma }}} </annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/82d7cd02561a0a341564c3407905db8653741588" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.577ex; height:3.343ex;" alt="{\displaystyle \rho {\vec {a}}=\rho {\vec {k}}+\operatorname {div} \;{\boldsymbol {\sigma }}}"> </dd> </dl> Hier ist ρ die <a href="https://de-m-wikipedia-org.translate.goog/wiki/Dichte?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Dichte">Dichte</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {a}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> a </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\vec {a}}} </annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/546e6615827e17295718741fd0b86f639a947f16" 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}}}"> die <a href="https://de-m-wikipedia-org.translate.goog/wiki/Beschleunigung?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Beschleunigung">Beschleunigung</a> des materiellen Punktes, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {k}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> k </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\vec {k}}} </annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ccd4b98d198d6538010ae815ee1199baabd3493" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.843ex;" alt="{\displaystyle {\vec {k}}}"> die <a href="https://de-m-wikipedia-org.translate.goog/wiki/Schwerebeschleunigung?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Schwerebeschleunigung">Schwerebeschleunigung</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {\sigma }}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> σ </mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\boldsymbol {\sigma }}} </annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e45fe1b9d8dcbc3103fc7805d69798bfe5ca5b16" 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 }}}"> der cauchysche <a href="https://de-m-wikipedia-org.translate.goog/wiki/Spannungstensor?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Spannungstensor">Spannungstensor</a> und div der <a href="https://de-m-wikipedia-org.translate.goog/wiki/Divergenz_eines_Vektorfeldes?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Divergenz eines Vektorfeldes">Divergenzoperator</a>. Die spezifische Impulsänderung bestimmt sich demnach aus der spezifischen Schwerkraft und dem Antrieb durch einen Spannungsanstieg. Alle Variablen in der Gleichung sind im Allgemeinen sowohl vom Ort als auch von der Zeit abhängig. Das zweite cauchy-eulersche Bewegungsgesetz entspricht dem lokal formulierten Drallsatz, der sich auf die Forderung nach der <a href="https://de-m-wikipedia-org.translate.goog/wiki/Symmetrische_Matrix?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Symmetrische Matrix">Symmetrie</a> des cauchyschen Spannungstensors reduziert: <dl> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {\sigma }}={\boldsymbol {\sigma }}^{\top }}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> σ </mi> </mrow> <mo> = </mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> σ </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> ⊤ </mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\boldsymbol {\sigma }}={\boldsymbol {\sigma }}^{\top }} </annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8f8a70fa32d7549270a3a5501e977e6874d017b9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.798ex; height:2.676ex;" alt="{\displaystyle {\boldsymbol {\sigma }}={\boldsymbol {\sigma }}^{\top }}"> </dd> </dl> Das Superskript „⊤“ markiert die <a href="https://de-m-wikipedia-org.translate.goog/wiki/Transponierte_Matrix?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Transponierte Matrix">Transposition</a>. Die Symmetrie entspricht dem Satz von der Gleichheit der zugeordneten <a href="https://de-m-wikipedia-org.translate.goog/wiki/Schubspannung?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Schubspannung">Schubspannungen</a>.<a href="https://de-m-wikipedia-org.translate.goog/wiki/Cauchy-eulersche_Bewegungsgesetze?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-szabo-1">[L 1]</a> <dl> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{ij}=\sigma _{ji},\quad i,j=1,2,3}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> σ </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> <mi> j </mi> </mrow> </msub> <mo> = </mo> <msub> <mi> σ </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> j </mi> <mi> i </mi> </mrow> </msub> <mo> , </mo> <mspace width="1em"></mspace> <mi> i </mi> <mo> , </mo> <mi> j </mi> <mo> = </mo> <mn> 1 </mn> <mo> , </mo> <mn> 2 </mn> <mo> , </mo> <mn> 3 </mn> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \sigma _{ij}=\sigma _{ji},\quad i,j=1,2,3} </annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cba3e1f1596767c86e4a07debad4d5ef70064725" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:23.512ex; height:2.843ex;" alt="{\displaystyle \sigma _{ij}=\sigma _{ji},\quad i,j=1,2,3}">. </dd> </dl> Bei großen Verschiebungen können beide Bewegungsgesetze in <a href="https://de-m-wikipedia-org.translate.goog/wiki/Lagrangesche_Betrachtungsweise?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Lagrangesche Betrachtungsweise">lagrangescher Betrachtungsweise</a> materiell oder <a href="https://de-m-wikipedia-org.translate.goog/wiki/Eulersche_Betrachtungsweise?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Eulersche Betrachtungsweise">eulerscher Betrachtungsweise</a> räumlich formuliert werden. Die Struktur der Gleichungen bleibt dabei erhalten, aber es kommt zu Modifikationen in den Abhängigkeiten oder im Spannungstensor. Die cauchy-eulerschen Bewegungsgesetze sind die Basis für die <a href="https://de-m-wikipedia-org.translate.goog/wiki/Eulersche_Gleichungen_(Str%C3%B6mungsmechanik)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Eulersche Gleichungen (Strömungsmechanik)">eulerschen Gleichungen der Strömungsmechanik</a>, der <a href="https://de-m-wikipedia-org.translate.goog/wiki/Navier-Stokes-Gleichungen?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Navier-Stokes-Gleichungen">Navier-Stokes-</a> und der <a href="https://de-m-wikipedia-org.translate.goog/wiki/Navier-Cauchy-Gleichungen?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Navier-Cauchy-Gleichungen">Navier-Cauchy-Gleichungen</a>. Eine der Grundgleichungen der <a href="https://de-m-wikipedia-org.translate.goog/wiki/Verschiebungsmethode?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Verschiebungsmethode">Verschiebungsmethode</a> in der <a href="https://de-m-wikipedia-org.translate.goog/wiki/Finite-Elemente-Methode?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Finite-Elemente-Methode">Finite-Elemente-Methode</a> ist das <a href="https://de-m-wikipedia-org.translate.goog/wiki/Cauchy-eulersche_Bewegungsgesetze?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Prinzip_von_d%E2%80%99Alembert">Prinzip von d’Alembert</a> in der lagrangeschen Fassung, das eine aus den cauchy-eulerschen Gesetzen folgende Aussage ist. Für Begriffsklärung empfiehlt sich die Lektüre des Artikels zur <a href="https://de-m-wikipedia-org.translate.goog/wiki/Kontinuumsmechanik?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Kontinuumsmechanik">Kontinuumsmechanik</a>. Die verwendeten Operatoren und Rechenregeln sind in den Formelsammlungen zur <a href="https://de-m-wikipedia-org.translate.goog/wiki/Formelsammlung_Tensoralgebra?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Formelsammlung Tensoralgebra">Tensoralgebra</a> und <a href="https://de-m-wikipedia-org.translate.goog/wiki/Formelsammlung_Tensoranalysis?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Formelsammlung Tensoranalysis">Tensoranalysis</a> aufgeführt. <div id="toc" class="toc" role="navigation" aria-labelledby="mw-toc-heading"> <input type="checkbox" role="button" id="toctogglecheckbox" class="toctogglecheckbox" style="display:none"> <div class="toctitle" lang="de" dir="ltr"> <h2 id="mw-toc-heading">Inhaltsverzeichnis</h2><label class="toctogglelabel" for="toctogglecheckbox"></label> </div> <ul> <li class="toclevel-1 tocsection-1"><a href="https://de-m-wikipedia-org.translate.goog/wiki/Cauchy-eulersche_Bewegungsgesetze?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Erstes_cauchy-eulersches_Bewegungsgesetz">1 Erstes cauchy-eulersches Bewegungsgesetz</a> <ul> <li class="toclevel-2 tocsection-2"><a href="https://de-m-wikipedia-org.translate.goog/wiki/Cauchy-eulersche_Bewegungsgesetze?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Impulssatz_am_Volumenelement">1.1 Impulssatz am Volumenelement</a></li> <li class="toclevel-2 tocsection-3"><a href="https://de-m-wikipedia-org.translate.goog/wiki/Cauchy-eulersche_Bewegungsgesetze?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Impulssatz_in_lagrangescher_Darstellung">1.2 Impulssatz in lagrangescher Darstellung</a></li> <li class="toclevel-2 tocsection-4"><a href="https://de-m-wikipedia-org.translate.goog/wiki/Cauchy-eulersche_Bewegungsgesetze?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Impulssatz_in_eulerscher_Darstellung">1.3 Impulssatz in eulerscher Darstellung</a></li> <li class="toclevel-2 tocsection-5"><a href="https://de-m-wikipedia-org.translate.goog/wiki/Cauchy-eulersche_Bewegungsgesetze?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Einfluss_von_Sprungstellen_im_Impulssatz">1.4 Einfluss von Sprungstellen im Impulssatz</a></li> </ul></li> <li class="toclevel-1 tocsection-6"><a href="https://de-m-wikipedia-org.translate.goog/wiki/Cauchy-eulersche_Bewegungsgesetze?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Zweites_cauchy-eulersches_Bewegungsgesetz">2 Zweites cauchy-eulersches Bewegungsgesetz</a> <ul> <li class="toclevel-2 tocsection-7"><a href="https://de-m-wikipedia-org.translate.goog/wiki/Cauchy-eulersche_Bewegungsgesetze?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Drallsatz_am_Volumenelement">2.1 Drallsatz am Volumenelement</a></li> <li class="toclevel-2 tocsection-8"><a href="https://de-m-wikipedia-org.translate.goog/wiki/Cauchy-eulersche_Bewegungsgesetze?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Drehimpulssatz_in_lagrangescher_Darstellung">2.2 Drehimpulssatz in lagrangescher Darstellung</a></li> <li class="toclevel-2 tocsection-9"><a href="https://de-m-wikipedia-org.translate.goog/wiki/Cauchy-eulersche_Bewegungsgesetze?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Drehimpulssatz_in_eulerscher_Darstellung">2.3 Drehimpulssatz in eulerscher Darstellung</a></li> <li class="toclevel-2 tocsection-10"><a href="https://de-m-wikipedia-org.translate.goog/wiki/Cauchy-eulersche_Bewegungsgesetze?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Einfluss_von_Sprungstellen_im_Drehimpulssatz">2.4 Einfluss von Sprungstellen im Drehimpulssatz</a></li> </ul></li> <li class="toclevel-1 tocsection-11"><a href="https://de-m-wikipedia-org.translate.goog/wiki/Cauchy-eulersche_Bewegungsgesetze?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Folgerungen_aus_den_Bewegungsgesetzen">3 Folgerungen aus den Bewegungsgesetzen</a> <ul> <li class="toclevel-2 tocsection-12"><a href="https://de-m-wikipedia-org.translate.goog/wiki/Cauchy-eulersche_Bewegungsgesetze?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Prinzip_von_d%E2%80%99Alembert">3.1 Prinzip von d’Alembert</a></li> <li class="toclevel-2 tocsection-13"><a href="https://de-m-wikipedia-org.translate.goog/wiki/Cauchy-eulersche_Bewegungsgesetze?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Bilanz_der_mechanischen_Energie">3.2 Bilanz der mechanischen Energie</a></li> <li class="toclevel-2 tocsection-14"><a href="https://de-m-wikipedia-org.translate.goog/wiki/Cauchy-eulersche_Bewegungsgesetze?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Energieerhaltungssatz">3.3 Energieerhaltungssatz</a></li> <li class="toclevel-2 tocsection-15"><a href="https://de-m-wikipedia-org.translate.goog/wiki/Cauchy-eulersche_Bewegungsgesetze?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Satz_von_Clapeyron">3.4 Satz von Clapeyron</a></li> </ul></li> <li class="toclevel-1 tocsection-16"><a href="https://de-m-wikipedia-org.translate.goog/wiki/Cauchy-eulersche_Bewegungsgesetze?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Fu%C3%9Fnoten">4 Fußnoten</a></li> <li class="toclevel-1 tocsection-17"><a href="https://de-m-wikipedia-org.translate.goog/wiki/Cauchy-eulersche_Bewegungsgesetze?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Literatur">5 Literatur</a></li> <li class="toclevel-1 tocsection-18"><a href="https://de-m-wikipedia-org.translate.goog/wiki/Cauchy-eulersche_Bewegungsgesetze?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Einzelnachweise">6 Einzelnachweise</a></li> </ul> </div> </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(1)"> <h2 id="Erstes_cauchy-eulersches_Bewegungsgesetz">Erstes cauchy-eulersches Bewegungsgesetz</h2> <a role="button" href="https://de-m-wikipedia-org.translate.goog/w/index.php?title=Cauchy-eulersche_Bewegungsgesetze&action=edit&section=1&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Abschnitt bearbeiten: Erstes cauchy-eulersches Bewegungsgesetz" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> Bearbeiten </a> </div> <section class="mf-section-1 collapsible-block" id="mf-section-1"> Das erste cauchy-eulersche Bewegungsgesetz folgt aus dem 1687 von <a href="https://de-m-wikipedia-org.translate.goog/wiki/Isaac_Newton?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Isaac Newton">Isaac Newton</a> formulierten und nach ihm benannten zweiten <a href="https://de-m-wikipedia-org.translate.goog/wiki/Newtonsche_Gesetze?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Newtonsche Gesetze">newtonschen Gesetz</a>, das dem <a href="https://de-m-wikipedia-org.translate.goog/wiki/Kinetik_(Mechanik)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Schwerpunktsatz_oder_Impulssatz" title="Kinetik (Mechanik)">Impulssatz</a> entspricht, demgemäß die Änderung des Impulses mit der Zeit gleich der auf einen Körper wirkenden äußeren Kräfte ist: <dl> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\mathrm {d} {\vec {p}}}{\mathrm {d} t}}={\vec {F}}_{v}+{\vec {F}}_{a}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> p </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> t </mi> </mrow> </mfrac> </mrow> <mo> = </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> F </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> v </mi> </mrow> </msub> <mo> + </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> F </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> a </mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\frac {\mathrm {d} {\vec {p}}}{\mathrm {d} t}}={\vec {F}}_{v}+{\vec {F}}_{a}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5dba1f2a7e337b9462ed3d95456a8fafceb61931" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:15.066ex; height:5.676ex;" alt="{\displaystyle {\frac {\mathrm {d} {\vec {p}}}{\mathrm {d} t}}={\vec {F}}_{v}+{\vec {F}}_{a}}"> </noscript><span class="lazy-image-placeholder" style="width: 15.066ex;height: 5.676ex;vertical-align: -2.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5dba1f2a7e337b9462ed3d95456a8fafceb61931" data-alt="{\displaystyle {\frac {\mathrm {d} {\vec {p}}}{\mathrm {d} t}}={\vec {F}}_{v}+{\vec {F}}_{a}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </dd> </dl> Der Vektor <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {p}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> p </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\vec {p}}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/84fee53c81592db54e0fe6c6f9eba002bb1dc74b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.415ex; height:2.676ex;" alt="{\displaystyle {\vec {p}}}"> </noscript><span class="lazy-image-placeholder" style="width: 1.415ex;height: 2.676ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/84fee53c81592db54e0fe6c6f9eba002bb1dc74b" data-alt="{\displaystyle {\vec {p}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  stellt den <a href="https://de-m-wikipedia-org.translate.goog/wiki/Impuls_(Mechanik)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Impuls (Mechanik)">Impuls</a> dar, dessen zeitliche Änderung sich aus volumenverteilten und oberflächig eingeleiteten Kräften <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {F}}_{v}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> F </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> v </mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\vec {F}}_{v}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/952406f765878c480c997a1948d652315b089750" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.801ex; height:3.176ex;" alt="{\displaystyle {\vec {F}}_{v}}"> </noscript><span class="lazy-image-placeholder" style="width: 2.801ex;height: 3.176ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/952406f765878c480c997a1948d652315b089750" data-alt="{\displaystyle {\vec {F}}_{v}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  bzw. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {F}}_{a}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> F </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> a </mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\vec {F}}_{a}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0f65ad46c96c6023160ca54f697e7e70f1d3c01f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.873ex; height:3.176ex;" alt="{\displaystyle {\vec {F}}_{a}}"> </noscript><span class="lazy-image-placeholder" style="width: 2.873ex;height: 3.176ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0f65ad46c96c6023160ca54f697e7e70f1d3c01f" data-alt="{\displaystyle {\vec {F}}_{a}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  ergibt. In dem die Kontinuumsmechanik den Körper als Punktmenge idealisiert, wird aus der obigen Gleichung eine <a href="https://de-m-wikipedia-org.translate.goog/wiki/Integralgleichung?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Integralgleichung">Integralgleichung</a>, in der der spezifische Impuls, die spezifische Schwerebeschleunigung und die oberflächig wirkenden Kräfte über das Volumen bzw. über die Oberfläche integriert werden. Bei kleinen Verformungen kann das erste cauchy-eulersche Bewegungsgesetz am Volumenelement hergeleitet werden. <div class="mw-heading mw-heading3"> <h3 id="Impulssatz_am_Volumenelement">Impulssatz am Volumenelement</h3> <a role="button" href="https://de-m-wikipedia-org.translate.goog/w/index.php?title=Cauchy-eulersche_Bewegungsgesetze&action=edit&section=2&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Abschnitt bearbeiten: Impulssatz am Volumenelement" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> Bearbeiten </a> </div> <figure class="mw-default-size" typeof="mw:File/Thumb"> <a href="https://de-m-wikipedia-org.translate.goog/wiki/Datei:Impscheibe.png?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-file-description"> <noscript> <img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/2b/Impscheibe.png/220px-Impscheibe.png" decoding="async" width="220" height="261" class="mw-file-element" data-file-width="596" data-file-height="707"> </noscript><span class="lazy-image-placeholder" style="width: 220px;height: 261px;" data-src="//upload.wikimedia.org/wikipedia/commons/thumb/2/2b/Impscheibe.png/220px-Impscheibe.png" data-width="220" data-height="261" data-srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/2b/Impscheibe.png/330px-Impscheibe.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/2b/Impscheibe.png/440px-Impscheibe.png 2x" data-class="mw-file-element"> </a> <figcaption> Spannungen an einem freigeschnittenen Scheibenelement </figcaption> </figure> Der zweidimensionale Fall im ebenen Spannungszustand lässt sich leichter veranschaulichen und soll daher vorangestellt werden. Dazu wird eine ebene Scheibe der Dicke h betrachtet, die durch in der Ebene wirkende Kräfte belastet wird, siehe oberen Bildteil. Aus dieser Scheibe wird gedanklich ein rechteckiges Stück (gelb) herausgeschnitten, parallel zu dessen Kanten ein <a href="https://de-m-wikipedia-org.translate.goog/wiki/Kartesisches_Koordinatensystem?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Kartesisches Koordinatensystem">kartesisches Koordinatensystem</a> definiert wird, in dem es die Breite dx und Höhe dy hat. Nach dem <a href="https://de-m-wikipedia-org.translate.goog/wiki/Schnittprinzip?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Schnittprinzip">Schnittprinzip</a> entstehen an den Schnittflächen Schnittspannungen, die an die Stelle des weggeschnittenen Teils treten, siehe <a href="https://de-m-wikipedia-org.translate.goog/wiki/Schnittreaktion?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Schnittreaktion">Schnittreaktion</a>. Bei einem (infinitesimal) kleinen Scheibenelement können die Schnittspannungen als über die Fläche konstant angenommen werden. Die Schnittspannungen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {t}}_{x}=\sigma _{xx}{\hat {e}}_{x}+\sigma _{xy}{\hat {e}}_{y}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> t </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> x </mi> </mrow> </msub> <mo> = </mo> <msub> <mi> σ </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> x </mi> <mi> x </mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> e </mi> <mo stretchy="false"> ^ </mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> x </mi> </mrow> </msub> <mo> + </mo> <msub> <mi> σ </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> x </mi> <mi> y </mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> e </mi> <mo stretchy="false"> ^ </mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> y </mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\vec {t}}_{x}=\sigma _{xx}{\hat {e}}_{x}+\sigma _{xy}{\hat {e}}_{y}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/507fbe1b490c132d369772fdd20dfea1b5069ad4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:19.961ex; height:3.509ex;" alt="{\displaystyle {\vec {t}}_{x}=\sigma _{xx}{\hat {e}}_{x}+\sigma _{xy}{\hat {e}}_{y}}"> </noscript><span class="lazy-image-placeholder" style="width: 19.961ex;height: 3.509ex;vertical-align: -1.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/507fbe1b490c132d369772fdd20dfea1b5069ad4" data-alt="{\displaystyle {\vec {t}}_{x}=\sigma _{xx}{\hat {e}}_{x}+\sigma _{xy}{\hat {e}}_{y}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  treten auf der Oberfläche mit der Normalen in x-Richtung auf und entsprechend operiert <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {t}}_{y}=\sigma _{yx}{\hat {e}}_{x}+\sigma _{yy}{\hat {e}}_{y}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> t </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> y </mi> </mrow> </msub> <mo> = </mo> <msub> <mi> σ </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> y </mi> <mi> x </mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> e </mi> <mo stretchy="false"> ^ </mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> x </mi> </mrow> </msub> <mo> + </mo> <msub> <mi> σ </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> y </mi> <mi> y </mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> e </mi> <mo stretchy="false"> ^ </mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> y </mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\vec {t}}_{y}=\sigma _{yx}{\hat {e}}_{x}+\sigma _{yy}{\hat {e}}_{y}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/940498dabfb6b713f5e2fb6805e98568a448812c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:19.591ex; height:3.509ex;" alt="{\displaystyle {\vec {t}}_{y}=\sigma _{yx}{\hat {e}}_{x}+\sigma _{yy}{\hat {e}}_{y}}"> </noscript><span class="lazy-image-placeholder" style="width: 19.591ex;height: 3.509ex;vertical-align: -1.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/940498dabfb6b713f5e2fb6805e98568a448812c" data-alt="{\displaystyle {\vec {t}}_{y}=\sigma _{yx}{\hat {e}}_{x}+\sigma _{yy}{\hat {e}}_{y}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  auf der Oberfläche mit der Normalen in y-Richtung. In der Komponente <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{ij}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> σ </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> <mi> j </mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \sigma _{ij}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/43acbf52cc4d4f83f187ceaa49f045114b71772e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.804ex; height:2.343ex;" alt="{\displaystyle \sigma _{ij}}"> </noscript><span class="lazy-image-placeholder" style="width: 2.804ex;height: 2.343ex;vertical-align: -1.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/43acbf52cc4d4f83f187ceaa49f045114b71772e" data-alt="{\displaystyle \sigma _{ij}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  bezieht sich der erste Index also auf die Flächennormale und der zweite Index auf die Wirkrichtung. Nach Voraussetzung gibt es keine Spannungen senkrecht zur Scheibenebene. An den Flächen, deren Normalen in positive Koordinatenrichtung weisen, ist das positive Schnittufer und die Spannungen wirken in positiver Richtung. An den Flächen, deren Normalen in negative Koordinatenrichtung weisen, ist das negative Schnittufer und die Spannungen wirken in negativer Richtung, siehe Bild. Sie sind im Gleichgewicht mit den Schnittspannungen an den benachbarten, weggeschnittenen Teilen des Körpers. Das zweite <a href="https://de-m-wikipedia-org.translate.goog/wiki/Newtonsche_Gesetze?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Newtonsche Gesetze">newtonsche Gesetz</a> besagt, dass die an dem Scheibenelement angreifenden Spannungen – multipliziert mit ihrer Wirkfläche – das Scheibenelement beschleunigen. An dem Scheibenelement führt das unter Berücksichtigung der Schwerebeschleunigung in x- und y-Richtung auf <dl> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{array}{ll}\mathrm {d} m\,a_{x}=\mathrm {d} m\,k_{x}\!\!\!\!\!\!&+\,\sigma _{xx}(x+\mathrm {d} x,y)h\,\mathrm {d} y+\sigma _{yx}(x,y+\mathrm {d} y)h\,\mathrm {d} x\\&-\,\sigma _{xx}(x,y)h\,\mathrm {d} y-\sigma _{yx}(x,y)h\,\mathrm {d} x\\\mathrm {d} m\,a_{y}=\mathrm {d} m\,k_{y}\!\!\!\!\!\!&+\,\sigma _{xy}(x+\mathrm {d} x,y)h\,\mathrm {d} y+\sigma _{yy}(x,y+\mathrm {d} y)h\,\mathrm {d} x\\&-\,\sigma _{xy}(x,y)h\,\mathrm {d} y-\sigma _{yy}(x,y)h\,\mathrm {d} x\end{array}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="left left" rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> m </mi> <mspace width="thinmathspace"></mspace> <msub> <mi> a </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> x </mi> </mrow> </msub> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> m </mi> <mspace width="thinmathspace"></mspace> <msub> <mi> k </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> x </mi> </mrow> </msub> <mspace width="negativethinmathspace"></mspace> <mspace width="negativethinmathspace"></mspace> <mspace width="negativethinmathspace"></mspace> <mspace width="negativethinmathspace"></mspace> <mspace width="negativethinmathspace"></mspace> <mspace width="negativethinmathspace"></mspace> </mtd> <mtd> <mo> + </mo> <mspace width="thinmathspace"></mspace> <msub> <mi> σ </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> x </mi> <mi> x </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <mi> x </mi> <mo> + </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> x </mi> <mo> , </mo> <mi> y </mi> <mo stretchy="false"> ) </mo> <mi> h </mi> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> y </mi> <mo> + </mo> <msub> <mi> σ </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> y </mi> <mi> x </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <mi> x </mi> <mo> , </mo> <mi> y </mi> <mo> + </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> y </mi> <mo stretchy="false"> ) </mo> <mi> h </mi> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> x </mi> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mo> − </mo> <mspace width="thinmathspace"></mspace> <msub> <mi> σ </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> x </mi> <mi> x </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <mi> x </mi> <mo> , </mo> <mi> y </mi> <mo stretchy="false"> ) </mo> <mi> h </mi> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> y </mi> <mo> − </mo> <msub> <mi> σ </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> y </mi> <mi> x </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <mi> x </mi> <mo> , </mo> <mi> y </mi> <mo stretchy="false"> ) </mo> <mi> h </mi> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> x </mi> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> m </mi> <mspace width="thinmathspace"></mspace> <msub> <mi> a </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> y </mi> </mrow> </msub> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> m </mi> <mspace width="thinmathspace"></mspace> <msub> <mi> k </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> y </mi> </mrow> </msub> <mspace width="negativethinmathspace"></mspace> <mspace width="negativethinmathspace"></mspace> <mspace width="negativethinmathspace"></mspace> <mspace width="negativethinmathspace"></mspace> <mspace width="negativethinmathspace"></mspace> <mspace width="negativethinmathspace"></mspace> </mtd> <mtd> <mo> + </mo> <mspace width="thinmathspace"></mspace> <msub> <mi> σ </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> x </mi> <mi> y </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <mi> x </mi> <mo> + </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> x </mi> <mo> , </mo> <mi> y </mi> <mo stretchy="false"> ) </mo> <mi> h </mi> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> y </mi> <mo> + </mo> <msub> <mi> σ </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> y </mi> <mi> y </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <mi> x </mi> <mo> , </mo> <mi> y </mi> <mo> + </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> y </mi> <mo stretchy="false"> ) </mo> <mi> h </mi> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> x </mi> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mo> − </mo> <mspace width="thinmathspace"></mspace> <msub> <mi> σ </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> x </mi> <mi> y </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <mi> x </mi> <mo> , </mo> <mi> y </mi> <mo stretchy="false"> ) </mo> <mi> h </mi> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> y </mi> <mo> − </mo> <msub> <mi> σ </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> y </mi> <mi> y </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <mi> x </mi> <mo> , </mo> <mi> y </mi> <mo stretchy="false"> ) </mo> <mi> h </mi> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> x </mi> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\begin{array}{ll}\mathrm {d} m\,a_{x}=\mathrm {d} m\,k_{x}\!\!\!\!\!\!&+\,\sigma _{xx}(x+\mathrm {d} x,y)h\,\mathrm {d} y+\sigma _{yx}(x,y+\mathrm {d} y)h\,\mathrm {d} x\\&-\,\sigma _{xx}(x,y)h\,\mathrm {d} y-\sigma _{yx}(x,y)h\,\mathrm {d} x\\\mathrm {d} m\,a_{y}=\mathrm {d} m\,k_{y}\!\!\!\!\!\!&+\,\sigma _{xy}(x+\mathrm {d} x,y)h\,\mathrm {d} y+\sigma _{yy}(x,y+\mathrm {d} y)h\,\mathrm {d} x\\&-\,\sigma _{xy}(x,y)h\,\mathrm {d} y-\sigma _{yy}(x,y)h\,\mathrm {d} x\end{array}}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e1f48a52c8752fc81c3f59229281cb6a5aa0adaf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.171ex; width:57.799ex; height:13.509ex;" alt="{\displaystyle {\begin{array}{ll}\mathrm {d} m\,a_{x}=\mathrm {d} m\,k_{x}\!\!\!\!\!\!&+\,\sigma _{xx}(x+\mathrm {d} x,y)h\,\mathrm {d} y+\sigma _{yx}(x,y+\mathrm {d} y)h\,\mathrm {d} x\\&-\,\sigma _{xx}(x,y)h\,\mathrm {d} y-\sigma _{yx}(x,y)h\,\mathrm {d} x\\\mathrm {d} m\,a_{y}=\mathrm {d} m\,k_{y}\!\!\!\!\!\!&+\,\sigma _{xy}(x+\mathrm {d} x,y)h\,\mathrm {d} y+\sigma _{yy}(x,y+\mathrm {d} y)h\,\mathrm {d} x\\&-\,\sigma _{xy}(x,y)h\,\mathrm {d} y-\sigma _{yy}(x,y)h\,\mathrm {d} x\end{array}}}"> </noscript><span class="lazy-image-placeholder" style="width: 57.799ex;height: 13.509ex;vertical-align: -6.171ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e1f48a52c8752fc81c3f59229281cb6a5aa0adaf" data-alt="{\displaystyle {\begin{array}{ll}\mathrm {d} m\,a_{x}=\mathrm {d} m\,k_{x}\!\!\!\!\!\!&+\,\sigma _{xx}(x+\mathrm {d} x,y)h\,\mathrm {d} y+\sigma _{yx}(x,y+\mathrm {d} y)h\,\mathrm {d} x\\&-\,\sigma _{xx}(x,y)h\,\mathrm {d} y-\sigma _{yx}(x,y)h\,\mathrm {d} x\\\mathrm {d} m\,a_{y}=\mathrm {d} m\,k_{y}\!\!\!\!\!\!&+\,\sigma _{xy}(x+\mathrm {d} x,y)h\,\mathrm {d} y+\sigma _{yy}(x,y+\mathrm {d} y)h\,\mathrm {d} x\\&-\,\sigma _{xy}(x,y)h\,\mathrm {d} y-\sigma _{yy}(x,y)h\,\mathrm {d} x\end{array}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </dd> </dl> Die Masse <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {d} m=\rho h\,\mathrm {d} x\,\mathrm {d} y}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> m </mi> <mo> = </mo> <mi> ρ </mi> <mi> h </mi> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> x </mi> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> y </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathrm {d} m=\rho h\,\mathrm {d} x\,\mathrm {d} y} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af3fe9edba5e5644807e5c3fb47684d2c546c60a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.817ex; height:2.676ex;" alt="{\displaystyle \mathrm {d} m=\rho h\,\mathrm {d} x\,\mathrm {d} y}"> </noscript><span class="lazy-image-placeholder" style="width: 14.817ex;height: 2.676ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af3fe9edba5e5644807e5c3fb47684d2c546c60a" data-alt="{\displaystyle \mathrm {d} m=\rho h\,\mathrm {d} x\,\mathrm {d} y}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  des Scheibenelements ergibt sich aus der Dichte ρ des Materials und dem Volumen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h\,\mathrm {d} x\,\mathrm {d} y}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> h </mi> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> x </mi> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> y </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle h\,\mathrm {d} x\,\mathrm {d} y} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d01d45ab6117a35dbae8e810a9f2495ae540e824" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.183ex; height:2.509ex;" alt="{\displaystyle h\,\mathrm {d} x\,\mathrm {d} y}"> </noscript><span class="lazy-image-placeholder" style="width: 7.183ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d01d45ab6117a35dbae8e810a9f2495ae540e824" data-alt="{\displaystyle h\,\mathrm {d} x\,\mathrm {d} y}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> . Division durch dieses Volumen liefert im Grenzwert <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {d} x\rightarrow 0}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> x </mi> <mo stretchy="false"> → </mo> <mn> 0 </mn> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathrm {d} x\rightarrow 0} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f1489a33a7dffc86c6c2bf60a73557e746c011f6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.399ex; height:2.176ex;" alt="{\displaystyle \mathrm {d} x\rightarrow 0}"> </noscript><span class="lazy-image-placeholder" style="width: 7.399ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f1489a33a7dffc86c6c2bf60a73557e746c011f6" data-alt="{\displaystyle \mathrm {d} x\rightarrow 0}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {d} y\rightarrow 0}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> y </mi> <mo stretchy="false"> → </mo> <mn> 0 </mn> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathrm {d} y\rightarrow 0} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a1ee68a2afb93f400e86881e967d6ef49c02ac1c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.225ex; height:2.509ex;" alt="{\displaystyle \mathrm {d} y\rightarrow 0}"> </noscript><span class="lazy-image-placeholder" style="width: 7.225ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a1ee68a2afb93f400e86881e967d6ef49c02ac1c" data-alt="{\displaystyle \mathrm {d} y\rightarrow 0}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  den lokalen Impulssatz in x- bzw. y-Richtung: <dl> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left.{\begin{array}{rcl}\rho a_{x}&=&\rho k_{x}+{\frac {\partial \sigma _{xx}}{\partial x}}+{\frac {\partial \sigma _{yx}}{\partial y}}\\\rho a_{y}&=&\rho k_{y}+{\frac {\partial \sigma _{xy}}{\partial x}}+{\frac {\partial \sigma _{yy}}{\partial y}}\end{array}}\right\}\leftrightarrow \quad \rho a_{i}=\rho k_{i}+\sum _{j=1}^{2}{\frac {\partial \sigma _{ji}}{\partial x_{j}}}\,,\quad i=1,2}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right center left" rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi> ρ </mi> <msub> <mi> a </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> x </mi> </mrow> </msub> </mtd> <mtd> <mo> = </mo> </mtd> <mtd> <mi> ρ </mi> <msub> <mi> k </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> x </mi> </mrow> </msub> <mo> + </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal"> ∂ </mi> <msub> <mi> σ </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> x </mi> <mi> x </mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal"> ∂ </mi> <mi> x </mi> </mrow> </mfrac> </mrow> <mo> + </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal"> ∂ </mi> <msub> <mi> σ </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> y </mi> <mi> x </mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal"> ∂ </mi> <mi> y </mi> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mi> ρ </mi> <msub> <mi> a </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> y </mi> </mrow> </msub> </mtd> <mtd> <mo> = </mo> </mtd> <mtd> <mi> ρ </mi> <msub> <mi> k </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> y </mi> </mrow> </msub> <mo> + </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal"> ∂ </mi> <msub> <mi> σ </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> x </mi> <mi> y </mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal"> ∂ </mi> <mi> x </mi> </mrow> </mfrac> </mrow> <mo> + </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal"> ∂ </mi> <msub> <mi> σ </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> y </mi> <mi> y </mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal"> ∂ </mi> <mi> y </mi> </mrow> </mfrac> </mrow> </mtd> </mtr> </mtable> </mrow> <mo> } </mo> </mrow> <mo stretchy="false"> ↔ </mo> <mspace width="1em"></mspace> <mi> ρ </mi> <msub> <mi> a </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo> = </mo> <mi> ρ </mi> <msub> <mi> k </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo> + </mo> <munderover> <mo> ∑ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> j </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal"> ∂ </mi> <msub> <mi> σ </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> j </mi> <mi> i </mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal"> ∂ </mi> <msub> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> j </mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mspace width="thinmathspace"></mspace> <mo> , </mo> <mspace width="1em"></mspace> <mi> i </mi> <mo> = </mo> <mn> 1 </mn> <mo> , </mo> <mn> 2 </mn> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \left.{\begin{array}{rcl}\rho a_{x}&=&\rho k_{x}+{\frac {\partial \sigma _{xx}}{\partial x}}+{\frac {\partial \sigma _{yx}}{\partial y}}\\\rho a_{y}&=&\rho k_{y}+{\frac {\partial \sigma _{xy}}{\partial x}}+{\frac {\partial \sigma _{yy}}{\partial y}}\end{array}}\right\}\leftrightarrow \quad \rho a_{i}=\rho k_{i}+\sum _{j=1}^{2}{\frac {\partial \sigma _{ji}}{\partial x_{j}}}\,,\quad i=1,2} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e65e412add80c3202b515621ac1b229e1414c688" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.999ex; margin-bottom: -0.173ex; width:68.861ex; height:9.509ex;" alt="{\displaystyle \left.{\begin{array}{rcl}\rho a_{x}&=&\rho k_{x}+{\frac {\partial \sigma _{xx}}{\partial x}}+{\frac {\partial \sigma _{yx}}{\partial y}}\\\rho a_{y}&=&\rho k_{y}+{\frac {\partial \sigma _{xy}}{\partial x}}+{\frac {\partial \sigma _{yy}}{\partial y}}\end{array}}\right\}\leftrightarrow \quad \rho a_{i}=\rho k_{i}+\sum _{j=1}^{2}{\frac {\partial \sigma _{ji}}{\partial x_{j}}}\,,\quad i=1,2}"> </noscript><span class="lazy-image-placeholder" style="width: 68.861ex;height: 9.509ex;vertical-align: -3.999ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e65e412add80c3202b515621ac1b229e1414c688" data-alt="{\displaystyle \left.{\begin{array}{rcl}\rho a_{x}&=&\rho k_{x}+{\frac {\partial \sigma _{xx}}{\partial x}}+{\frac {\partial \sigma _{yx}}{\partial y}}\\\rho a_{y}&=&\rho k_{y}+{\frac {\partial \sigma _{xy}}{\partial x}}+{\frac {\partial \sigma _{yy}}{\partial y}}\end{array}}\right\}\leftrightarrow \quad \rho a_{i}=\rho k_{i}+\sum _{j=1}^{2}{\frac {\partial \sigma _{ji}}{\partial x_{j}}}\,,\quad i=1,2}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </dd> </dl> wenn – wie üblich – die Koordinaten nach dem Schema x→1, y→2, z→3 durchnummeriert werden. In drei Dimensionen resultieren die gleichen Differentialgleichungen analog, nur wird von eins bis drei summiert: <dl> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho a_{i}=\rho k_{i}+\sum _{j=1}^{3}{\frac {\partial \sigma _{ji}}{\partial x_{j}}}\,,\quad i=1,2,3}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> ρ </mi> <msub> <mi> a </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo> = </mo> <mi> ρ </mi> <msub> <mi> k </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo> + </mo> <munderover> <mo> ∑ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> j </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 3 </mn> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal"> ∂ </mi> <msub> <mi> σ </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> j </mi> <mi> i </mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal"> ∂ </mi> <msub> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> j </mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mspace width="thinmathspace"></mspace> <mo> , </mo> <mspace width="1em"></mspace> <mi> i </mi> <mo> = </mo> <mn> 1 </mn> <mo> , </mo> <mn> 2 </mn> <mo> , </mo> <mn> 3 </mn> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \rho a_{i}=\rho k_{i}+\sum _{j=1}^{3}{\frac {\partial \sigma _{ji}}{\partial x_{j}}}\,,\quad i=1,2,3} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6831ce9e3bfeeba26afa1d49eda7433d7031f79d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:34.284ex; height:7.509ex;" alt="{\displaystyle \rho a_{i}=\rho k_{i}+\sum _{j=1}^{3}{\frac {\partial \sigma _{ji}}{\partial x_{j}}}\,,\quad i=1,2,3}"> </noscript><span class="lazy-image-placeholder" style="width: 34.284ex;height: 7.509ex;vertical-align: -3.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6831ce9e3bfeeba26afa1d49eda7433d7031f79d" data-alt="{\displaystyle \rho a_{i}=\rho k_{i}+\sum _{j=1}^{3}{\frac {\partial \sigma _{ji}}{\partial x_{j}}}\,,\quad i=1,2,3}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </dd> </dl> Multiplikation dieser Gleichungen mit dem Basisvektor êi der <a href="https://de-m-wikipedia-org.translate.goog/wiki/Standardbasis?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Standardbasis">Standardbasis</a> und Addition der resultierenden drei Gleichungen mündet in der Vektorgleichung <dl> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{array}{cccccccl}\underbrace {\displaystyle \sum _{i=1}^{3}\rho a_{i}{\hat {e}}_{i}} \!\!\!\!&=&\!\!\!\!\underbrace {\displaystyle \sum _{i=1}^{3}\rho k_{i}{\hat {e}}_{i}} \!\!\!\!&+&\!\!\!\!\underbrace {\displaystyle \sum _{k=1}^{3}{\hat {e}}_{k}{\frac {\partial }{\partial x_{k}}}} \!\!\!\!&\cdot &\!\!\!\!\underbrace {\displaystyle \sum _{i,j=1}^{3}\sigma _{ji}{\hat {e}}_{j}\otimes {\hat {e}}_{i}} \\\rho {\vec {a}}\!\!\!\!&=&\!\!\!\!\rho {\vec {k}}\!\!\!\!&+&\!\!\!\!\nabla \!\!\!\!&\cdot \!\!\!\!&{\boldsymbol {\sigma }}&=\;\rho {\vec {k}}+\operatorname {div} \;{\boldsymbol {\sigma }}\end{array}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="center center center center center center center left" rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <munder> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo> ∑ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 3 </mn> </mrow> </munderover> <mi> ρ </mi> <msub> <mi> a </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> e </mi> <mo stretchy="false"> ^ </mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> </mstyle> <mo> ⏟ </mo> </munder> </mrow> <mspace width="negativethinmathspace"></mspace> <mspace width="negativethinmathspace"></mspace> <mspace width="negativethinmathspace"></mspace> <mspace width="negativethinmathspace"></mspace> </mtd> <mtd> <mo> = </mo> </mtd> <mtd> <mspace width="negativethinmathspace"></mspace> <mspace width="negativethinmathspace"></mspace> <mspace width="negativethinmathspace"></mspace> <mspace width="negativethinmathspace"></mspace> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <munder> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo> ∑ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 3 </mn> </mrow> </munderover> <mi> ρ </mi> <msub> <mi> k </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> e </mi> <mo stretchy="false"> ^ </mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> </mstyle> <mo> ⏟ </mo> </munder> </mrow> <mspace width="negativethinmathspace"></mspace> <mspace width="negativethinmathspace"></mspace> <mspace width="negativethinmathspace"></mspace> <mspace width="negativethinmathspace"></mspace> </mtd> <mtd> <mo> + </mo> </mtd> <mtd> <mspace width="negativethinmathspace"></mspace> <mspace width="negativethinmathspace"></mspace> <mspace width="negativethinmathspace"></mspace> <mspace width="negativethinmathspace"></mspace> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <munder> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo> ∑ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 3 </mn> </mrow> </munderover> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> e </mi> <mo stretchy="false"> ^ </mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal"> ∂ </mi> <mrow> <mi mathvariant="normal"> ∂ </mi> <msub> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> </mrow> </msub> </mrow> </mfrac> </mrow> </mstyle> <mo> ⏟ </mo> </munder> </mrow> <mspace width="negativethinmathspace"></mspace> <mspace width="negativethinmathspace"></mspace> <mspace width="negativethinmathspace"></mspace> <mspace width="negativethinmathspace"></mspace> </mtd> <mtd> <mo> ⋅ </mo> </mtd> <mtd> <mspace width="negativethinmathspace"></mspace> <mspace width="negativethinmathspace"></mspace> <mspace width="negativethinmathspace"></mspace> <mspace width="negativethinmathspace"></mspace> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <munder> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo> ∑ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> <mo> , </mo> <mi> j </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 3 </mn> </mrow> </munderover> <msub> <mi> σ </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> j </mi> <mi> i </mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> e </mi> <mo stretchy="false"> ^ </mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> j </mi> </mrow> </msub> <mo> ⊗ </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> e </mi> <mo stretchy="false"> ^ </mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> </mstyle> <mo> ⏟ </mo> </munder> </mrow> </mtd> </mtr> <mtr> <mtd> <mi> ρ </mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> a </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mspace width="negativethinmathspace"></mspace> <mspace width="negativethinmathspace"></mspace> <mspace width="negativethinmathspace"></mspace> <mspace width="negativethinmathspace"></mspace> </mtd> <mtd> <mo> = </mo> </mtd> <mtd> <mspace width="negativethinmathspace"></mspace> <mspace width="negativethinmathspace"></mspace> <mspace width="negativethinmathspace"></mspace> <mspace width="negativethinmathspace"></mspace> <mi> ρ </mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> k </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mspace width="negativethinmathspace"></mspace> <mspace width="negativethinmathspace"></mspace> <mspace width="negativethinmathspace"></mspace> <mspace width="negativethinmathspace"></mspace> </mtd> <mtd> <mo> + </mo> </mtd> <mtd> <mspace width="negativethinmathspace"></mspace> <mspace width="negativethinmathspace"></mspace> <mspace width="negativethinmathspace"></mspace> <mspace width="negativethinmathspace"></mspace> <mi mathvariant="normal"> ∇ </mi> <mspace width="negativethinmathspace"></mspace> <mspace width="negativethinmathspace"></mspace> <mspace width="negativethinmathspace"></mspace> <mspace width="negativethinmathspace"></mspace> </mtd> <mtd> <mo> ⋅ </mo> <mspace width="negativethinmathspace"></mspace> <mspace width="negativethinmathspace"></mspace> <mspace width="negativethinmathspace"></mspace> <mspace width="negativethinmathspace"></mspace> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> σ </mi> </mrow> </mtd> <mtd> <mo> = </mo> <mspace width="thickmathspace"></mspace> <mi> ρ </mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> k </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo> + </mo> <mi> div </mi> <mspace width="thickmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> σ </mi> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\begin{array}{cccccccl}\underbrace {\displaystyle \sum _{i=1}^{3}\rho a_{i}{\hat {e}}_{i}} \!\!\!\!&=&\!\!\!\!\underbrace {\displaystyle \sum _{i=1}^{3}\rho k_{i}{\hat {e}}_{i}} \!\!\!\!&+&\!\!\!\!\underbrace {\displaystyle \sum _{k=1}^{3}{\hat {e}}_{k}{\frac {\partial }{\partial x_{k}}}} \!\!\!\!&\cdot &\!\!\!\!\underbrace {\displaystyle \sum _{i,j=1}^{3}\sigma _{ji}{\hat {e}}_{j}\otimes {\hat {e}}_{i}} \\\rho {\vec {a}}\!\!\!\!&=&\!\!\!\!\rho {\vec {k}}\!\!\!\!&+&\!\!\!\!\nabla \!\!\!\!&\cdot \!\!\!\!&{\boldsymbol {\sigma }}&=\;\rho {\vec {k}}+\operatorname {div} \;{\boldsymbol {\sigma }}\end{array}}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/013acd4ea4908090bece8729ed39fc35d533decf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.005ex; width:69.061ex; height:13.176ex;" alt="{\displaystyle {\begin{array}{cccccccl}\underbrace {\displaystyle \sum _{i=1}^{3}\rho a_{i}{\hat {e}}_{i}} \!\!\!\!&=&\!\!\!\!\underbrace {\displaystyle \sum _{i=1}^{3}\rho k_{i}{\hat {e}}_{i}} \!\!\!\!&+&\!\!\!\!\underbrace {\displaystyle \sum _{k=1}^{3}{\hat {e}}_{k}{\frac {\partial }{\partial x_{k}}}} \!\!\!\!&\cdot &\!\!\!\!\underbrace {\displaystyle \sum _{i,j=1}^{3}\sigma _{ji}{\hat {e}}_{j}\otimes {\hat {e}}_{i}} \\\rho {\vec {a}}\!\!\!\!&=&\!\!\!\!\rho {\vec {k}}\!\!\!\!&+&\!\!\!\!\nabla \!\!\!\!&\cdot \!\!\!\!&{\boldsymbol {\sigma }}&=\;\rho {\vec {k}}+\operatorname {div} \;{\boldsymbol {\sigma }}\end{array}}}"> </noscript><span class="lazy-image-placeholder" style="width: 69.061ex;height: 13.176ex;vertical-align: -6.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/013acd4ea4908090bece8729ed39fc35d533decf" data-alt="{\displaystyle {\begin{array}{cccccccl}\underbrace {\displaystyle \sum _{i=1}^{3}\rho a_{i}{\hat {e}}_{i}} \!\!\!\!&=&\!\!\!\!\underbrace {\displaystyle \sum _{i=1}^{3}\rho k_{i}{\hat {e}}_{i}} \!\!\!\!&+&\!\!\!\!\underbrace {\displaystyle \sum _{k=1}^{3}{\hat {e}}_{k}{\frac {\partial }{\partial x_{k}}}} \!\!\!\!&\cdot &\!\!\!\!\underbrace {\displaystyle \sum _{i,j=1}^{3}\sigma _{ji}{\hat {e}}_{j}\otimes {\hat {e}}_{i}} \\\rho {\vec {a}}\!\!\!\!&=&\!\!\!\!\rho {\vec {k}}\!\!\!\!&+&\!\!\!\!\nabla \!\!\!\!&\cdot \!\!\!\!&{\boldsymbol {\sigma }}&=\;\rho {\vec {k}}+\operatorname {div} \;{\boldsymbol {\sigma }}\end{array}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </dd> </dl> Der <a href="https://de-m-wikipedia-org.translate.goog/wiki/Nabla-Operator?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Nabla-Operator">Nabla-Operator</a> „ <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nabla }"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal"> ∇ </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \nabla } </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a3d0e93b78c50237f9ea83d027e4ebbdaef354b2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.936ex; height:2.176ex;" alt="{\displaystyle \nabla }"> </noscript><span class="lazy-image-placeholder" style="width: 1.936ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a3d0e93b78c50237f9ea83d027e4ebbdaef354b2" data-alt="{\displaystyle \nabla }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> “ liefert im <a href="https://de-m-wikipedia-org.translate.goog/wiki/Skalarprodukt?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Skalarprodukt">Skalarprodukt</a> die Divergenz div des cauchyschen Spannungstensors <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {\sigma }}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> σ </mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\boldsymbol {\sigma }}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e45fe1b9d8dcbc3103fc7805d69798bfe5ca5b16" 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 }}}"> </noscript><span class="lazy-image-placeholder" style="width: 1.594ex;height: 1.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e45fe1b9d8dcbc3103fc7805d69798bfe5ca5b16" data-alt="{\displaystyle {\boldsymbol {\sigma }}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> , der eine Summe von Dyaden ist, die mit dem <a href="https://de-m-wikipedia-org.translate.goog/wiki/Dyadisches_Produkt?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Dyadisches Produkt">dyadischen Produkt</a> „ <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \otimes }"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo> ⊗ </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \otimes } </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/de29098f5a34ee296a505681a0d5e875070f2aea" 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 }"> </noscript><span class="lazy-image-placeholder" style="width: 1.808ex;height: 2.176ex;vertical-align: -0.505ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/de29098f5a34ee296a505681a0d5e875070f2aea" data-alt="{\displaystyle \otimes }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> “ der Basisvektoren und seinen Komponenten <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{ji}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> σ </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> j </mi> <mi> i </mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \sigma _{ji}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1a2784210881b8d361b4ceb221dc5224dc2aeaaf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.804ex; height:2.343ex;" alt="{\displaystyle \sigma _{ji}}"> </noscript><span class="lazy-image-placeholder" style="width: 2.804ex;height: 2.343ex;vertical-align: -1.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1a2784210881b8d361b4ceb221dc5224dc2aeaaf" data-alt="{\displaystyle \sigma _{ji}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  gebildet werden. Die Vektorgleichung ist die koordinatenfreie Version des lokalen Impulssatzes, die in beliebigen Koordinaten eines <a href="https://de-m-wikipedia-org.translate.goog/wiki/Inertialsystem?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Inertialsystem">Inertialsystems</a> gilt. Der Schnittspannungsvektor <dl> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {t}}_{x}=\sigma _{xx}{\hat {e}}_{x}+\sigma _{xy}{\hat {e}}_{y}+\sigma _{xz}{\hat {e}}_{z}={\boldsymbol {\sigma }}^{\top }\cdot {\hat {e}}_{x}={\hat {e}}_{x}\cdot {\boldsymbol {\sigma }}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> t </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> x </mi> </mrow> </msub> <mo> = </mo> <msub> <mi> σ </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> x </mi> <mi> x </mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> e </mi> <mo stretchy="false"> ^ </mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> x </mi> </mrow> </msub> <mo> + </mo> <msub> <mi> σ </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> x </mi> <mi> y </mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> e </mi> <mo stretchy="false"> ^ </mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> y </mi> </mrow> </msub> <mo> + </mo> <msub> <mi> σ </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> x </mi> <mi> z </mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> e </mi> <mo stretchy="false"> ^ </mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> z </mi> </mrow> </msub> <mo> = </mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> σ </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> ⊤ </mi> </mrow> </msup> <mo> ⋅ </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> e </mi> <mo stretchy="false"> ^ </mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> x </mi> </mrow> </msub> <mo> = </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> e </mi> <mo stretchy="false"> ^ </mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> x </mi> </mrow> </msub> <mo> ⋅ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> σ </mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\vec {t}}_{x}=\sigma _{xx}{\hat {e}}_{x}+\sigma _{xy}{\hat {e}}_{y}+\sigma _{xz}{\hat {e}}_{z}={\boldsymbol {\sigma }}^{\top }\cdot {\hat {e}}_{x}={\hat {e}}_{x}\cdot {\boldsymbol {\sigma }}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5b0f154700a142b69f214f4cd088c3a8c0cadd49" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:47.547ex; height:3.509ex;" alt="{\displaystyle {\vec {t}}_{x}=\sigma _{xx}{\hat {e}}_{x}+\sigma _{xy}{\hat {e}}_{y}+\sigma _{xz}{\hat {e}}_{z}={\boldsymbol {\sigma }}^{\top }\cdot {\hat {e}}_{x}={\hat {e}}_{x}\cdot {\boldsymbol {\sigma }}}"> </noscript><span class="lazy-image-placeholder" style="width: 47.547ex;height: 3.509ex;vertical-align: -1.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5b0f154700a142b69f214f4cd088c3a8c0cadd49" data-alt="{\displaystyle {\vec {t}}_{x}=\sigma _{xx}{\hat {e}}_{x}+\sigma _{xy}{\hat {e}}_{y}+\sigma _{xz}{\hat {e}}_{z}={\boldsymbol {\sigma }}^{\top }\cdot {\hat {e}}_{x}={\hat {e}}_{x}\cdot {\boldsymbol {\sigma }}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </dd> </dl> in der Schnittfläche mit Normalenvektor in x-Richtung ist im cauchyschen Spannungstensor zeilenweise eingetragen, was sinngemäß auch für Schnittspannungsvektoren in y- und z-Richtung gilt. <div class="mw-heading mw-heading3"> <h3 id="Impulssatz_in_lagrangescher_Darstellung">Impulssatz in lagrangescher Darstellung</h3> <a role="button" href="https://de-m-wikipedia-org.translate.goog/w/index.php?title=Cauchy-eulersche_Bewegungsgesetze&action=edit&section=3&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Abschnitt bearbeiten: Impulssatz in lagrangescher Darstellung" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> Bearbeiten </a> </div> In der <a href="https://de-m-wikipedia-org.translate.goog/wiki/Lagrangesche_Betrachtungsweise?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Lagrangesche Betrachtungsweise">lagrangeschen Darstellung</a> lautet der globale Impulssatz <dl> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}\int _{V}\rho _{0}({\vec {X}})\,{\dot {\vec {\chi }}}({\vec {X}},t)\,\mathrm {d} V=\int _{V}\rho _{0}({\vec {X}})\,{\vec {k}}_{0}({\vec {X}},t)\,\mathrm {d} V+\int _{A}{\vec {t}}_{0}({\vec {X}},t)\,\mathrm {d} A}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> t </mi> </mrow> </mfrac> </mrow> <msub> <mo> ∫ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> V </mi> </mrow> </msub> <msub> <mi> ρ </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> X </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo stretchy="false"> ) </mo> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> χ </mi> <mo stretchy="false"> → </mo> </mover> </mrow> <mo> ˙ </mo> </mover> </mrow> </mrow> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> X </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo> , </mo> <mi> t </mi> <mo stretchy="false"> ) </mo> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> V </mi> <mo> = </mo> <msub> <mo> ∫ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> V </mi> </mrow> </msub> <msub> <mi> ρ </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> X </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo stretchy="false"> ) </mo> <mspace width="thinmathspace"></mspace> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> k </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> X </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo> , </mo> <mi> t </mi> <mo stretchy="false"> ) </mo> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> V </mi> <mo> + </mo> <msub> <mo> ∫ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> A </mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> t </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> X </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo> , </mo> <mi> t </mi> <mo stretchy="false"> ) </mo> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> A </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}\int _{V}\rho _{0}({\vec {X}})\,{\dot {\vec {\chi }}}({\vec {X}},t)\,\mathrm {d} V=\int _{V}\rho _{0}({\vec {X}})\,{\vec {k}}_{0}({\vec {X}},t)\,\mathrm {d} V+\int _{A}{\vec {t}}_{0}({\vec {X}},t)\,\mathrm {d} A} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/58911e68fee307c066a55fd76a6c872a700a36c7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:65.062ex; height:5.843ex;" alt="{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}\int _{V}\rho _{0}({\vec {X}})\,{\dot {\vec {\chi }}}({\vec {X}},t)\,\mathrm {d} V=\int _{V}\rho _{0}({\vec {X}})\,{\vec {k}}_{0}({\vec {X}},t)\,\mathrm {d} V+\int _{A}{\vec {t}}_{0}({\vec {X}},t)\,\mathrm {d} A}"> </noscript><span class="lazy-image-placeholder" style="width: 65.062ex;height: 5.843ex;vertical-align: -2.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/58911e68fee307c066a55fd76a6c872a700a36c7" data-alt="{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}\int _{V}\rho _{0}({\vec {X}})\,{\dot {\vec {\chi }}}({\vec {X}},t)\,\mathrm {d} V=\int _{V}\rho _{0}({\vec {X}})\,{\vec {k}}_{0}({\vec {X}},t)\,\mathrm {d} V+\int _{A}{\vec {t}}_{0}({\vec {X}},t)\,\mathrm {d} A}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </dd> </dl> die den materiellen Punkten (Partikel) zugeordnete physikalische Größen benutzt, siehe <a href="https://de-m-wikipedia-org.translate.goog/wiki/Kontinuumsmechanik?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Impulsbilanz" title="Kontinuumsmechanik">Impulsbilanz</a>. Die Partikel werden durch ihre materiellen Koordinaten <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {X}}\in V}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> X </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo> ∈ </mo> <mi> V </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\vec {X}}\in V} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0b96389a1d18d9ae781522869a1337e51b85ebce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.608ex; height:2.843ex;" alt="{\displaystyle {\vec {X}}\in V}"> </noscript><span class="lazy-image-placeholder" style="width: 6.608ex;height: 2.843ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0b96389a1d18d9ae781522869a1337e51b85ebce" data-alt="{\displaystyle {\vec {X}}\in V}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  in dem Volumen V des Körpers zu einer festgelegten Zeit t0 im Referenzzustand identifiziert, und auf diese Partikel bezieht sich die Impulsbilanz lokal. Die einem materiellen Punkt zugeordnete Dichte ρ0 ist auf Grund der <a href="https://de-m-wikipedia-org.translate.goog/wiki/Kontinuumsmechanik?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Massenbilanz" title="Kontinuumsmechanik">Massenbilanz</a> keine Funktion der Zeit. Der <a href="https://de-m-wikipedia-org.translate.goog/wiki/%C3%9Cberpunkt?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Als_wissenschaftliches_Symbol" title="Überpunkt">aufgesetzte Punkt</a> steht hier wie im Folgenden für die <a href="https://de-m-wikipedia-org.translate.goog/wiki/Substantielle_Ableitung?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Substantielle Ableitung">substantielle Ableitung</a>, also für die <a href="https://de-m-wikipedia-org.translate.goog/wiki/Zeitableitung?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Zeitableitung">Zeitableitung</a> bei festgehaltenem Partikel, denn die Gesetze der Mechanik beziehen sich auf die Partikel und nicht auf die Raumpunkte. Bei den Integralen oben ist das Integrationsgebiet materiell festgelegt, sodass es sich also mit dem Körper mitbewegt, ohne dass neue Partikel zum Gebiet hinzukommen oder wegfallen. Dies wird durch die Großschreibung V bzw. A der Integrationsgebiete symbolisiert. Weil das Referenzvolumen V somit nicht von der Zeit abhängt, kann die Zeitableitung des Integrals in den Integranden verschoben werden: <dl> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}\int _{V}\rho _{0}({\vec {X}})\,{\dot {\vec {\chi }}}({\vec {X}},t)\,\mathrm {d} V=\int _{V}\rho _{0}({\vec {X}})\,{\ddot {\vec {\chi }}}({\vec {X}},t)\,\mathrm {d} V}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> t </mi> </mrow> </mfrac> </mrow> <msub> <mo> ∫ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> V </mi> </mrow> </msub> <msub> <mi> ρ </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> X </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo stretchy="false"> ) </mo> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> χ </mi> <mo stretchy="false"> → </mo> </mover> </mrow> <mo> ˙ </mo> </mover> </mrow> </mrow> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> X </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo> , </mo> <mi> t </mi> <mo stretchy="false"> ) </mo> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> V </mi> <mo> = </mo> <msub> <mo> ∫ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> V </mi> </mrow> </msub> <msub> <mi> ρ </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> X </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo stretchy="false"> ) </mo> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> χ </mi> <mo stretchy="false"> → </mo> </mover> </mrow> <mo> ¨ </mo> </mover> </mrow> </mrow> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> X </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo> , </mo> <mi> t </mi> <mo stretchy="false"> ) </mo> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> V </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}\int _{V}\rho _{0}({\vec {X}})\,{\dot {\vec {\chi }}}({\vec {X}},t)\,\mathrm {d} V=\int _{V}\rho _{0}({\vec {X}})\,{\ddot {\vec {\chi }}}({\vec {X}},t)\,\mathrm {d} V} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fcf26800eda8612e55514821a8eb345c1cdaece0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:46.84ex; height:5.843ex;" alt="{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}\int _{V}\rho _{0}({\vec {X}})\,{\dot {\vec {\chi }}}({\vec {X}},t)\,\mathrm {d} V=\int _{V}\rho _{0}({\vec {X}})\,{\ddot {\vec {\chi }}}({\vec {X}},t)\,\mathrm {d} V}"> </noscript><span class="lazy-image-placeholder" style="width: 46.84ex;height: 5.843ex;vertical-align: -2.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fcf26800eda8612e55514821a8eb345c1cdaece0" data-alt="{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}\int _{V}\rho _{0}({\vec {X}})\,{\dot {\vec {\chi }}}({\vec {X}},t)\,\mathrm {d} V=\int _{V}\rho _{0}({\vec {X}})\,{\ddot {\vec {\chi }}}({\vec {X}},t)\,\mathrm {d} V}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </dd> </dl> Die von außen angreifenden, flächenverteilten Kräfte (Spannungen) <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {t}}_{0}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> t </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\vec {t}}_{0}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2c6d87b2de6589445294992c544f9848febf5b13" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.342ex; height:3.176ex;" alt="{\displaystyle {\vec {t}}_{0}}"> </noscript><span class="lazy-image-placeholder" style="width: 2.342ex;height: 3.176ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2c6d87b2de6589445294992c544f9848febf5b13" data-alt="{\displaystyle {\vec {t}}_{0}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  sind die mit dem Nennspannungstensor N transformierten <a href="https://de-m-wikipedia-org.translate.goog/wiki/Normaleneinheitsvektor?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Normaleneinheitsvektor">Normaleneinheitsvektoren</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {N}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> N </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\vec {N}}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/13e01cc1cec4201098af497311170ea68412c6b7" 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}}}"> </noscript><span class="lazy-image-placeholder" style="width: 2.064ex;height: 2.843ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/13e01cc1cec4201098af497311170ea68412c6b7" data-alt="{\displaystyle {\vec {N}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  an der Oberfläche A des Körpers: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {t}}_{0}=\mathbf {N} ^{\top }\cdot {\vec {N}}=\mathbf {P} \cdot {\vec {N}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> t </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mo> = </mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> N </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> ⊤ </mi> </mrow> </msup> <mo> ⋅ </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> N </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> P </mi> </mrow> <mo> ⋅ </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> N </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\vec {t}}_{0}=\mathbf {N} ^{\top }\cdot {\vec {N}}=\mathbf {P} \cdot {\vec {N}}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/100c3f16540761fa26577120c66eb768e2f37596" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:21.453ex; height:3.176ex;" alt="{\displaystyle {\vec {t}}_{0}=\mathbf {N} ^{\top }\cdot {\vec {N}}=\mathbf {P} \cdot {\vec {N}}}"> </noscript><span class="lazy-image-placeholder" style="width: 21.453ex;height: 3.176ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/100c3f16540761fa26577120c66eb768e2f37596" data-alt="{\displaystyle {\vec {t}}_{0}=\mathbf {N} ^{\top }\cdot {\vec {N}}=\mathbf {P} \cdot {\vec {N}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> . Darin ist P = N⊤ der erste-Piola-Kirchhoff Tensor und ⊤ bedeutet die <a href="https://de-m-wikipedia-org.translate.goog/wiki/Transponierte_Matrix?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Transponierte Matrix">Transponierung</a>. Das <a href="https://de-m-wikipedia-org.translate.goog/wiki/Oberfl%C3%A4chenintegral?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Oberflächenintegral">Oberflächenintegral</a> der Oberflächenspannungen wird mit dem <a href="https://de-m-wikipedia-org.translate.goog/wiki/Gaussscher_Integralsatz?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Gaussscher Integralsatz">gaußschen Integralsatz</a> in ein <a href="https://de-m-wikipedia-org.translate.goog/wiki/Volumenintegral?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Volumenintegral">Volumenintegral</a> umgewandelt: <dl> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\int _{A}{\vec {t}}_{0}\,\mathrm {d} A=&\int _{A}\mathbf {N} ^{\top }\cdot {\vec {N}}\,\mathrm {d} A=\int _{V}\nabla _{0}\cdot \mathbf {N} \,\mathrm {d} V\\=&\int _{A}\mathbf {P} \cdot {\vec {N}}\,\mathrm {d} A=\int _{V}\operatorname {DIV} (\mathbf {P} )\,\mathrm {d} V\end{aligned}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mo> ∫ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> A </mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> t </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> A </mi> <mo> = </mo> </mtd> <mtd> <mi></mi> <msub> <mo> ∫ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> A </mi> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> N </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> ⊤ </mi> </mrow> </msup> <mo> ⋅ </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> N </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> A </mi> <mo> = </mo> <msub> <mo> ∫ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> V </mi> </mrow> </msub> <msub> <mi mathvariant="normal"> ∇ </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mo> ⋅ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> N </mi> </mrow> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> V </mi> </mtd> </mtr> <mtr> <mtd> <mo> = </mo> </mtd> <mtd> <mi></mi> <msub> <mo> ∫ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> A </mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> P </mi> </mrow> <mo> ⋅ </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> N </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> A </mi> <mo> = </mo> <msub> <mo> ∫ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> V </mi> </mrow> </msub> <mi> DIV </mi> <mo> ⁡ </mo> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> P </mi> </mrow> <mo stretchy="false"> ) </mo> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> V </mi> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\begin{aligned}\int _{A}{\vec {t}}_{0}\,\mathrm {d} A=&\int _{A}\mathbf {N} ^{\top }\cdot {\vec {N}}\,\mathrm {d} A=\int _{V}\nabla _{0}\cdot \mathbf {N} \,\mathrm {d} V\\=&\int _{A}\mathbf {P} \cdot {\vec {N}}\,\mathrm {d} A=\int _{V}\operatorname {DIV} (\mathbf {P} )\,\mathrm {d} V\end{aligned}}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5bcffebac62e17b0b36c34e5c60e8b12fc8bc6dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.338ex; width:42.915ex; height:11.843ex;" alt="{\displaystyle {\begin{aligned}\int _{A}{\vec {t}}_{0}\,\mathrm {d} A=&\int _{A}\mathbf {N} ^{\top }\cdot {\vec {N}}\,\mathrm {d} A=\int _{V}\nabla _{0}\cdot \mathbf {N} \,\mathrm {d} V\\=&\int _{A}\mathbf {P} \cdot {\vec {N}}\,\mathrm {d} A=\int _{V}\operatorname {DIV} (\mathbf {P} )\,\mathrm {d} V\end{aligned}}}"> </noscript><span class="lazy-image-placeholder" style="width: 42.915ex;height: 11.843ex;vertical-align: -5.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5bcffebac62e17b0b36c34e5c60e8b12fc8bc6dc" data-alt="{\displaystyle {\begin{aligned}\int _{A}{\vec {t}}_{0}\,\mathrm {d} A=&\int _{A}\mathbf {N} ^{\top }\cdot {\vec {N}}\,\mathrm {d} A=\int _{V}\nabla _{0}\cdot \mathbf {N} \,\mathrm {d} V\\=&\int _{A}\mathbf {P} \cdot {\vec {N}}\,\mathrm {d} A=\int _{V}\operatorname {DIV} (\mathbf {P} )\,\mathrm {d} V\end{aligned}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </dd> </dl> Der Divergenzoperator DIV wird hier groß geschrieben und der <a href="https://de-m-wikipedia-org.translate.goog/wiki/Nabla-Operator?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Nabla-Operator">Nabla-Operator</a> wird mit einem Index 𝜵0 versehen, weil sie die materiellen Ableitungen nach den materiellen Koordinaten <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {X}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> X </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\vec {X}}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5dc35b5a0226cf11a2c3f2d2dbbac6ab5ade6036" 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}}}"> </noscript><span class="lazy-image-placeholder" style="width: 1.98ex;height: 2.843ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5dc35b5a0226cf11a2c3f2d2dbbac6ab5ade6036" data-alt="{\displaystyle {\vec {X}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  beinhalten. Es gilt für jedes Tensorfeld <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {T} ({\vec {X}})\colon \;\operatorname {DIV} \mathbf {T} =\nabla _{0}\cdot (\mathbf {T} ^{\top })}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> T </mi> </mrow> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> X </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo stretchy="false"> ) </mo> <mo> : </mo> <mspace width="thickmathspace"></mspace> <mi> DIV </mi> <mo> ⁡ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> T </mi> </mrow> <mo> = </mo> <msub> <mi mathvariant="normal"> ∇ </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mo> ⋅ </mo> <mo stretchy="false"> ( </mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> T </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> ⊤ </mi> </mrow> </msup> <mo stretchy="false"> ) </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbf {T} ({\vec {X}})\colon \;\operatorname {DIV} \mathbf {T} =\nabla _{0}\cdot (\mathbf {T} ^{\top })} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/82e42163c3d4a4e521144cd8020a4e2df5b884d8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.879ex; height:3.343ex;" alt="{\displaystyle \mathbf {T} ({\vec {X}})\colon \;\operatorname {DIV} \mathbf {T} =\nabla _{0}\cdot (\mathbf {T} ^{\top })}"> </noscript><span class="lazy-image-placeholder" style="width: 26.879ex;height: 3.343ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/82e42163c3d4a4e521144cd8020a4e2df5b884d8" data-alt="{\displaystyle \mathbf {T} ({\vec {X}})\colon \;\operatorname {DIV} \mathbf {T} =\nabla _{0}\cdot (\mathbf {T} ^{\top })}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> . Die Operatoren in dieser Gleichung sind von den räumlichen Operatoren div bzw. 𝜵 zu unterscheiden, die die räumlichen Ableitungen nach den räumlichen Koordinaten <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {x}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> x </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\vec {x}}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/db2dc6ced9cc3bc7e8b9f2707cbec033f6d3759c" 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}}}"> </noscript><span class="lazy-image-placeholder" style="width: 1.33ex;height: 2.343ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/db2dc6ced9cc3bc7e8b9f2707cbec033f6d3759c" data-alt="{\displaystyle {\vec {x}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  ausführen, und die in der eulerschen Darstellung benötigt werden. Mit den vorliegenden Ergebnissen kann die <a href="https://de-m-wikipedia-org.translate.goog/wiki/Cauchy-eulersche_Bewegungsgesetze?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Impulssatz_in_lagrangescher_Darstellung">Impulsbilanz</a> als verschwindendes Volumenintegral ausgedrückt werden: <dl> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{V}(\rho _{0}{\ddot {\vec {\chi }}}-\rho _{0}{\vec {k}}_{0}-\nabla _{0}\cdot \mathbf {N} )\,\mathrm {d} V=\int _{V}(\rho _{0}{\ddot {\vec {\chi }}}-\rho _{0}{\vec {k}}_{0}-\operatorname {DIV} \mathbf {P} )\,\mathrm {d} V={\vec {0}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mo> ∫ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> V </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <msub> <mi> ρ </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> χ </mi> <mo stretchy="false"> → </mo> </mover> </mrow> <mo> ¨ </mo> </mover> </mrow> </mrow> <mo> − </mo> <msub> <mi> ρ </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> k </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mo> − </mo> <msub> <mi mathvariant="normal"> ∇ </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mo> ⋅ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> N </mi> </mrow> <mo stretchy="false"> ) </mo> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> V </mi> <mo> = </mo> <msub> <mo> ∫ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> V </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <msub> <mi> ρ </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> χ </mi> <mo stretchy="false"> → </mo> </mover> </mrow> <mo> ¨ </mo> </mover> </mrow> </mrow> <mo> − </mo> <msub> <mi> ρ </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> k </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mo> − </mo> <mi> DIV </mi> <mo> ⁡ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> P </mi> </mrow> <mo stretchy="false"> ) </mo> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> V </mi> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mn> 0 </mn> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \int _{V}(\rho _{0}{\ddot {\vec {\chi }}}-\rho _{0}{\vec {k}}_{0}-\nabla _{0}\cdot \mathbf {N} )\,\mathrm {d} V=\int _{V}(\rho _{0}{\ddot {\vec {\chi }}}-\rho _{0}{\vec {k}}_{0}-\operatorname {DIV} \mathbf {P} )\,\mathrm {d} V={\vec {0}}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8cc3e2f260dea4f914f653e8a86cd53bb3686e28" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:64.649ex; height:5.676ex;" alt="{\displaystyle \int _{V}(\rho _{0}{\ddot {\vec {\chi }}}-\rho _{0}{\vec {k}}_{0}-\nabla _{0}\cdot \mathbf {N} )\,\mathrm {d} V=\int _{V}(\rho _{0}{\ddot {\vec {\chi }}}-\rho _{0}{\vec {k}}_{0}-\operatorname {DIV} \mathbf {P} )\,\mathrm {d} V={\vec {0}}}"> </noscript><span class="lazy-image-placeholder" style="width: 64.649ex;height: 5.676ex;vertical-align: -2.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8cc3e2f260dea4f914f653e8a86cd53bb3686e28" data-alt="{\displaystyle \int _{V}(\rho _{0}{\ddot {\vec {\chi }}}-\rho _{0}{\vec {k}}_{0}-\nabla _{0}\cdot \mathbf {N} )\,\mathrm {d} V=\int _{V}(\rho _{0}{\ddot {\vec {\chi }}}-\rho _{0}{\vec {k}}_{0}-\operatorname {DIV} \mathbf {P} )\,\mathrm {d} V={\vec {0}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </dd> </dl> Diese Gleichung gilt für jeden Körper und jeden seiner Teilkörper, sodass – Stetigkeit des Integranden vorausgesetzt – auf das erste cauchy-eulersche Bewegungsgesetz in der lagrangeschen Darstellung <dl> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho _{0}({\vec {X}})\,{\ddot {\vec {\chi }}}({\vec {X}},t)=\rho _{0}({\vec {X}})\,{\vec {k}}_{0}({\vec {X}},t)+\nabla _{0}\cdot \mathbf {N} ({\vec {X}},t)=\rho _{0}({\vec {X}})\,{\vec {k}}_{0}({\vec {X}},t)+\operatorname {DIV} \mathbf {P} ({\vec {X}},t)}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> ρ </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> X </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo stretchy="false"> ) </mo> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> χ </mi> <mo stretchy="false"> → </mo> </mover> </mrow> <mo> ¨ </mo> </mover> </mrow> </mrow> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> X </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo> , </mo> <mi> t </mi> <mo stretchy="false"> ) </mo> <mo> = </mo> <msub> <mi> ρ </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> X </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo stretchy="false"> ) </mo> <mspace width="thinmathspace"></mspace> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> k </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> X </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo> , </mo> <mi> t </mi> <mo stretchy="false"> ) </mo> <mo> + </mo> <msub> <mi mathvariant="normal"> ∇ </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mo> ⋅ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> N </mi> </mrow> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> X </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo> , </mo> <mi> t </mi> <mo stretchy="false"> ) </mo> <mo> = </mo> <msub> <mi> ρ </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> X </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo stretchy="false"> ) </mo> <mspace width="thinmathspace"></mspace> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> k </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> X </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo> , </mo> <mi> t </mi> <mo stretchy="false"> ) </mo> <mo> + </mo> <mi> DIV </mi> <mo> ⁡ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> P </mi> </mrow> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> X </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo> , </mo> <mi> t </mi> <mo stretchy="false"> ) </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \rho _{0}({\vec {X}})\,{\ddot {\vec {\chi }}}({\vec {X}},t)=\rho _{0}({\vec {X}})\,{\vec {k}}_{0}({\vec {X}},t)+\nabla _{0}\cdot \mathbf {N} ({\vec {X}},t)=\rho _{0}({\vec {X}})\,{\vec {k}}_{0}({\vec {X}},t)+\operatorname {DIV} \mathbf {P} ({\vec {X}},t)} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7fb249d2cf6b8022c44163a786feb3bd9951d9b1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:78.809ex; height:3.343ex;" alt="{\displaystyle \rho _{0}({\vec {X}})\,{\ddot {\vec {\chi }}}({\vec {X}},t)=\rho _{0}({\vec {X}})\,{\vec {k}}_{0}({\vec {X}},t)+\nabla _{0}\cdot \mathbf {N} ({\vec {X}},t)=\rho _{0}({\vec {X}})\,{\vec {k}}_{0}({\vec {X}},t)+\operatorname {DIV} \mathbf {P} ({\vec {X}},t)}"> </noscript><span class="lazy-image-placeholder" style="width: 78.809ex;height: 3.343ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7fb249d2cf6b8022c44163a786feb3bd9951d9b1" data-alt="{\displaystyle \rho _{0}({\vec {X}})\,{\ddot {\vec {\chi }}}({\vec {X}},t)=\rho _{0}({\vec {X}})\,{\vec {k}}_{0}({\vec {X}},t)+\nabla _{0}\cdot \mathbf {N} ({\vec {X}},t)=\rho _{0}({\vec {X}})\,{\vec {k}}_{0}({\vec {X}},t)+\operatorname {DIV} \mathbf {P} ({\vec {X}},t)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </dd> </dl> geschlossen werden kann. Das Vorkommen der materiellen Koordinaten und des Nennspannungstensors N bzw. des ersten-Piola-Kirchhoff’schen Spannungstensors P an Stelle des cauchyschen Spannungstensors berücksichtigt die Formänderung des bei der Betrachtung am Volumenelement oben herausgeschnittenen Teilkörpers bei großen Deformationen. Bei kleinen Verschiebungen ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {N} \approx \mathbf {P} \approx {\boldsymbol {\sigma }}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> N </mi> </mrow> <mo> ≈ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> P </mi> </mrow> <mo> ≈ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> σ </mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbf {N} \approx \mathbf {P} \approx {\boldsymbol {\sigma }}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/acac21387c95c042fc18ca482e363a8394bc2536" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.71ex; height:2.176ex;" alt="{\displaystyle \mathbf {N} \approx \mathbf {P} \approx {\boldsymbol {\sigma }}}"> </noscript><span class="lazy-image-placeholder" style="width: 11.71ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/acac21387c95c042fc18ca482e363a8394bc2536" data-alt="{\displaystyle \mathbf {N} \approx \mathbf {P} \approx {\boldsymbol {\sigma }}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  und zwischen den materiellen und räumlichen Koordinaten braucht nicht unterschieden zu werden, wodurch das eingangs angegebene Bewegungsgesetz entsteht. <div class="mw-heading mw-heading3"> <h3 id="Impulssatz_in_eulerscher_Darstellung">Impulssatz in eulerscher Darstellung</h3> <a role="button" href="https://de-m-wikipedia-org.translate.goog/w/index.php?title=Cauchy-eulersche_Bewegungsgesetze&action=edit&section=4&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Abschnitt bearbeiten: Impulssatz in eulerscher Darstellung" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> Bearbeiten </a> </div> In der <a href="https://de-m-wikipedia-org.translate.goog/wiki/Eulersche_Betrachtungsweise?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Eulersche Betrachtungsweise">eulerschen Darstellung</a> lautet der globale Impulssatz <dl> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}\int _{v}\rho ({\vec {x}},t){\vec {v}}({\vec {x}},t)\,\mathrm {d} v=\int _{v}\rho ({\vec {x}},t){\vec {k}}({\vec {x}},t)\,\mathrm {d} v+\int _{a}{\vec {t}}({\vec {x}},t)\,\mathrm {d} a}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> t </mi> </mrow> </mfrac> </mrow> <msub> <mo> ∫ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> v </mi> </mrow> </msub> <mi> ρ </mi> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> x </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo> , </mo> <mi> t </mi> <mo stretchy="false"> ) </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> v </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> x </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo> , </mo> <mi> t </mi> <mo stretchy="false"> ) </mo> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> v </mi> <mo> = </mo> <msub> <mo> ∫ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> v </mi> </mrow> </msub> <mi> ρ </mi> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> x </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo> , </mo> <mi> t </mi> <mo stretchy="false"> ) </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> k </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> x </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo> , </mo> <mi> t </mi> <mo stretchy="false"> ) </mo> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> v </mi> <mo> + </mo> <msub> <mo> ∫ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> a </mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> t </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> x </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo> , </mo> <mi> t </mi> <mo stretchy="false"> ) </mo> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> a </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}\int _{v}\rho ({\vec {x}},t){\vec {v}}({\vec {x}},t)\,\mathrm {d} v=\int _{v}\rho ({\vec {x}},t){\vec {k}}({\vec {x}},t)\,\mathrm {d} v+\int _{a}{\vec {t}}({\vec {x}},t)\,\mathrm {d} a} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d823070ebafb1ff3a22771fda6c7d696bf5cd44d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:57.159ex; height:5.843ex;" alt="{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}\int _{v}\rho ({\vec {x}},t){\vec {v}}({\vec {x}},t)\,\mathrm {d} v=\int _{v}\rho ({\vec {x}},t){\vec {k}}({\vec {x}},t)\,\mathrm {d} v+\int _{a}{\vec {t}}({\vec {x}},t)\,\mathrm {d} a}"> </noscript><span class="lazy-image-placeholder" style="width: 57.159ex;height: 5.843ex;vertical-align: -2.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d823070ebafb1ff3a22771fda6c7d696bf5cd44d" data-alt="{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}\int _{v}\rho ({\vec {x}},t){\vec {v}}({\vec {x}},t)\,\mathrm {d} v=\int _{v}\rho ({\vec {x}},t){\vec {k}}({\vec {x}},t)\,\mathrm {d} v+\int _{a}{\vec {t}}({\vec {x}},t)\,\mathrm {d} a}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </dd> </dl> Die räumlichen Punkte werden durch ihre räumlichen Koordinaten <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {x}}\in v}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> x </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo> ∈ </mo> <mi> v </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\vec {x}}\in v} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ce305701955b60bab06e8605008e879a1be5844" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.298ex; height:2.343ex;" alt="{\displaystyle {\vec {x}}\in v}"> </noscript><span class="lazy-image-placeholder" style="width: 5.298ex;height: 2.343ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ce305701955b60bab06e8605008e879a1be5844" data-alt="{\displaystyle {\vec {x}}\in v}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  in dem momentanen Volumen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> v </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle v} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e07b00e7fc0847fbd16391c778d65bc25c452597" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}"> </noscript><span class="lazy-image-placeholder" style="width: 1.128ex;height: 1.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e07b00e7fc0847fbd16391c778d65bc25c452597" data-alt="{\displaystyle v}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  zur Zeit t identifiziert, siehe <a href="https://de-m-wikipedia-org.translate.goog/wiki/Kontinuumsmechanik?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Impulsbilanz" title="Kontinuumsmechanik">Impulsbilanz</a>. Anders als in der lagrangeschen Darstellung sind die Integrationsgrenzen als Oberflächen des Körpers von der Zeit abhängig, was bei der Berechnung der Impulsänderung zu berücksichtigen ist. Nach dem <a href="https://de-m-wikipedia-org.translate.goog/wiki/Reynoldsscher_Transportsatz?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Reynoldsscher Transportsatz">reynoldsschen Transportsatz</a> gilt: <dl> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}{\frac {\mathrm {d} }{\mathrm {d} t}}\int _{v}\rho ({\vec {x}},t){\vec {v}}({\vec {x}},t)\,\mathrm {d} v=&\int _{v}{\frac {\mathrm {d} }{\mathrm {d} t}}(\rho {\vec {v}})\,\mathrm {d} v+\int _{a}\rho {\vec {v}}({\vec {v}}\cdot \mathrm {d} {\vec {a}})=\int _{v}\left[{\frac {\mathrm {d} }{\mathrm {d} t}}(\rho {\vec {v}})+\operatorname {div} ({\vec {v}})\rho {\vec {v}}\right]\,\mathrm {d} v\\=&\int _{v}{\big (}\underbrace {{\dot {\rho }}{\vec {v}}+\operatorname {div} ({\vec {v}})\rho {\vec {v}}} _{={\vec {0}}}+\rho {\dot {\vec {v}}}{\big )}\,\mathrm {d} v=\int _{v}\rho {\dot {\vec {v}}}\,\mathrm {d} v\end{aligned}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> t </mi> </mrow> </mfrac> </mrow> <msub> <mo> ∫ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> v </mi> </mrow> </msub> <mi> ρ </mi> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> x </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo> , </mo> <mi> t </mi> <mo stretchy="false"> ) </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> v </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> x </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo> , </mo> <mi> t </mi> <mo stretchy="false"> ) </mo> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> v </mi> <mo> = </mo> </mtd> <mtd> <mi></mi> <msub> <mo> ∫ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> v </mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> t </mi> </mrow> </mfrac> </mrow> <mo stretchy="false"> ( </mo> <mi> ρ </mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> v </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo stretchy="false"> ) </mo> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> v </mi> <mo> + </mo> <msub> <mo> ∫ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> a </mi> </mrow> </msub> <mi> ρ </mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> v </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> v </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo> ⋅ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> a </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo stretchy="false"> ) </mo> <mo> = </mo> <msub> <mo> ∫ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> v </mi> </mrow> </msub> <mrow> <mo> [ </mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> t </mi> </mrow> </mfrac> </mrow> <mo stretchy="false"> ( </mo> <mi> ρ </mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> v </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo stretchy="false"> ) </mo> <mo> + </mo> <mi> div </mi> <mo> ⁡ </mo> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> v </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo stretchy="false"> ) </mo> <mi> ρ </mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> v </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> </mrow> <mo> ] </mo> </mrow> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> v </mi> </mtd> </mtr> <mtr> <mtd> <mo> = </mo> </mtd> <mtd> <mi></mi> <msub> <mo> ∫ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> v </mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em"> ( </mo> </mrow> </mrow> <munder> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <munder> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> ρ </mi> <mo> ˙ </mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> v </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo> + </mo> <mi> div </mi> <mo> ⁡ </mo> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> v </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo stretchy="false"> ) </mo> <mi> ρ </mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> v </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> </mrow> <mo> ⏟ </mo> </munder> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mn> 0 </mn> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> </mrow> </munder> <mo> + </mo> <mi> ρ </mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> v </mi> <mo stretchy="false"> → </mo> </mover> </mrow> <mo> ˙ </mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em"> ) </mo> </mrow> </mrow> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> v </mi> <mo> = </mo> <msub> <mo> ∫ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> v </mi> </mrow> </msub> <mi> ρ </mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> v </mi> <mo stretchy="false"> → </mo> </mover> </mrow> <mo> ˙ </mo> </mover> </mrow> </mrow> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> v </mi> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\begin{aligned}{\frac {\mathrm {d} }{\mathrm {d} t}}\int _{v}\rho ({\vec {x}},t){\vec {v}}({\vec {x}},t)\,\mathrm {d} v=&\int _{v}{\frac {\mathrm {d} }{\mathrm {d} t}}(\rho {\vec {v}})\,\mathrm {d} v+\int _{a}\rho {\vec {v}}({\vec {v}}\cdot \mathrm {d} {\vec {a}})=\int _{v}\left[{\frac {\mathrm {d} }{\mathrm {d} t}}(\rho {\vec {v}})+\operatorname {div} ({\vec {v}})\rho {\vec {v}}\right]\,\mathrm {d} v\\=&\int _{v}{\big (}\underbrace {{\dot {\rho }}{\vec {v}}+\operatorname {div} ({\vec {v}})\rho {\vec {v}}} _{={\vec {0}}}+\rho {\dot {\vec {v}}}{\big )}\,\mathrm {d} v=\int _{v}\rho {\dot {\vec {v}}}\,\mathrm {d} v\end{aligned}}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6034e4d944d39058c3da1796baf8bec87e1868f4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.671ex; width:82.704ex; height:14.509ex;" alt="{\displaystyle {\begin{aligned}{\frac {\mathrm {d} }{\mathrm {d} t}}\int _{v}\rho ({\vec {x}},t){\vec {v}}({\vec {x}},t)\,\mathrm {d} v=&\int _{v}{\frac {\mathrm {d} }{\mathrm {d} t}}(\rho {\vec {v}})\,\mathrm {d} v+\int _{a}\rho {\vec {v}}({\vec {v}}\cdot \mathrm {d} {\vec {a}})=\int _{v}\left[{\frac {\mathrm {d} }{\mathrm {d} t}}(\rho {\vec {v}})+\operatorname {div} ({\vec {v}})\rho {\vec {v}}\right]\,\mathrm {d} v\\=&\int _{v}{\big (}\underbrace {{\dot {\rho }}{\vec {v}}+\operatorname {div} ({\vec {v}})\rho {\vec {v}}} _{={\vec {0}}}+\rho {\dot {\vec {v}}}{\big )}\,\mathrm {d} v=\int _{v}\rho {\dot {\vec {v}}}\,\mathrm {d} v\end{aligned}}}"> </noscript><span class="lazy-image-placeholder" style="width: 82.704ex;height: 14.509ex;vertical-align: -6.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6034e4d944d39058c3da1796baf8bec87e1868f4" data-alt="{\displaystyle {\begin{aligned}{\frac {\mathrm {d} }{\mathrm {d} t}}\int _{v}\rho ({\vec {x}},t){\vec {v}}({\vec {x}},t)\,\mathrm {d} v=&\int _{v}{\frac {\mathrm {d} }{\mathrm {d} t}}(\rho {\vec {v}})\,\mathrm {d} v+\int _{a}\rho {\vec {v}}({\vec {v}}\cdot \mathrm {d} {\vec {a}})=\int _{v}\left[{\frac {\mathrm {d} }{\mathrm {d} t}}(\rho {\vec {v}})+\operatorname {div} ({\vec {v}})\rho {\vec {v}}\right]\,\mathrm {d} v\\=&\int _{v}{\big (}\underbrace {{\dot {\rho }}{\vec {v}}+\operatorname {div} ({\vec {v}})\rho {\vec {v}}} _{={\vec {0}}}+\rho {\dot {\vec {v}}}{\big )}\,\mathrm {d} v=\int _{v}\rho {\dot {\vec {v}}}\,\mathrm {d} v\end{aligned}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </dd> </dl> Der aufgesetzte Punkt steht für die substantielle Ableitung und in der ersten Zeile wurde das Oberflächenintegral mit dem gaußschen Integralsatz in ein Volumenintegral überführt. Der unterklammerte Term trägt auf Grund der lokalen <a href="https://de-m-wikipedia-org.translate.goog/wiki/Kontinuumsmechanik?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Massenbilanz" title="Kontinuumsmechanik">Massenbilanz</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\dot {\rho }}+\rho \operatorname {div} {\vec {v}}=0}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> ρ </mi> <mo> ˙ </mo> </mover> </mrow> </mrow> <mo> + </mo> <mi> ρ </mi> <mi> div </mi> <mo> ⁡ </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> v </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo> = </mo> <mn> 0 </mn> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\dot {\rho }}+\rho \operatorname {div} {\vec {v}}=0} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e1a2243877581cd756c08dcb9ea91e369d5a14df" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.795ex; height:2.843ex;" alt="{\displaystyle {\dot {\rho }}+\rho \operatorname {div} {\vec {v}}=0}"> </noscript><span class="lazy-image-placeholder" style="width: 14.795ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e1a2243877581cd756c08dcb9ea91e369d5a14df" data-alt="{\displaystyle {\dot {\rho }}+\rho \operatorname {div} {\vec {v}}=0}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  in der eulerschen Darstellung nichts bei. Das Oberflächenintegral der von außen angreifenden Spannungen wird wie in der lagrangeschen Darstellung mit dem gaußschen Integralsatz in ein Volumenintegral überführt: <dl> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{a}{\vec {t}}\,\mathrm {d} a=\int _{a}{\boldsymbol {\sigma }}^{\top }\cdot {\vec {n}}\,\mathrm {d} a=\int _{v}\nabla \cdot {\boldsymbol {\sigma }}\,\mathrm {d} v=\int _{v}\operatorname {div} ({\boldsymbol {\sigma }})\,\mathrm {d} v}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mo> ∫ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> a </mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> t </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> a </mi> <mo> = </mo> <msub> <mo> ∫ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> a </mi> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> σ </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> ⊤ </mi> </mrow> </msup> <mo> ⋅ </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> n </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> a </mi> <mo> = </mo> <msub> <mo> ∫ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> v </mi> </mrow> </msub> <mi mathvariant="normal"> ∇ </mi> <mo> ⋅ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> σ </mi> </mrow> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> v </mi> <mo> = </mo> <msub> <mo> ∫ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> v </mi> </mrow> </msub> <mi> div </mi> <mo> ⁡ </mo> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> σ </mi> </mrow> <mo stretchy="false"> ) </mo> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> v </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \int _{a}{\vec {t}}\,\mathrm {d} a=\int _{a}{\boldsymbol {\sigma }}^{\top }\cdot {\vec {n}}\,\mathrm {d} a=\int _{v}\nabla \cdot {\boldsymbol {\sigma }}\,\mathrm {d} v=\int _{v}\operatorname {div} ({\boldsymbol {\sigma }})\,\mathrm {d} v} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d36b88c15ba8381dca696dc70be734f97a1366b4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:50.957ex; height:5.676ex;" alt="{\displaystyle \int _{a}{\vec {t}}\,\mathrm {d} a=\int _{a}{\boldsymbol {\sigma }}^{\top }\cdot {\vec {n}}\,\mathrm {d} a=\int _{v}\nabla \cdot {\boldsymbol {\sigma }}\,\mathrm {d} v=\int _{v}\operatorname {div} ({\boldsymbol {\sigma }})\,\mathrm {d} v}"> </noscript><span class="lazy-image-placeholder" style="width: 50.957ex;height: 5.676ex;vertical-align: -2.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d36b88c15ba8381dca696dc70be734f97a1366b4" data-alt="{\displaystyle \int _{a}{\vec {t}}\,\mathrm {d} a=\int _{a}{\boldsymbol {\sigma }}^{\top }\cdot {\vec {n}}\,\mathrm {d} a=\int _{v}\nabla \cdot {\boldsymbol {\sigma }}\,\mathrm {d} v=\int _{v}\operatorname {div} ({\boldsymbol {\sigma }})\,\mathrm {d} v}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </dd> </dl> denn der cauchysche Spannungstensor σ ist wegen des zweiten cauchy-eulerschen Bewegungsgesetzes unten <a href="https://de-m-wikipedia-org.translate.goog/wiki/Symmetrische_Matrix?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Symmetrische Matrix">symmetrisch</a>. Der <a href="https://de-m-wikipedia-org.translate.goog/wiki/Nabla-Operator?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Nabla-Operator">Nabla-Operator</a> 𝜵 und der <a href="https://de-m-wikipedia-org.translate.goog/wiki/Divergenz_eines_Vektorfeldes?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Divergenz eines Vektorfeldes">Divergenzoperator</a> div beinhalten die räumlichen Ableitungen nach den räumlichen Koordinaten <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {x}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> x </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\vec {x}}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/db2dc6ced9cc3bc7e8b9f2707cbec033f6d3759c" 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}}}"> </noscript><span class="lazy-image-placeholder" style="width: 1.33ex;height: 2.343ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/db2dc6ced9cc3bc7e8b9f2707cbec033f6d3759c" data-alt="{\displaystyle {\vec {x}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> . Mit den vorliegenden Ergebnissen kann die <a href="https://de-m-wikipedia-org.translate.goog/wiki/Cauchy-eulersche_Bewegungsgesetze?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Impulssatz_in_eulerscher_Darstellung">Impulsbilanz</a> als verschwindendes Volumenintegral ausgedrückt werden: <dl> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{v}(\rho {\dot {\vec {v}}}-\rho {\vec {k}}-\nabla \cdot {\boldsymbol {\sigma }})\,\mathrm {d} v={\vec {0}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mo> ∫ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> v </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <mi> ρ </mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> v </mi> <mo stretchy="false"> → </mo> </mover> </mrow> <mo> ˙ </mo> </mover> </mrow> </mrow> <mo> − </mo> <mi> ρ </mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> k </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo> − </mo> <mi mathvariant="normal"> ∇ </mi> <mo> ⋅ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> σ </mi> </mrow> <mo stretchy="false"> ) </mo> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> v </mi> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mn> 0 </mn> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \int _{v}(\rho {\dot {\vec {v}}}-\rho {\vec {k}}-\nabla \cdot {\boldsymbol {\sigma }})\,\mathrm {d} v={\vec {0}}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4e6598e0fe29a617c43e2546f845423697fc35ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:26.915ex; height:5.676ex;" alt="{\displaystyle \int _{v}(\rho {\dot {\vec {v}}}-\rho {\vec {k}}-\nabla \cdot {\boldsymbol {\sigma }})\,\mathrm {d} v={\vec {0}}}"> </noscript><span class="lazy-image-placeholder" style="width: 26.915ex;height: 5.676ex;vertical-align: -2.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4e6598e0fe29a617c43e2546f845423697fc35ba" data-alt="{\displaystyle \int _{v}(\rho {\dot {\vec {v}}}-\rho {\vec {k}}-\nabla \cdot {\boldsymbol {\sigma }})\,\mathrm {d} v={\vec {0}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </dd> </dl> Diese Gleichung gilt für jedes Volumen, sodass – Stetigkeit des Integranden vorausgesetzt – das erste cauchy-eulersche Bewegungsgesetz in der eulerschen Darstellung <dl> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho ({\vec {x}},t)\,{\dot {\vec {v}}}({\vec {x}},t)=\rho ({\vec {x}},t)\,{\vec {k}}({\vec {x}},t)+\nabla \cdot {\boldsymbol {\sigma }}({\vec {x}},t)}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> ρ </mi> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> x </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo> , </mo> <mi> t </mi> <mo stretchy="false"> ) </mo> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> v </mi> <mo stretchy="false"> → </mo> </mover> </mrow> <mo> ˙ </mo> </mover> </mrow> </mrow> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> x </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo> , </mo> <mi> t </mi> <mo stretchy="false"> ) </mo> <mo> = </mo> <mi> ρ </mi> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> x </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo> , </mo> <mi> t </mi> <mo stretchy="false"> ) </mo> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> k </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> x </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo> , </mo> <mi> t </mi> <mo stretchy="false"> ) </mo> <mo> + </mo> <mi mathvariant="normal"> ∇ </mi> <mo> ⋅ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> σ </mi> </mrow> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> x </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo> , </mo> <mi> t </mi> <mo stretchy="false"> ) </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \rho ({\vec {x}},t)\,{\dot {\vec {v}}}({\vec {x}},t)=\rho ({\vec {x}},t)\,{\vec {k}}({\vec {x}},t)+\nabla \cdot {\boldsymbol {\sigma }}({\vec {x}},t)} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cb67c8ea8d53c2914477197c1cd63594ed0e06cb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:41.81ex; height:3.343ex;" alt="{\displaystyle \rho ({\vec {x}},t)\,{\dot {\vec {v}}}({\vec {x}},t)=\rho ({\vec {x}},t)\,{\vec {k}}({\vec {x}},t)+\nabla \cdot {\boldsymbol {\sigma }}({\vec {x}},t)}"> </noscript><span class="lazy-image-placeholder" style="width: 41.81ex;height: 3.343ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cb67c8ea8d53c2914477197c1cd63594ed0e06cb" data-alt="{\displaystyle \rho ({\vec {x}},t)\,{\dot {\vec {v}}}({\vec {x}},t)=\rho ({\vec {x}},t)\,{\vec {k}}({\vec {x}},t)+\nabla \cdot {\boldsymbol {\sigma }}({\vec {x}},t)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </dd> </dl> abgeleitet werden kann. Hier ist die substantielle Zeitableitung der Geschwindigkeit bei festgehaltenem Partikel <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {X}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> X </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\vec {X}}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5dc35b5a0226cf11a2c3f2d2dbbac6ab5ade6036" 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}}}"> </noscript><span class="lazy-image-placeholder" style="width: 1.98ex;height: 2.843ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5dc35b5a0226cf11a2c3f2d2dbbac6ab5ade6036" data-alt="{\displaystyle {\vec {X}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  zu bilden, das sich zur Zeit t am Ort <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {x}}={\vec {\chi }}({\vec {X}},t)}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> x </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> χ </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> X </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo> , </mo> <mi> t </mi> <mo stretchy="false"> ) </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\vec {x}}={\vec {\chi }}({\vec {X}},t)} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9e92bff89d59a995104a9f1d246741c880d1b2b5" 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)}"> </noscript><span class="lazy-image-placeholder" style="width: 11.546ex;height: 3.343ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9e92bff89d59a995104a9f1d246741c880d1b2b5" data-alt="{\displaystyle {\vec {x}}={\vec {\chi }}({\vec {X}},t)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  befindet und dort die Geschwindigkeit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {v}}({\vec {x}},t)={\dot {\vec {\chi }}}({\vec {X}},t)}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> v </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> x </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo> , </mo> <mi> t </mi> <mo stretchy="false"> ) </mo> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> χ </mi> <mo stretchy="false"> → </mo> </mover> </mrow> <mo> ˙ </mo> </mover> </mrow> </mrow> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> X </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo> , </mo> <mi> t </mi> <mo stretchy="false"> ) </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\vec {v}}({\vec {x}},t)={\dot {\vec {\chi }}}({\vec {X}},t)} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4bc75511fedd7635904ff72ac3c7481227b21622" 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)}"> </noscript><span class="lazy-image-placeholder" style="width: 16.404ex;height: 3.343ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4bc75511fedd7635904ff72ac3c7481227b21622" data-alt="{\displaystyle {\vec {v}}({\vec {x}},t)={\dot {\vec {\chi }}}({\vec {X}},t)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  besitzt:<a href="https://de-m-wikipedia-org.translate.goog/wiki/Cauchy-eulersche_Bewegungsgesetze?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-Frechet-2">[F 1]</a> <dl> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}{\dot {\vec {v}}}({\vec {x}},t):=&\left.{\frac {\mathrm {d} }{\mathrm {d} t}}{\vec {v}}({\vec {\chi }}({\vec {X}},t),t)\right|_{{\vec {X}}\;{\text{fest}}}={\frac {\partial }{\partial {\vec {x}}}}{\vec {v}}({\vec {x}},t)\cdot {\dot {\vec {\chi }}}({\vec {X}},t)+{\frac {\partial }{\partial t}}{\vec {v}}({\vec {x}},t)\\=&\operatorname {grad} ({\vec {v}})\cdot {\vec {v}}+{\frac {\partial {\vec {v}}}{\partial t}}\end{aligned}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> v </mi> <mo stretchy="false"> → </mo> </mover> </mrow> <mo> ˙ </mo> </mover> </mrow> </mrow> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> x </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo> , </mo> <mi> t </mi> <mo stretchy="false"> ) </mo> <mo> := </mo> </mtd> <mtd> <msub> <mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> t </mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> v </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> χ </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> X </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo> , </mo> <mi> t </mi> <mo stretchy="false"> ) </mo> <mo> , </mo> <mi> t </mi> <mo stretchy="false"> ) </mo> </mrow> <mo> | </mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> X </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mspace width="thickmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mtext> fest </mtext> </mrow> </mrow> </msub> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal"> ∂ </mi> <mrow> <mi mathvariant="normal"> ∂ </mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> x </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> v </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> x </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo> , </mo> <mi> t </mi> <mo stretchy="false"> ) </mo> <mo> ⋅ </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> χ </mi> <mo stretchy="false"> → </mo> </mover> </mrow> <mo> ˙ </mo> </mover> </mrow> </mrow> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> X </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo> , </mo> <mi> t </mi> <mo stretchy="false"> ) </mo> <mo> + </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal"> ∂ </mi> <mrow> <mi mathvariant="normal"> ∂ </mi> <mi> t </mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> v </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> x </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo> , </mo> <mi> t </mi> <mo stretchy="false"> ) </mo> </mtd> </mtr> <mtr> <mtd> <mo> = </mo> </mtd> <mtd> <mi> grad </mi> <mo> ⁡ </mo> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> v </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo stretchy="false"> ) </mo> <mo> ⋅ </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> v </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo> + </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal"> ∂ </mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> v </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> </mrow> <mrow> <mi mathvariant="normal"> ∂ </mi> <mi> t </mi> </mrow> </mfrac> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\begin{aligned}{\dot {\vec {v}}}({\vec {x}},t):=&\left.{\frac {\mathrm {d} }{\mathrm {d} t}}{\vec {v}}({\vec {\chi }}({\vec {X}},t),t)\right|_{{\vec {X}}\;{\text{fest}}}={\frac {\partial }{\partial {\vec {x}}}}{\vec {v}}({\vec {x}},t)\cdot {\dot {\vec {\chi }}}({\vec {X}},t)+{\frac {\partial }{\partial t}}{\vec {v}}({\vec {x}},t)\\=&\operatorname {grad} ({\vec {v}})\cdot {\vec {v}}+{\frac {\partial {\vec {v}}}{\partial t}}\end{aligned}}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f92705ed12b1dcfcbad60aea7558c788bf79d37b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.988ex; margin-bottom: -0.183ex; width:64.053ex; height:11.509ex;" alt="{\displaystyle {\begin{aligned}{\dot {\vec {v}}}({\vec {x}},t):=&\left.{\frac {\mathrm {d} }{\mathrm {d} t}}{\vec {v}}({\vec {\chi }}({\vec {X}},t),t)\right|_{{\vec {X}}\;{\text{fest}}}={\frac {\partial }{\partial {\vec {x}}}}{\vec {v}}({\vec {x}},t)\cdot {\dot {\vec {\chi }}}({\vec {X}},t)+{\frac {\partial }{\partial t}}{\vec {v}}({\vec {x}},t)\\=&\operatorname {grad} ({\vec {v}})\cdot {\vec {v}}+{\frac {\partial {\vec {v}}}{\partial t}}\end{aligned}}}"> </noscript><span class="lazy-image-placeholder" style="width: 64.053ex;height: 11.509ex;vertical-align: -4.988ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f92705ed12b1dcfcbad60aea7558c788bf79d37b" data-alt="{\displaystyle {\begin{aligned}{\dot {\vec {v}}}({\vec {x}},t):=&\left.{\frac {\mathrm {d} }{\mathrm {d} t}}{\vec {v}}({\vec {\chi }}({\vec {X}},t),t)\right|_{{\vec {X}}\;{\text{fest}}}={\frac {\partial }{\partial {\vec {x}}}}{\vec {v}}({\vec {x}},t)\cdot {\dot {\vec {\chi }}}({\vec {X}},t)+{\frac {\partial }{\partial t}}{\vec {v}}({\vec {x}},t)\\=&\operatorname {grad} ({\vec {v}})\cdot {\vec {v}}+{\frac {\partial {\vec {v}}}{\partial t}}\end{aligned}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </dd> </dl> Der räumliche Operator grad berechnet den räumlichen <a href="https://de-m-wikipedia-org.translate.goog/wiki/Geschwindigkeitsgradient?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Geschwindigkeitsgradient">Geschwindigkeitsgradienten</a> mit Ableitungen nach den räumlichen Koordinaten x1,2,3. Der <a href="https://de-m-wikipedia-org.translate.goog/wiki/Konvektion?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Konvektion">konvektive Anteil</a> <dl> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {grad} ({\vec {v}})\cdot {\vec {v}}:=(\nabla \otimes {\vec {v}})^{\top }\cdot {\vec {v}}={\vec {v}}\cdot (\nabla \otimes {\vec {v}})=({\vec {v}}\cdot \nabla ){\vec {v}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> grad </mi> <mo> ⁡ </mo> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> v </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo stretchy="false"> ) </mo> <mo> ⋅ </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> v </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo> := </mo> <mo stretchy="false"> ( </mo> <mi mathvariant="normal"> ∇ </mi> <mo> ⊗ </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> v </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <msup> <mo stretchy="false"> ) </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> ⊤ </mi> </mrow> </msup> <mo> ⋅ </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> v </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> v </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo> ⋅ </mo> <mo stretchy="false"> ( </mo> <mi mathvariant="normal"> ∇ </mi> <mo> ⊗ </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> v </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo stretchy="false"> ) </mo> <mo> = </mo> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> v </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo> ⋅ </mo> <mi mathvariant="normal"> ∇ </mi> <mo stretchy="false"> ) </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> v </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \operatorname {grad} ({\vec {v}})\cdot {\vec {v}}:=(\nabla \otimes {\vec {v}})^{\top }\cdot {\vec {v}}={\vec {v}}\cdot (\nabla \otimes {\vec {v}})=({\vec {v}}\cdot \nabla ){\vec {v}}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/05e16be2f6ecab2ec54288ce9ed188538be34753" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:50.827ex; height:3.176ex;" alt="{\displaystyle \operatorname {grad} ({\vec {v}})\cdot {\vec {v}}:=(\nabla \otimes {\vec {v}})^{\top }\cdot {\vec {v}}={\vec {v}}\cdot (\nabla \otimes {\vec {v}})=({\vec {v}}\cdot \nabla ){\vec {v}}}"> </noscript><span class="lazy-image-placeholder" style="width: 50.827ex;height: 3.176ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/05e16be2f6ecab2ec54288ce9ed188538be34753" data-alt="{\displaystyle \operatorname {grad} ({\vec {v}})\cdot {\vec {v}}:=(\nabla \otimes {\vec {v}})^{\top }\cdot {\vec {v}}={\vec {v}}\cdot (\nabla \otimes {\vec {v}})=({\vec {v}}\cdot \nabla ){\vec {v}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </dd> </dl> in der substantiellen Beschleunigung berücksichtigt das Hindurchfließen des Materials durch das bei der Betrachtung am Volumenelement oben festgehaltene Volumen V bei großen Verschiebungen. Bei kleinen Verschiebungen kann der quadratische konvektive Anteil vernachlässigt werden, sodass mit <dl> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {a}}={\dot {\vec {v}}}={\frac {\partial {\vec {v}}}{\partial t}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> a </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> v </mi> <mo stretchy="false"> → </mo> </mover> </mrow> <mo> ˙ </mo> </mover> </mrow> </mrow> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal"> ∂ </mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> v </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> </mrow> <mrow> <mi mathvariant="normal"> ∂ </mi> <mi> t </mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\vec {a}}={\dot {\vec {v}}}={\frac {\partial {\vec {v}}}{\partial t}}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6c533bcde1b3fa7c75de461994c7a35730592890" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:11.966ex; height:5.509ex;" alt="{\displaystyle {\vec {a}}={\dot {\vec {v}}}={\frac {\partial {\vec {v}}}{\partial t}}}"> </noscript><span class="lazy-image-placeholder" style="width: 11.966ex;height: 5.509ex;vertical-align: -2.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6c533bcde1b3fa7c75de461994c7a35730592890" data-alt="{\displaystyle {\vec {a}}={\dot {\vec {v}}}={\frac {\partial {\vec {v}}}{\partial t}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </dd> </dl> das eingangs angegebene Bewegungsgesetz entsteht. <div class="mw-heading mw-heading3"> <h3 id="Einfluss_von_Sprungstellen_im_Impulssatz">Einfluss von Sprungstellen im Impulssatz</h3> <a role="button" href="https://de-m-wikipedia-org.translate.goog/w/index.php?title=Cauchy-eulersche_Bewegungsgesetze&action=edit&section=5&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Abschnitt bearbeiten: Einfluss von Sprungstellen im Impulssatz" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> Bearbeiten </a> </div> <figure class="mw-default-size" typeof="mw:File/Thumb"> <a href="https://de-m-wikipedia-org.translate.goog/wiki/Datei:Diskontinuitaet.png?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-file-description"> <noscript> <img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b6/Diskontinuitaet.png/220px-Diskontinuitaet.png" decoding="async" width="220" height="289" class="mw-file-element" data-file-width="367" data-file-height="482"> </noscript><span class="lazy-image-placeholder" style="width: 220px;height: 289px;" data-src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b6/Diskontinuitaet.png/220px-Diskontinuitaet.png" data-width="220" data-height="289" data-srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b6/Diskontinuitaet.png/330px-Diskontinuitaet.png 1.5x, //upload.wikimedia.org/wikipedia/commons/b/b6/Diskontinuitaet.png 2x" data-class="mw-file-element"> </a> <figcaption> Eine Sprungstelle auf der Fläche as trennt zwei Raumbereiche v+ und v− </figcaption> </figure> Die verlangte örtliche Stetigkeit der Integranden wird unter realen Verhältnissen verletzt, wenn beispielsweise <a href="https://de-m-wikipedia-org.translate.goog/wiki/Dichte?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Dichte">Dichte</a>sprünge an Materialgrenzen oder <a href="https://de-m-wikipedia-org.translate.goog/wiki/Sto%C3%9Fwelle?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Stoßwelle">Stoßwellen</a> auftreten. Solche flächigen Sprungstellen können jedoch berücksichtigt werden, wenn die Fläche selbst örtlich stetig differenzierbar ist und so in jedem ihrer Punkte einen <a href="https://de-m-wikipedia-org.translate.goog/wiki/Normalenvektor?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Normalenvektor">Normalenvektor</a> besitzt. Die Fläche – im Folgenden Sprungstelle genannt – muss keine materielle Fläche sein, kann sich also mit einer anderen Geschwindigkeit bewegen als die Masse selbst. Durch diese Fläche wird die Masse in zwei Stücke v+ und v− geteilt und es wird vereinbart, dass der Normalenvektor der Sprungstelle as in Richtung der Sprungstellengeschwindigkeit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {v}}_{s}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> v </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> s </mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\vec {v}}_{s}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf2e35aa3a72fe7aff65a3d0684235219174053e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.179ex; height:2.676ex;" alt="{\displaystyle {\vec {v}}_{s}}"> </noscript><span class="lazy-image-placeholder" style="width: 2.179ex;height: 2.676ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf2e35aa3a72fe7aff65a3d0684235219174053e" data-alt="{\displaystyle {\vec {v}}_{s}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  und das Volumen v+ weise, siehe Bild rechts. Dann lautet das Reynolds-Transport-Theorem mit Sprungstelle:<a href="https://de-m-wikipedia-org.translate.goog/wiki/Cauchy-eulersche_Bewegungsgesetze?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-3">[L 2]</a> <dl> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}{\frac {\mathrm {d} }{\mathrm {d} t}}\int _{v}\rho {\vec {v}}\,\mathrm {d} v=&\int _{v}{\frac {\partial (\rho {\vec {v}})}{\partial t}}\,\mathrm {d} v+\int _{a}\rho {\vec {v}}\;({\vec {v}}\cdot \,\mathrm {d} {\vec {a}})+\int _{a_{s}}[[\rho {\vec {v}}\;(({\vec {v}}_{s}-{\vec {v}})\cdot {\vec {n}})]]\mathrm {d} a_{s}\\=&\int _{v}\rho {\dot {\vec {v}}}\,\mathrm {d} v+\int _{a_{s}}[[\rho {\vec {v}}\;(({\vec {v}}_{s}-{\vec {v}})\cdot {\vec {n}})]]\mathrm {d} a_{s}\end{aligned}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> t </mi> </mrow> </mfrac> </mrow> <msub> <mo> ∫ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> v </mi> </mrow> </msub> <mi> ρ </mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> v </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> v </mi> <mo> = </mo> </mtd> <mtd> <mi></mi> <msub> <mo> ∫ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> v </mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal"> ∂ </mi> <mo stretchy="false"> ( </mo> <mi> ρ </mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> v </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo stretchy="false"> ) </mo> </mrow> <mrow> <mi mathvariant="normal"> ∂ </mi> <mi> t </mi> </mrow> </mfrac> </mrow> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> v </mi> <mo> + </mo> <msub> <mo> ∫ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> a </mi> </mrow> </msub> <mi> ρ </mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> v </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mspace width="thickmathspace"></mspace> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> v </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo> ⋅ </mo> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> a </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo stretchy="false"> ) </mo> <mo> + </mo> <msub> <mo> ∫ </mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi> a </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> s </mi> </mrow> </msub> </mrow> </msub> <mo stretchy="false"> [ </mo> <mo stretchy="false"> [ </mo> <mi> ρ </mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> v </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mspace width="thickmathspace"></mspace> <mo stretchy="false"> ( </mo> <mo stretchy="false"> ( </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> v </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> s </mi> </mrow> </msub> <mo> − </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> v </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo stretchy="false"> ) </mo> <mo> ⋅ </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> n </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo stretchy="false"> ) </mo> <mo stretchy="false"> ] </mo> <mo stretchy="false"> ] </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <msub> <mi> a </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> s </mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mo> = </mo> </mtd> <mtd> <mi></mi> <msub> <mo> ∫ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> v </mi> </mrow> </msub> <mi> ρ </mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> v </mi> <mo stretchy="false"> → </mo> </mover> </mrow> <mo> ˙ </mo> </mover> </mrow> </mrow> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> v </mi> <mo> + </mo> <msub> <mo> ∫ </mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi> a </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> s </mi> </mrow> </msub> </mrow> </msub> <mo stretchy="false"> [ </mo> <mo stretchy="false"> [ </mo> <mi> ρ </mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> v </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mspace width="thickmathspace"></mspace> <mo stretchy="false"> ( </mo> <mo stretchy="false"> ( </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> v </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> s </mi> </mrow> </msub> <mo> − </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> v </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo stretchy="false"> ) </mo> <mo> ⋅ </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> n </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo stretchy="false"> ) </mo> <mo stretchy="false"> ] </mo> <mo stretchy="false"> ] </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <msub> <mi> a </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> s </mi> </mrow> </msub> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\begin{aligned}{\frac {\mathrm {d} }{\mathrm {d} t}}\int _{v}\rho {\vec {v}}\,\mathrm {d} v=&\int _{v}{\frac {\partial (\rho {\vec {v}})}{\partial t}}\,\mathrm {d} v+\int _{a}\rho {\vec {v}}\;({\vec {v}}\cdot \,\mathrm {d} {\vec {a}})+\int _{a_{s}}[[\rho {\vec {v}}\;(({\vec {v}}_{s}-{\vec {v}})\cdot {\vec {n}})]]\mathrm {d} a_{s}\\=&\int _{v}\rho {\dot {\vec {v}}}\,\mathrm {d} v+\int _{a_{s}}[[\rho {\vec {v}}\;(({\vec {v}}_{s}-{\vec {v}})\cdot {\vec {n}})]]\mathrm {d} a_{s}\end{aligned}}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2440586bf727130838b65e9d94d67771ef358172" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.671ex; width:70.963ex; height:12.509ex;" alt="{\displaystyle {\begin{aligned}{\frac {\mathrm {d} }{\mathrm {d} t}}\int _{v}\rho {\vec {v}}\,\mathrm {d} v=&\int _{v}{\frac {\partial (\rho {\vec {v}})}{\partial t}}\,\mathrm {d} v+\int _{a}\rho {\vec {v}}\;({\vec {v}}\cdot \,\mathrm {d} {\vec {a}})+\int _{a_{s}}[[\rho {\vec {v}}\;(({\vec {v}}_{s}-{\vec {v}})\cdot {\vec {n}})]]\mathrm {d} a_{s}\\=&\int _{v}\rho {\dot {\vec {v}}}\,\mathrm {d} v+\int _{a_{s}}[[\rho {\vec {v}}\;(({\vec {v}}_{s}-{\vec {v}})\cdot {\vec {n}})]]\mathrm {d} a_{s}\end{aligned}}}"> </noscript><span class="lazy-image-placeholder" style="width: 70.963ex;height: 12.509ex;vertical-align: -5.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2440586bf727130838b65e9d94d67771ef358172" data-alt="{\displaystyle {\begin{aligned}{\frac {\mathrm {d} }{\mathrm {d} t}}\int _{v}\rho {\vec {v}}\,\mathrm {d} v=&\int _{v}{\frac {\partial (\rho {\vec {v}})}{\partial t}}\,\mathrm {d} v+\int _{a}\rho {\vec {v}}\;({\vec {v}}\cdot \,\mathrm {d} {\vec {a}})+\int _{a_{s}}[[\rho {\vec {v}}\;(({\vec {v}}_{s}-{\vec {v}})\cdot {\vec {n}})]]\mathrm {d} a_{s}\\=&\int _{v}\rho {\dot {\vec {v}}}\,\mathrm {d} v+\int _{a_{s}}[[\rho {\vec {v}}\;(({\vec {v}}_{s}-{\vec {v}})\cdot {\vec {n}})]]\mathrm {d} a_{s}\end{aligned}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </dd> </dl> Der zweite Term mit der <a href="https://de-m-wikipedia-org.translate.goog/wiki/Sprungklammer?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Sprungklammer">Sprungklammer</a> [[...]] kommt neu hinzu. Die Integrale über die von außen angreifenden Kräfte werden getrennt für die Volumina v+ und v− berechnet: <dl> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\left(\int _{v}\rho {\vec {k}}\;\mathrm {d} v+\int _{a}{\vec {t}}\,\mathrm {d} a\right)^{+}=&\int _{v^{+}}\rho {\vec {k}}\;\mathrm {d} v^{+}+\int _{a^{+}}{\boldsymbol {\sigma }}^{\top }\cdot {\vec {n}}\,\mathrm {d} a^{+}-\int _{a_{s}}{\boldsymbol {\sigma }}^{\top }\cdot {\vec {n}}\,\mathrm {d} a_{s}\\\left(\int _{v}\rho {\vec {k}}\;\mathrm {d} v+\int _{a}{\vec {t}}\,\mathrm {d} a\right)^{-}=&\int _{v^{-}}\rho {\vec {k}}\;\mathrm {d} v^{-}+\int _{a^{-}}{\boldsymbol {\sigma }}^{\top }\cdot {\vec {n}}\,\mathrm {d} a^{-}+\int _{a_{s}}{\boldsymbol {\sigma }}^{\top }\cdot {\vec {n}}\,\mathrm {d} a_{s}\end{aligned}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msup> <mrow> <mo> ( </mo> <mrow> <msub> <mo> ∫ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> v </mi> </mrow> </msub> <mi> ρ </mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> k </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mspace width="thickmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> v </mi> <mo> + </mo> <msub> <mo> ∫ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> a </mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> t </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> a </mi> </mrow> <mo> ) </mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo> + </mo> </mrow> </msup> <mo> = </mo> </mtd> <mtd> <mi></mi> <msub> <mo> ∫ </mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi> v </mi> <mrow class="MJX-TeXAtom-ORD"> <mo> + </mo> </mrow> </msup> </mrow> </msub> <mi> ρ </mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> k </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mspace width="thickmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <msup> <mi> v </mi> <mrow class="MJX-TeXAtom-ORD"> <mo> + </mo> </mrow> </msup> <mo> + </mo> <msub> <mo> ∫ </mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi> a </mi> <mrow class="MJX-TeXAtom-ORD"> <mo> + </mo> </mrow> </msup> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> σ </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> ⊤ </mi> </mrow> </msup> <mo> ⋅ </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> n </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <msup> <mi> a </mi> <mrow class="MJX-TeXAtom-ORD"> <mo> + </mo> </mrow> </msup> <mo> − </mo> <msub> <mo> ∫ </mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi> a </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> s </mi> </mrow> </msub> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> σ </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> ⊤ </mi> </mrow> </msup> <mo> ⋅ </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> n </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <msub> <mi> a </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> s </mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msup> <mrow> <mo> ( </mo> <mrow> <msub> <mo> ∫ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> v </mi> </mrow> </msub> <mi> ρ </mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> k </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mspace width="thickmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> v </mi> <mo> + </mo> <msub> <mo> ∫ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> a </mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> t </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> a </mi> </mrow> <mo> ) </mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo> − </mo> </mrow> </msup> <mo> = </mo> </mtd> <mtd> <mi></mi> <msub> <mo> ∫ </mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi> v </mi> <mrow class="MJX-TeXAtom-ORD"> <mo> − </mo> </mrow> </msup> </mrow> </msub> <mi> ρ </mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> k </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mspace width="thickmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <msup> <mi> v </mi> <mrow class="MJX-TeXAtom-ORD"> <mo> − </mo> </mrow> </msup> <mo> + </mo> <msub> <mo> ∫ </mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi> a </mi> <mrow class="MJX-TeXAtom-ORD"> <mo> − </mo> </mrow> </msup> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> σ </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> ⊤ </mi> </mrow> </msup> <mo> ⋅ </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> n </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <msup> <mi> a </mi> <mrow class="MJX-TeXAtom-ORD"> <mo> − </mo> </mrow> </msup> <mo> + </mo> <msub> <mo> ∫ </mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi> a </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> s </mi> </mrow> </msub> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> σ </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> ⊤ </mi> </mrow> </msup> <mo> ⋅ </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> n </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <msub> <mi> a </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> s </mi> </mrow> </msub> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\begin{aligned}\left(\int _{v}\rho {\vec {k}}\;\mathrm {d} v+\int _{a}{\vec {t}}\,\mathrm {d} a\right)^{+}=&\int _{v^{+}}\rho {\vec {k}}\;\mathrm {d} v^{+}+\int _{a^{+}}{\boldsymbol {\sigma }}^{\top }\cdot {\vec {n}}\,\mathrm {d} a^{+}-\int _{a_{s}}{\boldsymbol {\sigma }}^{\top }\cdot {\vec {n}}\,\mathrm {d} a_{s}\\\left(\int _{v}\rho {\vec {k}}\;\mathrm {d} v+\int _{a}{\vec {t}}\,\mathrm {d} a\right)^{-}=&\int _{v^{-}}\rho {\vec {k}}\;\mathrm {d} v^{-}+\int _{a^{-}}{\boldsymbol {\sigma }}^{\top }\cdot {\vec {n}}\,\mathrm {d} a^{-}+\int _{a_{s}}{\boldsymbol {\sigma }}^{\top }\cdot {\vec {n}}\,\mathrm {d} a_{s}\end{aligned}}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87bbb627413abf8da630baac1aad61ac9d338378" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.005ex; width:71.358ex; height:13.176ex;" alt="{\displaystyle {\begin{aligned}\left(\int _{v}\rho {\vec {k}}\;\mathrm {d} v+\int _{a}{\vec {t}}\,\mathrm {d} a\right)^{+}=&\int _{v^{+}}\rho {\vec {k}}\;\mathrm {d} v^{+}+\int _{a^{+}}{\boldsymbol {\sigma }}^{\top }\cdot {\vec {n}}\,\mathrm {d} a^{+}-\int _{a_{s}}{\boldsymbol {\sigma }}^{\top }\cdot {\vec {n}}\,\mathrm {d} a_{s}\\\left(\int _{v}\rho {\vec {k}}\;\mathrm {d} v+\int _{a}{\vec {t}}\,\mathrm {d} a\right)^{-}=&\int _{v^{-}}\rho {\vec {k}}\;\mathrm {d} v^{-}+\int _{a^{-}}{\boldsymbol {\sigma }}^{\top }\cdot {\vec {n}}\,\mathrm {d} a^{-}+\int _{a_{s}}{\boldsymbol {\sigma }}^{\top }\cdot {\vec {n}}\,\mathrm {d} a_{s}\end{aligned}}}"> </noscript><span class="lazy-image-placeholder" style="width: 71.358ex;height: 13.176ex;vertical-align: -6.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87bbb627413abf8da630baac1aad61ac9d338378" data-alt="{\displaystyle {\begin{aligned}\left(\int _{v}\rho {\vec {k}}\;\mathrm {d} v+\int _{a}{\vec {t}}\,\mathrm {d} a\right)^{+}=&\int _{v^{+}}\rho {\vec {k}}\;\mathrm {d} v^{+}+\int _{a^{+}}{\boldsymbol {\sigma }}^{\top }\cdot {\vec {n}}\,\mathrm {d} a^{+}-\int _{a_{s}}{\boldsymbol {\sigma }}^{\top }\cdot {\vec {n}}\,\mathrm {d} a_{s}\\\left(\int _{v}\rho {\vec {k}}\;\mathrm {d} v+\int _{a}{\vec {t}}\,\mathrm {d} a\right)^{-}=&\int _{v^{-}}\rho {\vec {k}}\;\mathrm {d} v^{-}+\int _{a^{-}}{\boldsymbol {\sigma }}^{\top }\cdot {\vec {n}}\,\mathrm {d} a^{-}+\int _{a_{s}}{\boldsymbol {\sigma }}^{\top }\cdot {\vec {n}}\,\mathrm {d} a_{s}\end{aligned}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </dd> </dl> Die Normale soll immer nach außen gerichtet sein und geht daher auf der Sprungstelle einmal mit positivem und einmal mit negativem Vorzeichen ein. Die Vereinigung der Oberflächen a+ und a− ergibt die Oberfläche a des gesamten Volumens v, zu dessen Oberfläche die innere Fläche as nicht gehört. Die Summe der drei Gleichungen führt nach Umformungen, wie sie oben bereits angegeben wurden, auf <dl> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{v}(\rho {\dot {\vec {v}}}-\rho {\vec {k}}-\nabla \cdot {\boldsymbol {\sigma }})\,\mathrm {d} v=\int _{a_{s}}[[\rho {\vec {v}}\;(({\vec {v}}-{\vec {v}}_{s})\cdot {\vec {n}})-{\boldsymbol {\sigma }}^{\top }\cdot {\vec {n}}]]\mathrm {d} a_{s}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mo> ∫ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> v </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <mi> ρ </mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> v </mi> <mo stretchy="false"> → </mo> </mover> </mrow> <mo> ˙ </mo> </mover> </mrow> </mrow> <mo> − </mo> <mi> ρ </mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> k </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo> − </mo> <mi mathvariant="normal"> ∇ </mi> <mo> ⋅ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> σ </mi> </mrow> <mo stretchy="false"> ) </mo> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> v </mi> <mo> = </mo> <msub> <mo> ∫ </mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi> a </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> s </mi> </mrow> </msub> </mrow> </msub> <mo stretchy="false"> [ </mo> <mo stretchy="false"> [ </mo> <mi> ρ </mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> v </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mspace width="thickmathspace"></mspace> <mo stretchy="false"> ( </mo> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> v </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo> − </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> v </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> s </mi> </mrow> </msub> <mo stretchy="false"> ) </mo> <mo> ⋅ </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> n </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo stretchy="false"> ) </mo> <mo> − </mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> σ </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> ⊤ </mi> </mrow> </msup> <mo> ⋅ </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> n </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo stretchy="false"> ] </mo> <mo stretchy="false"> ] </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <msub> <mi> a </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> s </mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \int _{v}(\rho {\dot {\vec {v}}}-\rho {\vec {k}}-\nabla \cdot {\boldsymbol {\sigma }})\,\mathrm {d} v=\int _{a_{s}}[[\rho {\vec {v}}\;(({\vec {v}}-{\vec {v}}_{s})\cdot {\vec {n}})-{\boldsymbol {\sigma }}^{\top }\cdot {\vec {n}}]]\mathrm {d} a_{s}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b938854dc5f35ab959f7537ba542b154e1e89018" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:59.978ex; height:6.009ex;" alt="{\displaystyle \int _{v}(\rho {\dot {\vec {v}}}-\rho {\vec {k}}-\nabla \cdot {\boldsymbol {\sigma }})\,\mathrm {d} v=\int _{a_{s}}[[\rho {\vec {v}}\;(({\vec {v}}-{\vec {v}}_{s})\cdot {\vec {n}})-{\boldsymbol {\sigma }}^{\top }\cdot {\vec {n}}]]\mathrm {d} a_{s}}"> </noscript><span class="lazy-image-placeholder" style="width: 59.978ex;height: 6.009ex;vertical-align: -2.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b938854dc5f35ab959f7537ba542b154e1e89018" data-alt="{\displaystyle \int _{v}(\rho {\dot {\vec {v}}}-\rho {\vec {k}}-\nabla \cdot {\boldsymbol {\sigma }})\,\mathrm {d} v=\int _{a_{s}}[[\rho {\vec {v}}\;(({\vec {v}}-{\vec {v}}_{s})\cdot {\vec {n}})-{\boldsymbol {\sigma }}^{\top }\cdot {\vec {n}}]]\mathrm {d} a_{s}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </dd> </dl> Jenseits der Sprungstelle verschwindet die rechte Seite und die lokale Impulsbilanz ohne Sprungstelle folgt. An der (flächigen) Sprungstelle ist dv=0 und die linke Seite kann vernachlässigt werden, sodass bei Stetigkeit des Integranden mit der Sprungklammer in der Fläche<a href="https://de-m-wikipedia-org.translate.goog/wiki/Cauchy-eulersche_Bewegungsgesetze?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-4">[L 3]</a> <dl> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [[\rho {\vec {v}}\;(({\vec {v}}_{s}-{\vec {v}})\cdot {\vec {n}})+{\boldsymbol {\sigma }}^{\top }\cdot {\vec {n}}]]=[[\rho {\vec {v}}\otimes ({\vec {v}}_{s}-{\vec {v}})+{\boldsymbol {\sigma }}^{\top }]]\cdot {\vec {n}}={\vec {0}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false"> [ </mo> <mo stretchy="false"> [ </mo> <mi> ρ </mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> v </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mspace width="thickmathspace"></mspace> <mo stretchy="false"> ( </mo> <mo stretchy="false"> ( </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> v </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> s </mi> </mrow> </msub> <mo> − </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> v </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo stretchy="false"> ) </mo> <mo> ⋅ </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> n </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo stretchy="false"> ) </mo> <mo> + </mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> σ </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> ⊤ </mi> </mrow> </msup> <mo> ⋅ </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> n </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo stretchy="false"> ] </mo> <mo stretchy="false"> ] </mo> <mo> = </mo> <mo stretchy="false"> [ </mo> <mo stretchy="false"> [ </mo> <mi> ρ </mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> v </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo> ⊗ </mo> <mo stretchy="false"> ( </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> v </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> s </mi> </mrow> </msub> <mo> − </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> v </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo stretchy="false"> ) </mo> <mo> + </mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> σ </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> ⊤ </mi> </mrow> </msup> <mo stretchy="false"> ] </mo> <mo stretchy="false"> ] </mo> <mo> ⋅ </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> n </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mn> 0 </mn> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle [[\rho {\vec {v}}\;(({\vec {v}}_{s}-{\vec {v}})\cdot {\vec {n}})+{\boldsymbol {\sigma }}^{\top }\cdot {\vec {n}}]]=[[\rho {\vec {v}}\otimes ({\vec {v}}_{s}-{\vec {v}})+{\boldsymbol {\sigma }}^{\top }]]\cdot {\vec {n}}={\vec {0}}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/040500c5001c02a217438c0f4abf13e4d562ea88" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:59.704ex; height:3.343ex;" alt="{\displaystyle [[\rho {\vec {v}}\;(({\vec {v}}_{s}-{\vec {v}})\cdot {\vec {n}})+{\boldsymbol {\sigma }}^{\top }\cdot {\vec {n}}]]=[[\rho {\vec {v}}\otimes ({\vec {v}}_{s}-{\vec {v}})+{\boldsymbol {\sigma }}^{\top }]]\cdot {\vec {n}}={\vec {0}}}"> </noscript><span class="lazy-image-placeholder" style="width: 59.704ex;height: 3.343ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/040500c5001c02a217438c0f4abf13e4d562ea88" data-alt="{\displaystyle [[\rho {\vec {v}}\;(({\vec {v}}_{s}-{\vec {v}})\cdot {\vec {n}})+{\boldsymbol {\sigma }}^{\top }\cdot {\vec {n}}]]=[[\rho {\vec {v}}\otimes ({\vec {v}}_{s}-{\vec {v}})+{\boldsymbol {\sigma }}^{\top }]]\cdot {\vec {n}}={\vec {0}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </dd> </dl> abgeleitet werden kann. Wenn die Sprungstelle eine materielle Fläche ist, wie beispielsweise an Materialgrenzen, dann ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {v}}_{s}={\vec {v}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> v </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> s </mi> </mrow> </msub> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> v </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\vec {v}}_{s}={\vec {v}}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2984e7378a6a2949141cdfefe282f17a86e45908" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.452ex; height:2.676ex;" alt="{\displaystyle {\vec {v}}_{s}={\vec {v}}}"> </noscript><span class="lazy-image-placeholder" style="width: 6.452ex;height: 2.676ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2984e7378a6a2949141cdfefe282f17a86e45908" data-alt="{\displaystyle {\vec {v}}_{s}={\vec {v}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  und es folgt: <dl> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [[{\boldsymbol {\sigma }}^{\top }]]\cdot {\vec {n}}={\vec {0}}\quad \Leftrightarrow \quad {{\boldsymbol {\sigma }}^{+}}^{\top }\cdot {\vec {n}}={\vec {t}}^{+}={{\boldsymbol {\sigma }}^{-}}^{\top }\cdot {\vec {n}}={\vec {t}}^{-}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false"> [ </mo> <mo stretchy="false"> [ </mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> σ </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> ⊤ </mi> </mrow> </msup> <mo stretchy="false"> ] </mo> <mo stretchy="false"> ] </mo> <mo> ⋅ </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> n </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mn> 0 </mn> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mspace width="1em"></mspace> <mo stretchy="false"> ⇔ </mo> <mspace width="1em"></mspace> <msup> <mrow class="MJX-TeXAtom-ORD"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> σ </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo> + </mo> </mrow> </msup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> ⊤ </mi> </mrow> </msup> <mo> ⋅ </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> n </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo> = </mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> t </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo> + </mo> </mrow> </msup> <mo> = </mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> σ </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo> − </mo> </mrow> </msup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> ⊤ </mi> </mrow> </msup> <mo> ⋅ </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> n </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo> = </mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> t </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo> − </mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle [[{\boldsymbol {\sigma }}^{\top }]]\cdot {\vec {n}}={\vec {0}}\quad \Leftrightarrow \quad {{\boldsymbol {\sigma }}^{+}}^{\top }\cdot {\vec {n}}={\vec {t}}^{+}={{\boldsymbol {\sigma }}^{-}}^{\top }\cdot {\vec {n}}={\vec {t}}^{-}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/71767aa85022b0f0b4d69f45c17a43d54588917c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:51.558ex; height:3.509ex;" alt="{\displaystyle [[{\boldsymbol {\sigma }}^{\top }]]\cdot {\vec {n}}={\vec {0}}\quad \Leftrightarrow \quad {{\boldsymbol {\sigma }}^{+}}^{\top }\cdot {\vec {n}}={\vec {t}}^{+}={{\boldsymbol {\sigma }}^{-}}^{\top }\cdot {\vec {n}}={\vec {t}}^{-}}"> </noscript><span class="lazy-image-placeholder" style="width: 51.558ex;height: 3.509ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/71767aa85022b0f0b4d69f45c17a43d54588917c" data-alt="{\displaystyle [[{\boldsymbol {\sigma }}^{\top }]]\cdot {\vec {n}}={\vec {0}}\quad \Leftrightarrow \quad {{\boldsymbol {\sigma }}^{+}}^{\top }\cdot {\vec {n}}={\vec {t}}^{+}={{\boldsymbol {\sigma }}^{-}}^{\top }\cdot {\vec {n}}={\vec {t}}^{-}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </dd> </dl> Die Schnittspannungen auf beiden Seiten einer materiellen Sprungstelle müssen gleich sein.<a href="https://de-m-wikipedia-org.translate.goog/wiki/Cauchy-eulersche_Bewegungsgesetze?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-5">[L 4]</a> </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(2)"> <h2 id="Zweites_cauchy-eulersches_Bewegungsgesetz">Zweites cauchy-eulersches Bewegungsgesetz</h2> <a role="button" href="https://de-m-wikipedia-org.translate.goog/w/index.php?title=Cauchy-eulersche_Bewegungsgesetze&action=edit&section=6&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Abschnitt bearbeiten: Zweites cauchy-eulersches Bewegungsgesetz" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> Bearbeiten </a> </div> <section class="mf-section-2 collapsible-block" id="mf-section-2"> Das zweite cauchy-eulersche Bewegungsgesetz folgt aus dem 1754 von Leonhard Euler aufgestellten <a href="https://de-m-wikipedia-org.translate.goog/wiki/Drallsatz?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Drallsatz">Drallsatz</a>, nach dem die zeitliche Änderung des Drehimpulses <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {L}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> L </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\vec {L}}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0c139fc28d6ca3873993892f44e7331e5ff18fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.843ex;" alt="{\displaystyle {\vec {L}}}"> </noscript><span class="lazy-image-placeholder" style="width: 1.583ex;height: 2.843ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0c139fc28d6ca3873993892f44e7331e5ff18fd" data-alt="{\displaystyle {\vec {L}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  gleich der von außen angreifenden <a href="https://de-m-wikipedia-org.translate.goog/wiki/Drehmoment?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Drehmoment">Drehmomente</a> ist: <dl> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\dot {\vec {L}}}={\vec {M}}_{v}+{\vec {M}}_{a}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> L </mi> <mo stretchy="false"> → </mo> </mover> </mrow> <mo> ˙ </mo> </mover> </mrow> </mrow> <mo> = </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> M </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> v </mi> </mrow> </msub> <mo> + </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> M </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> a </mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\dot {\vec {L}}}={\vec {M}}_{v}+{\vec {M}}_{a}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5c56cc5ddb0e8cd3046f25467573763a331c71f2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.538ex; height:3.843ex;" alt="{\displaystyle {\dot {\vec {L}}}={\vec {M}}_{v}+{\vec {M}}_{a}}"> </noscript><span class="lazy-image-placeholder" style="width: 14.538ex;height: 3.843ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5c56cc5ddb0e8cd3046f25467573763a331c71f2" data-alt="{\displaystyle {\dot {\vec {L}}}={\vec {M}}_{v}+{\vec {M}}_{a}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </dd> </dl> Der Vektor <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {M}}_{v}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> M </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> v </mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\vec {M}}_{v}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/765b67b0d3207ae60b0572cb1dc4d2f1d7942911" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.472ex; height:3.176ex;" alt="{\displaystyle {\vec {M}}_{v}}"> </noscript><span class="lazy-image-placeholder" style="width: 3.472ex;height: 3.176ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/765b67b0d3207ae60b0572cb1dc4d2f1d7942911" data-alt="{\displaystyle {\vec {M}}_{v}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  steht für das von volumenverteilten Kräften ausgehende Drehmoment und der Vektor <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {M}}_{a}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> M </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> a </mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\vec {M}}_{a}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c8d9d64abd1f4a8ee88a99c64b8f23aea8c21c7d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.544ex; height:3.176ex;" alt="{\displaystyle {\vec {M}}_{a}}"> </noscript><span class="lazy-image-placeholder" style="width: 3.544ex;height: 3.176ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c8d9d64abd1f4a8ee88a99c64b8f23aea8c21c7d" data-alt="{\displaystyle {\vec {M}}_{a}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  für das oberflächig eingeleitete Moment. <div class="mw-heading mw-heading3"> <h3 id="Drallsatz_am_Volumenelement">Drallsatz am Volumenelement</h3> <a role="button" href="https://de-m-wikipedia-org.translate.goog/w/index.php?title=Cauchy-eulersche_Bewegungsgesetze&action=edit&section=7&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Abschnitt bearbeiten: Drallsatz am Volumenelement" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> Bearbeiten </a> </div> <figure class="mw-default-size" typeof="mw:File/Thumb"> <a href="https://de-m-wikipedia-org.translate.goog/wiki/Datei:Drehimscheibe.png?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-file-description"> <noscript> <img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/cc/Drehimscheibe.png/220px-Drehimscheibe.png" decoding="async" width="220" height="367" class="mw-file-element" data-file-width="497" data-file-height="828"> </noscript><span class="lazy-image-placeholder" style="width: 220px;height: 367px;" data-src="//upload.wikimedia.org/wikipedia/commons/thumb/c/cc/Drehimscheibe.png/220px-Drehimscheibe.png" data-width="220" data-height="367" data-srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/cc/Drehimscheibe.png/330px-Drehimscheibe.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/cc/Drehimscheibe.png/440px-Drehimscheibe.png 2x" data-class="mw-file-element"> </a> <figcaption> Schnittspannungen an einem würfelförmigen Teilkörper </figcaption> </figure> Es wird ein belasteter Körper betrachtet, aus dem gedanklich ein <a href="https://de-m-wikipedia-org.translate.goog/wiki/W%C3%BCrfel_(Geometrie)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Würfel (Geometrie)">würfel</a>förmiger Teilkörper (im Bild gelb) herausgeschnitten wird, der die Kantenlänge 2L hat und in dessen <a href="https://de-m-wikipedia-org.translate.goog/wiki/Massenmittelpunkt?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Massenmittelpunkt">Massenmittelpunkt</a> ein zu den Würfelkanten parallel ausgerichtetes kartesisches Koordinatensystem gelegt wird. An den Würfelflächen entstehen dem Schnittprinzip zufolge Schnittspannungen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {t}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> t </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\vec {t}}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f793e16d925f6933447f6d57064554ef469e9d3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.287ex; height:2.843ex;" alt="{\displaystyle {\vec {t}}}"> </noscript><span class="lazy-image-placeholder" style="width: 1.287ex;height: 2.843ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f793e16d925f6933447f6d57064554ef469e9d3f" data-alt="{\displaystyle {\vec {t}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> , die an die Stelle des weggeschnittenen Teilkörpers treten und die nach dem <a href="https://de-m-wikipedia-org.translate.goog/wiki/Cauchysches_Fundamentaltheorem?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Cauchysches Fundamentaltheorem">cauchyschen Fundamentaltheorem</a> die mit dem cauchyschen Spannungstensor transformierten Normalenvektoren an die Schnittfläche sind. Bei infinitesimal kleinem Würfel können die Schnittspannungen als über die Fläche konstant angenommen werden und zu einer Resultierenden aufintegriert werden, die den Würfel aus Symmetriegründen in den Flächenmitten belasten. Für die in der Würfelmitte angreifenden Momente gilt: <ul> <li>Vom Schwerpunkt des Würfels weist der Vektor <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L{\hat {e}}_{x}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> L </mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> e </mi> <mo stretchy="false"> ^ </mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> x </mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle L{\hat {e}}_{x}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f4bd24aa61e5581d7573e5c659daada0ae9f903" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.047ex; height:2.509ex;" alt="{\displaystyle L{\hat {e}}_{x}}"> </noscript><span class="lazy-image-placeholder" style="width: 4.047ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f4bd24aa61e5581d7573e5c659daada0ae9f903" data-alt="{\displaystyle L{\hat {e}}_{x}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  zur Mitte der Schnittfläche am positiven Schnittufer mit Normale in +x-Richtung und die Schnittspannung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {t}}_{+x}={\hat {e}}_{x}\cdot {\boldsymbol {\sigma }}_{+x}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> t </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo> + </mo> <mi> x </mi> </mrow> </msub> <mo> = </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> e </mi> <mo stretchy="false"> ^ </mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> x </mi> </mrow> </msub> <mo> ⋅ </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> σ </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo> + </mo> <mi> x </mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\vec {t}}_{+x}={\hat {e}}_{x}\cdot {\boldsymbol {\sigma }}_{+x}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/80cfbcb9e66ee035fc0374039ac7673c9b4a12a5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.026ex; height:3.176ex;" alt="{\displaystyle {\vec {t}}_{+x}={\hat {e}}_{x}\cdot {\boldsymbol {\sigma }}_{+x}}"> </noscript><span class="lazy-image-placeholder" style="width: 15.026ex;height: 3.176ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/80cfbcb9e66ee035fc0374039ac7673c9b4a12a5" data-alt="{\displaystyle {\vec {t}}_{+x}={\hat {e}}_{x}\cdot {\boldsymbol {\sigma }}_{+x}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  wirkt dort auf der Fläche 4L².</li> <li>Das Moment der Schnittspannung am positiven Schnittufer lautet mit dem <a href="https://de-m-wikipedia-org.translate.goog/wiki/Kreuzprodukt?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Kreuzprodukt">Kreuzprodukt</a> „×“: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {M}}_{+x}=4L^{2}(L{\hat {e}}_{x})\times ({\hat {e}}_{x}\cdot {\boldsymbol {\sigma }}_{+x})}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> M </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo> + </mo> <mi> x </mi> </mrow> </msub> <mo> = </mo> <mn> 4 </mn> <msup> <mi> L </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo stretchy="false"> ( </mo> <mi> L </mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> e </mi> <mo stretchy="false"> ^ </mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> x </mi> </mrow> </msub> <mo stretchy="false"> ) </mo> <mo> × </mo> <mo stretchy="false"> ( </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> e </mi> <mo stretchy="false"> ^ </mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> x </mi> </mrow> </msub> <mo> ⋅ </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> σ </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo> + </mo> <mi> x </mi> </mrow> </msub> <mo stretchy="false"> ) </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\vec {M}}_{+x}=4L^{2}(L{\hat {e}}_{x})\times ({\hat {e}}_{x}\cdot {\boldsymbol {\sigma }}_{+x})} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/37dec054a7cbe0bc8f16f99d76e8674665908a47" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.486ex; height:3.343ex;" alt="{\displaystyle {\vec {M}}_{+x}=4L^{2}(L{\hat {e}}_{x})\times ({\hat {e}}_{x}\cdot {\boldsymbol {\sigma }}_{+x})}"> </noscript><span class="lazy-image-placeholder" style="width: 30.486ex;height: 3.343ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/37dec054a7cbe0bc8f16f99d76e8674665908a47" data-alt="{\displaystyle {\vec {M}}_{+x}=4L^{2}(L{\hat {e}}_{x})\times ({\hat {e}}_{x}\cdot {\boldsymbol {\sigma }}_{+x})}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> .</li> <li>Am negativen Schnittufer ist der Hebelarm <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -L{\hat {e}}_{x}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo> − </mo> <mi> L </mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> e </mi> <mo stretchy="false"> ^ </mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> x </mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle -L{\hat {e}}_{x}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48b10e8ca6ccb748dc389a58586c77775820df09" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.855ex; height:2.509ex;" alt="{\displaystyle -L{\hat {e}}_{x}}"> </noscript><span class="lazy-image-placeholder" style="width: 5.855ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48b10e8ca6ccb748dc389a58586c77775820df09" data-alt="{\displaystyle -L{\hat {e}}_{x}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  und die Schnittspannung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {t}}_{-x}=-{\hat {e}}_{x}\cdot {\boldsymbol {\sigma }}_{-x}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> t </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo> − </mo> <mi> x </mi> </mrow> </msub> <mo> = </mo> <mo> − </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> e </mi> <mo stretchy="false"> ^ </mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> x </mi> </mrow> </msub> <mo> ⋅ </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> σ </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo> − </mo> <mi> x </mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\vec {t}}_{-x}=-{\hat {e}}_{x}\cdot {\boldsymbol {\sigma }}_{-x}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4596322a369e4ba64fdbc0285610d9ae1d56b32d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.834ex; height:3.176ex;" alt="{\displaystyle {\vec {t}}_{-x}=-{\hat {e}}_{x}\cdot {\boldsymbol {\sigma }}_{-x}}"> </noscript><span class="lazy-image-placeholder" style="width: 16.834ex;height: 3.176ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4596322a369e4ba64fdbc0285610d9ae1d56b32d" data-alt="{\displaystyle {\vec {t}}_{-x}=-{\hat {e}}_{x}\cdot {\boldsymbol {\sigma }}_{-x}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  operiert auf der gleichen Fläche 4L²: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {M}}_{-x}=4L^{2}(-L{\hat {e}}_{x})\times (-{\hat {e}}_{x}\cdot {\boldsymbol {\sigma }}_{-x})}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> M </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo> − </mo> <mi> x </mi> </mrow> </msub> <mo> = </mo> <mn> 4 </mn> <msup> <mi> L </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo stretchy="false"> ( </mo> <mo> − </mo> <mi> L </mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> e </mi> <mo stretchy="false"> ^ </mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> x </mi> </mrow> </msub> <mo stretchy="false"> ) </mo> <mo> × </mo> <mo stretchy="false"> ( </mo> <mo> − </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> e </mi> <mo stretchy="false"> ^ </mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> x </mi> </mrow> </msub> <mo> ⋅ </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> σ </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo> − </mo> <mi> x </mi> </mrow> </msub> <mo stretchy="false"> ) </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\vec {M}}_{-x}=4L^{2}(-L{\hat {e}}_{x})\times (-{\hat {e}}_{x}\cdot {\boldsymbol {\sigma }}_{-x})} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/775a3e5ff4578f991775f16934b38fc13ba03b8a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:34.102ex; height:3.343ex;" alt="{\displaystyle {\vec {M}}_{-x}=4L^{2}(-L{\hat {e}}_{x})\times (-{\hat {e}}_{x}\cdot {\boldsymbol {\sigma }}_{-x})}"> </noscript><span class="lazy-image-placeholder" style="width: 34.102ex;height: 3.343ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/775a3e5ff4578f991775f16934b38fc13ba03b8a" data-alt="{\displaystyle {\vec {M}}_{-x}=4L^{2}(-L{\hat {e}}_{x})\times (-{\hat {e}}_{x}\cdot {\boldsymbol {\sigma }}_{-x})}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> .</li> <li>Die Momente der Schnittspannungen summieren sich zu <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {M}}_{x}=4L^{3}{\hat {e}}_{x}\times [{\hat {e}}_{x}\cdot ({\boldsymbol {\sigma }}_{+x}+{\boldsymbol {\sigma }}_{-x})]}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> M </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> x </mi> </mrow> </msub> <mo> = </mo> <mn> 4 </mn> <msup> <mi> L </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 3 </mn> </mrow> </msup> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> e </mi> <mo stretchy="false"> ^ </mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> x </mi> </mrow> </msub> <mo> × </mo> <mo stretchy="false"> [ </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> e </mi> <mo stretchy="false"> ^ </mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> x </mi> </mrow> </msub> <mo> ⋅ </mo> <mo stretchy="false"> ( </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> σ </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo> + </mo> <mi> x </mi> </mrow> </msub> <mo> + </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> σ </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo> − </mo> <mi> x </mi> </mrow> </msub> <mo stretchy="false"> ) </mo> <mo stretchy="false"> ] </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\vec {M}}_{x}=4L^{3}{\hat {e}}_{x}\times [{\hat {e}}_{x}\cdot ({\boldsymbol {\sigma }}_{+x}+{\boldsymbol {\sigma }}_{-x})]} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/625e9b4484781765ae437a7ab37e7b4058bb3b33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.995ex; height:3.343ex;" alt="{\displaystyle {\vec {M}}_{x}=4L^{3}{\hat {e}}_{x}\times [{\hat {e}}_{x}\cdot ({\boldsymbol {\sigma }}_{+x}+{\boldsymbol {\sigma }}_{-x})]}"> </noscript><span class="lazy-image-placeholder" style="width: 33.995ex;height: 3.343ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/625e9b4484781765ae437a7ab37e7b4058bb3b33" data-alt="{\displaystyle {\vec {M}}_{x}=4L^{3}{\hat {e}}_{x}\times [{\hat {e}}_{x}\cdot ({\boldsymbol {\sigma }}_{+x}+{\boldsymbol {\sigma }}_{-x})]}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> .</li> <li>In den anderen beiden Raumrichtungen ergibt sich entsprechend <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {M}}_{y}=4L^{3}{\hat {e}}_{y}\times [{\hat {e}}_{y}\cdot ({\boldsymbol {\sigma }}_{+y}+{\boldsymbol {\sigma }}_{-y})]}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> M </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> y </mi> </mrow> </msub> <mo> = </mo> <mn> 4 </mn> <msup> <mi> L </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 3 </mn> </mrow> </msup> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> e </mi> <mo stretchy="false"> ^ </mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> y </mi> </mrow> </msub> <mo> × </mo> <mo stretchy="false"> [ </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> e </mi> <mo stretchy="false"> ^ </mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> y </mi> </mrow> </msub> <mo> ⋅ </mo> <mo stretchy="false"> ( </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> σ </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo> + </mo> <mi> y </mi> </mrow> </msub> <mo> + </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> σ </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo> − </mo> <mi> y </mi> </mrow> </msub> <mo stretchy="false"> ) </mo> <mo stretchy="false"> ] </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\vec {M}}_{y}=4L^{3}{\hat {e}}_{y}\times [{\hat {e}}_{y}\cdot ({\boldsymbol {\sigma }}_{+y}+{\boldsymbol {\sigma }}_{-y})]} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/31f6c12bffaa47c0455fa206ae9310030145c1cf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:33.379ex; height:3.509ex;" alt="{\displaystyle {\vec {M}}_{y}=4L^{3}{\hat {e}}_{y}\times [{\hat {e}}_{y}\cdot ({\boldsymbol {\sigma }}_{+y}+{\boldsymbol {\sigma }}_{-y})]}"> </noscript><span class="lazy-image-placeholder" style="width: 33.379ex;height: 3.509ex;vertical-align: -1.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/31f6c12bffaa47c0455fa206ae9310030145c1cf" data-alt="{\displaystyle {\vec {M}}_{y}=4L^{3}{\hat {e}}_{y}\times [{\hat {e}}_{y}\cdot ({\boldsymbol {\sigma }}_{+y}+{\boldsymbol {\sigma }}_{-y})]}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {M}}_{z}=4L^{3}{\hat {e}}_{z}\times [{\hat {e}}_{z}\cdot ({\boldsymbol {\sigma }}_{+z}+{\boldsymbol {\sigma }}_{-z})]}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> M </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> z </mi> </mrow> </msub> <mo> = </mo> <mn> 4 </mn> <msup> <mi> L </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 3 </mn> </mrow> </msup> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> e </mi> <mo stretchy="false"> ^ </mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> z </mi> </mrow> </msub> <mo> × </mo> <mo stretchy="false"> [ </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> e </mi> <mo stretchy="false"> ^ </mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> z </mi> </mrow> </msub> <mo> ⋅ </mo> <mo stretchy="false"> ( </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> σ </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo> + </mo> <mi> z </mi> </mrow> </msub> <mo> + </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> σ </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo> − </mo> <mi> z </mi> </mrow> </msub> <mo stretchy="false"> ) </mo> <mo stretchy="false"> ] </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\vec {M}}_{z}=4L^{3}{\hat {e}}_{z}\times [{\hat {e}}_{z}\cdot ({\boldsymbol {\sigma }}_{+z}+{\boldsymbol {\sigma }}_{-z})]} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d9f1ef73f738de2a04048443e981bba067bab03a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.141ex; height:3.343ex;" alt="{\displaystyle {\vec {M}}_{z}=4L^{3}{\hat {e}}_{z}\times [{\hat {e}}_{z}\cdot ({\boldsymbol {\sigma }}_{+z}+{\boldsymbol {\sigma }}_{-z})]}"> </noscript><span class="lazy-image-placeholder" style="width: 33.141ex;height: 3.343ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d9f1ef73f738de2a04048443e981bba067bab03a" data-alt="{\displaystyle {\vec {M}}_{z}=4L^{3}{\hat {e}}_{z}\times [{\hat {e}}_{z}\cdot ({\boldsymbol {\sigma }}_{+z}+{\boldsymbol {\sigma }}_{-z})]}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> .</li> <li>Im infinitesimal kleinen Würfel kann von ortsunabhängiger Dichte ρ und ortsunabhängigem Schwerefeld ausgegangen werden, das daher in der Würfelmitte kein Moment verursacht.</li> </ul> Bei homogener Dichte hat der Würfel die Masse <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m=\rho (2L)^{3}=8\rho L^{3}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> m </mi> <mo> = </mo> <mi> ρ </mi> <mo stretchy="false"> ( </mo> <mn> 2 </mn> <mi> L </mi> <msup> <mo stretchy="false"> ) </mo> <mrow class="MJX-TeXAtom-ORD"> <mn> 3 </mn> </mrow> </msup> <mo> = </mo> <mn> 8 </mn> <mi> ρ </mi> <msup> <mi> L </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 3 </mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle m=\rho (2L)^{3}=8\rho L^{3}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/999bcd700cf9ddf8596cc2f2c680b7836fa552a6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.05ex; height:3.176ex;" alt="{\displaystyle m=\rho (2L)^{3}=8\rho L^{3}}"> </noscript><span class="lazy-image-placeholder" style="width: 20.05ex;height: 3.176ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/999bcd700cf9ddf8596cc2f2c680b7836fa552a6" data-alt="{\displaystyle m=\rho (2L)^{3}=8\rho L^{3}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  und den <a href="https://de-m-wikipedia-org.translate.goog/wiki/Liste_von_Tr%C3%A4gheitstensoren?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Platonische_K%C3%B6rper" title="Liste von Trägheitstensoren">Trägheitstensor</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {\Theta }}={\tfrac {m}{6}}(2L)^{2}\mathbf {1} ={\tfrac {16}{3}}\rho L^{5}\mathbf {1} }"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> Θ </mi> </mrow> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi> m </mi> <mn> 6 </mn> </mfrac> </mstyle> </mrow> <mo stretchy="false"> ( </mo> <mn> 2 </mn> <mi> L </mi> <msup> <mo stretchy="false"> ) </mo> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold"> 1 </mn> </mrow> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn> 16 </mn> <mn> 3 </mn> </mfrac> </mstyle> </mrow> <mi> ρ </mi> <msup> <mi> L </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 5 </mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold"> 1 </mn> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\boldsymbol {\Theta }}={\tfrac {m}{6}}(2L)^{2}\mathbf {1} ={\tfrac {16}{3}}\rho L^{5}\mathbf {1} } </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/192eb8d0f2b9731536e1f0a5a03341da7d0a020e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:25.155ex; height:3.676ex;" alt="{\displaystyle {\boldsymbol {\Theta }}={\tfrac {m}{6}}(2L)^{2}\mathbf {1} ={\tfrac {16}{3}}\rho L^{5}\mathbf {1} }"> </noscript><span class="lazy-image-placeholder" style="width: 25.155ex;height: 3.676ex;vertical-align: -1.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/192eb8d0f2b9731536e1f0a5a03341da7d0a020e" data-alt="{\displaystyle {\boldsymbol {\Theta }}={\tfrac {m}{6}}(2L)^{2}\mathbf {1} ={\tfrac {16}{3}}\rho L^{5}\mathbf {1} }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> , der proportional zum <a href="https://de-m-wikipedia-org.translate.goog/wiki/Einheitstensor?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Einheitstensor">Einheitstensor</a> 1 ist. Im Drallsatz <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {M}}={\dot {\vec {L}}}=({\boldsymbol {\Theta }}\cdot {\vec {\omega }}{\dot {)\,}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> M </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> L </mi> <mo stretchy="false"> → </mo> </mover> </mrow> <mo> ˙ </mo> </mover> </mrow> </mrow> <mo> = </mo> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> Θ </mi> </mrow> <mo> ⋅ </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> ω </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mo stretchy="false"> ) </mo> <mspace width="thinmathspace"></mspace> </mrow> <mo> ˙ </mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\vec {M}}={\dot {\vec {L}}}=({\boldsymbol {\Theta }}\cdot {\vec {\omega }}{\dot {)\,}}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/575ba36cdbaebcf44d4da626f53be01fc9f2a502" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.621ex; height:4.009ex;" alt="{\displaystyle {\vec {M}}={\dot {\vec {L}}}=({\boldsymbol {\Theta }}\cdot {\vec {\omega }}{\dot {)\,}}}"> </noscript><span class="lazy-image-placeholder" style="width: 17.621ex;height: 4.009ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/575ba36cdbaebcf44d4da626f53be01fc9f2a502" data-alt="{\displaystyle {\vec {M}}={\dot {\vec {L}}}=({\boldsymbol {\Theta }}\cdot {\vec {\omega }}{\dot {)\,}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  mit der <a href="https://de-m-wikipedia-org.translate.goog/wiki/Winkelgeschwindigkeit?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Winkelgeschwindigkeit">Winkelgeschwindigkeit</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {\omega }}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> ω </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\vec {\omega }}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f4e066a68ceb355e3314fb2b97f1c0c421ca6074" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.446ex; height:2.343ex;" alt="{\displaystyle {\vec {\omega }}}"> </noscript><span class="lazy-image-placeholder" style="width: 1.446ex;height: 2.343ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f4e066a68ceb355e3314fb2b97f1c0c421ca6074" data-alt="{\displaystyle {\vec {\omega }}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  ist die Massenträgheit demnach von fünfter Ordnung in den Abmessungen des ausgeschnittenen Teilkörpers während die Momente nur von dritter Ordnung sind, und das gilt auch bei einem nicht würfelförmigen Quader mit unterschiedlichen Dimensionen in x-, y- und z-Richtung, siehe <a href="https://de-m-wikipedia-org.translate.goog/wiki/Liste_von_Tr%C3%A4gheitstensoren?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Quader" title="Liste von Trägheitstensoren">Trägheitstensor eines Quaders</a>. Bei kleiner werdendem Teilkörper geht die Massenträgheit schneller gegen null als die Momente, woraus das <a href="https://de-m-wikipedia-org.translate.goog/wiki/Boltzmann-Axiom?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Boltzmann-Axiom">Boltzmann-Axiom</a> resultiert: <dl> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {M}}=4L^{3}\sum _{k=1}^{3}{\hat {e}}_{k}\times [{\hat {e}}_{k}\cdot ({\boldsymbol {\sigma }}_{+k}+{\boldsymbol {\sigma }}_{-k})]={\vec {0}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> M </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo> = </mo> <mn> 4 </mn> <msup> <mi> L </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 3 </mn> </mrow> </msup> <munderover> <mo> ∑ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 3 </mn> </mrow> </munderover> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> e </mi> <mo stretchy="false"> ^ </mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> </mrow> </msub> <mo> × </mo> <mo stretchy="false"> [ </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> e </mi> <mo stretchy="false"> ^ </mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> </mrow> </msub> <mo> ⋅ </mo> <mo stretchy="false"> ( </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> σ </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo> + </mo> <mi> k </mi> </mrow> </msub> <mo> + </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> σ </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo> − </mo> <mi> k </mi> </mrow> </msub> <mo stretchy="false"> ) </mo> <mo stretchy="false"> ] </mo> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mn> 0 </mn> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\vec {M}}=4L^{3}\sum _{k=1}^{3}{\hat {e}}_{k}\times [{\hat {e}}_{k}\cdot ({\boldsymbol {\sigma }}_{+k}+{\boldsymbol {\sigma }}_{-k})]={\vec {0}}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d95f02f71321c438c02d9560fd795b34ddbb2130" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:40.877ex; height:7.176ex;" alt="{\displaystyle {\vec {M}}=4L^{3}\sum _{k=1}^{3}{\hat {e}}_{k}\times [{\hat {e}}_{k}\cdot ({\boldsymbol {\sigma }}_{+k}+{\boldsymbol {\sigma }}_{-k})]={\vec {0}}}"> </noscript><span class="lazy-image-placeholder" style="width: 40.877ex;height: 7.176ex;vertical-align: -3.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d95f02f71321c438c02d9560fd795b34ddbb2130" data-alt="{\displaystyle {\vec {M}}=4L^{3}\sum _{k=1}^{3}{\hat {e}}_{k}\times [{\hat {e}}_{k}\cdot ({\boldsymbol {\sigma }}_{+k}+{\boldsymbol {\sigma }}_{-k})]={\vec {0}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </dd> </dl> Die inneren Kräfte in einem <a href="https://de-m-wikipedia-org.translate.goog/wiki/Kontinuum_(Physik)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Kontinuum (Physik)">Kontinuum</a> sind momentenfrei. Bei kleiner werdendem Teilkörper wird <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {\sigma }}_{+k}\approx {\boldsymbol {\sigma }}_{-k}\approx {\boldsymbol {\sigma }}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> σ </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo> + </mo> <mi> k </mi> </mrow> </msub> <mo> ≈ </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> σ </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo> − </mo> <mi> k </mi> </mrow> </msub> <mo> ≈ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> σ </mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\boldsymbol {\sigma }}_{+k}\approx {\boldsymbol {\sigma }}_{-k}\approx {\boldsymbol {\sigma }}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ceee98526e30348c66113d6e9e5d397c48fde02e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.715ex; height:2.009ex;" alt="{\displaystyle {\boldsymbol {\sigma }}_{+k}\approx {\boldsymbol {\sigma }}_{-k}\approx {\boldsymbol {\sigma }}}"> </noscript><span class="lazy-image-placeholder" style="width: 15.715ex;height: 2.009ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ceee98526e30348c66113d6e9e5d397c48fde02e" data-alt="{\displaystyle {\boldsymbol {\sigma }}_{+k}\approx {\boldsymbol {\sigma }}_{-k}\approx {\boldsymbol {\sigma }}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  und somit <dl> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}{\vec {0}}=&\sum _{k=1}^{3}{\hat {e}}_{k}\times ({\hat {e}}_{k}\cdot {\boldsymbol {\sigma }})=\sum _{k=1}^{3}{\hat {e}}_{k}\times \left({\hat {e}}_{k}\cdot \sum _{i,j=1}^{3}\sigma _{ij}{\hat {e}}_{i}\otimes {\hat {e}}_{j}\right)\\=&\sum _{i,j=1}^{3}\sigma _{ij}{\hat {e}}_{i}\times {\hat {e}}_{j}=(\sigma _{12}-\sigma _{21}){\hat {e}}_{3}+(\sigma _{23}-\sigma _{32}){\hat {e}}_{1}+(\sigma _{31}-\sigma _{13}){\hat {e}}_{2}\end{aligned}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mn> 0 </mn> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo> = </mo> </mtd> <mtd> <mi></mi> <munderover> <mo> ∑ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 3 </mn> </mrow> </munderover> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> e </mi> <mo stretchy="false"> ^ </mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> </mrow> </msub> <mo> × </mo> <mo stretchy="false"> ( </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> e </mi> <mo stretchy="false"> ^ </mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> </mrow> </msub> <mo> ⋅ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> σ </mi> </mrow> <mo stretchy="false"> ) </mo> <mo> = </mo> <munderover> <mo> ∑ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 3 </mn> </mrow> </munderover> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> e </mi> <mo stretchy="false"> ^ </mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> </mrow> </msub> <mo> × </mo> <mrow> <mo> ( </mo> <mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> e </mi> <mo stretchy="false"> ^ </mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> </mrow> </msub> <mo> ⋅ </mo> <munderover> <mo> ∑ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> <mo> , </mo> <mi> j </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 3 </mn> </mrow> </munderover> <msub> <mi> σ </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> <mi> j </mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> e </mi> <mo stretchy="false"> ^ </mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo> ⊗ </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> e </mi> <mo stretchy="false"> ^ </mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> j </mi> </mrow> </msub> </mrow> <mo> ) </mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mo> = </mo> </mtd> <mtd> <mi></mi> <munderover> <mo> ∑ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> <mo> , </mo> <mi> j </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 3 </mn> </mrow> </munderover> <msub> <mi> σ </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> <mi> j </mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> e </mi> <mo stretchy="false"> ^ </mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo> × </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> e </mi> <mo stretchy="false"> ^ </mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> j </mi> </mrow> </msub> <mo> = </mo> <mo stretchy="false"> ( </mo> <msub> <mi> σ </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 12 </mn> </mrow> </msub> <mo> − </mo> <msub> <mi> σ </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 21 </mn> </mrow> </msub> <mo stretchy="false"> ) </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> e </mi> <mo stretchy="false"> ^ </mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 3 </mn> </mrow> </msub> <mo> + </mo> <mo stretchy="false"> ( </mo> <msub> <mi> σ </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 23 </mn> </mrow> </msub> <mo> − </mo> <msub> <mi> σ </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 32 </mn> </mrow> </msub> <mo stretchy="false"> ) </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> e </mi> <mo stretchy="false"> ^ </mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> </mrow> </msub> <mo> + </mo> <mo stretchy="false"> ( </mo> <msub> <mi> σ </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 31 </mn> </mrow> </msub> <mo> − </mo> <msub> <mi> σ </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 13 </mn> </mrow> </msub> <mo stretchy="false"> ) </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> e </mi> <mo stretchy="false"> ^ </mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msub> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\begin{aligned}{\vec {0}}=&\sum _{k=1}^{3}{\hat {e}}_{k}\times ({\hat {e}}_{k}\cdot {\boldsymbol {\sigma }})=\sum _{k=1}^{3}{\hat {e}}_{k}\times \left({\hat {e}}_{k}\cdot \sum _{i,j=1}^{3}\sigma _{ij}{\hat {e}}_{i}\otimes {\hat {e}}_{j}\right)\\=&\sum _{i,j=1}^{3}\sigma _{ij}{\hat {e}}_{i}\times {\hat {e}}_{j}=(\sigma _{12}-\sigma _{21}){\hat {e}}_{3}+(\sigma _{23}-\sigma _{32}){\hat {e}}_{1}+(\sigma _{31}-\sigma _{13}){\hat {e}}_{2}\end{aligned}}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5252c81b2ef5bbe2ed2d62147452d6c4328acec4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.9ex; margin-bottom: -0.272ex; width:67.869ex; height:15.509ex;" alt="{\displaystyle {\begin{aligned}{\vec {0}}=&\sum _{k=1}^{3}{\hat {e}}_{k}\times ({\hat {e}}_{k}\cdot {\boldsymbol {\sigma }})=\sum _{k=1}^{3}{\hat {e}}_{k}\times \left({\hat {e}}_{k}\cdot \sum _{i,j=1}^{3}\sigma _{ij}{\hat {e}}_{i}\otimes {\hat {e}}_{j}\right)\\=&\sum _{i,j=1}^{3}\sigma _{ij}{\hat {e}}_{i}\times {\hat {e}}_{j}=(\sigma _{12}-\sigma _{21}){\hat {e}}_{3}+(\sigma _{23}-\sigma _{32}){\hat {e}}_{1}+(\sigma _{31}-\sigma _{13}){\hat {e}}_{2}\end{aligned}}}"> </noscript><span class="lazy-image-placeholder" style="width: 67.869ex;height: 15.509ex;vertical-align: -6.9ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5252c81b2ef5bbe2ed2d62147452d6c4328acec4" data-alt="{\displaystyle {\begin{aligned}{\vec {0}}=&\sum _{k=1}^{3}{\hat {e}}_{k}\times ({\hat {e}}_{k}\cdot {\boldsymbol {\sigma }})=\sum _{k=1}^{3}{\hat {e}}_{k}\times \left({\hat {e}}_{k}\cdot \sum _{i,j=1}^{3}\sigma _{ij}{\hat {e}}_{i}\otimes {\hat {e}}_{j}\right)\\=&\sum _{i,j=1}^{3}\sigma _{ij}{\hat {e}}_{i}\times {\hat {e}}_{j}=(\sigma _{12}-\sigma _{21}){\hat {e}}_{3}+(\sigma _{23}-\sigma _{32}){\hat {e}}_{1}+(\sigma _{31}-\sigma _{13}){\hat {e}}_{2}\end{aligned}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </dd> </dl> Hieraus folgt der eingangs erwähnte Satz von der Gleichheit der zugeordneten Schubspannungen.<a href="https://de-m-wikipedia-org.translate.goog/wiki/Cauchy-eulersche_Bewegungsgesetze?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-szabo-1">[L 1]</a> Die obige Summe kann mit der <a href="https://de-m-wikipedia-org.translate.goog/wiki/Vektorinvariante?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Vektorinvariante">Vektorinvariante</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {\mathrm {i} }}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> i </mi> </mrow> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\vec {\mathrm {i} }}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc27250111830f88f20bd6ece648ee8189f8c49c" 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 {\mathrm {i} }}}"> </noscript><span class="lazy-image-placeholder" style="width: 1.094ex;height: 2.843ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc27250111830f88f20bd6ece648ee8189f8c49c" data-alt="{\displaystyle {\vec {\mathrm {i} }}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  von Tensoren koordinatenfrei ausgedrückt werden: <dl> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {0}}=\sum _{i,j=1}^{3}\sigma _{ij}{\hat {e}}_{i}\times {\hat {e}}_{j}={\vec {\mathrm {i} }}\left(\sum _{i,j=1}^{3}\sigma _{ij}{\hat {e}}_{i}\otimes {\hat {e}}_{j}\right)={\vec {\mathrm {i} }}({\boldsymbol {\sigma }})}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mn> 0 </mn> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo> = </mo> <munderover> <mo> ∑ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> <mo> , </mo> <mi> j </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 3 </mn> </mrow> </munderover> <msub> <mi> σ </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> <mi> j </mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> e </mi> <mo stretchy="false"> ^ </mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo> × </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> e </mi> <mo stretchy="false"> ^ </mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> j </mi> </mrow> </msub> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> i </mi> </mrow> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mrow> <mo> ( </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> <mo> , </mo> <mi> j </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 3 </mn> </mrow> </munderover> <msub> <mi> σ </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> <mi> j </mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> e </mi> <mo stretchy="false"> ^ </mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo> ⊗ </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> e </mi> <mo stretchy="false"> ^ </mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> j </mi> </mrow> </msub> </mrow> <mo> ) </mo> </mrow> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> i </mi> </mrow> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> σ </mi> </mrow> <mo stretchy="false"> ) </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\vec {0}}=\sum _{i,j=1}^{3}\sigma _{ij}{\hat {e}}_{i}\times {\hat {e}}_{j}={\vec {\mathrm {i} }}\left(\sum _{i,j=1}^{3}\sigma _{ij}{\hat {e}}_{i}\otimes {\hat {e}}_{j}\right)={\vec {\mathrm {i} }}({\boldsymbol {\sigma }})} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6bc32f8542e7fabbe08fdb7c23517bc01f42834" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:48.373ex; height:7.676ex;" alt="{\displaystyle {\vec {0}}=\sum _{i,j=1}^{3}\sigma _{ij}{\hat {e}}_{i}\times {\hat {e}}_{j}={\vec {\mathrm {i} }}\left(\sum _{i,j=1}^{3}\sigma _{ij}{\hat {e}}_{i}\otimes {\hat {e}}_{j}\right)={\vec {\mathrm {i} }}({\boldsymbol {\sigma }})}"> </noscript><span class="lazy-image-placeholder" style="width: 48.373ex;height: 7.676ex;vertical-align: -3.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6bc32f8542e7fabbe08fdb7c23517bc01f42834" data-alt="{\displaystyle {\vec {0}}=\sum _{i,j=1}^{3}\sigma _{ij}{\hat {e}}_{i}\times {\hat {e}}_{j}={\vec {\mathrm {i} }}\left(\sum _{i,j=1}^{3}\sigma _{ij}{\hat {e}}_{i}\otimes {\hat {e}}_{j}\right)={\vec {\mathrm {i} }}({\boldsymbol {\sigma }})}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </dd> </dl> Denn in der Vektorinvariante ist das dyadische Produkt ⊗ durch das Kreuzprodukt × ausgetauscht. Nur der <a href="https://de-m-wikipedia-org.translate.goog/wiki/Schiefsymmetrische_Matrix?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Schiefsymmetrische Matrix">schiefsymmetrischer</a> Anteil des Tensors trägt zu seiner Vektorinvariante bei, die hier verschwindet, sodass die <a href="https://de-m-wikipedia-org.translate.goog/wiki/Symmetrische_Matrix?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Symmetrische Matrix">Symmetrie</a> des cauchyschen Spannungstensors folgt: <dl> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {\sigma }}={\boldsymbol {\sigma }}^{\top }}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> σ </mi> </mrow> <mo> = </mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> σ </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> ⊤ </mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\boldsymbol {\sigma }}={\boldsymbol {\sigma }}^{\top }} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8f8a70fa32d7549270a3a5501e977e6874d017b9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.798ex; height:2.676ex;" alt="{\displaystyle {\boldsymbol {\sigma }}={\boldsymbol {\sigma }}^{\top }}"> </noscript><span class="lazy-image-placeholder" style="width: 7.798ex;height: 2.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8f8a70fa32d7549270a3a5501e977e6874d017b9" data-alt="{\displaystyle {\boldsymbol {\sigma }}={\boldsymbol {\sigma }}^{\top }}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </dd> </dl> Diese Tensorgleichung, die in beliebigen <a href="https://de-m-wikipedia-org.translate.goog/wiki/Vektorraumbasis?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Vektorraumbasis">Vektorraumbasen</a> eines <a href="https://de-m-wikipedia-org.translate.goog/wiki/Inertialsystem?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Inertialsystem">Inertialsystems</a> gilt, ist die koordinatenfreie Version des lokalen Drallsatzes. <div class="mw-heading mw-heading3"> <h3 id="Drehimpulssatz_in_lagrangescher_Darstellung">Drehimpulssatz in lagrangescher Darstellung</h3> <a role="button" href="https://de-m-wikipedia-org.translate.goog/w/index.php?title=Cauchy-eulersche_Bewegungsgesetze&action=edit&section=8&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Abschnitt bearbeiten: Drehimpulssatz in lagrangescher Darstellung" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> Bearbeiten </a> </div> Der Drehimpulssatz lautet in globaler lagrangescher Formulierung: <dl> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}\int _{V}({\vec {\chi }}-{\vec {c}})\times \rho _{0}{\dot {\vec {\chi }}}\,\mathrm {d} V=\int _{V}({\vec {\chi }}-{\vec {c}})\times \rho _{0}{\vec {k}}_{0}\,\mathrm {d} V+\int _{A}({\vec {\chi }}-{\vec {c}})\times {\vec {t}}_{0}\,\mathrm {d} A}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> t </mi> </mrow> </mfrac> </mrow> <msub> <mo> ∫ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> V </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> χ </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo> − </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> c </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo stretchy="false"> ) </mo> <mo> × </mo> <msub> <mi> ρ </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> χ </mi> <mo stretchy="false"> → </mo> </mover> </mrow> <mo> ˙ </mo> </mover> </mrow> </mrow> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> V </mi> <mo> = </mo> <msub> <mo> ∫ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> V </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> χ </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo> − </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> c </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo stretchy="false"> ) </mo> <mo> × </mo> <msub> <mi> ρ </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> k </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> V </mi> <mo> + </mo> <msub> <mo> ∫ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> A </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> χ </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo> − </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> c </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo stretchy="false"> ) </mo> <mo> × </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> t </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> A </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}\int _{V}({\vec {\chi }}-{\vec {c}})\times \rho _{0}{\dot {\vec {\chi }}}\,\mathrm {d} V=\int _{V}({\vec {\chi }}-{\vec {c}})\times \rho _{0}{\vec {k}}_{0}\,\mathrm {d} V+\int _{A}({\vec {\chi }}-{\vec {c}})\times {\vec {t}}_{0}\,\mathrm {d} A} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8256e612ef50ef4eb51201a7098e1cb95147d838" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:69.064ex; height:5.843ex;" alt="{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}\int _{V}({\vec {\chi }}-{\vec {c}})\times \rho _{0}{\dot {\vec {\chi }}}\,\mathrm {d} V=\int _{V}({\vec {\chi }}-{\vec {c}})\times \rho _{0}{\vec {k}}_{0}\,\mathrm {d} V+\int _{A}({\vec {\chi }}-{\vec {c}})\times {\vec {t}}_{0}\,\mathrm {d} A}"> </noscript><span class="lazy-image-placeholder" style="width: 69.064ex;height: 5.843ex;vertical-align: -2.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8256e612ef50ef4eb51201a7098e1cb95147d838" data-alt="{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}\int _{V}({\vec {\chi }}-{\vec {c}})\times \rho _{0}{\dot {\vec {\chi }}}\,\mathrm {d} V=\int _{V}({\vec {\chi }}-{\vec {c}})\times \rho _{0}{\vec {k}}_{0}\,\mathrm {d} V+\int _{A}({\vec {\chi }}-{\vec {c}})\times {\vec {t}}_{0}\,\mathrm {d} A}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </dd> </dl> worin die physikalischen Größen zumeist sowohl vom Ort <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {X}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> X </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\vec {X}}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5dc35b5a0226cf11a2c3f2d2dbbac6ab5ade6036" 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}}}"> </noscript><span class="lazy-image-placeholder" style="width: 1.98ex;height: 2.843ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5dc35b5a0226cf11a2c3f2d2dbbac6ab5ade6036" data-alt="{\displaystyle {\vec {X}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  als auch von der Zeit t abhängen, was hier zwecks kompakter Darstellung unterschlagen wurde. Nur die Dichte ρ0 ist wegen der Massenerhaltung keine Funktion der Zeit und der beliebige Ortsvektor <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {c}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> c </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\vec {c}}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/965bd8710781b710cbfdb79da0b4e3b097bef506" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.223ex; height:2.343ex;" alt="{\displaystyle {\vec {c}}}"> </noscript><span class="lazy-image-placeholder" style="width: 1.223ex;height: 2.343ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/965bd8710781b710cbfdb79da0b4e3b097bef506" data-alt="{\displaystyle {\vec {c}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  ist ebenfalls zeitlich fixiert, siehe <a href="https://de-m-wikipedia-org.translate.goog/wiki/Kontinuumsmechanik?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Drehimpulsbilanz" title="Kontinuumsmechanik">Drehimpulsbilanz</a>. Die Zeitableitung des ersten Integrals kann wie beim Impulssatz in den Integranden verschoben werden: <dl> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}\int _{V}({\vec {\chi }}-{\vec {c}})\times \rho _{0}{\dot {\vec {\chi }}}\,\mathrm {d} V=\int _{V}[\underbrace {{\dot {\vec {\chi }}}\times \rho _{0}{\dot {\vec {\chi }}}} _{={\vec {0}}}+({\vec {\chi }}-{\vec {c}})\times \rho _{0}{\ddot {\vec {\chi }}}]\,\mathrm {d} V=\int _{V}({\vec {\chi }}-{\vec {c}})\times \rho _{0}{\ddot {\vec {\chi }}}\,\mathrm {d} V}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> t </mi> </mrow> </mfrac> </mrow> <msub> <mo> ∫ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> V </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> χ </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo> − </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> c </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo stretchy="false"> ) </mo> <mo> × </mo> <msub> <mi> ρ </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> χ </mi> <mo stretchy="false"> → </mo> </mover> </mrow> <mo> ˙ </mo> </mover> </mrow> </mrow> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> V </mi> <mo> = </mo> <msub> <mo> ∫ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> V </mi> </mrow> </msub> <mo stretchy="false"> [ </mo> <munder> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <munder> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> χ </mi> <mo stretchy="false"> → </mo> </mover> </mrow> <mo> ˙ </mo> </mover> </mrow> </mrow> <mo> × </mo> <msub> <mi> ρ </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> χ </mi> <mo stretchy="false"> → </mo> </mover> </mrow> <mo> ˙ </mo> </mover> </mrow> </mrow> </mrow> <mo> ⏟ </mo> </munder> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mn> 0 </mn> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> </mrow> </munder> <mo> + </mo> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> χ </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo> − </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> c </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo stretchy="false"> ) </mo> <mo> × </mo> <msub> <mi> ρ </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> χ </mi> <mo stretchy="false"> → </mo> </mover> </mrow> <mo> ¨ </mo> </mover> </mrow> </mrow> <mo stretchy="false"> ] </mo> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> V </mi> <mo> = </mo> <msub> <mo> ∫ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> V </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> χ </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo> − </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> c </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo stretchy="false"> ) </mo> <mo> × </mo> <msub> <mi> ρ </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> χ </mi> <mo stretchy="false"> → </mo> </mover> </mrow> <mo> ¨ </mo> </mover> </mrow> </mrow> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> V </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}\int _{V}({\vec {\chi }}-{\vec {c}})\times \rho _{0}{\dot {\vec {\chi }}}\,\mathrm {d} V=\int _{V}[\underbrace {{\dot {\vec {\chi }}}\times \rho _{0}{\dot {\vec {\chi }}}} _{={\vec {0}}}+({\vec {\chi }}-{\vec {c}})\times \rho _{0}{\ddot {\vec {\chi }}}]\,\mathrm {d} V=\int _{V}({\vec {\chi }}-{\vec {c}})\times \rho _{0}{\ddot {\vec {\chi }}}\,\mathrm {d} V} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a3366a958c7f9633c72e2f208870051106881007" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.838ex; width:82.098ex; height:8.343ex;" alt="{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}\int _{V}({\vec {\chi }}-{\vec {c}})\times \rho _{0}{\dot {\vec {\chi }}}\,\mathrm {d} V=\int _{V}[\underbrace {{\dot {\vec {\chi }}}\times \rho _{0}{\dot {\vec {\chi }}}} _{={\vec {0}}}+({\vec {\chi }}-{\vec {c}})\times \rho _{0}{\ddot {\vec {\chi }}}]\,\mathrm {d} V=\int _{V}({\vec {\chi }}-{\vec {c}})\times \rho _{0}{\ddot {\vec {\chi }}}\,\mathrm {d} V}"> </noscript><span class="lazy-image-placeholder" style="width: 82.098ex;height: 8.343ex;vertical-align: -4.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a3366a958c7f9633c72e2f208870051106881007" data-alt="{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}\int _{V}({\vec {\chi }}-{\vec {c}})\times \rho _{0}{\dot {\vec {\chi }}}\,\mathrm {d} V=\int _{V}[\underbrace {{\dot {\vec {\chi }}}\times \rho _{0}{\dot {\vec {\chi }}}} _{={\vec {0}}}+({\vec {\chi }}-{\vec {c}})\times \rho _{0}{\ddot {\vec {\chi }}}]\,\mathrm {d} V=\int _{V}({\vec {\chi }}-{\vec {c}})\times \rho _{0}{\ddot {\vec {\chi }}}\,\mathrm {d} V}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </dd> </dl> Das Oberflächenintegral wird wie gehabt mit dem gaußschen Integralsatz in ein Volumenintegral umgeschrieben: <dl> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\int _{A}({\vec {\chi }}-{\vec {c}})\times {\vec {t}}_{0}\,\mathrm {d} A=&\int _{A}({\vec {\chi }}-{\vec {c}})\times (\mathbf {N} ^{\top }\cdot {\vec {N}})\,\mathrm {d} A=\int _{A}[({\vec {\chi }}-{\vec {c}})\times \mathbf {N} ^{\top }]\cdot {\vec {N}}\,\mathrm {d} A\\=&-\int _{A}[\mathbf {N} \times ({\vec {\chi }}-{\vec {c}})]^{\top }\cdot {\vec {N}}\,\mathrm {d} A=-\int _{V}\nabla _{0}\cdot [\mathbf {N} \times ({\vec {\chi }}-{\vec {c}})]\,\mathrm {d} V\\=&-\int _{V}\left\{(\nabla _{0}\cdot \mathbf {N} )\times ({\vec {\chi }}-{\vec {c}})-{\vec {\mathrm {i} }}{\big (}[\nabla _{0}\otimes ({\vec {\chi }}-{\vec {c}})]^{\top }\cdot \mathbf {N} {\big )}\right\}\,\mathrm {d} V\\=&\int _{V}\left\{({\vec {\chi }}-{\vec {c}})\times (\nabla _{0}\cdot \mathbf {N} )+{\vec {\mathrm {i} }}{\big (}\mathbf {F} \cdot \mathbf {N} {\big )}\right\}\,\mathrm {d} V\end{aligned}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mo> ∫ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> A </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> χ </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo> − </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> c </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo stretchy="false"> ) </mo> <mo> × </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> t </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> A </mi> <mo> = </mo> </mtd> <mtd> <mi></mi> <msub> <mo> ∫ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> A </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> χ </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo> − </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> c </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo stretchy="false"> ) </mo> <mo> × </mo> <mo stretchy="false"> ( </mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> N </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> ⊤ </mi> </mrow> </msup> <mo> ⋅ </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> N </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo stretchy="false"> ) </mo> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> A </mi> <mo> = </mo> <msub> <mo> ∫ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> A </mi> </mrow> </msub> <mo stretchy="false"> [ </mo> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> χ </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo> − </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> c </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo stretchy="false"> ) </mo> <mo> × </mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> N </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> ⊤ </mi> </mrow> </msup> <mo stretchy="false"> ] </mo> <mo> ⋅ </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> N </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> A </mi> </mtd> </mtr> <mtr> <mtd> <mo> = </mo> </mtd> <mtd> <mi></mi> <mo> − </mo> <msub> <mo> ∫ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> A </mi> </mrow> </msub> <mo stretchy="false"> [ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> N </mi> </mrow> <mo> × </mo> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> χ </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo> − </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> c </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo stretchy="false"> ) </mo> <msup> <mo stretchy="false"> ] </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> ⊤ </mi> </mrow> </msup> <mo> ⋅ </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> N </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> A </mi> <mo> = </mo> <mo> − </mo> <msub> <mo> ∫ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> V </mi> </mrow> </msub> <msub> <mi mathvariant="normal"> ∇ </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mo> ⋅ </mo> <mo stretchy="false"> [ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> N </mi> </mrow> <mo> × </mo> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> χ </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo> − </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> c </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo stretchy="false"> ) </mo> <mo stretchy="false"> ] </mo> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> V </mi> </mtd> </mtr> <mtr> <mtd> <mo> = </mo> </mtd> <mtd> <mi></mi> <mo> − </mo> <msub> <mo> ∫ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> V </mi> </mrow> </msub> <mrow> <mo> { </mo> <mrow> <mo stretchy="false"> ( </mo> <msub> <mi mathvariant="normal"> ∇ </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mo> ⋅ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> N </mi> </mrow> <mo stretchy="false"> ) </mo> <mo> × </mo> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> χ </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo> − </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> c </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo stretchy="false"> ) </mo> <mo> − </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> i </mi> </mrow> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em"> ( </mo> </mrow> </mrow> <mo stretchy="false"> [ </mo> <msub> <mi mathvariant="normal"> ∇ </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mo> ⊗ </mo> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> χ </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo> − </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> c </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo stretchy="false"> ) </mo> <msup> <mo stretchy="false"> ] </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> ⊤ </mi> </mrow> </msup> <mo> ⋅ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> N </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em"> ) </mo> </mrow> </mrow> </mrow> <mo> } </mo> </mrow> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> V </mi> </mtd> </mtr> <mtr> <mtd> <mo> = </mo> </mtd> <mtd> <mi></mi> <msub> <mo> ∫ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> V </mi> </mrow> </msub> <mrow> <mo> { </mo> <mrow> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> χ </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo> − </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> c </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo stretchy="false"> ) </mo> <mo> × </mo> <mo stretchy="false"> ( </mo> <msub> <mi mathvariant="normal"> ∇ </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mo> ⋅ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> N </mi> </mrow> <mo stretchy="false"> ) </mo> <mo> + </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> i </mi> </mrow> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em"> ( </mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> F </mi> </mrow> <mo> ⋅ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> N </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em"> ) </mo> </mrow> </mrow> </mrow> <mo> } </mo> </mrow> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> V </mi> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\begin{aligned}\int _{A}({\vec {\chi }}-{\vec {c}})\times {\vec {t}}_{0}\,\mathrm {d} A=&\int _{A}({\vec {\chi }}-{\vec {c}})\times (\mathbf {N} ^{\top }\cdot {\vec {N}})\,\mathrm {d} A=\int _{A}[({\vec {\chi }}-{\vec {c}})\times \mathbf {N} ^{\top }]\cdot {\vec {N}}\,\mathrm {d} A\\=&-\int _{A}[\mathbf {N} \times ({\vec {\chi }}-{\vec {c}})]^{\top }\cdot {\vec {N}}\,\mathrm {d} A=-\int _{V}\nabla _{0}\cdot [\mathbf {N} \times ({\vec {\chi }}-{\vec {c}})]\,\mathrm {d} V\\=&-\int _{V}\left\{(\nabla _{0}\cdot \mathbf {N} )\times ({\vec {\chi }}-{\vec {c}})-{\vec {\mathrm {i} }}{\big (}[\nabla _{0}\otimes ({\vec {\chi }}-{\vec {c}})]^{\top }\cdot \mathbf {N} {\big )}\right\}\,\mathrm {d} V\\=&\int _{V}\left\{({\vec {\chi }}-{\vec {c}})\times (\nabla _{0}\cdot \mathbf {N} )+{\vec {\mathrm {i} }}{\big (}\mathbf {F} \cdot \mathbf {N} {\big )}\right\}\,\mathrm {d} V\end{aligned}}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/26e6cf4160f5bf8951c024394712762554a09e7b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -11.022ex; margin-bottom: -0.316ex; width:79.881ex; height:23.843ex;" alt="{\displaystyle {\begin{aligned}\int _{A}({\vec {\chi }}-{\vec {c}})\times {\vec {t}}_{0}\,\mathrm {d} A=&\int _{A}({\vec {\chi }}-{\vec {c}})\times (\mathbf {N} ^{\top }\cdot {\vec {N}})\,\mathrm {d} A=\int _{A}[({\vec {\chi }}-{\vec {c}})\times \mathbf {N} ^{\top }]\cdot {\vec {N}}\,\mathrm {d} A\\=&-\int _{A}[\mathbf {N} \times ({\vec {\chi }}-{\vec {c}})]^{\top }\cdot {\vec {N}}\,\mathrm {d} A=-\int _{V}\nabla _{0}\cdot [\mathbf {N} \times ({\vec {\chi }}-{\vec {c}})]\,\mathrm {d} V\\=&-\int _{V}\left\{(\nabla _{0}\cdot \mathbf {N} )\times ({\vec {\chi }}-{\vec {c}})-{\vec {\mathrm {i} }}{\big (}[\nabla _{0}\otimes ({\vec {\chi }}-{\vec {c}})]^{\top }\cdot \mathbf {N} {\big )}\right\}\,\mathrm {d} V\\=&\int _{V}\left\{({\vec {\chi }}-{\vec {c}})\times (\nabla _{0}\cdot \mathbf {N} )+{\vec {\mathrm {i} }}{\big (}\mathbf {F} \cdot \mathbf {N} {\big )}\right\}\,\mathrm {d} V\end{aligned}}}"> </noscript><span class="lazy-image-placeholder" style="width: 79.881ex;height: 23.843ex;vertical-align: -11.022ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/26e6cf4160f5bf8951c024394712762554a09e7b" data-alt="{\displaystyle {\begin{aligned}\int _{A}({\vec {\chi }}-{\vec {c}})\times {\vec {t}}_{0}\,\mathrm {d} A=&\int _{A}({\vec {\chi }}-{\vec {c}})\times (\mathbf {N} ^{\top }\cdot {\vec {N}})\,\mathrm {d} A=\int _{A}[({\vec {\chi }}-{\vec {c}})\times \mathbf {N} ^{\top }]\cdot {\vec {N}}\,\mathrm {d} A\\=&-\int _{A}[\mathbf {N} \times ({\vec {\chi }}-{\vec {c}})]^{\top }\cdot {\vec {N}}\,\mathrm {d} A=-\int _{V}\nabla _{0}\cdot [\mathbf {N} \times ({\vec {\chi }}-{\vec {c}})]\,\mathrm {d} V\\=&-\int _{V}\left\{(\nabla _{0}\cdot \mathbf {N} )\times ({\vec {\chi }}-{\vec {c}})-{\vec {\mathrm {i} }}{\big (}[\nabla _{0}\otimes ({\vec {\chi }}-{\vec {c}})]^{\top }\cdot \mathbf {N} {\big )}\right\}\,\mathrm {d} V\\=&\int _{V}\left\{({\vec {\chi }}-{\vec {c}})\times (\nabla _{0}\cdot \mathbf {N} )+{\vec {\mathrm {i} }}{\big (}\mathbf {F} \cdot \mathbf {N} {\big )}\right\}\,\mathrm {d} V\end{aligned}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </dd> </dl> Hier wurde die Produktregel <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nabla \cdot (\mathbf {T} \times {\vec {f}})=(\nabla \cdot \mathbf {T} )\times {\vec {f}}-{\vec {\mathrm {i} }}\left((\nabla \otimes {\vec {f}})^{\top }\cdot \mathbf {T} \right)}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal"> ∇ </mi> <mo> ⋅ </mo> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> T </mi> </mrow> <mo> × </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> f </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo stretchy="false"> ) </mo> <mo> = </mo> <mo stretchy="false"> ( </mo> <mi mathvariant="normal"> ∇ </mi> <mo> ⋅ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> T </mi> </mrow> <mo stretchy="false"> ) </mo> <mo> × </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> f </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo> − </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> i </mi> </mrow> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mrow> <mo> ( </mo> <mrow> <mo stretchy="false"> ( </mo> <mi mathvariant="normal"> ∇ </mi> <mo> ⊗ </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> f </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <msup> <mo stretchy="false"> ) </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> ⊤ </mi> </mrow> </msup> <mo> ⋅ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> T </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \nabla \cdot (\mathbf {T} \times {\vec {f}})=(\nabla \cdot \mathbf {T} )\times {\vec {f}}-{\vec {\mathrm {i} }}\left((\nabla \otimes {\vec {f}})^{\top }\cdot \mathbf {T} \right)} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dbebf7e6d11550e38cd1429aede6cdaf87b2b563" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:47.072ex; height:4.843ex;" alt="{\displaystyle \nabla \cdot (\mathbf {T} \times {\vec {f}})=(\nabla \cdot \mathbf {T} )\times {\vec {f}}-{\vec {\mathrm {i} }}\left((\nabla \otimes {\vec {f}})^{\top }\cdot \mathbf {T} \right)}"> </noscript><span class="lazy-image-placeholder" style="width: 47.072ex;height: 4.843ex;vertical-align: -1.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dbebf7e6d11550e38cd1429aede6cdaf87b2b563" data-alt="{\displaystyle \nabla \cdot (\mathbf {T} \times {\vec {f}})=(\nabla \cdot \mathbf {T} )\times {\vec {f}}-{\vec {\mathrm {i} }}\left((\nabla \otimes {\vec {f}})^{\top }\cdot \mathbf {T} \right)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> <a href="https://de-m-wikipedia-org.translate.goog/wiki/Cauchy-eulersche_Bewegungsgesetze?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-prodregel-6">[F 2]</a> und die Definition des <a href="https://de-m-wikipedia-org.translate.goog/wiki/Deformationsgradient?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Deformationsgradient">Deformationsgradienten</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {F} =\operatorname {GRAD} {\vec {\chi }}=(\nabla _{0}\otimes {\vec {\chi }})^{\top }}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> F </mi> </mrow> <mo> = </mo> <mi> GRAD </mi> <mo> ⁡ </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> χ </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo> = </mo> <mo stretchy="false"> ( </mo> <msub> <mi mathvariant="normal"> ∇ </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mo> ⊗ </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> χ </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <msup> <mo stretchy="false"> ) </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> ⊤ </mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbf {F} =\operatorname {GRAD} {\vec {\chi }}=(\nabla _{0}\otimes {\vec {\chi }})^{\top }} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3edc9b1f7bba30f1c53a23580bc255491d84551a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.381ex; height:3.176ex;" alt="{\displaystyle \mathbf {F} =\operatorname {GRAD} {\vec {\chi }}=(\nabla _{0}\otimes {\vec {\chi }})^{\top }}"> </noscript><span class="lazy-image-placeholder" style="width: 27.381ex;height: 3.176ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3edc9b1f7bba30f1c53a23580bc255491d84551a" data-alt="{\displaystyle \mathbf {F} =\operatorname {GRAD} {\vec {\chi }}=(\nabla _{0}\otimes {\vec {\chi }})^{\top }}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  eingesetzt. Die Operatoren 𝜵0 und GRAD bilden den materiellen <a href="https://de-m-wikipedia-org.translate.goog/wiki/Vektorgradient?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Vektorgradient">Vektorgradient</a> mit Ableitungen nach den materiellen Koordinaten X1,2,3, weshalb der Nabla-Operator 𝜵0 einen Index 0 und der Operator GRAD hier in Abgrenzung zum räumlichen Gradienten grad groß geschrieben wird. Mit den vorliegenden Ergebnissen kann die Drehimpulsbilanz als verschwindendes Volumenintegral ausgedrückt werden: <dl> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{V}[({\vec {\chi }}-{\vec {c}})\times (\underbrace {\rho _{0}{\ddot {\vec {\chi }}}-\rho _{0}{\vec {k}}_{0}-\nabla _{0}\cdot \mathbf {N} } _{={\vec {0}}})-{\vec {\mathrm {i} }}{\big (}\mathbf {F} \cdot \mathbf {N} {\big )}]\,\mathrm {d} V=-\int _{V}{\vec {\mathrm {i} }}{\big (}\mathbf {F} \cdot \mathbf {N} {\big )}\,\mathrm {d} V={\vec {0}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mo> ∫ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> V </mi> </mrow> </msub> <mo stretchy="false"> [ </mo> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> χ </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo> − </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> c </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo stretchy="false"> ) </mo> <mo> × </mo> <mo stretchy="false"> ( </mo> <munder> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <munder> <mrow> <msub> <mi> ρ </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> χ </mi> <mo stretchy="false"> → </mo> </mover> </mrow> <mo> ¨ </mo> </mover> </mrow> </mrow> <mo> − </mo> <msub> <mi> ρ </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> k </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mo> − </mo> <msub> <mi mathvariant="normal"> ∇ </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mo> ⋅ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> N </mi> </mrow> </mrow> <mo> ⏟ </mo> </munder> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mn> 0 </mn> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> </mrow> </munder> <mo stretchy="false"> ) </mo> <mo> − </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> i </mi> </mrow> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em"> ( </mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> F </mi> </mrow> <mo> ⋅ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> N </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em"> ) </mo> </mrow> </mrow> <mo stretchy="false"> ] </mo> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> V </mi> <mo> = </mo> <mo> − </mo> <msub> <mo> ∫ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> V </mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> i </mi> </mrow> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em"> ( </mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> F </mi> </mrow> <mo> ⋅ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> N </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em"> ) </mo> </mrow> </mrow> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> V </mi> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mn> 0 </mn> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \int _{V}[({\vec {\chi }}-{\vec {c}})\times (\underbrace {\rho _{0}{\ddot {\vec {\chi }}}-\rho _{0}{\vec {k}}_{0}-\nabla _{0}\cdot \mathbf {N} } _{={\vec {0}}})-{\vec {\mathrm {i} }}{\big (}\mathbf {F} \cdot \mathbf {N} {\big )}]\,\mathrm {d} V=-\int _{V}{\vec {\mathrm {i} }}{\big (}\mathbf {F} \cdot \mathbf {N} {\big )}\,\mathrm {d} V={\vec {0}}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/44077fc5072e7a23b70bd2b90e4c8f92573de9d4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.838ex; width:76.593ex; height:8.176ex;" alt="{\displaystyle \int _{V}[({\vec {\chi }}-{\vec {c}})\times (\underbrace {\rho _{0}{\ddot {\vec {\chi }}}-\rho _{0}{\vec {k}}_{0}-\nabla _{0}\cdot \mathbf {N} } _{={\vec {0}}})-{\vec {\mathrm {i} }}{\big (}\mathbf {F} \cdot \mathbf {N} {\big )}]\,\mathrm {d} V=-\int _{V}{\vec {\mathrm {i} }}{\big (}\mathbf {F} \cdot \mathbf {N} {\big )}\,\mathrm {d} V={\vec {0}}}"> </noscript><span class="lazy-image-placeholder" style="width: 76.593ex;height: 8.176ex;vertical-align: -4.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/44077fc5072e7a23b70bd2b90e4c8f92573de9d4" data-alt="{\displaystyle \int _{V}[({\vec {\chi }}-{\vec {c}})\times (\underbrace {\rho _{0}{\ddot {\vec {\chi }}}-\rho _{0}{\vec {k}}_{0}-\nabla _{0}\cdot \mathbf {N} } _{={\vec {0}}})-{\vec {\mathrm {i} }}{\big (}\mathbf {F} \cdot \mathbf {N} {\big )}]\,\mathrm {d} V=-\int _{V}{\vec {\mathrm {i} }}{\big (}\mathbf {F} \cdot \mathbf {N} {\big )}\,\mathrm {d} V={\vec {0}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </dd> </dl> Der unterklammerte Term trägt wegen der lokalen Impulsbilanz nichts bei. Das letzte Integral gilt für jeden beliebigen Teilkörper, sodass bei stetigem Integrand <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {\mathrm {i} }}{\big (}\mathbf {F} \cdot \mathbf {N} {\big )}={\vec {0}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> i </mi> </mrow> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em"> ( </mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> F </mi> </mrow> <mo> ⋅ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> N </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em"> ) </mo> </mrow> </mrow> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mn> 0 </mn> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\vec {\mathrm {i} }}{\big (}\mathbf {F} \cdot \mathbf {N} {\big )}={\vec {0}}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/160150a866f49086f5ed32a3a418657910c85b00" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.938ex; height:3.509ex;" alt="{\displaystyle {\vec {\mathrm {i} }}{\big (}\mathbf {F} \cdot \mathbf {N} {\big )}={\vec {0}}}"> </noscript><span class="lazy-image-placeholder" style="width: 12.938ex;height: 3.509ex;vertical-align: -1.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/160150a866f49086f5ed32a3a418657910c85b00" data-alt="{\displaystyle {\vec {\mathrm {i} }}{\big (}\mathbf {F} \cdot \mathbf {N} {\big )}={\vec {0}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  und – wie bei der Herleitung am Volumenelement – die Symmetrie von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {F\cdot N} }"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> F </mi> <mo> ⋅ </mo> <mi mathvariant="bold"> N </mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbf {F\cdot N} } </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a1ae0cd1101dafec014df705d90e622949eb1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.453ex; height:2.176ex;" alt="{\displaystyle \mathbf {F\cdot N} }"> </noscript><span class="lazy-image-placeholder" style="width: 5.453ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a1ae0cd1101dafec014df705d90e622949eb1c61" data-alt="{\displaystyle \mathbf {F\cdot N} }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  abgeleitet werden kann. Die lokale Drehimpulsbilanz in der lagrangeschen Darstellung reduziert sich demnach auf die Forderung <dl> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {F\cdot N} =(\mathbf {F\cdot N} )^{\top }=\mathbf {N^{\top }\cdot F^{\top }} }"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> F </mi> <mo> ⋅ </mo> <mi mathvariant="bold"> N </mi> </mrow> <mo> = </mo> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> F </mi> <mo> ⋅ </mo> <mi mathvariant="bold"> N </mi> </mrow> <msup> <mo stretchy="false"> ) </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> ⊤ </mi> </mrow> </msup> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi mathvariant="bold"> N </mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> ⊤ </mi> </mrow> </msup> <mo> ⋅ </mo> <msup> <mi mathvariant="bold"> F </mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> ⊤ </mi> </mrow> </msup> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbf {F\cdot N} =(\mathbf {F\cdot N} )^{\top }=\mathbf {N^{\top }\cdot F^{\top }} } </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/647a023bc418a8fe0a901c742afbc8da40b1bac1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.899ex; height:3.176ex;" alt="{\displaystyle \mathbf {F\cdot N} =(\mathbf {F\cdot N} )^{\top }=\mathbf {N^{\top }\cdot F^{\top }} }"> </noscript><span class="lazy-image-placeholder" style="width: 28.899ex;height: 3.176ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/647a023bc418a8fe0a901c742afbc8da40b1bac1" data-alt="{\displaystyle \mathbf {F\cdot N} =(\mathbf {F\cdot N} )^{\top }=\mathbf {N^{\top }\cdot F^{\top }} }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </dd> </dl> Multiplikation von links mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {F} ^{-1}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> F </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo> − </mo> <mn> 1 </mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbf {F} ^{-1}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f22670c5594281f4ec938aa0182280ce4b78967e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.015ex; height:2.676ex;" alt="{\displaystyle \mathbf {F} ^{-1}}"> </noscript><span class="lazy-image-placeholder" style="width: 4.015ex;height: 2.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f22670c5594281f4ec938aa0182280ce4b78967e" data-alt="{\displaystyle \mathbf {F} ^{-1}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  und von rechts mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {F} ^{\top -1}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> F </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> ⊤ </mi> <mo> − </mo> <mn> 1 </mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbf {F} ^{\top -1}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d99825abd03cf66b76c20e63ce2f4f27045fda06" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.294ex; height:2.676ex;" alt="{\displaystyle \mathbf {F} ^{\top -1}}"> </noscript><span class="lazy-image-placeholder" style="width: 5.294ex;height: 2.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d99825abd03cf66b76c20e63ce2f4f27045fda06" data-alt="{\displaystyle \mathbf {F} ^{\top -1}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  ergibt gleichbedeutend: <dl> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {F} ^{-1}\cdot \mathbf {F\cdot N\cdot F} ^{\top -1}=\mathbf {F} ^{-1}\cdot \mathbf {N^{\top }\cdot F^{\top }\cdot F} ^{\top -1}\;\leftrightarrow \;\mathbf {N\cdot F} ^{\top -1}=(\mathbf {N\cdot F} ^{\top -1})^{\top }}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> F </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo> − </mo> <mn> 1 </mn> </mrow> </msup> <mo> ⋅ </mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> F </mi> <mo> ⋅ </mo> <mi mathvariant="bold"> N </mi> <mo> ⋅ </mo> <mi mathvariant="bold"> F </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> ⊤ </mi> <mo> − </mo> <mn> 1 </mn> </mrow> </msup> <mo> = </mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> F </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo> − </mo> <mn> 1 </mn> </mrow> </msup> <mo> ⋅ </mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi mathvariant="bold"> N </mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> ⊤ </mi> </mrow> </msup> <mo> ⋅ </mo> <msup> <mi mathvariant="bold"> F </mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> ⊤ </mi> </mrow> </msup> <mo> ⋅ </mo> <mi mathvariant="bold"> F </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> ⊤ </mi> <mo> − </mo> <mn> 1 </mn> </mrow> </msup> <mspace width="thickmathspace"></mspace> <mo stretchy="false"> ↔ </mo> <mspace width="thickmathspace"></mspace> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> N </mi> <mo> ⋅ </mo> <mi mathvariant="bold"> F </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> ⊤ </mi> <mo> − </mo> <mn> 1 </mn> </mrow> </msup> <mo> = </mo> <mo stretchy="false"> ( </mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> N </mi> <mo> ⋅ </mo> <mi mathvariant="bold"> F </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> ⊤ </mi> <mo> − </mo> <mn> 1 </mn> </mrow> </msup> <msup> <mo stretchy="false"> ) </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> ⊤ </mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbf {F} ^{-1}\cdot \mathbf {F\cdot N\cdot F} ^{\top -1}=\mathbf {F} ^{-1}\cdot \mathbf {N^{\top }\cdot F^{\top }\cdot F} ^{\top -1}\;\leftrightarrow \;\mathbf {N\cdot F} ^{\top -1}=(\mathbf {N\cdot F} ^{\top -1})^{\top }} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4e30a46af9876934e48ff7dc64a108fa252ce35d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:71.814ex; height:3.676ex;" alt="{\displaystyle \mathbf {F} ^{-1}\cdot \mathbf {F\cdot N\cdot F} ^{\top -1}=\mathbf {F} ^{-1}\cdot \mathbf {N^{\top }\cdot F^{\top }\cdot F} ^{\top -1}\;\leftrightarrow \;\mathbf {N\cdot F} ^{\top -1}=(\mathbf {N\cdot F} ^{\top -1})^{\top }}"> </noscript><span class="lazy-image-placeholder" style="width: 71.814ex;height: 3.676ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4e30a46af9876934e48ff7dc64a108fa252ce35d" data-alt="{\displaystyle \mathbf {F} ^{-1}\cdot \mathbf {F\cdot N\cdot F} ^{\top -1}=\mathbf {F} ^{-1}\cdot \mathbf {N^{\top }\cdot F^{\top }\cdot F} ^{\top -1}\;\leftrightarrow \;\mathbf {N\cdot F} ^{\top -1}=(\mathbf {N\cdot F} ^{\top -1})^{\top }}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </dd> </dl> Der Tensor <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tilde {\mathbf {T} }}:=\mathbf {N\cdot F} ^{\top -1}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> T </mi> </mrow> <mo stretchy="false"> ~ </mo> </mover> </mrow> </mrow> <mo> := </mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> N </mi> <mo> ⋅ </mo> <mi mathvariant="bold"> F </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> ⊤ </mi> <mo> − </mo> <mn> 1 </mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\tilde {\mathbf {T} }}:=\mathbf {N\cdot F} ^{\top -1}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e6c41f747cb97c0aa3d5981b9e0a0e8ba7e11583" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:14.669ex; height:2.676ex;" alt="{\displaystyle {\tilde {\mathbf {T} }}:=\mathbf {N\cdot F} ^{\top -1}}"> </noscript><span class="lazy-image-placeholder" style="width: 14.669ex;height: 2.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e6c41f747cb97c0aa3d5981b9e0a0e8ba7e11583" data-alt="{\displaystyle {\tilde {\mathbf {T} }}:=\mathbf {N\cdot F} ^{\top -1}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  ist der zweite piola-kirchhoffsche Spannungstensor, dessen Symmetrie gemäß <dl> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tilde {\mathbf {T} }}={\tilde {\mathbf {T} }}^{\top }}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> T </mi> </mrow> <mo stretchy="false"> ~ </mo> </mover> </mrow> </mrow> <mo> = </mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> T </mi> </mrow> <mo stretchy="false"> ~ </mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> ⊤ </mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\tilde {\mathbf {T} }}={\tilde {\mathbf {T} }}^{\top }} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5e39f790569850d8258152615cea2fccb6fa118" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.328ex; height:3.176ex;" alt="{\displaystyle {\tilde {\mathbf {T} }}={\tilde {\mathbf {T} }}^{\top }}"> </noscript><span class="lazy-image-placeholder" style="width: 8.328ex;height: 3.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5e39f790569850d8258152615cea2fccb6fa118" data-alt="{\displaystyle {\tilde {\mathbf {T} }}={\tilde {\mathbf {T} }}^{\top }}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </dd> </dl> die Erfüllung der Drehimpulsbilanz sicherstellt. Bei kleinen Verschiebungen stimmen der zweite piola-kirchhoffsche und der cauchysche Spannungstensor näherungsweise überein: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tilde {\mathbf {T} }}\approx {\boldsymbol {\sigma }}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> T </mi> </mrow> <mo stretchy="false"> ~ </mo> </mover> </mrow> </mrow> <mo> ≈ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> σ </mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\tilde {\mathbf {T} }}\approx {\boldsymbol {\sigma }}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/896bb41fa7c39a198dc3208d776ecf483b90f2a5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.552ex; height:2.676ex;" alt="{\displaystyle {\tilde {\mathbf {T} }}\approx {\boldsymbol {\sigma }}}"> </noscript><span class="lazy-image-placeholder" style="width: 6.552ex;height: 2.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/896bb41fa7c39a198dc3208d776ecf483b90f2a5" data-alt="{\displaystyle {\tilde {\mathbf {T} }}\approx {\boldsymbol {\sigma }}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> . <div class="mw-heading mw-heading3"> <h3 id="Drehimpulssatz_in_eulerscher_Darstellung">Drehimpulssatz in eulerscher Darstellung</h3> <a role="button" href="https://de-m-wikipedia-org.translate.goog/w/index.php?title=Cauchy-eulersche_Bewegungsgesetze&action=edit&section=9&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Abschnitt bearbeiten: Drehimpulssatz in eulerscher Darstellung" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> Bearbeiten </a> </div> In globaler eulerscher Formulierung lautet der Drehimpulssatz: <dl> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}\int _{v}({\vec {x}}-{\vec {c}})\times \rho {\vec {v}}\,\mathrm {d} v=\int _{v}({\vec {x}}-{\vec {c}})\times \rho {\vec {k}}\,\mathrm {d} v+\int _{a}({\vec {x}}-{\vec {c}})\times {\vec {t}}\,\mathrm {d} a}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> t </mi> </mrow> </mfrac> </mrow> <msub> <mo> ∫ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> v </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> x </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo> − </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> c </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo stretchy="false"> ) </mo> <mo> × </mo> <mi> ρ </mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> v </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> v </mi> <mo> = </mo> <msub> <mo> ∫ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> v </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> x </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo> − </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> c </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo stretchy="false"> ) </mo> <mo> × </mo> <mi> ρ </mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> k </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> v </mi> <mo> + </mo> <msub> <mo> ∫ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> a </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> x </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo> − </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> c </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo stretchy="false"> ) </mo> <mo> × </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> t </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> a </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}\int _{v}({\vec {x}}-{\vec {c}})\times \rho {\vec {v}}\,\mathrm {d} v=\int _{v}({\vec {x}}-{\vec {c}})\times \rho {\vec {k}}\,\mathrm {d} v+\int _{a}({\vec {x}}-{\vec {c}})\times {\vec {t}}\,\mathrm {d} a} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2910cf682aca524a206770d97e2c06f875e6c6cc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:61.063ex; height:5.843ex;" alt="{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}\int _{v}({\vec {x}}-{\vec {c}})\times \rho {\vec {v}}\,\mathrm {d} v=\int _{v}({\vec {x}}-{\vec {c}})\times \rho {\vec {k}}\,\mathrm {d} v+\int _{a}({\vec {x}}-{\vec {c}})\times {\vec {t}}\,\mathrm {d} a}"> </noscript><span class="lazy-image-placeholder" style="width: 61.063ex;height: 5.843ex;vertical-align: -2.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2910cf682aca524a206770d97e2c06f875e6c6cc" data-alt="{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}\int _{v}({\vec {x}}-{\vec {c}})\times \rho {\vec {v}}\,\mathrm {d} v=\int _{v}({\vec {x}}-{\vec {c}})\times \rho {\vec {k}}\,\mathrm {d} v+\int _{a}({\vec {x}}-{\vec {c}})\times {\vec {t}}\,\mathrm {d} a}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </dd> </dl> worin die physikalischen Größen zumeist Funktionen sowohl vom Ort <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {x}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> x </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\vec {x}}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/db2dc6ced9cc3bc7e8b9f2707cbec033f6d3759c" 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}}}"> </noscript><span class="lazy-image-placeholder" style="width: 1.33ex;height: 2.343ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/db2dc6ced9cc3bc7e8b9f2707cbec033f6d3759c" data-alt="{\displaystyle {\vec {x}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  als auch von der Zeit t sind, was hier zwecks kompakter Darstellung unterschlagen wurde. Der Vektor <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {c}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> c </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\vec {c}}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/965bd8710781b710cbfdb79da0b4e3b097bef506" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.223ex; height:2.343ex;" alt="{\displaystyle {\vec {c}}}"> </noscript><span class="lazy-image-placeholder" style="width: 1.223ex;height: 2.343ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/965bd8710781b710cbfdb79da0b4e3b097bef506" data-alt="{\displaystyle {\vec {c}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  ist beliebig und zeitlich fixiert, und die räumlichen Koordinaten <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {x}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> x </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\vec {x}}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/db2dc6ced9cc3bc7e8b9f2707cbec033f6d3759c" 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}}}"> </noscript><span class="lazy-image-placeholder" style="width: 1.33ex;height: 2.343ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/db2dc6ced9cc3bc7e8b9f2707cbec033f6d3759c" data-alt="{\displaystyle {\vec {x}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  stellen Integrationsvariable dar, die daher auch nicht von der Zeit abhängen. Das erste Integral wird wie bei der Impulsbilanz mit dem reynoldsschen Transportsatz berechnet: <dl> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}{\frac {\mathrm {d} }{\mathrm {d} t}}\int _{v}({\vec {x}}-{\vec {c}})\times \rho {\vec {v}}\,\mathrm {d} v=&\int _{v}\left\{{\frac {\mathrm {d} }{\mathrm {d} t}}[({\vec {x}}-{\vec {c}})\times \rho {\vec {v}}]+\operatorname {div} ({\vec {v}})({\vec {x}}-{\vec {c}})\times \rho {\vec {v}}\right\}\,\mathrm {d} v\\=&\int _{v}({\vec {x}}-{\vec {c}})\times [\underbrace {{\dot {\rho }}\,{\vec {v}}+\operatorname {div} ({\vec {v}})\rho \,{\vec {v}}} _{={\vec {0}}}+\rho \,{\dot {\vec {v}}}]\,\mathrm {d} v=\int _{v}({\vec {x}}-{\vec {c}})\times \rho \,{\dot {\vec {v}}}\,\mathrm {d} v\end{aligned}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> t </mi> </mrow> </mfrac> </mrow> <msub> <mo> ∫ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> v </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> x </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo> − </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> c </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo stretchy="false"> ) </mo> <mo> × </mo> <mi> ρ </mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> v </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> v </mi> <mo> = </mo> </mtd> <mtd> <mi></mi> <msub> <mo> ∫ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> v </mi> </mrow> </msub> <mrow> <mo> { </mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> t </mi> </mrow> </mfrac> </mrow> <mo stretchy="false"> [ </mo> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> x </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo> − </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> c </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo stretchy="false"> ) </mo> <mo> × </mo> <mi> ρ </mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> v </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo stretchy="false"> ] </mo> <mo> + </mo> <mi> div </mi> <mo> ⁡ </mo> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> v </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo stretchy="false"> ) </mo> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> x </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo> − </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> c </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo stretchy="false"> ) </mo> <mo> × </mo> <mi> ρ </mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> v </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> </mrow> <mo> } </mo> </mrow> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> v </mi> </mtd> </mtr> <mtr> <mtd> <mo> = </mo> </mtd> <mtd> <mi></mi> <msub> <mo> ∫ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> v </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> x </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo> − </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> c </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo stretchy="false"> ) </mo> <mo> × </mo> <mo stretchy="false"> [ </mo> <munder> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <munder> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> ρ </mi> <mo> ˙ </mo> </mover> </mrow> </mrow> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> v </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo> + </mo> <mi> div </mi> <mo> ⁡ </mo> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> v </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo stretchy="false"> ) </mo> <mi> ρ </mi> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> v </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> </mrow> <mo> ⏟ </mo> </munder> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mn> 0 </mn> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> </mrow> </munder> <mo> + </mo> <mi> ρ </mi> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> v </mi> <mo stretchy="false"> → </mo> </mover> </mrow> <mo> ˙ </mo> </mover> </mrow> </mrow> <mo stretchy="false"> ] </mo> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> v </mi> <mo> = </mo> <msub> <mo> ∫ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> v </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> x </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo> − </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> c </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo stretchy="false"> ) </mo> <mo> × </mo> <mi> ρ </mi> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> v </mi> <mo stretchy="false"> → </mo> </mover> </mrow> <mo> ˙ </mo> </mover> </mrow> </mrow> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> v </mi> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\begin{aligned}{\frac {\mathrm {d} }{\mathrm {d} t}}\int _{v}({\vec {x}}-{\vec {c}})\times \rho {\vec {v}}\,\mathrm {d} v=&\int _{v}\left\{{\frac {\mathrm {d} }{\mathrm {d} t}}[({\vec {x}}-{\vec {c}})\times \rho {\vec {v}}]+\operatorname {div} ({\vec {v}})({\vec {x}}-{\vec {c}})\times \rho {\vec {v}}\right\}\,\mathrm {d} v\\=&\int _{v}({\vec {x}}-{\vec {c}})\times [\underbrace {{\dot {\rho }}\,{\vec {v}}+\operatorname {div} ({\vec {v}})\rho \,{\vec {v}}} _{={\vec {0}}}+\rho \,{\dot {\vec {v}}}]\,\mathrm {d} v=\int _{v}({\vec {x}}-{\vec {c}})\times \rho \,{\dot {\vec {v}}}\,\mathrm {d} v\end{aligned}}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a499e1dfd5282978a1a520a73f709bdbf055c7fe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.671ex; width:82.365ex; height:14.509ex;" alt="{\displaystyle {\begin{aligned}{\frac {\mathrm {d} }{\mathrm {d} t}}\int _{v}({\vec {x}}-{\vec {c}})\times \rho {\vec {v}}\,\mathrm {d} v=&\int _{v}\left\{{\frac {\mathrm {d} }{\mathrm {d} t}}[({\vec {x}}-{\vec {c}})\times \rho {\vec {v}}]+\operatorname {div} ({\vec {v}})({\vec {x}}-{\vec {c}})\times \rho {\vec {v}}\right\}\,\mathrm {d} v\\=&\int _{v}({\vec {x}}-{\vec {c}})\times [\underbrace {{\dot {\rho }}\,{\vec {v}}+\operatorname {div} ({\vec {v}})\rho \,{\vec {v}}} _{={\vec {0}}}+\rho \,{\dot {\vec {v}}}]\,\mathrm {d} v=\int _{v}({\vec {x}}-{\vec {c}})\times \rho \,{\dot {\vec {v}}}\,\mathrm {d} v\end{aligned}}}"> </noscript><span class="lazy-image-placeholder" style="width: 82.365ex;height: 14.509ex;vertical-align: -6.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a499e1dfd5282978a1a520a73f709bdbf055c7fe" data-alt="{\displaystyle {\begin{aligned}{\frac {\mathrm {d} }{\mathrm {d} t}}\int _{v}({\vec {x}}-{\vec {c}})\times \rho {\vec {v}}\,\mathrm {d} v=&\int _{v}\left\{{\frac {\mathrm {d} }{\mathrm {d} t}}[({\vec {x}}-{\vec {c}})\times \rho {\vec {v}}]+\operatorname {div} ({\vec {v}})({\vec {x}}-{\vec {c}})\times \rho {\vec {v}}\right\}\,\mathrm {d} v\\=&\int _{v}({\vec {x}}-{\vec {c}})\times [\underbrace {{\dot {\rho }}\,{\vec {v}}+\operatorname {div} ({\vec {v}})\rho \,{\vec {v}}} _{={\vec {0}}}+\rho \,{\dot {\vec {v}}}]\,\mathrm {d} v=\int _{v}({\vec {x}}-{\vec {c}})\times \rho \,{\dot {\vec {v}}}\,\mathrm {d} v\end{aligned}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </dd> </dl> Der unterklammerte Term trägt aufgrund der Massenbilanz nichts bei. Das Oberflächenintegral in der Drehimpulsbilanz wird analog zur lagrangeschen Darstellung mit dem gaußschen Integralsatz in ein Volumenintegral umgeschrieben:<a href="https://de-m-wikipedia-org.translate.goog/wiki/Cauchy-eulersche_Bewegungsgesetze?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-prodregel-6">[F 2]</a> <dl> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\int _{a}({\vec {x}}-{\vec {c}})\times {\vec {t}}\,\mathrm {d} a=&\int _{a}({\vec {x}}-{\vec {c}})\times ({\boldsymbol {\sigma }}^{\top }\cdot {\vec {n}})\,\mathrm {d} a=\int _{a}[({\vec {x}}-{\vec {c}})\times {\boldsymbol {\sigma }}^{\top }]\cdot {\vec {n}}\,\mathrm {d} a\\=&-\int _{a}[{\boldsymbol {\sigma }}\times ({\vec {x}}-{\vec {c}})]^{\top }\cdot {\vec {n}}\,\mathrm {d} a=-\int _{v}\nabla \cdot [{\boldsymbol {\sigma }}\times ({\vec {x}}-{\vec {c}})]\,\mathrm {d} v\\=&-\int _{v}\left\{(\nabla \cdot {\boldsymbol {\sigma }})\times ({\vec {x}}-{\vec {c}})-{\vec {\operatorname {i} }}\left([\nabla \otimes ({\vec {x}}-{\vec {c}})]^{\top }\cdot {\boldsymbol {\sigma }}\right)\right\}\,\mathrm {d} v\\=&\int _{v}\left\{({\vec {x}}-{\vec {c}})\times (\nabla \cdot {\boldsymbol {\sigma }})+{\vec {\operatorname {i} }}(\mathbf {1} \cdot {\boldsymbol {\sigma }})\right\}\,\mathrm {d} v\\=&\int _{v}\left[({\vec {x}}-{\vec {c}})\times (\nabla \cdot {\boldsymbol {\sigma }})+{\vec {\operatorname {i} }}({\boldsymbol {\sigma }})\right]\,\mathrm {d} v\end{aligned}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mo> ∫ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> a </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> x </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo> − </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> c </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo stretchy="false"> ) </mo> <mo> × </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> t </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> a </mi> <mo> = </mo> </mtd> <mtd> <mi></mi> <msub> <mo> ∫ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> a </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> x </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo> − </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> c </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo stretchy="false"> ) </mo> <mo> × </mo> <mo stretchy="false"> ( </mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> σ </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> ⊤ </mi> </mrow> </msup> <mo> ⋅ </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> n </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo stretchy="false"> ) </mo> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> a </mi> <mo> = </mo> <msub> <mo> ∫ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> a </mi> </mrow> </msub> <mo stretchy="false"> [ </mo> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> x </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo> − </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> c </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo stretchy="false"> ) </mo> <mo> × </mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> σ </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> ⊤ </mi> </mrow> </msup> <mo stretchy="false"> ] </mo> <mo> ⋅ </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> n </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> a </mi> </mtd> </mtr> <mtr> <mtd> <mo> = </mo> </mtd> <mtd> <mi></mi> <mo> − </mo> <msub> <mo> ∫ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> a </mi> </mrow> </msub> <mo stretchy="false"> [ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> σ </mi> </mrow> <mo> × </mo> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> x </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo> − </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> c </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo stretchy="false"> ) </mo> <msup> <mo stretchy="false"> ] </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> ⊤ </mi> </mrow> </msup> <mo> ⋅ </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> n </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> a </mi> <mo> = </mo> <mo> − </mo> <msub> <mo> ∫ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> v </mi> </mrow> </msub> <mi mathvariant="normal"> ∇ </mi> <mo> ⋅ </mo> <mo stretchy="false"> [ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> σ </mi> </mrow> <mo> × </mo> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> x </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo> − </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> c </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo stretchy="false"> ) </mo> <mo stretchy="false"> ] </mo> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> v </mi> </mtd> </mtr> <mtr> <mtd> <mo> = </mo> </mtd> <mtd> <mi></mi> <mo> − </mo> <msub> <mo> ∫ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> v </mi> </mrow> </msub> <mrow> <mo> { </mo> <mrow> <mo stretchy="false"> ( </mo> <mi mathvariant="normal"> ∇ </mi> <mo> ⋅ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> σ </mi> </mrow> <mo stretchy="false"> ) </mo> <mo> × </mo> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> x </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo> − </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> c </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo stretchy="false"> ) </mo> <mo> − </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="normal"> i </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mrow> <mo> ( </mo> <mrow> <mo stretchy="false"> [ </mo> <mi mathvariant="normal"> ∇ </mi> <mo> ⊗ </mo> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> x </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo> − </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> c </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo stretchy="false"> ) </mo> <msup> <mo stretchy="false"> ] </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> ⊤ </mi> </mrow> </msup> <mo> ⋅ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> σ </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> } </mo> </mrow> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> v </mi> </mtd> </mtr> <mtr> <mtd> <mo> = </mo> </mtd> <mtd> <mi></mi> <msub> <mo> ∫ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> v </mi> </mrow> </msub> <mrow> <mo> { </mo> <mrow> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> x </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo> − </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> c </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo stretchy="false"> ) </mo> <mo> × </mo> <mo stretchy="false"> ( </mo> <mi mathvariant="normal"> ∇ </mi> <mo> ⋅ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> σ </mi> </mrow> <mo stretchy="false"> ) </mo> <mo> + </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="normal"> i </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold"> 1 </mn> </mrow> <mo> ⋅ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> σ </mi> </mrow> <mo stretchy="false"> ) </mo> </mrow> <mo> } </mo> </mrow> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> v </mi> </mtd> </mtr> <mtr> <mtd> <mo> = </mo> </mtd> <mtd> <mi></mi> <msub> <mo> ∫ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> v </mi> </mrow> </msub> <mrow> <mo> [ </mo> <mrow> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> x </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo> − </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> c </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo stretchy="false"> ) </mo> <mo> × </mo> <mo stretchy="false"> ( </mo> <mi mathvariant="normal"> ∇ </mi> <mo> ⋅ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> σ </mi> </mrow> <mo stretchy="false"> ) </mo> <mo> + </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="normal"> i </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> σ </mi> </mrow> <mo stretchy="false"> ) </mo> </mrow> <mo> ] </mo> </mrow> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> v </mi> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\begin{aligned}\int _{a}({\vec {x}}-{\vec {c}})\times {\vec {t}}\,\mathrm {d} a=&\int _{a}({\vec {x}}-{\vec {c}})\times ({\boldsymbol {\sigma }}^{\top }\cdot {\vec {n}})\,\mathrm {d} a=\int _{a}[({\vec {x}}-{\vec {c}})\times {\boldsymbol {\sigma }}^{\top }]\cdot {\vec {n}}\,\mathrm {d} a\\=&-\int _{a}[{\boldsymbol {\sigma }}\times ({\vec {x}}-{\vec {c}})]^{\top }\cdot {\vec {n}}\,\mathrm {d} a=-\int _{v}\nabla \cdot [{\boldsymbol {\sigma }}\times ({\vec {x}}-{\vec {c}})]\,\mathrm {d} v\\=&-\int _{v}\left\{(\nabla \cdot {\boldsymbol {\sigma }})\times ({\vec {x}}-{\vec {c}})-{\vec {\operatorname {i} }}\left([\nabla \otimes ({\vec {x}}-{\vec {c}})]^{\top }\cdot {\boldsymbol {\sigma }}\right)\right\}\,\mathrm {d} v\\=&\int _{v}\left\{({\vec {x}}-{\vec {c}})\times (\nabla \cdot {\boldsymbol {\sigma }})+{\vec {\operatorname {i} }}(\mathbf {1} \cdot {\boldsymbol {\sigma }})\right\}\,\mathrm {d} v\\=&\int _{v}\left[({\vec {x}}-{\vec {c}})\times (\nabla \cdot {\boldsymbol {\sigma }})+{\vec {\operatorname {i} }}({\boldsymbol {\sigma }})\right]\,\mathrm {d} v\end{aligned}}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8f49bc93411a34b53e934514520d0b562cdbdcfa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -14.171ex; width:73.253ex; height:29.509ex;" alt="{\displaystyle {\begin{aligned}\int _{a}({\vec {x}}-{\vec {c}})\times {\vec {t}}\,\mathrm {d} a=&\int _{a}({\vec {x}}-{\vec {c}})\times ({\boldsymbol {\sigma }}^{\top }\cdot {\vec {n}})\,\mathrm {d} a=\int _{a}[({\vec {x}}-{\vec {c}})\times {\boldsymbol {\sigma }}^{\top }]\cdot {\vec {n}}\,\mathrm {d} a\\=&-\int _{a}[{\boldsymbol {\sigma }}\times ({\vec {x}}-{\vec {c}})]^{\top }\cdot {\vec {n}}\,\mathrm {d} a=-\int _{v}\nabla \cdot [{\boldsymbol {\sigma }}\times ({\vec {x}}-{\vec {c}})]\,\mathrm {d} v\\=&-\int _{v}\left\{(\nabla \cdot {\boldsymbol {\sigma }})\times ({\vec {x}}-{\vec {c}})-{\vec {\operatorname {i} }}\left([\nabla \otimes ({\vec {x}}-{\vec {c}})]^{\top }\cdot {\boldsymbol {\sigma }}\right)\right\}\,\mathrm {d} v\\=&\int _{v}\left\{({\vec {x}}-{\vec {c}})\times (\nabla \cdot {\boldsymbol {\sigma }})+{\vec {\operatorname {i} }}(\mathbf {1} \cdot {\boldsymbol {\sigma }})\right\}\,\mathrm {d} v\\=&\int _{v}\left[({\vec {x}}-{\vec {c}})\times (\nabla \cdot {\boldsymbol {\sigma }})+{\vec {\operatorname {i} }}({\boldsymbol {\sigma }})\right]\,\mathrm {d} v\end{aligned}}}"> </noscript><span class="lazy-image-placeholder" style="width: 73.253ex;height: 29.509ex;vertical-align: -14.171ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8f49bc93411a34b53e934514520d0b562cdbdcfa" data-alt="{\displaystyle {\begin{aligned}\int _{a}({\vec {x}}-{\vec {c}})\times {\vec {t}}\,\mathrm {d} a=&\int _{a}({\vec {x}}-{\vec {c}})\times ({\boldsymbol {\sigma }}^{\top }\cdot {\vec {n}})\,\mathrm {d} a=\int _{a}[({\vec {x}}-{\vec {c}})\times {\boldsymbol {\sigma }}^{\top }]\cdot {\vec {n}}\,\mathrm {d} a\\=&-\int _{a}[{\boldsymbol {\sigma }}\times ({\vec {x}}-{\vec {c}})]^{\top }\cdot {\vec {n}}\,\mathrm {d} a=-\int _{v}\nabla \cdot [{\boldsymbol {\sigma }}\times ({\vec {x}}-{\vec {c}})]\,\mathrm {d} v\\=&-\int _{v}\left\{(\nabla \cdot {\boldsymbol {\sigma }})\times ({\vec {x}}-{\vec {c}})-{\vec {\operatorname {i} }}\left([\nabla \otimes ({\vec {x}}-{\vec {c}})]^{\top }\cdot {\boldsymbol {\sigma }}\right)\right\}\,\mathrm {d} v\\=&\int _{v}\left\{({\vec {x}}-{\vec {c}})\times (\nabla \cdot {\boldsymbol {\sigma }})+{\vec {\operatorname {i} }}(\mathbf {1} \cdot {\boldsymbol {\sigma }})\right\}\,\mathrm {d} v\\=&\int _{v}\left[({\vec {x}}-{\vec {c}})\times (\nabla \cdot {\boldsymbol {\sigma }})+{\vec {\operatorname {i} }}({\boldsymbol {\sigma }})\right]\,\mathrm {d} v\end{aligned}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </dd> </dl> Abweichend von der lagrangeschen Darstellung tritt hier der cauchysche Spannungstensor an die Stelle des Nennspannungstensors und wegen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\nabla \otimes {\vec {x}})^{\top }=\operatorname {grad} {\vec {x}}=\mathbf {1} }"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false"> ( </mo> <mi mathvariant="normal"> ∇ </mi> <mo> ⊗ </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> x </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <msup> <mo stretchy="false"> ) </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> ⊤ </mi> </mrow> </msup> <mo> = </mo> <mi> grad </mi> <mo> ⁡ </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> x </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold"> 1 </mn> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle (\nabla \otimes {\vec {x}})^{\top }=\operatorname {grad} {\vec {x}}=\mathbf {1} } </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1fb2866425b76d47900f36ca043f164aae5dec98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.205ex; height:3.176ex;" alt="{\displaystyle (\nabla \otimes {\vec {x}})^{\top }=\operatorname {grad} {\vec {x}}=\mathbf {1} }"> </noscript><span class="lazy-image-placeholder" style="width: 23.205ex;height: 3.176ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1fb2866425b76d47900f36ca043f164aae5dec98" data-alt="{\displaystyle (\nabla \otimes {\vec {x}})^{\top }=\operatorname {grad} {\vec {x}}=\mathbf {1} }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  der Einheitstensor an die Stelle des Deformationsgradienten. Mit den vorliegenden Ergebnissen kann die Drehimpulsbilanz als verschwindendes Volumenintegral ausgedrückt werden: <dl> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{v}[({\vec {x}}-{\vec {c}})\times (\underbrace {\rho {\dot {\vec {v}}}-\rho {\vec {k}}-\nabla \cdot {\boldsymbol {\sigma }}} _{={\vec {0}}})-{\vec {\operatorname {i} }}({\boldsymbol {\sigma }})]\,\mathrm {d} v=-\int _{v}{\vec {\operatorname {i} }}({\boldsymbol {\sigma }})\,\mathrm {d} v={\vec {0}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mo> ∫ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> v </mi> </mrow> </msub> <mo stretchy="false"> [ </mo> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> x </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo> − </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> c </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo stretchy="false"> ) </mo> <mo> × </mo> <mo stretchy="false"> ( </mo> <munder> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <munder> <mrow> <mi> ρ </mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> v </mi> <mo stretchy="false"> → </mo> </mover> </mrow> <mo> ˙ </mo> </mover> </mrow> </mrow> <mo> − </mo> <mi> ρ </mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> k </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo> − </mo> <mi mathvariant="normal"> ∇ </mi> <mo> ⋅ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> σ </mi> </mrow> </mrow> <mo> ⏟ </mo> </munder> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mn> 0 </mn> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> </mrow> </munder> <mo stretchy="false"> ) </mo> <mo> − </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="normal"> i </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> σ </mi> </mrow> <mo stretchy="false"> ) </mo> <mo stretchy="false"> ] </mo> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> v </mi> <mo> = </mo> <mo> − </mo> <msub> <mo> ∫ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> v </mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="normal"> i </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> σ </mi> </mrow> <mo stretchy="false"> ) </mo> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> v </mi> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mn> 0 </mn> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \int _{v}[({\vec {x}}-{\vec {c}})\times (\underbrace {\rho {\dot {\vec {v}}}-\rho {\vec {k}}-\nabla \cdot {\boldsymbol {\sigma }}} _{={\vec {0}}})-{\vec {\operatorname {i} }}({\boldsymbol {\sigma }})]\,\mathrm {d} v=-\int _{v}{\vec {\operatorname {i} }}({\boldsymbol {\sigma }})\,\mathrm {d} v={\vec {0}}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/91ef84516ba263aa171c4bc9e686955e7b07b4cf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.838ex; width:60.897ex; height:8.176ex;" alt="{\displaystyle \int _{v}[({\vec {x}}-{\vec {c}})\times (\underbrace {\rho {\dot {\vec {v}}}-\rho {\vec {k}}-\nabla \cdot {\boldsymbol {\sigma }}} _{={\vec {0}}})-{\vec {\operatorname {i} }}({\boldsymbol {\sigma }})]\,\mathrm {d} v=-\int _{v}{\vec {\operatorname {i} }}({\boldsymbol {\sigma }})\,\mathrm {d} v={\vec {0}}}"> </noscript><span class="lazy-image-placeholder" style="width: 60.897ex;height: 8.176ex;vertical-align: -4.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/91ef84516ba263aa171c4bc9e686955e7b07b4cf" data-alt="{\displaystyle \int _{v}[({\vec {x}}-{\vec {c}})\times (\underbrace {\rho {\dot {\vec {v}}}-\rho {\vec {k}}-\nabla \cdot {\boldsymbol {\sigma }}} _{={\vec {0}}})-{\vec {\operatorname {i} }}({\boldsymbol {\sigma }})]\,\mathrm {d} v=-\int _{v}{\vec {\operatorname {i} }}({\boldsymbol {\sigma }})\,\mathrm {d} v={\vec {0}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </dd> </dl> Der unterklammerte Term trägt wegen der lokalen Impulsbilanz nichts bei und das letzte Integral gilt für jedes beliebige Volumen, sodass bei stetigem Integrand auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {\operatorname {i} }}({\boldsymbol {\sigma }})={\vec {0}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="normal"> i </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> σ </mi> </mrow> <mo stretchy="false"> ) </mo> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mn> 0 </mn> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\vec {\operatorname {i} }}({\boldsymbol {\sigma }})={\vec {0}}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/edee281f011ac949182873a75e3b5e5c0f69ff12" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.759ex; height:3.343ex;" alt="{\displaystyle {\vec {\operatorname {i} }}({\boldsymbol {\sigma }})={\vec {0}}}"> </noscript><span class="lazy-image-placeholder" style="width: 8.759ex;height: 3.343ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/edee281f011ac949182873a75e3b5e5c0f69ff12" data-alt="{\displaystyle {\vec {\operatorname {i} }}({\boldsymbol {\sigma }})={\vec {0}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  geschlossen werden kann. Analog zur lagrangeschen Darstellung reduziert sich die Drehimpulsbilanz in eulerscher Darstellung auf die Forderung nach der Symmetrie des cauchyschen Spannungstensors: <dl> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {\sigma }}={\boldsymbol {\sigma }}^{\top }}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> σ </mi> </mrow> <mo> = </mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> σ </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> ⊤ </mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\boldsymbol {\sigma }}={\boldsymbol {\sigma }}^{\top }} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8f8a70fa32d7549270a3a5501e977e6874d017b9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.798ex; height:2.676ex;" alt="{\displaystyle {\boldsymbol {\sigma }}={\boldsymbol {\sigma }}^{\top }}"> </noscript><span class="lazy-image-placeholder" style="width: 7.798ex;height: 2.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8f8a70fa32d7549270a3a5501e977e6874d017b9" data-alt="{\displaystyle {\boldsymbol {\sigma }}={\boldsymbol {\sigma }}^{\top }}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </dd> </dl> <div class="mw-heading mw-heading3"> <h3 id="Einfluss_von_Sprungstellen_im_Drehimpulssatz">Einfluss von Sprungstellen im Drehimpulssatz</h3> <a role="button" href="https://de-m-wikipedia-org.translate.goog/w/index.php?title=Cauchy-eulersche_Bewegungsgesetze&action=edit&section=10&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Abschnitt bearbeiten: Einfluss von Sprungstellen im Drehimpulssatz" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> Bearbeiten </a> </div> Analog zum ersten cauchy-eulerschen Bewegungsgesetz lautet das Reynolds-Transport-Theorem mit Sprungstelle hier: <dl> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}\int _{v}({\vec {x}}-{\vec {c}})\times \rho {\vec {v}}\,\mathrm {d} v=\int _{v}({\vec {x}}-{\vec {c}})\times \rho {\dot {\vec {v}}}\,\mathrm {d} v+\int _{a_{s}}[[({\vec {x}}-{\vec {c}})\times \rho {\vec {v}}\;(({\vec {v}}_{s}-{\vec {v}})\cdot {\vec {n}})]]\mathrm {d} a_{s}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> t </mi> </mrow> </mfrac> </mrow> <msub> <mo> ∫ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> v </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> x </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo> − </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> c </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo stretchy="false"> ) </mo> <mo> × </mo> <mi> ρ </mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> v </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> v </mi> <mo> = </mo> <msub> <mo> ∫ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> v </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> x </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo> − </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> c </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo stretchy="false"> ) </mo> <mo> × </mo> <mi> ρ </mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> v </mi> <mo stretchy="false"> → </mo> </mover> </mrow> <mo> ˙ </mo> </mover> </mrow> </mrow> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> v </mi> <mo> + </mo> <msub> <mo> ∫ </mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi> a </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> s </mi> </mrow> </msub> </mrow> </msub> <mo stretchy="false"> [ </mo> <mo stretchy="false"> [ </mo> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> x </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo> − </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> c </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo stretchy="false"> ) </mo> <mo> × </mo> <mi> ρ </mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> v </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mspace width="thickmathspace"></mspace> <mo stretchy="false"> ( </mo> <mo stretchy="false"> ( </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> v </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> s </mi> </mrow> </msub> <mo> − </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> v </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo stretchy="false"> ) </mo> <mo> ⋅ </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> n </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo stretchy="false"> ) </mo> <mo stretchy="false"> ] </mo> <mo stretchy="false"> ] </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <msub> <mi> a </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> s </mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}\int _{v}({\vec {x}}-{\vec {c}})\times \rho {\vec {v}}\,\mathrm {d} v=\int _{v}({\vec {x}}-{\vec {c}})\times \rho {\dot {\vec {v}}}\,\mathrm {d} v+\int _{a_{s}}[[({\vec {x}}-{\vec {c}})\times \rho {\vec {v}}\;(({\vec {v}}_{s}-{\vec {v}})\cdot {\vec {n}})]]\mathrm {d} a_{s}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a8f944a7db30029758f4755dc407602ce11535eb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:79.677ex; height:6.176ex;" alt="{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}\int _{v}({\vec {x}}-{\vec {c}})\times \rho {\vec {v}}\,\mathrm {d} v=\int _{v}({\vec {x}}-{\vec {c}})\times \rho {\dot {\vec {v}}}\,\mathrm {d} v+\int _{a_{s}}[[({\vec {x}}-{\vec {c}})\times \rho {\vec {v}}\;(({\vec {v}}_{s}-{\vec {v}})\cdot {\vec {n}})]]\mathrm {d} a_{s}}"> </noscript><span class="lazy-image-placeholder" style="width: 79.677ex;height: 6.176ex;vertical-align: -2.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a8f944a7db30029758f4755dc407602ce11535eb" data-alt="{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}\int _{v}({\vec {x}}-{\vec {c}})\times \rho {\vec {v}}\,\mathrm {d} v=\int _{v}({\vec {x}}-{\vec {c}})\times \rho {\dot {\vec {v}}}\,\mathrm {d} v+\int _{a_{s}}[[({\vec {x}}-{\vec {c}})\times \rho {\vec {v}}\;(({\vec {v}}_{s}-{\vec {v}})\cdot {\vec {n}})]]\mathrm {d} a_{s}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </dd> </dl> Der zweite Term mit der Sprungklammer [[...]] kommt neu hinzu. Die Integrale über die von außen angreifenden Kräfte werden getrennt für v+ und v− berechnet: <dl> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{array}{l}\displaystyle \left(\int _{v}({\vec {x}}-{\vec {c}})\times \rho {\vec {k}}\;\mathrm {d} v+\int _{a}({\vec {x}}-{\vec {c}})\times {\vec {t}}\,\mathrm {d} a\right)^{+}=\ldots \\\displaystyle \ldots =\int _{v^{+}}({\vec {x}}-{\vec {c}})\times \rho {\vec {k}}\;\mathrm {d} v^{+}+\int _{a^{+}}({\vec {x}}-{\vec {c}})\times {\boldsymbol {\sigma }}^{\top }\cdot {\vec {n}}\,\mathrm {d} a^{+}-\int _{a_{s}}({\vec {x}}-{\vec {c}})\times {\boldsymbol {\sigma }}^{\top }\cdot {\vec {n}}\,\mathrm {d} a_{s}\\\displaystyle \left(\int _{v}({\vec {x}}-{\vec {c}})\times \rho {\vec {k}}\;\mathrm {d} v+\int _{a}({\vec {x}}-{\vec {c}})\times {\vec {t}}\,\mathrm {d} a\right)^{-}=\ldots \\\displaystyle \ldots =\int _{v^{-}}({\vec {x}}-{\vec {c}})\times \rho {\vec {k}}\;\mathrm {d} v^{-}+\int _{a^{-}}({\vec {x}}-{\vec {c}})\times {\boldsymbol {\sigma }}^{\top }\cdot {\vec {n}}\,\mathrm {d} a^{-}+\int _{a_{s}}({\vec {x}}-{\vec {c}})\times {\boldsymbol {\sigma }}^{\top }\cdot {\vec {n}}\,\mathrm {d} a_{s}\end{array}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="left" rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow> <mo> ( </mo> <mrow> <msub> <mo> ∫ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> v </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> x </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo> − </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> c </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo stretchy="false"> ) </mo> <mo> × </mo> <mi> ρ </mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> k </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mspace width="thickmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> v </mi> <mo> + </mo> <msub> <mo> ∫ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> a </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> x </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo> − </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> c </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo stretchy="false"> ) </mo> <mo> × </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> t </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> a </mi> </mrow> <mo> ) </mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo> + </mo> </mrow> </msup> <mo> = </mo> <mo> … </mo> </mstyle> </mtd> </mtr> <mtr> <mtd> <mstyle displaystyle="true" scriptlevel="0"> <mo> … </mo> <mo> = </mo> <msub> <mo> ∫ </mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi> v </mi> <mrow class="MJX-TeXAtom-ORD"> <mo> + </mo> </mrow> </msup> </mrow> </msub> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> x </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo> − </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> c </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo stretchy="false"> ) </mo> <mo> × </mo> <mi> ρ </mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> k </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mspace width="thickmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <msup> <mi> v </mi> <mrow class="MJX-TeXAtom-ORD"> <mo> + </mo> </mrow> </msup> <mo> + </mo> <msub> <mo> ∫ </mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi> a </mi> <mrow class="MJX-TeXAtom-ORD"> <mo> + </mo> </mrow> </msup> </mrow> </msub> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> x </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo> − </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> c </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo stretchy="false"> ) </mo> <mo> × </mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> σ </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> ⊤ </mi> </mrow> </msup> <mo> ⋅ </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> n </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <msup> <mi> a </mi> <mrow class="MJX-TeXAtom-ORD"> <mo> + </mo> </mrow> </msup> <mo> − </mo> <msub> <mo> ∫ </mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi> a </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> s </mi> </mrow> </msub> </mrow> </msub> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> x </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo> − </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> c </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo stretchy="false"> ) </mo> <mo> × </mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> σ </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> ⊤ </mi> </mrow> </msup> <mo> ⋅ </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> n </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <msub> <mi> a </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> s </mi> </mrow> </msub> </mstyle> </mtd> </mtr> <mtr> <mtd> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow> <mo> ( </mo> <mrow> <msub> <mo> ∫ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> v </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> x </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo> − </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> c </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo stretchy="false"> ) </mo> <mo> × </mo> <mi> ρ </mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> k </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mspace width="thickmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> v </mi> <mo> + </mo> <msub> <mo> ∫ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> a </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> x </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo> − </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> c </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo stretchy="false"> ) </mo> <mo> × </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> t </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> a </mi> </mrow> <mo> ) </mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo> − </mo> </mrow> </msup> <mo> = </mo> <mo> … </mo> </mstyle> </mtd> </mtr> <mtr> <mtd> <mstyle displaystyle="true" scriptlevel="0"> <mo> … </mo> <mo> = </mo> <msub> <mo> ∫ </mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi> v </mi> <mrow class="MJX-TeXAtom-ORD"> <mo> − </mo> </mrow> </msup> </mrow> </msub> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> x </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo> − </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> c </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo stretchy="false"> ) </mo> <mo> × </mo> <mi> ρ </mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> k </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mspace width="thickmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <msup> <mi> v </mi> <mrow class="MJX-TeXAtom-ORD"> <mo> − </mo> </mrow> </msup> <mo> + </mo> <msub> <mo> ∫ </mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi> a </mi> <mrow class="MJX-TeXAtom-ORD"> <mo> − </mo> </mrow> </msup> </mrow> </msub> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> x </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo> − </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> c </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo stretchy="false"> ) </mo> <mo> × </mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> σ </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> ⊤ </mi> </mrow> </msup> <mo> ⋅ </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> n </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <msup> <mi> a </mi> <mrow class="MJX-TeXAtom-ORD"> <mo> − </mo> </mrow> </msup> <mo> + </mo> <msub> <mo> ∫ </mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi> a </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> s </mi> </mrow> </msub> </mrow> </msub> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> x </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo> − </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> c </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo stretchy="false"> ) </mo> <mo> × </mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> σ </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> ⊤ </mi> </mrow> </msup> <mo> ⋅ </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> n </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <msub> <mi> a </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> s </mi> </mrow> </msub> </mstyle> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\begin{array}{l}\displaystyle \left(\int _{v}({\vec {x}}-{\vec {c}})\times \rho {\vec {k}}\;\mathrm {d} v+\int _{a}({\vec {x}}-{\vec {c}})\times {\vec {t}}\,\mathrm {d} a\right)^{+}=\ldots \\\displaystyle \ldots =\int _{v^{+}}({\vec {x}}-{\vec {c}})\times \rho {\vec {k}}\;\mathrm {d} v^{+}+\int _{a^{+}}({\vec {x}}-{\vec {c}})\times {\boldsymbol {\sigma }}^{\top }\cdot {\vec {n}}\,\mathrm {d} a^{+}-\int _{a_{s}}({\vec {x}}-{\vec {c}})\times {\boldsymbol {\sigma }}^{\top }\cdot {\vec {n}}\,\mathrm {d} a_{s}\\\displaystyle \left(\int _{v}({\vec {x}}-{\vec {c}})\times \rho {\vec {k}}\;\mathrm {d} v+\int _{a}({\vec {x}}-{\vec {c}})\times {\vec {t}}\,\mathrm {d} a\right)^{-}=\ldots \\\displaystyle \ldots =\int _{v^{-}}({\vec {x}}-{\vec {c}})\times \rho {\vec {k}}\;\mathrm {d} v^{-}+\int _{a^{-}}({\vec {x}}-{\vec {c}})\times {\boldsymbol {\sigma }}^{\top }\cdot {\vec {n}}\,\mathrm {d} a^{-}+\int _{a_{s}}({\vec {x}}-{\vec {c}})\times {\boldsymbol {\sigma }}^{\top }\cdot {\vec {n}}\,\mathrm {d} a_{s}\end{array}}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c2f7a57af8d18bfc85ca402be46cc70f0a8682a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -12.225ex; margin-bottom: -0.279ex; width:80.369ex; height:26.176ex;" alt="{\displaystyle {\begin{array}{l}\displaystyle \left(\int _{v}({\vec {x}}-{\vec {c}})\times \rho {\vec {k}}\;\mathrm {d} v+\int _{a}({\vec {x}}-{\vec {c}})\times {\vec {t}}\,\mathrm {d} a\right)^{+}=\ldots \\\displaystyle \ldots =\int _{v^{+}}({\vec {x}}-{\vec {c}})\times \rho {\vec {k}}\;\mathrm {d} v^{+}+\int _{a^{+}}({\vec {x}}-{\vec {c}})\times {\boldsymbol {\sigma }}^{\top }\cdot {\vec {n}}\,\mathrm {d} a^{+}-\int _{a_{s}}({\vec {x}}-{\vec {c}})\times {\boldsymbol {\sigma }}^{\top }\cdot {\vec {n}}\,\mathrm {d} a_{s}\\\displaystyle \left(\int _{v}({\vec {x}}-{\vec {c}})\times \rho {\vec {k}}\;\mathrm {d} v+\int _{a}({\vec {x}}-{\vec {c}})\times {\vec {t}}\,\mathrm {d} a\right)^{-}=\ldots \\\displaystyle \ldots =\int _{v^{-}}({\vec {x}}-{\vec {c}})\times \rho {\vec {k}}\;\mathrm {d} v^{-}+\int _{a^{-}}({\vec {x}}-{\vec {c}})\times {\boldsymbol {\sigma }}^{\top }\cdot {\vec {n}}\,\mathrm {d} a^{-}+\int _{a_{s}}({\vec {x}}-{\vec {c}})\times {\boldsymbol {\sigma }}^{\top }\cdot {\vec {n}}\,\mathrm {d} a_{s}\end{array}}}"> </noscript><span class="lazy-image-placeholder" style="width: 80.369ex;height: 26.176ex;vertical-align: -12.225ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c2f7a57af8d18bfc85ca402be46cc70f0a8682a" data-alt="{\displaystyle {\begin{array}{l}\displaystyle \left(\int _{v}({\vec {x}}-{\vec {c}})\times \rho {\vec {k}}\;\mathrm {d} v+\int _{a}({\vec {x}}-{\vec {c}})\times {\vec {t}}\,\mathrm {d} a\right)^{+}=\ldots \\\displaystyle \ldots =\int _{v^{+}}({\vec {x}}-{\vec {c}})\times \rho {\vec {k}}\;\mathrm {d} v^{+}+\int _{a^{+}}({\vec {x}}-{\vec {c}})\times {\boldsymbol {\sigma }}^{\top }\cdot {\vec {n}}\,\mathrm {d} a^{+}-\int _{a_{s}}({\vec {x}}-{\vec {c}})\times {\boldsymbol {\sigma }}^{\top }\cdot {\vec {n}}\,\mathrm {d} a_{s}\\\displaystyle \left(\int _{v}({\vec {x}}-{\vec {c}})\times \rho {\vec {k}}\;\mathrm {d} v+\int _{a}({\vec {x}}-{\vec {c}})\times {\vec {t}}\,\mathrm {d} a\right)^{-}=\ldots \\\displaystyle \ldots =\int _{v^{-}}({\vec {x}}-{\vec {c}})\times \rho {\vec {k}}\;\mathrm {d} v^{-}+\int _{a^{-}}({\vec {x}}-{\vec {c}})\times {\boldsymbol {\sigma }}^{\top }\cdot {\vec {n}}\,\mathrm {d} a^{-}+\int _{a_{s}}({\vec {x}}-{\vec {c}})\times {\boldsymbol {\sigma }}^{\top }\cdot {\vec {n}}\,\mathrm {d} a_{s}\end{array}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </dd> </dl> Die Summe der drei Gleichungen führt nach Umformungen, wie sie oben bereits angegeben wurden, auf <dl> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{array}{l}\displaystyle \int _{v}[({\vec {x}}-{\vec {c}})\times (\rho {\dot {\vec {v}}}-\rho {\vec {k}}-\nabla \cdot {\boldsymbol {\sigma }})-{\vec {\operatorname {i} }}({\boldsymbol {\sigma }})]\,\mathrm {d} v=\ldots \\\qquad \qquad \ldots =\int _{a_{s}}[[({\vec {x}}-{\vec {c}})\times \rho {\vec {v}}\;(({\vec {v}}-{\vec {v}}_{s})\cdot {\vec {n}})-({\vec {x}}-{\vec {c}})\times {\boldsymbol {\sigma }}^{\top }\cdot {\vec {n}}]]\mathrm {d} a_{s}\end{array}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="left" rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mo> ∫ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> v </mi> </mrow> </msub> <mo stretchy="false"> [ </mo> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> x </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo> − </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> c </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo stretchy="false"> ) </mo> <mo> × </mo> <mo stretchy="false"> ( </mo> <mi> ρ </mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> v </mi> <mo stretchy="false"> → </mo> </mover> </mrow> <mo> ˙ </mo> </mover> </mrow> </mrow> <mo> − </mo> <mi> ρ </mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> k </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo> − </mo> <mi mathvariant="normal"> ∇ </mi> <mo> ⋅ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> σ </mi> </mrow> <mo stretchy="false"> ) </mo> <mo> − </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="normal"> i </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> σ </mi> </mrow> <mo stretchy="false"> ) </mo> <mo stretchy="false"> ] </mo> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> v </mi> <mo> = </mo> <mo> … </mo> </mstyle> </mtd> </mtr> <mtr> <mtd> <mspace width="2em"></mspace> <mspace width="2em"></mspace> <mo> … </mo> <mo> = </mo> <msub> <mo> ∫ </mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi> a </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> s </mi> </mrow> </msub> </mrow> </msub> <mo stretchy="false"> [ </mo> <mo stretchy="false"> [ </mo> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> x </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo> − </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> c </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo stretchy="false"> ) </mo> <mo> × </mo> <mi> ρ </mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> v </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mspace width="thickmathspace"></mspace> <mo stretchy="false"> ( </mo> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> v </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo> − </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> v </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> s </mi> </mrow> </msub> <mo stretchy="false"> ) </mo> <mo> ⋅ </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> n </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo stretchy="false"> ) </mo> <mo> − </mo> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> x </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo> − </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> c </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo stretchy="false"> ) </mo> <mo> × </mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> σ </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> ⊤ </mi> </mrow> </msup> <mo> ⋅ </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> n </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo stretchy="false"> ] </mo> <mo stretchy="false"> ] </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <msub> <mi> a </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> s </mi> </mrow> </msub> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\begin{array}{l}\displaystyle \int _{v}[({\vec {x}}-{\vec {c}})\times (\rho {\dot {\vec {v}}}-\rho {\vec {k}}-\nabla \cdot {\boldsymbol {\sigma }})-{\vec {\operatorname {i} }}({\boldsymbol {\sigma }})]\,\mathrm {d} v=\ldots \\\qquad \qquad \ldots =\int _{a_{s}}[[({\vec {x}}-{\vec {c}})\times \rho {\vec {v}}\;(({\vec {v}}-{\vec {v}}_{s})\cdot {\vec {n}})-({\vec {x}}-{\vec {c}})\times {\boldsymbol {\sigma }}^{\top }\cdot {\vec {n}}]]\mathrm {d} a_{s}\end{array}}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8659ac6120415488ec612cc770a7b5898ad9027a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.017ex; margin-bottom: -0.321ex; width:69.981ex; height:9.843ex;" alt="{\displaystyle {\begin{array}{l}\displaystyle \int _{v}[({\vec {x}}-{\vec {c}})\times (\rho {\dot {\vec {v}}}-\rho {\vec {k}}-\nabla \cdot {\boldsymbol {\sigma }})-{\vec {\operatorname {i} }}({\boldsymbol {\sigma }})]\,\mathrm {d} v=\ldots \\\qquad \qquad \ldots =\int _{a_{s}}[[({\vec {x}}-{\vec {c}})\times \rho {\vec {v}}\;(({\vec {v}}-{\vec {v}}_{s})\cdot {\vec {n}})-({\vec {x}}-{\vec {c}})\times {\boldsymbol {\sigma }}^{\top }\cdot {\vec {n}}]]\mathrm {d} a_{s}\end{array}}}"> </noscript><span class="lazy-image-placeholder" style="width: 69.981ex;height: 9.843ex;vertical-align: -4.017ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8659ac6120415488ec612cc770a7b5898ad9027a" data-alt="{\displaystyle {\begin{array}{l}\displaystyle \int _{v}[({\vec {x}}-{\vec {c}})\times (\rho {\dot {\vec {v}}}-\rho {\vec {k}}-\nabla \cdot {\boldsymbol {\sigma }})-{\vec {\operatorname {i} }}({\boldsymbol {\sigma }})]\,\mathrm {d} v=\ldots \\\qquad \qquad \ldots =\int _{a_{s}}[[({\vec {x}}-{\vec {c}})\times \rho {\vec {v}}\;(({\vec {v}}-{\vec {v}}_{s})\cdot {\vec {n}})-({\vec {x}}-{\vec {c}})\times {\boldsymbol {\sigma }}^{\top }\cdot {\vec {n}}]]\mathrm {d} a_{s}\end{array}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </dd> </dl> Jenseits der Sprungstelle verschwindet die rechte Seite und die Symmetrie des Spannungstensors folgt wie oben. An der (flächigen) Sprungstelle ist dv=0 und die linke Seite kann vernachlässigt werden, sodass bei Stetigkeit des Integranden mit der Sprungklammer in der Fläche <dl> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ({\vec {x}}-{\vec {c}})\times [[\rho {\vec {v}}\;(({\vec {v}}_{s}-{\vec {v}})\cdot {\vec {n}})+{\boldsymbol {\sigma }}^{\top }\cdot {\vec {n}}]]=({\vec {x}}-{\vec {c}})\times [[\rho {\vec {v}}\otimes ({\vec {v}}_{s}-{\vec {v}})+{\boldsymbol {\sigma }}^{\top }]]\cdot {\vec {n}}={\vec {0}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> x </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo> − </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> c </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo stretchy="false"> ) </mo> <mo> × </mo> <mo stretchy="false"> [ </mo> <mo stretchy="false"> [ </mo> <mi> ρ </mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> v </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mspace width="thickmathspace"></mspace> <mo stretchy="false"> ( </mo> <mo stretchy="false"> ( </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> v </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> s </mi> </mrow> </msub> <mo> − </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> v </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo stretchy="false"> ) </mo> <mo> ⋅ </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> n </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo stretchy="false"> ) </mo> <mo> + </mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> σ </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> ⊤ </mi> </mrow> </msup> <mo> ⋅ </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> n </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo stretchy="false"> ] </mo> <mo stretchy="false"> ] </mo> <mo> = </mo> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> x </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo> − </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> c </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo stretchy="false"> ) </mo> <mo> × </mo> <mo stretchy="false"> [ </mo> <mo stretchy="false"> [ </mo> <mi> ρ </mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> v </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo> ⊗ </mo> <mo stretchy="false"> ( </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> v </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> s </mi> </mrow> </msub> <mo> − </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> v </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo stretchy="false"> ) </mo> <mo> + </mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> σ </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> ⊤ </mi> </mrow> </msup> <mo stretchy="false"> ] </mo> <mo stretchy="false"> ] </mo> <mo> ⋅ </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> n </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mn> 0 </mn> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle ({\vec {x}}-{\vec {c}})\times [[\rho {\vec {v}}\;(({\vec {v}}_{s}-{\vec {v}})\cdot {\vec {n}})+{\boldsymbol {\sigma }}^{\top }\cdot {\vec {n}}]]=({\vec {x}}-{\vec {c}})\times [[\rho {\vec {v}}\otimes ({\vec {v}}_{s}-{\vec {v}})+{\boldsymbol {\sigma }}^{\top }]]\cdot {\vec {n}}={\vec {0}}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/93c7aa41b1e43856813661241ccb50967d2ee57d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:79.789ex; height:3.343ex;" alt="{\displaystyle ({\vec {x}}-{\vec {c}})\times [[\rho {\vec {v}}\;(({\vec {v}}_{s}-{\vec {v}})\cdot {\vec {n}})+{\boldsymbol {\sigma }}^{\top }\cdot {\vec {n}}]]=({\vec {x}}-{\vec {c}})\times [[\rho {\vec {v}}\otimes ({\vec {v}}_{s}-{\vec {v}})+{\boldsymbol {\sigma }}^{\top }]]\cdot {\vec {n}}={\vec {0}}}"> </noscript><span class="lazy-image-placeholder" style="width: 79.789ex;height: 3.343ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/93c7aa41b1e43856813661241ccb50967d2ee57d" data-alt="{\displaystyle ({\vec {x}}-{\vec {c}})\times [[\rho {\vec {v}}\;(({\vec {v}}_{s}-{\vec {v}})\cdot {\vec {n}})+{\boldsymbol {\sigma }}^{\top }\cdot {\vec {n}}]]=({\vec {x}}-{\vec {c}})\times [[\rho {\vec {v}}\otimes ({\vec {v}}_{s}-{\vec {v}})+{\boldsymbol {\sigma }}^{\top }]]\cdot {\vec {n}}={\vec {0}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </dd> </dl> abgeleitet werden kann, was wegen der Sprungbedingung im ersten cauchy-eulerschen Bewegungsgesetz identisch erfüllt ist. </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(3)"> <h2 id="Folgerungen_aus_den_Bewegungsgesetzen">Folgerungen aus den Bewegungsgesetzen</h2> <a role="button" href="https://de-m-wikipedia-org.translate.goog/w/index.php?title=Cauchy-eulersche_Bewegungsgesetze&action=edit&section=11&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Abschnitt bearbeiten: Folgerungen aus den Bewegungsgesetzen" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> Bearbeiten </a> </div> <section class="mf-section-3 collapsible-block" id="mf-section-3"> Aus den Bewegungsgesetzen können weitere, materialunabhängige, zu Prinzipien äquivalente Gleichungen gefolgert werden. Das erste cauchy-eulersche Bewegungsgesetz lautet: <dl> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}{\text{materiell:}}\quad &\rho _{0}\,{\ddot {\vec {\chi }}}\!\!\!\!\!\!\!\!\!\!&=&\;\rho _{0}\,{\vec {k}}_{0}+\nabla _{0}\cdot \mathbf {N} \\{\text{räumlich:}}\quad &\rho \,{\dot {\vec {v}}}\!\!\!\!\!\!\!\!\!\!&=&\;\rho \,{\vec {k}}+\nabla \cdot {\boldsymbol {\sigma }}\end{aligned}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext> materiell: </mtext> </mrow> <mspace width="1em"></mspace> </mtd> <mtd> <msub> <mi> ρ </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> χ </mi> <mo stretchy="false"> → </mo> </mover> </mrow> <mo> ¨ </mo> </mover> </mrow> </mrow> <mspace width="negativethinmathspace"></mspace> <mspace width="negativethinmathspace"></mspace> <mspace width="negativethinmathspace"></mspace> <mspace width="negativethinmathspace"></mspace> <mspace width="negativethinmathspace"></mspace> <mspace width="negativethinmathspace"></mspace> <mspace width="negativethinmathspace"></mspace> <mspace width="negativethinmathspace"></mspace> <mspace width="negativethinmathspace"></mspace> <mspace width="negativethinmathspace"></mspace> </mtd> <mtd> <mo> = </mo> </mtd> <mtd> <mspace width="thickmathspace"></mspace> <msub> <mi> ρ </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mspace width="thinmathspace"></mspace> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> k </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mo> + </mo> <msub> <mi mathvariant="normal"> ∇ </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mo> ⋅ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> N </mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext> räumlich: </mtext> </mrow> <mspace width="1em"></mspace> </mtd> <mtd> <mi> ρ </mi> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> v </mi> <mo stretchy="false"> → </mo> </mover> </mrow> <mo> ˙ </mo> </mover> </mrow> </mrow> <mspace width="negativethinmathspace"></mspace> <mspace width="negativethinmathspace"></mspace> <mspace width="negativethinmathspace"></mspace> <mspace width="negativethinmathspace"></mspace> <mspace width="negativethinmathspace"></mspace> <mspace width="negativethinmathspace"></mspace> <mspace width="negativethinmathspace"></mspace> <mspace width="negativethinmathspace"></mspace> <mspace width="negativethinmathspace"></mspace> <mspace width="negativethinmathspace"></mspace> </mtd> <mtd> <mo> = </mo> </mtd> <mtd> <mspace width="thickmathspace"></mspace> <mi> ρ </mi> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> k </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo> + </mo> <mi mathvariant="normal"> ∇ </mi> <mo> ⋅ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> σ </mi> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\begin{aligned}{\text{materiell:}}\quad &\rho _{0}\,{\ddot {\vec {\chi }}}\!\!\!\!\!\!\!\!\!\!&=&\;\rho _{0}\,{\vec {k}}_{0}+\nabla _{0}\cdot \mathbf {N} \\{\text{räumlich:}}\quad &\rho \,{\dot {\vec {v}}}\!\!\!\!\!\!\!\!\!\!&=&\;\rho \,{\vec {k}}+\nabla \cdot {\boldsymbol {\sigma }}\end{aligned}}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4a6cb5822addab3e040a90abdc0522226b617dca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:34.732ex; height:7.509ex;" alt="{\displaystyle {\begin{aligned}{\text{materiell:}}\quad &\rho _{0}\,{\ddot {\vec {\chi }}}\!\!\!\!\!\!\!\!\!\!&=&\;\rho _{0}\,{\vec {k}}_{0}+\nabla _{0}\cdot \mathbf {N} \\{\text{räumlich:}}\quad &\rho \,{\dot {\vec {v}}}\!\!\!\!\!\!\!\!\!\!&=&\;\rho \,{\vec {k}}+\nabla \cdot {\boldsymbol {\sigma }}\end{aligned}}}"> </noscript><span class="lazy-image-placeholder" style="width: 34.732ex;height: 7.509ex;vertical-align: -3.171ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4a6cb5822addab3e040a90abdc0522226b617dca" data-alt="{\displaystyle {\begin{aligned}{\text{materiell:}}\quad &\rho _{0}\,{\ddot {\vec {\chi }}}\!\!\!\!\!\!\!\!\!\!&=&\;\rho _{0}\,{\vec {k}}_{0}+\nabla _{0}\cdot \mathbf {N} \\{\text{räumlich:}}\quad &\rho \,{\dot {\vec {v}}}\!\!\!\!\!\!\!\!\!\!&=&\;\rho \,{\vec {k}}+\nabla \cdot {\boldsymbol {\sigma }}\end{aligned}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </dd> </dl> Diese Gleichungen werden mit einem Vektorfeld <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {q}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> q </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\vec {q}}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8a4e063f8ee7dae2488859c45a4e645db5148085" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.309ex; height:2.676ex;" alt="{\displaystyle {\vec {q}}}"> </noscript><span class="lazy-image-placeholder" style="width: 1.309ex;height: 2.676ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8a4e063f8ee7dae2488859c45a4e645db5148085" data-alt="{\displaystyle {\vec {q}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  skalar multipliziert, über das Volumen des Körpers integriert und umgeformt. Es entsteht: <dl> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}{\text{materiell:}}&\int _{V}\rho _{0}\,{\ddot {\vec {\chi }}}\cdot {\vec {q}}\,\mathrm {d} V+\int _{V}{\tilde {\mathbf {T} }}:\operatorname {sym} (\mathbf {F} ^{\top }\cdot \operatorname {GRAD} {\vec {q}})\,\mathrm {d} V=\ldots \\&\qquad \ldots =\int _{V}\rho _{0}\,{\vec {k}}_{0}\cdot {\vec {q}}\,\mathrm {d} V+\int _{A}{\vec {t}}_{0}\cdot {\vec {q}}\,\mathrm {d} A\\{\text{räumlich:}}&\int _{v}\rho \,{\dot {\vec {v}}}\cdot {\vec {q}}\,\mathrm {d} v+\int _{v}{\boldsymbol {\sigma }}:\operatorname {sym(grad} ({\vec {q}}))\,\mathrm {d} v=\ldots \\&\qquad \ldots =\int _{v}\rho \,{\vec {k}}\cdot {\vec {q}}\,\mathrm {d} v+\int _{a}{\vec {t}}\cdot {\vec {q}}\,\mathrm {d} a\end{aligned}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext> materiell: </mtext> </mrow> </mtd> <mtd> <mi></mi> <msub> <mo> ∫ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> V </mi> </mrow> </msub> <msub> <mi> ρ </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> χ </mi> <mo stretchy="false"> → </mo> </mover> </mrow> <mo> ¨ </mo> </mover> </mrow> </mrow> <mo> ⋅ </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> q </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> V </mi> <mo> + </mo> <msub> <mo> ∫ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> V </mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> T </mi> </mrow> <mo stretchy="false"> ~ </mo> </mover> </mrow> </mrow> <mo> : </mo> <mi> sym </mi> <mo> ⁡ </mo> <mo stretchy="false"> ( </mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> F </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> ⊤ </mi> </mrow> </msup> <mo> ⋅ </mo> <mi> GRAD </mi> <mo> ⁡ </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> q </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo stretchy="false"> ) </mo> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> V </mi> <mo> = </mo> <mo> … </mo> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mspace width="2em"></mspace> <mo> … </mo> <mo> = </mo> <msub> <mo> ∫ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> V </mi> </mrow> </msub> <msub> <mi> ρ </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mspace width="thinmathspace"></mspace> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> k </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mo> ⋅ </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> q </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> V </mi> <mo> + </mo> <msub> <mo> ∫ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> A </mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> t </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mo> ⋅ </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> q </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> A </mi> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext> räumlich: </mtext> </mrow> </mtd> <mtd> <mi></mi> <msub> <mo> ∫ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> v </mi> </mrow> </msub> <mi> ρ </mi> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> v </mi> <mo stretchy="false"> → </mo> </mover> </mrow> <mo> ˙ </mo> </mover> </mrow> </mrow> <mo> ⋅ </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> q </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> v </mi> <mo> + </mo> <msub> <mo> ∫ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> v </mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> σ </mi> </mrow> <mo> : </mo> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mi mathvariant="normal"> s </mi> <mi mathvariant="normal"> y </mi> <mi mathvariant="normal"> m </mi> <mo stretchy="false"> ( </mo> <mi mathvariant="normal"> g </mi> <mi mathvariant="normal"> r </mi> <mi mathvariant="normal"> a </mi> <mi mathvariant="normal"> d </mi> </mrow> <mo> ⁡ </mo> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> q </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo stretchy="false"> ) </mo> <mo stretchy="false"> ) </mo> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> v </mi> <mo> = </mo> <mo> … </mo> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mspace width="2em"></mspace> <mo> … </mo> <mo> = </mo> <msub> <mo> ∫ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> v </mi> </mrow> </msub> <mi> ρ </mi> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> k </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo> ⋅ </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> q </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> v </mi> <mo> + </mo> <msub> <mo> ∫ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> a </mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> t </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo> ⋅ </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> q </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> a </mi> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\begin{aligned}{\text{materiell:}}&\int _{V}\rho _{0}\,{\ddot {\vec {\chi }}}\cdot {\vec {q}}\,\mathrm {d} V+\int _{V}{\tilde {\mathbf {T} }}:\operatorname {sym} (\mathbf {F} ^{\top }\cdot \operatorname {GRAD} {\vec {q}})\,\mathrm {d} V=\ldots \\&\qquad \ldots =\int _{V}\rho _{0}\,{\vec {k}}_{0}\cdot {\vec {q}}\,\mathrm {d} V+\int _{A}{\vec {t}}_{0}\cdot {\vec {q}}\,\mathrm {d} A\\{\text{räumlich:}}&\int _{v}\rho \,{\dot {\vec {v}}}\cdot {\vec {q}}\,\mathrm {d} v+\int _{v}{\boldsymbol {\sigma }}:\operatorname {sym(grad} ({\vec {q}}))\,\mathrm {d} v=\ldots \\&\qquad \ldots =\int _{v}\rho \,{\vec {k}}\cdot {\vec {q}}\,\mathrm {d} v+\int _{a}{\vec {t}}\cdot {\vec {q}}\,\mathrm {d} a\end{aligned}}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/76cb964e96e71b45b58084eb84620ee8f48f1199" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -11.022ex; margin-bottom: -0.316ex; width:63.301ex; height:23.843ex;" alt="{\displaystyle {\begin{aligned}{\text{materiell:}}&\int _{V}\rho _{0}\,{\ddot {\vec {\chi }}}\cdot {\vec {q}}\,\mathrm {d} V+\int _{V}{\tilde {\mathbf {T} }}:\operatorname {sym} (\mathbf {F} ^{\top }\cdot \operatorname {GRAD} {\vec {q}})\,\mathrm {d} V=\ldots \\&\qquad \ldots =\int _{V}\rho _{0}\,{\vec {k}}_{0}\cdot {\vec {q}}\,\mathrm {d} V+\int _{A}{\vec {t}}_{0}\cdot {\vec {q}}\,\mathrm {d} A\\{\text{räumlich:}}&\int _{v}\rho \,{\dot {\vec {v}}}\cdot {\vec {q}}\,\mathrm {d} v+\int _{v}{\boldsymbol {\sigma }}:\operatorname {sym(grad} ({\vec {q}}))\,\mathrm {d} v=\ldots \\&\qquad \ldots =\int _{v}\rho \,{\vec {k}}\cdot {\vec {q}}\,\mathrm {d} v+\int _{a}{\vec {t}}\cdot {\vec {q}}\,\mathrm {d} a\end{aligned}}}"> </noscript><span class="lazy-image-placeholder" style="width: 63.301ex;height: 23.843ex;vertical-align: -11.022ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/76cb964e96e71b45b58084eb84620ee8f48f1199" data-alt="{\displaystyle {\begin{aligned}{\text{materiell:}}&\int _{V}\rho _{0}\,{\ddot {\vec {\chi }}}\cdot {\vec {q}}\,\mathrm {d} V+\int _{V}{\tilde {\mathbf {T} }}:\operatorname {sym} (\mathbf {F} ^{\top }\cdot \operatorname {GRAD} {\vec {q}})\,\mathrm {d} V=\ldots \\&\qquad \ldots =\int _{V}\rho _{0}\,{\vec {k}}_{0}\cdot {\vec {q}}\,\mathrm {d} V+\int _{A}{\vec {t}}_{0}\cdot {\vec {q}}\,\mathrm {d} A\\{\text{räumlich:}}&\int _{v}\rho \,{\dot {\vec {v}}}\cdot {\vec {q}}\,\mathrm {d} v+\int _{v}{\boldsymbol {\sigma }}:\operatorname {sym(grad} ({\vec {q}}))\,\mathrm {d} v=\ldots \\&\qquad \ldots =\int _{v}\rho \,{\vec {k}}\cdot {\vec {q}}\,\mathrm {d} v+\int _{a}{\vec {t}}\cdot {\vec {q}}\,\mathrm {d} a\end{aligned}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </dd> </dl> Je nach Vektorfeld <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {q}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> q </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\vec {q}}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8a4e063f8ee7dae2488859c45a4e645db5148085" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.309ex; height:2.676ex;" alt="{\displaystyle {\vec {q}}}"> </noscript><span class="lazy-image-placeholder" style="width: 1.309ex;height: 2.676ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8a4e063f8ee7dae2488859c45a4e645db5148085" data-alt="{\displaystyle {\vec {q}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  ergeben sich verschiedene Aussagen. <table class="wikitable mw-collapsible mw-collapsed"> <tbody> <tr> <td>Beweis</td> </tr> <tr> <td>Skalare Multiplikation des ersten cauchy-eulerschen Bewegungsgesetzes mit dem Vektorfeld <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {q}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> q </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\vec {q}}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8a4e063f8ee7dae2488859c45a4e645db5148085" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.309ex; height:2.676ex;" alt="{\displaystyle {\vec {q}}}"> </noscript><span class="lazy-image-placeholder" style="width: 1.309ex;height: 2.676ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8a4e063f8ee7dae2488859c45a4e645db5148085" data-alt="{\displaystyle {\vec {q}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  und Integration über das Volumen des Körpers liefert: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{array}{llcl}{\text{materiell:}}&\int _{V}\rho _{0}\,{\ddot {\vec {\chi }}}\cdot {\vec {q}}\,\mathrm {d} V&=&\int _{V}\rho _{0}\,{\vec {k}}_{0}\cdot {\vec {q}}\,\mathrm {d} V+\int _{V}(\nabla _{0}\cdot \mathbf {N} )\cdot {\vec {q}}\,\mathrm {d} V\\{\text{räumlich:}}&\int _{v}\rho \,{\dot {\vec {v}}}\cdot {\vec {q}}\,\mathrm {d} v&=&\int _{v}\rho \,{\vec {k}}\cdot {\vec {q}}\,\mathrm {d} v+\int _{v}(\nabla \cdot {\boldsymbol {\sigma }})\cdot {\vec {q}}\,\mathrm {d} v\end{array}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="left left center left" rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext> materiell: </mtext> </mrow> </mtd> <mtd> <msub> <mo> ∫ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> V </mi> </mrow> </msub> <msub> <mi> ρ </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> χ </mi> <mo stretchy="false"> → </mo> </mover> </mrow> <mo> ¨ </mo> </mover> </mrow> </mrow> <mo> ⋅ </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> q </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> V </mi> </mtd> <mtd> <mo> = </mo> </mtd> <mtd> <msub> <mo> ∫ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> V </mi> </mrow> </msub> <msub> <mi> ρ </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mspace width="thinmathspace"></mspace> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> k </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mo> ⋅ </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> q </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> V </mi> <mo> + </mo> <msub> <mo> ∫ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> V </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <msub> <mi mathvariant="normal"> ∇ </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mo> ⋅ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> N </mi> </mrow> <mo stretchy="false"> ) </mo> <mo> ⋅ </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> q </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> V </mi> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext> räumlich: </mtext> </mrow> </mtd> <mtd> <msub> <mo> ∫ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> v </mi> </mrow> </msub> <mi> ρ </mi> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> v </mi> <mo stretchy="false"> → </mo> </mover> </mrow> <mo> ˙ </mo> </mover> </mrow> </mrow> <mo> ⋅ </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> q </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> v </mi> </mtd> <mtd> <mo> = </mo> </mtd> <mtd> <msub> <mo> ∫ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> v </mi> </mrow> </msub> <mi> ρ </mi> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> k </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo> ⋅ </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> q </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> v </mi> <mo> + </mo> <msub> <mo> ∫ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> v </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <mi mathvariant="normal"> ∇ </mi> <mo> ⋅ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> σ </mi> </mrow> <mo stretchy="false"> ) </mo> <mo> ⋅ </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> q </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> v </mi> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\begin{array}{llcl}{\text{materiell:}}&\int _{V}\rho _{0}\,{\ddot {\vec {\chi }}}\cdot {\vec {q}}\,\mathrm {d} V&=&\int _{V}\rho _{0}\,{\vec {k}}_{0}\cdot {\vec {q}}\,\mathrm {d} V+\int _{V}(\nabla _{0}\cdot \mathbf {N} )\cdot {\vec {q}}\,\mathrm {d} V\\{\text{räumlich:}}&\int _{v}\rho \,{\dot {\vec {v}}}\cdot {\vec {q}}\,\mathrm {d} v&=&\int _{v}\rho \,{\vec {k}}\cdot {\vec {q}}\,\mathrm {d} v+\int _{v}(\nabla \cdot {\boldsymbol {\sigma }})\cdot {\vec {q}}\,\mathrm {d} v\end{array}}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c72c03e2a0c523d09bea661541d540f23e7b37a1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:67.685ex; height:7.843ex;" alt="{\displaystyle {\begin{array}{llcl}{\text{materiell:}}&\int _{V}\rho _{0}\,{\ddot {\vec {\chi }}}\cdot {\vec {q}}\,\mathrm {d} V&=&\int _{V}\rho _{0}\,{\vec {k}}_{0}\cdot {\vec {q}}\,\mathrm {d} V+\int _{V}(\nabla _{0}\cdot \mathbf {N} )\cdot {\vec {q}}\,\mathrm {d} V\\{\text{räumlich:}}&\int _{v}\rho \,{\dot {\vec {v}}}\cdot {\vec {q}}\,\mathrm {d} v&=&\int _{v}\rho \,{\vec {k}}\cdot {\vec {q}}\,\mathrm {d} v+\int _{v}(\nabla \cdot {\boldsymbol {\sigma }})\cdot {\vec {q}}\,\mathrm {d} v\end{array}}}"> </noscript><span class="lazy-image-placeholder" style="width: 67.685ex;height: 7.843ex;vertical-align: -3.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c72c03e2a0c523d09bea661541d540f23e7b37a1" data-alt="{\displaystyle {\begin{array}{llcl}{\text{materiell:}}&\int _{V}\rho _{0}\,{\ddot {\vec {\chi }}}\cdot {\vec {q}}\,\mathrm {d} V&=&\int _{V}\rho _{0}\,{\vec {k}}_{0}\cdot {\vec {q}}\,\mathrm {d} V+\int _{V}(\nabla _{0}\cdot \mathbf {N} )\cdot {\vec {q}}\,\mathrm {d} V\\{\text{räumlich:}}&\int _{v}\rho \,{\dot {\vec {v}}}\cdot {\vec {q}}\,\mathrm {d} v&=&\int _{v}\rho \,{\vec {k}}\cdot {\vec {q}}\,\mathrm {d} v+\int _{v}(\nabla \cdot {\boldsymbol {\sigma }})\cdot {\vec {q}}\,\mathrm {d} v\end{array}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> Der letzte Term auf der rechten Seite wird mit der Produktregel umgeformt: <dl> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{array}{rlcl}{\text{materiell:}}&\nabla _{0}\cdot (\mathbf {N} \cdot {\vec {q}})&=&(\nabla _{0}\cdot \mathbf {N} )\cdot {\vec {q}}+\mathbf {N} :(\nabla _{0}\otimes {\vec {q}})\\\rightarrow &(\nabla _{0}\cdot \mathbf {N} )\cdot {\vec {q}}&=&\nabla _{0}\cdot (\mathbf {N} \cdot {\vec {q}})-\mathbf {N} :(\nabla _{0}\otimes {\vec {q}})\\{\text{räumlich:}}&\nabla \cdot ({\boldsymbol {\sigma }}\cdot {\vec {q}})&=&(\nabla \cdot {\boldsymbol {\sigma }})\cdot {\vec {q}}+{\boldsymbol {\sigma }}:(\nabla \otimes {\vec {q}})\\\rightarrow &(\nabla \cdot {\boldsymbol {\sigma }})\cdot {\vec {q}}&=&\nabla \cdot ({\boldsymbol {\sigma }}\cdot {\vec {q}})-{\boldsymbol {\sigma }}:(\nabla \otimes {\vec {q}})\end{array}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left center left" rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext> materiell: </mtext> </mrow> </mtd> <mtd> <msub> <mi mathvariant="normal"> ∇ </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mo> ⋅ </mo> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> N </mi> </mrow> <mo> ⋅ </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> q </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo stretchy="false"> ) </mo> </mtd> <mtd> <mo> = </mo> </mtd> <mtd> <mo stretchy="false"> ( </mo> <msub> <mi mathvariant="normal"> ∇ </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mo> ⋅ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> N </mi> </mrow> <mo stretchy="false"> ) </mo> <mo> ⋅ </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> q </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo> + </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> N </mi> </mrow> <mo> : </mo> <mo stretchy="false"> ( </mo> <msub> <mi mathvariant="normal"> ∇ </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mo> ⊗ </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> q </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo stretchy="false"> ) </mo> </mtd> </mtr> <mtr> <mtd> <mo stretchy="false"> → </mo> </mtd> <mtd> <mo stretchy="false"> ( </mo> <msub> <mi mathvariant="normal"> ∇ </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mo> ⋅ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> N </mi> </mrow> <mo stretchy="false"> ) </mo> <mo> ⋅ </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> q </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> </mtd> <mtd> <mo> = </mo> </mtd> <mtd> <msub> <mi mathvariant="normal"> ∇ </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mo> ⋅ </mo> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> N </mi> </mrow> <mo> ⋅ </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> q </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo stretchy="false"> ) </mo> <mo> − </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> N </mi> </mrow> <mo> : </mo> <mo stretchy="false"> ( </mo> <msub> <mi mathvariant="normal"> ∇ </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mo> ⊗ </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> q </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo stretchy="false"> ) </mo> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext> räumlich: </mtext> </mrow> </mtd> <mtd> <mi mathvariant="normal"> ∇ </mi> <mo> ⋅ </mo> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> σ </mi> </mrow> <mo> ⋅ </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> q </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo stretchy="false"> ) </mo> </mtd> <mtd> <mo> = </mo> </mtd> <mtd> <mo stretchy="false"> ( </mo> <mi mathvariant="normal"> ∇ </mi> <mo> ⋅ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> σ </mi> </mrow> <mo stretchy="false"> ) </mo> <mo> ⋅ </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> q </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo> + </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> σ </mi> </mrow> <mo> : </mo> <mo stretchy="false"> ( </mo> <mi mathvariant="normal"> ∇ </mi> <mo> ⊗ </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> q </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo stretchy="false"> ) </mo> </mtd> </mtr> <mtr> <mtd> <mo stretchy="false"> → </mo> </mtd> <mtd> <mo stretchy="false"> ( </mo> <mi mathvariant="normal"> ∇ </mi> <mo> ⋅ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> σ </mi> </mrow> <mo stretchy="false"> ) </mo> <mo> ⋅ </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> q </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> </mtd> <mtd> <mo> = </mo> </mtd> <mtd> <mi mathvariant="normal"> ∇ </mi> <mo> ⋅ </mo> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> σ </mi> </mrow> <mo> ⋅ </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> q </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo stretchy="false"> ) </mo> <mo> − </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> σ </mi> </mrow> <mo> : </mo> <mo stretchy="false"> ( </mo> <mi mathvariant="normal"> ∇ </mi> <mo> ⊗ </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> q </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo stretchy="false"> ) </mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\begin{array}{rlcl}{\text{materiell:}}&\nabla _{0}\cdot (\mathbf {N} \cdot {\vec {q}})&=&(\nabla _{0}\cdot \mathbf {N} )\cdot {\vec {q}}+\mathbf {N} :(\nabla _{0}\otimes {\vec {q}})\\\rightarrow &(\nabla _{0}\cdot \mathbf {N} )\cdot {\vec {q}}&=&\nabla _{0}\cdot (\mathbf {N} \cdot {\vec {q}})-\mathbf {N} :(\nabla _{0}\otimes {\vec {q}})\\{\text{räumlich:}}&\nabla \cdot ({\boldsymbol {\sigma }}\cdot {\vec {q}})&=&(\nabla \cdot {\boldsymbol {\sigma }})\cdot {\vec {q}}+{\boldsymbol {\sigma }}:(\nabla \otimes {\vec {q}})\\\rightarrow &(\nabla \cdot {\boldsymbol {\sigma }})\cdot {\vec {q}}&=&\nabla \cdot ({\boldsymbol {\sigma }}\cdot {\vec {q}})-{\boldsymbol {\sigma }}:(\nabla \otimes {\vec {q}})\end{array}}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e0ee7a4d1f07abe7da1ec4efa806c6a89fc1b6e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.171ex; width:58.282ex; height:13.509ex;" alt="{\displaystyle {\begin{array}{rlcl}{\text{materiell:}}&\nabla _{0}\cdot (\mathbf {N} \cdot {\vec {q}})&=&(\nabla _{0}\cdot \mathbf {N} )\cdot {\vec {q}}+\mathbf {N} :(\nabla _{0}\otimes {\vec {q}})\\\rightarrow &(\nabla _{0}\cdot \mathbf {N} )\cdot {\vec {q}}&=&\nabla _{0}\cdot (\mathbf {N} \cdot {\vec {q}})-\mathbf {N} :(\nabla _{0}\otimes {\vec {q}})\\{\text{räumlich:}}&\nabla \cdot ({\boldsymbol {\sigma }}\cdot {\vec {q}})&=&(\nabla \cdot {\boldsymbol {\sigma }})\cdot {\vec {q}}+{\boldsymbol {\sigma }}:(\nabla \otimes {\vec {q}})\\\rightarrow &(\nabla \cdot {\boldsymbol {\sigma }})\cdot {\vec {q}}&=&\nabla \cdot ({\boldsymbol {\sigma }}\cdot {\vec {q}})-{\boldsymbol {\sigma }}:(\nabla \otimes {\vec {q}})\end{array}}}"> </noscript><span class="lazy-image-placeholder" style="width: 58.282ex;height: 13.509ex;vertical-align: -6.171ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e0ee7a4d1f07abe7da1ec4efa806c6a89fc1b6e4" data-alt="{\displaystyle {\begin{array}{rlcl}{\text{materiell:}}&\nabla _{0}\cdot (\mathbf {N} \cdot {\vec {q}})&=&(\nabla _{0}\cdot \mathbf {N} )\cdot {\vec {q}}+\mathbf {N} :(\nabla _{0}\otimes {\vec {q}})\\\rightarrow &(\nabla _{0}\cdot \mathbf {N} )\cdot {\vec {q}}&=&\nabla _{0}\cdot (\mathbf {N} \cdot {\vec {q}})-\mathbf {N} :(\nabla _{0}\otimes {\vec {q}})\\{\text{räumlich:}}&\nabla \cdot ({\boldsymbol {\sigma }}\cdot {\vec {q}})&=&(\nabla \cdot {\boldsymbol {\sigma }})\cdot {\vec {q}}+{\boldsymbol {\sigma }}:(\nabla \otimes {\vec {q}})\\\rightarrow &(\nabla \cdot {\boldsymbol {\sigma }})\cdot {\vec {q}}&=&\nabla \cdot ({\boldsymbol {\sigma }}\cdot {\vec {q}})-{\boldsymbol {\sigma }}:(\nabla \otimes {\vec {q}})\end{array}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </dd> </dl>In der materiellen Form wird noch der Deformationsgradient einmultipliziert: <dl> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\mathbf {N} :(\nabla _{0}\otimes {\vec {q}})=&(\mathbf {N} \cdot \mathbf {F} ^{\top -1}\cdot \mathbf {F} ^{\top }):(\nabla _{0}\otimes {\vec {q}})\\=&(\mathbf {N} \cdot \mathbf {F} ^{\top -1}):{\big (}(\nabla _{0}\otimes {\vec {q}})\cdot \mathbf {F} {\big )}\\=&{\tilde {\mathbf {T} }}:\operatorname {sym} (\mathbf {F} ^{\top }\cdot \operatorname {GRAD} {\vec {q}})\end{aligned}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> N </mi> </mrow> <mo> : </mo> <mo stretchy="false"> ( </mo> <msub> <mi mathvariant="normal"> ∇ </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mo> ⊗ </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> q </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo stretchy="false"> ) </mo> <mo> = </mo> </mtd> <mtd> <mi></mi> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> N </mi> </mrow> <mo> ⋅ </mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> F </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> ⊤ </mi> <mo> − </mo> <mn> 1 </mn> </mrow> </msup> <mo> ⋅ </mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> F </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> ⊤ </mi> </mrow> </msup> <mo stretchy="false"> ) </mo> <mo> : </mo> <mo stretchy="false"> ( </mo> <msub> <mi mathvariant="normal"> ∇ </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mo> ⊗ </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> q </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo stretchy="false"> ) </mo> </mtd> </mtr> <mtr> <mtd> <mo> = </mo> </mtd> <mtd> <mi></mi> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> N </mi> </mrow> <mo> ⋅ </mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> F </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> ⊤ </mi> <mo> − </mo> <mn> 1 </mn> </mrow> </msup> <mo stretchy="false"> ) </mo> <mo> : </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em"> ( </mo> </mrow> </mrow> <mo stretchy="false"> ( </mo> <msub> <mi mathvariant="normal"> ∇ </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mo> ⊗ </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> q </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo stretchy="false"> ) </mo> <mo> ⋅ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> F </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em"> ) </mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mo> = </mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> T </mi> </mrow> <mo stretchy="false"> ~ </mo> </mover> </mrow> </mrow> <mo> : </mo> <mi> sym </mi> <mo> ⁡ </mo> <mo stretchy="false"> ( </mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> F </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> ⊤ </mi> </mrow> </msup> <mo> ⋅ </mo> <mi> GRAD </mi> <mo> ⁡ </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> q </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo stretchy="false"> ) </mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\begin{aligned}\mathbf {N} :(\nabla _{0}\otimes {\vec {q}})=&(\mathbf {N} \cdot \mathbf {F} ^{\top -1}\cdot \mathbf {F} ^{\top }):(\nabla _{0}\otimes {\vec {q}})\\=&(\mathbf {N} \cdot \mathbf {F} ^{\top -1}):{\big (}(\nabla _{0}\otimes {\vec {q}})\cdot \mathbf {F} {\big )}\\=&{\tilde {\mathbf {T} }}:\operatorname {sym} (\mathbf {F} ^{\top }\cdot \operatorname {GRAD} {\vec {q}})\end{aligned}}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5bef6dd428961015343901632189aaae4400937f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.505ex; width:43.433ex; height:10.176ex;" alt="{\displaystyle {\begin{aligned}\mathbf {N} :(\nabla _{0}\otimes {\vec {q}})=&(\mathbf {N} \cdot \mathbf {F} ^{\top -1}\cdot \mathbf {F} ^{\top }):(\nabla _{0}\otimes {\vec {q}})\\=&(\mathbf {N} \cdot \mathbf {F} ^{\top -1}):{\big (}(\nabla _{0}\otimes {\vec {q}})\cdot \mathbf {F} {\big )}\\=&{\tilde {\mathbf {T} }}:\operatorname {sym} (\mathbf {F} ^{\top }\cdot \operatorname {GRAD} {\vec {q}})\end{aligned}}}"> </noscript><span class="lazy-image-placeholder" style="width: 43.433ex;height: 10.176ex;vertical-align: -4.505ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5bef6dd428961015343901632189aaae4400937f" data-alt="{\displaystyle {\begin{aligned}\mathbf {N} :(\nabla _{0}\otimes {\vec {q}})=&(\mathbf {N} \cdot \mathbf {F} ^{\top -1}\cdot \mathbf {F} ^{\top }):(\nabla _{0}\otimes {\vec {q}})\\=&(\mathbf {N} \cdot \mathbf {F} ^{\top -1}):{\big (}(\nabla _{0}\otimes {\vec {q}})\cdot \mathbf {F} {\big )}\\=&{\tilde {\mathbf {T} }}:\operatorname {sym} (\mathbf {F} ^{\top }\cdot \operatorname {GRAD} {\vec {q}})\end{aligned}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </dd> </dl>Im letzten Schritt wurde <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\nabla _{0}\otimes {\vec {q}})^{\top }=\operatorname {GRAD} {\vec {q}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false"> ( </mo> <msub> <mi mathvariant="normal"> ∇ </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mo> ⊗ </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> q </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <msup> <mo stretchy="false"> ) </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> ⊤ </mi> </mrow> </msup> <mo> = </mo> <mi> GRAD </mi> <mo> ⁡ </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> q </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle (\nabla _{0}\otimes {\vec {q}})^{\top }=\operatorname {GRAD} {\vec {q}}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e9d607545a7965b39af3eec708deffd578ff3276" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.307ex; height:3.176ex;" alt="{\displaystyle (\nabla _{0}\otimes {\vec {q}})^{\top }=\operatorname {GRAD} {\vec {q}}}"> </noscript><span class="lazy-image-placeholder" style="width: 22.307ex;height: 3.176ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e9d607545a7965b39af3eec708deffd578ff3276" data-alt="{\displaystyle (\nabla _{0}\otimes {\vec {q}})^{\top }=\operatorname {GRAD} {\vec {q}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  ausgenutzt, und dass im Skalarprodukt mit einem symmetrischen Tensor <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tilde {\mathbf {T} }}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> T </mi> </mrow> <mo stretchy="false"> ~ </mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\tilde {\mathbf {T} }}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5c9a3a6a6d1647ff631bf4ed995d6c1d53410666" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.859ex; height:2.676ex;" alt="{\displaystyle {\tilde {\mathbf {T} }}}"> </noscript><span class="lazy-image-placeholder" style="width: 1.859ex;height: 2.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5c9a3a6a6d1647ff631bf4ed995d6c1d53410666" data-alt="{\displaystyle {\tilde {\mathbf {T} }}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  nur die symmetrischen Anteile sym(·) etwas beitragen, was auch in der räumlichen Formulierung ausgenutzt wird: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {\sigma }}:(\nabla \otimes {\vec {q}})={\boldsymbol {\sigma }}:\operatorname {sym(grad} ({\vec {q}}))}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> σ </mi> </mrow> <mo> : </mo> <mo stretchy="false"> ( </mo> <mi mathvariant="normal"> ∇ </mi> <mo> ⊗ </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> q </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo stretchy="false"> ) </mo> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> σ </mi> </mrow> <mo> : </mo> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mi mathvariant="normal"> s </mi> <mi mathvariant="normal"> y </mi> <mi mathvariant="normal"> m </mi> <mo stretchy="false"> ( </mo> <mi mathvariant="normal"> g </mi> <mi mathvariant="normal"> r </mi> <mi mathvariant="normal"> a </mi> <mi mathvariant="normal"> d </mi> </mrow> <mo> ⁡ </mo> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> q </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo stretchy="false"> ) </mo> <mo stretchy="false"> ) </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\boldsymbol {\sigma }}:(\nabla \otimes {\vec {q}})={\boldsymbol {\sigma }}:\operatorname {sym(grad} ({\vec {q}}))} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/280e76f179540586b366f521361419e614abb23e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.592ex; height:2.843ex;" alt="{\displaystyle {\boldsymbol {\sigma }}:(\nabla \otimes {\vec {q}})={\boldsymbol {\sigma }}:\operatorname {sym(grad} ({\vec {q}}))}"> </noscript><span class="lazy-image-placeholder" style="width: 31.592ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/280e76f179540586b366f521361419e614abb23e" data-alt="{\displaystyle {\boldsymbol {\sigma }}:(\nabla \otimes {\vec {q}})={\boldsymbol {\sigma }}:\operatorname {sym(grad} ({\vec {q}}))}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> Das Volumenintegral des Divergenzterms wird mit dem gaußschen Integralsatz in ein Oberflächenintegral umgewandelt: <dl> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}{\text{materiell:}}&\int _{V}\nabla _{0}\cdot (\mathbf {N} \cdot {\vec {q}})\,\mathrm {d} V=\int _{A}(\mathbf {N} \cdot {\vec {q}})\cdot {\vec {N}}\,\mathrm {d} A=\int _{A}{\vec {q}}\cdot \mathbf {N} ^{\top }\cdot {\vec {N}}\,\mathrm {d} A=\int _{A}{\vec {q}}\cdot {\vec {t}}_{0}\,\mathrm {d} A\\{\text{räumlich:}}&\int _{v}\nabla \cdot ({\boldsymbol {\sigma }}\cdot {\vec {q}})\,\mathrm {d} v=\int _{a}({\boldsymbol {\sigma }}\cdot {\vec {q}})\cdot {\vec {n}}\,\mathrm {d} a=\int _{a}{\vec {q}}\cdot {\boldsymbol {\sigma }}^{\top }\cdot {\vec {n}}\,\mathrm {d} a=\int _{a}{\vec {q}}\cdot {\vec {t}}\,\mathrm {d} a\end{aligned}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext> materiell: </mtext> </mrow> </mtd> <mtd> <mi></mi> <msub> <mo> ∫ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> V </mi> </mrow> </msub> <msub> <mi mathvariant="normal"> ∇ </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mo> ⋅ </mo> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> N </mi> </mrow> <mo> ⋅ </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> q </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo stretchy="false"> ) </mo> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> V </mi> <mo> = </mo> <msub> <mo> ∫ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> A </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> N </mi> </mrow> <mo> ⋅ </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> q </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo stretchy="false"> ) </mo> <mo> ⋅ </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> N </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> A </mi> <mo> = </mo> <msub> <mo> ∫ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> A </mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> q </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo> ⋅ </mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> N </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> ⊤ </mi> </mrow> </msup> <mo> ⋅ </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> N </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> A </mi> <mo> = </mo> <msub> <mo> ∫ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> A </mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> q </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo> ⋅ </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> t </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> A </mi> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext> räumlich: </mtext> </mrow> </mtd> <mtd> <mi></mi> <msub> <mo> ∫ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> v </mi> </mrow> </msub> <mi mathvariant="normal"> ∇ </mi> <mo> ⋅ </mo> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> σ </mi> </mrow> <mo> ⋅ </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> q </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo stretchy="false"> ) </mo> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> v </mi> <mo> = </mo> <msub> <mo> ∫ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> a </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> σ </mi> </mrow> <mo> ⋅ </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> q </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo stretchy="false"> ) </mo> <mo> ⋅ </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> n </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> a </mi> <mo> = </mo> <msub> <mo> ∫ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> a </mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> q </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo> ⋅ </mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> σ </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> ⊤ </mi> </mrow> </msup> <mo> ⋅ </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> n </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> a </mi> <mo> = </mo> <msub> <mo> ∫ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> a </mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> q </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo> ⋅ </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> t </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> a </mi> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\begin{aligned}{\text{materiell:}}&\int _{V}\nabla _{0}\cdot (\mathbf {N} \cdot {\vec {q}})\,\mathrm {d} V=\int _{A}(\mathbf {N} \cdot {\vec {q}})\cdot {\vec {N}}\,\mathrm {d} A=\int _{A}{\vec {q}}\cdot \mathbf {N} ^{\top }\cdot {\vec {N}}\,\mathrm {d} A=\int _{A}{\vec {q}}\cdot {\vec {t}}_{0}\,\mathrm {d} A\\{\text{räumlich:}}&\int _{v}\nabla \cdot ({\boldsymbol {\sigma }}\cdot {\vec {q}})\,\mathrm {d} v=\int _{a}({\boldsymbol {\sigma }}\cdot {\vec {q}})\cdot {\vec {n}}\,\mathrm {d} a=\int _{a}{\vec {q}}\cdot {\boldsymbol {\sigma }}^{\top }\cdot {\vec {n}}\,\mathrm {d} a=\int _{a}{\vec {q}}\cdot {\vec {t}}\,\mathrm {d} a\end{aligned}}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3cd3d15ce5210cd7a51f0666890c33387d9b92a3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.046ex; margin-bottom: -0.292ex; width:84.064ex; height:11.843ex;" alt="{\displaystyle {\begin{aligned}{\text{materiell:}}&\int _{V}\nabla _{0}\cdot (\mathbf {N} \cdot {\vec {q}})\,\mathrm {d} V=\int _{A}(\mathbf {N} \cdot {\vec {q}})\cdot {\vec {N}}\,\mathrm {d} A=\int _{A}{\vec {q}}\cdot \mathbf {N} ^{\top }\cdot {\vec {N}}\,\mathrm {d} A=\int _{A}{\vec {q}}\cdot {\vec {t}}_{0}\,\mathrm {d} A\\{\text{räumlich:}}&\int _{v}\nabla \cdot ({\boldsymbol {\sigma }}\cdot {\vec {q}})\,\mathrm {d} v=\int _{a}({\boldsymbol {\sigma }}\cdot {\vec {q}})\cdot {\vec {n}}\,\mathrm {d} a=\int _{a}{\vec {q}}\cdot {\boldsymbol {\sigma }}^{\top }\cdot {\vec {n}}\,\mathrm {d} a=\int _{a}{\vec {q}}\cdot {\vec {t}}\,\mathrm {d} a\end{aligned}}}"> </noscript><span class="lazy-image-placeholder" style="width: 84.064ex;height: 11.843ex;vertical-align: -5.046ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3cd3d15ce5210cd7a51f0666890c33387d9b92a3" data-alt="{\displaystyle {\begin{aligned}{\text{materiell:}}&\int _{V}\nabla _{0}\cdot (\mathbf {N} \cdot {\vec {q}})\,\mathrm {d} V=\int _{A}(\mathbf {N} \cdot {\vec {q}})\cdot {\vec {N}}\,\mathrm {d} A=\int _{A}{\vec {q}}\cdot \mathbf {N} ^{\top }\cdot {\vec {N}}\,\mathrm {d} A=\int _{A}{\vec {q}}\cdot {\vec {t}}_{0}\,\mathrm {d} A\\{\text{räumlich:}}&\int _{v}\nabla \cdot ({\boldsymbol {\sigma }}\cdot {\vec {q}})\,\mathrm {d} v=\int _{a}({\boldsymbol {\sigma }}\cdot {\vec {q}})\cdot {\vec {n}}\,\mathrm {d} a=\int _{a}{\vec {q}}\cdot {\boldsymbol {\sigma }}^{\top }\cdot {\vec {n}}\,\mathrm {d} a=\int _{a}{\vec {q}}\cdot {\vec {t}}\,\mathrm {d} a\end{aligned}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </dd> </dl>Zusammenführung dieser Ergebnisse resultiert in den angegebenen Gleichungen.</td> </tr> </tbody> </table> <div class="mw-heading mw-heading3"> <h3 id="Prinzip_von_d’Alembert">Prinzip von d’Alembert</h3> <a role="button" href="https://de-m-wikipedia-org.translate.goog/w/index.php?title=Cauchy-eulersche_Bewegungsgesetze&action=edit&section=12&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Abschnitt bearbeiten: Prinzip von d’Alembert" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> Bearbeiten </a> </div> <div class="hauptartikel" role="navigation"> → Hauptartikel: <a href="https://de-m-wikipedia-org.translate.goog/wiki/D%E2%80%99Alembertsches_Prinzip?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="D’Alembertsches Prinzip">d’Alembertsches Prinzip</a> </div> Das Prinzip von d’Alembert hat eine grundlegende Bedeutung für die Lösung von Anfangsrandwertaufgaben der Kontinuumsmechanik, insbesondere der <a href="https://de-m-wikipedia-org.translate.goog/wiki/Verschiebungsmethode?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Verschiebungsmethode">Verschiebungsmethode</a> in der <a href="https://de-m-wikipedia-org.translate.goog/wiki/Finite-Elemente-Methode?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Finite-Elemente-Methode">Finite-Elemente-Methode</a>. Für das Vektorfeld <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {q}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> q </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\vec {q}}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8a4e063f8ee7dae2488859c45a4e645db5148085" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.309ex; height:2.676ex;" alt="{\displaystyle {\vec {q}}}"> </noscript><span class="lazy-image-placeholder" style="width: 1.309ex;height: 2.676ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8a4e063f8ee7dae2488859c45a4e645db5148085" data-alt="{\displaystyle {\vec {q}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  werden virtuelle Verschiebungen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta {\vec {u}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> δ </mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> u </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \delta {\vec {u}}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7055555a7d07ebbb35de9f00a6cfaf754c02a402" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.378ex; height:2.343ex;" alt="{\displaystyle \delta {\vec {u}}}"> </noscript><span class="lazy-image-placeholder" style="width: 2.378ex;height: 2.343ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7055555a7d07ebbb35de9f00a6cfaf754c02a402" data-alt="{\displaystyle \delta {\vec {u}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  eingesetzt, die vom Verschiebungsfeld <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {u}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> u </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\vec {u}}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/89c41e9cf70c5e5b56e2128a136985a75f90ba43" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:2.343ex;" alt="{\displaystyle {\vec {u}}}"> </noscript><span class="lazy-image-placeholder" style="width: 1.33ex;height: 2.343ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/89c41e9cf70c5e5b56e2128a136985a75f90ba43" data-alt="{\displaystyle {\vec {u}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  unabhängige, gedachte, weitgehend beliebige, differenzielle Verschiebungen sind und die mit den geometrischen Bindungen des Körpers verträglich sind. Die virtuellen Verschiebungen müssen verschwinden, wo immer Verschiebungsrandbedingungen des Körpers vorgegeben sind. Sei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {A}^{u}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi> A </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> u </mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {A}^{u}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9e0f7e703246fd77c570c75b6324a86432176063" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.916ex; height:2.343ex;" alt="{\displaystyle {A}^{u}}"> </noscript><span class="lazy-image-placeholder" style="width: 2.916ex;height: 2.343ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9e0f7e703246fd77c570c75b6324a86432176063" data-alt="{\displaystyle {A}^{u}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  der Teil der Oberfläche <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> A </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle A} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"> </noscript><span class="lazy-image-placeholder" style="width: 1.743ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" data-alt="{\displaystyle A}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  des Körpers, auf dem Verschiebungsrandbedingungen erklärt sind. Für ein Vektorfeld der virtuellen Verschiebungen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta {\vec {u}}_{0}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> δ </mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> u </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \delta {\vec {u}}_{0}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/faac1f44a898a8b5dcd1d24808acdd131e2ee30f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.433ex; height:2.676ex;" alt="{\displaystyle \delta {\vec {u}}_{0}}"> </noscript><span class="lazy-image-placeholder" style="width: 3.433ex;height: 2.676ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/faac1f44a898a8b5dcd1d24808acdd131e2ee30f" data-alt="{\displaystyle \delta {\vec {u}}_{0}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  ist dann <dl> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}{\text{materiell}}\quad &\delta {\vec {u}}_{0}({\vec {X}})={\vec {0}}\quad {\text{für alle}}\;{\vec {X}}{\in }{A}^{u}\\{\text{räumlich}}\quad &\delta {\vec {u}}({\vec {x}})={\vec {0}}\quad {\text{für alle}}\;{\vec {x}}{\in }{a}^{u}\end{aligned}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext> materiell </mtext> </mrow> <mspace width="1em"></mspace> </mtd> <mtd> <mi> δ </mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> u </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> X </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo stretchy="false"> ) </mo> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mn> 0 </mn> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mspace width="1em"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mtext> für alle </mtext> </mrow> <mspace width="thickmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> X </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo> ∈ </mo> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi> A </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> u </mi> </mrow> </msup> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext> räumlich </mtext> </mrow> <mspace width="1em"></mspace> </mtd> <mtd> <mi> δ </mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> u </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> x </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo stretchy="false"> ) </mo> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mn> 0 </mn> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mspace width="1em"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mtext> für alle </mtext> </mrow> <mspace width="thickmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> x </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo> ∈ </mo> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi> a </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> u </mi> </mrow> </msup> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\begin{aligned}{\text{materiell}}\quad &\delta {\vec {u}}_{0}({\vec {X}})={\vec {0}}\quad {\text{für alle}}\;{\vec {X}}{\in }{A}^{u}\\{\text{räumlich}}\quad &\delta {\vec {u}}({\vec {x}})={\vec {0}}\quad {\text{für alle}}\;{\vec {x}}{\in }{a}^{u}\end{aligned}}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24b4d86b7f8578a0e5604a034ad8f081dbba3952" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:40.255ex; height:7.843ex;" alt="{\displaystyle {\begin{aligned}{\text{materiell}}\quad &\delta {\vec {u}}_{0}({\vec {X}})={\vec {0}}\quad {\text{für alle}}\;{\vec {X}}{\in }{A}^{u}\\{\text{räumlich}}\quad &\delta {\vec {u}}({\vec {x}})={\vec {0}}\quad {\text{für alle}}\;{\vec {x}}{\in }{a}^{u}\end{aligned}}}"> </noscript><span class="lazy-image-placeholder" style="width: 40.255ex;height: 7.843ex;vertical-align: -3.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24b4d86b7f8578a0e5604a034ad8f081dbba3952" data-alt="{\displaystyle {\begin{aligned}{\text{materiell}}\quad &\delta {\vec {u}}_{0}({\vec {X}})={\vec {0}}\quad {\text{für alle}}\;{\vec {X}}{\in }{A}^{u}\\{\text{räumlich}}\quad &\delta {\vec {u}}({\vec {x}})={\vec {0}}\quad {\text{für alle}}\;{\vec {x}}{\in }{a}^{u}\end{aligned}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </dd> </dl> zu fordern. Auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {A}^{u}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi> A </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> u </mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {A}^{u}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9e0f7e703246fd77c570c75b6324a86432176063" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.916ex; height:2.343ex;" alt="{\displaystyle {A}^{u}}"> </noscript><span class="lazy-image-placeholder" style="width: 2.916ex;height: 2.343ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9e0f7e703246fd77c570c75b6324a86432176063" data-alt="{\displaystyle {A}^{u}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  können dann keine Oberflächenspannungen vorgegeben werden. Deshalb bezeichnet <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {A}^{\sigma }=A\setminus A^{u}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi> A </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> σ </mi> </mrow> </msup> <mo> = </mo> <mi> A </mi> <mo class="MJX-variant"> ∖ </mo> <msup> <mi> A </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> u </mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {A}^{\sigma }=A\setminus A^{u}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a16d826f014cb090d7c4706e2a5ecd8a0230059d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.867ex; height:2.843ex;" alt="{\displaystyle {A}^{\sigma }=A\setminus A^{u}}"> </noscript><span class="lazy-image-placeholder" style="width: 12.867ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a16d826f014cb090d7c4706e2a5ecd8a0230059d" data-alt="{\displaystyle {A}^{\sigma }=A\setminus A^{u}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  den Teil der Oberfläche des Körpers, auf dem Oberflächenspannungen wirken (können) was entsprechend auch in der räumlichen Formulierung definiert wird. So entsteht: <dl> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{array}{llcl}{\text{materiell:}}&\displaystyle \int _{V}\rho _{0}\,{\ddot {\vec {\chi }}}\cdot \delta {\vec {u}}_{0}\,\mathrm {d} V+\int _{V}{\tilde {\mathbf {T} }}:\delta \mathbf {E} \,\mathrm {d} V\!\!\!\!\!\!&=&\!\!\!\!\!\!\displaystyle \int _{V}\rho _{0}\,{\vec {k}}_{0}\cdot \delta {\vec {u}}_{0}\,\mathrm {d} V+\int _{A^{\sigma }}{\vec {t}}_{0}\cdot \delta {\vec {u}}_{0}\,\mathrm {d} A\\&{\text{für alle}}\;\delta {\vec {u}}_{0}\in {\mathcal {V}}_{0}\\{\text{räumlich:}}&\displaystyle \int _{v}\rho \,{\dot {\vec {v}}}\cdot \delta {\vec {u}}\,\mathrm {d} v+\int _{v}{\boldsymbol {\sigma }}:\delta {\boldsymbol {\varepsilon }}\,\mathrm {d} v\!\!\!\!\!\!&=&\!\!\!\!\!\!\displaystyle \int _{v}\rho \,{\vec {k}}\cdot \delta {\vec {u}}\,\mathrm {d} v+\int _{a^{\sigma }}{\vec {t}}\cdot \delta {\vec {u}}\,\mathrm {d} a\\&{\text{für alle}}\;\delta {\vec {u}}\in {\mathcal {V}}\end{array}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="left left center left" rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext> materiell: </mtext> </mrow> </mtd> <mtd> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mo> ∫ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> V </mi> </mrow> </msub> <msub> <mi> ρ </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> χ </mi> <mo stretchy="false"> → </mo> </mover> </mrow> <mo> ¨ </mo> </mover> </mrow> </mrow> <mo> ⋅ </mo> <mi> δ </mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> u </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> V </mi> <mo> + </mo> <msub> <mo> ∫ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> V </mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> T </mi> </mrow> <mo stretchy="false"> ~ </mo> </mover> </mrow> </mrow> <mo> : </mo> <mi> δ </mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> E </mi> </mrow> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> V </mi> <mspace width="negativethinmathspace"></mspace> <mspace width="negativethinmathspace"></mspace> <mspace width="negativethinmathspace"></mspace> <mspace width="negativethinmathspace"></mspace> <mspace width="negativethinmathspace"></mspace> <mspace width="negativethinmathspace"></mspace> </mstyle> </mtd> <mtd> <mo> = </mo> </mtd> <mtd> <mspace width="negativethinmathspace"></mspace> <mspace width="negativethinmathspace"></mspace> <mspace width="negativethinmathspace"></mspace> <mspace width="negativethinmathspace"></mspace> <mspace width="negativethinmathspace"></mspace> <mspace width="negativethinmathspace"></mspace> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mo> ∫ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> V </mi> </mrow> </msub> <msub> <mi> ρ </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mspace width="thinmathspace"></mspace> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> k </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mo> ⋅ </mo> <mi> δ </mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> u </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> V </mi> <mo> + </mo> <msub> <mo> ∫ </mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi> A </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> σ </mi> </mrow> </msup> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> t </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mo> ⋅ </mo> <mi> δ </mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> u </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> A </mi> </mstyle> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext> für alle </mtext> </mrow> <mspace width="thickmathspace"></mspace> <mi> δ </mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> u </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mo> ∈ </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script"> V </mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext> räumlich: </mtext> </mrow> </mtd> <mtd> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mo> ∫ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> v </mi> </mrow> </msub> <mi> ρ </mi> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> v </mi> <mo stretchy="false"> → </mo> </mover> </mrow> <mo> ˙ </mo> </mover> </mrow> </mrow> <mo> ⋅ </mo> <mi> δ </mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> u </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> v </mi> <mo> + </mo> <msub> <mo> ∫ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> v </mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> σ </mi> </mrow> <mo> : </mo> <mi> δ </mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> ε </mi> </mrow> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> v </mi> <mspace width="negativethinmathspace"></mspace> <mspace width="negativethinmathspace"></mspace> <mspace width="negativethinmathspace"></mspace> <mspace width="negativethinmathspace"></mspace> <mspace width="negativethinmathspace"></mspace> <mspace width="negativethinmathspace"></mspace> </mstyle> </mtd> <mtd> <mo> = </mo> </mtd> <mtd> <mspace width="negativethinmathspace"></mspace> <mspace width="negativethinmathspace"></mspace> <mspace width="negativethinmathspace"></mspace> <mspace width="negativethinmathspace"></mspace> <mspace width="negativethinmathspace"></mspace> <mspace width="negativethinmathspace"></mspace> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mo> ∫ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> v </mi> </mrow> </msub> <mi> ρ </mi> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> k </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo> ⋅ </mo> <mi> δ </mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> u </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> v </mi> <mo> + </mo> <msub> <mo> ∫ </mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi> a </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> σ </mi> </mrow> </msup> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> t </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo> ⋅ </mo> <mi> δ </mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> u </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> a </mi> </mstyle> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext> für alle </mtext> </mrow> <mspace width="thickmathspace"></mspace> <mi> δ </mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> u </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo> ∈ </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script"> V </mi> </mrow> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\begin{array}{llcl}{\text{materiell:}}&\displaystyle \int _{V}\rho _{0}\,{\ddot {\vec {\chi }}}\cdot \delta {\vec {u}}_{0}\,\mathrm {d} V+\int _{V}{\tilde {\mathbf {T} }}:\delta \mathbf {E} \,\mathrm {d} V\!\!\!\!\!\!&=&\!\!\!\!\!\!\displaystyle \int _{V}\rho _{0}\,{\vec {k}}_{0}\cdot \delta {\vec {u}}_{0}\,\mathrm {d} V+\int _{A^{\sigma }}{\vec {t}}_{0}\cdot \delta {\vec {u}}_{0}\,\mathrm {d} A\\&{\text{für alle}}\;\delta {\vec {u}}_{0}\in {\mathcal {V}}_{0}\\{\text{räumlich:}}&\displaystyle \int _{v}\rho \,{\dot {\vec {v}}}\cdot \delta {\vec {u}}\,\mathrm {d} v+\int _{v}{\boldsymbol {\sigma }}:\delta {\boldsymbol {\varepsilon }}\,\mathrm {d} v\!\!\!\!\!\!&=&\!\!\!\!\!\!\displaystyle \int _{v}\rho \,{\vec {k}}\cdot \delta {\vec {u}}\,\mathrm {d} v+\int _{a^{\sigma }}{\vec {t}}\cdot \delta {\vec {u}}\,\mathrm {d} a\\&{\text{für alle}}\;\delta {\vec {u}}\in {\mathcal {V}}\end{array}}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fc98c850b4b9cdacea05e8a3d5bd7ebb5413eb1c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -9.338ex; width:81.093ex; height:19.843ex;" alt="{\displaystyle {\begin{array}{llcl}{\text{materiell:}}&\displaystyle \int _{V}\rho _{0}\,{\ddot {\vec {\chi }}}\cdot \delta {\vec {u}}_{0}\,\mathrm {d} V+\int _{V}{\tilde {\mathbf {T} }}:\delta \mathbf {E} \,\mathrm {d} V\!\!\!\!\!\!&=&\!\!\!\!\!\!\displaystyle \int _{V}\rho _{0}\,{\vec {k}}_{0}\cdot \delta {\vec {u}}_{0}\,\mathrm {d} V+\int _{A^{\sigma }}{\vec {t}}_{0}\cdot \delta {\vec {u}}_{0}\,\mathrm {d} A\\&{\text{für alle}}\;\delta {\vec {u}}_{0}\in {\mathcal {V}}_{0}\\{\text{räumlich:}}&\displaystyle \int _{v}\rho \,{\dot {\vec {v}}}\cdot \delta {\vec {u}}\,\mathrm {d} v+\int _{v}{\boldsymbol {\sigma }}:\delta {\boldsymbol {\varepsilon }}\,\mathrm {d} v\!\!\!\!\!\!&=&\!\!\!\!\!\!\displaystyle \int _{v}\rho \,{\vec {k}}\cdot \delta {\vec {u}}\,\mathrm {d} v+\int _{a^{\sigma }}{\vec {t}}\cdot \delta {\vec {u}}\,\mathrm {d} a\\&{\text{für alle}}\;\delta {\vec {u}}\in {\mathcal {V}}\end{array}}}"> </noscript><span class="lazy-image-placeholder" style="width: 81.093ex;height: 19.843ex;vertical-align: -9.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fc98c850b4b9cdacea05e8a3d5bd7ebb5413eb1c" data-alt="{\displaystyle {\begin{array}{llcl}{\text{materiell:}}&\displaystyle \int _{V}\rho _{0}\,{\ddot {\vec {\chi }}}\cdot \delta {\vec {u}}_{0}\,\mathrm {d} V+\int _{V}{\tilde {\mathbf {T} }}:\delta \mathbf {E} \,\mathrm {d} V\!\!\!\!\!\!&=&\!\!\!\!\!\!\displaystyle \int _{V}\rho _{0}\,{\vec {k}}_{0}\cdot \delta {\vec {u}}_{0}\,\mathrm {d} V+\int _{A^{\sigma }}{\vec {t}}_{0}\cdot \delta {\vec {u}}_{0}\,\mathrm {d} A\\&{\text{für alle}}\;\delta {\vec {u}}_{0}\in {\mathcal {V}}_{0}\\{\text{räumlich:}}&\displaystyle \int _{v}\rho \,{\dot {\vec {v}}}\cdot \delta {\vec {u}}\,\mathrm {d} v+\int _{v}{\boldsymbol {\sigma }}:\delta {\boldsymbol {\varepsilon }}\,\mathrm {d} v\!\!\!\!\!\!&=&\!\!\!\!\!\!\displaystyle \int _{v}\rho \,{\vec {k}}\cdot \delta {\vec {u}}\,\mathrm {d} v+\int _{a^{\sigma }}{\vec {t}}\cdot \delta {\vec {u}}\,\mathrm {d} a\\&{\text{für alle}}\;\delta {\vec {u}}\in {\mathcal {V}}\end{array}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </dd> </dl> Die Menge <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {V}}_{(0)}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script"> V </mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> ( </mo> <mn> 0 </mn> <mo stretchy="false"> ) </mo> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\mathcal {V}}_{(0)}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/788f008af505351048f548cf1e98693698f7ef71" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:3.759ex; height:3.009ex;" alt="{\displaystyle {\mathcal {V}}_{(0)}}"> </noscript><span class="lazy-image-placeholder" style="width: 3.759ex;height: 3.009ex;vertical-align: -1.171ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/788f008af505351048f548cf1e98693698f7ef71" data-alt="{\displaystyle {\mathcal {V}}_{(0)}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  enthält alle zulässigen, materiellen bzw. räumlichen, virtuellen Verschiebungsfelder. Auf der linken Seite steht die virtuelle Arbeit der Trägheitskräfte und die virtuelle Deformationsarbeit und auf der rechten Seite die virtuelle Arbeit der äußeren Kräfte (volumen- und oberflächenverteilt.) In der materiellen Darstellung stehen die virtuelle Verzerrungen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta \mathbf {E} }"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> δ </mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> E </mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \delta \mathbf {E} } </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/60bc861a2f4add7bb1d104ba1449e7e3f5c1a118" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.806ex; height:2.343ex;" alt="{\displaystyle \delta \mathbf {E} }"> </noscript><span class="lazy-image-placeholder" style="width: 2.806ex;height: 2.343ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/60bc861a2f4add7bb1d104ba1449e7e3f5c1a118" data-alt="{\displaystyle \delta \mathbf {E} }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  für die Variation des green-lagrangeschen Verzerrungstensors: <dl> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta \mathbf {E} :={\frac {1}{2}}(\mathbf {F} ^{\top }\cdot \delta \mathbf {F} +\delta \mathbf {F} ^{\top }\cdot \mathbf {F} )=\operatorname {sym} (\mathbf {F} ^{\top }\cdot \delta \mathbf {F} )}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> δ </mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> E </mi> </mrow> <mo> := </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo stretchy="false"> ( </mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> F </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> ⊤ </mi> </mrow> </msup> <mo> ⋅ </mo> <mi> δ </mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> F </mi> </mrow> <mo> + </mo> <mi> δ </mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> F </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> ⊤ </mi> </mrow> </msup> <mo> ⋅ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> F </mi> </mrow> <mo stretchy="false"> ) </mo> <mo> = </mo> <mi> sym </mi> <mo> ⁡ </mo> <mo stretchy="false"> ( </mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> F </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> ⊤ </mi> </mrow> </msup> <mo> ⋅ </mo> <mi> δ </mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> F </mi> </mrow> <mo stretchy="false"> ) </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \delta \mathbf {E} :={\frac {1}{2}}(\mathbf {F} ^{\top }\cdot \delta \mathbf {F} +\delta \mathbf {F} ^{\top }\cdot \mathbf {F} )=\operatorname {sym} (\mathbf {F} ^{\top }\cdot \delta \mathbf {F} )} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c0c0fa80c130656f2ad51207758906f86ea3dd45" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:44.999ex; height:5.176ex;" alt="{\displaystyle \delta \mathbf {E} :={\frac {1}{2}}(\mathbf {F} ^{\top }\cdot \delta \mathbf {F} +\delta \mathbf {F} ^{\top }\cdot \mathbf {F} )=\operatorname {sym} (\mathbf {F} ^{\top }\cdot \delta \mathbf {F} )}"> </noscript><span class="lazy-image-placeholder" style="width: 44.999ex;height: 5.176ex;vertical-align: -1.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c0c0fa80c130656f2ad51207758906f86ea3dd45" data-alt="{\displaystyle \delta \mathbf {E} :={\frac {1}{2}}(\mathbf {F} ^{\top }\cdot \delta \mathbf {F} +\delta \mathbf {F} ^{\top }\cdot \mathbf {F} )=\operatorname {sym} (\mathbf {F} ^{\top }\cdot \delta \mathbf {F} )}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </dd> </dl> mit dem virtuellen Deformationsgradient <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta \mathbf {F} :=\operatorname {GRAD} \delta {\vec {u}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> δ </mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> F </mi> </mrow> <mo> := </mo> <mi> GRAD </mi> <mo> ⁡ </mo> <mi> δ </mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> u </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \delta \mathbf {F} :=\operatorname {GRAD} \delta {\vec {u}}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5eb2c2edce42737eb21f405e14b62e8913eada7c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:16.296ex; height:2.343ex;" alt="{\displaystyle \delta \mathbf {F} :=\operatorname {GRAD} \delta {\vec {u}}}"> </noscript><span class="lazy-image-placeholder" style="width: 16.296ex;height: 2.343ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5eb2c2edce42737eb21f405e14b62e8913eada7c" data-alt="{\displaystyle \delta \mathbf {F} :=\operatorname {GRAD} \delta {\vec {u}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> . In der räumlichen Darstellung bildet sich der virtuelle Verzerrungstensor <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta {\boldsymbol {\varepsilon }}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> δ </mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> ε </mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \delta {\boldsymbol {\varepsilon }}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7dcd145dd48190cbea329acfd863f5ba673a6790" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.278ex; height:2.343ex;" alt="{\displaystyle \delta {\boldsymbol {\varepsilon }}}"> </noscript><span class="lazy-image-placeholder" style="width: 2.278ex;height: 2.343ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7dcd145dd48190cbea329acfd863f5ba673a6790" data-alt="{\displaystyle \delta {\boldsymbol {\varepsilon }}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  aus dem räumlichen virtuellen <a href="https://de-m-wikipedia-org.translate.goog/wiki/Verschiebungsgradient?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Verschiebungsgradient">Verschiebungsgradient</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta \mathbf {h} :=\operatorname {grad} \delta {\vec {u}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> δ </mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> h </mi> </mrow> <mo> := </mo> <mi> grad </mi> <mo> ⁡ </mo> <mi> δ </mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> u </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \delta \mathbf {h} :=\operatorname {grad} \delta {\vec {u}}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c77d2ca81141dbbc979596690002326c53d4b1de" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.574ex; height:2.676ex;" alt="{\displaystyle \delta \mathbf {h} :=\operatorname {grad} \delta {\vec {u}}}"> </noscript><span class="lazy-image-placeholder" style="width: 13.574ex;height: 2.676ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c77d2ca81141dbbc979596690002326c53d4b1de" data-alt="{\displaystyle \delta \mathbf {h} :=\operatorname {grad} \delta {\vec {u}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> : <dl> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta {\boldsymbol {\varepsilon }}:=\operatorname {sym\,grad} \delta {\vec {u}}={\frac {1}{2}}(\delta \mathbf {h} +\delta \mathbf {h} ^{\top })}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> δ </mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> ε </mi> </mrow> <mo> := </mo> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mi mathvariant="normal"> s </mi> <mi mathvariant="normal"> y </mi> <mi mathvariant="normal"> m </mi> <mspace width="thinmathspace"></mspace> <mi mathvariant="normal"> g </mi> <mi mathvariant="normal"> r </mi> <mi mathvariant="normal"> a </mi> <mi mathvariant="normal"> d </mi> </mrow> <mo> ⁡ </mo> <mi> δ </mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> u </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo stretchy="false"> ( </mo> <mi> δ </mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> h </mi> </mrow> <mo> + </mo> <mi> δ </mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> h </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> ⊤ </mi> </mrow> </msup> <mo stretchy="false"> ) </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \delta {\boldsymbol {\varepsilon }}:=\operatorname {sym\,grad} \delta {\vec {u}}={\frac {1}{2}}(\delta \mathbf {h} +\delta \mathbf {h} ^{\top })} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ccac02222dc7ae2804502cbeea8dffa0bf432dba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:34.11ex; height:5.176ex;" alt="{\displaystyle \delta {\boldsymbol {\varepsilon }}:=\operatorname {sym\,grad} \delta {\vec {u}}={\frac {1}{2}}(\delta \mathbf {h} +\delta \mathbf {h} ^{\top })}"> </noscript><span class="lazy-image-placeholder" style="width: 34.11ex;height: 5.176ex;vertical-align: -1.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ccac02222dc7ae2804502cbeea8dffa0bf432dba" data-alt="{\displaystyle \delta {\boldsymbol {\varepsilon }}:=\operatorname {sym\,grad} \delta {\vec {u}}={\frac {1}{2}}(\delta \mathbf {h} +\delta \mathbf {h} ^{\top })}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </dd> </dl> <div class="mw-heading mw-heading3"> <h3 id="Bilanz_der_mechanischen_Energie">Bilanz der mechanischen Energie</h3> <a role="button" href="https://de-m-wikipedia-org.translate.goog/w/index.php?title=Cauchy-eulersche_Bewegungsgesetze&action=edit&section=13&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Abschnitt bearbeiten: Bilanz der mechanischen Energie" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> Bearbeiten </a> </div> Wenn für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {q}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> q </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\vec {q}}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8a4e063f8ee7dae2488859c45a4e645db5148085" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.309ex; height:2.676ex;" alt="{\displaystyle {\vec {q}}}"> </noscript><span class="lazy-image-placeholder" style="width: 1.309ex;height: 2.676ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8a4e063f8ee7dae2488859c45a4e645db5148085" data-alt="{\displaystyle {\vec {q}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  das Geschwindigkeitsfeld eingesetzt wird, folgt die Bilanz der mechanischen Energie: <dl> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{array}{llcl}{\text{materiell:}}&{\frac {\mathrm {d} }{\mathrm {d} t}}\int _{V}{\frac {\rho _{0}}{2}}\,{\dot {\vec {\chi }}}\cdot {\dot {\vec {\chi }}}\,\mathrm {d} V&=&\int _{V}\rho _{0}\,{\vec {k}}_{0}\cdot {\dot {\vec {\chi }}}\,\mathrm {d} V+\int _{A}{\vec {t}}_{0}\cdot {\dot {\vec {\chi }}}\,\mathrm {d} A-\int _{V}{\tilde {\mathbf {T} }}:{\dot {\mathbf {E} }}\,\mathrm {d} V\\{\text{räumlich:}}&{\frac {\mathrm {d} }{\mathrm {d} t}}\int _{v}{\frac {\rho }{2}}{\vec {v}}\cdot {\vec {v}}\,\mathrm {d} v&=&\int _{v}\rho \,{\vec {k}}\cdot {\vec {v}}\,\mathrm {d} v+\int _{a}{\vec {t}}\cdot {\vec {v}}\,\mathrm {d} a-\int _{v}{\boldsymbol {\sigma }}:\mathbf {d} \,\mathrm {d} v\end{array}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="left left center left" rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext> materiell: </mtext> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> t </mi> </mrow> </mfrac> </mrow> <msub> <mo> ∫ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> V </mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi> ρ </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mn> 2 </mn> </mfrac> </mrow> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> χ </mi> <mo stretchy="false"> → </mo> </mover> </mrow> <mo> ˙ </mo> </mover> </mrow> </mrow> <mo> ⋅ </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> χ </mi> <mo stretchy="false"> → </mo> </mover> </mrow> <mo> ˙ </mo> </mover> </mrow> </mrow> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> V </mi> </mtd> <mtd> <mo> = </mo> </mtd> <mtd> <msub> <mo> ∫ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> V </mi> </mrow> </msub> <msub> <mi> ρ </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mspace width="thinmathspace"></mspace> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> k </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mo> ⋅ </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> χ </mi> <mo stretchy="false"> → </mo> </mover> </mrow> <mo> ˙ </mo> </mover> </mrow> </mrow> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> V </mi> <mo> + </mo> <msub> <mo> ∫ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> A </mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> t </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mo> ⋅ </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> χ </mi> <mo stretchy="false"> → </mo> </mover> </mrow> <mo> ˙ </mo> </mover> </mrow> </mrow> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> A </mi> <mo> − </mo> <msub> <mo> ∫ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> V </mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> T </mi> </mrow> <mo stretchy="false"> ~ </mo> </mover> </mrow> </mrow> <mo> : </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> E </mi> </mrow> <mo> ˙ </mo> </mover> </mrow> </mrow> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> V </mi> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext> räumlich: </mtext> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> t </mi> </mrow> </mfrac> </mrow> <msub> <mo> ∫ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> v </mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi> ρ </mi> <mn> 2 </mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> v </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo> ⋅ </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> v </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> v </mi> </mtd> <mtd> <mo> = </mo> </mtd> <mtd> <msub> <mo> ∫ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> v </mi> </mrow> </msub> <mi> ρ </mi> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> k </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo> ⋅ </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> v </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> v </mi> <mo> + </mo> <msub> <mo> ∫ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> a </mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> t </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo> ⋅ </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> v </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> a </mi> <mo> − </mo> <msub> <mo> ∫ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> v </mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> σ </mi> </mrow> <mo> : </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> d </mi> </mrow> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> v </mi> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\begin{array}{llcl}{\text{materiell:}}&{\frac {\mathrm {d} }{\mathrm {d} t}}\int _{V}{\frac {\rho _{0}}{2}}\,{\dot {\vec {\chi }}}\cdot {\dot {\vec {\chi }}}\,\mathrm {d} V&=&\int _{V}\rho _{0}\,{\vec {k}}_{0}\cdot {\dot {\vec {\chi }}}\,\mathrm {d} V+\int _{A}{\vec {t}}_{0}\cdot {\dot {\vec {\chi }}}\,\mathrm {d} A-\int _{V}{\tilde {\mathbf {T} }}:{\dot {\mathbf {E} }}\,\mathrm {d} V\\{\text{räumlich:}}&{\frac {\mathrm {d} }{\mathrm {d} t}}\int _{v}{\frac {\rho }{2}}{\vec {v}}\cdot {\vec {v}}\,\mathrm {d} v&=&\int _{v}\rho \,{\vec {k}}\cdot {\vec {v}}\,\mathrm {d} v+\int _{a}{\vec {t}}\cdot {\vec {v}}\,\mathrm {d} a-\int _{v}{\boldsymbol {\sigma }}:\mathbf {d} \,\mathrm {d} v\end{array}}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/840f0038b1dabf380b0795483959f16301d17212" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:80.042ex; height:8.176ex;" alt="{\displaystyle {\begin{array}{llcl}{\text{materiell:}}&{\frac {\mathrm {d} }{\mathrm {d} t}}\int _{V}{\frac {\rho _{0}}{2}}\,{\dot {\vec {\chi }}}\cdot {\dot {\vec {\chi }}}\,\mathrm {d} V&=&\int _{V}\rho _{0}\,{\vec {k}}_{0}\cdot {\dot {\vec {\chi }}}\,\mathrm {d} V+\int _{A}{\vec {t}}_{0}\cdot {\dot {\vec {\chi }}}\,\mathrm {d} A-\int _{V}{\tilde {\mathbf {T} }}:{\dot {\mathbf {E} }}\,\mathrm {d} V\\{\text{räumlich:}}&{\frac {\mathrm {d} }{\mathrm {d} t}}\int _{v}{\frac {\rho }{2}}{\vec {v}}\cdot {\vec {v}}\,\mathrm {d} v&=&\int _{v}\rho \,{\vec {k}}\cdot {\vec {v}}\,\mathrm {d} v+\int _{a}{\vec {t}}\cdot {\vec {v}}\,\mathrm {d} a-\int _{v}{\boldsymbol {\sigma }}:\mathbf {d} \,\mathrm {d} v\end{array}}}"> </noscript><span class="lazy-image-placeholder" style="width: 80.042ex;height: 8.176ex;vertical-align: -3.505ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/840f0038b1dabf380b0795483959f16301d17212" data-alt="{\displaystyle {\begin{array}{llcl}{\text{materiell:}}&{\frac {\mathrm {d} }{\mathrm {d} t}}\int _{V}{\frac {\rho _{0}}{2}}\,{\dot {\vec {\chi }}}\cdot {\dot {\vec {\chi }}}\,\mathrm {d} V&=&\int _{V}\rho _{0}\,{\vec {k}}_{0}\cdot {\dot {\vec {\chi }}}\,\mathrm {d} V+\int _{A}{\vec {t}}_{0}\cdot {\dot {\vec {\chi }}}\,\mathrm {d} A-\int _{V}{\tilde {\mathbf {T} }}:{\dot {\mathbf {E} }}\,\mathrm {d} V\\{\text{räumlich:}}&{\frac {\mathrm {d} }{\mathrm {d} t}}\int _{v}{\frac {\rho }{2}}{\vec {v}}\cdot {\vec {v}}\,\mathrm {d} v&=&\int _{v}\rho \,{\vec {k}}\cdot {\vec {v}}\,\mathrm {d} v+\int _{a}{\vec {t}}\cdot {\vec {v}}\,\mathrm {d} a-\int _{v}{\boldsymbol {\sigma }}:\mathbf {d} \,\mathrm {d} v\end{array}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </dd> </dl> Auf der linken Seite steht die zeitliche Änderung der <a href="https://de-m-wikipedia-org.translate.goog/wiki/Kinetische_Energie?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Kinetische Energie">kinetischen Energie</a> und auf der rechten Seite steht die Leistung der äußeren Kräfte (volumen- und flächenverteilt) abzüglich der Verformungsleistung. Dieser Satz wird auch <a href="https://de-m-wikipedia-org.translate.goog/wiki/Arbeitssatz?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Arbeitssatz">Arbeitssatz</a><a href="https://de-m-wikipedia-org.translate.goog/wiki/Cauchy-eulersche_Bewegungsgesetze?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-7">[L 5]</a> oder „Satz von der geleisteten Arbeit“ (<a href="https://de-m-wikipedia-org.translate.goog/wiki/Englische_Sprache?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Englische Sprache">englisch</a> Theorem of work expended<a href="https://de-m-wikipedia-org.translate.goog/wiki/Cauchy-eulersche_Bewegungsgesetze?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-8">[L 6]</a>) genannt. <table class="wikitable mw-collapsible mw-collapsed"> <tbody> <tr> <td>Beweis</td> </tr> <tr> <td>Die Zeitableitung der kinetischen Energie ist gleich der Leistung der Trägheitskräfte <dl> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{array}{llcl}{\text{materiell:}}&{\frac {\mathrm {d} }{\mathrm {d} t}}\int _{V}{\frac {\rho _{0}}{2}}\,{\dot {\vec {\chi }}}\cdot {\dot {\vec {\chi }}}\,\mathrm {d} V&=&\int _{V}\rho _{0}\,{\dot {\vec {\chi }}}\cdot {\ddot {\vec {\chi }}}\,\mathrm {d} V\\{\text{räumlich:}}&{\frac {\mathrm {d} }{\mathrm {d} t}}\int _{v}{\frac {\rho }{2}}{\vec {v}}\cdot {\vec {v}}\,\mathrm {d} v&=&\int _{v}\left[{\frac {\mathrm {d} }{\mathrm {d} t}}\left({\frac {\rho }{2}}{\vec {v}}\cdot {\vec {v}}\right)+\operatorname {div} ({\vec {v}}){\frac {\rho }{2}}{\vec {v}}\cdot {\vec {v}}\right]\,\mathrm {d} v\\&&=&\int _{v}\left[{\frac {\dot {\rho }}{2}}{\vec {v}}\cdot {\vec {v}}+\rho {\vec {v}}\cdot {\dot {\vec {v}}}+\operatorname {div} ({\vec {v}}){\frac {\rho }{2}}{\vec {v}}\cdot {\vec {v}}\right]\,\mathrm {d} v\\&&=&\int _{v}{\Bigl [}(\underbrace {{\dot {\rho }}+\rho \operatorname {div} ({\vec {v}})} _{=0}){\frac {1}{2}}{\vec {v}}\cdot {\vec {v}}+\rho {\vec {v}}\cdot {\dot {\vec {v}}}{\Bigr ]}\,\mathrm {d} v=\int _{v}\rho {\vec {v}}\cdot {\dot {\vec {v}}}\,\mathrm {d} v\,,\end{array}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="left left center left" rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext> materiell: </mtext> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> t </mi> </mrow> </mfrac> </mrow> <msub> <mo> ∫ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> V </mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi> ρ </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mn> 2 </mn> </mfrac> </mrow> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> χ </mi> <mo stretchy="false"> → </mo> </mover> </mrow> <mo> ˙ </mo> </mover> </mrow> </mrow> <mo> ⋅ </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> χ </mi> <mo stretchy="false"> → </mo> </mover> </mrow> <mo> ˙ </mo> </mover> </mrow> </mrow> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> V </mi> </mtd> <mtd> <mo> = </mo> </mtd> <mtd> <msub> <mo> ∫ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> V </mi> </mrow> </msub> <msub> <mi> ρ </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> χ </mi> <mo stretchy="false"> → </mo> </mover> </mrow> <mo> ˙ </mo> </mover> </mrow> </mrow> <mo> ⋅ </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> χ </mi> <mo stretchy="false"> → </mo> </mover> </mrow> <mo> ¨ </mo> </mover> </mrow> </mrow> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> V </mi> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext> räumlich: </mtext> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> t </mi> </mrow> </mfrac> </mrow> <msub> <mo> ∫ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> v </mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi> ρ </mi> <mn> 2 </mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> v </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo> ⋅ </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> v </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> v </mi> </mtd> <mtd> <mo> = </mo> </mtd> <mtd> <msub> <mo> ∫ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> v </mi> </mrow> </msub> <mrow> <mo> [ </mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> t </mi> </mrow> </mfrac> </mrow> <mrow> <mo> ( </mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi> ρ </mi> <mn> 2 </mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> v </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo> ⋅ </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> v </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> + </mo> <mi> div </mi> <mo> ⁡ </mo> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> v </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo stretchy="false"> ) </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi> ρ </mi> <mn> 2 </mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> v </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo> ⋅ </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> v </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> </mrow> <mo> ] </mo> </mrow> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> v </mi> </mtd> </mtr> <mtr> <mtd></mtd> <mtd></mtd> <mtd> <mo> = </mo> </mtd> <mtd> <msub> <mo> ∫ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> v </mi> </mrow> </msub> <mrow> <mo> [ </mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> ρ </mi> <mo> ˙ </mo> </mover> </mrow> <mn> 2 </mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> v </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo> ⋅ </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> v </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo> + </mo> <mi> ρ </mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> v </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo> ⋅ </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> v </mi> <mo stretchy="false"> → </mo> </mover> </mrow> <mo> ˙ </mo> </mover> </mrow> </mrow> <mo> + </mo> <mi> div </mi> <mo> ⁡ </mo> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> v </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo stretchy="false"> ) </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi> ρ </mi> <mn> 2 </mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> v </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo> ⋅ </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> v </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> </mrow> <mo> ] </mo> </mrow> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> v </mi> </mtd> </mtr> <mtr> <mtd></mtd> <mtd></mtd> <mtd> <mo> = </mo> </mtd> <mtd> <msub> <mo> ∫ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> v </mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.623em" minsize="1.623em"> [ </mo> </mrow> </mrow> <mo stretchy="false"> ( </mo> <munder> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <munder> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> ρ </mi> <mo> ˙ </mo> </mover> </mrow> </mrow> <mo> + </mo> <mi> ρ </mi> <mi> div </mi> <mo> ⁡ </mo> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> v </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo stretchy="false"> ) </mo> </mrow> <mo> ⏟ </mo> </munder> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo> = </mo> <mn> 0 </mn> </mrow> </munder> <mo stretchy="false"> ) </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> v </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo> ⋅ </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> v </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo> + </mo> <mi> ρ </mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> v </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo> ⋅ </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> v </mi> <mo stretchy="false"> → </mo> </mover> </mrow> <mo> ˙ </mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.623em" minsize="1.623em"> ] </mo> </mrow> </mrow> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> v </mi> <mo> = </mo> <msub> <mo> ∫ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> v </mi> </mrow> </msub> <mi> ρ </mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> v </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo> ⋅ </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> v </mi> <mo stretchy="false"> → </mo> </mover> </mrow> <mo> ˙ </mo> </mover> </mrow> </mrow> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> v </mi> <mspace width="thinmathspace"></mspace> <mo> , </mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\begin{array}{llcl}{\text{materiell:}}&{\frac {\mathrm {d} }{\mathrm {d} t}}\int _{V}{\frac {\rho _{0}}{2}}\,{\dot {\vec {\chi }}}\cdot {\dot {\vec {\chi }}}\,\mathrm {d} V&=&\int _{V}\rho _{0}\,{\dot {\vec {\chi }}}\cdot {\ddot {\vec {\chi }}}\,\mathrm {d} V\\{\text{räumlich:}}&{\frac {\mathrm {d} }{\mathrm {d} t}}\int _{v}{\frac {\rho }{2}}{\vec {v}}\cdot {\vec {v}}\,\mathrm {d} v&=&\int _{v}\left[{\frac {\mathrm {d} }{\mathrm {d} t}}\left({\frac {\rho }{2}}{\vec {v}}\cdot {\vec {v}}\right)+\operatorname {div} ({\vec {v}}){\frac {\rho }{2}}{\vec {v}}\cdot {\vec {v}}\right]\,\mathrm {d} v\\&&=&\int _{v}\left[{\frac {\dot {\rho }}{2}}{\vec {v}}\cdot {\vec {v}}+\rho {\vec {v}}\cdot {\dot {\vec {v}}}+\operatorname {div} ({\vec {v}}){\frac {\rho }{2}}{\vec {v}}\cdot {\vec {v}}\right]\,\mathrm {d} v\\&&=&\int _{v}{\Bigl [}(\underbrace {{\dot {\rho }}+\rho \operatorname {div} ({\vec {v}})} _{=0}){\frac {1}{2}}{\vec {v}}\cdot {\vec {v}}+\rho {\vec {v}}\cdot {\dot {\vec {v}}}{\Bigr ]}\,\mathrm {d} v=\int _{v}\rho {\vec {v}}\cdot {\dot {\vec {v}}}\,\mathrm {d} v\,,\end{array}}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/60e5018ab1ab151cdd970b70bad6ce30b471d7db" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -10.338ex; width:85.818ex; height:21.843ex;" alt="{\displaystyle {\begin{array}{llcl}{\text{materiell:}}&{\frac {\mathrm {d} }{\mathrm {d} t}}\int _{V}{\frac {\rho _{0}}{2}}\,{\dot {\vec {\chi }}}\cdot {\dot {\vec {\chi }}}\,\mathrm {d} V&=&\int _{V}\rho _{0}\,{\dot {\vec {\chi }}}\cdot {\ddot {\vec {\chi }}}\,\mathrm {d} V\\{\text{räumlich:}}&{\frac {\mathrm {d} }{\mathrm {d} t}}\int _{v}{\frac {\rho }{2}}{\vec {v}}\cdot {\vec {v}}\,\mathrm {d} v&=&\int _{v}\left[{\frac {\mathrm {d} }{\mathrm {d} t}}\left({\frac {\rho }{2}}{\vec {v}}\cdot {\vec {v}}\right)+\operatorname {div} ({\vec {v}}){\frac {\rho }{2}}{\vec {v}}\cdot {\vec {v}}\right]\,\mathrm {d} v\\&&=&\int _{v}\left[{\frac {\dot {\rho }}{2}}{\vec {v}}\cdot {\vec {v}}+\rho {\vec {v}}\cdot {\dot {\vec {v}}}+\operatorname {div} ({\vec {v}}){\frac {\rho }{2}}{\vec {v}}\cdot {\vec {v}}\right]\,\mathrm {d} v\\&&=&\int _{v}{\Bigl [}(\underbrace {{\dot {\rho }}+\rho \operatorname {div} ({\vec {v}})} _{=0}){\frac {1}{2}}{\vec {v}}\cdot {\vec {v}}+\rho {\vec {v}}\cdot {\dot {\vec {v}}}{\Bigr ]}\,\mathrm {d} v=\int _{v}\rho {\vec {v}}\cdot {\dot {\vec {v}}}\,\mathrm {d} v\,,\end{array}}}"> </noscript><span class="lazy-image-placeholder" style="width: 85.818ex;height: 21.843ex;vertical-align: -10.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/60e5018ab1ab151cdd970b70bad6ce30b471d7db" data-alt="{\displaystyle {\begin{array}{llcl}{\text{materiell:}}&{\frac {\mathrm {d} }{\mathrm {d} t}}\int _{V}{\frac {\rho _{0}}{2}}\,{\dot {\vec {\chi }}}\cdot {\dot {\vec {\chi }}}\,\mathrm {d} V&=&\int _{V}\rho _{0}\,{\dot {\vec {\chi }}}\cdot {\ddot {\vec {\chi }}}\,\mathrm {d} V\\{\text{räumlich:}}&{\frac {\mathrm {d} }{\mathrm {d} t}}\int _{v}{\frac {\rho }{2}}{\vec {v}}\cdot {\vec {v}}\,\mathrm {d} v&=&\int _{v}\left[{\frac {\mathrm {d} }{\mathrm {d} t}}\left({\frac {\rho }{2}}{\vec {v}}\cdot {\vec {v}}\right)+\operatorname {div} ({\vec {v}}){\frac {\rho }{2}}{\vec {v}}\cdot {\vec {v}}\right]\,\mathrm {d} v\\&&=&\int _{v}\left[{\frac {\dot {\rho }}{2}}{\vec {v}}\cdot {\vec {v}}+\rho {\vec {v}}\cdot {\dot {\vec {v}}}+\operatorname {div} ({\vec {v}}){\frac {\rho }{2}}{\vec {v}}\cdot {\vec {v}}\right]\,\mathrm {d} v\\&&=&\int _{v}{\Bigl [}(\underbrace {{\dot {\rho }}+\rho \operatorname {div} ({\vec {v}})} _{=0}){\frac {1}{2}}{\vec {v}}\cdot {\vec {v}}+\rho {\vec {v}}\cdot {\dot {\vec {v}}}{\Bigr ]}\,\mathrm {d} v=\int _{v}\rho {\vec {v}}\cdot {\dot {\vec {v}}}\,\mathrm {d} v\,,\end{array}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </dd> </dl>was die ersten Terme auf den linken Seiten begründet. In der räumlichen Formulierung wurde der reynoldssche Transportsatz und die Massenbilanz angewendet.Der materielle Gradient der Geschwindigkeit ist die Zeitableitung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {GRAD} {\dot {\vec {\chi }}}={\dot {\mathbf {F} }}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> GRAD </mi> <mo> ⁡ </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> χ </mi> <mo stretchy="false"> → </mo> </mover> </mrow> <mo> ˙ </mo> </mover> </mrow> </mrow> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> F </mi> </mrow> <mo> ˙ </mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \operatorname {GRAD} {\dot {\vec {\chi }}}={\dot {\mathbf {F} }}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d14c9ebe0790586de6fa0b75da19826fe93a0771" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.677ex; height:3.176ex;" alt="{\displaystyle \operatorname {GRAD} {\dot {\vec {\chi }}}={\dot {\mathbf {F} }}}"> </noscript><span class="lazy-image-placeholder" style="width: 13.677ex;height: 3.176ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d14c9ebe0790586de6fa0b75da19826fe93a0771" data-alt="{\displaystyle \operatorname {GRAD} {\dot {\vec {\chi }}}={\dot {\mathbf {F} }}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  des Deformationsgradienten und der symmetrische Anteil des räumlichen <a href="https://de-m-wikipedia-org.translate.goog/wiki/Geschwindigkeitsgradient?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Geschwindigkeitsgradient">Geschwindigkeitsgradienten</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {l} :=\operatorname {grad} {\vec {v}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> l </mi> </mrow> <mo> := </mo> <mi> grad </mi> <mo> ⁡ </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> v </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbf {l} :=\operatorname {grad} {\vec {v}}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e16c300a14f80a6f18bcdbe933d0b3bf9b24978" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.579ex; height:2.676ex;" alt="{\displaystyle \mathbf {l} :=\operatorname {grad} {\vec {v}}}"> </noscript><span class="lazy-image-placeholder" style="width: 10.579ex;height: 2.676ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e16c300a14f80a6f18bcdbe933d0b3bf9b24978" data-alt="{\displaystyle \mathbf {l} :=\operatorname {grad} {\vec {v}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  ist der Verzerrungsgeschwindigkeitstensor d. Damit schreiben sich die Verformungsleistungen: <dl> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}{\text{materiell}}:&\int _{V}{\tilde {\mathbf {T} }}:\operatorname {sym} (\mathbf {F} ^{\top }\cdot {\dot {\mathbf {F} }})=:\int _{V}{\tilde {\mathbf {T} }}:{\dot {\mathbf {E} }}\,\mathrm {d} V\\{\text{räumlich}}:&\int _{v}{\boldsymbol {\sigma }}:\operatorname {sym} (\mathbf {l} )\,\mathrm {d} v=\int _{v}{\boldsymbol {\sigma }}:\mathbf {d} \,\mathrm {d} v\end{aligned}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext> materiell </mtext> </mrow> <mo> : </mo> </mtd> <mtd> <mi></mi> <msub> <mo> ∫ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> V </mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> T </mi> </mrow> <mo stretchy="false"> ~ </mo> </mover> </mrow> </mrow> <mo> : </mo> <mi> sym </mi> <mo> ⁡ </mo> <mo stretchy="false"> ( </mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> F </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> ⊤ </mi> </mrow> </msup> <mo> ⋅ </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> F </mi> </mrow> <mo> ˙ </mo> </mover> </mrow> </mrow> <mo stretchy="false"> ) </mo> <mo> =: </mo> <msub> <mo> ∫ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> V </mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> T </mi> </mrow> <mo stretchy="false"> ~ </mo> </mover> </mrow> </mrow> <mo> : </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> E </mi> </mrow> <mo> ˙ </mo> </mover> </mrow> </mrow> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> V </mi> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext> räumlich </mtext> </mrow> <mo> : </mo> </mtd> <mtd> <mi></mi> <msub> <mo> ∫ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> v </mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> σ </mi> </mrow> <mo> : </mo> <mi> sym </mi> <mo> ⁡ </mo> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> l </mi> </mrow> <mo stretchy="false"> ) </mo> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> v </mi> <mo> = </mo> <msub> <mo> ∫ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> v </mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> σ </mi> </mrow> <mo> : </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> d </mi> </mrow> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> v </mi> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\begin{aligned}{\text{materiell}}:&\int _{V}{\tilde {\mathbf {T} }}:\operatorname {sym} (\mathbf {F} ^{\top }\cdot {\dot {\mathbf {F} }})=:\int _{V}{\tilde {\mathbf {T} }}:{\dot {\mathbf {E} }}\,\mathrm {d} V\\{\text{räumlich}}:&\int _{v}{\boldsymbol {\sigma }}:\operatorname {sym} (\mathbf {l} )\,\mathrm {d} v=\int _{v}{\boldsymbol {\sigma }}:\mathbf {d} \,\mathrm {d} v\end{aligned}}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7446c3adbb51ed7c04f4e7c5e70c9a75df7c93e9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.046ex; margin-bottom: -0.292ex; width:46.963ex; height:11.843ex;" alt="{\displaystyle {\begin{aligned}{\text{materiell}}:&\int _{V}{\tilde {\mathbf {T} }}:\operatorname {sym} (\mathbf {F} ^{\top }\cdot {\dot {\mathbf {F} }})=:\int _{V}{\tilde {\mathbf {T} }}:{\dot {\mathbf {E} }}\,\mathrm {d} V\\{\text{räumlich}}:&\int _{v}{\boldsymbol {\sigma }}:\operatorname {sym} (\mathbf {l} )\,\mathrm {d} v=\int _{v}{\boldsymbol {\sigma }}:\mathbf {d} \,\mathrm {d} v\end{aligned}}}"> </noscript><span class="lazy-image-placeholder" style="width: 46.963ex;height: 11.843ex;vertical-align: -5.046ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7446c3adbb51ed7c04f4e7c5e70c9a75df7c93e9" data-alt="{\displaystyle {\begin{aligned}{\text{materiell}}:&\int _{V}{\tilde {\mathbf {T} }}:\operatorname {sym} (\mathbf {F} ^{\top }\cdot {\dot {\mathbf {F} }})=:\int _{V}{\tilde {\mathbf {T} }}:{\dot {\mathbf {E} }}\,\mathrm {d} V\\{\text{räumlich}}:&\int _{v}{\boldsymbol {\sigma }}:\operatorname {sym} (\mathbf {l} )\,\mathrm {d} v=\int _{v}{\boldsymbol {\sigma }}:\mathbf {d} \,\mathrm {d} v\end{aligned}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </dd> </dl>Die Leistungen der äußeren Kräfte ergeben sich durch Ersetzung des Vektors <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {q}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> q </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\vec {q}}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8a4e063f8ee7dae2488859c45a4e645db5148085" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.309ex; height:2.676ex;" alt="{\displaystyle {\vec {q}}}"> </noscript><span class="lazy-image-placeholder" style="width: 1.309ex;height: 2.676ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8a4e063f8ee7dae2488859c45a4e645db5148085" data-alt="{\displaystyle {\vec {q}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  durch den Geschwindigkeitsvektor.</td> </tr> </tbody> </table> <div class="mw-heading mw-heading3"> <h3 id="Energieerhaltungssatz">Energieerhaltungssatz</h3> <a role="button" href="https://de-m-wikipedia-org.translate.goog/w/index.php?title=Cauchy-eulersche_Bewegungsgesetze&action=edit&section=14&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Abschnitt bearbeiten: Energieerhaltungssatz" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> Bearbeiten </a> </div> In einem <a href="https://de-m-wikipedia-org.translate.goog/wiki/Konservatives_System?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Konservatives System">konservativen System</a> gibt es eine skalarwertige Funktion Wa, die <a href="https://de-m-wikipedia-org.translate.goog/wiki/Potentielle_Energie?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Potentielle Energie">potentielle Energie</a>, deren negative Zeitableitung gemäß <dl> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -{\frac {\mathrm {d} }{\mathrm {d} t}}W_{a}:=\int _{V}\rho _{0}\,{\vec {k}}_{0}\cdot {\dot {\vec {\chi }}}\,\mathrm {d} V+\int _{A}{\vec {t}}_{0}\cdot {\dot {\vec {\chi }}}\,\mathrm {d} A=\int _{v}\rho \,{\vec {k}}\cdot {\vec {v}}\,\mathrm {d} v+\int _{a}{\vec {t}}\cdot {\vec {v}}\,\mathrm {d} a}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo> − </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> t </mi> </mrow> </mfrac> </mrow> <msub> <mi> W </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> a </mi> </mrow> </msub> <mo> := </mo> <msub> <mo> ∫ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> V </mi> </mrow> </msub> <msub> <mi> ρ </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mspace width="thinmathspace"></mspace> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> k </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mo> ⋅ </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> χ </mi> <mo stretchy="false"> → </mo> </mover> </mrow> <mo> ˙ </mo> </mover> </mrow> </mrow> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> V </mi> <mo> + </mo> <msub> <mo> ∫ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> A </mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> t </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mo> ⋅ </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> χ </mi> <mo stretchy="false"> → </mo> </mover> </mrow> <mo> ˙ </mo> </mover> </mrow> </mrow> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> A </mi> <mo> = </mo> <msub> <mo> ∫ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> v </mi> </mrow> </msub> <mi> ρ </mi> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> k </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo> ⋅ </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> v </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> v </mi> <mo> + </mo> <msub> <mo> ∫ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> a </mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> t </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo> ⋅ </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> v </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> a </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle -{\frac {\mathrm {d} }{\mathrm {d} t}}W_{a}:=\int _{V}\rho _{0}\,{\vec {k}}_{0}\cdot {\dot {\vec {\chi }}}\,\mathrm {d} V+\int _{A}{\vec {t}}_{0}\cdot {\dot {\vec {\chi }}}\,\mathrm {d} A=\int _{v}\rho \,{\vec {k}}\cdot {\vec {v}}\,\mathrm {d} v+\int _{a}{\vec {t}}\cdot {\vec {v}}\,\mathrm {d} a} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d98c3c65f9d8c57ba992a416df586cd8cc31d7d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:68.329ex; height:5.843ex;" alt="{\displaystyle -{\frac {\mathrm {d} }{\mathrm {d} t}}W_{a}:=\int _{V}\rho _{0}\,{\vec {k}}_{0}\cdot {\dot {\vec {\chi }}}\,\mathrm {d} V+\int _{A}{\vec {t}}_{0}\cdot {\dot {\vec {\chi }}}\,\mathrm {d} A=\int _{v}\rho \,{\vec {k}}\cdot {\vec {v}}\,\mathrm {d} v+\int _{a}{\vec {t}}\cdot {\vec {v}}\,\mathrm {d} a}"> </noscript><span class="lazy-image-placeholder" style="width: 68.329ex;height: 5.843ex;vertical-align: -2.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d98c3c65f9d8c57ba992a416df586cd8cc31d7d" data-alt="{\displaystyle -{\frac {\mathrm {d} }{\mathrm {d} t}}W_{a}:=\int _{V}\rho _{0}\,{\vec {k}}_{0}\cdot {\dot {\vec {\chi }}}\,\mathrm {d} V+\int _{A}{\vec {t}}_{0}\cdot {\dot {\vec {\chi }}}\,\mathrm {d} A=\int _{v}\rho \,{\vec {k}}\cdot {\vec {v}}\,\mathrm {d} v+\int _{a}{\vec {t}}\cdot {\vec {v}}\,\mathrm {d} a}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </dd> </dl> die Leistung der äußeren Kräfte ist, und eine Formänderungsenergie Wi, deren Zeitableitung <dl> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}W_{i}:=\int _{V}{\tilde {\mathbf {T} }}:{\dot {\mathbf {E} }}\,\mathrm {d} V=\int _{v}{\boldsymbol {\sigma }}:\mathbf {d} \,\mathrm {d} v}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> t </mi> </mrow> </mfrac> </mrow> <msub> <mi> W </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo> := </mo> <msub> <mo> ∫ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> V </mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> T </mi> </mrow> <mo stretchy="false"> ~ </mo> </mover> </mrow> </mrow> <mo> : </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> E </mi> </mrow> <mo> ˙ </mo> </mover> </mrow> </mrow> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> V </mi> <mo> = </mo> <msub> <mo> ∫ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> v </mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> σ </mi> </mrow> <mo> : </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> d </mi> </mrow> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> v </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}W_{i}:=\int _{V}{\tilde {\mathbf {T} }}:{\dot {\mathbf {E} }}\,\mathrm {d} V=\int _{v}{\boldsymbol {\sigma }}:\mathbf {d} \,\mathrm {d} v} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/279c7969cef51646466d55ac82acfb59a3c5f353" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:35.535ex; height:5.843ex;" alt="{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}W_{i}:=\int _{V}{\tilde {\mathbf {T} }}:{\dot {\mathbf {E} }}\,\mathrm {d} V=\int _{v}{\boldsymbol {\sigma }}:\mathbf {d} \,\mathrm {d} v}"> </noscript><span class="lazy-image-placeholder" style="width: 35.535ex;height: 5.843ex;vertical-align: -2.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/279c7969cef51646466d55ac82acfb59a3c5f353" data-alt="{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}W_{i}:=\int _{V}{\tilde {\mathbf {T} }}:{\dot {\mathbf {E} }}\,\mathrm {d} V=\int _{v}{\boldsymbol {\sigma }}:\mathbf {d} \,\mathrm {d} v}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </dd> </dl> die Verformungsleistung ist. Mit der Abkürzung <dl> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K:=\int _{V}{\frac {\rho _{0}}{2}}\,{\dot {\vec {\chi }}}\cdot {\dot {\vec {\chi }}}\,\mathrm {d} V=\int _{v}{\frac {\rho }{2}}{\vec {v}}\cdot {\vec {v}}\,\mathrm {d} v}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> K </mi> <mo> := </mo> <msub> <mo> ∫ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> V </mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi> ρ </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mn> 2 </mn> </mfrac> </mrow> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> χ </mi> <mo stretchy="false"> → </mo> </mover> </mrow> <mo> ˙ </mo> </mover> </mrow> </mrow> <mo> ⋅ </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> χ </mi> <mo stretchy="false"> → </mo> </mover> </mrow> <mo> ˙ </mo> </mover> </mrow> </mrow> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> V </mi> <mo> = </mo> <msub> <mo> ∫ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> v </mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi> ρ </mi> <mn> 2 </mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> v </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo> ⋅ </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> v </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> v </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle K:=\int _{V}{\frac {\rho _{0}}{2}}\,{\dot {\vec {\chi }}}\cdot {\dot {\vec {\chi }}}\,\mathrm {d} V=\int _{v}{\frac {\rho }{2}}{\vec {v}}\cdot {\vec {v}}\,\mathrm {d} v} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1bc6134ee9e58b980453513a2ce770ac8d8fbbfe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:35.205ex; height:5.676ex;" alt="{\displaystyle K:=\int _{V}{\frac {\rho _{0}}{2}}\,{\dot {\vec {\chi }}}\cdot {\dot {\vec {\chi }}}\,\mathrm {d} V=\int _{v}{\frac {\rho }{2}}{\vec {v}}\cdot {\vec {v}}\,\mathrm {d} v}"> </noscript><span class="lazy-image-placeholder" style="width: 35.205ex;height: 5.676ex;vertical-align: -2.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1bc6134ee9e58b980453513a2ce770ac8d8fbbfe" data-alt="{\displaystyle K:=\int _{V}{\frac {\rho _{0}}{2}}\,{\dot {\vec {\chi }}}\cdot {\dot {\vec {\chi }}}\,\mathrm {d} V=\int _{v}{\frac {\rho }{2}}{\vec {v}}\cdot {\vec {v}}\,\mathrm {d} v}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </dd> </dl> für die kinetische Energie schreibt sich die Bilanz der mechanischen Energie: <dl> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}K+{\frac {\mathrm {d} }{\mathrm {d} t}}W_{i}+{\frac {\mathrm {d} }{\mathrm {d} t}}W_{a}={\frac {\mathrm {d} }{\mathrm {d} t}}(K+W_{i}+W_{a})=0\quad \Leftrightarrow K+W_{i}+W_{a}=E}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> t </mi> </mrow> </mfrac> </mrow> <mi> K </mi> <mo> + </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> t </mi> </mrow> </mfrac> </mrow> <msub> <mi> W </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo> + </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> t </mi> </mrow> </mfrac> </mrow> <msub> <mi> W </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> a </mi> </mrow> </msub> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> t </mi> </mrow> </mfrac> </mrow> <mo stretchy="false"> ( </mo> <mi> K </mi> <mo> + </mo> <msub> <mi> W </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo> + </mo> <msub> <mi> W </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> a </mi> </mrow> </msub> <mo stretchy="false"> ) </mo> <mo> = </mo> <mn> 0 </mn> <mspace width="1em"></mspace> <mo stretchy="false"> ⇔ </mo> <mi> K </mi> <mo> + </mo> <msub> <mi> W </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo> + </mo> <msub> <mi> W </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> a </mi> </mrow> </msub> <mo> = </mo> <mi> E </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}K+{\frac {\mathrm {d} }{\mathrm {d} t}}W_{i}+{\frac {\mathrm {d} }{\mathrm {d} t}}W_{a}={\frac {\mathrm {d} }{\mathrm {d} t}}(K+W_{i}+W_{a})=0\quad \Leftrightarrow K+W_{i}+W_{a}=E} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86f039b024dc107c664f832543f2e70c3c67032a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:73.959ex; height:5.509ex;" alt="{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}K+{\frac {\mathrm {d} }{\mathrm {d} t}}W_{i}+{\frac {\mathrm {d} }{\mathrm {d} t}}W_{a}={\frac {\mathrm {d} }{\mathrm {d} t}}(K+W_{i}+W_{a})=0\quad \Leftrightarrow K+W_{i}+W_{a}=E}"> </noscript><span class="lazy-image-placeholder" style="width: 73.959ex;height: 5.509ex;vertical-align: -2.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86f039b024dc107c664f832543f2e70c3c67032a" data-alt="{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}K+{\frac {\mathrm {d} }{\mathrm {d} t}}W_{i}+{\frac {\mathrm {d} }{\mathrm {d} t}}W_{a}={\frac {\mathrm {d} }{\mathrm {d} t}}(K+W_{i}+W_{a})=0\quad \Leftrightarrow K+W_{i}+W_{a}=E}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </dd> </dl> Die mechanische Gesamtenergie E, bestehend aus der kinetischen Energie, der Formänderungsenergie und der potentiellen Energie, ist mithin in einem konservativen System zeitlich konstant, was als <a href="https://de-m-wikipedia-org.translate.goog/wiki/Energieerhaltungssatz?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Energieerhaltungssatz">Energieerhaltungssatz</a> bekannt ist. <div class="mw-heading mw-heading3"> <h3 id="Satz_von_Clapeyron">Satz von Clapeyron</h3> <a role="button" href="https://de-m-wikipedia-org.translate.goog/w/index.php?title=Cauchy-eulersche_Bewegungsgesetze&action=edit&section=15&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Abschnitt bearbeiten: Satz von Clapeyron" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> Bearbeiten </a> </div> Wird bei kleinen Verformungen, linearer Elastizität und im statischen Fall für das Vektorfeld <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {q}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> q </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\vec {q}}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8a4e063f8ee7dae2488859c45a4e645db5148085" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.309ex; height:2.676ex;" alt="{\displaystyle {\vec {q}}}"> </noscript><span class="lazy-image-placeholder" style="width: 1.309ex;height: 2.676ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8a4e063f8ee7dae2488859c45a4e645db5148085" data-alt="{\displaystyle {\vec {q}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  das Verschiebungsfeld <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {u}}({\vec {X}},t):={\vec {\chi }}({\vec {X}},t)-{\vec {X}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> u </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> X </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo> , </mo> <mi> t </mi> <mo stretchy="false"> ) </mo> <mo> := </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> χ </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> X </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo> , </mo> <mi> t </mi> <mo stretchy="false"> ) </mo> <mo> − </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> X </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\vec {u}}({\vec {X}},t):={\vec {\chi }}({\vec {X}},t)-{\vec {X}}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/645584ac36778809518ef7e10a2f7d2532b6b9b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.676ex; height:3.343ex;" alt="{\displaystyle {\vec {u}}({\vec {X}},t):={\vec {\chi }}({\vec {X}},t)-{\vec {X}}}"> </noscript><span class="lazy-image-placeholder" style="width: 22.676ex;height: 3.343ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/645584ac36778809518ef7e10a2f7d2532b6b9b7" data-alt="{\displaystyle {\vec {u}}({\vec {X}},t):={\vec {\chi }}({\vec {X}},t)-{\vec {X}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  eingesetzt, dann ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {grad} {\vec {q}}=\operatorname {grad} {\vec {u}}=\mathbf {H} }"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> grad </mi> <mo> ⁡ </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> q </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo> = </mo> <mi> grad </mi> <mo> ⁡ </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> u </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> H </mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \operatorname {grad} {\vec {q}}=\operatorname {grad} {\vec {u}}=\mathbf {H} } </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35cadbe20879daff76a716c8a4e31fe2e912bd8c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:20.759ex; height:2.676ex;" alt="{\displaystyle \operatorname {grad} {\vec {q}}=\operatorname {grad} {\vec {u}}=\mathbf {H} }"> </noscript><span class="lazy-image-placeholder" style="width: 20.759ex;height: 2.676ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35cadbe20879daff76a716c8a4e31fe2e912bd8c" data-alt="{\displaystyle \operatorname {grad} {\vec {q}}=\operatorname {grad} {\vec {u}}=\mathbf {H} }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  der <a href="https://de-m-wikipedia-org.translate.goog/wiki/Verschiebungsgradient?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Verschiebungsgradient">Verschiebungsgradient</a> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|\mathbf {H} \|\ll 1}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false"> ‖ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> H </mi> </mrow> <mo fence="false" stretchy="false"> ‖ </mo> <mo> ≪ </mo> <mn> 1 </mn> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \|\mathbf {H} \|\ll 1} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2845a04fdd6e6b1f27dabc913818fa0a0927d635" 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 \|\mathbf {H} \|\ll 1}"> </noscript><span class="lazy-image-placeholder" style="width: 9.193ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2845a04fdd6e6b1f27dabc913818fa0a0927d635" data-alt="{\displaystyle \|\mathbf {H} \|\ll 1}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> , sodass alle Terme, die <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|\mathbf {H} \|}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false"> ‖ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> H </mi> </mrow> <mo fence="false" stretchy="false"> ‖ </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \|\mathbf {H} \|} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c40b2595632e195d2d1435cb721be9c7a3e3b615" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.416ex; height:2.843ex;" alt="{\displaystyle \|\mathbf {H} \|}"> </noscript><span class="lazy-image-placeholder" style="width: 4.416ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c40b2595632e195d2d1435cb721be9c7a3e3b615" data-alt="{\displaystyle \|\mathbf {H} \|}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  in höherer Ordnung als eins enthalten, vernachlässigt werden können. Es folgt: <dl> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\operatorname {GRAD} {\vec {u}}=\mathbf {H} =\operatorname {GRAD} {\vec {\chi }}-\mathbf {1} =\mathbf {F-1} \\\rightarrow \quad \operatorname {sym} (\mathbf {F} ^{\top }\cdot \operatorname {GRAD} {\vec {u}})=\operatorname {sym} (\mathbf {(1+H^{\top })\cdot H} )\approx \operatorname {sym} (\mathbf {H} )={\frac {1}{2}}(\mathbf {H+H^{\top }} )={\boldsymbol {\varepsilon }}\end{aligned}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi> GRAD </mi> <mo> ⁡ </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> u </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> H </mi> </mrow> <mo> = </mo> <mi> GRAD </mi> <mo> ⁡ </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> χ </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo> − </mo> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold"> 1 </mn> </mrow> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> F </mi> <mo mathvariant="bold"> − </mo> <mn mathvariant="bold"> 1 </mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mo stretchy="false"> → </mo> <mspace width="1em"></mspace> <mi> sym </mi> <mo> ⁡ </mo> <mo stretchy="false"> ( </mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> F </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> ⊤ </mi> </mrow> </msup> <mo> ⋅ </mo> <mi> GRAD </mi> <mo> ⁡ </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> u </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo stretchy="false"> ) </mo> <mo> = </mo> <mi> sym </mi> <mo> ⁡ </mo> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mo mathvariant="bold" stretchy="false"> ( </mo> <mn mathvariant="bold"> 1 </mn> <mo mathvariant="bold"> + </mo> <msup> <mi mathvariant="bold"> H </mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> ⊤ </mi> </mrow> </msup> <mo mathvariant="bold" stretchy="false"> ) </mo> <mo> ⋅ </mo> <mi mathvariant="bold"> H </mi> </mrow> <mo stretchy="false"> ) </mo> <mo> ≈ </mo> <mi> sym </mi> <mo> ⁡ </mo> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> H </mi> </mrow> <mo stretchy="false"> ) </mo> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> H </mi> <mo mathvariant="bold"> + </mo> <msup> <mi mathvariant="bold"> H </mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> ⊤ </mi> </mrow> </msup> </mrow> <mo stretchy="false"> ) </mo> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> ε </mi> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\begin{aligned}\operatorname {GRAD} {\vec {u}}=\mathbf {H} =\operatorname {GRAD} {\vec {\chi }}-\mathbf {1} =\mathbf {F-1} \\\rightarrow \quad \operatorname {sym} (\mathbf {F} ^{\top }\cdot \operatorname {GRAD} {\vec {u}})=\operatorname {sym} (\mathbf {(1+H^{\top })\cdot H} )\approx \operatorname {sym} (\mathbf {H} )={\frac {1}{2}}(\mathbf {H+H^{\top }} )={\boldsymbol {\varepsilon }}\end{aligned}}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/befd91451f60f4b0d9fcc181426b39fbd556f679" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.293ex; margin-bottom: -0.211ex; width:79.578ex; height:8.176ex;" alt="{\displaystyle {\begin{aligned}\operatorname {GRAD} {\vec {u}}=\mathbf {H} =\operatorname {GRAD} {\vec {\chi }}-\mathbf {1} =\mathbf {F-1} \\\rightarrow \quad \operatorname {sym} (\mathbf {F} ^{\top }\cdot \operatorname {GRAD} {\vec {u}})=\operatorname {sym} (\mathbf {(1+H^{\top })\cdot H} )\approx \operatorname {sym} (\mathbf {H} )={\frac {1}{2}}(\mathbf {H+H^{\top }} )={\boldsymbol {\varepsilon }}\end{aligned}}}"> </noscript><span class="lazy-image-placeholder" style="width: 79.578ex;height: 8.176ex;vertical-align: -3.293ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/befd91451f60f4b0d9fcc181426b39fbd556f679" data-alt="{\displaystyle {\begin{aligned}\operatorname {GRAD} {\vec {u}}=\mathbf {H} =\operatorname {GRAD} {\vec {\chi }}-\mathbf {1} =\mathbf {F-1} \\\rightarrow \quad \operatorname {sym} (\mathbf {F} ^{\top }\cdot \operatorname {GRAD} {\vec {u}})=\operatorname {sym} (\mathbf {(1+H^{\top })\cdot H} )\approx \operatorname {sym} (\mathbf {H} )={\frac {1}{2}}(\mathbf {H+H^{\top }} )={\boldsymbol {\varepsilon }}\end{aligned}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </dd> </dl> Der symmetrische Anteil des Verschiebungsgradienten ist der <a href="https://de-m-wikipedia-org.translate.goog/wiki/Verzerrungstensor?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Linearisierter_Verzerrungstensor" title="Verzerrungstensor">linearisierte Verzerrungstensor</a>. Der zweite piola-kirchhoffsche Spannungstensor geht bei kleinen Verformungen in den cauchyschen Spannungstensor über und es resultiert der Arbeitssatz<a href="https://de-m-wikipedia-org.translate.goog/wiki/Cauchy-eulersche_Bewegungsgesetze?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-9">[L 7]</a> <dl> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{v}{\boldsymbol {\sigma }}:{\boldsymbol {\varepsilon }}\,\mathrm {d} v=\int _{v}\rho \,{\vec {k}}\cdot {\vec {u}}\,\mathrm {d} v+\int _{a}{\vec {t}}\cdot {\vec {u}}\,\mathrm {d} a}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mo> ∫ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> v </mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> σ </mi> </mrow> <mo> : </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> ε </mi> </mrow> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> v </mi> <mo> = </mo> <msub> <mo> ∫ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> v </mi> </mrow> </msub> <mi> ρ </mi> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> k </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo> ⋅ </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> u </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> v </mi> <mo> + </mo> <msub> <mo> ∫ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> a </mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> t </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo> ⋅ </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> u </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> a </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \int _{v}{\boldsymbol {\sigma }}:{\boldsymbol {\varepsilon }}\,\mathrm {d} v=\int _{v}\rho \,{\vec {k}}\cdot {\vec {u}}\,\mathrm {d} v+\int _{a}{\vec {t}}\cdot {\vec {u}}\,\mathrm {d} a} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6a64137860bd5f7e14f1edbcc61a23b85bf410df" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:37.529ex; height:5.676ex;" alt="{\displaystyle \int _{v}{\boldsymbol {\sigma }}:{\boldsymbol {\varepsilon }}\,\mathrm {d} v=\int _{v}\rho \,{\vec {k}}\cdot {\vec {u}}\,\mathrm {d} v+\int _{a}{\vec {t}}\cdot {\vec {u}}\,\mathrm {d} a}"> </noscript><span class="lazy-image-placeholder" style="width: 37.529ex;height: 5.676ex;vertical-align: -2.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6a64137860bd5f7e14f1edbcc61a23b85bf410df" data-alt="{\displaystyle \int _{v}{\boldsymbol {\sigma }}:{\boldsymbol {\varepsilon }}\,\mathrm {d} v=\int _{v}\rho \,{\vec {k}}\cdot {\vec {u}}\,\mathrm {d} v+\int _{a}{\vec {t}}\cdot {\vec {u}}\,\mathrm {d} a}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </dd> </dl> Der Integrand auf der linken Seite ist das Doppelte der Formänderungsenergie <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {\sigma }}:{\boldsymbol {\varepsilon }}=2w}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> σ </mi> </mrow> <mo> : </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> ε </mi> </mrow> <mo> = </mo> <mn> 2 </mn> <mi> w </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\boldsymbol {\sigma }}:{\boldsymbol {\varepsilon }}=2w} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/17f32cc1014228fc4b4c06f7a72a2969b596f8ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.686ex; height:2.176ex;" alt="{\displaystyle {\boldsymbol {\sigma }}:{\boldsymbol {\varepsilon }}=2w}"> </noscript><span class="lazy-image-placeholder" style="width: 10.686ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/17f32cc1014228fc4b4c06f7a72a2969b596f8ab" data-alt="{\displaystyle {\boldsymbol {\sigma }}:{\boldsymbol {\varepsilon }}=2w}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  und es entsteht der Satz von Clapeyron<a href="https://de-m-wikipedia-org.translate.goog/wiki/Cauchy-eulersche_Bewegungsgesetze?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-10">[L 8]</a> <dl> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\int _{v}w\,\mathrm {d} v=\int _{v}\rho \,{\vec {k}}\cdot {\vec {u}}\,\mathrm {d} v+\int _{a}{\vec {t}}\cdot {\vec {u}}\,\mathrm {d} a}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn> 2 </mn> <msub> <mo> ∫ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> v </mi> </mrow> </msub> <mi> w </mi> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> v </mi> <mo> = </mo> <msub> <mo> ∫ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> v </mi> </mrow> </msub> <mi> ρ </mi> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> k </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo> ⋅ </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> u </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> v </mi> <mo> + </mo> <msub> <mo> ∫ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> a </mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> t </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo> ⋅ </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> u </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> a </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle 2\int _{v}w\,\mathrm {d} v=\int _{v}\rho \,{\vec {k}}\cdot {\vec {u}}\,\mathrm {d} v+\int _{a}{\vec {t}}\cdot {\vec {u}}\,\mathrm {d} a} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/82bf92f736d6d2e51d5cac6fe20608ad49151222" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:35.982ex; height:5.676ex;" alt="{\displaystyle 2\int _{v}w\,\mathrm {d} v=\int _{v}\rho \,{\vec {k}}\cdot {\vec {u}}\,\mathrm {d} v+\int _{a}{\vec {t}}\cdot {\vec {u}}\,\mathrm {d} a}"> </noscript><span class="lazy-image-placeholder" style="width: 35.982ex;height: 5.676ex;vertical-align: -2.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/82bf92f736d6d2e51d5cac6fe20608ad49151222" data-alt="{\displaystyle 2\int _{v}w\,\mathrm {d} v=\int _{v}\rho \,{\vec {k}}\cdot {\vec {u}}\,\mathrm {d} v+\int _{a}{\vec {t}}\cdot {\vec {u}}\,\mathrm {d} a}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </dd> </dl> </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(4)"> <h2 id="Fußnoten">Fußnoten</h2> <a role="button" href="https://de-m-wikipedia-org.translate.goog/w/index.php?title=Cauchy-eulersche_Bewegungsgesetze&action=edit&section=16&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Abschnitt bearbeiten: Fußnoten" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> Bearbeiten </a> </div> <section class="mf-section-4 collapsible-block" id="mf-section-4"> <ol class="references"> <li id="cite_note-Frechet-2"><a href="https://de-m-wikipedia-org.translate.goog/wiki/Cauchy-eulersche_Bewegungsgesetze?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-Frechet_2-0">↑</a> Die <a href="https://de-m-wikipedia-org.translate.goog/wiki/Fr%C3%A9chet-Ableitung?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Fréchet-Ableitung">Fréchet-Ableitung</a> einer Funktion <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> f </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle f} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" 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}"> </noscript><span class="lazy-image-placeholder" style="width: 1.279ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" data-alt="{\displaystyle f}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  nach <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> x </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle x} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" 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}"> </noscript><span class="lazy-image-placeholder" style="width: 1.33ex;height: 1.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" data-alt="{\displaystyle x}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  ist der beschränkte lineare Operator <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {A}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script"> A </mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\mathcal {A}}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/280ae03440942ab348c2ca9b8db6b56ffa9618f8" 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}}}"> </noscript><span class="lazy-image-placeholder" style="width: 1.903ex;height: 2.343ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/280ae03440942ab348c2ca9b8db6b56ffa9618f8" data-alt="{\displaystyle {\mathcal {A}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  der – sofern er existiert – in alle Richtungen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> h </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle h} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b26be3e694314bc90c3215047e4a2010c6ee184a" 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}"> </noscript><span class="lazy-image-placeholder" style="width: 1.339ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b26be3e694314bc90c3215047e4a2010c6ee184a" data-alt="{\displaystyle h}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  dem <a href="https://de-m-wikipedia-org.translate.goog/wiki/G%C3%A2teaux-Differential?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Gâteaux-Differential">Gâteaux-Differential</a> entspricht, also <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\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}}\;h}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script"> A </mi> </mrow> </mrow> <mo stretchy="false"> ( </mo> <mi> h </mi> <mo stretchy="false"> ) </mo> <mo> = </mo> <msub> <mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> s </mi> </mrow> </mfrac> </mrow> <mi> f </mi> <mo stretchy="false"> ( </mo> <mi> x </mi> <mo> + </mo> <mi> s </mi> <mi> h </mi> <mo stretchy="false"> ) </mo> </mrow> <mo> | </mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> s </mi> <mo> = </mo> <mn> 0 </mn> </mrow> </msub> <mo> = </mo> <munder> <mo movablelimits="true" form="prefix"> lim </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> s </mi> <mo stretchy="false"> → </mo> <mn> 0 </mn> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi> f </mi> <mo stretchy="false"> ( </mo> <mi> x </mi> <mo> + </mo> <mi> s </mi> <mi> h </mi> <mo stretchy="false"> ) </mo> <mo> − </mo> <mi> f </mi> <mo stretchy="false"> ( </mo> <mi> x </mi> <mo stretchy="false"> ) </mo> </mrow> <mi> s </mi> </mfrac> </mrow> <mspace width="1em"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mtext> für alle </mtext> </mrow> <mspace width="thickmathspace"></mspace> <mi> h </mi> </mstyle> </mrow> <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}}\;h} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c50d743aff9cbf01be79dc8aba5b1affab2a72f0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:60.727ex; 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}}\;h}"> </noscript><span class="lazy-image-placeholder" style="width: 60.727ex;height: 6.176ex;vertical-align: -2.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c50d743aff9cbf01be79dc8aba5b1affab2a72f0" data-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}}\;h}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  gilt. Darin ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s\in \mathbb {R} \,,f,x\,{\textsf {und}}\,h}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> s </mi> <mo> ∈ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck"> R </mi> </mrow> <mspace width="thinmathspace"></mspace> <mo> , </mo> <mi> f </mi> <mo> , </mo> <mi> x </mi> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext mathvariant="sans-serif"> und </mtext> </mrow> </mrow> <mspace width="thinmathspace"></mspace> <mi> h </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle s\in \mathbb {R} \,,f,x\,{\textsf {und}}\,h} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a9dbea5741432858342457ca6073c0098dfaf2ca" 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}"> </noscript><span class="lazy-image-placeholder" style="width: 16.391ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a9dbea5741432858342457ca6073c0098dfaf2ca" data-alt="{\displaystyle s\in \mathbb {R} \,,f,x\,{\textsf {und}}\,h}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  skalar-, vektor- oder tensorwertig aber <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> x </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle x} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" 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}"> </noscript><span class="lazy-image-placeholder" style="width: 1.33ex;height: 1.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" data-alt="{\displaystyle x}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> h </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle h} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b26be3e694314bc90c3215047e4a2010c6ee184a" 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}"> </noscript><span class="lazy-image-placeholder" style="width: 1.339ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b26be3e694314bc90c3215047e4a2010c6ee184a" data-alt="{\displaystyle h}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  gleichartig. Dann wird auch <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {A}}={\frac {\partial f}{\partial x}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script"> A </mi> </mrow> </mrow> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal"> ∂ </mi> <mi> f </mi> </mrow> <mrow> <mi mathvariant="normal"> ∂ </mi> <mi> x </mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\mathcal {A}}={\frac {\partial f}{\partial x}}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a5d851fc35445648bf933225a1d553ef67b9458f" 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}}}"> </noscript><span class="lazy-image-placeholder" style="width: 8.486ex;height: 5.676ex;vertical-align: -2.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a5d851fc35445648bf933225a1d553ef67b9458f" data-alt="{\displaystyle {\mathcal {A}}={\frac {\partial f}{\partial x}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  geschrieben.</li> <li id="cite_note-prodregel-6">↑ <a href="https://de-m-wikipedia-org.translate.goog/wiki/Cauchy-eulersche_Bewegungsgesetze?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-prodregel_6-0">a</a> <a href="https://de-m-wikipedia-org.translate.goog/wiki/Cauchy-eulersche_Bewegungsgesetze?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-prodregel_6-1">b</a> Beweis der Produktregel in <a href="https://de-m-wikipedia-org.translate.goog/wiki/Kartesische_Koordinaten?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Kartesische Koordinaten">kartesischen Koordinaten</a> mit <a href="https://de-m-wikipedia-org.translate.goog/wiki/Einsteinsche_Summenkonvention?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Einsteinsche Summenkonvention">einsteinscher Summenkonvention</a>: <dl> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\nabla \cdot (\mathbf {T} \times {\vec {f}})=&{\hat {e}}_{l}{\frac {\partial }{\partial x_{l}}}\cdot (\mathbf {T} \times {\vec {f}})=\left({\hat {e}}_{l}\cdot {\frac {\partial \mathbf {T} }{\partial x_{l}}}\right)\times {\vec {f}}+({\hat {e}}_{l}\cdot \mathbf {T} )\times {\frac {\partial {\vec {f}}}{\partial x_{l}}}\\=&(\nabla \cdot \mathbf {T} )\times {\vec {f}}-{\frac {\partial {\vec {f}}}{\partial x_{l}}}\times ({\hat {e}}_{l}\cdot \mathbf {T} )\\=&(\nabla \cdot \mathbf {T} )\times {\vec {f}}-{\vec {\mathrm {i} }}\left({\frac {\partial {\vec {f}}}{\partial x_{l}}}\otimes {\hat {e}}_{l}\cdot \mathbf {T} \right)\\=&(\nabla \cdot \mathbf {T} )\times {\vec {f}}-{\vec {\mathrm {i} }}\left((\nabla \otimes {\vec {f}})^{\top }\cdot \mathbf {T} \right)\end{aligned}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi mathvariant="normal"> ∇ </mi> <mo> ⋅ </mo> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> T </mi> </mrow> <mo> × </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> f </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo stretchy="false"> ) </mo> <mo> = </mo> </mtd> <mtd> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> e </mi> <mo stretchy="false"> ^ </mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> l </mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal"> ∂ </mi> <mrow> <mi mathvariant="normal"> ∂ </mi> <msub> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> l </mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo> ⋅ </mo> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> T </mi> </mrow> <mo> × </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> f </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo stretchy="false"> ) </mo> <mo> = </mo> <mrow> <mo> ( </mo> <mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> e </mi> <mo stretchy="false"> ^ </mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> l </mi> </mrow> </msub> <mo> ⋅ </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal"> ∂ </mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> T </mi> </mrow> </mrow> <mrow> <mi mathvariant="normal"> ∂ </mi> <msub> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> l </mi> </mrow> </msub> </mrow> </mfrac> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> × </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> f </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo> + </mo> <mo stretchy="false"> ( </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> e </mi> <mo stretchy="false"> ^ </mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> l </mi> </mrow> </msub> <mo> ⋅ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> T </mi> </mrow> <mo stretchy="false"> ) </mo> <mo> × </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal"> ∂ </mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> f </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> </mrow> <mrow> <mi mathvariant="normal"> ∂ </mi> <msub> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> l </mi> </mrow> </msub> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mo> = </mo> </mtd> <mtd> <mi></mi> <mo stretchy="false"> ( </mo> <mi mathvariant="normal"> ∇ </mi> <mo> ⋅ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> T </mi> </mrow> <mo stretchy="false"> ) </mo> <mo> × </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> f </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo> − </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal"> ∂ </mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> f </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> </mrow> <mrow> <mi mathvariant="normal"> ∂ </mi> <msub> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> l </mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo> × </mo> <mo stretchy="false"> ( </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> e </mi> <mo stretchy="false"> ^ </mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> l </mi> </mrow> </msub> <mo> ⋅ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> T </mi> </mrow> <mo stretchy="false"> ) </mo> </mtd> </mtr> <mtr> <mtd> <mo> = </mo> </mtd> <mtd> <mi></mi> <mo stretchy="false"> ( </mo> <mi mathvariant="normal"> ∇ </mi> <mo> ⋅ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> T </mi> </mrow> <mo stretchy="false"> ) </mo> <mo> × </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> f </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo> − </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> i </mi> </mrow> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mrow> <mo> ( </mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal"> ∂ </mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> f </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> </mrow> <mrow> <mi mathvariant="normal"> ∂ </mi> <msub> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> l </mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo> ⊗ </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> e </mi> <mo stretchy="false"> ^ </mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> l </mi> </mrow> </msub> <mo> ⋅ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> T </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mo> = </mo> </mtd> <mtd> <mi></mi> <mo stretchy="false"> ( </mo> <mi mathvariant="normal"> ∇ </mi> <mo> ⋅ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> T </mi> </mrow> <mo stretchy="false"> ) </mo> <mo> × </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> f </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mo> − </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> i </mi> </mrow> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <mrow> <mo> ( </mo> <mrow> <mo stretchy="false"> ( </mo> <mi mathvariant="normal"> ∇ </mi> <mo> ⊗ </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> f </mi> <mo stretchy="false"> → </mo> </mover> </mrow> </mrow> <msup> <mo stretchy="false"> ) </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> ⊤ </mi> </mrow> </msup> <mo> ⋅ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> T </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\begin{aligned}\nabla \cdot (\mathbf {T} \times {\vec {f}})=&{\hat {e}}_{l}{\frac {\partial }{\partial x_{l}}}\cdot (\mathbf {T} \times {\vec {f}})=\left({\hat {e}}_{l}\cdot {\frac {\partial \mathbf {T} }{\partial x_{l}}}\right)\times {\vec {f}}+({\hat {e}}_{l}\cdot \mathbf {T} )\times {\frac {\partial {\vec {f}}}{\partial x_{l}}}\\=&(\nabla \cdot \mathbf {T} )\times {\vec {f}}-{\frac {\partial {\vec {f}}}{\partial x_{l}}}\times ({\hat {e}}_{l}\cdot \mathbf {T} )\\=&(\nabla \cdot \mathbf {T} )\times {\vec {f}}-{\vec {\mathrm {i} }}\left({\frac {\partial {\vec {f}}}{\partial x_{l}}}\otimes {\hat {e}}_{l}\cdot \mathbf {T} \right)\\=&(\nabla \cdot \mathbf {T} )\times {\vec {f}}-{\vec {\mathrm {i} }}\left((\nabla \otimes {\vec {f}})^{\top }\cdot \mathbf {T} \right)\end{aligned}}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7f7bcfd064ce9a68899a0713030f305f676c2f37" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -12.224ex; margin-bottom: -0.281ex; width:67.24ex; height:26.176ex;" alt="{\displaystyle {\begin{aligned}\nabla \cdot (\mathbf {T} \times {\vec {f}})=&{\hat {e}}_{l}{\frac {\partial }{\partial x_{l}}}\cdot (\mathbf {T} \times {\vec {f}})=\left({\hat {e}}_{l}\cdot {\frac {\partial \mathbf {T} }{\partial x_{l}}}\right)\times {\vec {f}}+({\hat {e}}_{l}\cdot \mathbf {T} )\times {\frac {\partial {\vec {f}}}{\partial x_{l}}}\\=&(\nabla \cdot \mathbf {T} )\times {\vec {f}}-{\frac {\partial {\vec {f}}}{\partial x_{l}}}\times ({\hat {e}}_{l}\cdot \mathbf {T} )\\=&(\nabla \cdot \mathbf {T} )\times {\vec {f}}-{\vec {\mathrm {i} }}\left({\frac {\partial {\vec {f}}}{\partial x_{l}}}\otimes {\hat {e}}_{l}\cdot \mathbf {T} \right)\\=&(\nabla \cdot \mathbf {T} )\times {\vec {f}}-{\vec {\mathrm {i} }}\left((\nabla \otimes {\vec {f}})^{\top }\cdot \mathbf {T} \right)\end{aligned}}}"> </noscript><span class="lazy-image-placeholder" style="width: 67.24ex;height: 26.176ex;vertical-align: -12.224ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7f7bcfd064ce9a68899a0713030f305f676c2f37" data-alt="{\displaystyle {\begin{aligned}\nabla \cdot (\mathbf {T} \times {\vec {f}})=&{\hat {e}}_{l}{\frac {\partial }{\partial x_{l}}}\cdot (\mathbf {T} \times {\vec {f}})=\left({\hat {e}}_{l}\cdot {\frac {\partial \mathbf {T} }{\partial x_{l}}}\right)\times {\vec {f}}+({\hat {e}}_{l}\cdot \mathbf {T} )\times {\frac {\partial {\vec {f}}}{\partial x_{l}}}\\=&(\nabla \cdot \mathbf {T} )\times {\vec {f}}-{\frac {\partial {\vec {f}}}{\partial x_{l}}}\times ({\hat {e}}_{l}\cdot \mathbf {T} )\\=&(\nabla \cdot \mathbf {T} )\times {\vec {f}}-{\vec {\mathrm {i} }}\left({\frac {\partial {\vec {f}}}{\partial x_{l}}}\otimes {\hat {e}}_{l}\cdot \mathbf {T} \right)\\=&(\nabla \cdot \mathbf {T} )\times {\vec {f}}-{\vec {\mathrm {i} }}\left((\nabla \otimes {\vec {f}})^{\top }\cdot \mathbf {T} \right)\end{aligned}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </dd> </dl> Siehe auch <a href="https://de-m-wikipedia-org.translate.goog/wiki/Vektorinvariante?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Vektorinvariante">Vektorinvariante</a> und <a href="https://de-m-wikipedia-org.translate.goog/wiki/Formelsammlung_Tensoralgebra?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Kreuzprodukt_eines_Vektors_mit_einem_Tensor" title="Formelsammlung Tensoralgebra">Formelsammlung Tensoralgebra#Kreuzprodukt eines Vektors mit einem Tensor</a></li> </ol> </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(5)"> <h2 id="Literatur">Literatur</h2> <a role="button" href="https://de-m-wikipedia-org.translate.goog/w/index.php?title=Cauchy-eulersche_Bewegungsgesetze&action=edit&section=17&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Abschnitt bearbeiten: Literatur" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> Bearbeiten </a> </div> <section class="mf-section-5 collapsible-block" id="mf-section-5"> <ul> <li><a href="https://de-m-wikipedia-org.translate.goog/wiki/Holm_Altenbach?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Holm Altenbach">Holm Altenbach</a>: <cite style="font-style:italic">Kontinuumsmechanik</cite>. Springer, 2012, <a href="https://de-m-wikipedia-org.translate.goog/wiki/Spezial:ISBN-Suche/9783642241185?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="internal mw-magiclink-isbn">ISBN 978-3-642-24118-5</a>.<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:Cauchy-eulersche+Bewegungsgesetze&rft.au=Holm+Altenbach&rft.btitle=Kontinuumsmechanik&rft.date=2012&rft.genre=book&rft.isbn=9783642241185&rft.pub=Springer" style="display:none"> </li> <li>Ralf Greve: <cite style="font-style:italic">Kontinuumsmechanik</cite>. Ein Grundkurs für Ingenieure und Physiker. Springer, Berlin u. a. 2003, <a href="https://de-m-wikipedia-org.translate.goog/wiki/Spezial:ISBN-Suche/3540007601?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="internal mw-magiclink-isbn">ISBN 3-540-00760-1</a>.<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:Cauchy-eulersche+Bewegungsgesetze&rft.au=Ralf+Greve&rft.btitle=Kontinuumsmechanik&rft.date=2003&rft.genre=book&rft.isbn=3540007601&rft.place=Berlin+u.+a.&rft.pub=Springer" style="display:none"> </li> <li><a href="https://de-m-wikipedia-org.translate.goog/wiki/Morton_Gurtin?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Morton Gurtin">Morton E. Gurtin</a>: <cite style="font-style:italic">The Linear Theory of Elasticity</cite>. In: <a href="https://de-m-wikipedia-org.translate.goog/wiki/Clifford_Truesdell?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Clifford Truesdell">C. Truesdell</a> (Hrsg.): <cite style="font-style:italic">Festkörpermechanik : Teil 2</cite> (= <a href="https://de-m-wikipedia-org.translate.goog/wiki/Siegfried_Fl%C3%BCgge?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Siegfried Flügge">S. Flügge</a> [Hrsg.]: <cite style="font-style:italic">Handbuch der Physik</cite>. Band 6a, Teilbd. 2). Springer, Berlin/Heidelberg 1972, <a href="https://de-m-wikipedia-org.translate.goog/wiki/Spezial:ISBN-Suche/3540055355?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="internal mw-magiclink-isbn">ISBN 3-540-05535-5</a>, S. 1–295, <a href="https://de-m-wikipedia-org.translate.goog/wiki/Digital_Object_Identifier?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Digital Object Identifier">doi</a>:<a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://doi.org/10.1007/978-3-662-39776-3_1">10.1007/978-3-662-39776-3_1</a> (DOI der englischen Ausgabe).<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:Cauchy-eulersche+Bewegungsgesetze&rft.atitle=The+Linear+Theory+of+Elasticity&rft.au=Morton+E.+Gurtin&rft.btitle=Festk%C3%B6rpermechanik+%3A+Teil+2&rft.date=1972&rft.doi=10.1007%2F978-3-662-39776-3_1&rft.genre=book&rft.isbn=3540055355&rft.pages=1-295&rft.place=Berlin%2FHeidelberg&rft.pub=Springer&rft.series=Handbuch+der+Physik" style="display:none"> </li> <li>P. Haupt: <cite style="font-style:italic">Continuum Mechanics and Theory of Materials</cite>. Springer, 2002, <a href="https://de-m-wikipedia-org.translate.goog/wiki/Spezial:ISBN-Suche/9783642077180?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="internal mw-magiclink-isbn">ISBN 978-3-642-07718-0</a>, <a href="https://de-m-wikipedia-org.translate.goog/wiki/Digital_Object_Identifier?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Digital Object Identifier">doi</a>:<a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://doi.org/10.1007/978-3-662-04775-0">10.1007/978-3-662-04775-0</a>.<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:Cauchy-eulersche+Bewegungsgesetze&rft.au=P.+Haupt&rft.btitle=Continuum+Mechanics+and+Theory+of+Materials&rft.date=2002&rft.doi=10.1007%2F978-3-662-04775-0&rft.genre=book&rft.isbn=9783642077180&rft.pub=Springer" style="display:none"> </li> </ul> </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(6)"> <h2 id="Einzelnachweise">Einzelnachweise</h2> <a role="button" href="https://de-m-wikipedia-org.translate.goog/w/index.php?title=Cauchy-eulersche_Bewegungsgesetze&action=edit&section=18&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Abschnitt bearbeiten: Einzelnachweise" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> Bearbeiten </a> </div> <section class="mf-section-6 collapsible-block" id="mf-section-6"> <ol class="references"> <li id="cite_note-szabo-1">↑ <a href="https://de-m-wikipedia-org.translate.goog/wiki/Cauchy-eulersche_Bewegungsgesetze?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-szabo_1-0">a</a> <a href="https://de-m-wikipedia-org.translate.goog/wiki/Cauchy-eulersche_Bewegungsgesetze?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-szabo_1-1">b</a> <a href="https://de-m-wikipedia-org.translate.goog/wiki/Istv%C3%A1n_Szab%C3%B3_(Ingenieur)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="István Szabó (Ingenieur)">István Szabó</a>: <cite style="font-style:italic">Geschichte der mechanischen Prinzipien</cite>. Springer, 2013, <a href="https://de-m-wikipedia-org.translate.goog/wiki/Spezial:ISBN-Suche/9783034853019?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="internal mw-magiclink-isbn">ISBN 978-3-0348-5301-9</a> (<a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://books.google.de/books?id%3DSJOmBgAAQBAJ%26pg%3DPA27%23v%3Donepage">eingeschränkte Vorschau</a> in der Google-Buchsuche [abgerufen am 2. Mai 2021]).<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:Cauchy-eulersche+Bewegungsgesetze&rft.au=Istv%C3%A1n+Szab%C3%B3&rft.btitle=Geschichte+der+mechanischen+Prinzipien&rft.date=2013&rft.genre=book&rft.isbn=9783034853019&rft.pub=Springer" style="display:none"> </li> <li id="cite_note-3"><a href="https://de-m-wikipedia-org.translate.goog/wiki/Cauchy-eulersche_Bewegungsgesetze?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-3">↑</a> P. Haupt: <cite style="font-style:italic">Continuum Mechanics and Theory of Materials</cite>. Springer, 2002, <a href="https://de-m-wikipedia-org.translate.goog/wiki/Spezial:ISBN-Suche/9783642077180?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="internal mw-magiclink-isbn">ISBN 978-3-642-07718-0</a>, S. 141, <a href="https://de-m-wikipedia-org.translate.goog/wiki/Digital_Object_Identifier?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Digital Object Identifier">doi</a>:<a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://doi.org/10.1007/978-3-662-04775-0">10.1007/978-3-662-04775-0</a>.<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:Cauchy-eulersche+Bewegungsgesetze&rft.au=P.+Haupt&rft.btitle=Continuum+Mechanics+and+Theory+of+Materials&rft.date=2002&rft.doi=10.1007%2F978-3-662-04775-0&rft.genre=book&rft.isbn=9783642077180&rft.pages=141&rft.pub=Springer" style="display:none"> </li> <li id="cite_note-4"><a href="https://de-m-wikipedia-org.translate.goog/wiki/Cauchy-eulersche_Bewegungsgesetze?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-4">↑</a> P. Haupt: <cite style="font-style:italic">Continuum Mechanics and Theory of Materials</cite>. Springer, 2002, <a href="https://de-m-wikipedia-org.translate.goog/wiki/Spezial:ISBN-Suche/9783642077180?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="internal mw-magiclink-isbn">ISBN 978-3-642-07718-0</a>, S. 144, <a href="https://de-m-wikipedia-org.translate.goog/wiki/Digital_Object_Identifier?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Digital Object Identifier">doi</a>:<a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://doi.org/10.1007/978-3-662-04775-0">10.1007/978-3-662-04775-0</a>.<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:Cauchy-eulersche+Bewegungsgesetze&rft.au=P.+Haupt&rft.btitle=Continuum+Mechanics+and+Theory+of+Materials&rft.date=2002&rft.doi=10.1007%2F978-3-662-04775-0&rft.genre=book&rft.isbn=9783642077180&rft.pages=144&rft.pub=Springer" style="display:none"> </li> <li id="cite_note-5"><a href="https://de-m-wikipedia-org.translate.goog/wiki/Cauchy-eulersche_Bewegungsgesetze?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-5">↑</a> Ralf Greve: <cite style="font-style:italic">Kontinuumsmechanik</cite>. Ein Grundkurs für Ingenieure und Physiker. Springer, Berlin u. a. 2003, <a href="https://de-m-wikipedia-org.translate.goog/wiki/Spezial:ISBN-Suche/3540007601?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="internal mw-magiclink-isbn">ISBN 3-540-00760-1</a>, S. 74.<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:Cauchy-eulersche+Bewegungsgesetze&rft.au=Ralf+Greve&rft.btitle=Kontinuumsmechanik&rft.date=2003&rft.genre=book&rft.isbn=3540007601&rft.pages=74&rft.place=Berlin+u.+a.&rft.pub=Springer" style="display:none"> </li> <li id="cite_note-7"><a href="https://de-m-wikipedia-org.translate.goog/wiki/Cauchy-eulersche_Bewegungsgesetze?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-7">↑</a> W. H. Müller: <cite style="font-style:italic">Streifzüge durch die Kontinuumstheorie</cite>. Springer, 2011, <a href="https://de-m-wikipedia-org.translate.goog/wiki/Spezial:ISBN-Suche/9783642198694?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="internal mw-magiclink-isbn">ISBN 978-3-642-19869-4</a>, S. 72.<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:Cauchy-eulersche+Bewegungsgesetze&rft.au=W.+H.+M%C3%BCller&rft.btitle=Streifz%C3%BCge+durch+die+Kontinuumstheorie&rft.date=2011&rft.genre=book&rft.isbn=9783642198694&rft.pages=72&rft.pub=Springer" style="display:none"> </li> <li id="cite_note-8"><a href="https://de-m-wikipedia-org.translate.goog/wiki/Cauchy-eulersche_Bewegungsgesetze?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-8">↑</a> Ralf Sube: <cite style="font-style:italic">Wörterbuch Physik Englisch: German-English</cite>. Routledge, London 2001, <a href="https://de-m-wikipedia-org.translate.goog/wiki/Spezial:ISBN-Suche/9780415173384?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="internal mw-magiclink-isbn">ISBN 978-0-415-17338-4</a> (<a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://books.google.de/books?id%3Dvai7wiq5HIYC%26pg%3DPA1200%23v%3Donepage">eingeschränkte Vorschau</a> in der Google-Buchsuche [abgerufen am 17. März 2017]).<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:Cauchy-eulersche+Bewegungsgesetze&rft.au=Ralf+Sube&rft.btitle=W%C3%B6rterbuch+Physik+Englisch%3A+German-English&rft.date=2001&rft.genre=book&rft.isbn=9780415173384&rft.place=London&rft.pub=Routledge" style="display:none"> </li> <li id="cite_note-9"><a href="https://de-m-wikipedia-org.translate.goog/wiki/Cauchy-eulersche_Bewegungsgesetze?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-9">↑</a> <a href="https://de-m-wikipedia-org.translate.goog/wiki/Morton_Gurtin?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Morton Gurtin">Morton E. Gurtin</a>: <cite style="font-style:italic">The Linear Theory of Elasticity</cite>. In: <a href="https://de-m-wikipedia-org.translate.goog/wiki/Clifford_Truesdell?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Clifford Truesdell">C. Truesdell</a> (Hrsg.): <cite style="font-style:italic">Festkörpermechanik : Teil 2</cite> (= <a href="https://de-m-wikipedia-org.translate.goog/wiki/Siegfried_Fl%C3%BCgge?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Siegfried Flügge">S. Flügge</a> [Hrsg.]: <cite style="font-style:italic">Handbuch der Physik</cite>. Band 6a, Teilbd. 2). Springer, Berlin/Heidelberg 1972, <a href="https://de-m-wikipedia-org.translate.goog/wiki/Spezial:ISBN-Suche/3540055355?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="internal mw-magiclink-isbn">ISBN 3-540-05535-5</a>, S. 1–295, hier S. 60, <a href="https://de-m-wikipedia-org.translate.goog/wiki/Digital_Object_Identifier?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Digital Object Identifier">doi</a>:<a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://doi.org/10.1007/978-3-662-39776-3_1">10.1007/978-3-662-39776-3_1</a> (DOI der englischen Ausgabe).<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:Cauchy-eulersche+Bewegungsgesetze&rft.atitle=The+Linear+Theory+of+Elasticity&rft.au=Morton+E.+Gurtin&rft.btitle=Festk%C3%B6rpermechanik+%3A+Teil+2&rft.date=1972&rft.doi=10.1007%2F978-3-662-39776-3_1&rft.genre=book&rft.isbn=3540055355&rft.pages=1-295&rft.place=Berlin%2FHeidelberg&rft.pub=Springer&rft.series=Handbuch+der+Physik" style="display:none"> </li> <li id="cite_note-10"><a href="https://de-m-wikipedia-org.translate.goog/wiki/Cauchy-eulersche_Bewegungsgesetze?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-10">↑</a> Martin H. Sadd: <cite style="font-style:italic">Elasticity – Theory, applications and numerics</cite>. Elsevier Butterworth-Heinemann, 2005, <a href="https://de-m-wikipedia-org.translate.goog/wiki/Spezial:ISBN-Suche/0126058113?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="internal mw-magiclink-isbn">ISBN 0-12-605811-3</a>, S. 110.<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:Cauchy-eulersche+Bewegungsgesetze&rft.au=Martin+H.+Sadd&rft.btitle=Elasticity+-+Theory%2C+applications+and+numerics&rft.date=2005&rft.genre=book&rft.isbn=0126058113&rft.pages=110&rft.pub=Elsevier+Butterworth-Heinemann" style="display:none"> </li> </ol> </section> </div> <noscript> <img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1&useformat=mobile" alt="" width="1" height="1" style="border: none; position: absolute;"> </noscript> <div class="printfooter" data-nosnippet=""> Abgerufen von „<a dir="ltr" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://de.wikipedia.org/w/index.php?title%3DCauchy-eulersche_Bewegungsgesetze%26oldid%3D248111095">https://de.wikipedia.org/w/index.php?title=Cauchy-eulersche_Bewegungsgesetze&oldid=248111095</a>“ </div> </div> </div> <div class="post-content" id="page-secondary-actions"> </div> </main> <footer class="mw-footer minerva-footer" role="contentinfo"><a class="last-modified-bar" href="https://de-m-wikipedia-org.translate.goog/w/index.php?title=Cauchy-eulersche_Bewegungsgesetze&action=history&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB"> <div class="post-content last-modified-bar__content"> Zuletzt bearbeitet am 28. 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Cauchy-eulersche Bewegungsgesetze – Wikipedia