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\Blatt{3}{WS 2004/2005, 9.11.2004}
         {Mechanik der Kontinua}
         {Prof. J.~L.~van~Hemmen}

\begin{deflist}{20.}

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\item[9.]
{\bf Verzerrungstensor}
\par\smallskip
Man betrachte einen achsparallelen Quader mit kleinen Kantenlängen
$(l_1,l_2,l_3)$, der von den Vektoren $(l_1 \mathbf{e}_1, l_2
\mathbf{e}_2, l_3 \mathbf{e}_3)$ aufgespannt wird. Dieser erfahre eine
durch das Vektorfeld 
\begin{displaymath}
  \mathbf{u}: \mathbf{x} \mapsto
  \mathbf{u}(\mathbf{x})=\mathbf{v}(\mathbf{x}) \, \mathrm{d}t
\end{displaymath}
gegebene kleine Verrückung. Größen zweiter Ordnung in $\mathrm{d}t$
sind im Folgenden zu vernachlässigen.
\begin{enumerate}
\item Sei $\mathbf{r}$ ein Punkt im Quader vor der Verrückung und
  $\mathbf{r}'$ der entsprechende Punkt nach der Verrückung. Der
  Quader wird durch $\mathbf{u}$ transliert, gedreht und verzerrt. Man
  mache sich klar, dass in einem mit dem Quader mittranslierten und
  mitgedrehten Koordinatentensystem gilt
  \begin{displaymath}
    \mathbf{r}' \approx \mathbf{r} + D \mathbf{r} \,,
  \end{displaymath}
  wobei 
  \begin{displaymath}
    D=(d_{ij})=\frac{1}{2} \left(
      \frac{\partial u_i}{\partial x_j} + \frac{\partial u_j}{\partial
        x_i} \right) \,
  \end{displaymath}
  der Verzerrungstensor ist.
\item Zeigen Sie, dass für die relative Längenänderung einer Kante
  $\mathbf{r}_i=l_i \mathbf{e}_i$ des Quaders gilt
  \begin{displaymath}
    \frac{|\mathbf{r}'_i|-|\mathbf{r}_i|}{|\mathbf{r}_i|} \approx d_{ii} \,.
  \end{displaymath}
\item Die Kanten $\mathbf{r}_i$ und $\mathbf{r}_j$ des Quaders
  schließen vor der Deformation einen Winkel $\pi/2$ ein, nach der
  (kleinen) Deformation sei der Winkel $\pi/2-\delta_{ij}$. Zeigen
  Sie, dass gilt
  \begin{displaymath}
    \sin \delta_{ij} \approx 2 d_{ij} \,.
  \end{displaymath}
\item Zeigen Sie durch Berechnung der Determinante von $D+I$, dass für
  das Volumen $V'=\mathbf{r}'_1 \circ ( \mathbf{r}'_2 \times
  \mathbf{r}'_3)$ des verzerrten Quaders gilt
  \begin{displaymath}
    \frac{V'-V}{V} = \sum_{i=1}^3 d_{ii} \,.
  \end{displaymath}
\end{enumerate}


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\item[10.]
{\bf Totale Ableitung}
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Beweisen Sie für die Funktionen $f,g: \mathbb{R}^4 \to \mathbb{R};
(x,y,z,t) \mapsto f(x,y,z,t)$, $h: \mathbb{R} \to
\mathbb{R}$ und $\lambda \in \mathbb{R}$ folgende Eigenschaften der
totalen Ableitung $\mathrm{D}_t$:
\begin{enumerate}
\item Additivität
  \begin{displaymath}
    \mathrm{D}_t ( f+g ) = \mathrm{D}_t f + \mathrm{D}_t g
  \end{displaymath}
\item Homogenität
  \begin{displaymath}
    \mathrm{D}_t ( \lambda f ) = \lambda \, \mathrm{D}_t f
  \end{displaymath}
\item Produktregel
  \begin{displaymath}
     \mathrm{D}_t ( f \cdot g ) = f \, \mathrm{D}_t g + g \, \mathrm{D}_t f
  \end{displaymath}
\item Kettenregel
  \begin{displaymath}
    \mathrm{D}_t ( h \circ f ) = (h' \circ f) \, \mathrm{D}_t f
  \end{displaymath}
\end{enumerate}



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\footnotesize
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Besprechung der "Ubungen am Dienstag, den 23.11.2004 um 9.30 Uhr im Raum HS3
(Garching).\\
"Ubungsleitung: Moritz Franosch, mail@Franosch.org, http://www.physik.tu-muenchen.de/lehrstuehle/T35/.

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