\[\renewcommand{\u}[1]{\boldsymbol{#1}} \renewcommand{\b}[1]{\mathbb} \newcommand{\set}[2]{\left\{#1\,\big|\,#2\right\}} \newcommand{\pr}[1]{\left(#1\right)} \newcommand{\st}[2][]{#2_{\mathscr{s}#1}} \newcommand{\mt}[2][]{#2_{\mathscr{m}#1}} \newcommand{\lsc}[2][\mathscr{l}]{ { }^{#1}\!#2 } \newcommand{\norm}[1]{\lVert #1 \rVert}\]

The shrinking shrinking sphere

In the last class we derived that for the deformation mapping \(\u{\varphi}(\u{X},t)=e^{-\lambda t}\u{I}\u{X}\) the velocity and acceleration fields are, \(\begin{align} \u{V}&=-\lambda e^{-\lambda t} \u{I}\u{X}\\ \u{A}&=\lambda^2 e^{-\lambda t}\u{I} \u{X}\\ \u{v}&=-\lambda \u{I} \u{x}\\ \u{a}&=\lambda^2 \u{I}\u{x} \end{align}\)

However, the spatial acceleration was computed as $\u{a}(\u{x},t)=\u{A}(\u{\varphi}^{-1}(\u{x},t),t)$. That is we needed the material acceleration fields. Now we can compute the acceleration directly from the velocity field.

\[\begin{align} \text{grad}\u{v}&=-\lambda \u{I}\\ \text{grad}\u{v}\u{v}&=\lambda^2 \u{I}\u{x}\\ \frac{\partial \u{v}}{\partial t}&=\u{0}\\ \u{a}&=\text{grad}\u{v}\u{v}+\frac{\partial \u{v}}{\partial t}\\ &=\lambda^2 \u{x} \end{align}\]

For a material particle $\u{X}$ its trajectory is given by \(\u{\varphi}(\u{X},t)\) where \(\u{X}\) is treated as fixed and time \(t\) is the variable. Hence its velocity and accelerations are given as \(\begin{align} V_i(X_i,\tau)&=\frac{x_{i}(X_i,\tau)}{\partial \tau}\\ A_i(X_i,\tau)&=\frac{\partial^2 x_{i}(X_i,\tau)}{\partial \tau^2} \end{align}\)

Shrinking sphere with a given specific Point

Compute the material velocity and acceleration fields for the following motion. What is the velocity and acceleration of the material particle \(\u{X}_P=(\u{E}_1+\u{E}_2)/\sqrt{2}\) at \(t=1/2\). \(\begin{align} \u{\varphi}(\u{X},t)=\exp(-\lambda t)\u{X} \end{align}\) We get from the definitions of \(\u{V}(\u{X},t)\) and \(\u{A}(\u{X},t)\)that, \(\begin{align} \u{V}(\u{X},t)&=-\lambda \exp(-\lambda t)\u{X}\\ \u{A}(\u{X},t)&=\lambda^2 \exp(-\lambda t)\u{X} \end{align}\) The velocity and acceleration of \(\u{X}_p\) are \(\begin{align} \u{V}(\u{X}_P,1/2)&=-\lambda\exp(-\lambda/2)(\u{E}_1+\u{E}_2)/2\\ \u{A}(\u{X}_P,1/2)&=\lambda^2\exp(-\lambda/2)(\u{E}_1+\u{E}_2)/2 \end{align}\)