Simple motions

Let a subset $\kappa ^{(4)}$of $\mathcal{A}\times (0,T)$ and $\chi :(% \mathcal{A}\times (0,T))\backslash \kappa ^{(4)}\rightarrow \mathcal{E}$ be given. We call the pair $(\kappa ^{(4)},\chi )$ a simple motion of the piecewise fit region $\mathcal{A}$ during the time interval (0,T) if the pair $(\kappa ^{(4)},\mathsf{t}_\chi )$, with $\mathsf{t}_\chi$ the trajectory mapping defined in (6.6), is a ``space-time '' simple deformation, i.e., if $(\kappa ^{(4)},\mathsf{t}_\chi )\in$Sid $(% \mathcal{A}\times (0,T))$.

Example (``blinking motion''): Let $m\in \left\{ 1,2,3,\cdots \right\}$, $e\in \mathcal{V}$, $t\in (0,1)$, and let $\lfloor (mt)$ denote the greatest integer$\$less than or equal to mt. We define

 

$$\chi _m(x,t) =x+\frac{\lfloor (mt)}me \ {\rm for }\ (x,t)\in (\mathcal{A% }\times (0,1))\backslash \kappa _m^{(4)}$$ :

$$\kappa _m^{(4)} =\left\{ (x,\frac km)\,\mid \,x\in \mathcal{A},\,\,k\in \left\{ 1,2,\cdots ,m-1\right\} \right\} .$$ : (8)

In this simple motion, the points of the region $\mathcal{A}$ do not move during the time interval $(0,\frac 1m)$, but they suddenly all move at time $% \frac 1m$ to gain a displacement $\frac 1me$. The points again remain static during the interval $(\frac 1m,\frac 2m)$ and suddenly displace at time $% \frac 2m$, again by amount $\frac 1me$, and so on. Thus, if one happened to blink at each of the times $\frac 1m,\frac 2m,\cdots ,\frac{m-1}m,$ one would not observe any motion. Nevertheless, at the end of the time interval <tex2html_comment_mark> (0,1), the region would have displaced by an amount $\frac{m-1}me.$ Note that we have

$$\nabla ^{(4)}\mathsf{t}_{\chi _m}(x,t)=\left[ \begin{array}{cc} I & 0 \\ 0 & 1 \end{array}\right] =:I^{(4)}$$

for all $(x,t)\in (\mathcal{A}\times (0,1))\backslash \kappa _m^{(4)}$; this relation reflects the fact that in this example movement only occurs at a finite set of times and distances between points in the body never change.

We write Sim $(\mathcal{A}\times (0,T))$ for the collection of simple motions $(\kappa ^{(4)},\chi )$ of $\mathcal{A}$ during (0,T).