$2-6.$ Conservation of Mass in Planar Extension

Assume that a sheet of elastic material is being stretched continuously in one direction, its volume remaining constant. As shown in Fig. P$2-6,$ the surface at $x=0$ is fixed and that at $x=L(t)$ is pulled to the right at a constant speed $U.$ The instantaneous thickness is $2(t).$ The initial length and halfthickness are $L(0)=L_0$ and $(0)={}_0,$ respectively. The other dimension is indefinitely large and there is no motion in the $z$ direction.

$($a$)$ Assuming that the velocities in the elastic solid are of the form $v_x=v_x(x,t)$ and $v_y=v_y(y,t),$ show that

$v_x(x,t)=(t)x,v_y(y,t)=-(t)y,(t)=\frac{1}{L}\frac{dL}{dt}=-\frac{1}{}\frac{d}{dt}$

where $$ is the rate of strain.

Figure P$2-6.$ Planar extension of an elastic solid. The surface at $x=0$ is stationary and that at $x=L(t)$ is pulled to the right at a constant speed $U.$

$($b$)$ Show that

$\frac{(t)}{{}_0}=(1+\frac{Ut}{L_0}{)}^{-1}.$

Similarly, relate $L(t)$ and $(t)$ to $U$ and the initial dimensions.