Continuum Mechanics

Envision a rectangular bar stretched uniformly in the $e_1-$direction. This deformation results in a constant

strain field given by

$[lon]=\left[\begin{array}{ccc}{C}_1& 0& 0\\ 0& -C_2& 0\\ 0& 0& 0\end{array}\right]$

where $C_1$ and $C_2$ are constants. Assume infinitesimal deformations throughout this problem. Further,

assume that the displacement field depends only on $x_1$ and $x_2.$

$($a$)$ Integrate the strain$-$displacement relations to determine the displacement components $(u_1,u_2,u_3)$ and

identify any rigid$-$body motion terms in your displacement field.

$($b$)$ Show that the displacement field $u$ found in part $($a$)$ above can be written in the form

$u=C_1{x}_1{e}_1-C_2{x}_2{e}_2+Ax+b$

where $A$ is a skew tensor and $b$ is a constant vector. Finally, show that the infinitesimal rotation

$=skw(gradu)=A.$