Q$3.$ Consider a bar of cross$-$section A and length $L$ in fig$\mathrm{.}1.$ A uniform tensile load $P$ is applied

to the two ends of the rod; find the state of stress $(\{$:${}_x,{}_y,{}_z,{}_{yz},{}_{zx},{}_{xy})$ and strain $({}_x,$

${}_y,{}_z,{}_{yz},{}_{zx},{}_{xy}),$ and strain energy per unit volume of the body. Energy is defined as

the capacity to do work. In solid, deformable, elastic bodies under loads, the work done by

external loads is stored as recoverable strain energy. Assume that the rod is made of a

homogeneous isotropic material of Young's modulus, E$.$

Fig$\mathrm{.}1$: Cylindrical rod under uniform uniaxial load, P$.$

where the compliance matrix for an isotropic, homogeneous, and linearly elastic material is

defined as

$\left[\begin{array}{cccccc}\frac{1}{E}& -\frac{v}{E}& -\frac{v}{E}& 0& 0& 0\\ -\frac{v}{E}& \frac{1}{E}& -\frac{v}{E}& 0& 0& 0\\ -\frac{v}{E}& -\frac{v}{E}& \frac{1}{E}& 0& 0& 0\\ 0& 0& 0& \frac{1}{G}& 0& 0\\ 0& 0& 0& 0& \frac{1}{G}& 0\\ 0& 0& 0& 0& 0& \frac{1}{G}\end{array}\right]$