$(2).$ A torque is applied at the end of a circular cylinder $C=\{\begin{array}{c}x=x_i{e}_i\text{:}\sqrt[\phantom{2}]{x_1^2+x_2^2}=\\ x_{3}in\end{array}(0,L).$ Assume that each cross$-$section rotates about the $e_3$ axis through an angle that is proportional to the distance from the bottom of the bar $x_3,$ as show in Fig $1.$ Assume further that the displacement satisfies $U(x)*e_3=0$ for all $nC.$

Determine the deformation $(x)$ and displacement field of $U(x)$ of the cylinder.

Assuming $L1,$ linearize the displacement field $U(x),$ and find the small strain quantities $lon$ as a function of $x$ and $.$

Determine the maximum shear strain in the sample. Note, this is given by

${}_{\text{m}\text{a}\text{x}}=ma{x}_{nC}\frac{1}{2}({}_{\text{m}\text{a}\text{x}}(x)-{}_{\text{m}\text{i}\text{n}}(x))$

where ${}_i(i=1,2,3)$ are the eigenvalues of the strain tensor $lon($the principle strains$).$