$($a$)$ In certain deformation with small strain is given us

$[lon]=\left[\begin{array}{ccc}1& 3& -2\\ 3& 1& -2\\ -2& -2& 6\end{array}\right]{10}^{-6}$

Calculate the normal strain in the direction of vector $a=2$hat$(e{)}_1+2$hat$(e{)}_2+0$hat$(e{)}_3.$ Also, determine spherical and deviatoric strain, principal strain and their directions for the given strain.

$($b$)$ Consider the displacement

$u_1(x_1,x_2,x_3)=0$

$u_2(x_1,x_2,x_3)=k{x}_3$

$u_3(x_1,x_2,x_3)=k{x}_2$

where $k$ is a constant, be applied to the small cude of edge dimension $L($As shown in Fig. $3).$ Draw the deformed shape of the face ABGH of the cube and determine the difference $(L{)}_{\text{a}\text{f}\text{t}\text{e}\text{r}}-(L{)}_{\text{b}\text{e}\text{f}\text{o}\text{r}\text{e}}$ for the diagonals $AG,BH,OG$ of the cube.

$7$

$4.$ Suppose $e$ is the standard basis in $E.$ The matrix of the Cauchy stress is

$[{]}^e=\left[\begin{array}{ccc}1& 2& 3\\ 2& 4& 6\\ 3& 6& 1\end{array}\right]$Mpa

where $[]$e represent stress components are given with respect to orthonormal basis $e.$ Find the principal stress $($in MPa$)$ and principal directions of $[{]}^e.$ Let $f$ be another orthonormal basis for $E.$ Assume that the basis $f$ is aligned with the principal direction of $[{]}^e.$ Then verify that the Cauchy stress is represented in basis $f$ by a diagonal matrix $[{]}^f.$

$14$

$5.($a$)$ Derive the expressions for gauge factor of the electrical resistance strain gauge.

$4$

$($b$)$ The following observation are made with rectangular rosette mounted on a steel specimen $lo{n}_A=800\frac{m}{m},lo{n}_B=-100\frac{m}{m},lo{n}_C=-900\frac{m}{m}($as shown in figure $4).$

Determine the principal strain, principal stresses, principal angles ${}_1$ and ${}_2$ with respect to $x-$axis. For steel $E=200k\frac{N}{m}{m}^2$ and $v=0\mathrm{.}3.$

$10$