Components of a linear strain tensor ${}_{{?}_{?}}$ at a point with respect to the $(x,y,z)$ coordinate system are

$[]=\left[\begin{array}{ccc}0& 2& 2\\ 2& 0& 2\\ 2& 2& 0\end{array}\right]{10}^{-4}$

$($a$)$ Find the principal invariants of ${}_{?}.$

$($b$)$ Find the principal stresses ${}_i$ and the unit vectors hat$(e{)}_i$ along the principal directions. Arrange ${}_i$ such that

${}_1{}_2{}_3.$ Further, choose the senses of hat$(e{)}_i$ such that hat$(e{)}_1,$hat$(e{)}_2$hat$(e{)}_3=+1.$

$($c$)$ Find the maximum shear stress ${}_{n_1{n}_2}$ and the directions hat$(n{)}_{01}$ and hat$(n{)}_{02}$ associated with maximum shea

In plane strain rolling of a sheet, matrix of the stress tensor ${}_{{?}_{?}}$ with respect to the $(x,y,z)$ coordinate systen

at a particular point $A$ is given by

$[]=\left[\begin{array}{ccc}0\mathrm{.}21& 0& 0\\ 0& -0\mathrm{.}09& 0\\ 0& 0& 0\mathrm{.}06\end{array}\right]$ GPa

The state of stress at point A lies on the initial Tresca yield surface $($i$.$e$.$ the Tresca yield surface

corresponding to zero equivalent plastic strain$).$ On which side of the yield surface it lies?