Consider the following stress state

$=\left[\begin{array}{ccc}1-& & 0\\ & 0& 0\\ 0& 0& 0\end{array}\right]{}_0,$

where ${}_0$ is some stress value and $$ is a dimensionless parameter $in(0,1).$

$($a$)$ Calculate the principal invariants of the stress tensor: $I_1=tr,I_2=$

$.$

$($b$)$ Determine the deviatoric stress tensor $s.$

$($c$)$ Determine the deviatoric invariants $J_2$ ja $J_3$ and the Lode angle $$ and

draw the position of the stress on the deviatoric plane as a function of the

parameter $.$

$($d$)$ Determine traction vector $t$ on a plane whose normal points in direction

$(1,2,2{)}^T.$ Determine also the normal stress and shear stress $($and its

direction$)$ on this plane.

$($e$)$ Determine the principal stresses and the principal directions.

$($f$)$ Determine the principal values of the deviatoric stress tensor.