This is a task from continuation mechanics: In a Cartesian coordinate system x$,$ y$,$ z$,$ the stress tensor is given by $\left[\begin{array}{cccc}a& 0& c& ?\\ 0& b& 0& ?\\ c& 0& 0& ?\\ ?& ?& ?& ?\end{array}\right]$

where a$,$ b and c are constants. a$)$ Which a$,$ b$,$ and c give rise to valid stress tensors and why?

b$)$ Find the stress on $$a plane with the normal vector n $=($i $+$ j$)/\sqrt[\phantom{2}]{2}$where i and j are the unit vectors in the x and y direction respectively.

c$)$ Determine the normal stress and the tangential stress on the plane defined in b$).$

d$)$ Find the principal stresses and principal directions for the cases a $=1,$b $=2,$c $=3$ and a $=1,$b $=2,$c $=10.($Hint: Check the result by $$a using matlab$/$python $($sympy module$)$ to $$a solve See also "principal stresses and stress invariants" under stress $($mechanics$)$ on wikipedia for a solution method. $P=([a,0,c],[0,b,0],[c,0,0])$