do not copy from chat gpt

a$)$ In case of state of stress in $3$D$,$ develop the equations for normal and shear stress at any oblique plane and prove that,

$={}_x*l^2+{}_y*m^2+{}_z*n^2+2{}_{xy}*lm+2{}_{yz}*mn+2{}_{zz}*ln$ and

${}^2=({}_1-{}_2{)}^2*l^2{m}^2+({}_1-{}_3{)}^2*l^2{n}^2+({}_2-{}_3{)}^2*m^2{n}^2$

$l,m$ and $n$ are direction cosines and all other symbols have the conventional sense.

b$)$ With respect to yielding, explain the Distortion$-$Energy Criterion.

c$)$ What is Yield Locus? Prove that the yield locus of maximum shear stress $($MSS$)$ criterion falls in the von Mises yield ellipse.

d$)$ Explain the operation mechanisms of Frank$-$Read sources. Discuss the way they can contribute to hardening of metals.