Do Principal Stresses and Strains Align in Plane Stress States?

[mm391]

Homework Statement

For a plane stress state, show that the principal stresses and strains occur in the same direction?

Homework Equations

Hookes Law for PLane Stress States
Principal Stress is tan(2θ)=2δ/($\sigma$x-$\sigma$y)
Principal Strain is tan(2θ)=$\gamma$xy/($\epsilon$x-$\epsilon$y)
\gamma=\tau/G
G=E/(2(1+v))

The Attempt at a Solution

tan(2θ)*($\epsilon$x-$\epsilon$y)=tan(2θ)*($\sigma$x-$\sigma$y)2(1+v)/2E

I cannot see how to get rid of (1+v)/E, can anyone help me out please.

 

[Chestermiller]

What do the Hooke's law equations for each of the stresses explicitly in terms of the strains reduce to when σ_z is set equal to zero?