Proving equations for a plane stress condition

[a_hargy]

Homework Statement

For a plane stress condition (stress z-axis = 0 ), prove the following relations if strain x-axis and strain y-axis are determined by experiments.

σ_x=(∈_x+v∈_y/1-v^2)E

&

σ_y=(∈_y+v∈_x/1-v^2)E

where:
σ_x = stress in x-axis
σ_y = stress in y-axis
∈_x = strain in x-axis
∈_x = strain in y-axis
E = modulus of elasticity

Homework Equations

∈_z=-(∈_x+∈_y)(v/1-v)

Poission's ratio
∈_x=σ_x/E

∈_y=∈_z=-vσ_x/E

v=-lateral strain/axial strain

v=-∈_y/∈_x=-∈_z/∈_x

The Attempt at a Solution

I'm not sure which equations I should be using to solve the problem. Can I use Hooke's Law for multi-axial loading since it is a plane stress condition?

I have tried rearranging Poisson's ratio equation with no luck.

 

[Mapes]

Hooke's Law only applies to uniaxial loading of a rod. You might find generalized Hooke's Law useful:

$$\epsilon_x=\frac{\sigma_x}{E}-\frac{\nu\sigma_y}{E}-\frac{\nu\sigma_z}{E}$$

It applies in 3-D and in any isotropic material. More http://john.maloney.org/Papers/Generalized%20Hooke%27s%20Law%20%283-12-07%29.pdf" .