# Elastic Modulus in Arbitrary Direction for Orthotropic Material

Suppose I have a material with an elastic modulus of Ex in the x direction and an Elastic modulus of Ey in the y direction.

How would I find the effective elastic modulus in an arbitrary direction (say for an angle theta from the x axis)?

The elastic modulus can be shown to be: $$\frac{E_x*E_y}{\sqrt{E_x^2sin^2(\theta)+E_y^2cos^2(\theta)}}$$
$$\epsilon_x=\frac{\sigma cos(\theta)}{E_x}$$
$$\epsilon_y=\frac{\sigma sin(\theta)}{E_x}$$
$$\epsilon=\sqrt{\epsilon_x^2+\epsilon_y^2}=\sigma\sqrt{\frac{cos(\theta)}{E_x^2}+\frac{sin(\theta)}{E_y^2}}$$
$$E=\frac{\sigma}{\epsilon}=\frac{E_x*E_y}{\sqrt{E_x^2sin^2(\theta)+E_y^2cos^2(\theta)}}$$