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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)?

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

Here is the derivation:

$\epsilon_x=\frac{\sigma cos(\theta)}{E_x}$

$\epsilon_y=\frac{\sigma sin(\theta)}{E_x}$

$\epsilon=\sqrt{\epsilon_x^2+\epsilon_y^2}=\sigma\sqrt{\frac{cos(\theta)}{E_x^2}+\frac{sin(\theta)}{E_y^2}}$

$E=\frac{\sigma}{\epsilon}=\frac{E_x*E_y}{\sqrt{E_x^2sin^2(\theta)+E_y^2cos^2(\theta)}}$

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