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A question from a continuum mechanics beginner... If the stress $\sigma_{xx}$ is applied to an isotropic, three-dimensional body, the following strain tensor results:

$$\boldsymbol\epsilon=\left(\begin{matrix}\frac{1}{E}\sigma _{xx} & 0 & 0 \\0 & -\frac{\nu}{E}\sigma _{xx} & 0 \\0 & 0 & -\frac{\nu}{E}\sigma _{xx}\end{matrix}\right)$$

Now the tensor should be rotated in the xy-plane by the angle $\alpha$.

Transformation

How is the transformation matrix calculated to get the strain components in the x', y', z'-coordinate system?

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To transform a tensor from a frame to another you use

$$A' = Q A Q^T\, ,$$

where $Q$ is the transformation. In this case, your $Q$ should look something like

$$Q = \begin{bmatrix} \cos\alpha &\sin\alpha &0\\ -\sin\alpha &\cos\alpha &0\\ 0 &0 &1 \end{bmatrix}\, .$$

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