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The meaning of $\sigma_{ij}$ is force in direction $j$ applied in a surface whose normal is in the direction $i$. Therefore $\sigma_{xx}$ is an x-directed force applied in a surface whose normal is in the x direction, which we interpret as pressure. When $i\neq j$ we call it shear, but the idea is the same. This drawing from Wikipedia might be helpful:


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It is true they are in genereal both symmetric. The symmetry of the stress tensor is not only a matter of definition, it is a general property consecuence of angular momentum conservation. On the other hand, the strain tensor is found to be symmetric as a consecuence of its definition as a measure of $ds^2-ds_0^2$ and it is so not only in its linear ...


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The same formula does apply to 2D, 2D stress is just a particular case of 3D stress. However sometimes you don't want to consider the 3rd dimension in 2D stress and you can work with 2D matrices and tensors and in such cases you have to replace the number 3 in your formula by the number 2



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