The three Mohr's circles depict all the possible normal stress / shear stress combinations when material is taken and rotated along one of it's axes of principal stress. Nevertheless, it is clear that rotation around just one principal axis does not yield all the possible configurations. For any point on the Mohr's circle, one of the axes is aligned with one of the principal axes by definition - however, you can rotate a block in such a way so that no axes are aligned with the principal axes.

For example, aligned the axes $x$, $y$, $z$ with the principal stresses $\sigma_1$, $\sigma_2$, $\sigma_3$. If one rotates clockwise around $z$ by $30$ degrees giving the axes normal to the sides of the block $x'$, $y'$ and $z$, and then rotates about $x'$ by say 30 degrees to $x'$, $y''$, $z'$ none of the axes are aligned with the principal stresses anymore. Such a combination of normal and shear stresses does exist in the structure however.

My question is then: where would such a point fall on the coordinate axes of normal stress vs shear? Would it fall between Mohrs' circles? How do we know such a combination of rotations would not yeild a shear higher than that of the largest Mohr's circle?


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