General Stress State Rotated from the Principal Axes  System

This is a general state, we will rotate about the z-axis and compare the result to a general two-dimensional stress state.



Rotation about z-axis by angle θ



Transformation to general two-dimensional stress state coordinate system by rotating the principal system by θ around z-axis



Writing the same equation in a slightly different way...



Naming the coefficients of the rotated two-dimensional state:



Looking at the x-y components of stress (i.e, the upper-left  2×2 submatrix), notice that there are two invariants of the generalized two-dimensional  stress state: The trace and the determinant:





Do not depend on θ; thus illustrating the invariance of these quantities under rotation of coordinate rotations.

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