Suppose that we know the stress state of the soil on a particular plane, and we wish to determine the stress state on a different, rotated plane. This problem is equivalent to a transformation of the coordinate system. The components of the Cauchy stress tensor in the rotated coordinate space, , can be determined from the stress components in the reference state, , using the rotation matrix with components . Note that in soil mechanics, we often represent the shear stress components (i.e., the off-diagonal elements of the stress tensor) as instead of . Furthermore, the prime in usually denotes effective stress, but is used here to denote stresses in a rotated coordinate system.
In matrix form:
Figure 2.1.1. Stresses in a reference coordinate system (axes , , ), and rotated coordinate system (axes , , ). https://upload.wikimedia.org/wikipedia/commons/thumb/7/76/Stress_transformation_3D.svg/2000px-Stress_transformation_3D.svg.png
Note: The matrix must be orthogonal (i.e., the axes in the rotated coordinate system must form 90 degree angles with each other)
2 | 0 | 0 |
0 | 1 | 0 |
0 | 0 | 1 |