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, \(\hat{\sigma}_{ij}\), can be determined from the stress components in the reference state, \(\sigma_{ij}\), using the rotation matrix \(a\) with components \(a_{ij}\). Note that in soil mechanics, we often represent the shear stress components (i.e., the off-diagonal elements of the stress tensor) as \(\tau_{ij}\) instead of \(\sigma_{ij}\). Furthermore, the prime in \(\hat{\sigma}\) usually denotes effective stress, but is used here to denote stresses in a rotated coordinate system.
$$\hat{\sigma} = a \sigma a^T$$In matrix form:
\[\left[ \begin{matrix} \hat{\sigma}_{11} & \hat{\sigma}_{12} & \hat{\sigma}_{13} \\ \hat{\sigma}_{21} & \hat{\sigma}_{22} & \hat{\sigma}_{23} \\ \hat{\sigma}_{31} & \hat{\sigma}_{32} & \hat{\sigma}_{33} \\ \end{matrix} \right]=\left[ \begin{matrix} {{a}_{11}} & {{a}_{12}} & {{a}_{13}} \\ {{a}_{21}} & {{a}_{22}} & {{a}_{23}} \\ {{a}_{31}} & {{a}_{32}} & {{a}_{33}} \\ \end{matrix} \right]\left[ \begin{matrix} \sigma _{11}^{{}} & \sigma _{12}^{{}} & \sigma _{13}^{{}} \\ \sigma _{21}^{{}} & \sigma _{22}^{{}} & \sigma _{23}^{{}} \\ \sigma _{31}^{{}} & \sigma _{32}^{{}} & \sigma _{33}^{{}} \\ \end{matrix} \right]\left[ \begin{matrix} {{a}_{11}} & {{a}_{21}} & {{a}_{31}} \\ {{a}_{12}} & {{a}_{22}} & {{a}_{32}} \\ {{a}_{13}} & {{a}_{23}} & {{a}_{33}} \\ \end{matrix} \right]\]
Figure 2.1.1. Stresses in a reference coordinate system (axes \(x_1\), \(x_2\), \(x_3\)), and rotated coordinate system (axes \(x_1'\), \(x_2'\), \(x_3'\)). https://upload.wikimedia.org/wikipedia/commons/thumb/7/76/Stress_transformation_3D.svg/2000px-Stress_transformation_3D.svg.png
Note: The \(a\) 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 |