Stress Tensors

3D Stress Tensor

The overall stress of a component can be expressed with a 3D stress tensor:

$$\sigma =\left[\begin{array}{ccc}{\sigma }_{xx}& {\tau }_{xy}& {\tau }_{xz}\\ {\tau }_{yx}& {\sigma }_{yy}& {\tau }_{yz}\\ {\tau }_{zx}& {\tau }_{zy}& {\sigma }_{zz}\end{array}\right]$$

For equilibrium, the cross-shears are equal, such that:

$$\begin{array}{r}{\tau }_{yx}={\tau }_{xy}\\ {\tau }_{zy}={\tau }_{yz}\\ {\tau }_{xz}={\tau }_{zx}\end{array}$$

2D Stress Tensor

For a 2D component, we can use a 2D stress tensor:

$$\sigma =\left[\begin{array}{cc}{\sigma }_{xx}& {\tau }_{xy}\\ {\tau }_{yx}& {\sigma }_{yy}\end{array}\right]$$

For static equilibrium we once again need

$${\tau }_{xy}={\tau }_{xy}$$

The 2D stress tensor is symmetric, such that $\sigma ={\sigma }^T$.