Cauchy Stress Tensor

• Stress acting on a element has 9 components of Stress. This is because by definition, the stress is the response to the force acting per unit area.
• This means that the stress is dependant on the direction of the force, but also the orientation of the area (which has two components) with respect to the force.
• Hence we have three Normal stress components and 6 Shear stress components which can be represented by the Stress Tensor.

$$\left[\begin{array}{ccc}{T}_1^{(\mathbf{n})}& T_2^{(\mathbf{n})}& T_3^{(\mathbf{n})}\end{array}\right]=\left[\begin{array}{ccc}{n}_1& n_2& n_3\end{array}\right]\cdot \left[\begin{array}{ccc}{\sigma }_{11}& {\sigma }_{12}& {\sigma }_{13}\\ {\sigma }_{21}& {\sigma }_{22}& {\sigma }_{23}\\ {\sigma }_{31}& {\sigma }_{32}& {\sigma }_{33}\end{array}\right]$$