Cauchy's stress theorem ($\mathbf{t}(\mathbf{n})=\mathbf{\sigma }\mathbf{n}$)

Cauchy's stress theorem states that local contact traction depends linearly on the oriented unit normal vector: $t_i(\mathbf{n})={\sigma }_{ij}{n}_j$, with ${\sigma }_{ij}={\mathbf{e}}_i\cdot \mathbf{t}({\mathbf{e}}_j)$. It follows by balancing forces on a shrinking tetrahedron with three coordinate faces and one oblique face. Contact forces scale as the face area, while bounded body force and inertial contributions scale as its volume; dividing by the oblique face area and shrinking eliminates the latter. The coordinate-face projected areas are the oblique face area times $n_j$, giving the displayed linear map. Symmetry of the Cauchy stress tensor requires the additional balance of angular momentum in a continuum without independent couples.

The contact traction on an oriented material cut is assumed to depend locally on the point, time and unit normal, with no independent dependence on the cut's curvature or global geometry. Together with local momentum balance and regularity sufficient for shrinking-volume arguments, this implies linear normal dependence through a stress tensor. Referential traction is measured per reference area and yields the first Piola-Kirchhoff stress tensor; spatial traction is measured per current area and yields the Cauchy stress tensor. These are different representations and the nominal tensor is not generally symmetric.