Cauchy stress tensor ($\mathbf{t}=\mathbf{\sigma }\mathbf{n}$)

The stress tensor maps a surface normal to the traction exerted across that surface. Its divergence is the force density in local momentum balance. For an incompressible Newtonian fluid it combines pressure with twice the viscosity times the symmetric strain-rate tensor.

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 component of traction normal to a surface with unit normal $n$. For an incompressible Newtonian fluid, $\sigma =-pI+2\mu e$, so ${\sigma }_{nn}=-p+2\mu n\cdot en$. A free-interface normal-stress jump includes the curvature force from surface tension.

The principal stresses are the eigenvalues of the symmetric Cauchy stress tensor. In their orthonormal eigenbasis, each traction is normal to its corresponding principal plane.

A plane whose unit normal vector is an eigenvector of the stress tensor. Its traction has no tangential component, so the resolved shear stress vanishes. This is distinct from a principal plane in geometrical optics.

The difference between greatest and least principal stresses. It is twice the radius of the largest Mohr circle.