Principal stress (${\sigma }_i$)

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 differential stress is the difference between the greatest and least principal stresses, $\Delta \sigma ={\sigma }_{\max }-{\sigma }_{\min }\ge 0$. It measures the spread of the principal loading, rather than the isotropic part of the stress. Adding the same pressure contribution to every principal stress leaves it unchanged.

For a symmetric three-dimensional stress tensor, the largest possible resolved shear stress is $\Delta \sigma /2$. In a Mohr circle representation this is the radius of the outer circle. A uniaxial load with principal values $(S,0,0)$ therefore has differential stress $|S|$, while hydrostatic loading has zero differential stress even when the pressure is large. This distinction is useful in yield criterion: a threshold based on maximum shear stress depends on principal-stress differences, rather than simply on the magnitude of the largest compression.