Piola–Kirchhoff stress tensors

Summary

The Piola–Kirchhoff stress tensors express stress in terms of the undeformed reference configuration of a body. The first one, $P = J \sigma F^{-T}$, gives the actual force per unit undeformed area; the second one, $S = F^{-1} P$, is symmetric and fully referred to the reference configuration. Here $\sigma$ is the Cauchy stress tensor and $F$ the deformation gradient, with $J = \det F$; these tensors are the natural stress measures in large-deformation problems.