Piola–Kirchhoff stress tensors

Remarks

Nanson's formula $n\, da = J F^{-T} N\, dA$ relates an oriented area element $N\, dA$ of the reference configuration to its image $n\, da$. It gives $\sigma n\, da = P N\, dA$, so the tensor $P$ gives the current force per unit reference area. If $\sigma$ is symmetric, then $S$ is symmetric, while $P$ is in general not symmetric. For $F = I$, both tensors equal $\sigma$. For a rigid rotation $F = Q$ with $Q$ orthogonal and $\det Q = 1$, one gets $P = \sigma Q$ and $S = Q^T \sigma Q$: here $S$ is the Cauchy stress expressed in the rotated material frame. The tensor $S$ is work-conjugate to the Green–Lagrange strain, and $P$ is work-conjugate to $F$.