Second Piola-Kirchhoff stress tensor ($S=J{F}^{-1}\sigma F^{-T}=P{F}^{-T}$)

The second tensor has two material indices and is symmetric when the Cauchy stress tensor is symmetric. The displayed relation uses the material-first nominal stress tensor convention. In particular, $PF$ is generally not this tensor. These conventions agree with A. F. Bower's definitions of stress measures.