 First Piola-Kirchhoff stress tensor

The first Piola-Kirchhoff stress tensor expresses the current force transmitted through a material surface per unit reference area. In the spatial-first index convention used here,where $F$ is the deformation gradient, $J=\det F$, $\sigma$ the Cauchy stress tensor, and $\mathbf{N},\mathbf{n}$ the reference and current unit normal vectors. Nanson's formula supplies the transformation of oriented surface area.

Its first index refers to current force direction and its second to reference surface orientation, so it is generally nonsymmetric even when $\sigma$ is symmetric. The mechanical power per reference volume is $\mathcal{P}:\dot{F}$. In a uniform uniaxial test the relevant component is the axial force divided by original area, connecting this tensor to engineering stress. The material-first nominal stress tensor in this repository is $P={\mathcal{P}}^{\mathsf{T}}$; keeping this convention explicit prevents misplaced transposes.