Notes to myself on stress tensors

Cauchy stress is the true stress on a deformed configuration. It operates on normal vectors on deformed solid and gives traction on deformed solid.

First Piola-Kirchoff tensor operates to normal vectors on undeformed solid and gives traction on deformed solid. (This is also 3D generalization of "engineering stress" and unsymmetric in general.)

Second Piola-Kirchoff tensor operates to normal vectors on undeformed solid and gives the "pullback" of traction on undeformed solid. ("pullback" of traction = traction inverse mapped through deformation.)