In the case of finite deformations, the Piola–Kirchhoff stress tensors (named for Gabrio Piola and Gustav Kirchhoff) express the stress relative to the reference configuration.
For infinitesimal deformations and rotations, the Cauchy and Piola–Kirchhoff tensors are identical.
Describing the stress, strain and deformation either in the reference or current configuration would make it easier to define constitutive models (for example, the Cauchy Stress tensor is variant to a pure rotation, while the deformation strain tensor is invariant; thus creating problems in defining a constitutive model that relates a varying tensor, in terms of an invariant one during pure rotation; as by definition constitutive models have to be invariant to pure rotations).
It defines a family of tensors, which describe the configuration of the body in either the current or the reference state.
The force in the reference configuration is obtained via a mapping that preserves the relative relationship between the force direction and the area normal in the reference configuration.
If the material rotates without a change in stress state (rigid rotation), the components of the second Piola–Kirchhoff stress tensor remain constant, irrespective of material orientation.