Piola-Kirchhoff stress tensor

In the case of finite deformations, the Piola-Kirchhoff stress tensors are used to express the Gustav Kirchhoff.

1^st Piola-Kirchhoff stress tensor

Whereas the Cauchy stress tensor, τ_ij, relates forces in the present configuration to areas in the present configuration, the 1^st Piola-Kirchhoff stress tensor, K_Lj relates forces in the present configuration with areas in the reference ("material") configuration. K_Lj is given by

where J is the Jacobian, and X_L,i is the inverse of the deformation gradient.

Because it relates different coordinate systems, the 1^st Piola-Kirchhoff stress is a two-point tensor. In general, it is not symmetric. The 1^st Piola-Kirchhoff stress is the 3D generalization of the 1D concept of engineering stress.

If the material rotates without a change in stress state (rigid rotation), the components of the 1^st Piola-Kirchhoff stress tensor will vary with material orientation.

The 1^st Piola-Kirchhoff stress is energy conjugate to the deformation gradient.

2^nd Piola-Kirchhoff stress tensor

Whereas the 1^st Piola-Kirchhoff stress relates forces in the current configuration to areas in the reference configuration, the 2^nd Piola-Kirchhoff stress tensor S_IJ relates forces in the reference configuration to areas in the reference configuration. 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 current configuration.

This tensor is symmetric.

If the material rotates without a change in stress state (rigid rotation), the components of the 2^nd Piola-Kirchhoff stress tensor will remain constant, irrespective of material orientation.

The 2^nd Piola-Kirchhoff stress tensor is energy conjugate to the Green-Lagrange strain.

References

Introduction to the mechanics of a continuum medium, L. E. Malvern, Prentice-Hall, Englewood Cliffs, NJ, 1969.