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# Piola-Kirchhoff stress tensor

In the case of finite deformations, the Piola-Kirchhoff stress tensors are used to express the stress relative to the reference configuration. This is in contrast to the Cauchy stress tensor which expresses the stress relative to the present configuration. For infinitesimal deformations or rotations, the Cauchy and Piola-Kirchoff tensors are identical. These tensors take their names from Gabrio Piola and Gustav Kirchhoff.

## 1st Piola-Kirchhoff stress tensor

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

$K_{Lj}=J X_{L,i} \tau_{ij} \!$

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

Because it relates different coordinate systems, the 1st Piola-Kirchhoff stress is a two-point tensor. In general, it is not symmetric. The 1st 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 1st Piola-Kirchhoff stress tensor will vary with material orientation.

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

## 2nd Piola-Kirchhoff stress tensor

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

$S_{IJ}=J X_{I,j} \tau_{ij} X_{J,i} \!$

This tensor is symmetric.

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

The 2nd 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.