Alternative stress measures


In continuum mechanics, the most commonly used measure of stress is the Cauchy stress tensor, often called simply the stress tensor or "true stress". However, several alternative measures of stress can be defined:
  1. The Kirchhoff stress.
  2. The nominal stress.
  3. The Piola–Kirchhoff stress tensors
  4. #The first Piola–Kirchhoff stress. This stress tensor is the transpose of the nominal stress.
  5. #The second Piola–Kirchhoff stress or PK2 stress.
  6. The Biot stress

    Definitions

Consider the situation shown in the following figure. The following definitions use the notations shown in the figure.
In the reference configuration, the outward normal to a surface element is and the traction acting on that surface is leading to a force vector. In the deformed configuration, the surface element changes to with outward normal and traction vector leading to a force. Note that this surface can either be a hypothetical cut inside the body or an actual surface. The quantity is the deformation gradient tensor, is its determinant.

Cauchy stress

The Cauchy stress is a measure of the force acting on an element of area in the deformed configuration. This tensor is symmetric and is defined via
or
where is the traction and is the normal to the surface on which the traction acts.

Kirchhoff stress

The quantity,
is called the Kirchhoff stress tensor, with the determinant of. It is used widely in numerical algorithms in metal plasticity. It can be called weighted Cauchy stress tensor as well.

Piola–Kirchhoff stress

Nominal stress/First Piola–Kirchhoff stress

The nominal stress is the transpose of the first Piola–Kirchhoff stress and is defined via
or
This stress is unsymmetric and is a two-point tensor like the deformation gradient.
The asymmetry derives from the fact that, as a tensor, it has one index attached to the reference configuration and one to the deformed configuration.

Second Piola–Kirchhoff stress

If we pull back to the reference configuration we obtain the traction acting on that surface before the deformation assuming it behaves like a generic vector belonging to the deformation. In particular we have
or,
The PK2 stress is symmetric and is defined via the relation
Therefore,

Biot stress

The Biot stress is useful because it is energy conjugate to the right stretch tensor. The Biot stress is defined as the symmetric part of the tensor where is the rotation tensor obtained from a polar decomposition of the deformation gradient. Therefore, the Biot stress tensor is defined as
The Biot stress is also called the Jaumann stress.
The quantity does not have any physical interpretation. However, the unsymmetrized Biot stress has the interpretation

Relations

Relations between Cauchy stress and nominal stress

From Nanson's formula relating areas in the reference and deformed configurations:
Now,
Hence,
or,
or,
In index notation,
Therefore,
Note that and are not symmetric because is not symmetric.

Relations between nominal stress and second P–K stress

Recall that
and
Therefore,
or,
In index notation,
Alternatively, we can write

Relations between Cauchy stress and second P–K stress

Recall that
In terms of the 2nd PK stress, we have
Therefore,
In index notation,
Since the Cauchy stress is symmetric, the 2nd PK stress is also symmetric.
Alternatively, we can write
or,
Clearly, from definition of the push-forward and pull-back operations, we have
and
Therefore, is the pull back of by and is the push forward of.

Summary of conversion formula

Key:
Equation for