
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:
Definitions.
Consider the situation shown in the following figure. The following definitions use the notations shown in the figure. 
In the reference configuration formula_7, the outward normal to a surface element formula_8 is formula_9 and the traction acting on that surface is formula_10 leading to a force vector formula_11. In the deformed configuration formula_12, the surface element changes to formula_13 with outward normal formula_14 and traction vector formula_15 leading to a force formula_16. Note that this surface can either be a hypothetical cut inside the body or an actual surface. The quantity formula_17 is the deformation gradient tensor, formula_18 is its determinant.
Cauchy stress.
The Cauchy stress (or true 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 formula_15 is the traction and formula_14 is the normal to the surface on which the traction acts.
Kirchhoff stress.
The quantity, 
is called the Kirchhoff stress tensor, with formula_18 the determinant of formula_17. It is used widely in numerical algorithms in metal plasticity (where there
is no change in volume during plastic deformation). It can be called "weighted Cauchy stress tensor" as well.
Piola–Kirchhoff stress.
Nominal stress/First Piola–Kirchhoff stress.
The nominal stress formula_26 is the transpose of the first Piola–Kirchhoff stress (PK1 stress, also called engineering stress) formula_3 and is defined via
or
This stress is unsymmetric and is a two-point tensor like the deformation gradient.<br>
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 formula_16 to the reference configuration, we have
or, 
The PK2 stress (formula_5) 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 formula_36. The Biot stress is defined as the symmetric part of the tensor formula_37 where formula_38 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 formula_6 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 formula_2 and formula_3 are (generally) not symmetric because formula_17 is (generally) not symmetric.
Relations between nominal stress and second P–K stress.
Recall that
and
Therefore,
or (using the symmetry of formula_5),
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 (and hence the Kirchhoff 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, formula_5 is the pull back of formula_1 by formula_17 and formula_1 is the push forward of formula_5.