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Three stress measures are widely used in continuum mechanics (particularly in the computational context). These are
Consider the situation shown the following figure.

The following definitions use the information in the figure. In the reference configuration Ω_{0}, the outward normal to a surface element dΓ_{0} is and the traction acting on that surface is leading to a force vector . In the deformed configuration Ω, the surface element changes to dΓ 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 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 is the traction and is the normal to the surface on which the traction acts.
The nominal stress () is the transpose of the first PiolaKirchhoff stress (PK1 stress) () and is defined via
or
This stress is unsymmetric and is a two point tensor like the deformation gradient. This is because it relates the force in the deformed configuration to an oriented area vector in the reference configuration.
If we pull back to the reference configuration, we have
or,
The PK2 stress () is symmetric and is defined via the relation
Therefore,
Recall Nanson's formula relating areas in the reference and deformed configurations:
Now,
Hence,
or,
or,
In index notation,
Therefore,
The quantity is called the Kirchhoff stress tensor and is used widely in numerical algorithms in metal plasticity (where there is no change in volume during plastic deformation).
Note that and are not symmetric because is not symmetric.
Recall that
and
Therefore,
or (using the symmetry of ),
In index notation,
Alternatively, we can write
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 2n PK stress is also symmetric.
Alternatively, we can write
or,
Clearly, from definition of the pushforward and pullback operations, we have
and
Therefore, is the pull back of by and is the push forward of .
