The most commonly used measure of stress is the Cauchy stress tensor, often called simply the stress tensor or "true stress". However, several other measures of stress can be defined. Some such stress measures that are widely used in continuum mechanics, particularly in the computational context, are:
Contents
- Definitions of stress measures
- Cauchy stress
- Kirchhoff stress
- Nominal stressFirst Piola Kirchhoff stress
- Second Piola Kirchhoff stress
- Biot stress
- Relations between Cauchy stress and nominal stress
- Relations between nominal stress and second P K stress
- Relations between Cauchy stress and second P K stress
- References
- The Kirchhoff stress (
τ ). - The Nominal stress (
N ). - The first Piola-Kirchhoff stress (
P ). This stress tensor is the transpose of the nominal stress (P = N T - The second Piola-Kirchhoff stress or PK2 stress (
S ). - The Biot stress (
T )
Definitions of stress measures
Consider the situation shown in the following figure. The following definitions use the notations shown in the figure.
In the reference configuration
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
Kirchhoff stress
The quantity
Nominal stress/First Piola-Kirchhoff stress
The nominal stress
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.
Second Piola-Kirchhoff stress
If we pull back
or,
The PK2 stress (
Therefore,
Biot stress
The Biot stress is useful because it is energy conjugate to the right stretch tensor
The Biot stress is also called the Jaumann stress.
The quantity
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
Relations between nominal stress and second P-K stress
Recall that
and
Therefore,
or (using the symmetry of
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,