Nonlinear finite elements/Stress measures

Stress measuresEdit

Three stress measures are widely used in continuum mechanics (particularly in the computational context). These are

  1. The Cauchy stress ( ) or true stress.
  2. The Nominal stress ( ) (which is the transpose of the first Piola-Kirchhoff stress ( ).
  3. The second Piola-Kirchhoff stress or PK2 stress ( ).

Consider the situation shown the following figure.

 
Quantities used in the definition of stress measures

The following definitions use the information 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.

Cauchy stressEdit

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.

Nominal stress/First Piola-Kirchhoff stressEdit

The nominal stress ( ) is the transpose of the first Piola-Kirchhoff 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.

2nd Piola Kirchhoff stressEdit

If we pull back   to the reference configuration, we have

 

or,

 

The PK2 stress ( ) is symmetric and is defined via the relation

 

Therefore,

 

Relations between Cauchy stress and nominal stressEdit

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.

Relations between nominal stress and second P-K stressEdit

Recall that

 

and

 

Therefore,

 

or (using the symmetry of  ),

 

In index notation,

 

Alternatively, we can write

 

Relations between Cauchy stress and second P-K stressEdit

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 push-forward and pull-back operations, we have

 

and

 

Therefore,   is the pull back of   by   and   is the push forward of  .