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][2][3] Consider the situation shown in the following figure.

The following definitions use the notations shown in the figure.

, the outward normal to a surface element

and the traction acting on that surface (assuming it deforms like a generic vector belonging to the deformation) is

leading to a force vector

, 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.

is the deformation gradient tensor,

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 normal to the surface on which the traction acts.

The quantity, is called the Kirchhoff stress tensor, with

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.

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.

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.

) is symmetric and is defined via the relation Therefore, 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

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.

However, the unsymmetrized Biot stress has the interpretation From Nanson's formula relating areas in the reference and deformed configurations: Now, Hence, or, or, In index notation, Therefore, Note that

), 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 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,