Nonlinear finite elements/Objectivity of constitutive relations

Material frame indifferenceEdit

An important consider in nonlinear finite element analysis is material frame indifference or objectivity of the material response. The idea is that the position of the observer frame should not affect the constitutive relations of a material. You can find more details and a history of the idea in Truesdell and Noll (1992) - sections 17, 18, 19, and 19A. [1]

We have already talked about the objectivity of kinematic quantities and stress rates. Let us now discuss the same ideas with a particular constitutive model in mind.

Hyperelastic materialsEdit

A detailed description of thermoelastic materials can be found in Continuum mechanics/Thermoelasticity. In this discussion we will avoid the complications induced by including the temperature.

In the material configuration, a hyperelastic material satisfies two requirements:

  1. a stored energy function ( ) exists for the material.
  2. the stored energy function depends locally only on the deformation gradient.

Given these requirements, if   is the nominal stress (  is the first Piola-Kirchhoff stress tensor), then

 

ObjectivityEdit

The stored energy function   is said to be objective or frame indifferent if

 

where   is an orthogonal tensor with  .

This objectivity condition can be achieved only if (in the material configuration)

 

since  .

We can show that

Constitutive relations for hyperelastic materials

 


Proof:

The stress strain relation for a hyperelastic material is

 

The chain rule then implies that

 

for any second order tensor  .

Now, using the product rule of differentiation,

 

or,

 

where   is the fourth order identity tensor. Therefore,

 

Using the identity

 

we have

 

Therefore, invoking the arbitrariness of  , we have

 

Since   we have

 

which implies that

 

Recall the relations between the 2nd Piola-Kirchhoff stress tensor and the first Piola-Kirchhoff stress tensor (and the nominal stress tensor)

 

Therefore, we have

 

Also from the relation between the Cauchy stress and the 2nd Piola-Kirchhoff stress tensor

 

we have

 

We may also express these relations in terms of the Lagrangian Green strain

 

Then we have

 

Hence, we can write

 

The stored energy function   is objective if and only if the Cauchy stress tensor is symmetric, i.e., if the balance of angular momentum holds. Show this.

  1. C. Truesdell and W. Noll, 1992, The Nonlinear Field Theories of Mechanics:2nd ed., Springer-Verlag, Berlin