Nonlinear finite elements/Kinematics - objectivity

Objectivity of kinematic quantitiesEdit

Objectivity is one of the fundamental concepts of continuum mechanics. Objectivity is another name for frame indifference, i.e., the position of an observer should not affect any quantities of interest. The concept is an extension of the idea that rigid body motions should not affect the stress and strain tensors or the mechanical properties of a material.

A spatial strain tensor is said to transform objectively under rigid body motion if it transforms according to the transformation rules for second order tensors.

Example of an objective vectorEdit

Let us look at an example in the context of kinematics. Consider a vector   in the reference configuration that becomes   in the deformed configuration. Let us then rotate the vector by an orthogonal tensor   so that its rotated form is  .

Then,

 

The length of the vector   in terms of   is given by

 

Similarly,

 

Therefore, the vector   is objective under rigid body rotation.

Example of a non-objective vectorEdit

Let   and  . Then

 

If we compute the length of   we get

 

So the length of the velocity vector   changes if the rate of change of rotation is arbitrary (i.e., not constant or zero). Also the spatial velocity vector does not follow the standard vector transformation rule under rigid body rotation.

Therefore the spatial velocity vector violates objectivity.

Examples of objective 2nd-order tensorsEdit

Let now consider the effect of a rotation on the right Cauchy-Green deformation tensor  .

 

Then

 

Therefore   is an non-objective quantity. Similarly we can show that the Lagrangian Green strain tensor   is objective.

For the spatial deformation tensor   and the spatial strain tensor   we can show that

 

These tensors are objective because they follow the standard rules for tensor transformations under rigid body rotations.

The spatial rate of deformation tensor   transforms at

 

or,

 

or,

 

where  

Therefore   is an objective quantity.

Example of a non-objective 2nd-order tensorEdit

Let us now look at the velocity gradient tensor by itself. In that case

 

or,

 

Clearly the first term above is not zero for arbitrary rotations. Hence the spatial velocity gradient tensor is not objective.