Advanced elasticity/Objectivity in kinematics

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Objectivity of kinematic quantities[edit]

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 vector[edit]

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 vector[edit]

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 tensors[edit]

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

Then

Therefore is an 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 tensor[edit]

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.