On frame-indifference and form-invariance in constitutive theory

Summary

The purpose of this paper is to present an alternative formulation of the principles of Euclidean frame-indifference and indifference with respect to superimposed rigid body motions, i.e., rotations and translations. This is accomplished on the basis of the action of the proper Euclidean group on constitutive relationsinduced by the action of this group on the Euclidean tensors appearing in these relations. The resulting formulation of these concepts can then be used to show that the usual concept of material frame-indifference actually consists of twoindependent concepts, i.e., Euclidean frame-indifference andform-invariance, the latter being generally overlooked as an independent concept. On this basis, one can in addition show that any two of the concepts of Euclidean frame-indifference, form-invariance, and indifference with respect to superimposed rigid body motions, automatically implies the third. As an application of this formulation, we discuss the constitutive relations for a simple (elastic) material and a kinetic gas. In this context, it follows straighforwardly that the latter satisfy Euclidean frame-indifference, as shown by Murdoch [1], but not indifference with respect to superimposed rigid body motion, as shown by Müller [2]. As such, the current formation yields immediately that these are not form-invariant, and so not material frame-indifferent.