Material Transport of Sets and Fields

Abstract

We provide definitions for the “material transport”, or “comateriality”, of various mathematical objects of physical interest in continuum mechanics. Use is made of the concept of Lie dragging, i.e. of “pushing forward” and “pulling back” objects along path lines. Vectors in a Euclidean space can be comaterial in only two different senses, related to the behavior of their contravariant and covariant convected components; second-order tensors, regarded as linear transformations on a Euclidean vector space, can be comaterial in four different senses, related to their contravariant, covariant, and mixed representations; and, so on for tensors of higher order. We give necessary and sufficient conditions for comateriality. These lead immediately to other conditions that can be stated in terms of “comaterial rates”, which are Lie derivatives. Extensions and generalizations of the Helmholtz-Zorawski criterion are presented. The results are kinematical in nature and hold for all materials.