MATERIAL TRANSPORT OF SETS AND FIELDS
J. Casey and P. Papadopoulos Math. Mech. Sol., 7, pp. 647-676, (2002)

Abstract
Definitions are given 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'' of 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. Necessary and sufficient conditions for comateriality are given. 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.


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