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