Abstract
This paper studies the transformation properties of the spatial balance of energy equation for a dissipative material, under the superposition of arbitrary spatial diffeomorphisms. The study reveals that for a dissipative material the transformed energy balance equation has some non-standard terms in it. These terms are related to a system of microforces with its own balance equation. These microforces act during the superposition of the spatial diffeomorphism, because of the dissipative properties of the material. Moreover, it is shown that for the case in question the stress tensor is additively decomposed into a conventional part given by the standard Doyle-Ericksen formula and a non-conventional one which is related to changes in the material internal structure in the course of deformation. On the basis of the second law of thermodynamics and the integrability condition of a Pfaffian form it is shown that the non-conventional part of the stress tensor can be related not only to dissipative but also to conservative response. A further insight to this conservative response is provided by exploiting the invariance properties of the balance of energy equation within the context of the material intrinsic “physical” metric concept. In this case, it is shown that the assumption of spatial covariance yields the standard conservation and balance laws of classical mechanics but it does not yield the standard Doyle-Ericksen formula. In fact, the Doyle-Ericksen formula has an additional term in it, which is related directly to the evolution of the material internal structure, as it is determined by the (time) evolution of the material metric in the spatial configuration. A formal connection between this term and the Eshelby energy-momentum tensor is derived as well.
Highlights
Invariance principles play a fundamental role in several branches of mechanics and physics
The basic thrust of this paper resides in the determination of the transformation properties of the spatial balance of energy equation under the superposition of arbitrary spatial diffeomorphisms and the consequent repercussions of the second law of thermodynamics, as it is expressed by the local form of the ClausiusDuhem inequality, in the response of continua with internal structure
In particular in this paper: (a) We have derived the transformation formula of the spatial balance of energy equation by considering a microforce system b 0 and t 0, which acts during the spatial diffeomorphism by producing work
Summary
Invariance principles play a fundamental role in several branches of mechanics and physics. “Any theory (relativistic or nonrelativistic) that purports to be fundamental ought to be generalizable so that the underlying physical space is a manifold and not just Euclidean (or Newtonian) space.”, generalized the work of Green and Rivlin by replacing the (rigid) Euclidean ambient space by a Riemannian one These authors by postulating the covariance of the global energy balance equation, that is, its invariance under arbitrary superimposed spatial diffeomorphisms, derived the conservation and balance laws, and the mechanical state equation in the form of the. By following the suggestions given in these works, it is assumed that the internal structure of the material may be modeled macroscopically by the material intrinsic metric “as seen” in the spatial configuration, while its changes may by modeled by the time evolution of this metric In this case it is shown that the assumption of spatial covariance yields the standard conservation and balance laws of classical mechanics but does not yield the standard Doyle-Ericksen formula. A formal connection between this term and the Eshelby energy-momentum stress tensor “as seen” in the spatial configuration is derived as well
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