On the Representation of Energy and Momentum in Elasticity

TL;DR

This paper develops a generalized conservation framework for energy and momentum in finite elasticity, clarifying their forms and showing conditions under which classical specifications apply, with implications for hyperbolic system modeling.

Contribution

It introduces a generalized conservation format for energy and momentum in finite elasticity, relaxing prior assumptions and deriving conditions for classical representations.

Findings

Total energy separates into internal and kinetic parts.

Momentum is linear in velocity.

Classical energy and momentum suffice under strong ellipticity.

Abstract

In order to clarify common assumptions on the form of energy and momentum in elasticity, a generalized conservation format is proposed for finite elasticity, in which total energy and momentum are not specified a priori. Velocity, stress, and total energy are assumed to depend constitutively on deformation gradient and momentum in a manner restricted by a dissipation principle and certain mild invariance requirements. Under these assumptions, representations are obtained for energy and momentum, demonstrating that (i) the total energy splits into separate internal and kinetic contributions, and (ii) the momentum is linear in the velocity. It is further shown that, if the stress response is strongly elliptic, the classical specifications for kinetic energy and momentum are sufficient to give elasticity the standard format of a quasilinear hyperbolic system.