A certain counterpart in dissipative setting of the Noether theorem with no dissipation pseudo-potentials

Abstract

We look at a mechanical dissipation inequality differing from the standard one by what we call a relative power, a notion that is appropriate in the presence of material mutations. We prove that a requirement of structural invariance for such an inequality under the action of diffeomorphism-based changes of observers (covariance) implies (i) the representation of contact actions in terms of the first Piola–Kirchhoff stress, (ii) local balances of standard and configurational actions, (iii) a priori constitutive restrictions in terms of free energy, and (iv) a representation of viscous-type stress components. This article is part of the theme issue ‘Foundational issues, analysis and geometry in continuum mechanics’.

Proof of Straughan’s claim on Payne–Song’s and modified Guyer–Krumhansl’s equations

“…Remark 4.1. When ν is a scalar (or a pseudo-scalar), an appropriate procedure to derive the pertinent balance of microstructural interactions, instead of postulating it (in this specific case, a first postulate of such a scalar balance seems to have been suggested by Nunziato & Cowin [51]), is based on a requirement of invariance in structure for the second law, written in terms of Clausius-Duhem's inequality, under general diffeomorphism-based changes of observer, those described in Remark (2.1); this covariance principle for the second law of thermodynamics [42] requires, in short, that if an observer records a process as a dissipative one, any other observer related with it by diffeomorphisms must record the same dissipative character. To obtain a scalar balance of microstructural interactions, without postulating it, we could also adopt covariance of the first principle of thermodynamics (balance of energy), asking its covariance with respect to general diffeomorphism-based changes of observers, and adapting to the present case Marsden-Hughes' theorem [25]; however, we would not be able to obtain dissipative (macro and micro) stress components.…”

“…To obtain a scalar balance of microstructural interactions, without postulating it, we could also adopt covariance of the first principle of thermodynamics (balance of energy), asking its covariance with respect to general diffeomorphism-based changes of observers, and adapting to the present case Marsden-Hughes' theorem [25]; however, we would not be able to obtain dissipative (macro and micro) stress components. For them we need to refer to the covariance principle for the second law [42].…”

“…An approach based on measures over manifolds of maps has been proposed by Segev [41]. Eventually, it is possible to show that requiring invariance of the second law under general diffeomorphism-based changes of observers allows one to derive the representation of contact interactions and the balance laws, in addition to constitutive restriction (proof is in [42]). non-vanishing volume, under rigid-body-type changes of observer in the physical space and their effects on the space containing ν.…”

Foundational issues, analysis and geometry in continuum mechanics: introduction

“…Mariano [ 68 ] discusses the role of the second law of thermodynamics, written in terms of a mechanical dissipation inequality involving a power of actions that accounts for time-varying reference configurations. At variance of the traditional use, he shows that the representation of contact actions and the local balance can be deduced from the second law in addition to constitutive restrictions once we accept a covariance principle stating that if a given observer evaluates a process to be dissipative, any other observer should make the same evaluation.…”