invariance and covariance in mixtures of simple bodies

Abstract

We adapt to mixtures the procedure of invariance of external power under the action of SO (3) to deduce balance equations. The two classical axioms about the growths of momentum and moment of momentum are derived with the help of a rule on the structure of the total power. We discuss also the covariance of the balance equations of each constituent and the nature of the constituent stress. Key words Mixtures, invariance1 Commentary to the theory of mixtures of simple bodies: the rôle of the invarance of the power. Preliminary remarks.Let B 0 be the regular (in the sense of "fit") region of the three-dimensional Euclidean point space E 3 occupied by the mixture in a reference place. We call part of B 0 any regular subset b of it and denote with X its generic point. The regular region B occupied by the mixture in the current place is obtained from B 0 through a sufficiently smooth orientation preserving mapping f such that f (B 0 ) = B; for any x ∈B we have then x =f (X). Motions are sufficiently smooth one-parameter mappings f t , with t ∈ [t 0 , t 1 ] the time. The velocity in the spatial description is denoted with v, while the acceleration with a = ∂ t v + (gradv) v.At the introduction of Lecture 5 of his "Rational Thermodynamics", Truesdell [1] describes the mixtures by assuming as a crucial point of view that all constituents occupy B simultaneously. Such an assumption is not

Objective fluxes in a multi-scale continuum description of sparse medium dynamics

“…An already quoted theorem in Ref. [21] assigns to the counterpart of the Reynolds tensor H, appearing in the classical theory of mixtures, kinetic character within a derivation of balance equations from SO (3) invariance of an appropriate expression of the external power for the generic constituent of the mixture. An analogous analysis would be surely helpful in the present setting to determine the exact character of H, but we do not go along with it here.…”

“…We consider K and H as entities with kinetic nature-see Ref [21]. for a theorem justifying such an attribution also to H in the case of the standard theory of mixtures.…”

Electrically polarized mixtures of porous bodies

“…Differently from them, we try to derive such equations. Recently, Mariano (2005b) suggested a powerful tool to derive balance equations within the framework of the theory of mixtures. Once postulated the form of the external power for each phase, via an axiom of invariance of the power for observers, he finds global and microstructural balance laws.…”

“…We note that the term c c 1 arises naturally applying the mass balance law while it should be postulated when the approach of de Groot and Mazur (1984) is followed or predicted if we use the method proposed in Mariano (2005b).…”

Galilean relativistic fluid mechanics

“…The principle and also its formulation initiated a long and unfinished discussion among those who are interested in the fundamental aspects of continuum physics. Here we give an incomplete list of the most important looking related works [17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43].…”