Irreversible thermodynamics of nonlocal systems

Abstract

Global laws of balance of momentum, moment of momentum and energy, together with local conservation of mass are reduced to point statements involving localization residuals. A fundamental functional inequality is then obtained which reduces to the Clausius-Duhem inequality for local theories. For nonlocal theories, the functional inequality states that the total internal production of heat of a material body at any given time is non-negative. This inequality is reformulated in terms of a functional inequality on a Hilbert space of ordered collections of L 2 functions. The general solution of this functional inequality is obtained and this leads to all admissible constitutive relations. The existence of dissipation potentials and symmetry relations are established for material bodies with admissible constitutive relations. Nonlocal analogues of Maxwell's reciprocity relations are also obtained as well as a proof of consistency with the results of thermostatics. Satisfaction of nonlinear forms of Onsager's reciprocity relations are shown to be equivalent to the requirement that the operators generating the admissible constitutive relations be potential operators. It is also shown the certain functionals of curves in a function space are odd under time reversal if and only if the nonlinear form of Onsager's reciprocity relations are satisfied. Thus, Gurtin's results [10] for processes which may be approximated by linear departures from equilibrium are extended to all processes with admissible constitutive relations. Similar results are established for significantly less restrictive sets of histories than those used by Gurtin and for a wide class of generalizations of the time reversal operator on such histories. Indications are given that satisfaction of invariance under superimposed rigid body motions implies satisfaction of the zero mean conditions for all localization residuals.