Variational formulation of thermomechanical problems

Abstract

This article proposes that a 4D space-time continuum is used for building variational thermomechanical continuum models. In order to identify physical constants in reversible processes, physically justified hypotheses were formulated. They are the hypotheses of complementary shear stress, classical dependence of momentum on velocity, and heat flow potentiality (generalized Maxwell–Cattaneo law). The Duhamel–Neumann law was assumed to be classical. In the considered model, the generalized Maxwell–Cattaneo and Duhamel–Neumann laws were not introduced phenomenologically. They were derived from the compatibility equations by excluding thermal potential from the constitutive equations for temperature, heat flow, and pressure. Dissipation channels were considered as the simplest non-integrable variational forms, which are linear in the variations of arguments. As a result, a variational principle that generalizes L.I. Sedov’s principle was developed. It is a consequence of the virtual work principle and termed as the difference between the variation of the Lagrangian of reversible thermomechanical processes and the algebraic sum of dissipation channels. It was proved that for the classical thermomechanical processes, with second-order differential equations, there can only exist six dissipation channels. Two of them determine dissipation in an uncoupled system – in the equations of motion and heat balance. The remaining four channels define coupling effects in coupled problems of dissipative thermomechanics.