Variational-Lagrangian thermodynamics of non-linear thermorheology

A principle of virtual dissipation generalizing. d'Alembert's principle to non-linear irreversible thermodynamics provides a unifying foundation which leads to an extremely general variational - Lagrangian analysis of dissipative phenomena. Thus a synthesis is achieved between thermodynamics and classical mechanics. The present paper applies this principle to the non-linear thermomechanics of continua with dissipation and heat conduction. Field equations, constitutive equations and Lagrangian equations with generalized coordinates are derived for non-linear thermoviscoelasticity, non-linear thermoelasticity and heat conduction, plasticity, and compressible heat conducting fluids with Newtonian and non-Newtonian viscosity. The thermodynamics of instability is also analyzed from the same fundamental viewpoint.