On thermodynamics and the nature of the second law

Abstract The contents of this paper represent a new approach to continuum thermo­dynamics and are chiefly concerned with (a) a procedure for obtaining restrictions on constitutive equations, (b) an appropriate mathematical statement of the second law and (c) the nature of restrictions placed by the latter on thermo-mechanical behaviour of single phase continua. Our point of departure is the introduction of a balance of entropy and the use of the energy equation as an identity for all motions and all temperature distributions after the elimination of the external fields. This is in contrast to the approach adopted in most of the current literature on continuum ther­modynamics based on the use of the Clausius-Duhem inequality. In order to gain some insight into the nature of our procedure we first study the case of an elastic material, which includes that of an ideal fluid as a special case, before the consideration of the second law. We then go on to postu­late an inequality which reflects the fact that for every process associated with a dissipative material, a part of the mechanical work is always con­verted into heat and this cannot be withdrawn from the medium as mech­anical work. The restriction on the heat conduction vector is considered separately and is confined to equilibrium cases in which heat flow is steady. A restriction is also obtained for the internal energy when the body is in mechanical equilibrium subjected to spatially homogeneous temperature fields. Using the above approach, next we study the nature of thermodynamic restrictions on the thermo-mechanical response of a viscous fluid and simple materials with fading memory. A drawback to the Clausius-Duhem inequality is discussed by means of an example. For a class of rigid heat conductors in thermal equilibrium, the Clausius-Duhem inequality requires that if heat is added to the medium, the resulting spatially homogeneous temperature of the conductor decreases. Moreover, the in­-equality denies the possibility of propagation of heat in the conductor as a thermal wave with finite speed. The inequalities proposed in this paper do not suffer from these shortcomings.