On Nonequilibrium Thermodynamics of Viscoelasticity and Viscoplasticity

Abstract

The present paper considers macroscopic nonequilibrium thermodynamics of viscoelastic and viscoplastic materials where the nonlinear hereditary effects are accounted for by means of a set of internal variables. On the basis of: the energy balance equation, the assumption that a measurable empirical temperature exists at nonequilibrium states, and the fact that not all observable states in any given neighbourhood of a given state are adiabatically attainable, it is shown that, under some mild smoothness conditions

• (1) An empirical entropy exists at nonequilibrium states.

• (2) This entropy is extensive.

• (3) There exists an absolute temperature at nonequilibrium states, which depends on the empirical temperature only.

• (4) The entropy function serves as a potential in exactly the same manner as in the thermodynamics of equilibrium.

• (5) The entropy is not unique, inasmuch as it depends parametrically on the internal variables in a manner not deducible from the stated assumptions.

• (6) For materials with instantaneous elasticity, and certain other materials, the entropy function is monotone, say, nondecreasing, for all adiabatic transitions (which include transitions at constant strains and internal energy), so that the set of all entropy functions satisfies the Clausius—Planck inequality (note that this result is derived rather than assumed).

• (7) Because of (6), the set of entropy functions is convex and admits upper and lower bounds which can serve as entropy functions, thus rendering entropy unique.

• (8) Both of these bounds can serve as potentials in exactly the same manner as in (4), and hence these bounds can differ from entropy itself by additive functions of the internal variables only.

The final part of the chapter examines a generalisation of the classical approach in which an explicit dissipative mechanism is assumed for a given class of materials.