Miroslav Šilhavý

The Dynamic Response

For the nonequilibrium response the second law takes the form of the internal dissipation inequality (9.2.4) and the combination of the frame indifference with the symmetry leads to explicit representations of the nonequilibrium response functions. As these depend on more vectorial and tensorial variables (namely on F,θ D g or on F θ F G ), the representation theorems are necessarily more complicated. Since there are essentially no new ideas in them, and since some parts of the development are closely parallel to those concerning the equilibrium response, the material will be presented only briefly. The rest of the chapter deals with the nonequilibrium response from a broader point of view. First, the classical linear irreversible thermodynamics, based on Onsager’s relations, will be briefly described in view of its obvious historical importance. Then a possible nonlinear generalization of the Onsager relations, namely, the assumption of the existence of the convex dissipation potential, will be presented. Finally, the relaxation models including those studied by the extended linear irreversible thermodynamics will be examined from the point of view of the thermodynamics based on the Clausius—Duhem inequality.

Dynamical Thermoelastic and Adiabatic Theories

This part deals with aspects of behavior of thermoelastic bodies with heat conduction and viscosity in dynamical situations. The governing equations are the balance equations of Sects. 3.4–3.7 combined with the constitutive equations of Sect. 9. I; moreover, the constitutive equations must be compatible with the Clausius—Duhem inequality for smooth processes, see Sect. 9.2. In their full generality, the constitutive equations allow for a dissipation by the heat conduction and viscosity, but they also include as special cases the thermoelastic materials 9.1.3(1), where the heat conduction is present and the viscosity is absent, and the idealized dissipationless (‘adiabatic’) materials 9.1.3(2), where both the heat conduction and viscosity are absent. The behavior and interpretation of solutions strongly depend on whether the equations describe a material in which the viscosity and heat conduction are really present or whether they describe one of the two subcases mentioned above. Moreover, there are also situations when one can pass to isothermal or isentropic dynamic theories, thus omitting formally the thermal phenomena altogether. The latter two theories are often referred to by the common name ‘elastodynamics’. Their basic equations are formally identical, but for a given material, the isothermal elastodynamics uses the stress expressed as a function of the deformation gradient at a given fixed temperature, whereas the isentropic dynamics uses the stress expressed as a function of the deformation gradient at a fixed entropy.

Direct Methods in Equilibrium Theory

This chapter treats the equilibrium states by the direct methods of the calculus of variations. Given a class of admissible rest states Σ 0 satisfying various constraints (such as the boundary conditions), one seeks a state σ 0 for which the total canonical free energy P takes the least possible value on Σ 0. This state need not exist, depending on the free energy (bar y ) and on the class Σ 0. However, if the total energy is bounded from below on Σ 0, then P0 := inf {P(σ): σ ∈ Σ 0} is finite and one can always find a sequence σ k ∈ Σ 0 such that P(σ k ) → P0 as k → ∞, the minimizing sequence. Under the condition of coercivity, one can find a subsequence, still denoted by σ k , which converges weakly to some state σ 0 and we assume that σ 0 ∈ Σ 0. (To guarantee this inclusion, one has to admit states with a lower degree of smoothness than, say, continuous differentiability, see Sect. 21.2.) Since σ k converges weakly to σ 0 and P(σ 0) approaches P0, a natural question is whether P0 = P(σ 0). This will be the case if P is sequentially weakly lower semicontinuous (swlsc), i.e., if for every sequence σ k in Σ 0 converging weakly to some σ ∈ Σ 0, we have n n

The Equilibrium Response of Isotropic Bodies

mathop {lim inf {rm P}}limits_{k to infty } left( {{sigma _k}} right) geqslant {rm P}{left( sigma right)_ cdot }

Elements of Tensor Algebra and Analysis

n n.

The Principle of Material Frame Indifference

This chapter discusses elastic waves: Sect. 23.3 the plane, surface and acceleration waves, Sect. 23.5 the centered waves and Sects. 23.6–23.8 the shock waves. The exposition starts with the characteristic equation, which is important for all kinds of waves. Attention is concentrated especially on shock waves; here thermodynamics provides the entropy admissibility criterion. Some other proposed admissibility criteria are briefly reviewed and their relationship to the entropy criterion is established under additional assumptions. The Riemann problem is formulated and solved for small data in the strictly hyperbolic, genuinely nonlinear case. The nonuniqueness of solutions is explained.

A Local Approach to the Equilibrium of Solids

There are two classes of isotropic bodies: isotropic solids and fluids. In the thermodynamics of isotropic solids, there is a new issue in comparison with the isothermal case: the natural (unstressed) state depends on temperature. This calls for a treatment in which the stressed isotropic states are given the same status as the unstressed ones, and it is given below. For instance, Lame’s moduli are defined for general (possibly stressed) isotropic states. A compatibility relation for them is derived, and a free energy function is constructed fitting given (‘experimentally determined’) dependencies of the Lame moduli on temperature and specific volume.

The Equilibrium Response

The second law specifies conditions under which a cyclically operating body can do positive net work. The conditions are stated in terms of the exchange of heat of the body with its environment. In this respect the second law is similar to the first. The difference is that while for the formulation of the first law the single net gain of heat of the body suffices, the second law requires a much finer concept which distinguishes amounts of heat absorbed or emitted at different temperatures. The concepts of heating measure and accumulation function, to be introduced in Sect. 7.1, serve this purpose. It will be seen that the framework of heating measures enables one to remove the traditional ambiguities in the statements of the second law.