Entropy Identity and Material-Independent Equilibrium Conditions in Relativistic Thermodynamics

On the basis of the balance equations for energy-momentum, spin, particle and entropy density, an approach is considered which represents a comparatively general framework for special- and general-relativistic continuum thermodynamics. In the first part of the paper, a general entropy density 4-vector, containing particle, energy-momentum, and spin density contributions, is introduced which makes it possible, firstly, to judge special assumptions for the entropy density 4-vector made by other authors with respect to their generality and validity and, secondly, to determine entropy supply and entropy production. Using this entropy density 4-vector, in the second part, material-independent equilibrium conditions are discussed. While in literature, at least if one works in the theory of irreversible thermodynamics assuming a Riemann space-time structure, generally thermodynamic equilibrium is determined by introducing a variety of conditions by hand, the present approach proceeds as follows: For a comparatively wide class of space-time geometries the necessary equilibrium conditions of vanishing entropy supply and entropy production are exploited and, afterwards, supplementary conditions are assumed which are motivated by the requirement that thermodynamic equilibrium quantities have to be determined uniquely.Comment: Research Paper, 30 page

A unified thermodynamic theory for large deformations in elastic media and Kelvin (Voigt) media, and for viscous fluid flow

A theory is developed for large deformations in elastic media and in Kelvin (Voigt) media, and for viscous fluid flow, which is based on the thermodynamics of irreversible processes in continuous media. The results of the theory are formulated for the two cases when Lagrangian and when Eulerian coordinates are used. The basic assumption is that the entropy may be considered as a function of the internal energy and of the Lagrangian components of the strain tensor. According to the usual procedure of the thermodynamics of irreversible processes the expression for the entropy production, which is due to viscous flow and heat conduction, is derived. It is shown that if Lagrangian coordinates are employed, the tensor conjugate to the viscous pressure tensor in the expression for the entropy production may be written either as the substantial derivative with respect to time of the strain tensor or as the symmetric part of the covariant derivative of the velocity field. In case Eulerian coordinates are used, this tensor may be written either as the convected time flux of the strain tensor or as the symmetric part of the gradient of the velocity field. The phenomenological equations are formulated and the Onsager-Casimir reciprocity relations are discussed. Both distortional and volumetric phenomena are considered. Temperature effects are fully taken into account. A rheological equation for large deformations in anisotropic Kelvin media is derived. The case is discussed in which initially isotropic substances become anisotropic with respect to the irreversible processes due to straining. For this case rather general expressions, satisfying the Onsager relations, are given for the phenomenological tensors of heat conduction and viscous flow and the forms of the thermodynamic functions and of the equations of state are discussed. The special case in which an initially isotropic medium (for example a fluid) does not become anisotropic as a consequence of deformations is also considered. If no viscous phenomena occur the results of the theory are analogous to those obtained by Green and Adkins from their thermodynamic considerations of large thermoelastic deformations. In case the medium is non-elastic with respect to shear our equations reduce to those for ordinary (Newtonian) fluids with shear and volume viscosity

A thermodynamic derivation of the stress-strain relations for Burgers media and related substances

A generalization is given of the author's thermodynamic theory for mechanical phenomena in continuous media. The developments are based on the general methods of non-equilibrium thermodynamics. Temperature effects are fully taken into account. It is assumed that several microscopic phenomena occur which give rise to inelastic strains (for instance, slip, dislocations, etc.). The contributions of these phenomena to the inelastic strain tensor are introduced as internal degrees of freedom in the Gibbs relation. Moreover, it is assumed that a viscous flow phenomenon occurs which is analogous to the viscous flow of ordinary fluids. An explicit form for the entropy production is derived. The phenomenological equations (Fourier's law and generalizations of Lévy's law and of Newton's law for viscous fluid flow) are given, and the Onsager-Casimir reciprocity relations are formulated. It follows from the theory that several types of (macroscopic) stress fields may occur in a medium: A stress field teqaß which is of a thermoelastic nature, a stress fields tviaß which is analogous to the viscous stresses in ordinary fluids, and stress fields tkmaß which are probably connected with the microscopic stress fields surrounding imperfections in the medium. The stress field teqaß + tviaß is the mechanical stress field which occurs in the equations of motion and in the first law of thermodynamics, and stress fields of the type teqaß + tkmaß play the role of thermodynamic affinities in the phenomenological equations which are generalizations of Lévy's law. If the equations of state may be linearized (for example, Hooke's law and the Duhamel-Neumann law), and if the phenomenological coefficients may be regarded as constants, an explicit form for the stress-strain relation may be derived. In this case the relation for distortional phenomena in isotropic media has the form of a linear relation among the deviators of the mechanical stress tensor, the first n derivatives with respect to time of this tensor, the tensor of total strain (the sun of the elastic and inelastic strains), and the first n + 1 derivatives with respect to time of the tensor of total strain, where n is the number of phenomena that give rise to inelastic deformations. The well-known Burgers equation is a special case of this relation if n = 2. Moreover, the stress-strain relations for ordinary viscous fluids, for thermoelastic media, and for Maxwell, Kelvin, Jeffreys, and Poynting-Thomson media are also special cases of the more general relation mentioned above. In case the equations of state may be linearized explicit expressions are given for the free energy, the internal energy, and the entropy, both for isotropic and anisotropic media. If it is not permissible to linearize the equations of state and/or to regard the phenomenological coefficients as constants, the stress-strain relation is of a very complicated nature. Plasticity phenomena are left out of consideration

On vectorial internal variables and dielectric and magnetic relaxation phenomena

In a previous paper it has been shown by the author that a vectorial internal variable may give rise to dielectric relaxation phenomena and that if such a variable occurs the polarization P may be written in the form P = P(0) + P(1), where changes in P(0) are reversible processes and changes in P(1) are irreversible. In this paper we introduce a somewhat more general assumption concerning the entropy. This generalization leads to the possibility that both changes in P(0) and in P(1) are irreversible phenomena. In this way a formalism is obtained with two relaxation times for dielectric relaxation. In particular we investigate the linearized form of the theory. It is seen that in the linear case the relation between the electric field E and the polarization P has the form of a linear relation among E, P, the first derivatives with respect to time of E and P, and the second derivative with respect to time of P. Debye's equation for dielectric relaxation in polar liquids and the equation derived by De Groot and Mazur are special cases of the equation which has been obtained in this paper. Analogous results can be derived for magnetic relaxation phenomena. Snoek's equation and the equation obtained by De Groot and Mazur are special cases of the equation for magnetic relaxation which is derived in this paper