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

Abstract

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 τ eq αβ which is of a thermoelastic nature, a stress fields τ vi αβ which is analogous to the viscous stresses in ordinary fluids, and stress fields τ k mαβ which are probably connected with the microscopic stress fields surrounding imperfections in the medium. The stress field τ eq αβ + τ vi αβ 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 τ eq αβ + τ k mαβ 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.