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.