Thermodynamic generalization of the theory of thermoelastic diffusion for a medium with changing density

The theory of thermoelastic diffusion describes the interaction between temperature and concentration fields in deformable solids. Most of the known works devoted to the theory of thermoelastic diffusion are constructed by analogy with the theory of elasticity, including generalizations to media with relaxation of heat and mass. In this case, even when taking into account the dependences of properties on temperature and composition, the authors use linear constitutive relations linking the state parameters and physical variables that follow from thermodynamics in the approximation of constant density of the medium. This paper presents a generalization of the basic equations of the theory of thermoelastic diffusion for a medium whose density depends on the main state variables. This is taken into account when deriving the constitutive relations. The result leads to the system of generalized state equations in differential form, the coefficient matrix of which for the variable density loses symmetry. The derivation of relations is based on both the use of the Helmholtz potential and the use of the Gibbs energy as a potential. New mechanisms of heat and mass transfer are discovered. For example, the transfer of a component under the action of a deformation gradient is possible due to two "mechanisms". The first of them is connected with the difference in individual properties of the components (their molar volumes, through which the coefficients of concentration expansion are calculated). The second mechanism of transfer can be called the work of stresses along the deformation gradients. Moreover, in the case of an isotropic body, not only the invariants of the stress and deformation tensors but also their shear components participate in the interaction of fields of different nature. The equations obtained in different ways are outwardly different. However, in any case, they contain all the discovered interaction mechanisms. As in classical theories, the equivalence of the equations can be shown using the standard apparatus of thermodynamics of irreversible processes. In limiting cases, the formulas coincide with those obtained earlier.