Time-nonlocal six-phase-lag generalized theory of thermoelastic diffusion

Abstract

Lagging behavior is also expected in the process of mass diffusion similar to the process of heat transport during the ultra fast transient (D. Y. Tzou, Macro to microscale heat transfer: The lagging behavior. John Wiley and Sons, Ltd., 2015). We therefore propose a novel multi-field coupled mathematical model for a linear, isotropic and homogeneous thermoelastic diffusive continuum by taking into account the modified Fourier’s law of heat conduction, the modified Fick’s law of mass diffusion and extending the concept of phase lag from heat conduction to mass diffusion. The modified Fourier’s law includes temperature gradient and thermal displacement gradient among the constitutive variables with three different phase lags for the heat flux vector, the temperature gradient and the thermal displacement gradient. The modified Fick’s law includes chemical potential gradient and the chemical potential displacement gradient among the constitutive variables with three different phase lags for the mass flux vector, the chemical potential gradient and the chemical potential displacement gradient. The Fourier’s law of heat conduction is replaced by a fractional order approximation of the modified Fourier’s law whereas the Fick’s law of mass diffusion is replaced by a fractional order approximation of the modified Fick’s law. The proposed model includes some of the existing models in the field of thermoelastic diffusion as special cases. A variational principle is obtained and uniqueness of solution of the governing field equations under suitable conditions is proved. Finally, a reciprocity theorem is established for the proposed generalized thermoelastic diffusion model.