Thermo-Viscoelastic Interaction Subjected to Fractional Fourier law with Three-Phase-Lag Effects

In this paper, a new mathematical model of a Kelvin-Voigt type thermo-visco-elastic, infinite thermally conducting medium has been considered in the context of a new consideration of heat conduction having a non-local fractional order due to the presence of periodically varying heat sources. Three-phase-lag thermoelastic model, Green Naghdi models II and III (i.e., the models which predicts thermoelasticity without energy dissipation (TEWOED) and with energy dissipation (TEWED)) are employed to study the thermo-mechanical coupling, thermal and mechanical relaxation effects. In the absence of mechanical relaxations (viscous effect), the results for various generalized theories of thermoelasticity may be obtained as particular cases. The governing equations are expressed in Laplace-Fourier double transform domain. The inversion of the Fourier transform is carried out using residual calculus, where the poles of the integrand are obtained numerically in complex domain by using Laguerreʹs method and the inversion of the Laplace transform is done numerically using a method based on Fourier series expansion technique. Some comparisons have been shown in the form of the graphical representations to estimate the effect of the non-local fractional parameter and the effect of viscosity is also shown.