Thermoelastic interaction in a viscoelastic functionally graded half-space under three-phase-lag model

This paper aims at studying the thermo-viscoelastic interaction in a functionally graded, infinite, Kelvin–Voigt-type viscoelastic, thermally conducting medium due to the presence of periodically varying heat sources. Three-phase-lag thermoelastic model, Green–Naghdi model II (i.e. the model which predicts thermoelasticity without energy dissipation) and Green–Naghdi model III (i.e. the model which predicts thermoelasticity with energy dissipation) are employed to study thermomechanical coupling, thermal and mechanical relaxation effects. In the absence of mechanical relaxations (viscous effect), the results for various generalised theories of thermoelasticity may be obtained as particular cases. The governing equations are expressed in Laplace–Fourier double transform domain and are solved in that domain. The inversion of the Fourier transform is carried out using residual calculus, where the poles of the integrand are obtained numerically in the complex domain by using Laguerre’s method and the inversion of the Laplace transform is done numerically using a method based on the Fourier series expansion technique. The numerical estimates of the thermal displacement, temperature, stress and strain are obtained for a hypothetical material. A comparison of the results for different theories is presented and the effect of viscosity is also shown and the effect of non-homogeneity is also seen for different values of the non-homogeneity parameter.