The thermoelastic interaction for the three-phase-lag (TPL) heat equation in an isotropic infinite elastic body with a spherical cavity is studied by two-temperature generalized thermoelasticity theory (2TT). The heat conduction equation in the theory of TPL is a hyperbolic partial differential equation with a fourth-order derivative with respect to time. The medium is assumed to be initially quiescent. By the Laplace transformation, the fundamental equations are expressed in the form of a vector-matrix differential equation, which is solved by a state-space approach. The general solution obtained is applied to a specific problem, when the boundary of the cavity is subjected to the thermal loading (the thermal shock and the ramp-type heating) and the mechanical loading. The inversion of the Laplace transform is carried out by the Fourier series expansion techniques. The numerical values of the physical quantity are computed for the copper like material. Significant dissimilarities between two models (the two-temperature Green-Naghdi theory with energy dissipation (2TGN-III) and two-temperature TPL model (2T3phase)) are shown graphically. The effects of two-temperature and ramping parameters are also studied.

This paper is concerned with the determination of the thermoelastic displacement, stress, conductive temperature, and thermodynamic temperature in an infinite isotropic elastic body with a spherical cavity in the context of the two-temperature generalized thermoelasticity theory (2TT). The two-temperature Lord-Shulman (2TLS) model and two-temperature Green-Naghdi (2TGN) models of thermoelasticity are combined into a unified formulation introducing the unified parameters. The medium is assumed initially quiescent. The basic equations have been written in the form of a vector-matrix differential equation in the Laplace transform domain which is then solved by (a) the state-space approach and (b) the eigenvalue approach for any set of boundary conditions. The general solution obtained is applied to a specific problem when the boundary of the cavity is subjected to thermomechanical loading. The numerical inversion of the transform is carried out using Fourier-series expansion techniques. The computed results for thermoelastic stresses, conductive temperature, and thermodynamic temperature are shown graphically for the Lord Shulman model and for two models of Green-Naghdi and the effects of two temperatures are discussed. A comparative study of the two methods has also been carried out.

This paper deals with the problem of thermoelastic interactions in a functionally graded isotropic unbounded medium due to the presence of periodically varying heat sources in the context of the three-phase-lag thermoelastic models, GN ii (TEWOED) and GN iii (TEWED). The governing equations of three-phase-lag thermoelastic model (3P), generalized thermoelasticity without energy dissipation (GN ii) and with energy dissipation (GN iii) for a functionally graded material (i.e., a material with spatially varying material properties) are established. The governing equations are expressed in a Laplace–Fourier double transform domain and solved in that domain. Then the inversion of the Fourier transform is carried out by using residual calculus, where poles of the integrand are obtained numerically in a complex domain by using Laguerre's method and the inversion of the Laplace transform is done numerically using a method based on Fourier series expansion technique. The numerical estimates of the thermal displacement, temperature and thermal stress are obtained for a hypothetical material. The solution to the analogous problem for homogeneous isotropic material is obtained by taking a suitable non-homogeneity parameter. Finally, the results obtained are presented graphically to show the effect of non-homogeneity on thermal displacement, temperature and thermal stress. A comparison of the results for different theories (three-phase-lag model, GN ii and GN iii) is presented and the effect of non-homogeneity is also shown. In absence of non-homogeneity the results corresponding to the 3P model, GN ii and GN iii model agree with the results of the existing literature.

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