Introduction
During the past five decades, non-conventional thermodynamic theories have been formulated that include heat transfer equations of a hyperbolic-type permitting limited speeds for thermal waves. According to these theories, thermal propagation should be seen as a wave phenomenon rather than a propagation phenomenon. Different methods for obtaining wave-type heat conduction equations have been applied by different researchers. Lord and Shulman (LS) [1] introduced the generalized thermoelasticity theory by introducing one relaxation time in Fourier’s law of the heat conduction and, thus, converting the heat conduction equation into a hyperbolic type. The uniqueness of this theory proved under different conditions in [2–5]. Green and Lindsay (GL) [6] presented another theory, called a temperature-rate-dependent, involving two times of relaxation. In this model, Fourier's heat conduction is left unchanged, but the classical energy equation and the stress-strain-temperature relations are modified. These models have been applied to many problems in the field of thermoelasticity as stated in [7-14].
Green and Naghdi [15-17] established three new thermoelasticity models of a homogeneous isotropic material, which are labelled as GN models I, II, and III. If the respective models are linearized, the model I reduces to the classical heat conduction theory. The Linear model II predicts the limited speed of heat propagation including non-dissipation of energy. The third model III indicates the propagation of thermal waves with finite speed and allows energy dissipation. Problems involving to the generalized thermoelasticity theories relaxation times (LS and GL) and GN models with and without energy dissipation have been considered in [18–24].
The following generalization of the thermoelasticity theory is known as the dual-phase-lag model (DPL) established by Tzou [25, 26]. Chandrasekharaiah [27] extended the DPL model of heat conduction to a generalized thermoelasticity theory. Tzou [25] introduced two-phase lags to both the heat flux vector and the temperature gradient and considered a constitutive equation to describe the logging behavior in the heat conduction in solids. The phase-lag of the heat flux vector is interpreted as the time of relaxation for the rapid transient effects of the thermal inertia. The other phase-lag of the temperature gradient is interpreted as the delay time caused by the microstructural interactions. The stability and the qualitative aspects of this dual-phase-lag heat conduction were discussed in details in two papers by Quintanilla and Racke [28, 29]. Problems concerning with the generalized thermoelasticity proposed by Tzou were studied by many authors [30-34].
Extending the thermoelastic model introduced by Green–Naghdi [15, 17], Roychoudhuri [35] proposed a three-phase-lag heat (TPL) conduction theory that includes three-phase lags in the heat flux vector, the temperature gradient and in the thermal displacement gradient.
The aim of the present work is to establish a new generalized mathematical model of thermoelasticity that includes three-phase lags in the vector of heat flux, and in the thermal displacement and temperature gradients extending TPL model [35]. In this model, Fourier law of heat conduction is replaced using Taylor series expansions to a modification of the Fourier law with introducing three different phase lags for the heat flux vector, the temperature gradient, and the thermal displacement gradient and keeping terms up with suitable higher orders. The established high-order three-phase-lag heat conduction model (HTPL) reduces to the previous models of thermoelasticity as special cases.
To validate the accuracy of the current model, we have discussed a thermoelastic problem for an infinitely long annular cylinder whose boundary is subjected to sudden heating. Using the Laplace transform and numerical Laplace inversion, the problem is solved. The expressions of the studied variables are calculated under appropriate boundary and initial conditions. We deduce some particular cases of interest. The numerical results obtained have been tabulated and graphically illustrated. The results show that the analytic solutions are in good agreement with the numerical solutions. We also investigate the influences of phase-lags of high-order on the considered field variables. The results obtained in this work were found to be comparable with the results in the technical literature. It is believed that the analysis of this study will be useful for understanding the basic features of this new model for heat conduction.