State Space Approach to Thermoelastic Problem with Three-Phase-Lag Model
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International Applied Mechanics, Vol. 56, No. 2, March, 2020
STATE SPACE APPROACH TO THERMOELASTIC PROBLEM WITH THREE-PHASE-LAG MODEL
S. Biswas
A two-dimensional problem of generalized thermoelasticity is formulated using state space approach. In this formulation, the governing equations are transformed into a matrix differential equation whose solution enables to write the solution of two-dimensional problem in terms of the boundary conditions. The resulting formulation is applied to an isotropic half-space problem within three-phase-lag model of thermoelasticity. The bounding surface is traction free and subjected to a time dependent thermal shock. The solution for temperature distribution, displacements and stress components are obtained and presented graphically as well as a comparison with other thermoelastic models is made. Keywords: state space approach, normal mode analysis, thermal shock, three-phase-lag model 1. Introduction. The classical theory of thermoelasticity [1] suffers from the so-called paradox of heat conduction, i.e., the heat equations for both theories of a mixed parabolic-hyperbolic type, predicting infinite speeds of propagation for heat waves contrary to physical observations. To remove this paradox, the conventional theories of thermoelasticity has been generalized, where the generalization is in the sense that these theories involve a hyperbolic type heat transport equation supported by experiments, which exhibit the actual occurrence of wave type heat transport in solids, called second sound effect. To eliminate the second sound paradox of classical thermoelasticity theory, Lord and Shulman [2] established a generalized thermoelasticity theory which is often referred to as LS model and widely used in the case of heat flux and low temperature. Green and Lindsay [3] introduced one more theory, called GL theory, which involves two relaxation times. Later Green and Naghdi [4–6] developed three models for generalized thermoelasticity of homogeneous isotropic materials, which are labeled as G–N models I, II, III. Detailed information regarding these theories can be found in [7, 8]. The next generalization to the thermoelasticity is known as the dual-phase-lag model (DPL) developed by Tzou [9]. Tzou [9] considered micro-structural effects into the delayed response in time in macroscopic formulation by taking into account that increase of the lattice temperature is delayed due to phonon-electron interactions on the macroscopic level. Tzou [9] introduced two-phase lags to both the heat flux vector and the temperature gradient and considered as constitutive equation to describe the lagging behavior in the heat conduction in solids. Recently, Roychoudhuri [10] has established a generalized mathematical model of a coupled thermoelasticity theory that includes three-phase-lags in the heat flux vector, the temperature gradient and in the thermal displacement gradient. The more general model (TPL) established reduces to the previous models as special cases. In three-phase-lag heat conduction equat
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