Basic theorems and wave propagation in a piezothermoelastic medium with dual phase lag
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ORIGINAL PAPER
Basic theorems and wave propagation in a piezothermoelastic medium with dual phase lag R Kumar and P Sharma* Department of Mathematics, Kurukshetra University, Kurukshetra, Haryana 136119, India Received: 27 November 2018 / Accepted: 16 September 2019
Abstract: The basic governing equations for an anisotropic piezothermoelastic solid with dual phase lag are presented and used to study the problem. Some basic theorems like variational principle, uniqueness theorem and theorem of reciprocity are established for the assumed model. Also, we characterize an alternative formulation of the mixed initial boundary value problem. These theorems are also summarized for a special case of orthotropic piezothermoelastic solid with dual phase lag. We formulated the plane wave propagation in an orthotropic piezothermoelastic solid with dual-phase-lag model. The non-trivial solution of the system is insured by a quartic equation whose roots represent the complex velocities of four attenuating waves in the medium. Various characteristics of the waves like phase velocity, attenuation quality factor, specific heat loss and penetration depth are computed numerically and shown graphically for three different models with the direction of propagation in three-dimensional space. Some special cases are also deduced from the present investigation. Keywords: Piezothermoelastic; Orthotropic; Variational principle; Uniqueness; Plane waves; Phase velocity PACS Nos.: 62.20.Dc; 62.30.?d; 72.15.Jf; 74.25.Ld; 77.65.-j; 81.40.Jj
1. Introduction The exact nature of the earth is not known; therefore, one has to take the different mathematical model for the purpose of theoretical investigations. In general, experiment models are limited by size, cost, noise and many other laboratory uncertainties. Theoretical models can be more general; however, analytical solutions are restricted to relatively simple geometries and boundary conditions. The investigation of models of an elastic medium has been taken into account with growing interest under the influence of various physical fields such as thermal, electric and other fields. An impetus for such studies was the creation of many new materials possessing properties that are not characteristic of usual elastic bodies. In classical theory of thermoelasticity, Fourier’s heat conduction theory assumes that the thermal disturbances propagate at infinite speed which is unrealistic from the physical point of view. One of the generalizations of the
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classical theory of thermoelasticity has been developed which predicts only finite velocity of propagation for heat and displacement fields given by Lord and Shulman [1] which incorporates a flux rate term into the Fourier’s law of heat conduction and formulates a generalized theory admitting finite speed for thermal signals. Tzou [2] introduced a more generalized universal heat conduction model called the dual-phase-lag (DPL) model, relating the heat ~ h for an isoflux vector ~ q and the temperat
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