Three-phase-lag thermoelastic heat conduction model with higher-order time-fractional derivatives
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ORIGINAL PAPER
Three-phase-lag thermoelastic heat conduction model with higher-order time-fractional derivatives A E Abouelregal1,2* 1
Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt
2
Department of Mathematics, College of Science and Arts, Jouf University, Al-Qurayyat, Saudi Arabia Received: 23 April 2019 / Accepted: 06 September 2019
Abstract: In the last few years, the theory of fractional calculus has been successfully used in thermoelasticity theories and many models of thermoelasticity with fractional order are established by several authors. In the present article, a new model of three-phase-lag thermoelastic heat conduction of higher-order time-fractional derivatives has been derived based on fractional calculus. Using the approach of the Taylor series expansion of time-fractional order developed by Jumarie (Comput Math Appl 59:1142, 2010), an alternative construction model is established extending Ezzat and others (Arch Appl Mech 82:557, 2012) and Roychoudhuri (J Therm Stress 30:231, 2007) models. This new model includes high-order time-fractional derivative approximations of three-phase-lags in the heat flux vector, the temperature gradient and in the thermal displacement gradient. We applied the resulting formulation to an infinite non-homogeneous orthotropic thermoelastic functionally graded medium having a spherical cavity with a power-law distribution of material properties along the radial direction. The effects of high-order time-fractional derivative parameters and non-homogeneity index on various distributions are discussed in detail and represented graphically and tabular forms. Finally, to illustrate the validity and accuracy of the proposed model, a comparison was made with various previous models, which are considered as special cases of our model. Keywords: Fractional thermoelasticity; Three-phase-lags; Higher-order; Spherical cavity PACS Nos: 81.05.Ni; 46.25.Hf; 65.40.gh; 46.50.?a; 10.?i; 40.Jj
1. Introduction Biot [4] developed the coupled theory of thermoelasticity to deal with defeat of the uncoupled theory that mechanical cause has no effect on the temperature field. In this theory, the heat equation has a parabolic form which predicts an infinite speed for the propagation of mechanical waves. The theory of generalized thermoelasticity with one relaxation time was introduced by Lord and Shulman [5]. This theory was extended by Dhaliwal and Sherief [6]. In the theory, the Maxwell–Cattaneo law of heat conduction replaces the conventional Fourier’s law. For this theory, Ignaczak [7] studied the uniqueness of the solution. Green and Lindsay [8] presented another theory, called a temperature-rate-dependent, involving two times of relaxation.
In the last few years, fractional calculus was applied successfully in various areas to modify many existing models of physical processes, for example, chemistry, biology, modeling and identification, electronics, wave propagation and viscoelasticity. It has been examined that the use of fractional-order de
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