Document Type : Research Paper

**Authors**

Department of Applied Mathematics, University of Calcutta

**Abstract**

In this work, a new mathematical model of thermoelasticity theory has been considered in the context of a new consideration of heat conduction with fractional order theory. A functionally graded isotropic unbounded medium is considered subjected to a periodically varying heat source in the context of space-time non-local generalization of three-phase-lag thermoelastic model and Green-Naghdi models, in which the thermophysical properties are temperature dependent. The governing equations are expressed in 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 complex domain by using Laguerre’s method and the inversion of 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. Finally, the obtained results are presented graphically to show the effect of non-local fractional parameter on thermal displacement, temperature and thermal stress. A comparison of the results for different theories (three-phase-lag model, GN model II, GN model III) is presented and the effect of non-homogeneity is also shown. The results, corresponding to the cases, when the material properties are temperature independent, agree with the results of the existing literature.

**Keywords**

[2] Puri P., Kythe P.K., 1999, Non-classical thermal effects in stoke's problem, Acta Mechanica 112:1-9.

[3] Caputo M., 1967, Linear models of dissipation whose Q is almost frequently independent II, Geophysical Journal of the Royal Astronomical Society 13:529-539.

[4] Mainardi F., 1997, Fractional calculus: some basic problems in continuum and statistical mechanics, In: A. Carpinteri, Fractals and Fractional calculus in Continuum Mechanics, Springer, New York.

[5] Podlubny I., 1999, Fractional Differential Equations, Academic Press, New York.

[6] Kiryakova V., 1994, Generalized fractional calculus and applications, In: Pitman Research Notes in Mathematics Series, Longman-Wiley, New York.

[7] Mainardi F., Gorenflo R., 2000, On mittag-lettler-type function in fractional evolution processes, The Journal of Computational and Applied Mathematics 118:283-299.

[8] Kimmich R., 2002, Strange kinetics, porous media and NMR, The Journal of Chemical Physics 284:243-285.

[9] Fujita Y., 1990, Integrodifferential equation which interpolates the heat equation and wave equation (II), Osaka Journal of Mathematics 27:797-804.

[10] Povstenko Y.Z., 2004, Fractional heat conductive and associated thermal stress, Journal of Thermal Stresses 28:83-102.

[11] Povstenko Y.Z., 2011, Fractional catteneo-type equations and generalized thermoelasticity, Journal of Thermal Stresses 34:94-114.

[12] Sherief H.H., El-Said A., Abd El-Latief A., 2010, Fractional order theory of thermoelasticity, International Journal of Solids and Structures 47:269-275.

[13] Lord H., Shulman Y., 1967, A generalized dynamical theory of thermoelasticity, Journal of the Mechanics and Physics of Solids 15:299-309.

[14] Lebon G., Jou D., Casas-Vázquez J., 2008, Undersyanding Non-equilibrium Thermodynamics: Foundations, Applications Frontiers, Springer, Berlin.

[15] Youssef H., 2010, Theory of fractional order generalized thermoelasticity, Journal of Heat Transfer 132:1-7.

[16] Jumarie G., 2010, Derivation and solutions of some fractional black-scholes equations in coarse-grained space and time: application to merton's optimal portfolio, Computers & Mathematics with Applications 59: 1142-1164.

[17] El-Karamany A.S., Ezzat M.A., 2011, Convolutional variational principle, reciprocal and uniqueness theorems in linear fractional two-temperature thermoelasticity, Journal of Thermal Stresses 34(3):264-284.

[18] El-Karamany A.S., Ezzat M.A., 2011, On the fractional thermoelasticity, Mathematics and Mechanics of Solids 16(3): 334-346.

[19] El-Karamany A.S., Ezzat M.A., 2011, Fractional order theory of a prefect conducting thermoelastic medium, Canadian Journal of Physics 89(3):311-318.

[20] Sur A., Kanoria M., 2012, Fractional order two-temperature thermoelasticity with finite wave speed, Acta Mechanica 223(12):2685-2701.

[21] Green A.E., Naghdi P.M., 1991, A re-examination of the basic postulates of thermomechanics, Proceedings of the Royal Society of London, Series A 432:171-184.

[22] Green A.E., Naghdi P.M., 1992, On undamped heat waves in an elastic solid, Journal of Thermal Stresses 15:252-264.

[23] Green A.E., Naghdi P.M., 1993, Thermoelasticity without energy dissipation, Journal of Elasticity 31:189-208.

[24] Roychoudhuri S.K., 2007, On a thermoelastic three-phase-lag model, Journal of Thermal Stresses 30:231-238.

[25] Quintanilla R., Racke R.A., 2008, Note on stability in three-phase-lag heat conduction, International Journal of Heat and Mass Transfer 51:24-29.

[26] Quintanilla R., 2009, Spatial behavior of solutions of the three-phase-lag heat conduction, Applied Mathematics and Computation 213:153-162.

[27] Kar A., Kanoria M., 2000, Generalized thermoviscoelastic problem of a spherical shell with three-phase-lag effect, Applied Mathematical Modelling 33:3287-3298.

[28] Kumar R., Mukhopadhyay S., 2009, Effects of three-phase-lags on generalized thermoelsticity for an infinite medium with a cylindrical cavity, Journal of Thermal Stresses 32:1149-1165.

[29] Kumar R., Chawla V., 2011, A study of plane wave propagation in anisotropic three-phase-lag and two-phase-lag model, International Journal of Heat and Mass Transfer 38:1262-1268.

[30] Aboudi J., Pindera M.J., Arnold S.M., 1995, Thermo-inelastic response of functionally graded composites, International Journal of Solids and Structures 32:1675-1710.

[31] Wetherhold R.C., Wang S.S., 1996, The use of functionally graded materials to eliminate or control thermal deformation, Composites Science and Technology 28:1099-1104.

[32] Sugano Y., 1987, An expression for transient thermal stress in a nonhomogeneous plate with temperature variation through thickness, Ingenieur Archiv 57:147-156.

[33] Qian L.F., Batra R.C., 2004, Transient thermoelastic deformations of a thick functionally graded plate, Journal of Thermal Stresses 27:705-740.

[34] Ghosh M.K., Kanoria M., 2009, Analysis of thermoelastic response in a functionally graded spherically isotropic hollow spherebased on Green–Lindsay theory, Acta Mechanica 207:51-67.

[35] Kar A., Kanoria M., 2009, Generalized thermoelastic functionally graded orthotropic hollow sphere under thermal shock with three-phase-lag effect, European Journal of Mechanics A/Solids 28:757-767.

[36] Barik S.P., Kanoria M., Chaudhuri P.K., 2008, Steady-state thermoelastic contact problem in a functionally graded material, International Journal of Engineering Science 46:775-789.

[37] Honig G., Hirdes U., 1984, A method for the numerical inversion of Laplace transform, Journal of Computational and Applied Mathematics 10:113-132.

[38] Roychoudhuri S.K., Dutta P.S., 2005, Thermoelastic interaction without energy dissipation in an infinite solid with distributed periodically varying heat sources, International Journal of Solids and Structures 42:4192-4293.

[39] Roychoudhuri S.K., 2007, On a thermoelastic three-phase-lag model, Journal of Thermal Stresses 30:231-238.

[40] Quintanilla R., Racke R., 2008, A note on stability in three-phase-lag heat conduction, International Journal of Heat and Mass Transfer 51:24-29.

[41] Mallik S.H., Kanoria M., 2007, Generalized thermoelastic functionally graded solid with a periodically varying heat source, International Journal of Solids and Structures 44(22-23):7633-7645.

Winter 2014

Pages 54-69