Fractional-order generalized thermoelastic diffusion theory

doi:10.1007/s10483-017-2230-9

Abstract

Abstract:

The present work aims to establish a fractional-order generalized themoelastic diffusion theory for anisotropic and linearly thermoelastic diffusive media. To numerically handle the multi-physics problems expressed by a sequence of incomplete differential equations, particularly by a fractional equation, a generalized variational principle is obtained for the unified theory using a semi-inverse method. In numerical implementation, the dynamic response of a semi-infinite medium with one end subjected to a thermal shock and a chemical potential shock is investigated using the Laplace transform. Numerical results, i.e., non-dimensional temperature, chemical potential, and displacement, are presented graphically. The influence of the fractional order parameter on them is evaluated and discussed.

Key words: asymptotically regular mappings, p-uniformly convex Banach space, asymptotic center, fixed points, generalized thermoelastic diffusion, Laplace transform, fractional calculus, generalized variational principle

2010 MSC Number:

74F05

Chunbao XIONG, Yanbo NIU. Fractional-order generalized thermoelastic diffusion theory. Applied Mathematics and Mechanics (English Edition), 2017, 38(8): 1091-1108.

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References

[1] Oriani, R. A. Thermomigration in solid metals. J. Phys. Chem., 30, 339-351(1969)
[2] Nowacki, W. Dynamical problems of thermoelastic diffusion in elastic solids. Proc. Vib. Prob., 15, 105-128(1974)
[3] Kumar, R., Garg, S. K., and Ahuja, S. Propagation of plane waves at the interface of an elastic solid half-space and a microstretch thermoelastic diffusion solid half-space. Lat. Am. J. Solids Stru., 10, 1081-1108(2013)
[4] Sherief, H. H., Hamza, F. A., and Saleh, H. A. The theory of generalized thermoelastic diffusion. Int. J. Eng. Sci., 42, 591-608(2004)
[5] Load, H. W. and Shulman, Y. A generalized dynamic theory of thermoelasticity. J. Mech. Phys. Solids, 15, 299-309(1907)
[6] Sherief, H. H. and El-Maghraby, N. M. A thick plate problem in the theory of generalized thermoelastic diffusion. Int. J. Thermophys., 30, 2044-2057(2009)
[7] Sherief, H. H. and Hussein, E. M. Two-dimensional problem for a half-space with axi-symmetric distribution in the theory of generalized thermoelastic diffusion. Mech. Adv. Mater. Struc., 23, 216-222(2016)
[8] Xia, R. H., Tian, X. G., and Shen, Y. P. The influence of diffusion on generalized thermoelastic problems of infinite body with a cylindrical cavity. Int. J. Eng. Sci., 47, 669-679(2009)
[9] Othman, M. I. A., Atwa, S. Y., and Farouk, R. M. The effect of diffusion on two-dimensional problem of generalized thermoelasticity with Green-Naghdi theory. Int. Commun. Heat. Mass, 36, 857-864(2009)
[10] Ram, P., Sharma, N., and Kumar, R. Thermomechanical response of generalized thermoelastic diffusion with one relaxation time due to time harmonic sources. Iny. J. Therm. Sci., 36, 857-864(2009)
[11] Kumar, R. and Kansal, T. Propagation of Rayleigh waves on free surface of transversely isotropic generalized thermoelastic diffusion. Appl. Math. Mech. -Engl. Ed., 29(11), 1451-1462(2008) DOI 10.1007/s10483-008-1106-6
[12] Deswal, S. and Choudhary, S. Two-dimensional interactions due to moving load in generalized thermoelastic solid with diffusion. Appl. Math. Mech. -Engl. Ed., 29(2), 207-221(2008) DOI 10.1007/s10483-008-0208-5
[13] Aouadi, M. On thermoelastic diffusion thin plate theory. Appl. Math. Mech. -Engl. Ed., 36(5), 619-632(2015) DOI 10.1007/s10483-015-1930-7
[14] He, T. H., Li, C. L., Shi, S. H., and Ma, Y. B. A two-dimensional generalized thermoelastic diffusion problem for a half-space. Euro. J. Mech-A/Solids, 52, 37-43(2015)
[15] Li, C. L., Yu, Y. J., and Tian, X. G. Effect of rotation on plane waves of generalized electromagnetothermoelastic with diffusion for a half-space. J. Therm. Stress, 39, 27-43(2016)
[16] Li, C. L., Guo, H. L., Tian, X., and Tian, X. G. Transient response for a half-space with variable thermal conductivity and diffusivity under thermal and chemical shock. J. Therm. Stress, 40, 389-401(2017)
[17] Bai, Z. B. and Lu, H. S. Positive solutions for boundary value problem of nonlinear fractional differential equation. J. Math. Anal. Appl., 311, 495-505(2005)
[18] Deng, J. Q. and Ma, L. F. Existence and uniqueness of solutions of initial value problems for nonlinear fractional differential equations. Appl. Math. Lett., 23, 676-680(2010)
[19] Su, X. W. Boundary value problem for a coupled system of nonlinear fractional differential equations. Appl. Math. Lett., 22, 64-69(2009)
[20] Diethelm, K., Ford, N. J., Freed, A. D., and Luchko, Y. Algorithms for the fractional calculus:a selection of numerical methods. Comput. Methods Appl. Mech. Eng., 194, 743-773(2005)
[21] Chen, W. Time-space fabric underlying anomalous diffusion. Chaos Soliton. Fract., 28, 923-929(2006)
[22] Chen, W., Sun, H. G., Zhang, X. D., and Korosak, D. Anomalous diffusion modeling by fractal and fractional derivatives. Comput. Math. Appl., 59, 1754-1758(2010)
[23] Chen, W., Liang, Y. J., Hu, S. A., and Su, H. G. Fractional derivative anomalous diffusion equation modeling prime number distribution. Fract. Calc. Appl. Anal., 18, 789-798(2015)
[24] Suzuki, A., Fomin, S. A., Chugunov, V. A., Niibori, Y., and Hashida, T. Fractional diffusion modeling of heat transfer in porous and fractured media. Int. J. Heat Mass Tran., 103, 611-618(2016)
[25] Sweilam, N. H., Nagy, A. M., and El-Sayed, A. A. On the numerical solution of space fractional order diffusion equation via shifted Chebyshev polynomials of the third kind. Journal of King Saud University-Science, 28, 41-47(2016)
[26] Kumar, R. and Gupta, V. Uniqueness, reciprocity theorem, and plane waves in thermoelastic diffusion with a fractional order derivative. Chin. Phys. B, 22, 074601(2013)
[27] Liu, X. J., Wang, J. Z., Wang, X. M., and Zhou, Y. H. Exact solutions of multi-term fractional diffusion-wave equations with Robin type boundary conditions. Appl. Math. Mech. -Engl. Ed., 35(1), 49-62(2014) DOI 10.1007/s10483-014-1771-6
[28] El-Karamany, A. S. and Ezzat, M. A. On fractional thermoelastisity. Math. Mech. Solids, 16, 334-346(2011)
[29] El-Karamany, A. S. and Ezzat, M. A. Convolutional variational principle, reciprocal and uniqueness theorems in linear fractional two-temperature thermoelasticity. J. Therm. Stress, 34, 264-284(2011)
[30] Li, C. L., Guo, H. L., and Tian, X. G. A size-dependent generalized thermoelastic diffusion theory and its application. J. Therm. Stress., 40, 603-626(2017)
[31] Green, A. E. and Lindsay, K. E. Thermoelasticity. J. Elasticity, 2, 1-7(1972)
[32] Green, A. E. and Naghdi, P. M. On undamped heat waves in an elastic solid. J. Therm. Stress, 15, 252-264(1992)
[33] Green, A. E. and Naghdi, P. M. Thermoelasticity without energy dissipation. J. Elasticity, 31, 189-208(1993)
[34] Mainardi, F. and Gorenflo, R. On Mittag-Leffler-type functions in fractional evolution processes. J. Comput. Appl. Math., 118, 283-299(2000)
[35] Youssef, H. M. Theory of fractional order generalized thermoelasticity. ASME J. Heat Trans., 132, 6(2010)
[36] Mitra, K., Kumar, A., Vedavarz, A., and Moallemi, M. K. Experimental evidence of hyperbolic heat conduction in processed meat. J. Heat Transfer. Trans. ASME, 117, 568-573(1995)
[37] Ghazizadeh, H. R., Azimi, A., and Maerefat, M. An inverse problem to estimate relaxation parameter and order of fractionality in fractional single-phase-lag heat equation. Int. J. Heat Mass Trans., 55, 2095-2101(2009)
[38] Kuang, Z. B. Variational principles for generalized thermodiffusion theory in pyroelectricity. Acta Mech., 214, 275-289(2010)
[39] He, J. H. Semi-inverse method of establishing generalized variational principles for fluid mechanics with emphasis on turbomachinery aerodynamics. Int. J. Turbo. Jet. Eng., 14, 23-28(1997)
[40] Gurtin, M. E. Variational principles for linear elastodynamics. Arch. Ration. Mech. Anal., 16, 34-50(1964)
[41] He, J. H. Generalized variational principles for thermopiezoelectricity. Arch. Appl. Mech., 72, 248-256(2002)
[42] Honig, G. and Hirdes, U. A method for the numerical inversion of Laplace transforms. J. Comput. Appl. Math., 10, 113-132(1984)
[43] Sherief, H. H. and Saleh, H. A. A half-space problem in the theory of generalized thermoelastic diffusion. Int. J. Solids. Struc., 42, 4484-4493(2005)