Thermo-Viscoelastic Interaction Subjected to Fractional Fourier law with Three-Phase-Lag Effects
Subject Areas : Engineering
P Pal
1
(
Department of Applied Mathematics, University of Calcutta
)
A Sur
2
(
Department of Applied Mathematics, University of Calcutta
)
M Kanoria
3
(
Department of Applied Mathematics, University of Calcutta
)
Keywords: Generalized thermoelasticity, Kelvin-Voigt model, Three-phase-lag model, Modified riemann-liouville fractional derivatives, Fractional Taylor’s series,
Abstract :
In this paper, a new mathematical model of a Kelvin-Voigt type thermo-visco-elastic, infinite thermally conducting medium has been considered in the context of a new consideration of heat conduction having a non-local fractional order due to the presence of periodically varying heat sources. Three-phase-lag thermoelastic model, Green Naghdi models II and III (i.e., the models which predicts thermoelasticity without energy dissipation (TEWOED) and with energy dissipation (TEWED)) are employed to study the thermo-mechanical coupling, thermal and mechanical relaxation effects. In the absence of mechanical relaxations (viscous effect), the results for various generalized theories of thermoelasticity may be obtained as particular cases. The governing equations are expressed in Laplace-Fourier double transform domain. The inversion of the Fourier transform is carried out using residual calculus, where the poles of the integrand are obtained numerically in complex domain by using Laguerre's method and the inversion of the Laplace transform is done numerically using a method based on Fourier series expansion technique. Some comparisons have been shown in the form of the graphical representations to estimate the effect of the non-local fractional parameter and the effect of viscosity is also shown.
[1] Gross B., 1953, Mathematical Structure of the Theories of Viscoelasticity, Hermann, Paris.
[2] Staverman A.J., Schwarzl F., 1956, Die Physik der Hochpolymeren, Springer-Verlag, New York.
[3] Alfery T., Gurnee E.F., 1956, Theory and Applications, Academic Press, New York.
[4] Ferry J.D., 1970, Viscoelastic Properties of Polymers, John Wiley and Sons, New York.
[5] Bland D.R., 1960, The Theory of Linear Viscoelasticity, Pergamon Press, Oxford.
[6] Lakes R.S., 1998, Viscoelastic Solids, CRC Press, New York.
[7] Biot M.A., 1954, Theory of stress-strain relations in an isotropic viscoelasticity and relaxation phenomena, Journal of Applied Physics 25(11): 1385-1391.
[8] Biot M.A., 1955, Variational principal in irreversible thermodynamics with application to viscoelasticity, Physical Review 97(6): 1463-1469.
[9] Gurtin M.E., Sternberg E., 1962, On the linear theory of viscoelasticity, Archive for Rational Mechanics and Analysis 11: 291-356.
[10] Iiioushin A.A., Pobedria B.E., 1970, Mathematical Theory of Thermal Viscoelasticity, Nauka, Moscow.
[11] Tanner R.I., 1988, Engineering Rheology, Oxford University Press.
[12] Freudenthal A.M., 1954, Effect of rheological behaviour on thermal stress, Journal of Applied Physics 25: 1-10.
[13] Cattaneo C., 1958, Sur une forme de l'e'quation de la chaleur e'liminant le paradoxe d'une propagation instantane'e, Comptes Rendus de l'Académie des sciences 247: 431-433.
[14] Puri P., Kythe P.K., 1999, Non-classical thermal effects in Stoke's problem, Acta Mechanical 112: 1-9.
[15] Caputo M., 1967, Linear models of dissipation whose Q is almost frequently independent II, Geophysical Journal of the Royal Astronomical Society 13: 529-539.
[16] Podlubny I., 1999, Fractional Differential Equations, Academic Press, New York.
[17] Kiryakova V., 1994, Generalized Fractional Calculus and Applications, In: Pitman Research Notes in Mathematics Series, Longman-Wiley, New York.
[18] Miller K.S., Ross B., 1994, An Introduction to the Fractional Integrals and Derivatives-theory and Application, John Wiley & Sons Inc, New York.
[19] Samko S.G., Kilbas A.A., Marichev O.I., 1993, Fractional Integrals and Derivatives-Theory and Applications, Gordon and Breach, Longhorne.
[20] Oldman K.B., Spanier J., 1974, The Fractional Calculus, Academic Press, New York.
[21] Gorenflo R., Mainardi F., 1997, Fractional Calculus: Integral and Differential equations of fractional orders, Fractals and Fractional Calculus in Continuum Mechanics, Springer, Wien.
[22] Hilfer R., 2000, Applications of Fraction Calculus in Physics, World Scientific, Singapore.
[23] Khan M., Anjum A., Fetecau C., Haitao Q., 2010, Exact solutions for some oscillating motions of a fractional Burgers' fluid, Mathematical and Computer Modelling 51: 682-692.
[24] Hyder S., Haitao Q., 2010, Starting solutions for a viscoelastic fluid with Fractional Burgers' model in an annular pipe, Nonlinear Analysis 11: 547-554.
[25] Haitao G., Hui J., 2010, Unsteady helical flows of a genealized oldroyd-b fluid with fractional derivative, Nonlinear Analysis 10: 2700-2708.
[26] Saadatmandi A., Dehghan M., 2010, A new operational matrix for solving fractional-order differential equations, Computers & Mathematics with Applications 59: 1326-1336.
[27] Kimmich R., 2002, Strange kinetics, porous media, and NMR., Chemical Physics 284: 243-285.
[28] Fujita Y., 1990, Integrodifferential equation which interpolates the heat equation and wave equation (II), Osaka Journal of Mathematics 27: 797-804.
[29] Povstenko Y.Z., 2004, Fractional heat conductive and associated thermal stress, Journal of Thermal Stresses 28: 83-102.
[30] Povstenko Y.Z., 2011, Fractional catteneo-type equations and generalized thermoelasticity, Journal of Thermal Stresses 34: 94-114.
[31] 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.
[32] Lord H., Shulman Y., 1967, A generalized dynamical theory of thermoelasticity, Journal of the Mechanics and Physics of Solids 15: 299-309.
[33] Lebon G., Jou D., Casas-Vázquez J., 2008, Undersyanding Non-equilibrium Thermodynamics: Foundations, Applications Frontiers, Springer, Berlin.
[34] Jou D., Casas-Vázquez J., Lebon G., 1988, Extended irreversible thermodynamics, Reports on Progress in Physics 51: 1105-1179.
[35] Youssef H., 2010, Theory of fractional order generalized thermoelasticity, Journal of Heat Transfer 132: 1-7.
[36] 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.
[37] 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.
[38] El-Karamany A.S., Ezzat M.A., 2011, On the fractional Thermoelasticity, Mathematics and Mechanics of Solids 16(3): 334-346.
[39] Ezzat M.A., El Karamany A.S., Fayik M.A., 2012, Fractional order theory in thermoelastic solid with three-phase-lag heat transfer, Archive of Applied Mechanics 82(4): 557-572.
[40] 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.
[41] Sur A., Kanoria M., 2012, Fractional order two-temperature thermoelasticity with finite wave speed, Acta Mechanica 223(12): 2685-2701.
[42] 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.
[43] Tzou D.Y., 1995, A unified field approach for heat conduction from macro to micro scales, ASME Journal of Heat Transfer 117: 8-16.
[44] Ezzat M., 2010, Thermoelectric MHD non-newtonian fluid with fractional derivative heat transfer, Physica B: Condensed Matter 405: 4188-4194.
[45] Honig G., Hirdes U., 1984, A method for the numerical inversion of laplace transform, Journal of Computational and Applied Mathematics 10: 113-132.
[46] 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.
[47] Kanoria M., Mallik S.H., Generalized thermoviscoelastic interaction due to periodically varying heat source with three-phase-lag effect, European Journal of Mechanics - A/Solids 29: 695-703.