Thermo-Viscoelastic Interaction Subjected to Fractional Fourier law with Three-Phase-Lag Effects

Subject Areas : Engineering

P Pal ¹ , A Sur ² , M Kanoria ³

1 - Department of Applied Mathematics, University of Calcutta
2 - Department of Applied Mathematics, University of Calcutta
3 - Department of Applied Mathematics, University of Calcutta

Received: 2015-07-15 Accepted : 2015-09-20 Published : 2015-12-30

Keywords:

References:

[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.