Memory Response in Thermoelastic Plate with Three-Phase-Lag Model
Subject Areas : Engineering
1 - Department of Mathematics, University of North Bengal, Darjeeling-734013, India
Keywords: Three-phase-lag model, Laplace-Fourier transform, Memory-dependent derivative, State-space approach,
Abstract :
In this article, using memory-dependent derivative (MDD) on three-phase-lag model of thermoelasticity, a new generalized model of thermoelasticity theory with time delay and kernel function is constructed. The governing coupled equations of the new generalized thermoelasticity with time delay and kernel function are applied to two dimensional problem of an isotropic plate. The two dimensional equations of generalized thermoelasticity with MDD are solved using state space approach. Numerical inversion method is employed for the inversion of Laplace and Fourier transforms. The displacements, temperature and stress components for different thermoelastic models are presented graphically and the effect of different kernel and time delay on the considered parameters are observed.
[1] Biot M., 1956, Thermoelsticity and irreversible thermodynamics, Journal of Applied Physics 27: 240-253.
[2] Lord H., Shulman Y., 1967, A generalized dynamic theory of thermoelasticity, Journal of the Mechanics & Physics of Solids 15: 299-309.
[3] Green A. E., Lindsay K.A., 1972, Thermoelasticity, Journal of Elasticity 2: 1-7.
[4] Green A.E., Naghdi P.M., 1991, A re-examination of the basic properties of thermomechanics, Proceedings of Royal Society London Series A 432: 171-194.
[5] Green A.E., Naghdi P.M., 1992, On damped heat waves in an elastic solid, Journal of Thermal Stresses 15: 252-264.
[6] Green A.E., Naghdi P.M., 1993, Thermoelasticity without energy dissipation, Journal of Elasticity 31: 189-208.
[7] Tzou D.Y., 1995, A unique field approach for heat conduction from macro to micro scales, The Journal of Heat Transfer 117: 8-16.
[8] Roy Choudhuri S. K., 2007, On a thermoelastic three phase lag model, Journal of Thermal Stresses 30: 231-238.
[9] Wang J.L., Li H.F., 2011, Surpassing the fractional derivative: concept of the memory dependent derivative, Computers and Mathematics with Applications 62:1562-1567.
[10] El- Karamany A.S., Ezzat M.A., 2011, On fractional thermoelastivity, Mathematics and Mechanics of Solids 16: 334-346.
[11] Ezzat M.A., 2010, Thermoelastic MHD non-newtonian fluid with fractional derivative heat transfer, Physica B 405: 4188-4194.
[12] Ezzat M.A., El- Karamany A.S., 2011, Fractional order heat conduction law in magneto thermoelasticity involving two temperatures, Zeitschrift für Angewandte Mathematik und Physik 62: 937-952.
[13] Yu Y.-J., Hu W., Tian X.-G., 2014, A novel generalized thermoelasticity model based on memory dependent derivatives, International Journal of Engineering Sciences 81: 123-134.
[14] Ezzat M.A., El-Bary A.A., 2015, Memory dependent derivatives theory of thermo-viscoelasticity involving two temperatures, Journal of Mechanical Science and Technology 29(10):4273-4279.
[15] Ezzat M.A., El-Karamany A.S., El-Bary A.A., 2014, Generalized thermo-viscoelasticity with memory-dependent derivatives, International Journal of Mechanical Science 89: 470-475.
[16] Sherief H.H., Abd El-Latief A., 2013, Effect of variable thermal conductivity on a half-space under the fractional order theory of thermoelasticity, International Journal of Mechanical Science 74: 185-189.
[17] Ezzat M.A., El-Bary A.A., 2014, Two temperature theory of magneto thermo-viscoelasticity with fractional derivative and integral orders heat trasfes, Journal of Electromagnetic Waves and Applications 28: 1985-2004.
[18] Anwar M., Sherief H., 1988, State space approach to generalized thermoelasticity, Journal of Thermal Stresses 11: 353-365.
[19] Ezzat M.A., A.Othman M.I., El-Karamany A.S., 2002, State space approach to two-dimensional generalized thermoviscoelasticity with one relaxation time, Journal of Thermal Stresses 25: 295-316.
[20] Ezzat M.A., Othman M.I.A., Smaan A.A., 2001, State space approach to two-dimensional electro-magneto thermoelastic problem with two relaxation times, International Journal of Engineering Science 39: 1383-1404.
[21] Sherief H.H., El-Sayed A.M., 2014, State space approach to two-dimensional generalized micropolar thermoelasticity, Zeitschrift für Angewandte Mathematik und Physik 66: 1249-1265.
[22] Diethelm K., 2010, Analysis of Fractional Differential Equation: An Application Oriented Exposition Using Differential Operators of Caputo Type, Berlin, Heidelberg, Springer-Verlag.
[23] Sabatier J., Agarwal O.P., Tenreiro Machado J.A., 2007, Advances in Fractional Calculus Theoretical Developments and Applications in Physics and Engineering, Springer, Dordrecht.
[24] Hiffler R.,2000, Application of Fractional Calculus to Physics, World Scientific, Singapore.
[25] Atanackovic T.M., Pilipovic S., Stankovic B., Zorica D., 2014, Fractional Calculus with Application in Mechanics, Wiley, London.
[26] Biswas S., 2019, Modeling of memory-dependent derivatives in orthotropic medium with three-phase-lag model under the effect of magnetic field, Mechanics Based Design of Structures and Machines 47(3): 302-318.
[27] Honig G., Hirdes U., 1984, A method for the numerical inversion of the Laplace transforms, Journal of Computational and Applied Mathematics 10: 113-132.