A Generalized Thermo-Elastic Diffusion Problem in a Functionally Graded Rotating Media Using Fractional Order Theory
محورهای موضوعی : Engineering
1 - Department of Mathematics, Indian Institute of Engineering Science and Technology, Shibpur, Howrah-711103, India
2 - Department of Mathematics, Indian Institute of Engineering Science and Technology, Shibpur, Howrah-711103, India
کلید واژه: Functionally graded material, Thermo-elastic diffusion, Generalized thermo-elasticity, Fractional order,
چکیده مقاله :
A generalized thermo-elastic diffusion problem in a functionally graded isotropic, unbounded, rotating elastic medium due to a periodically varying heat source in the context of fractional order theory is considered in our present work. The governing equations of the theory for a functionally graded material with GNIII model are established. Analytical solution of the problem is derived in Laplace-Fourier transform domain. Finally, numerical inversions are used to show the effect of rotation, non-homogeneity and fractional parameter on stresses, displacement, chemical potential, mass distribution, temperature, etc. and those are illustrated graphically.
[1] Lord H.W., Shulman Y., 1967, A generalized dynamical theory of thermo-elasticity, Journal of the Mechanics and Physics of Solids 15(5) : 299-309.
[2] Green A.E., Lindsay K.A., 1972, Thermo-elasticity, Journal of Elasticity 2(1): 1-7.
[3] Green A.E., Naghdi P.M., 1991, A re-examination of the basic postulates of thermo-mechanics, Proceedings of the Royal Society of London A 432: 171-194.
[4] Green A.E., Naghdi P.M., 1993, Thermoelasticity without energy dissipation, Journal of Elasticity 31(3): 189-208.
[5] Green A.E., Naghdi P.M., 1992, On undamped heat waves in an elastic solid, Journal of Thermal Stresses 15(2): 253-264.
[6] Sherief H.H., Hamza F.A., Saleh H.A., 2004, The theory of generalized thermoelastic diffusion, International Journal of Engineering Science 42: 591-608.
[7] Elhagary M.A., 2011, Generalized thermoelastic diffusion problem for an infinitely long hollow cylinder for short times, Acta Mechanica 218: 205-215.
[8] Aouadi M., 2006, A generalized thermo-elastic diffusion problem for an infinitely long solid cylinder, International Journal of Mathematics and Mathematical Sciences 206: 1-15.
[9] Hetnarski R.B., Ignaczak J., 1994, Generalized thermoelasticity: response of semi-space to a short laser pulse, Journal of Thermal Stresses 17(3): 377-396.
[10] Bagri A., Eslami M.R., 2004, Generalized coupled thermo-elasticity of disks based on the Lord-Shulman Model, Journal of Thermal Stresses 27(8): 691-704.
[11] Kar A., Kanoria M., 2007, Thermo-elastic interaction with energy dissipation in an unbounded body with a spherical hole, International Journal of Solids and Structures 44(9): 2961-2971.
[12] Sherief H.H., Anwar M.N., 1989, A problem in generalized thermo-elasticity for an infinitely long annular cylinder composed of two different materials, ActaMechanica 80: 137-149.
[13] Sherief H.H., Hamza F.A., 1996, Generalized two-dimensional thermo-elastic problems in spherical regions under axisymmetric distributions, Journal of Thermal Stresses 19(1): 55-76.
[14] Suresh S., Mortesen A., 1998, Fundamentals of Functionally Graded Materials, Institute of Material Communications Ltd., London.
[15] Aboudi J., Pindera M.J., Arnold S.M., 1995, Thermo-inelastic response of functionally graded composites,International Journal of Solids and Structures 32(12): 1675-1710.
[16] Wetherhold R.C., Wang J., Seelman S., 1996, The use of functionally graded materials to eliminate or control thermal deformation, Composites Science and Technology 56(6): 1099-1104.
[17] Qian L.F., Batra R.C., 2004, Transient thermoelastic deformations of a thick functionally graded plate, Journal of Thermal Stresses 27(8): 705-740.
[18] Shao Z.S., Wang T.J., Ang K.K., 2007, Transient thermo-mechanical analysis of functionally graded hollow circular cylinders, Journal of Thermal Stresses 30(1): 81-104.
[19] 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.
[20] Bagri A., Eslami M.R., 2007, A unified generalized thermoelasticity formulation; application to thick functionally graded cylinders, Journal of Thermal Stresses 30(9-10): 911-930.
[21] Ootao Y., Tanigawa Y., 2007, Transient thermoelastic problem of a functionally graded cylindrical panel due to nonuniform heat supply, Journal of Thermal Stresses 30(5): 441-457.
[22] Kanoria M., Sur A., 2014, Fractional order generalized thermoelastic functionally graded solid with variable material properties, Journal of Solid Mechanics 6(1): 54-69.
[23] Pal P., Das P., Kanoria M., 2015, Magneto-thermoelastic response in a functionally graded rotating medium due to a periodically varying heat source, Acta Mechanica 226(7): 2103-2120.
[24] Nowacki W., 1974, Dynamical problems of thermo-diffusion in solids I, Bulletin de l’Academie Polonaise des Sciences, Serie des Sciences Techniques 22: 55-64.
[25] Nowacki W., 1974, Dynamical problems of thermo-diffusion in solids II, Bulletin de l’Academie Polonaise des Sciences, Serie des Sciences Techniques 22: 129-135.
[26] Nowacki W., 1974, Dynamical problems of thermo-diffusion in solids III, Bulletin de l’Academie Polonaise des Sciences, Serie des Sciences Techniques 22: 257-266.
[27] Nowacki W., 1974, Dynamical problems of thermo diffusion in elastic solids, Proceedings of Vibration Problems 15:105-128.
[28] Singh B., 2005, Reflection of P and SV waves from free surface of an elastic solid with generalized thermo-Diffusion, Journal of Earth System Science 114(2): 159-168.
[29] He T., Li C., 2014, Normal mode analysis to a two-dimensional generalized electromagnetothermoelastic problem with diffusion, Journal of Thermal Stresses 37(10): 1244-1263.
[30] Aouadi M., 2008, Generalized theory of thermoelastic diffusion for anisotropic media, Journal of Thermal Stresses 31: 270-285.
[31] Elhagary M.A., 2014, A two-dimensional generalized thermoelastic diffusion problem for a thick plate subjected to thermal loading due to laser pulse, Journal of Thermal Stresses 37: 1416-1432.
[32] He T., Li C., Shi H., Ha Y., 2015, A two-dimensional generalized thermoelastic diffusion problem for a half–space, European Journal of Mechanics A/Solids 52: 37-43.
[33] Amany M.E., 2016, A two-dimensional generalized thermoelastic diffusion for a half –space, Mathematics and Mechanics of Solids 21(9): 1045-1060.
[34] Li C., Yu Y., Tian X., 2016, Effect of rotation on plane waves of generalized electromegneticthermoelastics with diffusion for a half-space, Journal of Thermal Stresses 39(1): 27-43.
[35] Li C., Guo H., Tian X., 2017, A size-dependent generalized thermoelastic diffusion theory and its application, Journal of Thermal Stresses 40(5): 603-626.
[36] Youssefa H.M., Al-Lehaibib E.A.N., 2018, Three-dimensional generalized thermoelastic diffusion and application for a thermoelastic half-space subjected to rectangular thermal pulse, Journal of Thermal Stresses 41(8): 1008-1021.
[37] Paul K., Mukhopadhyay B., 2019, Two dimensional generalized megnetothermoelastic diffusion problem for a thick plate under laser pulse heating with three phase lag effect, Journal of Engineering Physics and Thermophysics 92(2): 516-527.
[38] Bagley R. L., Torvik P. J., 1983, A theoretical basis for the application of fractional calculus to viscoelasticity, Journal of Rheology 27(3): 201-210.
[39] Rossikhin Yu A., Shitikova M.V., 1997, Applications of fractional calculus to dynamic problems of linear and nonlinear heredity mechanics of solids, Applied Mechanics Reviews 50(1): 15-67.
[40] Muslih S.I., Baleanu D., 2005, Hamiltonian formulation of systems with linear velocities within riemann–liouville fractional derivatives, Journal of Mathematical Analysis and Applications 304(2): 599-606.
[41] Podlubny I., 1999, Fractional Differential Equations, Academic Press, New York.
[42] Povstenko Y.Z., 2005, Fractional heat conduction equation and associated thermal stresses, Journal of Thermal Stresses 28(1): 83-102.
[43] Sherief H.H., El-Sayed A.M.A., Abd El-Latief A.M., 2010, Fractional order theory of thermoelasticity, International Journal of Solids and Structures 47(2): 269-273.
[44] Youssef H.Y., 2010, Theory of fractional order generalized Thermoelasticity, Transaction of the American Society of Mechanical Engineers-Journal of Applied Mechanics 132(6): 1-7.
[45] Ezzat M.A., 2010, Thermoelectric MHD non-newtonian fluid with fractional derivative heat transfer, Physica B: Condensed Matter 405(19): 4188-4194.
[46] Ezzat M.A., 2011, Magneto-thermoelasticity with thermoelectric properties and fractional derivative heat transfer, Physica B: Condensed Matter 406(1): 30-35.
[47] Jumarie G., 2010, Derivation and solutions of some fractional black scholes equations in coarsegrained space and time, Application to Merton’s optimal portfolio, Computers & Mathematics with Applications 59(3): 1142-1164.
[48] Ezzat M.A., Mohsen A., 2011, Fractional order theory of thermoelastic diffusion, Journal of Thermal Stresses 34: 851-872.
[49] Pal P., Sur A., Kanoria M.,2015, Thermo-viscoelastic interaction subjected to fractional fourier law with three phase lag effects, Journal of Solid Mechanics 7(4): 400-415.
[50] Abbas I.A., 2015, Eigen value approach on fractional order theory of thermoelastic diffusion problem for an infinite elastic medium with a spherical cavity, Applied Mathematical Modelling 39: 6196-6206.
[51] Honig G., Hirdes U., 1984, A method for the numerical inversion of the Laplace transform, Journal of Computational and Applied Mathematics 10(1): 113-132.
[52] Sinha M., Bera R.K., 2003, Eigenvalue approach to study the effect of rotation and relaxation time in generalizedthermoelasticity, Computers & Mathematics with Applications 46: 783-792.
[53] Roychoudhuri S.K., 1985, Effect of rotation and relaxation times on plane waves in generalized Thermoelasticity, Journal of Elasticity 15(1): 59-68.
[54] Salgado B.F.M., Sampayo R.R., Hernandez A.T., Fuentes C., 2017, Application of Fractional Calculus to Oil Industry, The world’s Leading Publisher of Open Access Books, Built by Scientists.
[55] Hamza F., Abdou M., Abd El-Latief A.M., 2014, Generalized fractional thermoelasticity associate with two relaxation times, Journal of Thermal Stresses 37(9): 1080-1098.
