Abstract
The present work is concerned with the solution of a problem on thermoelastic interactions in a functional graded material due to thermal shock in the context of the fractional order three-phase lag model. The governing equations of fractional order generalized thermoelasticity with three-phase lag model for functionally graded materials (FGM) (i.e., material with spatially varying material properties) are established. The analytical solution in the transform domain is obtained by using the eigenvalue approach. The inversion of Laplace transform is done numerically. The graphical results indicate that the fractional parameter has significant effects on all the physical quantities. Thus, we can consider the theory of fractional order generalized thermoelasticity an improvement on studying elastic materials.
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ABEL N H. Solution of some problems in using integrales olefines [J]. Werke, 1823, 1: 10.
CAPUTO M. Linear model of dissipation whose Q is always frequency independent [J]. Geophysical Journal of the Royal Astronomical Society, 1967, 13: 529–539.
CAPUTO M. Vibrations on an infinite viscoelastic layer with a dissipative memory [J]. Journal of the Acoustic Society of America, 1974, 56: 897–904.
CAPUTO M, MAINARDI F. A new dissipation model based on memory mechanism [J]. Pure and Applied Geophysics, 1971, 91: 134–147.
CAPUTO M, MAINARDI F. Linear model of dissipation in an elastic solids [J]. Rivista del Nuovo Cimento, 1971, 1: 161–198.
POVSTENKO Y Z. Fractional heat conduction equation and associated thermal stresses [J]. J Therm Stress, 2005, 28: 83–102.
POVSTENKO Y Z. Thermoelasticity that uses fractional heat conduction equation [J]. Journal of Mathematical Stresses, 2009, 162: 296–305.
SHERIEF H H, EL-SAYED A M A, ABD EL-LATIEF A M. Fractional order theory of thermoelasticity [J]. Int J Solids Struct, 2010, 47: 269–273.
YOUSSEF H H. Theory of fractional order generalized thermoelasticity [J]. J Heat Transf (ASME), 2010, 132: 1–7.
EZZAT M A. Theory of fractional order in generalized thermoelectric MHD. Applied Mathematical Modelling, 2011, 35: 4965–4978.
EZZAT M A. Magneto-thermoelasticity with thermoelectric properties and fractional derivative heat transfer [J]. Phys B, 2011, 406: 30–35.
TZOU D. A unified field approach for heat conduction from macro-to micro-scales [J]. ASME J Heat Transfer, 1995, 117: 8–16.
CHANDRASEKHARAIAH D. Hyperbolic thermoelasticity: A review of recent literature [J]. Appl Mech Rev, 1998, 51: 705–729.
ROYCHOUDHURI S. On thermoelastic three-phase-lag model [J]. J Thermal Stresses, 2007, 30: 231–238.
EZZAT M, ELKARAMANY A, FAYIK M. Fractional order theory in thermoelastic solid with three-phase lag heat transfer [J]. Arch Appl Mech, 2012, 82: 557–572.
SURESH S, MORTENSEN A. Fundamentals of functionally graded materials [M]. London: Institute of Materials Communications Ltd, 1998.
MALLIK S H, KANORIA M. Generalized thermoelastic functionally graded solid with a periodically varying heat source [J]. International Journal of Solids and Structures, 2007, 44: 7633–7645.
DAS P, KANORIA M. Magneto-thermoelastic response in a functionally graded isotropic unbounded medium under a periodically varying heat source [J]. Int J Thermophys, 2009, 30: 2098–2121.
ABBAS I A. Three-phase lag model on thermoelastic interaction in an unbounded fiber-reinforced anisotropic medium with a cylindrical cavity [J]. Journal of Computational and Theoretical Nanoscience, 2014, 11(4): 987–992.
ABBAS I A, MOHAMED I. Generalized thermoelsticity of the thermal shock problem in an isotropic hollow cylinder and temperature dependent elastic moduli [J]. Chinese Physics B, 2012, 21(1): 4601.
ABBAS I A, YOUSSEF H M. A nonlinear generalized thermoelasticity model of temperature-dependent materials using finite element method [J]. International Journal of Thermophysics, 2012, 33(7): 1302–1313.
ABBAS I A. Generalized magneto-thermoelastic interaction in a fiber-reinforced anisotropic hollow cylinder [J]. International Journal of Thermophysics, 2012, 33(3): 567–579.
ABBAS I A, OTHMAN M I. Generalized thermoelastic interaction in a fiber-reinforced anisotropic half-space under hydrostatic initial stress [J]. Journal of Vibration and Control, 2012, 18(2): 175–182.
ZENKOUR A M, ABBAS I A. Magneto-thermoelastic response of an infinite functionally graded cylinder using the finite element model [J]. Journal of Vibration and Control, 2014, 20(12): 1907–1919.
ABBAS I A. A GN model based upon two-temperature generalized thermoelastic theory in an unbounded medium with a spherical cavity [J]. Applied Mathematics and Computation, 2014, 245: 108–115.
ABBAS I A. A GN model for thermoelastic interaction in an unbounded fiber-reinforced anisotropic medium with a circular hole [J]. Applied Mathematics Letters, 2013, 26(2): 232–239.
ABBAS I A, KUMAR R. Interaction due to a mechanical source in transversely isotropic micropolar media [J]. Journal of Vibration and Control, 2014, 20(11): 1607–1621.
ABBAS I A. Eigenvalue approach for an unbounded medium with a spherical cavity based upon two-temperature generalized thermoelastic theory [J]. Journal of Mechanical Science and Technology, 2014, 28(10): 4193–4198.
ABBAS I A. Fractional order GN model on thermoelastic interaction in an infinite fibre-reinforced anisotropic plate containing a circular hole [J]. Journal of Computational and Theoretical Nanoscience, 2014, 11(2): 380–384.
KUMAR R, GUPTA V, ABBAS I A. Plane deformation due to thermal source in fractional order thermoelastic media [J]. Journal of Computational and Theoretical Nanoscience, 2013, 10(10): 2520–2525.
OTHMAN M I A, ABBAS I A. Generalized thermoelasticity of thermal-shock problem in a non-homogeneous isotropic hollow cylinder with energy dissipation [J]. International Journal of Thermophysics, 2012, 33(5): 913–923.
ABBAS I A, ZENKOUR A M. LS model on electro-magneto-thermoelastic response of an infinite functionally graded cylinder [J]. Composite Structures, 2013, 96: 89–96.
ABBAS I A, KUMAR R. Deformation due to thermal source in micropolar thermo-elastic media with thermal and conductive temperatures [J]. Journal of Computational and Theoretical Nanoscience 2013, 10(9): 2241–2247.
ABBAS I A, ZENKOUR A M. The effect of rotation and initial stress on thermal shock problem for a fiber-reinforced anisotropic half-space using Green-Naghdi theory [J]. Journal of Computational and Theoretical Nanoscience, 2014, 11(2): 331–338.
ABBAS I A. Eigenvalue approach to fractional order generalized magneto-thermoelastic medium subjected to moving heat source [J]. Journal of Magnetism and Magnetic Materials, 2015, 377: 452–459.
STEHFEST H. Numerical inversion of Laplace transforms algorithm 368 [J]. Commun ACM, 1979, 13(1): 47–49.
GREEN A E, NAGHDI P M. On undamped heat waves in an elastic solid [J]. J Therm Stress, 1992, 15: 253–264.
GREEN A E, NAGHDI P M. Thermoelasticity without energy dissipation [J]. J Elast, 1993, 31: 189–208.
LAHIRI A, DAS B, DATTA B. Eigenvalue value approach to study the effect of rotation in three-dimensional problem of generalized thermoelasticity [J]. International Journal of Applied Mechanics and Engineering, 2010, 15: 99–120.
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Abbas, I.A. Generalized thermoelastic interaction in functional graded material with fractional order three-phase lag heat transfer. J. Cent. South Univ. 22, 1606–1613 (2015). https://doi.org/10.1007/s11771-015-2677-5
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DOI: https://doi.org/10.1007/s11771-015-2677-5