DOI QR코드

DOI QR Code

Modeling of fractional magneto-thermoelasticity for a perfect conducting materials

  • Ezzat, M.A. (Department of Mathematics, Faculty of Education, Alexandria University) ;
  • El-Bary, A.A. (Arab Academy for Science and Technology)
  • 투고 : 2016.03.11
  • 심사 : 2016.04.06
  • 발행 : 2016.10.25

초록

A unified mathematical model of the equations of generalized magneto-thermoelasticty based on fractional derivative heat transfer for isotropic perfect conducting media is given. Some essential theorems on the linear coupled and generalized theories of thermoelasticity e.g., the Lord- Shulman (LS) theory, Green-Lindsay (GL) theory and the coupled theory (CTE) as well as dual-phase-lag (DPL) heat conduction law are established. Laplace transform techniques are used. The method of the matrix exponential which constitutes the basis of the state-space approach of modern theory is applied to the non-dimensional equations. The resulting formulation is applied to a variety of one-dimensional problems. The solutions to a thermal shock problem and to a problem of a layer media are obtained in the present of a transverse uniform magnetic field. According to the numerical results and its graphs, conclusion about the new model has been constructed. The effects of the fractional derivative parameter on thermoelastic fields for different theories are discussed.

키워드

참고문헌

  1. Abbas I.A. (2015), "Eigenvalue approach to fractional order generalized magneto-thermoelastic medium subjected to moving heat source", J. Mag. Mag. Mat., 377(3), 452-459. https://doi.org/10.1016/j.jmmm.2014.10.159
  2. Adolfsson, K., Enelund, M. and Olsson, P. (2005), "On the fractional order model of visco- elasticity", Mech. Time-Depend. Mat., 9(1), 15-34. https://doi.org/10.1007/s11043-005-3442-1
  3. Biot, M. (1956), "Thermoelasticity and irreversible thermodynamics", J. Appl. Phys., 27 (3), 240-253. https://doi.org/10.1063/1.1722351
  4. Caputo, M. (1974), "Vibrations on an infinite viscoelastic layer with a dissipative memory", J. Acoust. Soc. Am., 56(3), 897-904. https://doi.org/10.1121/1.1903344
  5. Cattaneo, C. (1958), "Sur une forme de l'equation de la Chaleur eliminant le paradoxe d'une propagation instantaneee", C.R. Acad. Sci. Paris, 247 (3), 431-433.
  6. Chadwick, P. (1960), Thermoelasticity-The Dynamic Theory, (Eds. R. Hill and I.N. Sneddon), Progress in Solid Mechanics, 1, North-Holland Publishers, Amsterdam, 236.
  7. Chandrasekharaiah, D.S. (1998), "Hyperbolic thermoelasticity, A review of recent literature", Appl. Mech. Rev., 51 (12), 705-729. https://doi.org/10.1115/1.3098984
  8. Choudhuri, S. (1984), "Electro-magneto-thermo-elastic waves in rotating media with thermal relaxation", Int. J. Eng. Sci., 22(5), 519-530. https://doi.org/10.1016/0020-7225(84)90054-5
  9. Dreyer, W. and Struchtrup, H. (1993), "Heat pulse experiments revisited", Continuum. Mech. Therm., 5(1), 3-50. https://doi.org/10.1007/BF01135371
  10. Duhamel, J.H. (1937), "Second memoir sur les phenomenes thermomechanique", J. de L' Ecole Polytechnique, 15(25), 1-57.
  11. El-Karamany, A.S. and Ezzat, M.A. (2002), "On the boundary integral formulation of thermo-viscoelasticity theory", Int. J. Eng. Sci., 40(17), 1943-1956. https://doi.org/10.1016/S0020-7225(02)00043-5
  12. El-Karamany, A.S. and Ezzat, M.A. (2004a), "Discontinuities in generalized thermo- viscoelasticity under four theories", J. Therm. Stress., 27(12), 1187-1212. https://doi.org/10.1080/014957390523598
  13. El-Karamany, A.S. and Ezzat, M.A. (2004b), "Boundary integral equation formulation for the generalized thermoviscoelasticity with two relaxation times", Appl. Math. Comput., 151(2), 347-362. https://doi.org/10.1016/S0096-3003(03)00345-X
  14. El-Karamany, A.S. and Ezzat, M.A. (2004c), "Thermal shock problem in generalized thermo-viscoelasticity under four theories", Int. J. Eng. Sci., 42 (7), 649-671. https://doi.org/10.1016/j.ijengsci.2003.07.009
  15. El-Karamany, A.S. and Ezzat, M.A. (2005), "Propagation of discontinuities in thermopiezoelectric rod", J. Therm. Stress., 28 (10), 997-1030. https://doi.org/10.1080/01495730590964954
  16. El-Karamany, A.S. and Ezzat, M.A. (2009), "Uniqueness and reciprocal theorems in linear micropolar electro-magnetic thermoelasticity with two relaxation times", Mech. Time-Depend. Mater., 13(1), 93-115. https://doi.org/10.1007/s11043-008-9068-3
  17. El-Karamany, A.S. and Ezzat, M.A. (2011a), "On fractional thermoelastisity", Math. Mech. Solids, 16(3), 334-346. https://doi.org/10.1177/1081286510397228
  18. El-Karamany, A.S. and Ezzat, M.A. (2011b), "Convolutional variational principle, reciprocal and uniqueness theorems in linear fractional two-temperature thermoelasticity", J. Therm. Stress., 34(3), 264-284. https://doi.org/10.1080/01495739.2010.545741
  19. El-Karamany, A.S. and Ezzat, M.A. (2014), "On the dual-phase-lag thermoelasticity theory", Mecc., 49(1), 79-89. https://doi.org/10.1007/s11012-013-9774-z
  20. Ezzat, M.A. (1997), "State space approach to generalized magneto-thermoelasticity with two relaxation times in a medium of perfect conductivity", Int. J. Eng. Sci., 35(8), 741-752. https://doi.org/10.1016/S0020-7225(96)00112-7
  21. Ezzat, M.A. (2001), "Free convection effects on perfectly conducting fluid", Int. J. Eng. Sci., 39 (7), 799-819. https://doi.org/10.1016/S0020-7225(00)00059-8
  22. Ezzat, M.A. (2006), "The relaxation effects of the volume properties of electrically conducting viscoelastic material", Mat. Sci. Eng. B, 130(1-3), 11-23. https://doi.org/10.1016/j.mseb.2006.01.020
  23. Ezzat, M.A. (2010), "Thermoelectric MHD non-Newtonian fluid with fractional derivative heat transfer", Phys. B, 405 (19), 4188-4194. https://doi.org/10.1016/j.physb.2010.07.009
  24. Ezzat, M.A. (2011a), "Magneto-thermoelasticity with thermoelectric properties and fractional derivative heat transfer", Phys. B, 406 (1), 30-35. https://doi.org/10.1016/j.physb.2010.10.005
  25. Ezzat, M.A. (2011b), "Theory of fractional order in generalized thermoelectric MHD", Appl. Math. Model., 35(10), 4965-4978. https://doi.org/10.1016/j.apm.2011.04.004
  26. Ezzat, M.A. (2011c), "Thermoelectric MHD with modified Fourier's law", Int. J. Therm. Sci., 50(4), 449-455. https://doi.org/10.1016/j.ijthermalsci.2010.11.005
  27. Ezzat, M.A. (2012), "State space approach to thermoelectric fluid with fractional order heat transfer", Heat Mass Transf., 48(1), 71-82. https://doi.org/10.1007/s00231-011-0830-8
  28. Ezzat, M.A. and Abd Elaal, M.Z. (1997), "Free convection effects on a viscoelastic boundary layer flow with one relaxation time through a porous medium", J. Frank. Ins., 334(4), 685-706. https://doi.org/10.1016/S0016-0032(96)00095-6
  29. Ezzat, M.A. and Awad, E.S. (2010), "Constitutive relations, uniqueness of solution, and thermal shock application in the linear theory of micropolar generalized thermoelasticity involving two temperatures", J. Therm. Stress., 33(3), 226-250. https://doi.org/10.1080/01495730903542829
  30. Ezzat, M.A. and El-Bary, A.A. (2012), "MHD free convection flow with fractional heat conduction law", MHD, 48(4), 587-606.
  31. Ezzat, M.A. and El-Karamany, A.S. (2002a), "The uniqueness and reciprocity theorems for generalized thermoviscoelasticity for anisotropic media", J. Therm. Stress., 25(6), 507-522. https://doi.org/10.1080/01495730290074261
  32. Ezzat, M.A. and El-Karamany, A.S. (2002b), "The uniqueness and reciprocity theorems for generalized thermo-viscoelasticity with two relaxation times", Int. J. Eng. Sci., 40(11), 1275-1284. https://doi.org/10.1016/S0020-7225(01)00099-4
  33. Ezzat, M.A. and El-Karamany, A.S. (2003a), "On uniqueness and reciprocity theorems for generalized thermoviscoelasticity with thermal relaxation", Canad. J. Phys., 81(6), 823-833. https://doi.org/10.1139/p03-070
  34. Ezzat, M.A. and El-Karamany, A.S. (2003b), "Magnetothermoelasticity with two relaxation times in conducting medium with variable electrical and thermal conductivity", Appl. Math. Comput., 142 (2-3), 449-467. https://doi.org/10.1016/S0096-3003(02)00313-2
  35. Ezzat, M.A. and El-Karamany, A.S. (2006), "Propagation of discontinuities in magneto-thermoelastic half-space", J. Therm. Stress., 29(4), 331-358. https://doi.org/10.1080/01495730500360526
  36. Ezzat, M.A. and El-Karamany, A.S. (2011a), "Fractional order theory of a perfect conducting thermoelastic medium", Canad. J. Phys., 89(3), 311-318. https://doi.org/10.1139/P11-022
  37. Ezzat, M.A. and El-Karamany, A.S. (2011b), "Fractional order heat conduction law in magneto-thermoelasticity involving two temperatures", ZAMP, 62(5), 937- 952. https://doi.org/10.1007/s00033-011-0126-3
  38. Ezzat, M.A. and El-Karamany, A.S. (2011c), "Theory of fractional order in electro-thermoelasticity", Eur. J. Mech. A- Solid, 30(4), 491-500. https://doi.org/10.1016/j.euromechsol.2011.02.004
  39. Ezzat, M.A. and El-Karamany, A.S. (2012), "Fractional thermoelectric viscoelastic materials", J. Appl. Poly. Sci., 72(3), 2187-2199.
  40. Ezzat, M.A. and Othman, M.I. (2002), "State space approach to generalized magnetothermoelasticity with thermal relaxation in a medium of perfect conductivity", J. Therm. Stress., 25(5), 409-429. https://doi.org/10.1080/01495730252890168
  41. Ezzat, M.A. and Youssef, H.M. (2010), "Stokes' first problem for an electro-conducting micropolar fluid with thermoelectric properties", Canad. J. Phys., 88(1), 35-48. https://doi.org/10.1139/P09-100
  42. Ezzat, M.A., El-Karamany, A.S. and Ezzat, S.M. (2012), "Two-temperature theory in magneto-thermoelasticity with fractional order dual-phase-lag heat transfer", Nucl. Eng. Des., 252(11), 267- 277. https://doi.org/10.1016/j.nucengdes.2012.06.012
  43. Ezzat, M.A., El-Karamany, A.S., Zakaria, M.A. and Samaan, A.A. (2003), "The relaxation effects of the volume properties of viscoelastic material in generalized thermoelasticity with thermal relaxation", J. Therm. Stress., 26 (7), 671-690. https://doi.org/10.1080/713855997
  44. Ezzat, M.A., El-Karamany, S.A. and Smaan, A.A. (2004), "The dependence of the modulus of elasticity on reference temperature in generalized thermoelasticity with thermal relaxation", Appl. Math. Comput., 147 (1), 169-189. https://doi.org/10.1016/S0096-3003(02)00660-4
  45. Ezzat, M.A., Othman, M.I. and El-Karamany, A.S. (2002), "State space approach to two-dimensional generalized thermo-viscoelasticity with two relaxation times", Int. J. Eng. Sci., 40(11), 1251-1274. https://doi.org/10.1016/S0020-7225(02)00012-5
  46. Ezzat, M.A., Othman, M.I. and Helmy, K.A. (1999), "A problem of a micropolar magnetohydrodynamic boundary-layer flow", Canad. J. Phys., 77(10), 813-827. https://doi.org/10.1139/y99-083
  47. Glass, D.E. and Vick, B. (1985), "Hyperbolic heat conduction with surface radiation", Int. J. Heat Mass Transf., 28(10), 1823-1830. https://doi.org/10.1016/0017-9310(85)90204-2
  48. Green, A. and Lindsay, K. (1972), "Thermoelasticity", J. Elasticity, 2(1), 1-7. https://doi.org/10.1007/BF00045689
  49. Hamza, F., Abdou, M. and Abd El-Latief, A.M. (2014), "Generalized fractional thermoelasticity associated with two relaxation times", J. Therm. Stress., 37 (9), 1080-1093. https://doi.org/10.1080/01495739.2014.936196
  50. Hetnarski, R.B. and Eslami, M.R. (2009), Thermal stresses, advanced theory and Applications, New York (NY): Springer.
  51. Hetnarski, R.B. and Ignaczak, J. (1999), "Generalized thermoelasticity", J. Therm. Stress., 22(4-5), 451-476. https://doi.org/10.1080/014957399280832
  52. Honig, G. and Hirdes, U. (1984), "A method for the numerical inversion of the Laplace transform", Comput. Appl. Math., 10(1), 113-132. https://doi.org/10.1016/0377-0427(84)90075-X
  53. Horgan, C.O. and Quintanilla, R. (2005), "Spatial behaviour of solutions of the dual-phase-lag heat equation", Math. Meth. Appl. Sci., 28 (1), 43-57. https://doi.org/10.1002/mma.548
  54. Ignaczak, J. (1989), Generalized thermoelasticity and its applications, (Ed., Hetnarski, R.B.), Thermal Stresses III. Elsevier, New York.
  55. Ignaczak, J. and Ostoja-starzeweski, M. (2009), Thermoelasticity with finite wave speeds, Oxford University Press, Oxford, UK.
  56. Joseph, D.D. and Preziosi, L. (1989), "Heat waves", Rev. Mod. Phys., 61(1), 41-73. https://doi.org/10.1103/RevModPhys.61.41
  57. Joseph, D.D. and Preziosi, L. (1990), "Addendum to the paper: Heat waves", Rev. Mod. Phys., 62(2), 375-391. https://doi.org/10.1103/RevModPhys.62.375
  58. Jou, D. and Criado-Sancho, M. (1998), "Thermodynamic stability and temperature overshooting in dual-phase-lag heat transfer", Phys. Lett. A, 248(2-4), 172-178. https://doi.org/10.1016/S0375-9601(98)00573-8
  59. Kaliski, S. and Petykiewicz, J. (1959), "Equations of motion coupled with the field of temperatures in a magnetic field involving mechanical and electromagnetic relaxation for anisotropic bodies", Proc. of Vib. Prob., 4(3), 83- 101.
  60. Knopoff, L. (1955), "The interaction between elastic wave motion and a magnetic field in electrical conductors", J. Geophys. Res., 60 (4), 441-456. https://doi.org/10.1029/JZ060i004p00441
  61. Lord, H. and Shulman, Y. (1967), "A generalized dynamical theory of thermoelasticity", J. Mech. Phys. Solids, 15(5), 299-309. https://doi.org/10.1016/0022-5096(67)90024-5
  62. Nayfeh, A. and Nemat-Nasser, S. (1973), "Electromagneto-thermoelastic plane waves in solids with thermal relaxation", J. Appl. Mech. Ser. E, 39(1), 108-113.
  63. Nowinski, J.L. (1978), Theory of Thermoelasticity with Applications, Sijthoff & Noordhoff International, Alphen Aan Den Rijn.
  64. Ogata, K. (1967), State space analysis control system, Prentice-Hall, Englewood Cliffs, N.J. Chap. 6.
  65. Othman, M.I., Ezzat, M.A., Zaki, S.A. and El-Karamany, A.S. (2002), "Generalized thermo-viscoelastic plane waves with two relaxation times", Int. J. Eng. Sci., 40(12), 1329-1347. https://doi.org/10.1016/S0020-7225(02)00023-X
  66. Quintanilla, R. and Racke, R. (2006), "A note on stability in dual-phase-lag heat conduction", Int. J. Heat Mass Transf., 49 (7-8), 1209-1213. https://doi.org/10.1016/j.ijheatmasstransfer.2005.10.016
  67. Shereif, H.H. (1992), "Fundamental solution for thermoelasticity with two relaxation times", Int. J. Eng. Sci., 30(7), 861-870. https://doi.org/10.1016/0020-7225(92)90015-9
  68. Sherief, H.H. (1986), "Fundamental solution of generalized thermoelastic problem for short times", J. Therm. Stress., 9(2), 151-164. https://doi.org/10.1080/01495738608961894
  69. Sherief, H.H. and Ezzat, M.A. (1998), "A Problem in generalized magneto-thermoelasticity for an infinitely long annular cylinder", J. Eng. Math., 34 (1-4), 387-402. https://doi.org/10.1023/A:1004376014083
  70. Sherief, H.H., El-Said, A. and Abd El-Latief, A. (2010), "Fractional order theory of thermoelasticity", Int. J. Solids Struct., 47(2), 269-275. https://doi.org/10.1016/j.ijsolstr.2009.09.034
  71. Suhubi, E. (1975), Thermoelastic Solids, (Ed., A.C. Eringen), Cont. Phys., vol. II, Academic Press, New York, Chapter 21.
  72. Truesdell, C. and Muncaster, R.G. (1980), Fundamental of Maxwell's kinetic theory of a simple monatomic gas, Acad Press, NewYork.
  73. Tzou, D.Y. (1995), "A unified filed approach for heat conduction from macro to macroscales", J. Heat Transf. ASME, 117(1), 8-16. https://doi.org/10.1115/1.2822329
  74. Zencour, A.M. and Abbas, I.A. (2015), "Electro-magneto-thermo-elastic response of infinite functionally graded cylinders without energy dissipation", J. Mag. Mag. Mat., 395 (12), 123-129. https://doi.org/10.1016/j.jmmm.2015.07.038

피인용 문헌

  1. Fractional order theory to an infinite thermo-viscoelastic body with a cylindrical cavity in the presence of an axial uniform magnetic field vol.31, pp.5, 2017, https://doi.org/10.1080/09205071.2017.1285728
  2. Two-temperature theory in Green–Naghdi thermoelasticity with fractional phase-lag heat transfer vol.24, pp.2, 2018, https://doi.org/10.1007/s00542-017-3425-6
  3. Thermoelectric viscoelastic materials with memory-dependent derivative vol.19, pp.5, 2016, https://doi.org/10.12989/sss.2017.19.5.539
  4. Fractional magneto-thermoelastic materials with phase-lag Green-Naghdi theories vol.24, pp.3, 2017, https://doi.org/10.12989/scs.2017.24.3.297
  5. Rayleigh waves in anisotropic magnetothermoelastic medium vol.6, pp.3, 2016, https://doi.org/10.12989/csm.2017.6.3.317
  6. Variability of thermal properties for a thermoelastic loaded nanobeam excited by harmonically varying heat vol.20, pp.4, 2016, https://doi.org/10.12989/sss.2017.20.4.451
  7. Causes of uncertainty in thermoelasticity measurements of structural elements vol.20, pp.5, 2016, https://doi.org/10.12989/sss.2017.20.5.539
  8. Dynamic model of fractional thermoelasticity due to ramp-type heating with two relaxation times vol.44, pp.11, 2019, https://doi.org/10.1007/s12046-019-1197-7
  9. Analysis of Time-Fractional Heat Transfer and its Thermal Deflection in a Circular Plate by a Moving Heat Source vol.25, pp.3, 2020, https://doi.org/10.2478/ijame-2020-0040
  10. Transient memory response of a thermoelectric half-space with temperature-dependent thermal conductivity and exponentially graded modulii vol.38, pp.4, 2021, https://doi.org/10.12989/scs.2021.38.4.447
  11. Thermoelastic response of a nonhomogeneous elliptic plate in the framework of fractional order theory vol.91, pp.7, 2016, https://doi.org/10.1007/s00419-021-01962-w