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A novel model of fractional thermal and plasma transfer within a non-metallic plate

  • Ezzat, Magdy A. (Department of Mathematics, College of Science and Arts, Qassim University)
  • Received : 2020.03.06
  • Accepted : 2020.09.16
  • Published : 2021.01.25

Abstract

While in several publications the thermo-viscoelastic properties of solids have been documented, no attempt has been made to examine the action of coupled thermal and plasma wave in viscoelastic materials. In this paper, a new mathematical model for thermal and plasma transfer in an organic semiconductor was constructed with a time-fractional derivative of order α(0 < α ≤ 1) and a time-fractional integral of order β(0 < β ≤ 2), respectively. A two-dimensional problem is viewed for a half-space of viscoelastic thin-walled semiconductor whose surface is traction free and subjected to a heat flux with an exponentially decaying pulse. Laplace and Fourier's integral transforms are utilized. The carrier density, temperature, thermal stress, and viscoelastic displacement distributions have been obtained through the use of the theoretical model together with plasma and thermo-viscoelastic effects. The inversion technique for Fourier and Laplace transforms is carried out using a numerical technique based on Fourier series expansions. Comparisons are made with the results anticipated thru the coupled idea and generalized theory. The influence of the fractional-order parameter on all the regarded fields is examined.

Keywords

References

  1. Abbas, I.A. (2006), "Natural frequencies of a poroelastic hollow cylinder", Acta Mecc., 186(1-4), 229-237. https://doi.org/10.1007/s00707-006-0314-y
  2. Abbas, I.A. (2014), "Nonlinear transient thermal stress analysis of thick-walled FGM cylinder with temperature-dependent material properties", Mecc., 49(7), 1697-1708. https://doi.org/10.1007/s11012-014-9948-3
  3. Abbas, I.A. and Abo-Dahab, S.M. (2014), "On the numerical solution of thermal shock problem for generalized magneto-thermoelasticity for an infinitely long annular cylinder with variable thermal conductivity", J. Comput. Theor. Nanosci., 11(3), 607-618. https://doi.org/10.1166/jctn.2014.3402
  4. Abbas, I.A. and Kumar, R. (2016), "2D deformation in initially stressed thermoelastic half-space with voids", Steel Compos. Struct., Int. J., 20(5), 1103-1117. https://doi.org/10.12989/scs.2016.20.5.1103
  5. Abbas, I.A. and Marin, M. (2017), "Analytical solution of thermoelastic interaction in a half-space by pulsed laser heating", Phys. E., 87(3), 254-260. https://doi.org/10.1016/j.physe.2016.10.048
  6. 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
  7. Alzahrani, F.S. and Abbas, I.A. (2016), "The effect of magnetic field on a thermoelastic fiber-reinforced material under GN-III theory", Steel Compos. Struct., Int. J., 22(2), 369-386. https://doi.org/10.12989/scs.2016.21.4.791
  8. Alzahrani, F.S. and Abbas, I.A. (2018), "Generalized photo-thermo-elastic interaction in a semiconductor plate with tworelaxation times", Thin-Wall. Struct., 129, 342-348. https://doi.org/10.1016/j.tws.2018.04.011
  9. Bagley, R.L. and Torvik, P.J. (1983), "A theoretical basis for the application of fractional calculus to viscoelasticity", J. Rheol., 27(3), 201-210. https://doi.org/10.1122/1.549724
  10. Biot, M.A. (1956), "Thermoelasticity and irreversible thermodynamics", J. Appl. Mech. Tech. Phys., 27(3), 240-253. https://doi.org/10.1063/1.1722351
  11. Caputo, M. (1974), "Vibrations on an infinite viscoelastic layer with a dissipative memory", J. Acous. Soc. Am., 56(3), 897-904. https://doi.org/10.1121/1.1903344
  12. Caputo, M. and Mainardi, F. (1971), "A new dissipation model based on memory mechanism", Pure Appli. Geophys., 91, 134-147. https://doi.org/10.1007/BF00879562
  13. 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.
  14. 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
  15. 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
  16. Ezzat, M.A. (2006), "The relaxation effects of the volume properties of electrically conducting viscoelastic material", J. Mater. Sci. Eng. B., 130(1-3), 11-23. https://doi.org/10.1016/j.mseb.2006.01.020
  17. Ezzat, M.A. and El-Bary, A.A. (2012), "MHD free convection flow with fractional heat conduction law", MHD, 48(4), 587-606. https://doi.org/10.22364/mhd
  18. Ezzat, M.A. and El-Bary, A.A. (2016), "Modeling of fractional magneto-thermoelasticity for a perfect conducting materials", Smart Struct. Syst., Int. J., 18(4), 707-731. https://doi.org/10.12989/sss.2016.18.4.707
  19. Ezzat, M.A., Othman, M.I. and El-Karamany, A.S. (2001), "State space approach to generalized thermo-viscoelasticity with two relaxation times", Int. J. Eng. Sci., 40(3), 283-302. https://doi.org/10.1016/S0020-7225(01)00045-3
  20. Ezzat, M.A., Alsowayan, N.S., Al-Muhiameed, Z.I.A. and Ezzat, S.M. (2014), "Fractional modelling of Pennes' bioheat transfer equation", Heat Mass Trans., 50(7), 907-914. https://doi.org/10.1007/s00231-014-1300-x
  21. Fujita, Y. (1990), "Integrodifferential equation which interpolates the heat equation and wave equation (I)", Osaka J. Math., 27(2), 309-321. https://doi.org/10.18910/4060
  22. Gordon, J.P., Leite, R.C.C., Moore, R., Porto, S.P.S. and Whinnery, J.R. (1965), "Long-transient effects in lasers with inserted liquid samples", J. Appl. Phys., 36(1), 3-12. https://doi.org/10.1063/1.1713919
  23. Gross, B. (1953), Mathematical Structure of the Theories of Viscoelasticity, Hemann, Paris, France.
  24. Gurtin, M.E. and Sternberg, E. (1962), "On the linear theory of viscoelasticity", Arch. Rat. Mech. Anal., 11(1), 182-191. https://doi.org/10.1007/BF00253942
  25. Hobiny, A. and Abbas, I.A. (2018), "Analytical solutions of photo-thermo-elastic waves in a non-homogenous semiconducting material", Res. Phys., 10, 385-390. https://doi.org/10.1016/j.rinp.2018.06.035
  26. Honig, G. and Hirdes, U. (1984), "A method for the numerical inversion of the Laplace transform", J. Compos. Appl. Math., 10(1), 113-132. https://doi.org/10.1016/0377-0427(84)90075-X
  27. Ignaczak, J. (1989), Generalized Thermoelasticity and Its Applications. (Ed., Hetnarski, R.B.), Thermal Stresses III, Elsevier, New York, USA.
  28. Ilioushin, A.A. (1968), "The approximation method of calculating the constructors by linear thermal viscoelastic theory", Mekhanika. Polimerov, Riga, 2, 168-178.
  29. Ilioushin, A.A. and Pobedria, B.E. (1970), Mathematical Theory of Thermal Viscoelasticity, Nauka, Moscow, Russia.
  30. Kimmich, R. (2002), "Strange kinetics, porous media, and NMR", Chem. Phys., 284(1-2), 243-285. https://doi.org/10.1016/S0301-0104(02)00552-9
  31. Kiriyakova, V. (1994), "Generalized fractional calculus and applications", In: Pitman Research Notes in Mathematics Series, Volume 301, Longman-Wiley, New York, USA.
  32. Kumar, R. and Abbas, I.A. (2013), "Deformation due to thermal source in micropolar thermoelastic media with thermal and conductive temperatures", J. Comput. Theor. Nanosci., 10(9), 2241-2247. https://doi.org/10.1166/jctn.2013.3193
  33. Kumar, R. and Devi, S. (2017), "Thermoelastic beam in modified couple stress thermoelasticity induced by laser pulse", Comput. Concrete, 19(6), 701-709. https://doi.org/10.12989/cac.2017.19.6.701
  34. Kumar, R., Sharma, N. and Lata, P. (2016a), "Thermomechanical interactions in a transversely isotropic magnetothermoelastic with and without energy dissipation with combined effects of rotation, vacuum and two temperatures", Appl. Math. Model., 40(13-14), 6560-6575. https://doi.org/10.1016/j.apm.2016.01.061
  35. Kumar, R., Sharma, N. and Lata, P. (2016b), "Effects of Hall current in a transversely isotropic magnetothermoelastic two temperature medium with rotation and with and without energy dissipation due to normal force", Struct. Eng. Mech., Int. J., 57(1), 91-103. https://doi.org/10.12989/sem.2016.57.1.091
  36. Kumar, R., Sharma, N. and Lata, P. (2018), "Effects of Hall current and two temperatures in transversely isotropic magnetothermoelastic with and without energy dissipation due to Ramp type heat", Mech. Adv. Mater. Struct., 24(8), 625-635. https://doi.org/10.1080/15376494.2016.1196769
  37. Kreuzer, L.B. (1971), "Ultralow gas concentration infrared absorption spectroscopy", J. Appl. Phys., 42(7), 2934-2943. https://doi.org/10.1063/1.1660651
  38. Lata, P. (2018), "Effect of energy dissipation on plane waves in sandwiched layered thermoelastic medium", Steel Compos. Struct., Int. J., 27(4), 439-451. https://doi.org/10.12989/scs.2018.27.4.439
  39. Lata, P., Kumar, R. and Sharma, N. (2016), "Plane waves in an anisotropic thermoelastic", Steel Compos. Struct., Int. J., 22(3), 567-587. https://doi.org/10.12989/scs.2016.22.3.567
  40. 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
  41. Lotfy, K., Kumar, R., Hassan, W. and Gabr, M. (2018), "Thermomagnetic effect with microtemperature in a semiconducting photothermal excitation medium", Appl. Math. Mech., 39(5),783-796. https://doi.org/10.1007/s10483-018-2339-9
  42. Mainardi, F. and Gorenflo, R. (2000), "On Mittag-Lettler-type function in fractional evolution processes", J. Comput. Appl. Math., 118(1-2), 283-299. https://doi.org/10.1016/S0377-0427(00)00294-6
  43. Itu, C., Ochsner, A., Vlase, S. and Marin, M.I. (2019), "Improved rigidity of composite circular plates through radial ribs", Proceedings of the Institution of Mechanical Engineers, Part L: Journal of Materials: Design and Applications, 233(8), 1585-1593. https://doi.org/10.1177/1464420718768049
  44. Meyers, M.A. and Chawla, K.K. (1999), Mechanical Behavior of Materials. Prentice-Hall, NJ, USA, Volume 98, pp. 103.
  45. Mukhopadhyay, S. and Kumar, R. (2009), "Thermoelastic interactions on two-temperature generalized thermoelasticity in an infinite medium with a cylindrical cavity", J. Therm. Stress., 32(4), 341-360. https://doi.org/10.1080/01495730802637183
  46. Opsal, J. and Rosencwaig, A. (1985), "Thermal and plasma wave depth profiling in silicon", Appl. Phys. Lett., 47(5), 498-500. https://doi.org/10.1063/1.96105
  47. Othman, M.I., Fekry, M. and Marin, M. (2020), "Plane waves in generalized magneto-thermo-viscoelastic medium with voids under the effect of initial stress and laser pulse heating", Struct. Eng. Mech., Int. J., 73(6), 621-629. https://doi.org/10.12989/sem.2020.73.6.621
  48. Podlubny, I. (1999), Fractional Differential Equations, Academic: New York, USA.
  49. Povstenko, Y.Z. (2005), "Fractional heat conduction equation and associated thermal stress", J. Therm. Stress., 28(1), 83-102. https://doi.org/10.1080/014957390523741
  50. Press, W.H., Flannery, B.P., Teukolsky, S.A. and Vetterling, W.T. (1992), Numerical Recipes, (2nd ed.), Cambridge University Press, Cambridge, UK.
  51. Sharma, S. and Sharma, K. (2014), "Influence of heat sources and relaxation time on temperature distribution in tissues", Int. J. Appl. Mech. Eng., 19(2), 427-433. https://doi.org/10.2478/ijame-2014-0029
  52. Sharma, S., Sharma, K. and Bhargava, R. (2013), "Effect of viscosity on wave propagation in anisotropic thermoelastic with Green-Naghdi theory type-II and type-III", Mat. Phys. Mech., 16,144-158.
  53. Sherief, H.H. and Abd El-Latief, M. (2016), "Modeling of variable Lame's Modulii for a FGM generalized thermoelastic half space", Lat. Am. J. Solids Struct., 13(4), 715-730. https://doi.org/10.1590/1679-78252086
  54. Sherief, H.H. and Ezzat, M.A. (1994), "Solution of the generalized problem of thermoelasticity in the form of series of functions", J. Therm. Stress., 17(1), 75-95. https://doi.org/10.1080/01495739408946247
  55. Sherief, H.H. and Hussein, E.M. (2018), "Contour integration solution for a thermoelastic problem of a spherical cavity", Appl. Math. Comput., 320, 557-571. https://doi.org/10.1016/j.amc.2017.10.024
  56. Sherief, H.H., El-Said, A. and Abd El-Latief, A. (2010), "Fractional order theory of thermoelasticity", Int. J. Solid Struct., 47(2), 269-275. https://doi.org/10.1016/j.ijsolstr.2009.09.034
  57. Song, Y., Todorovic, D.M., Cretin, B. and Vairac, P. (2010), "Study on the generalized thermoelastic vibration of the optically excited semiconducting microcantilevers", Int. J. Solids Struct., 47(14-15), 1871-1875. https://doi.org/10.1016/j.ijsolstr.2010.03.020
  58. Song, Y., Bai, J. and Ren, Z. (2012), "Reflection of plane waves in a semiconducting medium under photothermal theory", Int. J. Thermophys., 33(7), 1270-1287. https://doi.org/10.1007/s10765-012-1239-4
  59. Sternberg, E. (1963), "On the analysis of thermal stresses in viscoelastic solids", Brown University Providence RI DIV of Applied Mathematics, 19, 213- 219.
  60. Tam, A.C. (1983), Ultrasensitive Laser Spectroscopy (Academic, New York), pp. 1-108.
  61. Tam, A.C. (1989), Photothermal Investigations in Solids and Fluids (Academic, Boston), pp. 1-33
  62. Todorovic, D. (2003), "Plasma, thermal, and elastic waves in semiconductors", Rev. Sci. Instrum, 74(1), 582-585. https://doi.org/10.1063/1.1523133
  63. Youssef, H. (2010), "Theory of fractional order generalized thermoelasticity", J. Heat Transf., 132(1), 1-7. https://doi.org/10.1115/1.4000705
  64. Yu, Y-J., Tian, X-G. and Tian, J-L. (2013), "Fractional order generalized electro-magnetothermo-elasticity", Eur. J. Mech., A/Solids, 42,188-202. https://doi.org/10.1016/j.euromechsol.2013.05.006
  65. Yu, Y-J., Hu, W. and Tian, X-G. (2014), "A novel generalized thermoelasticity model based on memory-dependent derivative", Int. J. Eng. Sci., 81, 123-134. https://doi.org/10.1016/j.ijengsci.2014.04.014.