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Effect of heat source and gravity on a fractional order fiber reinforced thermoelastic medium

  • Jain, Kavita (Department of Mathematics, Guru Jambheshwar University of Science and Technology) ;
  • Kalkal, Kapil Kumar (Department of Mathematics, Guru Jambheshwar University of Science and Technology) ;
  • Deswal, Sunita (Department of Mathematics, Guru Jambheshwar University of Science and Technology)
  • Received : 2017.11.08
  • Accepted : 2018.08.26
  • Published : 2018.10.25

Abstract

In this article, the theory of fractional order two temperature generalized thermoelasticity is employed to study the wave propagation in a fiber reinforced anisotropic thermoelastic half space in the presence of moving internal heat source. The whole space is assumed to be under the influence of gravity. The surface of the half-space is subjected to an inclined load. Laplace and Fourier transform techniques are employed to solve the problem. Expressions for different field variables in the physical domain are derived by the application of numerical inversion technique. Physical fields are presented graphically to study the effects of gravity and heat source. Effects of time, reinforcement, fractional parameter and inclination of load have also been reported. Results of some earlier workers have been deduced from the present analysis.

Keywords

References

  1. Abbas, I.A., Abd-Alla, A.N. and Othman, M.I.A. (2011), "Generalized magneto-thermoelasticity in a fiber reinforced anisotropic half-space", Int. J. Thermophys., 32(5), 1071-1085. https://doi.org/10.1007/s10765-011-0957-3
  2. Abd-Alla, A.N., Abo-Dahab, S.M. and Alotaibi, H.A. (2017), "Propagation of a thermoelastic wave in a half space of a homogeneous isotropic material subjected to the effect of gravity field", Arch. Civil Mech. Eng., 17(3), 564-573. https://doi.org/10.1016/j.acme.2016.10.012
  3. Ailawalia, P. and Narah, N.S. (2009), "Effect of rotation in generalized thermoelastic solid under the influence of gravity with an overlying infinite thermoelastic fluid", Appl. Math. Mech., 30(12), 1505-1518. https://doi.org/10.1007/s10483-009-1203-6
  4. Ailawalia, P. and Singla, S. (2015), "Disturbance due to internal heat source in thermoelastic solid using dual phase lag model", Struct. Eng. Mech., 56(3), 341-354. https://doi.org/10.12989/sem.2015.56.3.341
  5. Ailawalia, P., Khurana, G. and Kumar, S. (2009), "Effect of rotation in a generalized thermoelastic medium with two temperature under the influence of gravity", Int. J. Appl. Math. Mech., 5(5), 99-116.
  6. Baksi, A., Roy, B.K. and Bera, R.K. (2008), "Study of two dimensional viscoelastic problems in generalized thermoelastic medium with heat source", Struct. Eng. Mech., 29(6), 673-687. https://doi.org/10.12989/sem.2008.29.6.673
  7. Belfield, A.J., Rogers, T.G. and Spencer, A.J.M. (1983), "Stress in elastic plates reinforced by fibers lying in concentric circles", J. Mech. Phys. Sol., 31(1), 25-54. https://doi.org/10.1016/0022-5096(83)90018-2
  8. Bromwich, T.J.J.A. (1898), "On the influence of gravity on elastic waves and in particular on the vibrations of an elastic globe", Proc. Lond. Math. Soc., 30, 98-120.
  9. Chen, P.J. and Gurtin, M.E. (1968), "On a theory of heat conduction involving two temperatures", Z. Angew. Math. Phys., 19(4), 614-627. https://doi.org/10.1007/BF01594969
  10. Chen, P.J., Gurtin, M.E. and Williams, W.O. (1968), "A note on non-simple heat conduction", Z. Angew. Math. Phys., 19(6), 969-970. https://doi.org/10.1007/BF01602278
  11. Chen, P.J., Gurtin, M.E. and Williams, W.O. (1969), "On the thermodynamics of non-simple elastic material with two temperatures", Z. Angew. Math. Phys., 20(1), 107-112. https://doi.org/10.1007/BF01591120
  12. Danilovskaya, V.I. (1950), "Thermal stresses in an elastic semispace due to a sudden heating of its boundary", Prikl. Mat. Mech., 14, 316-318.
  13. Deswal, S. and Kalkal, K.K. (2013), "Fractional order heat conduction law in micropolar thermo-viscoelasticity with two temperature", Int. J. Heat Mass Transf., 66, 451-460. https://doi.org/10.1016/j.ijheatmasstransfer.2013.07.047
  14. Deswal, S., Kalkal, K.K. and Sheoran S.S. (2017), "A magneto-thermo-viscoelastic problem with fractional order strain under GN-II model", Struct. Eng. Mech., 63(1), 89-102. https://doi.org/10.12989/SEM.2017.63.1.089
  15. Eason, G. and Sneddon, I.N. (1959), "The dynamic stress produced in elastic body by uneven heating", Proc. Roy. Soc. Edin. Soc., 65, 143-176.
  16. El-Karamany, A.S. and Ezzat, M.A. (2011), "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
  17. 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
  18. Honig, G. and Hirdes, U. (1984), "A method for the numerical inversion of Laplace transforms", J. Comp. Appl. Math., 10(1), 113-132. https://doi.org/10.1016/0377-0427(84)90075-X
  19. Jumarie, G. (2010), "Derivation and solutions of some fractional Black-Scholes equations in coarse-grained space and time. Application to Merton's optimal portfolio", Comput. Math. Appl., 59(3), 1142-1164. https://doi.org/10.1016/j.camwa.2009.05.015
  20. Love, A.E.H. (1911), Some Problems of Geodynamics, Cambridge University Press, Cambridge.
  21. Nowacki, W. (1959), "Some dynamic problems of thermoelasticity", Arch. Mech. Stos., 9, 325-334.
  22. 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
  23. Rakshit, M. and Mukhopadhyay, B. (2007), "A two dimensional thermoviscoelastic problem due to instantaneous point heat source", Math. Comput. Modell., 46(11-12), 1388-1397. https://doi.org/10.1016/j.mcm.2006.11.036
  24. Said, S.M. and Othman, M.I.A. (2016), "Wave propagation in a two temperature fiber reinforced magnetothermoelastic medium with three phase lag model", Struct. Eng. Mech., 57(2), 201-220. https://doi.org/10.12989/sem.2016.57.2.201
  25. Sarkar, N., Atwa, S.Y. and Othman, M.I.A. (2016), "The effect of hydrostatic initial stress on the plane waves in a fiber reinforced magneto-thermoelastic medium with fractional derivative heat transfer", Int. Appl. Mech., 52(2), 203-216. https://doi.org/10.1007/s10778-016-0748-4
  26. Sengupta, P.R. and Nath, S. (2001), "Surface waves in fiber reinforced anisotropic elastic media", Sadhana, 26(4), 363-370. https://doi.org/10.1007/BF02703405
  27. Sherief, H.H., El-Sayed, A.M. and El-Latief, A.M. (2010), "Fractional order theory of thermoelasticity", Int. J. Sol. Struct., 47(2), 269-275. https://doi.org/10.1016/j.ijsolstr.2009.09.034
  28. Singh, B. and Singh, S.J. (2004), "Surface waves at the free surface of a fiber reinforced elastic half-space", Sadhana, 29(3), 249-257. https://doi.org/10.1007/BF02703774
  29. Yadav, R., Kalkal, K.K. and Deswal, S. (2017), "Two temperature theory of initially stressed electromicrostretch medium without energy dissipation", Microsyst. Technol., 23(10), 4931-4940. https://doi.org/10.1007/s00542-017-3323-y
  30. Youssef, H. (2013), "State-space approach to fractional order two-temperature generalized thermoelastic medium subjected to moving heat source", Mech. Adv. Mater. Struct., 20(1), 47-60. https://doi.org/10.1080/15376494.2011.581414