International Applied Mechanics

, Volume 52, Issue 2, pp 203–216 | Cite as

The Effect of Hydrostatic Initial Stress on the Plane Waves in a Fiber-Reinforced Magneto-Thermoelastic Medium with Fractional Derivative Heat Transfer

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The effect of three factors—the order of fractional derivative, the hydrostatic initial stress, and parameter of magnetic field—on the plane waves in a half-space made of fiber-reinforced material described by the theory of generalized magneto-thermoelasticity is studied. The problem is solved numerically using the normal mode analysis. The results correspond to the Lord–Shulman model and the model that uses fractional derivatives and are represented in the form of graphs. The findings show pronounced effect of the three factors. The results are compared with the case where the initial stress and magnetic field are absent

Keywords

generalized magneto-thermoelasticity fractional derivative fiber-reinforced material hydrostatic initial stress normal mode analysis 
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References

  1. 1.
    M. Biot, “Thermoelasticity and irreversible thermodynamics,” J. Appl. Phys., 27, 240 – 253 (1956).ADSMathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    W. Dreyer and H. Struchtrup, “Heat pulse experiments revisited,” Cont. Mech. Thermodynam., 5, 3–50 (1993).ADSMathSciNetCrossRefGoogle Scholar
  3. 3.
    J. Ignaczak and M. Ostoja-Starzewski, Thermoelasticity with Finite Wave Speeds, Oxford University Press, New York (2010).MATHGoogle Scholar
  4. 4.
    Y. Z. Povstenko, “Fractional heat conduction equation and associated thermal stress,” J. Thermal Stresses, 28, 83–102 (2005).MathSciNetCrossRefGoogle Scholar
  5. 5.
    M. Caputo, “Linear model of dissipation whose Q is almost frequency independent–II,” Geophys. J. Royal Astronomy Soc., 13, 529–39 (1967).ADSMathSciNetCrossRefGoogle Scholar
  6. 6.
    H. M. Youssef, “Theory of fractional order generalized thermoelasticity,” J. Heat. Trans., 132, 1–7 (2010).MathSciNetCrossRefGoogle Scholar
  7. 7.
    H. M. Youssef and E.Al-Lehaibi, “Fractional order generalized thermoelastic half space subjected to ramp type heating,” Mech. Res. Commun., 37, 448–452 (2010).Google Scholar
  8. 8.
    H. H. Sherief, A. El-Sayed, and A. El-Latief, “Fractional order theory of thermoelasticity,” Int. J. Solids Struct., 47, 269–275 (2010).CrossRefMATHGoogle Scholar
  9. 9.
    M. A. Ezzat and M. A. Fayik, “Fractional order theory of thermoelastic diffusion,” J. Thermal Stresses, 34, 851–872 (2011).CrossRefGoogle Scholar
  10. 10.
    A. J. Belfield, T. G. Rogers, and A. J. M. Spencer, “Stress in elastic plates reinforced by fibers lying in concentric circles,” J. Mech. Phys. Solids, 31, 25–54 (1983).ADSCrossRefMATHGoogle Scholar
  11. 11.
    A. J. M. Spencer, Deformation of Fibre-Reinforced Materials, Clarendon, Oxford (1941).Google Scholar
  12. 12.
    A. C. Pipkin, “Finite deformations of ideal fiber-reinforced composites,” in: G. P. Sendeckyi (ed), Composites Materials, Academic, New York (1973), pp. 251–308.Google Scholar
  13. 13.
    T. G. Rogers, “Finite deformations of strongly anisotropic materials,” in: J. F. Hutton, J. R. A. Pearson, and K. Walters (eds.), Theoretical Rheology, Applied Science Publication, London, (1975), pp. 141–168.Google Scholar
  14. 14.
    T. G. Rogers, “Anisotropic elastic and plastic materials,” in: P. Thoft-Christensen (ed.), Continuum Mechanics Aspects of Geodynamics and Rock Fracture. Mech., Reidel, (1975), pp. 177–200.Google Scholar
  15. 15.
    B. Singh and S. J. Singh, “Reflection of plane waves at the free surface of a fibre-reinforced elastic half-space,” Sadhana, 29, 249–257 (2004).CrossRefMATHGoogle Scholar
  16. 16.
    B. Singh, “Wave propagation in an incompressible transversely isotropic fibre-reinforced elastic media,” Arch. Appl. Mech., 77, 253–258 (2007).ADSCrossRefMATHGoogle Scholar
  17. 17.
    M. S. Craig and V. G. Hart, “The stress boundary-value problem for finite plane deformations of a fibre-reinforced material,” Quarterly J. Mech. Appl. Math., 32, 473–498 (1979).MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    P. R. Sengupta and S. Nath, “Surface waves in fibre-reinforced anisotropic elastic media,” Sadhana, 26, 363–370 (2001).CrossRefGoogle Scholar
  19. 19.
    R. Kumar and R. R. Gupta, “Study of wave motion in an anisotropic fiber-reinforced thermoelastic solid,” Int. J. Solid. Mech., 2, 91–100 (2010).Google Scholar
  20. 20.
    J. Nowinski, Theory of Thermoelasticity with Applications, Sijtho and Noordho Int. Publ. Alphen Aan Den Rijn (1978).Google Scholar
  21. 21.
    P. Chadwick, “Progress in solid mechanics,” in: R. Hill and I. N. Sneddon (eds.), North Holland, Amsterdam (1960).Google Scholar
  22. 22.
    N. Sarkar and A. Lahiri, “Eigenvalue approach to two-temperature magneto-thermoelasticity,” Vietnam J. Math., 40, 13–30 (2012).MathSciNetMATHGoogle Scholar
  23. 23.
    I. A. Abbas, A. N. Abd-alla, and M. I. A. Othman, “Generalized magneto-thermoelasticity in a fiber-reinforced anisotropic half-space,” Int. J. Thermophys., 32, 1071–1085 (2011).ADSCrossRefGoogle Scholar
  24. 24.
    M. A. Ezzat and H. M. Youssef, “Generalized magneto-thermoelasticity in a perfectly conducting medium,” Int. J. Solid. Struct., 42, 6319–6334 (2005).CrossRefMATHGoogle Scholar
  25. 25.
    M. A. Ezzat and H. M. Youssef, “Generation of generalized magneto-thermoelastic waves by thermal shock in a half-space of finite conductivity,” Italian J. Pure Appl. Math., 19, 9–26 (2005).MathSciNetMATHGoogle Scholar
  26. 26.
    H. M. Youssef, “Generalized magneto-thermoelasticity in a conducting medium with variable material properties,” Appl. Math. Comp., 173, 822–833 (2006).MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    M. A. Ezzat and M. Z. Abd Elall, “Generalized magneto-thermoelasticity with modified Ohm’s law,” Mech. Advan. Mater. Struct., 17, 74–84, (2010).CrossRefGoogle Scholar
  28. 28.
    Q. L. Xiong and X. G. Tian, “Transient magneto-thermoelastic response for a semi-infinite body with voids and variable material properties during thermal shock,” Int. J. Appl. Mech., 3, 891–902 (2011).CrossRefGoogle Scholar
  29. 29.
    M. I. A. Othman and K. Lotfy, “On the plane waves of generalized thermo-microstretch elastic half-space under three theories,” Int. Commun. Heat Mass Trans., 37, 192–200 (2010).CrossRefGoogle Scholar
  30. 30.
    M. I. A. Othman, K. Lotfy, and R. M. Farouk, “Generalized thermo-microstretch elastic medium with temperature dependent properties for different theories,” Eng. Anal. Boundary Element, 34, 229–237 (2010).MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    M. I. A. Othman, “Electrohydrodynamic instability of a rotating layer of a viscoelastic fluid heated from below,” J. Appl. Math. Phys., 55, 468–482 (2004).MathSciNetMATHGoogle Scholar
  32. 32.
    N. Sarkar and A. Lahiri, “A three-dimensional thermoelastic problem for a half-space without energy dissipation,” Int. J. Eng. Sci., 51, 310–325 (2012).MathSciNetCrossRefGoogle Scholar
  33. 33.
    H. W. Lord and Y. A. Shulman, “Generalized dynamical theory of thermoelasticity,” J. Mech. Phys. Solids, 15, 299–309 (1967).ADSCrossRefMATHGoogle Scholar
  34. 34.
    A. Montanaro, “On singular surface in isotropic linear thermoelasticity with initial stress,” J. Acoustic Soc. America, 106, 1586–1588 (1999).ADSCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Department of MathematicsThe University of BurdwanWest BengalIndia
  2. 2.Higher Institute of Engineer, Department of Engineering Mathematics and PhysicsShorouk AcademyShorouk CityEgypt
  3. 3.Department of Mathematics, Faculty of ScienceZagazig UniversityZagazigEgypt

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