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Dynamic response in two-dimensional transversely isotropic thick plate with spatially varying heat sources and body forces

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Abstract

This paper deals with a two-dimensional (2D) problem for a transverselyisotropic thick plate having heat sources and body forces. The upper surface of the plate is stress free with the prescribed surface temperature, while the lower surface of the plate rests on a rigid foundation and is thermally insulated. The study is carried out in the context of the generalized thermoelasticity proposed by Green and Naghdi. The governing equations for displacement and temperature fields are obtained in the Laplace-Fourier transform domain by applying the Laplace and Fourier transforms. The inversion of the double transform is done numerically. Numerical inversion of the Laplace transform is done based on the Fourier series expansion. Numerical computations are carried out for magnesium (Mg), and the results are presented graphically. The results for an isotropic material (Cu) are obtained numerically and presented graphically to be compared with those of a transversely isotropic material (Mg). The effect of the body forces is also studied.

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References

  1. Chadwick, P. Thermoelasticity, the Dynamic Theory, North-Holland, Amsterdam, 265 (1960)

    Google Scholar 

  2. Nowacki, W. Thermoelasticity, Pergamon Press, Oxford (1962)

    Google Scholar 

  3. Nowacki, W. Dynamic Problems of thermoelasticity, Noordhoff International Publishing, Leyden, the Netherlands (1975)

    Google Scholar 

  4. Lord, H. and Shulman, Y. A generalized dynamic theory of Thermoelasticity. Journal of the Mechanics and Physics of Solids, 15, 299–309 (1967)

    Article  MATH  Google Scholar 

  5. Green, A. E. and Lindsay, K. A. Thermoelasticity. Journal of Elasticity, 2, 1–7 (1972)

    Article  MATH  Google Scholar 

  6. Chandrasekharaiah, D. S. Thermoelasticity with second sound. Applied Mechanics Reviews, 39, 355–376 (1986)

    Article  MATH  Google Scholar 

  7. Chandrasekharaiah, D. S. A note on the uniqueness of solution in the theory of thermoelasticity without energy dissipation. Journal of Elasticity, 43, 279–283 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  8. Chandrasekharaiah, D. S. A uniqueness theorem in the theory of thermoelasticity without energy dissipation. Journal of Thermal Stresses, 19, 267–272 (1996)

    Article  MathSciNet  Google Scholar 

  9. Chandrasekharaiah, D. S. One dimensional wave propagation in the linear theory of thermoelasticity without energy dissipation. Journal of Thermal Stresses, 19, 695–710 (1996)

    Article  MathSciNet  Google Scholar 

  10. Chandrasekharaiah, D. S. and Srinath, K. S. Thermoelastic interactions without energy dissipation due to point heat sources. Journal of Elasticity, 50, 97–108 (1998)

    Article  MATH  Google Scholar 

  11. Sherief, H. H. On uniqueness and stability in generalized thermoelasticity. Quarterly of Applied Mathematics, 45, 773–778 (1987)

    MathSciNet  Google Scholar 

  12. Ignaczak, J. Uniqueness in generalized thermoelasticity. Journal of Thermal Stresses, 2, 171–175 (1979)

    Article  Google Scholar 

  13. Ignaczak, J. A note on uniqueness in thermoelasticity with one relaxation time. Journal of Thermal Stresses, 5, 257–263 (1982)

    Article  MathSciNet  Google Scholar 

  14. Ignaczak, J. Generalized thermoelasticity and its applications. Thermal Stresses (ed. Hetnarski, R. B.), Elsevier, Oxford (1989)

    Google Scholar 

  15. Ackerman, C. C. and Guyer, R. A. Temperature pulses in dielectric solids. Annals of Physics, 50, 128–185 (1968)

    Article  Google Scholar 

  16. Jackson, H. E., Walker, C. T., and McNelly, T. F. Second sound in NaF. Physical Review Letters, 25, 26–28 (1970)

    Article  Google Scholar 

  17. Jackson, H. E. and Walker, C. T. Thermal conductivity, second sound, and phonon-phonon interactions in NaF. Physical Review B, 3, 1428–1439 (1971)

    Article  Google Scholar 

  18. Rogers, S. J. Transport of heat and approach to second sound in some isotopically pure Alkali-Halide crystals. Physical Review B, 3, 1440–1457 (1971)

    Article  Google Scholar 

  19. Narayanmurti, V. and Dynes, R. C. Observation of second sound in bismuth. Physical Review Letters, 28, 1461–1465 (1972)

    Article  Google Scholar 

  20. Green, A. E. and Naghdi, P. M. A re-examination of the basic results of thermomechanics. Proceedings of the Royal Society of London — Series A: Mathematical and Physical Sciences, 432, 171–194 (1992)

    Article  MathSciNet  Google Scholar 

  21. Green, A. E. and Naghdi, P. M. On undamped heat waves in an elastic solid. Journal of Thermal Stresses, 15, 252–264 (1992)

    Article  MathSciNet  Google Scholar 

  22. Green, A. E. and Naghdi, P. M. Thermoelasticity without energy dissipation. Journal of Elasticity, 31, 189–208 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  23. Mallik, S. H. and Kanoria, M. Effect of rotation on thermoelastic interaction with and without energy dissipation in an unbounded medium due to heat sources — an eigenvalue approach. Far East Journal of Applied Mathematics, 23(2), 147–167 (2006)

    MathSciNet  MATH  Google Scholar 

  24. Mallik, S. H. and Kanoria, M. A two dimensional problem for a transversely isotropic generalized thermoelasticitic thick plate with spatially varying heat source. European Journal of Mechanics — A/Solids, 27, 607–621 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  25. Mallik, S. H. and Kanoria, M. A two dimensional problem in generalized thermoelasticity for a rotating orthotropic infinite medium with heat sources. Indian Journal of Mathematics, 49(I), 47–70 (2007)

    MathSciNet  MATH  Google Scholar 

  26. Mallik, S. H. and Kanoria, M. Generalized thermoelastic functionally graded solid with a periodically varying heat source. International Journal of Solids and Structures, 44(22–23), 7633–7645 (2007)

    Article  MATH  Google Scholar 

  27. Banik, S., Mallik, S. H., and Kanoria, M. Thermoelastic interaction with energy dissipation in an infinite solid with distributed periodically varying heat sources. International Journal of Pure and Applied Mathematics, 34(2), 231–245 (2007)

    MathSciNet  MATH  Google Scholar 

  28. Bondyopadhyay, N. and Roychoudhuri, S. K. Thermoelastic wave propagation without energy dissipation in an elastic half space. Bulletin of the Calcutta Mathematical Society, 97(6), 489–502 (2005)

    MathSciNet  Google Scholar 

  29. Roychoudhuri, S. K. and Dutta, P. S. Thermoelastic interaction without energy dissipation in an infinite solid with distributed periodically varying heat sources. International Journal of Solids and Structures, 42, 4192–4203 (2005)

    Article  MATH  Google Scholar 

  30. Verma, K. L. and Hasebe, N. Wave propagation in transversely isotropic plates in generalized thermoelasticity. Archive of Applied Mechanics, 72(6–7), 470–482 (2002)

    MATH  Google Scholar 

  31. Honig, G. and Hirdes, U. A method for the numerical inversion of the Laplace transform. Journal of Computational and Applied Mathematics, 10, 113–132 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  32. Dhaliwal, R. S. and Singh, A. Dynamic Coupled Thermoelasticity, Hindustan Pub. Corp., Delhi (1980)

    Google Scholar 

  33. El-Maghraby, N. M. Two dimensional problem in generalized thermoelasticity with heat sources. Journal of Thermal Stresses, 27, 227–239 (2004)

    Article  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, St. Xavier’s College (Evening), Kolkata, 700016, India

    M. Islam

  2. Department of Mathematics, Aliah University, Kolkata, 700091, India

    S. H. Mallik

  3. Department of Applied Mathematics, University of Calcutta, Kolkata, 700009, India

    M. Kanoria

Authors
  1. M. Islam
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  2. S. H. Mallik
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  3. M. Kanoria
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Correspondence to M. Kanoria.

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Islam, M., Mallik, S.H. & Kanoria, M. Dynamic response in two-dimensional transversely isotropic thick plate with spatially varying heat sources and body forces. Appl. Math. Mech.-Engl. Ed. 32, 1315–1332 (2011). https://doi.org/10.1007/s10483-011-1502-6

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  • DOI: https://doi.org/10.1007/s10483-011-1502-6

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