Relaxation effects on thermoelastic interactions for time-dependent moving heat source under a recent model of thermoelasticity

Abstract

In the present article, we investigate the thermal and elastic behaviour of an infinite thermoelastic material with a cylindrical cavity caused by a time-dependent moving heat source. The cavity surface is assumed to be subjected to a thermal shock. The formulation of the problem is applied for the recently proposed generalized thermoelastic model [Modified Green–Lindsay (MGL)] that takes into account the strain and temperature rates in the constitutive relations. We derive the governing equations in the present context along with the other two models, namely Lord–Shulman model and Green–Lindsay, by unifying governing equations under all these three models. The solutions are obtained for all three models in the Laplace transform domain, which are represented by unified expressions. The numerical computations for temperature, displacement and thermal stresses are carried out and depicted graphically. A detailed comparison is made amongst the results predicted by three models to highlight the effects of velocity of heat source and strain, temperature rate terms involved in the MGL model.

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References

  1. 1.

    Biot, M.A.: Thermoelasticity and irreversible thermodynamics. J. Appl. Phys. 27(3), 240–253 (1956)

    MathSciNet  Article  Google Scholar 

  2. 2.

    Peshkov, V.: The second sound in helium II. J. Phys. 8, 381 (1944)

    Google Scholar 

  3. 3.

    Ackerman, C.C., Bertman, B., Fairbank, H.A., Guyer, R.A.: Second sound in solid helium. Phys. Rev. Lett. 16(18), 789 (1966)

    Article  Google Scholar 

  4. 4.

    Bertman, B., Sandiford, D.J.: Second sound in solid helium. Sci. Am. 222(5), 92–103 (1970)

    Article  Google Scholar 

  5. 5.

    Chandrasekharaiah, D.S.: Hyperbolic thermoelasticity: a review of recent literature. Appl. Mech. Rev. 51(12), 705–729 (1998)

    Article  Google Scholar 

  6. 6.

    Lord, H.W., Shulman, Y.: A generalized dynamical theory of thermoelasticity. J. Mech. Phys. Solids 15(5), 299–309 (1967)

    Article  Google Scholar 

  7. 7.

    Green, A.E., Lindsay, K.A.: Thermoelasticity. J. Elast. 2(1), 1–7 (1972)

    Article  Google Scholar 

  8. 8.

    Green A.E., Naghdi, P.M.: A re-examination of the basic postulates of thermomechanics. In: Proc. R. S. Lond. A, vol. 432, pp. 171–194, The Royal Society (1991)

  9. 9.

    Green, A.E., Naghdi, P.M.: On undamped heat waves in an elastic solid. J. Therm. Stresses 15(2), 253–264 (1992)

    MathSciNet  Article  Google Scholar 

  10. 10.

    Green, A.E., Naghdi, P.M.: Thermoelasticity without energy dissipation. J. Elast. 31(3), 189–208 (1993)

    MathSciNet  Article  Google Scholar 

  11. 11.

    Tzou, D.Y.: A unified field approach for heat conduction from macro-to micro-scales. J. Heat Transf. 117(1), 8–16 (1995)

    Article  Google Scholar 

  12. 12.

    Roychoudhuri, S.K.: On a thermoelastic three-phase-lag model. J. Therm. Stresses 30(3), 231–238 (2007)

    Article  Google Scholar 

  13. 13.

    Hetnarski, R.B., Ignaczak, J.: Generalized thermoelasticity. J. Therm. Stresses 22(4–5), 451–476 (1999)

    MathSciNet  MATH  Google Scholar 

  14. 14.

    Straughan, B.: Heat Waves, vol. 177. Springer, Berlin (2011)

    Book  Google Scholar 

  15. 15.

    Ignaczak, J., Ostoja-Starzewski, M.: Thermoelasticity with Finite Wave Speeds. Oxford University Press, Oxford (2010)

    MATH  Google Scholar 

  16. 16.

    Yu, Y.J., Xue, Z.-N., Tian, X.-G.: A modified Green–Lindsay thermoelasticity with strain rate to eliminate the discontinuity. Meccanica 53(10), 2543–2554 (2018)

    MathSciNet  Article  Google Scholar 

  17. 17.

    Chandrasekharaiah, D.S., Keshavan, H.R.: Axisymmetric thermoelastic interactions in an unbounded body with cylindrical cavity. Acta Mech. 92(1–4), 61–76 (1992)

    Article  Google Scholar 

  18. 18.

    Singh, R.V., Mukhopadhyay, S.: An investigation on strain and temperature rate-dependent thermoelasticity and its infinite speed behavior. J. Therm. Stresses 43(4), 1–15 (2020)

    Article  Google Scholar 

  19. 19.

    Gupta, M., Mukhopadhyay, S.: Galerkin-type solution for the theory of strain and temperature rate-dependent thermoelasticity. Acta Mech. 230(10), 3633–3643 (2019)

    MathSciNet  Article  Google Scholar 

  20. 20.

    Shivay, O.N., Mukhopadhyay, S.: A complete Galerkin’s type approach of finite element for the solution of a problem on modified Green–Lindsay thermoelasticity for a functionally graded hollow disk. Eur. J. Mech. A Solids 80, 103914 (2020)

    MathSciNet  Article  Google Scholar 

  21. 21.

    Quintanilla, R.: Some qualitative results for a modification of the Green–Lindsay thermoelasticity. Meccanica 53(14), 3607–3613 (2018)

    MathSciNet  Article  Google Scholar 

  22. 22.

    Youssef, H.M.: Generalized thermoelastic infinite medium with cylindrical cavity subjected to moving heat source. Mech. Res. Commun. 36(4), 487–496 (2009)

    MathSciNet  Article  Google Scholar 

  23. 23.

    Stehfest, H.: Algorithm 368: numerical inversion of laplace transforms [D5]. Commun. ACM 13(1), 47–49 (1970)

    Article  Google Scholar 

  24. 24.

    Bateman, H., Erdélyi, A.: Tables of Integral Transforms, vol. 1. Book Company Inc, New York (1954)

    Google Scholar 

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Correspondence to Robin Vikram Singh.

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Singh, R.V., Mukhopadhyay, S. Relaxation effects on thermoelastic interactions for time-dependent moving heat source under a recent model of thermoelasticity. Z. Angew. Math. Phys. 72, 31 (2021). https://doi.org/10.1007/s00033-020-01462-x

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Keywords

  • Generalized thermoelasticity theory
  • TRDTE theory
  • Modified Green–Lindsay theory
  • Thermal shock
  • Moving heat source problem

Mathematics Subject Classification

  • 74F05