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Galerkin-type solution for the theory of strain and temperature rate-dependent thermoelasticity

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Abstract

Ultrafast heating has gained serious attention due to technological advancements in recent years. The thermoelastic process accounting for the finite speed of thermal signals is therefore becoming increasingly important. With the help of extended thermodynamics, Yu et al. (Meccanica 53(10):2543–2554, 2018) have established a thermoelastic model by including the strain rate and temperature rate in the constitutive equations. This model is also an attempt to remove the discontinuity in the displacement field under temperature rate-dependent thermoelasticity theory. The present work seeks to derive the representation of a Galerkin-type solution in the context of this recently proposed model in terms of elementary functions. The representation theorem of the Galerkin-type solution of the system of equations for steady oscillation is also proved. In accordance with this theorem, we finally establish a theorem which expresses the general solution of the system of homogeneous equations of steady oscillation in terms of metaharmonic functions.

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References

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

    Article  MathSciNet  Google Scholar 

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

    Article  Google Scholar 

  3. Cattaneo, C.: A form of heat-conduction equations which eliminates the paradox of instantaneous propagation. C. R. 247, 431–433 (1958)

    MATH  Google Scholar 

  4. Vernotte, P.: Les paradoxes de la théorie continue de l’équation de lachaleur. C. R. 246, 3154–3155 (1958)

    MathSciNet  MATH  Google Scholar 

  5. Vernotte, P.: Some possible complications in the phenomena of thermal conduction. C. R. 252, 2190–2191 (1961)

    Google Scholar 

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

    Article  Google Scholar 

  7. Chandrasekharaiah, D., Srikantiah, K.R.: Temperature-rate dependent thermoelastic waves in a half-space. Indian J. Technol. 24(2), 66–70 (1986)

    MATH  Google Scholar 

  8. Chandrasekharaiah, D.S., Srikantiah, K.R.: On temperature-rate dependent thermoelastic interactions in an infinite solid due to a point heat-source. Indian J. Technol. 25(1), 1–7 (1987)

    MATH  Google Scholar 

  9. Dhaliwal, R.S., Rokne, J.G.: One-dimensional thermal shock problem with two relaxation times. J. Therm. Stresses 12(2), 259–279 (1989)

    Article  MathSciNet  Google Scholar 

  10. Chatterjee, G., Roychoudhuri, S.K.: On spherically symmetric temperature-rate dependent thermoelastic wave propagation. J. Math. Phys. Sci. 24, 251–264 (1990)

    MATH  Google Scholar 

  11. Ignaczak, J., Mrówka-Matejewska, E.B.: One-dimensional Green’s function in temperature-rate dependent thermoelasticity. J. Therm. Stresses 13(3), 281–296 (1990)

    Article  MathSciNet  Google Scholar 

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

    Article  Google Scholar 

  13. Green, A.E., Naghdi, P.M.: A re-examination of the basic postulates of thermomechanics. Proc. R. Soc. Lond. A 432, 171–194 (1991)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  16. 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 

  17. Tzou, D.Y.: The generalized lagging response in small-scale and high-rate heating. Int. J. Heat Mass Transf. 38(17), 3231–3240 (1995)

    Article  Google Scholar 

  18. 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)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  20. Galerkin, B.: Contribution à la solution gènèrale du problème de la thèorie de ì èlasticite, dans le cas de trois dimensions. C. R. Acad. Sci. Paris 190, 1047–1048 (1930)

    MATH  Google Scholar 

  21. Iacovache, M.: O extindere a metogei lui Galerkin pentru sistemul ecuatiilor elsticittii. Bull. St. Acad. Rep. Pop Romane A 1, 593–596 (1949)

    MathSciNet  Google Scholar 

  22. Nowacki, W.: Green functions for the thermoelastic medium. Bull. Acad. Polon. Sci. Ser. Sci. Tech. 12, 465–472 (1964)

    MATH  Google Scholar 

  23. Nowacki, W.: On the completeness of potentials in micropolar elasticity. Arch. Mech. Stos. 21, 107–122 (1969)

    MathSciNet  MATH  Google Scholar 

  24. Sandru, N.: On some problems of the linear theory of the asymmetric elasticity. Int. J. Eng. Sci. 4(1), 81–94 (1966)

    Article  Google Scholar 

  25. Gurtin, M.E.: The linear theory of elasticity. In: Truesdell, C. (ed.) Encyclopedia of Physics, vol. VIa/2, pp. 1–295. Springer, Berlin (1972)

    Google Scholar 

  26. Nowacki, W.: Dynamic Problems in Thermoelasticity. Noordhoff, Leyden (1975)

    MATH  Google Scholar 

  27. Kupradze, V.D., Gegelia, T.G., Basheleishvili, M.O., Burchuladze, T.V.: Three-Dimensional Problems of the Mathematical Theory of Elasticity and Thermoelasticity. North-Holland, Amsterdam (1979)

    Google Scholar 

  28. Scalia, A., Svanadze, M.: On the representations of solutions of the theory of thermoelasticity with microtemperatures. J. Therm. Stresses 29(9), 849–863 (2006)

    Article  MathSciNet  Google Scholar 

  29. Chandrasekharaiah, D.: Complete solutions in the theory of elastic materials with voids. Q. J. Mech. Appl. Math. 40(3), 401–414 (1987)

    Article  MathSciNet  Google Scholar 

  30. Chandrasekharaiah, D.: Complete solutions in the theory of elastic materials with voids—II. Q. J. Mech. Appl. Math. 42(1), 41–54 (1989)

    Article  MathSciNet  Google Scholar 

  31. Ciarletta, M.: A solution of Galerkin type in the theory of thermoelastic materials with voids. J. Therm. Stresses 14(4), 409–417 (1991)

    Article  MathSciNet  Google Scholar 

  32. Ciarletta, M.: General theorems and fundamental solutions in the dynamical theory of mixtures. J. Elast. 39(3), 229–246 (1995)

    Article  MathSciNet  Google Scholar 

  33. Ciarletta, M.: A theory of micropolar thermoelasticity without energy dissipation. J. Therm. Stresses 22(6), 581–594 (1999)

    Article  MathSciNet  Google Scholar 

  34. Svanadze, M.Zh.: Representation of the general solution of the equation of steady-state oscillations of two component elastic mixtures. Prikladnaia Mekhanika 29, 22–29 (1993)

  35. Svanadze, M., de Boer, R.: On the representations of solutions in the theory of fluid-saturated porous media. Q. J. Mech. Appl. Math. 58(4), 551–562 (2005)

    Article  MathSciNet  Google Scholar 

  36. Mukhopadhyay, S., Kothari, S., Kumar, R.: On the representation of solutions for the theory of generalized thermoelasticity with three phase-lags. Acta Mech. 214, 305–314 (2010)

    Article  Google Scholar 

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

    Article  Google Scholar 

  38. Kothari, S., Mukhopadhyay, S.: On the representations of solutions in the theory of generalized thermoelastic diffusion. Math. Mech. Solids 17(2), 120–130 (2012)

    Article  MathSciNet  Google Scholar 

  39. Sherief, H.H., Hamza, F.A., Saleh, H.A.: The theory of generalized thermoelastic diffusion. Int. J. Eng. Sci. 42, 591–608 (2004)

    Article  MathSciNet  Google Scholar 

  40. Svanadze, M.M.: On the solutions of equations of the linear thermoviscoelasticity theory for Kelvin–Voigt materials with voids. J. Therm. Stresses 37(3), 253–269 (2014)

    Article  Google Scholar 

  41. Svanadze, M.M.: On the solutions in the linear theory of micropolar viscoelasticity. Mech. Res. Commun. 81, 17–25 (2017)

    Article  Google Scholar 

  42. Giorgashvili, L., Jaghmaidze, A., Karseladze, G., et al.: Boundary value problems of stationary oscillation of thermoelasticity of microstretch materials with microtemperatures. Georgian Math. J. 22(1), 57–70 (2015)

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematical Sciences, Indian Institute of Technology (BHU), Varanasi, 221005, India

    Manushi Gupta & Santwana Mukhopadhyay

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  1. Manushi Gupta
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  2. Santwana Mukhopadhyay
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Correspondence to Santwana Mukhopadhyay.

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Gupta, M., Mukhopadhyay, S. Galerkin-type solution for the theory of strain and temperature rate-dependent thermoelasticity. Acta Mech 230, 3633–3643 (2019). https://doi.org/10.1007/s00707-019-02482-z

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  • DOI: https://doi.org/10.1007/s00707-019-02482-z

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