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Galerkin-type solution for the Moore–Gibson–Thompson thermoelasticity theory

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Abstract

It is prominent that the Galerkin-type representation plays a dominant role in probing various challenges of mathematical physics, continuum mechanics and occupies an important place in the field of partial differential equations (PDEs). Thus, the contemporary analysis of different boundary value problems (BVPs) in thermoelasticity theory commonly begins by analyzing the Galerkin-type representation of the field equations in terms of elementary functions (harmonic, biharmonic, and metaharmonic, etc). This work is aimed at formulating the representation of a Galerkin-type solution by means of elementary functions for the recently developed Moore–Gibson–Thompson (MGT) thermoelasticity theory. The MGT theory is a generalized form of the Lord–Shulman (LS) model as well as of the Green–Naghdi (GN) thermoelastic model. Here, we establish a theorem and derive the Galerkin-type solution for the basic governing equations under this theory. Later, the Galerkin representation of a system of equations for steady oscillations is derived. Based on this representation, we finally establish the general solution (GS) for the system of homogeneous equations of stable oscillation, neglecting the extrinsic body force and extrinsic heat supply.

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References

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

    MathSciNet  MATH  Google Scholar 

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

    MATH  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)

    MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

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

    Google Scholar 

  8. Nayfeh, A.H., Nemat-Nasser, S.: Thermoelastic waves in solids with thermal relaxation. Acta Mech. 12, 53–59 (1971)

    MATH  Google Scholar 

  9. Puri, P.: Plane waves in generalized thermoelasticity. Int. J. Eng. Sci. 11, 735–744 (1973)

    MATH  Google Scholar 

  10. Agarwal, V.K.: On plane waves in generalized thermoelasticity. Acta Mech. 34, 185–198 (1979)

    MathSciNet  MATH  Google Scholar 

  11. 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 

  12. 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 

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

    MathSciNet  Google Scholar 

  14. 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 

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

    MathSciNet  Google Scholar 

  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  MATH  Google Scholar 

  17. Green, A.E., Naghdi, P.M.: A re-examination of the basic postulates of thermomechanics. Proc. R. Soc. Lond. 432(1885), 171–194 (1991)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

  21. Thompson, P.A.: Compressible-Fluid Dynamics. McGraw-Hill, New York (1972)

    MATH  Google Scholar 

  22. Quintanilla, R.: Moore–Gibson–Thompson thermoelasticity. Math. Mech. Solids 24(12), 4020–4031 (2019)

    MathSciNet  MATH  Google Scholar 

  23. Pellicer, M., Quintanilla, R.: On uniqueness and instability for some thermomechanical problems involving the Moore–Gibson–Thompson equation. Z. Angew. Math. Phys. 71, 84 (2020)

    MathSciNet  MATH  Google Scholar 

  24. Jangid, K., Mukhopadhyay, S.: A domain of influence theorem under the MGT thermoelasticity theory. Math. Mech. Solids (2020). https://doi.org/10.1177/1081286520946820

    Article  Google Scholar 

  25. Jangid, K., Mukhopadhyay, S.: A domain of influence theorem for a natural stress-heat-flux problem in the Moore–Gibson–Thompson thermoelasticity theory. Acta Mech. (2020). https://doi.org/10.1007/s00707-020-02833-1

    Article  Google Scholar 

  26. Bazarra, N., Fernandez, J.R., Quintanilla, R.: Analysis of a Moore–Gibson–Thompson thermoelasticity problem. J. Comput. Appl. Math. 382, 113058 (2021)

    MathSciNet  MATH  Google Scholar 

  27. Conti, M., Pata, V., Quintanilla, R.: Thermoelasticity of Moore–Gibson–Thompson type with history dependence in the temperature. Asymptot. Anal. 120(1–2), 1–21 (2020)

    MathSciNet  MATH  Google Scholar 

  28. Conti, M., Pata, V., Pellicer, M., Quintanilla, R.: On the analyticity of the MGT-viscoelastic plate with heat conduction. J. Differ. Equ. 269(10), 7862–7880 (2020)

    MathSciNet  MATH  Google Scholar 

  29. 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 

  30. Gurtin, M.E.: The linear theory of elasticity. In: Trusdell C.A. (ed.), Handbuch der Physik. 1, 1–295 (1972)

  31. Nowacki, W.: Dynamic Problems in Thermoelasticity. Noordhoff International Publications, Leyden (1975)

    MATH  Google Scholar 

  32. Nowacki, W.: Theory of Elasticity. Mir, Moscow (1975)

    MATH  Google Scholar 

  33. Kupradze, V.D., Gegelia, T.G., Basheleishvili, M.O., Burchuladze, T.V.: Three-dimensional Problems of the Mathematical Theory of Elasticity and Thermoelasticity, p. 222. North-Holland Publication Company, Amsterdam (1979)

    Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  Google Scholar 

  37. Eringen, A.C.: Foundations of micropolar thermoelasticity. Int. Cent. Mech. Stud. Course Lect. 23 (1970)

  38. Eringen, A.C.: Microcontinuum Field Theories I: Foundations and Solids. Springer, New York (1999)

    MATH  Google Scholar 

  39. Boschi, E., Iesan, D.: A generalized theory of linear micropolar thermoelasticity. Meccanica 7, 154–157 (1973)

    MATH  Google Scholar 

  40. Nowacki, W.: Theory of Asymmetric Elasticity. Pergamon, Oxford (1986)

    MATH  Google Scholar 

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

    Google Scholar 

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

    MathSciNet  Google Scholar 

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

    MathSciNet  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  46. 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)

    MATH  Google Scholar 

  47. 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  MATH  Google Scholar 

  48. Singh, B., Gupta, M., Mukhopadhyay, S.: On the fundamental solutions for the strain and temperature rate-dependent generalized thermoelasticity theory. J. Therm. Stress. 43(5), 650–664 (2020)

    Google Scholar 

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Funding

One of the authors, Bhagwan Singh, is grateful to the DST - INSPIRE Fellowship/2018/IF.170983.

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

    Bhagwan Singh & Santwana Mukhopadhyay

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  1. Bhagwan Singh
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  2. Santwana Mukhopadhyay
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Correspondence to Bhagwan Singh.

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Singh, B., Mukhopadhyay, S. Galerkin-type solution for the Moore–Gibson–Thompson thermoelasticity theory. Acta Mech 232, 1273–1283 (2021). https://doi.org/10.1007/s00707-020-02915-0

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  • DOI: https://doi.org/10.1007/s00707-020-02915-0

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