Problems of steady vibrations in the theory of Moore-Gibson-Thompson thermoviscoelasticity for materials with voids

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Authors

  • M.M. Svanadze Tbilisi State University, Georgia 0000-0002-0620-8502

Abstract

In this paper, the linear theory of Moore-Gibson-Thompson (MGT) thermoviscoelasticity for materials with voids is examined and the basic boundary value problems (BVPs) of steady vibrations are investigated. The governing equations of motion and steady vibrations are formulated. The fundamental solution to the system of steady vibration equations is constructed explicitly using four elementary functions, and its key properties are analyzed. Then, Green's first identity is established and the uniqueness theorems for classical solutions of the associated basic BVPs are proved. The surface and volume potentials are defined, and their essential properties are established. Singular integral operators are introduced, and their symbolic determinants and indices are calculated. Finally, existence theorems for classical solutions of the basic internal and external BVPs are established using the potential method.

Keywords:

thermoviscoelasticity, materials with voids, steady vibrations, potential method, existence and uniqueness theorems

References


  1. R.S. Lakes, Viscoelastic Materials, Cambridge University Press, Cambridge, UK, 2009.

  2. J. Park, R.S. Lakes, Biomaterials. An Introduction, 3rd ed., Springer, Berlin-Heidelberg-New York, 2007.

  3. J.W. Nunziato, S.C. Cowin, A nonlinear theory of elastic materials with voids, Archive for Rational Mechanics and Analysis, 72, 175–201, 1979.

  4. S.C. Cowin, J.W. Nunziato, Linear elastic materials with voids, Journal of Elasticity, 13, 125–147, 1983.

  5. D. Ieşan, A theory of thermoelastic materials with voids, Acta Mechanica, 60, 67–89, 1986.

  6. M. Ciarletta, D. Ieşan, Non-Classical Elastic Solids, Longman Scientific and Technical, John Wiley & Sons, New York, NY, Harlow, Essex, UK, 1993.

  7. D. Ieşan, Thermoelastic Models of Continua, Kluwer, Boston, 2004.

  8. M. Aouadi, A theory of thermoelastic diffusion materials with voids, Zeitschrift für Angewandte Mathematik und Physik, 61, 357–379, 2010.

  9. S. Chirita, A. Arusoaie, Thermoelastic waves in double porosity materials, European Journal of Mechanics – A/Solids, 86, 104177, 2021.

  10. M. Ciarletta, A. Scalia, On uniqueness and reciprocity in linear thermoelasticity of materials with voids, Journal of Elasticity, 32, 1–17, 1993.

  11. S. De Cicco, D. Ieşan, On the theory of thermoelastic materials with a double porosity structure, Journal of Thermal Stresses, 44, 1514–1533, 2021.

  12. R. Quintanilla, Impossibility of localization in linear thermoelasticity with voids, Mechanics Research Communications, 34, 522–527, 2007.

  13. M. Svanadze, Boundary value problems of steady vibrations in the theory of thermoelasticity for materials with a double porosity structure, Archives of Mechanics, 69, 4–5, 347–370, 2017.

  14. I. Tsagareli, Dynamic problems for elastic bodies with double voids, Zeitschrift für Angewandte Mathematik und Mechanik, 102, 4, e202000335, 2022.

  15. S.C. Cowin, The viscoelastic behavior of linear elastic materials with voids, Journal of Elasticity, 15, 185–191, 1985.

  16. D. Ieşan, On a theory of thermoviscoelastic materials with voids, Journal of Elasticity, 104, 369–384, 2011.

  17. A. Bucur, Spatial behavior in linear theory of thermoviscoelasticity with voids, Journal of Thermal Stresses, 38, 229–249, 2015.

  18. S. Chirita, On the spatial behavior of the steady-state vibrations in the thermoviscoelastic porous materials, Journal of Thermal Stresses, 38, 96–109, 2015.

  19. D. Ieşan, R. Quintanilla, Viscoelastic materials with a double porosity structure, Comptes Rendus Mécanique, 347, 124–130, 2019.

  20. R. Quintanilla, J.E.M. Rivera, P.X. Pamplona, On uniqueness and analyticity in thermoviscoelastic solids with voids, Journal of Applied Analysis and Computation, 1, 251–266, 2011.

  21. K. Sharma, P. Kumar, Propagation of plane waves and fundamental solution in thermoviscoelastic medium with voids, Journal of Thermal Stresses, 36, 94–111, 2013.

  22. M.M. Svanadze, Potential method in the linear theory of viscoelastic materials with voids, Journal of Elasticity, 114, 101–126, 2014.

  23. M.M. Svanadze, Potential method in the theory of thermoviscoelasticity for materials with voids, Journal of Thermal Stresses, 37, 905–927, 2014.

  24. M.M. Svanadze, Problems of steady vibrations in the coupled linear theory of double-porosity viscoelastic materials, Archives of Mechanics, 73, 4, 365–390, 2021.

  25. M.M. Svanadze, Potential method in the coupled theory of thermoviscoelastic materials with triple porosity, Applicable Analysis, 104, 3255–3277, 2025.

  26. S.K. Tomar, J. Bhagwan, H. Steeb, Time harmonic waves in a thermoviscoelastic material with voids, Journal of Vibration and Control, 20, 1119–1136, 2014.

  27. C. Cattaneo, Sulla conduzione del calore, Atti Del Seminario Matematico e Fisico Dell’Università di Modena, 3, 83–101, 1948.

  28. P. Vernotte, Les paradoxes de la théorie continue de l’équation de la chaleur, Comptes Rendus, 246, 3154–3155, 1958.

  29. H.W. Lord, Y. Shulman, A generalized dynamical theory of thermoelasticity, Journal of the Mechanics and Physics of Solids, 15, 299–309, 1967.

  30. A.E. Green, K.A. Lindsay, Thermoelasticity, Journal of Elasticity, 2, 1–7, 1972.

  31. A.E. Green, P.M. Naghdi, A re-examination of the basic postulates of thermomechanics, Proceedings of the Royal Society of London, Series A, 432, 1885, 171–194, 1991.

  32. A.E. Green, P.M. Naghdi, On undamped heat waves in an elastic solid, Journal of Thermal Stresses, 15, 253–264, 1992.

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

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

  35. B. Straughan, Heat Waves, New York, Springer Science Business Media, 2011.

  36. D.Y. Tzou, Macro- to Microscale Heat Transfer: The Lagging Behavior, 2nd ed., John Wiley & Sons, West Sussex, UK, 2015.

  37. D.S. Chandrasekharaiah, Thermoelasticity with second sound: A review, Applied Mechanics Reviews, 39, 355–376, 1986.

  38. D.S. Chandrasekharaiah, Hyperbolic thermoelasticity: a review of recent literature, Applied Mechanics Reviews, 51, 705–729, 1998.

  39. D.D. Joseph, L. Preziosi, Heat waves, Reviews of Modern Physics, 61, 41–73, 1989.

  40. D.D. Joseph, L. Preziosi, Addendum to the paper heat waves, Reviews of Modern Physics, 62, 375–391, 1990.

  41. F. Shakeriaski, M. Ghodrat, J. Escobedo-Diaz, M. Behnia, Recent advances in generalized thermoelasticity theory and the modified models: a review, Journal of Computational Design and Engineering, 8, 15–35, 2021.

  42. F.K. Moore, W.E. Gibson, Propagation of weak disturbances in a gas subject to relaxation effects, Journal of the Aerospace Sciences, 27, 2, 117–127, 1960.

  43. P.A. Thompson, Compressible Fluid Dynamics, McGraw-Hill, New York, 1972.

  44. R. Quintanilla, Moore–Gibson–Thompson thermoelasticity, Mathematics and Mechanics of Solids, 24, 4020–4031, 2019.

  45. N. Bazarra, J.R. Fernández, R. Quintanilla, Analysis of a Moore–Gibson–Thompson thermoelastic problem, Journal of Computational and Applied Mathematics, 382, 113058, 2021.

  46. N. Bazarra, J.R. Fernández, R. Quintanilla, On the decay of the energy for radial solutions in Moore–Gibson–Thompson thermoelasticity, Mathematics and Mechanics of Solids, 26, 1507–1514, 2021.

  47. O.A. Florea, A. Bobe, Moore–Gibson–Thompson thermoelasticity in the context of double porous materials, Continuum Mechanics and Thermodynamics, 33, 2243–2252, 2021.

  48. K. Jangid, M. Gupta, S. Mukhopadhyay, On propagation of harmonic plane waves under the Moore–Gibson–Thompson thermoelasticity theory, Waves in Random and Complex Media, 34, 1976–1999, 2024.

  49. K. Jangid, S. Mukhopadhyay, A domain of influence theorem under MGT thermoelasticity theory, Mathematics and Mechanics of Solids, 26, 285–295, 2021.

  50. M. Marin, M.I.A. Othman, A.R. Seadawy, C. Carstea, A domain of influence in the Moore–Gibson–Thompson theory of dipolar bodies, Journal of Taibah University for Science, 14, 653–660, 2020.

  51. R. Quintanilla, Moore–Gibson–Thompson thermoelasticity with two temperatures, Applications in Engineering Science, 1, 100006, 2020.

  52. B. Singh, S. Mukhopadhyay, Galerkin-type solution for the Moore–Gibson–Thompson thermoelasticity theory, Acta Mechanica, 232, 1273–1283, 2021.

  53. M. Svanadze, Potential method in the theory of Moore–Gibson–Thompson thermoporoelasticity, Archives of Mechanics, 77, 3–28, 2025.

  54. M. Svanadze, Uniqueness theorems in the steady vibration problems of the Moore–Gibson–Thompson thermoporoelasticity, Georgian Mathematical Journal, 32, 683–695, 2025.

  55. M. Svanadze, Uniqueness theorems in the steady vibration problems of the Moore–Gibson–Thompson thermoelasticity for materials with voids, Transactions of A. Razmadze Mathematical Institute, 179, 277–289, 2025.

  56. M.M. Svanadze, Plane waves and steady vibration problems in the Moore–Gibson–Thompson thermoviscoelasticity theory for Kelvin–Voigt materials, Journal of Thermal Stresses, 48, 12, 1792–1811, Special Issue for R.B. Hetnarski, 2025, https://doi.org/10.1080/01495739.2025.2517156

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

  58. M. Svanadze, Potential Method in Mathematical Theories of Multi-Porosity Media, Interdisciplinary Applied Mathematics, vol. 51, Springer Nature Switzerland, Cham, Switzerland, 2019.