Journal of Mathematical Sciences

, Volume 238, Issue 2, pp 139–153 | Cite as

Well-Posedness of the Lord–Shulman Variational Problem of Thermopiezoelectricity

  • V. V. Stelmashchuk
  • H. A. Shynkarenko

On the basis of the initial-boundary-value Lord–Shulman problem of thermopiezoelectricity, we formulate the corresponding variational problem in terms of the vector of elastic displacements, electric potential, temperature increment, and the vector of heat fluxes. By using the energy balance equation of the variational problem, we establish sufficient conditions for the regularity of input data of the problem and prove the uniqueness of its solution. To prove the existence of the general solution to the problem, we use the procedure of Galerkin semidiscretization in spatial variables and show that the limit of the sequence of its approximations is a solution of the variational problem of Lord–Shulman thermopiezoelectricity. This fact allows us to construct a reasonable procedure for the determination of approximate solutions to this problem.


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  1. 1.
    W. Nowacki, Effekty Elektro-Magnetyczne w Stalych Ciałach Odkształcalnych [in Polish], Państwowe Wyd-wo Nauk., Warszawa (1983).Google Scholar
  2. 2.
    Ya. S. Podstrigach and Yu. M. Kolyano, Generalized Thermomechanics [in Russian], Naukova Dumka, Kiev (1976).Google Scholar
  3. 3.
    V. Stelmashchuk and H. Shynkarenko, “Numerical simulation of the dynamic problems of pyroelectricity,” Visn. L’viv. Univ., Ser. Prykl. Mat. Inform., Issue 22, 92–107 (2014).Google Scholar
  4. 4.
    O. Fundak and H. Shynkarenko, “Barycentric representation of basis functions in the spaces of Raviart–Thomas approximations,” Visn. L’viv. Univ., Ser. Prykl. Mat. Inform., Issue 7, 102–114 (2003).zbMATHGoogle Scholar
  5. 5.
    H. A. Shynkarenko, “Projection-grid approximations for the variational problems of pyroelectricity. I. Statement of the problem and analysis of steady-state forced vibrations,” Differents. Uravn., 29, No. 7, 1252–1260 (1993).Google Scholar
  6. 6.
    H. A. Shynkarenko, “Projection-grid approximations for variational problems of pyroelectricity. IІ. Discretization and solvability of nonstationary problems,” Differents. Uravn., 30, No. 2, 317–326 (1994).Google Scholar
  7. 7.
    I. A. Chyr and H. A. Shynkarenko, “Well-posedness of the Green–Lindsay variational problem of dynamic thermoelasticity,” Mat. Met. Fiz.-Mekh. Polya, 58, No. 3, 15–25 (2015); English translation: J. Math. Sci., 226, No. 1, 11–27 (2017).Google Scholar
  8. 8.
    M. Aouadi, “Generalized theory of thermoelastic diffusion for anisotropic media,” J. Therm. Stresses, 31, No. 3, 270–285 (2008).CrossRefGoogle Scholar
  9. 9.
    M. H. Babaei and Z. T. Chen, “Transient thermopiezoelectric response of a one-dimensional functionally graded piezoelectric medium to a moving heat source,” Arch. Appl. Mech., 80, No. 7, 803–813 (2010).CrossRefzbMATHGoogle Scholar
  10. 10.
    D. S. Chandrasekharaiah, “A generalized linear thermoelasticity theory for piezoelectric media,” Acta Mech., 71, No. 1-4, 39–49 (1988).CrossRefzbMATHGoogle Scholar
  11. 11.
    D. S. Chandrasekharaiah, “Hyperbolic thermoelasticity: a review of recent literature,” Appl. Mech. Rev., 51, No. 12, 705–729 (1998).CrossRefGoogle Scholar
  12. 12.
    A. S. El-Karamany and M. A. Ezzat, “Propagation of discontinuities in thermopiezoelectric rod,” J. Therm. Stresses, 28, No. 10, 997–1030 (2005).CrossRefGoogle Scholar
  13. 13.
    R. B. Hetnarski and J. Ignaczak, “Generalized thermoelasticity,” J. Therm. Stresses, 22, No. 4-5, 451–476 (1999).MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    J. Ignaczak and M. Ostoja-Starzewski, Thermoelasticity with Finite Wave Speeds, Oxford Univ. Press, Oxford (2010).zbMATHGoogle Scholar
  15. 15.
    J. L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications, Vol. I, Springer-Verlag, Berlin etc. (1972);
  16. 16.
    H. W. Lord and Y. Shulman, “A generalized dynamical theory of thermoelasticity,” J. Mech. Phys. Solids, 15, No. 5, 299–309 (1967).CrossRefzbMATHGoogle Scholar
  17. 17.
    R. D. Mindlin, “On the equations of motion of piezoelectric crystals,” in: Problems of Continuum Mechanics: Contributions in Honor of the 70th Birthday of Academician N. I. Muskhelishvili, SIAM, Philadelphia, 282–290 (1961).Google Scholar
  18. 18.
    W. Nowacki, “Some general theorems of thermopiezoelectricity,” J. Therm. Stresses, 1, No. 2, 171–182 (1978).CrossRefGoogle Scholar
  19. 19.
    H. H. Sherief and A. M. Abd El-Latief, “Boundary element method in generalized thermoelasticity,” in: Encyclopedia of Thermal Stresses, Ed. R. B. Hetnarski, Springer, Dordrecht etc., Vol. 1, 407–415 (2014).Google Scholar
  20. 20.
    V. V. Stelmashchuk and H. A. Shynkarenko, “Numerical modeling of thermopiezoelectricity steady state forced vibrations problem using adaptive finite element method,” in: Advances in Mechanics: Theoretical, Computational and Interdisciplinary Issues, Eds. M. Kleiber et al. (Proc. 3rd Polish Congress of Mechanics (PCM) and 21st Int. Conf. on Computer Methods in Mechanics (CMM), Gdansk, Poland, 8-11 September 2015.), CRC Press, London (2016), pp. 547–550.Google Scholar
  21. 21.
    V. V. Stelmashchuk and H. A. Shynkarenko, “Numerical solution of Lord–Shulman thermopiezoelectricity forced vibrations problem,” Zh. Obchysl. Prykl. Matem., No. 2, 106–119 (2016); Scholar

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Authors and Affiliations

  • V. V. Stelmashchuk
    • 1
    • 2
  • H. A. Shynkarenko
    • 2
  1. 1.I. Franko Lviv National UniversityLvivUkraine
  2. 2.Opole University of TechnologyOpolePoland

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