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.
This is a preview of subscription content, access via your institution.
Buy single article
Instant access to the full article PDF.
Tax calculation will be finalised during checkout.
Subscribe to journal
Immediate online access to all issues from 2019. Subscription will auto renew annually.
Tax calculation will be finalised during checkout.
W. Nowacki, Effekty Elektro-Magnetyczne w Stalych Ciałach Odkształcalnych [in Polish], Państwowe Wyd-wo Nauk., Warszawa (1983).
Ya. S. Podstrigach and Yu. M. Kolyano, Generalized Thermomechanics [in Russian], Naukova Dumka, Kiev (1976).
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).
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).
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).
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).
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).
M. Aouadi, “Generalized theory of thermoelastic diffusion for anisotropic media,” J. Therm. Stresses, 31, No. 3, 270–285 (2008).
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).
D. S. Chandrasekharaiah, “A generalized linear thermoelasticity theory for piezoelectric media,” Acta Mech., 71, No. 1-4, 39–49 (1988).
D. S. Chandrasekharaiah, “Hyperbolic thermoelasticity: a review of recent literature,” Appl. Mech. Rev., 51, No. 12, 705–729 (1998).
A. S. El-Karamany and M. A. Ezzat, “Propagation of discontinuities in thermopiezoelectric rod,” J. Therm. Stresses, 28, No. 10, 997–1030 (2005).
R. B. Hetnarski and J. Ignaczak, “Generalized thermoelasticity,” J. Therm. Stresses, 22, No. 4-5, 451–476 (1999).
J. Ignaczak and M. Ostoja-Starzewski, Thermoelasticity with Finite Wave Speeds, Oxford Univ. Press, Oxford (2010).
J. L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications, Vol. I, Springer-Verlag, Berlin etc. (1972); http://www.springer.com/br/book/9783642651632.
H. W. Lord and Y. Shulman, “A generalized dynamical theory of thermoelasticity,” J. Mech. Phys. Solids, 15, No. 5, 299–309 (1967).
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).
W. Nowacki, “Some general theorems of thermopiezoelectricity,” J. Therm. Stresses, 1, No. 2, 171–182 (1978).
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).
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.
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); http://nbuv.gov.ua/UJRN/jopm_2016_2_11.
Published in Matematychni Metody ta Fizyko-Mekhanichni Polya, Vol. 59, No. 4, pp. 116–127, October–December, 2016.
About this article
Cite this article
Stelmashchuk, V.V., Shynkarenko, H.A. Well-Posedness of the Lord–Shulman Variational Problem of Thermopiezoelectricity. J Math Sci 238, 139–153 (2019). https://doi.org/10.1007/s10958-019-04224-x