Antiplane non-stationary motion of electromagnetic-elastic half-space with account for piezoelectric effects

Deformable body mechanics


Nguyen T. T.*, Tarlakovsky D. V.**

Moscow Aviation Institute (National Research University), 4, Volokolamskoe shosse, Moscow, А-80, GSP-3, 125993, Russia



The article considers homogeneous anisotropic unsteady electromagnetic-elastic motion related to the rectangular Cartesian coordinate system. The resolving system of equations includes equations of motion, Cauchy relations for deformations, Maxwell equations, as well as linearized generalized Ohm’s law and piezo-effects accounting for the physical relationships.

The article demonstrates that the antiplane movement is possible for the transversally isotropic medium in conditions of magnetic piezo-effects absence. It is assumed, that in this option the displacement and non-zero component of electric induction vector are set, and zero initial conditions and all components of stress and strain state are limited at the half-plane boundary.

The problem solving is being sought in the form of convolution functions of the specified displacement and electric induction with relevant surface Green’s functions. Laplace transformations in time and Fourier transformations in space coordinate are applied for their plotting. Analysis of images and characteristic equation revealed the intricacy of plotting the originals in an explicit form. Thus, the method of a small parameter, which is used as a coefficient, linking mechanical and electromagnetic fields, is employed. With this, it was marked that with the zero small parameter these fields are independent.

An explicit form of images of the first two coefficients of the corresponding power series was found. Their original are being found either by sequential Fourier and Laplace transforms inversion, or with the algorithm of joint Fourier and Laplace transforms inversion. In the most complicated case, the original is being presented in the form of definite integral, being calculated by a numerical procedure.

As a result, the problem solution is reduced to the two linear operators in convolution functions for the sought-for functions.

The examples of calculations are presented for the the material of a half-space in the form of quartz. The article demonstrates the dependencies on time and spatial coordinate of the resolving relations kernels, as well as electric induction and displacement for the concrete option of boundary conditions. Induction changes linearly along the coordinate, as the Heaviside function in time, while the displacement changes correspondingly as the Dirac delta function and linearly.

It is stated that single-error corrections introduced into the solution with due regard to piezoelectric properties of the medium have the order of the coefficient linking mechanical and electromagnetic fields.


antiplane process, piezoelectric effect, surface fundamental functions, Fourier and Laplace integral transformations, sequential inversion of transformations


  1. Nasedkin A.V. Mekhanika deformiruemykh tel. Mezhvuzovskii sbornik nauchnykh trudov, Rostov na- Donu, Donskoi gosudarstvennyi tekhnicheskii universitet, 1994, pp. 78 - 84.

  2. Bardzokas D.I., Senik N.A. Kontaktnye zadachi elektrouprugosti (Сontact problemы of electroelasticity), Moscow, Fizmatlit, 2001, pp. 583 - 606.

  3. Bardzokas D.I., Kudryavtsev B.A., Senik N.A. Rasprostranenie voln v elektromagnitouprugikh sredakh (Wave Propagation in Electromagnetic-Elastic Media), Moscow, Editorial, 2003, 336 p.

  4. Kirilyuk V.S. Prikladnaya mekhanika, 2006, vol. 42, no. 11, pp. 69 - 84.

  5. Kirilyuk V.S. Teoreticheskaya i prikladnaya mekhanika, 2007, no. 43, pp. 16 - 21.

  6. Kirilyuk V.S., Levchuk O.I. Prikladnaya mekhanika, 2006, vol. 42, no. 9, pp. 59 - 69.

  7. Vatul'yan A.O. Trudy 4-i Mezhdunarodnoi konferentsii “Sovremennye problemy mekhaniki sploshnoi sredy”, Rostov-na-Donu, Yuzhnyi federal'nyi nouniversitet, 1998, vol. 1, pp. 79 - 83.

  8. Vatul'yan A.O. Prikladnaya matematika i mekhanika, 1996, vol. 60, no. 2, pp. 309 - 312.

  9. Vatul'yan A.O., Dombrova O.B. Trudy 5-i Mezhdunarodnoi konferentsii “Sovremennye problemy mekhaniki sploshnoi sredy”, Rostov na-Donu, Izd-vo Yuzhnogo federal'nogo universiteta, 2000, vol. 2, pp. 48 - 52.

  10. Vestyak V.A., Tarlakovskii D.V. Metodi rozv'yazuvannya prikladnikh zadach mekhaniki deformivnogo tverdogo tila: Zbirnik naukovikh prats', Dnipropetrovs'k, IMA-pres, 2009, vol. 10, pp. 57 – 62.

  11. Tarlakovskii D.V., Vestyak V.A., Zemskov A.V. Dynamic Processes in Thermoelectromagnetoelastic and Thermoelastodiffusive Media, In: Encyclopedia of Thermal Stresses. Dordrecht, Heidelberg, New York, London: Springer, 2014, vol. 2, pp. 1064 - 1071.

  12. Vestyak V.A., Lemeshev V.A. Problemy vychislitel'noi mekhaniki i prochnosti konstruktsii: Sb. nauch. trudov, Dnepropetrovsk, DGU, 2009, vol. 13, pp. 24 - 30.

  13. 13.Vestyak V.A., Lemeshev V.A. Materialy XV Mezhdunarodnogo simpoziuma “Dinamicheskie i tekhnologicheskie problemy mekhaniki kontruktsii i sploshnykh sred” im. A.G. Gorshkova, Moscow, Paradiz, 2009, vol. 1, pp. 43.

  14. Vestyak V.A., Lemeshev V.A. Materialy XIV Mezhdunarodnogo simpoziuma “Dinamicheskie i tekhnologicheskie problemy mekhaniki kontruktsii i sploshnykh sred” im. A.G. Gorshkova, Moscow, ID MEDIAPRAKTIKA-M, 2008, vol. 1, pp. 59 – 60.

  15. Lai Tkhan' Tuan, Tarlakovskii D.V. Trudy MAI, 2012, no. 53, available at:

  16. Chan Le Tkhai, Tarlakovskii D.V. Trudy MAI, 2018, no. 102, available at:

  17. Slepyan L.I., Yakovlev Yu.S. Integral'nye preobrazovaniya v nestatsionarnykh zadachakh mekhaniki (Integral transformations in nonstationary problems of mechanics), Leningrad, Sudostroenie, 1980, 344 p.

  18. Fedorov F.I. Teoriya uprugikh voln v kristallakh (Theory of elastic waves in crystals), Moscow, Nauka. Glavnaya redaktsiya fiziko-matematicheskoi literatury, 1965, 388 p.

  19. Gradshtein I.S., Ryzhik I.M. Tablitsy integralov, summ, ryadov i proizvedenii (Tables of integrals, sums, series and products), Moscow, Nauka, 1971, 1108 p.

  20. Prudnikov A.P., Brychkov Yu.A., Marichev O.I. Integraly i ryady. T.1. Elementarnye funktsii (Integrals and series), Moscow, FIZMATLIT, 2002, 632 p.

  21. Grigor'ev I.S., Meilikhov E.Z. Fizicheskie velichiny. (Physical quantities), Moscow, Energoatomizdat, 1991, 1232 p.

  22. Nai Dzh. Fizicheskie svoistva kristallov (Physical properties of crystals), Moscow, Mir, 1967, 386 p.

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