Dynamic bending of an infinite electromagnetoelastic rod

Thong D. Pham ; D. V. Tarlakovskii

Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 20 (2020) no. 4, pp. 493-501.

Résumé

The problem of non-stationary bending of an infinite electromagnetoelastic rod is considered. It is assumed that the material of the rod is a homogeneous isotropic conductor. The closed-form system of process equations is constructed under the assumption that the desired functions depend only on the longitudinal coordinate and time using the corresponding relations for shells which take into account the initial electromagnetic field, the Lorentz force, Maxwell's equations, and the generalized Ohm's law. The desired functions are assumed to be bounded, and the initial conditions are assumed to be null. The solution of the problem is constructed in an integral form with kernels in the form of influence functions. Images of kernel are found in the space of Laplace transformations in time and Fourier transformations in spatial coordinates. It is noted that the images are rational functions of the Laplace transform parameter, which makes it quite easy to find the originals. However, for a general model that takes into account shear deformations, the subsequent inversion of the Fourier transform can be carried out only numerically, which leads to computational problems associated with the presence of rapidly oscillating integrals. Therefore, the transition to simplified equations corresponding to the Bernoulli – Euler rod and the quasistationary electromagnetic field is carried out. The method of a small parameter is used for which a coefficient is selected that relates the mechanical and electromagnetic fields. In the linear approximation, influence functions are found for which images and originals are constructed. In this case, the zeroth approximation corresponds to a purely elastic solution. Originals are found explicitly using transform properties and tables. Examples of calculations are given for an aluminum rod with a square cross section. It is shown that for the selected material the quantitative difference from the elastic solution is insignificant. At the same time, taking into account the connectedness of the process leads to additional significant qualitative effects.

Bibliographie

[1] Ambarcumyan S. A., Bagdasaryan G. E., Belubekyan M. V., Magnetoelasticity of thin shells and plates, Nauka, M., 1977, 272 pp. (in Russian)

[2] Bardzokas D. I., Zobnin A. I., Senik N. A., Fil'shtinskii M. L., Mathematical modeling in problems of mechanics of coupled fields, v. 2, Static and dynamic problems of electroelasticity for composite multiply connected bodies, KomKniga, M., 2005, 374 pp. (in Russian)

[3] Bardzokas D. I., Kudryavcev B. A., Senik N. A., Wave propagation in electromagnetoelastic media, URSS, M., 2003, 336 pp. (in Russian)

[4] Korotkina M. R., Electromagnetoelasticity, Moscow Univ. Press, M., 1988, 302 pp. (in Russian)

[5] Altay G., Dokmeci M. C., “On the fundamental equations of electromagnetoelastic media in variational form with an application to shell-laminae equations”, International Journal of Solids and Structures, 47:3–4 (2010), 466–492 | DOI | MR | Zbl

[6] Grinchenko V. T., Ulitko A. F., Shulga N. A., The mechanics of related fields in structural elements, in 5 vols, v. 5, Electroelasticity, Naukova dumka, Kiev, 1989, 280 pp. (in Russian)

[7] Kalinchuk V. V., Belyankova T. I., Dynamic contact tasks for prestressed electroelastic media, Fizmatlit, M., 2006, 273 pp. (in Russian)

[8] Novackij V., Electromagnetic effects in solids, Mir, M., 1986, 126 pp. (in Russian)

[9] Parton V. Z., Kudryavcev B. A., Electromagnetoelasticity of piezoelectric and electrically conductive bodies, Nauka, M., 1988, 470 pp. (in Russian)

[10] Khoroshun L., “Construction of dynamic equations of electromechanics of dielectrics and piezoelectrics based on two-continuum mechanics”, Fiz.-mat. modelyuv. inf. tekhnol., 2006, no. 3, 177–198 (in Russian)

[11] Wang Xiaomin, Shen Yapeng, “Some fundamental theory of electro-magneto-thermo-elastic media. I. The theory of dynamics”, Chinese Journal of Applied Mechanics, 1994, no. 3, 42–49

[12] Pao Y. H., Yeh C. S., “A linear theory for soft ferromagnetics elastic solids”, Int. J. Engng. Sci., 11 (1983), 415–436

[13] Vatulyan A. O., “Fundamental solutions in non-stationary problems of electroelasticity”, PMM, 60:2 (1996), 309–312 (in Russian) | Zbl

[14] Melnik V. N., “The existence and uniqueness theorems of a generalized solution for a class of non-stationary problems of coupled electroelasticity”, Soviet Math (Iz. VUZ), 35:4 (1991), 23–30 | MR | Zbl

[15] Jiang A., Ding H., “Analytical solutions to magneto-electro-elastic beams”, Structural Engineering and Mechanics, 18:2 (2004), 195–209 | DOI

[16] Cheban V. G., Fornya G. A., “Solution of the problem of the propagation of an electroelastic wave in a piezoceramic rod”, Izv. AN MSSR. Matematika, 1990, no. 1, 55–59 (in Russian) | Zbl

[17] Stepanov G. V., Babuckii A. I., Mameev I. A., “Nonstationary Stress-Strain State of a Long Rod Induced by a High-Density Electric Pulse”, Strength problems, 2004, no. 4, 60–67 (in Russian)

[18] Vestyak V. A., Tarlakovskii D. V., “The Model of Thin Electromagnetoelastic Shells Dynamics”, Proceedings of the Second International Conference on Theoretical, Applied and Experimental Mechanics. Structural Integrity, Springer, Nature Switzerland AG, 2019, 254–258 | DOI

[19] Gorshkov A. G., Medvedskii A. L., Rabinskii L. N., Tarlakovskii D. V., Waves in continuous media, Fizmatlit, M., 2004, 472 pp. (in Russian)

[20] Vestyak V. A., Gachkevich A. R., Musii R. S., Tarlakovskii D. V., Fedotenkov G. V., Two-dimensional non-stationary waves in electromagnetoelastic bodies, Fizmatlit, M., 2019, 288 pp. (in Russian)

[21] Prudnikov A. P., Brychkov Yu. A., Marichev O. I., Integrals and series, in 3 vols, v. 1, Elementary Functions, Fizmatlit, M., 2002, 623 pp. (in Russian) | MR

[22] Ditkin V. A., Prudnikov A. P., Handbook of operational calculus, Vysshaya shkola, M., 1965, 467 pp. (in Russian) | MR