Two-dimensional unsteady waves in an electromagnetoelastic sphere

Vestyak V.A.^a*, Tarlakovskii D.V.^a,b**

^a Moscow Aviation Institute (National Research University), Moscow, 125993 Russia

^b Research Institute of Mechanics, Moscow State University, Moscow, 119192 Russia

E-mail: ^* v.a.vestyak@mail.ru, ^** tdvhome@mail.ru

Received June 5, 2017

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Abstract

The propagation of unsteady kinematic or electromagnetic perturbations in an isotropic ball given on its surface has been studied. Along with the Maxwell equations and the linearized Ohm's law, we have studied the linear equations of motion for an elastic ball, the right side of whi ch includes the Lorentz force as the body force. The radial and tangential components of the unknown quantities expand into series of the Legendre and Gegenbauer polynomials. The initial-boundary value problem is solved by means of the Laplace transform by time and the expansion of the series coefficients into power series in a small parameter connecting the mechanical and electromagnetic characteristics of the continuum. The expansion in the power series allows to construct a recurrent sequence of boundary value problems with respect to the unknown components of the mechanical and electromagnetic fields. Each individual problem is solved by means of generalized convolutions of Green's functions with the unknown functions corresponding to the preceding terms of the recurrent sequence. The quasi-static analogs have been used as Green's functions for the electromagnetic field. For the mechanical field, the explicit form of Green's bulk functions found using computer algebra and complex analysis methods has been used.

Keywords: unsteady waves, electromagnetoelastic sphere, axial symmetry, coupled electromagnetic and mechanical fields, small parameter method, Laplace transform, Green's functions

Acknowledgments. The study was supported by the Russian Foundation for Basic Research (project no. 15-08-00788).

Figure Captions

Fig. 1. Variation of u₁ by radius r.

Fig. 2. Variation of ν₁ by radius r.

Fig. 3. Variation of H₁ ? 10¹¹ by radius r.

Fig. 4. Variation of E_b1 by radius r.

Fig. 5. Variation of u₁ in time τ.

Fig. 6. Variation of ν₁ in time τ.

Fig. 7. Variation of H₁ ? 10¹¹ by radius τ.

Fig. 8. Variation of E_b1 by radius τ.

Fig. 9. Series convergence by the small parameter.

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For citation: Vestyak V.A., Tarlakovskii D.V. Two-dimensional unsteady waves in an electromagnetoelastic sphere. Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, 2017, vol. 159, no. 3, pp. 306–317. (In Russian)

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