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IssuesArchive of Issues2002-5pp.27-37

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V. A. Babeshko and P. V. Syromyatnikov, "A method for the construction of the Fourier symbol of the Green matrix for multi-layered electroelastic half-space," Mech. Solids. 37 (5), 27-37 (2002)
Year 2002 Volume 37 Number 5 Pages 27-37
Title A method for the construction of the Fourier symbol of the Green matrix for multi-layered electroelastic half-space
Author(s) V. A. Babeshko (Krasnodar)
P. V. Syromyatnikov (Krasnodar)
Abstract A method developed previously for an isotropic multi-layered half-space with plane-parallel interfaces is generalized to the case of an electroelastic multi-layered half-space. The homogeneous electroelastic layers and the underlying half-space may possess arbitrary electroelastic anisotropy. The method is stable, in particular, in the high-frequency range. In the framework of this method, we compare the applicability and the correctness of the known radiation conditions: the Sommerfeld principle, the limit absorption principle, and a principle based on the analysis of slowness surfaces of spatial waves. In contrast to the isotropic case, the choice of the radiation condition may be critical.

In [1], a stable algorithm has been developed for the construction of the Fourier symbol of the Green matrix for a stratified isotropic half-space. This algorithm avoids growing exponents at all stages of the computation. The approach proposed here is a generalization of the said method to the case of electroelastic materials.

The questions considered here are important for both theoretical and numerical analyses of wave processes in multi-layered isotropic or anisotropic elastic and electroelastic media occurring in seismology (Seismic Anisotropy. Results, Problems, Possibilities. 2-nd International Workshop, Moscow, 1986.), acoustic electronics [2], design of composite materials, etc.

Seismic anisotropy reflects the presence of order in the medium, and this can be due to various directed processes in the earth lithosphere such as the effect of initial stresses, orientation of crystals and structures, cavities and fluid inclusions, etc. Observations show that seismic anisotropy has a more complex structure than that considered in the traditional transversely-isotropic model. An adequate description can be obtained in the framework of multi-layered orthotropic (orthorhombic), transversely-isotropic, and isotropic models.

In acoustic electronics, an interest in transducers with multi-layered structures [2, 3] is due to their recently discovered specific properties which differ from those of homogeneous substrates and look promising from the technological standpoint.

In Section 1 of the present paper, we give a detailed description of the algorithm for the construction of the symbol of the Green matrix for a homogeneous half-space with an arbitrary electroelastic anisotropy. This algorithm is based on the formalism similar to that proposed in [4, 5] but other than those of [1, 6, 7]. Special attention is given to the comparison of the familiar radiation principles: the Sommerfeld principle [1], the limit absorption principle [7], and the principle based on the analysis of slowness surfaces of spatial waves [8]. In Section 2, we describe the method for the construction of the Green matrix symbol for a multi-layered piezoelectric half-space. In Section 3, we give numerical examples for a homogeneous electroelastic half-space, and in Section 4, examples for a two-layer electroelastic and three-layer anisotropic half-spaces.
References
1.  V. A. Babeshko, E. V. Glushkov, and Zh. F. Zinchenko, Dynamics of Nonhomogeneous Linearly Elastic Media [in Russian], Nauka, Moscow, 1989.
2.  W. J. Chijsen and P. M. Van der Berg, "The computation of the acousto-electric field in multi-layered SAW devices," in IEEE Ultrason. Symp. Proc. Volume 1, pp. 198-202, N. Y., 1985.
3.  T. Shiosaki, Y. Mikamura, F. Takeda, and A. Kawabata, "High-coupling and highvelocity SAW using Zn4AlN films on glass substrate," IEEE Trans. Ultrason. Ferroelec. Frecq. Contr., Vol. 33, No. 3, pp. 324-330, 1986.
4.  A. N. Stroh, "Steady state problems in anisotropic elasticity," J. Math. Phys., Vol. 41, No. 2, pp. 77-103, 1969.
5.  K. A. Ingebrigtsen and A. Tonning, "Elastic surface waves in crystals," Phys. Rev., Vol. 184, No. 3, pp. 942-951, 1969.
6.  I. I. Vorovich, V. A. Babrshko, and O. D. Pryakhina, Dynamics of Massive Bodies and Resonance Phenomena in Deformable Media [in Russian], Nauchnyi Mir, Moscow, 1999.
7.  V. A. Babeshko, Generalized Factorization Method in 3D Dynamical Mixed Problems in Elasticity [in Russian], Nauka, Moscow, 1984.
8.  B. A. Auld, Acoustic Fields and Waves in Solids. Volume 1, Wiley, New York, 1973.
9.  E. Dieulesant and D. Rouiet, Elastic Waves in Solids [Russian translation], Nauka, Moscow, 1982.
10.  V. A. Babeshko, "On integral equations of electroelasticity in design of acousto-electronic devices," Doklady AN SSSR, Vol. 302, No. 4, pp. 812-815, 1988.
11.  J. G. Fryer and L. N. Frazer, "Seismic waves in stratified anisotropic media," Geophys. J. Roy. Astron. Soc., Vol. 78, No. 3, pp. 691-710, 1984.
12.  J. Garmany, "Some properties of elastodynamic eigensolutions in stratified media," Geophys. J. Roy. Astron. Soc., Vol. 75, pp. 565-569, 1983.
13.  M. P. Shaskol'skaya (Editor), Acoustic Crystals. Handbook [in Russian], Nauka, Moscow, 1982.
Received 23 May 2001
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