 | | Mechanics of Solids
A Journal of Russian Academy of Sciences | | Founded
in January 1966
Issued 6 times a year
Print ISSN 0025-6544
Online ISSN 1934-7936 |
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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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