Axisymmetric elastic waves in weakly coupled layered media of infinite radial extent

✍ Scribed by C. Cetinkaya; A.F. Vakakis; M. El-Raheb

• Publisher: Elsevier Science
• Year: 1995
• Tongue: English
• Weight: 852 KB
• Volume: 182
• Category: Article
• ISSN: 0022-460X
• DOI: 10.1006/jsvi.1994.0197

✦ Synopsis

The propagation of axisymmetric waves in layered periodic elastic media of infinite radial extent is investigated. Hankel and Laplace transforms are employed, to convert radial and time dependence of displacement and stress within a layer to frequency and radial wavenumber. Continuity of stress and displacement is imposed at the interface between layers yielding transfer matrices. The structure of propagation and attenuation zones (PZ's and AZ's) of the system with infinite number of layers is studied. When the ratios of shear and longitudinal mechanical impedances (tT , tL ) of the two layers are large, the PZ's of the layered system become narrow, approaching certain limiting curves in the frequency-wavenumber plane. For large (tT , tL ) an asymptotic analysis is employed analytically to approximate the width of the PZ's. Numerical computations of PZ's are given, which are in good agreement with the analytical predictions. The spacing of the resonance points of a layered system with finite number of layers is found to depend mainly on the structure of the PZ's of the corresponding system with an infinite number of layers. a surface in the frequency-wavenumber space. In references [5][6][7][8], Floquet theory was employed to study the properties of the dispersion surface of propagating SH-, P-and SV-waves. Herrmann and Hemami [9] considered plane strain (P-and SV-) wave propagation in a periodically layered half-space with layers parallel to the free surface. They found that Rayleigh waves propagating along the traction-free boundary are highly dispersive, and that some higher dispersion branches of the surface waves may be discontinuous. Rousseau [10] computed the propagation and attenuation zones in the frequency-wavenumber plane for oblique Floquet waves in a periodic medium of fluid and elastic layers. Rizzi and Doyle [11] developed a spectral method based on the Fast Fourier Transform to study transient waves in layered media. In reference [12], the structure of propagation and attenuation zones (PZ's and AZ's) for one-dimensional waves in layered media was examined. Parameters controlling the width of the PZ's were identified. The concept of ''fixed points'' relates the behavior of the propagation zones to the ratio of impedances of the layers in the limit when this ratio is large. Analytic approximations defining the PZ's were derived. Layered systems with weak disorder were also addressed in that work.

Numerical computations of the transient responses of layered systems were performed in additional works. Thomson [13] adopted a matrix formulation in treating plane waves in layered media. Haskell [14] extended Thomson's formulation, and derived the dispersion relation for Rayleigh and Love surface waves in bi-periodic and tri-periodic media. Transfer matrix methods suffer from possible numerical instabilities arising from evanescent waves (exponential dichotomy). Kundu and Mal [15] proposed a pole removal method to eliminate this numerical instability, and numerically computed the transient responses of a layered system forced by harmonic and impulsive point loads. This method was modified by Mal [16], and applied to transient waves in anisotropic layered systems. Recently, Tenenbaum and Zindeluk [17] derived an exact algebraic solution for computing the transient response of one-dimensional layered media excited by arbitrary input pulses.

In the field of anisotropic layered media, Anderson [18] studied harmonic waves in an elastic composite structure with transversely isotropic layers in the axial direction. Shah and Datta [19] used a stiffness method and Hamilton's princple to treat harmonic waves in periodically laminated infinite media. Braga [20] applied Floquet theory to waves in anisotropic layered media of infinite, semi-infinite and finite axial extent. In these studies, the governing sextic matrix equation excluded source and body forces. An exact solution for the dispersion relation of Floquet waves was obtained by applying Stroh's formalism [21].

This work examines axisymmetric waves in a bi-periodic medium of infinite extent in the radial direction. The formulation followed herein is similar to that employed by Miklowitz [22], who studied the axisymmetric response of an infinite elastic plate forced by two symmetrically placed point loads on its free surfaces. The main goal of the present study is to understand the structure of propagation and attenuation zones (PZ's and AZ's) for axisymmetric wave propagation in the system with infinite layers, and to derive asymptotic relations for the boundaries of the PZ's in the limit when the ratios of the shear and longitudinal mechanical impedances (t T , t L ) of the two layers become large. Furthermore, the concept of ''fixed points'' introduced in reference [12] is extended to the two-dimensional axisymmetric problem, to study the PZ's and AZ's of layered systems with large t T and t L . A secondary goal of the work is to investigate the resonances of a free-free system with a finite number of periodic sets, and to determine the design parameters which control the spacing of the resonance curves in the frequencywavenumber plane.