Effects of SH waves in a functionally graded plate

A computational method is presented to investigate SH waves in functionally graded material (FGM) plates. The FGM plate is first divided into quadratic layer elements (QLEs), in which the material properties are assumed as a quadratic function in the thickness direction. A general solution for the equation of motion governing the QLE has been derived. The general solution is then used together with the boundary and continuity conditions to obtain the displacement and stress in the wave number domain for an arbitrary FGM plate. The displacements and stresses in the frequency domain and time domain are obtained using inverse Fourier integration. Furthermore, a simple integral technique is also proposed for evaluating modified Bessel functions with complex valued order. Numerical examples are presented to demonstrate this numerical technique for SH waves propagating in FGM plates.     

Love waves in a two-layered piezoelectric/elastic composite plate with an imperfect interface

Based on the shear spring model, the propagation of Love wave in two-layered piezoelectric/elastic composite plates under the influence of interfacial defect is investigated. The piezoelectric layer is electrically shorted at both top and bottom surfaces. The wave form solutions of the piezoelectric and elastic layers are obtained, and the dispersion equation is derived by subjecting the boundary conditions and the continuity conditions to the obtained wave form solutions. Numerical results are performed for PZT4/aluminum composite plate. The phase velocities and the mode shapes of mechanical displacement and electric potential are illustrated graphically. The results show that both the interfacial defect and the thickness ratio between the piezoelectric and elastic layers have significant effect on the propagation characteristics of Love wave. One important feature is observed that the interfacial defect always decreases the phase velocities.     

The low frequency scatter of SH waves by a rough interface in an elastic plate

The study of the reflection and transmission of low frequency SH waves incident upon a rough interface in an elastic plate is undertaken by employing a theory of acoustic wave scattering from rough surfaces originally due to Biot and subsequently generalised to the case of elastic media. In this theory the interface is replaced by a distribution of voids/asperities whose individual size is small compared to the excitation wavelength. We plot the absolute values of the reflection and transmission coefficients versus frequency when a single symmetric SH plate mode is used as the input excitation. The different types of inclusions are used to simulate the rough surface are the hollow, fluid filled and aluminum spheres. Lastly, the loss of energy due to scattering is also estimated for the different inclusion distributions considered.     

Attenuation dispersion of Love waves in a two-layered half space

The attenuation of dispersive Love waves in an anelastic two-layered half space and in a simple symmetric homogeneous three-layered model has been investigated by introducing the complex propagation functions into the known explicit dispersion relation. A frequency-dependent attenuation relation is explicitly given assuming that the media quality factors are constant. The resultant quality factor of Love waves depends not only on the frequency, but also on the layer depth, via media quality factors. The attenuation coefficient of the Love waves becomes therefore a non-linear function of the frequency. The result is consistent with the results given in the literature on seismology and in-seam seismics. It is known that the spectral ratio method can only be used to estimate the frequency-independent quality factor. Therefore, a modification of the spectral ratio method is presented to invert the frequency-dependent quality factor of Love-waves. The modification facilitates investigations of frequency-dependent quality of the medium over previous methods.     

Buckling and cracking characteristics of two-layered plate in tension

This study is concerned with the loss of local stability of two-layered metal plates (steel–aluminium alloy) with a crack in tension. The critical stresses corresponding to the initiation of buckling are determined. The post-critical deformation and form of stability loss are investigated. The influence of buckling on the fracture characteristics is also estimated.     

Two-dimensional modulation and instabilities of flexural waves of a thin plate on nonlinear elastic foundation

In the present paper we intend to examine in detail the formation of localized modes and waves mediated by modulational instability in an elastic structure. The elastic composite structure consists of a nonlinear foundation coated with an elastic thin plate. The problem deals with flexural waves traveling on the plate. The attention is devoted to the behavior of nonlinear waves in the small-amplitude limit in view of deducing criteria of instability which produce localized waves. It is shown that, in the small-amplitude limit, the basic equation which governs the plate deflection is approximated by a two-dimensional nonlinear Schrödinger equation. The latter equation allows us to study the modulational instability conditions leading to different zones of instability. The examination of the instability provides useful information about the possible selection mechanism of the modulus of the carrier wave vector and growth rate of the instabilities taking place in both (longitudinal and transverse) directions of the plate. The mechanism of the self-generated nonlinear waves on the plate beyond the birth of modulational instability is numerically investigated. The numerics show that an initial plane wave is then transformed, through the instability process, into nonlinear localized waves which turn out to be particularly stable. In addition, the influence of the prestress on the nature of localized structures is also examined. At length, in the conclusion some other wave problems and extensions of the work are evoked.     

Propagation of plane surface waves over an underwater obstacle and a submerged plate

Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 55–62, May–June, 1991.     

Shear-horizontal (SH) surface waves propagating around stratified piezoelectric cylinders

Theoretical results describing the properties of SH (shear-horizontal) surface waves propagating around piezoelectric cylinders with superimposed layers are presented. Mode orthogonality, virial theorem and equality between group and energy velocities are proved. Integral formulas for the energy velocity are much more compact than the differential expressions for the group velocity, and this provides a convenient way for computing the latter. The results obtained are illustrated by numerical calculations and graphs for a set of typical cylindrical waveguides.     

Occurrence of standing surface gravity waves modulation in shallow water

Arising of modulations of surface gravity waves in a shallow water resonator under harmonic forcing is investigated in laboratory experiments. Different types of modulations are found, when the wave amplitude exceeds a certain threshold. Bifurcation diagram on the plane “amplitude of excitation – frequency of excitation” is determined. Numerical simulations of the Euler equations within the frameworks of the High-Order Spectral Method are performed with the purpose of reproducing the modulational regimes observed in the laboratory experiments. The simulations allowed us to determine physical mechanisms responsible for the occurrence of modulated waves.     

Non-linear longitudinal waves in bars

The finite amplitude longitudinal waves along a uniform bar are examined by using the method or multiple scales. The evolution of the complex amplitude of a quasi-monochromatic progressive wave is shown to be governed by a non-linear Schrödinger equation. The analysis reveals that the constant amplitude progressive waves are stable against modulation.     

Diffraction of surface waves on an inhomogeneous elastic plate

The oblique incidence of small-amplitude waves on an elastic semi-infinite composite plate floating on the free surface of finite-depth water is studied. The front part of a constant-width plate is highed to the basic part and has characteristics different from those of the basic part. The reflection and transmission coefficients of the waves and the vertical displacements of the plate are determined. It is shown that the heterogeneity of the plate material exerts a strong effect on surface-wave diffraction. Methods for decreasing the elastic deformations of the basic part of the plate are proposed. Lavrent'ev Institute of Hydrodynamics, Siberian Division, Russian Academy of Sciences, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 41, No. 4, pp. 42–48, July–August, 2000.     

Oblique incidence of surface waves on an elastic plate

Oblique incidence of small-amplitude waves on an elastic plate floating on a free liquid surface is studied. The reflection and transmission factors of the waves and the vertical displacements and strains of the plate are determined. It is shown that all these characteristics depend greatly on the incidence angle and frequency of the wave and the width of the plate. Approximate solutions for the reflection and transmission factors are obtained. Lavrent’ev Institute of Hydrodynamics, Siberian Division, Russian Academy of Sciences, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 40, No. 4, pp. 62–68, July–August, 1999.     

Low-frequency wave propagation in an elastic plate floating on a two-layered fluid

For some technical applications related to the ice–sea interaction, it is necessary to predict waveguide properties of elastic plates floating on a relatively thin layer of water, which has a non-uniform density distribution across its depth. The issue of particular concern is propagation of low-frequency waves in such a coupled waveguide. In the present paper, a stratified fluid is modelled as two homogeneous, inviscid and incompressible layers with slightly different densities. The lighter layer of fresh water lies on top of the heavier layer of salty water. The former one produces fluid loading at the pre-stressed plate, whereas the latter one is bounded by the sea bottom. The classical asymptotic methods are employed to identify significant regimes of wave motion in such a three-component waveguide. Dispersion diagrams obtained from approximate dispersion relations are compared with their exact counterparts. The phenomena of veering and generation of waves with zero group velocity induced by pre-stress are identified and quantified.     

Rayleigh-type waves in a water-saturated porous half-space with an elastic plate on its surface

The paper deals with the plane problem of steady-state time harmonic vibrations of an infinite elastic plate resting on a water-saturated porous solid. The displacements of the plate are described by means of the linear theory of small elastic oscillations. The motion of the two-phase medium is studied within the framework of Biot's linear theory of consolidation. The main interest is focused on the investigation of properties of the Rayleigh-type waves propagating alongside of the contact surface between the plate and the porous half-space. In particular, the dependence of the phase velocity and attenuation of the waves on the plate stiffness, mass coupling coefficient, and degree of saturation of the medium is studied. Besides, for the limiting case of an infinitely thin plate, the comparison of the wave characteristics is carried out with those of the pure Rayleigh waves.     

Non-linear interaction of waves in prestressed material

Interaction of two counterpropagating ultrasonic waves in an inhomogeneously prestressed elastic material (a structural element) with quadratic non-linearity is studied theoretically by using the perturbative formalism. The analytical solution that describes wave–wave, wave–material and wave–prestress non-linear interaction is derived. This rather cumbersome solution is studied in the case of a harmonic boundary condition in the material (specimen) subjected to two-parametric inhomogeneous prestressed state. The model problems are solved and the influence of the prestress on the wave interaction is cleared up. Resulting oscillations on two parallel boundaries of the material are prestress sensitive. It is proposed to use boundary oscillation data for non-destructive characterization of inhomogeneous prestressed state.     

SH波对浅埋裂纹的半圆形凹陷地形的散射

采用Green函数方法，研究浅埋裂纹和含有圆形凹陷的弹性半空间对入射SH波的散射。首先取含有半圆形凹陷的弹性半空间，任意一点承受时间谐和的出平面线源荷载作用时的位移函数基本解作为Green函数；然后求解含半圆形凹陷的弹性半空间对SH波的散射问题；最后在裂纹实际存在位置利用Green函数实施裂纹的人工切割以恢复存在的裂纹，给出浅埋裂纹的半圆形凹陷弹性空间内的位移函数，进而求解裂纹存在对地表位移的影响。     

Non-linear flexural oscillations of a partially tapered annular plate

The free finite amplitude axisymmetric oscillations of an isotropic annular plate with partially tapered thickness are investigated. The time variable is eliminated by a Ritz-Kantorovich averaging method. The von Karman plate equations are then reduced to two non-linear ordinary differential equations, which form a non-linear eigenvalue problem. Solutions to the problem are obtained by utilizing a direct computational method. The results reveal the effects of large amplitude upon the dynamic responses. Also, an annulus of constant thickness, which has the same boundary conditions and the same volume as the partially tapered one, is investigated. Their results, which may shed light on the optimal design of annular plates, are compared.     

Hydroelastic behavior of a floating plate in waves

The behavior of a floating semiinfinite plate in surface waves incident normally to the edge of the plate is studied. We used an analytical solution of this problem obtained earlier by the Wiener–Hopf technique. In this paper, we study the distributions of displacements, deformations, and pressure over the plate as functions of the dimensionless parameters of the problem (reduced rigidity and depth) and their asymptotic distributions for large and small wavelengths.