Multiple scattering of flexural waves by random configurations of inclusions in thin plates

Flexural waves are scattered by inclusions in a thin plate. For a single inclusion of arbitrary shape, reciprocity relations are obtained connecting coefficients in circular multipole expansions. Then, a formula for the effective wavenumber in a random arrangement of identical circular inclusions is derived, using the Lax quasi-crystalline approximation.     

DUAL RECIPROCITY BOUNDARY ELEMENT METHOD FOR FLEXURAL WAVES IN THIN PLATE WITH CUTOUT

The theoretical analysis and numerical calculation of scattering of elastic waves and dynamic stress concentrations in the thin plate with the cutout was studied using dual reciprocity boundary element method (DRM). Based on the work equivalent law, the dual reciprocity boundary integral equations for flexural waves in the thin plate were established using static fundamental solution. As illustration, numerical results for the dynamic stress concentration factors in the thin plate with a circular hole are given. The results obtained demonstrate good agreement with other reported results and show high accuracy.     

Scattering of water waves by inclined thin plate submerged in finite-depth water

Summary  The problem of water wave scattering by an inclined thin plate submerged in water of uniform finite depth is investigated here under the assumption of irrotational motion and linear theory. A hypersingular integral equation formulation of the problem is obtained by an appropriate use of Green's integral theorem followed by utilization of the boundary condition on the plate. This hypersingular integral equation involves the discontinuity in the potential function across the plate, which is approximated by a finite series involving Chebyshev polynomials. The coefficients of this finite series are obtained numerically by collocation method. The quantities of physical interest, namely the reflection and transmission coefficients, force and moment acting on the plate per unit width, are then obtained numerically for different values of various parameters, and are depicted graphically against the wavenumber. Effects of finite-depth water, angle of inclination of the plate with the vertical over the deep water and vertical plate results for these quantities are shown. It is observed that the deep-water results effectively hold good if the depth of the mid-point of the submerged plate below the free surface is of the order of one-tenth of the depth of the bottom. Received 30 November 2000; accepted for publication 26 June 2001     

Experimental slowing of flexural waves in dielectric elastomer films by voltage

Shape and physical properties of dielectric elastomers are changeable by voltage. Theoretical works show that these changes can be harnessed to tune the propagation of superposed elastic waves. We experimentally demonstrate this concept by manipulating waves in a dielectric elastomer film, focusing on the flexural mode at low frequencies. To this end, we design an experimental apparatus to pre-stretch, actuate, excite waves at low frequencies in a VHB™ 4910 film, and measure the velocity of the fundamental flexural mode. Our results show that the excited wave velocity is slowed down by the applied voltage, and provide experimental proof of concept for the application of deformable dielectrics as tunable waveguides.     

Scattering of water waves by a submerged thin vertical elastic plate

The problem of water wave scattering by a thin vertical elastic plate submerged in infinitely deep water is investigated here assuming linear theory. The boundary condition on the elastic plate is derived from the Bernoulli–Euler equation of motion satisfied by the plate. This is converted into the condition that the normal velocity of the plate is prescribed in terms of an integral involving the difference in velocity potentials (unknown) across the plate multiplied by an appropriate Green’s function. The reflection and transmission coefficients are obtained in terms of integrals involving combinations of the unknown velocity potential on the two sides of the plate and its normal derivative on the plate, which satisfy three simultaneous integral equations, solved numerically. These coefficients are computed numerically for various values of different parameters and are depicted graphically against the wave number for different situations. The energy identity relating these coefficients is also derived analytically by employing Green’s integral theorem. Results for a rigid plate are recovered when the parameters characterizing the elastic plate are chosen negligibly small.     

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.     

Multiple scattering of flexural waves in a semi-infinite thin plate with a cutout

The multiple scattering of flexural waves and dynamic stress concentration in a semi-infinite thin plate with a cutout are investigated, and the expressions of this problem are obtained. The analytical solutions of wave fields are expressed by employing the wave function expansion method and the expanded mode coefficients are solved by satisfying the boundary condition of the cutout. The image method is used to satisfy the traction free boundary condition of the plate. As an example, the numerical results of dynamic stress concentration factors are graphically presented and discussed. Numerical results show that the analytical results of the scattered waves and dynamic stress in semi-infinite plates are significantly different from those in infinite plates when the ratio of distance b/a is relatively little. In the region of low frequency and long wavelength, the maximum dynamic stress concentration factors occur on the illuminated side of the scattering body with θ = π, but not at the edge of the cutout with θ = π/2. As the incidence frequency increases (the wavelength becomes short), the dynamic stress on the illuminated side of the cutout decreases, however, the dynamic stress on the shadow side increases.     

Active control of flexural waves in a phononic crystal beam with staggered periodic properties

The band gaps of a phononic crystal beam with staggered periodic structure are investigated. The periodic system consists of a pure elastic (i.e. PMMA) matrix beam and some piezoelectric (i.e. PZT) patches with coupling between the mechanical–electrical components. The PZT patches connected by negative capacitance circuits are applied to function as the active control system. Based on the condition at the interface between adjacent unit cells, the transfer matrix and localization factor are derived. The influence of the degree of interlacing and negative capacitance circuits are discussed. The numerical results show that another band gap can be generated by the staggered periodic structure of PZT patches. The widths and locations of the band gaps can be changed by the degree of interlacing.     

Reflection and transmission of regular water waves by a thin,floating plate

Measurements of the wave fields reflected and transmitted by a thin floating plastic plate are reported for regular incident waves over a range of incident periods (producing wavelengths comparable to the plate length) and steepnesses (ranging from mild to storm-like). Two different plastics are tested, with different densities and mechanical properties, and three different configurations are tested. The configurations include freely floating plates, loosely moored plates (to restrict drift), and plates with edge barriers (to restrict waves overwashing the plates). The wave fields reflected and transmitted by plates without barriers are shown to become irregular, as the incident waves become steeper, particularly for the denser plastic and the moored plate. Further, the proportion of energy transmitted by the plates without barriers is shown to decrease as the incident wave becomes steeper, and this is related to wave energy dissipation.     

Scattering of flexural waves by a cracked Mindlin plate of soft ferromagnetic material in a uniform magnetic field

Consider the impingement of time harmonic flexural waves on a through crack in a soft ferromagnetic plate the surface of which is subjected to a uniform magnetic field at normal incidence. Mindlin's plate theory is used to account for the magneto-elastic interaction. For an incident wave that gives rise to moments symmetric about the crack plane, Fourier transforms are applied reducing the mixed boundary value problem to a Fredholm integral equation that can be solved numerically. The dynamic moment intensity factor versus frequency is computed to exhibit the influence of the magnetic field.     

Scattering of water waves by thin vertical plate submerged below ice-cover surface

The present paper is concerned with scattering of water waves from a vertical plate, modeled as an elastic plate, submerged in deep water covered with a thin uniform sheet of ice. The problem is formulated in terms of a hypersingular integral equation by a suitable application of Green's integral theorem in terms of difference of potential functions across the barrier. This integral equation is solved by a collocation method using a finite series involving Chebyshev polynomials. Reflection and transmission coefficients are obtained numerically and presented graphically for various values of the wave number and ice-cover parameter.     

含孔软铁磁材料Mindlin板中弹性波散射问题

基于考虑磁弹相互作用的Mindlin板弯曲波动方程，采用波函数展开法，分析研究 了含孔软铁磁材料Mindlin板中弹性波散射与动应力集中问题，给出了问题的分析 解和数值算例. 通过分析发现：磁感应强度对动弯矩集中系数和动剪力集中系数有 增加的作用，特别是在低频的情况下.     

Scattering of flexural waves and dynamic stress concentrations in Mindlin's thick plates with a cutout

Using the complex variable method and conformal mapping, scattering of flexural waves and dynamic stress concentrations in Mindlin's thick plates with a cutout have been studied. The general solution of the stress problem of the thick plate satisfying the boundary conditions on the contour of cutouts is obtained. Applying the orthogonal function expansion technique, the dynamic stress problem can be reduced into the solution of a set of infinite algebraic equations. As examples, numerical results for the dynamic stress concentration factor in Mindlin's plates with a circular, elliptic cutout are graphically presented in sequence. The project supported by the National Natural Science Foundation of China     

Scattering of flexural wave in a thin plate with multiple circular holes by using the multipole Trefftz method

The scattering of flexural wave by multiple circular holes in an infinite thin plate is analytically solved by using the multipole Trefftz method. The dynamic moment concentration factor (DMCF) along the edge of circular holes is determined. Based on the addition theorem, the solution of the field represented by multiple coordinate systems centered at each circle can be transformed into one coordinate system centered at one circle, where the boundary conditions are given. In this way, a coupled infinite system of simultaneous linear algebraic equations is derived as an analytical model for the scattering of flexural wave by multiple holes in an infinite plate subject to the incident flexural wave. The formulation is general and is easily applicable to dealing with the problem containing multiple circular holes. Although the number of hole is not limited in our proposed method, the numerical results of an infinite plate with three circular holes are presented in the truncated finite system. The effects of both incident wave number and the central distance among circular holes on the DMCF are investigated. Numerical results show that the DMCF of three holes is larger than that of one, when the space among holes is small and meanwhile the specified direction of incident wave is subjected to the plate.     

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.     

Scattering of obliquely incident waves by straight features in a plate

The scattering of obliquely incident waves by straight features in a plate is solved analytically. The reflection matrix of a free straight edge and the scattering matrix of a straight thickness step are obtained respectively by modal decomposition based on a real orthogonal relation. The formulas are illustrated through numerical examples. The matrices are found to be Hermitian for the propagating modes; thus, the mode conversions are reciprocal in terms of energy. The matrices can be used to determine the scattering from more complicated straight features if the cross sections are approximated as sequences of stairs and steps.     

Static flexural analysis of elliptic Reissner–Mindlin plates on a Pasternak foundation by the triangular differential quadrature method

In this paper, static flexural analysis of elliptic Reissner–Mindlin plates resting on a Pasternak foundation is conducted using the triangular differential quadrature method (TDQM). A triangular serendipity transformation is proposed to map a curvilinear domain onto a unit isosceles right triangle. The applicability of the TDQM to problems on domains with curvilinear boundaries is improved significantly since no inner points are needed in the transformation. In the case of thin circular plates, the results of the present method are in excellent agreement with those of the available exact solution, demonstrating the effectiveness of the present method. Elliptic plates with various aspect ratios and thickness-to-width ratios under uniformly distributed load are studied.