Perforated Elastic Plate Research Articles

Localizing Elastic Edge Waves via the Topological Rainbow Effect

We combine two different fields, topological physics and graded metamaterials, to design a topological metasurface to control and redirect elastic waves. We strategically design a two-dimensional crystalline perforated elastic plate, using a square lattice, that hosts symmetry-induced topological edge states. By concurrently allowing the elastic substrate to spatially vary in depth, we are able to convert the incident slow wave into a series of robust modes, with differing envelope modulations. This adiabatic transition localizes the incoming energy into a concentrated region where it can then be damped or extracted. For larger transitions, different behavior is observed; the incoming energy propagates along the interface before being partitioned into two disparate chiral beams. This ``topological rainbow'' effect leverages two main concepts, namely the quantum valley Hall effect and the rainbow effect usually associated with electromagnetic metamaterials. The topological rainbow effect transcends specific physical systems; hence, the phenomena we describe can be transposed to other wave physics. Because of the directional tunability of the elastic energy by geometry, our results have potential for applications such as switches, filters, and energy harvesters.

Underwater characterisation of acoustic modes supported by structured and unstructured elastic plates

Surface waves are supported by unstructured, or appropriately structured, elastic materials. Unstructured plates typically support a number of Surface Acoustic Waves (SAWs) that propagate at the boundary of the elastic materials—these can be bound or leaky. Whilst plates with appropriate structure can support bound Acoustic Surface Waves (ASWs) which propagate in the fluid at the fluid/solid boundary. In this talk, we present the underwater acoustic characterisation of both SAWs and ASWs supported by unstructured and structured aluminium plates. First, we measure the dispersion of the symmetric and antisymmetric leaky Lamb modes of an unstructured aluminium-alloy plate using a broadband ultrasound technique. Beaming of acoustic power over a narrow frequency band is observed. This is attributed to the first order symmetric leaky Lamb mode, which exhibits Negative Group Velocity (NGV) in a small region of its dispersion. Finally, we show that a perforated elastic plate supports a bound ASW. We characterize the dispersion of the supported modes using time-domain measurements of the surface wave pressure recorded as a function of position, and observe acoustic beaming of surface energy over a narrow frequency band. All results show close agreement with numerical models.

Numerical solution of acoustic scattering by finite perforated elastic plates.

We present a numerical method to compute the acoustic field scattered by finite perforated elastic plates. A boundary element method is developed to solve the Helmholtz equation subjected to boundary conditions related to the plate vibration. These boundary conditions are recast in terms of the vibration modes of the plate and its porosity, which enables a direct solution procedure. A parametric study is performed for a two-dimensional problem whereby a cantilevered perforated elastic plate scatters sound from a point quadrupole near the free edge. Both elasticity and porosity tend to diminish the scattered sound, in agreement with previous work considering semi-infinite plates. Finite elastic plates are shown to reduce acoustic scattering when excited at high Helmholtz numbers k0 based on the plate length. However, at low k0, finite elastic plates produce only modest reductions or, in cases related to structural resonance, an increase to the scattered sound level relative to the rigid case. Porosity, on the other hand, is shown to be more effective in reducing the radiated sound for low k0. The combined beneficial effects of elasticity and porosity are shown to be effective in reducing the scattered sound for a broader range of k0 for perforated elastic plates.

The effective properties of a perforated elastic plate Numerical optimization by genetic algorithm

We consider a perforated elastic plate in which identical traction-free holes form square or hexagonal periodic arrays. The optimization problem of finding the hole shape in this structure that maximizes either of its effective moduli is posed as a full scale inverse problem of elasticity. In this context, only limiting cases of a single hole and cellular solids with thin cell walls have been studied thus far. Here, we cover the gap between these cases numerically solving the problem by the genetic algorithm approach. A new time-saving scheme of the fitness evaluation provides reliable data even near the percolation limit. The presented numerical results comprehensively describe the optimal behavior of perforated plates in terms of equivalent homogeneous medium. Similar, though less sophisticated approach was used by the author Vigdergauz, (Int. J. Solids Struct. 38 (2001) 6851) to optimize an isolated inclusion in a plate.

Nonlinear mathematical theory of perforated viscoelastic thin plates with its applications

In this paper, the nonlinear mathematical theory of perforated viscoelastic thin plates, by the Kármán’s hypotheses of plates with large deflection and the Boltzmann’s constitutive law of linear viscoelastic materials, is established. One could see that the governing equations, boundary conditions and constraining conditions generally are of nonlinear integro-differential operator types and that they further generalize the mathematical theory for perforated elastic thin plates and could be also reduced to the existing mathematical theory of viscoelastic thin plates without holes. As an application, the nonlinear dynamical stability of a viscoelastic annular plate is analyzed and the effect of parameters on the stability is considered by using Galerkin averaging method and numerical methods in nonlinear dynamics. Some helpful conclusions are obtained. Specially, a new method calculating the Lyapunov exponent spectrum of dynamical systems excited periodically is suggested. By using this method, the computation time can be greatly reduced.

Sound generated by turbulence and discrete vortices interacting with a perforated elastic plate in low-mach-number flow

An analytical model is developed of the production of sound by turbulence and discrete vortex structures in low-Mach-number flow over single or multiple apertures in a thin elastic plate. The aperture dimension is comparable to or larger than the length scale of the unsteady flow, and in the undisturbed state an aperture is spanned by a vortex sheet. This sheet is assumed to be linearly disturbed from its planar form by the hydrodynamic pressure field of the flow. The strength of the radiation is dependent on the generalized Rayleigh conductivity K R (k,ω), which defines the volume flux through an aperture when a pressure disturbance of frequency ω and wavenumber κ in the plane of the plate is applied to the vortex sheet. Detailed results are given for rectangular apertures of large aspect ratio, whose conductivity is known in closed form, and application made to determine the sound produced when a line vortex is swept over an isolated aperture in a rigid or elastic plate. The problem of noise generation by boundary-layer flow over a thin, perforated elastic plate is investigated for large apertures, where the aperture diameter exceeds the boundary-layer displacement thickness. The predicted sound power produced by turbulent wall pressures whose length scales are smaller than the streamwise dimension of the apertures is shown to be significantly larger than traditional estimates that ignore the shear layer. Numerical results are given for a rigid perforated plate, and for a perforated steel plate in water.

ENERGY CONSERVATION AND THE DAMPING OF FLEXURAL WAVES BY VORTICITY PRODUCTION

An analysis is made of the transfer of energy between structural/acoustic vibrations and fluid kinetic energy in flows at very small Mach number. A general energy balance equation is discussed for a vibrating rigid body in either incompressible or low Mach number compressible mean flows. This equation can be used to calculate the growth or decay of structural or acoustic oscillations, and to locate regions of the flow where energy exchanges are significant. A similar general treatment for vibrating elasticbodies does not seem to be possible, except in simple cases involving thin elastic plates in parallel flow. Two such problems are discussed, involving the dissipation of structural vibrations by vorticity production (i) at the trailing edge of a large elastic plate, and (ii) in the circular apertures of a perforated elastic plate in a two-sided grazing mean flow. The interaction of boundary layer turbulence with the apertures of a perforated plate can be a particularly intense source of sound, but in applications where the characteristic frequencies are small, a grazing flow perforated screen has been shown to be an efficient sink of acoustic energy. In this paper predictions are given for damping of bending waves by the same mechanism. When the fluid loading is large, such as for a steel plate in water, these predictions indicate that the damping of resonant bending waves can exceed that normally achieved by coating the plate with elastomeric damping materials, at least over a restricted range of frequencies.

The damping of flexural and acoustic waves by a bias-flow perforated elastic plate

An analysis is made of the damping of sound and structural vibrations by vorticity production in the apertures of a bias flow, perforated elastic plate. Unsteady motion causes vorticity to be generated at the aperture edges; the vorticity and its energy are swept away by the bias flow and result in a net loss of acoustic and vibrational energy. In this paper we investigate the interaction of an arbitrary fluid-structure disturbance with a small circular aperture in the presence of a high Reynolds number, low Mach number bias flow. By considering the limit in which the aperture is small compared to the length scale of the impinging disturbance, it is shown that the effect of the interaction can be represented by a concentrated source in the plate bending wave equation consisting of a delta function and two of its axisymmetric derivatives. A generalized bending wave equation is then formulated for a plate perforated with an homogeneous distribution of small, bias flow circular apertures. This equation is used to predict the attenuation of sound and resonant bending waves by vorticity production. Acoustic damping is found to be significant provided the fluid loading is sufficiently small for the plate to be regarded as rigid (e.g. for an aluminium plate in air when the frequency is not too small). On the other hand, a bending wave is effectively damped only when the fluid loading is large enough for the wave to produce a substantial pressure drop across the plate; when this occurs the predicted attenuations are comparable with those usually achieved by the application of elastomeric damping materials. Numerical predictions are presented for steel and aluminium plates in air and water.

Damping of sound and vibration by flow nonlinearity in the apertures of a perforated elastic screen

An analysis is made of the influence of flow nonlinearity in the apertures of a perforated elastic plate on the damping of sound and flexural vibrations. Fluid is forced through the perforations by the pressure differential established across the plate by the incident disturbance. The Reynolds number is assumed to be sufficiently large that separation occurs, and the reciprocating aperture flows form 'jets' on alternate sides of the plate. The growth of these jets is modelled by means of a nonlinear equation proposed by Cummings (1986). This equation is solved simultaneously with a generalized bending wave equation derived by the author (Howe, 1995a) which governs motions of a perforated elastic plate whose lengths scales are large compared to the aperture spacing. It is shown that significant attenuations of large amplitude acoustic waves can occur except when the frequency is so small that the plate is acoustically transparent. Bending waves are also damped provided the amplitude of the plate surface velocity is not too large and the frequency is small enough to ensure the formation of substantial jets in the apertures. Numerical results are given for large amplitude sound waves incident on a perforated screen in air, and for bending waves propagating over aluminium and steel screens immersed in either air or water.

Optimal stiffening of holes under equibiaxial tension

In this paper, the optimal configuration—in a remote uniform tension field—is investigated for stiffening rings (made of a different material) in a perforated elastic plate. The complex variable approach and the Kolosov-Muskhelishvili potentials are used to determine the unknown shape of the multiconnected region in which we pose the equations of equilibrium. It turns out that, for any number and relative spacing of the holes, the ring boundaries should represent equalstrength contours. This finding is extended to the case of multilayered and composite materials with elastic moduli varying continuously in a given direction.

Least-weight design of perforated elastic plates for given compliance: Nonzero Poisson's ratio

In earlier papers [1, 2], least-weight solutions were derived for perforated elastic plates having (a) a constant thickness, (b) a prescribed compliance value, and (c) a Poisson's ratio of zero value. In view of earlier mathematical studies by Kohn, Strang, Lurie et al., the above results were based on a microstructure in which ribs of first- and second-order infinitesimal spacing run in the two principal directions. It was found, however, that the optimal solution for axially symmetric plates always consists of (i) unperforated regions or/and (ii) regions with ribs in only one (i.e. the radial) direction. In view of extensive recent study of this problem by leading mathematicians, the above conclusions have important implications. In the current paper, the foregoing results are extended to plates having a nonzero Poisson's ratio and the theory is illustrated with examples. Two solutions given are valid for both prescribed compliance and prescribed deflections. This investigation shows that in some unconstrained shape optimization problems the optimal solution cannot be obtained by conventional numerical (e.g. finite element) methods.

Least-weight design of perforated elastic plates—I

The paper investigates implications of some recent mathematical developments in the fields of shape optimization and relaxation of variational problems. Considering the least-weight design of perforated elastic plates in either flexure or plane stress for a prescribed compliance, it is shown that at low rib densities microstructures consisting of a combination of first- and second-order infinitesimal ribs is superior to those consisting of purely first-order infinitesimal ribs. Moreover, it is indicated that thin ribs of infinite length/thickness ratio do not contribute significantly to the stiffness in a direction normal to their plane. On the basis of this conclusion, a simple specific cost function is derived and then it is used in the design of circular, uniformly loaded perforated plates with zero value of Poisson's ratio. As a basis for comparison several intuitively selected topographies are optimized for the case of simply supported plates, and in Part II of this study a variational analysis is used to obtain the optimal solutions for plates with simply supported, clamped or loaded edges.

Least-weight design of perforated elastic plates—II

On the basis of a simple specific cost function for perforated, elastic plates with a microstructure of first- and second-order rib systems we derive optimal designs for axially symmetric plates with simply supported, clamped or loaded edges. The results are obtained by variational analysis and the conclusion of the analysis is that for the class of plates considered a first/second-order microstructure does not result in an improved economy as compared to the first-order microstructure.

The transverse flexure of thin perforated elastic plates supported at several points

ABSTRACTFollowing the method outlined in a previous paper, exact solutions in finite terms are obtained for the problem of an infinitely large plate with outer edge free and an inner free circular boundary, the plate being supported at any number of interior or boundary points and normally loaded over a circle. The load considered is symmetrical with respect to the centre of the circle.

Relaxation methods applied to engineering problems XIII. The flexure and extension of perforated elastic plates

The theory given in Part VIIA (1941) for a flat plate having a single external boundary is here extended to allow for internal boundaries (holes) at which, as well as at the external boundary, either tractions or displacements may be specified. Such holes entail appreciable complication, more especially when boundary tractions are specified and do not form an equilibrating system at each hole severally: ‘orthodox’ studies (e.g. by Bickley, Green, Howland, Jeffery) have been restricted to simple shapes (e.g. one or more circular holes in a plate having infinite extension). But any shape can be treated by Relaxation Methods, without excessive labour and with all the accuracy that has practical value. A worked example (stresses in a loaded tie-bar) shows that holes may entail a cyclic form for Airy’s stress-function X .