High-frequency asymptotics for microstructured thin elastic plates and platonics

T. Antonakakis, R. V. Craster


We consider microstructured thin elastic plates that have an underlying periodic structure, and develop an asymptotic continuum model that captures the essential microstructural behaviour entirely in a macroscale setting. The asymptotics are based upon a two-scale approach and are valid even at high frequencies when the wavelength and microscale length are of the same order. The general theory is illustrated via one- and two-dimensional model problems that have zero-frequency stop bands that preclude conventional averaging and homogenization theories. Localized defect modes created by material variations are also modelled using the theory and compared with numerical simulations.

1. Introduction

An important issue in the mechanics of materials is the creation of continuum equations on a macroscale from a non-trivial microstructure; there are numerous examples of such microstructures: metallic foams (Gibson & Ashby 1997), photonic crystals (Zolla et al. 2005), smart structures (Kalamkarov & Georgiades 2002), material science (Ostoja-Starzewski 2007) among others. The macroscale object of interest may be constructed from many hundreds, thousands or millions of microscale ‘cells’ and, in dynamic situations, high-frequency waves with wavelength of the order of the cell scale can pass through the structure. Unfortunately, the multiple scattering between cells precludes application of simple averaging, or homogenization, theories. It is none the less highly desirable to be able to replace this microstructured medium with a continuum model, described entirely upon the macroscale, that encapsulates the microscale even at high frequencies. Fortunately, such a model has recently been developed (Craster et al. 2010a) for wave propagation through microstructured media, governed on the scale of the cell by a scalar wave equation, and the theory has been generalized to grillage (lattice nets) created by cells of elastic strings (Nolde et al. 2011), and discrete mass-spring models relevant to solid-state physics (Craster et al. 2010b).

An important (and non-trivial) extension is to microstructured media created from elastic plates that allow for bending waves; miniaturized elastic plates are widely used in nano- and microscaled systems (Korvink & Paul 2006), and form the essential building blocks in micromechanical models of solids (Ostoja-Starzewski 2002, 2007). Elastic plates are not governed by a simple wave equation and instead, classical Kirchhoff–Love plate theory (Graff 1975) allows for bending moments and transverse shear forces to be present; a fourth-order equation emerges for the out-of-plane plate displacement u as Embedded Image 1.1 In this equation ρ, A, E, I are density, cross-sectional area, Young's modulus and cross-section moment of inertia, respectively. The study of wave propagation along infinite perfectly periodic joined or forced elastic plates has a long and distinguished history with significant contributions by Cremer & Leilich (1953), Ungar (1966), Mead (1970) and for grillage by Heckl (1964), with later contributions summarized by Mead (1996). Naturally, many underwater sonar applications involve sound scattering by submerged hulls with regular reinforcing ribs and the coupling of plates to a surrounding fluid has also been of much interest, see Mace (1980a,b) among many others, the extension to non-regular structures is non-trivial and some examples are covered in Spivack & Barbone (1994); jointed fluid-loaded plate structures with discontinuities in material properties have also been a popular canonical model (Brazier-Smith 1987; Howe 1994; Norris & Wickham 1995).

Complementary to the literature covering periodic reinforced plates (with and without fluid loading) is that of the so-called platonics (Movchan et al. 2007; McPhedran et al. 2009; Poulton et al. 2010) where ideas taken from photonics, in optics, are translated into the elastic plate application. It is well known in optics that periodic structures lead to dispersion diagrams with the possibility of stop bands (Zolla et al. 2005), i.e. bands in which free propagation of waves is not permitted; this and other features lead to control over light propagation (Joannopoulos et al. 2008). These topical applications can lead to ultra-refraction and effective negative refraction in structured elastic plates (Farhat et al. 2010a,c) and are related to the subject of optical metamaterials (Smith et al. 2004), and Pendry's perfect lens (Pendry 2000). In structural mechanics, a material can be constructed from a regular lattice of elastic plates or trusses and similar band-gap phenomena arise (Martinsson & Movchan 2003; Phani et al. 2006).

In both of these areas (reinforced or microstructured elastic plates and platonics), it would be highly attractive to replace the structured plate by an effective single continuous medium that would implicitly understand the precise presence of the structuring and its interaction with the wavefield. Numerical modelling currently requires every single element of the structure to be modelled individually and, in structural mechanics, this becomes particularly arduous for large structures with many reinforcements and impractical for microstructured media. At low frequencies, or for static media, a methodology for moving from the microscale to the macroscale exists in the form of classical homogenization theory which is well-developed (Bensoussan et al. 1978; Sanchez-Palencia 1980; Bakhvalov & Panasenko 1989; Panasenko 2005) and easily applied to elastic plates (Berdichevski 1983; Kolpakov 2004). Homogenization theory is poorly regarded in some of these areas as several of the most interesting features of wave propagation through structured media occur at frequencies inaccessible to standard homogenization theory, for instance the classical theory implicitly assumes that the wavelength is far larger than the dimension of a typical cell of the microstructure and totally ignores any possibility of multiple scattering between cell elements; at leading order, the assumption that the wavefield is spatially constant on the microscale removes fine scale, short wave, information. The theory we develop totally overcomes this restriction for materials with a microstructure that is periodic on the microscale, although the macrostructure need not be periodic or infinite, and the theory is versatile enough to deal with localization phenomena (Craster et al. 2010b) and localized forcing (Nolde et al. 2011) in the simpler wave equation modelling. Our aim in this article is to extend this entire approach to the fourth-order plate equations and thus enable the high-frequency homogenization theory of Craster et al. (2010a) to be applied in platonics, elastic plate theory and microstructured elastic plates.

The plan of this article is as follows: in §2, we develop the asymptotic high-frequency homogenization theory in both one and two dimensions in §2a and b, respectively: remarkably, despite the governing equation being fourth order, the asymptotic long-scale equation is second order. Crucially, there is no implication of low-frequencies, instead we argue that—on the scale of a microstructural cell—there exist standing waves with well-defined standing wave frequencies and displacements. This microstructural cell behaviour then encodes the microscale behaviour and the effect of multiple scattering by neighbouring cells. The standing wave displacements are then used to create integrated quantities such that a macroscale model emerges entirely on the macroscale with the microstructure encapsulated by these integrated quantities. The efficacy of this approach is then illustrated in §3 where plates that have no material variation but periodic and doubly periodic supports are treated; these have simple exact solutions in the context of Floquet–Bloch waves with which to compare and allow for interpretation. Floquet–Bloch systems are interesting and illustrative, but the asymptotic theory also provides insight in more general situations for which exact solutions are not forthcoming. More generally, material variation is considered in §3c and localized defect modes are found and described asymptotically in §3d. Finally, concluding remarks are drawn together in §4.

2. Theory

We operate in non-dimensional variables that is we choose constant reference values for the physical parameters E0,I0,ρ0,μ0,A0 and incorporate any spatial variation through Embedded Image defined via Embedded Image and Embedded Image. The spatial coordinate Embedded Image and, assuming time harmonic dependence Embedded Image (henceforth considered understood and suppressed) leads to a non-dimensional frequency Embedded Image from Embedded Image The governing equation (1.1) is now Embedded Image 2.1 the subscript on ∇ denotes that differentiation is now with respect to Embedded Image. Henceforth, we operate in the non-dimensional setting and drop the hat decoration, and the subscript for ∇.

For definiteness, we develop the asymptotic theory for a two-dimensional geometry and later state the results for one dimension. The typical geometry we have in mind is created from square cells repeating to fill the plane and thereby creating a composite plate; there is no requirement for each cell to be identical to its neighbour, but we take it to be so until §3d when we briefly consider localization created by slowly varying material properties. In one dimension, the geometries for which we provide explicit results in later sections are shown in figure 1. The piecewise continuous plate is composed of cells of length 2l; each cell is composed of two continuous plates each of different properties or more generally with properties periodic over a cell. As noted earlier, we are concerned with macrostructures, potentially composed of many hundreds or thousands of these cells, on a long-scale L (Ll and ε=l/L≪1).

Figure 1.

A one-dimensional periodic plate constructed from cells of size 2l. (a) A piecewise continuous variation. (b) A uniform simply supported plate.

In either one or two dimensions, we therefore adopt a multi-scale approach where l is the small lengthscale and L is a large lengthscale representing the size of the macrostructure. In order to apply this theory for finite macroscale problems, L needs to be large enough so that the solution is not affected in a general region far away from the boundaries. Adopting a multiple scales approach, we introduce two independent spatial variables: ξi=xi/l and Xi=xi/L for i=1,2, where ξi is the cell coordinate system so −1<ξi<1; as the material properties are periodic over the cell β=β(ξ) and μ=μ(ξ), where μ and β are the non-dimensional mass per unit length and the flexural rigidity of the plate, respectively.

We now consider (2.1) in the context of these two scales, one key idea of the asymptotic approach is that periodic, or nearly periodic, systems have the multiple scattering by neighbouring cells on the microscale encoded by standing wave solutions that occur at the edges of the Brillouin zone of a perfectly periodic infinite system; these have associated standing wave frequencies that need not be small. We will be looking for solutions emerging from standing waves (in-phase and out-of-phase across the microscale cell) that have periodic and anti-periodic conditions for u(ξ) on the microscale as Embedded Image 2.2 and Embedded Image 2.3 where i,j,k and m take the values of {1,2} and the +,− correspond to the periodic and anti-periodic cases, respectively. We expand u and ω in powers of ε with the ansatz Embedded Image 2.4 then substituting into (2.1) gives a hierarchy of equations in powers of ε. The hierarchy, starting from leading order in ε and up to second order, reads as Embedded Image 2.5 Embedded Image 2.6 and Embedded Image 2.7 It is clear now that this theory differs from classical homogenization in the order of the μω2 term. In classical homogenization μω2=O(ε2), whereas in our theory Embedded Image. The leading-order problem (2.5) is completely independent of the long-scale X and so has the solution u0=f0(X)U0(ξ;ω0); this solution consists of a possibly highly oscillatory microscale eigensolution U0, as a function of the short-scale ξ with frequency ω0, modulated by a long-scale function f0(X) that remains to be found. Indeed, the whole point of the theory we develop is to find an effective equation, a partial differential equation (PDE), that f0 satisfies entirely stated upon the long-scale X; all the microstructure must be encapsulated within coefficients of the PDE and the short-scale does not appear explicitly at all within the PDE. With this aim in mind, we progress through the hierarchy of equations order by order.

We now integrate over the cell the difference between the product of equation (2.6) and U0 and the product of equation (2.5) and u1/f0 to obtain the following, Embedded Image 2.8 After using the periodic, or anti-periodic, boundary conditions on the microscale, and integrating by parts, the integrals to the left of (2.8) vanish, thereby leading to ω1=0. Solving for u1(X,ξ) gives Embedded Image 2.9 where the first term just represents a solution of the homogeneous equation, and is absorbed into the leading-order solution and ignored, the second part is more interesting and involves a vector function U1 that can be written as U1k(ξ)=(V kξkU0) with the last term found explicitly; notably, V is not periodic or anti-periodic. The scalar function u1 must, however, respect the boundary conditions stated in equations (2.2) and (2.3). The vector V is introduced to restore the periodicity in u1, and has appropriate boundary conditions at the edge of a cell to ensure this. Each component of the vector function V satisfies the leading order equation (2.5), but now with non-periodic boundary conditions Embedded Image 2.10 Embedded Image 2.11 Embedded Image 2.12 and Embedded Image 2.13

We now have the solution to the first-order equation and move to second-order, our approach mirrors that of tackling the first-order problem: we multiply equation (2.7) by U0 and subtract the product of equation (2.5) by u2/f0. We then integrate over the whole cell to obtain, Embedded Image 2.14 Crucially, equation (2.14) is a PDE entirely upon the macroscale for f0(X) and is written concisely as Embedded Image 2.15 The tensor Tij is given by the ratio of double integrals over the micro-cell; therefore, the microscale is encapsulated by these integrated quantities. This PDE is the key result of the article, it is notable that we began with a fourth-order equation (2.1) but that the high-frequency homogenized PDE (2.15) is only second order. We now explore the validity and range of application of this PDE.

In one dimension, this PDE simplifies to an ordinary differential equation for f0(X), entirely on the macroscale, of the form Embedded Image 2.16 where the microstructure is now encoded by the single coefficient T given by an integral over the micro-cell as Embedded Image 2.17 At this point, we have said nothing about the macroscale problem in terms of the boundary conditions to be applied to either (2.15) or (2.16); in this article, we shall take the macroscale problem to be infinite and illustrate the efficacy of the homogenization approach to find asymptotic Bloch dispersion relations, or consider infinite lattices with varying material properties that have solutions decaying at infinity.

For the former situation of Bloch waves, we return to two dimensions and set Embedded Image to develop an asymptotic relation between frequency, ω, and Bloch wavenumber κ. In this macroscale notation, Kj=κjdj and dj=0,π/2ε,−π/2ε depending on the location we refer to in the reduced Brillouin zone. Equation (2.15) gives the asymptotic dispersion relation Embedded Image that can be written as Embedded Image 2.18 The generic Embedded Image denotes the standing wave frequency for the respective location in the Brillouin zone that is defined by the parameters di and dj.

In one dimension, a similar approach is taken for Bloch waves Embedded Image for the periodic case in ξ and (2.16) gives Embedded Image leading to Embedded Image 2.19 with a similar form for the out-of-phase (anti-periodic) case.

(a) Repeated roots

It has been implicit in the analysis, thus far, that each standing wave frequency is an isolated eigenvalue of the leading order equation (2.1). This is not always true and many examples, including those we see later, involve repeated eigenvalues. Assuming a pair of repeated eigenvalues, the general solution becomes Embedded Image 2.20 this can be extended in an obvious manner for eigenvalues of higher multiplicity; indeed, the two-dimensional example of §3c has repeated eigenvalues that appear as a triplet. Proceeding as before, we multiply equation (2.6) by Embedded Image then subtract Embedded Image, integrate over the cell, to obtain Embedded Image 2.21 for l=1,2; ultimately, this gives two coupled PDEs on the macroscale for Embedded Image and Embedded Image. In matrix form, this system yields Embedded Image 2.22 where Embedded Image. The entries of A(i) and B are Embedded Image 2.23 and Embedded Image 2.24 The non-diagonal entries of B, as well as the diagonal entries of A(i), are null for the respective reasons of orthogonal eigenfunctions and boundary conditions.

For Bloch waves, we set Embedded Image; a compact equation for the system is then Embedded Image 2.25 where Embedded Image, Embedded Image and Embedded Image for lm. For a non-trivial solution, the determinant of C must vanish giving the asymptotic dispersion relation.

Usually, the ω1 are non-zero; however, in some cases they may be zero. To then obtain the asymptotic form, one must go to next order using u1, Embedded Image 2.26 where the first two terms are again irrelevant.

In one dimension, this simplifies to get the system of ODEs Embedded Image 2.27 where Embedded Image and the elements of the matrices are Embedded Image 2.28 and Embedded Image 2.29

For Bloch waves taking Embedded Image we then obtain a fourth-order polynomial equation for ω1 Embedded Image 2.30 We then obtain two solutions for Embedded Image that will describe the linear relationship of both asymptotics emerging from the repeated eigenvalue ω0. They read Embedded Image 2.31 giving the linear behaviour of the two asymptotics at the repeated root for the periodic case as Embedded Image 2.32 with a similar result for the anti-periodic case.

3. Illustrative examples

Given the theory earlier, we now demonstrate its efficacy upon specific examples.

(a) Periodically pinned elastic plate

To fix ideas, we consider an example that can be completely and explicitly solved: a perfect elastic plate with constant material parameters resting upon periodic simple supports. To be precise Embedded Image 3.1 with supports at x=2n for Embedded Image. These supports impose un=u(2n)=0 and that ux,uxx are continuous at each support. Introducing a Bloch wavenumber κ such that Embedded Image the dispersion relation Embedded Image 3.2 emerges (Mace 1980a; Mead 1996), where Embedded Image; this has an associated explicit solution for the plate displacement Embedded Image 3.3 arbitrary up to a multiplicative constant, that can be summed as in Mace (1980a) if desired. This explicit solution rapidly follows using Bloch's theorem to factor out the phase shift and Fourier series to solve the resulting periodic problem. In any event, the dispersion curves are found (figure 2), and it is notable that there is a zero frequency stop band, i.e. there is no dispersion curve passing through the origin. This has the important consequence that long wave approximations at low frequencies are doomed to fail and the usual homogenization approach is useless. As we shall now demonstrate, the homogenization procedure developed herein has no such failings; we discuss the periodic case, asymptotics near κ=0, in detail and just present the anti-periodic results near κ=π/2. Figure 2 shows the dispersion relation plotted versus those from the asymptotics; log–log plots show the accuracy of the asymptotics detail of the quadratic behaviour of dispersion relation near the standing wave frequencies.

Figure 2.

The dispersion curves from (3.2), shown across the Brillouin zone (a). The solid line is from the numerical solution, the asymptotics for Graphic are the dashed curves and for Graphic are dotted. (b) Plots Graphic on log–log axes to illustrate the accuracy: numerics are given by the solid lines and the asymptotics are the dashed lines. (c) Plots Graphic on log–log axes to illustrate the accuracy: numerics are given by the solid lines and the asymptotics are the dotted lines.

Given the solution (3.3) one can extract the U0(ξ;ω0) and ω0 for the homogenization methodology immediately, in the two-scale notation ξ=x and X=εx, thus U0(ξ;ω0) is u(x) from (3.3) evaluated at ω0. Figure 3 plots U0s for the first three eigenfrequencies in the periodic and anti-periodic case, respectively, in panels (a) and (b). Indeed, U1 follows as Embedded Image 3.4 and, hence T as Embedded Image 3.5 We compute this integral to obtain Embedded Image 3.6 for the periodic case, with a similar formula for the anti-periodic case. Note the singularities at ω0=()2 that invalidate (3.6). If one inspects the dispersion relation in equation (3.2), then it has roots at ω0= and these singularities are in fact genuine solutions of the dispersion relations and arise owing to a degeneracy of the problem. For these special degenerate cases, we obtain trivial solutions, Embedded Image and V , for −1<ξ<0, Embedded Image 3.7 and for 0<ξ<1 Embedded Image 3.8 Having found V we compute U1, then perform the integral of equation (3.5), and obtain the T coefficients for the asymptotics. The integral can be performed exactly, or numerically, for both the periodic and anti-periodic cases and the resulting asymptotics are shown in figure 2. This not only verifies the theory, for this example, but it also provides a key ingredient for the localization of §3d, there are clearly stop bands in figure 2 and the behaviour near the edges of the Brillouin zone is completely characterized by the coefficients T; localized modes appear at the edges, so this information is vital.

Figure 3.

(a) and (b) plots, respectively, Graphic for i=1,2,3 and Graphic for i=1,2,3 which are the solutions to the leading order equation for the first three eigenfrequencies. (a) The periodic case; and (b) the antiperiodic case. The solid line represents U(1), the dotted line is for U(2) and the dashed line represents U(3).

(b) One-dimensional piecewise homogeneous plate

The preceding example has no material variation and the periodicity arises through the supports, it also has the advantage of there being simple exact solutions for comparison with both U0 and V. We now consider a piecewise homogeneous plate where both μ and β vary and for which the steps must be performed numerically; the general solution for the asymptotics developed in §2 holds. As shown in figure 1a each cell is of length 2l and is made of two parts each of different material and/or cross-sectional geometry. The varying coefficients that make this possible are β and μ that, respectively, denote the normalized flexural rigidity and the mass per unit length. The discontinuity of the material properties within the cell calls for continuity conditions across the join that is at ξ=0 (x=2n). Combining these conditions with the Bloch conditions at the edges of each cell leads to a well-defined boundary value problem posed on the cell. Note that β and μ show up in the homogeneous Bernoulli–Euler equation, but only β appears in the boundary conditions.

Taking into account the piecewise homogeneity of the structure, equations (2.17) and (2.32) are used to produce the asymptotic curves shown in figure 4, which shows the Bloch dispersion curves when μ1=1.5, μ2=7.7 and β1=1.065, β2=1.5 which are chosen as a repeated root then occurs for which the degeneracy of §2b occurs. The full dispersion curves are found numerically using a spectral collocation scheme as in, say, Adams et al. (2008). The spectral scheme is also used to find the required U0 and V for the asymptotics.

Figure 4.

The Bloch dispersion curves for piecewise homogeneous plates with μ1=1.5, μ2=7.7 and β1=1.065, β2=1.5. (a) Plots the Bloch dispersion curves as solid lines together with the associated asymptotics, as dashed lines and dotted lines for the respective periodic and anti-periodic cases. (b) Shows in detail the case of repeated roots seen in (a). Note the linear behaviour of dispersion curves for repeated roots. Solid lines and dashed lines represent respectively the dispersion relations and the asymptotics. (c) Plots in log–log axes the both the dispersion relation in solid lines and the asymptotic solution in dashed lines.

An additional point, when β and μ vary, is that two different dispersion modes can share the same standing wave frequency. Now there are repeated eigenvalues and the dispersion curves are linear near some standing wave frequencies as seen in figure 4b. The asymptotic expansions for this scenario are developed in §2a.

Another notable feature of the dispersion curves of figure 4 is that the zero frequency stop band has now been replaced by the more conventional situation of a dispersion curve passing through the origin; this is in contrast to figure 2. This low-frequency (acoustic) dispersion curve is slightly unusual as it has quadratic behaviour upon wavenumber, in most conventional settings, with the Helmholtz equation, the acoustic branch has linear dependence; the quadratic behaviour is due to the dispersive nature of waves in elastic plates.

(c) Two dimensions

Only a few examples of constrained plates in two dimensions are available in the literature: a grillage of line constraints as in Mace (1981) that is effectively two coupled one-dimensional problems, a periodic line array of point supports (Evans & Porter 2007) raises the possibility of Rayleigh–Bloch modes and for doubly periodic point supports there are exact solutions by Mace (1996) (simply supported points) and by Movchan et al. (2007) (clamped points). The simply supported case is accessible via Fourier series and we choose this as an illustrative example that is of interest in its own right; it is shown in figure 5a. In particular, the simply supported plate has a zero-frequency stop band and a non-trivial dispersion diagram.

Figure 5.

For the two-dimensional example, we show the geometry of the doubly periodic simply supported plate (the dots represent the simple supports) in (a) with the elementary cell shown by the dotted lines and in (b) the irreducible Brillouin zone with the lettering for wavenumber positions shown.

We consider a double periodic array of points at x1=2n1, x2=2n2, where u=0 (with the first and second derivatives continuous) and so the elementary cell is one in |x1|<1,|x2|<1 with u=0 at the origin (figure 5); Floquet–Bloch conditions are applied at the edges of the cell.

Applying Bloch's theorem and Fourier series, the displacement is readily found (Mace 1996) as Embedded Image 3.9 where N=(n1,n2), and enforcing the condition at the origin gives the dispersion relation Embedded Image 3.10 that converges nicely; when κ=0, it is immediate that ω=0 is not a solution and thus a zero-frequency stop band ensues. In this two-dimensional example, a Bloch wavenumber vector κ=(κ1,κ2) is used and the dispersion relation can be characterized completely by considering the irreducible Brillouin zone ABC shown in figure 5.

The dispersion diagram is shown in figure 6; the singularities of the summand in equation (3.10) correspond to solutions within the cell satisfying the Bloch conditions at the edges; in some cases, these singular solutions also satisfy the conditions at the support and are therefore true solutions to the problem, a similar situation occurs in the clamped case considered using multipoles in Movchan et al. (2007). Figure 6 shows both these singular solutions and the solutions directly from (3.10). Crosses in figure 6 label curves that are branches of the dispersion relation, notable features are the zero-frequency stop band and also crossings of branches at the edges of the Brillouin zone. Branches of the dispersion relation that touch the edges of the Brillouin zone singly fall into two categories: those with an additional two singular solutions (such as the lowest branch touching the left-hand side of the figure at B) and those that are completely alone (such as the second lowest branch on the left at B). The dispersion diagram was evaluated using the summation in (3.10), and the singularities. To confirm, this was indeed correct, and for some of the asymptotics, the entire PDE eigenvalue problem was performed with an independent numerical scheme based upon Fourier spectral collocation methods; the results are shown as the crosses in figure 6.

Figure 6.

The dispersion diagram for a doubly periodic array of point simple supports shown for the irreducible Brillouin zone of figure 5. The figure shows the singular solutions of D (3.10) as dotted lines, the solutions found directly from D as solid lines. In some cases, the singular solutions are actually true dispersion curves too, the dispersion curves are indicated by the crosses. Finally, as dashed lines, the asymptotic solutions from the high-frequency homogenization theory are shown.

To apply the asymptotics, we require U0, which follows from (3.9) evaluated at the edges of the reduced Brillouin zone. The first-order solution then comes from Embedded Image 3.11 cf. (2.6).

Let α2=(κ1πn1)2+(κ2πn2)2 and u1=f0,XpU1(p). Equation (3.11) reduces to Embedded Image 3.12 We look for a solution of the form of Embedded Image by introducing this form into equation (3.12) to obtain the relation for the coefficients A(p)n1n2 that reads as Embedded Image 3.13 We can now compute the integrals of equation (2.14), for instance, in the periodic-periodic case valid near the wavenumber labelled A we obtain the following coefficient Embedded Image 3.14 where the off-diagonal terms in the tensor Tij are zero as one naturally expects from the symmetry of the problem and T22=T11. There are similar expressions for the Tij at the edges of the Brillouin zone for the other cases. The formulae derived using Fourier series fail at the singularities of D, being degenerate for a similar reason to the one-dimensional simply supported case. In those cases, we proceed numerically by identifying the U0, U1 and performing a numerical integration, this is achieved with spectral accuracy using Fourier spectral collocation.

In any event, the asymptotics found are highly accurate as shown by the dashed curves in figure 6, indeed, they virtually cover the entire dispersion diagram and in some cases are almost indistinguishable from the exact dispersion curves. Interestingly, there are crossings of the dispersion curves at the edges of the Brillouin zone, these correspond to repeated eigenvalues and the resulting curves are linear (to leading order) and their asymptotics are given by the formulae of §2a. The usual homogenization theory is unable to capture any of these asymptotics. Importantly, with this asymptotic framework in place one can then accurately describe many phenomena for which exact solutions are no longer available, to illustrate this our final example is of this type.

(d) Localized defect modes

Thus far we have generated an asymptotic scheme and demonstrated its effectiveness and accuracy versus perfectly quasi-periodic solutions that are constructed analytically or numerically. The scheme is by no means limited to doing just this, let us now consider allowing spatial variation in μ through a function g(X) (on the long-scale) so Embedded Image 3.15 and the plate will be considered to be simply supported, so u(2n)=0 and ux,uxx are continuous at x=2n; this cannot be solved exactly and any localized eigensolution must be determined numerically. The eigensolutions found numerically are normalized so that max|u|=1.

The asymptotic scheme goes through as before, but we now pick up an additional term, and (2.16) becomes Embedded Image 3.16 localized defect modes require decay such that Embedded Image as Embedded Image.

For the simply supported case, the T and Embedded Image are known from §3a and, for the in-phase standing waves, the first six are given in table 1; notably, the sign of T alternates reflecting the change in group velocity as one moves from one stop band to the next.

View this table:
Table 1.

The first six standing wave frequencies for in-phase waves (cf. figure 2) together with associated values for T.

For definiteness, we use g(X)=sech2X to place (3.16) into the form of a differential eigenvalue equation given in Craster et al. (2010b) that then allows for an exact solution to (3.16) with Embedded Image 3.17 provided T and α are of the same sign; thus for T negative (positive) one requires α negative (positive) for localization to occur. Equation (3.17) then gives an explicit asymptotic estimate for the defect mode frequencies, and for more general variation in g(X), these estimates are easily found numerically.

For ε=0.25 and α=1, equation (3.17) gives ω=9.764 versus a direct numerical simulation of 80 cells with the same spatial variation for which ω=9.768. This localized eigenfrequency then lies within the second stop band, shown in figure 2, just below the standing wave frequency at ω0=9.870. For α=−1, the same calculation gives an eigenvalue just above ω0=5.593 with ω=5.644,5.646 for the asymptotics and numerics, respectively. The localized defect states for these two cases are in figures 7 and 8, which show the results of numerics versus the asymptotics. The detail of the states in figure 8 shows how the local form given by the U0(ξ), shown in figure 3, moves from one cell to the next but modulated by the decaying f0(X) on the longer scale. Notably, these computations are for a relatively large value of ε (ε=0.25 and the theory is for ε≪1) to enable the minor discrepancy to be actually seen.

Figure 7.

The localized defect states in the second stop band for (a) the eigenstate with α=−1 and (b) with α=+1; the dashed lines are f0 from the asymptotics (3.16) and, the solid lines from the full numerics.

Figure 8.

Detail of the localized defect states in the second stop band for (a) the eigenstate with α=−1 and (b) with α=+1, the dashed lines are f0 from the asymptotics, the solid lines from the full numerics and the dots are f0(X)U0(ξ).

4. Conclusions

The asymptotic two-scales approach of Craster et al. (2010a) is extended here to elastic plates, this has several consequences: it will enable asymptotic representations of plates that create ultra-refraction, all-angle negative refraction and localized defect states to be created in an analogous manner to optics (Craster et al. 2011); such effects have recently been considered in Farhat et al. (2010a,b). The technique is readily extended to grillage of elastic plates in an analogous fashion to that of a net formed from elastic strings (Nolde et al. 2011) which then has direct relevance to cellular structures (Phani et al. 2006). It breaks free of the usual restriction of two-scale analyses to low frequencies, homogenization theory is conventionally used to describe only the lowest dispersion branch when it emerges from the origin at zero frequency. To clearly illustrate the versatility of the asymptotic method, and its lack of any such restriction, the examples of simply supported plates are chosen as they have a zero-frequency stop band.

In terms of the physics, it is interesting to note that the fourth-order problem reduces, in its long-scale form, to a second-order PDE for f0 (2.15). This raises the issue of how one sets or finds the appropriate boundary conditions for the lower order equations. The boundary conditions of equation (2.15) need not be explicitly considered within the present paper, because the problems described are either quasi-periodic or exponentially decaying at infinity. Nevertheless, for a real finite problem defining the boundary conditions for equation (2.15) is not trivial and is under investigation. Despite the change in order, it is clear that this PDE accurately captures intricate behaviour in the dispersion diagram and localization effects. The PDE encapsulates the short-scale structure through integrated quantities over the microscale and so is entirely upon the long-scale; this makes this approach ideal for numerical simulations of wave propagation through large-scale microstructured media: it is anticipated that the high-frequency homogenization technique developed here will considerably ease such large-scale computations.


The authors gratefully acknowledge useful conversations with Sebastien Guenneau, Julius Kaplunov, Evgeniya Nolde and Aleksey Pichugin.

  • Received October 27, 2011.
  • Accepted January 11, 2012.


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