High frequency homogenization for structural mechanics

Journal of the Mechanics and Physics of Solids 59 (2011) 651–671

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High frequency homogenization for structural mechanics E. Nolde a, R.V. Craster b,, J. Kaplunov a a b

Department of Mathematical Sciences, Brunel University, Uxbridge, Middlesex UB8 3PH, UK Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Canada T6G 2G1

a r t i c l e in f o

abstract

Article history: Received 29 June 2010 Received in revised form 17 November 2010 Accepted 2 December 2010 Available online 10 December 2010

We consider a net created from elastic strings as a model structure to investigate the propagation of waves through semi-discrete media. We are particularly interested in the development of continuum models, valid at high frequencies, when the wavelength and each cell of the net are of similar order. Net structures are chosen as these form a general two-dimensional example, encapsulating the essential physics involved in the twodimensional excitation of a lattice structure whilst retaining the simplicity of dealing with elastic strings. Homogenization techniques are developed here for wavelengths commensurate with the cellular scale. Unlike previous theories, these techniques are not limited to low frequency or static regimes, and lead to effective continuum equations valid on a macroscale with the details of the cellular structure encapsulated only through integrated quantities. The asymptotic procedure is based upon a two-scale approach and the physical observation that there are frequencies that give standing waves, periodic with the period or double-period of the cell. A specific example of a net created by a lattice of elastic strings is constructed, the theory is general and not reliant upon the net being infinite, none the less the infinite net is a useful special case for which Bloch theory can be applied. This special case is explored in detail allowing for verification of the theory, and highlights the importance of degenerate cases; the specific example of a square net is treated in detail. An additional illustration of the versatility of the method is the response to point forcing which provides a stringent test of the homogenized equations; an exact Green’s function for the net is deduced and compared to the asymptotics. & 2010 Elsevier Ltd. All rights reserved.

Keywords: Stop bands Lattice materials Cellular structures Bloch waves Homogenization

1. Introduction Numerous engineering structures are composed from periodic cellular patterns made of bars, beams or strings, examples covering widely differing lengthscales include ceramic catalytic converters and filters (Hunt, 1993; Fleck and Qiu, 2007), space trusses and frames in structural dynamics (Weaver and Johnston, 1987; Bendiksen, 2000), hexagonal honeycomb cellular solids such as metallic or polymer foams (Gibson and Ashby, 1997), and micromechanical models of solids (Ostoja-Starzewski, 2002) amongst many others and these applications are of growing interest in the design of smart structures (Kalamkarov and Georgiades, 2002). The individual cell from which the material or structure is made may be relatively small with regard to the macro-structure that can be composed of many hundreds or thousands of cells. It is naturally desirable to replace these cells by a continuum model that accurately represents the effect of the microstructure.

 Corresponding author. Present address: Department of Mathematics, Imperial College London, London, SW7 2AZ, UK. Tel.: + 44 207 594 8517; fax: + 44 207 594 8483. E-mail address: [email protected] (R.V. Craster).

0022-5096/$ - see front matter & 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.jmps.2010.12.004

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Periodic lattice structures appear in many other branches of the natural sciences most notably in solid state physics where the atomic structure underlying crystalline solids has been of much interest (Kittel, 1996; Brillouin, 1953). That subject is dominated by discrete point mass-string models and for infinite periodic systems Floquet–Bloch theory allows one to reduce the analysis to a single-cell with quasi-periodicity assumed. The phase-shift across a cell is then related to the wave frequency by a dispersion relation, interestingly such dispersion diagrams are characterised by stop-bands, disallowing wave propagation for some frequencies, and flat regions where standing waves occur corresponding to regions of slow sound among other features. These characteristics are important as mechanical or spatial filters, Jensen (2003), and find applications in phononic, Sigmund and Jensen (2003), and photonic structures as described in Zolla et al. (2005). Other relevant applications for regular atomistic lattice structures in solid mechanics involve the dynamic fracture behaviour, on the microscale, as a crack breaks the bonds through the lattice. This too has attracted much attention (Slepyan, 1981; Marder and Gross, 1995; Slepyan, 2002) and it is of interest to relate this discrete fracture model on the microscale to continuum models on the macroscale (Slepyan, 2005). In all these examples and applications it is natural to wish to average, in some fashion, over these cells to create a continuum model that subsumes the microstructure and this motivates a substantial research area: homogenization theory (Sanchez-Palencia, 1980; Bakhvalov and Panasenko, 1989; Bensoussan et al., 1978; Mei et al., 1996; Panasenko, 2005), in which the static or quasi-static (low frequency, long wavelength relative to the cell) theory is now widely developed (Milton, 2002). The classical low frequency homogenization theory follows a standard algorithm and effective material properties readily emerge, see for instance Parnell and Abrahams (2008), for applications in elastodynamics for composites. However, as noted by recent authors, Phani et al. (2006), much less is known about the dynamic behaviour when the wavelength and cell scale is of similar order. Conventional homogenization theory is not capable of capturing the behaviour away from the low frequency regime and currently asymptotic methods for high frequencies that lead to continuum models are not available. This article rectifies this situation by developing a theory for high frequency waves passing through a cellular solid, for clarity the theory is developed for frames created by strings, which we henceforth call nets. The theory we develop is more generally valid and can be extended to space trusses and frames in structural dynamics at the expense of additional algebra. The authors, Craster et al. (2010a), have recently developed a related high frequency homogenization theory for continuous media created from periodic cells, say an elastic or electromagnetic medium punctured by a doubly-periodic array of holes or inclusions. A discrete version of the theory for point mass-string models from solid state physics is also available in Craster et al. (2010b). In both cases progress is made and effective continuum equations are found and their accuracy and versatility is verified by specializing to the infinite periodic Bloch wave cases. Furthermore localized modes created by local defects are identified from the homogenized models and compared to numerical solutions of the complete system in Craster et al. (2010b). The generalization and adaption of the techniques to frames presents additional difficulties and allows us to explore, and extend, the asymptotic procedure for another important class of problems. In particular, frames possess some features of discrete point media with the joints at regular discrete positions, but these joints are connected by strings individually having continuum governing equations. Ultimately we want an homogenized description completely in a continuum setting that encapsulates the discrete nature of the frame structure. Conversely, as we shall see later it is possible to completely couch the problem in a discrete setting and obtain exact solutions. For brevity and clarity we specialize to square frame lattices, but the fundamental ideas follow for all frame structures and to three-dimensions. We begin with an example of a planar net of overlapping strings, as shown in Fig. 1, with an underlying square lattice structure. These simple net structures form a quite general two-dimensional example, encapsulating the essential physics involved in the two-dimensional excitation of a lattice structure whilst retaining the simplicity of dealing with elastic strings. String models are sufficiently tractable that direct analytical progress can be made; special cases of this structure have been analyzed for Bloch waves in Martinsson and Movchan (2003) and Sparavigna (2007). We shall treat the more general situation of four different string densities creating a patchwork net of strings as this highlights further features of the problem, but in essence we are interested in taking a given microstructure of regular form and then generating macroscropic equations. The macroscopic equations must capture the microstructure, but involve only the macroscale variables with the microstructure built-in using averaged quantities. It is notable that the simpler situation of a one-dimensional piecewise

l

L Fig. 1. A macrostructure net of overlapping strings on scale L (left) constructed from an elementary cell of microscale l (right).

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constant string is treated in detail as an example in Craster et al. (2010a) and the general procedure is outlined there, as a result we do not re-derive the asymptotic procedure here. The macroscopic equations we derive are valid in general, but it is convenient to discuss them in relation to infinite Bloch wave problems. The dispersion relations between wavenumber and frequency are relatively straightforward to derive and thereby provide an illustrative check upon the analysis as well as being of interest in their own right. To further demonstrate the efficacy of the homogenized equations we consider point forcing of the net in Section 5.2, constructing both exact solutions to the full net problem and to the homogenized model which are then compared. 2. Bloch waves The situation of waves propagating through an infinite perfect net constructed from repeating cells can be formulated entirely upon the elementary cell shown in Fig. 2; more general nets formed from triangular and hexagonal cells are also commonplace (Brillouin, 1953; Martinsson and Movchan, 2003; Phani et al., 2006) and can be similarly treated. In solid state physics a periodic lattice created from elementary cells is the usual situation for waves in crystal structures as described in Kittel (1996) and, using Bloch’s theorem, so-called quasi-periodic Bloch conditions are imposed. We take the string displacements to be U(i) for i= 1y4 with the displacements dependent upon x ðZÞ for strings 1,3 (2,4), respectively. The Bloch conditions are that U ð1,3Þ ðx þ 2nÞ ¼ expðink1 ÞU ð1,3Þ ðxÞ,

U ð2,4Þ ðZ þ2mÞ ¼ expðimk2 ÞU ð2,4Þ ðZÞ, (a,b)

ð2:1Þ (i)

for n,m integer with jxjo 1 ðjZj o1Þ. The superscript notation U means U for i= a,b and similarly we shall use subscripts ra,b to denote ri for i= a,b. The phase-shift across the cell is characterized by the wavenumber vector j ¼ ðk1 , k2 Þ which is related to the frequency l, and dispersion relations are found relating wavenumber and frequency. From the periodicity of the net structure only a reduced set of wavenumbers need to be considered, represented by the reciprocal net in Fig. 2, in this case the wavenumbers along the triangle ABC suffice. The elementary cell for the structure is shown in Fig. 1 and each string has a different wavespeed (by varying the density). The numbering of the strings from the figure is used in the analysis; it is convenient to introduce the inverse wavespeed, ri, as ri =1/ci. The model, for Bloch waves, is that each string has governing equation 2

for 0 r x r1,

ð3Þ Uxx þ l r32 U ð3Þ ¼ 0

2

for 0 r Z r1,

ð4Þ UZZ þ l r42 U ð4Þ ¼ 0

ð1Þ þ l r12 U ð1Þ ¼ 0 Uxx ð2Þ UZZ þ l r22 U ð2Þ ¼ 0

2

for 1 r x r 0,

ð2:2Þ

2

for 1r Z r0:

ð2:3Þ

The displacements at the edges of the cell, from the Bloch conditions, are U ð1Þ ð1Þ ¼ expðik1 ÞU ð3Þ ð1Þ,

Uxð1Þ ð1Þ ¼ expðik1 ÞUxð3Þ ð1Þ,

ð2:4Þ

U ð2Þ ð1Þ ¼ expðik2 ÞU ð4Þ ð1Þ,

UZð2Þ ð1Þ ¼ expðik2 ÞUZð4Þ ð1Þ:

ð2:5Þ

This phase-shift is important in the analysis of the general structure as there are three special wavenumber vectors that automatically lead to standing waves across the structure: (i) at the wavenumber denoted by the point A, j ¼ ð0,0Þ, where waves are perfectly in-phase (periodic–periodic) as one passes from one cell to the next, (ii) at the point B, j ¼ ð0, pÞ η=1

Brillouin zone

String 2

C

κ2

2

String 3

0

String 1

B

A

ξ=−1

ξ=1

–2

–2

0 κ1

2

η=−1 String 4

Fig. 2. The reciprocal net (lattice) in j space is shown in the left panel together with the irreducible Brillouin zone given as the triangle ABC. On the right is the elementary cell showing the string numbering used in the text.

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(or equivalently ðp,0Þ), where the wave is perfectly in-phase in one direction and perfectly out-of-phase in the other (periodic, anti-periodic), and finally (iii) at the point C, j ¼ ðp, pÞ, with both waves perfectly out-of-phase (anti-periodic, antiperiodic) across the cell. The standing wave frequencies play a prominent role in the subsequent asymptotic analysis of Section 3. At the junction of the strings the displacements are coupled together, at the origin the displacement must be continuous so U ð1Þ ð0Þ ¼ U ð2Þ ð0Þ ¼ U ð3Þ ð0Þ ¼ U ð4Þ ð0Þ,

ð2:6Þ

and there is no jump in stress at the origin so Uxð1Þ ð0ÞUxð3Þ ð0Þ þUZð2Þ ð0ÞUZð4Þ ð0Þ ¼ 0:

ð2:7Þ

The Bloch dispersion curves follow from Q1 ðlÞQ2 ðlÞ½Q2 ðlÞS1 ðl, k1 Þ þ Q1 ðlÞS2 ðl, k2 Þ ¼ 0,

ð2:8Þ

where S1 ðl, k1 Þ ¼ 2r1 r3 ½cosk1 cosðlr1 Þcosðlr3 Þ þ ðr12 þ r32 Þsinðlr1 Þsinðlr3 Þ, Q1 ðlÞ ¼ r1 cosðlr1 Þsinðlr3 Þ þ r3 cosðlr3 Þsinðlr1 Þ,

ð2:9Þ ð2:10Þ

with S2 ðl, k2 Þ,Q2 ðlÞ identical with r1 ,r3 , k1 replaced by r2 ,r4 , k2 , respectively. The individual terms Si, Qi correspond to dispersion relations for special cases, that is, if a single piecewise constant string is taken only in x or Z then S1,2 ðl, kÞ ¼ 0 are the classical Kronig–Penney dispersion relations (Kronig and Penney, 1931). The Qi ðlÞ ¼ 0 solutions also correspond to single string eigensolutions that are constrained to have zero displacement at the origin, and so again the strings decouple, and are also in perfect phase at their endpoints; unsurprisingly these decoupling cases play a role later in degenerate situations. After some algebra, the displacements are given explicitly as U ð1Þ ðxÞ ¼ cosðlr1 xÞ þ P3 ðl, k1 Þsinðlr1 xÞ=Q1 ðlÞ, U ð3Þ ðxÞ ¼ cosðlr3 xÞP1 ðl, k1 Þsinðlr3 xÞ=Q1 ðlÞ, U ð2Þ ðZÞ ¼ cosðlr2 ZÞ þ P4 ðl, k2 Þsinðlr2 ZÞ=Q2 ðlÞ, U ð4Þ ðZÞ ¼ cosðlr4 ZÞP2 ðl, k2 Þsinðlr4 ZÞ=Q2 ðlÞ;

ð2:11Þ

where P1 ðl, k1 Þ ¼ r1 expðik1 Þ þ r3 sinðlr3 Þsinðlr1 Þr1 cosðlr1 Þcosðlr3 Þ,

ð2:12Þ

P3 ðl, k1 Þ ¼ r3 expðik1 Þ þ r1 sinðlr3 Þsinðlr1 Þr3 cosðlr1 Þcosðlr3 Þ,

ð2:13Þ

with P2, P4 following with r1 ,r3 , k1 replaced by r2 ,r4 , k2 , respectively. These displacements are normalised to have displacement unity at the origin, and assume both Q1, Q2 are non-zero; if these are zero then the displacements can decouple and can be zero at the origin. To whet the reader’s appetite we show a typical set of dispersion curves in Fig. 3(a) and displacements for some of the standing waves in Fig. 3(b, c) and 4, numbering the curves from lowest frequency upward, we point out notable features. First there are five frequency bands for which waves cannot propagate and this leads to the filtering properties noted in the introduction. These bands lie between the maximum of the first curve, the minimum of the second, between the maximum of the third curve and minimum of the fourth and so on, although one is not guaranteed a stop-band between two curves, for instance there is no stop-band between the second and third curves. The maxima and minima occur at the edges of the Brillouin zone, wavenumbers corresponding to A, B or C, which is typically the case, however counterexamples with extrema within the Brillouin zone can be constructed (Adams et al., 2008). Second, there are a wide variety of different types of curves: the first curve appears to have linear frequency-wavenumber dependence near the origin and the associated displacements at wavenumber point A are just constant; this is the case covered by conventional homogenization theory. All other curves have some non-constant displacement associated with them, although notably the fifth curve is apparently completely flat. On closer inspection it actually has a very gradual slope, this almost flat line corresponds to a disturbance with very small group velocity and the displacements at the origin in x,y at A are negligible; the displacements almost decouple horizontally and vertically, in this case Q1 and Q2 5 1. Other curves, say the fourth curve, seem to change character having a completely flat portion between wavenumbers A and C, but then having non-zero group velocity elsewhere. Other curves have strongly changing curvature for all wavenumbers, the first three, say, have this behaviour. One side-product of the asymptotic procedure and continuum equations that we derive is that we make sense of this sequence of events. Returning to Fig. 4 we note that the standing wave displacements at B and C take on the completely out-of-phase (antiperiodic) behaviour when necessary and that the displacements associated with the fifth curve are always zero at the origin, indicating the partial decoupling of the problem for this frequency. Looking at the displacements purely in x or y it may appear

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2.5

Frequency

2 1.5 1 0.5 0 –2

0

A

2

4

Wavenumber

2

2

1

1

0

0

U

U

B

–1 –2 –1

6

C

B

First Third Fifth

–1

–0.5

0

0.5

–2 –1

1

–0.5

ξ

0 η

0.5

1

2

2

1

1

0

0

U

U

pffiffiffi Fig. 3. Typical dispersion curves in panel (a) for ðr1 ,r2 ,r3 ,r4 Þ ¼ ð1:5, 2,2,4Þ. The displacements for the periodic–periodic case, at wavenumber A, are shown for the first, third and fifth standing wave frequencies (indicated by circles in panel (a)) in panels (b) we show U(1,3)(x) at A and in panel (c) U(2,4)(Z) at A.

–1 –2 –1

–1

–0.5

0

0.5

–2 –1

1

–0.5

ξ

0

0.5

1

0.5

1

η

1

2

0

0

U

U

2

–1 –2 –1

–2

–0.5

0 ξ

0.5

1

–1

–0.5

0 η

Fig. 4. The displacements for the antiperiodic–periodic case at wavenumber B (panels (a) and (b)) and for the antiperiodic–antiperiodic case at wavenumber C (panels (c) and (d)), are shown for the first, third and fifth standing wave frequencies (indicated by squares for wavenumber B and stars for C in Fig. 3(a)). (a) U(1,3)(x) at B, (b) U(2,4)(Z) at B, (c) U(1,3)(x) at C and (d) U(2,4)(Z) at C.

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that the displacement gradient is discontinuous, however, as indicated in (2.7) it is a combination of displacement gradients in both directions that must be zero. 3. Asymptotic theory The Bloch example is instructive showing a wide range of interesting physical phenomena. A purely numerical solution such as that shown in Figs. 3 and 4 yields little understanding of how the material properties affect the dispersion curves, apart from performing a large parametric study, and brings us no closer to our goal of generating effective continuum models for frame/net structures. However, several physical features, such as standing waves, are useful and we use these to our advantage. We take a large structure (not necessarily infinite) constructed from many cells, such as that shown in Fig. 1, and the strings have governing equations for the displacements u(1,3)(x,y) and u(2,4)(x,y) as 2

for 2n r x=l r 2n þ 1,

ð3:1Þ

2

for 1 þ2n rx=l r 2n,

ð3:2Þ

2

for 2m ry=l r2m þ 1,

ð3:3Þ

2

for 1 þ2mr y=l r 2m,

ð3:4Þ

2 ð1Þ l2 uð1Þ ¼0 xx þ l r1 u 2 ð3Þ l2 uð3Þ ¼0 xx þ l r3 u 2 ð2Þ l2 uð2Þ ¼0 yy þ l r2 u 2 ð4Þ l2 uð4Þ ¼0 yy þ l r4 u

with continuity of displacements and stress at the origin of each cell, that is uð1Þ jx=l ¼ 2n þ ¼ uð3Þ jx=l ¼ 2n ¼ uð2Þ jy=l ¼ 2m þ ¼ uð4Þ jy=l ¼ 2m ,

ð3:5Þ

ð3Þ ð2Þ ð4Þ uð1Þ x jx=l ¼ 2n þ ux jx=l ¼ 2n þ uy jy=l ¼ 2m þ uy jy=l ¼ 2m ¼ 0:

ð3:6Þ

and

There is also continuity of displacement and displacement gradient at x/l =  1+2n,1 + 2n, y/l =  1+2m,1 +2m for n,m integer. Notably we are now using displacements u(i)(x,y), that is, as functions of both x and y. In terms of the overall displacement u(x,y), in the continuum setting, we have uðx,yÞ ¼ dðy=l2mÞuð1,3Þ ðx,yÞ þ dðx=l2nÞuð2,4Þ ðx,yÞdðx=l2nÞdðy=l2mÞuð1,3Þ ðx,yÞ

ð3:7Þ

for integer n and m; the final term ensures that both strings do not simultaneously contribute at the crossing point. This introduction of delta functions allows us to move from the semi-discrete setting of Section 2 to a fully continuum setting. The overall structure has macroscale L and is constructed from elementary cells of scale l. The ratio of lengthscales E ¼ l=L5 1 is small and provides a natural parameter for asymptotic analysis. The microstructure is periodic in ðx, ZÞ ¼ ðx,yÞ=l but not necessarily on the macroscale with coordinates defined as (X,Y)=(x,y)/L. We adopt a multiple scales approach treating (X,Y) and ðx, ZÞ as new independent variables so uð1,3Þ ðx,yÞ ¼ uð1,3Þ ðX,Y, xÞ and uð2,4Þ ðx,yÞ ¼ uð2,4Þ ðX,Y, ZÞ. The asymptotics are developed for perturbations to the standing wave solutions that are periodic in both x and y directions; the final results will be given for all cases, that is, for perturbations about the other standing wave frequencies where the local behaviour can be out-of-phase across a cell. Using the independent variables gives the governing equation as 2

ð1,3Þ 2 ð1,3Þ 2 ð1,3Þ uð1,3Þ ¼ 0, xx þ2EuxX þ E uXX þ r1,3 l u

ð3:8Þ

in 0 r x r 1 for uð1Þ ðX,Y, xÞ, and in 1 r x r0 for uð3Þ ðX,Y, xÞ, and 2

ð2,4Þ 2 ð2,4Þ 2 ð2,4Þ uð2,4Þ ¼ 0, ZZ þ2EuZY þ E uYY þ r2,4 l u

ð3:9Þ

in 0 r Z r 1 for u ðX,Y, ZÞ, and in 1 r Z r 0 for u ðX,Y, ZÞ. Periodicity in x and Z explicitly enforces ð2Þ

ð4Þ

uð1Þ jx ¼ 1 ¼ uð3Þ jx ¼ 1 ,

uð2Þ jZ ¼ 1 ¼ uð4Þ jZ ¼ 1 ,

ð3:10Þ

ð3Þ uð1Þ x jx ¼ 1 ¼ ux jx ¼ 1 ,

ð4Þ uð2Þ Z jZ ¼ 1 ¼ uZ jZ ¼ 1 ,

ð3:11Þ

with coupling conditions at the origin uð1Þ jx ¼ 0 þ ¼ uð3Þ jx ¼ 0 ¼ uð2Þ jZ ¼ 0 ¼ uð4Þ jZ ¼ 0 þ ,

ð3:12Þ

ð1Þ ð3Þ ð3Þ ð2Þ ð4Þ ð2Þ ð4Þ ðuð1Þ x þ EuX Þjx ¼ 0 þ ðux þ EuX Þjx ¼ 0 þ ðuZ þ EuY ÞjZ ¼ 0 þ ðuZ þ EuY ÞjZ ¼ 0 ¼ 0:

ð3:13Þ

and

Thus a fundamental cornerstone of the asymptotic theory is that the boundary conditions on the microscale are explicitly set by the assumption that we are close to a standing wave frequency. We, however, say nothing about the boundary conditions

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on the macroscale (X,Y) and aim to develop continuum equations for the system on that scale, i.e. only in terms of (X,Y), that do not involve the microscale coordinates ðx, ZÞ explicitly. We now solve Eqs. (3.8) and (3.9) order-by-order in E where the ansatz ðX,Y, xÞ þ Euð1,3Þ ðX,Y, xÞ þ E2 uð1,3Þ ðX,Y, xÞ þ    uð1,3Þ ðX,Y, xÞ ¼ uð1,3Þ 0 1 2

ð3:14Þ

uð2,4Þ ðX,Y, ZÞ ¼ uð2,4Þ ðX,Y, ZÞ þ Eu1ð2,4Þ ðX,Y, ZÞ þ E2 uð2,4Þ ðX,Y, ZÞ þ    0 2

ð3:15Þ

l2 ¼ l20 þ El21 þ E2 l22 þ   

ð3:16Þ

is adopted. The essential difference between this expansion and that undertaken in conventional homogenization theory, Mei et al. (1996), is that the latter theory constrains the leading order displacement field to be constant on the microscale (as shown in Figs. 3 and 4 for the first line) so uð1,3Þ ðX,Y, xÞ, uð2,4Þ ðX,Y, ZÞ are independent of x, Z and this forces the frequency to 0 0 2 2 be just l  E2 l2 ; the latter constraint inherently limits the conventional theory to low frequencies. The ansatz (3.14)–(3.16) liberates us from both of these restrictions. At leading order Eqs. (3.8) and (3.9) are simply 2

2 uð1,3Þ þ r1,3 l0 uð1,3Þ ¼ 0, 0 0xx

2

ð2,4Þ 2 uð2,4Þ ¼ 0, 0ZZ þ r2,4 l0 u0

ð3:17Þ

in their respective regions of validity. Periodicity in x, Z is enforced by (3.10), (3.11), this together with the leading order conditions at the origin (3.13) and (3.12) set the boundary conditions. This leading order problem is then exactly the same as that for the Bloch waves of Section 2 for k1 ¼ k2 ¼ 0, so the leading order frequency l0 follows directly from the dispersion relation (2.8) as Q1 ðl0 ÞQ2 ðl0 Þ½Q2 ðl0 ÞS1 ðl0 ,0Þ þQ1 ðl0 ÞS2 ðl0 ,0Þ ¼ 0,

ð3:18Þ

and is the frequency of the standing wave. The leading order problem is completely independent of the macroscale X,Y, and the behaviour on that scale is captured by a multiplicative function f0(X,Y), with the string displacements being given by ðX,Y, xÞ ¼ f0 ðX,YÞU0ð1,3Þ ðxÞ, uð1,3Þ 0

uð2,4Þ ðX,Y, ZÞ ¼ f0 ðX,YÞU0ð2,4Þ ðZÞ, 0

ð3:19Þ

where f0(X,Y) remains to be determined. The functions U0ðiÞ are known from (2.11) with l ¼ l0 and k1 ¼ k2 ¼ 0. For instance, U0ð1Þ ðxÞ is U0ð1Þ ðxÞ ¼ cosðl0 r1 xÞ þ P3 ðl0 ,0Þsinðl0 r1 xÞ=Q1 ðl0 Þ,

ð3:20Þ

notably there is a division by Q1 ðl0 Þ. We shall assume that l0 is a simple eigenvalue and that neither Q1 ðl0 Þ ¼ 0, nor Q2 ðl0 Þ ¼ 0, and return to these degenerate cases in Sections 3.1 and 3.2. The function f0(X,Y) in (3.19) varies only upon the macroscale, and is currently unknown, and our ultimate aim is to find an equation for it. At first order in E equations (3.8) and (3.9) give 2

2

2

2

ð1,3Þ 2 2 þ r1,3 l0 uð1,3Þ ¼ ½2f0X U0ð1,3Þ , uð1,3Þ 1 x þr1,3 l1 f0 U0 1xx

ð3:21Þ

ð2,4Þ ð2,4Þ 2 2 ¼ ½2f0Y U0ð2,4Þ uð2,4Þ Z þr2,4 l1 f0 U0 : 1ZZ þ r2,4 l0 u1

ð3:22Þ u(i) 1 ,

with these holding in the respective regions of validity of each string. On the microscale the i =1y4, must satisfy periodicity in x and Z and the displacement and stress continuity conditions at the origin. We now employ a version of Green’s theorem, we take each leading order equation (3.17) and multiply by the corre(i) (i) sponding first order displacement u1 and subtract from each of these U0 times the first order equation from (3.21) or (3.22). The resulting expressions are integrated over the whole ‘‘cross’’ microcell, in a manner analogous to that employed in Craster et al. (2010a). Using periodicity it is found that " Z # Z Z Z

l21 f0 r12

1

0

ðU0ð1Þ Þ2 dx þ r32

0

1

ðU0ð3Þ Þ2 dx þr22

1

0

ðU0ð2Þ Þ2 dZ þ r42

0

1

ðU0ð4Þ Þ2 dZ ¼ 0,

ð3:23Þ 2

the term in the square brackets is non-zero and so the first correction to the eigenvalue is zero, that is, l1 ¼ 0. This then simplifies Eqs. (3.21) and (3.22) and we explicitly solve them to get ¼ f1 U0ð1,3Þ ðxÞ þf0X ½V1ð1,3Þ ðxÞxU0ð1,3Þ ðxÞ, uð1,3Þ 1

uð2,4Þ ¼ f1 U0ð2,4Þ ðZÞ þf0Y ½V1ð2,4Þ ðZÞZU0ð2,4Þ ðZÞ: 1

ð3:24Þ

This first order solution involves a homogeneous piece involving a new function f1(X,Y) that is unknown, but not required in the subsequent analysis, and an auxiliary set of functions V(i) 1 for i= 1y4. The auxiliary functions satisfy 2

ð1,3Þ 2 V1ð1,3Þ ¼ 0, xx þ r1,3 l0 V1

2

ð2,4Þ 2 V1ð2,4Þ ¼ 0, ZZ þr2,4 l0 V1

ð3:25Þ

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E. Nolde et al. / J. Mech. Phys. Solids 59 (2011) 651–671

(i) that is, the same equations as the leading order ones for U(i) 0 . Periodicity of u1 forces the following non-periodic boundary (i) (i) conditions upon the V1 , which also ensures that these functions are linearly independent of the U0 :

V1ð1Þ jx ¼ 1 V1ð3Þ jx ¼ 1 ¼ 2U0ð1Þ jx ¼ 1 ,

V1ð2Þ jZ ¼ 1 V1ð4Þ jZ ¼ 1 ¼ 2U0ð2Þ jZ ¼ 1 ,

ð3:26Þ

ð3Þ ð1Þ V1ð1Þ x jx ¼ 1 V1x jx ¼ 1 ¼ 2U0x jx ¼ 1 ,

ð4Þ ð2Þ V1ð2Þ Z jZ ¼ 1 V1Z jZ ¼ 1 ¼ 2U0Z jZ ¼ 1 ,

ð3:27Þ

at the origin we have the displacement and stress continuity conditions: f0X V1ð1Þ jx ¼ 0 þ ¼ f0X V1ð3Þ jx ¼ 0 ¼ f0Y V1ð2Þ jZ ¼ 0 þ ¼ f0Y V1ð4Þ jZ ¼ 0 ,

ð3:28Þ

ð3Þ ð2Þ ð4Þ f0X ½V1ð1Þ x jx ¼ 0 þ V1x jx ¼ 0  þ f0Y ½V1Z jZ ¼ 0 þ V1Z jZ ¼ 0  ¼ 0:

ð3:29Þ

and

The solution is that V1ð1,3Þ ðxÞ ¼

2r3,1 sinðl0 r1,3 xÞ, Q1 ðl0 Þ

2r4,2 sinðl0 r2,4 ZÞ: Q2 ðl0 Þ

V1ð2,4Þ ðZÞ ¼

ð3:30Þ

(i)

(i)

To recap, we have determined the leading order solution u0 in terms of a known function U0 , fully determined on the microscale, and a function f0(X,Y) as yet unknown on the macroscale. The first order term is known in terms of the leading order solution, together with a known auxiliary solution and an irrelevant homogeneous term characterized by a function f1(X,Y). To identify the continuum governing equation on the macroscale we now move to the next order. At order E2 from (3.8) and (3.9) we have 2

2

2

2

ð1,3Þ ð1,3Þ 2 2 þr1,3 l0 uð1,3Þ ¼ ½2u1ð1,3Þ , uð1,3Þ 2 xX þu0XX þr1,3 l2 u0 2xx

ð3:31Þ

ð2,4Þ ð2,4Þ ð2,4Þ 2 2 ¼ ½2u1ð2,4Þ uð2,4Þ 2ZZ þr2,4 l0 u2 ZY þu0YY þr2,4 l2 u0 :

ð3:32Þ

We again employ a version of Green’s theorem as in the discussion following (3.22), see also Craster et al. (2010a), and integrate over the microcell. Periodicity on the microscale is then utilised, substantial simplifications ensue, and a partial differential equation (PDE) for the unknown function f0(X,Y) emerges as 2

T11 f0XX þ T22 f0YY þ l2 f0 ¼ 0,

ð3:33Þ 2

that involves the frequency correction l2 . This continuum PDE is the main result of this article and is the homogenized equation that we have been searching for, it is defined only upon the macroscale and has no explicit dependence upon the microstructure. The microstructure is entirely contained within the coefficients T11, T22 which are integrals over the microcell. Specifically they are given as ! Z 0 Z 1 2 4r1 r3 l0 ð1Þ 2 ð3Þ ð3Þ ð1Þ ð1Þ ðU0 Þ jx ¼ 1 þ T11 ¼ , ð3:34Þ U0 V1x dx þ U0 V1x dx ¼ I Q1 ðl0 ÞI 1 0 2 ðU0ð2Þ Þ2 jZ ¼ 1 þ T22 ¼ I where I is the integral Z 1 Z I ¼ r12 ðU0ð1Þ Þ2 dx þ r32 0

Z

0 1

0 1

U0ð4Þ V1ð4Þ Z

Z

dZ þ

ðU0ð3Þ Þ2 dx þr22

1

0

Z 0

1

! U0ð2Þ V1ð2Þ Z

dZ

ðU0ð2Þ Þ2 dZ þ r42

¼

Z

4r2 r4 l0 , Q2 ðl0 ÞI

0

1

ðU0ð4Þ Þ2 dZ,

ð3:35Þ

ð3:36Þ

and Q1, Q2 are defined in, and after, (2.10). Importantly, we see that the standing wave frequencies, l0 , and the standing wave (i) (i) displacements, U0 , and auxiliary functions V1 play a key role, thus the intimate details of the microstructure, and wave structure on the microscale, are incorporated into these coefficients. We have explicitly presented the periodic–periodic case, that is, we assume that on the microscale we are close to a standing wave frequency and solution that is locally periodic in both x and Z. There are two other possible standing wave cases that must be considered: the antiperiodic–antiperiodic case where, in the Bloch wave language, j ¼ ðp, pÞ, and the antiperiodic–periodic case where j ¼ ðp,0Þ (or equivalently periodic–antiperiodic with j ¼ ð0, pÞ). We simply give the final results for these additional cases. It is convenient to introduce two index functions m1 and m2 such that m1 ¼ 1 ðperiodic in xÞ,

m1 ¼ 1 ðanti-periodic in xÞ,

m2 ¼ 1 ðperiodic in ZÞ,

m2 ¼ 1 ðanti-periodic in ZÞ,

ð3:37Þ

then T11 and T22 are T11 ¼

4m1 r1 r3 l0 , Q1 ðl0 ÞI

T22 ¼

4m2 r2 r4 l0 : Q2 ðl0 ÞI

ð3:38Þ

E. Nolde et al. / J. Mech. Phys. Solids 59 (2011) 651–671

659

It is crucial to note that the frequency l0 in these formulae, and that is used in Q1,Q2, is that associated with the standing wave at j ¼ ðp, pÞ or j ¼ ð0, pÞ (or j ¼ ðp,0Þ). Classical homogenization theory, as in say Mei et al. (1996), is only relevant for long waves valid around j ¼ ð0; 0Þ and l0  0, so both requires long waves and low frequencies. The asymptotics near this point are far more straightforward, the expansion is uð1,3Þ ðX,Y, xÞ ¼ f0 ðX,YÞ þ Euð1,3Þ ðX,Y, xÞ þ E2 uð1,3Þ ðX,Y, xÞ þ . . . ,

ð3:39Þ

uð2,4Þ ðX,Y, ZÞ ¼ f0 ðX,YÞ þ Euð2,4Þ ðX,Y, ZÞ þ E2 uð2,4Þ ðX,Y, ZÞ þ    ,

ð3:40Þ

with the frequency simply

l  E2 l22 þ   

ð3:41Þ

from which it rapidly emerges that 2½f0XX þ f0YY  2 þ l2 f0 ¼ 0: ðr12 þr22 þr32 þ r42 Þ

ð3:42Þ (i)

(i)

This can also be derived from (3.33) with the leading order displacements, U0 , just set to a non-zero constant and the V1 set to zero. Eq. (3.42) is only valid near ðk1 , k2 Þ ¼ ð0,0Þ and for l 5 1. As noted earlier the entire analysis thus far has assumed that the standing frequencies l0 are simple eigenvalues and that the Q1, Q2, given in (2.10), appearing in the denominators of the displacements (3.20) are non-zero. We now investigate the degenerate cases when one or both of the Q are zero. 3.1. Degenerate case (i) Q1 =0 (or Q2 =0) The leading order displacements decouple when Q1 =0 and U0ð1Þ ðxÞ ¼ Bð1Þ sinðl0 r1 xÞ,

U0ð3Þ ðxÞ ¼ Bð3Þ sinðl0 r3 xÞ,

U0ð2Þ ðZÞ ¼ U0ð4Þ ðZÞ ¼ 0,

ð3:43Þ

the displacement is clearly zero at the origin. The periodicity conditions and the conditions (displacement and stress continuity) at the origin require that these two equations r1 cosðl0 r1 Þsinðl0 r3 Þ þ r3 cosðl0 r3 Þsinðl0 r1 Þ ¼ 0,

ð3:44Þ

m1 cosðl0 r1 Þ ¼ cosðl0 r3 Þ,

are satisfied simultaneously (m1 is the index function from (3.37)); the coefficients B(1,3) satisfy r1 B(1) = r3 B(3). These are satisfied if l0 r1 ¼ np (or l0 r1 ¼ ð2n1Þp=2) for integer n. We treat the case l0 r1 ¼ np in more detail. Conditions (3.44) are satisfied if

l0 r3 ¼ kp provided ð1Þk ¼ m1 ð1Þn for integer k. The leading order U0ð1Þ ðxÞ ¼ r3 sinðl0 r1 xÞ, At order E, uð1,3Þ 1

l21

U0ð1,3Þ ðxÞ

ð3:45Þ

are simply

U0ð3Þ ðxÞ ¼ r1 sinðl0 r3 xÞ:

ð3:46Þ

is again zero and solving for the first order displacements the equivalent equations to (3.24) are

¼ f1 U0ð1,3Þ ðxÞ þf0X ½V1ð1,3Þ ðxÞxU0ð1,3Þ ðxÞ,

uð2,4Þ ¼ f1 U1ð2,4Þ ðZÞ: 1

ð3:47Þ

, forces their usual functional form to The main change is that the decoupling, and zero displacements at leading order for U(2,4) 0 appear at next order and   P4 ðl0 , kÞ sinðl0 r2 ZÞ , U1ð2Þ ðZÞ ¼ D cosðl0 r2 ZÞ þ Q2 ðl0 Þ

  P2 ðl0 , kÞ sinðl0 r4 ZÞ , U1ð4Þ ðZÞ ¼ D cosðl0 r4 ZÞ Q2 ðl0 Þ

ð3:48Þ (1,3)

for some constant D to be determined; the function f1 is now relevant and plays a role. The auxiliary functions V1 (3.25) and V1ð1,3Þ ¼ Hcosðl0 r1,3 xÞ þGð1,3Þ sinðl0 r1,3 xÞ,

still satisfy ð3:49Þ

where Gð1Þ r1 Gð3Þ r3 ¼ 2r1 r3 ,

ð3:50Þ

and H is to be determined; these solutions satisfy all the required periodicity conditions and some of the continuity conditions at the origin. The remaining continuity conditions relate D, H via f0X H ¼ f1 D,

2r1 r3 f0X þf1 D

S2 ðl0 , kÞ ¼ 0, Q2 ðl0 Þ

ð3:51Þ

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E. Nolde et al. / J. Mech. Phys. Solids 59 (2011) 651–671

giving 2r1 r3 Q2 ðl0 Þ : S2 ðl0 , kÞ

H¼

ð3:52Þ

We now evaluate the coefficients T11 and T22 to get T11 ¼

4l0 Q2 ðl0 Þ , S2 ðl0 , kÞ

T22 ¼ 0,

ð3:53Þ

with the highly significant result that T22 =0 and so the PDE (3.33) is, in this degenerate case, independent of Y. The situation of Q2 = 0 is identical, but with 1 and 2 interchanged in (3.53). The other possible solution, l0 r1 ¼ ð2n1Þp=2, is only realized if r1 ¼ r3

and

m1 ¼ 1:

ð3:54Þ

The algebra is very similar to that presented above and eventually T11 and T22 are again given by (3.53). 3.2. Degenerate case (ii) both Q1 and Q2 =0 We take both Q1 = 0 and Q2 = 0 but assume the displacement at the origin is not zero. This is only possible if the following conditions are satisfied simultaneously: m1 cosðl0 r1 Þcosðl0 r3 Þ ¼ 0,

m2 cosðl0 r2 Þcosðl0 r4 Þ ¼ 0,

m1 r1 sinðl0 r1 Þ þ r3 sinðl0 r3 Þ ¼ 0,

ð3:55Þ

m2 r2 sinðl0 r2 Þ þ r4 sinðl0 r4 Þ ¼ 0:

ð3:56Þ

These conditions can, and do, occur in practice. One set of conditions for them to be satisfied is that ri = nir and l0 r ¼ p for some integers ni with i=1y4 provided the index functions also satisfy m1 ¼ ð1Þn1 þ n3 ,

m2 ¼ ð1Þn2 þ n4 :

ð3:57Þ

l21 ,

is not The eigenvalues are no longer simple, but there are now repeated roots, the first order correction to the frequency, zero and the analysis must be modified. Repeated roots, or their analogues, also appeared in Craster et al. (2010a,b), and the leading order displacements are now the sum of the linearly independent leading order solutions to (3.17) namely uð1Þ 0 ¼ f0ð1Þ ðX,YÞcosðl0 r1 xÞ þ f0ð2Þ ðX,YÞr3 sinðl0 r1 xÞ,

0 r x r 1,

ð3:58Þ

uð3Þ 0 ¼ f0ð1Þ ðX,YÞcosðl0 r3 xÞ þ f0ð2Þ ðX,YÞr1 sinðl0 r3 xÞ,

1 r x r0,

ð3:59Þ

uð2Þ 0 ¼ f0ð1Þ ðX,YÞcosðl0 r2 ZÞ þf0ð3Þ ðX,YÞr4 sinðl0 r2 ZÞ,

0 r Z r1,

ð3:60Þ

uð4Þ 0 ¼ f0ð1Þ ðX,YÞcosðl0 r4 ZÞ þf0ð3Þ ðX,YÞr2 sinðl0 r4 ZÞ,

1 r Z r 0,

ð3:61Þ

for some functions f0(i), i= 1,2,3 to be determined. The application of the Green’s theorem approach gives three coupled equations for the macroscale functions f0(i) for i= 1,2,3 as 2

2l0 ðr1 r3 f0ð2ÞX þr2 r4 f0ð3ÞY Þ þ 12ðr12 þr22 þr32 þr42 Þl1 f0ð1Þ ¼ 0, 2

2l0 f0ð1ÞX r1 r3 l1 f0ð2Þ ¼ 0,

ð3:62Þ ð3:63Þ

2

2l0 f0ð1ÞY r2 r4 l1 f0ð3Þ ¼ 0,

ð3:64Þ

that involve constant coefficients and can be combined into a single equation, see (4.5), if desired. The conditions (3.55)–(3.56) are also satisfied if r1 = r3, r2 = r4, l0 r1 ¼ ð2n1Þp=2, l0 r2 ¼ ð2k1Þp=2 are furthermore m1 =m2 = 1; pursuing the analysis one finds that Eqs. (3.62)–(3.64) again emerge. The theory presented here has now generated homogenized PDEs, namely (3.33) and the coupled set (3.62)–(3.64), that capture the essential physics of wave motion near the standing wave frequencies of the structure; the PDEs only depend upon the macroscale variables X and Y. We now illustrate the theory. 4. Asymptotics applied to Bloch waves As noted earlier the infinite perfect net is an ideal vehicle upon which to test the theory, and to illustrate how knowledge of the coefficients T11, T22 can be used. Dispersion curves that illustrate all the salient points are given in Figs. 5–9, these show the frequency versus the wavenumber, as shown in the Brillouin zone of Fig. 2. First we need to translate the Bloch conditions into the multiple scales language, Eqs. (2.4), (2.5) become uð1,3Þ ðX þ 2E,Y, xÞ ¼ eiEk1 uð1,3Þ ðX,Y, xÞ,

uð2,4Þ ðX,Y þ2E, ZÞ ¼ eiEk2 uð2,4Þ ðX,Y, ZÞ

ð4:1Þ

E. Nolde et al. / J. Mech. Phys. Solids 59 (2011) 651–671

661

General case 4 3.5

Frequency

3 2.5 2 1.5 1 0.5 0 B

–2

0A

2

4 C

6

B

Wavenumber pffiffiffi pffiffiffi Fig. 5. The first six dispersion curves for the general case ðr1 ,r2 ,r3 ,r4 Þ ¼ ð1,1= 2,2, 2Þ with the full numerics as dashed curves and the solid lines from the asymptotics. The squares denote the degenerate cases where l0 r1 ¼ p (Q1 = 0). The dotted lines show the wavenumbers at points A, B and C.

6 5

Frequency

4 3 2 1 0 –2

B

0A

2

4 C

6

B

Wavenumber pffiffiffi pffiffiffi Fig. 6. The first six dispersion curves for the case ðr1 ,r2 ,r3 ,r4 Þ ¼ ð1,1= 2,1,1= 2Þ with the full numerics as dashed curves and the solid lines from the asymptotics. The squares (circles) denote degenerate cases where Q1 = 0 (Q2 = 0); those along C and B arise at l0 r1 ¼ p=2,3p=2 and that along A for l0 r1 ¼ p in line with the degeneracy discussed in the text (the Q2 = 0 cases have l0 r2 ¼ p, p=2).

3.5 3

Frequency

2.5 2 1.5 1 0.5 0 B

–2

0

A

2 Wavenumber

4

C

6

B

Fig. 7. The first six dispersion curves for the case (r1,r2,r3,r4) = (1,1,2,2) with the full numerics as dashed curves and the solid lines from the asymptotics. The star denotes the degenerate case where both Q1 and Q2 are zero with l0 r1 ¼ l0 r2 ¼ p and r3 = r4 = 2r1 (the straight lines for the second, fourth and sixth curve have both Q1 = Q2 = 0).

662

E. Nolde et al. / J. Mech. Phys. Solids 59 (2011) 651–671

3

Frequency

2.5 2 1.5 1 0.5 0 B

–2

0

A

2 Wavenumber

4

C

6

B

Fig. 8. The first six dispersion curves for the case (r1,r2,r3,r4)= (1,2,2, 2) with the full numerics as dashed curves and the solid lines from the asymptotics. The circles denote degenerate cases where Q2 = 0 (l0 r2 ¼ p=2,3p=2 along C and l0 r2 ¼ p on A, B) and the star where both Q1 and Q2 are zero ðl0 r1 ¼ p, r2 ¼ r3 ¼ r4 ¼ 2r1 Þ.

5

Frequency

4

3

2

1

0 B

–2

0A

2

4 C

6

B

Wavenumber Fig. 9. The first six dispersion curves for the case (r1,r2, r3,r4) =(1,1,1, 1) with the full numerics as dashed curves and the solid lines from the asymptotics. Each point where the curves cross is degenerate and Q1 = Q2 = 0 there; the case along A has l0 r1 ¼ p and m1 = m2, that along C has l0 r1 ¼ p=2,3p=2 and m1 =  m2. The star shows where both Q1 and Q2 are zero and the degenerate case of Section 3.2 occurs.

with ðk1 , k2 Þ ¼ Eðk1 ,k2 Þ. The periodicity of u(1,3) and u(2,4) in x and Z, respectively, force f0 ðX þ2E,Y þ2EÞ ¼ exp½iEðk1 X þ k2 YÞ=2f0 ðX,YÞ,

ð4:2Þ

and thus f0 ðX,YÞ ¼ exp½iðk1 X þ k2 YÞ=2:

ð4:3Þ

This is now used to extract the asymptotic dispersion relations from the PDE for f0 (3.33). In Figs. 5–9 the horizontal axis is the wavenumber given as the distance around the triangle shown for the reciprocal net in Fig. 2. Near the point labelled A where the standing waves are in-phase, f0(X,Y) is given in (4.3), and so using (3.33) ! ðT k2 þ T k2 Þ l  l0 1þ 11 1 2 22 2 , ð4:4Þ 8l0 so the frequency dependence upon the wavenumber is quadratic. This expression also shows that the local curvature is critically dependent upon the sign of T11 and T22 and (4.4) is valid provided both Q1 and Q2 are non-zero. For the zero Q cases, we recall that if Q1 = 0 (and Q2 a0) then we have the degenerate case of Section 3.1, (3.53) holds, and so T11 ¼ 4l0 Q2 =S2 , T22 = 0 and if Q2 =0 (and Q1 a0) T22 ¼ 4l0 Q1 =S1 , T11 = 0; locally the partial differential equation (3.33) becomes an ordinary differential equation and (4.4) is independent of either k2 or k1. If we consider the path along the triangle BACB shown as the horizontal axis in the dispersion curve figures then variation along BA has k2 =0, thus if Q2 = 0, forcing T11 =0, then Eq. (4.4) is completely independent of both k1 and k2 and the local dispersion relation is flat on BA near A (this is the situation

E. Nolde et al. / J. Mech. Phys. Solids 59 (2011) 651–671

663

near the circled point in Fig. 6 for the fifth curve along A). In the other direction away from A, that is, along AC we have k1 =k2 and so even if Q2 = 0 the dispersion curve remains locally quadratic in k1. If both Q1 and Q2 are zero then we have the degenerate situation given in Section 3.2 by the coupled equations (3.62)–(3.64) from which locally 4

f0ð1ÞXX þ f0ð1ÞYY þ

ðr12 þ r22 þr32 þr42 Þl1 2 8l0

f0ð1Þ ¼ 0,

ð4:5Þ

with f0(1)(X,Y) satisfying f0ð1Þ ðX,YÞ ¼ expði½k1 X þk2 Y=2Þ from the Bloch conditions. This PDE then involves the first order 2 2 correction l1 and not l2 and its solution gives jjj

l  l0 7 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ,

ð4:6Þ

2ðr12 þ r22 þ r32 þ r42 Þ

which is now linear in jjj; these linear curves are shown near the starred point in Fig. 9 along A (and also along B and C suitably displaced in Figs. 8 and 9). Eq. (4.4) is also invalid near the origin, where j ¼ ð0,0Þ and l 51, as the asymptotic theory is technically invalid there and should be replaced by classical homogenization theory (3.42) from which jjj

l  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi :

ð4:7Þ

2ðr12 þ r22 þ r32 þ r42 Þ

This is identical to the positive root in (4.6) displaced by a frequency l0 , this is not coincidental as the low frequency case can also be derived as a degenerate sub-case of the analysis leading to Eqs. (3.62)–(3.64). Notably this gives the locally linear dispersion curves near the origin in Figs. 5–9, so for low frequencies the net is represented in continuum terms as a membrane; Eq. (3.42) is just the usual membrane equation with inverse wavespeed squared given by the averaged inverse wavespeed squared of the strings. Near the point labelled as C where the standing waves are perfectly out-of-phase across the cell it is convenient to take Eðk1 ,k2 Þ as the wavenumbers relative to ðp, pÞ and so f0 ðX,YÞ ¼ expði½ðk1 p=EÞX þðk2 p=EÞY=2Þ,

ð4:8Þ

and thus

l  l0 1 þ

ðT11 ðk1 pÞ2 þ T22 ðk2 pÞ2 Þ 2

8l0

! ,

ð4:9Þ

where l0 is the frequency where j ¼ ðp, pÞ; importantly in these formulae one must use this frequency when calculating the T11, T22. Again there is the possibility of either Q1 = 0 and/or Q2 = 0 with the resulting degeneracies. Near the point labelled B the standing waves are periodic in the vertical, but out-of-phase in the horizontal so j ¼ ð 8 p,0Þ. pffiffiffi The sign change here reflects whether B is at p or 2p þ p in Figs. 5–9 and then f0 ðX,YÞ ¼ expði½ðk1 7 pÞX þ k2 Y=2Þ,

ð4:10Þ

and thus

l  l0 1 þ

ðT11 ðk1 7 pÞ2 þ T22 k22 Þ 2

8l0

! ,

ð4:11Þ

with l0 is the frequency where j ¼ ðp, pÞ; again one must use this frequency when calculating the T11, T22 and when either Q1 = 0 and/or Q2 = 0 the resulting degeneracies occur. Fig. 5 shows the first six dispersion curves for a typical set of parameter values, the dashed lines are Bloch curves determined numerically from the exact solution (2.8) and the solid lines from the asymptotics; for most curves the asymptotics almost completely cover the exact solution. Notably, there is the linear behaviour at low frequencies and zero wavenumber, that is, at A. Many curves, for instance the second curve along AC, feature nearly flat regions as Q1,Q2 are locally small, or completely flat regions such as for the fifth curve along CB where Q1 =0 and T22 = 0 so the dispersion curve is independent of k1,2; given the asymptotics we can predict or design the structure to have flat dispersion curves at specific frequencies just through the knowledge of T11, T22. Locally quadratic behaviour dominates near the standing wave frequencies. We now induce more degenerate behaviour by taking some of the ri equal. For overlapping different strings, r1 = r3 and r2 = r4, the resulting dispersion curves are shown in Fig. 6. The main features are similar to Fig. 5 but there are many more points at which either Q1 or Q2 are zero. An important point is that Q1 or Q2 =0 does not guarantee a locally flat dispersion curve. For instance the third curve has two degenerate points, but both have locally quadratic variation. For the degenerate point along A Q1 = 0 so T22 = 0 and locally l  l0 þ T11 k21 =8l0 and along BA and AC k1 varies and so the curve is quadratic. A notable success of the asymptotics is that it accurately represents subtle features such as the local change in curvature of a curve, say, the second curve at C where there is a local point of inflection. The other case of two equal strings, r1 = r2 and r3 = r4, is shown in Fig. 7, as noted in Martinsson and Movchan (2003) and from (2.8) a considerable simplification ensues (S1 = S2) and completely flat dispersion curves occur for which both Q1 and

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6

λ0 r2

λ0

6 4

2

2 0

4

1

3

2

0 0

0.5

1 r2

1.5

2

0

2

4

6

4

6

λ0 r1 5

T22

5

T11

4

0

–5

0

–5 0

2

4 λ0 r2

6

0

2 λ0 r2

Fig. 10. The variation of T11 and T22 with parameters for periodic standing waves and r1 = r3 = 1 and r2 = r4. Panel (a) shows the variation of the standing wave eigenvalue, l0 , with r2; the dashed curve is replotted in panel (b) as l0 r1 versus l0 r2 . Panel (b) shows colour-coded regions with solution in 1 having both T11, T22 negative, 2 has T11 negative and T22 positive, 3 has both T11, T22 positive and 4 has T11 positive and T22 negative. Panels (c) and (d) show the variation of T11 and T22 versus l0 r2 for solutions on the curve shown in panel (b). The dashed line is zero and the dotted lines show l0 r2 ¼ p=2, p,3p=2 with sign changes in T11, T22 occurring as predicted in panel (b). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Q2 = 0. The physics here is that the displacements of the strings decouple, with zero displacements at the origin, and perfect standing waves emerge. There are inherent geometrical symmetries in the problem that become manifest in the dispersion diagram; the first and fifth curves are translations of each other. The degeneracy of Q1 = Q2 = 0 leads to the local linear behaviour described by (4.6) at the starred point of the fifth curve. It is worthwhile noting that this example has none of the single Q zero degeneracies, this is as the material is locally isotropic, and one obtains T11 = T22 for the periodic–periodic cases. Perhaps surprisingly, taking three strings equal with one differing leads to a dispersion diagram with few obvious symmetries. Fig. 8 displays a wide variety of features: locally flat curves (third curve along AB created by Q2 = 0 degeneracy), or almost flat curves (second curve along AC), changes in curvature (fourth curve at A), a linear dispersion relation (sixth curve at B), all of these features are captured by the asymptotic procedure. The final set of dispersion curves, Fig. 9, has all strings equal and the curves are now completely degenerate and consist of either flat lines (Q1 = Q2 =0) or curves which cross the flat lines and have locally linear behaviour; from the geometrical symmetry of the net there are resulting symmetries in the curves. Notable is the fact that points where three dispersion curves cross, the starred point along C for the first three curves, separate as the string parameters change. The parameters r1, r2, r3, r4 in Figs. 5–9 are chosen to be close, often with just one or two values changing from the preceding or following figure so the reader can appreciate the quite dramatic changes that a single curve goes through as the parameters change; fortunately the asymptotics remove any mystery from the changes in behaviour which are related to sign changes, or zero values, of T11, T22. To illustrate the power of this approach we specialize to overlapping strings with r1 = r3 and r2 = r4 and consider the variation of T11 and T22 as shown in Fig. 10. As r1 and r2 vary, for the curve illustrated, it becomes clear that T11 and T22 alternate in sign, and alternations occur when l0 r2 is equal to integer multiples of p=2 and so one can predict when this happens and design a structure to have desirable properties.

5. Forcing The general theory given here can be used to do far more than simply generate asymptotics for Bloch waves, and we illustrate the generality here by considering point forcing of the frame structure. This is a very testing example for an asymptotic theory as the loading is concentrated, it is also an important example as it is effectively a Green’s function for the structure.

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5.1. Exact solution First, we note that there is an exact formulation of the frames using a discrete setting and this then connects with the point mass-string approach used in solid state physics for fully discrete periodic problems. The dynamics of the fully discrete lattice problem, that is, concentrated point masses connected by idealised massless strings, has a long history with the propagation of waves through perfect lattices presented in Born and Huang (1954); Lifshitz and Kosevich (1966); Maradudin et al. (1971) amongst others, with more recent work in Martinsson and Rodin (2002); Martin (2006), however, the fact that frames too have explicit Green’s solutions appears to have been overlooked. Let us consider a cell [n,m], we use the [n,m] notation to denote a cell, with strings of unit length, defined so 2n1 ox½n o 2n þ 1 and 2m1o y½m o 2m þ1 and we have material 1 in 2n o x½n o 2n þ 1, y[m] = 2m, material 2 in 2m o y½m o2mþ 1,x½n ¼ 2n, material 3 in 2n1 ox½n o 2n, y[m] = 2m and material 4 in 2m1o y½m o 2m, x[n] = 2n. The displacements in each piece of the cell can be written in a form entirely dependent upon the displacements at (x[n],y[m])=(2n  1,2m), (2n,2m), (2n+ 1,2m), (2n,2m+ 1), (2n,2m 1) which are written as u2n  1,2m, u2n,2m, u2n + 1,2m, u2n,2m + 1, u2n,2m  1, respectively. Namely in cell [n,m] uð1Þ ½n,m ¼

u2n,2m sin½lr1½n,m ðxð2n þ1ÞÞ þ u2n þ 1,2m sin½lr1½n,m ðx2nÞ , sinðlr1½n,m Þ

ð5:1Þ

uð3Þ ½n,m ¼

u2n,2m sin½lr3½n,m ðxð2n1ÞÞu2n1,2m sin½lr3½n,m ðx2nÞ , sinðlr3½n,m Þ

ð5:2Þ

uð2Þ ½n,m ¼

u2n,2m sin½lr2½n,m ðyð2m þ1ÞÞ þ u2n,2m þ 1 sin½lr2½n,m ðy2mÞ , sinðlr2½n,m Þ

ð5:3Þ

uð4Þ ½n,m ¼

u2n,2m sin½lr4½n,m ðyð2m1ÞÞu2n,2m1 sin½lr4½n,m ðy2mÞ , sinðlr4½n,m Þ

ð5:4Þ

where ri[n,m] (i=1y4) allows for the string constants to be different in each cell if required. The displacements are all continuous. The conditions at the origin of each cell, and where each cell connects to the next cell, lead to the following difference equations   r1½nm cosðlr1½n,m Þ r2½n,m cosðlr2½n,m Þ r3½n,m cosðlr3½n,m Þ r4½n,m cosðlr4½n,m Þ þ þ þ u2n,2m sinðlr1½n,m Þ sinðlr2½n,m Þ sinðlr3½n,m Þ sinðlr4½n,m Þ 

r1½n,m r2½n,m r3½n,m r4½n,m u2n þ 1,2m  u2n,2m þ 1  u2n1,2m  u2n,2m1 ¼ 0, sinðlr1½n,m Þ sinðlr2½n,m Þ sinðlr3½n,m Þ sinðlr4½n,m Þ

ð5:5Þ

  r1½n,m cosðlr1½n,m Þ r3½n þ 1,m cosðlr3½n þ 1,m Þ r3½n þ 1,m r1½n,m þ u2n þ 1,2m  u2n þ 2,2m  u2n,2m ¼ 0, sinðlr3½n þ 1,m Þ sinðlr1½n,m Þ sinðlr1½n,m Þ sinðlr3½n þ 1,m Þ

ð5:6Þ

  r2½n,m cosðlr2½n,m Þ r4½n,m þ 1 cosðlr4½n,m þ 1 Þ r4½n,m þ 1 r2½n,m þ u2n,2m þ 1  u2n,2m þ 2  u2n,2m ¼ 0, sinðlr4½n,m þ 1 Þ sinðlr2½n,m Þ sinðlr2½n,m Þ sinðlr4½n,m þ 1 Þ

ð5:7Þ

and this formulation can be used to consider either localized forcing or localized mass variation. This formulation then turns the continuum problem into a discrete one and one can then use the techniques used in Craster et al. (2010b) to look at localized defect modes, although it is interesting to note that the frequency, l, is embedded within the coefficients of the 2 difference equations (5.5)–(5.7) but is a simple factor of l in point mass-string problems. This reformulation is interpreted as concentrating the mass at the end-points and then having frequency dependent massless strings, the advantage is that this is now a fully discrete system of equations and thus we can proceed to solve it. We now set ri[n,m] = ri, that is, we make the string constants the same in each cell. We now consider forcing the net by a delta function located either where each cell joins to the next, or at the crossing point of the strings. A point forcing, as a delta function, is equivalent to a jump in displacement gradient across each joint and that then replaces the zero right-hand sides with a term 1=l in the appropriate difference equation. A delta function at the crossing point of the [n,m]th cell inserts this term into (5.5), at the joint at the right-hand side of the [n,m]th cell into (5.6) and into the top of the [n,m]th cell into (5.7). To begin with we retreat to a single isolated string, by omitting all terms involving r2[n,m], r4[n,m], and have a delta function at the right-hand side of the [n  1]th cell. An explicit solution is easily furnished using the semi-discrete Fourier transform pair Z p 1 X 1 ½u~ even ðaÞ, u~ odd ðaÞ ¼ ½u2n ,u2n þ 1 eian , ½u2n ,u2n þ 1  ¼ ½u~ even ðaÞ, u~ odd ðaÞeian da, ð5:8Þ 2 p p n ¼ 1

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~ denotes that there is one transform for even, and another for odd, numbered end-points in the subscript on the transforms, u, one-dimension (for the single string). In two-dimensions similarly one has the transform pair 1 X

½u~ cross ða, bÞ, u~ right ða, bÞ, u~ top ða, bÞ ¼

1 X

½u2n,2m ,u2n þ 1,2m ,u2n,2m þ 1 ei½an þ bm ,

ð5:9Þ

n ¼ 1 m ¼ 1

and inverse ½u2n,2m ,u2n þ 1,2m ,u2n,2m þ 1  ¼

Z pZ p

1 ð2pÞ2

p

p

½u~ cross ða, bÞ, u~ right ða, bÞ, u~ top ða, bÞei½an þ bm da db:

ð5:10Þ

Exact solutions readily emerge, for instance, for the single string with a delta function at x= 1: ½u~ even ðaÞ, u~ odd ðaÞ ¼

eia ½r sinðlr3 Þ þ eia r3 sinðlr1 Þ,Q1 ðlÞ, lS1 ðl, aÞ 1

ð5:11Þ

and numerical values of the end-point displacements are easily furnished through an application of the Fast Fourier Transform (FFT), or via explicit inversion of the transform using residue calculus. Given the end-point values the full solution is then reconstructed from (5.1)–(5.4). The emergence of the factor S1 ðl, aÞ (2.9) is not fortuitous as the Bloch dispersion relations can also be found using the end-point values (Martinsson and Movchan, 2003). In two-dimensions exact solutions again emerge, for instance for forcing at the origin ð3Þ ð2Þ ð4Þ uð1Þ x j0 þ ux j0 þ uy j0 þ uy j0 ¼ 1,

ð5:12Þ

which effectively inserts a term dn0 dm0 =l into the right-hand side of (5.5), and then we have ½u~ cross ða, bÞ, u~ right ða, bÞ, u~ top ða, bÞ ¼

1

lDðl, a, bÞ

½Q1 ðlÞQ2 ðlÞ,Q2 ðlÞðr1 sinðlr3 Þ þ r3 eia sinðlr1 ÞÞ,Q1 ðlÞðr2 sinðlr4 Þ þ r4 eib sinðlr2 ÞÞ,

ð5:13Þ with Dðl, a, bÞ ¼ Q2 ðlÞS1 ðl, aÞ þ S2 ðl, bÞQ1 ðlÞ:

ð5:14Þ

Inversion is easily achieved numerically using the FFT.

5.2. Asymptotic solution: one-dimension As noted earlier this is a testing example for an asymptotic theory. We illustrate the ideas using the one-dimensional string and put a delta function at a general point x = x0, equivalently x ¼ x0 for x, within the cell containing the delta function, and X =X0 (recalling that X ¼ Ex). The delta function at x= x0 is equivalent to demanding that u is continuous at x0 and ux(X0R)  ux(X0L) =1. We will use a subscript R,L to denote values to the right, left of the delta function. The asymptotic solution will involve the homogenized PDE for f0 (3.33) which for the string is an ODE 2

Tf 0XX þ l2 f0 ¼ 0,

ð5:15Þ 2

with solutions f0R ðXÞ ¼ expð þ ik0 XÞ, f0L ðXÞ ¼ expðik0 XÞ where k20 ¼ l2 =T. The sign of T is clearly important as it determines whether there are exponentially decaying, or propagating, solutions. 2 2 2 We recall that l ¼ l0 þ E2 l2 , and to apply the asymptotic approach we must identify l0 as the standing wave frequency 2 closest to l, we fix l2 to be either plus or minus one with the sign depending on whether we are above or below the standing wave frequency, and then E is uniquely determined. The coefficient T is given explicitly in Craster et al. (2010a), for r3 = 1, and in general is T¼

7 4l0 sinðl0 r1 Þsinðl0 r3 Þ , ðr1 sinðl0 r3 Þ 8 r3 sinðl0 r1 ÞÞðcosðl0 r3 Þ 8 cosðl0 r1 ÞÞ

ð5:16Þ

with the upper/lower signs depending whether we have a standing wave frequency for perfect in-phase or out-of-phase standing waves. The corresponding standing wave solutions, U0ð1,3Þ ðxÞ, are U0ð1Þ ðxÞ ¼

r3 sinðl0 r1 xÞ þ pcosðl0 r1 xÞ, r1

U0ð3Þ ðxÞ ¼ sinðl0 r3 xÞ þpcosðl0 r3 xÞ,

ð5:17Þ

with r3 sinðl0 r1 Þ r1 : cosðl0 r3 Þ 8 cosðl0 r1 Þ

sinðl0 r3 Þ 7 p¼

ð5:18Þ

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667

The auxiliary function V(1,3) , from (3.25), is also required with 1 V1ð1Þ ðxÞ ¼

r3 Gsinðl0 r1 xÞ, r1

V1ð3Þ ðxÞ ¼ Gsinðl0 r3 xÞ,

G¼ 7

2U0 ð1Þ , r3 sinðl0 r1 Þ r1

ð5:19Þ

sinðl0 r3 Þ 7

where again, the signs here are for whether the standing waves are in- or out-of-phase. Then from (3.19) and (3.24) we have uR ðxÞ ¼ B

eik0 X0 ½U0 ðxÞexpð þik0 XÞ þ E½f1R ðXÞU0 ðxÞ þik0 expð þik0 XÞ½V1 ðxÞxU0 ðxÞ, U0 ðx0 Þ

uRx ðxÞ ¼ B

ð5:20Þ

eik0 X0 ½U ðxÞexpð þ ik0 XÞ þ E½f1R ðXÞU0x ðxÞ þ ik0 expð þ ik0 XÞ½V1x ðxÞxU0x ðxÞ, U0 ðx0 Þ 0x

ð5:21Þ

where the leading order continuity of u at x0 has been assumed, a similar expression for uL and its derivative holds. At first order the continuity condition for u gives the useful result that   V1 ðx0 Þ eik0 X0 f1R ðX0 Þeik0 X0 f1L ðX0 Þ ¼ 2ik0 x0 : ð5:22Þ U0 ðx0 Þ The derivative jump is continuous at leading order, moving to the next order we obtain that   d V1  ¼ 1, 2ik0 EB dx U  0

ð5:23Þ

x ¼ x0

and thus to the asymptotic result that uðxÞ ¼ 7

Q1 ðl0 ÞU0 ðx0 Þ expðik0 jXX0 jÞU0 ðxÞ: 4ik0 El0 r1 r3 U02 ð1Þ

ð5:24Þ

The accuracy of the asymptotic technique is illustrated in Fig. 11. This figure shows the dispersion curves for the simple string, these are considerably simpler than those for a net (cf. Figs. 5–9) but still have stop-bands. Panels (b) and (c) of Fig. 11 show the solutions, displacement endpoints and asymptotics for a frequency in the stop-band. The chosen frequency is in a band set by the intercepts with the right-hand side of the Brillouin zone and therefore the standing wave solutions are locally out-of-phase from one cell to the next; this behaviour is clearly evident from the displacement endpoints shown in Fig. 11(b, c). The parameter E is chosen larger in panel (c) to emphasise any offset between exact and asymptotic solutions.

6 4 2 0 0

0.5

1

1.5

2

2.5

3

4

u

2 0 –2 –4 –25

–20

–15

–10

–5

0

5

10

15

20

25

–20

–15

–10

–5

0

5

10

15

20

25

2 1 0 –1 –2 –25

Fig. 11. The response for a delta function forcing at x =  1. Panel (a) shows the dispersion curves for r1 = 1/4, r3 = 1 and the star shows the position l ¼ 1:711. 2 Panel (b) shows u for a frequency in the second stop-band l ¼ 1:7112 þ E2 with E ¼ 0:1. The exact u is the solid line with the end-point displacements shown as circles. The dashed line is the asymptotic solution. Panel (c) shows the same functions, but with larger E: E ¼ 0:25.

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5.3. Alternative viewpoint A complementary approach to that of Section 5.1, that is appealing in one-dimension, is to utilize Floquet’s theorem (McLachlan, 1964). The exact solution deduced this way then directly connects with the asymptotic solution given in (5.24). Taking the localized forcing as dðxx0 Þ then we can present the displacement u in the form u þ ¼ C þ eik1 x P þ ðx, lÞ

for x 4x0 ,

u ¼ C eik1 x P ðx, lÞ

for x o x0 ,

ð5:25Þ

which follows from Floquet’s theorem. The functions P 7 are periodic functions defined on 1 rx r 1 as P 7 ¼ Pð1Þ 7 for 0 ox r 1 and P 7 ¼ Pð3Þ 7 for 1 ox r 0, which satisfy the differential equation 2

ð1,3Þ 2 2 ð1,3Þ P ð1,3Þ 7 xx 7 2ik1 P 7 x þðl r1,3 k1 ÞP 7 ¼ 0,

ð5:26Þ

ð3Þ ð1Þ ð3Þ and the periodicity conditions P ð1Þ 7 jx ¼ 1 ¼ P 7 jx ¼ 1 , P 7 x jx ¼ 1 ¼ P 7 x jx ¼ 1 . The solution can be written as   8 ik1 x r3 8 ik 1 x sinðlr1 xÞ þ p 7 cosðlr1 xÞ , P ð3Þ ½sinðlr3 xÞ þ p 7 cosðlr3 xÞ, P ð1Þ 7 ¼e 7 ¼e r1

ð5:27Þ

where we have replaced x with x, to make the relation to the asymptotic solution more transparent, and r3 sinðlr1 Þ r1 ¼ 7 2ik ; 1 cosðlr Þcosðlr Þ e 3 1 e 7 2ik1 sinðlr3 Þ þ

p7

ð5:28Þ

the dispersion relation S1 ðl, k1 Þ ¼ 0, S1 is given by (2.9), sets k1 in terms of l. Now to satisfy the two conditions at x = x0, continuity of u and the discontinuity in the first derivative, we assume that at x = x0, where x0 ¼ 2n þ x0 , 1 r x0 r1, we have P 7 jx ¼ x0 a0. Then we obtain C7 ¼

P þ jx ¼ x0 P jx ¼ x0 e 7 ik1 x0 Q1 ðlÞ : 2ilr1 r3 sinð2k1 Þ P þ jx ¼ 1 P jx ¼ 1 P 7 jx ¼ x0

ð5:29Þ

Thus the solution is u7 ¼

Q1 ðlÞP 8 jx ¼ x0 e 7 ik1 ðxx0 Þ P 7 ðx, lÞ, 2ilr1 r3 sinð2k1 ÞP 7 jx ¼ 1 P 8 jx ¼ 1

ð5:30Þ 2

2

2

where P 7 are given by (5.27), and this then reduces to (5.24) in the limit as l -l0 þ E2 l2 . 5.4. Asymptotic solution: two-dimensions The two-dimensional asymptotics are naturally harder. In particular, the issue where the microscale x, Z is semi-discrete and the macroscale X,Y is fully in the continuum setting comes to the fore. We specialize to the case where there is a forcing localized at the junction y= 1 on the string x= 0. The jump condition is ð2Þ uð4Þ y jy ¼ 1 þ uy jy ¼ 1 ¼ 1

at x ¼ 0,

or equivalently uy jy ¼ 1 þ uy jy ¼ 1 ¼ dðxÞ:

ð5:31Þ

The first expression has assumed the discrete nature of the problem, cf. (3.7), the second expression is in a sort of a continuum setting and the delta function is introduced to make it clear that the forcing is concentrated precisely on the string at x =0. The second expression (5.31) can be rewritten in the double-scaled form as ð4Þ ð2Þ ð2Þ ½ðuð4Þ Z þ EuY Þjy ¼ 1 þ ðuZ þ EuY Þjy ¼ 1 dðxÞ ¼ EdðXÞ,

ð5:32Þ

where uð2,4Þ ¼ uð2,4Þ ðX,Y, ZÞ. On integrating both side of this equation over a period in x (1 r x r1) we get a continuous condition in the macroscale X, ð4Þ ð2Þ ð2Þ ðuð4Þ Z þ EuY Þjy ¼ 1 þ ðuZ þ EuY Þjy ¼ 1 ¼ 2EdðXÞ:

ð5:33Þ

Below for brevity we consider strings such that r1 =r2 and r3 =r4, that is the case shown in Fig. 7; here T11 = T22 when we have frequencies close to the periodic–periodic standing waves, and so the homogenized Eq. (3.33) is isotropic. Next we use (continuous) Fourier transforms in the macroscale, that is, for f0(X,Y) Z 1 Z 1 1 f~ ða,YÞeiaX da, f~ 0 ða,YÞ ¼ f0 ðX,YÞeiaX dX, with inverse f0 ðX,YÞ ¼ ð5:34Þ 2p 1 0 1 and similarly for the other quantities. The asymptotics proceed by using Fourier transforms, in X, of uð2,4Þ ðX,Y, ZÞ where u~ ð2,4Þ ða,Y, ZÞ  AðaÞ½f~ 0 ða,YÞU0ð2,4Þ ðZÞ þ E½f~ 1 ða,YÞU0ð2,4Þ ðZÞ þ f~ 0Y ða,YÞ½V1ð2,4Þ ðZÞZU0ð2,4Þ ðZÞ,

ð5:35Þ

ð2,4Þ ð2,4Þ ð2,4Þ ~ ~ u~ ð2,4Þ þ Eu~ ð2,4Þ  AðaÞ½f~ 0 ða,YÞU0ð2,4Þ Z Y Z ðZÞ þ E½f 1 ða,YÞU0Z ðZÞ þ f 0Y ða,YÞ½V1Z ðZÞZU0Z ðZÞ:

ð5:36Þ

and

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2 Assume that l2 =T11 o 0. Then function f~ 0 ða,YÞ satisfies the equation

f~ 0YY ða2 þk20 Þf~ 0 ¼ 0, ð5:37Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 where k0 ¼ jl2 =T11 j. We are looking for a continuous function, differentiable everywhere except Y ¼ E (y= 0). Using the decay conditions at infinity the solution can be written as  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  f~ 0 ða,YÞ ¼ exp  a2 þk20 jYEj :

ð5:38Þ

One then uses the same sequence of ideas as in the one-dimensional case, the first order continuity condition at y= 1 gives ~ ð2Þ ~ ð4Þ ~ ð2Þ an expression for the jump in f~ 1 at y= 1 which when used in the derivative jump, ðu~ ð4Þ Z þ Eu Y Þjy ¼ 1 þ ðu Z þ Eu Y Þjy ¼ 1 ¼ 2E, gives AðaÞ, B AðaÞ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi , 2 a þk20

with

B¼

r2 sinðl0 r4 Þ þ r4 sinðl0 r2 Þ : 2l0 r2 r4

ð5:39Þ

The leading order solution is then  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðiÞ Z 1 exp  a2 þk2 jYEj cosðaXÞ BU 0 ðiÞ 0 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u  da, p 0 a2 þ k20

ð5:40Þ

and so (see Gradshteyn and Ryzhik, 2007 p. 491) 8 < U ð1,3Þ ðxÞ, for i ¼ 1,3, B 0 ðiÞ u ðx,yÞ  K0 ðk0 RÞ : U0ð2,4Þ ðZÞ, for i ¼ 2,4, p where R ¼

ð5:41Þ

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 X 2 þðYEÞ2 is the radial distance and K0 is the modified Bessel function which decays at infinity. If l2 =T11 4 0 (1)

then K0(k0R) is replaced by ðpi=2ÞH0ð1Þ ðk0 RÞ, where H0 is a Hankel function, Abramowitz and Stegun (1964, p. 358), which (1) corresponds to outgoing cylindrical waves. The K0 and H0 solutions occur for frequencies in the stop- and pass-bands, respectively.

0.2

u

0.1 0 –0.1 –0.2 –15

–10

–5

0 x

5

10

15

–10

–5

0 y

5

10

15

0.3 0.2

u

0.1 0 –0.1 –0.2 –0.3 –15

Fig. 12. The displacement of two strings (with the same parameter values as in Fig. 13). Panel (a) shows the displacement along the string lying in y =2, that is, in the x direction just above the forcing. Panel (b) shows the worst case scenario, the string along x= 0, in the y direction. In both panels the exact solution is the solid line, the dots are the displacements at the endpoints, and the dashed lines are from the asymptotic solution.

670

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s in x

String

directi

on

0.2 0.1 0 –0.1 –0.2

20 15 10 5 y 0 –5 –10 –15 –20

–15

–10

–5

0

5

10

15

20

x

Fig. 13. The displacement of the strings in the x direction for the net with r1 = r2 = 1, r3 = r4 = 2 and forcing of frequency l ¼ 1:9132 at x= 0, y= 1.

The solution has factored the macroscale behaviour into the Bessel function and the localized short-scale fine oscillations into the functions U(i) 0 in accordance with our intuition. The application of our long-wave model means that we formally only extract the far-field asymptotic structure. This is reflected in the solution, where notably the Bessel functions have a logarithmic singularity as R-0, suggesting the asymptotic method indeed breaks down within a region of order E in X (order unity in x) of the forcing; in practical terms this only effects the string that passes exactly through the delta function forcing (see Fig. 12(b)). A sample solution is shown in Fig. 13 for r1 = r2 = 1, r3 = r4 =2, that is for the values used in Fig. 7, and for a frequency qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi l ¼ l20 þ E2 where l0 ¼ 1:9106 and E ¼ 0:1. This is a frequency lying in the second stop-band close to a standing wave frequency corresponding to a periodic–periodic case. For clarity, Fig. 13 shows the displacement of only the strings in the x direction; the displacement near the forcing is naturally large with the oscillations dying away as one moves away from it. To illustrate the accuracy of the asymptotics we choose two strings, close to the forcing, and compare them to the exact solution in Fig. 12; Again it is notable that the asymptotic solution performs almost perfectly; importantly the decay behaviour is accurately captured and is available exactly in terms of T11 through k0. Panel (a) shows a string in the x direction lying just above the forcing; there is a minor discrepancy near x= 0, however, the vast majority of the behaviour is clearly captured. The displacement along the string that contains the forcing lies along x =0 (the delta function forcing is at y= 1) and is shown in panel (b). There is a discrepancy in a very small neighbourhood of the forcing, the asymptotic solution is logarithmically singular there whilst the exact solution remains finite, however, away from that region again the accuracy is admirable. Both panels show the end-point displacements which oscillate such that they are almost perfectly in-phase as one moves from one cell to the next; this is in line with the implicit assumption that we are close to a standing wave frequency with those features.

6. Discussion This article presents an homogenization theory that accurately captures the behaviour of a frame structure which vibrates at high frequencies, this then allows one to completely break free of the low frequency assumption that is implicit within classical homogenization theory. This now allows rapid calculation of localization behaviour, or as demonstrated here (in Section 5) the spatial decay of displacements if the forcing frequency is within the stop-band. Although illustrated in detail for the square net, and for a net of strings, similar ideas can be applied to other regular patterns (with more complicated algebra), to truss structures with beams rather than strings, or to three-dimensions. These extensions to the theory will be valuable, and are underway. As discussed in Craster et al. (2010a) the homogenization procedure, as developed here, is the counterpart of the longwave high frequency limit in the asymptotic theory of thin walled wave-guides. In the current situation the PDE (3.33) is identical to that governing the long-wave motion of an anisotropic elastic plate in the vicinity of thickness resonances (Kaplunov et al., 2000). At the same time the difference between the Bloch and Rayleigh–Lamb dispersion spectra affects

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the dynamic response caused by point loading. In particular, in the one-dimensional setup, in contrast to the results presented in Fig. 11, the solution of the analogous problem for an elastic strip is characterized by an extra boundary layer localized near the point force and also may contain a small amplitude propagating short-wave component (e.g. see Kaplunov and Markushevich, 1993). Although the main thrust of this article is toward developing a comprehensive and versatile homogenization theory, that will bypass lengthy computations with, say, finite elements, for huge microstructural cellular solids, it is important to note that we have developed an exact solution for frame structures using Fourier transforms (in Section 5); this approach can be used to generate exact Green’s functions for very complex frames and it is anticipated that this will be useful for numerical schemes such as the boundary integral method. Bloch waves, on infinite and perfect frames, were used as a vehicle to explain and present the asymptotic theory, however, these too are of interest in their own right. It is clear that specific string constants ri can be chosen to generate stop-bands at desired frequencies, or to create dispersion curves with zero, or very low, group velocities, and these properties are valuable in designing smart structures. We anticipate that high frequency homogenization will allow for the straightforward examination of the dispersion and vibration properties of many complex microstructures, at frequencies that were previously inaccessible except via largescale numerical simulation, and issues such as surface waves on lattice/frame structures can now be examined either exactly or through the asymptotic technique. It is notable that the two-scales approach, at high frequencies, employed here has been successful for continua, Craster et al. (2010a), and for completely discrete systems, Craster et al. (2010b), and it appears that the methodology is very general in its applicability.

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