## Abstract

This paper presents asymptotic and numerical modelling of elastic waves interacting with micro-structured solids containing multi-scale resonators. The resonators may be damaged by transverse cracks at their foundations. One particular feature of the resonators used here is the presence of low-frequency eigenmodes. Asymptotic estimates of low eigenfrequencies are included in the model. Hence, the overall periodic structure containing such resonators may support low-frequency standing waves and possess low-frequency stop bands. The analytical model is accompanied by numerical simulations illustrating dispersion, localization and focusing within the structured multi-scale solids.

## 1. Introduction

Waves in discrete lattice structures give a simple and efficient framework for the analysis of dynamic response of micro-structured solids. Fundamental work on dynamic discrete structures was published by Slepyan (2002), whose work also included the models of fracture and dissipation within structured media with discrete systems of bonds.

The discrete models have limitations, and they are efficient for low-frequency ranges. The asymptotic models combining discrete approximations with the analysis of continuous media were analysed in Jones *et al.* (2008). Boundary layer solutions were used to derive effective boundary conditions (junction conditions) for thin-walled solids with transverse cracks. Furthermore, these model solutions were used in the analysis of Bloch–Floquet waves in periodic media, in addition to standing waves and localization in structured solids with defects. For problems of vector elasticity, effective junction conditions for thin solids with cracks were studied in Gei *et al.* (2009), where both bending and tensile types of loads were considered for a junction region containing a transverse crack.

For flexural waves interacting with perforated plate structures, analytical and numerical modelling of apparent negative refraction was presented in Farhat *et al.* (2010). This also includes focusing of flexural waves by a structured interface. For a scalar formulation involving the Helmholtz equation, a thorough study of negative refraction and the corresponding analysis of dispersion properties of Bloch–Floquet waves in doubly periodic structures was published in McPhedran *et al.* (2004). For scalar problems of acoustics, which reduce to analysis of solutions of the Helmholtz equation, defect modes and focusing effects are discussed in Guenneau *et al.* (2007).

In the present paper, we extend the work presented in Gei *et al.* (2009) to the case when a continuum periodic structure contains multi-scale resonators, of the same nature as in Jones & Movchan (2007a,b). Analytical asymptotic analysis is applied to individual resonators, whereas the numerical study is performed for Bloch–Floquet waves in an infinite periodic structure.

Asymptotic models of multi-scale resonators lead to explicit analytical approximations of the first few eigenfrequencies and the corresponding eigenmodes. Furthermore, by introducing damage into the resonators, we estimate the change in the eigenfrequencies, and we outline the derivation of effective junction conditions for damaged resonators attached to the background elastic frame.

The analytical work is complemented by numerical simulations, which illustrate specially designed properties of micro-structured solids, such as apparent negative refraction, focusing by a flat-structured interface and suppression of lateral vibrations of a long micro-structured solid.

The structure of the paper can be described as follows. Section 2 contains asymptotic analysis of thin-legged resonators with cracks. Explicit asymptotic approximations of low-frequency vibration modes are derived for anti-plane shear, with cracked regions being replaced by ‘spring-like’ junction conditions. Furthermore, analysis of dispersion diagrams for Bloch–Floquet waves, localization and focusing in multi-scale periodic structures are presented in detail.

Filtering of anti-plane shear waves by a structured interface is analysed in §3. A certain range of frequencies is determined so that the structure possesses the property of apparent negative refraction. Both undamaged and damaged resonators are considered. It is shown that significant damage of resonator legs leads to a marginal reduction in frequencies corresponding to apparent negative refraction.

In §4, the models of damage for the multi-scale resonators have been extended to the vector case of plane strain elasticity. Compared with the scalar case, we have flexural and tensile modes of vibration around the junction region with a transverse crack. The analytical approximations of eigenfrequencies of standing waves within the micro-structured media are accompanied by the numerical study of dispersion properties of Bloch–Floquet waves. The asymptotic prediction for the fundamental modes of the multi-scale resonators agree well with the results of independent finite-element simulations. In §5, the analytical and numerical results of the paper are consolidated into a class of important engineering applications. This involves the demonstration of unusual transmission properties of a micro-structured elastic slab with embedded resonators. Within a selected frequency range, such a slab may lead to the focusing of an incident wave generated by a point source, despite the slab being flat. Finally, concluding remarks are included in §6.

## 2. Anti-plane vibration of a periodic structure with multi-scale resonators

Consider a periodic structure with the elementary cell shown in figure 1*a*. The cell incorporates a resonator consisting of a relatively large solid *Ω*_{1} and thin rods *Π*^{(j)} (*j* = 1,…,*N*), of equal length *l*, with *N* being a given positive integer. The thin rods are attached to the surrounding frame *Ω*_{2}. The cell of periodicity is denoted by . The interior boundary within the elementary cell is denoted by *Γ*, whereas *γ* represents the exterior boundary of the cell. Each of the thin rods *Π*_{1},…,*Π*_{N} has been damaged at its point of attachment to the frame. For Bloch–Floquet waves in such a structure, one can construct a dispersion diagram and show the presence of low-frequency standing waves.

### (a) Asymptotic approximations for the lowest eigenfrequency

Here, we give an asymptotic estimate of the lowest frequency standing wave and relate this to the compliance of the thin rods. Let *u* denote the amplitude of time-harmonic vibrations of radian frequency *ω*. Assume that such a vibration corresponds to a standing wave, i.e. a wave of zero group velocity. For such a mode, the resonator within the elementary cell will be moving with an amplitude much larger than that of the surrounding frame and hence, for a simple estimate, we can assume that the amplitude of the frame is negligibly small.

As shown in Kozlov *et al.* (1999), the leading term of the displacement amplitude in *Ω*_{1} is constant, say *A*. Assuming that the inertia of the thin rods can be neglected, the displacement amplitudes *u*_{j}(*x*_{1}) within *Π*_{j} can be approximated as linear functions of the local longitudinal variable *x*_{1}
2.1Ideal contact between a thin rod *Π*_{j} and the body *Ω* implies the continuity of the displacement, i.e.
2.2whereas at the damaged foundation of the thin rod, we have the spring-like Robin boundary condition:
2.3where *α*_{j} represents a measure of the effective stiffness of the damaged junction region. Hence, the function *u*_{j} takes the form
2.4Furthermore, the equation of motion for the large solid *Ω*_{1} of mass *M*, subjected to the forces *F*_{j} exerted by thin rods *Π*_{j} (*j* = 1,…,*N*) becomes
2.5where
2.6with *c*_{j} being the stiffness of the undamaged rod at the junction region connecting *Π*_{j} and *Ω*_{1}.

For the sake of simplicity, we assume that all undamaged rods are identical, i.e. *c*_{j} = *c*(*j* = 1,…,*N*). Then it follows from equations (2.4), (2.5) and (2.6) that the lowest eigenfrequency (the superscript ‘ap’ indicates ‘anti-plane’) for a standing wave is approximated by
2.7where *β*_{j} = (*α*_{j}*l*)^{−1} is the effective dimensionless compliance of the *j*th damaged junction.

In addition, we introduce the constraint on the overall effective stiffness of spring-like junctions at the foundations of the thin rods 2.8which implies 2.9

We will now show that the asymptotic approximation (2.7) with the constraint (2.8) leads to the conclusion that any non-uniformity in the distribution of damage between the rods will lower the first eigenfrequency of the structure shown in figure 1*b*, and correspondingly will reduce the frequency of the first standing wave mode for the periodic system in figure 1*a*.

This can be verified in a standard way. Namely, let
subject to the constraint
By considering
where *λ* is the Lagrange multiplier, we identify the extremal configuration by
Eliminating *λ* between the *i*th and the *j*th equation (*i* ≠ *j*), we deduce that
which yields, for the case of positive *β*_{j}, that
2.10In turn, the maximum value of the function *f* is

A particularly simple illustration is related to the case when *N* = 2. In this situation,
where *β*_{1} > 1/*p*. The graph of versus *β*_{1} is given in figure 2, and, as predicted, it takes its maximum value of 2*p*/(2 + *p*) at *β*_{1} = 2/*p*.

We note that for any admissible *β*_{1},*β*_{2}
2.11which delivers both the upper and the lower bounds for the first eigenfrequency of the two-legged resonator. This is also illustrated by the diagram in figure 2, where the limit corresponds to the case when leg 1 is broken, whereas the limit *β*_{1} → 1/*p* corresponds to the situation when leg 2 is broken (i.e. ).

We note that the estimate (2.11) can be generalized to the case of an *N*-legged resonator as follows:
2.12Namely, we have shown that there is only one interior point of extremum, defined by equation (2.10). The variable *β*_{N} can be eliminated by the constraint (2.9). Then, the infimum of across the admissible values of (*β*_{1},…,*β*_{N − 1}) can be achieved either at infinity (which means that all the legs 1,2,…,*N* − 1 are completely broken and the limit for gives (*c*/*Ml*)(*p*/(1 + *p*)) or along the curve . In turn, it follows by induction along such a curve is greater than or equal to (*c*/*Ml*)(*p*/(1 + *p*)), as required.

It is also noted that, for a fixed value of the constant *p*, the upper bound in equation (2.12) increases monotonically with increase in *N*.

We summarize by saying that the upper bound for the first eigenfrequency corresponds to the case when the damage is distributed uniformly across all legs within the resonator, and the lower bound corresponds to the situation when all the legs except for one are completely broken. The maximum value of increases monotonically when we increase *N*, provided *p* remains fixed.

### (b) Dispersion diagrams

We consider anti-plane shear Bloch waves in the periodic structure with an elementary cell *Ω* shown in figure 1*a*. Using the notation *u*(**x**) for the anti-plane displacement amplitude, we set the standard spectral problem for the Bloch–Floquet fields by imposing the Helmholtz equation
2.13with homogenous traction boundary conditions on the interior boundary
2.14and the Bloch–Floquet quasi-periodicity conditions on the exterior boundary of the elementary cell:
2.15where **e**_{1},**e**_{2} are the basis unit vectors of the Cartesian coordinate system, *d* is the length of the elementary cell of the doubly periodic structure and **K** = (*K*_{1},*K*_{2}) is the Bloch vector.

We solve the spectral problem numerically; given the components of the Bloch vector **K**, we determine the radian frequency *ω*. This is done using the finite-element package COMSOL. It is assumed that **K** characterizes the positions of points within the first Brillouin zone shown in figure 4*b* below. Using the notation *ω*_{j}(*K*_{1},*K*_{2}) (*j* = 1,2,…) for the corresponding eigenfrequencies, the surfaces *ω* = *ω*_{j}(*K*_{1},*K*_{2}) are plotted in figure 3*a*. There is evidence of standing waves which may be expected to exist within such a periodic structure. There is a low-frequency band gap between the first two dispersion surfaces. This is more clearly shown in figure 3*b* where the first two low-frequency surfaces are shown.

As *ω* → 0, we can apply long-wave homogenization and consider an effective medium as isotropic for the present case of a square elementary cell. As the frequency increases, the presence of built-in resonators becomes pronounced, and the group velocity for the waves propagating through the structured medium depends on the direction. For the low-frequency regime, we consider the first two dispersion surfaces (referred to as acoustic and optical surfaces), shown in figure 3*b*.

#### (i) The sectional dispersion diagram and low-frequency modes

Since it may be easier to work with sectional plots, we have included these in figure 4*a*, which shows dispersion curves along the segments *LN*, *NM* and *LM* of the Brillouin zone shown in figure 4*b*. We compare the sectional dispersion diagram of figure 4*a* with the dispersion diagram constructed for a frame without resonators, shown in figure 4*c*. We can see that the main changes are in the low-frequency range, where an additional dispersion curve appears accompanied by the low-frequency stop band.

Low-frequency standing waves near the boundaries of the band gap appear to be important in a range of applications. One of the examples concerns the interaction of an incident wave with a finite-width slab consisting of vertical arrays of multi-scale resonators as shown in figure 7 below. For a specially chosen frequency of the incident wave, we observe the effect of apparent negative refraction and consequently show that the structured interface focuses the wave produced by a localized source. Although such an effect is counterintuitive, for scalar problems of electro-magnetism and finite-width slabs consisting of circular dielectric inclusions, it has been observed and discussed in an earlier paper (McPhedran *et al.* 2004).

Here, we illustrate the influence of the multi-scale resonators, and later address a similar configuration for a full vector problem of elasticity (§4).

#### (ii) The group velocity as a vector function of Bloch vector components for the acoustic dispersion surface

We plot the acoustic dispersion surface in figure 5*a*. Although it coincides with the acoustic surface of figure 3*b*, we choose a different orientation in order to emphasize that the points (*K*_{1}*d*,*K*_{2}*d*) = (0,*π*) and (*K*_{1}*d*,*K*_{2}*d*) = (*π*,0) are saddle points, whereas (*K*_{1}*d*,*K*_{2}*d*) = (*π*,*π*) is a maximum point for the acoustic dispersion surface extended to all admissible values of (*K*_{1}*d*,*K*_{2}*d*). This is a common feature, noted for example in McPhedran *et al.* (2004) (in particular section C and fig. 7) and Callaway (1974). Compared with McPhedran *et al.* (2004), where the periodic structure includes a square array of circular holes and the acoustic dispersion surface is symmetric with respect to *K*_{1} = *K*_{2}, we do not have equivalence of the directions *K*_{1} = 0 and *K*_{2} = 0 since our structure includes a different type of resonator with a well-defined orientation. The effect of the directional preference is especially pronounced for the problem of vector elasticity discussed in the text below. For the current situation involving the scalar Helmholtz equation, the directional preference is visible near the edges of the Brillouin zone. In particular, figure 5*a* for the acoustic dispersion surface shows that *ω*(*π*,0) < *ω*(0,*π*) < *ω*(*π*,*π*), which correspond to the frequency values at the corner points of the Brillouin zone. Obviously, the group velocity is equal to zero at these stationary points, and the corresponding vibration modes represent standing waves within the periodic structure. In particular, the frequency *ω*(*π*,*π*) on the acoustic dispersion surface is approximated by the first eigenfrequency of the thin-legged resonator estimated in §2*a*. We also note that *ω*(*π*,*π*) determines the lower bound of the first stop band on the dispersion diagram (figure 5*a*), and hence the position of the first stop band can be estimated analytically via the first eigenfrequency of a thin-legged resonator derived in §2*a*. From equation (2.7) with *β* = 0, the value of the asymptotic estimate for the steel resonator vibrating in this mode is 960 Hz. At point L in figure 4*a*, the first non-zero eigenfrequency (where the resonator is vibrating in this mode) is 897 Hz, showing good agreement. The resonator moves in this way in the standing waves on either side of the band gap. The difference between these standing waves is in the motion of the frame. The asymptotic estimate is closer to the standing wave on the higher side of the band gap because there is less motion of the frame in this mode than in that on the lower side.

The computations, produced in §3 for interaction of waves with a finite-thickness structured slab, refer to a frequency close to 700 Hz, which is in the neighbourhood of the saddle point corresponding to (*K*_{1}*d*,*K*_{2}*d*) = (0,*π*). For such a frequency, the oblique direction is ‘preferable’ compared with the direction of the *K*_{1} axis. The magnitude of the group velocity |**∇***ω*(**K**)| at all points of the acoustic dispersion surface is shown in figure 6, which indicates that the largest magnitude is attained at the origin, whereas the zero values of the group velocity are linked to the corner points of the Brillouin zone. For the purpose of illustration, we also include in figure 5*b* the contour diagram for the first dispersion surface. Five level curves between 680 and 720 Hz are highlighted as these indicate the variation of the group velocity within the Brillouin zone for frequencies located near the saddle point.

The implication of such a choice of frequency is the phenomenon of apparent negative refraction in the problem of interaction between the point source of waves and a finite-thickness structured slab. From the current analysis, it follows that wave propagation is slowed down in the direction of the *K*_{1} axis compared with oblique directions. This in turn results in a phase shift of the waves arriving at the right-hand boundary of the slab. The phase difference between ‘secondary sources’ on the right-hand boundary of the slab leads to the effect of apparent negative refraction, as described in the §3.

## 3. Filtering versus dispersion properties of anti-plane shear Bloch–Floquet waves

In this section, we switch attention from an infinite periodic structure to a finite-thickness structured interface. Strictly, a wave generated by a point source and then interacting with the structured slab is not a Bloch–Floquet wave. However, as discussed in the earlier work by Lekner (1987), the analysis of Bloch–Floquet waves can be used in qualitative description of wave propagation through finite-size structures. The analysis of dispersion relations for Bloch–Floquet waves proves to be particularly useful in the present case of an anti-plane shear wave interacting with a finite-width slab, which contains an array of multi-scale resonators described in §2. Below, we consider and compare the cases of slabs where resonators are damaged and undamaged.

### (a) Undamaged interface

Here, we consider an illustration of an interaction between a finite-width structured slab, containing built-in undamaged resonators, and a wave generated by a localized source placed at a close distance to the interface boundary. The configuration used in the computational model is shown in figure 7*a*. We also adopt perfectly matched layers (PMLs) at the exterior boundary, apart from the slab, to suppress reflection from the boundary of the computational window. In figure 7, these are shown as layers adjacent to the domains and are tuned accordingly to reduce significantly reflections from the boundary.

The frequency chosen here is 720 Hz, which is close to the lower edge of the band gap between the acoustic and optical dispersion surfaces. For the horizontal direction, across the slab, the group velocity of the waves is small, which is consistent with the computation produced in figure 4*a* for Bloch–Floquet waves in an infinite structure. On the other hand, the group velocity in the oblique direction is positive, which enables the waves to propagate, as seen in figure 7*a*. This anisotropy results in the phase shift between the waves reaching the other side of the structured slab. The result is the apparent negative refraction and focusing of the image of the source clearly seen on the right-hand side from the structured slab.

### (b) Damaged interface

A similar simulation (figure 7*b*) has been performed for an interface with all resonators possessing damaged legs, with transverse cracks, namely with the zone connecting legs and the frame (figure 1*a*) equal to 5 per cent of the total width of each leg. Such a change of geometry results in a small change of dispersion curves at low frequencies. Otherwise, the dispersion diagram for the corresponding infinite structure is very similar to that for the undamaged case, figure 4*a*, with the change at low frequencies being of order 20–30 Hz. As a result, a focusing effect, although not identical to figure 7*a*, is still present at a slightly reduced frequency of 694 Hz. This finding of very little change when the resonators are damaged could be considered surprising. However, it can be explained by recalling that, for the anti-plane problem (Zalipaev *et al.* 2007), the damaged junction performs similar to an undamaged junction unless the size of the uncracked ligament is exponentially small compared with the thickness of the thin leg of the resonator.

## 4. The vector problem of plain strain

We now consider the case of plain strain in the same infinite, periodic elastic structure, whose elementary cell is shown in figure 1*a*. We will concentrate on resonators with three legs only. The elastic displacement is a time-harmonic vector function, whose amplitude **u**(**x**) satisfies the following equations:
4.1and
4.2where **u** = (*u*_{1},*u*_{2}), and *σ*^{(n)} is the vector of tractions with entries with *σ*_{jk} being the components of the Cauchy stress tensor and (*n*_{1},*n*_{2}) being the outward unit normal on the interior boundary *Γ*.

The Bloch–Floquet quasi-periodicity conditions are also imposed on the vector function **u**(**x**) as follows:
4.3similarly to those imposed in equation (2.15).

### (a) Asymptotic approximations for the fundamental translational and rotational modes

Two fundamental eigenmodes are shown in figure 8. In figure 8*a*, a translational eigenmode is shown corresponding to a standing wave within the periodic structure such that the subdomain *Ω*_{1} (*Ω*_{1} = {(*x*_{1},*x*_{2}):*l* < *x*_{1} < *l* + *a*,|*x*_{2}| < *b*/2}) moves in the *x*_{2} direction, with a low frequency. The other interesting vibration mode is a rotational mode, shown in figure 8*b*, also corresponding to a standing wave within the periodic structure. For the case when the junction between the leg and the frame is undamaged, asymptotic estimates of the frequencies of these two modes have been given in Jones & Movchan (2007*b*). These estimates were based on the observation that the subdomains *Ω*_{1} and *Ω*_{2} move like rigid solids (figure 8). Further, the legs were assumed to be thin and their inertia was neglected. Full details are given in Jones & Movchan (2007*b*).

For the case when the junction of the leg and the frame is damaged by two edge cracks perpendicular to the main axis of the leg (figure 9), effective spring-like junction conditions have been derived for the cases of longitudinal displacement loading (Zalipaev *et al.* 2007) and bending in Gei *et al.* (2009) and Ostachowicz & Krawczuk (1991).

For the case of longitudinal loading, the effective boundary condition is
4.4where is the first-order approximation to *u*_{1}, the longitudinal displacement of leg. The leg thickness is *h* and the ligament length is *δ*. This is valid for situations of large damage with a small remaining ligament, i.e. small values of *δ*/*h*. The relation (4.4) also defines the first-order approximation of the effective compliance *k*_{l} of the junction region incorporating two cracks and a thin bridge. We note that in this first-order approximation, the ‘spring effect’ becomes significant when the thickness of the bridge is exponentially small. This implies that for a finite-size bridge, the second term in the left-hand side of equation (4.4) is small, and this relation can be replaced by the homogeneous boundary condition of the Dirichlet type.

For the case of bending, the model in Gei *et al.* (2009) is accurate for large damage situations and that in Ostachowicz & Krawczuk (1991) is accurate for small damage with a large remaining ligament. This is discussed in Gei *et al.* (2009) and is summarized as
4.5where is the first-order approximation to *u*_{2}, the transverse displacement of leg.

The ‘spring constant’ *k*_{b} is given by
4.6where the function *q*(*α*) is
4.7and *ν* is Poisson’s ratio of a leg.

These conditions may be used for the derivations of the translational and rotational eigenfrequencies in the case of damaged junctions. For the translational mode, the legs are predominantly loaded in bending. The derivation follows closely that given in Jones & Movchan (2007*b*) for the undamaged resonator and for the anti-plane shear situation derived in §2. However, the undamaged junction boundary condition is replaced by the boundary condition (4.5), and this leads to an asymptotic estimate for the translational (*ω*_{t}) mode radian frequency as:
4.8where *ρ*_{1} and *ρ*_{2} are the densities of *Ω*_{1} and *Ω*_{2}, respectively, while 𝒜(*Ω*_{2}) corresponds to the area of the frame *Ω*_{2} that has been taken into account here.

For the rotational mode, the rotation of *Ω*_{1} is caused predominantly by longitudinal motion of the legs (Jones & Movchan 2007*b*). The derivation for the damaged junction in this case follows closely that in Jones & Movchan (2007*b*) except that the boundary condition (4.4) is used. This leads to an asymptotic estimate for the rotational (*ω*_{r}) mode radian frequency as:
4.9where 2*d* is the distance between the outer two legs.

The approximations (4.8) and (4.9) have been compared with finite-element simulations using the COMSOL finite-element package. A frequency response analysis has been carried out for a cell of unit extent and arm width of 0.1 m containing the resonator (as in figure 1*a*) in the case of them both being made up of AISI steel 4340 (*E* = 205 GPa, *ν* = 0.28, *ρ*_{1} = *ρ*_{2} = 7850 kg m^{−3}). The other parameter values are *l* = 0.2 m, *a* = 0.14 m, *b* = 0.3 m and *d* = 0.06 m. The comparisons are shown for the translational mode in figure 10*a* where the frequency is plotted as a function of damage measured by the variable *δ*/*h*. Two sets of comparisons are shown for *h* = 0.01 m and *h* = 0.005 m, representing width to length values of the legs as 1 : 20 and 1 : 40, respectively.

It can be seen that the degree of agreement between the asymptotic approximation and the finite-element computation is better for smaller values of *h*/*l*, as is expected. Corresponding results for the rotational mode are shown in figure 10*b*.

### (b) Dispersion diagrams

The dispersion diagram, showing the first six eigenfrequencies of the undamaged periodic structure discussed above, is shown in figure 11 together with the path in the first Brillouin zone. There are clear standing waves at approximately 70, 720 and 1270 Hz, and these are significant features for all values of (*k*_{1}, *k*_{2}) in the first Brillouin zone. The first two standing waves correspond to the fundamental translational and rotational modes, respectively, discussed earlier and the third represents the large relative motion of the resonator legs. In fact, the fifth and sixth modes are close to the 1270 Hz frequency near to point ‘M’ in the Brillouin zone and the seventh mode, which is not shown, is also close to these, having an eigenfrequency of 1295 Hz at point M. The clustering of the fifth, sixth and seventh modes represents various large relative motions of the resonator legs.

## 5. Applications of thin-legged resonators in filtering and localization of vibrations

Here we consider a type of applications where dispersion properties of Bloch–Floquet waves are used to design elastic structures that filter waves within a certain frequency range. The examples in the text below involve a structured interface of finite thickness that possesses apparent negative refraction properties and hence can be used for ‘imaging’ of individual sources of waves.

Let elastic waves interact with an elastic slab (also referred to as the structured interface) shown in figures 12 and 13. The incident wave is generated by a single point source or a periodic array of point sources, in the vertical direction, placed close to the boundary of the elastic slab which is of infinite extent in the vertical direction. The structure of the slab is consistent with the models of §4, i.e. it incorporates thin-legged resonators of the same dimensions and material properties in plane strain as that of §4. The displacement field is time-harmonic, and the boundary conditions on the left-hand side and on the right-hand side of the domain are modelled by PMLs.

Firstly, we consider the case when periodicity conditions are set on the upper and lower surfaces of the computational domain. The ‘source’ is modelled by a small rigid, square inclusion being moved harmonically in the vertical direction, i.e. parallel to the structured interface. In figures 12 and 13, we show the modulus of the displacement amplitude for forced frequencies of the source of 660 and 999 Hz, respectively. At this stage, we refer to the dispersion diagrams (figure 11) constructed earlier for the elementary cell with a thin-legged resonator. Although the transmission problem is formally different from the Bloch–Floquet spectral problem, we can use the dispersion diagram for prediction of the reflection/transmission pattern. Namely, for frequencies within any stop bands, the interface will reflect the substantial part of the energy of the incident wave. Here, we draw attention to the branch with negative group velocity (figure 11*a*) within the range of frequencies (800–1250 Hz). It is suggested that within this range, the wave pattern may lead to the effect of apparent negative refraction.

The image in figure 12 shows predominantly plane wave fronts with no obvious evidence of focusing. At this frequency of 600 Hz, there is no evidence of negative group velocity displayed on the dispersion diagram (figure 11) in the horizontal (LN) direction. However, the focusing effect is visible in figure 13 where the frequency of the source oscillation is within the 800–1250 Hz range.

The next numerical experiment involves a single source, rather than a periodic array of sources, interacting with the structured interface. The computational domain is shown in figures 14 and 15. The periodicity conditions are maintained on the upper and lower boundaries of the structured interface. On the remaining parts of the horizontal boundaries, the periodicity conditions are replaced by the PMLs absorbing the outgoing waves and hence removing reflection from the boundary of the computational domain. In the physical configuration, this corresponds to a single source placed on the left from the infinitely long, structured interface. The source is moving harmonically in time in the vertical direction, i.e. parallel to the structured interface. Choosing a forcing frequency of 950 Hz, within the interval corresponding to a branch with negative group velocity on the dispersion diagram, we expect to see the focusing effect. This is confirmed by the computation, where the image of the source is clearly visible on the right side from the structured interface in figure 14.

For this particular orientation of thin-legged resonators within the interface, a simple explanation of the focusing effect can be given. At the selected frequency of 950 Hz, according to the dispersion diagram in figure 11, the group velocity for the wave propagating across the layer in the horizontal direction is smaller compared with the waves propagating in the diagonal directions. Hence, this induces a phase shift for waves initiated at the right-hand boundary. Assuming that every junction region within the interface can be considered as a source of waves emanating into the right-hand medium and coupling this with the phase shift, we arrive at apparent plane waves intersecting with each other, as shown in figure 14 (and also in figure 13 for a periodic array of sources). This replicates the effect of focusing. This effect is frequency dependent and, as the frequency is reduced to 800 Hz, we observe a narrow beam being created on the image side as shown in figure 15.

The presence of resonators with damaged legs (as in §2) can lower the frequency at which focusing occurs. This is illustrated in figure 16 where the left-hand first vertical layer of resonators has been damaged. All the legs of all of these resonators in the layer have been equally damaged at the foundation with *δ*/*h* = 0.05 (figure 9). The configuration is otherwise the same as in figure 14. The frequency at which focusing takes place has been reduced from 950 to 945 Hz.

## 6. Concluding remarks

In this paper, we have demonstrated a non-trivial connection between three classes of models involving multi-scale resonators: (a) a model for a single multi-legged resonator with damage or otherwise, (b) a model of Bloch-Floquet waves interacting with a doubly periodic structure whose elementary cell contains a multi-scale resonator, and (c) a dynamic response of a finite-width structured interface with built-in multi-scale resonators.

The first model allows for an analytical asymptotic treatment that leads to an estimate of frequencies corresponding to a low-frequency stop band for Bloch–Floquet waves in a doubly periodic structure. Furthermore, the frequency response analysis for a single elementary cell yields dispersion surfaces and accurate information about the group velocities.

Problems of optimal design for finite-size micro-structures can be addressed by treating them as waveguides and hence using the properties of Bloch–Floquet waves analysed in the class (b) of the models mentioned. Illustrations include interaction of waves with a finite-width structured interface. The apparent negative refraction is one of the interesting outcomes of the model, which has been fully explained in classical terms within the framework of the proposed approach.

## Acknowledgements

The financial support of EPSRC grant no. EP/H018514/1 and of MIUR-PRIN grant no. 2007YZ3B24 is gratefully acknowledged.

- Received June 30, 2010.
- Accepted September 10, 2010.

- This journal is © 2011 The Royal Society