## Abstract

This paper considers the interaction of elastic waves with materials with microstructure. The paper presents a mathematical model of elastic waves within a lattice system incorporating rotational motions and interaction between different lattice elements through elastic links. The waves are dispersive and the lattice system itself is heterogeneous, i.e. the elastic stiffness and/or mass are non-uniformly distributed. For such systems, one can identify stop bands, representing the intervals of frequencies of waves, which become evanescent and cannot propagate through the structure. Filtering properties of such lattices are studied in this paper. Defect modes are created by removing a periodic array of elastic links, which leads to localization within a macro-cell. Special attention is given to the evaluation of the effective group velocities and to the study of standing waves within the system. Analytical estimates are accompanied by numerical simulations and analysis of dispersion surfaces. We also consider an example showing the focusing and the creation of an *image point* by a flat elastic ‘lens’ formed from a finite micropolar lattice system.

## 1. Introduction

The emphasis of this work is on a micropolar dynamic interaction between inertial elements in a discrete lattice structure. Here, we consider a vector problem of two-dimensional elasticity, with several types of rotational interactions defined within a lattice system. The structure is heterogeneous and elastic waves are dispersive. The study of two-dimensional elastic discrete systems, accompanied by the analysis of dispersion properties of waves, was included in Martinsson & Movchan (2003). It was shown that it is possible to control the position of low-frequency stop bands by redistributing the mass across the junctions of the lattice structure. The paper by Jones & Movchan (2007) includes a model of dynamic defects within an elastic system induced by thermal pre-stress. In this case, temperature was used as a control parameter and the pre-stressed elastic system responded by changing its filtering properties with respect to the elastic waves propagating through the system. Analysis of dispersion properties of waves in periodic solids with pre-stress was also presented in Gei *et al.* (2009).

In the present paper, we show the rotational defect modes formed in the configuration where a periodic array of elastic ligaments has been removed; the latter can be created as a result of a thermal buckling within a heterogeneous system. Micropolar interactions are shown to be important for such systems.

Kunin (1982), Morozov (1984), Nazarov & Paukshto (1984) and Maz’ya *et al.* (2000) have studied two-dimensional static lattices and introduced a potential of torsional interaction between elastic ligaments at the junction points. The problem was reduced to a finite difference system of equations, and for a special class of triangular and hexagonal lattices a connection has been made with the homogenized isotropic continuum. The homogenized Lamé coefficients have been evaluated and a rotational micropolar interaction has been identified. The effects of micropolar interactions in the continuum have been discussed by Eringen (1966) and Kafadar & Eringen (1971). The lattice model involving extensible and flexible rods and derivation of the long-wave approximation for homogenized equations of motion of the micropolar medium have been discussed by Askar & Cakmak (1968).

In this paper, we consider the effect of dynamic micropolar interactions on the dynamic response of discrete inertial systems outside the standard homogenization regime. We describe several types of interaction and also present a comparison with the earlier work of Morozov (1984) and Maz’ya *et al.* (2000). Explicit analytical formulae have been derived for the effective group velocities in the low-frequency limit.

The structure of the paper is as follows. The main notations and the governing equations are introduced in §2. The geometry of the discrete system is introduced in §3. Section 3 also deals with the low-frequency approximations for micropolar heterogeneous lattices and includes asymptotic formulae for the effective group velocities. This section also presents a comparison with the results of the earlier work. The study of dispersion properties, control of stop band width and localization in a lattice with defects are discussed in §4. We finalize the presentation by considering an example of the interaction between a wave generated by a point source and the structured interface constructed from a micropolar lattice. Emphasis is placed on the effect of focusing by the flat interface for waves within a certain frequency range.

## 2. Notations and formulation of the dispersion equation

A general algorithm for analysing the spectral properties of a periodic lattice was established by Martinsson & Movchan (2003). Using the algorithm, the topology of a general lattice may be specified by the following objects.

**The irreducible cell.**The irreducible cell is the parallelogram spanned by the cell lattice vectors , with*i*=1,2. The cell lattice vectors are written as the columns of the translation matrix such that its components have the form . We then refer to the set*Ω*^{(n)}=*Ω*+*T***n**as the**n**th cell, with .**Nodes.**We specify the position nodes of the reference cell in the list . The position of the*κ*th node in the**n**th cell is then referenced by**x**^{(n,κ)}=**x**^{κ}+*T***n**.**Lattice links.**Each node is connected to several neighbouring nodes by a series of lattice links. The list specifies the lattice links that are numbered*j*=1,…,*b*. Specifically, each element references the node within the irreducible cell from which the member starts**x**^{κj}and the node at which it terminates .

The variable **u**^{(n,κ)} is used to denote the generalized displacement of the node (**n**,*κ*). It is emphasized that the interpretation of this *generalized displacement* depends on the context. For example, in problems relating to heat conduction, it represents the temperature at the node. In mechanical lattices, it represents both translational and rotational displacements. Assuming that the lattice links behave in a Hookean fashion, the equilibrium equation for a lattice member of type (*κ*,**m**,*Λ*) can be written as
2.1
Here, the variables **f**^{(i)} are the generalized force acting at the ends of the member. By means of the discrete Fourier transform
2.2
Newton's second law for the *κ*th node in the irreducible cell, written in the frequency domain, has the form (Martinsson & Movchan 2003),
2.3
The quantity *M*^{κ} is a matrix corresponding to the inertial properties of the node (**n**,*κ*). The set is the set of all indices **n** such that the *κ*th node in the irreducible cell connects to the *Λ*th node in the **n**th cell. For the case of harmonic waves, the term can simply be replaced by . Additionally, the forcing term is assumed to vanish. We introduce the inertia matrix and the stress matrix *σ*(**k**,*ω*) with *κΛ* blocks of
2.4
Finally, we collect the generalized displacements in a single column vector , which reduces equation (2.3) to
2.5
The solvability condition for equation (2.5) then yields the dispersion equation for the lattice
2.6
We note that if the inertia of the lattice links is neglected then the matrix function *σ* is *ω*-independent.

## 3. The effective group velocities in the low-frequency regime

We consider an infinite triangular diatomic lattice in , as shown in figure 1. The elementary cell consists of two nodes: one with mass *m*_{1}≡1 and non-dimensional polar mass moment of inertia , and a second with non-dimensional mass and non-dimensional polar mass moment of inertia . The bars indicate the usual dimensional quantities, which have been normalized for convenience. The lattice vectors are given by **t**^{(1)}=[2*l*,0]^{T} and . Such structures are often referred to as *truss structures* since the axial stiffness of the lattice links entirely dominates the problem. Conventionally, the bending interactions and any rotations are neglected. Here, we consider three distinct classes of interactions:

*The truss interaction*, where the lattice nodes are modelled as pin-joints, connected via springs. Only central interactions are considered.*The truss and torsional spring interaction*, as for (1), with an additional interaction based on the angle between the lattice links. Physically, this corresponds to the case where the lattice links are rigid in the transverse direction and linear torsional springs are located at the nodes.*The Euler–Bernoulli beam interaction.*Here, the lattice nodes are connected by Euler–Bernoulli beams. The angles at which the beams meet at the nodes are fixed.

For each type of interaction, we consider two classes of lattice links: (i) *massless links* and (ii) *inertial links*. In the former class, the links connecting the lattice are massless with the entire mass of the lattice being concentrated at the nodes. In the latter class, the mass is distributed both over the lattice nodes and along the lattice links. According to the class of interaction, the lattice links are assigned an axial rigidity *c*=*SE*/*l* and a flexural rigidity *d*=*EI*. Here, *S*, *E* and *l* denote the cross-sectional area, Young's modulus and member length, respectively. The second moment of inertia of the links is denoted by *I*. In the case of inertial links, the lattice links have a non-dimensional density . For convenience, we introduce the non-dimensional parameter *β*=2*d*/*cl*^{3}. In the case of the truss and torsional spring interaction, we introduce the non-dimensional parameter *ξ*=*τ*/*cl*^{2}, where *τ* is stiffness of the torsional spring.

The effective group velocity in the low-frequency regime is understood as ∇*ω*(**k**) in the limit as both *ω* and **k** tend to zero. For each class of interaction described above, and for small *ω* and **k**, the dispersion equation (2.6) may formally be expanded in the form
3.1
where we sum over the multi-index and . Keeping only terms up to |*α*|=4 and solving for non-negative *ω*=*ω*(**k**) yields the estimate for *ω*=*ω*(**k**) in the small |**k**| and low-frequency regime. Finally, taking the (non-dimensional) gradient *l*^{1}∇*ω*(**k**) produces an estimate for the effective group velocity in the low-frequency limit. The non-dimensional radian frequency and group velocities are normalized as and .

The derivation of the governing equations in the periodic lattice is routine, and it follows the same pattern as in Martinsson & Movchan (2003). The equations needed here are summarized in appendix A. The effective group velocities are evaluated via the asymptotic expansion of the dispersion equation for Bloch–Floquet waves for small *ω* and |**k**|. We note that in the general case of the diatomic system the homogenized material is anisotropic, and hence ∂*ω*/∂**k** may depend on the direction chosen.

### (a) The truss interaction

Consider a linearly elastic spring of non-dimensional density *ϱ*, aligned with the **e**_{1} direction. Neglecting any transverse stiffness, the equilibrium equations for the ends of the beam take the form of equation (2.1) with
3.2
Here, denotes the adjoint matrix of *A*_{12}. The non-dimensional parameter characterizes the natural frequency of the lattice member. The corresponding matrices for the case of massless lattice links can be recovered by taking the limit in equation (3.2). The forces and displacements in equation (2.1) have also been normalized: and . The dispersion equation for the system takes the same form as equation (2.6) with the inertia matrix . The components of the matrix function *σ*=*σ*(**k**,*ω*) are given by equations (A1) and (A2) in appendix A, with *ξ*=0. The effective group velocities in the low-frequency limit are then
3.3
The vector is the unit Bloch vector. The subscript ‘T’ denotes that the velocities correspond to the *truss interaction*. We note that the effective group velocities for the lattice with massless links can be obtained by choosing *ϱ*=0 in equation (3.3). It is also noted that the total mass in the irreducible cell is 1+*m*_{2}+6*ϱ*, the sum of the mass at the two nodes and the mass of the six lattice links (cf. figure 1b). However, the inertia term appears as 1+*m*_{2}+3*ϱ* in equation (3.3). We observe an apparent morphological change in the effective group velocities of the lattice when the inertia is distributed along the lattice links.

### (b) The truss and torsional spring interaction

This class is similar to that discussed above, with an additional interaction resulting from torsional springs located at the lattice nodes. Located at each lattice node there is a torsional spring of stiffness *τ*. Let *θ* be the angle between two lattice links at a particular node. The torsional spring then exerts a torque of magnitude *T*=−*τθ* on the two lattice links. Consider a thin elastic beam, rigid in the transverse direction, oriented along the **e**_{1} axis with torsional springs located at the origin. If the ends of the beam are subjected to small axial and transverse displacements then, in the linear regime, the equilibrium equations are of the form (2.1). The corresponding matrices are then
3.4
The force and displacement vectors and mass matrix remain the same as in §3*a* above. The equivalent matrices for the case of massless lattice links can be recovered by taking the limit in equation (3.4). The matrix function *σ* is given by equations (A1) and (A2) in appendix A. The effective group velocities in the low-frequency limit are then
3.5
In this case, the subscript ‘TS’ denotes that the velocities correspond to the *truss and torsional spring interaction*. We once again note that the corresponding group velocities for the lattice with massless links can be obtained by choosing *ϱ*=0 in equation (3.5). Moreover, we observe the same morphological change in the effective group velocities as for the truss interaction. Neglecting the contribution from the torsional springs by setting *ξ*=0 recovers the case of the truss interaction.

### (c) The effective group velocity for a fine periodic grid

Maz’ya *et al.* (2000) developed a homogenization method for treating fine periodic grids. This included the construction of asymptotic solutions to difference equations with rapidly oscillating coefficients. In particular, they considered a crystalline grid of bars with longitudinal rigidity *K* and ‘transverse rigidity’ *L*. The problem was then reduced to a finite difference system of equations on the irreducible cell. Similar work has been carried out by Morozov (1984) and Nazarov & Paukshto (1984), where the interaction, force per unit mass, between two points **x** and **y** in the lattice is given as
where (**y**−**x**)^{⊥} denotes the vector perpendicular to **y**−**x**. The first term is interpreted as the longitudinal interaction, while the second is the transverse interaction. We note that the ‘transverse rigidity’ *L* in the above formula is not equivalent to the flexural rigidity of an elastic bar used in the engineering literature. This type of interaction is that of the truss and torsional spring discussed above. In this work, the elastic constants *K* and *L* are identified as normalized longitudinal and rotational stiffnesses per unit area. For the special case of a very fine isotropic triangular lattice, Maz’ya *et al.* (2000, ch. 20) made the connection with the homogenized isotropic continuum, computing the effective Lamé parameters as
3.6
Based on this, the shear and pressure wave speeds, and , respectively, may by computed as
3.7a
and
3.7b
where *P* is the mass of the elementary cell. We observe that equations (3.7*a*,*b*) are fully consistent with the asymptotic formulae (3.5) derived for the lattice with massless links with *ϱ*=0,*m*_{2}=1,*P*=1. However, redistributing the inertia along the lattice links results in a morphological change in the effective group velocity in the low-frequency limit.

### (d) The Euler–Bernoulli beam interaction

Consider a Euler–Bernoulli beam of non-dimensional density *ϱ*≥0, oriented along the **e**_{1} direction. Prescribing small rotations and displacements at the ends of the beam results in an equilibrium equation of the form (2.1) with
3.8
The non-dimensional parameter characterizes the natural frequencies of the flexural waves in the beam. The corresponding matrices for the case of massless lattice links are obtained by taking the limit in equation (3.8). Indeed, the matrices corresponding to massless links are given explicitly in Martinsson & Movchan (2003) and are consistent with the limiting case, , of those in equation (3.8). The non-dimensional forces and displacement take the form and . In each case, the first two components are the linear forces and displacements, while the third is the torque and polar rotation, respectively. The components of the matrix function *σ*(**k**,*ω*) are specified in equations (A3*a*,*b*) in appendix A. The inertia matrix in equation (2.5) has the form . For this class of interaction, the effective group velocities in the low-frequency regime are
3.9
As usual, the subscript ‘EB’ denotes the class of interaction under consideration, in this case the *Euler–Bernoulli beam interaction*. In contrast to the previous two classes of interactions, it is observed that distributing the inertia along the lattice links causes no morphological change in the effective group velocities. We note that the case of massless lattice links can be recovered by choosing *ϱ*=0 in equation (3.9).

It should also be noted that neglecting the contribution of beam bending (letting in equation (3.8)) does not necessarily reduce the problem to that of the truss interaction. We emphasize that *λ*=*λ*(*β*) with as *β*→0^{+} and the elements of equation (3.8) do not converge in the limit *β*→0^{+}. For the case of massless lattice links, the bending moment and shear forces vanish for all strains in the limit , that is, letting *β* approach zero is equivalent to neglecting the contributions from beam bending. However, in the case of massive beams, the dynamic Euler–Bernoulli equation is singularly perturbed for small values of *β*.

The finite-element software COMSOL Multiphysics was used to perform numerical simulations of a lattice with the Euler–Bernoulli interaction and inertial links. The eigenfrequencies were computed for given values of the Bloch vector over the irreducible Brillouin zone. Newton's difference quotient was then used to compute the numerical gradient at the origin. The material parameters are chosen as *E*=200 GPa, kg m^{−3}, *S*=2120 mm^{2}, *l*=1 m, *I*=349×10^{−8} m^{4}, kg, kg, kg m^{2} and kg m^{2}. The asymptotic estimates yield the numerical values m s^{−1} and m s^{−1}, which are in good agreement with the results obtained from the finite-element analysis: m s^{−1} and m s^{−1}.

We note that, in the low-frequency regime, the triangular diatomic lattice is isotropic for all three of the interactions discussed above. However, it can be seen from the dispersion diagrams presented in §4 that the lattice is highly anisotropic for higher frequencies.

## 4. Dispersion surfaces, standing waves and defect modes

In this section, we present a set of dispersion diagrams for the Euler–Bernoulli beam interaction. Given the spectral problem of the form (2.5) and the corresponding multivariate function *g*(**k**,*ω*), we plot the level curves {(*k*_{1},*k*_{2},*ω*)|*g*(*k*_{1},*k*_{2},*ω*)=0} over the region . Formally, the primitive reciprocal lattice cell is the parallelogram bounded by the reciprocal lattice vectors, and . However, it is convenient to plot the rectangular region , which contains the irreducible reciprocal lattice cell. The dispersion surfaces are shown in figures 2 and 3 for *the lattice with massless links* and *the lattice with inertial links*, respectively. For the former case, the matrix function *σ* is independent of *ω*, and MATLAB is used to solve the generalized eigenvalue problem (2.5) numerically. In the latter case, the matrix *σ* is a function of **k** and *ω*. The finite-element software COMSOL Multiphysics is then used to solve the spectral problem and determine *ω*=*ω*(**k**) numerically. In figures 2 and 3, the dispersion surfaces are projected on the frequency axes as grey bars.

### (a) Standing waves

As expected with periodic systems, the dispersion surfaces show evidence of standing waves which are indicated by flat regions where ∂*ω*/∂**k**=0. In particular, we note that the first two dispersion surfaces in both diagrams are almost flat. Numerical simulations in both MATLAB and COMSOL Multiphysics indicate that these surfaces correspond to standing rotational waves. In the case of the lattice with massless links, we derive explicit analytical estimates for the rotational modes. Figure 4a illustrates one such rotational mode. It is observed that the translational displacements of the nodes are much smaller than the rotational components. Hence, for a simple estimate, the translational displacement of the nodes may be neglected. For purely rotational motion, the equations of motion for the nodes in the irreducible cell of the lattice reduce to
4.1
where we have chosen **k**=**0** for convenience. Here, *θ*^{(i)} and *τ*^{(i)} represent the non-dimensional angular displacement and torque, respectively. We search for time-harmonic waves and therefore set ∂_{tt}*θ*^{(i)}=−*ω*^{2}*θ*^{(i)} and *τ*^{(i)}=0. The system (4.1) has non-trivial solutions if and only if
4.2
The positive solutions for *ω* then yield the estimates for the frequencies of the standing rotational modes,
4.3
Taking *J*_{1}=2, *J*_{2}=6, *β*=0.001 yields numerical estimates of *ω*^{(rot)}_{+}=0.0853 and *ω*^{(rot)}_{−}=0.0454, which are in good agreement with the numerical solutions to the full spectral problem. In the case of lattices with inertial links, we observe that, for low frequencies and sufficiently small values of 2*ϱ*/*β*, the equations of motion for pure rotations approximately reduce to the form (4.1). For the values of the parameters used above together with *ϱ*=1 the results of the finite-element computations are in good agreement with the estimates, *ω*^{(rot)}_{FE+}=0.0845 and *ω*^{(rot)}_{FE−}=0.0452. It is noted that, usually, this triangular lattice is treated as a truss and bending moments are neglected. However, these bending moments yield low-frequency rotational standing waves, even in truss-like structures.

In addition to the rotational standing waves, there exist a number of standing modes of other types. In particular, there exists a standing mode at the origin of the fifth dispersion surface (labelled by the star symbol in figure 2), which provides a convenient estimate for the upper boundary of the stop band shown in figure 2. The standing mode, illustrated in figure 4b, involves the relative translation of the two nodes within the elementary cell with ∂*ω*/∂**k**=0. Both lattice nodes move in an approximately linear fashion along the direction . Writing the nodal displacement amplitude vectors as **u**^{(i)}=*u*_{i}**a**, the dispersion equation (2.5) reduces to the scalar equation
4.4
where *σ*_{11} and *σ*_{12} are blocks of the *σ* matrix given by equation (A3*a*,*b*) in appendix A, in the limit . From the finite-element computations, it can be determined that the relative amplitudes are *u*_{2}/*u*_{1}=−0.1, leading to the estimate
4.5
With *β*=0.001 and *m*_{2}=10, the estimate is *ω*^{(trans)}_{1}=1.058, which is in good agreement with the numerical solution of the full spectral problem. This estimate also gives the location of the upper boundary of the band gap seen in figure 2.

Comparing figures 2 and 3, we observe that a significant difference is the presence of several additional dispersion surfaces in figure 3. These additional surfaces appear in the region occupied by the band gap in figure 2 and include several almost flat surfaces, grouped closely together and labelled by the star symbol in figure 3. Finite-element simulations indicate that these densely packed flat surfaces correspond to standing waves with no nodal displacements; these modes are related to the natural frequencies of the lattice links. Figure 5 illustrates a mode which is representative of these standing waves. For the fundamental modes of the lattice links, the nodal displacements vanish and the problem reduces to that of a Euler–Bernoulli beam with clamped ends. Such systems have been treated extensively in the literature (e.g. Graff (1975)). A brief discussion is included here for completeness.

Consider the boundary value problem for the non-dimensional time-harmonic deflection of a Euler–Bernoulli beam
4.6a
and
4.6b
where *λ*^{4}=2*ω*^{2}*ϱ*/*β*. The solution to the boundary value problem results in the equation . The first of an infinite family of solutions is *λ*_{1}≈4.69409. Thus, the estimate for the lattice member standing waves is
4.6c
For the parameter values used to produce figure 3, we obtain , which gives the approximate location for the five additional flat dispersion surfaces labelled by the star symbol in figure 3.

### (b) Defect modes

In this section, we introduce a periodic array of defects into the lattice in the form of buckled links. Such buckling could occur as a result of pre-stress. For example, selected links could have a higher thermal expansion coefficient than the ambient lattice and thus buckle under thermal load. For the purposes of illustration, we consider the case of a lattice with massless links and Euler–Bernoulli beams, where some of the links connecting two similar nodes buckle to the extent to which they may be neglected entirely (figure 6). The dispersion surfaces for the defect lattice are shown in figure 7 and were produced in the same manner as those described earlier. Here, the primitive cell in the reciprocal space is again a parallelogram bounded by the vectors and . However, we plot over the rectangular region .

Numerical simulations with Matlab reveal a number of standing waves. In particular, we focus on the two modes illustrated in figure 8. These modes lie on the lowest two dispersion surfaces of figure 7. For convenience, we label the nodes 1–4, as indicated in figure 8. In the case of figure 8*a*, nodes 1 and 4 rotate, while 2 and 3 remain almost stationary. Neglecting the motion of nodes 2 and 3 entirely and choosing **k**=**0**, the equations of motion for nodes 1 and 4 become
4.7
For harmonic motion, we set *τ*^{(i)}=0 and ∂_{tt}*θ*^{(i)}=−*ω*^{2}*θ*^{(i)}. The numerical simulations indicate that the two nodes are approximately in anti-phase, that is, *θ*^{(1)}≈−*θ*^{(4)}. The estimate for the frequency of the standing wave is
4.8
Choosing *β*=0.001 and *J*_{1}=2 in accordance with the numerical computations yields an estimate of *ω*^{(defect)}_{1}=0.0707, which is in good agreement with the numerical solution. Figure 8*b* depicts a second rotational mode where nodes 2 and 3 rotate, while the remaining two nodes remain almost stationary. The numerical simulations indicate that the two oscillating nodes approximately move in anti-phase. In this case, we obtain the following estimate for the frequency of the standing wave:
4.9
Using the same numerical values for *β* and *J*_{2} as before, we obtain *ω*^{(defect)}_{1}=0.0387, which agrees with the numerical solution for the full spectral problem.

In addition to these rotational modes, there exists a standing mode on the eighth dispersion surface (labelled by the diamond symbol in figure 7), which provides an estimate for the lower boundary of the band gap illustrated in figure 7. The standing mode involves the relative translation of the lattice nodes within the elementary cell with ∂*ω*/∂**k**=0. It is observed that the displacements of the white nodes in figure 6 are small when compared with those of the black nodes. Hence, for a simple estimate, the displacement of the white nodes may be neglected entirely. Moreover, the rotational motion is small in comparison with the translational motion and can also be neglected. The equations of motion for the two black nodes are then (choosing **k**=**0** for convenience)
4.10
with *i*,*j*=2,3 and *i*≠*j*. Here, and **F**^{(i)} are the non-dimensional displacements and forces, respectively. The matrix *M*^{(i)} is the mass matrix. For this particular mode, the numerical computations suggest that the lattice nodes oscillate in anti-phase. Therefore, for harmonic waves the equation of motion becomes
4.11
The positive solution for *ω* then yields the estimate for the frequency of the standing wave
4.12
Choosing *β*=0.001 and *m*_{2}=10 results in an estimate of *ω*^{(defect)}_{3}=0.549, which differs from the numerical solution of *ω*=0.490, but still gives a good estimate of the lower boundary of the band gap in figure 7.

The dispersion surface labelled by the star symbol in figure 7 characterizes several standing waves, and is of special interest. One of the standing waves is similar in character to that depicted in figure 4*b*. In this case, the white nodes in figure 6 undergo a relative translation along the direction . The displacement of the black nodes is negligibly small. Writing the nodal displacement amplitude vectors as **u**^{(i)}=*u*_{i}**a** the equations of motion reduce to the scalar equation
4.13
where *σ*_{11} and *σ*_{12} are blocks of the *σ* matrix given by equation (A3*a*,*b*), in the limit . From the finite-element computations it can be determined that the ratio *u*_{4}/*u*_{1}=−0.1, leading to the estimate
4.14
With *β*=0.001 the estimate is *ω*^{(trans)}_{4}=1.058, which is in good agreement with the results of the numerical computations. Moreover, *ω*^{(trans)}_{4} is approximately where the upper boundary of the band gap lies.

## 5. The elastic lens, focusing and filtering of elastic waves

We consider applications of the dispersive properties of Bloch–Floquet waves within discrete systems. In particular, we present applications relating to the effects of filtration and focusing of elastic waves by a ‘flat lens’ for certain frequencies. The effects of focusing and filtering for solutions of the Helmholtz equation have already been demonstrated in the literature (e.g. McPhedran *et al.* 2004). More recently, Jones *et al.* (2010) have analysed the effect for the case of vector elasticity in a structured continuum. Here, we discuss the effects of focusing and filtration of elastic waves in discrete structures.

Consider a finite triangular lattice of the same geometry as in §3. Let the ambient lattice be monatomic and homogeneous. Within the ambient lattice, we embed a finite slab of heterogeneous diatomic lattice of the same geometry. Both the ambient lattice and interface (finite slab) are lattices with inertial links, formed from Euler–Bernoulli beams of unit length, density , Young's modulus 200 Gpa, cross-sectional area *S*=2120 mm^{2} and area moment of inertia *I*=349×10^{−8} m^{4}. The nodes in the ambient lattice have mass *m*=91.531 kg, and polar mass moment of inertia *J*=66.568 kg m^{2}. The nodes in the heterogeneous diatomic interface have masses *m*_{1}=16.642 kg and *m*_{2}=166.42 kg, and polar mass moments of inertia *J*_{1}=33.284 kg m^{2} and *J*_{2}=99.852 kg m^{2}. A schematic diagram of the ambient and interface lattices is shown in figure 9.

Let elastic waves travel through the ambient lattice and interact with the structured interface, as shown in figure 10*a*. The wave is generated by a single point source: a time-harmonic displacement of amplitude 10^{−6} m in the horizontal direction is prescribed at one of the lattice nodes. To eliminate reflection from the outer boundary of the computational domain, we apply damping to the beams in the neighbourhood of the fixed boundary nodes.

We now refer to the dispersion surfaces for the elementary cell of the structured interface, shown in figure 11*a*. The transmission problem is formally distinct from the Bloch–Floquet spectral problem. However, we can use the dispersion diagram to predict the reflection and transmission patterns. Figure 10*b* shows the magnitude of the displacement field when the forcing frequency is 100 Hz. A similar wave pattern can clearly be observed on both sides of the interface layer, indicating that the low-frequency response of the structured interface is very close to that of the ambient medium. We also observe a similar wave pattern in figure 10*a* where the structured interface has been removed entirely. In contrast, figure 12 shows the magnitude of the displacement field when the forcing frequency is 700 Hz, which lies in the band gap of figure 11*a*. In this case, the incoming wave is reflected with very little transmission.

It has been suggested that the phenomenon of focusing by a flat interface is linked to the presence of saddle points and regions of negative group velocities (McPhedran *et al.* 2004; Jones *et al.* 2010). Referring to the dispersion surfaces for the heterogeneous interface lattice (figure 11*a*), it is observed that the surface labelled by the star symbol possesses a saddle point and regions where the group velocity is negative. The dispersion contour for the sixth eigenmode, plotted for *k*_{1}*l*∈[0,*π*] and *k*_{2}=0, is shown in figure 11*b*. Here, we note that the directional group velocity ∂*ω*/∂*k*_{1} is negative in the frequency range 622–661 Hz. In contrast, the group velocity in the *k*_{2} direction is positive in the same frequency range. Figure 13 shows a plot of the absolute value of the displacement field when the forcing frequency is 642.5 Hz. The choice of the frequency was determined by the position of a saddle point on the corresponding dispersion surface. The effect described here is typical for neighbourhoods of saddle points. A clear directional preference can be observed within the interface. In addition, one can see the *secondary source* on the right-hand side of the interface. Figure 13 shows the preferential direction of propagation and the effect of focusing. This feature of the waves will persist in a small interval containing 642.5 Hz.

Finally, in figure 14, we present a computation where the source has been shifted away from the interface region. In this case, the forcing frequency is 654.4 Hz, which again is within the region where there is a preferential direction of propagation. Moreover, where the beams intersect on the right-hand side, we can see the formation of the *image point*. This effect is strongly frequency dependent.

Animated versions of these computations are available from the online multimedia appendix (http://pcwww.liv.ac.uk/~mf0u60af/Daniel_Colquitt/Animations.html).

## 6. Concluding remarks

In this paper, we have examined the dynamic interactions between inertial elements in a heterogeneous elastic lattice structure. Several classes of interactions were analysed, including the dispersion properties, standing modes and control of stop band width. The non-trivial connection between the effective group velocities, in the long-wave limit, and the distribution of mass between the lattice links and nodes was demonstrated. In particular, it was shown that distributing the inertia of the lattice along the links results in a morphological change in the effective group velocities for some classes of interaction.

A periodic array of defects was also considered, with an emphasis on rotational modes and a micropolar interaction. The resulting structure was also periodic, with an enlarged elementary cell of periodicity. Defects were created by the removal of some of the elastic links in the Euler–Bernoulli system—in practice, problems like this may occur as a result of fracture of elastic links or buckling of some of the beams. A particularly important feature of such a system is the formation of a class of standing waves, ‘concentrated’ in the defect regions. This also leads to the formation of new stop bands on the dispersion diagram, which in turn affects the filtering properties of the microstructured medium.

The dispersion properties of Bloch–Floquet waves in an infinite lattice structure can be used in problems of optimal design for finite size microstructures. In particular, we illustrated the interaction of waves with a heterogeneous diatomic lattice of finite width. Special attention is drawn to the range of frequencies in the neighbourhood of saddle points on the dispersion diagram. The corresponding regime shows directional preferences for waves interacting with the structured medium. The apparent focusing and the creation of an *image point* by a flat elastic ‘lens’ is one of the interesting outcomes of the paper.

## Acknowledgements

D.J.C. gratefully acknowledges the support of an EPSRC research studentship (grant EP/H018514/1).

## Appendix A

#### (a) The stiffness matrix functions

For convenience, we introduce the quantities and . The matrix functions *σ*(**k**,*ω*) defined in this appendix correspond to the case of inertial lattice links. For the *truss* and *truss and torsional spring* interactions, the case of massless lattice links can be recovered by taking the limit . For the case of Euler–Bernoulli interactions, the case of massless lattice links can be obtained by taking the limit .

#### (b) The stiffness matrix for the lattice with truss and torsional spring interactions

Below is detailed the stiffness matrix functions for the *truss and torsional spring interactions*. The corresponding matrices for the *truss interaction* may be obtained by setting *ξ*=0. The components of *σ*_{11}=*σ*_{22} are
A1
The elements of are given by
A2

#### (c) The stiffness matrix for the lattice with Euler–Bernoulli interactions

For the case of Euler–Bernoulli beam interactions, the full matrix function *σ*(**k**,*ω*) is most conveniently written in terms of the matrix blocks *A*_{11} and *A*_{12} as defined in equation (3.8) and the rotation matrix
where *t*=0,1,…,5. The blocks of the stress matrix may then be written
A3a
and
A3b

- Received February 22, 2011.
- Accepted April 12, 2011.

- This journal is © 2011 The Royal Society