A semi-analytical approach towards plane wave analysis of local resonance metamaterials using a multiscale enriched continuum description

This work presents a novel multiscale semi-analytical technique for the acoustic plane wave analysis of (negative) dynamic mass density type local resonance metamaterials with complex micro-structural geometry. A two step solution strategy is adopted, in which the unit cell problem at the micro-scale is solved once numerically, whereas the macro-scale problem is solved using an analytical plane wave expansion. The macro-scale description uses an enriched continuum model described by a compact set of differential equations, in which the constitutive material parameters are obtained via homogenization of the discretized reduced order model of the unit cell. The approach presented here aims to simplify the analysis and characterization of the effective macro-scale acoustic dispersion properties and performance of local resonance metamaterials, with rich micro-dynamics resulting from complex metamaterial designs. First, the dispersion eigenvalue problem is obtained, which accurately captures the low frequency behavior including the local resonance bandgaps. Second, a modified transfer matrix method based on the enriched continuum is introduced for performing macro-scale acoustic transmission analyses on local resonance metamaterials. The results obtained at each step are illustrated using representative case studies and validated against direct numerical simulations. The ∗Corresponding author. Email address: v.g.kouznetsova@tue.nl (V.G. Kouznetsova) Preprint submitted to International Journal of Mechanical Sciences August 5, 2017 methodology establishes the required scale bridging in multiscale modeling for dispersion and transmission analyses, enabling rapid design and prototyping of local resonance metamaterials.


Introduction
Acoustic metamaterials can be used to engineer systems that are capable of advanced manipulation of elastic waves, such as band-stop filtering, redirection, channeling, multiplexing etc. which is impossible using ordinary materials [1,2,3,4]. This can naturally lead to many potential applications in vari-5 ous fields, for example medical technology, civil engineering, defense etc. The extraordinary properties of these materials are a result of either one or a combination of two distinct phenomena, namely local resonance and Bragg scattering.
Bragg scattering is exhibited by periodic lattices in the high frequency regime where the propagating wavelength is of the same order as the lattice constant. 10 Local resonance, on the other hand, is a low frequency/long wavelength phenomena and, in general, does not require periodicity. "Local resonance acoustic metamaterial" (LRAM) is therefore the term used here to distinguish the subclass of acoustic metamaterials based solely on local resonance. The present work is only concerned with the modeling and analysis of LRAMs restricted to 15 linear elastic material behavior in the absence of damping.
Based on the primary medium of the wave propagation, LRAMs can be further classified as solid (e.g. [5]) or fluid/incompressible media (e.g. [6]) based. This paper is concerned only with the modeling of solid type LRAMs. A typical representative cell of such LRAMs is characterized by a relatively stiff matrix 20 containing a softer and usually heavier inclusion. The micro-inertial effects resulting from the low frequency vibration modes of the inclusion induce the local resonance action. The complexity of the geometry of the inclusion plays an important role in the response of the LRAM. For instance multi-coaxial cylindrical inclusions have been proposed [7], which exhibit more pronounced micro-inertial effects due to the larger number of local resonance vibration modes, hence leading to more local resonance bandgaps. The symmetries of the inclusion also play a key role. A 'total' (or omni-directional) bandgaps, where any wave polarization along any given direction is attenuated within a given frequency range is observed in micro-structures exhibiting mode multiplicity (or degenerate eigen-30 modes) resulting from a combined plane and 4-fold (w.r.t. 90 degree rotation) symmetry. For geometries with only plane symmetry, 'selective' (or directional) bandgaps, which only attenuate certain wave modes in a given frequency range can be observed. Furthermore, if the inclusion is not plane symmetric with respect to the direction of wave propagation, hybrid wave modes, which are a 35 combination of compressive and shear wave modes can be observed. In general, a more extensive micro-dynamic behavior results from an increased complexity in the LRAM design.
A plethora of approaches are available for modeling LRAMs. For steady state analysis of periodic structures, the Floquet-Bloch theory [8] gives the gen-40 eral solution that reduces the problem to the dispersion analysis of a single unit cell. The unit cell problem can then be further discretized and solved using appropriate solution methodologies. A popular technique highly suited to composites with radially symmetric inclusions is given by Multiple Scattering theory (MST) [9,10]. However, MST has not been successfully applied to 45 more complex unit cell designs. The finite element (FE) method provides a robust approach for modeling arbitrary and complex unit cell designs. Such an approach, also known as the wave finite element (WFE) method, has been extensively employed in literature for analyzing periodic structures including acoustic metamaterials [11,12,13,14]. 50 Since LRAMs operate in the low frequency, long wave regime, it is appropriate to exploit this fact towards approximating the general solution, leading to simpler methodologies. This is equivalent to ignoring nonlocal scattering effects resulting from reflections and refractions at the interfaces of the heterogeneities. This is the formal assumption made in many dynamic homogenization/effective 55 medium theories [15,16,17,18,19,20,21] which recover the classical balance of momentum, but where the local resonance phenomena manifests in terms of dynamic (frequency dependent) constitutive material parameters. The most striking feature of the effective parameters is that they can exhibit negative values, which indicates the region of existence of a bandgap. The specific macro-60 scale coupling of the local resonance phenomena is dictated by the vibration mode type of the inclusion [15]. A negative mass density is obtained in LRAMs exhibiting dipolar resonances e.g., [22]. Similarly, a negative bulk and shear modulus are obtained in LRAMs exhibiting monopolar and quadrupolar resonances , e.g. [23], though not further considered in this work. The homogenized 65 models accurately capture the dispersion spectrum of LRAMs provided that the condition on the separation of scales is satisfied, i.e. the wavelength of the applied loading is much larger compared to the size of the inclusion. This can be ensured in the local resonance frequency regime by employing a sufficiently stiff matrix material compared to that of the inclusion.

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The expression for the effective parameters of materials with radially symmetric inclusions has been obtained using analytical methods using the Coherent Potential Approximation (CPA) approach [15,16]. A generalization towards ellipsoidal inclusions has also been presented in [17,24], thereby extending the method of Eshelby [25] to the dynamic case. Analytical models for unit cells 75 composed discrete elements (e.g. trusses) have also been derived [19,26,27].
For arbitrary complex micro-structures, it is necessary to adopt a computational homogenization approach. A FE based multiscale methodology in the framework of an extended computational homogenization theory was presented in [20,21].

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The present work builds upon the computational homogenization framework introduced in [21]. A first-order multiscale analysis is combined with model order reduction techniques to obtain an enriched continuum model, i.e. a compact set of partial differential equations governing the macro-scale behavior of LRAMs. The reduced order basis is constructed through the superposition of 85 the quasi-static and the micro-inertial contribution, where the latter is repre-sented by a set of local resonance eigenmodes of the inclusion. In the homogenization process, the generalized amplitudes associated to the local resonance modes emerge as additional kinematic field quantities, enriching the macro-scale continuum with micro-inertia effects in a micromorphic sense as initially defined 90 by Eringen [28]. An equivalent approach for modeling LRAM was derived using asymptotic homogenization theory in [18], but was not further elaborated and demonstrated as a computational technique.
In this work, the enriched continuum of [21] is exploited to develop an ultrafast semi-analytical technique method for performing dispersion and transmis-95 sion analysis of LRAMs with arbitrary complex micro-structure geometries that retains the accuracy of WFE methods at a fraction of the computational cost.
A modified transfer matrix method is developed based on the enriched continuum that can be used to derive closed form solutions for wave transmission problems involving LRAMs at normal incidence. Furthermore, the dispersion 100 characteristics of the enriched continuum and their connection to various symmetries of the inclusion are discussed in detail. In order to ensure the accuracy of the method in the frequency range of the analysis, a procedure for verifying the homogenizability of a given LRAM is introduced. Although the enriched continuum predicts both negative dynamic mass and elastic modulus effects, 105 the latter exhibits a significant coupling only at frequencies close to and beyond the applicability (homogenizability) limit of the present approach. Since such effects cannot be robustly modeled, they will not be considered in the present paper.
The structure of the paper is as follows. The relevant details of the enriched 110 multiscale methodology are briefly recapitulated in Section 2. In Section 3, a plane wave transform is applied to obtain the dispersion eigenvalue problem of the enriched continuum. The influence of the inclusion symmetry on the dispersion characteristics is highlighted. Based on the obtained dispersion spectrum, a procedure for checking the applicability (homogenizability) of the problem in 115 the frequency range of analysis is elaborated that provides a reasonable estimation of its validity. In Section 4, a general plane wave expansion is applied to derive a modified transfer matrix method for performing macro-scale transmission analyses of LRAMs at normal wave incidence. The results obtained in each of the sections are illustrated with numerical case studies and validated against 120 direct numerical simulations (DNS). The conclusions are presented in Section 5.
The following notation is used throughout the paper to represent different quantities and operations. Unless otherwise stated, scalars, vectors, second, third and fourth-order Cartesian tensors are generally denoted by a (or A), a, A, A (3) and A (4) respectively; n dim denotes the number of dimensions of the 125 problem. A right italic subscript is used to index the components of vectorial and tensorial quantities. The Einstein summation convention is used for all vector and tensor related operations represented in index notation. The standard operations are denoted as follows for a given basis e p , p = 1, ..n dim , dyadic product: a ⊗ b = a p b q e p ⊗ e q , dot product: A · b = A pq b q e p and double contraction: Matrices of any type of quantity are in general denoted by (•) except for a column matrix, which is denoted by (• ). A left superscript is used to index quantities belonging to a group and to denote sub-matrices of a matrix for instance for a and b , by mn a and m b , respectively. The transpose of a second order tensor is defined as follows: for A = A pq e p e q , A T = A qp e p e q . The 135 transpose operation also simultaneously yields the transpose of a matrix when applied to one. The first and second time derivatives are denoted by (•) and (•) respectively. The zero vector is denoted as 0.
2. An enriched homogenized continuum of local resonance metamaterials 140 This section summarizes the essential features of the enriched continuum model introduced in [21]. The framework is based on the extension of the computational homogenization approach [29] to the transient regime [20]. The classical (quasi-static) homogenization framework relies on the assumption of a vanishingly small micro-structure in comparison to the macroscopic wavelength.

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No micro-inertial effects are recovered in this limit. The effective mass density obtained in this case is a constant and is merely equal to the volume average of the micro-structure densities, unlike the frequency dependent quantity observed in metamaterials.
The key aspect rendering the classical computational homogenization method 150 applicable to LRAMs is the introduction of a relaxed scale separation principle.
Making use of the typical local confinement of the resonators in LRAM, the long wave approximation is still assumed for the matrix (host medium), while for the heterogeneities (resonators), full dynamical behavior is considered. Note, that the long wavelength approximation poses a restriction on the properties of 155 the matrix which has to be relatively stiff in order to ensure that it behaves quasi-statically in the local resonance frequency regime. i.e. captures all the representative micro-mechanical and local resonance effects at that scale. For a periodic system, the RVE is simply given by the periodic unit. The homogenization method can be applied to a random distribution of 165 inclusions provided that there is sufficient spacing between the inclusions (i.e. excluding the interaction between the resonators).
The Hill-Mandel condition [30], which establishes the energy consistency between the two scales, is generalized by taking into account the contribution of the momentum into the average energy density at both scales. As such, the resulting computational homogenization framework is valid for general nonlinear problems. Restricted to linear elasticity, a compact set of closed form macro-scale continuum equations can be obtained. To this end, exploiting linear superposition, the solution to the micro-scale problem is expressed as the sum of the quasi-static response, which recovers the classical homogenized model, and the micro-inertial contribution spanned by a reduced set of mass normalized vibration eigenmodes s φ, of the inclusion (shown in continuum form) as follows, Further details of the derivation of this framework can be found in [21].
In the following, only the final equations describing the homogenized enriched continuum are given.

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The macro balance of momentum The reduced micro balance of momentum (to be solved at the macro-scale) The homogenized constitutive relations Here, σ represents the macro-scale Cauchy stress tensor and p the macroscale momentum density vector. The terms C (4) and ρ are, respectively, the ef- They are obtained through discretization and model reduction (See [21] for details). Specifically, the parameters s j and s H are obtained by projecting the computed inertial force per unit amplitude of the local resonance eigenmodes onto the rigid body and the static macro-strain deformation mode respectively.
The explicit expressions for these effective parameters in continuum form are given as follows, where ρ( x m ) represents the mass density of the heterogeneous micro-structure. Subsequently, the system (2)-(4) can be solved at the macro-scale.
In the following sections, the analysis will be restricted to dipolar local resonances, i.e. the coefficient s H will be disregarded in the sequel. The justification for this is as follows. Since the primary concern of the present paper is to de- compressive) to at least 100 times higher than that of the inclusion. In this case, the boundary of the inclusion is effectively rigid, thereby suppressing the action of monopolar and quadrupolar eigenmodes, which necessarily require a deformable boundary [15]. This fact has also been evidenced in [18] where a homogenized model has been derived via asymptotic homogenization. Indeed,

Dispersion analysis
In this section, a plane wave analysis is carried out on an infinite enriched continuum, i.e. without taking into account macro-scale boundary conditions.

Theoretical development
The plane wave transform of any continuous field variable can be expressed as, where the (•) represents the transformed variable, k denotes the wave vector, ω the frequency and i the imaginary unit. The wave vector can be represented by its magnitude and direction as k = k e θ , where k = 2π λ is the wave number, λ the corresponding wavelength and e θ a unit vector in the direction of wave propagation. The above expression provides the plane wave transform in an infinite medium. Applying the transformation (6) to equations (2)-(4) while disregarding s H as discussed before (see the last paragraph in Section 2) yields, Eliminatingσ andˆ p by substituting equations (7c) and (7d) into equation (7a) and combining it with equation (7b) results in the following parameterized eigenvalue problem in ω, where The above eigenvalue problem is the so-called k − ω form, in which ω is the eigenvalue and k and e θ are parameters. The problem consists of n dim + N Q variables, which gives the number of dispersion branches predicted by the model.
The alternative ω − k form in which k is the eigenvalue and ω and e θ are the parameters can be obtained by eliminatingη from the system of equations (8) resulting in where where s J = 1 V s j ⊗ s j is a measure of the translational inertia associated to 220 each local resonant eigenmode, termed modal mass densities and ρ(ω) is the effective dynamic mass density tensor which is a well known quantity in the literature [18,22,31,32,33]. In most works, however, the expression for the dynamic mass density is derived for a particular unit cell design often after some simplifications and approximations. Here, on the contrary, equation (10) holds 225 for arbitrary geometries, for which it is obtained numerically.
The solution to equation (9) at a given ω yields n dim eigen wave numbers k p (ω), p = 1..n dim and corresponding eigen wave vectors denoted by υ p (ω).
These vectors are also termed polarization modes. Let υ p (ω) be normalized with respect to C θ , i.e.
In the present context, only positive real or imaginary eigenvalue solutions are considered to avoid ambiguity in the direction of wave propagation or decay respectively. If υ p (ω) is parallel to e θ (i.e. υ p × e θ = 0) for a given p, the corresponding wave mode is purely compressive. On the other hand if υ p (ω) is 230 orthogonal to e θ (i.e. e θ · υ p = 0), the resulting mode is purely shear. All other combinations represent hybrid modes.
For an arbitrary RVE geometry and wave direction, the dispersion problem (8) and (9) can be solved numerically which, owing to the reduced nature of the enriched continuum problem, is computationally much cheaper in comparison 235 to standard Bloch analysis techniques, especially in the 3D case. It is arguably cheaper even compared to reduced order Bloch analysis techniques [34,35] since the spectrum of the homogenized problem is much smaller than the Bloch spectrum leading to a more compact dispersion equation.
If Q accounts for all dipolar modes, the following relation holds due to the mass orthogonality of eigenmodes where µ inc is the static mass fraction of the inclusion defined as the ratio of the where, s µ Jp = (ρ) −1 e p · s J · e p , s ∈ Q is the modal mass fraction of the local 240 resonant eigenmode along e p for p = 1..n dim . In practice, it is sufficient if the above expression is satisfied in the approximation as the local resonance modes in general will never fully be able to capture all the mass of the inclusion. The tolerance can vary based on the design requirements but a good measure would be 5% of the inclusion mass fraction. Thus, Q should be the smallest set of low 245 frequency eigenmodes that satisfies equation (13) within the given tolerance. semi-definite, a "selective" bandgap is formed that attenuates some of the wave modes. Two bandgaps will overlap only if a pair of eigenfrequencies s ω, r ω, s, r ∈ Q, s = r are sufficiently close enough and the s j is orthogonal to r j.
In the special case of a system exhibiting mode multiplicity, i.e. s ω = r ω and s j = r j , the bandgaps will overlap perfectly. Such a situation is guaranteed for an inclusion domain with isotropic constituents that possesses the following symmetries; plane symmetry with respect to 2 mutually orthogonal axes, and a 4-fold (discrete 90 degree) rotational symmetry about the normal to these axes (exhibited for e.g. by a square, cross, cylinder, hexagonal prism etc.) . In this case, all r j vectors will be aligned along one of the symmetry axis. For a 3D where the fact n dim p=1 e p ⊗ e p = I has been used. Thus, the mass density tensor becomes isotropic and diagonal in this case. Such a tensor is always either 280 positive or negative definite at any given frequency. Therefore, an inclusion with plane and 4-fold symmetry only exhibits total bandgaps and no hybrid wave solutions.
For inclusions with only plane symmetry with respect to e p , p = 1, ..n dim (but not the 4-fold symmetry), for e.g. an ellipsoid, rectangle etc., degenerate eigenmodes are no longer guaranteed, however the s j vectors will still align along e p . Now the set Q is divided into n dim subsets p Q defined as p Q = {s ∈ Q| e p · s J · e p = 0} for p = 1..n dim . Using equation (10), the mass density tensor can now be re-written for the plane symmetry case as follows, where s µ J = (ρV ) −1 s j 2 . Thus the mass density tensor is now orthotropic, or diagonal with respect to the axis defined by e p , p = 1, ..n dim . If the wave propagates along one of the symmetry axis, i.e. e θ = e p for some p = 1, ..n dim , then no hybrid wave mode solutions will be obtained at all frequencies due to local resonance. Furthermore, if C θ can be diagonalized with respect to e p , which is the case if the effective stiffness tensor is either isotropic or orthotropic with the orthotropy axis aligned with the symmetry axis of the inclusion, equation (9) can be fully decoupled giving n dim independent scalar equations, where, Here, C θpp = e p · C θ · e p and ρ pp (ω) = e p · ρ(ω) · e p . The solution for the wave number derived from equations (16) and (17) is given as where, represents the effective wave speed in the quasi-static limit of the system. In accordance with the normalization of υ p with respect to C θ given by equation (11) , the expressions for the wave polarization modes are Hence, in the plane symmetric case, one compression mode (with υ p || e θ for a given value of p) and two shear (one in the 2D case) modes (with υ q ⊥ e θ for 285 p = q) are always observed without formation of hybrid wave modes, for wave propagation along the symmetry axis.
Since the combined plane and 4-fold symmetry is a special case of plane symmetry, the scalar dispersion relations (18) also hold in this case.

Homogenizability limit 290
A criterion on the applicability limit of the developed semi-analytical analysis can be obtained heuristically. The relaxed scale separation principle (stated above in Section 2 ) no longer applies when the wavelength of the macroscopic wave in the matrix approaches the size of the heterogeneities. At that limit, higher order scattering effects start to play a role which is not accounted for by 295 the present homogenization theory. A safe estimate for the scale separation limit is when the matrix wavelength is at least 10 times the relevant microstructural dimension, i.e. k p < 0.2 π , where is the relevant dimension for all p = 1, ..n dim .
For the local resonance frequencies to lie within this limit, the effective stiffness of the matrix structure (both shear and compressive) has to be at least 300 100 times higher than the effective stiffness of the inclusion. The high stiffness of the matrix structure also implies that the local resonances will be internally contained, i.e. preventing any interaction between neighboring inclusions. This also imposes a constraint on the maximum volume fraction of the inclusions, which is largely determined by the material properties of the matrix and the

Numerical case study and validation
In order to validate the proposed semi-analytical methodology and to illustrate the results presented thus far, the dispersion properties of three 2D LRAM RVE designs shown in Figure 1 are studied. A periodic micro-structure is assumed, hence the RVEs considered here represent the periodic unit cell.

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The periodicity is assumed for the sake of simplicity of analysis since the distribution is captured by a single inclusion. It also allows Bloch analysis [14] to be performed in order to compute the reference solution. However, the results obtained for a periodic unit can to a large extent be applied to a random distribution of the inclusions as well, provided that the average volume of the  Table 1a and Table 1b,   The homogenized enriched continuum parameters are extracted from the fi-360 nite element models by applying the method described in [21]. The values of the computed static effective parameters are given in Table 2a. Due to the high compliance of the rubber coating in comparison to the epoxy, the overall effective stiffness of each unit cell only differs up to 1%, and hence this difference is neglected in the Table. The effective mass densities of all the designs are identical. The dispersion spectra for the considered unit cells are computed using equation (8) and validated against the spectra computed using Bloch analysis [14] as shown in Figure 2. An excellent match is observed between the spectra computed using the two methods for all designs within the displayed frequency 390 range. The group velocities observed at approximately 0.2 times the distance from Γ to X for the three unit cells are all close to zero, hence confirming again that within the considered frequency range, the analysis lies well within the homogenizability limit discussed in Section 3.3. To summarize this section, the various local resonance effects resulting from complex micro-structure designs such as total and selective bandgaps and hybrid wave modes have been illustrated. The semi-analytical model accurately 425 Figure 3: The general macroscopic acoustic boundary value problem for normal wave incidence.
captures all effects in the given frequency regime, while being computationally much more efficient and faster compared to a full Bloch analysis.

Transmission analysis
In this section, a transmission analysis framework is derived for wave propagation in an enriched media at normal incidence. It is a generalization of the 430 standard transfer matrix method [36] used in the analysis of plane wave propagation through a layered arrangement of several dissimilar materials, which can in general be distinct locally resonant metamaterials, described by enriched effective continuum. The technique can be extended to oblique wave incidence, but this is beyond the scope of the present paper. The framework is applied 435 to analyze the steady state responses of systems constructed using PlySym and UC3 unit cell designs including the influence of the finite size of the structure and the boundary conditions.

Theoretical development
The general problem can be described as a serial connection of m layered (enriched) media with the first and the last layer being semi-infinite as shown in Figure 3. Since only normal wave incidence is considered, the wave direction vector e θ defines the 1D macro-scale coordinate axis. Let x represent the corresponding spatial coordinate. The general plane wave solution at x corresponding to medium r can be expressed as a sum of the forward and backward propagating components represented by rˆ u f and rˆ u b respectively. Hence, The bounding semi-infinite media will posses only a forward wave or a backward wave depending whether its boundary is on the left or the right side, respectively.
Each component of the total displacement is composed of the individual wave where, rξ fp and rξ bp , p = 1, ..n dim are the wave mode amplitudes of the forward and backward waves, respectively. These can be found by making use of the normalization condition (11), giving Next, the traction-displacement relation needs to be setup. Let rˆ t + (x) denote the macro-scale traction vector in the r th medium with respect to e θ (a "−" superscript is used to indicate traction with respect to − e θ ). This can be determined from the effective homogenized constitutive relation (4a), disregarding the last term as discussed before (see the last paragraph in Section 2), i.e. rˆ t + (x) = e θ · r σ(x) = e θ · r C (4) .
The displacement gradient is expressed in terms of the plane wave solution by making use of equations (21), (22) and (23), where r κ p (ω) = ω −1r k p (ω) has been introduced. Substituting equation (25) in (24) gives, where, is the effective impedance of the considered metamaterial medium, defined as the respectively is obtained by applying equation (22) and (23), where, is the effective transfer operator. Finally, the continuity conditions at the media interfaces are established. Let r x give the coordinate of the r th and (r+1) th interface. Applying equations (21) and (26), the traction and displacement continuity conditions between the r and (r + 1) th medium r x can respectively be written as, Therefore any general problem can be solved by applying equations (28) and (30) for all media with the appropriate constraints/boundary conditions. Hz in order to capture the first two compressive wave bandgaps and the first three shear wave bandgaps (see Figure 2b). An excellent match between the 485 semi-analytical model and DNS is obtained.

Numerical case study and validation
One of the interesting observations is that the shear transmission bandgaps are more pronounced (deeper) than those of the compressive wave. This can be attributed to the fact that the effective shear stiffness of the unit cell is much lower than its compressive stiffness. Due to the requirements of the scale 490 separation, the semi-analytical model is only valid for a relatively stiff matrix material. Beyond this limit, Bragg scattering effects start to play a role, leading to hybridization of Bragg and local resonance effects [37]. This effect does enhance the attenuation performance of the metamaterial but at a cost of overall structural stiffness, hence there is a tradeoff. LRAMs can therefore be employed 495 in applications where a higher structural stiffness are required.
Next, the transmission analysis is performed exactly with UC3 as the LRAM medium ( Figure 5b). Again, the results obtained from the semi-analytical approach match very well with DNS. The UC3 response is similar to UC2 except for the reduced attenuation rate and some additional cross coupling of trans-500 mission spectra at the local resonance frequencies. This is due to the fact that a propagating hybrid wave mode will always be excited in UC3 for pure horizontal or vertical excitations at the local resonance frequencies, leading to some residual transmission. The effects due to the hybrid wave modes occurring in UC3 metamaterial are further illustrated in Figure 6, where the absolute verti-505 cal displacement at the right interface is shown, for the unit horizontal applied displacement. The frequency now ranges from 300 − 700Hz in order to capture the effects due to the first local resonance at 508Hz. Comparing the results with DNS shows, once again, a perfect match between the two approaches. The peak vertical displacements in the LRAM medium is observed exactly at the 510 Figure 6: The transmission ratio for applied horizontal excitation and measured vertical displacement in the example macro-scale problem using UC3 as the LRAM medium computed using the semi-analytical approach (SA) and direct numerical simulation (DNS).
first local resonance frequency and drops rapidly away from it, indicating the formation of hybrid modes due to the dynamic anisotropy of the mass density tensor at the resonance frequency. This characteristic response of the UC3 LRAM can lead to interesting applications. As a consequence of horizontal applied displacements on the LRAM, a shear wave will be excited in the ho-515 mogeneous medium bounding the LRAM. This enables the design of selective mode converters that convert an incident wave mode into another mode upon transmission at particular frequencies for normal wave incidence.

Conclusion
A multiscale semi-analytical technique was presented for the acoustic plane Applying the boundary conditions on the plane wave solution given by equation (30) yields, Note that the overbar is not applied on medium 1 and 3 to indicate that they are not homogenized quantities. Solving for 2ˆ u b ( 2 x) with respect to 2ˆ u f ( 2 x) using equations (A.1c) and (A.1d) gives where, is the effective reflection coefficient tensor at the given interface. Applying the transfer operation as defined by equation (28) gives 2ˆ u f ( 2 x) = 2 T(n ) · 2ˆ u f ( 1 x) and 2ˆ u b ( 1 x) = 2 T(n ) · 2ˆ u b ( 2 x). Combining the result with equation (A.2) gives where A Tr is the effective transmission coefficient between the applied displacement and the transmitted wave in medium 3, expressed as A Tr = I + A Re · 2 T(n ) · I + 2 T(n ) · A Re · 2 T(n ) −1 . (A.8)