A semi-analytical approach towards plane wave analysis oflocal resonance metamaterials using a multiscale enrichedcontinuum descriptionCitation for published version (APA):Sridhar, A., Kouznetsova, V., & Geers, M. G. D. (2017). A semi-analytical approach towards plane wave analysisof local resonance metamaterials using a multiscale enriched continuum description. International Journal ofMechanical Sciences, 133, 188-198. https://doi.org/10.1016/j.ijmecsci.2017.08.027

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International Journal of Mechanical Sciences 133 (2017) 188–198

Contents lists available at ScienceDirect

International Journal of Mechanical Sciences

journal homepage: www.elsevier.com/locate/ijmecsci

A semi-analytical approach towards plane wave analysis of local resonance

metamaterials using a multiscale enriched continuum description

A. Sridhar, V.G. Kouznetsova

∗ , M.G.D. Geers

Eindhoven University of Technology, Department of Mechanical Engineering, P.O. Box 513, Eindhoven 5600 MB, The Netherlands

a r t i c l e i n f o

Keywords:

Local resonance

Acoustic metamaterials

Enriched continuum

Semi-analytical

Multiscale

Acoustic analysis

a b s t r a c t

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 methodology establishes the required scale bridging in

multiscale modeling for dispersion and transmission analyses, enabling rapid design and prototyping of local

resonance metamaterials.

© 2017 The Authors. Published by Elsevier Ltd.

This is an open access article under the CC BY license. ( http://creativecommons.org/licenses/by/4.0/ )

1

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. Introduction

Acoustic metamaterials can be used to engineer systems that are ca-

able of advanced manipulation of elastic waves, such as band-stop fil-

ering, redirection, channeling, multiplexing etc. which is impossible

sing ordinary materials [1–4] . This can naturally lead to many poten-

ial applications in various fields, for example medical technology, civil

ngineering, defense etc. The extraordinary properties of these materi-

ls are a result of either one or a combination of two distinct phenom-

na, namely local resonance and Bragg scattering. Bragg scattering is

xhibited by periodic lattices in the high frequency regime where the

ropagating wavelength is of the same order as the lattice constant. Lo-

al resonance, on the other hand, is a low frequency/long wavelength

henomena and, in general, does not require periodicity. “Local reso-

ance acoustic metamaterial ” (LRAM) is therefore the term used here

o distinguish the subclass of acoustic metamaterials based solely on lo-

al resonance. The present work is only concerned with the modeling

nd analysis of LRAMs restricted to linear elastic material behavior in

he absence of damping.

∗ Corresponding author.

E-mail address: v.g.kouznetsova@tue.nl (V.G. Kouznetsova).

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ttp://dx.doi.org/10.1016/j.ijmecsci.2017.08.027

eceived 20 April 2017; Received in revised form 4 August 2017; Accepted 10 August 2017

vailable online 13 August 2017

020-7403/© 2017 The Authors. Published by Elsevier Ltd. This is an open access article under

Based on the primary medium of the wave propagation, LRAMs can

e 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

ype LRAMs. A typical representative cell of such LRAMs is character-

zed by a relatively stiff matrix containing a softer and usually heavier

nclusion. The micro-inertial effects resulting from the low frequency

ibration modes of the inclusion induce the local resonance action. The

omplexity of the geometry of the inclusion plays an important role in

he response of the LRAM. For instance multi-coaxial cylindrical inclu-

ions have been proposed [7] , which exhibit more pronounced micro-

nertial effects due to the larger number of local resonance vibration

odes, hence leading to more local resonance bandgaps. The symme-

ries of the inclusion also play a key role. A ‘total ’ (or omni-directional)

andgaps, where any wave polarization along any given direction is at-

enuated within a given frequency range is observed in micro-structures

xhibiting mode multiplicity (or degenerate eigenmodes) resulting from

combined plane and 4-fold (w.r.t. 90° rotation) symmetry. For geome-

ries with only plane symmetry, ‘selective ’ (or directional) bandgaps,

hich only attenuate certain wave modes in a given frequency range

an be observed. Furthermore, if the inclusion is not plane symmetric

ith respect to the direction of wave propagation, hybrid wave modes,

hich are a combination of compressive and shear wave modes can be

the CC BY license. ( http://creativecommons.org/licenses/by/4.0/ )

A. Sridhar et al. International Journal of Mechanical Sciences 133 (2017) 188–198

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bserved. In general, a more extensive micro-dynamic behavior results

rom an increased complexity in the LRAM design.

A plethora of approaches are available for modeling LRAMs. For

teady state analysis of periodic structures, the Floquet–Bloch theory

8] gives the general solution that reduces the problem to the disper-

ion analysis of a single unit cell. The unit cell problem can then be fur-

her discretized and solved using appropriate solution methodologies. A

opular technique highly suited to composites with radially symmetric

nclusions is given by Multiple Scattering theory (MST) [9,10] . How-

ver, MST has not been successfully applied to more complex unit cell

esigns. The finite element (FE) method provides a robust approach for

odeling arbitrary and complex unit cell designs. Such an approach,

lso known as the wave finite element (WFE) method, has been exten-

ively employed in literature for analyzing periodic structures including

coustic metamaterials [11–14] .

Since LRAMs operate in the low frequency, long wave regime, it is

ppropriate to exploit this fact towards approximating the general so-

ution, leading to simpler methodologies. This is equivalent to ignor-

ng nonlocal scattering effects resulting from reflections and refractions

t the interfaces of the heterogeneities. This is the formal assumption

ade in many dynamic homogenization/effective medium theories [15–

1] which recover the classical balance of momentum, but where the lo-

al resonance phenomena manifests in terms of dynamic (frequency de-

endent) constitutive material parameters. The most striking feature of

he effective parameters is that they can exhibit negative values, which

ndicates the region of existence of a bandgap. The specific macro-scale

oupling of the local resonance phenomena is dictated by the vibration

ode type of the inclusion [15] . A negative mass density is obtained

n LRAMs exhibiting dipolar resonances e.g., [22] . Similarly, a negative

ulk and shear modulus are obtained in LRAMs exhibiting monopolar

nd quadrupolar resonances, e.g. [23] , though not further considered

n this work. The homogenized models accurately capture the disper-

ion spectrum of LRAMs provided that the condition on the separation

f scales is satisfied, i.e. the wavelength of the applied loading is much

arger compared to the size of the inclusion. This can be ensured in the

ocal resonance frequency regime by employing a sufficiently stiff ma-

rix material compared to that of the inclusion.

The expression for the effective parameters of materials with radi-

lly symmetric inclusions has been obtained using analytical methods

sing the Coherent Potential Approximation (CPA) approach [15,16] .

generalization towards ellipsoidal inclusions has also been presented

n [17,24] , thereby extending the method of Eshelby [25] to the dy-

amic case. Analytical models for unit cells composed discrete elements

e.g. trusses) have also been derived [19,26,27] . For arbitrary complex

icro-structures, it is necessary to adopt a computational homogeniza-

ion approach. A FE based multiscale methodology in the framework

f an extended computational homogenization theory was presented in

20,21] .

The present work builds upon the computational homogenization

ramework introduced in [21] . A first-order multiscale analysis is com-

ined with model order reduction techniques to obtain an enriched con-

inuum model, i.e. a compact set of partial differential equations gov-

rning the macro-scale behavior of LRAMs. The reduced order basis is

onstructed through the superposition of the quasi-static and the micro-

nertial contribution, where the latter is represented by a set of local res-

nance eigenmodes of the inclusion. In the homogenization process, the

eneralized amplitudes associated to the local resonance modes emerge

s additional kinematic field quantities, enriching the macro-scale con-

inuum with micro-inertia effects in a micromorphic sense as initially

efined by Eringen [28] . An equivalent approach for modeling LRAM

as derived using asymptotic homogenization theory in [18] , but was

ot further elaborated and demonstrated as a computational technique.

In this work, the enriched continuum of [21] is exploited to develop

n ultra-fast semi-analytical technique method for performing disper-

ion and transmission analysis of LRAMs with arbitrary complex micro-

tructure geometries that retains the accuracy of WFE methods at a frac-

189

ion of the computational cost. A modified transfer matrix method is

eveloped based on the enriched continuum that can be used to derive

losed form solutions for wave transmission problems involving LRAMs

t normal incidence. Furthermore, the dispersion characteristics of the

nriched continuum and their connection to various symmetries of the

nclusion are discussed in detail. In order to ensure the accuracy of the

ethod in the frequency range of the analysis, a procedure for verify-

ng the homogenizability of a given LRAM is introduced. Although the

nriched continuum predicts both negative dynamic mass and elastic

odulus effects, the latter exhibits a significant coupling only at fre-

uencies close to and beyond the applicability (homogenizability) limit

f the present approach. Since such effects cannot be robustly modeled,

hey will not be considered in the present paper.

The structure of the paper is as follows. The relevant details of the

nriched multiscale methodology are briefly recapitulated in Section 2 .

n Section 3 , a plane wave transform is applied to obtain the dispersion

igenvalue problem of the enriched continuum. The influence of the in-

lusion symmetry on the dispersion characteristics is highlighted. Based

n the obtained dispersion spectrum, a procedure for checking the ap-

licability (homogenizability) of the problem in the frequency range of

nalysis is elaborated that provides a reasonable estimation of its valid-

ty. In Section 4 , a general plane wave expansion is applied to derive a

odified transfer matrix method for performing macro-scale transmis-

ion analyses of LRAMs at normal wave incidence. The results obtained

n each of the sections are illustrated with numerical case studies and

alidated against direct numerical simulations (DNS). The conclusions

re presented in Section 5 .

The following notation is used throughout the paper to represent dif-

erent quantities and operations. Unless otherwise stated, scalars, vec-

ors, second, third and fourth-order Cartesian tensors are generally de-

oted by a (or A ), 𝑎 , A , 𝔸

(3) and 𝔸

(4) respectively; n dim

denotes the

umber of dimensions of the problem. A right italic subscript is used

o index the components of vectorial and tensorial quantities. The Ein-

tein summation convention is used for all vector and tensor related

perations represented in index notation. The standard operations are

enoted as follows for a given basis 𝑒 𝑝 , 𝑝 = 1 , ..𝑛 𝑑𝑖𝑚 , dyadic product:

⊗ �� = 𝑎 𝑝 𝑏 𝑞 𝑒 𝑝 ⊗ 𝑒 𝑞 , dot product: 𝐀 ⋅ �� = 𝐴 𝑝𝑞 𝑏 𝑞 𝑒 𝑝 and double contraction:

∶ 𝐁 = 𝐴 𝑝𝑞 𝐵 𝑞𝑝 . Matrices of any type of quantity are in general denoted

y ( •) except for a column matrix, which is denoted by ( ∙˜ ) . A left su-

erscript is used to index quantities belonging to a group and to denote

ub-matrices of a matrix for instance for a and 𝑏 ˜ , by mn a and 𝑚 𝑏

˜ , respec-

ively. The transpose of a second order tensor is defined as follows: for

= 𝐴 pq 𝑒 𝑝 ⊗ 𝑒 𝑞 , 𝐀

T = 𝐴 qp 𝑒 𝑝 ⊗ 𝑒 𝑞 . The transpose operation also simulta-

eously yields the transpose of a matrix when applied to one. The first

nd second time derivatives are denoted by ( ∙) and ( ∙) respectively. The

ero vector is denoted as 0 .

. An enriched homogenized continuum of local resonance

etamaterials

This section summarizes the essential features of the enriched contin-

um model introduced in [21] . The framework is based on the extension

f the computational homogenization approach [29] to the transient

egime [20] . The classical (quasi-static) homogenization framework re-

ies on the assumption of a vanishingly small micro-structure in com-

arison to the macroscopic wavelength. No micro-inertial effects are

ecovered in this limit. The effective mass density obtained in this case

s a constant and is merely equal to the volume average of the micro-

tructure densities, unlike the frequency dependent quantity observed

n metamaterials.

The key aspect rendering the classical computational homogeniza-

ion method applicable to LRAMs is the introduction of a relaxed scale

eparation principle . Making use of the typical local confinement of the

esonators in LRAM, the long wave approximation is still assumed for

he matrix (host medium), while for the heterogeneities (resonators),

ull dynamical behavior is considered. Note, that the long wavelength

A. Sridhar et al. International Journal of Mechanical Sciences 133 (2017) 188–198

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pproximation poses a restriction on the properties of the matrix which

as to be relatively stiff in order to ensure that it behaves quasi-statically

n the local resonance frequency regime.

With this relaxed separation of scales, the multiscale formulation

or LRAM is established. The full balance of linear momentum equa-

ions are considered at both the micro and macro-scales. The domain

f the micro-scale problem associated to each point on the macro-scale

omain, termed the Representative Volume Element (RVE), is selected

uch that it is statistically representative, i.e. captures all the represen-

ative micro-mechanical and local resonance effects at that scale. For

periodic system, the RVE is simply given by the periodic unit. The

omogenization method can be applied to a random distribution of in-

lusions 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 consis-

ency between the two scales, is generalized by taking into account the

ontribution of the momentum into the average energy density at both

cales. As such, the resulting computational homogenization framework

s valid for general non-linear problems. Restricted to linear elasticity,

compact set of closed form macro-scale continuum equations can be

btained. To this end, exploiting linear superposition, the solution to

he micro-scale problem is expressed as the sum of the quasi-static re-

ponse, which recovers the classical homogenized model, and the micro-

nertial contribution spanned by a reduced set of mass normalized vi-

ration eigenmodes 𝑠 𝜙, of the inclusion (shown in continuum form) as

ollows,

m

( 𝑥 m

) = 𝑢 + 𝕊 (3) ( 𝑥 m

) ∶ ∇ 𝑢 +

∑𝑠 ∈

𝑠 𝜙( 𝑥 m

) 𝑠 𝜂 , (1)

here 𝑢 m

and �� m

represent the micro-scale displacements and position

ector respectively, 𝑢 the macro-scale displacement and s 𝜂 the general-

zed (modal) amplitude of the s th local resonance eigenmode of the inclu-

ion indexed by the set . The size of denoted by 𝑁( ) gives the num-

er of modal degrees of freedom. The third order tensor 𝕊 (3) ( 𝑥 m

) repre-

ents the quasi-static response of the RVE to applied macro-scale strain

nder periodic boundary conditions. The eigenmodes are obtained via

n eigenvalue analysis of a single inclusion unit under fixed displace-

ent boundary condition. For LRAMs, these eigenmodes naturally rep-

esent the local resonance vibration modes of the system. Only one in-

lusion needs to be considered here even for random distributions since

ll relevant micro-inertial properties can be obtained from it. The gener-

lized degrees of freedom associated to these eigenmodes then emerge

t the macro-scale as additional internal field variables, thus resulting

n an enriched continuum description. FE is used to discretize and solve

or the quasi-static response and the eigenvalue problem.

Further details of the derivation of this framework can be found in

21] . In the following, only the final equations describing the homoge-

ized enriched continuum are given.

The macro balance of momentum

⋅ 𝝈T −

𝑝 = 0 . (2)

The reduced micro balance of momentum (to be solved at the macro-

cale)

𝜔

2 res

𝑠 𝜂 +

𝑠 �� = −

𝑠 𝑗 ⋅ 𝑢 −

𝑠 𝐇 ∶ ∇ 𝑢 , 𝑠 ∈ . (3)

The homogenized constitutive relations

= ℂ

(4) ∶ ∇ 𝑢 +

1 𝑉

∑𝑠 ∈

𝑠 𝐇

𝑠 ��, (4a)

= 𝜌 𝑢 +

1 𝑉

∑𝑠 ∈

𝑠 𝑗 𝑠 ��. (4b)

Here, 𝝈 represents the macro-scale Cauchy stress tensor and 𝑝 the

acro-scale momentum density vector. The terms ℂ

(4) and 𝜌 are, re-

pectively, the effective linear (static) elastic stiffness tensor and the

190

ffective mass density of the RVE (the over-bar ( ∙) is used to distinguish

n effective material property from its respective counterpart of a homo-

eneous material). These are the classical quantities obtained under the

uasi-static approximation of the micro-structure and are not to be con-

used with their dynamic counterparts [15] . The term

𝑠 𝜔 res represents

he eigenfrequency (in radians per second), of the s th local resonance

igenmode of the inclusion. The formulation of depends on the mini-

um number of eigenmodes, required to sufficiently accurately capture

he dispersive behavior of the system in the desired frequency regime. A

ore precise mode selection criterion is discussed in Section 3 . The set

is indexed in the order of increasing eigenfrequencies, i.e. for 𝑟, 𝑠 ∈ ,

< s implies 𝑟 𝜔 res ≤

𝑠 𝜔 res . The coupling between the macro and reduced

icro-scale balance equations is represented by the vector 𝑠 𝑗 and the

ensor 𝑠 𝐇 . The vector 𝑠 𝑗 describes the coupling of dipolar local reso-

ance modes, whereas 𝑠 𝐇 gives the coupling of monopolar, torsional

nd quadrupolar modes. Finally, in Eq. (4), V represents the volume of

he RVE. The coefficients ℂ

(4) , 𝜌, 𝑠 𝑗 , 𝑠 𝐇 and 𝑠 𝜔 res ( 𝑠 ∈ ) constitute the

et of effective material parameters characterizing an arbitrary LRAM

VE. They are obtained through discretization and model reduction (See

21] for details). Specifically, the parameters 𝑠 𝑗 and 𝑠 𝐇 are obtained by

rojecting the computed inertial force per unit amplitude of the local

esonance eigenmodes onto the rigid body and the static macro-strain

eformation mode respectively. The explicit expressions for these effec-

ive parameters in continuum form are given as follows,

𝑗 = ∫𝑉

𝜌( 𝑥 m

) 𝑠 𝜙( 𝑥 m

) d 𝑉 , (5a)

𝐇 = ∫𝑉

𝑠 𝜙( 𝑥 m

) ⋅ ( 𝜌( 𝑥 m

) 𝕊 (3) ( 𝑥 m

))d 𝑉 , (5b)

here 𝜌( 𝑥 m

) represents the mass density of the heterogeneous micro-

tructure. Subsequently, the system (2) –(4) can be solved at the macro-

cale.

In the following sections, the analysis will be restricted to dipolar

ocal resonances, i.e. the coefficient 𝑠 𝐇 will be disregarded in the se-

uel. The justification for this is as follows. Since the primary concern

f the present paper is to develop an accurate methodology compara-

le to standard numerical methods, the analysis is restricted to the deep

ub-wavelength regimes, where the applicability of the method (i.e. ho-

ogenizability of the problem) is guaranteed. The details of the homog-

nizability criterion are elaborated in Section 3.3 , where it is required

or the effective stiffness of the matrix structure (both shear and com-

ressive) to at least 100 times higher than that of the inclusion. In this

ase, the boundary of the inclusion is effectively rigid, thereby suppress-

ng the action of monopolar and quadrupolar eigenmodes, which nec-

ssarily require a deformable boundary [15] . This fact has also been

videnced in [18] where a homogenized model has been derived via

symptotic homogenization. Indeed, within the homogenizability limit

f the proposed method, the bandgaps due to negative stiffness effects

ave negligible bandwidths and appear as flat branches (see for e.g. the

ranch corresponding to mode 7 in Fig. 2 a representing a monopolar

esonance). Since only dipolar resonances are dominant in the regime

onsidered here, it is therefore justified to neglect the term

𝑠 𝐇 in further

erivations for the sake of simplification.

. Dispersion analysis

In this section, a plane wave analysis is carried out on an infinite en-

iched continuum, i.e. without taking into account macro-scale bound-

ry conditions. First, a plane wave (Fourier) transform is applied on

qs. (2) –(4) to obtain the dispersion eigenvalue problem valid for an

rbitrary LRAM RVE design. The relation between the emergent dis-

ersion characteristics and the properties of the dynamic mass density

ensor and the geometric symmetries of the inclusion is then discussed.

procedure for checking the homogenizability of the approach is then

A. Sridhar et al. International Journal of Mechanical Sciences 133 (2017) 188–198

(a) (b) (c)

Γ

Fig. 1. The unit cell designs used for the case studies (a) UC1, (b) UC2 and (c) UC3.

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stablished via dispersion analysis. Three example periodic unit cell de-

igns are introduced as case studies to illustrate the influence of the

nclusion geometry on the general dispersion characteristics of LRAMs

nd to validate the methodology against standard Bloch analysis.

.1. Theoretical development

The plane wave transform of any continuous field variable can be

xpressed as,

∙) =

(∙)𝑒 𝑖 𝑘 ⋅�� − 𝑖𝜔𝑡 , (6)

here the ( ∙) represents the transformed variable, �� denotes the wave

ector, 𝜔 the frequency and i the imaginary unit. The wave vector can

e represented by its magnitude and direction as 𝑘 = 𝑘 𝑒 𝜃, where 𝑘 =

2 𝜋𝜆

s the wave number, 𝜆 the corresponding wavelength and 𝑒 𝜃 a unit vec-

or in the direction of wave propagation. The above expression provides

he plane wave transform in an infinite medium. Applying the transfor-

ation (6) to Eqs. (2) –(4) while disregarding 𝑠 𝐇 as discussed before (see

he last paragraph in Section 2 ) yields,

𝑖 𝑘 ⋅ 𝝈T + 𝑖𝜔 𝑝 = 0 , (7a)

𝜔

2 res

𝑠 �� − 𝜔

2 𝑠 �� = 𝜔

2 𝑠 𝑗 ⋅ 𝑢 , 𝑠 ∈ (7b)

𝝈 = 𝑖 (( ℂ )

(4) ⋅ ��

)⋅ 𝑢 , (7c)

= − 𝑖𝜔

(

𝜌 𝑢 +

1 𝑉

∑𝑠 ∈

𝑠 𝑗 𝑠 ��

)

. (7d)

Eliminating 𝝈 and 𝑝 by substituting Eq. (7c) and (7d) into equation

7a) and combining it with Eq. (7b) results in the following parameter-

zed eigenvalue problem in 𝜔 ,

[

𝑘 2 𝐂 𝜃 ��

˜ T

��

˜ 𝜔

2 res

]

− 𝜔

2 ⎡ ⎢ ⎢ ⎣ 𝜌𝐈 1

𝑉 𝑗 ˜

T

𝑗 ˜

𝐼

⎤ ⎥ ⎥ ⎦ ⎞ ⎟ ⎟ ⎠ [

⋅ 𝑢 ��˜

]

=

[

0 𝑂

˜

]

, (8)

here 𝐂 𝜃 = 𝑒 𝜃 ⋅ ℂ

(4) ⋅ 𝑒 𝜃, 𝜔

res is a 𝑁 ×𝑁 diagonal matrix of the eigen-

requencies, ��˜

and 𝑗 ˜

are column matrices of size 𝑁 containing the

odal amplitudes s 𝜂 and the coupling vectors 𝑠 𝑗 , respectively; ��

˜ and

˜ are column matrices of size 𝑁 with all entries equal to the zero vec-

or 0 and scalar 0, respectively. The above eigenvalue problem is the

o-called 𝑘 − 𝜔 form, in which 𝜔 is the eigenvalue and k and 𝑒 𝜃 are pa-

ameters. The problem consists of 𝑛 𝑑𝑖𝑚 + 𝑁 variables, which gives the

umber of dispersion branches predicted by the model. The alternative

− 𝑘 form in which k is the eigenvalue and 𝜔 and 𝑒 𝜃 are the parameters

an be obtained by eliminating ��˜

from the system of Eqs. (8) resulting

n

𝑘 2 𝐂 𝜃 − 𝜔

2 𝝆( 𝜔 )

)⋅ 𝑢 = 0 , (9)

here

( 𝜔 ) = 𝜌𝐈 +

∑𝑠 ∈

𝜔

2

𝑠 𝜔

2 res − 𝜔

2 𝑠 𝐉 , (10)

t

191

here 𝑠 𝐉 =

1 𝑉

𝑠 𝑗 ⊗ 𝑠 𝑗 is a measure of the translational inertia associated

o each local resonant eigenmode, termed modal mass densities and 𝝆( 𝜔 )s the effective dynamic mass density tensor which is a well known quan-

ity in the literature [18,22,31,32,33] . In most works, however, the ex-

ression for the dynamic mass density is derived for a particular unit

ell design often after some simplifications and approximations. Here,

n the contrary, Eq. (10) holds for arbitrary geometries, for which it is

btained numerically.

The solution to Eq. (9) at a given 𝜔 yields n dim

eigen wave num-

ers k p ( 𝜔 ), 𝑝 = 1 ..𝑛 𝑑𝑖𝑚 and corresponding eigen wave vectors denoted

y 𝜐𝑝 ( 𝜔 ) . These vectors are also termed polarization modes. Let 𝜐𝑝 ( 𝜔 ) be

ormalized with respect to 𝐂 𝜃 , i.e.

𝑝 ⋅ 𝐂 𝜃 ⋅ ��𝑞 = 𝛿𝑝𝑞 . (11)

n the present context, only positive real or imaginary eigenvalue solu-

ions are considered to avoid ambiguity in the direction of wave prop-

gation or decay respectively. If ��𝑝 ( 𝜔 ) is parallel to 𝑒 𝜃 (i.e. ��𝑝 × 𝑒 𝜃 = 0 )or a given p , the corresponding wave mode is purely compressive. On

he other hand if 𝜐𝑝 ( 𝜔 ) is orthogonal to 𝑒 𝜃 (i.e. 𝑒 𝜃 ⋅ ��𝑝 = 0 ), the resulting

ode is purely shear. All other combinations represent hybrid modes.

For an arbitrary RVE geometry and wave direction, the dispersion

roblem (8) and (9) can be solved numerically which, owing to the re-

uced nature of the enriched continuum problem, is computationally

uch cheaper in comparison to standard Bloch analysis techniques, es-

ecially in the 3D case. It is arguably cheaper even compared to reduced

rder Bloch analysis techniques [34,35] since the spectrum of the ho-

ogenized problem is much smaller than the Bloch spectrum leading to

more compact dispersion equation.

If accounts for all dipolar modes, the following relation holds due

o the mass orthogonality of eigenmodes

∑ ∈

𝑠 𝐉 = 𝜇inc 𝜌𝐈 , (12)

here 𝜇inc is the static mass fraction of the inclusion defined as the

atio of the mass of the inclusion (including the central mass and the

ompliant support) and the total mass of the RVE. The above relation

an be used to formulate a mode selection criterion for setting up .

rojecting Eq. (12) along some arbitrary set of orthonormal basis vectors

𝑝 , 𝑝 = 1 ..𝑛 𝑑𝑖𝑚 gives a set of scalar equations,

∑ ∈

𝑠 𝜇J 𝑝 = 𝜇inc . (13)

here, 𝑠 𝜇J 𝑝 = ( 𝜌) −1 𝑒 𝑝 ⋅ 𝑠 𝐉 ⋅ 𝑒 𝑝 , 𝑠 ∈ is the modal mass fraction of the lo-

al resonant eigenmode along 𝑒 𝑝 for 𝑝 = 1 ..𝑛 𝑑𝑖𝑚 . In practice, it is suf-

cient if the above expression is satisfied in the approximation as the

ocal resonance modes in general will never fully be able to capture all

he mass of the inclusion. The tolerance can vary based on the design

equirements but a good measure would be 5% of the inclusion mass

raction. Thus, should be the smallest set of low frequency eigenmodes

hat satisfies Eq. (13) within the given tolerance.

A. Sridhar et al. International Journal of Mechanical Sciences 133 (2017) 188–198

Fig. 2. The dispersion spectra of the considered unit cells, (a) UC1, (b) UC2 and (c) UC3

computed using Bloch analysis (black dashed lines) and the presented semi-analytical

(SA) approach (colored lines). The mode shapes of some of the local resonant eigenmodes

and their associated dispersion branches are also shown. The topmost colorbar represents

the cosine of 𝜐𝑝 , 𝑝 = 1 , 2 , with respect to 𝑒 𝜃 , red indicating a pure compression wave and

blue a pure shear wave, intermediate colors represent hybrid wave modes. The colorbar

below indicates the norm of the displacements of the local resonance eigenmodes. (For

interpretation of the references to colour in this figure legend, the reader is referred to the

web version of this article.)

3

d

b

c

e

F

a

f

p

o

m

o

I

t

o

r

e

r

𝝆

a

i

e

h

w

d

t

4

(

t

F

i

4

p

q

s

t

{

t

W

p

t

𝝆

w

t

w

f

b

1

e

𝑠

s

i

p

𝝆

w

n

.2. Relation between the dispersion characteristics, the dynamic mass

ensity tensor and geometrical symmetries of the inclusion

The dispersion characteristics of a LRAM are primarily determined

y the properties of 𝝆( 𝜔 ) . It is anisotropic in general and its components

an take all values from [ −∞, +∞] at different frequencies. A bandgap

192

xists at 𝜔 when at least one of the eigenvalues of 𝝆( 𝜔 ) is negative.

rom Eq. (10) , it can be concluded that a bandgap is always initiated

t s 𝜔 , 𝑠 ∈ , where the determinant of 𝝆 jumps from ∞ to −∞. For

requencies close to s 𝜔 , the dispersion is strongly determined by the

roperties of the corresponding 𝑠 𝑗 vector due to the large magnitude

f its coefficient in Eq. (10) . Around that frequency, the polarization

odes 𝜐𝑝 , 𝑝 = 1 , ..𝑛 𝑑𝑖𝑚 obtained from the dispersion problem (9) become

riented with respect to 𝑠 𝑗 , either along the vector or orthogonal to it.

f 𝑠 𝑗 is not parallel or orthogonal to 𝑒 𝜃, hybrid wave solutions are ob-

ained. Note, that hybrid wave modes can also result from the anisotropy

f the effective static stiffness tensor. However, only the hybrid modes

esulting from local resonances are considered here.

A “total ” bandgap is formed in the frequency range where all the

igenvalues of 𝝆 are negative (i.e. it is negative definite). Within this

ange, all waves regardless of the direction will be attenuated. When

is only negative semi-definite, a “selective ” bandgap is formed that

ttenuates some of the wave modes. Two bandgaps will overlap only

f a pair of eigenfrequencies s 𝜔 , r 𝜔 , 𝑠, 𝑟 ∈ , 𝑠 ≠ 𝑟 are sufficiently close

nough and 𝑠 𝑗 is orthogonal to 𝑟 𝑗 . In the special case of a system ex-

ibiting mode multiplicity, i.e. 𝑠 𝜔 =

𝑟 𝜔 and ∥𝑠 𝑗 ∥= ∥𝑟 𝑗 ∥, the bandgaps

ill overlap perfectly. Such a situation is guaranteed for an inclusion

omain with isotropic constituents that possesses the following symme-

ries; plane symmetry with respect to 2 mutually orthogonal axes, and a

-fold (discrete 90°) rotational symmetry about the normal to these axes

exhibited for e.g. by a square, cross, cylinder, hexagonal prism etc.). In

his case, all 𝑟 𝑗 vectors will be aligned along one of the symmetry axis.

or a 3D inclusion, a mode multiplicity of 3 is observed provided the

nclusion has plane symmetry about 3 mutually orthogonal axes and

-fold rotational symmetry about these axes.

For such a highly symmetric inclusion domain, the set can be re-

arameterized by gathering all the 𝑠 𝑗 vectors with duplicate eigenfre-

uencies and labeling them using the same number. Let 𝑆 represent

uch a set. Here 𝑁( 𝑆 ) = 𝑛 −1 𝑑𝑖𝑚

𝑁( ) . Let 𝑒 𝑝 , 𝑝 = 1 , ..𝑛 𝑑𝑖𝑚 be the unit vec-

ors representing the symmetry axis. For every 𝑠 ∈ 𝑆 , a set of vectors

𝑠 𝑗 𝑝 =

√𝑠 𝜇J 𝜌𝑉 𝑒 𝑝 , 𝑝 = 1 , ..𝑛 𝑑𝑖𝑚 } is obtained where 𝑠 𝜇J = ( 𝜌𝑉 ) −1 ‖𝑠 𝑗 𝑝 ‖2 , is

he modal mass fraction associated to that degenerate eigenmode group.

ith this, the expression for the effective mass density tensor (10) for

lane + 4-fold symmetry case can be re-written by gathering all the

erms under the duplicate eigenfrequencies as follows,

( 𝜔 ) = 𝜌𝐈 +

∑𝑠 ∈ 𝑆

𝜔

2

𝑠 𝜔

2 res − 𝜔

2 1 𝑉

𝑛 𝑑𝑖𝑚 ∑𝑝 =1

𝑠 𝑗 𝑝 ⊗𝑠 𝑗 𝑝 ,

= 𝜌(1 +

∑𝑠 ∈ 𝑆

𝜔

2

𝑠 𝜔

2 res − 𝜔

2 𝑠 𝜇J ) 𝐈 , (14)

here the fact ∑𝑛 𝑑𝑖𝑚

𝑝 =1 𝑒 𝑝 ⊗ 𝑒 𝑝 = 𝐈 has been used. Thus, the mass density

ensor becomes isotropic and diagonal in this case. Such a tensor is al-

ays either positive or negative definite at any given frequency. There-

ore, an inclusion with plane and 4-fold symmetry only exhibits total

andgaps and no hybrid wave solutions.

For inclusions with only plane symmetry with respect to 𝑒 𝑝 , 𝑝 = , ..𝑛 𝑑𝑖𝑚 (but not the 4-fold symmetry), for e.g. an ellipsoid, rectangle

tc., degenerate eigenmodes are no longer guaranteed, however the

𝑗 vectors will still align along 𝑒 𝑝 . Now the set is divided into n dim

ubsets 𝑝

′ defined as 𝑝

′ = { 𝑠 ∈ | 𝑒 𝑝 ⋅ 𝑠 𝐉 ⋅ 𝑒 𝑝 ≠ 0} for 𝑝 = 1 ..𝑛 𝑑𝑖𝑚 . Us-

ng Eq. (10) , the mass density tensor can now be re-written for the

lane symmetry case as follows,

( 𝜔 ) = 𝜌

(

𝐈 +

𝑛 𝑑𝑖𝑚 ∑𝑝 =1

∑𝑠 ∈𝑝 ′

𝜔

2

𝑠 𝜔

2 res − 𝜔

2 𝑠 𝜇J 𝑒 𝑝 ⊗ 𝑒 𝑝

)

, (15)

here 𝑠 𝜇J = ( 𝜌𝑉 ) −1 ‖𝑠 𝑗 ‖2 . Thus the dynamic mass density tensor is

ow orthotropic, or diagonal with respect to the axis defined by 𝑒 𝑝 ,

A. Sridhar et al. International Journal of Mechanical Sciences 133 (2017) 188–198

𝑝

𝑒

b

𝐂

t

a

d

𝑘

w

𝜌

H

w

𝑘

w

𝑐

r

t

b

𝜐

H

f

𝜐

m

p

c

3

a

c

o

e

r

A

l

𝑘

t

t

s

r

b

m

b

o

o

S

w

o

t

t

b

Table 1

The geometric and material parameters of the considered unit cell designs.

(a) Geometric parameters of the unit cell

𝐷 in Diameter of lead inclusion 10 mm

𝐷 out Outer diameter of rubber coating 15 mm

w Width of hard rubber insert 2 mm

𝓁 Length of the unit cell ( 𝓁 1 = 𝓁 2 ) 20 mm

V Volume of unit cell 400 ×10 3 mm

3

(b) Istropic linear elastic material parameters

Material 𝜌 [kg/m

3 ] E [MPa] 𝜈

Epoxy 1180 3.6 ×10 3 0.368

Lead 11600 4.082 ×10 4 0.37

Soft rubber 1300 0.1175 0.469

Hard rubber 1300 11.75 0.469

w

i

g

m

h

a

r

3

t

t

m

t

i

a

r

t

w

m

s

e

s

s

c

t

T

a

e

c

i

h

p

T

𝑒

s

b

w

o

l

l

a

r

r

m

w

t

1 The material properties of the two rubber materials used here are hypothetical and

do not target a particular rubber material.

= 1 , ..𝑛 𝑑𝑖𝑚 . If the wave propagates along one of the symmetry axis, i.e.

𝜃 = 𝑒 𝑝 for some 𝑝 = 1 , ..𝑛 𝑑𝑖𝑚 , then no hybrid wave mode solutions will

e obtained at all frequencies due to local resonance. Furthermore, if

𝜃 can be diagonalized with respect to 𝑒 𝑝 , which is the case if the effec-

ive stiffness tensor is either isotropic or orthotropic with the orthotropy

xis aligned with the symmetry axis of the inclusion, Eq. (9) can be fully

ecoupled giving n dim

independent scalar equations,

2 𝑝 𝐶 𝜃𝑝𝑝 − 𝜔

2 𝜌𝑝𝑝 ( 𝜔 ) = 0 , ( no summation on 𝑝 ) , (16)

here,

𝑝𝑝 ( 𝜔 ) = 𝜌

(

1 +

∑𝑠 ∈𝑝 ′

𝜔

2

𝑠 𝜔

2 res − 𝜔

2 𝑠 𝜇J

)

. (17)

ere, 𝐶 𝜃𝑝𝑝 = 𝑒 𝑝 ⋅ 𝐂 𝜃 ⋅ 𝑒 𝑝 and 𝜌pp ( 𝜔 ) = 𝑒 𝑝 ⋅ 𝝆( 𝜔 ) ⋅ 𝑒 𝑝 . The solution for the

ave number derived from equations (16) and (17) is given as

𝑝 ( 𝜔 ) =

𝜔

𝑐 0 𝜃𝑝

√ √ √ √

(

1 +

∑𝑠 ∈𝑝 ′

𝜔

2

𝑠 𝜔

2 res − 𝜔

2 𝑠 𝜇J

)

, (18)

here,

0 𝜃𝑝 =

√

𝐶 𝜃𝑝𝑝

𝜌, (19)

epresents the effective wave speed in the quasi-static limit of the sys-

em. In accordance with the normalization of 𝜐𝑝 with respect to 𝐂 𝜃 given

y Eq. (11) , the expressions for the wave polarization modes are

𝑝 =

1 √

𝐶 𝜃𝑝𝑝

𝑒 𝑝 . (20)

ence, in the plane symmetric case, one compression mode (with 𝜐𝑝 || 𝑒 𝜃or a given value of p ) and two shear (one in the 2D case) modes (with

𝑞 ⟂ 𝑒 𝜃 for p ≠ q ) are always observed without formation of hybrid wave

odes, for wave propagation along the symmetry axis.

Since the combined plane and 4-fold symmetry is a special case of

lane symmetry, the scalar dispersion relations (18) also hold in this

ase.

.3. Homogenizability limit

A criterion on the applicability limit of the developed semi-analytical

nalysis can be obtained heuristically. The relaxed scale separation prin-

iple (stated above in Section 2 ) no longer applies when the wavelength

f the macroscopic wave in the matrix approaches the size of the het-

rogeneities. At that limit, higher order scattering effects start to play a

ole which is not accounted for by the present homogenization theory.

safe estimate for the scale separation limit is when the matrix wave-

ength is at least 10 times the relevant microstructural dimension, i.e.

𝑝 < 0 . 2 𝜋𝓁 , where 𝓁 is the relevant dimension for all 𝑝 = 1 , ..𝑛 𝑑𝑖𝑚 .

For the local resonance frequencies to lie within this limit, the effec-

ive stiffness of the matrix structure (both shear and compressive) has

o be at least 100 times higher than the effective stiffness of the inclu-

ion. The high stiffness of the matrix structure also implies that the local

esonances will be internally contained, i.e. preventing any interaction

etween neighboring inclusions. This also imposes a constraint on the

aximum volume fraction of the inclusions, which is largely determined

y the material properties of the matrix and the desired frequency limit

f analysis. A stiffer matrix material can internally contain the local res-

nances at higher volume fractions compared to a compliant matrix.

imilar arguments apply to the cases of random inclusion distributions

here the inclusions can be arbitrarily close to one another, which can

stensibly result in some interaction. Hence it is assumed here that, in

he case of random inclusion distributions, there is still sufficient dis-

ance between any two neighboring inclusions.

The approximate frequency limit of the homogenization method can

e estimated by solving the dispersion Eq. (8) at 𝑘 = 0 . 2 𝜋 along a given

𝓁

193

ave direction. The smallest eigenfrequency for which the correspond-

ng group velocity is non-negligible, indicating a matrix dominant wave,

ives a conservative estimation of the homogenizability limit of the

ethod. Local resonance modes occurring beyond this limit might ex-

ibit significant nonlocal interactions between neighboring inclusions,

t which point the present homogenization method will become inaccu-

ate and ultimately fail.

.4. Numerical case study and validation

In order to validate the proposed semi-analytical methodology and

o illustrate the results presented thus far, the dispersion properties of

hree 2D LRAM RVE designs shown in Fig. 1 are studied. A periodic

icro-structure is assumed, hence the RVEs considered here represent

he periodic unit cell. The periodicity is assumed for the sake of simplic-

ty of analysis since the distribution is captured by a single inclusion. It

lso allows Bloch analysis [14] to be performed in order to compute the

eference solution. However, the results obtained for a periodic unit can

o a large extent be applied to a random distribution of the inclusions as

ell, provided that the average volume of the matrix material over the

acroscopic domain remains the same and that the inclusions are well

paced between each other, i.e. the localized resonances are not influ-

nced. This is because the effective dynamic mass density tensor, which

olely determines the bandgap properties (i.e. its position and size) re-

ulting from local resonance, is unaffected by the distribution of the in-

lusions. The only difference between a periodic and a random distribu-

ion is the degree of anisotropy of the resulting effective static stiffness.

hus, only the phase speeds of the propagating waves along oblique

ngles with respect to the lattice vectors will be different in this case.

The considered unit cell designs are based on that of Liu et al. [5] ,

ach consisting of a square epoxy matrix with an embedded rubber

oated cylindrical lead inclusion. The difference between the designs

s in the configuration of the rubber coating such that each unit cell ex-

ibits a different degree of symmetry. The first design in Fig. 1 a is both

lane symmetric with respect 𝑒 1 and 𝑒 2 and 4-fold symmetric about 𝑒 3 .

he second design in Fig. 1 b is only plane symmetric with respect to the

1 − 𝑒 3 and 𝑒 2 − 𝑒 3 planes, in which a patch of hard rubber replaces the

oft rubber as shown in the figure. The third design in Fig. 1 c is obtained

y rotating the inclusion in the second design 22.5° counterclockwise

ith respect to 𝑒 3 . These three examples will exemplify the influence

f the dynamic mass anisotropy on the formation of total bandgaps, se-

ective bandgaps and hybrid wave modes. The designs are henceforth

abeled as UC1, UC2 and UC3, respectively. Plane strain kinematics is

ssumed in the 𝑒 1 − 𝑒 2 plane. The values of the geometric and mate-

ial parameters of the three designs are given in Table 1 a and Table 1 b,

espectively 1 . A finite element discretization is used to obtain the nu-

erical models of the unit cells. Four-node quadrilateral finite elements

ere used with a maximum mesh size restricted to 0.57 mm in the ma-

rix and 0.2 mm in the rubber coating. The models comprise on average

A. Sridhar et al. International Journal of Mechanical Sciences 133 (2017) 188–198

Table 2

Homogenized enriched continuum material properties of the considered unit cells.

(a) Effective static material properties. Only non-zero components of ℂ (4)

are shown.

Design UC1 & UC2 & UC3

ℂ (4)

[GPa] 𝐶 1111 = 𝐶 2222 = 1 . 955 𝐶 1122 = 𝐶 2211 = 0 . 546 𝐶 1212 = 𝐶 2121 = 𝐶 1221 = 𝐶 2112 = 0 . 26

𝜌 [kg m

-3 ] 3201.6

(b) Enriched continuum effective properties associated to the local resonance eigenmodes. Here 𝑒 ′1 and 𝑒 ′2 rotated by 22.5 o with respect to 𝑒 1 and 𝑒 2 , respectively

Design UC1 UC2 & UC3

Mode 𝑠 𝑗 = √

𝑠 𝜇J 𝜌 𝑒 𝑟 𝑠 𝑗 =

√𝑠 𝜇J 𝜌 𝑒 1

𝑠 𝑗 = √

𝑠 𝜇J 𝜌 𝑒 2

𝑟 = 1 𝑟 = 2 ( 𝑠 𝑗 = √

𝑠 𝜇J 𝜌 𝑒 ′1 ) ( 𝑠 𝑗 =

√𝑠 𝜇J 𝜌 𝑒

′2 )

2 5 3 6 1 6 3 4 9 s 𝜔 [Hz] 355.7 1239 355.7 1239 508 1615 936 1263 2053 s 𝜇J 0.724 0.0468 0.724 0.0468 0.75 0.0157 0.704 0.0675 0.0013

8

s

f

[

i

p

d

e

r

m

s

o

U

a

s

c

f

U

c

r

o

i

i

a

t

v

s

r

o

w

c

i

y

s

p

0

t

r

i

a

r

r

a

h

w

a

fi

w

a

a

o

m

s

s

i

m

t

t

s

s

1

e

i

e

t

t

s

t

a

r

c

4

p

t

p

s

m

b

t

r

s

b

4

e

a

w

L

000 elements and 15,000 degrees of freedom. Convergence of the re-

ults with respect to the discretization size has been verified.

The homogenized enriched continuum parameters are extracted

rom the finite element models by applying the method described in

21] . The values of the computed static effective parameters are given

n Table 2 a. Due to the high compliance of the rubber coating in com-

arison to the epoxy, the overall effective stiffness of each unit cell only

iffers up to 1%, and hence this difference is neglected in the Table. The

ffective mass densities of all the designs are identical. The first 15 local

esonance eigenmodes are determined and the corresponding enriched

aterial parameters are computed for each case. The mode shapes of

ome of the modes are displayed in Fig. 2 for reference. Among these,

nly modes 2, 3, 5 and 6 of UC1 and modes 1, 3, 4, 6 and 9 of UC2 and

C3 unit cells are dipolar and possess a sufficient modal mass fraction

long one of the considered directions. Modes 1 and 4 of UC1 repre-

ent the torsional resonances, involving the central inclusion and the

oating, respectively. Mode 7 of UC1 is of monopolar type as evidenced

rom the fact that it is plane symmetric. The dipolar modes 2 and 3 in

C1 and 1 and 3 in UC2 and UC3 represent the vibration of the central

ore whereas modes 5 and 6 in UC1 and 4, 6 and 9 in UC2 and UC3

epresent localized vibration in the coating without any involvement

f the central inclusion. The properties of the dipolar modes are given

n Table 2 b. The sum of the modal mass fractions of the dipolar modes

s almost 98% of the total mass fraction of the inclusion ( 𝜇inc = 0 . 787 )long any given direction, hence forming a sufficient basis that satisfies

he mode selection criterion given by Eq. (13) . However, for the sake of

alidation, all 15 modes were considered in the analysis. Furthermore,

imilar to the static properties, the effective properties describing local

esonance modes are identical for the UC2 and UC3 modes, with the

nly difference being the orientation of the coupling vector 𝑠 𝑗 of UC3,

hich would be rotated by 22.5° counter-clockwise with respect to the

orresponding ones of UC2.

The dispersion spectra for the considered unit cells are computed us-

ng Eq. (8) and validated against the spectra computed using Bloch anal-

sis [14] as shown in Fig. 2 . An excellent match is observed between the

pectra computed using the two methods for all designs within the dis-

layed frequency range. The group velocities observed at approximately

.2 times the distance from Γ to X for the three unit cells are all close

o zero, hence confirming again that within the considered frequency

ange, the analysis lies well within the homogenizability limit discussed

n Section 3.3 .

Only total bandgaps are observed in the case of UC1 as expected

nd mostly selective bandgaps are observed for UC2 and UC3. Two nar-

ow total bandgaps are seen for UC2 and UC3 in a narrow frequency

ange above the eigenfrequncy of the 3rd and 6th eigenmode at 936

nd 1615 Hz respectively as shown in Fig. 2 b and 2 c. The formation of

ybrid wave modes is clearly visible in UC2 and UC3 along Γ-M and X-M,

hen the wave direction is not aligned along the respective symmetry

xis of the inclusion. The hybrid wave effects are localized around the

194

rst local resonance frequency. The loss of hybrid wave mode effects

hen the wave propagation is perfectly oriented along the symmetry

xis can also be observed. Furthermore, the bandgaps of UC2 and UC3

ppear at exactly the same frequency locations, which is a consequence

f the fact that both designs possess the same set of eigenfreqencies and

odal mass fractions. Several flat branches are also observed in both

pectra corresponding to the modes with zero modal densities (i.e. tor-

ional, monopolar modes etc.). Hence it is confirmed that these modes

ndeed do not influence the dispersive properties of the system and the

odes indicated by the proposed selection criterion are adequate.

Finally, it should be emphasized that the total computational cost of

he reduced dispersion problem including the offline cost of computing

he enriched material parameters which involves meshing, matrix as-

embly and model reduction for a given unit cell design, is significantly

maller compared to Bloch analysis on the full FE model (even with all

5 modes included). The Bloch and the homogenized analysis were both

xecuted using a MATLAB script on a desktop computer with an Intel

7-3630 QM core and 6GB memory. The Bloch analysis took on an av-

rage 230 s for each unit cell, while the homogenized analysis took a

otal time (including offline and online costs) of about 4 s, indicating a

remendous gain ( ∼50 times in this case) in computational speed.

To summarize this section, the various local resonance effects re-

ulting from complex micro-structure designs such as total and selec-

ive bandgaps and hybrid wave modes have been illustrated. The semi-

nalytical model accurately captures all effects in the given frequency

egime, while being computationally much more efficient and faster

ompared to a full Bloch analysis.

. Transmission analysis

In this section, a transmission analysis framework is derived for wave

ropagation in an enriched media at normal incidence. It is a generaliza-

ion of the standard transfer matrix method [36] used in the analysis of

lane wave propagation through a layered arrangement of several dis-

imilar materials, which can in general be distinct locally resonant meta-

aterials, described by enriched effective continuum. The technique can

e extended to oblique wave incidence, but this is beyond the scope of

he present paper. The framework is applied to analyze the steady state

esponses of systems constructed using PlySym and UC3 unit cell de-

igns including the influence of the finite size of the structure and the

oundary conditions.

.1. Theoretical development

The general problem can be described as a serial connection of m lay-

red (enriched) media with the first and the last layer being semi-infinite

s shown in Fig. 3 . Since only normal wave incidence is considered, the

ave direction vector 𝑒 𝜃 defines the 1D macro-scale coordinate axis.

et x represent the corresponding spatial coordinate. The general plane

A. Sridhar et al. International Journal of Mechanical Sciences 133 (2017) 188–198

Fig. 3. The general macroscopic acoustic boundary value problem for normal wave incidence.

w

o

𝑟

𝑢

T

a

r

c

𝑟

𝑟

w

f

i

𝑟

𝑟

d

t

T

l

p

𝑟

T

l

∇

w

(

𝑟

w

𝑟

i

fi

t

a

w

t

Fig. 4. The acoustic transmission problem.

f

e

t

v

e

𝑟

(

𝑟

w

𝑟

i

l

i

t

a

𝑢

(

c

4

p

m

𝑥

d

d

a

ave solution at x corresponding to medium r can be expressed as a sum

f the forward and backward propagating components represented by 𝑢 f and 𝑟 𝑢 b respectively. Hence,

( 𝑥 ) =

𝑟 𝑢 f ( 𝑥 ) +

𝑟 𝑢 b ( 𝑥 ) . (21)

he bounding semi-infinite media will posses only a forward wave or

backward wave depending whether its boundary is on the left or the

ight side, respectively. Each component of the total displacement is

omposed of the individual wave modes

𝑢 f ( 𝑥 ) =

𝑛 𝑑𝑖𝑚 ∑𝑝 =1

𝑟 ��𝑝 𝑟 𝜉f 𝑝 𝑒

𝑖 𝑟 𝑘 𝑝 𝑥 , (22a)

𝑢 b ( 𝑥 ) =

𝑛 𝑑𝑖𝑚 ∑𝑝 =1

𝑟 ��𝑝 𝑟 𝜉b 𝑝 𝑒

− 𝑖 𝑟 𝑘 𝑝 𝑥 , (22b)

here, 𝑟 𝜉f 𝑝 and 𝑟 𝜉b 𝑝 , 𝑝 = 1 , ..𝑛 𝑑𝑖𝑚 are the wave mode amplitudes of the

orward and backward waves, respectively. These can be found by mak-

ng use of the normalization condition (11) , giving

𝜉f 𝑝 =

𝑟 𝐂 𝜃 ⋅𝑟 ��𝑝 ⋅

𝑟 𝑢 f ( 𝑥 ) 𝑒 − 𝑖 𝑟 𝑘 𝑝 𝑥 , (23a)

𝜉b 𝑝 =

𝑟 𝐂 𝜃 ⋅𝑟 ��𝑝 ⋅

𝑟 𝑢 b ( 𝑥 ) 𝑒 𝑖 𝑟 𝑘 𝑝 𝑥 . (23b)

Next, the traction–displacement relation needs to be setup. Let 𝑟 �� + ( 𝑥 )enote the macro-scale traction vector in the r th medium with respect

o 𝑒 𝜃 (a “ - ” superscript is used to indicate traction with respect to − 𝑒 𝜃).

his can be determined from the effective homogenized constitutive re-

ation (4a) , disregarding the last term as discussed before (see the last

aragraph in Section 2 ), i.e.

�� + ( 𝑥 ) = 𝑒 𝜃 ⋅

𝑟 𝝈( 𝑥 ) = 𝑒 𝜃 ⋅

𝑟 ( ℂ ) (4)

∶ ∇ 𝑢 ( 𝑥 ) . (24)

he displacement gradient is expressed in terms of the plane wave so-

ution by making use of Eqs. (21) –(23),

𝑢 ( 𝑥 ) = 𝑖 𝑟 𝑘 𝑝

𝑛 𝑑𝑖𝑚 ∑𝑝 =1

(𝑒 𝜃 ⊗

𝑟 ��𝑝 ⊗𝑟 𝐂 𝜃 ⋅ 𝑟 ��𝑝 ⋅ 𝑟 𝑢 f ( 𝑥 ) − 𝑒 𝜃 ⊗ 𝑟 ��𝑝 ⊗

𝑟 𝐂 𝜃 ⋅ 𝑟 ��𝑝 ⋅ 𝑟 𝑢 b ( 𝑥 ) ),

= 𝑖𝜔 𝑟 𝜅𝑝 𝑛 𝑑𝑖𝑚 ∑𝑝 =1

(𝑒 𝜃 ⊗

𝑟 ��𝑝 ⊗𝑟 𝐂 𝜃 ⋅ 𝑟 ��𝑝 ⋅ 𝑟 𝑢 f ( 𝑥 ) − 𝑒 𝜃 ⊗ 𝑟 ��𝑝 ⊗

𝑟 𝐂 𝜃 ⋅ 𝑟 ��𝑝 ⋅ 𝑟 𝑢 b ( 𝑥 ) ),

(25)

here 𝑟 𝜅𝑝 ( 𝜔 ) = 𝜔 −1 𝑟 𝑘 𝑝 ( 𝜔 ) has been introduced. Substituting Eq. (25) in

24) gives,

�� + ( 𝑥 ) = 𝑖𝜔 𝑟 𝐙 ( 𝜔 ) ⋅ 𝑟 𝑢 f − 𝑖𝜔 𝑟 𝐙 ( 𝜔 ) ⋅ 𝑟 𝑢 b , (26)

here,

𝐙 ( 𝜔 ) = 𝑛 𝑑𝑖𝑚 ∑𝑝 =1

𝑟 𝐂 𝜃 ⋅ ( 𝑟 𝜅𝑝 ( 𝜔 ) 𝑟 ��𝑝 ( 𝜔 ) ⊗ 𝑟 𝐂 𝜃 ⋅ 𝑟 ��𝑝 ( 𝜔 )) , (27)

s the effective impedance of the considered metamaterial medium, de-

ned as the constitutive parameter relating the macro-scale traction to

he velocity ( 𝑖𝜔 𝑟 𝑢 f or 𝑖𝜔 𝑟 𝑢 b ) at a given interface. Note from Eq. (26) that

positive sign is assigned to the impedance with respect to the forward

ave component of the total velocity and a negative sign is assigned

o the impedance with respect to the backward wave component. It is

195

requency dependent unlike the impedance of ordinary homogeneous

lastic materials.

Another relation that is needed for solving the boundary problem is

he transfer operation, which, given a quantity known at x , returns its

alue at another point 𝑥 + Δ𝑥 within the corresponding medium. The

xpression for components 𝑟 ( ∙) f ( 𝑥 + Δ𝑥 ) and 𝑟 ( ∙) b ( 𝑥 + Δ𝑥 ) in terms of ( ∙) f ( 𝑥 ) and 𝑟 ( ∙) b ( 𝑥 ) respectively is obtained by applying Eqs. (22) and

23),

𝑟 ( ∙) f ( 𝑥 + Δ𝑥 ) = 𝑛 𝑑𝑖𝑚 ∑𝑝 =1

(𝑟 ��𝑝 ⊗

𝑟 𝐂 𝜃 ⋅ 𝑟 ��𝑝 𝑒 𝑖 𝑟 𝑘 𝑝 Δ𝑥

)⋅ 𝑟 ( ∙) f ( 𝑥 ) = 𝑟 𝐓 (Δ𝑥 ) ⋅ 𝑟 ( ∙) f ( 𝑥 ) ,

( ∙) b ( 𝑥 + Δ𝑥 ) = 𝑛 𝑑𝑖𝑚 ∑𝑝 =1

(𝑟 ��𝑝 ⊗

𝑟 𝐂 𝜃 ⋅ 𝑟 ��𝑝 𝑒 − 𝑖 𝑟 𝑘 𝑝 Δ𝑥

)⋅ 𝑟 ( ∙) b ( 𝑥 ) = 𝑟 𝐓

−1 (Δ𝑥 ) ⋅ 𝑟 ( ∙) b ( 𝑥 ) ,

(28)

here,

𝐓 (Δ𝑥 ) = 𝑛 𝑑𝑖𝑚 ∑𝑝 =1

𝑟 ��𝑝 ⊗𝑟 𝐂 𝜃 ⋅ 𝑟 ��𝑝 𝑒 𝑖

𝑟 𝑘 𝑝 Δ𝑥 , (29)

s the effective transfer operator.

Finally, the continuity conditions at the media interfaces are estab-

ished. Let r x give the coordinate of the r th and ( 𝑟 + 1) 𝑡ℎ interface. Apply-

ng Eqs. (21) and (26) , the traction and displacement continuity condi-

ions between the r and ( 𝑟 + 1) 𝑡ℎ medium

r x can respectively be written

s,

𝑟 �� + ( 𝑟 𝑥 ) +

𝑟 +1 �� − ( 𝑟 𝑥 ) = 0

𝑖𝜔

𝑟 𝐙 ( 𝜔 ) ⋅ ( 𝑟 𝑢 f ( 𝑟 𝑥 ) −

𝑟 𝑢 b ( 𝑟 𝑥 )) − 𝑖𝜔

𝑟 +1 𝐙 ( 𝜔 ) ⋅ ( 𝑟 +1 𝑢 f ( 𝑟 𝑥 ) −

𝑟 +1 𝑢 b ( 𝑟 𝑥 )) = 0 ,

(30a)

( 𝑟 𝑥 ) =

𝑟 𝑢 f ( 𝑟 𝑥 ) +

𝑟 𝑢 b ( 𝑟 𝑥 ) =

𝑟 +1 𝑢 f ( 𝑟 𝑥 ) +

𝑟 +1 𝑢 b ( 𝑟 𝑥 ) . (30b)

Therefore any general problem can be solved by applying Eqs.

28) and (30) for all media with the appropriate constraints/boundary

onditions.

.2. Numerical case study and validation

The acoustic transmission analysis framework is applied on the sim-

le macro-scale case study shown in Fig. 4 . The system consists of a

etamaterial medium made of 𝑛 = 10 unit cells starting at 𝑥 = 0 till = 𝑛 𝓁, where 𝓁 is the size of the unit cell. The 1D macro-scale coor-

inate axis and the wave propagation direction are taken along 𝑒 1 as

efined in the unit cell Fig. 1 . An acoustic actuation source is applied

t 𝑥 = 0 , which prescribes a given displacement at the interface. The

A. Sridhar et al. International Journal of Mechanical Sciences 133 (2017) 188–198

(a) (b)

Fig. 5. The transmission ratio for applied horizontal (vertical) excitation and measured horizontal (vertical) displacement in the example macro-scale problem using (a) UC2 and (b)

UC3 as the LRAM medium computed using the semi-analytical approach (SA) and direct numerical simulation (DNS).

m

m

a

a

m

a

t

F

t

d

p

i

a

m

d

D

c

p

i

d

s

T

a

t

p

d

n

z

r

b

c

b

w

n

t

o

s

a

t

s

a

t

a

i

Fig. 6. The transmission ratio for applied horizontal excitation and measured vertical dis-

placement in the example macro-scale problem using UC3 as the LRAM medium computed

using the semi-analytical approach (SA) and direct numerical simulation (DNS).

c

c

a

l

e

f

t

m

t

r

a

s

a

n

T

p

L

i

c

p

5

t

I

a

R

etamaterial medium is bounded on the right side by a semi-infinite ho-

ogeneous medium and on the left by an actuation source that applies

prescribed displacement. The impedances of the bounding medium

nd the actuation source are matched with the impedance of the matrix

aterial within the LRAM. The solution to this problem using the semi-

nalytical approach is derived in A.1 . To verify the semi-analytical solu-

ion, a direct numerical simulation (DNS) is carried out using a standard

E software package (COMSOL). The full waveguide structure used in

he DNS is built by serially repeating the FE unit cell model used for the

ispersion analysis in Section 3.4 . Periodic boundary conditions are ap-

lied to the top and bottom edges to mimic an infinitely large structure

n the vertical direction. An acoustic impedance boundary condition is

dded at both the ends to account for the impedance of the bounding

edia and the actuation source. The assembled system has a number of

egrees of freedom in the order of 10 5 .

The transmission analysis using the semi-analytical approach and

NS are first carried out for the LRAM medium consisting of UC2 unit

ells. The results are shown in Fig. 5 a. The analysis is performed for ap-

lied unit horizontal and vertical displacements separately (while fix-

ng the displacement in the other direction). The absolute value of the

isplacement is measured at the right interface which gives the corre-

ponding transmission coefficient with respect to the applied excitation.

he expression for the transmission coefficient obtained using the semi-

nalytical approach is given by Eq. (A.8) . Due to the plane symmetry of

he UC2 along 𝑒 1 , no hybrid wave solutions exist and a horizontal ap-

lied displacement will excite a pure compressive wave and a vertical

isplacement will excite a pure shear wave. Hence, only the compo-

ent of the transmission coefficient corresponding to that of the non-

ero applied displacement is shown in the figure. The applied frequency

ange is 350–2100 Hz in order to capture the first two compressive wave

andgaps and the first three shear wave bandgaps (see Fig. 2 b). An ex-

ellent match between the semi-analytical model and DNS is obtained.

One of the interesting observations is that the shear transmission

andgaps are more pronounced (deeper) than those of the compressive

ave. This can be attributed to the fact that the effective shear stiff-

ess of the unit cell is much lower than its compressive stiffness. Due

o the requirements of the scale separation, the semi-analytical model is

nly valid for a relatively stiff matrix material. Beyond this limit, Bragg

cattering effects start to play a role, leading to hybridization of Bragg

nd local resonance effects [37] . This effect does enhance the attenua-

ion performance of the metamaterial but at a cost of overall structural

tiffness, hence there is a tradeoff. LRAMs can therefore be employed in

pplications where a higher structural stiffness are required.

Next, the transmission analysis is performed exactly with UC3 as

he LRAM medium ( Fig. 5 b). Again, the results obtained from the semi-

nalytical approach match very well with DNS. The UC3 response is sim-

lar to UC2 except for the reduced attenuation rate and some additional

196

ross coupling of transmission spectra at the local resonance frequen-

ies. This is due to the fact that a propagating hybrid wave mode will

lways be excited in UC3 for pure horizontal or vertical excitations at the

ocal resonance frequencies, leading to some residual transmission. The

ffects due to the hybrid wave modes occurring in UC3 metamaterial are

urther illustrated in Fig. 6 , where the absolute vertical displacement at

he right interface is shown, for the unit horizontal applied displace-

ent. The frequency now ranges from 300 to 700 Hz in order to capture

he effects due to the first local resonance at 508 Hz. Comparing the

esults with DNS shows, once again, a perfect match between the two

pproaches. The peak vertical displacements in the LRAM medium is ob-

erved exactly at the first local resonance frequency and drops rapidly

way from it, indicating the formation of hybrid modes due to the dy-

amic anisotropy of the mass density tensor at the resonance frequency.

his characteristic response of the UC3 LRAM can lead to interesting ap-

lications. As a consequence of horizontal applied displacements on the

RAM, a shear wave will be excited in the homogeneous medium bound-

ng the LRAM. This enables the design of selective mode converters that

onvert an incident wave mode into another mode upon transmission at

articular frequencies for normal wave incidence.

. Conclusion

A multiscale semi-analytical technique was presented for the acous-

ic plane wave analysis of (negative) dynamic mass density type LRAMs.

t enables an efficient and accurate computation of the dispersion char-

cteristics and acoustic performance of LRAM structures with complex

VE geometries in the low frequency regime. The technique uses a two

A. Sridhar et al. International Journal of Mechanical Sciences 133 (2017) 188–198

s

u

m

e

t

m

e

m

l

a

s

a

m

m

t

a

a

g

m

i

i

d

w

g

A

l

T

t

m

c

t

d

i

p

i

w

a

i

A

t

F

[

A

A

S

F

m

3

s

(

d

r

d

r

b

1

1

𝑢

0

t

s

2

w

𝐀

i

p

2

E

2

S

2

N

o

2

F

t

𝑢

w

d

𝐀 = 𝐈 + 𝐀 ⋅ 𝐓 ( 𝑛 𝓁) ⋅ 𝐈 + 𝐓 ( 𝑛 𝓁) ⋅ 𝐀 ⋅ 𝐓 ( 𝑛 𝓁) . (A.8)

cale solution ansatz in which the micro-scale problem is discretized

sing FE (capturing the detailed micro-dynamic effects of a particular

icro-structure) and where an analytical plane wave solution is recov-

red at the macro-scale analysis. This approach offers two main advan-

ages over traditional Bloch based approaches for dispersion and trans-

ission analysis of LRAMs.

• First, as mentioned earlier, it is computationally remarkably cheap,

even in comparison to the reduced order Bloch analysis approaches.

The computation of the homogenized enriched material coefficients

forms the biggest part of the overall numerical cost which involves

meshing, matrix assembly and model reduction. For a particular RVE

design, this computation has to be performed only once and there-

fore constitutes an offline cost. The dispersion spectrum is then com-

puted very cheaply due to the highly reduced nature of the corre-

sponding eigenvalue problem. The mode selection criterion ensures

that the resulting model is as compact as possible without sacrificing

the accuracy of the solution. • Second, it permits an intuitive and insightful characterization of

LRAMs and its dispersive properties in terms of a compact set of

enriched effective material parameters. The only limitation of the

method is that it is only applicable towards the analysis of LRAMs

in the low frequency regime.

The primary result of the work is the derivation of the dispersion

igenvalue problem of the enriched continuum, which gives the funda-

ental plane wave solution in an infinite LRAM medium. The prob-

em can be conveniently framed in the 𝑘 − 𝜔 or 𝜔 − 𝑘 form, the latter

llowing evanescent wave solutions to be computed. Furthermore, the

imple structure of the dispersion model leads to clarifying qualitative

nd quantitative insights. For instance, the relation between the geo-

etric symmetries, the dynamic anisotropy of the resulting dynamic

ass tensor and the nature of the dispersive behavior becomes clear, i.e.

he formation of selective/total bandgaps and/or hybrid wave modes

t certain frequencies. Several case studies were used to illustrate the

pproach. The effective continuum description was combined with a

eneral plane wave expansion resulting in a modified transfer matrix

ethod on enriched media. The analysis was restricted to normal wave

ncidence, but it can easily be extended to the general case of an oblique

ncidence.

As discussed earlier, complex designs induce extended micro-

ynamics, which manifests itself at the macro-scale in two important

ays. First, in the proliferation of local resonance modes, which trig-

er the formation of additional bandgaps in the dispersion spectrum.

nd second, in the dynamic anisotropy of the material response which

eads to the formation of selective bandgaps and hybrid wave modes.

hese effects can be exploited towards developing more advanced fil-

ers and frequency multiplexers, which target specific wave modes at

ultiple desired frequencies. The developed framework was applied to

ase studies revealing interesting phenomena such as pronounced shear

ransmission bandgaps compared to the compressive ones (due to the

ifference in the elastic stiffness). The localized hybrid mode formation

n considered LRAM unit cell was used to demonstrate a potential ap-

lication towards selective mode conversion, where the energy of the

ncident normal wave mode can be channeled into a different mode

ithin a certain frequency range. Hence, the presented semi-analytical

pproach serves as a valuable tool for optimization and rapid prototyp-

ng of LRAMs for specific engineering applications.

cknowledgments

The research leading to these results has received funding from

he European Research Council under the European Union ’s Seventh

ramework Programme (FP7/2007-2013) / ERC grant agreement no

339392 ].

197

ppendix A

1. Solution to the acoustic transmission case study problem

The acoustic transmission analysis framework discussed in

ection 4 is applied to obtain the results described in Section 4.2 and

ig. 4 . The actuation source on the left is indexed as 1, the LRAM

edium in the middle as 2 and the homogeneous medium on the right

. Note that medium 3 has only forward traveling waves since it is

emi-infinite. The coordinates of the left ( 𝑥 = 0) and the right interface

𝑥 = 𝑛 𝓁) are represented by 1 x and 2 x , respectively. Let the prescribed

isplacement acting on the interface at 1 x be denoted as 1 𝑢 app and the

esulting reactive traction be denoted as 1 �� app . The transmitted wave

isplacement into medium 3, 𝑢 ( 2 𝑥 ) as a function of 1 𝑢 app gives the main

esult of this section.

Applying the boundary conditions on the plane wave solution given

y Eq. (30) yields,

𝑢 app =

2 𝑢 f ( 1 𝑥 ) +

2 𝑢 b ( 1 𝑥 ) , (A.1a)

�� app = − 𝑖𝜔

1 𝐙 ⋅ 1 𝑢 app − 𝑖𝜔

2 𝐙 ⋅(2 𝑢 f ( 1 𝑥 ) −

2 𝑢 b ( 1 𝑥 ) ), (A.1b)

( 2 𝑥 ) =

3 𝑢 f ( 2 𝑥 ) =

2 𝑢 f ( 2 𝑥 ) +

2 𝑢 b ( 2 𝑥 ) , (A.1c)

= 𝑖𝜔

2 𝐙 ⋅(2 𝑢 f ( 2 𝑥 ) −

2 𝑢 b ( 2 𝑥 ) )− 𝑖𝜔

3 𝐙 ⋅ 3 𝑢 f ( 2 𝑥 ) . (A.1d)

Note that the overbar is not applied on medium 1 and 3 to indicate

hat they are not homogenized quantities. Solving for 2 𝑢 b ( 2 𝑥 ) with re-

pect to 2 𝑢 f ( 2 𝑥 ) using Eqs. (A.1c) and (A.1d) gives

𝑢 b ( 2 𝑥 ) = 𝐀 Re ⋅2 𝑢 f ( 2 𝑥 ) , (A.2)

here,

Re = ( 2 𝐙 +

3 𝐙 ) −1 ⋅ ( 2 𝐙 −

3 𝐙 ) , (A.3)

s the effective reflection coefficient tensor at the given interface. Ap-

lying the transfer operation as defined by Eq. (28) gives 2 𝑢 f ( 2 𝑥 ) = 𝐓 ( 𝑛 𝓁) ⋅ 2 𝑢 f ( 1 𝑥 ) and 2 𝑢 b ( 1 𝑥 ) =

2 𝐓 ( 𝑛 𝓁) ⋅ 2 𝑢 b ( 2 𝑥 ) . Combining the result with

q. (A.2) gives

𝑢 b ( 1 𝑥 ) =

2 𝐓 ( 𝑛 𝓁) ⋅ 𝐀 Re ⋅2 𝐓 ( 𝑛 𝓁) ⋅ 2 𝑢 f ( 1 𝑥 ) . (A.4)

ubstituting the above expression into Eq. (A.1a) and solving for 2 𝑢 f ( 1 𝑥 )

𝑢 f ( 1 𝑥 ) =

(𝐈 +

2 𝐓 ( 𝑛 𝓁) ⋅ 𝐀 Re ⋅2 𝐓 ( 𝑛 𝓁)

)−1 ⋅ 1 𝑢 app . (A.5)

ow, applying the transfer operation to Eq. (A.5) , the solution 2 𝑢 f ( 2 𝑥 ) isbtained as

𝑢 f ( 2 𝑥 ) =

2 𝐓 ( 𝑛 𝓁) ⋅(𝐈 +

2 𝐓 ( 𝑛 𝓁) ⋅ 𝐀 Re ⋅2 𝐓 ( 𝑛 𝓁)

)−1 ⋅ 1 𝑢 app . (A.6)

inally, using Eqs. (A.2) and (A.1c) and the fact that 𝑢 f ( 3 𝑥 ) =

𝑢 ( 2 𝑥 ) gives

he desired result

f ( 3 𝑥 ) =

(𝐈 + 𝐀 Re

)⋅ 2 𝐓 ( 𝑛 𝓁) ⋅

(𝐈 +

2 𝐓 ( 𝑛 𝓁) ⋅ 𝐀 Re ⋅2 𝐓 ( 𝑛 𝓁)

)−1 ⋅ 1 𝑢 app

= 𝐀 Tr ⋅1 𝑢 app , ( A.7)

here 𝐀 Tr is the effective transmission coefficient between the applied

isplacement and the transmitted wave in medium 3, expressed as ( )2

(2 2

)−1

Tr Re Re

A. Sridhar et al. International Journal of Mechanical Sciences 133 (2017) 188–198

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