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: [email protected] (V.G. Kouznetsova).
c
w
w
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
𝓁193
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 ReA. Sridhar et al. International Journal of Mechanical Sciences 133 (2017) 188–198
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