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Extreme Mechanics Letters
journal homepage:www.elsevier.com/locate/emlDynamic homogenization theory for nonlocal acoustic metamaterials
Marie-Fraise Ponge
a,∗, Olivier Poncelet
a, Daniel Torrent
b aUniversité de Bordeaux - Institut de Mécanique et d’Ingénierie, CNRS UMR 5295 - Talence, France bUniversité de Bordeaux - Centre de recherche Paul Pascal, CNRS UPR 8641 - Pessac, Francea r t i c l e i n f o
Article history:
Received 14 April 2016 Received in revised form 21 July 2016 Accepted 8 October 2016 Available online xxxx Keywords: Homogenization Metamaterial Phononic crystal Nonlocal materials
a b s t r a c t
We present a homogenization method for periodic acoustic composites based on the Plane Wave Expansion (PWE) method. We show that the description of periodic acoustic composites needs constitutive parameters which depend on frequency and wavenumber, meaning that the effective material is resonant and nonlocal. Also, an anisotropic mass density and an additional constitutive parameter, called the Willis term in analogy to its counterpart in elasticity, are found. Numerical calculations compare the present method with the traditional multiple scattering method, showing a good agreement between both theories. However, the method presented here overcomes the limitations of the multiple scattering method, where the incorporation of anisotropy and non-locality implies solving the scattering problem of anisotropic objects with additional boundary-conditions. A final example showing the importance of nonlocal effects is provided. This work shows that acoustic metamaterials are nonlocal materials in general, and provides a tool for the proper modeling of the nonlocal constitutive parameters.
© 2016 Elsevier Ltd. All rights reserved.
1. Introduction
The study of the propagation of acoustic waves through complex media has received a new insight within the last ten years with the advent of the so called ‘‘acoustic metamaterials’’ [1]. These structures consist essentially in arrangements (periodic or not) of resonators designed in such a way that, when the wavelength of the acoustic wave is larger than the typical distance between scattering units, the structure behaves as an effective material with frequency-dependent constitutive parameters. Then, metamaterials present resonant constitutive parameters so that they can behave like materials with negative mass density [1], bulk modulus [2] or both simultaneously [3– 5]. Additionally, it was also shown that acoustic metamaterials required for their proper description an anisotropic mass density, even in the quasi-static limit [6]. These unusual constitutive parameters offer as well a wide variety of applications, like hyperlenses [7,8], cloaking devices [6,9] or omnidirectional absorbers [10,11].
Describing micro-structured materials at the macro-scale by means of homogenization methods not only shortens the sim-ulation time, but it also facilitates the design of metamaterials
∗Corresponding author.
E-mail address:[email protected](M.-F. Ponge).
by a description in terms of effective parameters. Additionally, it also gives new ways to describe the physical behavior of hetero-geneous micro-structured media. However, the development of mathematical tools for the dynamical description of these com-posites is a complex task [12]. Although homogenization theo-ries for fluid or solid composites in the quasi-static limit are well known [13,14], finite-frequency methods have been recently de-veloped based on the coherent potential approximation [15,16] or the multiple scattering theory [17,18], which have allowed the in-clusion of a resonant-like behavior of the constitutive parameters. Recent methods allow also for the description of composites even in the high frequency limit [19], where the discussion about the meaning of the constitutive parameters is even more complex.
In the previously referenced works, the underlying idea for the homogenization of metamaterials is that a combination of materi-als gives a composite with frequency-dependent constitutive pa-rameters. However, the pioneering work of Willis [20] on the ho-mogenization of elastic composites revealed that in the dynamic regime, composites could change the nature of the constituent materials, in the sense that new constitutive parameters would emerge as a consequence of the averaging process. In this context, equations for spatially averaged fields are of the Willis form [21– 23]: the mass density becomes tensorial and momentum and strain are coupled by means of the so called ‘‘Willis tensor’’, which is a new constitutive parameter not necessarily found in the individ-ual materials forming the composite. The exact expressions of fre-quency–wavenumber dependent effective parameters were first
http://dx.doi.org/10.1016/j.eml.2016.10.006 2352-4316/©2016 Elsevier Ltd. All rights reserved.
derived by Willis for a periodic laminate [22] in term of the Green’s function of the media. Efforts have then been made to find analyti-cal and easier analyti-calculable expressions, first in one-dimensional [24– 26] periodic composites, and later in three-dimensional [27–29] periodic structures. The philosophy of Willis work is combined with the Bloch–Floquet decomposition and the Plane Wave Expan-sion method [28,29], the monodromy matrix formalism [24,30], or the comparison with a reference medium [27].
In this context, we present a homogenization theory, based on the Plane Wave Expansion (PWE) method, which defines a set of generalized (non-local) effective parameters for periodic acoustic composites. As mentioned before, it is shown that both the effective mass density and bulk modulus of these composites are frequency dependent and non-local, with the additional complexity that the mass density is also anisotropic. Moreover, it is found that an additional constitutive parameter, called the Willis term in analogy to its counterpart in elasticity, is needed for their description. It will be introduced here but not deeply analyzed, since the present work focuses its attention on the non-local properties of the effective parameters.
The paper is organized as follows: Section2outlines the homog-enization model used to calculate the constitutive parameters of
the propagation. Section3compares the present method with the
classical multiple-scattering homogenization method. Section4
shows a numerical example consisting of a rectangular lattice of cylinders which behaves as an anisotropic metamaterial with non-local effective parameters. Finally, the conclusions and the per-spectives of this work are reported in Section5.
2. Homogenization theory for periodic acoustic composites We present a homogenization theory for sonic crystals based on the PWE method [31–33], and similar in methodology to that previously developed for phononic crystals [29]. The starting point is the equation of motion for an inhomogeneous fluid, in which a
harmonic time dependence of frequency
ω
has been assumed,∇ · [
ρ
−1(
r)∇
P(
r)] = −
B−1(
r)ω
2P(
r),
(1) withρ(
r)
and B(
r)
being the position dependent mass density and bulk modulus, respectively, and P(
r)
, the pressure field. For a homogeneous mediumρ(
r) = ρ
band B(
r) =
Bb, the dispersionrelation is obtained by assuming plane wave propagation with wave vector k
=
kn (with n the direction of the propagation),k2
=
ω
2ρ
b/
Bb.
(2)In an infinite periodic structure,
ρ(
r)
and B(
r)
are periodic functions of r, then they can be expanded in a Fourier series on the reciprocal lattice vector G. Under this periodicity condition, Bloch theorem states that the pressure field can be expanded as P(
r) =
eik·r
G
PGeiG·r
,
(3)which inserted into the wave equation becomes
G′
i(
k+
G)
iρ
G−−1G′(
k+
G ′)
iPG′=
ω
2
G′ B−G−1G′PG′,
(4) withρ
G−1and B−G1, the Fourier components of the inverse of the mass density and inverse of the bulk modulus, respectively. The index i indicates a sum over the three components of k, G and G′vectors. In matrix form, Eq.(4)is expressed as
G′ MGG′PG′=
ω
2
G′ NGG′PG′ (5) with MGG′=
i(
k+
G)
iρ
G−−1G′(
k+
G ′)
i (6) and NGG′=
B−1 G−G′.
(7)Eq.(5)is a generalized eigenvalue equation, which for a given
k returns a set of eigenfrequencies
ω
. The eigenfrequencies yield the band structure or dispersion curveω = ω(
k)
of the periodic medium. The description of the phononic crystal as a homogeneous material is made by assuming that the effective pressure field propagates as a purely Bloch wave, that isPeff
= ⟨P⟩e
ik·r,
(8)with
⟨P⟩
being the spatial average of the pressure field. Then,finding an equation for
⟨P⟩
and its relationship withω
and kwill give a definition of an effective medium and will allow an identification of the different constitutive parameters. Since the material is periodic, the average of the pressure field will be equal to the average in the unit cell, so that from Eq.(3)we find that
⟨
P⟩ =
PG=0.
(9)The average of the pressure field is then given by the G
=
0component of PG, which can be obtained by expressing Eq.(5)as
ω
2 N00P0+
ω
2
G′̸=0 N0G′PG′=
M00P0+
G′̸=0 M0G′PG′ (10a)ω
2N G0P0+
ω
2
G′̸=0 NGG′PG′=
MG0P0+
G′̸=0 MGG′PG′.
(10b)Hereafter, it is considered that matrix elements labeled with G do
not include the term G
=
0, which is extracted from the abovedecomposition. We can now solve from the second equation for PG, PG′
= −
G̸=0χ
G′G(ω,
k)(
MG0−
ω
2NG0)
P0 (11) whereχ
G′G(ω,
k) = (
MG′G−
ω
2NG′G)
−1 G′G,
(12)and insert it into the first one (Eq.(10a)), obtaining the following equation
ω
2N 00−
G,G′̸=0ω
2N 0G′χ
G′G(
MG0−
ω
2NG0) −
M00+
G,G′̸=0 M0G′χ
G′G(
MG0−
ω
2NG0)
P0=
0.
(13)Eq.(13)is formally the same equation as(5), however it is not an eigenvalue equation, but a secular equation for P0similar to Eq.(2), where the solutions
ω = ω(
k)
are obtained from the zeros of the functionΓ, Γ=
ω
2N00−
G,G′̸=0ω
2N 0G′χ
G′G(
MG0−
ω
2NG0) −
M00+
G,G′̸=0 M0G′χ
G′G(
MG0−
ω
2NG0).
(14)In the above equation the coefficients are in general functions
of both
ω
and k, what makes it less suitable for band structurecalculation than Eq.(5)but more suitable for the description of the sonic crystal as an effective material. Indeed, we can see that the
elements of the N00, N0G′and NG0do not depend explicitly on the wave vector k, N00
= ¯
B−1,
(15) N0G′=
B−1 −G′,
(16) NG0=
B−G1,
(17) withB¯
−1=
B−1G=0the average in the unit cell of the reciprocal of the bulk modulus. On the contrary, M00, M0G′and MG0contain this
dependence with the wave vector, since
M00
=
kiρ
¯
−1,
(18) M0G′=
kiρ
−1 −G′(
k+
G ′)
i,
(19) MG0=
(
k+
G)
iρ
G−1ki,
(20) withρ
¯
−1=
ρ
−1G=0 the average in the unit cell of the mass
density. The dependence with the wave vector and frequency can be reorganized and the secular equation becomes
Γ
=
ω
2B−1−
k2niρ
−ij1nj−
ω
k(
niSi+
SiĎni),
(21)where the coefficients B−1,
ρ
−1ij , Siand SiĎare given by:
B−1
(ω,
k) = ¯
B−1+
ω
2
G,G′̸=0 B−−1G′χ
G′G(ω,
k)
B−G1,
(22a)ρ
−1 ij(ω,
k) = ¯ρ
−1δ
ij−
G,G′̸=0ρ
−1 −G′(
k+
G ′)
iχ
G′G(ω,
k)
×
(
k+
G)
jρ
G−1,
(22b) Si(ω,
k) = ω
G,G′̸=0ρ
−1 −G′(
k+
G ′)
iχ
G′G(ω,
k)
B−G1,
(22c) SiĎ(ω,
k) = ω
G,G′̸=0 B−−1G′χ
G′G(ω,
k)ρ
G−1(
k+
G)
i.
(22d)Eq.(21)is similar to Eq.(2), but the constitutive parameters that describe the sonic crystal are more general. This equation shows that the sonic crystal is similar to a non-local Willis medium [21, 28] in which the mass density is a tensorial quantity and with the presence of the coupling field Si. The above expressions are
valid at any frequency and wavenumber. Recently, Torrent et al. have demonstrated that local resonances make solid phononic crystals behave like a Willis medium with resonant and non-local parameters even at the low-frequency limit [29]. It will be shown here that this is also the case for acoustic sonic crystals.
3. Validation of the homogenization method
In this section, the homogenization method is validated by means of two ways. First, the effective parameters computed by the present theory are compared with the effective parameters obtained using the multiple scattering method described in [17]. Second, the dispersion curve obtained via the effective parameters in the local approximation is compared with the full band structure. The example to be analyzed consists of an infinite, two dimensional, square lattice of fluid cylindrical scatterers of radius Ra
=
0.
1a, with a being the lattice constant of the arrangement.The associated effective medium is thus isotropic. The mass density and bulk modulus of the scatterers are
ρ
a=
ρ
b/
2, Ba=
0.
02Bb,with
ρ
b the mass density of the background and Bb its bulkmodulus.
Effective parameters. The effective mass density and bulk modulus are presented inFig. 1. They are normalized with respect to the mass density and the bulk modulus in the background. Dark gray dashed–dotted line corresponds to the calculation with the
Fig. 1. Normalized effective mass densityρ∗/ρ
band bulk modulus B∗/Bbversus normalized frequency fa/cbcomputed by means of the periodic homogenization method in the local approximation (dark gray dashed–dotted line) and the multiple scattering method (light gray dashed line).
periodic homogenization method in the local approximation, and light gray dashed line corresponds to the calculation with the multiple scattering method in the long-wavelength limit.
In the static limit, both methods fit perfectly, the effective medium behaves as a homogenized composite in the long-wavelength limit: [13] 1 B∗
=
1−
f Bb+
f Ba,
(23)ρ
∗=
ρ
bρ
a(
1+
f) + ρ
b(
1−
f)
ρ
a(
1−
f) + ρ
b(
1+
f)
.
(24)Given that the lattice is square, the effective metamaterial is isotropic
ρ
11∗=
ρ
22∗. In the reduced frequency range [0.2,0.45], the effective bulk modulus and mass density are negative and positive, respectively, so that no propagation is expected in this frequency range, which will lead to a gap in the band structure, as will be shown later.Fig. 1shows that there is a small discrepancy between the resonances of the effective parameters. The mass densities and bulk moduli calculated by both the periodic homogenization method and the multiple scattering method differ only by 2% and 3%, which shows that both methods describe the same effective metamaterial.Band structure. Let us consider the propagation of acoustic waves in
this metamaterial in the local approximation (K
+
G≈
G). Giventhat the effective medium is isotropic, the frequency-dependent
effective wavenumber k∗
(ω)
can be computed by means of theeffective mass density
ρ
∗(ω)
and bulk modulus B∗(ω)
ask∗
(ω) = ω
ρ
∗
(ω)
B∗
(ω)
.
(25)The Willis terms S and SĎdisappear from the local version of the dispersion curve (Eq.(21)) since these quantities are complex
Fig. 2. Dispersion curves along theΓX direction computed by the PWE method
(black dots), by the periodic homogenization method in the local approximation (dark gray dashed–dotted line) and by the multiple scattering homogenization method (light gray dashed line) for an infinite square array of cylinders embedded in a fluid background so thatρa=ρb/2, Ba=0.02Bband Ra=0.1a. The curves are presented in terms of normalized wave vector ka/(2π)and normalized frequency
fa/cb.
conjugates, and it can be shown that in the local approximation they are purely imaginary, so that their addition cancels in this approximation [28,29].
Fig. 2compares the normalized dispersion curves along theΓX direction computed by means of the PWE method (black dots), the
periodic homogenization method in the local approximation (K
+
G
≈
G) (dark gray dashed–dotted line) and a multiple scattering homogenization method described in Ref. [17] (light gray dashed line). It is shown that the metamaterial presents a gap between the normalized frequencies 0.2 and 0.45. The agreement between the three methods is very good for the first branch for[0
−
0.
1]fa/
cb.For higher frequencies, the first branch is correctly approximated by the periodic homogenization method up to ka
/(
2π) =
0.
35. As expected, at kx=
0, the frequencies of the second branch of thegap computed by means of the PWE method and by the periodic homogenization coincide exactly at fa
/
cb=
0.
447. They differonly by 4% from the frequency obtained by the multiple scattering method. However, for this second branch, there is an agreement only for small wavenumbers. The local description of the material is not accurate and the dependence on the wave vector of the effective parameters has to be taken into account [29].
4. Anisotropic and nonlocal acoustic metamaterial
In this section the propagation of acoustic waves in an anisotropic metamaterial made of a rectangular lattice of cylinders is investigated. The unit cell of the lattice is a rectangle of size a and b
=
2a. The side of length a is oriented along the x axis. The mass density and bulk modulus of the scatterers and background medium are the same as in the previous section, but the radiusof the cylinders is Ra
=
0.
49a. The dispersion curves along thedirections x and y are depicted inFig. 3(b) and (a), respectively. The anisotropy of the metamaterial yields an anisotropic mass density with the same principal axis of the crystal, thus in the x–y plane the dispersion curve is
ρ
−1 xx(ω)
k 2 x+
ρ
−1 yy(ω)
k 2 y=
ω
2B−1(ω).
(26)Fig. 3(b) and (a) show the dispersion curves along the x and y
directions (ky
=
0 and kx=
0, respectively) using the aboveexpression. It is shown that the branches at low wavenumber are very well predicted by the homogenization method, however as we move towards the edge of the Brillouin zone the deviation from the local approximation are more important, and the anisotropy of the crystal becomes more evident given the high difference between the two directions of propagation. This deviation from the local description is even more drastic in the normalized frequency range [0.14–0.165], where along the x direction, two modes are found for each frequency, while the local approximation predicts no propagating solution at all, since the bulk modulus is negative and the mass density is positive (seeFig. 4(a) and (c)) at kx
=
0.This behavior cannot be predicted by a dispersion relation like that shown in Eq.(26)if the coefficients
ρ
xx−1, ρ
yy−1and B are functions of the frequency only, since then for the same frequency there is only one mode. Thus, nonlocal effects, that is, the dependence of these coefficients with the wavenumber, need to be included.Fig. 4, panel (a), plots the mathematical expression of the non-local effective bulk modulus B
(ω,
kx)
, Eq.(22a), for ten frequenciesin the range [0.14–0.165] along the x direction, that is, in the region where the second branch in the dispersion curve is found (see panel (b)). Panels (c) and (d) show the same plots for
ρ
xx−1 andρ
−1yy (Eq.(22b)), respectively. We see that the dependence
of these expressions with the wavenumber is strong, since they are resonant not only in frequency but also in wavenumber. For this geometry, we have shown by additional calculations that the contribution of the Willis term is negligible.
It is numerically found that the descending branch corresponds
to the resonances of Eq.(22a) associated to the effective bulk
modulus, that is, for those pairs of
(ω,
kx)
where B(ω,
kx) → ∞
.These solutions are obtained from the dispersion relation (26)
along the x direction (ky
=
0) in its nonlocal version,ρ
−1xx
(ω,
kx)
k2x=
ω
2B−1
(ω,
kx
).
(27)We can see then inFig. 4, panel (a), that before the resonant wavenumber, the bulk modulus is negative, so that the above equation has no propagating solution, but at a resonance it covers all the possible positive values in a very sharp wavenumber-region. Then it is easy to find a solution for the dispersion curve at this wavenumber. Consequently, the dependence on the wavenumber of Eq.(22a)yields the existence of this solution for the dispersion curve.
Surprisingly, the ascending branch corresponds to the reso-nances of the expression of
ρ
−yy1(ω,
kx)
parameter, which isspe-cially intriguing given that the propagation is along the x direction, thus ky
=
0 and, according to Eq.(26),ρ
yy−1plays no role in thedis-persion curve. To understand this behavior, let us multiply Eq.(26) by
ρ
yy(ω,
kx)
, we have then, for ky=
0,ρ
yy(ω,
kx)[ρ
xx−1(ω,
kx)
k2x−
ω
2B−1
(ω,
kx
)] =
0.
(28)Eq.(28)shows that for the pairs
(ω,
kx)
such thatρ
yy−1(ω,
kx) →
∞, that is,
ρ
yy(ω,
kx) →
0, we have an additional solution ofthe dispersion curve, consequence of the non-local nature of the effective mass density.
Therefore, the description of metamaterials as locally resonant structures is clearly incomplete, since, even at low frequency, it is found an incomplete prediction of the dispersion curve for large
Fig. 3. Normalized dispersion curves in the directions of propagation x (b) and y (a) calculated by the PWE method (black dots) and the periodic homogenization in the local
approximation. The full dark gray dots present the homogenized wave number calculated withρ∗
xxand the empty light gray dots present the homogenized wave number calculated withρ∗
yy.
Fig. 4. Variations of the mathematical expressions of the non local effective parameters as a function ofωand kx(Eq.(22)): B/Bb(a), dispersion curve given by the plane wave expansion method (b),ρ−1
xxρb(c) andρyy−1ρb(d) for ten normalized frequencies.
wavenumbers and, even more important, the inclusion of nonlocal effects explains the existence of solutions in frequency regions where the local approximation predicts evanescent waves only. It is clear that a nonlocal theory for metamaterials is required, where the constitutive parameters are computed by means of expressions (22).
5. Summary
In summary, we have presented an effective medium theory for periodic acoustic metamaterials based on the Plane Wave Expansion method. This method extracts the average fields from the generalized eigenvalue equation, and after identification of the
different coefficients we can define a set of generalized constitutive parameters. In the local approximation, a good agreement with the classical multiple scattering approach is found, while it is also found that the local approximation is not always valid. Then, an anisotropic system is presented in which additional solutions are found for a frequency range in which the local approximation predicts no solution for propagating waves, which shows that the expressions of the non-local parameters have to be employed to properly predict the behavior of the metamaterial. It is clear then that metamaterials are in general non-local composites, and further developments of both theoretical and practical aspects will require the use of an advanced theory for their description. Acknowledgment
This work was supported by LabEx AMADEus (ANR-10-LABX-42) in the framework of IdEx Bordeaux (ANR-10-445IDEX-03-02), France.
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