Personales.upv.Es ~Jsanched ReyesAyonaE JASA132 2896 12-Homogenization Theory

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Homogenization theory for periodic distributions of elasticcylinders embedded in a viscous fluid

Edgar Reyes-Ayona, Daniel Torrent, and Jos e S anchez-Dehesa a)

Grupo de Fen omenos Ondulatorios, Universitat Polit ecnica de Valencia, Camino de Vera s.n. (Edicio 7F), ES-46022 Valencia, Spain

(Received 23 October 2011; revised 30 March 2012; accepted 10 April 2012)

A multiple-scattering theory is applied to study the homogenization of clusters of elastic cylindersdistributed in a isotropic lattice and embedded in a viscous uid. Asymptotic relations are derivedand employed to obtain analytical formulas for the effective parameters of homogenized clusters inwhich the underlying lattice has a low lling fraction. It is concluded that such clusters behave, inthe low frequency limit, as an effective elastic medium. Particularly, it is found that the effectivedynamical mass density follows the static estimate; i.e., the homogenization procedure does notrecover the non-linear behavior obtained for the inviscid case. Moreover, the longitudinal and trans-versal sound speeds do not show any dependence on uid viscosity. Numerical simulationsperformed for clusters made of brass cylinders embedded in glycerin support the reliability of theeffective parameters resulting from the homogenization procedure reported here.VC 2012 Acoustical Society of America . [http://dx.doi.org/10.1121/1.4744933]

PACS number(s): 43.20.Bi, 43.35.Mr, 43.35.Bf [ANN] Pages: 2896–2908

I. INTRODUCTION

The propagation of acoustic waves in inhomogeneousmedia has attracted much attention over the years. Recently,there has been a growing interest in a special type of inho-mogeneous materials named phononic crystals, whose elas-tic parameters vary periodically in space. The interest inthese materials arises from the possibility of having fre-quency regions, known as absolute band gaps, where thepropagation of elastic waves is forbidden, whatever their polarization and wave vector. 1 – 5 In addition to their abilityto behave like perfect mirrors, these structures can proveparticularly useful for applications requiring a spatial con-nement of acoustic waves and can hence be used as acous-tic lters or very efcient waveguides. 5 A class of phononiccrystals where the solid background is replaced by a uid or a gas are usually called sonic crystals. 6 – 8

The theoretical description of sonic and phononic crys-tals has employed two main strategies. One consists of bandstructure calculations 1 – 4 ,6 – 8 of the corresponding innite sys-tem in order to match the gaps in the dispersion relation withthe attenuations found in the transmission spectra. The other one calculates the transmission spectra by different algo-rithms such as the transfer matrix method, 9 nite differen-

ces,10 – 12

or by multiple scattering.13 – 15

The latter is the mostsuitable for describing nite-size structures. Experimentalobservations have recently demonstrated that phononic crys-tals can also be employed in the frequency region well belowthe bandgap as acoustic lenses for sound focusing as wellas acoustic interferometers that work similarly to their opti-cal counterparts. 16 ,17 Inspired by these ndings, severalapproaches have been presented in order to give a descrip-tion of phononic crystals in the low frequency limit or long

wavelengths. 18 – 28 Kafesaki et al. 18 analyzed the phenom-enon of the sound speed of drops in mixtures by means of aneffective medium obtained by using the coherent potentialapproximation. Krokhin et al. 19 employed plane waveexpansions to derive analytical expressions for the speed of sound valid for arbitrary lling fraction and different geome-tries of the inclusions. More recently Mei et al. 24 and Torrentet al. 25 ,26 applied the multiple scattering method to developa homogenization theory to simultaneously obtain the effec-tive sound speed as well as the effective density of nitecluster of cylinders embedded in air. Kutsenko et al. 27 intro-

duced a new analytical approach based on the monodromymatrix for the two dimensional case to derive the effectiveshear speed in two dimensional phononic crystals. Finally,the results of Wu and co-workers 28 are also of interest to thiswork though they report an effective medium theory appliedto the case of elastic inclusions embedded in a different elas-tic medium.

Despite of the extensive studies in sonic/phononic crys-tals the effect of losses at low frequencies has been scarcelyconsidered. Only a few works have been devoted to this topic.A related problem was rstly tackled by Einstein in his classi-cal 1905 paper, where he studied the effective viscosity of rigid spheres in a viscous medium. 29 Afterwards, Batchelor and Green 30 tried to determine the bulk stress in a suspensionof spherical particles up the second order in the lling frac-tion and Sprik and Wegdam 31 showed that shear viscosity inthe liquid constituent may lead to gap formation in solid-liquid systems. For the case of periodic distributions, Psaro-bas et al. 32 ,33 developed an on-shell multiple scatteringmethod in order to incorporate the effect of viscoelastic lossesin the band structure calculation by means of a complex anddispersive Lam e’s constant. Hussein 34 ,35 introduced a modi-ed nite element method (reduced Bloch mode expansion)to calculate the dispersion relation using nite-number set of Bloch mode eigenvectors at each wave-vector point.

a) Author to whom correspondence should be addressed. Electronic mail: [email protected]

2896 J. Acoust. Soc. Am. 132 (4), Pt. 2, October 2012 0001-4966/2012/132(4)/2896/13/$30.00 VC 2012 Acoustical Society of America



It was found that in sonic crystals the existence of com-plete band gap is more difcult due to suppression of trans-versal sound in the liquid background. However, viscosity inliquids may lead to gap formation in solid-liquid systems.Shear viscosity in the liquid introduces a new length scaleassociated with the penetration of shear stress into the viscousliquid. When the viscous penetration depth d ¼ ffiffiffiffiffiffiffiffiffiffiffiffið2g=qbx Þp becomes comparable to the structural length scale in the com-posite, viscous effects cannot be ignored in describing theacoustical properties.

In this work we analyze the scattering of acoustic wavesby clusters of elastic cylinders distributed in a periodic lat-tice and immersed in a viscous medium. We are here inter-ested in the homogenization of this type of systems and their properties are obtained using an effective medium theory. Inother words, we are considering clusters interacting withsound wavelengths much larger than the distance betweencylinders and larger also than the cylinder’s diameter size. Interms of frequencies, we are dealing with very low frequen-cies; i.e., in a frequency region very far from the frequencyat which the rst acoustic band gap of the underlying lattice

appears. Asymptotic relations are derived and employed toformulate a method of homogenization based on the scatter-ing properties of the cluster. By using the multiple-scatteringmethod we have demonstrated that, in the long wavelengthlimit, a distribution of elastic cylinders embedded in a vis-cous uid effectively behaves like a single elastic cylinder with parameters that can be analytically obtained. Semi-analytical formulas for the effective parameters (i.e., effec-tive longitudinal sound speed, effective transversal soundspeed, and effective density) are obtained in the case of dilute (low lling fraction) structures. We have obtained thatthe effective mass density follows the linear static estimate,that is, the homogenization procedure does not recover an

effective mass density with a non-linear dependence on thelling fraction, as was the result for the inviscid case. More-over, it is worth mentioning that there is no explicit depend-ence on the uid viscosity for the mass density or for thelongitudinal and transversal sound speeds. However, viscos-ity effects are implicitly involved in the resulting effectiveparameters through the boundary conditions employed in thehomogenization procedure.

This article is organized as follows. In Sec. II the mathe-matical formulation for a single elastic cylinder in a viscousbackground is introduced and the generalization to the caseof a cluster by using the multiple scattering method is alsobriey described. In Sec. III the concept of effective t matrixis introduced and the effective parameters are obtained for ageneral elastic-viscous uid composite. Numerical resultsfor a single and a cluster of elastic cylinders embedded in aviscous uid are reported and discussed in Sec. IV, wherethe limit of wavelength to reach homogenization behavior inthese systems is also studied as a function of the frequency.Finally, the work is summarized in Sec. V.

II. MATHEMATICAL FORMULATION

There are a wide variety of models in the literature thatdeal with viscous effects in acoustic wave propagation. The

model most commonly used, which is also followed here, isbased on the solution of the mechanical equations; i.e., allthe terms in the linearized Navier–Stokes equations are takeninto account. This particular treatment of viscosity compli-cates the analysis because the uid medium can uphold shear and compressional modes; both must be accounted for satis-fying the boundary conditions at the interfaces. First, wetheoretically examine the two-dimensional (2 D) scatteredacoustic eld produced by an impinging plane wave on a sin-gle elastic solid cylinder in order to obtain the t matrix. Weassume, without loss of generality, that the cylinder has acircular cross-section and that it is innitely long along thez-axis. We also consider that the incident wave propagatesnormally to the cylinder axis. Then we generalize the resultto the case of multiple scatterers of arbitrary cross section.

A. t matrix of an elastic cylinder embeddedin a viscous medium

Let us consider the standard acoustic equations for lin-ear ows in a homogeneous viscous uid medium; i.e., theequation of continuity

@ q@ t þqbr v ¼0; (1)

the momentum equation

r p þqb@ v@ t ¼gr 2v þ n þ

13

g rðr vÞ; (2)

and the equation of state

dpd q ¼c2

b; (3)

in which v is the complex uid velocity vector, qb is the am-bient uid density, q is the density perturbation, p is the uidpressure, cb is the sound speed, and g and n are the shear andbulk viscosities, respectively.

Equations (1) – (3) can be combined to obtain the equa-tion for the velocity vector v:

qb@ 2v@ t 2

qbc2brðr vÞ¼gr 2 @ v

@ t þ nþ13

g r r@ v@ t :

(4)

The velocity eld can be expressed as a superposition of lon-

gitudinal and transversal vector components by using theHelmholtz decomposition theorem,

v ¼ ru þ r w; (5)

where u and w are the scalar and vector potentials, respec-tively. For 2D periodicity the vector potential has onlyz-component, w ¼ ð0; 0; wÞ. Now, substituting this decom-position into Eq. (4) , we get the following two equations:

ðr2 þk 2‘ Þu ¼0;

ðr2 þk 2t Þw ¼0; (6)

J. Acoust. Soc. Am., Vol. 132, No. 4, Pt. 2, October 2012 Reyes-Ayona et al. : Homogenization of rods in a viscous fluid 2897



where k ‘ and k t are the longitudinal and transversal wavenumbers, respectively, in the viscous uid and are given by

k ‘ ¼ xcb

1 þ i x2qbc2

b

43

g þ n ;

k t ¼ ð1 þ iÞ ffiffiffiffiffiffiffiffi ffixq b

2gr : (7)

The solutions of Eq. (6) are obtained by expanding the longi-tudinal and transversal vector components in series of Bessel(Hankel) functions as follows:

u ext ¼X1

q¼ 1 A‘

q J qðk ‘ r Þeiqh; (8)

and

u scatt ¼X1

q¼ 1 B‘

q H qðk ‘ r Þeiqh;

wscatt ¼X1

q¼ 1 B

t q H qðk t r Þe

iqh:

(9)

The eld u ext represents the incident (external) eld over thecylinder while u scatt and wscatt are the respective componentsof the scattered elds. The expansion over Hankel functionsonly guarantees that the scattered waves are outgoing. Wealso assume that the external eld is pure longitudinal; i.e.,the background considered here is a viscous uid that, in ab-sence of scatterers, supports propagation of longitudinalwaves only.

On the other hand, sound wave in the elastic cylinder isdescribed by the equation for the displacement vector (Nav-ier’s equation of elasticity):

qa@ 2u@ t 2 ¼ ðka þ l aÞrðruÞ þl ar 2u ; (10)

where qa , ka , and l a are the density and the Lam e’s con-stants of the cylinder, respectively. The displacement vector is related to the velocity vector by means of the temporal de-rivative, v ¼@ u=@ t ¼ ix u . Introducing scalar and vector potential for velocity v, we obtain

v r ¼ ix @ /@ r þ

1r

@ f@ h ;

v h ¼ix 1r

@ /@ h þ@ f

@ r : (11)

Here the potentials / and n can be analogously expandedover Bessel functions

/ int ¼X1

q¼ 1C‘

q J qðk a‘ r Þeiqh;

f int ¼X1

q¼ 1Ct

q J qðk at r Þeiqh; (12)

where k a‘ ¼x =c‘ and k a‘ ¼x =ct are the longitudinal andtransversal wave vectors, respectively, and c‘ and ct their corresponding velocities, which are given in terms of massdensity and Lam e’s constants.

The coefcients of expansions (9) and (12) are obtainedby imposing the continuity of the normal and tangentialcomponents of the velocity and stresses, respectively, at thecylinder surface. After some algebra, we get the t matrix thatrelates the coefcients A‘

q to B‘q and Bt

q is (see Appendix A)

tq ¼ ½Q q R q O 1q Nq

1

½Pq R q O 1q Mq : (13)

In general the t -matrix is a block diagonal matrix whose di-agonal elements tq are 2 2 matrices

tq ¼t ‘‘q t ‘t

q

t t ‘q t tt q0@ 1A

; (14)

where t ‘‘q and t t ‘q involved Bessel and Hankel functions, andt ‘t q and t tt q are equal to zero (see Appendix A ). Note that even

when determinant of the t -matrix is zero the ux conserva-tion is satised. Elasticity involves the existence of twopolarization modes so that the conservation of ux impliesthe conservation of the total ux; i.e., the addition of the lon-gitudinal and transversal uxes instead of conservation of individual components.

Let us recall that the problem of a single elastic cylinder in a viscous uid has been already considered in Refs. 36 and37 for circular and elliptic cross sections, respectively. In thesestudies the coefcients B‘

q and Bt q are obtained from A‘

q bysolving a system of linear equations. However, here we havedeveloped a t matrix formalism in order to tackle the problemof multiple elastic cylinders in a cluster. For a comprehensivedescription of this formalism and its application to problemsof multiple scattering the readers are addressed to the book byVaradan and Varadan, 38 and the references therein.

B. Sound scattering by a cluster of elastic cylindersembedded in a viscous medium

Now, let us consider a cluster of N cylinders located atpositions Rb ðb ¼1; 2; :::; N Þ, Rb is a vector in the XY plane.The cylinders will be assumed of arbitrary cross section. Thegeometry of the problem and the denitions of the variableemployed are shown in Fig. 1.

If an external wave u ext with temporal dependence e ix t

impinges the cluster, the total eld around a cylinder a is asuperposition of the external eld and the radiation scatteredby the rest of the cylinders b:

u aðr ; hÞ ¼u extðr ; hÞ þX N

b6¼a

u scattb ðr ; hÞ;

waðr ; hÞ ¼X N

b6¼a

wscattb ðr ; hÞ; (15)

where u scattb and wscatt

b are the elds scattered by the b cylin-der. Without loss of generality, those elds can be expanded

2898 J. Acoust. Soc. Am., Vol. 132, No. 4, Pt. 2, October 2012 Reyes-Ayona et al. : Homogenization of rods in a viscous fluid



as a combination of Bessel and Hankel functions centered atthe cylinder position. If the expansion coefcients are ðCaÞ

‘q,

ð AaÞ‘q, ð BbÞ

‘q , ðCaÞ

t q, and ð BbÞ

t q , for u a , u ext , u scatt

b , wa , and

wscattb , respectively, the expressions (15) can be cast into the

following relations between the coefcients:

ðCaÞ‘q ¼ ð AaÞ‘

q þX N

b6¼a X1

s¼ 1ðGab ðk ‘ÞÞqsð BbÞ

ls;

ðCaÞt q ¼

X

N

b¼1

X

1

s¼ 1ðGab ðk t ÞÞqsð BbÞt

s; (16)

ðGab ðk iÞÞqs being the propagator from b to a whose compo-nents are

ðGab ðk iÞÞqs ¼ ð1 dab Þeiðs qÞU ab H q sðk ir ab Þ; (17)

for k i ¼k ‘ ; k t .Note that coefcients ð AaÞ‘

q in Eq. (16) are known, but

ðCaÞ‘q, ð BbÞ

‘q , ðCaÞ

t q, and ð BbÞ

t q are not. To reduce the com-

plexity of the following calculations it is convenient toexpress Eq. (16) in a more compact way, i.e.,

ðcaÞq ¼ ða aÞq þX N

b¼1 X1

s¼ 1ðGab ÞqsðbbÞs; (18)

with

ðcaÞq ¼ ðCaÞ‘q

ðCaÞt q

!; ðbbÞs ¼ ð BbÞ‘s

ð BbÞt s !;

ða aÞq ¼ ð AaÞ‘q

0 !and

ðG ab Þqs ¼ðGab ðk ‘ÞÞqs 0

0 ðGab ðk t ÞÞqs0@ 1A

: (19)

As in the case of a single cylinder, the boundary conditionsat the cylinder surface allow us to calculate the t matrixwhose components ðtaÞqs relate the coefcients ðcaÞq and

ðbbÞs:

ðb aÞs ¼XqðtaÞqsðcaÞs: (20)

Introducing the coefcients (20) in Eq. (18) , after straight-forward calculation we get

ðb aÞq X N

b¼1 X1

s¼ 1ðta G ba ÞqsðbbÞs ¼ ðta a aÞq: (21)

By truncating the sum over s within jsj< qmax , this equationis reduced to a linear equation where the dimension of the

relevant matrix is 2 N ð2qmax þ1Þ 2 N ð2qmax þ1Þ. Thus, inmatrix form MB ¼ S , where B and S are column blockmatrices with elements ðb1Þq , ðb2Þq,…, ðb N Þq , and ðt1a1Þq,

ðt2a2Þq ,…, ðt N a N Þq , respectively. Mis a N N block matrixwhere each element is a matrix of dimension 2 ð2qmax þ 1Þ2ð2qmax þ 1Þ. In short, the matrix elements can beexpressed by

ðM ab Þqs ¼dab dqs I ðta G ab Þqs : (22)

Finally, the unknown coefcients B can be easily obtainedby a matrix inversion, B ¼ M1S :

ðb aÞq ¼X N

b¼1 X1s¼ 1

ðM 1ab Þqsðtb abÞs: (23)

Thus the solution for a given cluster is obtained in terms of the block matrix ðM 1

ab Þqs and the t matrix of the individualcylinders. As explained before, the t matrix is diagonal as inthe inviscid case but now has a higher dimensionality sinceinvolves two different modes of propagation. This result isvery general and valid for cylinders of any cross section, anylling fraction and any frequency. However, it is worth tomention that the above statement must be restricted, in thecase of complex cross sections, to the possibility of evaluat-

ing the t matrix correctly and efciently. Despite that we canassume that it is valid in general and it is particularly validfor low frequencies, which is the regime studied in the nextsection.

III. HOMOGENIZATION OF A CLUSTER OF ELASTICCYLINDERS EMBEDDED IN A VISCOUS MEDIUM

The purpose here is to introduce the effective t matrix of a cluster. This t -matrix will relate the coefcients of the inci-dent eld to the coefcients of the eld scattered by the clus-ter. The total scattered eld around the cluster will be a

FIG. 1. Coordinate systems and denition of variables employed in theequations of the multiple-scattering.




superposition of the elds scattered by all the cylinders, thatis,

Pscatt ðr ; hÞ ¼qbc2

b

ix r vscatt ðr ; hÞ

¼ P0X N

a¼1 X1

q¼ 1ð BaÞ‘

q H qðk ‘r aÞeiqha ; (24)

where H qð Þ is the q-th order Hankel function of rst kindand ðr a ; haÞ are the polar coordinates with the origin trans-lated to the center of the a-cylinder, i.e., r a ¼r R a , asshown in Fig. 1.

The last equation can be also expressed as a function of coordinates centered at the cluster origin using the Graf’saddition theorem 39 as

Pscatt ðr ; hÞ ¼ P0 X1

p¼ 1 Bsc

p H pðk ‘ r Þeiph; (25)

where

Bsc p ¼X

N

a¼1 X1

q¼ 1 J p qðk ‘ RaÞeiðq pÞU að BaÞ

‘q: (26)

These coefcients are related to ð AbÞls , or more precisely to

ðabÞs, through the expression in Eq. (23) since ð BaÞ‘q is the

longitudinal projection of ðb aÞq, i.e., ð BaÞ‘q ¼‘ ðbaÞq .

Therefore,

Bsc p ¼‘ Xa ; b Xq; r

J p qðk ‘ RaÞeiðq pÞU a ðM 1ab Þqr ðtb abÞr :

(27)

The relationship between coefcients ðabÞs and those of theexternal eld aq can be obtained as follows. First, we assumea generic incident eld that can be expanded as a sum of Bessel functions

Pextðr ; hÞ ¼ P0 X1

q¼ 1aq J qðk ‘r Þeiqh; (28)

which can be expressed in terms of coordinates with originat the a cylinder as follows:

Pextðr a ; haÞ ¼ P0X1

s¼ 1 X1

q¼ 1 J q sðk ‘ RaÞeiðq sÞU a aq0@ 1A

J sðk ‘ r aÞeisha ; (29)

and, therefore, the coefcients ða aÞs are

ða aÞs ¼X1

q¼ 1 J q sðk ‘ RaÞeiðq sÞU a aq: (30)

Now, the elements of the effective t matrix can be obtainedin the following form:

t eff ps ¼‘ Xa ; b Xq; r ; t

J p qðk ‘ RaÞeiðq pÞU aðM 1ab Þqr

ðtbÞrt J s t ðk ‘ RbÞeiðs t ÞU b ; (31)

which can also be cast as

t eff ps ¼‘

Xa ; b

Xq; r ; t

ðJ aÞ pqðM 1ab Þqr ðtbÞrt ðJ 0bÞst ; (32)

where

ðJ aÞ pq ¼ J p qðk ‘ RaÞeiðq pÞU a ;

ðJ 0bÞst ¼ J s t ðk ‘ RbÞeiðs t ÞU b : (33)

When the underlying lattice of the cluster is isotropic (squareor hexagonal) the cluster behaves, in the long wavelengthlimit, as a single homogeneous and isotropic cylinder. 40 – 42

Moreover, it also has been shown that the homogenizationcondition for a cluster of cylinders immersed in a invisciduid is given by 25 ,26

t eff pq ¼t cyl

pq ; 8 p; q; (34)

where t pq are the k -independent coefcients of the lower order terms in the k -expansion of the corresponding t -matrixelements. It is straightforward to show that, due to the formof the effective t matrix in Eq. (32) , the homogenization pro-cedure in our case of elastic cylinder in a viscous mediumalso leads to Eq. (34) . However, as will be seen below, theeffective parameters are quite different from those obtainedfor inviscid case.

Following the procedure employed for the inviscidcase, 26 the effective parameters for a cluster of elastic cylin-ders embedded in a viscous uid and distributed in a latticewith a low lling fraction are (see Appendix B):

qeff ¼ f q a þ ð1 f Þqb; (35a)

1 Beff ¼ f

1 Ba þ ð1 f Þ

1 Bb

; (35b)

l eff ¼ f l a ; (35c)

where f is the fraction of volume occupied by the cylindersin the clusters and Ba ¼ka þ l a is the 2D bulk modulus of the elastic cylinder. Note that the effective parameters do not

show any explicit dependence on the viscosity for the casehere considered of clusters with low f . However, the expres-sions have embedded the viscosity through the boundaryconditions employed in their derivation. It is remarkable thatthe expression for q eff in Eq. (38) is similar to that obtainedfrom the homogenization of composites consisting of elasticinclusions embedded in another elastic medium. 28 This is aninteresting result indicating that, though the boundary condi-tions in our system are slightly different to that of two elasticmedia, we arrive to similar expressions for qeff .

Let us remark that our homogenization procedure cannotrecover the effective mass density obtained for the inviscid




case, which for the case of dilute systems is qeff ¼qb

ðq að1 þ f Þþqbð1 f ÞÞ=ðq að1 f Þ qbð1 þ f ÞÞ.26 ,43 – 45 Thiscan be explained as due to the fact that the inviscid case(g ! 0) is a singular limit in the sense of the asymptoticexpressions derived for the viscous medium. In order torecover the results for the inviscid case we have to return tothe equation and apply the corresponding boundary conditionsat the cylinders’ surface. It can be concluded that the suppres-sion of the non-linear behavior of the mass density in compo-sites made of elastic inclusions in an elastic background andelastic bodies in a viscous background is directly related withthe boundary conditions at the corresponding interfaces.

From Eq. (35) we conclude that the homogenized cylin-der behaves as an effective elastic medium. Their effectivelongitudinal and transversal phase velocities are derived byusing 2D elasticity; i.e., c‘; eff ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffið Beff þl eff Þ=q eff p andct ; eff ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffil eff =q eff p . Their expressions are

c2‘; eff ¼

Ba

ð1 f Þ Ba þ f þ f l a

Bb 1q a f þð1 f Þ ; (36a)

c2t ; eff ¼ f l

a= Bb

q a f þ ð1 f Þ ; (36b)

where the bar symbols indicate that the magnitudes are nor-malized to the corresponding values of the background. Inother words, the velocities are divided by cb , and the bulkmodulus and density by Bb and qb, respectively. As it isshown later, these simple expressions are valid for values f as large as 0.6. A similar conclusion was reported by Parnellet al. 45 who analyzed the validity of dilute estimates in a va-riety of homogenization schemes applied to elastic compo-sites. Note that an explicit dependence on the viscosity willappear for the case (not studied here) of larger f values,

through the terms containing the cylinders’ interaction. Thecase of highly compact clusters will be the object of a futurework that will be published elsewhere.

Figures 2 – 4 depict the effective parameters (normal-ized to the background) as a function of lling fraction.They are obtained by analyzing clusters of N cylindersmade of brass, iron (Fe), and aluminum (Al), respectively,in three different uid backgrounds: water in Fig. 2, oliveoil in Fig. 3, and glycerin in Fig. 4. We have taken N ¼151cylinders as representative volume element necessary toguarantee that the homogenization method produces reli-able effective parameters, as it was demonstrated in theinviscid case. 25 ,26 ,39 The acoustic parameters of the threebackgrounds employed in the calculations are given inTable I. The material parameters of elastic cylinders are takento be qa ¼8500 Kg/m 3 , c‘ ¼4305 m/s, ct ¼2152 m/s for brass; qa ¼7870 Kg/m 3 , c‘ ¼5778 m/s, ct ¼3137 m/sfor iron; qa ¼2699 kg/m 3 , c‘ ¼6507 m/s, ct ¼3044 m/s for aluminum. 47 Numerical calculations are made by using theseparameters normalized to that of the background, and theyare summarized in Table II. The cylinders in the cluster weredistributed in a hexagonal lattice whose lling fraction f hex ¼ ð2p= ffiffiffi3p Þð Ra =aÞ

2 , where a is the lattice constant. Thehexagonal lattice was selected because is an isotropic lattice(like the square lattice) and for comparison purposes with the

inviscid case, where results were obtained using this lat-tice. 25 ,26 ,48 For the calculations we have taken a

¼794 l m

while the cylinders’ radius have been changed from Ra ¼0to Ra ¼0:5a , which corresponds to 0 f hex 0:907 (closepacking condition). The radius Reff of the homogenized cylin-der is calculated by imposing the condition

f cls ¼ N ðp R2

aÞp R2

eff ¼ f hex ; (37)

where f cls is the fraction of volume occupied by the N cylinders in the cluster. For the structure under study Reff

¼ ffiffiffiffiffiffiffiffiffið N ffiffiffi3p =2pÞq a ¼6:452 a .

Figures 2 – 4 show that c‘; eff is always larger than ct ; eff

and that for f ! 0 the background properties are recovered;i.e., cl; eff ! cb and ct ; eff ! 0.

Figure 5 shows an alternative representation of thephase velocities given in Figs. 2 – 4. Each point in the dia-gram corresponds to a lling fraction of the underlaying hex-agonal lattice f hex . It is observed that the effective mediahave ratios between longitudinal and transversal phasevelocities c‘=ct that are not approximately equal or greater than 2 as in ordinary solids. 49 In fact, the ratios c‘=ct arelower than 2 and can be tailored by adjusting f hex . With thesenew materials it is possible to get Poisson’s ratios higher than 1 =2 for lling fractions small enough. Recall that

FIG. 2. (Color online) Effective parameters of composites made of cylindersof brass (continuous lines), iron (dashed lines) and aluminum (dotted lines),respectively, embedded in water as function of the lling fraction. The pa-rameters are normalized to those of water (see Table I). The shadowedregions dened the lling fraction where the values obtained from Eqs. (35)and (36) are not reliable.




Poisson’s ratio is r ¼1 2ðct =c‘Þ2 for 2D systems and thatvalues r larger than 0.5 corresponds to a modulus of rigiditythat is small compared with the compression modulus. Val-ues to the right of the arrows correspond to f 0:6, whereEq. (36) is not reliable. The velocities of the material cylin-ders are represented by symbols. Note that, despite of thedilute medium approach, the effective velocities shown inFig. 5 tend to that of their material constituents as f ! 0:9,the close packing condition of the hexagonal lattice. Notethat effective elastic media with ratios c‘=ct not found inordinary solids is direct a consequence of the backgroundviscosity here considered.

Although effective parameters in Eq. (35) do not showany explicit dependence on viscosity g, the next section willshow that viscosity produces observable effects in the result-ing effective medium.

IV. VISCOSITY EFFECTS: RESULTSAND DISCUSSION

Here, we study two selected structures: a single elasticcylinder with circular cross-section and a cluster made of cir-cular elastic cylinders having an external circular shape. Bothstructures are considered to be immersed in a viscous uid.First, in Sec. IV A, we analyze the angular distribution of thescattered pressure at the far eld for the case in which the vis-

cous uid is glycerin. Afterwards, in Sec. IV B, we analyzethe relative difference of the forward elds by the effectivemedium and the circular cluster, respectively, in order toestablish the homogenization limit. We conclude that viscos-ity produces observable effects at the far eld as well as inthe frequency cutoff determining the homogenization limit.

A. Viscosity effects at the far field

The scattered eld by a cluster of cylinders is

Pscatt ðr ; hÞ ¼ P0

X N

a

¼1

X1

q

¼ 1

ð BaÞ‘q H qðk ‘ r aÞeiqha : (38)

FIG. 3. (Color online) Effective parameters, as in Fig. 2, with olive oil as abackground. The effective parameters are normalized to the properties of the uid background.

FIG. 4. (Color online) Effective parameters, as in Fig. 2, with glycerin as auid background. The effective parameters are normalized to the propertiesof the uid background.

TABLE I. Parameters of the uids used in the calculations. They are takenfrom Ref. 46.

Water Olive oil Glycerine

qb (Kg/m 3) 998 920 1259cb (m/s) 1486 1430 1909g (Pa s) 0.001 0.084 0.95

TABLE II. Material properties of the elastic cylinders used in the numericalexamples. The elastic parameters are normalized to the corresponding uidbackground properties.

Water Olive oil Glycerine

Brass Fe Al Brass Fe Al Brass Fe Al

q a 8.52 7.89 2.70 9.24 8.55 2.93 6.75 6.25 2.14c ‘ 2.90 3.89 4.38 2.25 4.04 4.55 2.26 3.03 3.41c t 1.45 2.11 2.05 1.51 2.19 2.13 1.13 1.64 1.59




By using the asymptotic form of the Hankel functions for large arguments, 39 the acoustic pressure at the far eld is

r ðk ; hÞ limr !1 j ffiffir p Pscatt ðr ; hÞj

¼ P0 X N

a¼1 X1

q¼ 1ð iÞ

q

ð BaÞ‘qe ik ‘ Ra cosðh haÞeiqh :

(39)

For the case of a single cylinder this expression is simpliedto36

r ðk ; hÞ ¼X1

q¼ 1ð iÞ

q B‘qeiqh : (40)

Numerical simulations are performed by considering thecase of (i) a brass cylinder with circular cross section and ra-dius Ra ¼794 l m immersed in glycerin and (ii) a circular cluster of 151 brass cylinders distributed in a hexagonal lat-tice with parameter a ¼794 l m and with radii Ra ¼0:3a

The cluster radius is Rcls ¼6:452 a , which is obtained byusing Eq. (37) .Figure 6 represents the pressure at the far eld of both

cases for several frequencies. For the single cylinder case itis observed that viscosity only modies the forward andbackward pressure patterns of the pressure when the fre-quency of the incident wave is increased. A similar resultwas previously reported by Lin and Raptis. 36 However, notethat our result at frequency kRa ¼5 slightly differs from that

FIG. 5. (Color online) Phase diagram of the effective phase velocities. Thevelocities are calculated using expressions obtained for the case of dilutedstructures (low lling fractions). Values to the right of the downward arrowscorrespond to lling fractions higher that 0.6, where the calculated valuesare not reliable. The symbols represent the values of the materials employedin the different structures.

FIG. 6. (Color online) The scattered far eld for a brass cylinder immerse in a glycerin background for different frequencies: (a) kRa ¼0:1, (b) kRa ¼1, and(c) kRa ¼5, and the scattered far eld for a cluster of 151 cylinders of brass in glycerin (d) kRcls ¼0:1, (e) kRcls ¼1, and (f) kRcls ¼5; red line (viscous result)and black line (inviscid result).




in Fig. 2(c) of Ref. 36 . We attribute this difference to better convergence of data obtained here. In our calculations, wehave determined that convergence is achieved by includingangular momenta up to a certain qmax , which is obtainedfrom the following relationship:

qmax ¼q0 þ k N

X

N

i

¼1

Ri; (41)

with N and Ri being the number of cylinders and their radii,respectively. For the frequencies employed in this study ithas been found that q0 ¼3 ensures good convergence. Then,for the case of a single cylinder N ¼1 and kRa ¼5 theresulting qmax ¼9.

Regarding the pressure behavior at the far eld for thecluster [see Figs. 6(d) – 6(f) ], the strongest effect of the vis-cosity is due to additive contributions of losses produced atthe surfaces of cylinders. 49 For the frequencies used in thecalculation we get a signicant viscous penetration depth d.We can conclude that viscosity plays non-negligible role inthe modication of the pressure at the far eld, especiallyalong the backward and forward directions. This result isexpected since the main amount of energy is essentially con-centrated in both directions and, therefore, they are moresensitive to quantify the losses due to viscosity. We haveemployed this result to consider the forward direction as areference in order to establish the condition determining thelong wavelength limit (homogenization limit) of a circular cluster of elastic cylinders.

B. Homogenization limit for a cluster of elasticcylinders in viscous medium

A wavelength of around four times the lattice parameter has been found as a minimum wavelength in order to reachthe homogenization of a cluster of rigid cylinders in an invis-cid background. 25 ,26 However, due to the presence of viscos-ity in the background as well as the elastic properties of the

cylinders it is expected that this cutoff could be slightly dif-ferent. In fact, it has been shown above how the backgroundviscosity modies the pressure at the far eld at differentfrequencies.

In order to establish the new limit of homogenizationfor the viscous-elastic structures under study here we ana-lyze the pressure eld distributions produced by a cluster of brass cylinders and its corresponding homogenized cylinder.Particularly, we calculate the forward scattering cross sec-tions as a function of the frequency by using the expressionsin Eqs. (39) and (40) . Figure 7 reports the relative difference

between the forward scattering cross section calculated for the cluster and the corresponding homogenized cylinder r eff

for three different lling fractions; f ¼0:2, 0:4, and 0 :6. The

FIG. 7. (Color online) The relative difference between the forward scatter-ing cross section calculated for a cluster of 151 brass cylinders r cls and itscorresponding homogenized cylinder r eff for three lling fraction: f ¼0:2(dotted line), f ¼0:4 (continuous line), and f ¼0:6 (dashed line).

TABLE III. Effective parameters of hexagonal distributions of brasscylinders embedded in glycerin for several values of lling fraction. Theyare normalized to the corresponding values of glycerin (see also Fig. 4).

f q eff c ‘; eff c t ; eff

0.2 2.15 1.172 0.89340.4 3.30 1.239 1.01980.6 4.45 1.301 1.0755

FIG. 8. (Color online) Pressure map (amplitude) representing the scatteringof a sound plane wave of wavelength k ¼4a interacting with (a) a cluster made of 151 brass cylinders arranged in a hexagonal lattice with lling frac-tion f ¼0:2 and embedded in glycerin and (b) the corresponding homoge-nized cylinder also embedded in glycerin.




values of effective parameters employed to calculate r eff aregiven in Table III. Results in this gure establish that thenew homogenization cutoff is around k ¼6a , where a rela-

tive error less than 6% is obtained within the homogeniza-tion region for the lower lling fractions.

To support our previous nding, Figs. 8 and 9 depictpressure maps of the scattering of a plane sound wave by acircular cluster of brass cylinders distributed in a hexagonallattice with f hex ¼0:2 and the corresponding homogenizedcylinder for k ¼4a and k ¼6a , respectively. From Fig. 7 itis observed that the wavelength k ¼4a (the homogenizationlimit in an inviscid background) is not enough to get a goodagreement between both pressure patterns. Unlike this, mapsin Fig. 8, which correspond to k ¼ 6a , show a fairly goodagreement.

Based on these results we conclude that viscosityincreases the wavelength cutoff above which the homogeni-zation is achieved.

V. SUMMARY

We have employed the multiple scattering method toanalyze the scattering of acoustic waves by a cluster of elas-tic inclusions embedded in a viscous uid host. Particularly,we have obtained asymptotic relations of the correspondingt matrix elements in the long wavelengths limit. They areused to derive analytical formulas for the parameters of theeffective homogeneous solid representing the cluster in such

a limit. We demonstrated that the effective mass density aswell as the effective longitudinal and transversal soundspeed depend on the structural distribution of the cylinders,their physical parameters, and the embedded medium. Theresulting effective homogenized solid is a kind of elasticmetamaterial whose elastic properties can be tailored bychanging the cylinders’ lling fraction in the cluster. Wereported numerical simulations for several elastic-viscousuid composites that support the validity of the formulasobtained for the effective parameters. It is also found thatviscosity produces an increase of minimum wavelength for which the homogenization is achieved in comparison withthe case of an inviscid host. We concluded that articial elas-tic materials with parameters not found in nature can beobtained by homogenization of clusters made of solid cylin-ders in a viscous host.

ACKNOWLEDGMENTS

This work was partially supported by the U.S. Ofce of Naval Research (Award No. N000140910554) and by theSpanish Ministry of Science and Innovation under Contracts

Nos. TEC2010-19751 and CSD2008-66 (CONSOLIDERprogram). D.T. acknowledges a research grant provided bythe program “Campus de Excelencia 2010 UPV.”

APPENDIX A: t -MATRIXFrom Eq. (5) , the components of the velocity in the vis-

cous uid can be expresses in terms of their associatedpotentials as

v r ¼@ u@ r þ

1r

@ w@ h

; (A1a)

v

h ¼1

r

@ u

@ h @ w

@ r ; (A1b)

and those corresponding to the elastic scatterers are

v r ¼ ix @ /@ r þ

1r

@ f@ h ;

v h ¼ix 1

r @ /@ h þ

@ f@ r : (A2)

These components satisfy the following boundaryconditions:

v

þr ð RaÞ ¼v

r ð RaÞ;v þh ð RaÞ ¼v h ð RaÞ;r þrr ð RaÞ ¼r rr ð RaÞ;r þr hð RaÞ ¼r r hð RaÞ; (A3)

where Ra is the radius of the elastic cylinder.By introducing the expansions of u ext , u scatt , wscatt , / int ,

and f int in the velocity and after applying the boundary con-ditions to the resulting expressions, we obtain from the rsttwo conditions in Eq. (A3) that

FIG. 9. (Color online) Similar to Fig. 7 but for an impinging wavelengthk ¼6a . Note the good agreement between both pressure maps.




k ‘ A‘q J 0qðk ‘ RaÞ þ B‘

q H 0qðk ‘ RaÞ þ iq Ra

Bt q H qðk t RaÞ

¼ ix k a‘ C‘q J 0qðk a‘ RaÞ þ

iq Ra

Ct q J qðk at RaÞ ;

iq Ra

A‘q J qðk ‘ RaÞ þ B‘

q H qðk ‘ RaÞ k t Bt q H 0qðk t RaÞ

¼ ix iq

RaC‘

q J qðk a‘ RaÞ k at Ct q J 0qðk at RaÞ

; (A4)

which can be cast in matrix form as

M qaq þNqbq ¼O qcq; (A5)

where

M q ¼ 1 Ra

ðk ‘ RaÞ J 0qðk ‘ RaÞ 0

iqJ qðk ‘ RaÞ 0 ; (A6)

Nq ¼ 1 Ra

ðk ‘ RaÞ H 0qðk ‘ RaÞ iqH qðk t RaÞiqH qðk ‘ RaÞ ðk t RaÞ H 0qðk t RaÞ !; (A7)

O q ¼ix

Ra

ðk a‘ RaÞ J 0qðk a‘ RaÞ iqJ qðk at RaÞiqJ qðk a‘ RaÞ ðk at RaÞ J 0qðk at RaÞ !; (A8)

and

aq ¼ A‘

q

00B@

1CA; bq ¼

B‘q

Bt q

0B@1CA

; cq ¼C‘

q

C t q

0B@1CA

: (A9)

Similarly, from the last two boundary conditions in Eq. (A3)we obtain

D k 2‘ A‘q J 00qðk ‘ RaÞ þ B‘

q H 00qðk ‘ RaÞ þiqk t Ra

Bt q H 0qðk t RaÞ

iq R2

a

Bt q H qðk t RaÞ

þ !

Ra

q2

Ra A‘

q J qðk ‘ RaÞ þ B‘q H qðk ‘ RaÞ iqk t Bt

q H 0qðk t RaÞ k ‘ A‘q J 0qðk ‘ RaÞ þ B‘

q H 0qðk ‘ RaÞ þ iq Ra

Bt q H qðk t RaÞ

¼D a ðk a‘ Þ2C‘

q J 00q ðk a‘ RaÞ þ iqk at

Ra J 0qðk at RaÞ

iq R2

a J qðk at RaÞ Ct

q þ

ka

Ra

q2

RaC‘

q J qðk a‘ RaÞ iqk at Ct q J 0qðk at RaÞ k a‘ C‘

q J 0qðk a‘ RaÞ þ iq Ra

C t q J qðk at RaÞ (A10)

and

g k 2t

Bt

q H 0

qðk t Ra

Þiqk ‘

Ra A‘

q J 0

qðk ‘ Ra

Þ þ B‘

q H 0

qðk ‘ Ra

Þ þ iq

R2a A‘

q J q

ðk ‘ Ra

Þ þ B‘

q Hq

ðk ‘ Ra

Þ 1 Ra

q2

Ra Bt

q H qðk t RaÞ þ iqk ‘ A‘q J 0qðk ‘ RaÞ þ B‘

q H 0qðk ‘ RaÞ k t Bt q H 0qðk t RaÞ

iq Ra

A‘q J qðk ‘ RaÞ þ B‘

q H qðk ‘ RaÞ ¼l a ðk at Þ

2Ct q J 00qðk at RaÞ

iqk a‘ Ra

J 0qðk a‘ RaÞ iq R2

a J qðk a‘ RaÞ C‘

q

1 Ra

q2

RaC t

q J qðk at r Þ þ iqk a‘ C‘q J 0qðk a‘ RaÞ k at C

t q J 0qðk at RaÞ

iq Ra

C‘q J qðk a‘ RaÞ ; (A11)

where

D ¼n þ43

g þiBx

; ! ¼D 2g; D a ¼ka þ2l a ;

(A12)

with g, n, ka , l a , and B, being the viscosity constants, theLam e’s coefcients, and the bulk modulus, respectively.

Equations (A10) and (A11) can also be cast in matrixform as

Pqaq þQ qbq ¼R qcq; (A13)

where

Pq ¼ 1 R2

a

D ðk ‘ RaÞ2 J 00qðk ‘ RaÞ þ! ½ðk ‘ RaÞ J 0qðk ‘ RaÞ q2 J qðk ‘ RaÞ 0

2igq½ðk ‘ RaÞ J 0qðk ‘ RaÞ J qðk ‘ RaÞ 0 !; (A14)




Q q ¼ 1 R2

a

Dðk ‘ RaÞ2 H 00qðk ‘ RaÞþ! ½ðk ‘ RaÞ H 0qðk ‘ RaÞ q2 H qðk ‘ RaÞ 2igq½ðk t RaÞ H 0qðk t RaÞ H qðk t RaÞ

2igq½ðk ‘ RaÞ H 0qðk ‘ RaÞ H qðk ‘ RaÞ g½ðk t RaÞ2 H 00qðk t RaÞ ðk t RaÞ H 0qðk t RaÞþq2 H qðk t RaÞ !;

(A15)

R q ¼ 1 R2

a

D aðk a‘ RaÞ2 J 00qðk a‘ RaÞþka½ðk a‘ RaÞ J 0qðk a‘ RaÞ q2 J qðk a‘ RaÞ 2il a q½k at Ra J 0qðk at RaÞ J qðk at RaÞ2il a q

½ðk a

‘ Ra

Þ J 0

qðk a

‘ Ra

Þ J q

ðk a

‘ Ra

Þ l a

½ðk a

t Ra

Þ2 J 00

q ðk a

t Ra

Þ ðk a

t Ra

Þ J 0

qðk a

t Ra

Þþq2 J q

ðk a

t Ra

Þ !:

(A16)

It is straightforward to obtain the t -matrix, which relates thecoefcients bq to aq , as

tq ¼ ½Q q R q O 1q Nq

1

½P q R q O 1q Mq ; (A17)

where ½1 means matrix inversion. Note that t q is a 2 2

matrix that has the following form:

t q ¼t ‘‘q 0

t t ‘q 00@ 1A: (A18)

Let us point out that the inverse matrix ½1 in Eq. (A17)

presents singularities associated to its functional dependenceon the Hankel functions, which can take values close to zerofor certain parameters and frequency regions, where thet matrix cannot be obtained.

APPENDIX B: LOWER ORDER ELEMENTS OF THEk -EXPANSION OF THE t -SCATTERING MATRIX

The previous appendix has shown that the elements of the t -matrix can be calculated by using the matrix expression(A17) , where tsq ¼tqq dsq . Now, by using the power seriesexpansions of Hankel and Bessel functions for small argu-ments, is tedious but straightforward to show that the lower order terms of this matrix are

limk ! 0

t ‘‘00 ¼ip R2

a

4 Bb Ba

Ba1 þ ick ð B2

bg B2a aÞ

B2b Ba B2

a Bb k 2

þ#ðk 4Þ; (B1)

limk ! 0

t ‘‘11 ¼

ip R2a

4

qb qa

qb ½ Bb ick a k 2

þ#

ðk 4

Þ; (B2)

limk ! 0

t ‘‘22 ¼ip R2

a

4 l a

Bb þ2ick a2 B2

b k 2 þ#ðk 4Þ; (B3)

where Ba ¼ka þ l a and a ¼43 g þ n.

In the expressions above we have only considered thelongitudinal projection of the tsq matrix (i.e., ‘ tsq ), whichwill be compared with the effective matrix of a cluster of cylinders.

The k -independent factors of lower terms in the power expansion are dened by

t ‘‘qq limk ! 0

t ‘‘qq

k 2 ; q ¼0; 1; 2: (B4)

If we consider clusters of low lling fractions of the inclu-sions (i.e., the volume occupied by the cylinders is small)the diagonal terms of the effective t matrix representingthe homogenized cluster accomplish that

t eff qq

N ‘ t qq ; q

¼0; 1; 2; (B5)

where N is the number of cylinder and ‘ t qq are the lower terms in the power expansion mentioned above. This resultis independent of the external shape of the cluster. It is wellknown that for a cylinder of arbitrary shape, the isotropicelement t cyl

00 is given by

t cyl00 ¼

iAcyl

4 Bb

Bcyl1 : (B6)

This expression is employed here to describe the isotropicterm of an arbitrarily shaped cylinder representing the ho-mogenized cluster with area Aeff , and bulk modulus Beff . Inother words, it is possible to write Eq. (B5) as

iAeff

4 Bb

Beff 1 ¼ N

ip R2a

4 Bb

Ba1 ; (B7)

by introducing the lling fraction f as

f N ðp R2

aÞ Aeff

; (B8)

and after some simplications, the effective bulk modulus of the homogenized cluster is nally obtained as

1 Beff ¼ f

Ba þ 1 f Bb

: (B9)

In what follows we determine qeff by using the next diagonalterm in the power expansion, which is easily obtained fromEqs. (34) and (B2) :

t eff 11 ¼

ip R2eff

4qb q eff

qb N ‘ t 11 : (B10)

Now, inserting the denition of f [see Eq. (B8) ] allows oneto merge the last equation in




qb q eff

qb ¼ f qb q a

qb : (B11)

Solving for qeff we get

qeff ¼ f qa þ ð1 f Þqb: (B12)

Finally, for the third diagonal term, the low lling fractionlimit implies

t eff 22 N ‘ t 22 ; (B13)

and from Eq. (B3) it is clear that

l eff ¼ f l a : (B14)

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