On the sonic composites without/with defectsVeturia Chiroiu, Cornel Brian, Mihaela Popescu, Iulian Girip, and Ligia Munteanu Citation: Journal of Applied Physics 114, 164909 (2013); doi: 10.1063/1.4828475 View online: http://dx.doi.org/10.1063/1.4828475 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/114/16?ver=pdfcov Published by the AIP Publishing Advertisement:

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On the sonic composites without/with defects

Veturia Chiroiu,1,a) Cornel Brisan,2,b) Mihaela Popescu,1 Iulian Girip,1

and Ligia Munteanu1,a)

1Institute of Solid Mechanics, Romanian Academy, Bucharest, Romania2Department of Mechatronics and System Dynamics, Technical University of Cluj-Napoca,Cluj-Napoca, Romania

(Received 2 September 2013; accepted 7 October 2013; published online 30 October 2013)

The sound attenuation in a sonic composite without/with point defects is studied using a new method

that combines the features of the cnoidal method and the genetic algorithm. Acoustic scatterers are

composed by piezoceramic hollow spheres of functionally graded materials—the Reddy and cosine

graded hollow spheres. This method enables to obtain the dispersion relation for defect modes, and the

prediction of the evanescent nature of the modes inside the band-gaps. VC 2013 AIP Publishing LLC.

[http://dx.doi.org/10.1063/1.4828475]

I. INTRODUCTION

A sonic composite is a finite size periodic array com-

posed of scatterers embedded in a homogeneous material.1–3

A great number of applications based on the sonic compo-

sites are explained by the existence of the band-gaps into the

acoustic filters,4 acoustic barriers,5 or wave guides.6,7 The

generation of large band-gaps is due to the superposition of

multiple reflected waves within the array according to the

Bragg’s theory, and, consequently, it is connected with a

large acoustic impedance ratio between the scatterers’ mate-

rial and the matrix’ material, respectively.

The band-gaps correspond to the Bragg reflections that

occur at different frequencies inversely proportional to the central

distance between two scatterers. The waves are reflected com-

pletely from this periodic array in the frequency range where all

partial band-gaps for the different periodical directions overlap.

This makes sharp bends of the wave-guide in the sonic compos-

ite. The evanescent waves characterized by complex wave num-

bers8 are distributed across the boundary of the waveguide into

the surrounding composite by several times the lattice constant.

Recent experimental and theoretical results9–11 show that the

presence of defects in sonic composite is related to the generation

of localized modes in the vicinity of the point defect with a sig-

nificant evanescent behavior of the waves outside the defect

point. This means that the evanescent modes are related to the ex-

istence of the band-gaps where no real wave number exists. The

authors have revealed that the level of the sound is higher inside

the vicinity of the defect point than into the composite. A review

focusing on the fundamental physical aspects of the sonic materi-

als and the evanescent modes was reported by Miyashita.12

Recent works show the calculation of complex band struc-

tures for phononic crystals13,14 (phononic crystals which exhibit

elastic band gaps are the acoustic analogue of the photonic crys-

tals which exhibit electromagnetic band gaps), using the explicit

matrix formulation and the approximation of the supercells.

This technique enables to be extended to the 2D complete sonic

composites, as well as in sonic composites with point defects.9

An important issue is related to the interfaces between

defect and matrix which may occur as deformations or loadings

continue to increase, and may affect the load-carrying capacity

and overall stress-strain behavior of the composites.15

The primary goal of this paper is to propose an alternative

method for obtaining the band structures of the 3D sonic com-

posites without/with point defects. The point defects are

vacancies or foreign interstitial atoms which are supported by

the interfaces between the hollow spheres and the matrix. The

proposed method is used to simulate a sonic plate composed

of an array of acoustic scatterers which are piezoceramic hol-

low spheres embedded in an epoxy matrix. The scatterers are

made from functionally graded materials with radial polariza-

tion, which support the Reddy and cosine laws.16–18 Readers

are also referred to Ref. 19 for vibrations of solid spheres of

functionally graded materials, and to Refs. 20 and 21 for the

wave propagation in functionally graded materials.

The proposed approach is based on the theory of piezoelec-

trics coupled with the cnoidal method and a genetic algorithm.

Such sonic plates have not been investigated in the literature as

yet. For a single sphere made from a functionally graded mate-

rial, the free vibration problem was analyzed in Refs. 22–26.

The paper is organized as follows: In Sec. II, the theory

and the formulation of the problem are presented. The cnoi-

dal method and the genetic algorithm for solving the nonlin-

ear equations are described in Sec. III. Results are reported

in Sec. IV. Finally, Sec. V contains the conclusions.

II. THE THEORY

The sonic composite is consisting of an array of acoustic

scatterers embedded in an epoxy matrix. The acoustic scat-

terers are hollow spheres made from a nonlinear isotropic

piezoelectric ceramic, while the matrix is made from a non-

linear isotropic epoxy resin (Fig. 1). The sonic plate consists

of 72 local resonators of diameter a. A rectangular coordi-

nate system Ox1x2x3 is employed. The origin of the coordi-

nate system Ox1x2x3 is located at the left end, in the middle

plane of the sample, with the axis Ox1 in-plane and normal

a)Electronic addresses: ligia_munteanu@hotmail.com and veturiachiroiu@

yahoo.comb)E-mail: Cornel.Brisan@mmfm.utcluj.ro

0021-8979/2013/114(16)/164909/9/$30.00 VC 2013 AIP Publishing LLC114, 164909-1

JOURNAL OF APPLIED PHYSICS 114, 164909 (2013)

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128.143.23.241 On: Thu, 31 Oct 2013 08:23:15

to the layers and the axis Ox3 out-plane and normal to the

plate. The length of the plate is l, its width is d, while the di-

ameter of the hollow sphere is a and its thickness is e > a.

In order to avoid unphysical reflections from the boun-

daries of the specimen, we have implemented the absorbing

boundary conditions in the x1-direction, at x1 ¼ 0 and x1 ¼ l.A transducer and a receiver are located at x1 ¼ b and

x1 ¼ l� b, respectively. The role of the transducer is to

inject into the plate the plane monochromatic waves propa-

gating in the x1-direction.

The displacements are registered at both sides of the plate.

The sound attenuation coefficient is obtained from the ratio of

the displacements at the receiver and at the input transducer.

The center of anisotropy is the same with the origin of the

spherical coordinate system ðr; h;uÞ. An index followed by a

comma represents partial differentiation with respect to space

variables, while a superposed dot indicates differentiation with

respect to time. Throughout the paper, repeated indices denote

summation over the range (1, 2, 3). The constitutive equations

for the piezoelectric hollow sphere are given by22–25

Rhh ¼ rrhh ¼ C11Shh þ C12Suu þ C13Srr þ f31r/r;

Ruu ¼ rruu ¼ C12Shh þ C11Suu þ C13Srr þ f31r/;r;

Rrr ¼ rrrr ¼ C13Shh þ C13Suu þ C33Srr þ f33r/;r;

Rrh ¼ rrrh ¼ 2C44Srh þ f15/;h;

Rru ¼ rrru ¼ 2C44Sru þ f15 csc h/;u;

Rhu ¼ rrhu ¼ 2C66Shu;

Kh ¼ rDh ¼ 2C15Srh � f11/;h;

Ku ¼ rDu ¼ 2f15Sru � f11 csc h/;u;

Kr ¼ rDr ¼ f31Shh þ f31Suu þ f33Srr � f33r/;r;

(2.1)

where rij is the stress tensor, / is the electric potential, Di is the

electric displacement vector, Cij are the elastic constants,

C66¼ðC11�C12Þ=2, fij are the piezoelectric constants fij, fij are

the dielectric constants, and i¼ r;h;u. The elastic, piezoelectric,

and dielectric constants are arbitrary functions of the radial coor-

dinate r. On denoting the components of the strain tensor and

displacement vector by eij and ui, i¼ r;h;u, respectively, the

quantities Sij related to the strain tensor eij are defined as

Srr ¼ rerr ¼ rur;r;

Shh ¼ rehh ¼ uh;h þ ur;

Suu ¼ reuu ¼ csc huu;u þ ur þ uh cot h;

2Srh ¼ 2rerh ¼ ur;h þ ruh;r � uh;

2Sru ¼ 2reru ¼ csc hur;u þ ruu;r � uu;

2Shu ¼ 2rehu ¼ csc huh;u þ uu;h � uu cot h:

(2.2)

Denoting the density of the material byq, which is assumed

to be an arbitrary function of r, the equations of motion become

rRrh;r þ csc hRuh;u þ Rhh;h þ 2Rrh

þ ðRhh � RuuÞcot h ¼ qr2€uh;

rRru;r þ csc hRuu;u þ Rhu;h þ 2Rru

þ 2Rhu cot h ¼ qr2€uu;

rRrr;r þ csc hRru;u þ Rrh;h þ Rrr � Rhh � Ruu

þ Rrh cot h ¼ qr2 €ur: (2.3)

The charge equation of electrostatics is given by

rKr;r þ Kr þ csc hðKh sin hÞ;h þ csc hKu;u ¼ 0: (2.4)

The Chen functions F, G, and w, and stress functions R1

and R2 defined as

uh ¼ �csc hF;u � G;h uu ¼ F;h � csc hG;u ur ¼ w;

Rrh ¼ �csc hR1;u � R2;h; Rru ¼ R1;h � csc hR2;u;

are used in order to simplify Eqs. (2.1)–(2.4). Therefore,

these equations can be separated into two independent sets

of equations

rA;r ¼ MA; rB;r ¼ PB; (2.5)

where

B ¼ ½Rrr; R2;G;w;Kr;/�T ;

M ¼ �2 �C66ðr2 þ 2Þ þ r2q@2

@t2C�1

44 1

24

35;

with r2 ¼ @2

@h2 þ cot h @@hþ csc 2h @2

@u2. It should be noted that

the first Eq. (2.5) is related to two state variables, namely,

A ¼ ½R1; F�T , while the second Eq. (2.5) is related to six

state variables Rrr; R2;G;w;Kr;/ (Ref. 2).

The nonzero components of the matrix P are given by

P11 ¼ 2b� 1; P12 ¼ r2; P13 ¼ k1r2;

P14 ¼ �2k1 þ r2q@2

@t2;

FIG. 1. Sketch of the sonic composite.

164909-2 Chiroiu et al. J. Appl. Phys. 114, 164909 (2013)

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P15 ¼ 2P25 ¼ �P64 ¼ 2c; P21 ¼ b; P22 ¼ �2;

P23 ¼ k2r2 � 2C66 þ r2q@2

@t2;

P24 ¼ �k1; P32 ¼ C�144 ; P33 ¼ P34 ¼ �P55 ¼ 1;

P36 ¼ C�144 f15; P41 ¼ a�1f33; P43 ¼ br2;

P44 ¼ �2b; P45 ¼ a�1 f33; P52 ¼ C�244 f15r2;

P56 ¼ k3r2; P61 ¼ a�1 f33; P63 ¼ cr2 ;

P65 ¼ �a�1C33;

where

a ¼ C33f33 þ f 233; b ¼ a�1ðC13f33 þ f31 f33Þ;

c ¼ a�1ðC13f33 � C33 f31Þ;

k1 ¼ 2ðC13bþ f31cÞ � ðC11 þ C12Þ; k2 ¼ 0:5k1 � C66;

k3 ¼ f11 þ f 215C�1

44 :

Consider now two piezoceramic hollow spheres with the

ratio of the inner and outer radii n0. Two laws represent the

functionally graded property of the material. The first one is

the Reddy law16–18 given by

M ¼ Mplk þMzð1� lkÞ; (2.6)

where l is the gradient index,23 Mp and Mz are material con-

stants of two materials, namely, PZT-4 and ZnO.27,28 The

case l ¼ 0 corresponds to a homogeneous PZT-4 hollow

sphere and l!1, to a homogeneous ZnO hollow sphere.

The second law is expressed as

M ¼ Mp coslþMzð1� cos lÞ: (2.7)

The constitutive equations for epoxy-resin material are

tij ¼ keekkdij þ 2leeij þ Aeeilejl þ 3Beekkeij þ Cee2kkdij; (2.8)

where tij is the stress tensor, eij is the strain tensor, ke and le

are the Lam�e elastic constants, and Ae; Be and Ce are the

second-order elastic constants. The motion equations are

written as

qe €ui ¼ tij:j; (2.9)

where qe is density of the epoxy material and u is the dis-

placement vector.

At the interfaces between the hollow spheres and the

matrix, sharp periodic boundary conditions for the displace-

ment and traction vectors are added29 for sonic composites

without/with defects. In addition, in the case of the point

defects situated in the interfaces between the hollow spheres

and the matrix, a new equation must be added to reflect the

dynamic of the concentration of the defects.

The distribution of the defects of the concentration cðrÞ is

characterized by the diffusive contribution �Drc, where D is

the diffusion coefficient, and the forced contribution gcFwhere g is the mobility and F the driving force which acts

upon the defects. The mobility and the diffusivity are related

through the Nernst-Einstein relation D ¼ gkT where k is the

Boltzmann’s constant and T the absolute temperature.

The rate of the change of the concentration in the pres-

ence of the source Sðr; tÞ is30

@c

@t¼ �rð�DrcÞ þ Sðr; tÞ; (2.10)

or

@c

@t¼ D r2c� 1

kTðFrcþ crFÞ

� �þ Sðr; tÞ: (2.11)

The source Sðr; tÞ represents the rate at which the defects

are created at any point. The elastic force acting on a defect

depends on the symmetry of the defect itself. This is because

the force results from the interaction of the applied strain field

with that produced by the defect itself. Because of the very

small size of the defect it can be considered that the surround-

ing medium is infinite. Also, we suppose that the surrounding

medium is isotope. Eshelby31 defines the force acting on a

point defect by treating the defect as a Somigliana dislocation.

If the applied stress is r and the displacement field of the

defect in any closed surface which encloses the point defect Ris u, the force is given by the surface integral30

FR ¼ðR

ðurr� rruÞds: (2.12)

The point defect is supposed to be a sphere of radius r0,

of volume V which is related to the volume of the surrounding

medium V0 through the relation V=V0 ¼ 1þ a. The intersti-

tial atom is a centre of dilatation for which a > 0, and the

vacancy is a centre of contraction with a < 0, respectively.

This surface has the symmetry of the displacement field. So,

Eq. (2.12) is written with respect to the normal stress

FR ¼4

3par3

0rðrrr þ rhh þ ruuÞ: (2.13)

To take into account the lowest possible interaction

between defects, the displacement field around a point defect

is considered as the derivative of the Lam�e strain potential

uðrÞ ¼ �DVðrÞ; VðrÞ ¼ A

r; (2.14)

with A an arbitrary constant. The strain is the symmetric

derivatives of the displacement

eii ¼A

r31� 3

r2i

r2

� �; eij ¼ �

3

2

A

r3

rirj

r2; (2.15)

where ri ¼ ðr; h;uÞ, i ¼ 1; 2; 3. From Eq. (2.15) it follows

err ¼ � 2Ar3 and ehh ¼ euu ¼ A

r3, in consequence TrðeÞ ¼ 0.

Therefore, the strain field due to the defect is a pure shear,

which causes no dilatation of the surrounding medium. The

equation, which reflects the dynamics of the defects, is writ-

ten as

164909-3 Chiroiu et al. J. Appl. Phys. 114, 164909 (2013)

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@c

@t¼ D r2c� 1

kTðFRrcþ crFRÞ

� �þ Sðr; tÞ; (2.16)

with FR given by Eq. (2.13).

In the following, we add that two simple porous absorb-

ing layers with the flow resistivity re are considered at

x1 ¼ 0 and x1 ¼ l. The flow resistivity of the absorbing

layers has a significant role in the modeling and stability of

the computational scheme. The appropriate selection of the

absorbing coefficient is necessary not only to achieve no

reflections but also to have a stable algorithm. It is easy to

observe that reflection is reduced when resistivity is high,

but the thickness of the absorbing layer can be small since

the wave is quickly damped inside a high resistivity layer.

When the resistivity is low, the reflection can also be

reduced, but the thickness of the absorbing layer has to be

large in order to damp the wave inside the absorbing layer.

Otherwise, the remaining wave can reflect at the end of the

absorbing layer and still propagates back to the medium do-

main. The selection of the values re is made in order to

accommodate the requirement of reducing unphysical reflec-

tions from the boundaries of the plate with a reasonable

absorbing-layer thickness.

III. CNOIDAL METHOD AND THE GENETICALGORITHM

In the 1D case, the set of Eqs. (2.5)–(2.9) and (2.16) can be

reduced to equations similar to the Weierstrass equation32–34

_h2 ¼ A1h þ A2h

2 þ A3h3 þ A4h

4 þ A5h5; (3.1)

with h a generic function of time, and Ai > 0 constants. The

dot means differentiation with respect to the variable

t! x� ct, c a constant. It is well-known that Eq. (3.1)

admits a particular solution expressed as an elliptic

Weierstrass function that is reduced, in this case, to the cnoi-

dal function cn35

h � f ðtÞ ¼ e2 � ðe2 � e3Þcn2� ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðe1 � e3Þ

pt; m�; (3.2)

where e1; e2; e3 are the real roots of the equation 4y3 � g1y�g2 ¼ 0 with e1 > e2 > e3, and g1; g2 2 R are expressed in

terms of the constants Ai, i¼ 1, 2…5, and satisfy the condi-

tion g31 � 27g2

2 > 0. The modulus m of the Jacobian elliptic

function is m ¼ e2�e3

e1�e3. For arbitrary initial conditions

hð0Þ ¼ h0; _hð0Þ ¼ hp0; (3.3)

the solution of Eq. (3.1) can be written as a linear superposi-

tion of cnoidal vibrations of the form (Eq. (3.2))

hlin ¼ 2Xn

k¼0

akcn2ðxkt; mkÞ; (3.4)

where the moduli 0 � mk � 1, frequencies xk and ampli-

tudes ak depend on h0, hp0, and Ak.

For equations that can be reduced to the Weierstrass

equation with polynomials of higher-order

_h2 ¼

Xp

j¼0

AjðtÞhj; (3.5)

the solutions can be according to following theorem:

Theorem: The general solution of Eqs. (2.5)–(2.9) and

(2.16) with boundary and initial conditions can be written in

the form

h ¼ hlin þ hnonlin; (3.6)

where the linear and nonlinear terms express the linear and

nonlinear superposition of cnoidal vibrations, respectively,

hlin ¼ 2Xn

k¼0

akcn2ðxkt; mkÞ; hnonlin ¼

Xn

k¼0

bkcn2ðxkt; mkÞ

1þXn

k¼0

ckcn2ðxkt;mkÞ:

(3.7)

In Eq. (3.7), the moduli 0 � mk � 1, bk, ck and xk

depend on the initial conditions and the coefficients which

appear in AjðtÞ.Proof. The theta-function representation of solutions is

used for deriving Eq. (3.7).36,37 The derivation is a compli-

cated affair drawing on many beautiful results in the theory

of Hill’s equation and Riemann’s solution of the Jacob’s

inversion problem.

Let us introduce the following dependent-variable

transformation:

h ¼ 2d2

dt2log HnðtÞ: (3.8)

Introducing Eq. (3.8) into Eq. (3.5), a bilinear differen-

tial equation is obtained

D2t ð1þ D2

t ÞHn �Hn ¼ 0; (3.9)

where the operator Dt is defined as Dmt a � b ¼ ð@t � @t0 Þm

aðtÞbðt0Þjt¼t0 . The function Hn is given in terms of an asymp-

totic expansion of the type (near-identity)

Hn ¼ 1þ eHð1Þn þ e2Hð2Þn þ :::; (3.10)

with Hð1Þn ¼Pni¼1

expðixitÞ. For n ¼ 1 we have

H1 ¼ 1þ expðix1tþ B11Þ;

H2 ¼ 1þ expðix1tþ B11Þ þ expðix2tþ B22Þþ expðx1 þ x2 þ B12Þ;

H3 ¼ 1þ expðix1tþ B11Þ þ expðix2tþ B22Þ

þ expðix3tþ B33Þ þ expðx1 þ x2 þ B12Þ

þ expðx1 þ x3 þ B13Þ þ expðx2 þ x3 þ B23Þ

þ expðx1 þ x2 þ x3 þ B12 þ B13 þ B23Þ:

164909-4 Chiroiu et al. J. Appl. Phys. 114, 164909 (2013)

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For an arbitrary n, the following expression is obtained:

Hn ¼X

M¼0;1

exp

�iXn

i¼1

Mixitþ1

2

Xn

i<j

BijMiMj

�; (3.11)

where exp Bij ¼ xi�xj

xiþxj

� �2, exp Bii ¼ x2

i . In Eq. (3.11) M ¼½M1;M2� is the vector of integer indices (0 and 1), x ¼½x1;x2; :::;xn� the frequency vector, and n is the finite num-

ber of degrees of freedom for a particular solution. The

matrix B is written as a sum of a diagonal matrix and an off-

diagonal matrix B ¼ Dþ O. The solution (Eq. (3.8)) can be

written under the form

h ¼ 2d2

dt2log HnðtÞ ¼ 2

d2

dt2log GðtÞ þ 2

d2

dt2log 1þ FðtÞ

GðtÞ

� �;

GðtÞ ¼X

M

exp iMxtþ 1

2MTDM

� �;

FðtÞ ¼X

M

ðexp MTOM � 1Þexp iMxtþ 1

2MTDM

� �:

Consider now the linear part of the solution only. The

function G(t) can be written in the product form

GðtÞ ¼Qn

k¼1 GkðtÞ, where Gk is the classical theta function

given by the series38

GkðtÞ ¼X1

Mk¼�1exp iMkxkntþ 1

2M2

k Dkk

� �:

Consequently, we obtain for the linear part of the

solution

hlin ¼Xn

k¼0

hklin; hklin ¼ 2d2

dt2log GkðtÞ; (3.12)

hlin ¼Xn

l¼1

al2p

Klffiffiffiffiffimlp

X1k¼0

qkþ1=2l

1þ q2kþ1l

cosð2k þ 1Þpxlt

2Kl

!224

35;

where we recognize the linear expression (3.7) (Ref. 39)

q ¼ exp ð�pK0=KÞ; K0ðm1Þ ¼ KðmÞ; mþ m1 ¼ 1;

K ¼ KðmÞ þðp=2

0

duffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� m sin2up :

The nonlinear term of the solution can be written in the

form

2d2

dt2log 1þ FðtÞ

GðtÞ

� ��

Xn

k¼0

bkcn2ðxkt; mkÞ

1þXn

k¼0

ckcn2ðxkt; mkÞ; (3.13)

with 0 � mk � 1. For mk ¼ 0 and mk ¼ 1, relation (3.13) is

satisfied. A numerical verification of Eq. (3.13) was

performed for different values of mk with a maximum error

of 5� 10�7.

To couple the cnoidal method with a genetic algorithm, we

express the state variables of the problem in the form (Eq. (3.6))

R1 ¼ bC044

Xn

k¼1

cn2ðnk1; mkÞ þ

Xn

k¼0

bk1cn2ðnk1; mkÞ

1þXn

k¼0

ck1cn2ðnk1; mkÞ;

nk1 ¼ R11knþ R12khþ R13ku� xt;

F ¼ bXn

k¼1

cn2ðnk2; mkÞ þ

Xn

k¼0

bk2cn2ðnk2; mkÞ

1þXn

k¼0

ck2cn2ðnk2; mkÞ;

nk2 ¼ F1knþ F2khþ F3ku� xt;

Rrr ¼ bC044

Xn

k¼1

cn2ðnk3; mkÞ þ

Xn

k¼0

bk3cn2ðnk3; mkÞ

1þXn

k¼0

ck3cn2ðnk3; mkÞ;

nk3 ¼ Rr1knþ Rr2khþ Rr3ku� xt;

R2 ¼ bC044

Xn

k¼1

cn2ðnk4; mkÞ þ

Xn

k¼0

bk3cn2ðnk4; mkÞ

1þXn

k¼0

ck3cn2ðnk4; mkÞ;

nk4 ¼ R21knþ R22khþ R23ku� xt;

G ¼ bXn

k¼1

cn2ðnk5; mkÞ þ

Xn

k¼0

bk3cn2ðnk5; mkÞ

1þXn

k¼0

ck3cn2ðnk5; mkÞ;

nk5 ¼ G1knþ G2khþ G3ku� xt;

w ¼ bXn

k¼1

cn2ðnk6; mkÞ þ

Xn

k¼0

bk6cn2ðnk6; mkÞ

1þXn

k¼0

ck6cn2ðnk6; mkÞ;

nk6 ¼ w1knþ w2khþ w3ku� xt;

Kr ¼ bf 033

Xn

k¼1

cn2ðnk7; mkÞ þ

Xn

k¼0

bk7cn2ðnk7; mkÞ

1þXn

k¼0

ck7cn2ðnk7; mkÞ;

nk7 ¼ K1knþ K2khþ K3ku� xt;

/ ¼ bf 033

f033

Xn

k¼1

cn2ðnk8; mkÞ þ

Xn

k¼0

bk8cn2ðnk8; mkÞ

1þXn

k¼0

ck8cn2ðnk8; mkÞ;

nk8 ¼ /1knþ /2khþ /3ku� xt; (3.14)

164909-5 Chiroiu et al. J. Appl. Phys. 114, 164909 (2013)

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where n ¼ 2r=a, x is the circular frequency, C044 ¼ C44j2r¼a,

f 033 ¼ f33j2r¼a, f0

33 ¼ f33j2r¼a. The unknowns Vm ¼ fR11k;R12k;R13k; :::;/1k;/2k;/3kg, m ¼ 24n, are determined from

a genetic algorithm. The goal of the genetic algorithm is to

determine the set Vm, m ¼ 24n from the minimization of the

residuals

rAj;r �MijAj ¼ Ri; i; j ¼ 1; 2; rBl;r � PklBl ¼ Rk;

k; l ¼ 1; 2; :::; 6: (3.15)

These residuals evaluate the verification of the motion

equations. The fitness function is expressed as

FðVmÞ ¼X8

j¼1

R2j þ d2; (3.16)

where d is a measure of fitting the boundary and initial

conditions.

We use a binary vector with m genes representing the real

values of Vm, m ¼ 24n. The length of the vector depends on

the required precision, which in this case is of order Oð10�6Þ.The domain of the parameters Vm 2 ½�aj; aj� with length 2aj

is divided into at least 15 000 segments of the same length.

That means that each parameter Vm,m ¼ 24n, is represented

by a string of 22 bits (221 < 3000000 � 222). One individual

consists of the row of 24n genes, that is, a binary vector with

22 � 24 n components.

The genetic algorithm is linking to the problem is solved

through the fitness function (Eq. (3.16)), which measures

how well an individual satisfies the requirements. From one

generation to the next one, the genetic algorithm usually

decreases the fitness function of the best model and the aver-

age fitness of the population. The starting population (with Kindividuals) is generating randomly. Then, new descendant

populations are iteratively created, with the goal of an over-

all fitness function decrease from one generation to the next

one.

IV. RESULTS

Consider a plate with the length l ¼ 18 cm and

widthd ¼ 11 cm, while the diameter of the hollow sphere

and its thickness are a ¼ 10:5 mm and e ¼ 12 mm, respec-

tively, and n0 ¼ 0:3. The numerical results are carried out

for the following constants:2

for PZT-4 C11¼13:9�1010N=m2, C12¼7:8�1010N=m2,

C13¼7:4�1010 N=m2, C33¼11:5�1010 N=m2, C44¼2:56

�1010 N=m2, f15¼12:7C=m2, f31¼�5:2C=m2, f33

¼15:1C=m2, f11¼650�10�11F=m, f33¼560�10�11F=m,

q¼7500kg=m3, for ZnO C11¼20:97�1010 N=m2, C12

¼12:11�1010 N=m2, C13¼10:51�1010 N=m2, C33¼21:09

�1010 N=m2, C44¼4:25�1010 N=m2, f15¼�0:59C=m2,

f31¼�0:61C=m2, f33¼1:14C=m2, f11¼7:38�10�11 F=m,

f33¼7:83�10�11 F=m, q¼5676kg=m3, and for epoxy-resin

ke¼42.31�109 N=m2, le¼3.76�109 N=m2, Ae¼2:8�109 N=m2, Be¼ 9.7�109 N=m2, Ce¼ �5.7�109 N=m2, and

qe¼1170 kg=m3.

The independent sets of Eq. (2.5) yield two independent

classes of free vibrations. The first class does not involve the

piezoelectric or dielectric parameters, being identical to the

one for the corresponding spherically isotropic elastic

sphere. The second class depends on the piezoelectric or

dielectric parameters. With the increase of the gradient index

l, the natural frequencies increase for all modes and func-

tionally graded laws, the variation being more significant

when l � 10. For l!1 the variation of natural frequen-

cies is not significant with respect to those of l ¼ 10. It is

seen that for a piezoceramic hollow sphere, the piezoelectric

effect consists of increasing the values for the natural fre-

quencies in both classes of vibrations. If n ¼ 2r=a increases,

the natural frequencies increase for the first class of vibra-

tions and decrease for the second class.

The propagation of sound is characterized by the superpo-

sition of multiply reflected waves. Featuring of the length scale

a, the structure of the full band-gap can be better understood

by representing the linear band structure (dispersion curve).

The simulation is carried out for a variation of the

defects concentration represented in Fig. 2 with respect to

the radius of R which encloses the point defect (vacancies or

foreign interstitial atoms), for 300 K. In this figure only the

effect of diffusion is shown, without any stress gradient, after

1000 s.

Fig. 3 plots the dispersion curve including the first par-

tial band-gaps for the composite without defects and for the

composite with defects (c ¼1.5 �1021 cm�3). The reduced

units for the frequency are xa=2pc0, with c0 the speed of

sound in air. We see that the point defects confine acoustic

waves in localized modes and in consequence the band-gaps

are larger than in the case of the complete composite.

The guided waves are accompanied by evanescent

waves which extend to the periodic array of the scatterers

surrounding the wave-guide. Using the Joannopoulus repre-

sentation8 for the bad-gap structure, Fig. 4 presents the band

structure with the evanescent modes with exponential decay

for the sonic composite without defects. The modes present

FIG. 2. Variation of the concentration with respect to the radius of R which

encloses the point defect.

164909-6 Chiroiu et al. J. Appl. Phys. 114, 164909 (2013)

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purely imaginary wave vectors. The central grey region is

the full band-gap ranged between 8.02 kHz and 8.72 kHz,

given by the real part of the wave vector constrained in the

first Brillouin zone for each frequency. The left region repre-

sents the imaginary part of the wave vector for longitudinal

direction frequency (tension/compression), while the right

region is the imaginary part of the wave vector for transverse

direction frequency (shear). The red lines represent the imag-

inary part of the wave vector of the evanescent modes inside

the bad-gap.

If we want to have a full band-gap, we must have struc-

tures with band-gaps for both longitudinal and transverse

waves in the same frequency region.

The difference in the sound velocities between trans-

verse and longitudinal modes causes partial gaps at different

frequencies. If the mechanical contrast is small, these partial

gaps are narrow and do not overlap. As mechanical contrast

increases, the partial gaps widen and begin to overlap in the

same frequency region leading to the appearance of a full

band-gap independent of the polarization.

It is strongly expected that mode coupling waves arise

between adjacent wave-guides. The output of the coupled

modes is compared with the input waves, as shown in Fig. 5

in the case of Reddy law and re ¼ 2:2 kPasm�2. Fig. 6

presents the band structure for the sonic composite with

defects (c ¼1.5 �1021 cm�3) in the case of the Reddy law.

We observe that this time the portion widens from 7.95 kHz

to 8.76 kHz. The coupled and the input waves are shown in

Fig. 7 for re ¼ 2:2 kPasm�2. The difference from the com-

posite without defects consists only in the size and structure

of the full band-gaps.

Fig. 8 presents the band structure for the sonic composite

with defects (c ¼1.5 �1021 cm�3) in the case of the cosine

law. We observe that the full band-gap has the same length but

has undergone a translation, i.e., to 7.67 kHz and 8.48 kHz.

The length of the full band-gap as function of the con-

centration of the point defects is represented in Fig. 9, for

FIG. 3. Linear dispersion for sonic composite without/with defects.

FIG. 4. Band structure for the sonic composite without defects in the case of

Reddy law.

FIG. 5. The input and coupled waves for sonic composite without defects in

the case of Reddy law.

FIG. 6. Band structure for the sonic composite with defects

(c ¼1.5�1021 cm�3) in the case of Reddy law.

164909-7 Chiroiu et al. J. Appl. Phys. 114, 164909 (2013)

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both Reddy law and cosine law, respectively. For Reddy

law, the length increases with concentration and shows a flat

portion starting to the concentration of 1.5 with a slight

decrease of concentration at around 0.88 kHz. For the cosine

law, the length shows two flat zones and a visible drop of

concentration around 0.64 kHz.

V. CONCLUSIONS

Analytical and numerical solutions and verification to

real results are presented in this paper for propagation of

waves in sonic composites without/with defects. The point

defects are vacancies or foreign interstitial atoms which are

situated in the interfaces between the hollow spheres and the

matrix. It is shown that the point defects confine acoustic

waves in localized modes, and the defect modes are created

within the band-gaps of the composite. The localization of

the sound in the defect regions leads to increase in the inten-

sity of the acoustic wave’s interactions. Such localizations

increase the length of the full band-gap frequency for sonic

composites with defects by 15%–20% compared with the

values for similar composites without defects.

The scattering problem inside the sonic composite is

solved by a method which combines the cnoidal method with

a genetic algorithm. The reason for choosing the cnoidal

method lies in the facts that the governing equations are

reduced to Weierstrass equations with polynomials of higher-

order, for a change of variable t! x� ct, with c a constant.

The solutions are expressed as a sum of the linear and the non-

linear superposition of cnoidal vibrations, respectively.

ACKNOWLEDGMENTS

The authors gratefully acknowledge the financial sup-

port of the National Authority for Scientific Research

ANCS/UEFISCDI through the project PN-II-ID-PCE-2012-

4-0023, Contract No. 3/2013.

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