Multiple scattering of antiplane shear waves in a fiber-reinforced composite medium with interfacial layers

Int. J. SolidsStructuresVol. 35, No. 7 8, pp. 733-745, 1998 © 1997ElsevierScienceLtd All rightsreserved.Printedin Great Britain 0020-7683/98 $19.00 + .00

Pergamon

PII: S0020-7683(97)00074-7

MULTIPLE SCATTERING OF ANTIPLANE SHEAR WAVES IN A FIBER-REINFORCED COMPOSITE MEDIUM WITH INTERFACIAL LAYERS YASUHIDE SHINDO, NAOKI NIWA and RYUICHI TOGAWA Department of Materials Processing, Faculty of Engineering, Tohoku University, Aramaki Aza-Aoba, Aoba-ku, Sendai 980, Japan (Received 11 September 1996; in revised f o r m 18 February 1997)

Abstract--This study considers the multiple scattering of antiplane shear waves in a metal matrix composite reinforced by fibers with interfacial layers. We assume same-size cylindrical inclusion and same-thickness interface layers with nonhomogeneous elastic properties. First, the problem of the scattering of plane axial shear waves by a large number N of fibers, arbitrarily distributed in an infinite matrix, is considered. The resulting equations are then averaged, considering the positions of the fibers to be random. The averaged equations are solved by using Lax's quasicrystalline approximation. Numerical calculations for an SiC-fiber-reinforced Al composite are carried out and the effect of interface properties on the phase velocity and attenuation of coherent plane wave, and the effective shear modulus is shown graphically. © 1997 Elsevier Science Ltd

1. I N T R O D U C T I O N

Much current practical interest exists concerning wave propagation through a composite medium with a random distribution of inclusions with interface layers (Shindo et al., 1995). The theoretical investigation of the dynamic properties of such composites is a prerequisite to the design of a composite with high strength and high damping. Recently, Shindo and Niwa (1996) analyzed the scattering of antiplane shear waves by a cylindrical inclusion with thick nonhomogeneous interface layer, and applied the results of the single scattering problem to coherent plane wave in a fiber-reinforced metal matrix composite with interface layers. While the scattering of plane elastic waves by a single inclusion has been the subject of much research in the past, the wave propagation problem for composite materials including the effect of multiple scattering by several inclusions has been rather sparsely discussed. The purpose of this study is to analyze the effects of interface layers and multiple scattering by a distribution of inclusions on the wave propagation of time-harmonic axial shear waves in a fiber-reinforced metal matrix composite. The interface layer is modeled by any number of homogeneous layers, which may be chosen to approximate any nonhomogeneous material properties. The composite medium contains a random distribution of cylindrical inclusions of same size with interface layers of same thickness. The problem of the scattering of plane axial shear waves by a large number N of cylindrical inclusions with interfaces, arbitrarily distributed in an infinite matrix, is analyzed and the resulting equations are then averaged, considering the positions of the inclusions to be random (Bose and Mal, 1973, 1974). The averaged equations are solved by using Lax's quasicrystalline approximation to yield the propagation characteristics of the average wave (Lax, 1952). The particular case when the pair correlation function has an exponential form, is examined in detail. The phase velocity and attenuation of coherent elastic shear wave, and the effective shear modulus are obtained numerically for an SiC fiber-reinforced A1 composite and the numerical values are shown in graphs for various interface properties at designated frequencies. The results in the dependence on concentration at wavelength comparable to scatter size here are valid for thick nonhomogeneous interface layers and a wide range of frequencies. 733

734

Y. S h i n d o et ~d.

yi

(rio, 0jo, r~

Matrix ~, p, v

h Xi

oi (no, 0io, z)

Inclusion

/

,.

~/,

IXo,po, vo

hi.n, pn, Vn

y

Cp

~

",,,__~_~ v2

~x 0 Fig. 1. A cylindrical inclusion with an interface layer and coordinate systems.

2. STATEMENT OF THE PROBLEM AND SCATTERING OF ANTIPLANE SHEAR WAVES

BY N INCLUSIONS We suppose the identical cylindrical inclusions of radius a0 to be located within a large region S in an infinite matrix. Let #, p, v be the shear modulus, the mass density, the Poisson's ratio of the matrix, and #0, P0, v0 those of the inclusions. We assume that thick layers of uniform thickness h with variable material properties are presented at the interfaces separating the matrix from each cylinder. Let the inclusion be separated from the matrix by n layers. The geometry is depicted in Fig. 1 where (x, y, z) is the Cartesian coordinate system with origin at o and (r, 0, z) is the corresponding cylindrical coordinate system. The layer is subdivided into several thick-walled shells and the material properties within each shell of inner radius a~_ ~, outer radius at(l = 1 ~ n) and uniform thickness ht = a / - a~_ t are #l, Pl, vz. Labelling the inclusions by suffixes i = 1,2 . . . . . N and taking suitable coordinate axes in a transverse plane, let the boundaries of the ith cylindrical inclusion and the shells be denoted by C~ (l = 0 ~ n) and the Cartesian and cylindrical coordinates of those center oi (rio, Oio, z) be (xi, Yi, z) and (ri, Oi, z), respectively. The displacement in the z-direction w satisfies the wave equation 1 O2w V2w = - 2 ~t 2 ' Csh

(1)

Multiple scattering of antiplane shear waves

735

where V~ = a2/0r2+ (I/r)(O/Or)+ (1/r2)(02/a02) is the two-dimensional Laplacian operator, t is the time and Cshis the shear wave speed in the matrix,

The stress component z,~ is found as ~W

Z,z =/~ ~-r'

(3)

We consider a plane shear (longitudinal shear, SH) wave polarized in the z-direction and propagating in the positive x-direction. Thus,

wi =

W 0

exp [i(kshX--a~t)],

(4)

where a superscript i stands for the incident component, 09 is the circular frequency of the wave and w0 is the amplitude of the incident SH wave. ksh is the wave number of the SH wave in the matrix, tD k~

-

.

(5)

Csh

In what follows, the time factor exp(-ioot) will be omitted from all the field quantities. The displacement fields in the matrix, the lth layer of the ith cylindrical inclusion and the ith cylindrical inclusion may be expressed in the forms N

ws= Ew~ i=l

w~ =

~ rn

w~ =

=

AjmH,,(k,hr~) exp(imO~)

~,, [B~H,,,(ktshri)exp(imO~)+C~..Jm(U~hr,)exp(imOi)] m~

(6)

--oo

(/=l.-.n)

(7)

--oo

W~= ~ OimJm(k°shri)exp(imOi), rn

=

--

(8)

o~

where superscripts s, t and l (l = 1 ~ n) denote the scattered component within a matrix, the transmitted component within a cylindrical inclusion and the field quantity within an /th layer, Aim, Bim, t fire t and Dim a r e the unknowns to be solved, H,,() is the mth order Hankel function of the first kind and arm() is the mth order Bessel function of the first kind (Watson, 1966). The wave numbers ~h (l = 1 ~ n) in the lth layer and ks° in the cylindrical inclusion are given by (0

Ush=-- (l= 1 ~n) c~s~ k~° -

(o 0 '

Csh

(9)

736

Y. S h i n d o e t a l .

.0 in the cylindrical where the longitudinal shear wave speeds C~h in the /th layer and C~h inclusion are

(1 = 1 ~ n)

C~h = \ P d o = ( # o ~ ''2

(lO)

The b o u n d a r y conditions on C~ (l = 0 ~ n) are wn, = w ~ + w e,

Wj

l

.W j l+1 ~

~

i

"Crzj ~

,t

14, j ~

Wj

1~

z~..j" = zS~z+Z~ l+1

"[rT:j

(rj

~ = z rzlI.

"~rzj

(r s = a,)

(11)

at, l = 1 ~ n - 1 )

(12)

(rj

(13)

a0),

where Z~zs(Z~j, t t z s. . . .re . . ) is the stress c o m p o n e n t corresponding to w ~(w ~ j~w , we). The condition of continuity of displacement at P~ (a., 0j, z) on C 7 gives

[B~mHm(ks~a.) exp( imO fl + C j"~Jm(~'ha.) exp( imOs)] m=

- oz

= [woexp(ik~hX)+~ ~ AemH,.(k~hr~)exp(imOs)]. i= 1 m = --~

(14)

-Ie7

Multiplying by exp [ - ivOi] and integrating from 0 to 2re, we have

Bi"vHv(ks~a.) + Cj~,,Jv(ks~an) = woiVJv(kshan) exp(ik~hrj.o cos Ojo) + ~

Zimgijmv

i= I m=

,

(15)

--~

where

1 12~ [Hm(kshre)exp(imOe)]~exp(-ivOj)dOj

Kej,,,v - f n L

= H~(ksha,)6m,.

(i ~ j )

(i = j ) ,

(16)

bm~ is the Kronecker delta. Using the addition theorem of Hankel functions (Bose and Mal, 1973), we get

Keim~ - -

1 2r~ )<

/4.,

exp(imOjs)(-- 1) m s=

m

( - 1)'J~(k~ha~)

(ksh rje) exp [is(Oj -- 0je)] } exp( -- ivO/) dot

= J~ (k~ha.)nm_v (k~hrje) exp [i(m -- v)Oje]

(i ~ j).

(17)

where (rje, Ose) are the polar coordinates of oj referred to o~ as origin. Thus, eqn (15) becomes

Multiple scattering of antiplane shear waves

737

B j~H~(k~ha.) + C j%J~(k~shan) = AjvHv(ksha.) +J~(k~a.) Wot"exp(ik~hrjo cos Ojo) + ~ ' i=l

Ai.m+~Hm(k~hrj3exp(imOj3 , m=

(18)

--oo

where Z' denotes the sum over all cylindrical inclusions except the jth. The conditions o f continuity o f displacement at P~ (a,, 0j, z) (! = 1 ~ n - 1 ) on C~ and pO (a0, 0j, z) on C ° give t t ~ (k~hat) t + 114 tvt+ i at) + '~-'jv pt+ 1ro v evt+ ~at) B j~Hv (ksha~) + C s~Jv =w a.,jv ;t • v \ n ' s h I.tVsh

(19)

Dj.J~(kOhao)= B j~H~(k~hao) 1 , t 1hao). + C s~J~(k,

(20)

The conditions o f continuity of shear stress at PT, P~ (l = 1 ~ n - 1) and po similarly give

-~[BT~o-~H~(~ha,)+C~o-~J~(~ha,)l=Agv~H~(ksha,) + ~a~ J~(ksha") w°i~exp(ik~hrj°c°sOj°)+,=l ~ ' m= -oo A~,,~+~H~(k~hrj~)exp(imOji)

#~ B~atH~(U~na,)+Cj~

J~(ktshat) = Bj~ ~aH~(~h a , ) + C~ ( j v+1

l'tl + 1

(21)

J~(Us+lat) (22)

/~o

O

o

c~

1

~ Djv ~ao J.(k,hao) = B), o~o n.(k2hao) + C)~ ~ao J,(k~nao).

(23)

F r o m eqns (18)-(23), the u n k n o w n Ajv is found to be

Ajv = A;Fjv,

(24)

where

Hv(~ha.) . J~(~ha.) A" = Xv Hv(ksha.~) + rv Hv(ksha.)

J~(ksha.) H~(k~ha.)

(25)

N

Fjv = Wot~ exp(ikshrjo COSOjo) + Z ' i=l

A'm+vF~,m+vHm(kshrji)exp(imOj3. m=

(26)

--cxD

In eqn (25), Xv and Y~ are given in the Appendix.

3. THE AVERAGE FIELD FOR A R A N D O M DISTRIBUTION OF CYLINDRICAL

INCLUSIONS We consider the positions o f the cylindrical inclusions to be r a n d o m . If we denote the position vector o f oi by rio and the probability density o f the r a n d o m variable (rio, r2o,. •., rNo) by p(r~o, r2 . . . . . . rNo), then due to the indistinguishability o f the cylindrical inclusions, it is symmetric in its arguments and we have ( W a t e r m a n and Truell, 1961)

p(rlo, r2 . . . . . . rNo) = p(rio)p(rlo, rz . . . . . . ' . . . . . rNolrio) = p(r~o)p(rjo Ir~o)p(rlo, rz . . . . . . ' . . . . . ' . . . . . rNo IUo, r~o) p(r/o) = p(rlo),

p(rjolr~o) = p(rzo[rlo)

(i ~ j),

(27)

Y. Shindo et al.

738

where the vertical lines in the arguments denote the usual conditional probabilities. A prime in the first part of eqn (27) means rio is absent, while two primes in the second part of eqn (27) mean both rio and r/,, are absent. For a uniform composite, the positions of a single cylindrical inclusion are equally probable within a large region S of a cross-section of the material and, hence, its distribution is uniform with density 1

p(rio)=~

rioES

= 0

rio ¢ S.

(28)

If now o~, well within S, is held fixed, the distribution of the cylindrical inclusions around the cylindrical inclusion will be circularly symmetrical. Thus, p(rjolrio) is a function of r(j alone, and we can write 1

P(rjolrio) = ~ [1 -g(r~i)] = 0

rjoeS (29)

rjo¢S,

where the pair correlation function g(rij) ~< 1 is a decreasing function of rij. The normalization condition gives, in the limit as S ~ l

R

i g(rii)rii nl~i m -R 2 Jo

drij = O.

(30)

Due to the impossibility of interpenetration of the cylindrical inclusions and their independence when they are infinitely apart, we have

g(rij) = 1 rij < 2a, = 0

r~j--* ~ .

(31)

A function satisfying these conditions is

g(r~j) = 1 rij < 2a,, = Vexp(-r~i/L)

r!j >~ 2a,

(32)

where V (0 < V ~< exp [2a./L]) is the coefficient and L > 0 is the correlation length. We denote the conditional expectations of a statistical quantity f when either oi or o~ and oj together are held fixed as

(f)g=f.'.ffP(rl .... '.,rNolrio)dZ~..'.dzN (f)~i=f.'.'.ffp(rlo,.'

'.,rNolrio, rjo) dZl.'.'.dZN,

(33)

where dzi (i = 1 ~ N) is the volume element at rio. To determine (Fi~)i of eqn (26) we take the conditional expectation to obtain

Multiple scattering of antiplane shear waves

(Fi~)i = woi" exp(ik~hrio cos 0~o)+no 1 --

739

A;,+~ m

cJo

[1 -g(rij)](Fj,,,+~}~jH~(k~hrij) exp(imOj~) dzi,

x

(34)

dri o , r j o ~ S where no = N/S = e/rta~ is the number of cylindrical inclusions per unit area and c is the volume concentration of inclusions in the matrix. Equation (34) involves the conditional expectation with two cylindrical inclusions held fixed. If we take the conditional expectation of eqn (26) with two cylindrical inclusions held fixed, the resulting equation will contain the conditional expectation with three cylindrical inclusions held fixed, and so on. We shall eliminate this hierarchy by assuming Lax's quasicrystalline approximation (Lax, 1952), which involves the two-inclusion correlation function and implies (Fi~)ij = (Fi~)i,

i Cj.

(35)

According to the extinction theorem when S and N become infinitely large (Lax, 1952), the incident wave is extinguished on entering the composite, so that the corresponding term in eqn (34) can be dropped. Thus, this equation reduces to

(Fim)i

=

f

no ~ A'm+~ v = -- oo

Jlrj o_riol

[1-9(rji)](Fj,m+v)jH~(k~hrji)exp(iv0ji)

dzj.

(36)

> 2a n

Assuming the existence of an average plane wave, we try, for eqn (36), the solution

(F~m)i = i'F,, exp(iK~hxio),

Xio

=

riocos Oio,

(37)

where Fm is a constant and K~his the wave number of the effective SH wave. Making use of the Green's theorem and the plane wave expansion

exp(iK~hXjo) = exp(iKshXio) ~ i-sJ~(K~hrji)e x p ( - isOji), s=

(38)

-oo

we find that the first integral appearing in eqn (36) becomes

f,jo

exp(iKshXjo)Hv(kshrji) exp(ivOji) dTj 2a n

_

1

k s2h-- K~

I

'io-r,oI>~a,

{V2[exp(iKshXjo)]Hv(kshrji)exp(ivOji)

-- exp( iK~hXjo)V 2[Hv(k~hrji) exp( ivOji)]} dzj

_[

o

3

= exp(iK~hxio) 2~a"i-v J~(2K~ha.) H~(2ksha.)-Hv(2k~ha.) ~ a A(2K~ha.) .

(39)

The second integral in eqn (36) can be also simplified by using the above expansion and eqn (36) reduces to the system of equations F m = 2r~n0 ~ v=

A'm+vF,,+v

-0o

x{~IJv(2Kshan)~--~Hv(2kshan)--Hv(2kshan)o-~Jv(2Ksha,) ]

740

Y. Shindo et al.

The elimination of Fm from the above equations yields a determinantal equation of infinite order for K~h. The effects of multiple scattering on the coherent wave are of great practical importance for the concentration c = 0.01 ~ 0.4. At very low concentrations (c < 0.01) multiple scattering can be neglected and each scatterer can be treated as independent. Assuming kshao and L to be sufficiently small compared to the wavelength, we obtain by expanding the Bessel and Hankel functions and retaining the lowest order terms for

h/ao = 0.0

....

rm

tG)

l-

l, ) i 1 ~sh 2 "}- 7 k2hlv

(41)

where P0

A;

= --

-1

P #--#o A"_+ I - -

#0 + #

Am = 0,

(Iml /> 2)

(42)

I0~-rt

I1

~-

i

V L 2 Ksh

~Z i

ksh

K 2 VL2 (,Xsh~

ImP--O, m>~3.

(43)

The effective shear modulus #*~ can be easily obtained from the phase velocity Re(k~h/K~h) of the effective SH wave as follows :

#*"=#

7-

Re

--

t,Ksh)J

(44)

,

where the average mass density p* is

p*=p

1-c

1+

+poc+

#=l

ptc

2+

n

.

(45)

4. N U M E R I C A L RESULTS A N D DISCUSSIONS

To examine the effect of interface properties on the phase velocity and attenuation of coherent plane wave through the composite medium, for a given value of kshao, A'v is computed. Next, the complex coefficient matrix M corresponding to Fm [eqn (40)] is formed. The complex determinant of the coefficient matrix is computed using standard Gauss elimination techniques. For a given kshao, the root of the equation det M = 0 is searched in the complex Ksh plane using Muller's method. G o o d initial guesses are provided by eqn

Multiple scattering of antiplane shear waves

741

Table 1. Material properties of SiC and A1 Po (kg m -j)

P-0 (GPa)

v0

3181

188.1

0.17

SiC

p (kg m -j)

# (GPa)

v

2706

26.7

0.34

AI

(41) at low values of ksha0 and these can be used systematically to obtain quick convergence of roots at increasingly higher values of kshao. The considered composite was an SiC-AI composite. The constituent properties are given in Table 1. Three special cases of interface material are considered. The elastic properties of Case I, II and III are given by

Case I /a,(r) =

P~ -

#+~t0 2

P+Po 2 (ao <~r<~ao+h).

(46)

Case H r--

a0

r--ao

(ao <<.r<<.ao+h).

(47)

Case III ,,

pi.(r)

=

. I - r - (ao -t- h/2)-] 3

L-

j +.+.02

4(p_po)Ir_ -(ah_+hl2)]3

+ P+Po ~

(a0 ~
(48)

The material properties of the layers given above are calculated at the midpoint of each layer assuming variations of Cases I, II and III from the boundary of the inclusion to the matrix medium. Figure 2 shows the variation of the phase velocity Re(ksh/Ksh) of the effective SH wave with the number of layers n for Case II and c = 0.3, h/ao = 0.1, aoOg/Csh= 1.0. Case II refers to the case of the interface material through which the elastic properties vary linearly from those of the inclusions to those of the matrix. It is found that the truncation after n = 30 gives practically adequate results for Case II. The effect of the interface layer on Re(ksh/K~h) at aoaUc,h= 1.0 for c = 0.3 is shown in Fig. 3. The figure shows that the phase velocity Re(k~h/K~h)increases with the h/ao ratio, and depends on the constituents and the nature of the interface layer. The phase velocity for Cases II and III, which was evaluated by taking n = 30 and 32, agreed to at least three decimal places. Thus, it may be said that the result for n = 30 is, from a practical view point, quite satisfactory. In Fig. 4, the phase velocity Re(k~h/Ksh) of the effective SH wave is plotted as function of the frequency aoOg/c~hfor c = 0.3. The dashed curve refers to the case h/ao = 0.0 and the solid curve refers to h/ao = 0.1. The interface material for Case III is considered. The phase velocity increases with the frequency, reaches a maximum, and then decreases, and the interface effect increases the phase velocity. Figure 5 shows the variation of the attenuation Im(ksh/ksh ) of the effective SH wave with the frequency aotg/Csh for Case III and c = 0.3, h/ao = 0.0, 0.1. The attenuation decreases with the frequency, reaches a minimum, and then increases, and the interface effect increases the attenuation. The computations carried out

Y. Shindo et a/

742

I

1.208

I

SiC-AI c=0.3

h/ao=O.1 aOOffCsh=l'OU°'°°~x'x InclusionI

1.206

I ITMa,mP.x

~ly~

r=ao

r=ao+h

~'-~'--_2C-_~-C-C--~.~-C---

I

,

10

0

A-_

I

,

,

20

I

,

30

40

n

Fig. 2. Phase velocityvs n.

1.24

........... ....... : ~

Case I Case II Case III

,," , ~

,~/,"" s J j ' f ~JJ s /, /

SiC-A1

7_,-,2./." .-""

-

1.21 "_ aoOa/Csh:l. 0 . ~ . ' c=O.3

. oot,. 1.18

/l p _ . I ~erf.d~l "'° ,~:luslO. I ~ty:aCe'\X~M.mx t

- -

,/" _

. . . .

0.0

I

, '--?.

0.1

] 0.2

h/a o Fig. 3. Phase velocity vs h/ao. reveal that the truncation after n = 30 gives practically adequate results at any desired finite frequency for Cases II and III. Figure 6 shows the variation of the effective shear modulus/~*z of SiC-A1 with the frequency for Case III and c = 0.3, = 0.0, 0.1. The effective shear modulus increases with the frequency, reaches a maximum, and then decreases, and the interface effect increases the effective shear modulus. As the dynamic effective shear modulus tends to the static solution. Using the Eshelby method, we obtain the effective shear modulus Yx*zfor = 0.0 as Wakashima (1976)

aoe~/Csh

h/ao ao~o/csh--*O,

h/ao

(~0-~) 2 /~*~ = (1 - c ) # + C#o - c ( 1 - c)

Making use of the law of mixture, we also have

(1 - c)/~0 + (1 + c)/~"

(49)

Multiple scattering of antiplane shear waves

743

I

f

1.25

N

Inclusionl Layer

~

,-'-",

1.20

:....... ~u,p i Matrix r=a° r=ao+h I

~

SiC-A1 "-.,. ~ c=0.3 ""-....

-'--......

Case III

"~ %

1.15

......... h/a 0 = 0.0

-...

h/a 0 = 0.1

"'I

0.0

1.0

2.0

a00~/Csh Fig. 4. Effectof interfaceon phase velocityvs frequency.

!

,

"~

1 fir,

.2

,

l S

v/"

|

,

i

,

,

¢s /

//o, P o

10 -3

E V

r = - S i C . A ; =a°+h

1

c=0.3 Case III

1 0-4~ ......... h/a 0 = 0.0

F

0.0

h/ao = 0.1 I

I

I

I

I

I

I

I

I

1.0

2.0

a0(O/Csh Fig. 5. Effectof interfaceon attenuation vs frequency.

u/~0

/~*= - (1 - c ) / ~ 0 +c/~"

(50)

Figure 7 shows the variation of the static effective shear modulus #% with the volume concentration c for h/ao= 0.0. A comparison of the static effective shear modulus is made in aoCO/Csh = 0, Eshelby method and law of mixture. The results agree much better with those obtained from the Eshelby method in the lower concentration. As mentioned earlier, the Lax's quasicrystalline approximation is valid for the small concentration (c < 0.4) or to second-order in the concentration c. In conclusion, the multiple scattering of antiplane shear waves by cylindrical inclusions with thick nonhomogeneous interface layers was analyzed. The interface effect can increase phase velocity, attenuation of coherent plane wave in a metal matrix composite and effective

Y. Shindo eta/.

744 i

i

i

I

'

i

#o, Po

4b.U -

In lu ,on ~

'

~ p

¢8

N

:~ 4 0 . 0

-

"-,

h/a0 = 0.1

-

I

I

I

I

[

0.0

I

I

I

I

1.0

2.0

a00ffCsh Fig. 6. Effect of interface on effective shear modulus vs frequency.

I

a00~ / Csh = 0

200

r,.3

/

/

Eshelby Method

/IY

Law of Mixture

/,/'//i

. . . . . . . . .

/

/f',/ //

100

,~s

I

I

I

,/,

./

SiC-A1

0

.i

I

I

I

I

I

0.5

I

.0

¢ Fig. 7. Static effective shear modulus vs concentration.

elastic constant, and depends on the frequency and the material properties of the interface layers. The numerical results at the volume concentration of inclusions c = 0.3 were obtained for any given finite frequency, and layers with nonhomogeneous elastic properties of any desired finite thickness. REFERENCES Bose, S. K. and Mal, A. K. (1973) Longitudinal shear waves in a fiber-reinforced composite. International Journal of Solids and Structures, 9, 1075-1085. Bose, S. K. and Mal, A. K. (1974a) Axial shear waves in a medium with randomly distributed cylinders. Journal of the Acoustic Society of America, 55, 519-523. Bose, S. K. and Mal, A. K. (1974b) Elastic waves in a fiber-reinforced composite. Journal of the Mechanics and Physics of Solids, 22, 217-229. Lax, M. (1952) Multiple scattering of waves. II. The effective field in dense systems. Physical Review, 85, 621-629.

Multiple scattering of antiplane shear waves

745

Shindo, Y., Nozaki, H. and Datta, S. K. (1995) Effect of interface layers on elastic wave propagation in a metal matrix composite reinforced by particles. ASME Journal of Applied Mechanics, 62, 178-185. Shindo, Y. and Niwa, N. (1996) Scattering of antiplane shear waves in a fiber-reinforced composite medium with interfacial layers. Acta Mechanica, 117, 181-190. Wakashima, K. (1976) Macroscopic mechanical properties of composite materials part II. Elastic moduli and thermal expansion coefficients. Japanese Society for Composite Materials, 2, 161-167. Waterman, P. C. and Truell, R. (1961) Multiple scattering of waves. Journal of Mathematical Physics, 2, 512-537. Watson, G. N. (1966) A Treatise on the Theory of Bessel Functions, 2nd edn. Cambridge University Press, Cambridge.

APPENDIX X,, Y~in eqn (25) are G~Q'~

X~

E~Q7 - U,P"~

-G~P"~ , E~Q 7 -- U~P~

Y,

(AI)

where

o o #. E~ = H, (~han) ~nanH~(k~han) - H,(k,ha.) ~a. H,(~han) ~O

O

O

0

It,

U~ = J. (~han) ~a. H.(ksha.) - Hv(ksha.) ~a. J~(k~shan)"~" (A2)

G~ = J.(k~ha.) ~nanH~(kshan) - U~(k~han) ~an J~(ksha.). The recurrence formula for PT, Q7 are given by e~+ ~ =-~K~+Mv~ P~v I

/

U Q~

Q': ' = ~ IJ, + N'~ (1= 1 ~ n - l )

(A3)

= H~(k~hao)-~j~aoJ,(k~hao)--J,(k~hao)d~oH~(k~hao) Q~

l Ito 0 o o 0 i = Jv(kshao) ~ ~aoJv(kshao) -J~(kshao) ~aoJv(kshao).

(A4)

Ineqn(A3), Kv,i L ,l M , t Nvi (I= 1 ~ n - l ) are I

r

.

l

+ I

'+ a,)--Hv(~h

a

a,)~+~atJ~(~hat) ] Itl

d

Itt ~atH.(~ha, d /[j.(~hat) ~+~ ) --H.(lC~hat) . ~--~+l Itt ~atJ~(k'shat)J c~ . "]

L'~ = [J~(Ushat)~aiJ~(U~h at)-J~(k~h at) ~+l ~a tJ,(~ha,)] It1 0

/ [ , J (~ha,) ~+l~at - ~(k/sha/) H 0

l Itt 0 H~(ksha,) p~+l~at J~(~hat)]

+~

t+~

It~ C~

t

M'~ = [H~(~ha,)-~atH.(~h a,)-H~(k~ h at)--~+l ~aiH~(ksha,)]

/[

H,(~ha,) ~t. .~j~(~hat)_j,(~ha,) It' o~tH,(~ha,) 1 Its+ 1 va t Itt+l

L

at) -J~(ksh

--

) Itt oat Itt+ i /rH~(k:hat) #t+#t 1 9.~_j~(klshat)_J~(klha, L

H~(U~hal) ~atH,(k~ha,)l

(A5)