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AD-A036 431

DISPERSION RELATIONS AND MODE SHAPES FOR

WAVES IN LAMINATED VISOELASTiC COMPOSITES

BY VARIATIONAL METHODS

STANFORD UNIVERSITY

CALIFORNIA

OCTOBER 1976

FINAL REPORTN

ONR Scientific Report OFContract report No. 26 APPLIED

MECHANICS

Q DISPERSION RELATIONS AND MODE SHAPES FORDEPARTMENT

WAVES IN LAMINATED VISCOELASTIC COMPOSITES OF

BY VARIATIONAL METHODS MECHANICALENGINEERING

BY

Subrata Mukher,;ee and E. H. Lee

STANFORDOffice of Naval Research UNIVERSITYGrant N00014-75-C-0698

STANFORD,CALIFORNIA94305

SUDAM '6-4

October 1976

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DISPERSION RELATIONS AND MODE SHAPES FOR FINAL REPORTWAVES IN LAMINATED VISCOELASTIC COMPOSITES 6 PERFORMING ORG. REPORT NM;MERBY VARIATIONAL METHODS SUDAN No. 76-4

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Subrata Mukherjee and E. H. Lee N00014-75-C-0698

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-0-

19. KEY WORDS (Continue on reverse id.e it n cees.,y and Identity by block number)

Viscoelasticity , Floquet wavesComposite material, Disport-ion relationsWave propagation

20ABSTRACT (Continue an reverse side It nocssow and Identify by block number)The propagation of oscillatory waves through periodic elastic compositeshas been analyzed on the basis of Floquet theory. This leads to self-adjointdifferential equation systems which it has proved convenient to solve by var-ational methods. Many composites, such as the light-weight high-strengthboron-epoxy material, consist of strong reinforcing components in a plasticmatrix. The latter can exhibit viscoelastic properties which can have a sig-nificant influence on wave propagation characteristics.

cont. next page

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20. continued

Replacement of the elastic constant by the viscoelastic complex

modulus changes the mathematical structure so that the differentialequation system is no longer self-adjoint. However, a modification ofthe variational principles is suggested which retains formal self-adjointness, and yields variational principles which contain additional

boundary terms. These are applied to the determination of wave speedand mode shapes for a laminated composite made of homogeneous elasticreinforcing plates in a homogeneous viscoelastic matrix for plane waves

propagating normally to the reinforcing plates. These results agreewell with the exact solution which can be evaluated in this simple case.The variational principles permit solutions for periodic, but otherwisearbitrary variation of material properties.

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F

Dispersion Relations and Mode Shapes for

Waves in Laminated Viscoelastic Composites by

Variational Methods

Subrata MukherjeeAssistant Professor

Department of Theoretical and Applied MechanicsCornell UniversityIthaca, New York

and

E.H. LeeProfessor

Division of Applied MechanicsStanford UniversityStanford, California

00I

Abstract

The propagation of oscillatory waves through periodic elastic com-

posites has been analyzed on the basis of Floquet theory. This leads to

self-adjoint differential equation systems which it has proved conveni-

ent to solve by variational methods. Many composites, such as the light-

weight high-strength boron-epoxy material, consist of strong reinforcing

components in a plastic matrix. The latter can exhibit ¢iscoelaslic pro-

perties which can have a significant influence on wave propagation charac-

teristics.

Replacement of the elastic constant by the viscoelastic complex

modulus changes the mathematical structure so that the differential equa-

tion system is no longer self-adjoint. However, a modification of the

variational principles is suggested which retains formal self-adjointness,

and yields variational principles which contain additional boundary terms.

These are applied to the determination of wave speeds and mode shapes for

a laminated composite made of homogeneous elastic reinforcing plates in

a homogeneous viscoelastic matrix for plane waves propagating normally

to the reinforcing plates. These results agree well with the exact solu-

tion which can be evaluated in this simple case. The variational principles

permit solutions for periodic, but otherwise arbitrary variation of ma-

terial properties.

- iv -

*I'

4 p

INTRODUCTION

A composite medium typically consists of a matrix material with an

embedded reinforcing material in the form of fibers or laminations. A

laminated composite can usually be modelled as alternating layers of

matrix and filament arranged in a periodic manner, while a fiber rein-

forced composite can often be represented by a homogenious matrix in

which a two dimensional doubly periodic array of filaments is embedded.

A convenient method for analysing wave propagation in composite

media is the use of Floquet theory. This approach has been recently

used by several authors [1-7 to study steady state wave propagation in

periodic elastic composites. Of these, some [1-5] used variational methods

while others [6-7] used direct numerical methods involving discretiza-

tion of the governing differential equations and associated quasi-peri-

odic boundary conditions.

In practice, the matrix in a composite is often a polymer which exhi-

bits viscoelastic properties. This leads to dissipation as well as dis-

persion. Free wave propagation in an infinite laminated viscoelastic

composite was studied by the present authors in [8] using Floquet theory

and finite difference methods. In the viscoelastic case, forced steady

wave propagation leads to real frequencies and complvx wave numbers

(attenuating waves) while free waves with no applied tractions have com-

plex frequenies and real wave numbers (damped waves) [8].

Consideration of viscoelastic properties of the matrix leads to

complex viscoelastic moduli which are functions of the frequency that

can be real or complex. Even in the elastic case [1] it is convenient

to utilize complex analysis to incorporate the changing phase as the

-1I-

-2-

wave traverses the periodic medium, so that the displacement is expressed

by

= u(x)elWt

where w is the frequency, t is time and u(x) is complex. Applica-

tion of variational methods involves integrals of the type

a/2 dul d22I(UlU 2 ) f {-n(x)-- + p(x)W UlU2 }dx

-a/2 dx dx

where n(x) is the elastic modulus variation, p(x) that of the delsity,

a is the periodic cell length and * denotes the complex conjugate

For ela Acity with n afend frequency w real, the integrals are

Hermitian: I(UU 2) I*(u2,uI ) and self-adjoint differential operator

systems result for determining the displacement or stress fields. For

viscoelasticity, when 1 and w are complex, the integrals are not*

Hermitian since n r, and the resulting differential operators are

non-self-adjoint. However, if one formally bases the theory on real

analysis, so that the complex conjugate sign is removed from the integral

even though u(x) is complex, the integral is then symmetric: I(ulu 2)

I(u2,u1) , and hence formally self-adjoint differential operators are gen-

erated. This is so, with u(x) complex, even though the integral I(u,u) is

then no longer physically an enorgy integral and the variational princi-

ple no longer expresses Hamilton's principle. This formulation introduces

an additional boundery term, so that the system is formally self adjoint:

a self-adjoint cuerator but not with th,, appropriate boundary conditions.

These boundary terms can be included to generate variational principles,

and the ones presented here are formulated in this way.

-3-

In this pape-r, the strain energy [1], complimentary energy [2] and

Hellinger-Reissner [3-5] variational principles are extended so that

they can be used in the viscoelastic case. As an illustration, the ex-

tended version of the Hellinger-Reissner [4] variational principle is

applied to the problem of free waves in an infinite one dimensional visco-

elastic composite (8]. This principle is chosen because it gave very

accurate results in the elast.c case (4]. A composite with two homogene-

ous layers per cell is studied with the filament elastic and the matrix

modelled as a three element solid. The Rayleigh-Ritz method is used to

obtain dispersion relations and mode shapes and the results compared with

the exact solution which exists in this case. For two homogeneous com-

ponents the elastic solution has been obtained in closed form (9,2] and

this can be adapted to the case of linear viscoelastic components. The

question of rapidity of convergence to the exact solution is discussed.

The present method can be used to study wave motions in genera] one-

dimensional periodic viscoelastic compositeswhich otherwise exhibit arbitr-

ary variations of material properties. Waves in fiber reinforced visco-

elastic composites can be studied by using similar variational principlep

in two dimensions.

0o

GOVERNING EQUATIONS

We consider wave propagation in a one dimensional laminated visco-

elastic composite, and, in particular, choose a two material composite

as showr in Fig. 1. This example is chosen because the variational

principles to be presented in the next section can then be easily com-

pared with those in the elastic case [1-5]. Variational principles

for any general one-dimensional inhomogenious periodic viscoelastic

medium can be easily obtained by suitable modification of these princi-

ples.

The composite covers the full space - < x < -. We study one-

dimensional strain waves propagating in a direction x normal to the

interface planes. For harmonic waves, the stress and displacement are

iwt , iwtof the form 6 = a(x)e and t = u(x)e . The quantities a and

u are, in general, complex.

Within each cell, the governing differential equations are

d + P(x)w2u = 0 (1)

dxx= d (2)

where n = X + 2P in terms of complex Lam6 viscoelastic moduli of

the constituent materials. The quantities n and p are periodic with

period a

= ,(x;w) p(x+a) = p(x)

and within each cell they ax discontinuous functions defined by

-- 4

- I , - . . . ..

T1l(w)' Pi -a/2 < x < -b/2

n(x;w) , p(x) = I2 (W), P2 -b/2 < x < b/2

i ( W) , P 1 b/2 < x < a/2

We note that 1 and T2 are, in general, complex functions of the

possibly complex frequency w.

By Floquet theory [i the displacement and stress satisfy the quasi-

periodic boundary conditions across each cell

u(a/2) = u(-a/2 )eiqa (3)

a(a/2) = (-a/ 2 )eiqa (4)

where q is the wave number.

The interfaces are assumed perfectly bonded so that u and a are

continuous across them

U(xo+) = U(x-) x = ±b/2 (5)

= +) a(xo-) x = b/2 • (6)

VARIATIONAL PRINCIPLES

1. Strain energy principle

Let us choose a pair of sectionally continuous and differentiable

functions u and a which satisfy the constitutive equation (2), the

quasi-periodic boundary conditions (3) and (4) and continuity condition

(5). The solution (u.c) which also satisfies the equation of motion (1)

and the continuity condition (6) is given by the variational equation

6I(u) + a(a/2)Su(a/2)(1-e-qa - (7)

wherea/2 2

I(u) nf Wn (x ) + P(x)Wu dx2 2dx

-a/2 '

Proof:

Taking variations with respect to u and after carrying out the

necessary integrations, we obtain

a/22i0u dj u0.~~w61(u) + (a/ 2 u(a/ 2 )(l-e2 iqa) = (n())+P(X)2 6udx

f ~dx J-a/2(

+ Ti(a/2)-nlU(a/2 6u(a/2)

+ nl dx(-a/2 u(-a/2)-a(a/2)6u(a/2)e- 2 q

+ < n(b/2)-(b/2)6u(b/2) >dx

+ < ,n(-b/2)dLU(-b/2)6u(-b/2) >+ " dx'

Here we use the notation

< g(x) > = x - g(x )0 0 -

-7-

If u and a are restricted to the subclass of functions which

satisfy equations(2), (3), (4) and (5), this expression reduces to

a/2

(-+p(x)w 2u)u dx +< o(b/2) > 6u(b/2) + < o(-b/2) > 6u(-b/2)

-a/2

This expression vanishes for arbitrary Su if and only if u and

a satisfy equations (1) and (6).

2. Complementary energy principle

Let us choose a pair of sectionally continuous and differentiable

functions u and a which satisfy the equation of motion (1), the

quasi-periodic boundary conditions (3) and (4) and continuity condition

(6). The solution (uo ) which also satisfies the constitutive equation

(2) and the ccntinuity condition (5) is given by the variational equation

6J(a) + u(a/2)So.a/2)(l-e 2 iq&a) = 0 (8)

where

a/2 2 (02J(O) =2nllx) + 2xw2 dx

-a/22()2

and the prime denotes differentiation with respect to x.

Proof:

Taking variations with respect to a and after carrying out the

necessary integrations we obtain

&~ $ - - ~ .- 8 -- ~-V 4 l }

a/2

J(o) + u(a/2)o(a/2)[l-e 2i q a ] I= (_o + dx' 2 6a dx

ap (x)2

+ (u(a/2) + (a/2))6a(a/2)P1

- ,(-a/2) 6a(-a/2)-u(a/2)6o(a/2)e-2iqa2

P 1W

p (b/2)w

- < a'(-b/2) 6(-b/2) >

p(-b/2)w2

If u and a are restricted to the subclass of functions which

satisfy equations (1), (3), (4) and (6), this expression reduces to

a/2

.du - x,- )) , + < u(b/2)> 6 o(b/2) + < u(-b/2) > 6((-b/2).

-a/2

This expression vanishes for arbitrary So if and only if u and

o satisfy equations (2) and (5).

3. hellnger-Reisiner principle.

Let us choose a pair of sectionally continuous and differentiable

functions u and a which satisfy the quasi-periodic boundary condi-

tions (3) and (4) and continuity condition (5). The solution (u,a)which

also satisf'ies the equation of motion (1), the constitutive equation (2)

and continuity condition (6) is given by the variational equation

,.-2iqa()K(u,c) + (a/2)ou(a/2),-e (9)

- - 9 -- _ - -

a/2 du_ _

where K(u,o) = a a+L + 2) + P(x l dxf dx 2ndx) 2/

-a/2

Proof

Taking variations with respect to u and a and after carrying

out the necessary integrations we obtain

a/2 a/2

6K(u,o) o = f (doP (x)W 2 u)6u dx + f (--y - du)6ad dxf dx f n (x) dx

-a/2 -a/2

+ o(-a/2)6u(-a/2) - a(a/2)6u(a/2)e- 2 iq a

+ < c(b/2)6u(b/2) > + < o(-b/2)6u(-b/2) >

If u and a are restricted to the subclass of functions which satisfy

equations (3), (4) and (5), this expression reduces to

a/2 a/2(j q + Px)w2u6u dx + u(-T--- ,ii)6a dx + < a(b/2) > 6u(b/2)

-a/2 -a/2

+ < a(-b/2) > 6u(-b/2).

This expression vanishes for arbitrary 6u and 6a if and only if

u and a satisfy equations (1), (2) and (6).

4. A general variational principle

The variational equation

6[K(u,a)+Lb+L-b] + (a/2){16u(a/2)-u(-a/2)e

+ {u(a/2)e-iqau(-a/2)}6a(a/2) =0 (]0)

- 10 -

where

1S= -12--o(b-/2)+o(b+/2)] < u(b/2) >

L b = - -a(-b-/2)+o(-b+/2)] < u(-b/2) >

and K(u,a) is defined in equation (9) is completely equivalent to the

original boundary-value problem (1)-(6).

Proof.

Taking variations with respect to u and a and after carrying

out the necessary integrations, we obtain

a/2 a/2

)6 do: 2+ - duf (+p(x)u)6udx+ f (x) )6odx

-a/2 -a/2

+ {u(a/2)e-iqau(-a/2 ))6o(a/2 ) -{a(a/2)e qa(-a/2)6u(-a/2)

1 < u(b/2) > {6a(b+/2)+6o(b-/2)} - < u(-b/2) > [6a(-b /2)+6(-b/2)121 1

+ " < -(b/2) > {6u(b+/2) + 6u(b-/2)}+ 1- < o(-b/2) > {6u(-b+/2)+6u(-b-/2)}

This expression vanishes for arbitrary 6u and 6a if and only if

u and a satisfy the complete set of equations (1)-(6).

A NUMICAL EXAMPLE

1. The problem an-' method of solution.

As an illustration, ye consider an application of the extended

version of the Hellinger-Reissner principle (9). This principle offers

greater flexibility than the strain energy or complementary energy

principles and Neat-Nasser (] used its elastic analog to obtain very

accurate results for wave propagation in elastic composites. The gen-

eral variational principle (10) is not us -l since, in the one-dimen-

sional case, it is easy to choose test functions for u and o which

are continuous and satisfy the quasi-periodic boundary conditions (3)

and (4). The two-dimensional version of (10) should prove very useful

for study of wave propagation in fiber reinforced viscoelastic composites.

The exact dispersion relation for this problem was obtained by Sun,

Achenbach and Herrmann [9]

cosqa = Cos( Cl )cos( 2 ) - +2 sin(q a-- )sin() 11)

C c 1s ) cs~ +2 2p wC-1 Ca)2

where c2 = n=/P 2 /P and p 2 P n /P n

1 1 1 2 n2 2 2fl2 P1 1

The exact equation for the mode shape is given in [2].

We note that since we are studying free waves in an infinite

viscoelastic composite nI is a complex function of the complex

frequency w. The wave number q is real.

We obtain approximate solutions for various numbers of terms

in series expansions for u and a and compare these with the exact

solution. This tests the efficiency of the variational principle (9).

-1ii-

-12-

We define dimensionless quantities

X = x/a , U(x) = u(x)/a , S(x) = 2

Q = qa , t(US) = K(uq)/n 2 a.

= 2 2 ,e, n 'Q= a P I/,W A 2 e P2 /P 1 n 2/, Al

a = b/a

In ters of these, equation (9) becomes

6K(U,S) + S(I/2)SU(1/2)(1-e- 2 iQ) = 0 (12)

where

1/2 2 2

R(us) = (-s d, 2+ S (x) )dx

-1/2

We expand U and S in series

n n

U = C cU(x) = 5 ei(Q+2wj)X

J-n -n

n n

S Z k~(X) I ~D e(Q+21r)X (13)

k=-r -n

It is clear that U and S satisfy (3), (4), (5) and (6).

Substituting (13) into (12) and after performing the necessary inte-

grations, we obtain

-13-

n

k(u,s) -Ci'(cI+ D1 CI"U)L.. ~ k jk k jk 3 k 3k

n

S(1/2)6U(1/2)(1-e-iQ) = 6 J 5CJ (14i)

J,k=-

where

1/2

i uf PX)1 2 J(X)Uk(X)dX-1/2

QU 1L-) sinQ+(e-1)sin{a(Q+ir(j+k))LIQ#-~jk2 ( Q+,(3+k))

-2if Q =.rjk

12

s3 f/ 2~ s MXsk(X)dX-1/2

3 +k-sif{a(Q+1(+k))±FL(1) sinQ-sinci(Q+ir(j+k))}] if Q ~ -i(J+Ik)

2(Q+yr(j+k))

=~~i Q~Ci-x -ir(j+k)2

1/2

Ijk f dX Skx)dx-1/2

(-)1+ki (Q+2j)sinQ

Q+(J+k) if Q -w(j+k)

= i(Q+2wj) if Q = -w(j+k)

su = -2iQ i(Q+w(j+k))*(1-e- )e

Taking variations and equating the coefficients of 6C and 6Dk

to zero, we obtain the matrix equation

21 u I SU -1Isu C 0

su T 21sD

Thematrices , , and I are each of order m x m

(where m 2n+l). The elements of Iu and Is are functions of the

complex frequency w through Q and n. Thus, for any given wave number

Q, the frequency w can be obtainad by solving the equation

det[N(w,q)) = 0 (16)

where N is a 2m x 2m matrix

~1

2Iu Jsu_ su

su.T 21

Once an eigenvalue w is known, the correpsonding eigenvector and

hence the mode shapes U and S can be easily calculated.

-15-

In order to proceed further, we must assume a functional form for C(W).

We assume, as in[8] that the matrix is a standard linear solid and the

filament is elastic, so that

R GT i + s "i GT11)- l 1+, n 2(s) 2 (17)

where s = W, n11 and n l are the "rubbery" (at long time) and "glassy"

(at short time) moduli respectively and T is the relaxation time of the

matrix material. The filament i3 elastic so that n2 is independent of w

and is real.

The standard linear solid is used as an example. The method applies

to any linear viscoelastic material as long as l(w) and n2( ) are known.

We define some further dimensionless quantities

G A G 6 R G

(18)

G , s' sa/CG

= -i sh_'nG

is a reference phase velocity.G )

1 11Gwhere C1 = ,i eeec hs eoiy

In terms of these

n1 (l+s 'tIC(S) =t(19)

= G

Q()= -(s')2/

-16 -

We now rewrite the matrix N(s) in order to show the frequency

dependence explicitly. Thus, equation (16) becomes

,2 B

det ' (20)(+s't )- I GF - zst

L (61 + 1~

where , B, E, F, G are (m x m) matrices given by

[(-1)JksinQ+(0-1)sin{a(Q+i(J+k))} ] if Q $ -r(J+k)

= - (l+a(O-1)) if Q = -w(j+k)

G

BU j U

jk Jk jk

E I= sjk k

G

F Ti [(-l1)+k sinQ-sinlc(Q+w(J+k))}] if Q -(j+k)jk Q+7t(j+k)

= G(-) if Q = -r(J+k)

G sin(q(Q+ir(j+k))} if Qsjk Q+7r(j+k)

= if Q = -n(j+k)

The elements of A, B, E, F and G are known once the wave number

and the other parameters of the problem are specified. Equation (20) re-

presents a nonlinear eignevalue problem since many of the elements of N

-17-

are nonlinear functions of s. It is possible to reduce this to a standard

eigenvalue problem so that s' is an eigenvalue of a new matrix M of order

3m x 3m. This increases the size of the problem but we can now use any

of the standard computer algorithms for determination of eigenvalues of a

matrix.

We observe that

-- M

1 -1det - -= det( M [ I M2-MI IM 1I! 2]

221121 1-

M21 -22

L

if the matrices are conformable and ut exists, so that equation (20)

can be expanded as

det[I s'3+P s,2+H s'+YH = 0 (21)where

S= tlR- A-I-R :t(G+F),

and I is the unit matrix.

Now s' is an eigenvalue of a 3m x 3m matriy M defined as

M 0 0 ' (22)

- e -!, H.

-18-

This can le easily proved by expanding the equation

det[1-I s'] = 0 . (23)

2. Results and conclusions.

Numerical calculations are carried out for the following values of the

parameters [8]

n = 4 and 50 O = 3

61 .70 t = .0455 a =.5

m 7 and 11

The wave number Q is specified and only the range 0 < Q < w needs

to be studied (10] because the Floquet form of the solution is not uniquely

determined.

The dispersion curves are obtained by calculating the eigenvalues of

M (equation (22)) using an IBM 360/67 computer. A standard QR algorithm

is used. Once the frequency is known for a certain Q we calculate the

corresponding eigenvector of the matrix N (equations 15, 20). Now the

mode shape U and S are obtained from equation (13).

Figures 2, 3, 4 and 5 show the real and imaginary parts of the

frequency w as a function of the wave number Q for the first 5 modes.

The! approximate solutions for m = 7 and 11 are compared with the exact

solution obtained from (11). The agreement is excellent with m = 11

for the first two modes but the approximate solutions become progressively

inaccurate for higher modes. The maximum error in the real part of the

frequency for m = 11 is of the order of 10% and occurs for the fifth

-39 -

mode for Q = 1. The error is much higher for the imaginary part of the

frequency for the fifth mode. We recall that determines the rate

of exponential decay in time.

A comparison with [8] shows that in this case the finite difference

scheme is more efficient. Also, a comparison with [4] shows that the

Hellinger-Reissner principle does not work as efficiently as it does in

the elastic case. The fundamental mode, however, is obtained very accur-

ately with m 7 and this mode dominates since the higher modes decay

faster.

Mode shapes are shown in Figs. 6 and 7. The real parts of the dis-

placement and stress are shown for tha fundamental mode for Q = Tr/2.

The normalzing constant is chosen as in [8] so that U(o) = 1 + 0i. The

results for m = 11 are in exceilent agreement with the exact solution

even though the stress and displacement are expanded in series of smooth

functions (13) while the solutions contain sharp discontinuities in slope

at the interfaces. The stress solution is more accurate than in [8] because

here it is obtained independently rather than being obtained from U by

numerical differentiation.

The Hellinger-Reissner principle enables us to obtain continuity of

both stress and displacement across the interface while if the strain

energy principle is used together with a smooth displacement field a

stress discontinuity results [1].

To ,um up, this paper presents some varational principles which can

be used to study wave propagation in laminated viscoelastic composites. A

numerical example gives encouraging results. These principles can be

easily extended to higher dimensions to study wave propagation in fiber

reinforced viscoelastic composites.

ACKNOWLEDGMENT

This research was supported by the Office of Naval Research under

Contract N0001475-C-0698 with Stanford University, Stanford, California.

- 20 -

-- Par--

REFERENCES

1. 7. Kohn, J.A. Krumhansl and E.H. Lee, Variational methods for dis-persion relations and elastic properties of composite materias.J. App1. Mech. 39, 327 (1972).

2. E.H. Lee, A survey of variational methods for elastic wave propagationanalysis in composites with periodic structures, Dynamics of CompositeMaterials, ASME 122 (1972).

3. S. Nemat Nasser, General variational methods for waves in elasticcomposites, Jn. Elasticity 2, 73 (1972).

4. S. Nemat Nasser, Harmonic waves in layered composites. J. Appl.Mech. 39, 850 (1972).

5. S. Nemat Nasser and F.C.L. Fu, Harmonic waves in layered composites:Bounds on frequencies. J. Appl. Mech. 41, 288 (1974).

6. W.H. Yang and E.H. Lee, Modal analysis of Floquet waves in compositematerial, Jn. Appl. Mech. 41, 429 (1974).

7. G.H. Golub L. Jenning and W.H. Yang, Waves in periodically structuredmedium. To appear.

8. S. Mukherjee and E.H. Lee, Dispersion relations and mode shapes forwaves in laminated viscoelastic composites by finite differencemethods. Jn. Computers and Structures, 5, 279 (1975).

9. C.T. Sun, J.D. Achenbach and G. Herrmann, Continuum theory for alaminated medium, Jn. Appl. Mech. 35, 467 (1968).

10. E.H. Lee and W. Yang, On waves in composite materials with periodicstructure, SIAM Jn. Appl. Math. 25, 492 (1973).

-21 -

MATRIX FILAMENT MATRIX

-a/2 -/ / /

I' bF a

Figure 1. Composite Cell.

15

- .-.

5

I~zZZZIIIZ..__...................___0 1 2 3 7

EXACT SOLUTION---------------1 PLANE WAVES

------- 7 PLANE WAVES

Figure~ 2. Dispersion Curves w R versusQ

G 3 =. 1 05n 4 = 3 ci . 6, .,T t, 045

-23--

I

i

- I/i

L 54I

I/

4-I ii '

AR SOLUTIN.6 ---- -*--~ - * - . . i "'"5 ..

4.. . -/.. . .. . . . . .----. 4 I

Fg 3. D*-.-erio C w R ve3-us

2 2

o I 2 37"

EXACT SOLUTION

------ II PLANE WAVES

7 PLANE WAVES

Figure 3. Dispersion Curves WR versus Q

n = 50 e = 3 = .5 61 .7 t

0.

EXACT SOLUTION

11 PLANE WAVES

7 PLANE WAVES

igure . D mPing curv s i 'versus .045

n G3 .5

1.0 /

0.8-

0.4

0.22

0 I2 3

-EXACT SOLUTION---------- 11 PLANE WAVES

7 PLANE WVES

Figure 5. Damping Curves wversusQ

G= 50 e 3 a= .5 61 = T7 tj .05

| I

MATRIX I FILAMENT I MATRIX1. -I I

~DISPLACEMENT

0.8-

S0.6I

II0.4

0.0.-II

II

II

-0.2-

EXACT SOLUTIO 9

-------- 11 PLANE WAVES

7 PLANE WAVES

Figure 6. Displacement and Stress Distribution for the First Mode0 G

n r04 e=3 2 .5 Q =n/2

I i

MATRIX FILAMENT , MATRIX

DISPLZEMEN'r"T

0.8-

0.6-

I IN

04-

0.2- STRESS

0-0.50 -0.5 0 0.25I\ I

POSITION-0.2-I I I

I I-0.4

-0.6 I

EXACT SOLUTION

II PLANE WAVES

7 PLANE WAVES

Figure 7. Displacement and Stress Distribution for the First Mode

G = 50 e - 3 a = .5 Q = /2