dispersive plane wave propagation in periodically layered anisotropic media

Dispersive Plane Wave Propagation in Periodically Layered Anisotropic MediaAuthor(s): Andrew NorrisSource: Proceedings of the Royal Irish Academy. Section A: Mathematical and PhysicalSciences, Vol. 92A, No. 1 (Apr., 1992), pp. 49-67Published by: Royal Irish AcademyStable URL: http://www.jstor.org/stable/20489401 .

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DISPERSIVE PLANE WAVE PROPAGATION IN PERIODICALLY LAYERED ANISOTROPIC MEDIA

By ANDREW NORRIS

Rutgers University, Piscataway, New Jersey

(Communicated by M.A. Hayes, M.R.l.A.)

[Received 10 July 1990. Read 27 March 1991. Published 24 April 1992.]

ABSTRACT

Plane wave propagation through a periodic layering of anisotropic elastic materials is considered in the low frequency or long wavelength limit. To lowest order the laminate acts as an effectively homogeneous elastic medium with density equal to the average and elastic moduli which can be derived from a purely static analysis. Higher order dispersive effects are obtained by expanding the frequency for the fundamental Bloch waves as a regular asymptotic series in the dimension less wavenumber. The first set of equations in the frequency expansion yields the

phase speed for the homogenised medium, and the next order implies that the term of second degree in wavenumber is identically zero. The first dispersive effects are of third order in wavenumber and can be expressed in terms of averages of material properties over the periodic cell. The theory is valid for plane waves propagating in any direction through arbitrarily inhomogeneous anisotropic periodic media. Explicit, simple expressions are given for the case of SH waves propagating through isotropic elastic layers.

1. Introduction

The low frequency, or long wavelength, propagation of waves through a periodically layered medium is governed, to a first approximation, by the static effective moduli of the medium. At this level of approximation the effective

medium is a homogeneous and non-dispersive anisotropic solid. At the same time, wave propagation through periodically layered elastic materials can be described precisely in terms of the Bloch waves of the system. These are defined by the roots of the Floquet dispersion relation over a unit period of the medium. Although the exact finite frequency character of plane waves is contained in this dispersion equation, its solution is not necessary in order to find the first correction to the homogenised medium. It turns out to be sufficient to consider only the fundamental Bloch waves, which are defined as the modes that have

dispersion curves at zero frequency. In general, for an elastic medium there can be at most three such Bloch modes, corresponding to the three types of elastic wave propagation in the uniform effective medium.

Plane wave propagation through periodically layered anisotropic solids is considered in this paper. The low frequency expansion of the dispersion relation of the fundamental Bloch waves is obtained using a regular perturbation proce dure for frequency as a function of dimensionless wavenumber. The theory is developed for arbitrary periodic layering with no specific material symmetries

Proc. R. Ir. Acad. Vol. 92A, No. 1, 49-67 (1992)



50 Proceedings of the Royal Irish Academy

assumed for the constituent layers. The results show how the static effective medium drops out as the first approximation, and a general expression is obtained

for the first dispersive correction. The simplest case of all is for shear horizontally (SH) polarised waves propagating through layers of isotropic solids, and the explicit formulae obtained shed some light on the more general results for lower

symmetries. There is quite an extensive literature on the propagation of time harmonic

waves in layered and fibre-reinforced media. Many of these studies have assumed either that the constituent layers are isotropic or that the waves propagate normal

to the layers. However, the exact form of the dispersion equation for plane waves

in any direction in a periodically layered anisotropic material can be expressed in terms of the 6 x 6 propagator matrices of the unit period. Details of this

procedure can be found in recent papers by Braga and Herrmann [2] and Ting

and Chadwick [10]. The dispersion equation for wave propagation normal to the layering of a bilaminated medium can be reduced to the determinant of a 3 x 3

matrix [9; 11]. The purpose of this paper is not to reproduce the work of these

and other papers but rather to extract some useful results for long wavelength signals, and thus to help our understanding of the general problem of waves propagating in arbitrary directions in multiphase composites.

The well-known sextic formalism [3] will be used to a great extent and it is

reviewed in section 3. The main asymptotic results are derived in section 4 and the static effective moduli are discussed in section 5. Section 6 contains some simplifications to the matrices and expressions obtained in section 4, and exam ples are presented in section 7. We begin with some definitions and matrix identities.

2. Notation and preliminaries

A good deal of the later analysis will be concerned with identities involving d x d matrices where d is the dimension of the problem under consideration. For

instance, SH waves through layered isotropic composites reduces to a d = 2

problem. In-plane longitudinal and transverse motion through similar media requires d = 4. The most general case of plane wave propagation through fully

anisotropic composites with no overall symmetry requires d = 6. In general, d is

even and equal to 2, 4 or 6.

Define the determinant of a square matrix A as JAI, and the cofactor matrix associated with A as A. These are related by the identities

Al = d tr{AA} , A1 = IAK'A (1)

where the latter only makes sense if the inverse to A exists. The cofactor matrix

can be expressed explicitly as

_ 1

Aim = (d-1)! eqi .. klemn. .pqAjn .. kp AkAq (2)

where ei. kl is the permutation tensor of order d on the integers 1, 2, . , d,



NoRRis - Waves in layered media 51

e.g. e12 d =1, etc., and the summation convention on repeated suffices is understood in (2) and subsequent equations. Let A be a differentiable function of some scalar variable and let ' denote the derivative of a quantity with respect to this variable. We shall be interested in the first and second derivatives of JAI, and in expressing these derivatives in terms of derivatives of A. The first two derivatives are

Al' = tr{ trAA"} + tr{[A]'A'} , (3)

where [A]' is the derivative of the cofactor. It will be shown later that it is not actually necessary to compute [Aj' for the problem at hand.

Define the d x d matrix K as

K =4 j, K =1. (4)

A d x d matrix A is defined to be of type (i) if

KAK =AT (5)

and of type (ii) if

T KAK=-AT. (6)

It is easy to see that these definitions are equivalent to saying that A is of type (i)

if and only if it is of the form

[AlA21 A= , A2=A[,A3=A(7)

L3 A[

and A is of type (ii) if and only if

_FA1 A21

A=[ AA] A=-A,3-3 (8) A- I I~ A= -A 2A3 A3

LA3 --A[Jl '3 A 8

If A is of type (i) and B is of type (ii) then

tr{AB} = 0. (9)

The proof follows from repeated use of the identity K2 = I. Thus,

tr{AB} = tr{KAKKBK} (10)

= -tr{AB}

It is shown in Appendix A that if A is of type (i) or (ii), then the cofactor A is of

the same type. Combining these two results implies that if A is of type (i) and B is




of type (ii) then

tr{AB}=O. (11)

3. Equations of motion

The tensor of elastic moduli is C with components Cijkl relative to an

orthonormal basis (e1, e2, e3) where e3 is in the direction of layering. The usual

symmetries are assumed to apply,

Cijkl =C,iki Cklij ' (12)

where i, j, k, 1 assume the values 1, 2 and 3. Define the tensors Q(6, 4) and

R(6, 4) of order a as [3]

Qik(ol 4) = Cijklnjnl, Rik(6, Cijk,minl , (13)

where

n(6, 4)cos 6 cos 4, e1 + cos 6 sin e2+ sin e3,

(14) m(O, 4) = sin 0 cos 4, el + sin 6 sin 4, e2 - cos 6 e3.

The matrices associated with Q and R reduce to x 2 matrices, i.e. Q and R

become scalars for the simplest SH problem. We note the representation formulae

Q(6, 4) - cos2 OQ(O, 4) - sin 0 cos 6[R(0,4) + RT(O, 4)] + sin2 OQ(e3), (15)

R(6, 4) cos2 OR(O, 4) + sin 0 cos 6[Q(O, 4) - Q(e3)]- sin2 ORT(O, 4),

where Q(e3) Q( 2, 4) is independent of 4, and also

R( + 2 4 ) = -R"(0, 4) (16)

The tensor Q(6, 4) is symmetric and positive definite for all 0 and 4,

Q,j(0, 4)eej > O, for all e # 0. (17)

Define the tensor N(4,) of order d as

-N1(j) N2(4)

N(4) - N3(O) N[ (4) (18)

where

N1(4) = Q 1(e3)R(O, 4),



NORRIS - Waves in layered media 53

N2(0) = N () = Q(e3) (19)

N3(0) = N3 (0) = RT(0, 4)Q1(e3)R(O, 4) - Q(O, 4).

tIhe tensor N(4) is known as the fundamental elasticity tensor [3] and its d x d

matrix is obviously of type (i). This property will be used repeatedly below. Let u(x, t) be the displacement vector at any point and oij(x, t) be the

components of the stress tensor. We assume a plane wave solution of the form

ui(x, t) = (ik)lVi(X3)e[k1(COS #x1+sin rx2)-wtj

(20) cr13(X, t) = Ti(X3)ei[k1(cos

' xi?sin cx2)-wtl

where w > 0 is the frequency and k1 the horizontal wavenumber. By assumption, the composite medium is layered in the X3 direction with period h. The frequency and wavenumber are not independent but are related through the Floquet condition that the displacement and traction satisfy the periodicity condition

V(X3 + h) =V(x3) e ik3h (21)

where V(X3) is the d-vector

V(x3) = V(X3)) (22)

Equation (21) is the usual relation required of a Bloch wave in a periodic

medium. Define the phase angle 6 and the absolute wavenumber k by

k =kcosO, k3 =ksinO. (23)

The two angles 0 and 4 together define the phase direction n(0, 4). The formulation of the problem is completed by noting that the equations of

motion may be written succinctly as

dx (X3) = ikPV(x3), (24)

where P is a type (i) tensor of order d,

[cos ON1(4' N2(4') ] 2

P cos2ON3(4) cos ONT(4)

+ kp

L (25)

p is the mass density per unit volume, and

L[= fl (26)

The dispersion equations for the infinite set of Bloch waves then follow by finding




Ct = -(k) such that (21), (22) and (24) admit non-trivial solutions for given 0 andl

4. The asymptotic expansion

The propagation equation (24) can be formally solved in a Neumann or Peano series [6] which is guaranteed to converge for any distance of propagation.

However, the rate of convergence becomes painfully slow for any appreciable distance since the correct solution is of exponential form. For the present problem the Neumann expansion is perfectly adequate on account of the low frequency. In particular, the expansion of the solution for asymptotically small values of kx3 is

V(x3)= [I+ (ik) P+ (ik)2 PfPP+ (ik) PJ P{P?+ V(O), (27)

where

PJ P(yl)3dy, P fP tP(yl))dy P(Y2)dY2, (28)

f pfpfP-X P(yl)dylf P(y2)dy2f P(y3)dy3, . (29)

Without any loss in generality, the problem simplifies by taking h = 1, so that the Floquet condition (21) can be written as

det [I + (ik) P + (ik)2 P P + (ik)3 P JPJ P(+30

-I [1 + (ik) sin o + 2 (ik)2 sin2 O +6 (ik)1 sino3 0 + *J=o

The upper limit of 1 on the integrals indicates that they are taken over the unit

cell.

4.1. The first three terms Our objective is to find solutions to the dispersion equation (30) for asymp

totically small values of the dimensionless wavenumber k, i.e. k < 1. We assume

the ansatz 2

where fl is the phase speed of the fundamental Bloch wave of the system. The subsequent terms, f22 U3, etc., describe the dispersion of the fundamental Bloch

wave at low frequencies. Each of I?,, Q12, A3, ... , is independent of k but

depends upon 6, 4 and the material composition. We will content ourselves with finding the first three terms indicated. It will emerge shortly that n2 is always zero, and therefore 12l and 123 are sufficient to define the phase speed and the first




dispersive effects for each possible fundamental Bloch wave. This level of approximation is also sufficient to determine the long-time response of an impulsive, planar, source [9], or to describe the evolution for all time of a sufficiently broad initial pulse with a plane wavefront [7].

We could have assumed initially that D2 = 0 since it can be shown using the

conservation of total energy for the assumed lossless elasticity equations that w)2 iS an even function of k [1]. However, it is interesting and also serves as a check to

find that the same result falls out of the asymptotic analysis. Substitution of (31) into the Floquet condition (30) yields

IS(ik)l = 0, (32)

where

S(ik) = SO + ikS1 + (ik)2S2 + *, (33)

and each of SO, SI,... , is independent of k. The first term is

S=o PO(y)dy-sin d, (34)

where

PO lim P

cos ON1QO) N2(Q)

L cos2 0N3(k) cos ON[(;) +pQlL. (35)

The next two terms are

S fI P fPOr- 4sin2 ol+ m2L p, (36)

S2= PI Pf PJ -6 sin 3 o1+f2[f PoJpL?f pLJPO]+Qt3Lf p. 6 ~~~~~~~~~~~~~(37)

We may simplify Si by noting that for any square matrix A(x3),

JA JA+ (AT JAT) = (JA) (38)

and in particular

J ?PJO2[ Po] +T, (39)




where

T=4 [ Pf PO- ( PO PO)] (40)

It can be directly verified that T is a matrix of type (ii). Eliminating the integral of

PO in favour of SO using (34) gives

S1 = 2 + sin0 So+T+ n2L p. (41)

We note that

SoSI = SoM [4 SO + sin oi] + SOT + 22SOL p, (42)

and therefore since SO is of type (i) and T is of type (ii) it follows from (11) that

tr{SoSt} = |SoI[sin Gd + 2 tr{So}] + 2tr{SL} fO p (43)

The Floquet condition now becomes, using (3), (33) and (43),

jSOj + ik{ so [sin Od + 4 tr{so}] + Q2tr{S0L} f p

+ (ik)2[tr{g0S2} + 4 tr{ISs}] ? - 0, (44)

where Sn is the derivative with respect to ik of the cofactor of S(ik) at k = 0. We will discuss this term in detail shortly. The dispersion relation (44) must hold for all k and therefore each of the coefficients in the different powers of k must

vanish separately. The first order term implies that ?2, satisfies

jso =o. (45)

It will be demonstrated in the next section that this is precisely the secular

equation for the phase speed f2l of plane waves propagating in the direction

n(G, q) in the equivalent static effective medium. The next order term in (44) gives

Q2tr{S0L} = 0 (46)

An explicit expression for SO will be derived later where it will be shown that in

general tr{S0L} #0 O. We therefore conclude that, as expected [1,

2 = 0. (47)

The 0(k2) term in (44) simplifies by virtue of (47) to give, using (37),



NORRIS Waves in layered media 57

93 [tr{ 0L)f p] sin otr{g0}-tr{ 0f POJPOJPoJ-A] (48)

where

A 2tr{SS1}. (49)

The efficient evaluation of A requires some matrix identities, discussed next.

4.2. Some matrix identities According to the definition of the cofactor in (2) we have

1 A = Si.S,/qeij klemn . O.. p0sojn

. SOkp (50) 2(d - 2)!iiSie..keff..pq

Substitute the expression for SI from (41), noting that (22 0 O. There are several

terms in the resulting form for A, corresponding to the different combinations of the three terms in SI. The terms in (50) that arise from combinations of So only

are each zero since they are coefficients in the expansion of the determinant jSO + zS2I as a function of z. This determinant is identically zero for all z, and

therefore each term in its expansion vanishes. There are also terms in (50) that arise from cross-products of SO and S2 with T, but these can also be shown to

vanish because of the property that So is of type (i) and T is of type (ii). The

proof is very similar to that presented in Appendix A and will not be given here.

We are finally left with

A 2(d 2)! Ti mT eij ... klemn . pq. Ojn.p (51)

Consider the expansion of the determinant of the sum A = B + zD as a

function of z, where B and D are arbitrary constant matrices of order d.

B + zDj = |BI + z tr{BD} + j tr{A6D} + + 2 tr{BD} + zflDI .

Now let B = So and D = T, then the first term vanishes on account of (45), while

the second and all terms in odd powers of z vanish because SO is of type (i) and T

is of type (ii). The proof follows the same line of reasoning given in Appendix A. The third term is related to A, and so we have that for any z

ISu + zTj - z2A + (terms of even order in z) + - . + zd TI. (52)

Since d is either 2, 4 or 6, this identity can be used to express A in terms of

determinants. If d = 2, then the first and ultimate terms on the RHS of (52) are the same,

implying

A = ITI, (d = 2). (53)




For d = 4, (52) becomes

1SO + zTl = z2A + z4lTJ (d = 4). (54)

Since this is true for all z, we can thus express A in terms of two determinants of 4 x 4 matrices, without computing any cofactor matrices separately. If, for instance, we choose z = 1, then

A=1S +TI-ITI, (d=4). (55)

Finally, when d = 6, (52) becomes

IS, + zTl=Z2A + Z4X+ z6lTj, (d == 6) (56)

where x is some constant. This can be eliminated by evaluating both sides for two

values of z, z1 andz 2, to give

A= z2z2 1 [ISO+ZiTl_ JSO +?z2Tl]+ ITI4 (d=6). (57)

For instance, taking z1 = 1/2 and Z2 = I gives

16 1 1 1 A= 3 S+ -

T--3 ISO+TI + - ITI, (d=6). (58) 3 023 44

This type of algorithm is to be preferred for purposes of numerical implementa tion since it only requires calling a function that calculates determinants.

It is also possible to simplify the terms in (48) that are of the form tr{SOA}, where A is either I, L or J P0 5 P0 5 P0. For arbitrary A define the function f(z)

f(z) = ISO + zAl; (59)

then it is clear that f is a d-nominal in z with the zo coefficient equal to zero. The

coefficient of Z1 is tr{S0A} and the coefficient of Zd is JAj. The other terms may be eliminated by choosing several values of z. If d = 2, we have simply

f(z) = z tr{S0A} + z2JAl, (d = 2), (60)

and therefore selecting z 1 yields

tr{SOA} = ISO + Al - A, (d = 2). (61)

Things are a bit more complicated for d = 4, since f(z) contains two terms that are

not of interest, i.e. the terms in Z and ZA. These can be removed by taking three

values of z, zl, Z2 and Z3, to give




tr{SOA}=

JZ 2 Z1 (z33)f(zZ) + Z2l(Z3-z1)f(z2) ?2(zi - Z2)f(z1) + Al

Z2z3 ; 2 (Zl Z2)(Z2 -

Z3)(Z3 - Z1)

(d = 4). (62)

For instance, the choice (z1 Z2 z3) = (1 2, 3 ) gives

tr{SOA} =ISO+AI-9 So+ 2

A +9 So+ 4 A |IAI|, (d =4). (63)

This provides a consistent method for evaluating all the terms of the form tr{SOA} in the expression for f23 using only the determinant function. It is not necessary to find SO explicitly. The benefits of this method are marginal for the most general case of d = 6 because the formulae analogous to (62) and (63) are quite a bit

more complicated and it may be just as easy to compute the cofactor matrix

explicitly, which is done in section 6.

5. The static effective medium

Equation (45) determines the phase speed D1 for given values of c, 6 and 4. It

may be shown that the same equation arises for the equivalent static effective medium, which is homogeneous with effective moduli C kl and effective density p*. These follow by setting Po =P*, where P* is defined as the equivalent

tensor PO for the effective parameters. Since the equality holds for any value of 0, we have p* f= p, so that the effective inertia is simply the average. Let Q*(Q, 4) and R*(0, 4) be the appropriate tensors for the effective moduli. Then, for a given azimuthal angle 4), it follows from the definition of P* that

Q*(e3)= [f Q(e3)] (64)

R*(, ) Q*(e3)f Ql(e3)R(O, 4)), (65)

Q*(O 4) f) Q(O 4) - RT(O, O)Q1'(e3)R(O, 0)

+ R* T(O, 4))Q* (e3)R*(O, 4) . (66)

The equation for Q*(e3) defines the six effective moduli c33, c4*, C35, c:4, c5 and

c4 . If, for instance, 4 - 0, then the equation for R*(O, 0) defines the six

additional moduli c<3, c'4 c(K c;6 c6 and c4 . The Q*(O, 0) equation gives three more, c4 c4 and c4 , and hence a total of 15 effective moduli. The six remaining

moduli can only be determined by selecting a different value of 4. The tensor Q*(e3) is independent of 4, and so it always yields the same six moduli. For

example, if 4 = 2, then the additional moduli not already found which derive from R*(O, are c* c*4, c4 , ct c46 and c* , and Q*(O, ff) yields the final three

cl2, 2 and 466




The explicit form of the 21 effective moduli can be expressed in a more

concise manner. Define the matrices

F C15 C25 C56 Cll C12 C16

W = C14 C24 C46 Z ZTZ f C12 C22 C26 . (67) LC13 C23 C36 _J L C16 C26 C66_

Six of the effective moduli are given by equation (64) for Q*(e3), where

[C55 C45 C35

Q*(e3) = C45 C44 C34 ,

(68) C35 C34 C33j

and the other 15 follow from

W* = Q*(e3) Q1(e3)W, (69)

Z*= Z- WTQ-I(e3)W + W* TQ* (e3)W* (70)

This prescription for the effective moduli is identical to that of Pagano [5] and

Schoenberg and Muir [8]. These derivations are based upon static considerations, viz. in purely static deformation, the three stresses a63, i = 1, 2, 3, are constants, as are the strains E1l, 622 and C12. It is then a matter of algebra to express the

remaining stresses and strains in each layer in terms of these six constants. The effective moduli are then defined by relating the spatially averaged strains to the

averaged stresses. This procedure is formulated in a coordinate-invariant manner

by Norris [4]. The average elastic constants were first derived by Behrens [11 in a manner

similar to that of the present paper. He looked at the dispersion equation for long wavelengths and from it deduced an expression for the effective moduli. Behrens did not, however, pursue the method to derive the first dispersive effects, i.e. the constant 03.

6. The cofactor matrix and some simplifications

From its definition in (34), and using the transformation property (15), the tensor SO can be expressed as

?= cos OR * T(O O ?)Q *-Il (e 3) I -I - sin 0I ]

x Q(,)- Q2 (71) -cos OR*(O, 0)-sin OQ*(e3) I]

where the * superscript indicates that the quantity is evaluated for the static

effective medium. The determinant is therefore



NORRS -Waves in layered media 61

ISOI = IQ*(e3)V IQ*(o, ))- p , (72)

and since the determinant of Q(e3) is positive on account of (17), the condition (45) implies that

IQ*(o, 4) ) p*f2j1 = 0. (73)

This is precisely the equation that Ql must satisfy if it is to be the phase speed of a

wave with phase vector in the direction of n(O, 4). Thus Q2 is the phase speed in

the effective medium. The cofactor SO can now be determined from (71). Define the 2 2matrices F

and Y:

F= T = cof{Q*(O, 4) _ p*f2I}, (74)

Y = sin OQ*(e3) - cos OR*(O, 4)). (75)

Then using the property of the cofactor that cof{AB} = cof{B} cof{A}, it can be shown that

0 =IQ*(e3)V Y] (76)

It is clear that SO is of type (i) as expected. Let vl, v2 and V be the phase speeds

of waves with phase vectors in the direction of n(O, 4), and let the corresponding polarisations, normalised to unity, be pa, a = 1, 2, 3. The spectral decomposition of Q*(O, 4)) is thus

3 Q*(9 4)) = E va pa ? pG (77)

a-1

Suppose that 121 = V; then

3

Q*(0Q 4) _p*12 = p 3 (Va2 - V12)Pa pa (78)

a=2

and the cofactor is obviously

F=p*2(v 12 V22)(u1l2 V32)p ?pl . (79)

Define the traction vectors Ta, a =1, 2, 3, as

Ir= Ci3kpkn,, sta =1, 2, 3, (80)

where n = n(Q, 4). Note that a pa is simply related to the vertical component of

the wave velocity vector ca,




a 1c'paa a 123 C1= a~j qa j

n PiJ 2i3 (81)

Cl p * v ijkiP, Pk I 9 9 v (1

The cofactor may now be written in the form

So-*2 (U V v22)(V V -2) [p10T1 P12P] (82)

IQ*(e3)I L70T 'O

In particular, it is evident that the factor in equation (46) is

i2(V2 _ 22)( 12 _ 32)

tr{SoL} =

_p IQ*(e3)I (83)

which is non-zero as long as the root f21 = v' is distinct. This is generally the case

in anisotropic media, although there may be specific directions for which the roots are degenerate. These correspond to directions in which the slowness sheets intersect. For these directions we can appeal to the fact that there are always directions arbitrarily close for which the roots are distinct. The effective medium can then be slightly perturbed to make the root (2l distinct, and we again deduce that f22-O.

Finally, we note that tr{SO} also follows from (82), allowing us to simplify one of the terms in the general expression (48) for (23:

[tr{SoL} f P] -sin3 0 tr{0} =o sin30 vtc? (84)

7. An example

7.1. SH wave propagation through isotropic layers This is the simplest case, for which the response is independent of the

azimuthal angle 4. The velocity and stress vectors, V(X3) and T(X3), are each

oriented in the e. direction, which is perpendicular to the plane spanned by n(0, 4) and m(0, 4). The vectors can therefore be replaced by their scalar components in the el, direction, implying d = 2, and

N=[O ] P P2 _ j COS2 0 (85)

Let (.) denote the average over the unit cell, e.g. (p) = f p(y)dy. Then the effective medium condition (45) becomes

(p)Q22= (pCOS2 0+(K) sin2 0. (86)

The effective medium is anisotropic with

c*4c=4 C=

_ ()1, c6*

=

(iu),

and

p= (p). The dispersion parameter then follows from (48) and (53),




-(PA)( 3(O)=cos4 6[ A(?)a + b0 + d- e]

+2 sin2 Ocos2 [a+bb2- (tty1K?7 e] (87)

+ sin4O[(p,<1 ?)a+b2-],

where

a=(p)-2(A ) ( )(()f Pffp+(p) fifpf), (88)

b i p _ ( ) - 1 ( 1 > - 1 1 I 1( 8 9 )

d= ILJ + (O fi f J (91)

(14 f +if (p<1( f f A9

+(p)y (pj)( pf pL + If fJ fp) (92)

Santosa and Symes [7] derived a formula for 123 for waves propagating in the

direction normal to the layers. This corresponds to the present case with 0 2 and when this is substituted into (87) we find that the resulting equation is in

agreement with their equation (37).

7.2. A dynamic effective medium

For simplicity, let the SH displacement u(x1, X3, t) be in the x2 direction. The

dispersion equation (31), with fi2 of (86), f2 = 0, and 123 of (87), is then

consistent with the following effective equation of motion for a homogeneous dispersive medium,

a2u adu 4u p at2 = D, x dx + Dijkl dx9 dx8 dXI 3)

where the only non-zero elements D,j, Dijki, are

D11 = (pj ), D33=(-), (94)




D1111 = h2K?) [ObK(A )a+b2 +d-e] (95)

D3333 =h (a+b (96)

D?133 =D331 =h (i) a + bb2 - () 1-) ] (97)

We have included the explicit dependence on the length h of the basic period. 24 Equation (93) gives the same dispersion relation for Cw, correct to order k4, as

the perturbation analysis. It is not, however, unique in that the same low

frequency dispersion relation is obtained from the equation

a2u a4u a2u a4____

p*2+p1*Eq + (Dil +DiEkl) a

at at 2 D axx U +xi axi Xk ax,

(98)

where E is an arbitrary constant matrix with dimensions of (length)2. One could use this non-uniqueness to advantage. For example, E can be chosen so as to

make the effective equation of motion strictly hyperbolic.

7.3. Constant impedance and constant wave speed Define the impedance z and shear wave speed c,

z =-p , c == AIp. (99)

The expression for 03 simplifies somewhat if either z or c is independent of X3. Consider the case of constant impedance. Some of the double and triple

integrals in the formula for 923 can be removed by the use of identities

fl f2 =(l f)(2 f2 fL (100)

J flt f2 f3 = ( 2) f A ff3 ft f1 f3- f t f f3 tl (101)

where fi, f2 and f3 are scalar functions. After a bit of simplification, we find

13(0)= -cos4 0K) [(3)f cf c fc+ (c)f 2 fcf I

(102)

If c - (c ) f f c-i (c)2(




Note that this is identically zero for waves propagating normal to the direction of layering (6 = ff), which is to be expected since the lack of any impedance contrast

between the layers means that no waves are reflected, and hence the signal travels without dispersion. If the wave propagates obliquely the reflection coefficients between layers are not zero but depend upon the angles of incidence and transmission on either side of an interface. Hence we would expect 93 # 0 for 6 # 2, in general. Equation (102) implies, for the medium of constant impedance at least, that if 93(6) is non-zero for one value of 0 then it is non-zero for all

o $ 2. Also, the dispersion goes as cos4 0, and so is greatest for waves travelling parallel to the layers.

Alternatively, if c is constant it turns out that

93(0)=-sin4c (z<Q2(Z)-3- ( fzf ! fz+(zj !zf

(fi ! f)2 ( )Kz) f1z f - Z (Z)2K (103)

Note the similarity of this expression to (102). In this case the dispersion vanishes when the wave travels parallel to the layers, and is proportional to sin4 6 otherwise. The fact that 93(0) =0 is not surprising, since the wave propagates with the same speed in each layer and there is no tendency for the signal to spread

out in the time domain.

8. Conclusion

We have derived the first three terms in the low frequency expansion of the dispersion relation for the fundamental Bloch waves. The general form follows from (31) and (47) as

w9==1ik- k3 + * D (104) 29i

where Q1 is the phase speed of waves propagating in the direction n(6, k) in the equivalent homogeneous static effective medium. The effective moduli are de fined by the average of the fundamental elasticity tensor [3], and the effective density is simply the average. The first dispersive term is of order k3 and depends upon the quantity 93 defined in (48). These results are valid for any periodic layering of anisotropic elastic solids.

We note in closing that this level of approximation to the dispersion relation is suitable for modelling the long-time behaviour of initially localised disturbances

with a plane wavefront [9]. Alternatively, if the initial disturbance is long wavelength relative to the unit period, then an extension of the results of Santosa and Symes [71 indicates that the same dispersion relation, (104), governs the evolution of the planar pulse for all subsequent time. Furthermore, the numerical simulations of Santosa and Symes [7] for SH waves propagating in the vertical direction in a two-phase composite show quite clearly that the dispersion can




significantly alter the pulse shape after propagation through many layers. The same phenomenon would be expected for waves travelling at oblique angles through anisotropic laminates, with the additional effect that the degree of pulse spreading will depend upon the type of wave and the direction of propagation.

Appendix A

We will demonstrate that the cofactor of a matrix of type (i) or (ii) is of the same type. The elements of the tensor K are

Kii 5iq ijd

+ - I. d

j-2

where 5 is the Kronecker delta. It then follows that

e0i klKipKj ... KkrKS ep ... rsed + 1dd+2, d,12, d epq 2 '2 d 21(Al)

epq ... rs

Suppose that A is of type (i), i.e.

Aip =KiiKPP APi,

then we can use this identity to write the cofactor of A in a seemingly complicated manner,

Ai - 1)! eij ... klepq rsKjj Aq j Kqq * Kkk,Ark KrrlKlZA sKss

(d 1!

(d -)! e epq K.rsK

. Kkk KII Kqq. Krr Kss Aq'j'

A .

k'A (d 0 ii .. k pq .. pq is qj ik si

-Kii Kpp, (dl)! eij< kIl'ep q ... r's Aq * j Ar'k As, X

where (Al) has been used to obtain the final relation, which is just

Aip =

Kfi1-KPP,AP i

and therefore A is also of type (i). A similar analysis shows that if A is of type (ii) then A is also.

There is in fact a much simpler way of proving these results. Suppose that a matrix A is of type (i) and that its inverse exists. Then

KA IK = [KAKf ' =[ATK-I

= [A-11T




Since the cofactor of A is proportional to A-1 it follows that the cofactor is also of type (i). This proof apparently breaks down when the determinant of A is zero.

However, every singular matrix is arbitrarily close to a non-singular matrix, and therefore by using a perturbation analysis it is possible to extend the proof to

matrices of zero determinant. We have presented the lengthier proof using permutation tensors since it is easily generalised to provide the proofs for the results stated but not proved in section 4.

ACKNOWLEDGEMENT

This work has benefited greatly from discussions with Fadil Santosa, to whom I am grateful.

References

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dispersive plane wave propagation in periodically layered anisotropic media

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