- Email: [email protected]
Wave propagation condition in linear anisotropic viscoelastic media
Int. J. Engng Sci. Vol. 33, No. 7, pp. 1059-1074, 1995
Pergamon
0020-722504) E0036-I
Copyright t~) 1995 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0020-7225/95 $9.50+ 0.00
W A V E P R O P A G A T I O N C O N D I T I O N IN L I N E A R A N I S O T R O P I C VISCOELASTIC M E D I A G. C A V I G L I A Department of Mathematics, Via Alberti 4, 16132 Genova, Italy
A. M O R R O DIBE, University of Genoa, Via Opera Pia lla, 1-16145 Genova, Italy (Communicated by G. A. MAUGIN) Abstract--Time-harmonic wave propagation in anisotropic, homogeneous linear, viscoelastic solids is investigated. By generalizing a procedure which traces back to Barnett and Lothe, and was envisaged for surface waves, a six-dimensional description of wave motion is shown to result in a complex eigenvalue problem. Next an integral formalism is established which simplifies the evaluation of the eigenvalues. This in turn allows the determination of fundamental solutions. Wave propagation in a transversely isotropic solid is examined in detail.
1. I N T R O D U C T I O N Wave propagation in linear, homogeneous, anisotropic solids is a subject of interest in many respects. Owing to linearity and Fourier analysis great attention is addressed to time-harmonic plane waves. In this regard, expository accounts of the subject are given in books (e.g. [1, 2]) and review articles (e.g. [3-5]). Unlike the theory for isotropic solids, difficulties occur even in connection with propagation modes in that the determinantal condition is a sextic equation that in general symmetries is not amenable to explicit solutions, which means that only numerical estimates are possible [6]. Hence, any operative approach has to circumvent the sextic in the determination of the propagation modes. In the last decade, Barnett, Chadwick, Lothe and co-workers (cf. [7, 8] and [9, 10] and references therein) developed a procedure which avoids the application of the sextic in connection with the study of surface waves in anisotropic elastic solids. As we show, the same scheme can be applied successfully to the determination of the propagation modes. This paper faces new aspects of wave propagation in anisotropic solids. First, the dissipative properties of the body are incorporated. Dissipation is a basic feature in several applications as, for example, in seismology. Here the solid is modelled as linearly viscoelastic with an arbitrary relaxation function. Secondly, solutions are sought in analytical form. Specifically, a class of fundamental solutions are determined which are a basis for the family of waves with fixed components of the wave vector in a given direction. Fundamental solutions are important in many respects such as the description of wave phenomena at interfaces. Thirdly, no use is made of the slowness surface. Though a slowness surface might still be defined, no practical consequence is likely to be derived because of the complex-valued nature of the wave vector and the elasticities. In essence, the paper models time-harmonic wave propagation through a six-dimensional eigenvalue problem. We fix a direction, say a unit vector n, in the plane spanned by the real and imaginary parts of the wave vector, and regard as unknowns the components of the displacement and those of the traction at planes orthogonal to n. Accordingly the wave motion is described by a first order system of ordinary differential equations, which ultimately results in a six-dimensional complex eigenvalue problem, where the eigenvalue is the component of the wave vector along n, while the component orthogonal to n plays the role of a parameter. The corresponding eigenvectors are associated with fundamental solutions. A suitable 1-parameter 1059
1060
G. CAVIGLIA and A. MORRO
problem and an "integral formalism" provide the way to determine the eigenvalues and eigenvectors without solving the sextic determinantal condition. Specifically, a method for the determination of fundamental solutions is envisaged where the original sixth-order eigenvalue problem reduces to two lower-order problems. At first sight the complex nature of wave propagation results in a rather involved approach. While it is so in many respects, it is remarkable that some peculiar features of the real case disappear such as the classification in subsonic-transonic intervals [5]. In addition, when applied to elastic solids, the approach provides a unified treatment of homogeneous and inhomogeneous waves, This in turn allows a systematic investigation of reflection and transmission at an interface in full analogy with the case of isotropic solids. As an application of the procedure, wave modes in transversely isotropic viscoelastic solids are determined by letting n be in the direction of the symmetry axis. Following the integral formulation the sixth-order eigenvalue problem splits into two third-order problems. In addition the third-order determinantal equation factorizes, thus making the search for eigenvalues and eigenvectors simple also from the computational viewpoint.
2. P R O P A G A T I O N
CONDITION FOR INHOMOGENEOUS
WAVES
Throughout we follow the standard notation of continuum mechanics (cf., e.g., [11-13]) and consider Cartesian tensors. In the absence of body forces, the linearized equation of motion is written in the form pii = V. T,
(2.1)
where p is the constant mass density, u is the displacement vector and T is the Cauchy stress. We consider time-harmonic displacement fields, whereby time dependence is always factorized as exp(-&ot). The stress-strain relation is given as x -- 51 c I V u + (vu)T],
where the dependence on the position vector x and the time t is omitted and a superscript T means transpose. Here C is a complex-valued, frequency dependent, fourth-order tensor defined by
C = Co + f~/C'(s)exp(i~os) ds where Co and C' have values in the space of fourth-order tensors and C' ~ LI(~+). Indeed we let C be endowed with the usual symmetry conditions; in suffix notation Cqhk = Cjihk = Cjikh =
Ckhji" We look for solutions to the equation of motion in the form of inhomogeneous waves, namely we let the displacement vector be represented in the exponential form u(x, t) = A exp[i(k • x - oJt)].
(2.2)
Here A is the amplitude vector and k denotes the wave vector. Once u is introduced into the definition of T and the resulting expression is substituted into the equation of motion we find the propagation condition [kCk - p~o213]A = 0,
(2.3)
the contraction of C with k being on the first and the fourth index: (2.3) is invariant under the
Wave propagation in linear anisotropic viscoelastic media
1061
transformation k---~ - k . H e n c e f o r t h In, with n positive and integer, denotes the n x n identity matrix. The system (2.3) admits non-trivial solutions for A if and only if the secular equation det[kCk
-
po92|3]
=
0
(2.4)
holds. The determinantal equation (2.4) is viewed as a condition for the admissible wave vectors k. Owing to the complex-valuedness of C, it follows that k is generally complex-valued. Moreover, Re k and Im k need not be parallel. According to (2.2), Re k is orthogonal to the planes of constant phase and I m k is orthogonal to the planes of constant amplitude. The corresponding solutions are called inhomogeneous waves in that the amplitude varies in planes of constant phase [14]. O f course, (2.4) is the condition for the existence of a non-vanishing amplitude vector A. In general the vector A turns out to be complex-valued and the physical meaning of displacement is ascribed to the real part of n.
3. T H E E I G E N V A L U E
PROBLEM
The propagation condition (2.3) involves the unknown vector A as eigenvector, but po92 is known while k is to be determined. We can give (2.3) the true form of an eigenvalue problem by following the six-dimensional Stroh formalism [5, 15]. However, the complex-valuedness of k m a k e s the application non-trivial. We consider an o r t h o n o r m a l reference pair (m, n) such that the real and imaginary parts of k belong to the plane spanned by m and n. Specifically we let
k = K(m + p n )
(3.1)
where both K and p are complex-valued and non-zero. The case when one c o m p o n e n t of k vanishes, which requires in particular that Re k and I m k are parallel to each other, is not considered here. The constant K entering the representation of the wave vector is regarded as a given p a r a m e t e r while p is unknown. In other words, we look for solutions n of the form (2.3) to the equation of motion where the projection k . m = K is given, while k . n = Kp is unknown. If In and n are chosen as unit vectors of the x and z axes of a Cartesian system then u = u(x, z) and the dependence of u on x has been fixed while that on z is to be determined through (2.4). If needed, the arbitrariness in the choice of n and m allows n to be taken as the unit vector of the imaginary part of k, thus making K real. The reference pair may be completed to an o r t h o n o r m a l basis m, n, t where the unit vector t is defined by t = m x n. U p o n substitution for k, the propagation condition (2.3) can be given the form of an equation for p and A as M A = 0,
(3.2)
where the matrix
po92
M = p 2 0 o + p ( a + R T) + O., - --2 -I3 is symmetric, and Qm = m C m ,
Qn = nCn,
R = mCn.
Although the Q's and R are tensor quantities, we prefer to look at (3.2) as a matrix equation, where A is regarded as a column vector and the entries of the matrices Q and R are given by the c o m p o n e n t s of the related tensors in the given orthonormal basis. The symmetry of C implies that R r = nCm. M o r e o v e r the matrices Q are symmetric but not Hermitian. It follows
1062
G. CAVIGLIA and A. MORRO
from thermodynamic considerations that Re Q and Im Q are positive and negative definite, respectively. The notation (ram), (ran), (nn) in place of Qm, R, Qn is used in [8]. In a moment we need the use of the inverse of Q,,. In this regard, assume that Ill, H2 are symmetric, non-singular matrices and ask for the possibility that ill + i a 2 is non-singular. Hence we look for matrices Dl, i12 such that I = (D1 + iD2)(H1 + ill2) = DiH1 - D2H2 + i ( D 2 H , + O l H l ) .
The vanishing of the imaginary part results in D2 = -D1H,_H(I. Substitution gives D1Hi - D 2 H : = D I H I ( I + HIIH2Hi-IH2). Hence D1 satisfies D~H1 - D2H2 = I provided only that I + HI-IH2HIIH2 is non-singular. Such is the case if H1 and H2 commute thus making (HigH2) r = Hi~H2. Otherwise, chosen any norm I[']l, a sufficiently small value of I[HflH2[I allows 1 + H ~ H 2 H ~ H 2 , and hence Ha + ill2, to be non-singular. The condition det M = 0 yields a sextic equation for the component p of the wave vector. Hence, once the corresponding roots are found, the related amplitudes follow. However, the sextic equation is in general unsolvable in radicals, elementary transcendental functions, or elliptic modular functions for tetragonal, orthorhombic, monoclinic, and triclinic symmetry [6]. As an alternative to numerical investigations, we determine the propagation modes in viscoelastic media by adapting and generalizing the approach by Barnett and Lothe to the analysis of surface waves. To avoid inessential complications it is assumed that the sextic equation has distinct roots. The condition (3.2) is restated as a genuine eigenvalue problem as follows. Let L exp{i[sc(m + p n ) . x - cot]} : - T n be the traction force acting through the plane surface of normal n as a result of the deformation induced by the displacement u. Direct calculation yields L = -iK(R r +pQ,,)A.
(3.3)
Insertion into (3.2) yields the equation
[ tc(RQ,,'Rr - Q,,,) + ~ I~I A - iRQ~IL = ipL.
(3.4)
The system (3.3)-(3.4) is equivalent to the propagation condition (3.2). If (3.3) is solved with respect to ipA, the resulting system of linear equations may be expressed in the form of an eigenvalue problem parameterized by K. namely ( N - i p k ) ~ = O,
(3.5)
where A)
~=
(3.6)
L"
The 6 × 6 matrix N has the block structure
N =
Nl
N2)
N3
N4
Wave propagation in linear anisotropic viscoelasticmedia
1063
and is parameterized by K in that Nl = - i Q Z 1Rr, N 2 = --Qn-1/K, N3 : t~(RQn IRT - Qm) + (P°)2/K)][3, N4 = - i R Q S I. For later convenience we establish some properties about the matrix N and the eigenvectors ~. A b o u t the determinantal equation we have det(N - ipI6) = 0
¢:> det[peQ. + p ( R + R r) + Qm -- (Pt02/K2)|3] = 0.
By the symmetry of Qn and Qm it follows that the matrices N1 . . . . . N4 satisfy N 2 ~-- N27~
N 3 = N T,
N 4 = N IT.
Since the roots of the sextic are taken to be distinct, the eigenvalues of N are distinct, the matrix is simple, and there exist six linearly independent eigenvectors. We denote by ~ the eigenvectors of (3.5) and by p,, the corresponding eigenvalues, where the suffix a varies from 1 to 6. Of course, the eigenvectors are defined up to a multiplicative constant. This freedom is used to construct a convenient basis of C 6. Preliminarily we establish a further property of the matrix N. For convenience, let K be the block matrix defined by
Since K is symmetric, we have (KN) r = N r K . Hence, because (KN) r = N r K = NN, it follows that KN is symmetric. Now, left multiplication of (3.5) by ~ K yields
~KN~
=
ip,~K~,~.
Interchange of ~ and g~, subtraction, and use of the symmetry of KN and K leads to 0 = (p~ - p~)~K~.
If a # / 3 it follows that ~ r K ~ = 0. If instead a =/3, we assume that ~rK~, ~ 0 ; then the undetermined constant in the definition of ~, is so chosen that ~rK~, = 1. With reference to (3.6) and the definition of K, the restriction placed on ~, is expressed as 2 A , . L~ = 1. Accordingly we can write
~SK~¢ = 6,¢, where 6~t~ is the Kronecker symbol. For later use and by analogy with [5], we consider the vectors ~, = K~. Comparison with (3.6) shows that
~1~= A~ ES 33:6-J
(3.7)
1064
G. CAVIGLIA and A. MORRO
and "q~ = ~ K . Hence, by (3.7) we have =
(3.8)
Now, for any w E C 6, let w = Et~ wt3~e be the representation of w in the basis {~t~}; it is understood that summation on Greek subscripts runs from 1 to 6. As an immediate consequence of (3.8) and the representation of w we find that ~q~w = w,, whence it follows that w = 5", (nSw)
o.
a
On taking the v-component of both sides we find that
This identity may be written in compact tensor notation as
w-(?,o®.o)w, holding for any vector w. Accordingly we have the completeness relation ~ @ ~1~ = I6.
(3.9)
On substituting into (3.9) the representation (3.6) for ~ and the explicit expression of ~1~ we find that (3.9) is equivalent to the three matrix relations ~] A~ ®L~ = 13, ot
~ A~®A,~ = 0 ,
~L~®L~
o~
=0.
(3.10)
ol
Moreover, with the use of (3.9) and (3.5) we find the chain of identities
yielding the spectral representation of the matrix N. Comparison with the block expression of N yields N, = i ~ p , ~ A ~ ®L~,
N2=i~p~A~®A~,
N3 = i ~ p ~ L , ~ ® L , .
c~
(3.11)
ot
The physical interpretation of each eigenvector {, and the related eigenvalue p, comes from the observation that the column vector
(u(x,t) t)n!]
exp{i[K(m + p n ) - x - tot]} = \ - T ( x ,
(3.12)
is in fact the stress-displacement vector [3] and describes wave propagation, with wave vector of the form (3.1) and parameterized by K. Any such wave results from a linear combination of the six waves associated with the eigenvectors of (3.5). Indeed, if m and n are identified with the unit vectors of the x- and z-axes, then the problem of solving the equation of motion at fixed exponential dependence of u on x has been turned into an eigenvalue problem for the traction-displacement vector, thus providing the dependence of the solution on z. Equation (3.5) characterizes a basis for the set of solutions of the system of ordinary differential equations describing propagation of the traction-displacement vector in the direction of the z-axis. Such a basis has been subject to a set of "orthonormality" conditions as in (3.7). This has led to the determination of the completeness relation (3.9) or (3.10) and the spectral representation (3.11) of N.
Wave propagation in linear anisotropic viscoelastic media
1065
T h e p r o p a g a t i o n condition is n o w in the genuine f o r m (3.5) of an eigenvalue p r o b l e m . Y e t the u n k n o w n p is given by the d e t e r m i n a n t a l e q u a t i o n which is a sextic. T h e p r o c e d u r e of Sections 4 - 6 allows the d e t e r m i n a t i o n of the eigenvectors ~ to be i n d e p e n d e n t of the evaluation of p.
4. C H A N G E
OF THE REFERENCE
PAIR
C o n s i d e r the o r t h o r n o r m a l basis, say ~ , fi, i, that is o b t a i n e d by rotating the r e f e r e n c e pair m, n a b o u t the t axis of an angle 4,. Specifically we let = cos 4, m + sin 4,n,
fi = - s i n 4,m + cos 4,n,
i = t.
(4.1)
D e n o t e by t2 the c o r r e s p o n d i n g o r t h o g o n a l matrix, viz.
sin !)
o_-/-So, cos,0 of course f ~ r
= 13. L e t
~
= a n , c n , n ~,
A direct e v a l u a t i o n yields ~C~ lllCn
= cos 2 4,Qm + sin 4, cos 4,(R + R T) + sin 2 4,Qn, =
COS2 4,R - sin 2 4,R v + sin 4, cos 4,(Q~ - Qm),
fiCfi = sin 2 4,Qm - sin 4, cos 4,(R + R r ) + cos 2 4,Q.. F o r c o n v e n i e n c e let 0(4,) = sin z 4,Q,, - sin 4, cos 4,(R + R r ) + cos 2 4,Q,,
(4.2)
8 ( 4 , ) = cos 2 4,R - sin 2 4,R r + sin 4, cos 4,(Q, - Q,,).
(4.3)
W e have
0(4,) = 0~ ( 4, ), 0(o) = Q.,
0(to/2) = Q,,,,
I~(o) = R,
and
( ~ = D.~(4, + x/2)f~ r,
R = f ~ ( 4 , ) O r,
( ~ = D~2(4,)~ r,
(4.4)
0(4, + 7r/2) = n T O , ~ n ,
8(4,) = n~n,
0(4,) = nT(~.n.
(4.5)
or
O f course the periodicity p r o p e r t y of 0 and 8 , n a m e l y 0(4, + 7r) = 0(4,), holds.
8 ( 4 , + ~) = ~ ( 4 , ) ,
(4.6)
1066
G. CAVIGLIA and A. MORRO
Let
and .~, = f~A so that the propagation condition reads MA
(4.7)
= 0.
Consistent with (3.1), let ~k = [
- = ,~
(n~
+ p~),
where ff and/~ are to be determined. By (4.1) we obtain ff = K(cos ~b + p sin ~b),
ffp = K(-sin ~b + p cos ~b).
These conditions relate the eigenvalue p, in the new basis 61, fi, to the eigenvalue p in the initial basis m, n. Substitution of (4.4) into (4.7) and use of (4.2), (4.3) reduces (4.7) to
~M~
= o.
Of course, p and ,~ can be related to the solution of an eigenvalue problem in complete analogy with (3.5).
5. A O N E - P A R A M E T E R F A M I L Y OF E I G E N V A L U E P R O B L E M S For ease of comparison we write equation (2.3) in the form
(mh q-pl'lh)(Chqrs P°92mh6qrm,)(ms+pn~)Zr=O. K2
(5.1)
To avoid the direct investigation of (5.1) we consider an auxiliary one-parameter eigenvalue problem obtained by replacing the vector K(m + pn) with K(r~ + p~fi), namely
(r~lh+ P~nh)( Chqrs P£°-~ K mh6qrm.,)(ffts + Pe~fi~)[A+]r =0.
(5.2)
Since 61 and fi depend on the angle ~b, the unknown eigenvalue Pc, and the eigenvector A~ are expected to depend on 4). If 4)=0, (5.2) reduces to the component form (5.1) of the propagation condition. To give evidence to common features with the propagation condition, it is convenient to insert the matrices I),~, fi, t)~, which allow equation (5.2) to be written in the compact form ~r[p24,Q ~ + p~(fi
+ fi T) + Q,~ - ~P 0")2 (p62 sm- 2 4, -
2p~ sin 4) cos 4) + cOS2 ~b)13]f~A,~ = 0.
In a purely formal way let f2A+ = . ~ . Left multiplication by the orthogonal matrix f~ yields [p~0~ + p~(fi + f i r ) + 0,~ - p°J---~zK2(p~ sin 2 4) - 2p~ sin 4) cos 4) + cos 2 4))13]A~. = 0.
(5.3)
It is worth observing that (5.3) is equivalent to (4.7). We can prove that (5.3) is merely the propagation condition in the rotated basis fin, ft. For the time being it is simply a relation for the new unknowns A~ and p+. However we recognize at once that if ~o = 0, as in statics, (5.3) is the rotated form of (5.1) with p6 = p . In an analysis of the equilibrium equation, Ting [16] concludes that, under a rotation of the coordinate system about the z-axis, the (Stroh) eigenvectors transform according to the law of vectors. Our remark shows that Ting's statement can be generalized to the dynamic case.
Wave propagation in linear anisotropic viscoelastic media
1067
Comparison with (4.5) shows that equation (5.3) can be written in the equivalent form {p~Q~(4,) + p,[R~(4,) + Rr(4,)] + Q~(4, + Jr/2)}A~ = 0,
(5.4)
where P to2 • 2
P 0)2
Q,(4,) = 0(4,) - --T-Ksin 4,I3,
RK(4,) = ~(4,) + ~
sin 4, COS 4,13.
(5.5)
In view of (4.6) and (5.5) we have Q,,(4, + ;,r) = Q,,(4,),
R,,(4, + n:) = R,,(4,).
Consider the one-parameter "traction" L , = -iK[p,~Q~(4,) + Rr(4,)]A,
(5.6)
and let Q~(4,) be non-singular. Hence (5.4) is equivalent to the eigenvalue problem (5.7)
[N(4,) - ip , I 6 ] ~ , = 0 where ~'~=
L,~"
The 6 x 6 matrix N(4,) is given by
(N,(4,) N2(4,) 1%1(4,) = \ N 3 ( 4 , )
N4(4,)/,
where NI(&) = -iQ~-~(4,)R~(4,),
N2(&) = --Q~-](4,)/K
N3(4,) = K[R,,(4,)Q~(4,)R~r(4,) - Q,,(4, + ~/2)],
N4(4,) = -iR,,(4,)Q~-](4,).
The symmetry of QK(4,) implies that N2(4,) = N2r(4,),
N3(4,) = N3r(4,),
N4(4, ) = Nlr(4,).
(5.8)
Incidentally, the eigenvalue problem (5.7) reduces to the propagation condition (3.5) in the special case 4, = 0. To investigate the relationship between the eigenvalues p and p , is convenient to go back to the formulation (5.2) and to make a comparison with (3.2). In this regard we represent l~ and fi in (5.2) in terms of m and n as given in (4.1). Upon multiplication by 1/(cos 4, - p , sin 4,)2 and appropriate substitutions, we can write (5.2) in the form M , A , = 0, where (sin 4, + p~ cos 4,~2A sin 4, + p , cos 4, ( R + R r ) + Q m - Po21 K2 3. M4' = \cos 4, - p , sin 4,/ t~n + cos 4, - p,~ sin 4, Comparison with the definition (3.2) of M shows that the roots of the determinantal equation det M(p) = 0 are related with those of the equation det M , ( p , ) = 0 by p-
sin 4, + p , cos 4, cos 4, - p~, sin 4,
with inverse p, =
- s i n 4' + P cos 4,
cos 4, + p sin 4,
(5.9)
Accordingly, without any loss in generality we let A~ = A. In addition, substitution of p , from
1068
G. CAVIGLIA and A. MORRO
(5.9) in (5.6) and comparison with (3.3) shows that L,~ = L. Hence, in view of (5.7) we conclude that ~,~ = ~,
(5.10)
whatever the value of the parameter 4'. In words, the eigenvectors of N(4') are independent of the parameter 4' and then coincide with those of N. As an aside we observe that the invariance of the eigenvector is proved in [5] via a different procedure. The matrix N and the eigenvalue p,~ are shown to satisfy the differential equations dp6/d4' = - 1 _ p 2 ,
dN(4')/d4'
=
--16
--
N2(4'),
and hence the invariance of ~ is derived. Our approach, instead, is purely algebraic and, to our mind, simpler.
6. I N T E G R A L F O R M A L I S M Now we take advantage of (5.10) and obtain a very simple eigenvalue problem. Consider the expression (5.9) of p+. The integral of p6 from 0 to 21r is evaluated as follows. We let exp(i4') = z in the expression (5.9) of p~ so that the integral on [0, 21r] is changed to an integral over the unit circle, with centre at the origin, in the complex plane. Use of the residue theorem provides the result, viz.
£
~~p6 d4' = 2z¢i sgn(Im(p)).
(6.1)
Accordingly, integration of (5.7) with respect to 4' on [0, 2to] yields S~ = -sgn(Im(p))~,
(6.2)
where S is parameterized by K and
(s, S=
$3
S
'
with S,
21r Jo N,(4') d4',
i = 1. . . . . 4.
(6.3)
Observe that Q~(4') may be singular at isolated values of 4' and the same may occur to Ni(4'). Hence the integral (6.3) is meant as the principal value. Of course, by (5.8) S inherits the algebraic properties S 2 = S T,
S 3 : S T,
S4 = S T
(6.4)
Equation (6.2) represents essentially an average of the eigenvalue problem (5.7) and takes into account the fact that the eigenvectors are independent of the parameter 4'. The matrix S is parameterized by the value K of the wave vector k in the direction of m. The result (6.2) is important in that allows the determination of eigenvectors and eigenvalues of the propagation condition (3.5) without any recourse to the sextic equation for p. The matrix S is semisimple. Specifically, there exist only the multiple eigenvalues + 1 and - 1 , and the two corresponding eigenspaces have the same dimension as the multiplicity of the eigenvalues. This follows as an ultimate consequence of (5.10) and the assumption that the propagation condition (3.5) admits six distinct eigenvalues and six linearly independent eigenvectors.
Wave propagation in linear anisotropic viscoelastic media
1069
Denote by ~ , ot = 1 , . . . , 6, six linearly independent eigenvectors corresponding to the eigenvalues +1, - 1 . In principle, the ~ ' s may coincide with the eigenvectors of (3.5). In any case comparison with (6.2) shows that S2~o = S ( S ~ ) = ~o, = 1. . . . . 6. Since the eigenvectors constitute a basis for C 6 we conclude that S 2 -- 16.
(6.5)
Substitution into (6.5) of the block expression for S(K) and account for the relations (6.4) yields the identities SiS1 q- 8283 =
13,
S1S 2 -[- 82 ST = 0, 8381 q- s T s 3 : 0.
(6.6)
If S2 is non-singular, the first equation of (6.6) provides S 3 = S21(13 -- 82),
(6.7)
which is a useful property in practical applications whenever the explicit form of S is required.
7. F U N D A M E N T A L SYSTEM OF S O L U T I O N S Mathematically we now determine a basis for the time-harmonic solutions (2.2) to the equation of motion (2.1). In other words, we establish the set of propagation modes whose linear combinations generate any time-harmonic wave propagating in the unbounded anisotropic solid. Any such basis is said to be a fundamental system of solutions. The matrix S is known. Operatively, the first step is to determine the eigenvectors 4, of S. Since the eigenvalues of S are known a priori (+1), we determine the eigenvectors without any trouble to the evaluation of the eigenvalues. The second step is to determine the eigenvectors and the eigenvalues p of N that enter the wave solutions under consideration. To fix ideas, consider the eigenvalue + 1 of S and denote by {~7} the associated eigenvectors. For definiteness, we assume that the associated linear space is three-dimensional, so that j varies from 1 to 3. It is not restrictive to let (~7)TK~ = ~jh,
(7.1)
with j, h = 1, 2, 3. Incidentally, by (6.2), we are considering the eigenvectors associated with values of p with I m p < 0 . Indeed, the eigenvectors {~7} are required to generate the eigenvectors of N with eigenvalues p subject to I m p < 0. Consider a linear combination ~3=1 ajar, with complex coefficients aj, and require that it be an eigenvector of N with eigenvalue p. The coefficients aj are then subject to 3
3
j=l
j=l
Left multiplication of both sides by (~h~)rK and use of (7.1) yields 3
(~-~)rKN~;aj = ipah. j=l
This condition may be given the form of an eigenvalue problem as Aa = ipa
(7.2)
1070
G. CAVIGLIA and A. MORRO
where a = (a 1, a2, a3) T
is an unknown column vector and the entries of the matrix A are given by Ahj = ( ~ ) r K N ~ . The advantage of the formulation (7.2) is that the determinantal condition det[A - ipI3] = 0 in the unknown eigenvalue p of (3.5) is a complex polynomial of third degree. Hence the three roots can be found explicitly and the coefficients a l, a2, a3 are derived from (7.2) thus determining the eigenvectors ~ of the propagation condition (3.5) with I m p < 0. The first three components of ~ constitute the amplitude vector A. The derivation of the eigenvectors associated with I m p > 0 proceeds along the same lines. In conclusion, for every fixed value of K we determine the eigenvalues Px, Pz . . . . . P6 and the amplitude vectors. Hence the general plane-wave solution to (2.1) is a linear combination of the propagation modes, viz. 6
u(x, t) = ~
b,~A~ exp{i[K(m + p ~ n ) • x - o~t]}.
oe--I
The coefficients {b~} are determined, e.g. by boundary conditions at interfaces or initial values. For instance, in a reflection-transmission problem the incident wave is given and the six-tuple {b,~} for reflected and transmitted waves is determined by requiring that the continuity conditions at the interface hold. In that case, via a proper choice of m, account for Snell's law (cf. [14]) results in a common value of K for all waves involved.
8. W A V E S IN T R A N S V E R S E L Y
ISOTROPIC SOLIDS
Wave propagation in transversely isotropic solids is of interest with regard to many practical situations. In addition, it offers a remarkable example of application of the present theory. We observe that waves in transversely isotropic solids have been investigated by Chadwick et al. [5, 17]. Besides assuming that the solid is elastic, they let the wave vector k belong to the basal plane. Here we consider a viscoelastic solid and take the complex-valued wave vector k in a zonal plane. Although this is not the general case, it certainly provides a decisive generalization of the previous investigations. The complex-valued components of the tensor C are represented by using Voigt's notation, namely by considering a 6 × 6 symmetric matrix {C~,v} related to {Ciik/} by the correspondence rule 11 - 1, 22 - 2, 33 - 3, 23 - 4, 13 - 5, 12 - 6. The non-zero entries are given by
(C,~) =
Cll
C12
C13
C12 C~3
Cll C~3
C13 C33
(8.1)
C44 C44 C66
Wave propagation in linear anisotropic viscoelasticmedia the entries C~,~ being parametrized by oJ and C66 = ( e l l (3.2) can be given the form
(p2f 4p~c~O--ci4~2/K2
0 p2C44 + C66 -
C12)[2. The propagation condition p(C13 + C44)
p¢.02/1¢.2
0
1071
)A= O.
0 + C44-- p¢-02/l<2
p2C33
(8.2)
The resulting secular equation for p factorizes as ¢'02 2 "0)2"~ [ 2 (p2C44+666 P~)I(pC44-~-Cll-~)~Pf33JvC44
~0)2~ C44)2 ] = O, (8.3) K2 ] - p 2 ( C 1 3 -~-
as found first by Synge for transversely isotropic elastic solids [18]. Unlike the general case, the secular equation can be explicitly solved. To this end it is convenient to multiply (8.3) by K 6 and let/32 = K2p2;/3 is the component of the wave vector k along n. Hence (8.3) becomes [(p¢o 2 -- C44/3 2 -- E l l K2)(ptO 2 -- C33/3 2 - C44 K2) -- (C13 -{- C44)2K2/32](pto 2 -- C44/3 2 -- C66/(2) : 0,
for the new unknown/3. The vanishing of the first factor reads /34 __ (0/1 + 0/2 + r)/3 2 + 0/10/2 = 0,
where po) 2 0/1 -- - -
Cll
C33
C33
F ~--- --K21/ e l l \ C44
poJ 2
K2~
0/2 --
C~3
2 C1~3
Ca,
C33 C44
C33
K2~
ell
1|.\
C33
/
Letting = (O/1 "[- 0/2 "+ 1-')2 -- 40/10/2,
we can write the solutions for /32 as / 3 L = (0/, + 0/2 + r • v
r)/2,
where V~ is chosen with positive real part. We assume that /32 and /32 are distinct and non-zero. The vanishing of the second factor in (8.3) gives a third solution for/32 as /32 - PrO2 C44
C66 K2; C44
we l e t / 3 3 ~ 0 and/3~ #/3~" #/3~.2 By the definitions (4.2) and (4.3) we obtain I CI~ sin 2 ~b + C44 cos 2 4~ ~(4~) =
0 ~1( C ,t - C , z ) s i n e ~b
0
+ C44 cos 2 (~
C44 sin 2 ¢k + C33 c0s2 t~
0
C44 - CII~ in 4' cos
IC44 _ 1 (C,l - C12)]sin ~C44 cos 2 (~ - C13 sin z ~b
0
0
-(C~3 + C44)sin 6 cos 4'
8(6)=
--(el3 -t- C44)sin ~ cos t~
0
C13 cos2 ~ - C44 sin 2 ¢h
cos ~b (C33 -
Hence we have Q . = 0(0),
Qm = 00r/2),
R = ~(0).
0 1 C44)sin ~b cos ~b
1072
G. CAVIGLIA and A. MORRO
T h e matrices Q=(th) and RK(~b) are o b t a i n e d t h r o u g h substitution into (5.5). Meanwhile the matrices Ni are given by
0//
N1 = N4r = -iC13/C33 N 2 ----diag
( 1 KC44 ,
0
0
0
0
1
, t
KC44 ,
"" {\ {C23 \%3 N3=o,ag/K/~---Cu
)
+
,
K~33 ,
p~°2 -KC66 + po~ poJ21 K
K
K
/
T h e list of eigenvalues of N m a y then be written as {Kp,, a = 1 . . . . .
6} --- {/3,, a = 1 . . . . .
6} = {/31,/32, -/31, -/32,/33, -/33},
where/31, /32, /33 a r e chosen with positive imaginary part. Correspondingly, the eigenvectors of N are the o r d e r e d c o l u m n s of the matrix bI
b2
b]
b2
0
0
0
0
0
1
0
1
KC l
KC 2
--KC 1
- - KC 2
0
0
-dl
-d2
dl
d2
0
0
0
0
0
0
-i/33644
i/33C44
--Ka~
--Ka 2
--Kal
--Ka 2
0
0
(8.4)
where al,2 = i( C13Poa2 - C13C44 K2 Jr C33644/32,2),
bl.2 = pro 2 - C44 K2 - C33/32,2, C1,2 =/31,2(C13 -]- C44 ) dl,2 = i/31,2C44(pt02 + C13K2 _ C33/31,2).2
It is w o r t h r e m a r k i n g that here the d e t e r m i n a t i o n of the eigenvalues is p e r f o r m e d in a simple way thanks to the f a c t o r e d f o r m of the secular equation. Should the factorization not take place, the p r o c e d u r e b e c o m e s the s t a n d a r d one. Since we n e e d the matrices N(~b) and S we evaluate Q~l(4, ) and R~(~b)Q~I(th). W e find that Q~1(4, ) can be given in the f o r m
o!)
C
,
0 the constants
a, b, c, d, e being
given by
ira = - ( p r o 2 - C44 K2 - C33~2)K2/sin 2 q~, o'b = o'd = (C13 + C44)~'r3/sin 2 ~b, c = K2/[C44 sin 2 ~b(ff2 -/32)], (re = - ( p r o 2 - C]I K2 - C44sr2)K2/sin 2 q~ where Or = C ~ C 4 4 ( ~
~ -- ~ ) ( ~
-- ~ ) ,
and t" = K cot ¢. Similarly, we find that
R,~(~b)Q~-~(~)
=
(i
g
0
0
,
Wave propagation in linear anisotropic viscoelasticmedia
1073
where 0-~ = [(C13 q- C44)K4(C13/sin 2 q~ - Cl 3 - C44) - (p(..o2 - 644 K2 - C33~2)(p0) 2 q- C44K 2 - Cl 1K2)]~/K, 0-/~ = (C13 q- C44)(P 0)2 - Cll K2)(ff 2 q- K2) -- C13(P £02 - Cll K2 -- C44~2)K2/sin2 6,
0-t~-
0092 -- C66 K2 -~ C44 K2 C44(~2 _ /32) K'
o'd = (C13 --[-C44)(po) 2 - C44K2)(~ 2 q- K 2) -- C44(Po) 2 - C44 K2 - C33ff2)K2/sin2 4~, o'~ = (Cl3 q- C44)tc4(f44/sin 2 ~ - C13 - C44 ) - (poJ 2 - C44 K2 --[-C33K2)(po) 2 - Cll K2 - C44~2).
For formal simplicity we now let K be real and Im/3 > 0. By (6.3), the matrix S is determined through integration of N(~b) with respect to 4). It follows that Sl = S4r =
0 0
2
S1~13)
S1,31 0
where
=s1'13
K
.( C13(poJ2-- C44 K2)
633C44(7~1 -[-/32) \
/31/32
C33 C44 ) -
'
,
S1'31
C33(/3~ -[- /32) \
/31/32
are the roots of the secular equation with positive imaginary part. and /3, i = 1 , 2 , 3 , Analogously it follows that S~l 1 S2 =
0
0 )
$2,22
0
,
0
$2,33
where
=. -$2'11
(P°)2-C44 K2 C33C44(/31 q-/32) \
i
/31/32
) '}- C33 '
(pro 2 - Cl1K2 C44)"
$2'33 = -- 633644(~1 q-/32) \
/31/32
q-
Finally, S3 is evaluated by substitution into (6.7) of the expressions for $1 and $2. We find that
83 =
(1 -- S1,13S1,31)/$2,11 0 0 1/s2,22 0
0
0 ) 0 . (1 - s1,13s1,31)/$2,33
A direct check shows that the columns of the matrix (8.4) are eigenvectors of S. We now reverse the procedure and show that the eigenvalues and eigenvectors of the matrix N are recovered from the eigenvectors of S. A direct evaluation yields the three eigenvectors of S associated with the eigenvalue + 1 in the form ~- = 0-1(1, O, O, 1/$2,11 , O, --S1,31/$2,33)T, ~2~ = 0"2(0, 0, 1, -s~,,3/s2.,1, O,
1/S2,33) T,
~- = 0"3(0, 1, 0, 0, i/33644, 0) T.
They meet the normalization condition (3.7) provided the pertinent constants 0" are taken to be given by
o21 = s2,1~/2,
0-2 = S2,33/2,
0-2 = 1/(2i/33C44)"
1074
G. CAVIGLIA and A. MORRO
The linear system for the coefficients of the superposition eigenvectors of N takes the form
of
(A1. A2 0)(al) A021
A22 - ip
0
a2
0
A33 - ip
a3
the
vectors
t~÷
yielding
the
=0
where 2[- [ C23 All = O'I/K|---k \ C33
Cll
[
0(.02 613s1 31 + ..... + 2i ] /¢ C33 $2,33
.C13 1
i
A12 = A21 = o'1o'2~- l -C33s2,33
A22 = 0"2 0o)2 \
2i s1'13
K -]-
1
x
$2.11
K644 $2,11
~- 1 s1,13
1
1
s2311
-
~
I
/£C33 $2.33 ]
S1,31~
- - T -+ 2 ' KC44s2,11 KC33s2,33]
$2,11
1 sin3 2 K644S2,11
1
1 1 K C33 s~733 , '
A33 = --i[33/K.
It follows immediately that there exists an eigenvalue, say P3, given by P3 =
--~3/K
which belongs to the eigenvector of components (0, 1, 0, 0 , / ~ 3 C 4 4 , 0) T
and corresponds to letting al = a 2 = 0 , and a 3 = 1. The determination of the other two eigenvectors is straightforward in principle but requires involved calculations. However, we already known that the eigenvectors are given by the third and the fourth columns of (8.4). Hence it is easily seen that they follow from combinations of ~? and ~ with coefficiems b~, --KCl, and b2, - KC2, respectively. Acknowledgements--The authors are grateful to Professors P. Chadwick and J. Lothe for providing an unpublished article (P,C.) and a reprint of papers which were not easy to get (J.L.).
REFERENCES
[1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18]
M. J. P. MUSGRAVE, Crystal Acoustics. Holden-Day, Calif. (1970). F. I. FEDOROV, Theory o f Elastic Waves in Crystals. Plenum Press, New York (1968). S. CRAMPIN, Wave Motion 3, 343 (1981). P. CHADWICK, Recent developments in the theory of elastic surface and interracial waves. In Elastic Wave Propagation (Edited by M. F. McCARTHY and M. A. HAYES), pp. 3-16. Elsevier, Amsterdam (1989). P. CHADWICK and G. D. SMITH, Adv. Appl. Mech. 17, 303 (1977). A. K. HEAD, J. Elasticity 9, 9 (1979). D. M. BARNETT and J. LOTHE, Physica Norvegica 7, 13 (1973). J. LOTHE and D. B. BARNETT, J. Appl. Phys. 47, 428 (1976). J. LOTHE, Uniformly moving dislocations; surface waves. In Elastic Strain Fields and Dislocation Mobility (Edited by V. L. INDENBOM and J. LOTHE), pp. 447-487. Elsevier, Amsterdam (1992). P. CHADWICK and N. J. WILSON, Proc. R, Soc. Lond. A 438, 225 (1992). M. E. GURTIN, An Introduction to Continuum Mechanics. Academic Press, New York (1981). A. J. M. SPENCER, Continuum Mechanics. Longman, London (1980). G. A. MAUGIN, Continuum Mechanics o f Electromagnetic Solids. North-Holland, Amsterdam (1988). G. CAVIGL1A and A. MORRO, lnhomogeneous Waves in Solids and Fluids. World Scientific, Singapore (1992). A. N. STROH, J. Math. Phys. 41, 77 (1962). T. C. T. TING, Int. J. Solids Structures 18, 139 (1982). V. I. ALSHITS and J. LOTHE, Wave Motion 3, 297 (1981). J. L. SYNGE, J. Math. Phys. 35, 323 (1956). (Received 8 March 1994; accepted 11 March 1994)
Pergamon
0020-722504) E0036-I
Copyright t~) 1995 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0020-7225/95 $9.50+ 0.00
W A V E P R O P A G A T I O N C O N D I T I O N IN L I N E A R A N I S O T R O P I C VISCOELASTIC M E D I A G. C A V I G L I A Department of Mathematics, Via Alberti 4, 16132 Genova, Italy
A. M O R R O DIBE, University of Genoa, Via Opera Pia lla, 1-16145 Genova, Italy (Communicated by G. A. MAUGIN) Abstract--Time-harmonic wave propagation in anisotropic, homogeneous linear, viscoelastic solids is investigated. By generalizing a procedure which traces back to Barnett and Lothe, and was envisaged for surface waves, a six-dimensional description of wave motion is shown to result in a complex eigenvalue problem. Next an integral formalism is established which simplifies the evaluation of the eigenvalues. This in turn allows the determination of fundamental solutions. Wave propagation in a transversely isotropic solid is examined in detail.
1. I N T R O D U C T I O N Wave propagation in linear, homogeneous, anisotropic solids is a subject of interest in many respects. Owing to linearity and Fourier analysis great attention is addressed to time-harmonic plane waves. In this regard, expository accounts of the subject are given in books (e.g. [1, 2]) and review articles (e.g. [3-5]). Unlike the theory for isotropic solids, difficulties occur even in connection with propagation modes in that the determinantal condition is a sextic equation that in general symmetries is not amenable to explicit solutions, which means that only numerical estimates are possible [6]. Hence, any operative approach has to circumvent the sextic in the determination of the propagation modes. In the last decade, Barnett, Chadwick, Lothe and co-workers (cf. [7, 8] and [9, 10] and references therein) developed a procedure which avoids the application of the sextic in connection with the study of surface waves in anisotropic elastic solids. As we show, the same scheme can be applied successfully to the determination of the propagation modes. This paper faces new aspects of wave propagation in anisotropic solids. First, the dissipative properties of the body are incorporated. Dissipation is a basic feature in several applications as, for example, in seismology. Here the solid is modelled as linearly viscoelastic with an arbitrary relaxation function. Secondly, solutions are sought in analytical form. Specifically, a class of fundamental solutions are determined which are a basis for the family of waves with fixed components of the wave vector in a given direction. Fundamental solutions are important in many respects such as the description of wave phenomena at interfaces. Thirdly, no use is made of the slowness surface. Though a slowness surface might still be defined, no practical consequence is likely to be derived because of the complex-valued nature of the wave vector and the elasticities. In essence, the paper models time-harmonic wave propagation through a six-dimensional eigenvalue problem. We fix a direction, say a unit vector n, in the plane spanned by the real and imaginary parts of the wave vector, and regard as unknowns the components of the displacement and those of the traction at planes orthogonal to n. Accordingly the wave motion is described by a first order system of ordinary differential equations, which ultimately results in a six-dimensional complex eigenvalue problem, where the eigenvalue is the component of the wave vector along n, while the component orthogonal to n plays the role of a parameter. The corresponding eigenvectors are associated with fundamental solutions. A suitable 1-parameter 1059
1060
G. CAVIGLIA and A. MORRO
problem and an "integral formalism" provide the way to determine the eigenvalues and eigenvectors without solving the sextic determinantal condition. Specifically, a method for the determination of fundamental solutions is envisaged where the original sixth-order eigenvalue problem reduces to two lower-order problems. At first sight the complex nature of wave propagation results in a rather involved approach. While it is so in many respects, it is remarkable that some peculiar features of the real case disappear such as the classification in subsonic-transonic intervals [5]. In addition, when applied to elastic solids, the approach provides a unified treatment of homogeneous and inhomogeneous waves, This in turn allows a systematic investigation of reflection and transmission at an interface in full analogy with the case of isotropic solids. As an application of the procedure, wave modes in transversely isotropic viscoelastic solids are determined by letting n be in the direction of the symmetry axis. Following the integral formulation the sixth-order eigenvalue problem splits into two third-order problems. In addition the third-order determinantal equation factorizes, thus making the search for eigenvalues and eigenvectors simple also from the computational viewpoint.
2. P R O P A G A T I O N
CONDITION FOR INHOMOGENEOUS
WAVES
Throughout we follow the standard notation of continuum mechanics (cf., e.g., [11-13]) and consider Cartesian tensors. In the absence of body forces, the linearized equation of motion is written in the form pii = V. T,
(2.1)
where p is the constant mass density, u is the displacement vector and T is the Cauchy stress. We consider time-harmonic displacement fields, whereby time dependence is always factorized as exp(-&ot). The stress-strain relation is given as x -- 51 c I V u + (vu)T],
where the dependence on the position vector x and the time t is omitted and a superscript T means transpose. Here C is a complex-valued, frequency dependent, fourth-order tensor defined by
C = Co + f~/C'(s)exp(i~os) ds where Co and C' have values in the space of fourth-order tensors and C' ~ LI(~+). Indeed we let C be endowed with the usual symmetry conditions; in suffix notation Cqhk = Cjihk = Cjikh =
Ckhji" We look for solutions to the equation of motion in the form of inhomogeneous waves, namely we let the displacement vector be represented in the exponential form u(x, t) = A exp[i(k • x - oJt)].
(2.2)
Here A is the amplitude vector and k denotes the wave vector. Once u is introduced into the definition of T and the resulting expression is substituted into the equation of motion we find the propagation condition [kCk - p~o213]A = 0,
(2.3)
the contraction of C with k being on the first and the fourth index: (2.3) is invariant under the
Wave propagation in linear anisotropic viscoelastic media
1061
transformation k---~ - k . H e n c e f o r t h In, with n positive and integer, denotes the n x n identity matrix. The system (2.3) admits non-trivial solutions for A if and only if the secular equation det[kCk
-
po92|3]
=
0
(2.4)
holds. The determinantal equation (2.4) is viewed as a condition for the admissible wave vectors k. Owing to the complex-valuedness of C, it follows that k is generally complex-valued. Moreover, Re k and Im k need not be parallel. According to (2.2), Re k is orthogonal to the planes of constant phase and I m k is orthogonal to the planes of constant amplitude. The corresponding solutions are called inhomogeneous waves in that the amplitude varies in planes of constant phase [14]. O f course, (2.4) is the condition for the existence of a non-vanishing amplitude vector A. In general the vector A turns out to be complex-valued and the physical meaning of displacement is ascribed to the real part of n.
3. T H E E I G E N V A L U E
PROBLEM
The propagation condition (2.3) involves the unknown vector A as eigenvector, but po92 is known while k is to be determined. We can give (2.3) the true form of an eigenvalue problem by following the six-dimensional Stroh formalism [5, 15]. However, the complex-valuedness of k m a k e s the application non-trivial. We consider an o r t h o n o r m a l reference pair (m, n) such that the real and imaginary parts of k belong to the plane spanned by m and n. Specifically we let
k = K(m + p n )
(3.1)
where both K and p are complex-valued and non-zero. The case when one c o m p o n e n t of k vanishes, which requires in particular that Re k and I m k are parallel to each other, is not considered here. The constant K entering the representation of the wave vector is regarded as a given p a r a m e t e r while p is unknown. In other words, we look for solutions n of the form (2.3) to the equation of motion where the projection k . m = K is given, while k . n = Kp is unknown. If In and n are chosen as unit vectors of the x and z axes of a Cartesian system then u = u(x, z) and the dependence of u on x has been fixed while that on z is to be determined through (2.4). If needed, the arbitrariness in the choice of n and m allows n to be taken as the unit vector of the imaginary part of k, thus making K real. The reference pair may be completed to an o r t h o n o r m a l basis m, n, t where the unit vector t is defined by t = m x n. U p o n substitution for k, the propagation condition (2.3) can be given the form of an equation for p and A as M A = 0,
(3.2)
where the matrix
po92
M = p 2 0 o + p ( a + R T) + O., - --2 -I3 is symmetric, and Qm = m C m ,
Qn = nCn,
R = mCn.
Although the Q's and R are tensor quantities, we prefer to look at (3.2) as a matrix equation, where A is regarded as a column vector and the entries of the matrices Q and R are given by the c o m p o n e n t s of the related tensors in the given orthonormal basis. The symmetry of C implies that R r = nCm. M o r e o v e r the matrices Q are symmetric but not Hermitian. It follows
1062
G. CAVIGLIA and A. MORRO
from thermodynamic considerations that Re Q and Im Q are positive and negative definite, respectively. The notation (ram), (ran), (nn) in place of Qm, R, Qn is used in [8]. In a moment we need the use of the inverse of Q,,. In this regard, assume that Ill, H2 are symmetric, non-singular matrices and ask for the possibility that ill + i a 2 is non-singular. Hence we look for matrices Dl, i12 such that I = (D1 + iD2)(H1 + ill2) = DiH1 - D2H2 + i ( D 2 H , + O l H l ) .
The vanishing of the imaginary part results in D2 = -D1H,_H(I. Substitution gives D1Hi - D 2 H : = D I H I ( I + HIIH2Hi-IH2). Hence D1 satisfies D~H1 - D2H2 = I provided only that I + HI-IH2HIIH2 is non-singular. Such is the case if H1 and H2 commute thus making (HigH2) r = Hi~H2. Otherwise, chosen any norm I[']l, a sufficiently small value of I[HflH2[I allows 1 + H ~ H 2 H ~ H 2 , and hence Ha + ill2, to be non-singular. The condition det M = 0 yields a sextic equation for the component p of the wave vector. Hence, once the corresponding roots are found, the related amplitudes follow. However, the sextic equation is in general unsolvable in radicals, elementary transcendental functions, or elliptic modular functions for tetragonal, orthorhombic, monoclinic, and triclinic symmetry [6]. As an alternative to numerical investigations, we determine the propagation modes in viscoelastic media by adapting and generalizing the approach by Barnett and Lothe to the analysis of surface waves. To avoid inessential complications it is assumed that the sextic equation has distinct roots. The condition (3.2) is restated as a genuine eigenvalue problem as follows. Let L exp{i[sc(m + p n ) . x - cot]} : - T n be the traction force acting through the plane surface of normal n as a result of the deformation induced by the displacement u. Direct calculation yields L = -iK(R r +pQ,,)A.
(3.3)
Insertion into (3.2) yields the equation
[ tc(RQ,,'Rr - Q,,,) + ~ I~I A - iRQ~IL = ipL.
(3.4)
The system (3.3)-(3.4) is equivalent to the propagation condition (3.2). If (3.3) is solved with respect to ipA, the resulting system of linear equations may be expressed in the form of an eigenvalue problem parameterized by K. namely ( N - i p k ) ~ = O,
(3.5)
where A)
~=
(3.6)
L"
The 6 × 6 matrix N has the block structure
N =
Nl
N2)
N3
N4
Wave propagation in linear anisotropic viscoelasticmedia
1063
and is parameterized by K in that Nl = - i Q Z 1Rr, N 2 = --Qn-1/K, N3 : t~(RQn IRT - Qm) + (P°)2/K)][3, N4 = - i R Q S I. For later convenience we establish some properties about the matrix N and the eigenvectors ~. A b o u t the determinantal equation we have det(N - ipI6) = 0
¢:> det[peQ. + p ( R + R r) + Qm -- (Pt02/K2)|3] = 0.
By the symmetry of Qn and Qm it follows that the matrices N1 . . . . . N4 satisfy N 2 ~-- N27~
N 3 = N T,
N 4 = N IT.
Since the roots of the sextic are taken to be distinct, the eigenvalues of N are distinct, the matrix is simple, and there exist six linearly independent eigenvectors. We denote by ~ the eigenvectors of (3.5) and by p,, the corresponding eigenvalues, where the suffix a varies from 1 to 6. Of course, the eigenvectors are defined up to a multiplicative constant. This freedom is used to construct a convenient basis of C 6. Preliminarily we establish a further property of the matrix N. For convenience, let K be the block matrix defined by
Since K is symmetric, we have (KN) r = N r K . Hence, because (KN) r = N r K = NN, it follows that KN is symmetric. Now, left multiplication of (3.5) by ~ K yields
~KN~
=
ip,~K~,~.
Interchange of ~ and g~, subtraction, and use of the symmetry of KN and K leads to 0 = (p~ - p~)~K~.
If a # / 3 it follows that ~ r K ~ = 0. If instead a =/3, we assume that ~rK~, ~ 0 ; then the undetermined constant in the definition of ~, is so chosen that ~rK~, = 1. With reference to (3.6) and the definition of K, the restriction placed on ~, is expressed as 2 A , . L~ = 1. Accordingly we can write
~SK~¢ = 6,¢, where 6~t~ is the Kronecker symbol. For later use and by analogy with [5], we consider the vectors ~, = K~. Comparison with (3.6) shows that
~1~= A~ ES 33:6-J
(3.7)
1064
G. CAVIGLIA and A. MORRO
and "q~ = ~ K . Hence, by (3.7) we have =
(3.8)
Now, for any w E C 6, let w = Et~ wt3~e be the representation of w in the basis {~t~}; it is understood that summation on Greek subscripts runs from 1 to 6. As an immediate consequence of (3.8) and the representation of w we find that ~q~w = w,, whence it follows that w = 5", (nSw)
o.
a
On taking the v-component of both sides we find that
This identity may be written in compact tensor notation as
w-(?,o®.o)w, holding for any vector w. Accordingly we have the completeness relation ~ @ ~1~ = I6.
(3.9)
On substituting into (3.9) the representation (3.6) for ~ and the explicit expression of ~1~ we find that (3.9) is equivalent to the three matrix relations ~] A~ ®L~ = 13, ot
~ A~®A,~ = 0 ,
~L~®L~
o~
=0.
(3.10)
ol
Moreover, with the use of (3.9) and (3.5) we find the chain of identities
yielding the spectral representation of the matrix N. Comparison with the block expression of N yields N, = i ~ p , ~ A ~ ®L~,
N2=i~p~A~®A~,
N3 = i ~ p ~ L , ~ ® L , .
c~
(3.11)
ot
The physical interpretation of each eigenvector {, and the related eigenvalue p, comes from the observation that the column vector
(u(x,t) t)n!]
exp{i[K(m + p n ) - x - tot]} = \ - T ( x ,
(3.12)
is in fact the stress-displacement vector [3] and describes wave propagation, with wave vector of the form (3.1) and parameterized by K. Any such wave results from a linear combination of the six waves associated with the eigenvectors of (3.5). Indeed, if m and n are identified with the unit vectors of the x- and z-axes, then the problem of solving the equation of motion at fixed exponential dependence of u on x has been turned into an eigenvalue problem for the traction-displacement vector, thus providing the dependence of the solution on z. Equation (3.5) characterizes a basis for the set of solutions of the system of ordinary differential equations describing propagation of the traction-displacement vector in the direction of the z-axis. Such a basis has been subject to a set of "orthonormality" conditions as in (3.7). This has led to the determination of the completeness relation (3.9) or (3.10) and the spectral representation (3.11) of N.
Wave propagation in linear anisotropic viscoelastic media
1065
T h e p r o p a g a t i o n condition is n o w in the genuine f o r m (3.5) of an eigenvalue p r o b l e m . Y e t the u n k n o w n p is given by the d e t e r m i n a n t a l e q u a t i o n which is a sextic. T h e p r o c e d u r e of Sections 4 - 6 allows the d e t e r m i n a t i o n of the eigenvectors ~ to be i n d e p e n d e n t of the evaluation of p.
4. C H A N G E
OF THE REFERENCE
PAIR
C o n s i d e r the o r t h o r n o r m a l basis, say ~ , fi, i, that is o b t a i n e d by rotating the r e f e r e n c e pair m, n a b o u t the t axis of an angle 4,. Specifically we let = cos 4, m + sin 4,n,
fi = - s i n 4,m + cos 4,n,
i = t.
(4.1)
D e n o t e by t2 the c o r r e s p o n d i n g o r t h o g o n a l matrix, viz.
sin !)
o_-/-So, cos,0 of course f ~ r
= 13. L e t
~
= a n , c n , n ~,
A direct e v a l u a t i o n yields ~C~ lllCn
= cos 2 4,Qm + sin 4, cos 4,(R + R T) + sin 2 4,Qn, =
COS2 4,R - sin 2 4,R v + sin 4, cos 4,(Q~ - Qm),
fiCfi = sin 2 4,Qm - sin 4, cos 4,(R + R r ) + cos 2 4,Q.. F o r c o n v e n i e n c e let 0(4,) = sin z 4,Q,, - sin 4, cos 4,(R + R r ) + cos 2 4,Q,,
(4.2)
8 ( 4 , ) = cos 2 4,R - sin 2 4,R r + sin 4, cos 4,(Q, - Q,,).
(4.3)
W e have
0(4,) = 0~ ( 4, ), 0(o) = Q.,
0(to/2) = Q,,,,
I~(o) = R,
and
( ~ = D.~(4, + x/2)f~ r,
R = f ~ ( 4 , ) O r,
( ~ = D~2(4,)~ r,
(4.4)
0(4, + 7r/2) = n T O , ~ n ,
8(4,) = n~n,
0(4,) = nT(~.n.
(4.5)
or
O f course the periodicity p r o p e r t y of 0 and 8 , n a m e l y 0(4, + 7r) = 0(4,), holds.
8 ( 4 , + ~) = ~ ( 4 , ) ,
(4.6)
1066
G. CAVIGLIA and A. MORRO
Let
and .~, = f~A so that the propagation condition reads MA
(4.7)
= 0.
Consistent with (3.1), let ~k = [
- = ,~
(n~
+ p~),
where ff and/~ are to be determined. By (4.1) we obtain ff = K(cos ~b + p sin ~b),
ffp = K(-sin ~b + p cos ~b).
These conditions relate the eigenvalue p, in the new basis 61, fi, to the eigenvalue p in the initial basis m, n. Substitution of (4.4) into (4.7) and use of (4.2), (4.3) reduces (4.7) to
~M~
= o.
Of course, p and ,~ can be related to the solution of an eigenvalue problem in complete analogy with (3.5).
5. A O N E - P A R A M E T E R F A M I L Y OF E I G E N V A L U E P R O B L E M S For ease of comparison we write equation (2.3) in the form
(mh q-pl'lh)(Chqrs P°92mh6qrm,)(ms+pn~)Zr=O. K2
(5.1)
To avoid the direct investigation of (5.1) we consider an auxiliary one-parameter eigenvalue problem obtained by replacing the vector K(m + pn) with K(r~ + p~fi), namely
(r~lh+ P~nh)( Chqrs P£°-~ K mh6qrm.,)(ffts + Pe~fi~)[A+]r =0.
(5.2)
Since 61 and fi depend on the angle ~b, the unknown eigenvalue Pc, and the eigenvector A~ are expected to depend on 4). If 4)=0, (5.2) reduces to the component form (5.1) of the propagation condition. To give evidence to common features with the propagation condition, it is convenient to insert the matrices I),~, fi, t)~, which allow equation (5.2) to be written in the compact form ~r[p24,Q ~ + p~(fi
+ fi T) + Q,~ - ~P 0")2 (p62 sm- 2 4, -
2p~ sin 4) cos 4) + cOS2 ~b)13]f~A,~ = 0.
In a purely formal way let f2A+ = . ~ . Left multiplication by the orthogonal matrix f~ yields [p~0~ + p~(fi + f i r ) + 0,~ - p°J---~zK2(p~ sin 2 4) - 2p~ sin 4) cos 4) + cos 2 4))13]A~. = 0.
(5.3)
It is worth observing that (5.3) is equivalent to (4.7). We can prove that (5.3) is merely the propagation condition in the rotated basis fin, ft. For the time being it is simply a relation for the new unknowns A~ and p+. However we recognize at once that if ~o = 0, as in statics, (5.3) is the rotated form of (5.1) with p6 = p . In an analysis of the equilibrium equation, Ting [16] concludes that, under a rotation of the coordinate system about the z-axis, the (Stroh) eigenvectors transform according to the law of vectors. Our remark shows that Ting's statement can be generalized to the dynamic case.
Wave propagation in linear anisotropic viscoelastic media
1067
Comparison with (4.5) shows that equation (5.3) can be written in the equivalent form {p~Q~(4,) + p,[R~(4,) + Rr(4,)] + Q~(4, + Jr/2)}A~ = 0,
(5.4)
where P to2 • 2
P 0)2
Q,(4,) = 0(4,) - --T-Ksin 4,I3,
RK(4,) = ~(4,) + ~
sin 4, COS 4,13.
(5.5)
In view of (4.6) and (5.5) we have Q,,(4, + ;,r) = Q,,(4,),
R,,(4, + n:) = R,,(4,).
Consider the one-parameter "traction" L , = -iK[p,~Q~(4,) + Rr(4,)]A,
(5.6)
and let Q~(4,) be non-singular. Hence (5.4) is equivalent to the eigenvalue problem (5.7)
[N(4,) - ip , I 6 ] ~ , = 0 where ~'~=
L,~"
The 6 x 6 matrix N(4,) is given by
(N,(4,) N2(4,) 1%1(4,) = \ N 3 ( 4 , )
N4(4,)/,
where NI(&) = -iQ~-~(4,)R~(4,),
N2(&) = --Q~-](4,)/K
N3(4,) = K[R,,(4,)Q~(4,)R~r(4,) - Q,,(4, + ~/2)],
N4(4,) = -iR,,(4,)Q~-](4,).
The symmetry of QK(4,) implies that N2(4,) = N2r(4,),
N3(4,) = N3r(4,),
N4(4, ) = Nlr(4,).
(5.8)
Incidentally, the eigenvalue problem (5.7) reduces to the propagation condition (3.5) in the special case 4, = 0. To investigate the relationship between the eigenvalues p and p , is convenient to go back to the formulation (5.2) and to make a comparison with (3.2). In this regard we represent l~ and fi in (5.2) in terms of m and n as given in (4.1). Upon multiplication by 1/(cos 4, - p , sin 4,)2 and appropriate substitutions, we can write (5.2) in the form M , A , = 0, where (sin 4, + p~ cos 4,~2A sin 4, + p , cos 4, ( R + R r ) + Q m - Po21 K2 3. M4' = \cos 4, - p , sin 4,/ t~n + cos 4, - p,~ sin 4, Comparison with the definition (3.2) of M shows that the roots of the determinantal equation det M(p) = 0 are related with those of the equation det M , ( p , ) = 0 by p-
sin 4, + p , cos 4, cos 4, - p~, sin 4,
with inverse p, =
- s i n 4' + P cos 4,
cos 4, + p sin 4,
(5.9)
Accordingly, without any loss in generality we let A~ = A. In addition, substitution of p , from
1068
G. CAVIGLIA and A. MORRO
(5.9) in (5.6) and comparison with (3.3) shows that L,~ = L. Hence, in view of (5.7) we conclude that ~,~ = ~,
(5.10)
whatever the value of the parameter 4'. In words, the eigenvectors of N(4') are independent of the parameter 4' and then coincide with those of N. As an aside we observe that the invariance of the eigenvector is proved in [5] via a different procedure. The matrix N and the eigenvalue p,~ are shown to satisfy the differential equations dp6/d4' = - 1 _ p 2 ,
dN(4')/d4'
=
--16
--
N2(4'),
and hence the invariance of ~ is derived. Our approach, instead, is purely algebraic and, to our mind, simpler.
6. I N T E G R A L F O R M A L I S M Now we take advantage of (5.10) and obtain a very simple eigenvalue problem. Consider the expression (5.9) of p+. The integral of p6 from 0 to 21r is evaluated as follows. We let exp(i4') = z in the expression (5.9) of p~ so that the integral on [0, 21r] is changed to an integral over the unit circle, with centre at the origin, in the complex plane. Use of the residue theorem provides the result, viz.
£
~~p6 d4' = 2z¢i sgn(Im(p)).
(6.1)
Accordingly, integration of (5.7) with respect to 4' on [0, 2to] yields S~ = -sgn(Im(p))~,
(6.2)
where S is parameterized by K and
(s, S=
$3
S
'
with S,
21r Jo N,(4') d4',
i = 1. . . . . 4.
(6.3)
Observe that Q~(4') may be singular at isolated values of 4' and the same may occur to Ni(4'). Hence the integral (6.3) is meant as the principal value. Of course, by (5.8) S inherits the algebraic properties S 2 = S T,
S 3 : S T,
S4 = S T
(6.4)
Equation (6.2) represents essentially an average of the eigenvalue problem (5.7) and takes into account the fact that the eigenvectors are independent of the parameter 4'. The matrix S is parameterized by the value K of the wave vector k in the direction of m. The result (6.2) is important in that allows the determination of eigenvectors and eigenvalues of the propagation condition (3.5) without any recourse to the sextic equation for p. The matrix S is semisimple. Specifically, there exist only the multiple eigenvalues + 1 and - 1 , and the two corresponding eigenspaces have the same dimension as the multiplicity of the eigenvalues. This follows as an ultimate consequence of (5.10) and the assumption that the propagation condition (3.5) admits six distinct eigenvalues and six linearly independent eigenvectors.
Wave propagation in linear anisotropic viscoelastic media
1069
Denote by ~ , ot = 1 , . . . , 6, six linearly independent eigenvectors corresponding to the eigenvalues +1, - 1 . In principle, the ~ ' s may coincide with the eigenvectors of (3.5). In any case comparison with (6.2) shows that S2~o = S ( S ~ ) = ~o, = 1. . . . . 6. Since the eigenvectors constitute a basis for C 6 we conclude that S 2 -- 16.
(6.5)
Substitution into (6.5) of the block expression for S(K) and account for the relations (6.4) yields the identities SiS1 q- 8283 =
13,
S1S 2 -[- 82 ST = 0, 8381 q- s T s 3 : 0.
(6.6)
If S2 is non-singular, the first equation of (6.6) provides S 3 = S21(13 -- 82),
(6.7)
which is a useful property in practical applications whenever the explicit form of S is required.
7. F U N D A M E N T A L SYSTEM OF S O L U T I O N S Mathematically we now determine a basis for the time-harmonic solutions (2.2) to the equation of motion (2.1). In other words, we establish the set of propagation modes whose linear combinations generate any time-harmonic wave propagating in the unbounded anisotropic solid. Any such basis is said to be a fundamental system of solutions. The matrix S is known. Operatively, the first step is to determine the eigenvectors 4, of S. Since the eigenvalues of S are known a priori (+1), we determine the eigenvectors without any trouble to the evaluation of the eigenvalues. The second step is to determine the eigenvectors and the eigenvalues p of N that enter the wave solutions under consideration. To fix ideas, consider the eigenvalue + 1 of S and denote by {~7} the associated eigenvectors. For definiteness, we assume that the associated linear space is three-dimensional, so that j varies from 1 to 3. It is not restrictive to let (~7)TK~ = ~jh,
(7.1)
with j, h = 1, 2, 3. Incidentally, by (6.2), we are considering the eigenvectors associated with values of p with I m p < 0 . Indeed, the eigenvectors {~7} are required to generate the eigenvectors of N with eigenvalues p subject to I m p < 0. Consider a linear combination ~3=1 ajar, with complex coefficients aj, and require that it be an eigenvector of N with eigenvalue p. The coefficients aj are then subject to 3
3
j=l
j=l
Left multiplication of both sides by (~h~)rK and use of (7.1) yields 3
(~-~)rKN~;aj = ipah. j=l
This condition may be given the form of an eigenvalue problem as Aa = ipa
(7.2)
1070
G. CAVIGLIA and A. MORRO
where a = (a 1, a2, a3) T
is an unknown column vector and the entries of the matrix A are given by Ahj = ( ~ ) r K N ~ . The advantage of the formulation (7.2) is that the determinantal condition det[A - ipI3] = 0 in the unknown eigenvalue p of (3.5) is a complex polynomial of third degree. Hence the three roots can be found explicitly and the coefficients a l, a2, a3 are derived from (7.2) thus determining the eigenvectors ~ of the propagation condition (3.5) with I m p < 0. The first three components of ~ constitute the amplitude vector A. The derivation of the eigenvectors associated with I m p > 0 proceeds along the same lines. In conclusion, for every fixed value of K we determine the eigenvalues Px, Pz . . . . . P6 and the amplitude vectors. Hence the general plane-wave solution to (2.1) is a linear combination of the propagation modes, viz. 6
u(x, t) = ~
b,~A~ exp{i[K(m + p ~ n ) • x - o~t]}.
oe--I
The coefficients {b~} are determined, e.g. by boundary conditions at interfaces or initial values. For instance, in a reflection-transmission problem the incident wave is given and the six-tuple {b,~} for reflected and transmitted waves is determined by requiring that the continuity conditions at the interface hold. In that case, via a proper choice of m, account for Snell's law (cf. [14]) results in a common value of K for all waves involved.
8. W A V E S IN T R A N S V E R S E L Y
ISOTROPIC SOLIDS
Wave propagation in transversely isotropic solids is of interest with regard to many practical situations. In addition, it offers a remarkable example of application of the present theory. We observe that waves in transversely isotropic solids have been investigated by Chadwick et al. [5, 17]. Besides assuming that the solid is elastic, they let the wave vector k belong to the basal plane. Here we consider a viscoelastic solid and take the complex-valued wave vector k in a zonal plane. Although this is not the general case, it certainly provides a decisive generalization of the previous investigations. The complex-valued components of the tensor C are represented by using Voigt's notation, namely by considering a 6 × 6 symmetric matrix {C~,v} related to {Ciik/} by the correspondence rule 11 - 1, 22 - 2, 33 - 3, 23 - 4, 13 - 5, 12 - 6. The non-zero entries are given by
(C,~) =
Cll
C12
C13
C12 C~3
Cll C~3
C13 C33
(8.1)
C44 C44 C66
Wave propagation in linear anisotropic viscoelasticmedia the entries C~,~ being parametrized by oJ and C66 = ( e l l (3.2) can be given the form
(p2f 4p~c~O--ci4~2/K2
0 p2C44 + C66 -
C12)[2. The propagation condition p(C13 + C44)
p¢.02/1¢.2
0
1071
)A= O.
0 + C44-- p¢-02/l<2
p2C33
(8.2)
The resulting secular equation for p factorizes as ¢'02 2 "0)2"~ [ 2 (p2C44+666 P~)I(pC44-~-Cll-~)~Pf33JvC44
~0)2~ C44)2 ] = O, (8.3) K2 ] - p 2 ( C 1 3 -~-
as found first by Synge for transversely isotropic elastic solids [18]. Unlike the general case, the secular equation can be explicitly solved. To this end it is convenient to multiply (8.3) by K 6 and let/32 = K2p2;/3 is the component of the wave vector k along n. Hence (8.3) becomes [(p¢o 2 -- C44/3 2 -- E l l K2)(ptO 2 -- C33/3 2 - C44 K2) -- (C13 -{- C44)2K2/32](pto 2 -- C44/3 2 -- C66/(2) : 0,
for the new unknown/3. The vanishing of the first factor reads /34 __ (0/1 + 0/2 + r)/3 2 + 0/10/2 = 0,
where po) 2 0/1 -- - -
Cll
C33
C33
F ~--- --K21/ e l l \ C44
poJ 2
K2~
0/2 --
C~3
2 C1~3
Ca,
C33 C44
C33
K2~
ell
1|.\
C33
/
Letting = (O/1 "[- 0/2 "+ 1-')2 -- 40/10/2,
we can write the solutions for /32 as / 3 L = (0/, + 0/2 + r • v
r)/2,
where V~ is chosen with positive real part. We assume that /32 and /32 are distinct and non-zero. The vanishing of the second factor in (8.3) gives a third solution for/32 as /32 - PrO2 C44
C66 K2; C44
we l e t / 3 3 ~ 0 and/3~ #/3~" #/3~.2 By the definitions (4.2) and (4.3) we obtain I CI~ sin 2 ~b + C44 cos 2 4~ ~(4~) =
0 ~1( C ,t - C , z ) s i n e ~b
0
+ C44 cos 2 (~
C44 sin 2 ¢k + C33 c0s2 t~
0
C44 - CII~ in 4' cos
IC44 _ 1 (C,l - C12)]sin ~C44 cos 2 (~ - C13 sin z ~b
0
0
-(C~3 + C44)sin 6 cos 4'
8(6)=
--(el3 -t- C44)sin ~ cos t~
0
C13 cos2 ~ - C44 sin 2 ¢h
cos ~b (C33 -
Hence we have Q . = 0(0),
Qm = 00r/2),
R = ~(0).
0 1 C44)sin ~b cos ~b
1072
G. CAVIGLIA and A. MORRO
T h e matrices Q=(th) and RK(~b) are o b t a i n e d t h r o u g h substitution into (5.5). Meanwhile the matrices Ni are given by
0//
N1 = N4r = -iC13/C33 N 2 ----diag
( 1 KC44 ,
0
0
0
0
1
, t
KC44 ,
"" {\ {C23 \%3 N3=o,ag/K/~---Cu
)
+
,
K~33 ,
p~°2 -KC66 + po~ poJ21 K
K
K
/
T h e list of eigenvalues of N m a y then be written as {Kp,, a = 1 . . . . .
6} --- {/3,, a = 1 . . . . .
6} = {/31,/32, -/31, -/32,/33, -/33},
where/31, /32, /33 a r e chosen with positive imaginary part. Correspondingly, the eigenvectors of N are the o r d e r e d c o l u m n s of the matrix bI
b2
b]
b2
0
0
0
0
0
1
0
1
KC l
KC 2
--KC 1
- - KC 2
0
0
-dl
-d2
dl
d2
0
0
0
0
0
0
-i/33644
i/33C44
--Ka~
--Ka 2
--Kal
--Ka 2
0
0
(8.4)
where al,2 = i( C13Poa2 - C13C44 K2 Jr C33644/32,2),
bl.2 = pro 2 - C44 K2 - C33/32,2, C1,2 =/31,2(C13 -]- C44 ) dl,2 = i/31,2C44(pt02 + C13K2 _ C33/31,2).2
It is w o r t h r e m a r k i n g that here the d e t e r m i n a t i o n of the eigenvalues is p e r f o r m e d in a simple way thanks to the f a c t o r e d f o r m of the secular equation. Should the factorization not take place, the p r o c e d u r e b e c o m e s the s t a n d a r d one. Since we n e e d the matrices N(~b) and S we evaluate Q~l(4, ) and R~(~b)Q~I(th). W e find that Q~1(4, ) can be given in the f o r m
o!)
C
,
0 the constants
a, b, c, d, e being
given by
ira = - ( p r o 2 - C44 K2 - C33~2)K2/sin 2 q~, o'b = o'd = (C13 + C44)~'r3/sin 2 ~b, c = K2/[C44 sin 2 ~b(ff2 -/32)], (re = - ( p r o 2 - C]I K2 - C44sr2)K2/sin 2 q~ where Or = C ~ C 4 4 ( ~
~ -- ~ ) ( ~
-- ~ ) ,
and t" = K cot ¢. Similarly, we find that
R,~(~b)Q~-~(~)
=
(i
g
0
0
,
Wave propagation in linear anisotropic viscoelasticmedia
1073
where 0-~ = [(C13 q- C44)K4(C13/sin 2 q~ - Cl 3 - C44) - (p(..o2 - 644 K2 - C33~2)(p0) 2 q- C44K 2 - Cl 1K2)]~/K, 0-/~ = (C13 q- C44)(P 0)2 - Cll K2)(ff 2 q- K2) -- C13(P £02 - Cll K2 -- C44~2)K2/sin2 6,
0-t~-
0092 -- C66 K2 -~ C44 K2 C44(~2 _ /32) K'
o'd = (C13 --[-C44)(po) 2 - C44K2)(~ 2 q- K 2) -- C44(Po) 2 - C44 K2 - C33ff2)K2/sin2 4~, o'~ = (Cl3 q- C44)tc4(f44/sin 2 ~ - C13 - C44 ) - (poJ 2 - C44 K2 --[-C33K2)(po) 2 - Cll K2 - C44~2).
For formal simplicity we now let K be real and Im/3 > 0. By (6.3), the matrix S is determined through integration of N(~b) with respect to 4). It follows that Sl = S4r =
0 0
2
S1~13)
S1,31 0
where
=s1'13
K
.( C13(poJ2-- C44 K2)
633C44(7~1 -[-/32) \
/31/32
C33 C44 ) -
'
,
S1'31
C33(/3~ -[- /32) \
/31/32
are the roots of the secular equation with positive imaginary part. and /3, i = 1 , 2 , 3 , Analogously it follows that S~l 1 S2 =
0
0 )
$2,22
0
,
0
$2,33
where
=. -$2'11
(P°)2-C44 K2 C33C44(/31 q-/32) \
i
/31/32
) '}- C33 '
(pro 2 - Cl1K2 C44)"
$2'33 = -- 633644(~1 q-/32) \
/31/32
q-
Finally, S3 is evaluated by substitution into (6.7) of the expressions for $1 and $2. We find that
83 =
(1 -- S1,13S1,31)/$2,11 0 0 1/s2,22 0
0
0 ) 0 . (1 - s1,13s1,31)/$2,33
A direct check shows that the columns of the matrix (8.4) are eigenvectors of S. We now reverse the procedure and show that the eigenvalues and eigenvectors of the matrix N are recovered from the eigenvectors of S. A direct evaluation yields the three eigenvectors of S associated with the eigenvalue + 1 in the form ~- = 0-1(1, O, O, 1/$2,11 , O, --S1,31/$2,33)T, ~2~ = 0"2(0, 0, 1, -s~,,3/s2.,1, O,
1/S2,33) T,
~- = 0"3(0, 1, 0, 0, i/33644, 0) T.
They meet the normalization condition (3.7) provided the pertinent constants 0" are taken to be given by
o21 = s2,1~/2,
0-2 = S2,33/2,
0-2 = 1/(2i/33C44)"
1074
G. CAVIGLIA and A. MORRO
The linear system for the coefficients of the superposition eigenvectors of N takes the form
of
(A1. A2 0)(al) A021
A22 - ip
0
a2
0
A33 - ip
a3
the
vectors
t~÷
yielding
the
=0
where 2[- [ C23 All = O'I/K|---k \ C33
Cll
[
0(.02 613s1 31 + ..... + 2i ] /¢ C33 $2,33
.C13 1
i
A12 = A21 = o'1o'2~- l -C33s2,33
A22 = 0"2 0o)2 \
2i s1'13
K -]-
1
x
$2.11
K644 $2,11
~- 1 s1,13
1
1
s2311
-
~
I
/£C33 $2.33 ]
S1,31~
- - T -+ 2 ' KC44s2,11 KC33s2,33]
$2,11
1 sin3 2 K644S2,11
1
1 1 K C33 s~733 , '
A33 = --i[33/K.
It follows immediately that there exists an eigenvalue, say P3, given by P3 =
--~3/K
which belongs to the eigenvector of components (0, 1, 0, 0 , / ~ 3 C 4 4 , 0) T
and corresponds to letting al = a 2 = 0 , and a 3 = 1. The determination of the other two eigenvectors is straightforward in principle but requires involved calculations. However, we already known that the eigenvectors are given by the third and the fourth columns of (8.4). Hence it is easily seen that they follow from combinations of ~? and ~ with coefficiems b~, --KCl, and b2, - KC2, respectively. Acknowledgements--The authors are grateful to Professors P. Chadwick and J. Lothe for providing an unpublished article (P,C.) and a reprint of papers which were not easy to get (J.L.).
REFERENCES
[1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18]
M. J. P. MUSGRAVE, Crystal Acoustics. Holden-Day, Calif. (1970). F. I. FEDOROV, Theory o f Elastic Waves in Crystals. Plenum Press, New York (1968). S. CRAMPIN, Wave Motion 3, 343 (1981). P. CHADWICK, Recent developments in the theory of elastic surface and interracial waves. In Elastic Wave Propagation (Edited by M. F. McCARTHY and M. A. HAYES), pp. 3-16. Elsevier, Amsterdam (1989). P. CHADWICK and G. D. SMITH, Adv. Appl. Mech. 17, 303 (1977). A. K. HEAD, J. Elasticity 9, 9 (1979). D. M. BARNETT and J. LOTHE, Physica Norvegica 7, 13 (1973). J. LOTHE and D. B. BARNETT, J. Appl. Phys. 47, 428 (1976). J. LOTHE, Uniformly moving dislocations; surface waves. In Elastic Strain Fields and Dislocation Mobility (Edited by V. L. INDENBOM and J. LOTHE), pp. 447-487. Elsevier, Amsterdam (1992). P. CHADWICK and N. J. WILSON, Proc. R, Soc. Lond. A 438, 225 (1992). M. E. GURTIN, An Introduction to Continuum Mechanics. Academic Press, New York (1981). A. J. M. SPENCER, Continuum Mechanics. Longman, London (1980). G. A. MAUGIN, Continuum Mechanics o f Electromagnetic Solids. North-Holland, Amsterdam (1988). G. CAVIGL1A and A. MORRO, lnhomogeneous Waves in Solids and Fluids. World Scientific, Singapore (1992). A. N. STROH, J. Math. Phys. 41, 77 (1962). T. C. T. TING, Int. J. Solids Structures 18, 139 (1982). V. I. ALSHITS and J. LOTHE, Wave Motion 3, 297 (1981). J. L. SYNGE, J. Math. Phys. 35, 323 (1956). (Received 8 March 1994; accepted 11 March 1994)