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A theory of pulse propagation in anisotropic elastic solids
WAVE MOTION 9 (1987) 509-532 NORTH-HOLLAND
A THEORY
OF PULSE
509
PROPAGATION
IN ANISOTROPIC
ELASTIC
SOLIDS
A. N. N O R R I S Department of Mechanics and Materials Science, Rutgers University, Piscataway, NJ 08855-0909, U.S.A.
Received 9 March 1987, Revised 24 June 1987
A theory is described for the propagation of pulses in anisotropic elastic media. The pulse is initially defined by a harmonically modulated Gaussian envelope. As it propagates the pulse remains Gaussian, its spatial form characterized by a complex-valued envelope tensor. The center of the pulse follows the ray path defined by the initial velocity direction of the pulse. Relatively simple expressions are presented for the evolution of the amplitude and phase of the pulse in terms of the wave velocity, the phase slowness and unit displacement vectors. The spreading of the pulse is characterized by a spreading matrix. Explicit equations are given for this matrix in a transversely isotropic material. The rate of spreading can vary considerably, depending upon the direction of propagation. New reflected and transmitted pulses are created when a pulse strikes an interface of material discontinuity. Relations are given for the new envelope tensors in terms of the incident pulse parameters. The theory provides a convenient method to describe the evolution and change of shape of an ultrasonic pulse as it traverses a piecewise homogeneous solid. Numerical simulations are presented for pulses in a strongly anisotropic fiber reinforced composite.
I. Introduction M a n y f u n d a m e n t a l p r o b l e m s h a v e b e e n s o l v e d for w a v e m o t i o n in a n i s o t r o p i c elastic m e d i a . A t h o r o u g h review o f p l a n e wave p r o p a g a t i o n a n d reflection is given b y M u s g r a v e [1]. S o l u t i o n s f o r r a d i a t i o n f r o m p o i n t forces in full a n d h a l f - s p a c e s are also k n o w n [2, 3]. The m o r e recent m o n o g r a p h o f P a y t o n [4] c o n s i d e r s t r a n s v e r s e l y i s o t r o p i c m e d i a in d e t a i l a n d also lists m o r e recent references. H o w e v e r , t h e r e is a g r o w i n g n e e d to u n d e r s t a n d a n d m o d e l the p r o p a g a t i o n a n d scattering o f finite-sized p u l s e s t h r o u g h a n i s o t r o p i c materials. O n e a r e a o f a p p l i c a t i o n , for e x a m p l e , is in u l t r a s o n i c i n s p e c t i o n o f f i b e r - r e i n f o r c e d c o m p o s i t e s for p u r p o s e s o f n o n d e s t r u c t i v e e v a l u a t i o n . T h e p u l s e s c o n s i d e r e d h e r e are a s s u m e d to be l o c a l i z e d in the f o r m o f h a r m o n i c a l l y m o d u l a t e d G a u s s i a n e n v e l o p e s . T h e centers o f the G a u s s i a n w a v e p a c k e t s p r o a g a t e a l o n g the u s u a l rays or c h a r a c t e r i s t i c curves. T h e p r e s e n t t h e o r y is an e x t e n s i o n o f the a n a l y s i s o f N o r r i s [5] on G a u s s i a n wave p a c k e t s in i n h o m o g e n e o u s a c o u s t i c m e d i a . The initial a n s a t z u s e d h e r e is s i m i l a r for that o f t i m e h a r m o n i c G a u s s i a n b e a m s ; see [6] for a d i s c u s s i o n o f this in i s o t r o p i c elastic solids. T h e G a u s s i a n b e a m s o l u t i o n s are o f infinite extent in the d i r e c t i o n o f p r o p a g a t i o n . The p r e s e n t t h e o r y i n c l u d e s an extra d e g r e e o f f r e e d o m in t h a t the p u l s e is o f finite length. I n the limit as this length b e c o m e s infinite, it r e d u c e s to time h a r m o n i c G a u s s i a n b e a m theory. It is i m p o r t a n t to n o t e t h a t the w a v e p a c k e t s o f this p a p e r a n d the p a p e r ' s references [5, 7] are n o t s i m p l y the F o u r i e r t r a n s f o r m o f a t i m e h a r m o n i c G a u s s i a n b e a m . T h e d i s t i n c t i o n is e v i d e n t f r o m the e x a m p l e s in [5] o f a w a v e p a c k e t in an a c o u s t i c m e d i u m o f c o n s t a n t v e l o c i t y gradient. S o m e o f the cases t h e r e c a n n o t be e x p l a i n e d b y G a u s s i a n b e a m s . Specifically, if the wave p a c k e t is i n i t i a l l y e x t e n d e d in the d i r e c t i o n o f p r o p a g a t i o n , t h e n as the c e n t e r o f the p a c k e t follows the c u r v e d ray p a t h the b u l k o f the p a c k e t rotates relative to the r a y direction. G a u s s i a n b e a m t h e o r y b a s e d u p o n a single b e a m w o u l d p r e d i c t the s o l u t i o n to be s p a t i a l l y s y m m e t r i c a b o u t the r a y p a t h . 0165-2125/87/$3.50 O 1987 Elsevier Science Publishers B.V. (North-Holland)
A.N. Norris / Pulses in anisotropic solids
510
Norris [7] considered Gaussian wave packets in inhomogeneous isotropic elastic media and provided explicit formulae for the jump conditions at interfaces. Elastic wave propagation in anisotropic media is complicated by the fact that the wave and phase velocity vectors are distinct for any given wave mode. This leads to the curious result that wavefronts are not orthogonal to the direction of propagation. The general equations of rays and wavefront curvatures in inhomogeneous anisotropic media have been discussed previously, e.g. [8,9]. The present theory includes these curvature evolution equations as a special case. However, for simplicity only piecewise homogenous media are considered here. The generalization to smoothly varying media is not difficult, but the significance of the results can be obscured by the required amount of excessive notation. The basic equations are described in Section 2. The general Gaussain wave packet solution is derived and discussed in Section 3. A complete analysis of the reflection/transmission problem for a Gaussian wave packet incident upon an interface is given in Section 4.
2. Eikonal and transport equations
The equations of motion for the displacement field u(x, t) in a homogeneous, anisotropic linearly elastic solid are Cok~Uk, jl--pUi,, = 0,
(1)
where Cokt = Ck,j = Cj~kt is the elastic modulus tensor, p the density, and the summation convention on repeated subscripts is assumed. Let [5],
u(x, t) = C(x, t) e i~(x''),
(2)
p=V6,
(3)
then (1) becomes, ( ito )2[ C ~ktpjpl-- pdp2 8ik] Uk
+ (ito)[ C~jkl(pjUk, t + p l U k j + &,jt Uk) -- 2pqb, U~,, - pcb, Ui] + [Ci~kt Uk,~t -- P Ui,,,] = 0.
(4)
Now consider the asymptotic limit of to >>1. Here to denotes the center frequency of the solution. Assuming the ansatz co
U ( x , t ) = ~ (ito)-nU(")(x, t)
(5)
n=0
equation (4) reduces to a sequence of asymptotic equations. The first of these is 2
(o) _
( C OktP~Pt -- pgb, 8~k) Uk -- O.
(6)
A necessary condition for the existence of a nontrivial solution to (6) is det( C ijkt pjpt - Ptb, 2 6,k) = 0.
(7)
In general, there are three independent solutions to (7), of the form [1] cb,2 - f , , (2 p )
= 0,
m = 1, 2, 3.
(8)
A.N. Norris / Pulses in anisotropic solids
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Each function fro(P) defines a slowness surface. In the case of an isotropic medium, two of the slowness surfaces coalesce and the explicit formulae are f , ( p ) = CLIP" p]'/2,
f2(P) = Cx[p " p],/2
(9)
where CL and cT are the longitudinal and transverse wave speeds. The three separate identities in (8) are known as the anisotropic eikonal equations. The next in the sequence of asymptotic equations is ,-'k 2
, - ~-e~,'-'
~,, - p ' C ' , , U I ° ) ]
(1)
+ (Coklpspl --pdp,Sik) Uk = O.
(10)
Multiply this by UI °) and sum over i, using (6), to obtain the anisotropic transport equation
C okt( pylO) u(kO)) j - t '~,., ~ w t ~,,(o), i t~iT(o)~Lt=O
(11)
3. The Gaussian wave packet solution and evolution equations 3.1. The ray velocity
Consider the following root of the eikonal equation (8), ok, + f ( p ) = 0,
(12)
where f ( p ) is any one of f i, f2 or f3- Equation (12) is a first-order partial differential equation in ~b of the form H ( x , t,p, 4o,, dp)=- ckt + f ( p ) =0.
(13)
This admits of solutions along characteristic curves in space-time. Let time be the ray parameter along these curves, defined by x = ~(t), and let an overdot represent the total derivative with respect to t along a ray. Then the Hamilton-Jacobi equations are i = aH/ack, = 1, = -on/ox
= O,
~ = aH/ap = af(p)/ap, ~k, = - O H ~ o r = O.
(14a, b) (14c, d)
Equation (14d) implies tk, is constant on a ray. Without loss o f generality, this constant can be taken as - 1 , so that equation (12) for the slowness surface becomes f ( p ) = 1.
(15)
Equation (14c) show that the slowness vector is constant on a ray, and hence, from (14b), the rays are straight lines. These equations are easily generalized to inhomogeneous media (see [8]). Define the wave velocity vector c as c=af(p)/Op.
(16)
Thus, c is perpendicular to the slowness surface at p [1]. A more explicit formulation of equation (16) is given in Appendix A. In general (see Appendix A), p. c = 1
(17)
512
A.N. Norris / Pulses in anisotropic solids
and (18)
p^c~O.
Define the wave and phase speeds, c and v, respectively, as c = ( e . c) 1 / 2 ,
v = ( p . p) -1/:
(19)
Therefore, by (17) and (19),
c/> v
(20)
with equality in an isotropic medium, in which case the wave velocity and slowness vectors are parallel (p ^ c = 0). Finally, note that the ray equation in a homogeneous medium is ~ ( t ) = :~(o) + tc.
(21)
3.2. Second-order derivatives o f the phase
So far, we have derived equations for the evolution along a ray of V~b = p and 4~,, both of which are constant. Next consider the matrix of second-order spatial derivatives, M ( t ) = VV~b(~(t), t).
(22)
Along a ray, both = V qbt + M ~ = V qb, + M c = O
(23)
and ~, = 4~, + c. V4,, = 0 .
(24)
It then follows that V 4~, = - M e ,
(25)
c~t, = cT Mc.
(26)
Thus, all the second-order derivatives of the phase can be written in terms of M ( t ) . Let
A(t)=
O~( t ) Oa '
B(t)-
Op( t ) dot '
(27)
where a is some vector independent of t, e.g., a = ~(0), the initial ray position. Then, R ( t ) =- M -l = A B -1
(28)
R ( t ) = A B - ' - A B - ' B B -~.
(29)
and
The values of A and B follow by taking the variation of equations (14b) and (14c) with respect to a. Thus, A = NB,
B = 0,
(30a, b)
A.N. Norris / Pulses in anisotropic solids
513
where
N o =-a2f(p)/Op, Opj,
(31)
Equations (29)-(31) imply (32)
R=N. Since N is constant, equations (28) and (32) give
M ( t ) = M ( 0 ) [ I + tNM(O) ]-'.
(33)
An explicit expression for N of (31) is given in Appendix A (equations (A.12), (A.13) and (A.23)). It is shown in (A.15) and (A.19) that the real-valued matrix N projects onto the plane perpendicular to p, and so one of its eigenvalues is always zero. The other two eigenvalues are related to the principal curvatures of the slowness surface f ( p ) = 1. Define normal points on the slowness surface as those points at which these two eigenvalues are nonzero. Points that are not normal are inflection points and correspond to cusps on the wave surface. The regions between the cusps define lids on the wave surfaces [2-4]. The areas on the slowness surface associated with the lids are concave. In these regions, the matrix N possesses one or two negative eigenvalues. The two eigenvalues are positive in regions where the slowness surface is convex. The present theory is valid in normal regions of the slowness surface, which thus includes convex and concave parts of the surface. For example, in transversely isotropic materials, points o f inflection can only occur on the quasi-transverse (qT) slowness surface, and then only for certain combinations of moduli [4]. These features are illustrated by example below. We are now able to approximate the phase ~b(x, t) locally in space and time about the ray position ~(r) at time r. Let Ax = x-.~(~'),
At = t - r .
(34)
Then a second-order Taylor series approximation yields, using (3), (22), (25), (26), (34) and ~b, = - 1 , ~b(x, t) = ~bo+ (p" A x - At) + ½(Ax-- c A t ) T M ( r ) ( A x - cat),
(35)
where 4)0 is a constant. Since p and c are real vectors, it follows that Im[~b(x, t) - ~b(£(r), ~')] = ½(Ax- c A t ) r Im M(~')(Ax - c At),
(36)
It is clear from (2) and (36) that the solution is in the form of a localized Gaussian wave packet (GWP) if and only if Im M is positive definite. It can be shown, using (30) and the methods of [5, Appendix B] that Im M ( r ) is positive definite for all I" if Im M ( 0 ) is positive definite. The other necessary conditions, which are physically obvious, are that M ( 0 ) is symmetric and det A(0) ~ 0. Note from (35) that the linear correction to the phases, ( p . A x - At), propagates with the phase velocity p/I p 12. However, the center of the GWP is defined as the point at which the quantity in equation (36) is zero. Therefore, the center propagates with the wave velocity c. This is also apparent from (21). This discrepancy between the two velocities is characteristic of anisotropic media.
3.3. The transport equation Let g be the unit eigenvector of (6), so that U(°)= u(O)g.
(37)
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Since g is constant, the transport equation (11) reduces to aU~°)2 aU<°)2 [ ( ag,g~) ] U~0)~CqklgigkPj c3X------~--Pqbt--"~+ Cijkl gigkMjl+pj~pm Mml -pMijciej -0.
(38)
Equations (22) and (26) have been used in deriving (38) from (11). Let O<°)(t) be the Value of U ~°) along the ray. Then, since tht = - 1 and from the identities (A.8) and (A.11), equation (38) reduces to ( ~ t + e • V)0<0)2 + NoMo U --- ¢',~o)2= ~,a
(39)
d log[ 0 ~°)2]+ Tr[NM( t)] -- 0.
(40)
or
Noting from (28) and (30a) that d
log det A(t) = Tr(NM),
(41)
equation (40) can then be solved to get
[~-(°)(t)
~-
rdet
L~e~~OJ
(42) "
In this form, det A(t) can be identified as the Jacobian of the mapping ~(0)-*2(t). Alternatively, since M ( t ) is known explicitly from (33), equation (40) can be integrated to yield O(°)(t) = 0(°)(0)/[det(l + tNM(O)] ;/2
(43)
It can also be shown that d e t A ( t ) # 0 if (1) d e t A ( 0 ) # 0 , (2) M(0) is symmetric, and (3) Im M(0) is positive definite. These are the same conditions that ensure Im M ( t ) as positive definite. The method of proof follows that procedure outlined in [5, Appendix B].
3.4. The Gaussian wave packet The Gaussian wave packet solution follows from (2), (35) and (43) as
u(x, t)=
U<0),0 , [det M ( t ) ] l / 2 ~ )g[~et--~J exp{ito[p. (X--~)+½(X--~)TMq t)(x--2)]},
(44)
where 2 = ~ ( t ) is given in (21), and M(t) in (33).
3.5. Discussion Given an initial phase slowness vector p and the ray velocity vector c of (16), the center of the GWP propagates according to (21). The shape of the GWP about its center is determined by the tensor M(t), which evolves according to (33). It is shown in (A.15) and (A.19) that N projects onto the plane perpendicular to the phase slowness direction. Therefore, in order to understand the evolution of M, it is useful to define an orthonormal triad {dl, d2, d3} such that d3 = vp, and dl, (/2 are the principal directions of N. With respect to this basis,
N = SldlaT+ S2d2a~,
(45)
A.N. Norris / Pulses in anisotropic solids
515
where S1 and $2 have dimensions of (speed) 2, and shall be called the "spreading factors." Obviously, S~ and $2 depend upon the direction of propagation. They may be negative for certain directions in anisotropic media, as illustrated in the example below. By comparison, in an isotropic medium the directions d~ and d2 are orthogonal to the ray direction, and $1 = $2 = c 2, where c is the wave speed. Define h4tij, i, j = 1, 2, 3, as the components of M in the basis {d~, d2, d3}, or M = )(l~did~.
(46)
In particular, let M(0) have its principal directions coincide with this basis, so that/~r(0) is diagonal. Then h4t(t) remains diagonal for t > 0, and from (33), (45) and (46) / ~ l l ( t ) = / ~ l l ( 0 ) / [ 1 + tS~/~tH(0)] ,
(47a)
h4t22(t) = 2~t22(0)/[ 1 + tS22VI22(O)],
(47b)
)l,]raa(t) = h4"33(0).
(47c)
The form of the GWP remains constant in the phase direction, but it broadens in the orthogonal directions d~ and d2. The widths in these directions depends upon Im hT/l~(t) and I m / ~ t 2 2 ( t ) , respectively. The rate of broadening depends upon the values of S~ and $2. Since these generally differ, the GWP will broaden faster in one direction than in the other. 3.6. Numerical example f o r a transversely isotropic composite
Consider a transversely isotropic composite of parallel cylindrical glass fibers in epoxy matrix, with the fibers 60% of the bulk volume. The composite may be approximated for quasistatic deformation as a homogeneous transversely isotropic solid. The quasistatic approximation should be valid for dynamic wave propagation in the composite if the wave length is much longer than the fiber thickness. This is often the case for ultrasonic studies, and will be assumed here. The constituent parameters are (A,/.t, p) = (13.3, 29.9, 2.55) for glass, and (0.89, 1.28, 1.25) for epoxy, where A and/~ are Lain6 moduli, p density and the units are GPa and gm/cm 3, respectively. Assuming the fibers are aligned in the x~-direction, this direction is the symmetry axis of the transversely isotropic medium. The effective moduli follow from, for example, the theory of [10] as ( C ~ , C33 , C13 , C66 , p ) : (43.4, 10.7, 2.2, 4.4, 2.0), in the same units as before. Figures 1 and 2 show the slowness and wave surfaces of this composite for wave motion polarized in the x~x3-plane, i.e. for qL and qT modes of propagation. These surfaces are defined in Appendix B. Note that the qT surface possesses points of inflection. The qT wave surface has corresponding cusp points. The spreading matrix N is defined explicitly in Appendix B for the three wave types. For wave motion polarized in the plane of XlX3, the eigenvalues S~ and $2 of N are $2 = N22 (see (B.17)), and S 1 is the eigenvalue of the 2 x 2 symmetric matrix of elements NH, NI3 and N33 (see (B.14)-(B.16)). Thus, S~ is the spreading factor in the X~Xa-plane. It is zero at the points of inflection on the qT slowness surface, and becomes negative between these points. Everywhere else it is positive. A plot of S~ is shown in Fig. 3. The striking feature of this plot is the large variation in the value of S~. The corresponding plot for an isotropic solid is two circular arcs, signifying equal spreading in all directions. The present theory is incomplete at the points of inflection on the qT slowness surface. For these discrete directions, the spreading is zero, and the theory predicts no change in the GWP, apart from the spreading associated with $2. The pulse then acts like a collimated beam. However, this is not correct for long times. The correct behavior requires a more sophisticated analysis than presently considered. Specifically, we need to look at cubic terms in the phase and higher-order transport terms (see [11] for a relevant discussion).
516
A.N. Norris / Pulses in anisotropic solids co
c3-
P~-
o
Q i
0.0
0.2
i
0.4
i
0.6
0.8
Pl
Fig. 1. Slowness surfaces of the qL (inner) and qT (outer) waves in a transversely isotropic solid. The material models a fiberreinforced, epoxy matrix composite. The fibers are in the xt-direction, and the units of slowness are p.s/mm.
Cs
]
3~
2
4i
5I
61 Fig. 2. The wave velocity surfaces corresponding to the slowness surfaces of Fig. 1. Units are mm/~s.
T h e large v a r i a t i o n in s p r e a d i n g is b e s t i l l u s t r a t e d b y e x a m p l e . C o n s i d e r a flat u l t r a s o n i c t r a n s d u c e r b o n d e d to a s a m p l e o f the c o m p o s i t e such t h a t the n o r m a l to the t r a n s d u c e r face is in the XlX3-plane, a n d m a k e s an angle ~ with the x3-axis. T h e t r a n s d u c e r has a c e n t e r f r e q u e n c y o f 5 M H z , a n d emits a G a u s s i a n p u l s e o f initial w i d t h 5 mm. The initial s h a p e o f the p u l s e is best defined b y the m a t r i x / ~ i n t r o d u c e d in (60) b e l o w . In this c o n f i g u r a t i o n , we have /17/(0) = diag(/(/~](0), ]~/1](0),/~/33(0)).
A.N. Norris / Pulses in anisotropic solids
518
~:
900
W : 20*
t :0
t :
~:0
20
~':90
°
t:20
°
t =20
Fig. 4. Plots of a qL pulse propagating in a transversely isotropic medium, approximating a fibre/epoxy composite. The direction of propagation is defined by ~0,the angle the phase velocity makes with the x3-axis. The initial pulse is shown for qJ= 90°. The initial center is at (0, 0). The units are mm and p.s. The plots show the envelope, defined as u(x, t).g exp[-ioJp- (x-~(t))]. The phase factor exp[-io~p • ( x - i ( t ) ) ] removes the oscillations in the p-direction (see (44)) and is included to improve the visual appearance.
Let the i n c i d e n t G W P , d e n o t e d b y u (]~, b e o f the f o r m given in (44). It is c h a r a c t e r i z e d b y its s l o w n e s s v e c t o r p(]), with c o r r e s p o n d i n g d i s p l a c e m e n t v e c t o r g(~). The a m p l i t u d e a n d s h a p e o f u ~]) at i n c i d e n c e are d e t e r m i n e d b y U(°)I(to) a n d M(x)(to). The b o u n d a r y c o n d i t i o n s o f c o n t i n u i t y o f d i s p l a c e m e n t a n d n o r m a l t r a c t i o n s in the n e i g h b o r h o o d o f (Xo, to) are satisfied by i n t r o d u c i n g reflected a n d t r a n s m i t t e d G W P s . In g e n e r a l , t h e r e will be three o f each, o n e c o r r e s p o n d i n g to the q u a s i - l o n g i t u d i n a l (qL) wave a n d t h e o t h e r two to the q u a s i - t r a n s v e r s e waves (qT1 a n d qT2) p o s s i b l e o n either side o f S. Let the reflected a n d t r a n s m i t t e d fields be respectively, u (R~)
(49)
U (Tc~),
(50)
= qL,qTl,qT2
and = qL,qTl,qT2
w h e r e each o f these six G W P s has its a s s o c i a t e d p, g, U(°)(to) a n d M(to).
A.N. Norris / Pulses in anisotropic solids
519
4.2. Determining the slowness vectors Each of the reflected and transmitted GWPs is of the form of equation (2), or more specifically, equation (44). Define the phase ~b(x, t) of each of these GWPs to equal the incident phase at (x, to). Then, the phases must remain continuous in the neighborhood of (Xo, to) on S. This neighborhood is given by (48), therefore there are three first-order conditions:
ckt,
O~blO~i, i = 1, 2
are continuous at (Xo, to).
(51)
These conditions simplify, using (3) and (49), to give that O~
3x
3 -V4~.--~j=p.tj,
j=1,2
are continuous at (xo, to).
(52)
These two equations determine two components of the slowness vectors. The third component follows from the facts that ~bt = - 1 is continuous and that each p satisfies an equation of the form (12). The sign of the third component o f p, i . e . p , t3, is obtained by requiring that the associated wave velocity vector c have the appropriate direction. Thus, c. t3 should be of the same sign as c (~). t3 for the transmitted GWPs, but of opposite sign for the reflected GWPs.
4.3. Determining the displacement vectors and amplitudes Having found the slowness vector p for each new GWP, the associated wave velocity vector c and unit displacement vector g follow from (6) and (A.8). The continuity of total displacement at (Xo, to) then implies
U(°)X( to)g~I) +
~,
U~°)R~(to)g ~R~) =
a = q L , q T 1, q T 2
~
U(°)T'( to)g (TcO.
(53)
a = q L,qTI ,qT2
Similarly, the continuity of normal traction conditions are
r
CO)" ijkl ~3j L UW)I( to)g(kl)p~1)+
•
u(O)R~( to)g(kR~)pl R~) ] /
ot = q L , q T l , q T 2
= r~(2), ,-- Uk,'3j
Y.
u(O)T~(to)g(kT~)plT~),
i = 1, 2, 3,
(54)
ot = q L , q T l , q T 2
where C (1) and C ~2) are the elastic moduli tensors on either side of S. The six amplitudes follow from the six equations in (53) and (54).
4.4. The M-tensors of the reflected and transmitted GWPs The six elements of the complex, symmetric M-tensors follow from the six second-order conditions that r~,, 02qb/3t 3~j and 32t.b/3~:i 35, i, j = 1, 2 are all continuous. The first of these, ~ , continuous, implies using (26) that
cTMc is continuous.
(55)
The second two conditions simplify using (25), (48) and 32~ =V~b, 3x
at 3~
"37
(56)
A.N. Norris / Pulses in anisotropic solids
520 to give eTMtj,
j=l,2
(57)
are continuous. The final three conditions become, using c924~
[ c~x\ T
[ ~x \
~2 x
(58)
(3), (22) and (48), the three conditions tTMtj + 2D,jp" t3,
£ k = 1, 2
are continuous.
(59)
Further simplification of these conditions requires that M be referrred to a specific basis. Previously it was shown that the basis {dl, d2, d3} is the natural one for considering the evolution of M. In addition, a ray coordinate system may be defined by the orthonormal triad {el, e2, e3}, where e3 is the unit vector in the direction of c. Obviously, these triads coincide for isotropy. There is also the basis {tl, tz, t3} defined by the surface S at Xo. It is not obvious which basis is most advantageous for simplifying the jump conditions. The answer in fact is that none of these is the best, but rather that M be referred to a n o n - o r t h o g o n a l basis defined by the triad {~1, e2, e3}, where ~ = e~, ~2 = e2 and e3 = d3 = vp. Thus, define the 3 x 3 symmetric complex matrix If/u, i, j = 1, 2, 3 by M = hT/~,~T.
(60)
The reason for choosing the non-orthogonal basis {el, e2, e3} will become apparent presently. The conditions in (55), (57) and (59) indicate that the hT/-matrix transforms discontinuously from the incident hT/ to the transmitted and reflected matrices. Thus, the individual elements suffer jumps, the values of which are determined by (55), (57) and (59). The first jump condition, (55), implies, using (17), (19) and (60), that vZhT/33 is continuous.
(61)
The two jump conditions of (57) imply that vlQIi3e i • t2, j = 1, 2
are continuous.
(62)
These conditions simplify, using (51), (52) and (61), to v Y. hT/i3~i"tj, j = l , 2 i=1,2
are continuous.
(63)
For each of the bases { ~ , e2, e3}, define the 2 × 2 transformation matrix Q by Qq=ti.~=ti.ej,
i, j = 1, 2.
(64)
Thus, det Q = t3 • e3
(65)
A.N. Norris/Pulses in anisotropicsolids
521
The jump conditions (63) now become
v ~, Qij~3,
i=1,2
j~1,2
(66)
are continuous. The remaining three jump conditions of (59), are, using (64), that
~.
Q,kQ;,]VIk,+(6" ~3) ~. Qj,MI,3+(t~" ~3) ~
k.I = 1,2
1 = 1,2
+2(t3.p)Du,
Q,kMlk3+(t," ~3)(tj" ~3)h7/33
k = 1.2
i , j = 1,2 (67)
are continuous. These relations simplify further upon noting
t,. ~3 = vt,. p
(68)
and using the previous conditions of (51), (52), (61) and (66), to give
~.
Q,kQ~,IVIki+2--Dot3. ~3,
k , l = l,2
i,j=1,2
I)
(69)
are continuous.
Let a denote any one of the six reflected and transmitted GWPs. Then the six jump conditions can be solved explicitly, from (61), (66) and (69), to give the matrices AT/(~)= AT/c~)(to) as DO) 2
g,t(~) - V~( a ) 2 ~(~) ~va 3 3 , .L.a 33
(70a)
- -
i3 = v ( . ) k~l 2 ~'U •,
=
,:
Vjk 1" k3,
i = l, 2,
(70b)
,
[") ( a ) - l f ) (or) - 1 / ' ~ ( I ) ~ ( 1 )
~'A ( ! )
k,l,m,n = l,2
+2(t3.p('-t3.p
<")) ~
QI~)-'Q)7)-'D~,
i,j=l,2.
(70c)
k,/=l,2
It should now be apparent why we originally chose to represent M in the non-orthogonal basis {~, e2, ea}. By so doing, relatively simple equations have been obtained for the new 217/-matrices. These equations are decoupled as much as possible. Any other choice of basis would result in greater coupling between the coefficient hT/o. Note that the form of (70) is very similar to those derived previously for an isotropic medium [7].
4.5. Simplification and localization properties The six jump conditions of (70) can be written succinctly in matrix form as ]$~(~) = (S(~)-'S(I))l~l(O(S(~)-~s(l))T + 2t3 " (p(I)_p(~))Q(~)-IDQ(~)-~+ '
(71)
where S (') is the 3 x 3 matrix associated with the 2 x 2 matrix Q(~), (72)
522
A.N. Norris / I~lses in anisotropic solids
Also, the last term in (71), which represents a 2 x 2 matrix must be understood to apply only to the i, j = 1, 2 elements ofhT/~). This term is identically zero if the interface is locally flat at the point of incidence. Note that it is preferable to express M in the basis {d~, d2, d3} for the purposes of calculating the evolution of the M-tensors before and after incidence. The required transformations hdr <-->]17/is
lfl= V~IV T,
(73)
where the 3 x 3 matrix V is V~j= di" ~.
(74)
Using the facts that c =
V-l=
ce3 and p = v-~d3, it is easy to deduce from (17) and (74) that det V = v/c, and
- d 2 " el d I • e3
d 1 • e 1
0
d 2 • e3
v/c
.
(75)
Now combine (71) and (74) to obtain an expression for the six j u m p conditions h7/
(76)
Several useful results are apparen t from this relation. First, it is clear that M ~) will be symmetric if M ~) is symmetric. Secondly, since (76) represents a linear transformation, and the matrices in square brackets in (76) are all real, it is clear that the real matrices I m h] r ~ satisfy similar transformations. In fact, the imaginary part of (76) represents the correct transformation Im hqr~t) ~ Im h4# ~) even for curved interfaces, since the extra terms necessary in this equation are real. Thus, I m / ~ ) is positive definite if Im h]r~t) is positive definite. But the latter is a necessary condition, expressing the fact that the incident G W P is localized. Hence, the reflected and transmitted GWPs are also localized. Finally, define the angle 0 ~) that the ray direction of the G W P of type a makes with the interface normal, COS 0 (~)= t 3 " e~3~). Equation (77), along with (64), (72) and det V ~) = curved interface, r C(I) COS 0(I).72 det Im M ~I) - Lc-~.~) cos o ~ ) J "
(77)
v~'~)/c~'~, implies that for incidence upon an arbitrarily
det Im M ~)
(78)
The ratio of complex numbers, det M ~ ) / d e t M ~) is equal to the real ratio in (78) if the interface is locally flat.
4.6. Two-dimensional simplification The general results simplify considerably if the transmission/reflection process possesses a plane of symmetry. This occurs if the incident pulse, material anisotropies and interface are such that d~~) = e~~) = t2 for each c~. The local interface geometry is depicted in Fig. 5, which shows the ray and phase vectors for the incident and a single transmitted GWP. The local radii of curvature of the interface are al in the plane of Fig. 5, and a 2 in the plane of t 2 and ta. The matrix D of (48) is D = diag(1/2al, 1/2a2). It will be ,-,~i) assumed, for further simplicity, that ~,aA'-At(I)12 ----~vJA'-Ar(I)23 ----0, and hence, Jw ~2 = ~r<~)= ,-, 23 0. The results presented here will be used in the example of Section 4.8.
523
A.N. Norris / Pulses in anisotropic solids
+ I I I I I
8a
.( x") p_(x)
I
d3~
I
(T)
.(X)
i_ ¢"' /~01_ ~
or) (
T)
ilmm
ta tl
a
0
+> Fig. 5. A two-dimensional illustration of the reflection/transmission of a Gaussian wave packet from an interface. The incident pulse and one of the transmitted pulses are considered.
The relations in (70) reduce to ATe(T) ~.. 12 -. . .ATf(T) . 23 = 0, and /~(T)
'0 (I)2 A+,I(1)
(79a)
33 -~-"V--'*(~ l w 33 ~
~ ( T ) _ v°) cos 0 °) t3
+ (x)
(79b)
V(T) COS 0 (T) M 1 3 '
(COSI]/(I)
A~,f(T ) __ C O S 2 0 (I)
• -- ]1 -
cos ~ o (T> ~ , ) ~
a~ COS12 0 (T) ~k
__ t-3(1)
(cos,,+,'" ++-+~'++
++ c o s +
oc,>X, ~<+
,+,c,> ]"
t3(T)
J'
(79c)
(79d)
Following the notation of Musgrave [1], let d be the angle between the ray and phase directions: A=0--~.
(80)
The matrix V of (74) is now
V=
0 -sin A
1 0
,
(81)
524
A.N. Norris / Pulses in anisotropic solids
The relation between the matrices h~to) and A~t
/)(I)
O~ = COS A (I) COS 0 (T) '
(82a, b)
fl -- /)(T),
COS 0 (I) [ 3' - cos A (I) sin ~b°-) sin A~l)cossin0
sin A (r) sin 0 (T)]
co--~0(T--~ j,
~ = s e c 2 0(T)(CO~(~(1' COsO(T).] V(T)
(82c) (82d)
]'
then 2 M11' -- (1" + - t~ xv~A'-At(T)11 ---- a -
al
cos 2 A(T) '
(83a)
A~t~) = afl~r~l)+ a3'M~ll)--~ sin A (T) cos A(T), al
(83b)
~ [ ( T33) --- ~ 2 ~ I )
(83c)
.4_ 2 ~ 3 " ~ 1 )
+ 32~
~I1)_~_k
al - (i, + - 8- cos2 ,va~'-Ar(T)22 ----M22'
o(T).
sin 2
A (T),
(83d)
a2
Note that A is zero in an isotropic medium. Hence, y---0 of both materials are isotropic, in which case the coupling between .~r-elements disappears [5, 7]. 4. 7. Discussion
The general results in (79) and (83) for the two-dimensional configuration are for an arbitrary incident Gaussian wave packet. The latter result, (83), is defined in the phase slowness vector system {dl, dE, d3} which, as discussed in Subsection 3.4 above, is the natural basis for considering the evolution of the M-tensors on either side of the interface. However, the jump conditions are best understood by considering (79) for M referred to the non-orthogonal basis {el, e2, d3}. To see this, consider two extreme limits for the incident pulse. The first is that of a very short pulse for which Im z~,at~r(1)33)'> I m a~'/~I1), Im ~,-A~I[(I)13 • This is equivalent to a delta-function wavefront, oriented perpendicular to the phase slowness direction d~I). The transmitted pulse, from (79), also satisfies Im ,,, 33 " " 11 , Im ,,~7~(T)13 • Hence, it is also a propagating singularity oriented perpendicular to its phase direction. This limit is analogous to considering wavefronts of infinitely wide plane waves [1]. The other extreme is an incident pulse that is very long in the wave velocity direction e~3", i.e., I m M~I1 ~ ) >>Im I,,A~I(1)33,Im ,..~'f(1)13,and thus the transmitted pulse is a long, pencil-shaped object oriented in the new wave direction e~7). For these two limiting cases, the principal directions of both the incident and scattered pulses are defined by the basis {dl, d2, d3} of the slowness vector in the first instance, and by the basis {el, e2, e3} of the wave velocity vector in the second. This relative simplicitly disappears for an incident pulse that is neither very short nor very long. For example, if the incident pulse is oriented in the directions dl, and d3 ( -~t(I) , 13 = 0 in (83)) and the interface is flat (a~ 1= a21= 0), then (83) shows that the transmitted M
525
A.N. Norris / Pulses in anisotropic solids
is
,V/O~'?
o
a ~ W"~") l 11
0
A-/t(1) .tv~ 22
0
c t y M-- l(I) l
0
~"At(l) -Ip0 2 ~,-33-
] (84)
y2j~I
1)
The principal directions of the envelope of the transmitted pulse are defined by the eigenvectors of Im M (T), which from (84), are obviously dependent upon a and y. Note that for isotropic media [5, 7], y ~ 0, and so the matrix M(T)___ ]~7/(-r) is diagonal. 4.8. Example: Transmission f r o m a fluid into a transversely isotropic solid
As an illustration of transmission consider a wave packet originating in water, incident upon the transversely isotropic glass/epoxy composite material discussed in Section 3.6. Using the same coordinate axes as there, let the solid occupy x3 > 0, the fluid x3 < 0, with the flat interface on x3 = 0. The initial packet is the same, with the initial packet center at a distance of 15 m m from the origin. Since the sound speed is 1.5 mm/izs, the packet center is incident upon the interface at t = 10~s, and the point of incidence is the origin. The initial direction of propagation is in the xlxa-plane, and makes an angle 0 with the interface normal, see Fig. 5. The results of Section 4.6 apply here. Two transmitted packets are generated, a qL and a qT, each with its propagation direction in the xlx3-plane. These directions can be estimated for a given 0 from Fig. 1. To do this use Pl =2 sin 0, the ray directions then correspond to the directions of the normals to the slowness surfaces. Figures 6-8 show the transmitted packets at t = 20 jzs, for 0 = 5 °, 10 ° and 15 °. The qL and qT packets are shown separately for clarity. It is clear from the positions of the packet centers in these figures that the wave speeds vary with propagation direction, as expected. Note the orientation ol~ the packets, in agreement with the discussion above for a short pulse. Probably the most interesting feature in these figures is the large variation in the spreading. This may be understood on the basis of the spreading factor of Fig. 3. For the material considered, the spreading factor is a very sensitive function of the propagation direction, and this sensitivity is reflected in the results of Figs. 6-8.
qL
# -- 5 °
qT
#----5"
0.0
0
"1(3
Fig. 6. The qL and qT transmitted pulses for 0 = 5°, 10 ~,s after incidence at xt = x3 = 0 into the transversely isotropic material. The plotted surfaces show the envelopes, defined as u(x, t) • g exp[-iwp. (x-~(t))] for the separate wave packets.
526
A.N. Norris / Pulses in anisotropic solids
qL
~=100
~
qT
= 10 °
,¢a"~'t""
-.g
Fig. 7. The same as Fig. 6, but for 0 = 10°. Note the increased lateral spreading in comparison with Fig. 6.
qL
~=15"
qT
0 = 15 °
~
0.o
10.~
"10
~'10
Fig. 8. The same as Figs. 6 and 7, but for 0 = 15°.
5. Conclusions
A general theory has been given for the propagation and scattering of compact, Gaussian shaped pulses in piecewise homogeneous anisotropic solids. The general solution is described by (44), where the evolution of the envelope tensor M is given ,in (33). The form o f this pulse solution is relatively simple, considering that it is explicitly time-dependent and localized. This simplicity makes the theory ideal for modeling the propagation of ultrasonic pulses. The evolution of the propagating pulse is characterized by a spreading matrix N. The eigenvalues of N define the rate of broadening, and these rates can vary significantly with direction, as shown by example. Therefore, it is necessary to characterize N for a given material in order to understand the propagation of pulses through it. Explicit equations have been given for the elements of this matrix. The reflection/transmission problem of a pulse incident upon a smooth interface has also been discussed and the necessary j u m p conditions for M derived. The properties of the transmitted and reflected pulses are fundamentally different than for isotroloic media.
A.N. Norris / Pulses in anisotropi¢ solids
527
Acknowledgment
Thanks to B. White for useful discussions. This work was supported by the National Science Foundation, through Grant No. M D M 85-16256.
Appendix A. The wave velocity vector and related quantities
Define the tensor (A.1)
rij = P - 1 C kifl PkPI"
The secular equation (7) can then be written, using (8), as
D~,(r~, _fZ~k,) = 0,
(A.2)
where 1
2
D O= ~eiktejmn(Fkm - - f t~km)(FIn --f2~ln) and eOk is the third-order alternating tensor, el23 :
(A.3) 1,
et32 = - 1 , el22 =
0,
etc. Differentiating (A.2) gives,
~_~ D~n OF~n 2f
- Dkk
OPi
(A.4)
The wave velocity vector c then follows from (15), (16) and (A.4) as
ci = p--lCoklPt~k/ Dm,~,
(A.5)
where Do =- Do ( f = 1). It then follows from (15), (A.1), (A.2) and (A.5) that
p~c~= Fuff)o/19,,,,,, = 1.
(A.6)
A more transparent expression exists for c. Let g be the unit eigenvector of (6). It then follows from (A.2) that
g~gj = Do/ Dkk.
(A.7)
The wave speed may now be written from (A.5) as
ci = p-1C oklgjgkpt.
(A.8)
An expression for N of (31) follows by rewriting (A.4) using (A.7) as
Of Opi
1 c3Fkl -- -~ gkg, • C3pi
(g.9)
Then,
02f 1 02Fkt 1 Of Of 10Fkl Ogkgi Op, Opj -~gkgtop~Opj fOpi Opj 2 f Opi Opj
(A.10)
Putting f = 1 in (A.10), and using (16) and (A.1), 0 Nij = 19-1Cikljgkg! -- CiCj+ CmkljPk 7 (g,,gt). opi
(A.11)
A.N. Norris / Pulses in anisotropic solids
528
Let us rewrite this as N = N (1) + N (2),
(A.12)
NI])= p -1 C ikjlgkgt - CiCj,
(A.13)
10Ckt Ogkgt 20pi Opj
(A.14)
where
NI~)
It is clear that N (1) is symmetric. Also, from (A.6), (A.8) and (A.13),
Nl))pj =0.
(A.15)
The symmetry of N (2) is not immmediately apparent, but will be demonstrated shortly. Noting
OFk~ p,--~p~ = 2rk,,
(A.16)
it follows that
Nl2) ,j pi = Fkl Ogk& ~ •
(A.17)
Differentiating (A.2), using (A.7), gives
(Fk, _f23kl ) 3gk&+ (gk& OFkl 3pj
\
Opj
2cj~ = 0.
]
(A.18)
Equations (A.18), (A.17), (A.1) and (A.8), along with gkgk = 1, imply
N(,f)pi = 0.
(A.19)
Thus, both N (1) and N (2), and hence N, are projection tensors onto the plane perpendicular to the phase slowness vector p. The equation for N ~1) in (A.13) can be made explicit through the use of (A.7) and (A.8), once the slowness surface is determined from (A.2). The matrix N ~2) still needs to be simplified. From its definition in (A.14), and (A.7), (A.8), o
1 [ao °
2Dqq l 3pt
ar~ aD,.,.-!I 3pi -2ci ,,vt .T| f=,
(A.20)
From (A.3) and the definition L)o = D o ( f = 1), 0D.,. y=_r919,,,. ~Pj aPt
OD,.. OF.,,, f=, 3pi Opi
2[(Fqq_2)F.,_F,..]ct, 3D,,,,, OF,,,,, 2ct 3D,,,,,, 3pt cOpi cOpi
(A.21) (A.22)
Equations (A.20)-(A.22) give an explicit formula for N(2): N ~ ) = L2[10/),,, 0 F , , , ~3p~ pj
\ (c'C31Dmm-b~cO' OmrnJ~'+]ci(jOpi Fmm-3)]~j/q]qO' pj
The symmetry of N (2) is apparent from (A.23).
(A.23)
A.N. Norris / Pulses in anisotropic solids
529
A matrix related to N occurs in the paper by Cerveny [9]. An examination of that paper shows that the matrix F ~ ) / 2 D in [9, eq. (34)] should be identical to N 0. However, using his [9, eq. (35)], it can be shown that
F~)/2D=No+c,cj+[CjaDr.m 4 "] / -Op, -:'cArm~-3)J/oqq"
(A.24)
where the right-hand side of this equation uses the present notation. The discrepancy between the two results is due to Cerveny's failure to properly include the implicit derivatives, as in (A.21) and (A.22).
Appendix B. Transverse isotropy Let the x r a x i s be the zonal axis or axis of symmetry. The five independent elastic moduli are CH, C33, C~3, (?23 and (?66. See Musgrave [1] and Payton [4] for a complete discussion of the slowness and wave surfaces. Without loss of generality, we take P2 = 0. The purely transverse T H wave is polarized in the x2-direction, i.e., g = (0, 1, 0). Then
f ( p ) = (A2p21+ B2p~),/2,
(B.1)
A = (C66/p),/2,
(B.2)
B = (C44/p) '/2,
(B.3)
C44 .~ 1( C33 _ C23) '
(B.4)
where
and f ( p ) = 1 defines the slowness surface, in this case a spheroid. Explicit differentiation of (B.1) gives the wave velocity form (16) as
c = (A2p,, O, B2p3)
(B.5)
and the spreading matrix, from (31), as
N = A2B 2
p2
0
-P,P3]
0
A -2
0
|.
-Pl P3
0
p2
3
(B.6)
The quasi-longitudinal (qL) and quasi-transverse (qT) waves have their displacement eigenvectors in the x,x3-plane and are defined by f÷ = 1 and f_ = 1, respectively, where f±(p) =
[oo: 1+--+ 2
2
(D+2-2Ep2-2Fp]-4Gp,p3)
l,-
1/2
(B.7)
and
(B.8)
D = [(C,, + C66)P~ + (C33+ C66)p2]/p - 2 , E = {[2C11 C66p2 + (C1, C33 +
F
=
(C,, + C66)/p}/O ,
(B.9)
{[2C22C66p2+ (C,,C33+ C26)p~]/p2-(C33+ C66)/p}/D ,
(B.10)
O = (C,3 "t- C66)2plp3/p2D.
C26)p2]/192 -
(B.11)
530
A.N. Norris / Pulses in anisotropic solids
The wave velocity vector is c = (c~, 0, c3), where c1 = p 1 E +p3 G,
(B.12)
c3 = p l G + paF.
(B.13)
The elements of the matrix N follow from Appendix A: N~I = E - c 2 A - { [ a f l l f 6 6 p 2 - ( C13 + C66)2p~]/ p 2 - a c l p 1 ( C114- C66)/ p h-4c2}/ D,
(B.14)
N33 = F - c23+{[4C33C66P 2 - (C134- C66)2p~]/ p 2 - 4 c a p 3 ( C33 + C66)/p + 4 c 2 } / D ,
(B.15)
N13=2G_clc3+2{plP3(CllC13_4_
2 2 C66)/P
-- [clP3(C33+ C66) "]- c3PI( CI1 + C66)]/p + 2c~c3} / D,
(B.16)
N22 = c3/P3 ,
(B.17)
N12 = N21 = 0.
(B.18)
Several simplifying cases should be noted for (B.14)-(B.18). First, let p 3 = 0 for the qL wave. Then c3 = O, c1 = 1/pl = ( C l l / p ) ~/2 and N = diag(0, N22, N22), where N22 = [C66 "~-(C13-~- C 6 6 ) 2 / ( C l l - C66)]/p.
(B.19)
This represents the spreading of the pure L-wave propagating in the direction of the zonal axis. Second, let P3 = 0 for the qT wave. Then this is a pure transverse wave with c3 = 0 , c l - - 1 / p l = (C66/P) 1/2 and N = diag(0, N22, N22), where N22 = [C33- (C~3+
C66)2/(Cll- C66)]/P
(B.20)
Note that for both the qL and qT waves in the xrdirection, the matrix N has degenerate eigenvectors in the x2x3-plane. This is to be expected from the symmetry. However, the N matrix for the qT wave is not the same as N for the TH m o d e of (B.6), which is N = diag(0, B 2, B2). Thus, while the transverse wave slowness surfaces coalesce at P2 = P3 = 0, the local curvatures of the surfaces are different. The qL and qT modes also become pure modes if p~ = 0 . For the qL mode, we then have c1= 0, C3 = l i p 3 = (C33/P) 1/2 and N = d i a g ( N l l , N22, 0) with NI~ = [C66+ (C~3+ C66)2/(C33 - C66)]/p,
(B.21)
N22 = C33/P.
(B.22)
For the qT mode, c1= 0, c3 = 1/p3 = (C66/P) 1/2, N = d i a g ( N n , N22,0) and NI, = [ C n - (C~3+
C66)2/(C33-C66)]/P,
N22 = C66/p.
(B.23) (B.24)
Appendix C. Transverse isotropy transmission coefficients Reflection and transmission coefficients for a wave incident from an inviscid fluid upon a transversely isotropic solid are presented. The situation is as depicted in Fig. 5, with medium 1 the fluid, and medium 2 the transverse solid with the axis of symmetry in the xl-direction. One reflected wave is generated in the fluid. A quasi-transverse (qT) and quasi-longitudinal (qL) pair of waves are transmitted into the solid.
A.N. Norris/Pulses in anisotropic solids
531
Referring to Section 4.3, the incident and reflected waves are defined by g(l) = (sin
0 (I),
0,
(C.1)
COS 0 ( ! ) ) ,
p") = c?l(sin 0 (1), 0, cos 0°)),
(C.2)
g(R) = (sin 0 "), 0, --cos 0")),
(C.3)
p(R) = c~-l(sin 0(i), 0, - c o s 0")),
(C.4)
where cf is the fluid sound speed. The transmitted slownesses are
P(T~)=(p~T~),O,p(3T~)),
a = qL, qT,
(C.5)
where Snelrs law dictates that (C.6)
p~TqL) = p~TqT) = p ~ l ) ..~_P l "
The slowness components p~T~), a = qL, qT, then follow from Appendix B. The transmitted unit displacement vectors are
g(T~) = (G~), 0, Gg~))/EG~)~+ G(3Ta)2] 1/2,
t~ =
qL, qT,
(C.7)
where G~T~)=
p-1(C,3+ C66)plp~T'~), p
n2
G~Ta) = 1 --p-1~tlF1
~--1/,',
a = qL, qT,
,(Ta) 2
~, ~.~66p3
,
(c.8)
a = qL, qT.
(c.9)
The reflected and transmitted amplitudes are then (see Section 4.3),
[ u(°,R 1
Fc°s o(" TqL, g Tq )l_,[cos o(" 1 o(qL) Q(qT,] L 0
]
(c.lo)
where P(~) - C --
Q(a) ~
~ t~(T°t)"~ - / ~
13//1/51
~(Ta)o(Ta)..L
--F3
$1
~(Tu)o(Toz)
~'~33F3 s 3
~ ~(Tt~)
--Fl~;3
,
,
a = q L , qT,
a = qL, qT,
(C.11) (C.12)
and pf is the fluid density. Equation (C.10) follows from the continuity of 0-13, 0"33and u3 at the interface.
References [1] M.J.P. Musgrave, Crystal Acoustics, Holden Day, San Francisco, CA (1970). [2] G.F.D. Duff, "The Cauchy problem for elastic waves in an anisotropic medium", Philos. Trans. Roy. Soc. London A 252, 31-273 (1960). [3] E.A. Kraut, "Advances in the theory of anisotropic elastic wave propagation", Rev. Geophys. I, 401-447 (1963). [4] R.G. Payton, Elastic Wave Propgation in Transversely Isotropic Media, Nijhoff, The Hague (1983). [5] A.N. Norris, B.S. White and J.R. Schrieffer, "Gaussian wave packets in inhomogeneous media with curved interfaces", Proc. Roy. Soc. London A 412, 93-123 (1987). [6] V. Cerveny and I. Psencik, "Gaussian beams and paraxial ray approximation in three-dimensional elastic inhomogeneous media", J. Geophys. 53, 1-15 (1983). [7] A.N. Norris, "Elastic Gaussian wave packets in isotropic media", Acta Mech. (1987), to appear.
532
A.N. Norris / Pulses in anisotropic solids
[8] V. Cerveny, I.A. Molotkov and I. Psencik, Ray Method in Seismology, University of Karlova, Prague (1977). [9] V. Cerveny, "Seismic rays and ray intensities in inhomogeneous anisotropic media", J. Geophys. 28, 1-13 (1972). [10] H. Murakami and G.A. Hegemier, "A mixture model for unidirectionally fiber-reinforced composites", J. Appl. Mech. 53, 765-773 (1986). [11] V.T. Buchwald, "Elastic Waves in Anisotropic Media", Proc. Roy. Soc. London A 253, 563-580 (1959).
A THEORY
OF PULSE
509
PROPAGATION
IN ANISOTROPIC
ELASTIC
SOLIDS
A. N. N O R R I S Department of Mechanics and Materials Science, Rutgers University, Piscataway, NJ 08855-0909, U.S.A.
Received 9 March 1987, Revised 24 June 1987
A theory is described for the propagation of pulses in anisotropic elastic media. The pulse is initially defined by a harmonically modulated Gaussian envelope. As it propagates the pulse remains Gaussian, its spatial form characterized by a complex-valued envelope tensor. The center of the pulse follows the ray path defined by the initial velocity direction of the pulse. Relatively simple expressions are presented for the evolution of the amplitude and phase of the pulse in terms of the wave velocity, the phase slowness and unit displacement vectors. The spreading of the pulse is characterized by a spreading matrix. Explicit equations are given for this matrix in a transversely isotropic material. The rate of spreading can vary considerably, depending upon the direction of propagation. New reflected and transmitted pulses are created when a pulse strikes an interface of material discontinuity. Relations are given for the new envelope tensors in terms of the incident pulse parameters. The theory provides a convenient method to describe the evolution and change of shape of an ultrasonic pulse as it traverses a piecewise homogeneous solid. Numerical simulations are presented for pulses in a strongly anisotropic fiber reinforced composite.
I. Introduction M a n y f u n d a m e n t a l p r o b l e m s h a v e b e e n s o l v e d for w a v e m o t i o n in a n i s o t r o p i c elastic m e d i a . A t h o r o u g h review o f p l a n e wave p r o p a g a t i o n a n d reflection is given b y M u s g r a v e [1]. S o l u t i o n s f o r r a d i a t i o n f r o m p o i n t forces in full a n d h a l f - s p a c e s are also k n o w n [2, 3]. The m o r e recent m o n o g r a p h o f P a y t o n [4] c o n s i d e r s t r a n s v e r s e l y i s o t r o p i c m e d i a in d e t a i l a n d also lists m o r e recent references. H o w e v e r , t h e r e is a g r o w i n g n e e d to u n d e r s t a n d a n d m o d e l the p r o p a g a t i o n a n d scattering o f finite-sized p u l s e s t h r o u g h a n i s o t r o p i c materials. O n e a r e a o f a p p l i c a t i o n , for e x a m p l e , is in u l t r a s o n i c i n s p e c t i o n o f f i b e r - r e i n f o r c e d c o m p o s i t e s for p u r p o s e s o f n o n d e s t r u c t i v e e v a l u a t i o n . T h e p u l s e s c o n s i d e r e d h e r e are a s s u m e d to be l o c a l i z e d in the f o r m o f h a r m o n i c a l l y m o d u l a t e d G a u s s i a n e n v e l o p e s . T h e centers o f the G a u s s i a n w a v e p a c k e t s p r o a g a t e a l o n g the u s u a l rays or c h a r a c t e r i s t i c curves. T h e p r e s e n t t h e o r y is an e x t e n s i o n o f the a n a l y s i s o f N o r r i s [5] on G a u s s i a n wave p a c k e t s in i n h o m o g e n e o u s a c o u s t i c m e d i a . The initial a n s a t z u s e d h e r e is s i m i l a r for that o f t i m e h a r m o n i c G a u s s i a n b e a m s ; see [6] for a d i s c u s s i o n o f this in i s o t r o p i c elastic solids. T h e G a u s s i a n b e a m s o l u t i o n s are o f infinite extent in the d i r e c t i o n o f p r o p a g a t i o n . The p r e s e n t t h e o r y i n c l u d e s an extra d e g r e e o f f r e e d o m in t h a t the p u l s e is o f finite length. I n the limit as this length b e c o m e s infinite, it r e d u c e s to time h a r m o n i c G a u s s i a n b e a m theory. It is i m p o r t a n t to n o t e t h a t the w a v e p a c k e t s o f this p a p e r a n d the p a p e r ' s references [5, 7] are n o t s i m p l y the F o u r i e r t r a n s f o r m o f a t i m e h a r m o n i c G a u s s i a n b e a m . T h e d i s t i n c t i o n is e v i d e n t f r o m the e x a m p l e s in [5] o f a w a v e p a c k e t in an a c o u s t i c m e d i u m o f c o n s t a n t v e l o c i t y gradient. S o m e o f the cases t h e r e c a n n o t be e x p l a i n e d b y G a u s s i a n b e a m s . Specifically, if the wave p a c k e t is i n i t i a l l y e x t e n d e d in the d i r e c t i o n o f p r o p a g a t i o n , t h e n as the c e n t e r o f the p a c k e t follows the c u r v e d ray p a t h the b u l k o f the p a c k e t rotates relative to the r a y direction. G a u s s i a n b e a m t h e o r y b a s e d u p o n a single b e a m w o u l d p r e d i c t the s o l u t i o n to be s p a t i a l l y s y m m e t r i c a b o u t the r a y p a t h . 0165-2125/87/$3.50 O 1987 Elsevier Science Publishers B.V. (North-Holland)
A.N. Norris / Pulses in anisotropic solids
510
Norris [7] considered Gaussian wave packets in inhomogeneous isotropic elastic media and provided explicit formulae for the jump conditions at interfaces. Elastic wave propagation in anisotropic media is complicated by the fact that the wave and phase velocity vectors are distinct for any given wave mode. This leads to the curious result that wavefronts are not orthogonal to the direction of propagation. The general equations of rays and wavefront curvatures in inhomogeneous anisotropic media have been discussed previously, e.g. [8,9]. The present theory includes these curvature evolution equations as a special case. However, for simplicity only piecewise homogenous media are considered here. The generalization to smoothly varying media is not difficult, but the significance of the results can be obscured by the required amount of excessive notation. The basic equations are described in Section 2. The general Gaussain wave packet solution is derived and discussed in Section 3. A complete analysis of the reflection/transmission problem for a Gaussian wave packet incident upon an interface is given in Section 4.
2. Eikonal and transport equations
The equations of motion for the displacement field u(x, t) in a homogeneous, anisotropic linearly elastic solid are Cok~Uk, jl--pUi,, = 0,
(1)
where Cokt = Ck,j = Cj~kt is the elastic modulus tensor, p the density, and the summation convention on repeated subscripts is assumed. Let [5],
u(x, t) = C(x, t) e i~(x''),
(2)
p=V6,
(3)
then (1) becomes, ( ito )2[ C ~ktpjpl-- pdp2 8ik] Uk
+ (ito)[ C~jkl(pjUk, t + p l U k j + &,jt Uk) -- 2pqb, U~,, - pcb, Ui] + [Ci~kt Uk,~t -- P Ui,,,] = 0.
(4)
Now consider the asymptotic limit of to >>1. Here to denotes the center frequency of the solution. Assuming the ansatz co
U ( x , t ) = ~ (ito)-nU(")(x, t)
(5)
n=0
equation (4) reduces to a sequence of asymptotic equations. The first of these is 2
(o) _
( C OktP~Pt -- pgb, 8~k) Uk -- O.
(6)
A necessary condition for the existence of a nontrivial solution to (6) is det( C ijkt pjpt - Ptb, 2 6,k) = 0.
(7)
In general, there are three independent solutions to (7), of the form [1] cb,2 - f , , (2 p )
= 0,
m = 1, 2, 3.
(8)
A.N. Norris / Pulses in anisotropic solids
511
Each function fro(P) defines a slowness surface. In the case of an isotropic medium, two of the slowness surfaces coalesce and the explicit formulae are f , ( p ) = CLIP" p]'/2,
f2(P) = Cx[p " p],/2
(9)
where CL and cT are the longitudinal and transverse wave speeds. The three separate identities in (8) are known as the anisotropic eikonal equations. The next in the sequence of asymptotic equations is ,-'k 2
, - ~-e~,'-'
~,, - p ' C ' , , U I ° ) ]
(1)
+ (Coklpspl --pdp,Sik) Uk = O.
(10)
Multiply this by UI °) and sum over i, using (6), to obtain the anisotropic transport equation
C okt( pylO) u(kO)) j - t '~,., ~ w t ~,,(o), i t~iT(o)~Lt=O
(11)
3. The Gaussian wave packet solution and evolution equations 3.1. The ray velocity
Consider the following root of the eikonal equation (8), ok, + f ( p ) = 0,
(12)
where f ( p ) is any one of f i, f2 or f3- Equation (12) is a first-order partial differential equation in ~b of the form H ( x , t,p, 4o,, dp)=- ckt + f ( p ) =0.
(13)
This admits of solutions along characteristic curves in space-time. Let time be the ray parameter along these curves, defined by x = ~(t), and let an overdot represent the total derivative with respect to t along a ray. Then the Hamilton-Jacobi equations are i = aH/ack, = 1, = -on/ox
= O,
~ = aH/ap = af(p)/ap, ~k, = - O H ~ o r = O.
(14a, b) (14c, d)
Equation (14d) implies tk, is constant on a ray. Without loss o f generality, this constant can be taken as - 1 , so that equation (12) for the slowness surface becomes f ( p ) = 1.
(15)
Equation (14c) show that the slowness vector is constant on a ray, and hence, from (14b), the rays are straight lines. These equations are easily generalized to inhomogeneous media (see [8]). Define the wave velocity vector c as c=af(p)/Op.
(16)
Thus, c is perpendicular to the slowness surface at p [1]. A more explicit formulation of equation (16) is given in Appendix A. In general (see Appendix A), p. c = 1
(17)
512
A.N. Norris / Pulses in anisotropic solids
and (18)
p^c~O.
Define the wave and phase speeds, c and v, respectively, as c = ( e . c) 1 / 2 ,
v = ( p . p) -1/:
(19)
Therefore, by (17) and (19),
c/> v
(20)
with equality in an isotropic medium, in which case the wave velocity and slowness vectors are parallel (p ^ c = 0). Finally, note that the ray equation in a homogeneous medium is ~ ( t ) = :~(o) + tc.
(21)
3.2. Second-order derivatives o f the phase
So far, we have derived equations for the evolution along a ray of V~b = p and 4~,, both of which are constant. Next consider the matrix of second-order spatial derivatives, M ( t ) = VV~b(~(t), t).
(22)
Along a ray, both = V qbt + M ~ = V qb, + M c = O
(23)
and ~, = 4~, + c. V4,, = 0 .
(24)
It then follows that V 4~, = - M e ,
(25)
c~t, = cT Mc.
(26)
Thus, all the second-order derivatives of the phase can be written in terms of M ( t ) . Let
A(t)=
O~( t ) Oa '
B(t)-
Op( t ) dot '
(27)
where a is some vector independent of t, e.g., a = ~(0), the initial ray position. Then, R ( t ) =- M -l = A B -1
(28)
R ( t ) = A B - ' - A B - ' B B -~.
(29)
and
The values of A and B follow by taking the variation of equations (14b) and (14c) with respect to a. Thus, A = NB,
B = 0,
(30a, b)
A.N. Norris / Pulses in anisotropic solids
513
where
N o =-a2f(p)/Op, Opj,
(31)
Equations (29)-(31) imply (32)
R=N. Since N is constant, equations (28) and (32) give
M ( t ) = M ( 0 ) [ I + tNM(O) ]-'.
(33)
An explicit expression for N of (31) is given in Appendix A (equations (A.12), (A.13) and (A.23)). It is shown in (A.15) and (A.19) that the real-valued matrix N projects onto the plane perpendicular to p, and so one of its eigenvalues is always zero. The other two eigenvalues are related to the principal curvatures of the slowness surface f ( p ) = 1. Define normal points on the slowness surface as those points at which these two eigenvalues are nonzero. Points that are not normal are inflection points and correspond to cusps on the wave surface. The regions between the cusps define lids on the wave surfaces [2-4]. The areas on the slowness surface associated with the lids are concave. In these regions, the matrix N possesses one or two negative eigenvalues. The two eigenvalues are positive in regions where the slowness surface is convex. The present theory is valid in normal regions of the slowness surface, which thus includes convex and concave parts of the surface. For example, in transversely isotropic materials, points o f inflection can only occur on the quasi-transverse (qT) slowness surface, and then only for certain combinations of moduli [4]. These features are illustrated by example below. We are now able to approximate the phase ~b(x, t) locally in space and time about the ray position ~(r) at time r. Let Ax = x-.~(~'),
At = t - r .
(34)
Then a second-order Taylor series approximation yields, using (3), (22), (25), (26), (34) and ~b, = - 1 , ~b(x, t) = ~bo+ (p" A x - At) + ½(Ax-- c A t ) T M ( r ) ( A x - cat),
(35)
where 4)0 is a constant. Since p and c are real vectors, it follows that Im[~b(x, t) - ~b(£(r), ~')] = ½(Ax- c A t ) r Im M(~')(Ax - c At),
(36)
It is clear from (2) and (36) that the solution is in the form of a localized Gaussian wave packet (GWP) if and only if Im M is positive definite. It can be shown, using (30) and the methods of [5, Appendix B] that Im M ( r ) is positive definite for all I" if Im M ( 0 ) is positive definite. The other necessary conditions, which are physically obvious, are that M ( 0 ) is symmetric and det A(0) ~ 0. Note from (35) that the linear correction to the phases, ( p . A x - At), propagates with the phase velocity p/I p 12. However, the center of the GWP is defined as the point at which the quantity in equation (36) is zero. Therefore, the center propagates with the wave velocity c. This is also apparent from (21). This discrepancy between the two velocities is characteristic of anisotropic media.
3.3. The transport equation Let g be the unit eigenvector of (6), so that U(°)= u(O)g.
(37)
A.N. Norris / Pulses in anisotropic solids
514
Since g is constant, the transport equation (11) reduces to aU~°)2 aU<°)2 [ ( ag,g~) ] U~0)~CqklgigkPj c3X------~--Pqbt--"~+ Cijkl gigkMjl+pj~pm Mml -pMijciej -0.
(38)
Equations (22) and (26) have been used in deriving (38) from (11). Let O<°)(t) be the Value of U ~°) along the ray. Then, since tht = - 1 and from the identities (A.8) and (A.11), equation (38) reduces to ( ~ t + e • V)0<0)2 + NoMo U --- ¢',~o)2= ~,a
(39)
d log[ 0 ~°)2]+ Tr[NM( t)] -- 0.
(40)
or
Noting from (28) and (30a) that d
log det A(t) = Tr(NM),
(41)
equation (40) can then be solved to get
[~-(°)(t)
~-
rdet
L~e~~OJ
(42) "
In this form, det A(t) can be identified as the Jacobian of the mapping ~(0)-*2(t). Alternatively, since M ( t ) is known explicitly from (33), equation (40) can be integrated to yield O(°)(t) = 0(°)(0)/[det(l + tNM(O)] ;/2
(43)
It can also be shown that d e t A ( t ) # 0 if (1) d e t A ( 0 ) # 0 , (2) M(0) is symmetric, and (3) Im M(0) is positive definite. These are the same conditions that ensure Im M ( t ) as positive definite. The method of proof follows that procedure outlined in [5, Appendix B].
3.4. The Gaussian wave packet The Gaussian wave packet solution follows from (2), (35) and (43) as
u(x, t)=
U<0),0 , [det M ( t ) ] l / 2 ~ )g[~et--~J exp{ito[p. (X--~)+½(X--~)TMq t)(x--2)]},
(44)
where 2 = ~ ( t ) is given in (21), and M(t) in (33).
3.5. Discussion Given an initial phase slowness vector p and the ray velocity vector c of (16), the center of the GWP propagates according to (21). The shape of the GWP about its center is determined by the tensor M(t), which evolves according to (33). It is shown in (A.15) and (A.19) that N projects onto the plane perpendicular to the phase slowness direction. Therefore, in order to understand the evolution of M, it is useful to define an orthonormal triad {dl, d2, d3} such that d3 = vp, and dl, (/2 are the principal directions of N. With respect to this basis,
N = SldlaT+ S2d2a~,
(45)
A.N. Norris / Pulses in anisotropic solids
515
where S1 and $2 have dimensions of (speed) 2, and shall be called the "spreading factors." Obviously, S~ and $2 depend upon the direction of propagation. They may be negative for certain directions in anisotropic media, as illustrated in the example below. By comparison, in an isotropic medium the directions d~ and d2 are orthogonal to the ray direction, and $1 = $2 = c 2, where c is the wave speed. Define h4tij, i, j = 1, 2, 3, as the components of M in the basis {d~, d2, d3}, or M = )(l~did~.
(46)
In particular, let M(0) have its principal directions coincide with this basis, so that/~r(0) is diagonal. Then h4t(t) remains diagonal for t > 0, and from (33), (45) and (46) / ~ l l ( t ) = / ~ l l ( 0 ) / [ 1 + tS~/~tH(0)] ,
(47a)
h4t22(t) = 2~t22(0)/[ 1 + tS22VI22(O)],
(47b)
)l,]raa(t) = h4"33(0).
(47c)
The form of the GWP remains constant in the phase direction, but it broadens in the orthogonal directions d~ and d2. The widths in these directions depends upon Im hT/l~(t) and I m / ~ t 2 2 ( t ) , respectively. The rate of broadening depends upon the values of S~ and $2. Since these generally differ, the GWP will broaden faster in one direction than in the other. 3.6. Numerical example f o r a transversely isotropic composite
Consider a transversely isotropic composite of parallel cylindrical glass fibers in epoxy matrix, with the fibers 60% of the bulk volume. The composite may be approximated for quasistatic deformation as a homogeneous transversely isotropic solid. The quasistatic approximation should be valid for dynamic wave propagation in the composite if the wave length is much longer than the fiber thickness. This is often the case for ultrasonic studies, and will be assumed here. The constituent parameters are (A,/.t, p) = (13.3, 29.9, 2.55) for glass, and (0.89, 1.28, 1.25) for epoxy, where A and/~ are Lain6 moduli, p density and the units are GPa and gm/cm 3, respectively. Assuming the fibers are aligned in the x~-direction, this direction is the symmetry axis of the transversely isotropic medium. The effective moduli follow from, for example, the theory of [10] as ( C ~ , C33 , C13 , C66 , p ) : (43.4, 10.7, 2.2, 4.4, 2.0), in the same units as before. Figures 1 and 2 show the slowness and wave surfaces of this composite for wave motion polarized in the x~x3-plane, i.e. for qL and qT modes of propagation. These surfaces are defined in Appendix B. Note that the qT surface possesses points of inflection. The qT wave surface has corresponding cusp points. The spreading matrix N is defined explicitly in Appendix B for the three wave types. For wave motion polarized in the plane of XlX3, the eigenvalues S~ and $2 of N are $2 = N22 (see (B.17)), and S 1 is the eigenvalue of the 2 x 2 symmetric matrix of elements NH, NI3 and N33 (see (B.14)-(B.16)). Thus, S~ is the spreading factor in the X~Xa-plane. It is zero at the points of inflection on the qT slowness surface, and becomes negative between these points. Everywhere else it is positive. A plot of S~ is shown in Fig. 3. The striking feature of this plot is the large variation in the value of S~. The corresponding plot for an isotropic solid is two circular arcs, signifying equal spreading in all directions. The present theory is incomplete at the points of inflection on the qT slowness surface. For these discrete directions, the spreading is zero, and the theory predicts no change in the GWP, apart from the spreading associated with $2. The pulse then acts like a collimated beam. However, this is not correct for long times. The correct behavior requires a more sophisticated analysis than presently considered. Specifically, we need to look at cubic terms in the phase and higher-order transport terms (see [11] for a relevant discussion).
516
A.N. Norris / Pulses in anisotropic solids co
c3-
P~-
o
Q i
0.0
0.2
i
0.4
i
0.6
0.8
Pl
Fig. 1. Slowness surfaces of the qL (inner) and qT (outer) waves in a transversely isotropic solid. The material models a fiberreinforced, epoxy matrix composite. The fibers are in the xt-direction, and the units of slowness are p.s/mm.
Cs
]
3~
2
4i
5I
61 Fig. 2. The wave velocity surfaces corresponding to the slowness surfaces of Fig. 1. Units are mm/~s.
T h e large v a r i a t i o n in s p r e a d i n g is b e s t i l l u s t r a t e d b y e x a m p l e . C o n s i d e r a flat u l t r a s o n i c t r a n s d u c e r b o n d e d to a s a m p l e o f the c o m p o s i t e such t h a t the n o r m a l to the t r a n s d u c e r face is in the XlX3-plane, a n d m a k e s an angle ~ with the x3-axis. T h e t r a n s d u c e r has a c e n t e r f r e q u e n c y o f 5 M H z , a n d emits a G a u s s i a n p u l s e o f initial w i d t h 5 mm. The initial s h a p e o f the p u l s e is best defined b y the m a t r i x / ~ i n t r o d u c e d in (60) b e l o w . In this c o n f i g u r a t i o n , we have /17/(0) = diag(/(/~](0), ]~/1](0),/~/33(0)).
A.N. Norris / Pulses in anisotropic solids
518
~:
900
W : 20*
t :0
t :
~:0
20
~':90
°
t:20
°
t =20
Fig. 4. Plots of a qL pulse propagating in a transversely isotropic medium, approximating a fibre/epoxy composite. The direction of propagation is defined by ~0,the angle the phase velocity makes with the x3-axis. The initial pulse is shown for qJ= 90°. The initial center is at (0, 0). The units are mm and p.s. The plots show the envelope, defined as u(x, t).g exp[-ioJp- (x-~(t))]. The phase factor exp[-io~p • ( x - i ( t ) ) ] removes the oscillations in the p-direction (see (44)) and is included to improve the visual appearance.
Let the i n c i d e n t G W P , d e n o t e d b y u (]~, b e o f the f o r m given in (44). It is c h a r a c t e r i z e d b y its s l o w n e s s v e c t o r p(]), with c o r r e s p o n d i n g d i s p l a c e m e n t v e c t o r g(~). The a m p l i t u d e a n d s h a p e o f u ~]) at i n c i d e n c e are d e t e r m i n e d b y U(°)I(to) a n d M(x)(to). The b o u n d a r y c o n d i t i o n s o f c o n t i n u i t y o f d i s p l a c e m e n t a n d n o r m a l t r a c t i o n s in the n e i g h b o r h o o d o f (Xo, to) are satisfied by i n t r o d u c i n g reflected a n d t r a n s m i t t e d G W P s . In g e n e r a l , t h e r e will be three o f each, o n e c o r r e s p o n d i n g to the q u a s i - l o n g i t u d i n a l (qL) wave a n d t h e o t h e r two to the q u a s i - t r a n s v e r s e waves (qT1 a n d qT2) p o s s i b l e o n either side o f S. Let the reflected a n d t r a n s m i t t e d fields be respectively, u (R~)
(49)
U (Tc~),
(50)
= qL,qTl,qT2
and = qL,qTl,qT2
w h e r e each o f these six G W P s has its a s s o c i a t e d p, g, U(°)(to) a n d M(to).
A.N. Norris / Pulses in anisotropic solids
519
4.2. Determining the slowness vectors Each of the reflected and transmitted GWPs is of the form of equation (2), or more specifically, equation (44). Define the phase ~b(x, t) of each of these GWPs to equal the incident phase at (x, to). Then, the phases must remain continuous in the neighborhood of (Xo, to) on S. This neighborhood is given by (48), therefore there are three first-order conditions:
ckt,
O~blO~i, i = 1, 2
are continuous at (Xo, to).
(51)
These conditions simplify, using (3) and (49), to give that O~
3x
3 -V4~.--~j=p.tj,
j=1,2
are continuous at (xo, to).
(52)
These two equations determine two components of the slowness vectors. The third component follows from the facts that ~bt = - 1 is continuous and that each p satisfies an equation of the form (12). The sign of the third component o f p, i . e . p , t3, is obtained by requiring that the associated wave velocity vector c have the appropriate direction. Thus, c. t3 should be of the same sign as c (~). t3 for the transmitted GWPs, but of opposite sign for the reflected GWPs.
4.3. Determining the displacement vectors and amplitudes Having found the slowness vector p for each new GWP, the associated wave velocity vector c and unit displacement vector g follow from (6) and (A.8). The continuity of total displacement at (Xo, to) then implies
U(°)X( to)g~I) +
~,
U~°)R~(to)g ~R~) =
a = q L , q T 1, q T 2
~
U(°)T'( to)g (TcO.
(53)
a = q L,qTI ,qT2
Similarly, the continuity of normal traction conditions are
r
CO)" ijkl ~3j L UW)I( to)g(kl)p~1)+
•
u(O)R~( to)g(kR~)pl R~) ] /
ot = q L , q T l , q T 2
= r~(2), ,-- Uk,'3j
Y.
u(O)T~(to)g(kT~)plT~),
i = 1, 2, 3,
(54)
ot = q L , q T l , q T 2
where C (1) and C ~2) are the elastic moduli tensors on either side of S. The six amplitudes follow from the six equations in (53) and (54).
4.4. The M-tensors of the reflected and transmitted GWPs The six elements of the complex, symmetric M-tensors follow from the six second-order conditions that r~,, 02qb/3t 3~j and 32t.b/3~:i 35, i, j = 1, 2 are all continuous. The first of these, ~ , continuous, implies using (26) that
cTMc is continuous.
(55)
The second two conditions simplify using (25), (48) and 32~ =V~b, 3x
at 3~
"37
(56)
A.N. Norris / Pulses in anisotropic solids
520 to give eTMtj,
j=l,2
(57)
are continuous. The final three conditions become, using c924~
[ c~x\ T
[ ~x \
~2 x
(58)
(3), (22) and (48), the three conditions tTMtj + 2D,jp" t3,
£ k = 1, 2
are continuous.
(59)
Further simplification of these conditions requires that M be referrred to a specific basis. Previously it was shown that the basis {dl, d2, d3} is the natural one for considering the evolution of M. In addition, a ray coordinate system may be defined by the orthonormal triad {el, e2, e3}, where e3 is the unit vector in the direction of c. Obviously, these triads coincide for isotropy. There is also the basis {tl, tz, t3} defined by the surface S at Xo. It is not obvious which basis is most advantageous for simplifying the jump conditions. The answer in fact is that none of these is the best, but rather that M be referred to a n o n - o r t h o g o n a l basis defined by the triad {~1, e2, e3}, where ~ = e~, ~2 = e2 and e3 = d3 = vp. Thus, define the 3 x 3 symmetric complex matrix If/u, i, j = 1, 2, 3 by M = hT/~,~T.
(60)
The reason for choosing the non-orthogonal basis {el, e2, e3} will become apparent presently. The conditions in (55), (57) and (59) indicate that the hT/-matrix transforms discontinuously from the incident hT/ to the transmitted and reflected matrices. Thus, the individual elements suffer jumps, the values of which are determined by (55), (57) and (59). The first jump condition, (55), implies, using (17), (19) and (60), that vZhT/33 is continuous.
(61)
The two jump conditions of (57) imply that vlQIi3e i • t2, j = 1, 2
are continuous.
(62)
These conditions simplify, using (51), (52) and (61), to v Y. hT/i3~i"tj, j = l , 2 i=1,2
are continuous.
(63)
For each of the bases { ~ , e2, e3}, define the 2 × 2 transformation matrix Q by Qq=ti.~=ti.ej,
i, j = 1, 2.
(64)
Thus, det Q = t3 • e3
(65)
A.N. Norris/Pulses in anisotropicsolids
521
The jump conditions (63) now become
v ~, Qij~3,
i=1,2
j~1,2
(66)
are continuous. The remaining three jump conditions of (59), are, using (64), that
~.
Q,kQ;,]VIk,+(6" ~3) ~. Qj,MI,3+(t~" ~3) ~
k.I = 1,2
1 = 1,2
+2(t3.p)Du,
Q,kMlk3+(t," ~3)(tj" ~3)h7/33
k = 1.2
i , j = 1,2 (67)
are continuous. These relations simplify further upon noting
t,. ~3 = vt,. p
(68)
and using the previous conditions of (51), (52), (61) and (66), to give
~.
Q,kQ~,IVIki+2--Dot3. ~3,
k , l = l,2
i,j=1,2
I)
(69)
are continuous.
Let a denote any one of the six reflected and transmitted GWPs. Then the six jump conditions can be solved explicitly, from (61), (66) and (69), to give the matrices AT/(~)= AT/c~)(to) as DO) 2
g,t(~) - V~( a ) 2 ~(~) ~va 3 3 , .L.a 33
(70a)
- -
i3 = v ( . ) k~l 2 ~'U •,
=
,:
Vjk 1" k3,
i = l, 2,
(70b)
,
[") ( a ) - l f ) (or) - 1 / ' ~ ( I ) ~ ( 1 )
~'A ( ! )
k,l,m,n = l,2
+2(t3.p('-t3.p
<")) ~
QI~)-'Q)7)-'D~,
i,j=l,2.
(70c)
k,/=l,2
It should now be apparent why we originally chose to represent M in the non-orthogonal basis {~, e2, ea}. By so doing, relatively simple equations have been obtained for the new 217/-matrices. These equations are decoupled as much as possible. Any other choice of basis would result in greater coupling between the coefficient hT/o. Note that the form of (70) is very similar to those derived previously for an isotropic medium [7].
4.5. Simplification and localization properties The six jump conditions of (70) can be written succinctly in matrix form as ]$~(~) = (S(~)-'S(I))l~l(O(S(~)-~s(l))T + 2t3 " (p(I)_p(~))Q(~)-IDQ(~)-~+ '
(71)
where S (') is the 3 x 3 matrix associated with the 2 x 2 matrix Q(~), (72)
522
A.N. Norris / I~lses in anisotropic solids
Also, the last term in (71), which represents a 2 x 2 matrix must be understood to apply only to the i, j = 1, 2 elements ofhT/~). This term is identically zero if the interface is locally flat at the point of incidence. Note that it is preferable to express M in the basis {d~, d2, d3} for the purposes of calculating the evolution of the M-tensors before and after incidence. The required transformations hdr <-->]17/is
lfl= V~IV T,
(73)
where the 3 x 3 matrix V is V~j= di" ~.
(74)
Using the facts that c =
V-l=
ce3 and p = v-~d3, it is easy to deduce from (17) and (74) that det V = v/c, and
- d 2 " el d I • e3
d 1 • e 1
0
d 2 • e3
v/c
.
(75)
Now combine (71) and (74) to obtain an expression for the six j u m p conditions h7/
(76)
Several useful results are apparen t from this relation. First, it is clear that M ~) will be symmetric if M ~) is symmetric. Secondly, since (76) represents a linear transformation, and the matrices in square brackets in (76) are all real, it is clear that the real matrices I m h] r ~ satisfy similar transformations. In fact, the imaginary part of (76) represents the correct transformation Im hqr~t) ~ Im h4# ~) even for curved interfaces, since the extra terms necessary in this equation are real. Thus, I m / ~ ) is positive definite if Im h]r~t) is positive definite. But the latter is a necessary condition, expressing the fact that the incident G W P is localized. Hence, the reflected and transmitted GWPs are also localized. Finally, define the angle 0 ~) that the ray direction of the G W P of type a makes with the interface normal, COS 0 (~)= t 3 " e~3~). Equation (77), along with (64), (72) and det V ~) = curved interface, r C(I) COS 0(I).72 det Im M ~I) - Lc-~.~) cos o ~ ) J "
(77)
v~'~)/c~'~, implies that for incidence upon an arbitrarily
det Im M ~)
(78)
The ratio of complex numbers, det M ~ ) / d e t M ~) is equal to the real ratio in (78) if the interface is locally flat.
4.6. Two-dimensional simplification The general results simplify considerably if the transmission/reflection process possesses a plane of symmetry. This occurs if the incident pulse, material anisotropies and interface are such that d~~) = e~~) = t2 for each c~. The local interface geometry is depicted in Fig. 5, which shows the ray and phase vectors for the incident and a single transmitted GWP. The local radii of curvature of the interface are al in the plane of Fig. 5, and a 2 in the plane of t 2 and ta. The matrix D of (48) is D = diag(1/2al, 1/2a2). It will be ,-,~i) assumed, for further simplicity, that ~,aA'-At(I)12 ----~vJA'-Ar(I)23 ----0, and hence, Jw ~2 = ~r<~)= ,-, 23 0. The results presented here will be used in the example of Section 4.8.
523
A.N. Norris / Pulses in anisotropic solids
+ I I I I I
8a
.( x") p_(x)
I
d3~
I
(T)
.(X)
i_ ¢"' /~01_ ~
or) (
T)
ilmm
ta tl
a
0
+> Fig. 5. A two-dimensional illustration of the reflection/transmission of a Gaussian wave packet from an interface. The incident pulse and one of the transmitted pulses are considered.
The relations in (70) reduce to ATe(T) ~.. 12 -. . .ATf(T) . 23 = 0, and /~(T)
'0 (I)2 A+,I(1)
(79a)
33 -~-"V--'*(~ l w 33 ~
~ ( T ) _ v°) cos 0 °) t3
+ (x)
(79b)
V(T) COS 0 (T) M 1 3 '
(COSI]/(I)
A~,f(T ) __ C O S 2 0 (I)
• -- ]1 -
cos ~ o (T> ~ , ) ~
a~ COS12 0 (T) ~k
__ t-3(1)
(cos,,+,'" ++-+~'++
++ c o s +
oc,>X, ~<+
,+,c,> ]"
t3(T)
J'
(79c)
(79d)
Following the notation of Musgrave [1], let d be the angle between the ray and phase directions: A=0--~.
(80)
The matrix V of (74) is now
V=
0 -sin A
1 0
,
(81)
524
A.N. Norris / Pulses in anisotropic solids
The relation between the matrices h~to) and A~t
/)(I)
O~ = COS A (I) COS 0 (T) '
(82a, b)
fl -- /)(T),
COS 0 (I) [ 3' - cos A (I) sin ~b°-) sin A~l)cossin0
sin A (r) sin 0 (T)]
co--~0(T--~ j,
~ = s e c 2 0(T)(CO~(~(1' COsO(T).] V(T)
(82c) (82d)
]'
then 2 M11' -- (1" + - t~ xv~A'-At(T)11 ---- a -
al
cos 2 A(T) '
(83a)
A~t~) = afl~r~l)+ a3'M~ll)--~ sin A (T) cos A(T), al
(83b)
~ [ ( T33) --- ~ 2 ~ I )
(83c)
.4_ 2 ~ 3 " ~ 1 )
+ 32~
~I1)_~_k
al - (i, + - 8- cos2 ,va~'-Ar(T)22 ----M22'
o(T).
sin 2
A (T),
(83d)
a2
Note that A is zero in an isotropic medium. Hence, y---0 of both materials are isotropic, in which case the coupling between .~r-elements disappears [5, 7]. 4. 7. Discussion
The general results in (79) and (83) for the two-dimensional configuration are for an arbitrary incident Gaussian wave packet. The latter result, (83), is defined in the phase slowness vector system {dl, dE, d3} which, as discussed in Subsection 3.4 above, is the natural basis for considering the evolution of the M-tensors on either side of the interface. However, the jump conditions are best understood by considering (79) for M referred to the non-orthogonal basis {el, e2, d3}. To see this, consider two extreme limits for the incident pulse. The first is that of a very short pulse for which Im z~,at~r(1)33)'> I m a~'/~I1), Im ~,-A~I[(I)13 • This is equivalent to a delta-function wavefront, oriented perpendicular to the phase slowness direction d~I). The transmitted pulse, from (79), also satisfies Im ,,, 33 " " 11 , Im ,,~7~(T)13 • Hence, it is also a propagating singularity oriented perpendicular to its phase direction. This limit is analogous to considering wavefronts of infinitely wide plane waves [1]. The other extreme is an incident pulse that is very long in the wave velocity direction e~3", i.e., I m M~I1 ~ ) >>Im I,,A~I(1)33,Im ,..~'f(1)13,and thus the transmitted pulse is a long, pencil-shaped object oriented in the new wave direction e~7). For these two limiting cases, the principal directions of both the incident and scattered pulses are defined by the basis {dl, d2, d3} of the slowness vector in the first instance, and by the basis {el, e2, e3} of the wave velocity vector in the second. This relative simplicitly disappears for an incident pulse that is neither very short nor very long. For example, if the incident pulse is oriented in the directions dl, and d3 ( -~t(I) , 13 = 0 in (83)) and the interface is flat (a~ 1= a21= 0), then (83) shows that the transmitted M
525
A.N. Norris / Pulses in anisotropic solids
is
,V/O~'?
o
a ~ W"~") l 11
0
A-/t(1) .tv~ 22
0
c t y M-- l(I) l
0
~"At(l) -Ip0 2 ~,-33-
] (84)
y2j~I
1)
The principal directions of the envelope of the transmitted pulse are defined by the eigenvectors of Im M (T), which from (84), are obviously dependent upon a and y. Note that for isotropic media [5, 7], y ~ 0, and so the matrix M(T)___ ]~7/(-r) is diagonal. 4.8. Example: Transmission f r o m a fluid into a transversely isotropic solid
As an illustration of transmission consider a wave packet originating in water, incident upon the transversely isotropic glass/epoxy composite material discussed in Section 3.6. Using the same coordinate axes as there, let the solid occupy x3 > 0, the fluid x3 < 0, with the flat interface on x3 = 0. The initial packet is the same, with the initial packet center at a distance of 15 m m from the origin. Since the sound speed is 1.5 mm/izs, the packet center is incident upon the interface at t = 10~s, and the point of incidence is the origin. The initial direction of propagation is in the xlxa-plane, and makes an angle 0 with the interface normal, see Fig. 5. The results of Section 4.6 apply here. Two transmitted packets are generated, a qL and a qT, each with its propagation direction in the xlx3-plane. These directions can be estimated for a given 0 from Fig. 1. To do this use Pl =2 sin 0, the ray directions then correspond to the directions of the normals to the slowness surfaces. Figures 6-8 show the transmitted packets at t = 20 jzs, for 0 = 5 °, 10 ° and 15 °. The qL and qT packets are shown separately for clarity. It is clear from the positions of the packet centers in these figures that the wave speeds vary with propagation direction, as expected. Note the orientation ol~ the packets, in agreement with the discussion above for a short pulse. Probably the most interesting feature in these figures is the large variation in the spreading. This may be understood on the basis of the spreading factor of Fig. 3. For the material considered, the spreading factor is a very sensitive function of the propagation direction, and this sensitivity is reflected in the results of Figs. 6-8.
qL
# -- 5 °
qT
#----5"
0.0
0
"1(3
Fig. 6. The qL and qT transmitted pulses for 0 = 5°, 10 ~,s after incidence at xt = x3 = 0 into the transversely isotropic material. The plotted surfaces show the envelopes, defined as u(x, t) • g exp[-iwp. (x-~(t))] for the separate wave packets.
526
A.N. Norris / Pulses in anisotropic solids
qL
~=100
~
qT
= 10 °
,¢a"~'t""
-.g
Fig. 7. The same as Fig. 6, but for 0 = 10°. Note the increased lateral spreading in comparison with Fig. 6.
qL
~=15"
qT
0 = 15 °
~
0.o
10.~
"10
~'10
Fig. 8. The same as Figs. 6 and 7, but for 0 = 15°.
5. Conclusions
A general theory has been given for the propagation and scattering of compact, Gaussian shaped pulses in piecewise homogeneous anisotropic solids. The general solution is described by (44), where the evolution of the envelope tensor M is given ,in (33). The form o f this pulse solution is relatively simple, considering that it is explicitly time-dependent and localized. This simplicity makes the theory ideal for modeling the propagation of ultrasonic pulses. The evolution of the propagating pulse is characterized by a spreading matrix N. The eigenvalues of N define the rate of broadening, and these rates can vary significantly with direction, as shown by example. Therefore, it is necessary to characterize N for a given material in order to understand the propagation of pulses through it. Explicit equations have been given for the elements of this matrix. The reflection/transmission problem of a pulse incident upon a smooth interface has also been discussed and the necessary j u m p conditions for M derived. The properties of the transmitted and reflected pulses are fundamentally different than for isotroloic media.
A.N. Norris / Pulses in anisotropi¢ solids
527
Acknowledgment
Thanks to B. White for useful discussions. This work was supported by the National Science Foundation, through Grant No. M D M 85-16256.
Appendix A. The wave velocity vector and related quantities
Define the tensor (A.1)
rij = P - 1 C kifl PkPI"
The secular equation (7) can then be written, using (8), as
D~,(r~, _fZ~k,) = 0,
(A.2)
where 1
2
D O= ~eiktejmn(Fkm - - f t~km)(FIn --f2~ln) and eOk is the third-order alternating tensor, el23 :
(A.3) 1,
et32 = - 1 , el22 =
0,
etc. Differentiating (A.2) gives,
~_~ D~n OF~n 2f
- Dkk
OPi
(A.4)
The wave velocity vector c then follows from (15), (16) and (A.4) as
ci = p--lCoklPt~k/ Dm,~,
(A.5)
where Do =- Do ( f = 1). It then follows from (15), (A.1), (A.2) and (A.5) that
p~c~= Fuff)o/19,,,,,, = 1.
(A.6)
A more transparent expression exists for c. Let g be the unit eigenvector of (6). It then follows from (A.2) that
g~gj = Do/ Dkk.
(A.7)
The wave speed may now be written from (A.5) as
ci = p-1C oklgjgkpt.
(A.8)
An expression for N of (31) follows by rewriting (A.4) using (A.7) as
Of Opi
1 c3Fkl -- -~ gkg, • C3pi
(g.9)
Then,
02f 1 02Fkt 1 Of Of 10Fkl Ogkgi Op, Opj -~gkgtop~Opj fOpi Opj 2 f Opi Opj
(A.10)
Putting f = 1 in (A.10), and using (16) and (A.1), 0 Nij = 19-1Cikljgkg! -- CiCj+ CmkljPk 7 (g,,gt). opi
(A.11)
A.N. Norris / Pulses in anisotropic solids
528
Let us rewrite this as N = N (1) + N (2),
(A.12)
NI])= p -1 C ikjlgkgt - CiCj,
(A.13)
10Ckt Ogkgt 20pi Opj
(A.14)
where
NI~)
It is clear that N (1) is symmetric. Also, from (A.6), (A.8) and (A.13),
Nl))pj =0.
(A.15)
The symmetry of N (2) is not immmediately apparent, but will be demonstrated shortly. Noting
OFk~ p,--~p~ = 2rk,,
(A.16)
it follows that
Nl2) ,j pi = Fkl Ogk& ~ •
(A.17)
Differentiating (A.2), using (A.7), gives
(Fk, _f23kl ) 3gk&+ (gk& OFkl 3pj
\
Opj
2cj~ = 0.
]
(A.18)
Equations (A.18), (A.17), (A.1) and (A.8), along with gkgk = 1, imply
N(,f)pi = 0.
(A.19)
Thus, both N (1) and N (2), and hence N, are projection tensors onto the plane perpendicular to the phase slowness vector p. The equation for N ~1) in (A.13) can be made explicit through the use of (A.7) and (A.8), once the slowness surface is determined from (A.2). The matrix N ~2) still needs to be simplified. From its definition in (A.14), and (A.7), (A.8), o
1 [ao °
2Dqq l 3pt
ar~ aD,.,.-!I 3pi -2ci ,,vt .T| f=,
(A.20)
From (A.3) and the definition L)o = D o ( f = 1), 0D.,. y=_r919,,,. ~Pj aPt
OD,.. OF.,,, f=, 3pi Opi
2[(Fqq_2)F.,_F,..]ct, 3D,,,,, OF,,,,, 2ct 3D,,,,,, 3pt cOpi cOpi
(A.21) (A.22)
Equations (A.20)-(A.22) give an explicit formula for N(2): N ~ ) = L2[10/),,, 0 F , , , ~3p~ pj
\ (c'C31Dmm-b~cO' OmrnJ~'+]ci(jOpi Fmm-3)]~j/q]qO' pj
The symmetry of N (2) is apparent from (A.23).
(A.23)
A.N. Norris / Pulses in anisotropic solids
529
A matrix related to N occurs in the paper by Cerveny [9]. An examination of that paper shows that the matrix F ~ ) / 2 D in [9, eq. (34)] should be identical to N 0. However, using his [9, eq. (35)], it can be shown that
F~)/2D=No+c,cj+[CjaDr.m 4 "] / -Op, -:'cArm~-3)J/oqq"
(A.24)
where the right-hand side of this equation uses the present notation. The discrepancy between the two results is due to Cerveny's failure to properly include the implicit derivatives, as in (A.21) and (A.22).
Appendix B. Transverse isotropy Let the x r a x i s be the zonal axis or axis of symmetry. The five independent elastic moduli are CH, C33, C~3, (?23 and (?66. See Musgrave [1] and Payton [4] for a complete discussion of the slowness and wave surfaces. Without loss of generality, we take P2 = 0. The purely transverse T H wave is polarized in the x2-direction, i.e., g = (0, 1, 0). Then
f ( p ) = (A2p21+ B2p~),/2,
(B.1)
A = (C66/p),/2,
(B.2)
B = (C44/p) '/2,
(B.3)
C44 .~ 1( C33 _ C23) '
(B.4)
where
and f ( p ) = 1 defines the slowness surface, in this case a spheroid. Explicit differentiation of (B.1) gives the wave velocity form (16) as
c = (A2p,, O, B2p3)
(B.5)
and the spreading matrix, from (31), as
N = A2B 2
p2
0
-P,P3]
0
A -2
0
|.
-Pl P3
0
p2
3
(B.6)
The quasi-longitudinal (qL) and quasi-transverse (qT) waves have their displacement eigenvectors in the x,x3-plane and are defined by f÷ = 1 and f_ = 1, respectively, where f±(p) =
[oo: 1+--+ 2
2
(D+2-2Ep2-2Fp]-4Gp,p3)
l,-
1/2
(B.7)
and
(B.8)
D = [(C,, + C66)P~ + (C33+ C66)p2]/p - 2 , E = {[2C11 C66p2 + (C1, C33 +
F
=
(C,, + C66)/p}/O ,
(B.9)
{[2C22C66p2+ (C,,C33+ C26)p~]/p2-(C33+ C66)/p}/D ,
(B.10)
O = (C,3 "t- C66)2plp3/p2D.
C26)p2]/192 -
(B.11)
530
A.N. Norris / Pulses in anisotropic solids
The wave velocity vector is c = (c~, 0, c3), where c1 = p 1 E +p3 G,
(B.12)
c3 = p l G + paF.
(B.13)
The elements of the matrix N follow from Appendix A: N~I = E - c 2 A - { [ a f l l f 6 6 p 2 - ( C13 + C66)2p~]/ p 2 - a c l p 1 ( C114- C66)/ p h-4c2}/ D,
(B.14)
N33 = F - c23+{[4C33C66P 2 - (C134- C66)2p~]/ p 2 - 4 c a p 3 ( C33 + C66)/p + 4 c 2 } / D ,
(B.15)
N13=2G_clc3+2{plP3(CllC13_4_
2 2 C66)/P
-- [clP3(C33+ C66) "]- c3PI( CI1 + C66)]/p + 2c~c3} / D,
(B.16)
N22 = c3/P3 ,
(B.17)
N12 = N21 = 0.
(B.18)
Several simplifying cases should be noted for (B.14)-(B.18). First, let p 3 = 0 for the qL wave. Then c3 = O, c1 = 1/pl = ( C l l / p ) ~/2 and N = diag(0, N22, N22), where N22 = [C66 "~-(C13-~- C 6 6 ) 2 / ( C l l - C66)]/p.
(B.19)
This represents the spreading of the pure L-wave propagating in the direction of the zonal axis. Second, let P3 = 0 for the qT wave. Then this is a pure transverse wave with c3 = 0 , c l - - 1 / p l = (C66/P) 1/2 and N = diag(0, N22, N22), where N22 = [C33- (C~3+
C66)2/(Cll- C66)]/P
(B.20)
Note that for both the qL and qT waves in the xrdirection, the matrix N has degenerate eigenvectors in the x2x3-plane. This is to be expected from the symmetry. However, the N matrix for the qT wave is not the same as N for the TH m o d e of (B.6), which is N = diag(0, B 2, B2). Thus, while the transverse wave slowness surfaces coalesce at P2 = P3 = 0, the local curvatures of the surfaces are different. The qL and qT modes also become pure modes if p~ = 0 . For the qL mode, we then have c1= 0, C3 = l i p 3 = (C33/P) 1/2 and N = d i a g ( N l l , N22, 0) with NI~ = [C66+ (C~3+ C66)2/(C33 - C66)]/p,
(B.21)
N22 = C33/P.
(B.22)
For the qT mode, c1= 0, c3 = 1/p3 = (C66/P) 1/2, N = d i a g ( N n , N22,0) and NI, = [ C n - (C~3+
C66)2/(C33-C66)]/P,
N22 = C66/p.
(B.23) (B.24)
Appendix C. Transverse isotropy transmission coefficients Reflection and transmission coefficients for a wave incident from an inviscid fluid upon a transversely isotropic solid are presented. The situation is as depicted in Fig. 5, with medium 1 the fluid, and medium 2 the transverse solid with the axis of symmetry in the xl-direction. One reflected wave is generated in the fluid. A quasi-transverse (qT) and quasi-longitudinal (qL) pair of waves are transmitted into the solid.
A.N. Norris/Pulses in anisotropic solids
531
Referring to Section 4.3, the incident and reflected waves are defined by g(l) = (sin
0 (I),
0,
(C.1)
COS 0 ( ! ) ) ,
p") = c?l(sin 0 (1), 0, cos 0°)),
(C.2)
g(R) = (sin 0 "), 0, --cos 0")),
(C.3)
p(R) = c~-l(sin 0(i), 0, - c o s 0")),
(C.4)
where cf is the fluid sound speed. The transmitted slownesses are
P(T~)=(p~T~),O,p(3T~)),
a = qL, qT,
(C.5)
where Snelrs law dictates that (C.6)
p~TqL) = p~TqT) = p ~ l ) ..~_P l "
The slowness components p~T~), a = qL, qT, then follow from Appendix B. The transmitted unit displacement vectors are
g(T~) = (G~), 0, Gg~))/EG~)~+ G(3Ta)2] 1/2,
t~ =
qL, qT,
(C.7)
where G~T~)=
p-1(C,3+ C66)plp~T'~), p
n2
G~Ta) = 1 --p-1~tlF1
~--1/,',
a = qL, qT,
,(Ta) 2
~, ~.~66p3
,
(c.8)
a = qL, qT.
(c.9)
The reflected and transmitted amplitudes are then (see Section 4.3),
[ u(°,R 1
Fc°s o(" TqL, g Tq )l_,[cos o(" 1 o(qL) Q(qT,] L 0
]
(c.lo)
where P(~) - C --
Q(a) ~
~ t~(T°t)"~ - / ~
13//1/51
~(Ta)o(Ta)..L
--F3
$1
~(Tu)o(Toz)
~'~33F3 s 3
~ ~(Tt~)
--Fl~;3
,
,
a = q L , qT,
a = qL, qT,
(C.11) (C.12)
and pf is the fluid density. Equation (C.10) follows from the continuity of 0-13, 0"33and u3 at the interface.
References [1] M.J.P. Musgrave, Crystal Acoustics, Holden Day, San Francisco, CA (1970). [2] G.F.D. Duff, "The Cauchy problem for elastic waves in an anisotropic medium", Philos. Trans. Roy. Soc. London A 252, 31-273 (1960). [3] E.A. Kraut, "Advances in the theory of anisotropic elastic wave propagation", Rev. Geophys. I, 401-447 (1963). [4] R.G. Payton, Elastic Wave Propgation in Transversely Isotropic Media, Nijhoff, The Hague (1983). [5] A.N. Norris, B.S. White and J.R. Schrieffer, "Gaussian wave packets in inhomogeneous media with curved interfaces", Proc. Roy. Soc. London A 412, 93-123 (1987). [6] V. Cerveny and I. Psencik, "Gaussian beams and paraxial ray approximation in three-dimensional elastic inhomogeneous media", J. Geophys. 53, 1-15 (1983). [7] A.N. Norris, "Elastic Gaussian wave packets in isotropic media", Acta Mech. (1987), to appear.
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A.N. Norris / Pulses in anisotropic solids
[8] V. Cerveny, I.A. Molotkov and I. Psencik, Ray Method in Seismology, University of Karlova, Prague (1977). [9] V. Cerveny, "Seismic rays and ray intensities in inhomogeneous anisotropic media", J. Geophys. 28, 1-13 (1972). [10] H. Murakami and G.A. Hegemier, "A mixture model for unidirectionally fiber-reinforced composites", J. Appl. Mech. 53, 765-773 (1986). [11] V.T. Buchwald, "Elastic Waves in Anisotropic Media", Proc. Roy. Soc. London A 253, 563-580 (1959).