WAVES IN ANISOTROPIC ELASTIC MEDIA
E. L. Roetman
MathMechanics, 3016 67th Ave. SE, Mercer Island, WA 98040, USA
ABSTRACT. Reformulation of the elastodynamics problem as a system of first-order partial
differential equations is reviewed and preliminary results are presented for isotropic and anisotropic
materials that suggest that the approach will extend in a reasonable way to analysis of more general
materials. The propagating fields are expressed as a sum of characteristic propagation modes of an
expanded acoustic matrix, which include the stress terms as well as the displacements. Boundary
projection operators for the reflection and refraction coefficients are determined and the classic
Frennel coefficients are recovered as particular examples within a general theory.
INTRODUCTION
The propagation of plane waves of infinitesimal amplitude in an isotropic material is
determined by solving the linear second order Navier equations for the displacements
putt
= A + //)grad(divw + //V 2 w
(1)
The potential methods that are traditionally used decompose the displacement field into a
scalar potential term and a vector potential term, which thereby decompose the vector
dynamical system into uncoupled systems for the individual fields [1]. The time harmonic
case given by
u =Aex$(jk(K'x)-ja>t),
|H| = 1
(2)
g =pa>2^
(3)
leads to the acoustic matrix
[(1+H}K®K
+(ju-g)3]A
= 0,
which provides the usual signal speeds
C\ = (A + 2ju)p-1,
C2T
=np-1.
(4)
The disadvantages of working with system include the fact that there is difficulty
applying some boundary conditions, and it is very difficult to analyze an anisotropic
CP657, Review of Quantitative Nondestructive Evaluation Vol. 22, ed. by D. O. Thompson and D. E. Chimenti
© 2003 American Institute of Physics 0-7354-0117-9/03/S20.00
141
problem by using the method of potentials as the anisotropic system does not uncouple
with respect to the potentials in any reasonable way for the more general anisotropic case
modeled with
pumnt
=DikmUj,kl.
(5)
The level that can be achieved is illustrated in [2].
FIRST ORDER SYSTEM
Reformulation of the infinitesimal, linear problem to that of a system of nine first-order
partial differential equations suggests that the above objections can be mitigated. Set
(Love notation) sl=tll , s2=t22 , s 3 =f 3 3 , s4=t23 » ^5=^13 > S6=tu \and e\=u\n >
e2=u2,2 , e 3 =w 3 , 3 , e 4 =w 2 , 3 +w 3 , 2 , e 5 =w 1 J 3 +M 3 , 1 9 e6= u },2 +u 29l (where the commas
mean differentiation with respect to the indicated variable) to get
s =Me = T
M
il ° 1\e
M2
and
e = Es =\T
E
i ° 1\s.
l
O
E
(6)
The continuum form of Newton's laws can now be written in terms of the velocity 3vector v (vk=uk) and the stress 6- vector s in the form
pv=div T +F 9 Es=e=f(v).
(7)
We introduce a stress- velocity nine- vector as a partitioned column vector, and develop
the system (7) as a 9x9 system of first order, partial differential equations. Write then a
nine- vector z as
ZT
=(sl9s2,s3,s4,s5,s69vl9v2,v3) .
(8)
The mechanical system (7) now becomes a system with the standard form
Bz,, =lLB*z,,
(9)
where the commas mean differentiation with respect to the indicated variable. The
coefficient matrices will be explicitly exhibited for the isotropic case with comments on
the procedure for the anisotropic. Thus, B becomes a partitioned matrix made up of 6x6
and 3x3 blocks as
B =
v; i
with E defined in (6) and E 3 =/?3 , a diagonal, scaling matrix. The matrices of
coefficients for the spatial derivatives will be symmetric and built up in steps as
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B*
=
0
0
T
Q\
0
0
Q\
Q\
Q?
o
k= 1,2,3
(11)
where the symbol 0 again represents 3x3 sub-matrices of zeroes and the border submatrices are incidence matrices defined as diagonal and symmetric matrices (we give only
the value of the non zero term)
Q\ = diag[l 00]
Q\ = diag[0 10]
Q\ =[Q\ 32 =i]
Q\ =[Q\ 31 =i]
Q\ = diag[0 0 1]
(12)
Q\ =[Q\ 21 =!]•
This system is the foundation of the development that follows. The usual 3dimensional acoustic matrix is buried in this larger system, and we will determine the
propagation modes and speeds from this form. Moreover, we will get additional
information for stress and strain interaction since both the stress terms and the
displacement velocities are determined simultaneously. We will also get improved
understanding of the boundary interactions since the stress and displacements will appear
in a natural way at the boundary, incorporated through projection operators generated by
the boundary relationships which are to be developed below. The remainder of this article
is devoted to drawing information about the propagation of elastic waves and their
interaction with a plane boundary or interface using this formulation of the elastodynamic
system. The reader may want to compare this discussion with that in v. d. Hijden [3] and
S. Nielsen [4].
The approach through first order, partial differential equations taken here may be
different enough that it may be useful to consider the one-dimensional problem where the
space variation of the parameters is in the x coordinate only. The system (9) becomes now
Bz,,
=Elz9l.
(13)
Since only x variation is considered the rows of zeroes in Q\ and Q\ mean that three
of the nine equations of the system have no spatial derivatives on the right hand side. That
is, the 9x9 matrix B1 has only 6 non zero elements in the positions indicated - the
remaining terms all being zero
B 1 = [B\1=Bl9=Bl62=Blll=Bl6=Bl95=l\
which means that the second, third and fourth equations in (13) give three equations of
compatibility from the matrix B that are solved to obtain
When these relations are substituted into the remaining six equations, the system reduces
to three uncoupled systems of two equations each including a stress term and
corresponding velocity. The longitudinal wave satisfies the equation
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and transverse waves satisfy
Each of the three systems has the form of transmission line equations for which solutions
are readily obtained. Clearly, the transverse waves have the same signal speed.
To illustrate the extension to composite material consider a material with elastic
properties consistent with a graphite-epoxy lay-up with fibers in the z direction having an
elasticity matrix in psi units
1.0132 -.3954 -.009
0
0
0
-.3954 1.0132 -.009
0
0
0
-.009 -.009 .0372
0
0
0
0
0
0
0
2.8153
0
0
0
0
0
2.8153
0
0
0
0
1.3895
0
0
(14)
and density /? = .0561b/in3 yielding the partitioned matrix
~E
:
0
B=
0
The second, third and fourth equations in B1 again vanish giving compatibility conditions
and reduced systems of equations with distinct propagation velocities as
.8547(lO> w =v l f I
.056Vp, -sl9l
for the longitudinal wave and for the transverse waves
2.8153(10^)^,, =v 3J1
1.3895(lO- 6 )* 6 ,,=v 2 , 1
0.056 v 3 ,,=s 5 , 1
0.056 v 2J/ =s6n
PROPAGATION MODES
The general space and time harmonic representation for the stress-velocity
9-vector is given by an expression generalizing (2)
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When substituted into the first order system of partial differential equations (9) describing
the elastodynamics of the material, this leads to the system of algebraic equations for the
amplitude vector A given by
k(Z w *- w B w
+afi)A
=0
(16)
where k is the wave number and a =6>k~ 1 . This is a 9-dimensional eigenvector
problem with dependence of the eigenvalues and eigenvectors on the propagation direction
vector K.. The system (16) provides what we will call the propagation matrix, a 9x9
matrix
so that the system (4-2) becomes kC(#;/r)A =0 . The matrix C(0;/r) is a
partitioned matrix that can be reduced to an upper block diagonal form by premultiplication by a partitioned matrix so that there holds the relation
(18)
^J[ 0
c^;rjj
where we have in general the matrix relation
c(<2;/r) =QT(JC)MQ(K')-pa2 3
(19)
which in the isotropic case is just the familiar acoustic matrix (3). The previous discussion
of the eigenvalues and eigenvectors of the acoustic matrix is again applicable, but needs to
be expanded to reflect the stress components that are associated with the eigenvalues of the
system. The eigenvectors are now 9-dimensional with 3 components of velocity and 6
components of stress. The basis vectors un (n = 1, 2, 3) of the kernel of c(a',/c) are
eigenvectors of
QT(K}MQ(K}
=0
(20)
(/i =1,2,3)
with corresponding propagation speeds an (n - 1, 2, 3) determined by the eigenvalues.
Use of the partitioned form of the mode amplitude vector An =[/?„
in system (18) yields that it is required that the 6-vector satisfy
p, =a-lMQ(r)un,
145
(»=1,2,3).
u n]T
(n=l,2, 3)
(21)
The complete algebraic description of the stress and velocity components will not be
possible in this case, but numerically the above steps can be reproduced so that specific
anisotropic problems can be investigated. Thus, determination of the eigenvalues and
eigenvectors of an appropriate problem again yields the propagation structure of the
general anisotropic elastic problem.
A plane wave incident on a plane interface is reflected and transmitted, at angles to be
determined, dependent upon the nature of the materials or conditions at the interface. The
engineering problem that we wish to solve is that of an incident wave coming from infinity
in the second quadrant of the (x19 x2) coordinate plane, and then reflecting into the first
quadrant and possibly being transmitted into the fourth quadrant. Certainly the system can
be analyzed without this reduction to two coordinates, but it is much more cumbersome.
The general first order system (9) reduces now to the system with only two space
derivative and the plane wave representation has the usual form obtained from (16) by
simply setting *T3 = 0 .
The results of the previous sections do apply and the propagation modes that result for
isotropic material will include the usual longitudinal wave, and two transverse shear waves
as in the one-dimensional example discussed earlier. Associated with each propagation
mode determined above is a time harmonic fundamental solution to the first order system
(9). We will establish here fixed representation formulae for each of these fundamental
solutions to be available for the solution. The mode amplitudes will not be normalized to
one, but will reflect the magnitude of the 6-vector induced from the unit propagation
direction K. With reference to the discussion above, the longitudinal mode fundamental
solution is
Z/(kL,jr,jc) =^/exp(yk L (^-jc ))
where k|_ is the associated longitudinal wave number and the amplitude 9- vector is
Al = \_PL(K}
^ ] • Also, the transverse wave has a fundamental solution given by
Z/z(k T ,/f ;jc) =^exp(yk T (^-;c))
where ky is the transverse wave number and the corresponding amplitude 9-vector is
Ah = [qH(ic,vH)
VH~\
Also, the other transverse wave has a fundamental
with amplitude 9-vector Av =[# F (/f,v F ) vv ] . Since these are the only propagation
modes consistent with the eigenvalues of the acoustic matrix and the 9 x 9 acoustic
propagation matrix, the propagation of disturbances can be expressed entirely in terms of
these basis modes.
The time harmonic field representations for the reflected and incident waves in the
material containing the in-coming wave is given in terms of the fundamental solutions
defined in the previous section as (zr(jc) + zz(jc)) and for the transmitted waves in the
second material it is zt(x) where the spatial dependence terms are (including the
146
possibility of distinct propagation directions)
zr(x)
=AlZh^(^K(l\x}+A2Zv^(^K"^
(22)
zt(x) =5
where the coefficients A], A 2, A$, Bj, 82, B$ are determined by the conditions at the
boundary. The incident wave is also expressible in terms of these fundamental modes
with known coefficients. It will be given as a single mode incident wave as any multimode incident solution can be synthesized by addition. Thus, write
At the boundary, some part of the total stress-velocity 9-vector that specifies the field
will be influenced by the material properties of the region. The interaction is expressed in
terms of projection matrices that pick out the pertinent components. It is found that the
interaction at X2 = 0 can be expressed as:
BPzr-BPPz/ = -BPz/ ,
(23)
where BP and BPP are appropriate incidence matrices made up of zeroes and ones
determined below in specific examples. In any case, at the interface each term will have
the form mQxp(jkfCl jcj. That the relations (23) are to hold for all values of xi it is
necessary that the phase values must be equal, which leads to the familiar Snell's Law.
Since the spatial phase factors can be cancelled from the expression, (23) leads to
(1)
+ A2 Av(1)
+ B2 Av(2)
} Ah
(2)
-EPP(Bj Ah
= -BP ai
By creating new 9x3 matrices using the three mode amplitude 9-vectors as columns, we
obtain the matrices
Mr
= [Ah(l'
Av^
M,
= \Ah(2)
Av(2]
With these matrices the system (6-10) becomes
~B,
BPM r
- BPP Mt
= - BP ai
Further consolidation is achieved if the system is written as a partitioned system for the
single partitioned 6-vector composed of the coefficients AI, ... , BS . One can reorder the
rows and columns to obtain the modified system
147
"a 4i
1
0
0
0
0
044 0
0
- 1 0 0
0 0 2 2 a23
0 a62 a63
0 a72 a73
0 a 8 2 a 83
0
0 " "V
0 0 *i
a25 a26 ^ 2
a65 a66 ^ 3
a75 a16 B2
a 85 # 8 6 _ A _
~ai~
ai9
ai2
= Ai
ai6
ai7
ai,_
For the incident wave
ai = 0 0 0 ^ - cos# -A sin ft
0 0 0 1
the reflection coefficient R defined by Aj =RAi is the Fresnel coefficient
^
COS
#
The details of the analysis require much more space than is available here, but the ability to
assess propagation and plane boundary scattering for anisotropic, composite materials is
established.
REFERENCES
1. Achenbach, J.D., Wave Propagation in Elastic Solids. North-Holland, Amsterdam,
1975.
2. Musgrave, M.J.P., Elastic Waves in Anisotropic Media, Progress in Solid Mechanics,
Vol. II. (J.N.Sneddon and R. Hill, editors). North-Holland, Amsterdam, 1961.
3. van der Hijden, J.H.M.T., Propagation of Transient Waves in Stratified Anisotropic
Media. North-Holland, Amsterdam, 1987.
4. Nielsen, S.A., "Elastic wave modeling using a multidomain Chebyshev collocation
method," in Review of Quantitative Nondestructive Evaluation vol. 27, edited by D.O.
Thompson and D.E. Chimenti, AIP Press, 2002, p. 35-42.
148