A computationally efficient representation for propagation of elastic waves in anisotropic solids

A computationally efficient representation for propagation of elastic waves in anisotropic solids

V.K. Tewary and C. M. Fortunko Materials Reliability Division, National Institute of Standards and Technology, Boulder, Colorado 80303

(Received 14 June 1991; accepted for publication 4 January 1992)

A new closed-form representation is developed for the exact solution of the Christoffel equation for wave propagation in solids. The new representation is numerically more efficient than the traditional representations based on the use of Fourier and Laplace transforms. Using the new representation, the retarded Green's functions are derived for an infinite anisotropic solid and an anisotropic half-space. The method is applied to calculate the elastic-wave response of an anisotropic cubic solid to highly localized delta function and step function type impulses. Both surface and bulk wave responses have been calculated. The effect of anisotropy is discussed by considering cubic solids with different anisotropy parameters. Interestingly, it is found that, for certain values of the anisotropy parameter, two distinct longitudinally polarized components can be observed to propagate along an axis of cubic symmetry. One of the signals is the normal longitudinal wave signal while the other results from the concave shape of the transverse slowness surface.

PACS numbers: 43.20. Gp, 43.20.Jr, 43.35.Cg, 43.40.Ph

INTRODUCTION

In this paper, we describe a new method of obtaining closed-form solutions of the Christoffel equation for elastic- wave propagation in solids. Our method is particularly useful for studying the space-time dependence of time-limited signals that propagate in anisotropic solids. We represent the solutions in terms of functions 1/(q.x + t)", where x and t are the space and time coordinates, q is a slowness vector, and n is an integer.

Our method is computationally more efficient than the traditional methods based on Fourier (or Laplace) transforms (see, for example, Refs. 1-3). The Fourier transform over space and time involves 4-D (four-dimensional) integrations. For anisotropic solids, these integrations have to be evaluated numerically, except possibly over one coordinate. On the other hand, our method involves 3-D integrations. In general, integration over one coordinate can be carried out analytically. Thus, only a 2-D integration needs to be done numerically. Moreover, the Fourier integrations have to be carried out over oscillatory functions that are poorly conver- gent. In our representation, the basis functions have a smooth behavior that helps the convergence.

Our study has been motivated by specific experimental design needs arising from the use of ultrasonic nondestructive evaluation methods to study the mechanical properties of polymer composites of very high elastic anisotropies. The anisotropy can have a large effect on the spatial and tempo- ral characteristics of the ultrasonic signals used to study the properties of the composites. Whereas the traditional Four- ier transform methods 1-3 are suitable for isotropic solids, their application to anisotropic material systems results in complicated representations, which are not convenient for interpretation of the experimental results.

Most of the analytical calculations done in the past on elastic wave propagation are confined to isotropic solids. The published calculations on anisotropic solids are either confined to 2-D models 4 or based upon certain approxima- tions 5 or numerical schemes that may be useful only in spe- cial cases. Various general characteristics of elastic-wave propagation in anisotropic solids have been given in Refs. 6- 10, which deal with the plane waves in infinite solids. Even purely numerical studies (see, for example, Refs. 11 and 12) are mostly limited to 2-D models since a 3-D numerical calculation requires unduly large CPU time. Our calculations should be useful for interpreting the experimental results on elastic pulse propagation in modern composite materials that are highly anisotropic and cannot be modelled as 2-D solids.

In this paper, we use our representation to examine in detail the general properties of solutions that represent signals originating at source points located either inside an infinite or on a semi-infinite ("half-space") anisotropic solid. We study the evolution in time and space of two types of elastic-wave signals generated separately by two localized pulses: a delta function and a step function in time. Both are assumed to be localized as delta functions in space. The delta function gives the space-time Green's function, which can be used to calculate the solution for any integrable force distribution in space and time.

We illustrate our method by applying it to infinite and half-space cubic solids. We show that by varying the value of the anisotropy parameter, it is possible to observe elastic- wave responses that would not be generally expected in isotropic media. We also show that our representation yields the familiar results in the isotropic limit.

We find an interesting result for anisotropic cubic solids. For certain values of the anisotropy parameter, and for a

1888 J. Acoust. Soc. Am. 91 (4), Pt. 1, April 1992 1888

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longitudinal delta function source, we find two longitudinally polarized signals in a high symmetry direction. One signal travels with the usual longitudinal wave velocity, whereas the other travels with a velocity close to that of the transverse wave. This result arises due to the concave nature of the slowness surface.

Our representation for the solution of the Christoffel equation for an infinite anisotropic solid, is described in Sec. I. The effect of a free surface and elastic-wave propagation in a half-space solid are calculated in Sec. II. The application of the formulation to cubic anisotropic solids and the effect of anisotropy are discussed in Sec. III. A brief summary of the results and the conclusion are given in Sec. IV. Some mathematical properties of our representation and an inversion formula are described in Appendix A. The method is used to reproduce the known results for isotropic solids in Appendix B.

I. BASIC FORMULATION FOR AN INFINITE SOLID

In this section we describe the basic formulation for

elastic-wave propagation in an anistropic infinite solid. We assume a Cartesian frame of reference as shown in Fig. 1. The space and time variables will be denoted by x and t. The X, Y, and Z components of a vector will be labeled by indices 1, 2, and 3. Summation convention over repeated indices will be used unless indicated otherwise.

The matrix form of the equations of motion (called Christoffel equation) is

• (x,t)u(x,t) = F(x,t), ( 1 ) where

•2 •2

•qfij(X,t) • Cikjl •Xk •Xl --p o•t2 , (2) u is the displacement field, c is the elastic constant tensor, p is the density of the solid, F is the applied force or the source strength, x and t are space and time coordinates, and the

Free Surfoce

FIG. 1. The coordinate axes. The Yaxis is normal to the plane of the paper. The free surface is taken on the plane x• = 0. For calculation of the Green's function, the source is taken at R and virtual forces are applied just outside the free surface.

indices i, j, k, and I label the Cartesian coordinates. We assume p = 1, which defines the units of c and F. For later reference, the homogeneous Christoffel equation is

•6• (x,t)u(x,t) --0. (3) We solve Eq. ( 1 ) for two types of applied forces:

(i) DF (delta function) pulse source,

F i (x,t) =f•(x)•(t) (4)

and (ii) HF (step function) pulse source,

F, (x,t) =fS(x)H(t), (5)

where f is a constant vector, and 8 and H are, respectively, the Dirac delta function and the Heaviside step function.

The solution ofEq. ( 1 ) for the DF pulse with unit f gives the space-time Green's function for the Christoffel equation. Our interest is in the retarded Green's function that is 0 for t < 0. The retarded Green's function can also be constructed

by solving the Cauchy problem for Eq. (3), subject to the initial conditions,

Ga (x,0) =0 (6) and

c•G, (x,t = 0) = •(x), (7)

c9t

where G• (x,t) is the antisymmetric Green's function that is a solution of Eq. (3). The solution for other initial conditions can be written in terms of G• by using the Stokes- Duhamel formula. 6'? The retarded Green's function G(x,t) is given by

G(x,t) = H(t)G• (x,t). ( 8 )

The solution ofEq. ( 1 ) i•or any integrable force distribution F'(x,t) can be written in terms of the Green's function as

u(x,t) = f G(x- x',t-- t')F'(x',t')dx' dt', (9) where the integration is over the entire range of x and t. If F'(x,t) is 0 inside the space of the solid, Eq. (9) will be a solution of Eq. (3) in the space of the solid.

We solve Eq. (1) for a HF pulse subject to the initial condition given by Eq. (6). We also solve Eq. (3) subject to Eqs. (6) and (7), which gives the solution of Eq. ( 1 ) for a DF pulse. We consider the solution of Eq. ( 1 ) only for and omit the factor H(t) in Eq. (8). For both DF and HF pulses, we require

u(x,t)-•0 as x-• c• (10)

Consider the set of functions

mnq (x,t) = 1/[q'x + t ]n, ( 1 1 ) where n is a positive integer and q is a slowness vector. All three components of q vary between -- oo and + c•. For any n, Eq. ( 11 ) will be a solution of Eq. (3), if q is a solution of

Det L(q) = 0, (12) where

L(q) --A(q) --I, (13)

I is the unit matrix, and A (q) is the Christoffel matrix,

1889 J. Acoust. Soc. Am., Vol. 91, No. 4, Pt. 1, April 1992 V.K. Tewary and C. M. Fortunko: Elastic waves in solids 1889


Aij (q) =Cijklqkql. (14)

The functions m nq (X,t) provide a representation for the general solution of Eq. ( 1 ). We attempt to write a solution of Eq. (1) in the form

u(x,t) = (2rr) -4p f fi(q)mn q (x,t)dq, (15) where the integration is over the entire 3-D q space and P denotes the Cauchy principal value. In Eq. (15), for any n, the functions mnq (x,t) transform a function fi (q) of one variable q into a function u(x,t) of two variables x and t. This is similar to a wavelet representation. 13 For n ---- 1, the imaginary part of the rhs in Eq. (15) will give the Radon transform.

Our objective is to determine fi(q) and suitable combinations of the m functions so that Eq. (15) gives the solution of Eq. ( 1 ) and satisfies the boundary and initial conditions. We define the Green's function matrix in q space as

•,(q) = [A(q) -- I] -l, (16) which is analagous to the Green's function in Fourier space. 14 From Eqs. (A15) and (A21) of Appendix A, we see that the solution of Eq. ( 1 ) for the DF and HF pulses are (for t>0) as follows: (i) DF pulse

u(x,t)-- 2 G(q) . 1 (2rr) 4 oo (q.x + t +/;•)2

1 ) (q.x -- t + re) 2' dq, (17) (ii) HF pulse

u(x,t) = 1 G(q)f '(q.x + t + re) (2•') 4 - oo

1 )del. (18) (q.x- t +

The integrals in Eqs. (17) and (18) are to be interpreted a•s Cauchy principal value integrals at the singularities of G(q). For this purpose, it is convenient to replace I in Eq. (16) and onward by I ( 1 + •e), where • = x/( - 1 ) and e = + 0 in the limit. The actual displacement field is given by the real part of the expressions.

Equations (17) and (18) give the required solutions of the Christoffel equation. Both solutions satisfy Eqs. (6) and (10). Equation (17) also satisfies Eq. (7), which explains the difference of the factor 2 between the two equations. As shown in Appendix B, these solutions reduce to the well- known solutions in the isotropic case. In general, the 3-D integration in Eqs. (17) or ( 18 ) has to be done numerically. However, as shown in Sec. III, it is possible to integrate analytically over one component of q.

The solution ofEq. ( 1 ) for a DF pulse could be obtained by taking the time derivative of the solution for a HF pulse given by Eq. (18). However, it would not satisfy Eq. (6). The solution given by Eq. (17) satisfies Eq. (6) but is a solution of Eq. (3). Equation (17) also satisfies the initial momentum condition given by Eq. (7). It is, therefore, a

solution of the Cauchy problem. 6 The retarded Green's function is then given by Eq. (8) with Ga given by Eq. (17) and f replacedby a unit vector. The presence of H(t) term in the solution gives rise to 6 (t) term when Eq. ( 8 ) is substitut- ed in Eq. ( 1 ).

Physically, this feature of the DF solution arises from the fact that a DF pulse gives a finite instantaneous momentum to the solid but not displacement at t = 0. The initial momentum at t = 0 is expressed by the initial condition given by Eq. (7). After t = 0, the solid is not subjected to a body force. The Cauchy problem consists of solving Eq. (3) subject to the initial conditions given by Eqs. (6) and (7). Since our interest is only in the time domain t>0, in what follows, the factor H(t) will not be written in the solution for a DF pulse unless explicitly needed.

II. EFFECT OF A FREE SURFACE

In this section, we consider the effect of a free surface on elastic-wave propagation. We model the solid as an elastic half-space extending from x• = 0 to x• = oo in the X direction and - oo to + oo in the Yand Z directions as shown in

Fig. 1. We consider two cases: a buried point source and a surface point source.

We first consider a buried point source at R inside the half-space solid with a DF or a HF time dependence. The DF source will give the space-time Green's function for the solid. We solve Eq. ( 1 ) subject to Eq. (6) and the surface boundary condition

•'il (Xg-,X3,t) = O, (19)

at x I = 0. In Eq. (19), v is the stress tensor defined by

8u• (20) 'J'il '• Ciljk •X k ß To satisfy Eq. (20), we apply a virtual force distribution

outside the free surface. The solution obtained by using this force function in Eq. (9) will be a solution of Eq. (3). We then write the total solution of Eq. ( 1 ) as

u(x,t) -- 2 G(q) [fM2q (x -- R,t)H(t) (2•/') 4

+ h(q•_,q3;R)M%(x + e,t) ]dq, (21) where h denotes the contribution of the virtual force.

Following the steps given in Appendix A, we see that the second term on the rhs of Eq. (21 ) is a solution of Eq. ( 3 ) in the half-space xl >0. Our object is to determine h such that u (x,t) satisfies Eq. (19). Using Eqs. (19) and Eq. (20) on Eq. (21 ), we get an integral equation for h. This equation is solved by using the integral representation given by Eq. (A6). Finally, for t>0,

u(x,t) -- 2 G(q) (2•/') 4 - •o

X (M2q (x- R,t)tt(t) --exp(t•ql ) f_, X [S(q2,q3 ) ] -1 {r(q[,q2,q 3 )G(q[,q2,q 3 )

X P2 (q,q; ,x,R,t) dq; )f dq, (22)

1890 d. Acoust. Soc. Am., Vol. 91, No. 4, Pt. 1, April 1992 V.K. Tewary and C. M. Fortunko: Elastic waves in solids 1890


where

S(q2,q3 ) • o'(ql ,q2,q3 )G(ql ,q2,q3 )

X exp(teql )dql,

ai• (q) = ciu•q•, and

P• (q,q[ ,x,R,t)

(23)

(24)

1 ) [{l'(X -- R) + R 1 (ql --q•) -- t -[- t•:] n' ' (25) with n = 2 in Eq. (22). As discussed at the end of Sec. II, the first term in Eq. (22) contains the factor H(t).

Equation (22) gives the 3-D space-time Green's function for an anisotropic half-space solid. Similarly, the solution of Eq. ( 1 ) for a HF pulse is

u(x,t)

1 G(q) Mlq (x -- R,t) -- exp(teql ) (2n') 4 - •o

X [S(q2,q3 ) ] -l tr(q•,q2,q3 )G(q• ,q2,q3 )

XP• (q,q• ,x,R,t)dq• )f dq, (26) where P• is given by Eq. (25) for n ---- 1.

Now we consider a surface point source that can be represented by imposing the boundary condition 2'3

•'i! (X2,X3,t) -- fi6(x)F(t), (27)

at Xl = 0, where F(t) = H(t) for a HF stress pulse. We represent a DF pulse in terms of a localized 6(t) force at the surface and not in terms of stress. The physical reason for this representation will be discussed at the end of this section. We solve Eq. (27) for f = 6• or 6i3 corresponding to applied •'• or r3•. The initial condition is given by Eq. (6).

Following the technique used for the buried point source as given above, we obtain the following results: (i) for HF pulse

u(x,t) = _ t• G(q) [S(q2,q3)] -• (2T;) 3 oo

X exp(tEq• )Mlq (x,t) f dq, and (ii) for DF pulse:

u(x,t) -- 2 G(q)[S(q2,q• ) ] --1 (2•) 3 - oo

X exp(tEq• )M2q (x,t) f dq.

(28)

(29)

It can be verified by direct substitution and following the steps given in Appendix A, that Eq. (28) satisfies Eq. (27) forxl =0. Both Eqs. (28) and (29) satisfy Eq. (3) in the half-space x• >•0. To prove it, we use the integral representation given by Eq. (A6). We substitute u(x,t) in Eq. (3), use Eq. (16) and integrate over q. Since S is indepen-

dent of ql, the integral over ql will be 0 for xl >0 in view of Eq. (A13).

The solution given by Eq. (29) along with the factor H(t) corresponds to a 6(t) force at the surface and satisfies the free surface condition. As discussed at the end of Sec. II, a DF force gives an instantaneous momentum to the solid at t = 0 but no instantaneous displacements and, therefore, zero stress at t - 0. After an infinitesimal time the stress is

localized as 6(x) in space in accordance with Eq. (27) and decreases rapidly with time. The solution given by Eq. (29), therefore, does not exactly represent a DF stress pulse.

III. ELASTIC WAVE PROPAGATION IN CUBIC SOLIDS

In this section, we apply the formalism developed in the preceding sections to cubic solids. For cubic solids, the matrix elements of L(q) as defined by Eq. (13) reduce to

Lii (q) = (Oil -- c4.4 )qi 2 + c44q 2 -- 1, for i =j, and

L/i (q) = (c12 .-Jr- c4. 4 )qiq• for i•=j. The anisotropy parameter is defined by

T] = (Cll -- Cl2 )/2c44.

(3O)

(31)

(32)

For isotropic solids r/= 1. The anisotropy corresponding to r/> 1 is called positive anisotropy and to r/< 1, the negative anisotropy.

In general, we can write G (q) in the form,

G(q) = C(q)/D(q), (33)

where D(q) is the determinant of L(q), and C(q) is the matrix of cofactors of L(q). The elements of C(q) have no singularities. Hence, the singularities of G(q) are at the zeros olD(q). Further, the matrix elements of C(q) are poly- nomials of degree 4 in q, whereas D(q) is of degree 6. Each element of G(q) will therefore, approach 0 at infinity in q space as l/q2. We shall use this fact in the evaluation of the q integrals in u (x,t).

We first consider elastic-wave propagation in infinite cubic solids. To study the effect of anisotropy on wave propagation, we consider three solids corresponding to r/= 1.25 (positive anisotropy), 1 (isotropic), and 0.75 ( negative anisotropy). For all the three solids, we take C ll = 4 and c44 = 1. The corresponding values of the longitudinal and the transverse wave velocities in the (0,0,1 ) directions are 2 and l, respectively. The value of Ca2 for each solid is given by Eq. (32).

We calculate u(x,t) at the plane Xl = 0 for different values of t/x by using Eqs. (17) and (18). The slowness surfaces and the wave surfaces for cubic solids for plane- wave propagation have been discussed in detail by Mus- grave. 9 We, therefore, restrict our attention to pulse propagation.

We evaluate the integrals in Eq. (17) and (18) analytically over ql by using contour integration in the complex ql plane. For x• = 0, the only poles in the two integrands will 'be at the zeros of D(q). It is convenient to introduce a factor exp(tq• •) in the numerator of the integrand, where • ap-



proaches q- 0 in the limit. We can then take a semicircular contour in the upper half-plane. The contribution of the se- micircle vanishes in the limit and the integral is given by the standard Cauchy formula.

For a cubic solid, D(q) is a polynomial of degree 3 in q•2 (i = 1-3). Equation (12) can be solved analytically for the six roots of q l. These roots give the poles of the integrand in the complex q l plane. After the q l integral is evaluated analytically, we evaluate the integrals over q2 and q3 numerically.

The numerical choice of e in Eqs. (17) and (18) is im- portant for the convergence of numerical integration process. Since these are principal value integrals, the value of e should be chosen to be equal to the integration interval. For better convergence, we normalize the integrals by the numerical value of N where

N = Im f_• (y + re) -1 dy. (34) The integral in Eq. (34) is calculated over the same limits as chosen for q2 and q3 and using the same numerical value of e. Theoretically, the value of N is rr in the limit e = q- 0.

The calculated values of [ z 2 u3 (x,t) ] and [ z u3 (x,t) ] in the (0,0,1 ) direction, that is, for x = x3 = z, are shown in Figs. 24 for the three solids for DF and HF pulses. In these calculations, f = (0,0,1 ), so the curves give the longitudinal displacement in the Z direction, which is one of the high- symmetry directions. Figure 2 shows the result for an isotropic solid. Although it could be obtained analytically, we calculated it numerically by using the general program for the purpose of checking.

We first examine the known results for the isotropic solid, shown in Fig. 2. The maximum for the DF pulse occurs at t/z = VL, the longitudinal wave velocity. The HF pulse gives almost a constant value of (zu 3 ) for large t showing the well-known TM 1/z behavior of the static displacement field. The displacement field for the DF pulse becomes 0 for large t. The result for the DF source shows that, in addition to the DF pulse, a linearly varying displacement field, which we

32.0

28.8

25.6

22.4

19.2

5'16.o c)

12.8

9.6

6.4

3.2

0.0 0.00 0.22 0.44 0.66 0.88 1.10 1.32 1.54 1.76 1.98 2.20

t/z

FIG. 2. The longitudinal displacement field (u 1 ) plotted against t/z for a delta function (DF) and a step function (HF) pulse source with the wave traveling along the Z axis in an isotropic cubic solid. For the DF source, C = z 2 and for the HF source, C = z. The source is at (0,0,0) with the force in the Z direction.

27.0

24.3

2! 6

!89

!62

_•!35

408

81

54

' HF

27 / / O0 --•-.• .•- ,•r• , , , , , , , ,••, ,. , ,

0.00 0.22 0.44 0.66 0.88 1.10 1.32 1.54 1.76 1.98 2.20 t/z

FIG. 3. Same as Fig. 2 for a cubic solid with negative anisotropy.

call the secondary displacement field, is superposed on the DF displacement. This gives a weak second maximum in the curve at about t/z = Vt.

The fact that the displacement for the DF pulse is not exactly a •5 function, or that for the HF pulse is not exactly an H function, is because of the numerical approximation, since e was not 0 and the limits of integration were not o•. Similar- ly, the finite relative height of the delta function maximum compared to the second maximum is also due to the numerical approximation. For a delta function, this ratio would be

However, the finite ratio of the maxima in the DF case is significant from the practical point of view because an ideal DF pulse can not be generated experimentally. A nonzero value of e corresponds to a pulse with a finite duration. The linear rise in the displacement is included in the Stokes formula 2 and has also been obtained by Bresse and Hutchins. 15

A rather dramatic effect of anisotropy occurs on the second maximum as shown in Fig. 3. For negative anisotropy (•/< 1), the secondary displacement field varies faster than linear. As a result, the second maximum in the displacement field for the DF pulse is exaggerated as shown in Fig. 3. Depending upon the value of •/, the second maximum can be comparable to the first maximum. In an experiment, it would appear as though two longitudinally polarized pulses were traveling in the Z direction--one with the usual longitudinal wave velocity and the other close to the transverse wave velocity. This result is apparent in the theory of Duff 6 but has not been explicitly noted earlier.

The result for positive anisotropy (•1 > 1 ) is shown in Fig. 4. In this case the variation of the secondary displacement field is slower than linear and its contribution is small.

Hence, the second maximum in the curve is absent. The shape of the pulse becomes more like the incident delta function.

The physical process responsible for the secondary displacement field has been discussed in detail by Duff. 6 The existence of the second maximum in Fig. 3 can be under- stood by considering the shape of the slowness surfaces near a symmetry direction. Figure 5 shows the calculated YZ cross sections of the quasilongitudinal and quasitransverse slowness surfaces for the three cubic solids corresponding to



45.0

40.5

36.0

31.5

27.0 DF 22.5

18.0

15.5

9.0

4.5

0.0 0.00 0.22 0.44 0.66 0.88 1.10 1.,32 1.54 1.76 1.98 2.20

t/z

FIG. 4. Same as Fig. 2 for a cubic solid with positive anisotropy.

•7 = 1, 1.25, and 0.75. These surfaces, as is well known, 9 become purely longitudinal and transverse in the directions of high symmetry.

The direction of wave propagation is given by the normal at the slowness surface. Consider the contribution of the

two slowness surfaces to the displacement field at a point on the Z axis. The first contribution comes from the longitudinal wave which is the faster of the two. This gives the first peak in the curves in Figs. 2-4 at the longitudinal wave velocity.

After the longitudinal wave passes, contributions traveling at slower speeds come from those points on the transverse surface where the surface normals can cross the Z axis.

By symmetry, the displacement components normal to the Z axis cancel, so the resultant displacement on the Z axis re- mains longitudinal. These contributions will be 0 after the transverse wave has passed. The secondary displacement field, therefore, exists only for t/z in the range V• to V, and is longitudinally polarized.

The slowness surfaces are spherical for isotropic solids (Fig. 5). The contribution to the secondary displacement field comes from the points at, or infinitesimally near the Z axis on the transverse slowness surface. This contribution is

small and gives the weak second maximum or the hump in Fig. 2. In the ideal theoretical case, this contribution will be infinitesimal. For solids with •7 > 1, the transverse slowness surface is convex near the Z axis (Fig. 5 ). Hence, the surface normals do not intersect the Z axis and the contribution to

the secondary displacement field is 0. This explains why there is no second maximum in the curve for a solid with

positive anisotropy in Fig. 4. In solids with negative anisotropy, the transverse slow-

ness surface is concave near the Z axis as shown in Fig. 5. The normal to these points will pass through the Z axis. Hence, these points make a substantial contribution to the displacement field. This results in a strong second maximum as seen in Fig. 3. For strong anisotropy, the second maximum can be even higher than the first.

To confirm the explanation given above, we calculated the displacement field in the (0,1,1 ) direction. In this direction, as shown in Fig. 5, the situation is opposite of that in the (0,0,1 ) direction. The quasitransverse slowness curve is convex for solids with negative anisotropy and concave for solids with positive anisotropy. In this direction, we found a strong second maximum for solids with positive anisotropy and no maximum for solid with negative anisotropy. These results are not given here because they are similar to those shown in Figs. 2-4.

Now, we give our results for the effect of a free surface on the elastic-wave propagation in the three model solids for •7 = 1, 1.25, and 0.75. We assume a surface source, located at the origin and defined in terms of the boundary condition given by Eq. (27). We have calculated the displacement field for both HF and DF pulses. However, in this paper, we show our results only for the DF pulse that gives the Green's function for the half-space.

The free surface is taken to be the ( 1,0,0) plane as shown in Fig. 1. The values of the displacement field for waves propagating in the (0,0,1 ) direction on the free surface, obtained by using Eqs. (28) and (29), are shown in Figs. 6-9. In Eq. (27), f was taken to be (1,0,0) and (0,0,1), corre-

1.25

1.00

z

_o 0.75

• 0.50

0.25 [ 0.00 0.00 0.25 0.50 0.75 1.00 1.25

Z-DIRECTION

FIG. 5. Cross sections of slowness surfaces on the YZplane. The outer three curves are for quasitransverse waves, and the inner three are for the quasilongitudinal waves. Isotropic solid: dotted curves; solid with negative anisotropy: dashed curves; solid with positive anisotropy: solid curves.

42.0

36.5

31.0

25.5

20.0

14.5

9.0

3.5

-2.0

-7.5 0.0

47.5 - .

.

0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

t/z

FIG. 6. X component of the displacement field (u 1 ) for wave propagation in the Z direction at the free surface in half-space cubic solids for DF source at (0,0,0) along the X axis. The strong peaks at t/z • 1 indicate the contribution of the Rayleigh wave. Isotropic solid: dotted curves; solid with negative anisotropy: dashed curves; solid with positive anisotropy: solid curves.



6.0

2.4

-1.2

-4.8

-8.4

3-12.0

-1 5.6

-19.2

-22.8

-26.4

-30.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

FIG. 7. Same as Fig. 6 for u 3 .

45.0

40.5

36.0

31.5

27.0

22.5

!

0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 'r./z

18.o / "'i•. ' ß •3.5 /' ,. 9.0

4.5 -.

0.0 • ' •'• 0.0

FIG. 9. Same as Fig. 7 for the force in the Z direction. The peaks at t/z•0.5 show the longitudinal pulse. The peaks at t/z• 1 denote the contribution of the Rayleigh wave and the secondary displacement field.

sponding to, respectively, 7-• and 7'13. The integration over q• was done analytically, and those over q2 and q3 were done numerically.

The results shown in Figs. 6-9 are as expected physically. In each case a strong peak is observed that shows the existence of a Rayleigh or a surface wave. The velocity of the surface wave in each case is slightly less than the transverse wave velocity. Because of anisotropy, the velocity of the surface waves will be different in different directions. In Figs. 8 and 9, where f is in the (0,0,1 ) direction, the contribution of the secondary displacement field overlaps with that of the surface wave.

Although not shown here, our results for HF pulse in isotropic solids agree with those obtained analytically by Pe- keris •6 (see also Refs. 1 and 2). The only difference between our results and those of Pekeris is that the Rayleigh peak in our results is finite and the discontinuity near the Rayleigh peak is smoothed. This reflects the inherent inability of a numerical integration process to reproduce exactly the dis- continuities of a function.

IV. SUMMARY AND CONCLUSIONS

A representation for the space-time Green's function of the Christoffel equation is developed in terms of functions of

32

28

24

20

16

=12 N

8

4

0

-8 , ! , i i i i i i i i i i i i I i i i J

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 t/z

FIG. 8. Same as Fig. 6 for the force along the Z axis.

the form 1/(q.x + t)". This representation is particularly convenient for calculating the solution of the Christoffel equation for anisotropic solids.

Expressions for the space-time Green's function and the displacement field (wave amplitudes) are derived for general anisotropic solids for localized pulse excitations corresponding to the delta function and the step function time dependence. Results are given for an infinite as well as a half- space solid containing a free surface.

Detailed numerical results are given for pulse propagation in anisotropic cubic solids. The effect of a free surface and the contribution of the Rayleigh wave on pulse propagation in a half-space solid is also calculated. For infinite cubic solids, it is shown that for a longitudinal pulse source, and for certain values of the anisotropy parameter, two longitudinally polarized pulses can be observed in high-symmetry directions. One of these travels with the usual longitudinal wave velocity and the other travels with a velocity close to that of the transverse wave.

ACKNOWLEDGMENTS

The authors thank Dr. L. J. Bond and Dr. D. W. Fitting for useful suggestions and discussions. This work was sup- ported in part through a reasearch contract between the Ohio State University and the National Institute of Stan- dards and Technology.

APPENDIX A: SOME MATHEMATICAL PROPERTIES OF THE m FUNCTIONS

In this Appendix we consider the orthogonality of the m functions, derive the inversion relation for the transforma- tion given by Eq. (15) and obtain solutions ofEq. ( 1 ) for HF and DF pulses. Consider the transform [ Eq. (15) for n = 1 ]

u(x,t) = (2•r) -• fi(q)m•q (x,t)dq, (A1)

where

m•q (x,t) = 1/(q.x + t + •e), (A2)



and t is positive. In order to take the principal value of the integral at the singularities in the m functions, we have intro- duced a small imaginary part in the denominator. It is un- derstood that we have to take the real part of Eq. (A 1 ) and take the limit as e approaches + 0.

We shall be interested in those cases in which fi(q) is symmetric in q space, that is,

fi(-q) = fi(q). (A3) We can then write Eq. (A 1 ) as the real part of

u(x,t) - 1 fi(q)M•q (x,t)dq, (A4) (2•r) 4

where

M•q (x,t) -- m•q (x,t) -- m•q (X, -- t). (A5) We derive the inverse transform of Eq. (A4) by relating fi (q) to the Fourier transform of u (x,t). Using the integral representation

mlq (x,t) = -- t exp[tco(q.x + t + te) ]do (A6)

in Eq. (A4), we obtain

u(x,t) -- t do fi co- 3 {exp[t (k.x (27r) 4 + cot + re) ] -- exp[t (k.x -- cot + re) ])dk,

(A7)

where we have used the fact that co in Eq. (A6) is positive and k = qco. Using Eq. (A3), we rewrite Eq. (A6) for e = 0, as

u(x,t) -- t fi co-3 (27r) 4 _ • - •

X exp[t (k.x + cot) ]dcodk. (A8)

Equation (A8) is in the form of a Fourier integral. This gives fi(k/co) = tco3uF(k,co), (A9)

where the superscript F denotes the space-time Fourier transform. Equation (A9) also shows which type of functions can be represented in the form of Eq. (A4).

We now consider the orthogonality of the M, functions over the variable t. We define the integral

P(qx,q'x') = Re Mq (x,t)Mq, (x',t)dt, (A10)

where the overbar denotes complex conjugate. From Eqs. (A5) and (A 10), and extending t to the entire space ( - oo to oo ), we get

P(qx,q'x') = [m•q(X,t)•lq,(X',t) - m•q(X,t)

X•q, (x', - t) ]dr. (All) Using Eq. (A6) in Eq. (A 11 ), we obtain

P(q.x,q'.x') = 2•r2tS(q.x - q'.x'), (A12) where we have used the representation

rS(p) = (2•') -• exp(tpt)dt

= (•r) - • Im 1/(p -- re) (A13)

for any real scalar p. For later use, we also write

•5(px) - •5(x)/lpl 3

= (2•-) - 3 exp(tpq.x)dq. (A14)

Equation (A 12) gives the partial orthogonality proper- ty of the M functions over t. We could not derive a similar simple orthogonality relation over the variables q or x.

We now show that a solution of Eq. ( 1 ) for the HF pulse can be written as

u(x,t) -- 2 Re G(q)fm•q(X, + t)dq, (2T/') 4 oo -- (A15)

where G(q) is defined by Eq. (16). We consider only the positive sign of t. The proof for the negative t follows the same steps. Using Eqs. (A 15 ) and ( 16 ), we obtain for the lhs of Eq. (1)

•0•(x,t)u(x,t) _ 4 f (x,t)dq. (A16) (2T/.) 4 m3q Using the relation

mnq (x,t) = ( 1 )n-- 1 1 c• n- 1 -- m 1 (x,t) (n- 1)! c•t n-1 q '

(A17)

and the integral representation given by Eq. (A6), we obtain •o• (x,t)u(x,t)

2L

(2T/') 4 --• f dco exp [tco(q.x + t + re) ]dq.

O•t 2 - oo (A18)

Using Eqs. (A 13 ) and (A 14), Eq. (A 18 ) reduces to

•ø•(x,t)u(x,t) = ftS(x)H(t), (A19) where we have used the representation

•rH(t) -- Im fo © lexp[tco(t + re) ]dco. (A20) Similarly, we can show that a solution of Eq. ( 1 ) for the DF pulse is given by the real part of

u(x,t) -- 1 Re G(q)fm2q(X, + t)dq. (A21) (2•r) 4 oo --

The solutions given by Eqs. (17) and (18) are formed by linear combinations of the m functions that satisfy Eq. (6).

APPENDIX B: APPLICATION TO 3-D ISOTROPIC CASE

In this Appendix we apply our representation to reproduce the known result for a 3-D isotropic solid (r/= 1 ). We consider a DF pulse with a unit force in the Z direction corresponding to f = (0,0,1 ) in Eq. (4). In view of the iso- tropy of the solid, we choose the Z axis along x and carry out the volume integration in Eq. (17) over a sphere. For illus- tration, we calculate only the Z component of the displacement, which is given by



U3 (x,t)

where

and

2 q2 dq G3 3 (q) . 1 (2•)3X 2 (q COS 0 -• T) 2

1 .) sin 0 dO, (q cos 0 - T) 2 (B1)

T-- t( 1 q-

G33 (q) = (G 1 q- G2 q- G3 )/Cll,

G1 = -- sin 2 0/(q2 __ A 2),

G 2 =c• sin 2 0/c44 (q2_ B2),

G3 = 1/(q 2 -- A 2),

A 2= (1 -- •e)2/c•,

(B2)

(B3)

(B4)

(BS)

(B6)

(B7)

B 2 = (1 -- t6)2/c44 . (B8)

The integrand in Eq. (B 1 ) does not depend upon the azimuthal angle that has been integrated out. Then we carry out the integral over 0. The integral of the (73 term in Eq. (B3), denoted by 13, reduces to

t -q Im .q q I3 = 27y3;llX2 q2 ,42 ½ r - (B9)

Using the second of Eq. (A13), and the fact that • (q + t/x) and (5 (q + A ) are zero for positive values of q, t, and x, we obtain for real part of 13 in the limit e -- q- 0,

1 6(q - ,4)6 q - dq 4•Cl 1

•1 6(•--A) (BlO) 4rrc 11 .,¾2 ' which is the familiar term 1,2 showing the pulse propagation at the longitudinal wave Velocity 1/,4.

Now we consider the integral of the G• term which will be denoted by I1. After carrying out the 0 integral, we obtain

Ii= t_t. fo © 1 im[ln(l+_•t +t•) rr3CllX3 q ( q2 _,4 2) qx

--ln(--l+ t + te) ] dq. (Bll) qx

We define the branches of the log function in Eq. (B 11 ) in the usual manner by taking the argument of the complex number to vary between -rr and rc with the cut on the negative real axis. The imaginary part of the first log term is 0 in the limit • - 0. Using Eqs. (A 13 ) and (A20), we obtain

I1 I t 6(q--A)H q-- dq 2•rx 3

2•rx 3

Similarly, the integral of the G2 term is found to be

12 -- ( t /2rrx3)H(B -- t/x). (B13)

Finally, the total longitudinal displacement along x for a DF pulse is given by

U 3 (x,t) = • Cll X 2

ß (B14)

For a general time-dependent applied force F(t) (5 (x), the displacement field uF3 (x,t) is given by the convolution

uF3 (x,t) - F(t-- t')U 3 (x,t')dt'. (B15)

Using Eqs. (B 12) and (B 13),

UF3(X,t)

_ 1 (1 F(t_Ax)+ 2J• • ) 4rr c• x •-• t 'F( t -- t ')dt ' , (B16)

which is the well known Stokes' result. •'2

•J. D. Achenbach, Wave Propagation in Elastic Solids (North-Holland, Amsterdam, 1973). j. Miklowitz, The Theory of Elastic Waves and Wave Guides (North-Hol- land, Amsterdam, 1978). R. G. Payton, Elastic Wave Propagation in Transversely Isotropic Media (Nijhoff, The Hague, 1983). Do M. Barnett and J. Lothe, "Surface Waves in Anisotropic Elastic Half Spaces," Proc. R. Soc. London Ser. A 402, 135-152 (1985). A. N. Norris, "A Theory of Pulse Propagation in Anisotropic Elastic Sol- ids," Wave Motion 9, 509-532 (1987). G. F. D. Duff, "The Cauchy Problem for Elastic Waves in an Anisotropic Medium," Philos. Trans. R. Soc. London Ser. A 252, 249-273 (1960). E. A. Kraut, "Advances in the Theory of Anisotropic Elastic Wave Prop- agation," Rev. Geophys. 1, 401448 (1963).

8V. T. Buchwald, "Elastic Waves in Anisotropic Media," Proc. R. Soc. London Ser. A 253, 563-580 (1959). M. J.P. Musgrave, Crystal Acoustics ( Holden Day, San Francisco, 1970).

2o A. H. Nayfeh, "The General Problem of Elastic Wave Propagation in Anisotropic Media," J. Acoust. Soc. Am. 89, 1521-1531 (1991). L. J. Bond, "Numerical Techniques and Their Use to Study Wave Propa- gation and Scattering", in Elastic Waves and Ultrasonic Nondestructive Evaluation, edited by S. K. Datta, J. D. Achenbach, and Y. S. Rajapakse (Elsevier, Amsterdam, 1990), pp. 17-27. W. Lord, R. Ludwig, and Z. You, "Developments in Ultrasonic Model- ing with Finite Element Analysis," J. Nondestr. Eval. 9, 129-143 (1990). Wavelets, Proceedings of the International Conference, Marseille, France, 1987, edited by J. M. Combes, A. Grossmann, and Ph. Tchamit- chian (Springer-Verlag, Berlin, 1989).

•4V. K. Tewary, "Green's Function Method for Lattice Statics," Adv. Phys. 22, 757-810 (1973). L. F Bresse and D. A. Hutchins, "Transient Generation of Elastic Waves in Solids by a Disk-Shaped Normal Force Source," J. Acoust. Soc. Am. 86, 810-817 (1989). C. L. Pekeris, "The Seismic Surface Pulse," Proc. Natl. Acad. Sci. USA 41, 469-480 (1955).



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A computationally efficient representation for propagation of elastic waves in anisotropic solids