Transient scattering of Rayleigh waves by surface-breaking and sub-surface cracks

Pergamon

Int. J. Engng Sci. Vol. 34, No. 9, pp. 1059-1075, 1996

0020-7225(95)00143-3

Copyright ~) 1996 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0020-7225/96 $15.00 + 0.00

TRANSIENT SCATTERING OF R A Y L E I G H WAVES BY SURFACE-BREAKING AND SUB-SURFACE CRACKS SHAW-WEN LIU Department of Mechanical Engineering, Chinese Air Force Academy, P.O. Box 90277-4, Kangsan, Kaohsiung, Taiwan, Republic of China

JEN-CHUN SUNG and CHING-SEN CHANG Department of Civil Engineering, National Cheng Kung University, Tainan, Taiwan, 70101, Republic of China Abstract--The goal of this paper is to enhance the theoretical understanding of the mechanism of Rayleigh wave scattering by surface-breaking and sub-surface cracks. The problem is analyzed in frequency domain by a hybrid method combining the finite element discretization of the near-field with the boundary integral representation of the far-field. Time histories are then obtained by inverting the response spectra using a fast Fourier transform routine. The surface displacements, crack opening displacements and stress intensity factors are presented. The results provide essential features about the scattering of Rayleigh waves by cracks. This problem is of interest both for ultrasonic non-destruct:Lye evaluation and for fracture mechanics studies. Copyright Q 1996 Elsevier Science Ltd

1. INTRODUCTION Detection of a crack in an elastic medium by non-destructive testing techniques is a problem of considerable practical significance. A reliable method of quantitative non-destructive evaluation (QNDE) is needed not only to detect the presence and the location of a crack but also to determine its size, shape and orientation. One of the most promising QNDE methods is based on the scattering of elastic (ultrasonic) waves by cracks. The scattered waves may provide a great deal of information for the inverse problem of crack characterization. Furthermore, the stress intensity f~Lctorsobtained from the crack opening displacements can be used to determine the stress field near the crack tip. This is a prerequisite to the study of crack propagation under dynamic loading. It is well known that Rayleigh waves traveling along the free surface of an elastic half-space are non-dispersive and the associated displacements decay exponentially with the distance from the free surface. Since Rayleigh waves dominate the elastodynamic field near the free surface, they are easily accessible to measurement with a transducer on the surface. Thus, the studies of interaction between Rayleigh surface waves and a surface-breaking or a near-surface crack are of considerable interests in the fields of QNDE and fracture mechanics. The dynamic analysis of the scattering problem is complicated by the presence of the free surface of the medium in addition to the crack surfaces and the associated sharp corners (tips). Unfortunately, tlhere are few exact closed-form analytical solutions to elastodynamic scattering problems. Often the best that can be done is to reduce the mathematical formulation to a form which is suitable for numerical work. Closed-form solutions can be obtained at high frequencies (Kirchhoff approximation or geometrical diffraction theory) or at low frequencies (Rayleigh approximation). However, these approximations are of limited value for scattering in the mid-frequency range in which the characteristic wavelength of an incident pulse is of the same order of magnitl/de as a characteristic length of the flaw. More general and realistic problems must be treated by numerical method. Achenbach attd Brind [1] and Achenbach et al. [2] used an integral equation approach to study the scattering of surface waves by near-surface cracks. Tittmann et al. [3] interpreted the frequency dependence of diffracted Rayleigh wave intensity in terms of the crack dimensions. The interaction between Rayleigh waves and a surface-breaking crack was investigated 1059

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SHAW-WEN LIU et al.

experimentally by Hirao e t al. [4] and Yew et aL [5]. The transient response of two interface cracks, in a layered half-space subjected to anti-plane stress fields, was studied by Kundu [6] using a couple set of integral equations. Based on the finite difference methods, the numerical simulations of transient elastic wave scattering have been carried out by Scandrett and Achenbach [7], Saffari and Bond [8] and Harumi and Uchida [9]. In this paper the transient scattering of Rayleigh surface waves by surface-breaking and sub-surface cracks perpendicular to the free surface of an elastic half-space is analyzed by a combined boundary integral and finite element technique. The boundary integral representation derived from Betti's reciprocal theorem and Green's tensor is used in the far-field outside a contour completely enclosing the scatterer (crack). This is then coupled with the finite element solutions in the near-field. This method can utilize the versatility of the finite element method for the detailed modeling in the near-field and the effectiveness of the boundary element method in the far-field. Another advantage of using this method is that the boundary integral is on a different boundary than the scatterer. Thus it is independent of the shape and number of the scatterers. The computation of the Green's functions, which is the most time consuming part in the boundary methods, need not be redone at each change in the geometry of the scatterer. Similar techniques were used by Liu e t al. [10] and Liu and Datta [11] to study the transient scattering of impact waves by cracks in a glass plate. Numerical results for the stress intensity factors, crack opening displacements and displacements on the free surface are presented for incident Rayleigh waves with the shape of a Ricker pulse. Some essential features can be observed from this study. It is helpful to the understanding of the mechanism of Rayleigh wave scattering by cracks.

2. P R O B L E M F O R M U L A T I O N A homogeneous, isotropic, linearly elastic half-space (z -> 0), with a surface-breaking crack or a sub-surface crack, is subjected to an incident Rayleigh wave pulse as shown in Fig. 1. The cracks are perpendicular to the free surface (z -- 0), and the crack faces are free of tractions. It is assumed that the problem is independent of the y-coordinate. To solve this problem by the hybrid method considered here, two artificial boundaries (B and C) and two regions (R~ and Re) are defined as shown in Fig. 1. Waves propagating in this medium are scattered by the crack, which is completely enclosed by the boundary C. The interior region R~ is bounded by the boundary B. The exterior region R e is bounded inside by the contour C. The interior region is discretized with finite elements and an integral representation over C for the displacements on the boundary B is introduced to solve for the

/\

Rayl¢igh wave

,x

°

d

\/

23I

Z

Fig. 1. Configurationof the problem. A half-space containing a surface-breaking(sub-surface) crack is subjected to an incident Rayleighpulse.

Fig. 2. Crack-tipelements.

Transient scatteringof Rayleighwaves

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scattered field. First the problem is solved in the frequency domain, then the transient response is obtained by Fourier inversion of the spectrum. Let the displacement u(x, z, t) be time-harmonic of the form u(x, z)e -i°', where to is the circular frequency. Then the governing equation in frequency domain is given by ilzV2U +

(A "F ] z ) V V

° U "F f =

--pto2U,

(1)

where A and/x are Lam6 constants, p is the mass density, f is the body force per unit volume and the factor e -~'°' has been dropped.

2.1 Boundary integral representation The total fields generated by interaction of the incident wave and the crack can be expressed as u = n °) + u (~),

(2)

where u°)(x,z) is the incident field and u(~)(x, z) denotes the scattered field. For the study considered here, the incident field can be expressed in an analytical form by solving the problem of the flawless half-space subjected to an incident Rayleigh surface wave. The incident field is given in Appendix A. The scattered field can be represented by a line integral based on Betti's reciprocal theorem. In order to do that we need to solve the Green's problem which is a flawless half-space subjected to a concentrated line load. For a load at the source point (x', z'), the equation of Green's problem is given by

Zqk.k + pto2Gij = - a i j a ( x - x ' ) t 3 ( z - z'),

(3)

where ] stands for the displacement direction and i stands for the force direction, and Gij and ]~ijk are the Green's displacement tensors and the corresponding stresses. Expressions for Green's displacements and stresses are given in Appendix B. Since this study is invariant in the y-direction, the following two-dimensional Betti's reciprocal theorem is introduced

f fA (f " v - g " w) dA = ~c(W " S - V " t) dC,

(4)

where (w, t, f) and (v, s, g) are two solution states, w and t represent the displacement and surface traction caused by body force f while v and s are the displacement and surface traction produced by body force g. First, this theorem is applied to the exterior region. The scattered field is referred as the first state and the second state is the Green's elastodynamic state with source on boundary B. Using Betti's reciprocal theorem these two states can be related in the following manner,

- u~)(x ',z ')= ~ (u~" x~j~- c,~))(-n~) dC,

(5)

where tr)~) denotes the stress tensor of scattered field and nk is the component of the outward unit normal vector to contour C. Secondly, we apply (4) to the region bounded by contour C. Two states considered here are the Green's field with its source outside this region and the incident field. Thus, neither field has sources here. This yields /. 0 = d~

Jr

(u~~):~,j~- Gij(rjk "))nk dC,

(6)

SHAW-WEN LIU et al.

1062

where o-}~,) denotes the stress tensor associated with the incident field. Combining (2), (5) and (6), the displacement at point (x', z') is obtained:

Ui(X ~, Z) = ~(. (U s Yijk -- Gisoik)nk dC + u~i)(x', z ').

(7)

Equation (7) is the integral representation of the total field at any point in the exterior region. However, the evaluation of the contour integration is impossible at the moment because the scattered field at contour C is still unknown. This can be achieved with the aid of the finite element technique.

2.2 Finite element method The interior region, which encloses all the inhomogeneities, is discretized by the finite element technique. The finite element equations are derived in the frequency domain using a variational principle. In order to have a correct simulation for the stress field near the crack tips, six-node triangular quarter-point singular elements are used around the crack tips. The equation of motion for each element can be written in the following form Kedc - toZMede -- re

(8)

where K~ and Me are the stiffness and mass matrices, and de and re are the nodal displacement and force. Equation (8) can be written in a simple form Sede = r e

(9)

where the matrix Sc is the elemental impedance matrix. Then the equations of motion for all elements can be assembled into a global equation of motion, Sd = r

(10)

where d and r represent all the nodal displacements and forces, respectively. The nodal displacement dB at boundary B can be separated from the interior nodal displacement d~. Thus (10) is partitioned into the following form,

SBB SBi]fdB~ = {rB} Sm

Sil/tdiJ

(11)

rl '

where rB represents the nodal force at the boundary B and r~ the interior nodal force. Since there are no external forces on the interior nodes, r~ = 0, then (11) becomes

s.. s,.

.

2.3 Coupling of finite element and boundary integral representation In order to obtain the solution we have to combine the equations from boundary integral representation with those from the finite element method. Now evaluating the integrals at all the nodes on the boundary B and separating dB from dj, (7) gives dB= ABBdH + Amdi + d~ )

(13)

where ABB and Aat are complex matrices and d~ ) is the incident field displacement solution at the boundary B. Combining (13) with the second equation of (12), we obtain

1- ABB --AB,]~dB] __{d~) } Sm

Sil J(dl J

(14) "

The total field solutions at the nodes in R~ at a certain frequency can be obtained by solving (14). Note that the matrix on the left-hand side is a large sparse unsymmetric complex matrix.

Transient scattering of Rayleigh waves

1063

Once the nodal displacements are solved from (14), the total displacements at any point in the exterior region can be easily calculated by using (7). For a vertical crack as shown in Fig. 1, the crack opening displacements (CODs) (7, and Cz can be obtained by the following form C, = u,(0 +, z) - ux(0-, z),

(15)

Cz = u~(O +, z) - uz(O-, z).

(16)

The stress intensity factors K~ and Kn are calculated from the COD near a crack tip by the relations [12] --

(1 ~'/2~A,~

{KK',} 4(1~_v ) \ ~ ]

LA2J,

(17)

where A 1 = 4(u,6 - / ' / x 4 ) - (Ux3 -

Ux2),

Az = 4(Uz6 - Uz4) - (Uz3 - - UZ2), and v is Poisson's ratio. In the above equation u,i and uzi are the displacement components at the ith node. The nodes 2, 3, 4 and 6 along with L are shown in Fig. 2.

3. SOLUTION SCHEME The hybrid method combining the finite element method with the boundary integral representation is indeed a fairly versatile and powerful tool, but the computational time and memory requirements are quite substantial when the total degrees of freedom of the problem become large. This is true especially for more realistic transient problems. The total degrees of freedom must be large enough to capture the characteristics of waves of the highest frequency. Thus an economic: method is needed to reduce both computational time and required storage. Now the problem is to solve (14) which involves a large sparse unsymmetric complex matrix. Direct methods usually produce a lot of fill-ins which destroy the sparsity of the matrix and increase the storage needs. In this study, a compacted data structure [13] is used to store only non-zero terms of the sparse matrix in a column list scheme. This not only reduces the storage needs a great deal but also can be used directly for iterative methods. Then the biconjugate gradient method (see Appendix C) is used to solve (14) and provides satisfactory solutions. The method used in this paper has quite general applications. Results for a sample case of transient scattering of Rayleigh waves by cracks are considered in this study. Two vertical cracks of length el are considered here: one surface-breaking crack and one sub-surface crack which is d / 4 deeper than the surface-breaking crack. Poisson's ratio of the half-space is assumed to be 1/3. All the variables used in this study are presented in dimensionless form. The spatial coordinates are normalized with respect to the length (d) of the crack. The non-dimensional frequency (f) and the non-dimensional time (t--) are defined as f =fd

(18)

7 = t/~ d'

(19)

and

where f and/3 are frequency and shear wave velocity, respectively. The calculated responses are normalized relative to the amplitude of the incident wave. The shape of the incident pulse is a Ricker wavelet [14] which has the expression u(t) = (2~rzf~t 2 - 1)e -"'t~'2

(20)

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SHAW-WEN LIU et aL

where fc is the characteristic frequency of the pulse. The Ricker pulse is suited for the examination of the wave propagation phenomena for different frequency ranges because of its narrow band in both time and frequency domains. In this paper, only characteristic frequency = 1.0 is considered for transient studies, in which the wavelength of the shear wave is equal to the length (d) of the crack. In order to obtain the transient responses, the frequency range of 0.0-
4. R E S U L T S AND D I S C U S S I O N To show the validity of the hybrid method and to confirm the accuracy of the numerical results, the zero-scattering test has been checked. Then the surface displacements, CODs and stress intensity factors were presented to illustrate the responses of a cracked half-space. These results are helpful to the observation and understanding of the physical phenomena of the transient elastic wave scattering. In order to have a better observation, the time when the center of the Ricker pulse arrives at the location of cracks (z = 0) has been shifted to t-= 3.2 for all of the transient responses shown in this paper.

4.1 Surface displacements Figures 3, 4 and 5, 6 show the displacement responses at the free surface of a cracked half-space containing a surface-breaking crack and a sub-surface crack, respectively. There are 70 observation points along the free surface from x / d = - 2 to 2 for the case of a surface-breaking crack, and 69 points for the case of a sub-surface crack. Figures 3(a)-6(a) show the spectra caused by an incident Rayleigh delta pulse. The corresponding time histories due to an incident Ricker pulse are shown in Figs 3(b)-6(b), in which the thicker curves are used to indicate the location of the cracks. Note that the amplitude of an incident Rayleigh wave is normalized by its horizontal component at the free surface, which gives the vertical amplitude of 1.56. It is obvious from both frequency domain and time domain responses shown in Figs 3 - 6 that the blocking effect of the surface-breaking crack to the Rayleigh wave considered here is much stronger than that of the sub-surface crack. On the contrary, the case of the sub-surface crack has noticeable transmitted Rayleigh waves. From all of the time histories shown in Figs 3(b)-6(b), the velocity of Rayleigh waves can be estimated to be about 0.93 times as fast as the velocity of shear waves. When a surface wave reaches a discontinuity such as a corner or tip, part of the energy will be radiated as P and S waves into the body of the solid and part will be reflected back as a surface wave, leaving the remainder to bend around the corner and continue as a surface wave. The Rayleigh wave traveling around the surface of the crack to the other side of the crack in the forward direction can be used for measuring the depth of the crack. Figures 3(b)-6(b) show that the remarkable diffracted Rayleigh waves can be observed in the back-scattered region (x < 0). For the case of the sub-surface crack shown in Figs 5(b) and 6(b), the diffracted and transmitted Rayleigh waves are mixed together in the forward-scattered region (x > 0). The diffracted P waves can also be observed in the horizontal responses shown in Figs 3(b) and 5(b). However, the amplitude of diffracted P waves is much smaller than that

Transient scattering of Rayleigh waves

1065

(a)

(b) -2

-I

0

UX

2-

I0 -2

0

2 '

I

2

'

I

4

'

I

6

'

8

Fig. 3. (a) Spectral amplitudes and (b) time histories of horizontal displacements on the free surface of a half-space with a surface-breaking crack.

of the Rayleigh wave because most of the energy of a Rayleigh wave is confined to the free surface while the energy of diffracted P waves (or body waves) is spread over the entire half-space. Unlike the diffracted P waves, the diffracted S waves cannot be observed in all these figures. It is. believed that this is because the velocity of the S wave is close to that of the

1066

SHAW-WEN LIU et al.

(a)

(b) --2

.-1

-0

UZ

1 210-|-2

0

2 ;

I

2

'

I

4

'

I

6

'

8

tlJ/d Fig. 4. (a) Spectral amplitudes and (b) time histories of vertical displacements on the free surface of a half-space with a surface-breaking crack.

Rayleigh wave so that the diffracted S waves with small amplitudes are not easy to distinguish from the dominant Rayleigh waves. All of the scattering results discussed above are mainly due to the presence of either the crack mouth of a surface-breaking crack or the upper crack tip of a sub-surface crack. In this study, the Rayleigh wave experienced less scattering by the lower crack tips owing to the short central band wavelength (AR "~ 0.93d) of the incident wave considered here. This phenomenon

Transient scattering of Rayleigh waves

1067

(a)

3

(b) -2

--1

UX

2I

0

'

I

2

'

I

4

'

I

6

'

I

8

t /d Fig. 5. (a) Spectral amplitudes and (b) time histories of horizontal displacements on the free surface of a half-space with a sub-surface crack.

illustrates that the Rayleigh surface wave does not penetrate much deeper into a half-space than approximate]Iy one wavelength. A significant result is found from the responses of the surface-breaking crack shown in Figs 3(b) and 4(b). The Rayleigh wave, created at the crack mouth, traveling along one of the crack surfaces to the crack tip and then to the other surface of the crack in the forward-scattered region produces a small disturbance at ?-~5.5. Most of the energy of the Rayleigh wave

1068

SHAW-WEN LIU et al.

(a)

(b)

I

--2

--1

0

UZ

2 1-

-I

0

'

I

2

'

I

4

'

I

6

'

I

8

t /d Fig. 6. (a) Spectral amplitudes and (b) time histories of vertical displacements on the free surface of a half-space with a sub-surface crack.

propagating around the crack has been converted to bulk waves at the crack tip and crack mouth so that only a small disturbance can be observed on the free surface. The delay time of this surface wave can be used to estimate the crack depth [e.g. crack depth ~ 0.93d × ( 5 . 5 3.2)/2 ~ 1.07d]. The result observed here can also be confirmed by the following studies of C O D s and stress intensity factors.

Transient scattering of Rayleigh waves

1069

(a)

Cx

'

I

'

I

2

0

'

I

4

'

6

8

t /d (b)

Cz

A

I

'

0

I

2

'

I

4

'

I

'

6

tfl/d Fig. 7. Time histories of (a) horizontal and (b) vertical CODs for the surface-breaking crack.

1070

SHAW-WEN LIU et al.

0.25

Cz

1.25

I

I

0

2

'

I

4

'

I

6

8

t[J/d Fig. 8. Time histories of vertical CODs for the sub-surfacecrack. 4.2 Crack opening displacements Figures 7(a) and 7(b) show the horizontal and vertical transient CODs along the whole crack length of the surface-breaking crack, respectively. When the incident wave arrives at the crack, it is found that the amplitude of CODs is very large at the crack mouth and become smaller along the crack depth. The reason for the decrease in amplitude is related to the exponential decay of Rayleigh waves with depth. After the incident wave arrives at the crack, it is seen that a Rayleigh wave is created at the crack mouth and then travels along the crack surface to the crack tip. At the crack tip, most of the energy of this surface wave is radiated as body waves into the half-space and the remainder continues as a surface wave traveling upward to the crack mouth. This phenomenon is consistent with what we observed in the surface responses shown in Figs 3(b) and 4(b). Figure 8 show the vertical component of COD for the sub-surface crack. The corresponding horizontal component is not shown here due to its small amplitude. Unlike the surfacebreaking crack studied above, the surface wave traveling around the crack is not observed here. 4.3 Stress intensity factors The stress intensity factors, which are important for fracture mechanics considerations are shown in Figs 9-11. Figures 9 ( a ) a n d 9(b) show the amplitude spectra and the corresponding time histories of stress intensity factors (k~ and/(',,) for the surface-breaking crack. It can be seen from Fig. 9(b) that the maxima of both stress intensity f a c t o r s / ~ and /~II do not occur at the time (?-~ 3.2) when the direct incident wave arrives but happen some time later (7~ 4.3) when the surface wave propagating along the crack surface arrives at the crack tip. These results again are in accordance with our previous studies of CODs shown in Fig. 7 and surface displacements shown in Figs 3(b) and 4(b). From the observation in the above studies, it is quite interesting to note that the stress intensity factors for a surface-breaking crack are significantly influenced by the Rayleigh waves traveling around the crack. Finally, the stress intensity factors/(', and k,, for the sub-surface crack are shown in Figs 10 and 11, respectively. In these figures, it is found that the overall nature of the stress intensity

Transient scattering of Rayleigh waves

1071

(a) 8

. ,

,

/

i

t%.'

! ~. %/ I I

i

\

A/

/

/i

N I f ' ,

/ I

t

\

/

I

I

v

i~ -

~ x

t

\/

/

\

../

,p I

~i

---'~

I

____~

\~.

I"

~

/ I

I V

,..i I

0

'

I

'

I

1

0

'

I

2

'

4

3

fd/

(b) 2

/ r, I

/

~ /'X / I ',

\/

-2

'

0

I

2

/,x\\

I

~z.'

O.

i/

'

1

~\

/X

I

I xd

I

4

'

I

6

'

8

tp/d Fig. 9. (a) Spectral amplitudes and (b) time histories of stress intensity factors (K~ and Ku) for the surface-breaking crack.

ES 34-9-D

1072

SHAW-WEN LIU et al.

(a) 4

Top-tip Bottom-tip

K I 2-

/

Z

\

I It

~

A

I

',..i

. '

I

'

1

0

I

'

2

I

'

3

fd/

4

(b)

Top-tip Bottom-tip

!

KI

o-

I I

I I I !

/

-1"I 0

'

I

2

'

I

4

'

I

'

6

t /a Fig. 10. (a) Spectral amplitudes and (b) time histories of stress intensity factors (Kt) for the sub-surface crack.

8

Transient scattering of Rayleigh waves

1073

(a) 8

Top-tip Bottom-Up

Kn

4-

"X\

O_ o

I"N.

'

~.~

I

1

'

I

2

'

I

3

4

fdl[3 (b) 4

~

Top-tip Bottom~tip

K n o-

0

2

4

6

t /a Fig. 11. (a) Spectral amplitudes and (b) time histories of stress intensity factors (KH) for the sub-surface crack.

8

1074

S H A W - W E N LIU et aL

factors for the upper tip and lower tip are similar, but the upper tip usually has larger values. This can be expected because the surface wave experiences more scattering by the upper crack tip. It is believed that the appearance of distinct peaks in the amplitude spectra at some specific frequencies shown in Fig. 10(a) may provide useful information for the characterization of a crack. In contrast to the lower tip, the upper tip has larger stress intensity factors which occur when the incident wave arrives. Comparing Figs 9 and 10, it is found that the stress intensity factors/¢~ for the surface-breaking crack are larger than those for the sub-surface crack.

5. C O N C L U S I O N

By using the hybrid method described in this paper, the stress intensity factors, CODs and surface displacements for surface-breaking and sub-surface cracks in a half-space subjected to incident Rayleigh waves have been obtained. The cracks can not only be detected but can also be distinguished by the numerical results presented here. The propagation of elastic waves can be clearly observed and all information at each point at any time is available. Thus, the mechanism of elastic wave scattering in a solid can be understood further.

REFERENCES [1] J. D. A C H E N B A C H and R. J. BRIND, J. Sound Vibr. 76, 43 (1981). [2] J. D. A C H E N B A C H , W. LIN and L. M. KEER, I E E E Trans. Sonics Ultrasonics SU-30, 270 (1983). [3] B. R. TITTMANN, O. BUCK, L. A H L B E R G , M. DE BILLY, F. C O H E N - T E N O U D J 1 , A. J U N G M A N and G. Q U E N T I N , J. AppL Phys. 51, 142 (1980). 141 M. H I R A O , H. F U K U O K A and Y. M I U R A , J. Acoust. Soc. Am. 72, 602 (1982). [5] C. H. YEW, K. G. C H E N and D. L. W A N G , J. Acoust. Soc. Am. 75, 189 (1984). [6] T. K U N D U , Int. J. Engng Sci. 25, 1427 (1987). [7] C. L. S C A N D R E T T and J. D. A C H E N B A C H , Wave Motion 9, 171 (1987). ]8] N. S A F F A R I and L. J. BOND, J. Nondestr. EvaL 6, 1 (1987). [9] K. H A R U M I and M. UCH1DA, J. Nondestr. Eval. 9, 81 (1990). [10] S. W. LIU, S. K. D A T T A and T. H. JU, J. Nondestr. Eval. 10, 111 (1991). [11] S. W. LIU and S. K. D A T T A , J. AppL Mech. 60, 352 (1993). [12] A. H. SHAH, K. C. W O N G and S. K. D A T T A , Int. J. Solids Structures 22, 845 (1986). [13] B. N O U R - O M I D and R. L. T A Y L O R , Engng Comput. 1, 312 (1984). [14] N. H. RICKER, Transient Waves in Visco-Elastic Media. Elsevier, Amsterdam (1977). (Received 30 November 1994; accepted 26 September 1995)

APPENDIX

A

The displacement components of a Rayleigh wave can be written as

u!,!)=Z[e z,,:+ \ (2k2 k,.-l)e

"/2~]e'k"x

(A1)

where ,/j = X / ~ , y~ = ~ / ~ , A is a constant; k~, k2 and k,. are the wave numbers of longitudinal, shear and Rayleigh waves, respectively, k r can be obtained by solving the Rayleigh equation R(k) = 0, where

R(k) = (2k 2 - k~) 2 - 4k2V(U - ~ ) ( k 2 - k~).

APPENDIX

(A3)

B

Expressions of Green's displacements and stresses are given here.

y_'~.

'Y'i.r.r =

[l(k, z)le ikt.... ') dk

A 4p,(A + p,)ikGir, A -}-2/z Y/r: + ,~ + 2~

i = x,z

(BI)

(B2)

Transient scattering of Rayleigh waves

1075

where

1

['-i I° -Yl

[l(k, z)l = [S±(k, z)l + 41rto2Ol.t2R(k)

i

i

e

i:

-

0

e-~'2:

- - Or2 ..I

[<'~ -,,lr=, -',lr ~-~'': o o Oil JL 17

1

-

-

1 (B3)

,,,<,,>,~,] -'

and

ik

[S~:(k,z)i=4_~p

:t:y I ik /ro-:'"=-:"

±~, 17

where S ÷ for z>z', S- for R(k) is given by (A3).

±Y27 -n / L

o lr-(ik/,,/,) e-v~lz-='lJL =F 1

o

/ q7 O/2 _ ]

z
±1

1

-(ik/y2)j'

(B4)

a~=2itxk'Y~, a E = 2 i # k y 2 , ~ = / ~ ( 2 k 2 - k 2 ) and

APPENDIX

C

The algorithm of biconjugate gradient method used to solve Ax = b is listed as follows ro = b - Ar~l = wo = t~ =

Po

ro

(c1) (c2)

where the overbar denotes the conjugate of a complex number and xo is an initial guess. ak

(r,; q,) (Ap,: w,)

(C3)

x,+ I = x , + a , p k

(ca)

a, A p ,

(c5)

rk+ I = r, -

qk÷t = q, - akA*wk (r,+j; q,+l)

ck

(r,:qk >

Pk+l =rk+l Wk+l =qk+l

(C6) (C7)

+ckpk

(ca)

+ CkWk .

(C9)

In the above equations, * denotes the conjugate transpose. The iteration is terminated once the following error criterion is satisfied IIAxk - bll ~ 10-6. Ilbll

(C10)