Transfer Matrix Method of Wave Propagation in a Layered Medium With Multiple Interface Cracks: Antiplane Case Y.-S. Wang Institute of Engineering Mechanics, Northern Jiaotong University, Beijing 100044, P. R. China e-mail:
[email protected]
D. Gross Institute of Mechanics, TU Darmstadt, Hochschulstr. 1, D-64289 Darmstadt, Germany
The paper develops a universal method for SH-wave propagation in a multilayered medium with an arbitrary number of interface cracks. The method makes use of the transfer matrix and Fourier integral transform techniques to cast the mixed boundary value problem to a set of Cauchy singular integral equations of the first type which can be solved numerically. The paper calculates the dynamic stress intensity factors for some simple but typical examples. 关DOI: 10.1115/1.1360180兴
2
Transfer Matrix and Dual Integral Equations
Consider the problem shown in Fig. 1. An elastic material is composed of N⫹1 layers bonded through N interfaces. Multiple Griffith cracks are distributed on m(⭐N) interfaces. The interface between the rth and (r⫹1)th layer is denoted as the rth interface. We suppose that there are n(p) collinear cracks on r(p)th interface. Take the x-axis along the 1st interface, and denote the x-coordinates of the tips of crack L pq as a pq and b pq 共p⫽1⬃m, q⫽1⬃n(p)兲 and the y-coordinate of the rth interface as y⫽h r . This paper will consider the propagation of harmonic SH waves with frequency in such a layered medium. The harmonic term e ⫺i t will be omitted throughout paper. We decompose the total displacement and stress wave fields 兵u, 其 as the sum of the fields without cracks 兵 u ( 0 ) , ( 0 ) 其 and those due to the scattering of the cracks 兵 u ( s ) , ( s ) 其 , i.e., 兵 u, 其 ⫽ 兵 u ( 0 ) , ( 0 ) 其 ⫹ 兵 u ( s ) , ( s ) 其 , where 兵 u ( 0 ) , ( 0 ) 其 can be obtained by the classical transfer matrix method or by other methods 共cf. 关20兴兲. The following analysis will be focused on the solution of 兵 u ( s ) , ( s ) 其 . Without confusion, we omit the superscript 共s兲. The Helmholtz equation for SH-wave motion in the rth layer is 2 ⵜ 2 w r ⫹K Tr w r ⫽0,
r⫽1⬃N⫹1
(1)
where w r is the displacement component in z-direction; K Tr ⫽ /C Tr with C Tr ⫽ 冑 r / r is the shear wave velocity; r and r are, respectively, the shear modulus and mass density. We denote the displacement discontinuity on the pth interface as ⌬w p which may be expressed as n共 p 兲
ⵜw p ⫽
1
Introduction
Wave propagation in a layered medium is of both theoretical and practical importance in such fields as composite materials, geophysics, etc. Since the 1970s, the problems of wave scattering from an interface crack between two bonded elastic solids have been widely investigated by many authors, for instance, Loeber and Sih 关1,2兴, Takai, Shindo, and Atsumi 关3兴 Srivastava, Palaiya, and Karaulia 关4兴, and Bostro¨m 关5兴 for a mode III Griffith or penny-shaped interface crack between two half-spaces; Srivastava, Gupta, and Palaiya 关6,7兴, and Qu 关8,9兴 for a Mode I or II interface crack; and Neerhaff 关10兴 Kundu 关11兴, Li and Tai 关12兴, Yang and Bogy 关13兴, and Gracewski and Bogy 关14,15兴 for a Griffith interface crack of Mode I, II, and III in a layered plate or a layered half-space. However, one may note that only a few papers have considered multiple interface cracks. The published results are limited to some simple cases. Kundu 关16兴 first discussed the interaction between two interface cracks in a layered half-space under antiplane transient loading, and then in a three-layered plate 共关17兴兲. Zhang 关18,19兴 analyzed the SH-wave propagating through a periodic array of interface cracks between two bonded halfspaces. If the medium is composed of multiple layers and, furthermore, if cracks may occur in any interface with an arbitrary number, the associated wave propagation problems will become more difficult. Even by using numerical methods such as the finite element method and boundary element method, the problems cannot be solved easily. Here, in the present paper, we develop a universal method for wave propagation in a multilayered medium with multiple cracks distributed in different interfaces. The method makes use of the transfer matrix and singular integral equation techniques. As a preliminary analysis, we treat the SH-wave motion in this paper. But the method can be extended to the in-plane case in a straightforward manner. Contributed by the Applied Mechanics Division of THE AMERICAN SOCIETY OF MECHANICAL ENGINEERS for publication in the ASME JOURNAL OF APPLIED MECHANICS. Manuscript received by the ASME Applied Mechanics Division, Sept. 15, 1999; final revision, Aug. 10, 2000. Associate Editor: R. C. Benson.
Journal of Applied Mechanics
兺 ⌬w
q⫽1
pq 关 H 共 x⫺a pq 兲 ⫹H 共 x⫺b pq 兲兴 ,
with ⌬w pq being the unknown tearing displacement of the crack L pq 共q⫽1⬃n(p), p⫽1⬃m兲 and H( ) the Heaviside function. Then the boundary conditions may be written as
yzr ⫽0,
y⫽h 0 ,h N⫹1 ,
yzr ⫺ yzr⫹1 ⫽0,
y⫽h r ,
w r ⫺w r⫹1 ⫽⌬w p ␦ rr 共 p 兲 , 0兲 0兲 yzr ⫽ yzr⫹1 ⫽⫺ 共yzr ⫽⫺ 共yzr⫹1 ,
r⫽1,N⫹1
(2)
r⫽1⬃N
(3)
y⫽h r ,
r⫽1⬃N
x苸L pq ,
y⫽h r ,
(4) r⫽r 共 p 兲 (5)
where ␦ rr ( p ) is Kroneck symbol. Applying Fourier integral transform to 共1兲 with respect to x, we obtain its solution in the transformed space, which is written in the matrix form as
兵 S r 其 ⫽ 关 T r 共 y 兲兴 兵 C r 其 ,
r⫽1⬃N⫹1
(6)
where ¯ r ,¯ yzr 其 , 兵 S r其 ⫽ 兵 w with 关 T r0 兴 ⫽
冋
兵 C r 其 ⫽ 兵 C 1r ,C 2r 其 T , 1
1
⫺ r r
r r
册
,
关 T r 共 y 兲兴 ⫽ 关 T r0 兴关 E r 共 y 兲兴 ,
关 E r 共 y 兲兴 ⫽
冋
e ⫺ry 0
0 e
ry
册
2 1/2 and  r ⫽(s 2 ⫺K Tr ) of which the branch should be determined such that 2 1/2  r ⫽ 共 s 2 ⫺K Tr 兲 ,
2 兩 s 兩 ⭓K Tr ;  r ⫽⫺i 共 K Tr ⫺s 2 兲 1/2,
兩 s 兩 ⬍K Tr . (7)
The bars appearing in above equations indicate the Fourier integral transforms, s is the parameter of the integral transforms, and C 1r ,C 2r are undetermined functions of s. The integral transforms of boundary conditions 共2兲–共4兲 can be written as
兵 C 1 其 ⫽ 兵 B 其 C 21 ,
Copyright © 2001 by ASME
兵 C N⫹1 其 ⫽ 兵 X 其 C 1,N⫹1 ,
(8)
MAY 2001, Vol. 68 Õ 499
m
¯ yzr ⫽
兺m
p⫽1
¯p rr 共 p 兲 ⌬w
.
(15)
Inserting the inverse Fourier transform of 共15兲 into the boundary condition 共5兲, we obtain 1 2
冕
⬁
n共 p 兲
m
兺 兺 mˆ
⫺⬁ p⫽1 q⫽1
¯ pq e k p ⌬w
⫺isx
ds⫽⫺ 共yz0 兲 共 x,h r 共 k 兲 兲 ,
(16)
where m ˆ k p ⫽m r ( k ) r ( p ) 兩 y⫽h r ( k ) , x苸L k j , j⫽1⬃n(k), k⫽1⬃m. It is straightforward that the following relation holds: 1 2
冕
⬁
⫺⬁
⌬w ¯ k j e ⫺isx ds⫽0,
x苸L k j .
(17)
Equations 共16兲 and 共17兲 are dual integral equations which will be transformed to a set of Cauchy singular integral equations in the next section. Fig. 1 A multilayered medium with multiple interface cracks
3 兵 S r 其 ⫺ 兵 S r⫹1 其 ⫽ 兵 ⌬S r 其 ,
y⫽h r ,
r⫽1⬃N
(9)
Singular Integral Equations and Numerical Solution Introduce the dislocation density function of the crack L pq ,
pq 共 x 兲 ⫽
where
兵B其⫽兵e
21h0
,1其 , T
兵 X 其 ⫽ 兵 1,e
¯ p ␦ rr 共 p 兲 ,0其 , 兵 ⌬S r 其 ⫽ 兵 ⌬w T
⫺2  N⫹1 h N⫹1 T
其 ,
兺
E r 共 p 兲 兴 兵 ⌬S r 共 p 兲 其 , 关¯
(10)
m
兵 C r其 ⫽
兺 共关 ¯L
p⫽1
p⫽1⬃m
(18)
⌬w ¯ pq ⫽is ⫺1
冕
pq 共 u 兲 e isu du,
(19)
L pq
which, when substituted to 共16兲 and 共17兲, yields
m
p⫽1
q⫽1⬃n 共 p 兲 ,
By considering the differential properties of the Fourier transform, it is not difficult to obtain
p⫽1⬃m.
The above equations involve the cases of h 0 →⫺⬁ and/or h N⫹1 →⫹⬁. Equation 共9兲 is a recurrence relation. Substituting 共6兲 into this relation, we can express 兵 C r 其 with 兵 ⌬S r ( p ) 其 as
兵 C 1其 ⫽
共 ⌬w pq 兲 , x
i 2
rr 共 p 兲 兴 ⫹ 关 K rr 共 p 兲 兴 H 共 r⫺r 共 p 兲 ⫺1 兲兲 兵 ⌬S r 共 p 兲 其 ,
¯
r⫽2⬃N⫹1
冕
⬁
n共 p 兲
m
兺兺s
⫺1
⫺⬁ p⫽1 q⫽1
m ˆ kp
冕
pq 共 u 兲 e is 共 u⫺x 兲 duds
L pq
⫽⫺ 共yz0 兲 共 x,h r 共 k 兲 兲 ,
冕
(11)
where
(20)
k j 共 u 兲 du⫽0
(21)
Lk j
¯ r 兴 ⫺1 关 ¯E i 兴 ; 关 K ¯ ri 兴 ⫽⫺ 关 W ¯ r 兴 ⫺1 关 L i 兴 ; 关 ¯L ri 兴 ⫽ 关 W
where x苸L k j , j⫽1⬃n(k), k⫽1⬃m. It is found that s ⫺1 m ˆ k p is an antisymmetric function of s and has the following asymptotic behavior as s→⫹⬁,
¯ 兴 ⫺1 关 L i 兴 ; 关 L i 兴 ⫽ 关 W ¯ i 兴关 T i 共 h i 兲兴 ⫺1 ; E i 兴 ⫽ 兵 B 其 关 1,0兴关 W 关¯ ¯ 兴 ⫽ 兵 B 其 关 1,0兴 ⫺ 关 W ¯ N⫹1 兴 兵 X 其 关 0,1兴 ; 关W ¯ r兴⫽关 W2兴¯关 Wr兴, 关W
s
¯ 1 兴 ⫽ 关 I兴 ; r⬎1; 关 W
关 W r⫹1 兴 ⫽ 关 T r 共 h r 兲兴 ⫺1 关 T r⫹1 共 h r 兲兴 .
Substitution of 共10兲 and 共11兲 into 共6兲 yields m
兵 S r其 ⫽
兺 关M
p⫽1
rr 共 p 兲 兴 兵 ⌬S r 共 p 兲 其 ,
r⫽1⬃N⫹1
(12)
⫺1
m ˆ kp→
冕
r⫽1
¯ rr共p兲兴H共r⫺r共 p兲⫺1兲兲, 关M rr共p兲兴⫽关Tr共y 兲兴共关¯Lrr共p兲兴⫹关K
which is the transfer matrix of the multiple layered medium with interface cracks. Write the matrix 关 M rr ( p ) 兴 as 关 M rr 共 p 兲 兴 ⫽
冋
*
*
册
, (14) * where *s are the other elements of the matrix 关 M rr ( p ) 兴 which are of no use in the following analysis. Then we have 500 Õ Vol. 68, MAY 2001
m rr 共 p 兲
⬁
⫺⬁
(13) r⬎1
0,
k⫽p .
(22)
k⫽p
This result can be easily proved 共e.g., by using Mathematica, Version 3.0兲 for fixed values of N, m, and p. Due to 共22兲, special care must be taken in interchanging the two integrations in 共20兲. However, if we consider the following relation
where we have denoted E r共 p 兲兴 , 关 M rr 共 p 兲 兴 ⫽ 关 T 1 共 y 兲兴关 ¯
再
r 共 k 兲 r 共 k 兲 ⫹1 ,␥k , r 共 k 兲 ⫹ r 共 k 兲 ⫹1
⫺
sgn共 s 兲 e is 共 u⫺x 兲 ds⫽
2i , u⫺x
(23)
and denote P k p 共 u,x 兲 ⫽
i 2
⫽⫺
1
冕 冕
⬁
⫺⬁ ⬁
ˆ k p ⫺ ␥ k ␦ k p sgn共 s 兲兴 e is 共 u⫺x 兲 ds 关 s ⫺1 m
ˆ k p ⫺ ␥ k ␦ k p 兴 sin关 s 共 u⫺x 兲兴 ds, (24) 关 s ⫺1 m
0
we can transform 共20兲 into Cauchy singular integral equations: Transactions of the ASME
兺冕
n共 k 兲
⫺
␥k q⫽1
兺兺冕 n共 p 兲
m
kq 共 u 兲 du⫹ p⫽1 L kq u⫺x
q⫽1
pq 共 u 兲 P kp 共 u,x 兲 du
L pq
⫽⫺ 共yz0 兲 共 x,h r 共 k 兲 兲 .
(25)
P p (u,x) is a Fredhelm kernel which has no singularity except ˆ p (n) becomes infinite in some cases which we will diswhen M cuss later. By introducing the substitutions
冦
u⫽c kq ⫹d kq
x⫽c k j ⫹d k j ,
⌽ pq 共 兲 ⫽ pq 共 c pq ⫹d pq 兲 1 Q pq 共 , 兲 ⫽⫺ P kp 共 c pq ⫹d pq ,c k j ⫹d k j 兲 ␥k
(26)
␦ kp 共 1⫺ ␦ q j 兲 1 ⫹ , c kq ⫺c k j ⫹d kq ⫺d k j
with c k j ⫽(b k j ⫺a k j )/2 and d k j ⫽(a k j ⫹b k j )/2, Eq. 共25兲 can be further converted to standard Cauchy singular integral equations 1
冕
m
⌽ k j共 兲 d⫹ p⫽1 ⫺1 ⫺ 1
n共 p 兲
兺兺
⫽
q⫽1
冕
1
⫺1
c pq ⌽ pq 共 兲 Q pq 共 , 兲 d
1 共0兲 共 c ⫹d k j ,h r 共 k 兲 兲 . ␥ k yz k j
1
⫺1
(27)
⌽ k j 共 兲 d ⫽0.
(28)
The above equations can be solved numerically by the method developed by Erdogan and Gupta 关21兴. Set ⌽ k j共 兲⫽
F k j共 兲
冑1⫺ 2
.
(29)
Then 共27兲 and 共28兲 reduce to 1 M
M
兺
s⫽1
⫽
冋
m
F k j共 s 兲 ⫹ s⫺ t p⫽1
n共 p 兲
兺兺c q⫽1
pq Q pq 共 s
1 共0兲 共 c ⫹d ,h 兲 , ␥ k yz k j t k j r 共 k 兲
M
, t 兲 F pq 共 s 兲
册 (30)
M
兺F
s⫽1
k j 共 s 兲 ⫽0,
The method developed above is quite general and can be applied to many complicated problems. In this brief note, we only present numerical results for some simple but typical examples. Our attention is focused on the dynamic stress intensity factors which are defined as
再
K k⫹j ⫽ lim 关 冑2 共 x⫺b k j 兲 yz 共 x,h r 共 k 兲 兲兴 ⫹
K k⫺j ⫽
x→b k j
lim 关 冑2 共 a k j ⫺x 兲 yz 共 x,h r 共 k 兲 兲兴
.
(32)
⫺ x→a k j
The numerical results may be obtained by the following formula 共cf. 关23兴兲: K k⫾j ⫽⫺ ␥ k 冑c k j F k j 共 ⫾1 兲 .
(33)
Example 1 Two Bonded Half-Spaces. In order to verify the validity and accuracy of our solution, we calculate the simplest case—an interface crack of length 2c lies between two dissimilar half-spaces and compare our results to those of Loeber and Sih 关1兴. The material constants are taken as 1 / 2 ⫽2 and 1 / 2 ⫽1. A harmonic SH-wave of the general form (34)
strikes the interface normally ( 0 ⫽0 deg) from medium 1 共see the sketch in Fig. 2兲. A 0 and 0 are the amplitude and incident angle, respectively. In computation we choose M ⫽30 in Eq. 共30兲. The variation of the dynamic stress intensity factor with normalized frequency K T1 c is shown in Fig. 2, where the dynamic stress intensity factor is normalized by ¯ 0 冑c with ¯ 0 being the shearing stress along the interface without the crack. The results of Loeber and Sih 关1兴 are also plotted in Fig. 2. Good agreement between our results and theirs is observed. Next we consider a more complex example—three cracks with the same length 2c and the same distance 2d between cracks lie on the interface 共see the sketch in Fig. 3兲. The ratio of c and d is set to 1:1.25. The results are shown in Fig. 3, where the dynamic stress intensity factors are normalized by 0 冑c with 0 ⫽ 1 A 0 K T1 . Resonance is observed at the lower frequencies, and the resonance peaks for the inner crack tips 共tips 1 and 2兲 are more pronounced than that for the outer ones 共tips 3兲. The later oscillates more pronouncedly at the higher frequencies. In this example we also take M ⫽30. In order to check the convergence of the solution we calculate the dynamic stress intensity factor at K T1 c ⫽1 for the crack tip 1 by choosing different values of M and list the results in Table 1 共see the first line for the present example兲. It is shown that M ⫽30 can give good accuracy.
(31)
where s ⫽cos((2s⫺1)/2M ), t ⫽cos(t/M), t⫽1⬃M ⫺1; M is the number of the discrete points of F k j ( ) in 共⫺1,1兲; and j⫽1 ⬃n(k), k⫽1⬃m. It is noted that difficulties may arise in evaluation of the semiinfinite integrals 共24兲 because of the possible simple poles of the integrands along the integral path and located between min(KTr) and max(KTr). These poles correspond to the general Love-type surface waves. One should note that the path of the integration along the real s-axis is indeed the limit of the path as it approaches the real s-axis from below 共关14,15兴兲. Based on this fact, two different ways have been developed for dealing with these poles in the integrations. Kundu 关22兴 developed a technique of removing the singularities from the integrands. In result, the original integrals are divided into two parts—the residues of the integrands at the poles and Cauchy principal integrals. The other technique deforms the contour of integration below the real axis so that no poles occur on the path of integration 共cf. 关13–15兴兲. In this brief note we will employ the second method because it is easy for calculation. Journal of Applied Mechanics
Examples
w 共 i 兲 ⫽A 0 e iK T1 共 x sin 0 ⫹y cos 0 兲 ⫺i t ,
Meanwhile, 共21兲 becomes
冕
4
Fig. 2 Dynamic stress intensity factor for one interface crack in a system of two bonded half-spaces
MAY 2001, Vol. 68 Õ 501
Fig. 3 Dynamic stress intensity factor for three interface cracks in a system of two bonded half-spaces Table 1 Normalized dynamic stress intensity factors, K III Õ 0 冑c , at K T 1 c Ä1 for crack tips 1 in the three examples by choosing different values of M
Example 1 Example 2 Example 3
M ⫽20
M ⫽30
M ⫽40
M ⫽60
0.804341 0.684742 0.910096
0.804396 0.684793 0.910223
0.804422 0.684805 0.910245
0.804443 0.684822 0.910267
Example 2 A Layered Half-Space. Again consider the above example, but the upper half-space is of finite thickness h 共see the sketch in Fig. 4兲. Figure 4 illustrates the normalized dynamic stress intensity factors K III / 0 冑c versus the normalized frequency K T1 c for h/c⫽1. The effects of the free surface can be observed in the figure. As the frequency increases from zero the dynamic stress intensity factors also increase from zero and reach resonant peak values at rather low frequencies. Contrary to Example I, the outer crack tips involve more pronounced peaks in the present case. It is worthy of note that a zero value of the dynamic stress intensity factors appears at a higher frequency (k T1 c⬇2.24). All these features may be explained by the reflection of waves between the free surface and the interface. Example 3 Two Half-Spaces Bonded Through a Layer Consider two half-spaces bonded through an interlayer of thickness h 共see the sketch in Fig. 5兲. There are two cracks lying on
Fig. 4 Dynamic stress intensity factor for three interface cracks in a system of a layered half-space
502 Õ Vol. 68, MAY 2001
Fig. 5 Dynamic stress intensity factor for four interface cracks in a system of two half-spaces bonded through a layer
each interface. The crack size and distribution are the same as those in Example 1. The material constants are taken as 1 : 2 : 3 ⫽2:1:2 and 1 ⫽ 2 ⫽ 3 ; and h:c is set to 1 : 1. The incident SH-wave with the form 共34兲 propagates normally ( 0 ⫽0 deg) to the interlayer in material 1. The normalized dynamic stress intensity factors K III / 0 冑c for the four cracks are plotted versus K T1 c in Fig. 5. As the frequency increases the dynamic stress intensity factors first decrease and then increase to peak values. Generally the cracks on the lower interface have higher dynamic stress intensity factors than those on the upper interfaces. In the last two examples, we take M ⫽30 as in the first example. The convergence is shown in Table 1. Finally we mention that the method developed in this brief note is universal and can be used to solve many complex problems. However, we only give some simple examples. A lot of topics based on this piece of work are left for further investigation. The in-plane case that is more complicated will be explored in subsequent works.
Acknowledgment The work was finished during the first author’s stay in TUDarmstadt. Support of the Alexander von Humboldt Foundation is gratefully acknowledged.
References 关1兴 Loeber, J. F., and Sih, G. C., 1973, ‘‘Transmission of Anti-Plane Shear Waves past an Interface Crack in Dissimilar Media,’’ Eng. Fract. Mech., 5, pp. 699– 725. 关2兴 Loeber, J. F., and Sih, G. C., 1973, ‘‘Torsional Waves Scattering About a Penny-Shaped Crack Lying on a Bimaterial Interface,’’ Dynamic Crack Propagation, G. C. Sih, ed., Noordhoff, Leydon, pp. 513–528. 关3兴 Takai, M., Shindo, Y., and Atsumi, A., 1982, ‘‘Diffraction of Transient Horizontal Shear Waves by a Finite Crack at the Interface of Two Bonded Dissimilar Elastic Solids,’’ Eng. Fract. Mech., 16, pp. 799–807. 关4兴 Srivastava, K. N., Palaiya, R. M., and Karaulia, D. S., 1980, ‘‘Interaction of Antiplane Shear Waves by a Griffith Crack at the Interface of Two Bonded Dissimilar Elastic Half-Spaces,’’ Int. J. Fract., 16, pp. 349–358. 关5兴 Bostro¨m, A., 1987, ‘‘Elastic Wave Scattering From an Interface Crack: AntiPlane Strain,’’ ASME J. Appl. Mech., 54, pp. 503–508. 关6兴 Srivastava, K. N., Gupta, O. P., and Palaiya, R. M., 1978, ‘‘Interaction of Elastic Waves in Two Bonded Dissimilar Elastic Half-Spaces Having Griffith Crack at Interface—I,’’ Int. J. Fract., 14, pp. 145–154. 关7兴 Srivastava, K. N., Palaiya, R. M., and Gupta, O. P., 1979, ‘‘Interaction of Longitudinal Wave With a Penny-Shaped Crack at the Interface of Two Bonded Dissimilar Elastic Solids—II,’’ Int. J. Fract., 15, pp. 591–599. 关8兴 Qu, J., 1994, ‘‘Interface Crack Loaded by a Time-Harmonic Plane Wave,’’ Int. J. Solids Struct., 31, pp. 329–345. 关9兴 Qu, J., 1995, ‘‘Scattering of Plane Waves from an Interface Crack,’’ Int. J. Eng. Sci., 33, pp. 179–194. 关10兴 Neerhaff, F. L., 1979, ‘‘Diffraction of Love Waves by a Stress-Free Crack of Finite Width in the Plane Interface of a Layered Composite,’’ Appl. Sci. Res., 35, pp. 237–249.
Transactions of the ASME
关11兴 Kundu, T., 1986, ‘‘Transient Response of an Interface Crack in a Layered Plate,’’ ASME J. Appl. Mech., 53, pp. 579–786. 关12兴 Li, D. H., and Tai, W. H., 1991, ‘‘Elastodynamic Response of an Interface Crack in a Layered Composite Under Anti-Plane Shear Impact Load,’’ Eng. Fract. Mech., 39, pp. 687–693. 关13兴 Yang, H. J., and Bogy, D. B., 1985, ‘‘Elastic Waves Scattering From an Interface Crack in a Layered Half-Space,’’ ASME J. Appl. Mech., 52, pp. 42–50. 关14兴 Gracewski, S. M., and Bogy, D. B., 1986, ‘‘Elastic Wave Scattering From an Interface Crack in a Layered Half-Space Submerged in Water: Part I. Applied Tractions at the Liquid-Solid Interface,’’ ASME J. Appl. Mech., 53, pp. 326– 332. 关15兴 Gracewski, S. M., and Bogy, D. B., 1986, ‘‘Elastic Wave Scattering From an Interface Crack in a Layered Half-Space Submerged in Water: Part II. Incident Plane Waves and Bounded Beams,’’ ASME J. Appl. Mech., 53, pp. 333–338. 关16兴 Kundu, T., 1987, ‘‘Transient Response Between Two Interface Cracks at the Interface of a Layered Half Space,’’ Int. J. Eng. Sci., 25, pp. 1427–1439. 关17兴 Kundu, T., 1988, ‘‘Dynamic Interaction Between Two Interface Cracks in a Three-Layered Plate,’’ Int. J. Solids Struct., 24, pp. 27–39. 关18兴 Zhang, Ch., 1991, ‘‘Dynamic Stress Intensity Factors for Periodically Spaced Collinear Antiplane Shear Cracks Between Dissimilar Media,’’ Theor. Appl. Fract. Mech., 15, pp. 219–277. 关19兴 Zhang, Ch., 1991, ‘‘Reflection and Transmission of SH Wave by a Periodic Array of Interface Cracks,’’ Int. J. Eng. Sci., 29, pp. 481–491. 关20兴 Kennett, B. L. N., 1983, Seismic Wave Propagation in Stratified Media, Cambridge University Press, Cambridge, UK. 关21兴 Erdogan, F., and Gupta, G. D., 1972, ‘‘On the Numerical Solution of Singular Integral Equations,’’ Q. J. Appl. Math., 29, pp. 525–539. 关22兴 Kundu, T., 1985, ‘‘Elastic Waves in a Multilayered Solid due to a Dislocation Source,’’ Wave Motion, 7, pp. 459–471. 关23兴 Wang, Y. S., and Wang, D., 1996, ‘‘Scattering of Elastic Waves by a Rigid Cylindrical Inclusion Partially Debonded From Its Surrounding Matrix—I. SH case,’’ Int. J. Solids Struct., 33, pp. 2789–2815.
Journal of Applied Mechanics
Copyright © 2001 by ASME
MAY 2001, Vol. 68 Õ 503
