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Scattering of SH waves by embedded cavities
WAVE MOTION 4 (1982) 265-283 NORTH-HOLLAND PUBLISHING COMPANY
265
S C A T T E R I N G OF SH W A V E S BY E M B E D D E D CAVITIES S.K. DATI'A Department o/Mechanical Engineering, University of Colorado, Boulder, CO 80309, USA
A.H. SHAH Department of Civil Engineering, University of Manitoba, Winnipeg, Canada R3T 2N2 Received 31 August 1981, Revised 25 January 1982
Scattering of plane SH waves by sub-surface circular cavities and thin slits in a semi-infinite elastic medium is analyzed in this paper. Two methods of solution are used to obtain the displacements on the free-surface. One of these is a method of matched asymptotic expansion that is very effective when the wavelength is long compared to the dimensions of the cavity (or the crack). The other is a combined finite element and analytical technique, which is useful in the long to intermediate wavelength range. The results obtained by these two techniques are shown to agree quite well for long wave-lengths. Numerical results for the surface displacements for various incident wave angles are seen to depend significantly on the depth and size of the cavity and the crack. In the latter case the orientation of the crack has a significant influence on the scattered field.
1. Introduction Scattering of elastic waves by cavities and cracks has been the subject of numerous investigations in the last decade. Most of these studies have dealt with the case of an infinite medium. Only recently [1-5] has the problem of scattering in a semi-infinite medium been given adequate attention. It can be seen from these papers that the presence of the free boundary influences the scattered field in a significant manner. This influence must be properly taken into account for the purposes of QNDE. In [6, 7] the problem of scattering of SH-waves by surface depressions (canyons) and edge-cracks was considered. In [6] a solution was presented also for the problem of multiple scattering by a triangular canyon and a circular cavity. The technique used in both these papers was a combined finite element and analytical technique. The advantage of this is that it is quite versatile. Problems of multiple scattering by a cluster of scatterers or by scatterers embedded in a different material from the surrounding matrix can be dealt with as easily as the simpler problem of a single scatterer in a homogeneous material. In [7] results are presented for an edge crack in an anisotropic insert embedded in a homogeneous isotropic material. In this paper we have considered the case when the cavity or the crack is embedded in a semi-infinite medium. The solution to this problem is obtained by two different techniques. The first of these is the combined finite-element and analytical technique similar to that used in [6, 7]. In this the cavity (or the crack) is enclosed by a circular contour. The region interior to this contour is represented by finite elements. The solution in the exterior region is expressed in the exact eigenfunction series satisfying the boundary conditions on the free surface. Continuity conditions for the displacement and traction on the circular contour are imposed to find the constants multiplying the eigenfunctions in the series. In order 0165-2125/82/0000-0000/$02.75 © 1982 North-Holland
266
S.K. Datta, A.H. Shah / Scattering by embedded cavities
to ascertain the accuracy of the solution thus obtained comparison is made with the results obtained by a method of matched-asymptotics (MAE). Application of MAE to solve scattering problems in bounded and unbounded media has been demonstrated in earlier papers [1, 8, 9]. It may be noted that the problem of scattering of SH-waves by an embedded object can be reduced to the problem of scattering by two similar objects in an infinite medium. This latter problem of multiple scattering can be solved in different ways. One is to expand the solution in the form of a multipolar expansion and then satisfy the boundary conditions on one scatterer. This leads to an infinite set of algebraic equations which can be solved after truncation. This approach has been taken in this paper for expressing the solution outside the circular contour (see also [8] and [10]). An alternative approach is to use successive reflections and was adapted in [11-14]. In [12] the authors studied the problem of scalar wave scattering by two parallel cylinders and derived expansions for the far-field. This problem is similar to the one being studied here. Using the combined finite element and wave function expansion we are able to obtain numerical results everywhere in the half-space.
2. Numerical-analytical technique
"
Figure 1 shows a portion of the semi-infinite xy-plane occupied by a homogeneous isotropic elastic material of rigidity modulus,/z and density, p. A cavity (or crack) bounded by the contour C is located in this plane at a depth h from the free surface. The cavity will be assumed to be enclosed by a circle B of radius b (
h
r
l ..........
R2
O'
13
R1 y, Y
Fig. 1. Geometry of the problem.
For the propagation of harmonic SH waves in region R2 the displacement in the Z-direction, w (2), satisfies the Helmholtz equation -2 (2) 192W(2)OX 2 -1"-d a - ~ - ' F k 2 w (2) = 0 (1) where k 2 = pto2/~, ¢o being the circular frequency.
S.K. Datta, A.H. Shah / Scatteringby embedded cavities
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The boundary condition for w (2) on C is that 0W (2) --=0
On
on C where O/On represents normal derivative. Furthermore let w (2)= wB on B, where wB is a known function to be defined later (see Eq. (16)). Then the variational equivalence to this boundary value problem is the minimization of the quadratic functional [15-17], F = f (Vw (2)" ~--~(2)_ k2w(2)l~(2)) dx dy. JR 2
(2)
with the condition that w <2)= w8 on B. Here the bar denotes the complex conjugate. RE is subdivided into finite elements having N~ +NB number of nodes where NB is the number of nodes on B. The displacement field within each element is represented in terms of shape functions Lj(x, y) and nodal values wj as w(2)(x, y) = ~, Ljwj
(3)
where the shape function Lj is 1 at the jth node and is zero at all the other nodes. The subscript / refers to node position, some internal (subscripted I) and some on the boundary B (subscripted B). Then (3) may be written as w(2)(x, y) = {L,}T{W,}+ {LB}T{WB}
(4)
where {L} and {w} are column vectors and superscript T represents transpose. Defining the matrices SIt, SIB, Sin, and SBB as SMN = IR2 [{VLN}T{VLN}- k2{LM}T{LN}] dx dy,
(5)
the functional F of (2) can be rewritten as F = {#1 }TES,~]{w,}+ { #, }T[$ZB]{WB} + {#B}T[$m]{W,}+ {#B}T[sBB]{Ws}.
(6)
Treating the boundary condition {WB}as known, but yet unspecified, and taking the variation aG
--~
O~
it is found that w~ is related to WB by the equation [Su]{wl} = --[SIs]{WB}.
(7)
In order now to solve for w at the internal and boundary nodal points it is necessary to match the solution in R2 with the solution in the exterior region R1. For this purpose the solution for w satisfying eq. (1) and the boundary condition on y = 0, viz. Ow(1) --=0 0y
ony=0
(8)
will have to be found. This solution can be formally written as w (I)= w (°)+ w (s~ w h e r e w (°) is the total incident field near the cavity a n d w (') is the scattered field.
(9)
S.K. Datta, A.H. Shah / Scatteringby embeddedcavities
268
The solution w (s) can be written as
w(~)= ~ [A.(Hn(kr)cosnO+H.(kR)cosnO)+B~(H~(kr)sinnO+H.(kR)sinnO)]
(10)
n=0
where H . is the Hankel function of the first kind and the coordinates (r, 0), (R, O) are shown in Fig. 1, Since H~ cos nO = H.(2kh)+
~. {H.+,~(2kh)+(-1)mH.-r~(2kh)}Jr~(kr)cos toO, m=l
(11) H~ sin nO = ~ {H.+m(2kh)-(-1)mHn-,.(2kh)}J,.(kr)
sin toO,
(10) can be rewritten as
(12)
w ~s~= ~ [Fro cos mO +Gm sin toO] m=O
where
Fm= A.,H,. (kr) + ~ A. {H. +,. (2kh) + ( - 1 ) ' H . _ . , (2 kh)}Jm (kr),
m~0,
n=O
=AoHo(kr) + ~ A.H.(2kh)Jo(kr),
m = O,
n=O
and
Gm = B.,H.,(kr) + ~ B.{H.÷m ( 2 k h ) - ( - 1 ) " H . _ . , (2kh)}J.,(kr). n=0
For the purpose of numerical solution only a finite number of terms of the series will be kept. Choosing the number of nodes, NB, on the boundary B to be even, (12) will be written as
NB/2-1 w (')=
Y~ F . , c o s m 0 + m~0
N~/2 ~ G,.sinmO
(13)
m=l
with
Fm =AmHm(kr)+
An{H.+m(2kh)+(-1)r~H._.,(2kh)}J,.(kr),
m SO,
n=0
=AoHo(kr) + N ~ - I A . H . (2kh),
m = 0,
n=0
NB/2 Gr.=B.,Hm(kr)+ ~. B.{H.+,.(2kh)-(-1)'H._,.(2kh)}J,.(kr). n=l
Thus there are NB unknown constants A . and B.. Guided by the expansion (12), we assume that the interior field can be written in the form Ns/2
w(2)(r, 0) = Nn -1 Dnun~2)(r,O)+ ~ E . .v(2)(r .. 0) n=0
n=l
(14)
S.K. Datta, A.H. Shah / Scattering by embedded cavities
269
where D., E . are unknown constants and, u~ ) and v~ ) are unknown functions. Continuity conditions require that on r = b,
w(2)(b,O)=
No~2-1 N8/2 Y. D , cosnO+ ~. E~sinnO n=O
(15)
n=l
tr, 0) and v~)(r, O) at the interior nodal points by In order to find w(:)(r, O) we first solve for u .(2)" assuming that the nodal values of u-(2),L. w, 0) and v~)(b, O) are given by the values of cos nO and sin nO, respectively, at the boundary nodes. Let
{Us}. =cos nOo, n = 0 . . . . . ½ N o - l ,
{Vo}~=sinnOo,
n = l . . . . . ½No.
(16)
Using (16) in (7) we solve for {Ut}. and {Vr}.. Then two Nz x ½No matrices can be constructed with the nodal values of U and V in the interior, viz. [Ur] = -[Srl]-l[Sro][Uo],
[VI] = -[Szl]-l[Sro][Vs].
(17)
Eq. (13) can now be written in the matrix form as {w ~z>}= [ W]{D}
(18)
where {w ¢z)}is an (Nz + Nn) × 1 column matrix, [ w ] is an (NI + No) × No matrix and {D } is an NB × 1 column matrix. Note that
The next task is to determine the 2NB unknown constants A~, B., D., and E.. To this end the normal derivatives of w(E)(r, 0) at the boundary nodes are calculated as {w ~)'} = [ Wo']iD}.
(19)
Since the matrix [ W] is known, [ We'] can be obtained from (4). The continuity conditions
w(1)(b, 0) = w(2)(b, 0),
Ow (1)
0w (2) 0--7- (b, 0) = T (b, 0)
(20)
are now used to obtain a system of 2No equations in the 2No unknown constants. The interior and boundary nodal values of w (2) are then found from (18) and the scattered field in the exterior region is given by (10). The results of these computations are discussed in Section 4 along with those obtained by the method of matched asymptotic expansion (MAE), which is described in the next section.
3. Long wavelength solution by MAE In our recent work it has been shown that if the wavelength is long compared to the dimensions of the scatterer, the scattered field anywhere in the medium can be obtained by matched-asymptotics. In this technique the scattered field is expanded into two asymptotic series, one valid in an interior region containing the scatterer and the other valid away from the scatterer. The complete solution is obtained by matching these series in some overlap region.
S.K. Datta, A . H . Shah / Scattering by embedded cavities
270
A n outline of the m e t h o d as applied to deep circular cavities was given in [8]. The case of an e m b e d d e d elliptic cylindrical inclusion was briefly discussed in [5]. In this section the solution for an elliptical cavity is given. As limiting cases the results for circular cavity and crack are obtained. Let C represent the ellipse with center at 0', and m a j o r and minor axes along 0'x' and 0'y', respectively (Fig. 1). The local coordinates of a point P outside C may be represented in parametric form by x' = a cosh ~ cos r/,
y = a sinh ~:sin r/,
0~< r/~<2~r, ~>~:o.
(21)
In this system ~ = ~:o represents points on C. The polar coordinates of P referred to O'x'y' are r, ~b, such that r cos ~b = x',
r sin ~b = y'.
(22)
Clearly, 4~ = 3"rr- a + O
(23)
where o~ is the angle m a d e by x'-axis with the x-axis. Let the incident field, w (1), be given by w ( i ) = e i k ( x cos v - y sin ",/)
This wave will be reflected from the free surface y = 0 giving rise to the reflected field, w ( r ) ~ e i k ( x cos v + y sin v)
Thus the total field seen by the cavity is w(O)= 2elk ....
v
cos(ky sin y)
(24)
which m a y be expanded in Fourier-Bessel series as w (°)=
~
a,,Jm(kr)e i"~.
(25)
m ~--oo
H e r e am = 2i" e i"~ cos(kh sin y - my). As indicated before, the scattered field w (s) m a y formally be taken in the f o r m given by (10). It will be m o r e convenient to write (10) as w(S)=
~
an(Hn(kr) ei"°+Hn(kR)ein~9) = ~
n =--oo
s~nHn(kr) eine'+ ~. s~Jm(kr)e im~
n ~--oo
m
(26)
~--oo
where •~ n
= i" e ina A.,
p .m t s~,~=l Ame
irnot
,
A ,p. =
A.H,,+m(2kh). n = -oo
T h e object here is to obtain expansions for w (s) valid near and away from C when e = ka is small. For this purpose it is necessary to define an inner region (RE) containing the cavity, C, and an outer region (R1), which is far away f r o m C (kr = O(1)). In the inner region we define the inner variables as ~, = _x', a
~7' =-Y , a
f = _r.
(27)
a
The outer variables defined in the outer region are
£' = e~',
~'= e;',
~ = e~.
(28)
S.K. Datta, A.H. Shah / Scattering by embedded cavities
271
It is now assumed that w ~2) may be expanded as (29)
W (2) = WO "~ $.LI(E)W1-1-/-L2(E)W2 "~- " " "
where lim~-.o/X,+l//xn = 0. Similarly, an expansion for w m is taken as (30)
w ° ) = w (°) + v l ( e ) W I + v 2 ( e ) W 2 + • • •
where l i m , - . o V n+ d v , = O. The equations to be satisfied by wn and W, are obtained in the following manner. After making the transformation of variables given by (27) the equation satisfied by w ~2) (Eq. (1)) is 02W (2) 02W (2) "-----'~aX + ---2"S7y+ d" y "
(31)
EEw(2) ---- 0.
Substituting the series expansion for w ~2) in (31) and equating the like order terms to zero, a set of equations is obtained to solve for wm (m = 0, 1, 2 . . . . ). The boundary condition on w,, is that C~Wm
--=0
on C
(32)
On
where n is the outward normal to C. Similarly, making the transformation (28) the equation satisfied by w tl) is found to be t~2W(1)
~2 (1) .0 W ,
a£' t - - ~ - - - ~ - w
(1)
=0.
(33)
Thus each Wm satisfies (33). Further, the boundary condition on W,, is aWm =0, ay
y=0.
(34)
In addition, W,, must satisfy the radiation condition as P-} oo. The procedure for finding Wr~ and Wm satisfying the conditions stated above and the matching conditions has been discussed in [5, 8, 9]. So it will not be repeated here. It can be shown that / X l ( E ) ---- e,
/-t2(e) ---- 8 2
In
e,
/ . t 3 ( e ) = 8 2,
/z4(e) = 6 3
In e,
/zs(e) = e 3.
(35)
The functions, win, are W0 = a o ,
wl = ½ ( a l - a - O f ' + ½ i ( a l + a _ O ~ ' + e - e ( D n D n = ~(al + a - l ) cosh ~:oe e°,
sin r / + E n cos r/),
E n = ½(al - a - l ) sinh ~:o e ~°,
W2 ----D 2 o , 1
w3 = -zaor
-2
+
1
-r2
~(a2 + a-2)(x
(36) - ;,2) + ¼i(a2- a_2)f'37'
- t - O 3 0 - t - C 3 0 ~ -I- e - E f T ( D 3 2
D32
=
sin 277 q - E 3 2
~i(a2 - a-2) e 2e° cosh 2~eo,
(?30 = ¼ao sinh 2s%,
1)2o
=
C3o.
cos
2r/),
E32 = ~(a2 + a-2) e 2e° sinh 2~eo,
S.K. Datta, A.It. Shah / Scattering by embedded cavities
272
The constant D3o is connected with the coefficients of the far-field expansion to O(e2). It is found ~1(~) =
that
~
and WI = ~¢o(H1 (r') + Ho(/~)) + (~¢1 - ~¢_l)Hx(r') cos 4~ + i(~/a + ~¢-I)HI(D sin ~b+ (A x e ie - A - I e-iO)Hl(/~).
(37)
Matching the outer expansion to O(e 2) of the inner expansion to O(e 2) with the inner expansion to O(e 2) of the outer expansion to O(e 2) one obtains ~2~X __ j 2 ~ _ 1 = 4-1~Elx, X.
~x +~¢-x
= z1~ D x l ,
6~o
• = - ~1.r C 3 o = - ~1.r a 0 smh 2~¢o.
(38)
Finally, D3o is obtained as D3o = C3o(~ - 2 In 2 - ½i'rr)+ sgoHo(2kh) + ~xH1 (2kh) -
~I-IH1 (2kh)
where @ is Euler's constant. Continuing this process it is found that
w4=i ( s g 1 - ~ - O c o s h f c o s n - l (~ll+~l-Osinh~sinrt+e-~(D41sinn+E41cosrl), 'IT
(39)
~"
D4x = --1- (~¢x + ~1-) cosh ~¢o,
E4I
= i ( ~ 1 -- 6~-1)
'IT
sinh ~¢o,
'IT 1.
t
ws = ½(~¢1- ~¢-1) cosh f cos 17+ ~(~¢x + ~¢'-x ) sinh f sin rt -
6~[(ax - a-0{cos 77 (cosh 3~: + 2 cosh ~:) + cosh ~: cos
3rt}
+ i(al + a_l){sin rt (sinh 3 f - 2 sinh ~) +sinh 3 f sin 3~7}] + 1-~z[(a3- a_3)(cosh 3~¢ cos 3rt +3 cosh f cos ~7) +i(a3 + a_3)(sinh 3 f sin 3rt + 3 sinh 7/sin f)] + (C51-¼Dll~e) sinh ~ sin 77 + (Fsl -¼El 1~) cosh ~: cos 7/+ (D51e - e - 3~D113-se) sin rt + (Dss e - s e - 3~Dll e -e) sin 3 7/ + (Esx e -e - ~ E I 1 e -s~) cos n + (Ess e -se - 3~Ell e -e) cos 3n,
(40)
D51 = ee°[(C~l- ¼Dxx~:o)cosh ~:o+ 3~DH e -3~° X.
!
+ ~1(~/1 + M'-I) cosh ~¢o- 6~i(al + a-1)(3 cosh 3~¢o- 2 cosh ~¢o)+~4i(a3 + a-s) cosh ~:o], D53 = 9~DH e 2~°- ~ eSe°[i(al + a-l) cosh ~o- i(as + a-s) cosh 3~:o], E~I = e~°[(F51-1Ellfo) sinh f o + ~ E x x e -se° 1
+ ~(~/1 -~¢'-1) sinh fo-~n(al - a - 0 ( 3 sinh 3 f o + 2 sinh ~¢o)+ 6-~(as- a-3) sinh ~¢o], E53 = 1 E l l e 2~°+ 19-L~[(a3- a-s) sinh 3~Co- (al - a-a) sinh ~:o]e 3~°. The constants C~1 and Fs~ are obtained by matching the outer expansion to O(e 2) of the inner expansion to O(e s) with the inner expansion to O(e s) of the outer expansion to O(e~). They are Csx
=---
x 1. 1 (M1 +M_I)[@- 2 In 2 - ~ - 2~'rr],
i F51 = - (~tl - ~ t - 0 [ @ - 2 In 2 - ½- ½i~r].
S.K. Datta, A.H. Shah / Scattering by embedded cavities
273
It is also found that the next order t e r m in the outer expansion must be O(e4). Because the algebra gets fairly complicated we will refrain f r o m calculating the O(e 4) terms in the outer and inner expansions. The limiting cases of a circular cavity and a thin slit (crack) can be obtained from the results presented above. In the latter case it is only necessary to put ~:0 = 0. Then one finds
Dll=½i(al+a_l),
1.
D3o=~r(al+a-1)Ht(2kh)cosot,
D32=izl(a2-a-2),
1.
D41=-~(al+a-1),
D s t = (at + a-t){-3~i - ~rH2(2kh) + ~'trHo(2kh) cos 2a} + 6~i(a3 + a-3), I •
D53=lg~a(a3+a-3),
(41)
1. Cst = - ~ ( a l + a _ t ) ( ~ / - 2 In 2 - ½ - ½i~r), M _ t = M t = ~ i ( a t + a _ t ) .
All the other coefficients are zero. For the calculation of the stress-intensity factor it is necessary to obtain expressions for the displacement near the crack-tips. It is easily shown that on the crack face the scattered displacement field takes the form
w = ex/2(a - r)/a [Dtt + 2D32 +
e 2 In e D41 + e 2(D51 + 3Ds3 - ~ D l t ) ]
(42)
1
where the + signs correspond to 0 < r / < ½,tr and ~'tr < rl < ~r, respectively. Coefficients for the inner and outer fields for a circular cavity can be obtained by taking the limit a -* 0, ~:o~ oo such that a cosh ~:o = a. The coefficients for the far field are found to be Ao = -~'rra0, 1.
At
= ¼"rr e
-ia a t ,
A - 1 = -¼xr e'~a_t. "
(43)
4. Numerical results and discussion T h e displacement w at the nodal points within and on B were c o m p u t e d by the m e t h o d outlined in Section 2. Also, the coefficients An and Bn in the series expansion (10) were found. This enabled the computation of w at all points outside B. Particular attention was focused on the distribution of w on the surface y = 0 for various angles of incidence 3' as well as for different orientations of the crack. The w a v e n u m b e r k was varied in the range 0 < k ~<7.0. For the e m b e d d e d crack the radius of the circle B was chosen to be 5 times the half crack-length. Total n u m b e r of nodes on B was 32 and the n u m b e r of nodes within B was 144 (Fig. 2). N e a r the crack tips six-node isoparametric triangular quarter point elements [18, 19] were chosen to obtain the correct square root singularity at the tips. These were surrounded by a layer of 7-node transitional elements [20]. Constant strain triangular elements were used elsewhere. Shape functions for each of these different types of elements are given in the appendix. For the e m b e d d e d circular cavity the ratio of the cavity radius to the radius of the circle B was taken as 1.5. The total n u m b e r of b o u n d a r y and interior elements were 40 and 160, respectively. Only c o n s t a n t strain triangular elements were used in this case. Scattered displacement fields on the free surface y = 0 for grazing incidence due to an e m b e d d e d vertical crack and a circular cavity are c o m p a r e d in Fig. 3. Note that 0 represents the angle between the radius f r o m the center of the cavity (or crack) to the observation point and the vertical line. It is observed that the backscattered signal in the region - 4 0 ° ~< 0 <~0 ° due to a circular cavity is m u c h larger than a crack. The difference between the backscattered signals f r o m two circular cavities of different radii are also appreciable in this region. For the cracks, however, large differences are observed in regions somewhat farther away f r o m the origin. It is also of interest to note the m a r k e d differences between the backscattered signals f r o m cracks and cavities. Fig. 4 shows the scattered field on y = 0 due to a crack at two different depths. It is seen that the signals are not much different as long as the cracks are fairly deep. The scattered signals from a vertical
S.K. Datta, A.H. Shah / Scattering by embedded cavities
274
Fig. 2. Geometry of the finite elements.
- - - - - - VERTICAL CRACK CIRCULAR CAVITY
,= 1 . 0 - ~
2.0
~
hla =1.83
m
1.0 '
\\
~ /
\\ 0 -80*
I -40*
\\
/
I//
)~1/ O*
//
x-~=l-O
\
-~-~=0.6 I 40*
80*
8 Fig. 3. Scattered displacement amplitudes on the free surface due to a buried circular cavity and a vertical crack.
crack are compared with those from a 45*-inclined crack in Fig. 5. This figure brings out the marked differences between the signals received from cracks of different orientations. The most noticeable distinction is that for a 45°-inclined crack most of the energy is scattered in the forward direction, whereas for a vertical crack the energy is scattered much more uniformly in both directions. Fig. 6 depicts the scattered signals from cracks of different orientations for various incidence angles. As in Fig. 4 it is seen
S.K. Datta, A.H. Shah I Scattering by embedded cavities
275
2.(
e:l.O y:O*
hla =I .
~
O*
40*
t.(
0 -80*
-40*
80*
O Fig. 4. Scattered displacement amplitudes on the free surface due to a vertical crack embedded at different depths.
Ct = I I / 2
------
Ct = I I / 4
1.0
/,.,.o_.-\ \/
,-.°:,.---7...,. \ ,," / ",,, , /,:,.oi,,
0.5
/// < . 0 -80*
/
/al
/ /
,__o.ol-,
~b"
I
I
-40"
O*
40*
80*
O Fig. 5. Scattered displacement amplitudes due to buried cracks of different orientations.
276
S.K. Datta, A.H. Shah / Scattering by embedded cavities
1.0
U" 3
0.5
0
-8
3*
0 Fig. 6. Scattered displacement amplitudes due to buried cracks of different orientations and for different angles of incidence.
that scattered signals are significantly influenced by the orientations of the crack. As observed in Fig. 5 most of the scattered energy due to a 45°-inclined crack is confined in the acute angled region formed by the crack and the free surface. It is interesting to observe that the scattered fields due to 45 ° and 135 ° incidences are nearly the same. However, it has been found (not shown in this figure) that as 3' becomes closer to 0 ° the scattered field drops rapidly in amplitude. Comparison of the backscattered fields from deeply embedded cracks and cavities, and edge cracks [7] is made in Fig. 7. It is found that the backscattered signals from the circular cavity increase more rapidly with frequency than those from either the edge crack or the deep crack. Also the maximum amplitude is attained at a lower frequency for a circular cavity than either for an edge crack or for a deep crack. Fig. 8 shows the maximum backscattered amplitudes for an edge crack and a deep crack. It is quite interesting to note that the backscattered amplitude for a deep crack increases more slowly with frequency but reaches almost twice the maximum value for an edge crack at a frequency, which is also twice that for an edge crack. Note that l is the length of the crack. Figs. 9 and 10 depict the scattered displacement fields on y = 0 for a deep circular cavity. It is seen from these figures that the field is strongly dependent on the incidence angles as well as on the frequency. The last Figs. 11, 12, and 13 show the nature of the displacement field near the cavity and the crack. The displacement distribution on the cavity wall is shown in Fig. 10. It is noted that the displacement amplitudes are somewhat higher on the side of the cavity nearer the free surface than on the far side. It is also noticed that the distribution of the local maxima occur on the illumination side of the cavity wall and local minima are on the shadow side. The stress intensity factors for deep cracks of different orientation angles are plotted in Figs. 12 and 13. Also shown in these figures are the stress intensity factors predicted by (42). For the purpose of computing the stress intensity factors from the nodal values of the displacement, use was made of the relation between the values at the nodes of the singular crack-tip
277
S.K. Datta, A.H. Shah / Scattering by embedded cavities 1.5 NORMAL EDGE CRACK DEEP NORMAL CRACK (h/~=0.92) DEEPNORMAL CRACK ( h / l = 1.25) ~ DEEP CIRCULAR /~7"~~._ CAVITY . f f ~ . (h/a = 1.83) ,7 J
-----• 1.0 3=
0.5 -
/
/
.
.
.
"
//
///
__,~ /
1=0*
-~-
/C
"~
2.0
1.0
Fig. 7. Comparisonof the back scattered displacementamplitudesdue to a buried crack and a circularcavity, and an edge crack. element on the crack face near the tip and.the displacement at any point within this singular element. It can be shown that the stress intensity factor is approximately given by Kin
1
/zatlZ = 2~/~---a-a~ {4(w6 - w4) - ( w 3 - w2)}
(44)
where L and the nodes 2, 3, 4, 6 are defined in Fig. 14. It is noted that (44) predicts somewhat lower value of the stress intensity factor than that calculated from (42), although the difference is not large.
p----
NORMAL EDGE CRACK DEEP NORMAL CRACK (h//_, = 0.92)
2.0
7"-'0° m
1.0 /
0
/
/
/
~
I
I
I
1.0
2.0
3.0
4.0
2~ Fig. 8. Comparison of the back scattered displacement amplitudes due to a buried vertical crack and a vertical edge crack.
S.K. Datta, A.H. Shah / Scattering by embedded cavities
278
3"01
-80 °
0°
h/o = 1.5
~
-40 °
0o
40 °
80 °
e Fig. 9. Scattered displacement amplitudes on y = 0 [or a buried circular cavity for various angles of incidence.
1.5
1.0
0.5
0 -80 °
I -40 °
I 0o
I 40 °
80 °
0 Fig. 10. Scattered displacement amplitudes on y = 0 for different waves n u m b e r s due to a buried circular cavity.
S.K. Datta, A.H. Shah / Scattering by embedded cavities
279
3.0
~0
m
4-5 °
90*
I
I
135 °
I
515"
270"
225*
180"
I
5.0
Fig. 11. Displacement of the cavity wall.
2.2 ~
h/a=t.83(MAE} h/a = 1.83(FEM)
)(
2.0
h/a = 2.5 (FEM) / / / /
fire
(2 W
a : 90",
x=O *
/
1.8
t.4 0
_
/
/ /
I 0.5
I 1.0 E
Fig. 12. Stress intensity factors for deep vertical cracks. Also, (42) predicts nearly the same value even when the depth is changed somewhat. It also predicts almost the same value at both the tips. It is believed that this is because the crack is deep inside the half-space. Eq. (44) also predicts nearly the same value at both the tips. However, at small values of e it predicts lower stress intensity factors with larger depths. But this is reversed as e increases. Fig. 13 shows the variation in K m for inclined cracks. It is found that in this case both (42) and (44) predict different stress intensity factors at the two tips when the frequency is large. It is quite interesting to note that the F E M predicts that the stress intensity factor at the lower tip becomes larger than that at the upper tip when the frequency becomes large.
S.K. Datta, A.H. Shah / Scattering by embedded cavities
280 1.4
hla
=1.83, a=45", T=O* /
f
t
11
/ /
&--
t.2 O
v
~ 1.0
-
•-.~--~'- UPPER ~ TIPM (A E ) -LOWERTIP (MAE) UPPERTIP (FEM) LOWERTIP (FEM)
~ \ \ \ 't
o
I 0.5
I 1.0
Fig. 13. Stress intensity factors for 450 inclined embedded cracks.
-~
I
I
t I
; I
I
o
~
LI4
3
1
Yi t t
I I
t a
i
_1 -I
Fig. 14. Crack-tip elements.
Finally, in Table 1 we show the comparison between the predictions for the scattered displacement amplitudes on y = 0 by M A E and FEM techniques for a buried circular cavity. It may be seen that values of e 2 W1, which is the scattered field correct to O(e2), agree quite well with FEM calculations when eh/a is O(1) and e is small.
$. Conclusion A numerical method combining exact analytical solutions wlth the finite element formulation is used here to study scattering of SH waves by deeply embedded cracks and circular cavities in a half-space. It is shown that the scattered displacement field is significantly influenced by the orientation of the cracks as well as by the incidence angles. Also noticeable is the difference between the scattered fields due to
S.K. Datta, A.H. Shah / Scattering by embedded cavities
281
Table 1 Scattered displacement amplitudes at the surface of the half-space due to a buried circular cavity. Comparison of MAE and FEM (~,= o 0)
#
80* 60* 40 ° 20* 0* -20 ° -40* -60* -80*
e=O.l,a/h=0.133
e=O.5, a/h=0.67
e=O.5, a/h=0.33
e=l.O,a/h=0.67
FEM
MAE
FEM
MAE
FEM
MAE
FEM
MAE
0.01447 0.02321 0.02072 0.01359 0.02981 0.0528 0.06637 0.06345 0.03984
0.01211 0.01941 0.0172 0.01205 0.02749 0.03819 0.05997 0.05726 0.03596
0.5130 0.8990 0.9567 0.7709 0.8905 1.354 1.575 1.417 0.8640
0.30283 0.48536 0.42998 0.30115 0.68727 1.1987 1.4991 1.4316 0.89891
0.2358 0.3329 0.2442 0.2230 0.5343 0.8426 0.997 0.9300 0.5818
0.20867 0.28928 0.19796 0.17289 0.50182 0.83846 1.0346 0.99528 0.6338
0.5287 0.8493 0.8823 1.409 2.325 2.7806 2.5731 2.0026 1.1455
0.83466 1.1571 0.79184 0.69157 2.0073 3.3538 4.1385 3.9811 2.5352
embedded cracks, circular cavities and edge cracks. Comparison has been made between the predictions of this numerical technique and the method of matched asymptotic expansions. The comparison is found to be favorable as long as the wavelength is long. The advantage of the numerical technique is that it is applicable to any arbitrary shapes and inhomogeneities. The limitation is that at very short wavelengths it would become quite expensive to implement. For very short wavelengths numerical solution of the appropriate singular integral equation governing the COD seems to be most effective [4, 21].
Appendix
The shape functions corresponding to the constant strain triangular elements, 6-node crack-tip singular elements and 7-node transition elements are given here for easy reference. The geometries of the elements are shown in Fig. 15. For the constant strain triangular elements the shape functions are [22] L1 = 2-~ {x2Y3 -- x3Y2 + X (Y2 -- Y3) + Y (X2 -- X3)}, L2 = 1
{x3yl - xly3 + x
1 La = ~
(Y3 - yl) + y (xl - x3)},
{xly2 - x2yl + x (Yl - Y2) + Y (x2 - Xl)},
where A is the area of the element and (xl, yl), (x2, y2), (x3, Ya) are the coordinates of the vertices. For the six-node and seven-node elements the shape functions are expressed in terms of the local (~, ~/) coordinates as follows. 6-node singular element: L1 = 0.5~:(~¢- 1),
L2 = 0.25(1 + ~)(1 - se)(se- ~ - 1),
L4 = 0.5(1-~:2)(1- r/),
Ls = 0.5(1 + s¢)(1- 7/2),
L3 = 0.25(1 + ~)(1 + r/)(~:+ r / _ 1), L6 = 0 . 5 ( 1 - s¢2)(1+ 7/);
S.K. Datta, A.H. Shah / Scattering by embedded cavities
282
6-NODE CRACK-TIP SINGULAR ELEMENT
(0 i)
2(I,-I)
T-NODE TRANSITION ELEMENT
(-I,-I) (O,-t)
4'-
I
c-',,
(l,-i)
5
,,o,
I (-U)
I (O,t) O,I)
CONSTANT STRAIN TRIANGULAR ELEMENT 2
Fig. 15. Geometries of the elements.
7 - n o d e transition element: L t = -0.25~:(1 - ~:)(1 - n),
L2=O.25(l+~)(1-n)(~-n-1),
L3 = 0.25(1 + ~)(1 + n ) ( ~ + n - 1), L5 = 0.5(1 - ~:2)(1- r/),
L4 = - 0 . 2 5 ~ ( 1 - ~ ) ( 1
L6 = 0.5(1 + ~ : ) ( ! - n2),
+ n),
L7 = 0.5(1 - ~:2)(1 + n).
Acknowledgment The work of Professor Shah was carried out when he was a visiting faculty associate at the University of Colorado. The authors are grateful for the support received from the National Science Foundation under the grant CME78-24179 and from the Natural Science and Engineering Research Council of Canada under the grant A-7988. The authors wish to thank Mr. Roy Swanson for help in the numerical computation.
References [1] S.K. Datta, "Diffraction of SH-waves by an edge crack", J. AppL Mech. 46, 101-106 (1979). [2] S.F. Stone, M.L. Ghosh and A.K. Mal, "Diffraction of antiplane shear waves by an edge crack", J. Appl. Mech. 47, 359-362 (1980). [3] D.A. Mendelsohn, J.D. Achenbach and L.M. Keer, "Scattering of elastic waves by a surface-breaking crack", Wave Motion 2, 277-292 (1980). [4] J.D. Achenbach and R.J. Brind, "Scattering of surface waves by a sub-surface crack", Z Sound and Vib. 76, 43-56 (1981). [5] S.K. Datta and N. EI-Akily, "Diffraction of elastic waves in a half-space. I: Integral representation and matched asymptotic expansions", in: Modern Problems in Elastic Wave Propagation, ed. J. Miklowitz and J.D. Achenbach, Wiley, New York (1978), 197-218. [6] A.H. Shah, K.C. Wong and S.K. Datta, "Diffraction of plane SH-waves in a half-space", Earthq. Engg. Sir. Dyn., to appear. [7] S.K. Datta, A.H. Shah and C.M. Fortunko, "Diffraction of medium and long wavelength SH-waves by edge cracks", J. Appl. Phys., to appear. [8] S.K. Datta and N. EI-Akily, "Diffraction of elastic waves by cylindrical cavity in a half-space", J. Acoust. Soc. Amer. 64, 1692-1699 (1978). [9] S.K. Datta, "Diffraction of SH-waves by an elliptic elastic cylinder", Int. ]. Solids Structures I0, 123-133 (1974).
S.K. Datta, A.H. Shah / Scattering by embedded cavities
283
[10] R.D. Gregory, "The propagation of waves in an elastic half-space containing a cylindrical cavity", Proc. Camb. Phil. Soc. 67, 689-710 (1970).
[11] V.R. Thiruvenkatachar and K. Viswanathan, "Dynamic response of an elastic half-space with cylindrical cavity to timedependent surface tractions over the boufidary of the cavity", J. Math. Mech. 14, 541-571 (1965). [12] N. Zitron and S.N. Karp, "Higher-order approximations in multiple scattering. I. Two-dimensional scalar case", J. Math. Phys. 2, 394-402 (1961). [13] N. El-Akily and S.K. Datta, "Response of a circular cylindrical shell to disturbances in a half-space", Earthq. Engg. Str. Dyn. 8, 469-477 (1980). [14] S. Sancar and Y.H. Pap, "Spectral analysis of elastic pulses backscattered from two cylindrical cavities in a solid. Part I", 3". Acoust. Soc. Amer. 69, 1591-1596 (1981). [15] O. C. Zienkiewicz, Finite Element Method, McGraw-Hill, London (1977). [16] K. Washizu, Variational Methods in Elasticity and Plasticity, Pergamon Press, New York (1968). [17] R. Courant and D. Hilbert, Methods o[ Mathematical Physics, Vol. 1, Wiley, New York (1953). [18] R.S. Barsoum, "On the use of isoparametric finite elements in linear fracture mechanics", Int. J. Num. Methods Engg. 10, 25-38 (1976). [19] R.S. Barsoum, "Triangular quarter-point elements as elastic and perfectly plastic crack tip elements", Int. J. Num. Methods Engg. 11, 85-98 (1977). [20] P.P. Lynn and A.R. Ingraffia, "Transition elements to be used with quarter-point crack-tip elements", Int. J. Num. Methods Engg. 12, 1031-1036 (1978). [21] A.K. Mal, "Diffraction of SH waves by a near surface crack", presented at the Air Force/DARPA Review of Progress in Quantitative NDE, University of Colorado, Boulder, August 2-7 (1981). [22] R.D. Cook, Concepts and Applications o[Finite Element Analysis, Wiley, New York (1974).
265
S C A T T E R I N G OF SH W A V E S BY E M B E D D E D CAVITIES S.K. DATI'A Department o/Mechanical Engineering, University of Colorado, Boulder, CO 80309, USA
A.H. SHAH Department of Civil Engineering, University of Manitoba, Winnipeg, Canada R3T 2N2 Received 31 August 1981, Revised 25 January 1982
Scattering of plane SH waves by sub-surface circular cavities and thin slits in a semi-infinite elastic medium is analyzed in this paper. Two methods of solution are used to obtain the displacements on the free-surface. One of these is a method of matched asymptotic expansion that is very effective when the wavelength is long compared to the dimensions of the cavity (or the crack). The other is a combined finite element and analytical technique, which is useful in the long to intermediate wavelength range. The results obtained by these two techniques are shown to agree quite well for long wave-lengths. Numerical results for the surface displacements for various incident wave angles are seen to depend significantly on the depth and size of the cavity and the crack. In the latter case the orientation of the crack has a significant influence on the scattered field.
1. Introduction Scattering of elastic waves by cavities and cracks has been the subject of numerous investigations in the last decade. Most of these studies have dealt with the case of an infinite medium. Only recently [1-5] has the problem of scattering in a semi-infinite medium been given adequate attention. It can be seen from these papers that the presence of the free boundary influences the scattered field in a significant manner. This influence must be properly taken into account for the purposes of QNDE. In [6, 7] the problem of scattering of SH-waves by surface depressions (canyons) and edge-cracks was considered. In [6] a solution was presented also for the problem of multiple scattering by a triangular canyon and a circular cavity. The technique used in both these papers was a combined finite element and analytical technique. The advantage of this is that it is quite versatile. Problems of multiple scattering by a cluster of scatterers or by scatterers embedded in a different material from the surrounding matrix can be dealt with as easily as the simpler problem of a single scatterer in a homogeneous material. In [7] results are presented for an edge crack in an anisotropic insert embedded in a homogeneous isotropic material. In this paper we have considered the case when the cavity or the crack is embedded in a semi-infinite medium. The solution to this problem is obtained by two different techniques. The first of these is the combined finite-element and analytical technique similar to that used in [6, 7]. In this the cavity (or the crack) is enclosed by a circular contour. The region interior to this contour is represented by finite elements. The solution in the exterior region is expressed in the exact eigenfunction series satisfying the boundary conditions on the free surface. Continuity conditions for the displacement and traction on the circular contour are imposed to find the constants multiplying the eigenfunctions in the series. In order 0165-2125/82/0000-0000/$02.75 © 1982 North-Holland
266
S.K. Datta, A.H. Shah / Scattering by embedded cavities
to ascertain the accuracy of the solution thus obtained comparison is made with the results obtained by a method of matched-asymptotics (MAE). Application of MAE to solve scattering problems in bounded and unbounded media has been demonstrated in earlier papers [1, 8, 9]. It may be noted that the problem of scattering of SH-waves by an embedded object can be reduced to the problem of scattering by two similar objects in an infinite medium. This latter problem of multiple scattering can be solved in different ways. One is to expand the solution in the form of a multipolar expansion and then satisfy the boundary conditions on one scatterer. This leads to an infinite set of algebraic equations which can be solved after truncation. This approach has been taken in this paper for expressing the solution outside the circular contour (see also [8] and [10]). An alternative approach is to use successive reflections and was adapted in [11-14]. In [12] the authors studied the problem of scalar wave scattering by two parallel cylinders and derived expansions for the far-field. This problem is similar to the one being studied here. Using the combined finite element and wave function expansion we are able to obtain numerical results everywhere in the half-space.
2. Numerical-analytical technique
"
Figure 1 shows a portion of the semi-infinite xy-plane occupied by a homogeneous isotropic elastic material of rigidity modulus,/z and density, p. A cavity (or crack) bounded by the contour C is located in this plane at a depth h from the free surface. The cavity will be assumed to be enclosed by a circle B of radius b (
h
r
l ..........
R2
O'
13
R1 y, Y
Fig. 1. Geometry of the problem.
For the propagation of harmonic SH waves in region R2 the displacement in the Z-direction, w (2), satisfies the Helmholtz equation -2 (2) 192W(2)OX 2 -1"-d a - ~ - ' F k 2 w (2) = 0 (1) where k 2 = pto2/~, ¢o being the circular frequency.
S.K. Datta, A.H. Shah / Scatteringby embedded cavities
267
The boundary condition for w (2) on C is that 0W (2) --=0
On
on C where O/On represents normal derivative. Furthermore let w (2)= wB on B, where wB is a known function to be defined later (see Eq. (16)). Then the variational equivalence to this boundary value problem is the minimization of the quadratic functional [15-17], F = f (Vw (2)" ~--~(2)_ k2w(2)l~(2)) dx dy. JR 2
(2)
with the condition that w <2)= w8 on B. Here the bar denotes the complex conjugate. RE is subdivided into finite elements having N~ +NB number of nodes where NB is the number of nodes on B. The displacement field within each element is represented in terms of shape functions Lj(x, y) and nodal values wj as w(2)(x, y) = ~, Ljwj
(3)
where the shape function Lj is 1 at the jth node and is zero at all the other nodes. The subscript / refers to node position, some internal (subscripted I) and some on the boundary B (subscripted B). Then (3) may be written as w(2)(x, y) = {L,}T{W,}+ {LB}T{WB}
(4)
where {L} and {w} are column vectors and superscript T represents transpose. Defining the matrices SIt, SIB, Sin, and SBB as SMN = IR2 [{VLN}T{VLN}- k2{LM}T{LN}] dx dy,
(5)
the functional F of (2) can be rewritten as F = {#1 }TES,~]{w,}+ { #, }T[$ZB]{WB} + {#B}T[$m]{W,}+ {#B}T[sBB]{Ws}.
(6)
Treating the boundary condition {WB}as known, but yet unspecified, and taking the variation aG
--~
O~
it is found that w~ is related to WB by the equation [Su]{wl} = --[SIs]{WB}.
(7)
In order now to solve for w at the internal and boundary nodal points it is necessary to match the solution in R2 with the solution in the exterior region R1. For this purpose the solution for w satisfying eq. (1) and the boundary condition on y = 0, viz. Ow(1) --=0 0y
ony=0
(8)
will have to be found. This solution can be formally written as w (I)= w (°)+ w (s~ w h e r e w (°) is the total incident field near the cavity a n d w (') is the scattered field.
(9)
S.K. Datta, A.H. Shah / Scatteringby embeddedcavities
268
The solution w (s) can be written as
w(~)= ~ [A.(Hn(kr)cosnO+H.(kR)cosnO)+B~(H~(kr)sinnO+H.(kR)sinnO)]
(10)
n=0
where H . is the Hankel function of the first kind and the coordinates (r, 0), (R, O) are shown in Fig. 1, Since H~ cos nO = H.(2kh)+
~. {H.+,~(2kh)+(-1)mH.-r~(2kh)}Jr~(kr)cos toO, m=l
(11) H~ sin nO = ~ {H.+m(2kh)-(-1)mHn-,.(2kh)}J,.(kr)
sin toO,
(10) can be rewritten as
(12)
w ~s~= ~ [Fro cos mO +Gm sin toO] m=O
where
Fm= A.,H,. (kr) + ~ A. {H. +,. (2kh) + ( - 1 ) ' H . _ . , (2 kh)}Jm (kr),
m~0,
n=O
=AoHo(kr) + ~ A.H.(2kh)Jo(kr),
m = O,
n=O
and
Gm = B.,H.,(kr) + ~ B.{H.÷m ( 2 k h ) - ( - 1 ) " H . _ . , (2kh)}J.,(kr). n=0
For the purpose of numerical solution only a finite number of terms of the series will be kept. Choosing the number of nodes, NB, on the boundary B to be even, (12) will be written as
NB/2-1 w (')=
Y~ F . , c o s m 0 + m~0
N~/2 ~ G,.sinmO
(13)
m=l
with
Fm =AmHm(kr)+
An{H.+m(2kh)+(-1)r~H._.,(2kh)}J,.(kr),
m SO,
n=0
=AoHo(kr) + N ~ - I A . H . (2kh),
m = 0,
n=0
NB/2 Gr.=B.,Hm(kr)+ ~. B.{H.+,.(2kh)-(-1)'H._,.(2kh)}J,.(kr). n=l
Thus there are NB unknown constants A . and B.. Guided by the expansion (12), we assume that the interior field can be written in the form Ns/2
w(2)(r, 0) = Nn -1 Dnun~2)(r,O)+ ~ E . .v(2)(r .. 0) n=0
n=l
(14)
S.K. Datta, A.H. Shah / Scattering by embedded cavities
269
where D., E . are unknown constants and, u~ ) and v~ ) are unknown functions. Continuity conditions require that on r = b,
w(2)(b,O)=
No~2-1 N8/2 Y. D , cosnO+ ~. E~sinnO n=O
(15)
n=l
tr, 0) and v~)(r, O) at the interior nodal points by In order to find w(:)(r, O) we first solve for u .(2)" assuming that the nodal values of u-(2),L. w, 0) and v~)(b, O) are given by the values of cos nO and sin nO, respectively, at the boundary nodes. Let
{Us}. =cos nOo, n = 0 . . . . . ½ N o - l ,
{Vo}~=sinnOo,
n = l . . . . . ½No.
(16)
Using (16) in (7) we solve for {Ut}. and {Vr}.. Then two Nz x ½No matrices can be constructed with the nodal values of U and V in the interior, viz. [Ur] = -[Srl]-l[Sro][Uo],
[VI] = -[Szl]-l[Sro][Vs].
(17)
Eq. (13) can now be written in the matrix form as {w ~z>}= [ W]{D}
(18)
where {w ¢z)}is an (Nz + Nn) × 1 column matrix, [ w ] is an (NI + No) × No matrix and {D } is an NB × 1 column matrix. Note that
The next task is to determine the 2NB unknown constants A~, B., D., and E.. To this end the normal derivatives of w(E)(r, 0) at the boundary nodes are calculated as {w ~)'} = [ Wo']iD}.
(19)
Since the matrix [ W] is known, [ We'] can be obtained from (4). The continuity conditions
w(1)(b, 0) = w(2)(b, 0),
Ow (1)
0w (2) 0--7- (b, 0) = T (b, 0)
(20)
are now used to obtain a system of 2No equations in the 2No unknown constants. The interior and boundary nodal values of w (2) are then found from (18) and the scattered field in the exterior region is given by (10). The results of these computations are discussed in Section 4 along with those obtained by the method of matched asymptotic expansion (MAE), which is described in the next section.
3. Long wavelength solution by MAE In our recent work it has been shown that if the wavelength is long compared to the dimensions of the scatterer, the scattered field anywhere in the medium can be obtained by matched-asymptotics. In this technique the scattered field is expanded into two asymptotic series, one valid in an interior region containing the scatterer and the other valid away from the scatterer. The complete solution is obtained by matching these series in some overlap region.
S.K. Datta, A . H . Shah / Scattering by embedded cavities
270
A n outline of the m e t h o d as applied to deep circular cavities was given in [8]. The case of an e m b e d d e d elliptic cylindrical inclusion was briefly discussed in [5]. In this section the solution for an elliptical cavity is given. As limiting cases the results for circular cavity and crack are obtained. Let C represent the ellipse with center at 0', and m a j o r and minor axes along 0'x' and 0'y', respectively (Fig. 1). The local coordinates of a point P outside C may be represented in parametric form by x' = a cosh ~ cos r/,
y = a sinh ~:sin r/,
0~< r/~<2~r, ~>~:o.
(21)
In this system ~ = ~:o represents points on C. The polar coordinates of P referred to O'x'y' are r, ~b, such that r cos ~b = x',
r sin ~b = y'.
(22)
Clearly, 4~ = 3"rr- a + O
(23)
where o~ is the angle m a d e by x'-axis with the x-axis. Let the incident field, w (1), be given by w ( i ) = e i k ( x cos v - y sin ",/)
This wave will be reflected from the free surface y = 0 giving rise to the reflected field, w ( r ) ~ e i k ( x cos v + y sin v)
Thus the total field seen by the cavity is w(O)= 2elk ....
v
cos(ky sin y)
(24)
which m a y be expanded in Fourier-Bessel series as w (°)=
~
a,,Jm(kr)e i"~.
(25)
m ~--oo
H e r e am = 2i" e i"~ cos(kh sin y - my). As indicated before, the scattered field w (s) m a y formally be taken in the f o r m given by (10). It will be m o r e convenient to write (10) as w(S)=
~
an(Hn(kr) ei"°+Hn(kR)ein~9) = ~
n =--oo
s~nHn(kr) eine'+ ~. s~Jm(kr)e im~
n ~--oo
m
(26)
~--oo
where •~ n
= i" e ina A.,
p .m t s~,~=l Ame
irnot
,
A ,p. =
A.H,,+m(2kh). n = -oo
T h e object here is to obtain expansions for w (s) valid near and away from C when e = ka is small. For this purpose it is necessary to define an inner region (RE) containing the cavity, C, and an outer region (R1), which is far away f r o m C (kr = O(1)). In the inner region we define the inner variables as ~, = _x', a
~7' =-Y , a
f = _r.
(27)
a
The outer variables defined in the outer region are
£' = e~',
~'= e;',
~ = e~.
(28)
S.K. Datta, A.H. Shah / Scattering by embedded cavities
271
It is now assumed that w ~2) may be expanded as (29)
W (2) = WO "~ $.LI(E)W1-1-/-L2(E)W2 "~- " " "
where lim~-.o/X,+l//xn = 0. Similarly, an expansion for w m is taken as (30)
w ° ) = w (°) + v l ( e ) W I + v 2 ( e ) W 2 + • • •
where l i m , - . o V n+ d v , = O. The equations to be satisfied by wn and W, are obtained in the following manner. After making the transformation of variables given by (27) the equation satisfied by w ~2) (Eq. (1)) is 02W (2) 02W (2) "-----'~aX + ---2"S7y+ d" y "
(31)
EEw(2) ---- 0.
Substituting the series expansion for w ~2) in (31) and equating the like order terms to zero, a set of equations is obtained to solve for wm (m = 0, 1, 2 . . . . ). The boundary condition on w,, is that C~Wm
--=0
on C
(32)
On
where n is the outward normal to C. Similarly, making the transformation (28) the equation satisfied by w tl) is found to be t~2W(1)
~2 (1) .0 W ,
a£' t - - ~ - - - ~ - w
(1)
=0.
(33)
Thus each Wm satisfies (33). Further, the boundary condition on W,, is aWm =0, ay
y=0.
(34)
In addition, W,, must satisfy the radiation condition as P-} oo. The procedure for finding Wr~ and Wm satisfying the conditions stated above and the matching conditions has been discussed in [5, 8, 9]. So it will not be repeated here. It can be shown that / X l ( E ) ---- e,
/-t2(e) ---- 8 2
In
e,
/ . t 3 ( e ) = 8 2,
/z4(e) = 6 3
In e,
/zs(e) = e 3.
(35)
The functions, win, are W0 = a o ,
wl = ½ ( a l - a - O f ' + ½ i ( a l + a _ O ~ ' + e - e ( D n D n = ~(al + a - l ) cosh ~:oe e°,
sin r / + E n cos r/),
E n = ½(al - a - l ) sinh ~:o e ~°,
W2 ----D 2 o , 1
w3 = -zaor
-2
+
1
-r2
~(a2 + a-2)(x
(36) - ;,2) + ¼i(a2- a_2)f'37'
- t - O 3 0 - t - C 3 0 ~ -I- e - E f T ( D 3 2
D32
=
sin 277 q - E 3 2
~i(a2 - a-2) e 2e° cosh 2~eo,
(?30 = ¼ao sinh 2s%,
1)2o
=
C3o.
cos
2r/),
E32 = ~(a2 + a-2) e 2e° sinh 2~eo,
S.K. Datta, A.It. Shah / Scattering by embedded cavities
272
The constant D3o is connected with the coefficients of the far-field expansion to O(e2). It is found ~1(~) =
that
~
and WI = ~¢o(H1 (r') + Ho(/~)) + (~¢1 - ~¢_l)Hx(r') cos 4~ + i(~/a + ~¢-I)HI(D sin ~b+ (A x e ie - A - I e-iO)Hl(/~).
(37)
Matching the outer expansion to O(e 2) of the inner expansion to O(e 2) with the inner expansion to O(e 2) of the outer expansion to O(e 2) one obtains ~2~X __ j 2 ~ _ 1 = 4-1~Elx, X.
~x +~¢-x
= z1~ D x l ,
6~o
• = - ~1.r C 3 o = - ~1.r a 0 smh 2~¢o.
(38)
Finally, D3o is obtained as D3o = C3o(~ - 2 In 2 - ½i'rr)+ sgoHo(2kh) + ~xH1 (2kh) -
~I-IH1 (2kh)
where @ is Euler's constant. Continuing this process it is found that
w4=i ( s g 1 - ~ - O c o s h f c o s n - l (~ll+~l-Osinh~sinrt+e-~(D41sinn+E41cosrl), 'IT
(39)
~"
D4x = --1- (~¢x + ~1-) cosh ~¢o,
E4I
= i ( ~ 1 -- 6~-1)
'IT
sinh ~¢o,
'IT 1.
t
ws = ½(~¢1- ~¢-1) cosh f cos 17+ ~(~¢x + ~¢'-x ) sinh f sin rt -
6~[(ax - a-0{cos 77 (cosh 3~: + 2 cosh ~:) + cosh ~: cos
3rt}
+ i(al + a_l){sin rt (sinh 3 f - 2 sinh ~) +sinh 3 f sin 3~7}] + 1-~z[(a3- a_3)(cosh 3~¢ cos 3rt +3 cosh f cos ~7) +i(a3 + a_3)(sinh 3 f sin 3rt + 3 sinh 7/sin f)] + (C51-¼Dll~e) sinh ~ sin 77 + (Fsl -¼El 1~) cosh ~: cos 7/+ (D51e - e - 3~D113-se) sin rt + (Dss e - s e - 3~Dll e -e) sin 3 7/ + (Esx e -e - ~ E I 1 e -s~) cos n + (Ess e -se - 3~Ell e -e) cos 3n,
(40)
D51 = ee°[(C~l- ¼Dxx~:o)cosh ~:o+ 3~DH e -3~° X.
!
+ ~1(~/1 + M'-I) cosh ~¢o- 6~i(al + a-1)(3 cosh 3~¢o- 2 cosh ~¢o)+~4i(a3 + a-s) cosh ~:o], D53 = 9~DH e 2~°- ~ eSe°[i(al + a-l) cosh ~o- i(as + a-s) cosh 3~:o], E~I = e~°[(F51-1Ellfo) sinh f o + ~ E x x e -se° 1
+ ~(~/1 -~¢'-1) sinh fo-~n(al - a - 0 ( 3 sinh 3 f o + 2 sinh ~¢o)+ 6-~(as- a-3) sinh ~¢o], E53 = 1 E l l e 2~°+ 19-L~[(a3- a-s) sinh 3~Co- (al - a-a) sinh ~:o]e 3~°. The constants C~1 and Fs~ are obtained by matching the outer expansion to O(e 2) of the inner expansion to O(e s) with the inner expansion to O(e s) of the outer expansion to O(e~). They are Csx
=---
x 1. 1 (M1 +M_I)[@- 2 In 2 - ~ - 2~'rr],
i F51 = - (~tl - ~ t - 0 [ @ - 2 In 2 - ½- ½i~r].
S.K. Datta, A.H. Shah / Scattering by embedded cavities
273
It is also found that the next order t e r m in the outer expansion must be O(e4). Because the algebra gets fairly complicated we will refrain f r o m calculating the O(e 4) terms in the outer and inner expansions. The limiting cases of a circular cavity and a thin slit (crack) can be obtained from the results presented above. In the latter case it is only necessary to put ~:0 = 0. Then one finds
Dll=½i(al+a_l),
1.
D3o=~r(al+a-1)Ht(2kh)cosot,
D32=izl(a2-a-2),
1.
D41=-~(al+a-1),
D s t = (at + a-t){-3~i - ~rH2(2kh) + ~'trHo(2kh) cos 2a} + 6~i(a3 + a-3), I •
D53=lg~a(a3+a-3),
(41)
1. Cst = - ~ ( a l + a _ t ) ( ~ / - 2 In 2 - ½ - ½i~r), M _ t = M t = ~ i ( a t + a _ t ) .
All the other coefficients are zero. For the calculation of the stress-intensity factor it is necessary to obtain expressions for the displacement near the crack-tips. It is easily shown that on the crack face the scattered displacement field takes the form
w = ex/2(a - r)/a [Dtt + 2D32 +
e 2 In e D41 + e 2(D51 + 3Ds3 - ~ D l t ) ]
(42)
1
where the + signs correspond to 0 < r / < ½,tr and ~'tr < rl < ~r, respectively. Coefficients for the inner and outer fields for a circular cavity can be obtained by taking the limit a -* 0, ~:o~ oo such that a cosh ~:o = a. The coefficients for the far field are found to be Ao = -~'rra0, 1.
At
= ¼"rr e
-ia a t ,
A - 1 = -¼xr e'~a_t. "
(43)
4. Numerical results and discussion T h e displacement w at the nodal points within and on B were c o m p u t e d by the m e t h o d outlined in Section 2. Also, the coefficients An and Bn in the series expansion (10) were found. This enabled the computation of w at all points outside B. Particular attention was focused on the distribution of w on the surface y = 0 for various angles of incidence 3' as well as for different orientations of the crack. The w a v e n u m b e r k was varied in the range 0 < k ~<7.0. For the e m b e d d e d crack the radius of the circle B was chosen to be 5 times the half crack-length. Total n u m b e r of nodes on B was 32 and the n u m b e r of nodes within B was 144 (Fig. 2). N e a r the crack tips six-node isoparametric triangular quarter point elements [18, 19] were chosen to obtain the correct square root singularity at the tips. These were surrounded by a layer of 7-node transitional elements [20]. Constant strain triangular elements were used elsewhere. Shape functions for each of these different types of elements are given in the appendix. For the e m b e d d e d circular cavity the ratio of the cavity radius to the radius of the circle B was taken as 1.5. The total n u m b e r of b o u n d a r y and interior elements were 40 and 160, respectively. Only c o n s t a n t strain triangular elements were used in this case. Scattered displacement fields on the free surface y = 0 for grazing incidence due to an e m b e d d e d vertical crack and a circular cavity are c o m p a r e d in Fig. 3. Note that 0 represents the angle between the radius f r o m the center of the cavity (or crack) to the observation point and the vertical line. It is observed that the backscattered signal in the region - 4 0 ° ~< 0 <~0 ° due to a circular cavity is m u c h larger than a crack. The difference between the backscattered signals f r o m two circular cavities of different radii are also appreciable in this region. For the cracks, however, large differences are observed in regions somewhat farther away f r o m the origin. It is also of interest to note the m a r k e d differences between the backscattered signals f r o m cracks and cavities. Fig. 4 shows the scattered field on y = 0 due to a crack at two different depths. It is seen that the signals are not much different as long as the cracks are fairly deep. The scattered signals from a vertical
S.K. Datta, A.H. Shah / Scattering by embedded cavities
274
Fig. 2. Geometry of the finite elements.
- - - - - - VERTICAL CRACK CIRCULAR CAVITY
,= 1 . 0 - ~
2.0
~
hla =1.83
m
1.0 '
\\
~ /
\\ 0 -80*
I -40*
\\
/
I//
)~1/ O*
//
x-~=l-O
\
-~-~=0.6 I 40*
80*
8 Fig. 3. Scattered displacement amplitudes on the free surface due to a buried circular cavity and a vertical crack.
crack are compared with those from a 45*-inclined crack in Fig. 5. This figure brings out the marked differences between the signals received from cracks of different orientations. The most noticeable distinction is that for a 45°-inclined crack most of the energy is scattered in the forward direction, whereas for a vertical crack the energy is scattered much more uniformly in both directions. Fig. 6 depicts the scattered signals from cracks of different orientations for various incidence angles. As in Fig. 4 it is seen
S.K. Datta, A.H. Shah I Scattering by embedded cavities
275
2.(
e:l.O y:O*
hla =I .
~
O*
40*
t.(
0 -80*
-40*
80*
O Fig. 4. Scattered displacement amplitudes on the free surface due to a vertical crack embedded at different depths.
Ct = I I / 2
------
Ct = I I / 4
1.0
/,.,.o_.-\ \/
,-.°:,.---7...,. \ ,," / ",,, , /,:,.oi,,
0.5
/// < . 0 -80*
/
/al
/ /
,__o.ol-,
~b"
I
I
-40"
O*
40*
80*
O Fig. 5. Scattered displacement amplitudes due to buried cracks of different orientations.
276
S.K. Datta, A.H. Shah / Scattering by embedded cavities
1.0
U" 3
0.5
0
-8
3*
0 Fig. 6. Scattered displacement amplitudes due to buried cracks of different orientations and for different angles of incidence.
that scattered signals are significantly influenced by the orientations of the crack. As observed in Fig. 5 most of the scattered energy due to a 45°-inclined crack is confined in the acute angled region formed by the crack and the free surface. It is interesting to observe that the scattered fields due to 45 ° and 135 ° incidences are nearly the same. However, it has been found (not shown in this figure) that as 3' becomes closer to 0 ° the scattered field drops rapidly in amplitude. Comparison of the backscattered fields from deeply embedded cracks and cavities, and edge cracks [7] is made in Fig. 7. It is found that the backscattered signals from the circular cavity increase more rapidly with frequency than those from either the edge crack or the deep crack. Also the maximum amplitude is attained at a lower frequency for a circular cavity than either for an edge crack or for a deep crack. Fig. 8 shows the maximum backscattered amplitudes for an edge crack and a deep crack. It is quite interesting to note that the backscattered amplitude for a deep crack increases more slowly with frequency but reaches almost twice the maximum value for an edge crack at a frequency, which is also twice that for an edge crack. Note that l is the length of the crack. Figs. 9 and 10 depict the scattered displacement fields on y = 0 for a deep circular cavity. It is seen from these figures that the field is strongly dependent on the incidence angles as well as on the frequency. The last Figs. 11, 12, and 13 show the nature of the displacement field near the cavity and the crack. The displacement distribution on the cavity wall is shown in Fig. 10. It is noted that the displacement amplitudes are somewhat higher on the side of the cavity nearer the free surface than on the far side. It is also noticed that the distribution of the local maxima occur on the illumination side of the cavity wall and local minima are on the shadow side. The stress intensity factors for deep cracks of different orientation angles are plotted in Figs. 12 and 13. Also shown in these figures are the stress intensity factors predicted by (42). For the purpose of computing the stress intensity factors from the nodal values of the displacement, use was made of the relation between the values at the nodes of the singular crack-tip
277
S.K. Datta, A.H. Shah / Scattering by embedded cavities 1.5 NORMAL EDGE CRACK DEEP NORMAL CRACK (h/~=0.92) DEEPNORMAL CRACK ( h / l = 1.25) ~ DEEP CIRCULAR /~7"~~._ CAVITY . f f ~ . (h/a = 1.83) ,7 J
-----• 1.0 3=
0.5 -
/
/
.
.
.
"
//
///
__,~ /
1=0*
-~-
/C
"~
2.0
1.0
Fig. 7. Comparisonof the back scattered displacementamplitudesdue to a buried crack and a circularcavity, and an edge crack. element on the crack face near the tip and.the displacement at any point within this singular element. It can be shown that the stress intensity factor is approximately given by Kin
1
/zatlZ = 2~/~---a-a~ {4(w6 - w4) - ( w 3 - w2)}
(44)
where L and the nodes 2, 3, 4, 6 are defined in Fig. 14. It is noted that (44) predicts somewhat lower value of the stress intensity factor than that calculated from (42), although the difference is not large.
p----
NORMAL EDGE CRACK DEEP NORMAL CRACK (h//_, = 0.92)
2.0
7"-'0° m
1.0 /
0
/
/
/
~
I
I
I
1.0
2.0
3.0
4.0
2~ Fig. 8. Comparison of the back scattered displacement amplitudes due to a buried vertical crack and a vertical edge crack.
S.K. Datta, A.H. Shah / Scattering by embedded cavities
278
3"01
-80 °
0°
h/o = 1.5
~
-40 °
0o
40 °
80 °
e Fig. 9. Scattered displacement amplitudes on y = 0 [or a buried circular cavity for various angles of incidence.
1.5
1.0
0.5
0 -80 °
I -40 °
I 0o
I 40 °
80 °
0 Fig. 10. Scattered displacement amplitudes on y = 0 for different waves n u m b e r s due to a buried circular cavity.
S.K. Datta, A.H. Shah / Scattering by embedded cavities
279
3.0
~0
m
4-5 °
90*
I
I
135 °
I
515"
270"
225*
180"
I
5.0
Fig. 11. Displacement of the cavity wall.
2.2 ~
h/a=t.83(MAE} h/a = 1.83(FEM)
)(
2.0
h/a = 2.5 (FEM) / / / /
fire
(2 W
a : 90",
x=O *
/
1.8
t.4 0
_
/
/ /
I 0.5
I 1.0 E
Fig. 12. Stress intensity factors for deep vertical cracks. Also, (42) predicts nearly the same value even when the depth is changed somewhat. It also predicts almost the same value at both the tips. It is believed that this is because the crack is deep inside the half-space. Eq. (44) also predicts nearly the same value at both the tips. However, at small values of e it predicts lower stress intensity factors with larger depths. But this is reversed as e increases. Fig. 13 shows the variation in K m for inclined cracks. It is found that in this case both (42) and (44) predict different stress intensity factors at the two tips when the frequency is large. It is quite interesting to note that the F E M predicts that the stress intensity factor at the lower tip becomes larger than that at the upper tip when the frequency becomes large.
S.K. Datta, A.H. Shah / Scattering by embedded cavities
280 1.4
hla
=1.83, a=45", T=O* /
f
t
11
/ /
&--
t.2 O
v
~ 1.0
-
•-.~--~'- UPPER ~ TIPM (A E ) -LOWERTIP (MAE) UPPERTIP (FEM) LOWERTIP (FEM)
~ \ \ \ 't
o
I 0.5
I 1.0
Fig. 13. Stress intensity factors for 450 inclined embedded cracks.
-~
I
I
t I
; I
I
o
~
LI4
3
1
Yi t t
I I
t a
i
_1 -I
Fig. 14. Crack-tip elements.
Finally, in Table 1 we show the comparison between the predictions for the scattered displacement amplitudes on y = 0 by M A E and FEM techniques for a buried circular cavity. It may be seen that values of e 2 W1, which is the scattered field correct to O(e2), agree quite well with FEM calculations when eh/a is O(1) and e is small.
$. Conclusion A numerical method combining exact analytical solutions wlth the finite element formulation is used here to study scattering of SH waves by deeply embedded cracks and circular cavities in a half-space. It is shown that the scattered displacement field is significantly influenced by the orientation of the cracks as well as by the incidence angles. Also noticeable is the difference between the scattered fields due to
S.K. Datta, A.H. Shah / Scattering by embedded cavities
281
Table 1 Scattered displacement amplitudes at the surface of the half-space due to a buried circular cavity. Comparison of MAE and FEM (~,= o 0)
#
80* 60* 40 ° 20* 0* -20 ° -40* -60* -80*
e=O.l,a/h=0.133
e=O.5, a/h=0.67
e=O.5, a/h=0.33
e=l.O,a/h=0.67
FEM
MAE
FEM
MAE
FEM
MAE
FEM
MAE
0.01447 0.02321 0.02072 0.01359 0.02981 0.0528 0.06637 0.06345 0.03984
0.01211 0.01941 0.0172 0.01205 0.02749 0.03819 0.05997 0.05726 0.03596
0.5130 0.8990 0.9567 0.7709 0.8905 1.354 1.575 1.417 0.8640
0.30283 0.48536 0.42998 0.30115 0.68727 1.1987 1.4991 1.4316 0.89891
0.2358 0.3329 0.2442 0.2230 0.5343 0.8426 0.997 0.9300 0.5818
0.20867 0.28928 0.19796 0.17289 0.50182 0.83846 1.0346 0.99528 0.6338
0.5287 0.8493 0.8823 1.409 2.325 2.7806 2.5731 2.0026 1.1455
0.83466 1.1571 0.79184 0.69157 2.0073 3.3538 4.1385 3.9811 2.5352
embedded cracks, circular cavities and edge cracks. Comparison has been made between the predictions of this numerical technique and the method of matched asymptotic expansions. The comparison is found to be favorable as long as the wavelength is long. The advantage of the numerical technique is that it is applicable to any arbitrary shapes and inhomogeneities. The limitation is that at very short wavelengths it would become quite expensive to implement. For very short wavelengths numerical solution of the appropriate singular integral equation governing the COD seems to be most effective [4, 21].
Appendix
The shape functions corresponding to the constant strain triangular elements, 6-node crack-tip singular elements and 7-node transition elements are given here for easy reference. The geometries of the elements are shown in Fig. 15. For the constant strain triangular elements the shape functions are [22] L1 = 2-~ {x2Y3 -- x3Y2 + X (Y2 -- Y3) + Y (X2 -- X3)}, L2 = 1
{x3yl - xly3 + x
1 La = ~
(Y3 - yl) + y (xl - x3)},
{xly2 - x2yl + x (Yl - Y2) + Y (x2 - Xl)},
where A is the area of the element and (xl, yl), (x2, y2), (x3, Ya) are the coordinates of the vertices. For the six-node and seven-node elements the shape functions are expressed in terms of the local (~, ~/) coordinates as follows. 6-node singular element: L1 = 0.5~:(~¢- 1),
L2 = 0.25(1 + ~)(1 - se)(se- ~ - 1),
L4 = 0.5(1-~:2)(1- r/),
Ls = 0.5(1 + s¢)(1- 7/2),
L3 = 0.25(1 + ~)(1 + r/)(~:+ r / _ 1), L6 = 0 . 5 ( 1 - s¢2)(1+ 7/);
S.K. Datta, A.H. Shah / Scattering by embedded cavities
282
6-NODE CRACK-TIP SINGULAR ELEMENT
(0 i)
2(I,-I)
T-NODE TRANSITION ELEMENT
(-I,-I) (O,-t)
4'-
I
c-',,
(l,-i)
5
,,o,
I (-U)
I (O,t) O,I)
CONSTANT STRAIN TRIANGULAR ELEMENT 2
Fig. 15. Geometries of the elements.
7 - n o d e transition element: L t = -0.25~:(1 - ~:)(1 - n),
L2=O.25(l+~)(1-n)(~-n-1),
L3 = 0.25(1 + ~)(1 + n ) ( ~ + n - 1), L5 = 0.5(1 - ~:2)(1- r/),
L4 = - 0 . 2 5 ~ ( 1 - ~ ) ( 1
L6 = 0.5(1 + ~ : ) ( ! - n2),
+ n),
L7 = 0.5(1 - ~:2)(1 + n).
Acknowledgment The work of Professor Shah was carried out when he was a visiting faculty associate at the University of Colorado. The authors are grateful for the support received from the National Science Foundation under the grant CME78-24179 and from the Natural Science and Engineering Research Council of Canada under the grant A-7988. The authors wish to thank Mr. Roy Swanson for help in the numerical computation.
References [1] S.K. Datta, "Diffraction of SH-waves by an edge crack", J. AppL Mech. 46, 101-106 (1979). [2] S.F. Stone, M.L. Ghosh and A.K. Mal, "Diffraction of antiplane shear waves by an edge crack", J. Appl. Mech. 47, 359-362 (1980). [3] D.A. Mendelsohn, J.D. Achenbach and L.M. Keer, "Scattering of elastic waves by a surface-breaking crack", Wave Motion 2, 277-292 (1980). [4] J.D. Achenbach and R.J. Brind, "Scattering of surface waves by a sub-surface crack", Z Sound and Vib. 76, 43-56 (1981). [5] S.K. Datta and N. EI-Akily, "Diffraction of elastic waves in a half-space. I: Integral representation and matched asymptotic expansions", in: Modern Problems in Elastic Wave Propagation, ed. J. Miklowitz and J.D. Achenbach, Wiley, New York (1978), 197-218. [6] A.H. Shah, K.C. Wong and S.K. Datta, "Diffraction of plane SH-waves in a half-space", Earthq. Engg. Sir. Dyn., to appear. [7] S.K. Datta, A.H. Shah and C.M. Fortunko, "Diffraction of medium and long wavelength SH-waves by edge cracks", J. Appl. Phys., to appear. [8] S.K. Datta and N. EI-Akily, "Diffraction of elastic waves by cylindrical cavity in a half-space", J. Acoust. Soc. Amer. 64, 1692-1699 (1978). [9] S.K. Datta, "Diffraction of SH-waves by an elliptic elastic cylinder", Int. ]. Solids Structures I0, 123-133 (1974).
S.K. Datta, A.H. Shah / Scattering by embedded cavities
283
[10] R.D. Gregory, "The propagation of waves in an elastic half-space containing a cylindrical cavity", Proc. Camb. Phil. Soc. 67, 689-710 (1970).
[11] V.R. Thiruvenkatachar and K. Viswanathan, "Dynamic response of an elastic half-space with cylindrical cavity to timedependent surface tractions over the boufidary of the cavity", J. Math. Mech. 14, 541-571 (1965). [12] N. Zitron and S.N. Karp, "Higher-order approximations in multiple scattering. I. Two-dimensional scalar case", J. Math. Phys. 2, 394-402 (1961). [13] N. El-Akily and S.K. Datta, "Response of a circular cylindrical shell to disturbances in a half-space", Earthq. Engg. Str. Dyn. 8, 469-477 (1980). [14] S. Sancar and Y.H. Pap, "Spectral analysis of elastic pulses backscattered from two cylindrical cavities in a solid. Part I", 3". Acoust. Soc. Amer. 69, 1591-1596 (1981). [15] O. C. Zienkiewicz, Finite Element Method, McGraw-Hill, London (1977). [16] K. Washizu, Variational Methods in Elasticity and Plasticity, Pergamon Press, New York (1968). [17] R. Courant and D. Hilbert, Methods o[ Mathematical Physics, Vol. 1, Wiley, New York (1953). [18] R.S. Barsoum, "On the use of isoparametric finite elements in linear fracture mechanics", Int. J. Num. Methods Engg. 10, 25-38 (1976). [19] R.S. Barsoum, "Triangular quarter-point elements as elastic and perfectly plastic crack tip elements", Int. J. Num. Methods Engg. 11, 85-98 (1977). [20] P.P. Lynn and A.R. Ingraffia, "Transition elements to be used with quarter-point crack-tip elements", Int. J. Num. Methods Engg. 12, 1031-1036 (1978). [21] A.K. Mal, "Diffraction of SH waves by a near surface crack", presented at the Air Force/DARPA Review of Progress in Quantitative NDE, University of Colorado, Boulder, August 2-7 (1981). [22] R.D. Cook, Concepts and Applications o[Finite Element Analysis, Wiley, New York (1974).