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Reflection and transmission of antiplane surface waves by a surface-breaking crack in a layered elastic solid
ELSEVIER
Wave Motion 20 (1994) 371-383
Reflection and transmission of antiplane surface waves by a surface-breaking crack in a layered elastic solid Y.C. Angel Department of Mechanical Engineering and Materials Science, Rice University, Houston, TX 77251-1892, USA Received 8 March 1994; revised 10 May 1994
Abstract The reflection and transmission of antiplane surface waves (Love waves) by a surface-breaking crack in a layered elastic solid is investigated. The crack is normal to the free surface, and breaks into the lower half-space solid. The formulation of the problem is reduced to a singular integral equation of the Cauchy type. In this equation, the unknown function, which is the slope of the crack-face displacement, is discontinuous at the interface between the two solids. It is shown that the magnitude of the discontinuity is related to the ratio of the shear moduli. A Gaussian numerical method is used to obtain the solution of the singular integral equation. At some distance from the plane of the crack, the wave motion is the superposition of a finite number of Lovewave modes. The amplitudes of these modes are readily evaluated in terms of the slope of the crack-face displacement. Curves are presented for the reflection coefficients corresponding to the first three modes and forthe transmission coefficient as functions of the dimensionless frequency.
1. Introduction The reflection and transmission of surface waves by cracks in homogeneous elastic solids has been investigated for various configurations. For example, there exist solutions for embedded cracks and surface-breaking cracks in two and three dimensions when the incident wave is a Rayleigh wave. The cracks can be oriented in any direction with respect to the free surface, and the solutions are obtained by using integral-transform methods or boundaryelement methods. An account of these investigations is available in Refs. [ 1] and [ 21. In cracked layered elastic solids, which can be used to model a large class of composite materials and also the earth crust, it is more difficult to obtain accurate analytical and numerical solutions for the reflection and transmission of surface waves. This is because the propagation of elastic waves becomes dispersive. One solution of this type was obtained by Keer and Luong [ 31. These authors consider the propagation of Love waves in a cracked layer of finite thickness bonded to a half-space; they calculate the stress-intensity factors and the ratio of the reflection coefficient to the transmission coefficient, taking into account only the first-order mode. In a later analysis by Angel [ 41, the higher-order reflection coefficients are calculated. Other solutions for layered solids have been obtained by Neerhoff [ 51 and Yang and Bogy [ 61. In both cases, the crack lies in the interface between the two solids. In this paper, we consider the reflection and transmission of antiplane surface waves (Love waves) by a surfacebreaking crack in a layered elastic solid. The crack is normal to the free surface, and breaks into the lower halfspace solid. We derive a singular integral equation for the slope of the crack-face displacement. We show that, at 0165-2125/94/$07.00 0 1994 Elsevier Science B.V. All rights reserved SSDIO165-2125(94)00035-2
372
Y.A. Angel/ Wave Motion 20 (1994) 371-383
the interface between the two solids, this slope is discontinuous, and the magnitude of the discontinuity is related to the ratio of the shear moduli. The method of derivation of the integral equation, which is based on the formulas (3.8) and (3.9)) is new and can be extended to the case of a surface-breaking crack that breaks into two or more elastic solids. In fact, no elastodynamic solution corresponding to a crack breaking through two or more solids appears to be available. The numerical solution of the singular integral equation is obtained by using the Gaussian method of Erdogan and Gupta [ 71. In our application of the method, the number of Gaussian points along the portion of the crack contained in the layer is equal to that along the portion of the crack contained in the lower half-space. The formulation of the problem is presented in Section 2. In Section 3 we derive a singular integral equation of the first kind for the slope of the crack-face displacement. In Section 4 we show that, at some distance from the plane of the crack, the wave motion is the superposition of a finite number of Love-wave modes. The numerical method of solution is discussed in Section 5. Finally, we discuss in Section 6 the numerical results for the reflection coefficients corresponding to the first three modes, and those for the transmission coefficient.
2. Formulation Consider a layered half-space made of two different materials perfectly bonded to each other, as shown in Fig. 1. Both materials are linearly elastic, homogeneous, and isotropic. The layered half-space contains a surface-breaking crack of depth a normal to the free surface, and the depth a is greater than the thickness h of the layer. The crack, which extends to infinity in the +x, directions, lies in the plane xi = 0. Let p, /_L,s, {p’, p’, s; ] denote the mass density, the shear modulus, and the slowness of transverse waves in the layer {in the half-space}, and assume that s+ < s,. One has $=pfp,
SF
=p'Ip'
.
A time-harmonic antiplane surface wave (Love wave) is incident on the crack. Let u,,, w, and s denote an amplitude factor, the frequency, and the slowness of the incident wave. The time factor exp( - iwt), which is common to all field variables in a steady-state regime, is omitted throughout this paper. We now define the following quantities m-p’lp, Co = uola ,
E=S;/ST h= hla,
)
l-j=sIsT)
a(u)=(l-z#‘2,
cK’(U) = (u2-
E2) 1’2 ,
ii= h/h,,
FT = ws,a ,
(2.1)
where AT = 29r/ ( osT) is the wavelength of transverse waves in the layer. Then, the displacement an incident Love wave can be written in the form
uit(x, y) = I.& exp( ifT 71x) cosr~Y4rl)l
{ COS[~&.X(~)]
7
O
exp[-5r(y-h)cr’(rl)l,
h
Love Wave
Fig. 1. Incidence of a Love wave in a layered half-space on a surface-breaking crack of depth a.
u’;” generated by
(2.2)
313
Y.A. Angel/ Wave Motion 20 (1994) 371-383
where u’i;(x, y) = ui;(ar, uy) /a )
x,=ax,
x2 =ay .
The value of 71lies in the interval [ e, 1) and is a real root of the frequency equation tan[27rGCY(77)]-mcu’(T))l(Y(?I) =o.
(2.3)
Eq. (2.3) can be solved by defining an auxiliary variable w such that w=4W( 1- 772)*‘2. Then, (2.3) becomes tan(mv/2)-rn[(l-e2)(4W)2-~2]1’2/W=0.
(2.4)
Eq. (2.4) has at least one real root, which lies in the interval (0, l), for all positive frequencies. The number of roots increases as the frequency increases. In fact, given the frequency (3, the total number of roots is the unique integer k such that k-1<2cj(l-e2)i’2
(2.5)
and the root of order 1 ( 1 G 1d k) lies in the interval [ 21- 2,21- 1). On the crack faces, the stress u?~ associated with the displacement field of Eq. (2.2) takes the form &CO, ay) =ij.G&77 cos[FTya(q)l cr’,“,(O,uy) =i~‘C&~
cos[QG~(~)]
,
O
(2.6)
exp[ -&(y-K)cx’(~~)]
,
h
1.
(2.7)
The linearity of the governing equations allows us to decompose the total field in the solid into the sum of the incident field in an untracked solid and the scattered field in the cracked solid. The scattered field is generated by crack-face tractions that are opposite to the tractions of (2.6) and (2.7). The scattered field is physically antisymmetric with respect to the plane xi = 0. Thus, the components u3 and ~23 are odd functions of x1, whereas gi3 is an even function of xi, and we need only consider the quarter-space region defined by x, > 0 and x2 & 0. The boundary conditions for the scattered problem are V23=O,
X1.0,
X2=0,
,&3=c;3,
u3=u;,
x,.0,
x2=h,
c~~=-(T~~,
CT;,=-UZ,
x,=0,
h
x,>,O, x,=h,
u;=o,
x, =0, x1=0,
(2.8a, b)
O
(2.8c, d)
a
(2.8e, f)
where the unprimed quantities refer to the layer, and the primed ones to the underlying half-space. The stress components 0’1” in (2.8d) and (2.8e) are defined, respectively, in (2.6) and (2.7). In the next section, we reduce the formulation of the problem to a singular integral equation of the first kind.
3. Singular integral equation For steady-state antiplane motions, the displacement equations of motion are [ 8, p. 2181 u,~,+u3,22+(@.f&~~3=0, O
+4,22
+(~;)~uj=O,
h
(3.1) (3.2)
Using integral transform methods as in Ref. [ 41, one can write general solutions of ( 3.1) and ( 3.2) (in the quarter space x1 2 0, x2 & 0) in the form
374
Y.A. Angel/
Wave Motion 20 (1994) 371-383
cc
(7a “*u3 =
I
m
[Be- PC&2 + Ce P(Rx2] sin( &,) dl+
0
I
EePPCnx’ COS(&~) d&,
0gx2
G/Z,
(3.3)
0
m (42)
l/Zul _ 3-
Ae - P’(ox2sin( &xi) d[+ I 0
I
De-acnX’cos(&2)
dt,
hgx2,
(3.4)
0
where p( # and p’( .!J are defined by PYR=5‘*-(+)*,
Im(P)
P’*(0=5*-(ws;)*,
90,
Im(P’> GO.
The quantities A, B, C, D, E are functions of 5 to be determined from the boundary conditions (3.3) and (3.4)) one finds that the displacement at x, = 0 is such that
(3.5) (2.8). From Eqs.
m
( 7r/2)“%3(o,
x2) =
Ecos(&,)
d5,
0~x2
(3.6)
D cos(&)
dt,
h
(3.7)
0 m
(QT/2)“2U;(O,
x2) =
J
0
Now we define a function b in the interval (0, h) and a function c in the interval (h, a) such that
5(7r/2)“*E=
I b(s)
sin(&)
ds+
5(n/2)i’*o=
1 c(s) sin(&)
ds,
(3.8)
II
0
i c(s) sin(&)
J
(3.9)
ds.
h
Since the functions b and c are defined in two different intervals, and are therefore independent of each other, it follows from (3.8) and (3.9) that E and D are, in general, independent. Substituting (3.8) into (3.6) and (3.9) into (3.7), interchanging the order of integration, and evaluating the infinite sine integrals, one finds that
u,(O,&)=
jb(s)ds+
jc(s)ds, h
X2
O
(3.10)
(3.11)
Eqs. (3.10) and (3.11) show that the displacement at xi = 0 is continuous in [ 0, a) for all integrable functions and c. In addition, the boundary condition (2.8f) is satisfied. According to (2.8b), the stress component ~23 must be continuous at xi = 0 and x2 = h. Thus, one has ~23(O, h-)
=ai,(O,
h+)
.
(3.12)
b
Y.A. Angel/Wave
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315
Differentiating (3.10) and (3.11) with respect to x2, and assuming that the functions b and c are bounded at x2 = h, one infers from Hooke’s law and ( 3.12) that /.f&h-)=/L’c(h+).
(3.13)
Since the shear moduli /_Land /.L’are not equal in general, it follows from (3.13) that the functions b and c are not equal at x2 = h. Thus, the crack-face displacement has a discontinuous slope at x2 = h. It follows from the displacement (3.3), Hooke’s law, and the boundary condition (2.8a) that (3.14)
B=C.
Next, we consider the boundary conditions (2.8b, c), which specify that the displacement and the stress are continuous along the interface x2 = h. These boundary conditions, together with (3.8), (3.9)) and (3.14)) yield a linear system of two equations for the two unknowns A and B. Using integral formulas Eq. (3.914) of Ref. [9, p. 4821 and Eq. (43) of Ref. [ 10, p. 1121, one can write A and B as a sum of two integrals. The first integral is evaluated in (0, h) and contains the function b; the second integral is in (h, a) and contains c. The five quantities A, B, C, D, E can thus be expressed in terms of the functions b and c. To determine b and c, we apply the boundary conditions (2.8d) and (2.8e). Using (3.3)) (3.4)) (3.8)) (3.9)) (3.14)) and the expressions of A and B in terms of b and c, we find that (2.8d) and (2.8e) yield two integral equations for the functions b and c. Invoking the result -
a
[I c(s) sin(ts>ds] COS(~X~> dt=
j
0
CCS$&+-&I*,
i i
2
h
h
(3.15)
2
(where the integral in the right-hand side is defined in the principal-value sense if h < 1x2 1< a), and an analogous result with c replaced by b, one can write these integral equations in the form 1
0
1 -+Y-Y
1 Y+Y
+G(xY)
1dy=iGgl(Y) ,
1 WY-6 0
+
1
WY-6
Y-Y
+G(Y,Y)
Y+Y
(3.16)
O
dy=i&g2(y),
i
(3.17)
The integrals in (3.16) and (3.17) are defined in the principal-value sense, H denotes the Heaviside step function, and e(r) =
The
Way) , o< y
(3.18)
gl(Y)=m;-77cos[&Y417)1
7
(3.19)
g2(y)=~~l7cos[~~~~(r))l
exp[-G(y-~)a’(q)l.
(3.20)
= 1,2), are defined by quantities S,, 7, h; zio,(Y,cy’are given in (2.1). The functions G,( y, y) , ( CY
m G,( x Y) = - 2s;
I
0
where
m G,~(Y, Y, u) du-G
I
0
G&Y, Y, u) du,
(3.21)
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Y.A. Angel/Wave
Motion 20 (1994) 371-383
G,,(y,y,u)=[l-y]sin(FTv)cos(j,yu),
G2,(y, y, u) =H(
y-h)[l-
GZ(X
Y, ~1 =u[~‘I(Y,
GAY,
Y, u> = @,(Y,
A(u) =mp’(u)[P,(h; p2(u)=u2--1,
+]
sin(FTv)
u)I&(Y, u)/A(u)
u> +P,(Y, uV%(x
u)lA(u)
u) +ll
cos(FTyu)
,
7
u) - l] ,
-j(u)[Pi(h;
p(U)=U24,
,
h(P)
GO,
Im(j’) GO,
(3.22)
and the functions B,, B2, PI, P2, P3, and P4 are given in the Appendix. The integral of Gcr2 in (3.21) is not a regular integral. In fact, the portion J of this integral over the interval 1] can be written in the form
[E,
1
F(% Y, u) L(u) c
I
J(Y,Y)=
where F is a function L(u) = iexp[
du
integrable
(3.23)
’ in [ E, 11, and =rncd(u)
-i&&z(u)]A(u)
cos[FTh;u(u)]
-a(u)
sin[&hcu(u)]
.
(3.24)
Comparing the equation L(u) = 0 with the frequency equation (2.3)) and recalling that FTh= 21r&j, we conclude that both equations have the same set of k simple roots, where the integer k is determined by Eq. (2.5). Hence, the integrandofEq. (3.23) has ksimplepoles TV, (Z=l, . . . . k), suchthat •<~I~
J(y, y)
=ir
k F(Y, Y, 71) + F(Y, Y, u) C L’(77*) L(u) c
I
1-1
where the integral is performed in the principal-value is
L’(d=s,Q,
cos[~~~~(~~)l/[(1--77:)a’(rlr)l
du
(3.25)
’
sense, and L’(u) is the derivative of L( u) . The value L’( ~7~)
,
(3.26)
where Q,=m(l-~~)+S~hLY’(~~)[1-(m~)~+(m~~)~-~~].
(3.27)
Observe that the pole 7~~ of order k is such that vk= E for all frequencies W that satisfy the relation 2W(l-e2)1’2= k - 1 with k > 2. These frequencies require a special treatment, which is not included in this paper.
4. Displacement
field
The displacement field is obtained in terms of the function e of (3.18) by substituting into Eqs. (3.3) and (3.4) the expressions (3.8)) (3.9)) (3.14)) and the appropriate expressions for A and B. In dimensionless form, one has
Y.A. Angel/Wave
ms3(-%Y) =
311
Motion 20 (1994) 371-383
e(y)Hl(y,
y, x) dy,
O
(4.1)
e(y)Wy,
y, x) dy,
h
(4.2)
0
n-4(x, Y) = 0
where x, =ax,
&(x, Y) = udw
x2=ay,
ay) /a,
40, y) = uXax, ay)la ,
m
ml ; Gr2( ‘y,y, u) sin(F-xu) du + 2
I
Hl(Y,Y94=
M( 7, y, u) e -*Po’)Xdu
,
I 0
0
m
H2( 77
Y, x) =
-1
I
~G~~(y,y,u)
sin(Srxu)du+2H(y-h)
M(y,y,u)
e-aB'c")xdu,
I 0
0
M( 7, y, u) = i sin(FT 3/u) cos(f-yu)
(4.3)
,
and the functions G,2 and G22 are defined in (3.22). The functions HI and H2 contain integrals of the type (3.23). Thus, the integrations should be interpreted as in (3.25). Eq. (3.25) allows us to decompose the displacements (4.1) and (4.2) into two parts: the first part oscillates, without decay, as a function of x; and the second part decays as rapidly as lx I- 1’2as 1x 1+ ~0.The oscillating part u 7 of the displacements (4.1) and (4.2) is k uT(x, y) =
C
U,
cos[&ya(q)l
exp(&qx)
,
(4.4)
O
I=1
uS(x,y) = i
Ul cos[f44~1)l
exp[ -My-h)a’(q,)l
exp(i&qlx)
,
ky ,
(4.5)
1=1
where 1 u,= e(y)&(y, L’( 771)0
(4.6)
vl) dy,
and L’( q) is defined by (3.26), whereas the function Br (-y, u) is given in the Appendix. The displacement field (4.4) and (4.5) is the superposition of k Love-wave modes of the type (2.2). At a large distance from the plane of the crack ( 1x1+ w), the wave motion in the layer is dominated by these k Love-wave modes. Therefore, it is meaningful to define the reflected and transmitted displacements uy and u:, respectively, by m&Y)=-u:(-xrY),
uT(x, Y) =$Xx,
x +Gk
Y) ,
(4.7) x>o
1
where zi? is defined by (2.2), and the antisymmetry with respect to the plane x1 = 0 has been used.
(4.8)
378
Y.A. Angel/Wave
Motion 20 (1994) 371-383
Here and in the rest of the paper, the incident field is assumed to be the first-order Love-wave mode, which exists for all frequencies. Thus, the parameter 77in (2.2) takes the value 77, of the first root of L( u) = 0, where L(U) is defined by (3.24). We may now define the first-order reflection coefficient R,, the transmission coefficient T,, and the higher-order reflection coefficients R, (l> 2) by R, = U,/u, ,
7-1= 1+ lJ,ll.& ,
R, = lJ,/&, .
(4.9)
In general, the amplitudes U, of (4.6) are complex numbers. Hence, the reflected and transmitted the incident wave both in phase and in modulus.
waves differ from
5. Method of solution The singular integral equations (3.16) and (3.17) are solved numerically by using the method of Erdogan and Gupta [ 71. First, we change the variables y and y into the variables 5 and z, respectively. The formulas for these changes are (O
y=hz
y=h5,
O
y=(l-h)z+h
r=(l-h)l+h,
(5.1)
(h
h
(5.2)
With (5.1), the interval of integration (0,h) in y becomes (0, 1) , and with (5.2) the interval of integration in y becomes also (0, 1) . Thus, (3.16) and (3.17) can now be written, respectively, in the form
1
+
1 (l_h)l+&_hz
I
1 + (1-h)l+h+hz
- +G,[(l-h)[+h,hz]
0
=iUog,(hz,)
,
O
(6, 1)
1
(l-h)d5
(5.3)
1, l-h (l-h)(l+z)+2h
+(l-h)G,[(l-h)[+h,
(1-h)z+h]
1 dl
1 +
I
e(&)G,[&,
(l-h)z+i]idl=iCog,[(l-h)z+h] ,
O
(5.4)
0
Recall that the function e( -y) has a square-root singularity at the crack tip ( y= 1). Also, it can be shown, by considering the limit of (3.16) as y + 0, that e vanishes at the crack mouth. Thus, one can define a function $ and a function 4 such that e(@) = (c1(5) 9
e[(l-h)L+hl=
-
N5) ~l_~2~112~ O
(5.5)
The functions $ and C$are bounded in the interval [ 0, 11, and in addition $( 0) = 0. The discontinuity (3.13) at h, together with (3.18) and the definition m = /.L’/F, implies that e(h-)
=me(h+)
.
(5.6)
Y.A. Angel/Wave Motion 20 (1994) 371-383
379
Thus, we infer from (5.5) and (5.6) that +(l)-mf#J(O)=O.
(5.7)
An equation similar to (5.6) is given in Ref. [ 12, Rq. (32), p. 2321 for the static antiplane problem of a finite crack crossing an interface. To obtain this result, Erdogan and Cook consider the limit of their integral equation near the interface. We have taken the limit of ( 3.16) as y approaches 6 from below and the limit of (3.17) as y approaches 6 from above. This procedure requires a careful examination of the terms in G, and Gz that become singular near y = hand y = h: After some lengthy calculations, we have found that the dominant terms near y = h of Eqs. (3.16) and (3.17)) respectively, are 1
K
fi
dy,
Sl(Y) = 1
/i
O
(5.8)
1
h
S,(Y) =
1.
(5.9)
Using (5.8), (5.9)) and the method of Muskhelishvili [ 131, one can show that the unknown function e( y) does not have a power singularity near y= h: On the other hand, if one assumes that the function e(y) is bounded near y = 6, then one infers from (5.8) and (5.9) that the discontinuity of e is given by (5.6). Thus, we can say that Eqs. (3.16) and (3.17) are consistent with the discontinuity that was obtained earlier in (3.13). We have also investigated the possible existence of a logarithmic singularity at the interface. In this analysis, we write that e has the form e(r)=&(r)
h&L-y),
O
(5.10)
e(y)=+d~)
Wy-61,
h
(5.11)
Substituting (5.10) and (5.11) into (5.8) and (5.9), and using the results of Gakhov [ 141 (p. 62), one finds that the bounded function +i and & of (5.10) and (5.11) must satisfy the condition (5.12)
&(K) = m&(h) .
We conclude that Eqs. (3.16) and (3.17) are also consistent with a logarithmic singularity. In the next section, however, we have performed the numerical calculations under the assumption that the function e is bounded at the interface and satisfies (5.6). The numerical results do not indicate that the function e tends to become infinite near the interface. Now we approximate Eqs. (5.3) and (5.4) by using (5.5) and the method of Erdogan and Gupta [7]. In this method, the Gaussian points tj and yr are given for an odd number II of quadrature points by rj=cos[a(2j--1)/(2n)],
y,=cos(mln),
r
r=l,2,
. . .. (n-1)/2.
(5.13)
Then, we have the following approximations
I -[1
e(h5) 0
1 l-yr
1 + 5+yr
1
dc= ;
(n--I)/2
C j=l
~(~)(l-f;)1’2
1 -+G-Y,
1 ri+yr
1 ’
(5.14)
380
Y.A. Angel/Wave
1
5-y, 1
I
+ [+y,+& 1
Motion
20 (1994)
371-383
1 7T (n-1)/2 d5=;
j;
+($I -+ 1
(5.15) 1
-h)I+hlWL
e[(l I 0
y,) d4’= fen-f’*
4(tjM$,
Y,) + E
dWK(O,
YA ,
(5.16)
j=1
where K represents any continuous function, h^is a real parameter, and F, = il evaluated by using the (n - 1) /2 Lagrangian polynomials Ak(x) [ 151. One has
[yr( 6 + y,)
1. The value +( 0) is
(n-1)/2 WV
=
C
+(04(O)
(5.17)
9
k=l
Ak(x) = ; (x-tl) D=(fk-fl)*.
. +~~X-tk-1~~X-fk+1~~~~~~-t(,-1)/2~
(5.19)
~~~k-tk-l~~fk-tk+l~~~~~~k-~(n-1)/2~~
Formulas similar to (5.14)-( 5.16) have been used to approximate that (5.3) and (5.4) yield the following system of equations (n-1)/2 C
all the terms of (5.3) and (5.4). Then, one finds
(n-1)/2 +($Fl($,
YJ +
C,
j=l
+($FAti,
YJ =iU,~g,(~Y,)
(5.20)
,
j=l
(n-1)/2 C
(5.18)
9
(n$4tj)K~(+,yr)+
1)/2 C
j=l
d(r,)K,(rj,y,)=ils,~g2[(l-~)yr+~l
.
(5.21)
j=l
In EIqs. (5.20) and (5.21) , the subscript r takes the values r = 1, 2, . . ., (n - 1) /2. Thus, these equations represent a linear system of (n - 1) equations for the (it - 1) complex-valued unknowns $( 9) and & $) . The functions Ki (i= 1,2, 3,4) can be easily obtained by using (5.3), (5.4) and (5.14)-(5.19). Observe that formulas (5.15) and (5.16) are written in terms of c$(0). This is because +( 0) does not vanish in general, and the Gaussian formulas of Erdogan and Gupta [7] can be applied in the interval [ - 1, 1] only to continuous functions. Thus, when the Gaussian formulas are applied in the interval [0, 11, one must include an additional term that contains +( 0). We now consider the discontinuity condition (5.7). Using the Lagrangian polynomials (5.18)) one infers from (5.7) that (n-
(n-l)/2 C j=l
Ilr(t,)Aj(l)-m
1)/2 C
+(Q>Aj(O)=O
*
(5.22)
j=l
The system (5.20) and (5.21) is solved numerically, subject to the constraint (5.22). In order to use the constraint, we eliminate the equation of (5.21) that corresponds to r = 1, and we replace it with (5.22). Thus, we eliminate the equation that corresponds to the position yr closest to the crack tip. The result of this replacement is that we solve a system of (II - 1) equations for the (n - 1) complex-valued unknowns #( 5) and c$(5). It suffices to prescribe the values of five dimensionless parameters: E= s; Is=, m = p’/p, 6= h/a, tie = u,/a, W= h/h,; where AT =2?r/(ws,) is the wavelength of transverse waves in the layer. All other dimensionless parameters can be written in terms of these five basic parameters. The integral of Eq. (4.6) for the amplitude of the far-field displacements can also be evaluated by using the method of Erdogan and Gupta [ 71.
Y.A.Angel/Wave Motion 20 (1994) 371-383
381
6. Discussion of numerical results
Numerical results are presented in Figs. 2-5. These results have been computed for a choice of parameter values such that E= 0.8, m = 1.75, and I= 0.5 and 0.9. The dimensionless frequency Wvaries in the range [ 0,3], and the increment is 0.01. Thus, the curves of Figs. 2-5 are the result of 300 computations. For each frequency, we have solved the linear system (5.20)-( 5.22) with II = 15. The solution of the system has been substituted into (4.6) and (4.9) in order to obtain the coefficients R,, T,, R2, and R3. The values of E and m that we have chosen correspond approximately to those of the uniform and single-layered earth model of Neerhoff [ 51. The choice of n = 15 was made after solving the system (5.20)-( 5.22) for W= 0.5 and n = 15,21, and 31. It was found that the real parts and the imaginary parts of the unknown functions I++ and 4 take values that do not change appreciably for these three values of n. Thus, we conclude that the value n = 15 yields convergent numerical results. For n = 15, Eqs. (5.20)-( 5.22) represent a system of 28 real-valued equations for 28 real-valued unknowns. The (5.13). 28 unknowns are the seven values of Re( $) , Re ( 4) , Im( +) , and Im( 4) at ti , fz, . . .. t,,where$isdefinedin With E= 0.8, we infer from Eq. (2.5) that the Love-wave mode of order k emerges at a frequency c;i= 5( k - 1) / 6. In the frequency range of interest, the relevant values are W=O.8333, 1.667, and 2.5, which correspond to the emergence of the second, third, and fourth modes, respectively. Fig. 2 {3) shows the modulus of the coefficient R, { Tl} versus the frequency for two values &= 0.5 and 0.9. When i= 0.5, the length of the crack is equal to two times the layer thickness h. When i= 0.9, the crack length a is a = h/0.9, which exceeds by a small amount the layer thickness. In the limit as 0 approaches zero, the coefficient R, vanishes, whereas T, is equal to unity. For frequencies greater than 1.5, 1RI 120.996 and 1T, 1~0.004. Thus, the crack acts as a nearly perfect reflector for all frequencies 0 > 1.5. Observe that the solid curve (h= 0.5) in Fig. 2 lies above the dashed curve (h= 0.9) for all frequencies. Thus, as expected, the reflection increases as the length of the crack increases. Fig. 4 (5) shows the moduli of the coefficient R2 {R3] versus the frequency. The peak values of I R2 I are 0.0273 at&i=099 (&=0.5),and0.0506at(5=1.04 (i=0.9).Thepeakvaluesof lR31 areO.O367at(3=1.76 (h=OS), 1.2
IT, I
Fig. 2. Modulus of the reflection coefficient of order one versus the dimensionless frequency; incident Love-wave mode of order one; s$/ s,=O.8, &I/L= 1.75, hla=0.9 (dashed) and 0.5 (solid). Fig. 3. Modulus of the transmission coefficient versus the dimensionless frequency; incident Love-wave mode of order one; s+/sT= 0.8, p’/ /~=1.75,h/a=0.9 (dashed) and0.5 (solid).
382
Y.A. Angel/Wave
0.06
I
IR,I
Motion 20 (1994) 371-383
0.06
IRJ
O-
0
1.5
0
3
0
-i _1 0
1.5
0
3
Fig. 4. Modulus of the reflection coefficient of order two versus the dimensionless frequency; incident Love-wave mode of order one; s;/ s,=O.S, p’/k= 1.75, hla=0.9 (dashed) and 0.5 (solid). Fig. 5. Modulus of the reflection coefficient of order three versus the dimensionless frequency; incident Love-wave mode of order one; s;/ s,=O.S, p’/p= 1.75, h/a=0.9 (dashed) and 0.5 (solid).
and 0.0164 at W= 1.78 (h= 0.9). Thus, the coefficients R2 and R3 are much smaller than the coefficient RI. In fact, R, and R3 are almost negligible in comparison with R,. In Ref. [ 41, where the crack is contained entirely in the layer, several sharp spikes were observed in R,, T,, R,, and RJ. Here, however, the results of Figs. 2-5 do not show the presence of sharp spikes. The results of this work suggest that the sharp spikes of Ref. [4] are caused by resonance effects that take place inside the layer. Indeed, in Ref. [4] the crack tip lies inside the layer and multiple wave reflections take place between the crack faces and the boundaries of the layer. When the crack terminates exactly at the interface (a =h), one infers from (5.8) and from the method of Muskhelishvili [ 131 that the function e( 7) has a power singularity at the crack tip ( y= a = h) with a real-valued exponent j3. The value of p is given by the equation cos(p7r)=-(1-m)l(l+m).
(6.1)
Eq. (6.1) is identical to Eq. (20b) of Erdogan and Cook [ 12, p. 2301. Numerical results for the case of a crack terminating at the interface can be obtained by using Gauss-Jacobi integration formulas [ 161. The behavior of the numerical solution near the emergence frequencies of the Love-wave modes (G= 0.8333 and 1.667) shows some signs of numerical instability. Indeed, one can see in Figs. 4 and 5 that R2 and R, vary noticeably for small frequency changes near the emergence frequencies.
Acknowledgements
This work has been supported by NSF Grant MSM 900039P through an allocation of service units on the CRAY YMP computer of the Pittsburgh Supercomputing Center. The author would like to acknowledge the contribution of Y.K. Koba who has generated the numerical results of this paper.
383
Y.A. Angel/ Wave Motion 20 (1994) 371-383
Appendix
The functions B1 and B2 of Eq. (3.22) are B,(y, u)= -mu+
[I-Pl(y,
+2m~H(y-h‘)[l-P,(y,u)l+ &(~u)=-
u)ll
u>-sgn(i-y)[l-P,(y,
;WA-Wy,u)l,
(A.11
$[Pl(L,u)-ll[l-P,(y,u)-sgn(h-y)[l-P,(y,u)]]
+u- p$ Mh; + ;
u) - VW--6[2-P,h
u) -P~(Y, u)l
Mh; u>+ 11[P,(Y, u) -P1(y, u)l
-m;U’,Vi u) +W-W-6[h(y,
(A-2)
u) -P~(Y, u)l ,
where sgn denotes the sign function, and P~(Y, u) =exp[ -G(K+r)P(u)l
PAY, u) =ew[ -fd~+y)P’(u)l j”(u) = u*- 1 ,
Pdx
,
,
p!‘(u) =u*-E2)
u) =exp[ -G
Ii-yIP(
,
~4(~,~)=exp[-~I~-yJP’(u)l, InG)GO,
Im(P’) 90.
(A.3)
References [l] J.D. Achenbach, Y.C. Angel and W. Lin, “Scattering from surface-breaking and near-surface cracks”, in: G.C. Johnson, ed., Waoe Propagation in Homogeneous Media and Ultrasonic Nondestructive Evaluation, AMD, Vol. 62, American Society of Mechanical Engineers, New York ( 1984). [2] J.D. Achenbach, “From ultrasonics to failure prevention”, in: SK. Datta, J.D. Achenbach and Y.S. Rajapakse, eds., Elastic Waves and Ultrasonic Non-destructive Evaluation, North-Holland, Amsterdam ( 1990). [ 31 L.M. Keer and W.C. Luong, “Diffraction of waves and stress intensity factors in a cracked layered composite”, J. Appl. Mech. 56,16811686 (1974). [4] Y.C. Angel, “Scattering of Love-waves by a surface-breaking crack”, J. Appl. Mech 53.587-592 ( 1986).
[5] F.L. Neerhoff, “DitTraction of Love waves by a stress-free. crack of finite width in the plane interface of a layered composite”, Appl.Sci. Res. 35,265-315 (1979). [6] H.J. Yang and D.B. Bogy, “Elastic wave scattering from an interface crack in a layered half space”, J. Appl. Mech. 52,42-50 (1985). [ 71 F. Erdogan and G.D. Gupta, “On the numerical solution of singular integral equations”, Quart. Appl. Math. 30.525-534 ( 1972). [ 81 J.D. Achenbach, Wave Propagation in Elastic Solids, North-Holland, Amsterdam ( 1973). [9] LS. Gradshteyn and LM. Ryzhik, Table oflntegrals Series and Products, Academic Press, New York (1965). [lo] A. Erdelyi, Tables oflntegral Transforms, Vol. 1, McGraw-Hill, New York ( 1954). [ 111 H. Lamb, “On the propagation of tremors over the surface of an elastic solid”, Philos. Trans. R. Sot. London, Series A, 203,142 ( 1904). [ 121 F. Erdogan and T.S. Cook, “Antiplane shear crack terminating at and going througha bimaterialinterface”, Int. J. Fracture 10.227-240 (1974). [ 131 N.I. Muskhelishvili, Singular Integral Equations, Noordhoff, Groningen (1977). [ 141 F.D. Gakhov, Bounaby Value Problems, Pergamon, New York (1966). [ IS] F.B. Hi&brand, Introduction to Numerical Analysis, 2nd edition, McGraw-Hill, New York ( 1974). [ 161 F. Erdogan, Treatise on Continuum Physics, A.C. Eringen, ed., Vol. II, Part 111.3,Academic Press, New York ( 1975).
Wave Motion 20 (1994) 371-383
Reflection and transmission of antiplane surface waves by a surface-breaking crack in a layered elastic solid Y.C. Angel Department of Mechanical Engineering and Materials Science, Rice University, Houston, TX 77251-1892, USA Received 8 March 1994; revised 10 May 1994
Abstract The reflection and transmission of antiplane surface waves (Love waves) by a surface-breaking crack in a layered elastic solid is investigated. The crack is normal to the free surface, and breaks into the lower half-space solid. The formulation of the problem is reduced to a singular integral equation of the Cauchy type. In this equation, the unknown function, which is the slope of the crack-face displacement, is discontinuous at the interface between the two solids. It is shown that the magnitude of the discontinuity is related to the ratio of the shear moduli. A Gaussian numerical method is used to obtain the solution of the singular integral equation. At some distance from the plane of the crack, the wave motion is the superposition of a finite number of Lovewave modes. The amplitudes of these modes are readily evaluated in terms of the slope of the crack-face displacement. Curves are presented for the reflection coefficients corresponding to the first three modes and forthe transmission coefficient as functions of the dimensionless frequency.
1. Introduction The reflection and transmission of surface waves by cracks in homogeneous elastic solids has been investigated for various configurations. For example, there exist solutions for embedded cracks and surface-breaking cracks in two and three dimensions when the incident wave is a Rayleigh wave. The cracks can be oriented in any direction with respect to the free surface, and the solutions are obtained by using integral-transform methods or boundaryelement methods. An account of these investigations is available in Refs. [ 1] and [ 21. In cracked layered elastic solids, which can be used to model a large class of composite materials and also the earth crust, it is more difficult to obtain accurate analytical and numerical solutions for the reflection and transmission of surface waves. This is because the propagation of elastic waves becomes dispersive. One solution of this type was obtained by Keer and Luong [ 31. These authors consider the propagation of Love waves in a cracked layer of finite thickness bonded to a half-space; they calculate the stress-intensity factors and the ratio of the reflection coefficient to the transmission coefficient, taking into account only the first-order mode. In a later analysis by Angel [ 41, the higher-order reflection coefficients are calculated. Other solutions for layered solids have been obtained by Neerhoff [ 51 and Yang and Bogy [ 61. In both cases, the crack lies in the interface between the two solids. In this paper, we consider the reflection and transmission of antiplane surface waves (Love waves) by a surfacebreaking crack in a layered elastic solid. The crack is normal to the free surface, and breaks into the lower halfspace solid. We derive a singular integral equation for the slope of the crack-face displacement. We show that, at 0165-2125/94/$07.00 0 1994 Elsevier Science B.V. All rights reserved SSDIO165-2125(94)00035-2
372
Y.A. Angel/ Wave Motion 20 (1994) 371-383
the interface between the two solids, this slope is discontinuous, and the magnitude of the discontinuity is related to the ratio of the shear moduli. The method of derivation of the integral equation, which is based on the formulas (3.8) and (3.9)) is new and can be extended to the case of a surface-breaking crack that breaks into two or more elastic solids. In fact, no elastodynamic solution corresponding to a crack breaking through two or more solids appears to be available. The numerical solution of the singular integral equation is obtained by using the Gaussian method of Erdogan and Gupta [ 71. In our application of the method, the number of Gaussian points along the portion of the crack contained in the layer is equal to that along the portion of the crack contained in the lower half-space. The formulation of the problem is presented in Section 2. In Section 3 we derive a singular integral equation of the first kind for the slope of the crack-face displacement. In Section 4 we show that, at some distance from the plane of the crack, the wave motion is the superposition of a finite number of Love-wave modes. The numerical method of solution is discussed in Section 5. Finally, we discuss in Section 6 the numerical results for the reflection coefficients corresponding to the first three modes, and those for the transmission coefficient.
2. Formulation Consider a layered half-space made of two different materials perfectly bonded to each other, as shown in Fig. 1. Both materials are linearly elastic, homogeneous, and isotropic. The layered half-space contains a surface-breaking crack of depth a normal to the free surface, and the depth a is greater than the thickness h of the layer. The crack, which extends to infinity in the +x, directions, lies in the plane xi = 0. Let p, /_L,s, {p’, p’, s; ] denote the mass density, the shear modulus, and the slowness of transverse waves in the layer {in the half-space}, and assume that s+ < s,. One has $=pfp,
SF
=p'Ip'
.
A time-harmonic antiplane surface wave (Love wave) is incident on the crack. Let u,,, w, and s denote an amplitude factor, the frequency, and the slowness of the incident wave. The time factor exp( - iwt), which is common to all field variables in a steady-state regime, is omitted throughout this paper. We now define the following quantities m-p’lp, Co = uola ,
E=S;/ST h= hla,
)
l-j=sIsT)
a(u)=(l-z#‘2,
cK’(U) = (u2-
E2) 1’2 ,
ii= h/h,,
FT = ws,a ,
(2.1)
where AT = 29r/ ( osT) is the wavelength of transverse waves in the layer. Then, the displacement an incident Love wave can be written in the form
uit(x, y) = I.& exp( ifT 71x) cosr~Y4rl)l
{ COS[~&.X(~)]
7
O
exp[-5r(y-h)cr’(rl)l,
h
Love Wave
Fig. 1. Incidence of a Love wave in a layered half-space on a surface-breaking crack of depth a.
u’;” generated by
(2.2)
313
Y.A. Angel/ Wave Motion 20 (1994) 371-383
where u’i;(x, y) = ui;(ar, uy) /a )
x,=ax,
x2 =ay .
The value of 71lies in the interval [ e, 1) and is a real root of the frequency equation tan[27rGCY(77)]-mcu’(T))l(Y(?I) =o.
(2.3)
Eq. (2.3) can be solved by defining an auxiliary variable w such that w=4W( 1- 772)*‘2. Then, (2.3) becomes tan(mv/2)-rn[(l-e2)(4W)2-~2]1’2/W=0.
(2.4)
Eq. (2.4) has at least one real root, which lies in the interval (0, l), for all positive frequencies. The number of roots increases as the frequency increases. In fact, given the frequency (3, the total number of roots is the unique integer k such that k-1<2cj(l-e2)i’2
(2.5)
and the root of order 1 ( 1 G 1d k) lies in the interval [ 21- 2,21- 1). On the crack faces, the stress u?~ associated with the displacement field of Eq. (2.2) takes the form &CO, ay) =ij.G&77 cos[FTya(q)l cr’,“,(O,uy) =i~‘C&~
cos[QG~(~)]
,
O
(2.6)
exp[ -&(y-K)cx’(~~)]
,
h
1.
(2.7)
The linearity of the governing equations allows us to decompose the total field in the solid into the sum of the incident field in an untracked solid and the scattered field in the cracked solid. The scattered field is generated by crack-face tractions that are opposite to the tractions of (2.6) and (2.7). The scattered field is physically antisymmetric with respect to the plane xi = 0. Thus, the components u3 and ~23 are odd functions of x1, whereas gi3 is an even function of xi, and we need only consider the quarter-space region defined by x, > 0 and x2 & 0. The boundary conditions for the scattered problem are V23=O,
X1.0,
X2=0,
,&3=c;3,
u3=u;,
x,.0,
x2=h,
c~~=-(T~~,
CT;,=-UZ,
x,=0,
h
x,>,O, x,=h,
u;=o,
x, =0, x1=0,
(2.8a, b)
O
(2.8c, d)
a
(2.8e, f)
where the unprimed quantities refer to the layer, and the primed ones to the underlying half-space. The stress components 0’1” in (2.8d) and (2.8e) are defined, respectively, in (2.6) and (2.7). In the next section, we reduce the formulation of the problem to a singular integral equation of the first kind.
3. Singular integral equation For steady-state antiplane motions, the displacement equations of motion are [ 8, p. 2181 u,~,+u3,22+(@.f&~~3=0, O
+4,22
+(~;)~uj=O,
h
(3.1) (3.2)
Using integral transform methods as in Ref. [ 41, one can write general solutions of ( 3.1) and ( 3.2) (in the quarter space x1 2 0, x2 & 0) in the form
374
Y.A. Angel/
Wave Motion 20 (1994) 371-383
cc
(7a “*u3 =
I
m
[Be- PC&2 + Ce P(Rx2] sin( &,) dl+
0
I
EePPCnx’ COS(&~) d&,
0gx2
G/Z,
(3.3)
0
m (42)
l/Zul _ 3-
Ae - P’(ox2sin( &xi) d[+ I 0
I
De-acnX’cos(&2)
dt,
hgx2,
(3.4)
0
where p( # and p’( .!J are defined by PYR=5‘*-(+)*,
Im(P)
P’*(0=5*-(ws;)*,
90,
Im(P’> GO.
The quantities A, B, C, D, E are functions of 5 to be determined from the boundary conditions (3.3) and (3.4)) one finds that the displacement at x, = 0 is such that
(3.5) (2.8). From Eqs.
m
( 7r/2)“%3(o,
x2) =
Ecos(&,)
d5,
0~x2
(3.6)
D cos(&)
dt,
h
(3.7)
0 m
(QT/2)“2U;(O,
x2) =
J
0
Now we define a function b in the interval (0, h) and a function c in the interval (h, a) such that
5(7r/2)“*E=
I b(s)
sin(&)
ds+
5(n/2)i’*o=
1 c(s) sin(&)
ds,
(3.8)
II
0
i c(s) sin(&)
J
(3.9)
ds.
h
Since the functions b and c are defined in two different intervals, and are therefore independent of each other, it follows from (3.8) and (3.9) that E and D are, in general, independent. Substituting (3.8) into (3.6) and (3.9) into (3.7), interchanging the order of integration, and evaluating the infinite sine integrals, one finds that
u,(O,&)=
jb(s)ds+
jc(s)ds, h
X2
O
(3.10)
(3.11)
Eqs. (3.10) and (3.11) show that the displacement at xi = 0 is continuous in [ 0, a) for all integrable functions and c. In addition, the boundary condition (2.8f) is satisfied. According to (2.8b), the stress component ~23 must be continuous at xi = 0 and x2 = h. Thus, one has ~23(O, h-)
=ai,(O,
h+)
.
(3.12)
b
Y.A. Angel/Wave
Motion20(1994)371-383
315
Differentiating (3.10) and (3.11) with respect to x2, and assuming that the functions b and c are bounded at x2 = h, one infers from Hooke’s law and ( 3.12) that /.f&h-)=/L’c(h+).
(3.13)
Since the shear moduli /_Land /.L’are not equal in general, it follows from (3.13) that the functions b and c are not equal at x2 = h. Thus, the crack-face displacement has a discontinuous slope at x2 = h. It follows from the displacement (3.3), Hooke’s law, and the boundary condition (2.8a) that (3.14)
B=C.
Next, we consider the boundary conditions (2.8b, c), which specify that the displacement and the stress are continuous along the interface x2 = h. These boundary conditions, together with (3.8), (3.9)) and (3.14)) yield a linear system of two equations for the two unknowns A and B. Using integral formulas Eq. (3.914) of Ref. [9, p. 4821 and Eq. (43) of Ref. [ 10, p. 1121, one can write A and B as a sum of two integrals. The first integral is evaluated in (0, h) and contains the function b; the second integral is in (h, a) and contains c. The five quantities A, B, C, D, E can thus be expressed in terms of the functions b and c. To determine b and c, we apply the boundary conditions (2.8d) and (2.8e). Using (3.3)) (3.4)) (3.8)) (3.9)) (3.14)) and the expressions of A and B in terms of b and c, we find that (2.8d) and (2.8e) yield two integral equations for the functions b and c. Invoking the result -
a
[I c(s) sin(ts>ds] COS(~X~> dt=
j
0
CCS$&+-&I*,
i i
2
h
h
(3.15)
2
(where the integral in the right-hand side is defined in the principal-value sense if h < 1x2 1< a), and an analogous result with c replaced by b, one can write these integral equations in the form 1
0
1 -+Y-Y
1 Y+Y
+G(xY)
1dy=iGgl(Y) ,
1 WY-6 0
+
1
WY-6
Y-Y
+G(Y,Y)
Y+Y
(3.16)
O
dy=i&g2(y),
i
(3.17)
The integrals in (3.16) and (3.17) are defined in the principal-value sense, H denotes the Heaviside step function, and e(r) =
The
Way) , o< y
(3.18)
gl(Y)=m;-77cos[&Y417)1
7
(3.19)
g2(y)=~~l7cos[~~~~(r))l
exp[-G(y-~)a’(q)l.
(3.20)
= 1,2), are defined by quantities S,, 7, h; zio,(Y,cy’are given in (2.1). The functions G,( y, y) , ( CY
m G,( x Y) = - 2s;
I
0
where
m G,~(Y, Y, u) du-G
I
0
G&Y, Y, u) du,
(3.21)
316
Y.A. Angel/Wave
Motion 20 (1994) 371-383
G,,(y,y,u)=[l-y]sin(FTv)cos(j,yu),
G2,(y, y, u) =H(
y-h)[l-
GZ(X
Y, ~1 =u[~‘I(Y,
GAY,
Y, u> = @,(Y,
A(u) =mp’(u)[P,(h; p2(u)=u2--1,
+]
sin(FTv)
u)I&(Y, u)/A(u)
u> +P,(Y, uV%(x
u)lA(u)
u) +ll
cos(FTyu)
,
7
u) - l] ,
-j(u)[Pi(h;
p(U)=U24,
,
h(P)
GO,
Im(j’) GO,
(3.22)
and the functions B,, B2, PI, P2, P3, and P4 are given in the Appendix. The integral of Gcr2 in (3.21) is not a regular integral. In fact, the portion J of this integral over the interval 1] can be written in the form
[E,
1
F(% Y, u) L(u) c
I
J(Y,Y)=
where F is a function L(u) = iexp[
du
integrable
(3.23)
’ in [ E, 11, and =rncd(u)
-i&&z(u)]A(u)
cos[FTh;u(u)]
-a(u)
sin[&hcu(u)]
.
(3.24)
Comparing the equation L(u) = 0 with the frequency equation (2.3)) and recalling that FTh= 21r&j, we conclude that both equations have the same set of k simple roots, where the integer k is determined by Eq. (2.5). Hence, the integrandofEq. (3.23) has ksimplepoles TV, (Z=l, . . . . k), suchthat •<~I~
J(y, y)
=ir
k F(Y, Y, 71) + F(Y, Y, u) C L’(77*) L(u) c
I
1-1
where the integral is performed in the principal-value is
L’(d=s,Q,
cos[~~~~(~~)l/[(1--77:)a’(rlr)l
du
(3.25)
’
sense, and L’(u) is the derivative of L( u) . The value L’( ~7~)
,
(3.26)
where Q,=m(l-~~)+S~hLY’(~~)[1-(m~)~+(m~~)~-~~].
(3.27)
Observe that the pole 7~~ of order k is such that vk= E for all frequencies W that satisfy the relation 2W(l-e2)1’2= k - 1 with k > 2. These frequencies require a special treatment, which is not included in this paper.
4. Displacement
field
The displacement field is obtained in terms of the function e of (3.18) by substituting into Eqs. (3.3) and (3.4) the expressions (3.8)) (3.9)) (3.14)) and the appropriate expressions for A and B. In dimensionless form, one has
Y.A. Angel/Wave
ms3(-%Y) =
311
Motion 20 (1994) 371-383
e(y)Hl(y,
y, x) dy,
O
(4.1)
e(y)Wy,
y, x) dy,
h
(4.2)
0
n-4(x, Y) = 0
where x, =ax,
&(x, Y) = udw
x2=ay,
ay) /a,
40, y) = uXax, ay)la ,
m
ml ; Gr2( ‘y,y, u) sin(F-xu) du + 2
I
Hl(Y,Y94=
M( 7, y, u) e -*Po’)Xdu
,
I 0
0
m
H2( 77
Y, x) =
-1
I
~G~~(y,y,u)
sin(Srxu)du+2H(y-h)
M(y,y,u)
e-aB'c")xdu,
I 0
0
M( 7, y, u) = i sin(FT 3/u) cos(f-yu)
(4.3)
,
and the functions G,2 and G22 are defined in (3.22). The functions HI and H2 contain integrals of the type (3.23). Thus, the integrations should be interpreted as in (3.25). Eq. (3.25) allows us to decompose the displacements (4.1) and (4.2) into two parts: the first part oscillates, without decay, as a function of x; and the second part decays as rapidly as lx I- 1’2as 1x 1+ ~0.The oscillating part u 7 of the displacements (4.1) and (4.2) is k uT(x, y) =
C
U,
cos[&ya(q)l
exp(&qx)
,
(4.4)
O
I=1
uS(x,y) = i
Ul cos[f44~1)l
exp[ -My-h)a’(q,)l
exp(i&qlx)
,
ky ,
(4.5)
1=1
where 1 u,= e(y)&(y, L’( 771)0
(4.6)
vl) dy,
and L’( q) is defined by (3.26), whereas the function Br (-y, u) is given in the Appendix. The displacement field (4.4) and (4.5) is the superposition of k Love-wave modes of the type (2.2). At a large distance from the plane of the crack ( 1x1+ w), the wave motion in the layer is dominated by these k Love-wave modes. Therefore, it is meaningful to define the reflected and transmitted displacements uy and u:, respectively, by m&Y)=-u:(-xrY),
uT(x, Y) =$Xx,
x
Y) ,
(4.7) x>o
1
where zi? is defined by (2.2), and the antisymmetry with respect to the plane x1 = 0 has been used.
(4.8)
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Y.A. Angel/Wave
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Here and in the rest of the paper, the incident field is assumed to be the first-order Love-wave mode, which exists for all frequencies. Thus, the parameter 77in (2.2) takes the value 77, of the first root of L( u) = 0, where L(U) is defined by (3.24). We may now define the first-order reflection coefficient R,, the transmission coefficient T,, and the higher-order reflection coefficients R, (l> 2) by R, = U,/u, ,
7-1= 1+ lJ,ll.& ,
R, = lJ,/&, .
(4.9)
In general, the amplitudes U, of (4.6) are complex numbers. Hence, the reflected and transmitted the incident wave both in phase and in modulus.
waves differ from
5. Method of solution The singular integral equations (3.16) and (3.17) are solved numerically by using the method of Erdogan and Gupta [ 71. First, we change the variables y and y into the variables 5 and z, respectively. The formulas for these changes are (O
y=hz
y=h5,
O
y=(l-h)z+h
r=(l-h)l+h,
(5.1)
(h
h
(5.2)
With (5.1), the interval of integration (0,h) in y becomes (0, 1) , and with (5.2) the interval of integration in y becomes also (0, 1) . Thus, (3.16) and (3.17) can now be written, respectively, in the form
1
+
1 (l_h)l+&_hz
I
1 + (1-h)l+h+hz
- +G,[(l-h)[+h,hz]
0
=iUog,(hz,)
,
O
(6, 1)
1
(l-h)d5
(5.3)
1, l-h (l-h)(l+z)+2h
+(l-h)G,[(l-h)[+h,
(1-h)z+h]
1 dl
1 +
I
e(&)G,[&,
(l-h)z+i]idl=iCog,[(l-h)z+h] ,
O
(5.4)
0
Recall that the function e( -y) has a square-root singularity at the crack tip ( y= 1). Also, it can be shown, by considering the limit of (3.16) as y + 0, that e vanishes at the crack mouth. Thus, one can define a function $ and a function 4 such that e(@) = (c1(5) 9
e[(l-h)L+hl=
-
N5) ~l_~2~112~ O
(5.5)
The functions $ and C$are bounded in the interval [ 0, 11, and in addition $( 0) = 0. The discontinuity (3.13) at h, together with (3.18) and the definition m = /.L’/F, implies that e(h-)
=me(h+)
.
(5.6)
Y.A. Angel/Wave Motion 20 (1994) 371-383
379
Thus, we infer from (5.5) and (5.6) that +(l)-mf#J(O)=O.
(5.7)
An equation similar to (5.6) is given in Ref. [ 12, Rq. (32), p. 2321 for the static antiplane problem of a finite crack crossing an interface. To obtain this result, Erdogan and Cook consider the limit of their integral equation near the interface. We have taken the limit of ( 3.16) as y approaches 6 from below and the limit of (3.17) as y approaches 6 from above. This procedure requires a careful examination of the terms in G, and Gz that become singular near y = hand y = h: After some lengthy calculations, we have found that the dominant terms near y = h of Eqs. (3.16) and (3.17)) respectively, are 1
K
fi
dy,
Sl(Y) = 1
/i
O
(5.8)
1
h
S,(Y) =
1.
(5.9)
Using (5.8), (5.9)) and the method of Muskhelishvili [ 131, one can show that the unknown function e( y) does not have a power singularity near y= h: On the other hand, if one assumes that the function e(y) is bounded near y = 6, then one infers from (5.8) and (5.9) that the discontinuity of e is given by (5.6). Thus, we can say that Eqs. (3.16) and (3.17) are consistent with the discontinuity that was obtained earlier in (3.13). We have also investigated the possible existence of a logarithmic singularity at the interface. In this analysis, we write that e has the form e(r)=&(r)
h&L-y),
O
(5.10)
e(y)=+d~)
Wy-61,
h
(5.11)
Substituting (5.10) and (5.11) into (5.8) and (5.9), and using the results of Gakhov [ 141 (p. 62), one finds that the bounded function +i and & of (5.10) and (5.11) must satisfy the condition (5.12)
&(K) = m&(h) .
We conclude that Eqs. (3.16) and (3.17) are also consistent with a logarithmic singularity. In the next section, however, we have performed the numerical calculations under the assumption that the function e is bounded at the interface and satisfies (5.6). The numerical results do not indicate that the function e tends to become infinite near the interface. Now we approximate Eqs. (5.3) and (5.4) by using (5.5) and the method of Erdogan and Gupta [7]. In this method, the Gaussian points tj and yr are given for an odd number II of quadrature points by rj=cos[a(2j--1)/(2n)],
y,=cos(mln),
r
r=l,2,
. . .. (n-1)/2.
(5.13)
Then, we have the following approximations
I -[1
e(h5) 0
1 l-yr
1 + 5+yr
1
dc= ;
(n--I)/2
C j=l
~(~)(l-f;)1’2
1 -+G-Y,
1 ri+yr
1 ’
(5.14)
380
Y.A. Angel/Wave
1
5-y, 1
I
+ [+y,+& 1
Motion
20 (1994)
371-383
1 7T (n-1)/2 d5=;
j;
+($I -+ 1
(5.15) 1
-h)I+hlWL
e[(l I 0
y,) d4’= fen-f’*
4(tjM$,
Y,) + E
dWK(O,
YA ,
(5.16)
j=1
where K represents any continuous function, h^is a real parameter, and F, = il evaluated by using the (n - 1) /2 Lagrangian polynomials Ak(x) [ 151. One has
[yr( 6 + y,)
1. The value +( 0) is
(n-1)/2 WV
=
C
+(04(O)
(5.17)
9
k=l
Ak(x) = ; (x-tl) D=(fk-fl)*.
. +~~X-tk-1~~X-fk+1~~~~~~-t(,-1)/2~
(5.19)
~~~k-tk-l~~fk-tk+l~~~~~~k-~(n-1)/2~~
Formulas similar to (5.14)-( 5.16) have been used to approximate that (5.3) and (5.4) yield the following system of equations (n-1)/2 C
all the terms of (5.3) and (5.4). Then, one finds
(n-1)/2 +($Fl($,
YJ +
C,
j=l
+($FAti,
YJ =iU,~g,(~Y,)
(5.20)
,
j=l
(n-1)/2 C
(5.18)
9
(n$4tj)K~(+,yr)+
1)/2 C
j=l
d(r,)K,(rj,y,)=ils,~g2[(l-~)yr+~l
.
(5.21)
j=l
In EIqs. (5.20) and (5.21) , the subscript r takes the values r = 1, 2, . . ., (n - 1) /2. Thus, these equations represent a linear system of (n - 1) equations for the (it - 1) complex-valued unknowns $( 9) and & $) . The functions Ki (i= 1,2, 3,4) can be easily obtained by using (5.3), (5.4) and (5.14)-(5.19). Observe that formulas (5.15) and (5.16) are written in terms of c$(0). This is because +( 0) does not vanish in general, and the Gaussian formulas of Erdogan and Gupta [7] can be applied in the interval [ - 1, 1] only to continuous functions. Thus, when the Gaussian formulas are applied in the interval [0, 11, one must include an additional term that contains +( 0). We now consider the discontinuity condition (5.7). Using the Lagrangian polynomials (5.18)) one infers from (5.7) that (n-
(n-l)/2 C j=l
Ilr(t,)Aj(l)-m
1)/2 C
+(Q>Aj(O)=O
*
(5.22)
j=l
The system (5.20) and (5.21) is solved numerically, subject to the constraint (5.22). In order to use the constraint, we eliminate the equation of (5.21) that corresponds to r = 1, and we replace it with (5.22). Thus, we eliminate the equation that corresponds to the position yr closest to the crack tip. The result of this replacement is that we solve a system of (II - 1) equations for the (n - 1) complex-valued unknowns #( 5) and c$(5). It suffices to prescribe the values of five dimensionless parameters: E= s; Is=, m = p’/p, 6= h/a, tie = u,/a, W= h/h,; where AT =2?r/(ws,) is the wavelength of transverse waves in the layer. All other dimensionless parameters can be written in terms of these five basic parameters. The integral of Eq. (4.6) for the amplitude of the far-field displacements can also be evaluated by using the method of Erdogan and Gupta [ 71.
Y.A.Angel/Wave Motion 20 (1994) 371-383
381
6. Discussion of numerical results
Numerical results are presented in Figs. 2-5. These results have been computed for a choice of parameter values such that E= 0.8, m = 1.75, and I= 0.5 and 0.9. The dimensionless frequency Wvaries in the range [ 0,3], and the increment is 0.01. Thus, the curves of Figs. 2-5 are the result of 300 computations. For each frequency, we have solved the linear system (5.20)-( 5.22) with II = 15. The solution of the system has been substituted into (4.6) and (4.9) in order to obtain the coefficients R,, T,, R2, and R3. The values of E and m that we have chosen correspond approximately to those of the uniform and single-layered earth model of Neerhoff [ 51. The choice of n = 15 was made after solving the system (5.20)-( 5.22) for W= 0.5 and n = 15,21, and 31. It was found that the real parts and the imaginary parts of the unknown functions I++ and 4 take values that do not change appreciably for these three values of n. Thus, we conclude that the value n = 15 yields convergent numerical results. For n = 15, Eqs. (5.20)-( 5.22) represent a system of 28 real-valued equations for 28 real-valued unknowns. The (5.13). 28 unknowns are the seven values of Re( $) , Re ( 4) , Im( +) , and Im( 4) at ti , fz, . . .. t,,where$isdefinedin With E= 0.8, we infer from Eq. (2.5) that the Love-wave mode of order k emerges at a frequency c;i= 5( k - 1) / 6. In the frequency range of interest, the relevant values are W=O.8333, 1.667, and 2.5, which correspond to the emergence of the second, third, and fourth modes, respectively. Fig. 2 {3) shows the modulus of the coefficient R, { Tl} versus the frequency for two values &= 0.5 and 0.9. When i= 0.5, the length of the crack is equal to two times the layer thickness h. When i= 0.9, the crack length a is a = h/0.9, which exceeds by a small amount the layer thickness. In the limit as 0 approaches zero, the coefficient R, vanishes, whereas T, is equal to unity. For frequencies greater than 1.5, 1RI 120.996 and 1T, 1~0.004. Thus, the crack acts as a nearly perfect reflector for all frequencies 0 > 1.5. Observe that the solid curve (h= 0.5) in Fig. 2 lies above the dashed curve (h= 0.9) for all frequencies. Thus, as expected, the reflection increases as the length of the crack increases. Fig. 4 (5) shows the moduli of the coefficient R2 {R3] versus the frequency. The peak values of I R2 I are 0.0273 at&i=099 (&=0.5),and0.0506at(5=1.04 (i=0.9).Thepeakvaluesof lR31 areO.O367at(3=1.76 (h=OS), 1.2
IT, I
Fig. 2. Modulus of the reflection coefficient of order one versus the dimensionless frequency; incident Love-wave mode of order one; s$/ s,=O.8, &I/L= 1.75, hla=0.9 (dashed) and 0.5 (solid). Fig. 3. Modulus of the transmission coefficient versus the dimensionless frequency; incident Love-wave mode of order one; s+/sT= 0.8, p’/ /~=1.75,h/a=0.9 (dashed) and0.5 (solid).
382
Y.A. Angel/Wave
0.06
I
IR,I
Motion 20 (1994) 371-383
0.06
IRJ
O-
0
1.5
0
3
0
-i _1 0
1.5
0
3
Fig. 4. Modulus of the reflection coefficient of order two versus the dimensionless frequency; incident Love-wave mode of order one; s;/ s,=O.S, p’/k= 1.75, hla=0.9 (dashed) and 0.5 (solid). Fig. 5. Modulus of the reflection coefficient of order three versus the dimensionless frequency; incident Love-wave mode of order one; s;/ s,=O.S, p’/p= 1.75, h/a=0.9 (dashed) and 0.5 (solid).
and 0.0164 at W= 1.78 (h= 0.9). Thus, the coefficients R2 and R3 are much smaller than the coefficient RI. In fact, R, and R3 are almost negligible in comparison with R,. In Ref. [ 41, where the crack is contained entirely in the layer, several sharp spikes were observed in R,, T,, R,, and RJ. Here, however, the results of Figs. 2-5 do not show the presence of sharp spikes. The results of this work suggest that the sharp spikes of Ref. [4] are caused by resonance effects that take place inside the layer. Indeed, in Ref. [4] the crack tip lies inside the layer and multiple wave reflections take place between the crack faces and the boundaries of the layer. When the crack terminates exactly at the interface (a =h), one infers from (5.8) and from the method of Muskhelishvili [ 131 that the function e( 7) has a power singularity at the crack tip ( y= a = h) with a real-valued exponent j3. The value of p is given by the equation cos(p7r)=-(1-m)l(l+m).
(6.1)
Eq. (6.1) is identical to Eq. (20b) of Erdogan and Cook [ 12, p. 2301. Numerical results for the case of a crack terminating at the interface can be obtained by using Gauss-Jacobi integration formulas [ 161. The behavior of the numerical solution near the emergence frequencies of the Love-wave modes (G= 0.8333 and 1.667) shows some signs of numerical instability. Indeed, one can see in Figs. 4 and 5 that R2 and R, vary noticeably for small frequency changes near the emergence frequencies.
Acknowledgements
This work has been supported by NSF Grant MSM 900039P through an allocation of service units on the CRAY YMP computer of the Pittsburgh Supercomputing Center. The author would like to acknowledge the contribution of Y.K. Koba who has generated the numerical results of this paper.
383
Y.A. Angel/ Wave Motion 20 (1994) 371-383
Appendix
The functions B1 and B2 of Eq. (3.22) are B,(y, u)= -mu+
[I-Pl(y,
+2m~H(y-h‘)[l-P,(y,u)l+ &(~u)=-
u)ll
u>-sgn(i-y)[l-P,(y,
;WA-Wy,u)l,
(A.11
$[Pl(L,u)-ll[l-P,(y,u)-sgn(h-y)[l-P,(y,u)]]
+u- p$ Mh; + ;
u) - VW--6[2-P,h
u) -P~(Y, u)l
Mh; u>+ 11[P,(Y, u) -P1(y, u)l
-m;U’,Vi u) +W-W-6[h(y,
(A-2)
u) -P~(Y, u)l ,
where sgn denotes the sign function, and P~(Y, u) =exp[ -G(K+r)P(u)l
PAY, u) =ew[ -fd~+y)P’(u)l j”(u) = u*- 1 ,
Pdx
,
,
p!‘(u) =u*-E2)
u) =exp[ -G
Ii-yIP(
,
~4(~,~)=exp[-~I~-yJP’(u)l, InG)GO,
Im(P’) 90.
(A.3)
References [l] J.D. Achenbach, Y.C. Angel and W. Lin, “Scattering from surface-breaking and near-surface cracks”, in: G.C. Johnson, ed., Waoe Propagation in Homogeneous Media and Ultrasonic Nondestructive Evaluation, AMD, Vol. 62, American Society of Mechanical Engineers, New York ( 1984). [2] J.D. Achenbach, “From ultrasonics to failure prevention”, in: SK. Datta, J.D. Achenbach and Y.S. Rajapakse, eds., Elastic Waves and Ultrasonic Non-destructive Evaluation, North-Holland, Amsterdam ( 1990). [ 31 L.M. Keer and W.C. Luong, “Diffraction of waves and stress intensity factors in a cracked layered composite”, J. Appl. Mech. 56,16811686 (1974). [4] Y.C. Angel, “Scattering of Love-waves by a surface-breaking crack”, J. Appl. Mech 53.587-592 ( 1986).
[5] F.L. Neerhoff, “DitTraction of Love waves by a stress-free. crack of finite width in the plane interface of a layered composite”, Appl.Sci. Res. 35,265-315 (1979). [6] H.J. Yang and D.B. Bogy, “Elastic wave scattering from an interface crack in a layered half space”, J. Appl. Mech. 52,42-50 (1985). [ 71 F. Erdogan and G.D. Gupta, “On the numerical solution of singular integral equations”, Quart. Appl. Math. 30.525-534 ( 1972). [ 81 J.D. Achenbach, Wave Propagation in Elastic Solids, North-Holland, Amsterdam ( 1973). [9] LS. Gradshteyn and LM. Ryzhik, Table oflntegrals Series and Products, Academic Press, New York (1965). [lo] A. Erdelyi, Tables oflntegral Transforms, Vol. 1, McGraw-Hill, New York ( 1954). [ 111 H. Lamb, “On the propagation of tremors over the surface of an elastic solid”, Philos. Trans. R. Sot. London, Series A, 203,142 ( 1904). [ 121 F. Erdogan and T.S. Cook, “Antiplane shear crack terminating at and going througha bimaterialinterface”, Int. J. Fracture 10.227-240 (1974). [ 131 N.I. Muskhelishvili, Singular Integral Equations, Noordhoff, Groningen (1977). [ 141 F.D. Gakhov, Bounaby Value Problems, Pergamon, New York (1966). [ IS] F.B. Hi&brand, Introduction to Numerical Analysis, 2nd edition, McGraw-Hill, New York ( 1974). [ 161 F. Erdogan, Treatise on Continuum Physics, A.C. Eringen, ed., Vol. II, Part 111.3,Academic Press, New York ( 1975).