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Elliptic arc crack subjected to anti-plane shear wave
Engineering Fracture Mechanics Vol. 48, No. 2, pp. 289-291, 1994 Copyright 0 1994 Elsevierscience Ltd Printedin Great Britain. All rights reserved 0013-79441% $7.00+ 0.00
Pergamon
ELLIPTIC ARC CRACK SUBJECTED ANTI-PLANE SHEAR WAVE
TO
WANG YUE-SHENG and WANG DUO Department of Engineering Mechanics, P.O. Box 344, Harbin Institute of Technology, Harbin 150006, P.R.C. Abstract-An elliptic arc crack subjected to an anti-plane shear wave is considered in this paper. The problem is first reduced to a set of simultaneous dual series equations by using the wave function expansion method. Then, a dislocation density function is introduced to transform these equations to a Hilbert singular integral equation which is further converted to a Cauchy singular integral equation. The dynamic stress intensity factors (DSIFs) at crack tips are calculated by solving the integral equation numerically. The results show a low frequency resonance in the DSIF, especially for cracks of large angular width. The resonance frequency and the peak value of the DSIF depend on many factors, such as the ratio of long and short radii, the crack size and the incident angle of waves.
1. INTRODUCTION THE STUDY of cracks subjected to elastic waves is of practical importance to the application of dynamic fracture mechanics and has therefore received considerable attention. Although many investigations have been published on this problem, very few of them have considered the case of curved cracks because of the mathematical complexity. Fil’shtinskii [I] first attacked the problem of interaction of elastic waves with curved cracks. He reduced the problem to a set of singular integro-differential equations for crack opening displacements (CODS) and solved them numerically to obtain the dynamic stress intensity factors (DSIFs). Later, various dynamic problems of curved cracks were solved using the same method (see ref. [2]). This method is very complicated algebraically since it involves the use of a Green function. Recently, Yang and Norris [3] considered a circular arc interface crack subjected to SH waves. They used the wave function expansion method to obtain an integral equation for the COD. The COD was subsequently expanded in terms of Chebyshev polynomials and an infinite system of linear algebraic equations was derived for the coefficients of the polynomials. In ref. [4], we presented a new method to calculate the DSIF of a circular arc crack subjected to anti-plane shear waves. In that method, a dislocation density function was introduced to reduce the problem to a singular integral equation which can be easily solved numerically. The method is very simple in mathematical processing and, unlike those in refs [l-3], where the singularity of the local stress near the crack tips was assumed to be - l/2, it can give this singularity directly from the derived singular integral equation. In this paper, the method is extended to an elliptic arc crack subjected to anti-plane shear waves. Special attention is devoted to the determination of DSIFs.
2. DESCRIPTION OF THE PROBLEM AND DERIVATION OF THE DUAL SERIES EQUATIONS Consider an arc crack lying on an ellipse with long radius a and short radius b, as shown in Fig. 1, where an elliptic coordinate system (5, q) is used and the angular coordinates of the crack tips are aI and a2. All motion is time harmonic of frequency o, and the term eViw’will be omitted for simplicity. The incident anti-plane harmonic shear wave propagates in the direction 0, with the form
w”j({ 3 q) = A eiK~r&oshCcosqcos8~+ 289
sinh (sin qsin 8,)
9
(1)
WANG YUE-SHENG and WANG DUO
290
where KT= w/C, is the wave number, CT= ,/(p/lp) is the shear wave velocity, ~1is the shear modulus, p is mass density and A is the amplitude. Equation (1) can be further expressed as w(‘)(<,rj) = 2A
f imce,(BO)Mc~)(t)ce,(?) i m-0
+ f ??I=1
~“~e,(RW~!%%e,(
rl)
,
(2)
I
were ce,, se,, MC;) and it&r:) are angular and radial Mathieu functions defined in ref. [5]. The stress component ryi for an incident wave is imce,(8,)Mc~)‘(5)ce,(tl)
+ 2
i”se,(80)Ms~)‘(r)se,(q)
, (3)
rn=l
where J = J[(cosh
25 - cos 2~)/2]
(da)
r. = ,/(a’ - b2).
(W
Due to the existence of the crack, the incident wave will be scattered. The total out-of-plane displacement can be represented as w(& ?) =
~(“(5, V) + wY(<, tt) i w(~)(<,q) + w?)(5, q)
for for
5 > to l < c0 '
(5)
where to = tanh-‘(b/a) is the ellipse on which the crack is located, and w$) and wp) are the scattered fields, which satisfy the following Helmholtz equation in ellipse coordinates [6]:
(6) and the radiation condition at infinity. By the use of the wave function expansion method [6], ~5”)and WY)can be written as
The corresponding
stress components r& (j = 0, 1) are
In eqs (7a,b) and (8a,b), MC:) and MS:) are radial Mathieu functions of the third kind, and A,,,, B,,,, a,,, and B,,, are unknown coefficients.
Let the discontinuity of displacement at 5 = lo be Aw(q), which is equal to the crack opening displacement (COD) when 1 E (a,, az) or zero when ?I 4:(a,, a2). The continuity conditions at r = to for the displacement and stress are w6”‘(50,rt) - w’?(ro, tl) = Aw(tt) rtJ&50, rl) =
7t"l',Go, tth
Pa) W9
We express Aw(u) as an infinite series in cc,(q) and se,(q), i.e. -
Aw(q) = g Aw,ce,,,(rl)+ m=O
2 AGse,(tl), m-l
(10)
291
Elliptic arc crack subjected to anti-plane shear wave
Fig. 1. An elliptic arc crack subjected to an anti-plane shear wave.
where a2
-
Aw, = L
m=0,1,2,...
Aw(q)ce,,,(q)dq
= s aI
(114
a2
AG=r
m = 1,2,. . .
Aw(~)ce,(~)d~
(lib)
7r s #I
Substituting eqs (7a,b), (8a,b) and (10) into eqs (9a,b) yields
where
DC,,, = Mc~‘(&,)Mc~“(&,) - Mc~“(&,)Mc~‘(&,)
(W
hf.p(e,)hh:y~,j.
W)
Ds,
= Msg’(g,)Msp(&J -
The condition that the total stress vanishes on the crack faces is +(L~)=#(Lrl)=
-+1(50,~)
rlE(6,a~).
(14)
k
r’
2
2
E g 1
0 0.0
0.3
0.6
0.9
1.2
1.5
Fig. 2. The normalized DSIFs for cracks symmetric about the major axis with various angular width 2~
WANG YUE-SHENG and WANG DUO
292
-0.0
0.3
0.6
0.9
1.2
1.5
%a
Fig. 3. The normalized DSIFs for cracks symmetric about the minor axis with various angular width 2a.
From eqs @a,b), (12a,b) and (14), we can obtain the following series equation:
f
Dc,‘Mc~“(C,)Mc~)‘(5,)ce,(?)Aw,
-
where Jo(q) is J defined by eq. (4a), in which t; = &,. That Aw(q) = 0 when q $ (LX,,a2) implies f Alice,+ PI=0
f
A%%,,(~)=0
(16)
~$(u,,u,).
l?l=I
Up to now, we have reduced the problem to a set of simultaneous dual series equations (15) and (16).
0.5 -
0.0 0.0
I 0.3
I 0.6
I 0.9
I 1.2
I IS
KTa
Fig. 4. The normalized DSIFs for cracks symmetric about the major axis with various incident angle 0,.
Elliptic arc crack subjected to anthplam shear wave
293
e
Fig. 5. The napalm
3. DER~A~ON
DSlFs fox cracks symmetric about the minor axis with various incident angle @,.
AND SOL~UN
OF THE S~G~AR
EGRAL EQUA~ON
We introduce an au~lia~ function:
In fact, &r(g)has distinct physical meaning. It is called the dislocation density function. From eq. (17), we have
where z(q) =&.9 PI:
=ce&>,
x(q> ~-se&).
Co~side~ng the f~ilowing expressions of c~~~~~and
M= t-42,.1*
m = 2,4, . . . in which q = f&C2 Tr**, and A 5’@(q)and B!“‘)(q) are defined in ref. [Sj, we have
WANG YUE-SHENG and WANG DUO
294
@lb)
Substituting eqs (i9a,b) into eq. (IS) yieIds
where (23a) (23b) Equation (16) implies (24) The Mathieu functions have the following properties as m --, f 00 [5]: mtj
ce,(q)-rcos w, ce,(l)-yg-
sin rn{
+--
2
cos
(25)
0.0
0.0
I
I
I
I
1
0.3
0.6
0.9
1.2
1.5
%a
Fig. 6. The normalized DSIFs for cracks symmetric about the major axis with various b/u.
Elliptic arc crack subjected to anti-plane shear wave
*.oi 0.0
I
I
1
0.3
0.6
0.9
295
I 1.2
I
1.5
%a
Fig. 7, The normalized DSIFs for cracks symmetric about the minor axis with various bfa.
and therefore m -+ + 00 M2(m)se,(~)~(C)~~rcosmr;
sinmq
t2W
t2W
m-*-too.
Supposing that
and noting the relation mz, (sin rnc cos mq - cos rnc sin mq) = icot( we can reduce eq, (15) to a Wilbert singular integral equation:
P(5, q), defined by eq. (27), is a regular integral kernel which has no singularity. By the use of the following substitutions: ~=cfSd,
[=cz+d
&(t) = J&t + d), @(T)=#(cT L(?, t) = ;P(cr
&(t, = J&r + d)
-t-d) +d,cr
+d)+-s--
1 Z(f)
27~JO(t) Cot
C(? - 1) --
2
I
nc(z - t)’
(30)
where c = (a2 - al)/2 and d = (cl,f a,)/2, eq. (29) is further converted to a standard Cauchy singular integral equation of the first kind: (31) EFM
WZ-J
296
WANG
YUE-SHENG
and WANG
DUO
and eq. (24) is converted to I
dz = 0.
&(z)@(t)
s -I
(32)
So far, the dual series equations (15) and (16) have been reduced to the Cauchy singular integral equation (31) and the restrictive equation (32). The general theory of singular integral equations [7] shows that G(r) has - l/2 singularity at + 1. Thus, letting (33) and using the numerical method described in ref. [8], we obtain a system of linear algebraic equations from eqs (31) and (32):
where rj = cos(rr/2N)(2j - I), fk = cos(kn/N), points of F(q) between - 1 and + 1.
k = l-N-1,
and N is the number of the discrete
4. DYNAMIC STRESS INTENSITY
FACTORS
The DSIFs at crack tips ai and a2 are defined as &ii1 = lim [J[2r0 J&Hal
4-V
-
tt)l~e(L?)I
Wa)
where
Using eq. (30), the principal part of rtr(&, q) as q +a ; and ai [or rcz(&,, ct + d) as t -+ - 1 and I+] can be written as (37) Representing F(q) in (33) by a series of Chebyshev polynomials F(rl)=
f
T,(q), i.e. (38)
Ajq(?),
j-0
and substituting it into (36) yields Kill = -$ J(roc)F(-
G2 = --:
(394
1)
J(roc)fY1),
W’b)
where we have used the relation 1
’ (1 - r12)-“2q(rl) dtt = K’-
71 -I -s
q-t
(_ly+'(52_
lV2 - rY 1)1'2
151'
l*
(40)
Elliptic arc crack subjected to anti-plane shear wave
297
5. NUMERICAL RESULTS AND DISCUSSION In this section, the DSIFs for various cases have been calculated numerically and displayed in Figs 2-7, where the DSIFs are normalized by r,,&(r ,, = PA&) and tl represents the half angular width of the crack. The variation of the normalized DSIF with the dimensionless wave number Kra for b/a = 0.5 and different a is shown in Figs 2 (for a crack symmetric about the major axis) and 3 (for a crack symmetric about the minor axis). A resonance appears at low frequency with a peak value of the DSIF at the resonance frequency. As the crack size is increased, the peak becomes more pronounced and occurs at lower frequency. This resonance at low frequency was also demonstrated in refs [3] and [4] for a circular arc crack. Also note that the peak value of the DSIF appears sharper in Fig. 3 than that in Fig. 2 for larger cracks. However, it is different for smaller cracks. Figures 2 and 3 also show us the effect of the incident angle 6,. This effect is further displayed in Figs 4 and 5. In Figs 6 and 7, numerical results are presented for the normalized DSIF vs the dimensionless wave number K,a for tl = 90” and different b/u. As b/a becomes smaller, both the peak value of the DSIF and the resonance frequency increase for the crack symmetric about the major axis (see Fig. 6), while for the crack symmetric about the minor axis (see Fig. 7) the peak value of the DSIF decreases and the resonance frequency increases slightly as b/a becomes smaller. As b/a+l, we obtain results for a circular arc crack which agree with those given in ref. [4]. In conclusion, a distinguishing feature of this problem is that a strong resonance occurs at low frequency, especially for a larger crack. The peak value of the DSIF and the resonance frequency depend on the ratio of long and short radii b/a, the incident angle 6,, the crack size ~1,crack position, etc. Finally, we mention here that the present approach can also be applied to the dynamic problem of an arc interface crack between an elliptic inclusion and its surrounding matrix, where a similar resonance phenomenon is expected. REFERENCES [I] L. A. Fil’shtinskii, Dynamic problem in the theory of elasticity for a medium with curvilinear cuts (antiplane shear strain). Dokl. Akad. Nauk SSSR 236, 1327-1330 (1977). [2] V. Z. Parton and V. G. Boriskovsky, Dynamic Fracture Mechanics, Vol. 1. Hemisphere, U.S.A. (1989). [3] Y. Yang and A. N. Norris, Shear wave scattering from a debonded fiber. J. Me&. Phys. Solirls 39.273-294 (1991). [4] Y. S. Wang and D. Wang, Dynamic stress intensity factor of a circular arc-shaped crack subjected to anti-plane shear wave. Int. J. Fracture 59, R33-38 (1993). [S] M. Abramowitz and I. A. &gun, Handbook of Mathematical Functions. Dover, New York (1965). [6] Y. H. Pao and C. C. Mow, Dtfiaction of Elastic Waoes and Dynamic Stress Concentrations. Crane and Russak, New York (1973). [7] N. I. Muskhelishvili, Singufar Integral Equutions. Noordhoff, Leyden (1953). [8] F. Erdogan and G. D. Gopta, On the numerical solution of singular integral equations. Q. appl. Math. 29, 525-539 (1972). (Received 23 October 1992)
Pergamon
ELLIPTIC ARC CRACK SUBJECTED ANTI-PLANE SHEAR WAVE
TO
WANG YUE-SHENG and WANG DUO Department of Engineering Mechanics, P.O. Box 344, Harbin Institute of Technology, Harbin 150006, P.R.C. Abstract-An elliptic arc crack subjected to an anti-plane shear wave is considered in this paper. The problem is first reduced to a set of simultaneous dual series equations by using the wave function expansion method. Then, a dislocation density function is introduced to transform these equations to a Hilbert singular integral equation which is further converted to a Cauchy singular integral equation. The dynamic stress intensity factors (DSIFs) at crack tips are calculated by solving the integral equation numerically. The results show a low frequency resonance in the DSIF, especially for cracks of large angular width. The resonance frequency and the peak value of the DSIF depend on many factors, such as the ratio of long and short radii, the crack size and the incident angle of waves.
1. INTRODUCTION THE STUDY of cracks subjected to elastic waves is of practical importance to the application of dynamic fracture mechanics and has therefore received considerable attention. Although many investigations have been published on this problem, very few of them have considered the case of curved cracks because of the mathematical complexity. Fil’shtinskii [I] first attacked the problem of interaction of elastic waves with curved cracks. He reduced the problem to a set of singular integro-differential equations for crack opening displacements (CODS) and solved them numerically to obtain the dynamic stress intensity factors (DSIFs). Later, various dynamic problems of curved cracks were solved using the same method (see ref. [2]). This method is very complicated algebraically since it involves the use of a Green function. Recently, Yang and Norris [3] considered a circular arc interface crack subjected to SH waves. They used the wave function expansion method to obtain an integral equation for the COD. The COD was subsequently expanded in terms of Chebyshev polynomials and an infinite system of linear algebraic equations was derived for the coefficients of the polynomials. In ref. [4], we presented a new method to calculate the DSIF of a circular arc crack subjected to anti-plane shear waves. In that method, a dislocation density function was introduced to reduce the problem to a singular integral equation which can be easily solved numerically. The method is very simple in mathematical processing and, unlike those in refs [l-3], where the singularity of the local stress near the crack tips was assumed to be - l/2, it can give this singularity directly from the derived singular integral equation. In this paper, the method is extended to an elliptic arc crack subjected to anti-plane shear waves. Special attention is devoted to the determination of DSIFs.
2. DESCRIPTION OF THE PROBLEM AND DERIVATION OF THE DUAL SERIES EQUATIONS Consider an arc crack lying on an ellipse with long radius a and short radius b, as shown in Fig. 1, where an elliptic coordinate system (5, q) is used and the angular coordinates of the crack tips are aI and a2. All motion is time harmonic of frequency o, and the term eViw’will be omitted for simplicity. The incident anti-plane harmonic shear wave propagates in the direction 0, with the form
w”j({ 3 q) = A eiK~r&oshCcosqcos8~+ 289
sinh (sin qsin 8,)
9
(1)
WANG YUE-SHENG and WANG DUO
290
where KT= w/C, is the wave number, CT= ,/(p/lp) is the shear wave velocity, ~1is the shear modulus, p is mass density and A is the amplitude. Equation (1) can be further expressed as w(‘)(<,rj) = 2A
f imce,(BO)Mc~)(t)ce,(?) i m-0
+ f ??I=1
~“~e,(RW~!%%e,(
rl)
,
(2)
I
were ce,, se,, MC;) and it&r:) are angular and radial Mathieu functions defined in ref. [5]. The stress component ryi for an incident wave is imce,(8,)Mc~)‘(5)ce,(tl)
+ 2
i”se,(80)Ms~)‘(r)se,(q)
, (3)
rn=l
where J = J[(cosh
25 - cos 2~)/2]
(da)
r. = ,/(a’ - b2).
(W
Due to the existence of the crack, the incident wave will be scattered. The total out-of-plane displacement can be represented as w(& ?) =
~(“(5, V) + wY(<, tt) i w(~)(<,q) + w?)(5, q)
for for
5 > to l < c0 '
(5)
where to = tanh-‘(b/a) is the ellipse on which the crack is located, and w$) and wp) are the scattered fields, which satisfy the following Helmholtz equation in ellipse coordinates [6]:
(6) and the radiation condition at infinity. By the use of the wave function expansion method [6], ~5”)and WY)can be written as
The corresponding
stress components r& (j = 0, 1) are
In eqs (7a,b) and (8a,b), MC:) and MS:) are radial Mathieu functions of the third kind, and A,,,, B,,,, a,,, and B,,, are unknown coefficients.
Let the discontinuity of displacement at 5 = lo be Aw(q), which is equal to the crack opening displacement (COD) when 1 E (a,, az) or zero when ?I 4:(a,, a2). The continuity conditions at r = to for the displacement and stress are w6”‘(50,rt) - w’?(ro, tl) = Aw(tt) rtJ&50, rl) =
7t"l',Go, tth
Pa) W9
We express Aw(u) as an infinite series in cc,(q) and se,(q), i.e. -
Aw(q) = g Aw,ce,,,(rl)+ m=O
2 AGse,(tl), m-l
(10)
291
Elliptic arc crack subjected to anti-plane shear wave
Fig. 1. An elliptic arc crack subjected to an anti-plane shear wave.
where a2
-
Aw, = L
m=0,1,2,...
Aw(q)ce,,,(q)dq
= s aI
(114
a2
AG=r
m = 1,2,. . .
Aw(~)ce,(~)d~
(lib)
7r s #I
Substituting eqs (7a,b), (8a,b) and (10) into eqs (9a,b) yields
where
DC,,, = Mc~‘(&,)Mc~“(&,) - Mc~“(&,)Mc~‘(&,)
(W
hf.p(e,)hh:y~,j.
W)
Ds,
= Msg’(g,)Msp(&J -
The condition that the total stress vanishes on the crack faces is +(L~)=#(Lrl)=
-+1(50,~)
rlE(6,a~).
(14)
k
r’
2
2
E g 1
0 0.0
0.3
0.6
0.9
1.2
1.5
Fig. 2. The normalized DSIFs for cracks symmetric about the major axis with various angular width 2~
WANG YUE-SHENG and WANG DUO
292
-0.0
0.3
0.6
0.9
1.2
1.5
%a
Fig. 3. The normalized DSIFs for cracks symmetric about the minor axis with various angular width 2a.
From eqs @a,b), (12a,b) and (14), we can obtain the following series equation:
f
Dc,‘Mc~“(C,)Mc~)‘(5,)ce,(?)Aw,
-
where Jo(q) is J defined by eq. (4a), in which t; = &,. That Aw(q) = 0 when q $ (LX,,a2) implies f Alice,+ PI=0
f
A%%,,(~)=0
(16)
~$(u,,u,).
l?l=I
Up to now, we have reduced the problem to a set of simultaneous dual series equations (15) and (16).
0.5 -
0.0 0.0
I 0.3
I 0.6
I 0.9
I 1.2
I IS
KTa
Fig. 4. The normalized DSIFs for cracks symmetric about the major axis with various incident angle 0,.
Elliptic arc crack subjected to anthplam shear wave
293
e
Fig. 5. The napalm
3. DER~A~ON
DSlFs fox cracks symmetric about the minor axis with various incident angle @,.
AND SOL~UN
OF THE S~G~AR
EGRAL EQUA~ON
We introduce an au~lia~ function:
In fact, &r(g)has distinct physical meaning. It is called the dislocation density function. From eq. (17), we have
where z(q) =&.9 PI:
=ce&>,
x(q> ~-se&).
Co~side~ng the f~ilowing expressions of c~~~~~and
M= t-42,.1*
m = 2,4, . . . in which q = f&C2 Tr**, and A 5’@(q)and B!“‘)(q) are defined in ref. [Sj, we have
WANG YUE-SHENG and WANG DUO
294
@lb)
Substituting eqs (i9a,b) into eq. (IS) yieIds
where (23a) (23b) Equation (16) implies (24) The Mathieu functions have the following properties as m --, f 00 [5]: mtj
ce,(q)-rcos w, ce,(l)-yg-
sin rn{
+--
2
cos
(25)
0.0
0.0
I
I
I
I
1
0.3
0.6
0.9
1.2
1.5
%a
Fig. 6. The normalized DSIFs for cracks symmetric about the major axis with various b/u.
Elliptic arc crack subjected to anti-plane shear wave
*.oi 0.0
I
I
1
0.3
0.6
0.9
295
I 1.2
I
1.5
%a
Fig. 7, The normalized DSIFs for cracks symmetric about the minor axis with various bfa.
and therefore m -+ + 00 M2(m)se,(~)~(C)~~rcosmr;
sinmq
t2W
t2W
m-*-too.
Supposing that
and noting the relation mz, (sin rnc cos mq - cos rnc sin mq) = icot( we can reduce eq, (15) to a Wilbert singular integral equation:
P(5, q), defined by eq. (27), is a regular integral kernel which has no singularity. By the use of the following substitutions: ~=cfSd,
[=cz+d
&(t) = J&t + d), @(T)=#(cT L(?, t) = ;P(cr
&(t, = J&r + d)
-t-d) +d,cr
+d)+-s--
1 Z(f)
27~JO(t) Cot
C(? - 1) --
2
I
nc(z - t)’
(30)
where c = (a2 - al)/2 and d = (cl,f a,)/2, eq. (29) is further converted to a standard Cauchy singular integral equation of the first kind: (31) EFM
WZ-J
296
WANG
YUE-SHENG
and WANG
DUO
and eq. (24) is converted to I
dz = 0.
&(z)@(t)
s -I
(32)
So far, the dual series equations (15) and (16) have been reduced to the Cauchy singular integral equation (31) and the restrictive equation (32). The general theory of singular integral equations [7] shows that G(r) has - l/2 singularity at + 1. Thus, letting (33) and using the numerical method described in ref. [8], we obtain a system of linear algebraic equations from eqs (31) and (32):
where rj = cos(rr/2N)(2j - I), fk = cos(kn/N), points of F(q) between - 1 and + 1.
k = l-N-1,
and N is the number of the discrete
4. DYNAMIC STRESS INTENSITY
FACTORS
The DSIFs at crack tips ai and a2 are defined as &ii1 = lim [J[2r0 J&Hal
4-V
-
tt)l~e(L?)I
Wa)
where
Using eq. (30), the principal part of rtr(&, q) as q +a ; and ai [or rcz(&,, ct + d) as t -+ - 1 and I+] can be written as (37) Representing F(q) in (33) by a series of Chebyshev polynomials F(rl)=
f
T,(q), i.e. (38)
Ajq(?),
j-0
and substituting it into (36) yields Kill = -$ J(roc)F(-
G2 = --:
(394
1)
J(roc)fY1),
W’b)
where we have used the relation 1
’ (1 - r12)-“2q(rl) dtt = K’-
71 -I -s
q-t
(_ly+'(52_
lV2 - rY 1)1'2
151'
l*
(40)
Elliptic arc crack subjected to anti-plane shear wave
297
5. NUMERICAL RESULTS AND DISCUSSION In this section, the DSIFs for various cases have been calculated numerically and displayed in Figs 2-7, where the DSIFs are normalized by r,,&(r ,, = PA&) and tl represents the half angular width of the crack. The variation of the normalized DSIF with the dimensionless wave number Kra for b/a = 0.5 and different a is shown in Figs 2 (for a crack symmetric about the major axis) and 3 (for a crack symmetric about the minor axis). A resonance appears at low frequency with a peak value of the DSIF at the resonance frequency. As the crack size is increased, the peak becomes more pronounced and occurs at lower frequency. This resonance at low frequency was also demonstrated in refs [3] and [4] for a circular arc crack. Also note that the peak value of the DSIF appears sharper in Fig. 3 than that in Fig. 2 for larger cracks. However, it is different for smaller cracks. Figures 2 and 3 also show us the effect of the incident angle 6,. This effect is further displayed in Figs 4 and 5. In Figs 6 and 7, numerical results are presented for the normalized DSIF vs the dimensionless wave number K,a for tl = 90” and different b/u. As b/a becomes smaller, both the peak value of the DSIF and the resonance frequency increase for the crack symmetric about the major axis (see Fig. 6), while for the crack symmetric about the minor axis (see Fig. 7) the peak value of the DSIF decreases and the resonance frequency increases slightly as b/a becomes smaller. As b/a+l, we obtain results for a circular arc crack which agree with those given in ref. [4]. In conclusion, a distinguishing feature of this problem is that a strong resonance occurs at low frequency, especially for a larger crack. The peak value of the DSIF and the resonance frequency depend on the ratio of long and short radii b/a, the incident angle 6,, the crack size ~1,crack position, etc. Finally, we mention here that the present approach can also be applied to the dynamic problem of an arc interface crack between an elliptic inclusion and its surrounding matrix, where a similar resonance phenomenon is expected. REFERENCES [I] L. A. Fil’shtinskii, Dynamic problem in the theory of elasticity for a medium with curvilinear cuts (antiplane shear strain). Dokl. Akad. Nauk SSSR 236, 1327-1330 (1977). [2] V. Z. Parton and V. G. Boriskovsky, Dynamic Fracture Mechanics, Vol. 1. Hemisphere, U.S.A. (1989). [3] Y. Yang and A. N. Norris, Shear wave scattering from a debonded fiber. J. Me&. Phys. Solirls 39.273-294 (1991). [4] Y. S. Wang and D. Wang, Dynamic stress intensity factor of a circular arc-shaped crack subjected to anti-plane shear wave. Int. J. Fracture 59, R33-38 (1993). [S] M. Abramowitz and I. A. &gun, Handbook of Mathematical Functions. Dover, New York (1965). [6] Y. H. Pao and C. C. Mow, Dtfiaction of Elastic Waoes and Dynamic Stress Concentrations. Crane and Russak, New York (1973). [7] N. I. Muskhelishvili, Singufar Integral Equutions. Noordhoff, Leyden (1953). [8] F. Erdogan and G. D. Gopta, On the numerical solution of singular integral equations. Q. appl. Math. 29, 525-539 (1972). (Received 23 October 1992)