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The stress field in the vicinity of two collinear cracks subject to antiplane shear in a strip of finite width
Pergamon
Medaanics Research Communications, Vol. 25, No. 2, pp. 183-188, 1998 Copyright © 1998 E l z v i r r Science Ltd Printed in the USA. All rights re~rved 0093-6413/98 $19.00 + .00
PII S0093-6413(98)00023-g
The stress field in the vicinity of two collinear cracks subject to antiplane shear in a strip o f finite width
Z h e n - G o n g Zhou, B i a o - W a n g and Shan-Yi Du P.O.Box 2147, Center for composite materials, Harbin Institute of Technology Harbin 150001, P.R.China (Received 8 July 1997: accepted for print 3 December 1997)
I. Introduction
From an engineering point of view, the crack problems of the strip are of particular interest. From this reason, the strip problem was treated by many researchers by using different methods, for example, [1-5]. Such as, Irwin [1] gave an approximation by periodic crack solution to determining the distribution of the stress in the neighborhood of two collinear edge cracks. The stress field of a Griffith crack in a strip of finite width has been discussed by Sneddon [2] and Gupta [3] using the integral transform method. The problem of two collinear edge cracks in the strip of finite width had been treated by Bethem [4] and Yamamoto [5] using asymptotic approximations and the finite element method, respectively. In the present paper, the stress intensity factor of two collinear symmetrical cracks subjects to antiplane shear in an infinitely long strip is considered by using a new method, namely Schmidt's method. It is a simple and convenient method for solving this problem. Fourier transform is applied and a mixed boundary value problem is reduced to a set of triple integral equations. In solving the triple integral equations, the crack surface displacement is expanded in a series using Jacobi's polynomials and Schmidt's method [6,7] is used. This process is quite different from that adopted in references [1-5]. The form of solution is easy to understand. Numerical calculations are carried out for the stress intensity factor.
183
184
Z.-G. ZHOU, B. WANG and S.-Y. DU
2. F o r m u l a t i o n
of the problem
We consider a strip o f material - h _ 1 ), with two collinear symmetrical cracks located in the interior o f the material on the line y=0, - I < x -< - a , a _ b > a > 0)). The corresponding stress field consists o f two shear stresscs are as follows: re: = ~ - . ~ , r _ = l l cv"
(1)
While all other components vanish. In equations(l), g stands for the shear module of thc clasticity of the solid medium. Substituting equations(l) into the equation o f elasticity in the Zdirection renders (7 2 w
c~ 2
e7~2 w
+ ~]-~ = 0
(2)
The boundary conditions on the crack laces a t ) ' = 0 may be stated as Ibllows: r , : ( x , 0 ) = -r<, , a _< ix:-< 1 w(x,O)=O,
(3)
O_
r , : ( + h , y ) = 0,
-oo{y(~
in which r 0 is a constant having the dimension o f stress. Y +
1
h
e
x
®® 4
(4) (5)
1
f Fig. 1. An elastic strip containing two collinear cracks
COLLINEAR CRACKS UNDER SHEAR
185
3. Analysis
A Fourier cosine transform [8] is applied on equation (2) and result is ( y _>0 ) w = 2 ~ A ( ) s e -'y cos ( sx )ds + 2 ~ 1 3 (s) cos h( sx ) sin (sy)ds
(6)
Because of symmetry, it suffices to consider the problem in the first quadrant only. Making use of equations (1) and (6), equations (3) and (4) render a set of triple integral equations
0 ~0 < X <_a, 1 < X < h
~:A(s)cos(sx)ds
(7)
~: sA(s)cos(sx)ds = [® sB(s)cosh(sx)ds + :rr°
a -< x < 1, (8) J0 2p' The relationship between the function A and B is obtained by applying a Fourier sine transform [8] to equation (5): _ 2 ~, _ . . sin(r/h). B(s) sinh(sh) - ~ J0 r/At r/) ~ ar/
(9)
To solve integral equations (7) and (8), we represent displacement w by the following series: ®
, ', ixI l + a
w(x'O)= ~'~a~P2~'~ ( l - a 2 )(1 n~O
=0
--'2
,
for
(Ixl_ _f)2
,
(1_. a)2 ) i ' for
a_
2
0---Ixl-
(10)
where a n are unknown coefficients to be determined and P~YV::2)(x) is a Jacobi polynomial [8]. The Fourier cosine transform for equation (10) is [9]
A(s) = ~(s,O) = Ea~G.(s)B.J~÷,[s(
)]s-'.
(1 1)
n-O
I
(-1) 2 cos(s where G , ( s ) = / n.t .+ [ ( - I ) 2 sin(s~---~),n---1,3,5,7 .... ....
Bn =2x/-~
l-(n+ 1 + -) n!
where F(x) and Jn (x) are the Gamma and Bessel functions, respectively. Substitution equation (11) into equations (7) and (8), respectively, the equation (7) has been automatically satisfied by using the Fourier transform. Then the remaining equation (8) reduces to the form after integration with respect to x for a < x -< 1
~,anBn ~=0
G,(s)Jn.l(
s)s-'[sin(sx)-sin(as)]ds=-~B ( x - a ) +
186
Z.-G. ZHOU, B. WANG and S.-Y. DU ®
..o~a"B"f£
®2[s;ph(sr) - sinh(aS)]dse~G j 1 - a sin(r/h) ~rsinn~sh) J0 "(r/) " " ( ~ - - - r / ) - ~ - ~ s 2drl
(12)
For a large s, the integrands of the double semi-infinite integral in the equation (12) almost all exponentially, so the double semi-infinite integral in equation (12) can be evaluated numerically by Filon's method[9]. The first semi-infinite integral in equation (12) can be easily obtained by using the integral formula (see [8]). Thus equation (12) can be solved for coefficients a. by the Schmidt's method [6,7]. For brevity, we have rewritten equation (12) as oC
~a.E.(x)
= U(x).
a
(13)
n~0
where E.(x)
and U(x) are known functions and coefficients a. are unknown and to be
determined. A set of functions P. (x) which satisfy the orthogonality condition et
J l~ (x)P. (x)dx = ,¥.d . . . . . .
I
V = f~ P" (x)dx
(14)
can be constructed from the function, E. (x), such that n
P.(x) = ~-~ M " E (x) ,=0
Mnn
(15)
'
where M,. is the cofactor of the element d,. of D., which is defined as d l 0 , d t t d°~.d:2' , d r 2 d°2 ...... . . . . do. d°°d2o,d:~' dz.d°"- ,
D. =
d,. = ~jE,(x)E.(x)dx
.........................
d.o,d.~,d
(16)
i
: ..... d
j
Using equations (13-16). we obtain a. = ~--~qj M.j with q, = ]=n
l~'(x)Pj(x)dx
(17)
MM
4. Stress Intensity Factor and Numerical Results
When coefficients a, are known, the entire stress field is obtainable. However, in fracture mechanics, it is of importance to determine stress r~ in the vicinity of the crack's tips. r~.. at
y= 0 is given by r~ -
a.B. {~iG. ( s)J.., ( •"*/" n = 0
s) cos( sx)ds
COLLINEAR CRACKS UNDER SHEAR
187
2seosh(sx) _ r* . . . . . 1 - a , sin(r/h) j , ~ "-":-TT~, , , , aS l. ~ , l.rl)J,+, l . r l - - Z - ) ~ r l ~ , (18) trsmn~sn) -. z r1 + s The singular portion o f the stress field results can be obtained by using the integral formulas (see [8]). At the left end o f the right crack, we obtain the stress intensity factor K L as K L = lim x / 2 x ( a - x) - o = / l
a.B.(-1)"
(19)
x--~a
At the right end o f the right crack, we obtain the stress intensity factor K R as K R = lira 2 x / ~ - 1 ) . o " x-w
.[
2
=/~ - - ~ - ' , . B .
-
(20)
V x ( l - a ) ..0 10
For a check of the accuracy, the values of ~ ' a , E , ( x )
and U ( x ) a r e given in table 1 for
n=O
a = 05,h = 1.8. Hence, it is clearly that the Schrnidt's method is carried out satisfactorily. The
variation with a o f the stress intensity factors K L and K R are given in table 2---4. For the case of a --->0, the present problem reduces to that for a single crack [2]. For the case o f h > 3, the results in this paper show a similar tendency the results of two Griffith cracks in an infinite medium. The results o f the stress intensity factors K L and K R become higher with the increase of the crack length. The stress intensity factors K L and Ku become lower with the increase of the strip width. For the case h = i.2,a > 0.3, the stress intensity factor K L is bigger than the stress intensity factor K R . For the case a --~ 1, K L and K R approach 0. For the case a ~ 0, K L increases more
quickly than K~.
~"a.E.(x).Eo 10
T a b l e 1. v a l u e s o f and
/(-~-~)
U(x)/(~)
for a = 0.5,h = 1.8
10
0.5 0.6 0.7 0.8 0.9
0.0 0.1 0.2 0.3 0.4
0.0 0.1 0.2 0.3 0.4
T a b l e 2. v a r i a t i o n w i t h a o f the intensity factor
K L and
K R for h = l . 2
a
KLIro
KRIr o
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
2.66708 2.15620 1.87970 1.65365 1.45074 1.25911 1.06747 0.856926 0.59819
2.49919 2.15586 1.92477 1.69832 1.48102 1.27729 1.07703 0.860411 0.598695 (Table continued)
188
Z.-G. ZHOU, B. WANG and S.-Y. DU
Table 2. variation with a of the intensity i'at:iur K L and K~ for h=l.8
Table 3. variation with a of the intensity factor A L and K R for h=3.5
a
KL / r0
KR / r0
a
KL / r0
KR / r0
0.005 0.02 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.93 0.97
5.29970 3.60718 2.35343 1.95718 1.72716 1.54558 !.38023 1.21621 1.04247 0.845288 0.595302 0.497724 0.325661
2.56963 2.41415 2.10011 1.86601 1.69710 1.53477 1.37663 1.21531 1.04238 0.845310 0.595310 0.497726 0.325661
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.99
2.21146 1.86532 1.66466 1.50385 1.35356 1.20049 1.03445 0.842234 0.594728 0.187998
1.96223 1.77971 1.62986 1.48927 1.34767 !.19834 1.03380 0.842101 0.594718 0.187998
Acknowledgment--This work was supported by the National Foundation for Excellent Young Investigators. References:
[1] [2]
[3] [4]
[5]
[6] [7] [8] [9]
Irwin, G.R., "Analysis of stresses and strains near the end of a crack transversing a plate," Journal of Applied Mechanics, vol. 24, 1957, pp361. Sneddon, I.N. and Srivastav, R.P., "'The stress field in the vicinity of a Griffith crack in a strip of finite width", International Journal of Engineering Science. Vol. 9, 1971, p 479488. Gupta, G.D., "'An integral equation approach to the semi-infinite strip problem", Journal of Applied Mechanics. December, 1973, p948. Bethem, J.P. and Koiter, W.T., "Asymptotic approximations to crack problems," Methods of Analysis and Solutions of Crack problems, Mechanics of Fracture I,(edited by Sih, G.C), 1973, Noordhoff, Holland. Yamamoto, Y. and Tokuda, N., "Stress Intensity Factors in Plate Structures Calculated by the Finite Element Method," Journal Society of Naval Arch, Japan, Vol. 130, 1971. pp 219233. Morse,P.M., and Feshbach,H., "Methods of Theoretical Physics," vol.l, McGraw-Hill, New York, 1958. ltou,S., "Transient response of a finite crack in a strip with stress free edges," Journal of Applied Mechanics, vol. 47, 1980, p801-805. Gradshteyn,I.S., and Ryzhik,I.M., "Table of integral , series and products," Academic Press, New York, 1980. Amemiya, A., and Taguchi,T., Numerical Analysis and Fortran, Maruzen, Tokyo, 1969.
Medaanics Research Communications, Vol. 25, No. 2, pp. 183-188, 1998 Copyright © 1998 E l z v i r r Science Ltd Printed in the USA. All rights re~rved 0093-6413/98 $19.00 + .00
PII S0093-6413(98)00023-g
The stress field in the vicinity of two collinear cracks subject to antiplane shear in a strip o f finite width
Z h e n - G o n g Zhou, B i a o - W a n g and Shan-Yi Du P.O.Box 2147, Center for composite materials, Harbin Institute of Technology Harbin 150001, P.R.China (Received 8 July 1997: accepted for print 3 December 1997)
I. Introduction
From an engineering point of view, the crack problems of the strip are of particular interest. From this reason, the strip problem was treated by many researchers by using different methods, for example, [1-5]. Such as, Irwin [1] gave an approximation by periodic crack solution to determining the distribution of the stress in the neighborhood of two collinear edge cracks. The stress field of a Griffith crack in a strip of finite width has been discussed by Sneddon [2] and Gupta [3] using the integral transform method. The problem of two collinear edge cracks in the strip of finite width had been treated by Bethem [4] and Yamamoto [5] using asymptotic approximations and the finite element method, respectively. In the present paper, the stress intensity factor of two collinear symmetrical cracks subjects to antiplane shear in an infinitely long strip is considered by using a new method, namely Schmidt's method. It is a simple and convenient method for solving this problem. Fourier transform is applied and a mixed boundary value problem is reduced to a set of triple integral equations. In solving the triple integral equations, the crack surface displacement is expanded in a series using Jacobi's polynomials and Schmidt's method [6,7] is used. This process is quite different from that adopted in references [1-5]. The form of solution is easy to understand. Numerical calculations are carried out for the stress intensity factor.
183
184
Z.-G. ZHOU, B. WANG and S.-Y. DU
2. F o r m u l a t i o n
of the problem
We consider a strip o f material - h _
(1)
While all other components vanish. In equations(l), g stands for the shear module of thc clasticity of the solid medium. Substituting equations(l) into the equation o f elasticity in the Zdirection renders (7 2 w
c~ 2
e7~2 w
+ ~]-~ = 0
(2)
The boundary conditions on the crack laces a t ) ' = 0 may be stated as Ibllows: r , : ( x , 0 ) = -r<, , a _< ix:-< 1 w(x,O)=O,
(3)
O_
r , : ( + h , y ) = 0,
-oo{y(~
in which r 0 is a constant having the dimension o f stress. Y +
1
h
e
x
®® 4
(4) (5)
1
f Fig. 1. An elastic strip containing two collinear cracks
COLLINEAR CRACKS UNDER SHEAR
185
3. Analysis
A Fourier cosine transform [8] is applied on equation (2) and result is ( y _>0 ) w = 2 ~ A ( ) s e -'y cos ( sx )ds + 2 ~ 1 3 (s) cos h( sx ) sin (sy)ds
(6)
Because of symmetry, it suffices to consider the problem in the first quadrant only. Making use of equations (1) and (6), equations (3) and (4) render a set of triple integral equations
0 ~0 < X <_a, 1 < X < h
~:A(s)cos(sx)ds
(7)
~: sA(s)cos(sx)ds = [® sB(s)cosh(sx)ds + :rr°
a -< x < 1, (8) J0 2p' The relationship between the function A and B is obtained by applying a Fourier sine transform [8] to equation (5): _ 2 ~, _ . . sin(r/h). B(s) sinh(sh) - ~ J0 r/At r/) ~ ar/
(9)
To solve integral equations (7) and (8), we represent displacement w by the following series: ®
, ', ixI l + a
w(x'O)= ~'~a~P2~'~ ( l - a 2 )(1 n~O
=0
--'2
,
for
(Ixl_ _f)2
,
(1_. a)2 ) i ' for
a_
2
0---Ixl-
(10)
where a n are unknown coefficients to be determined and P~YV::2)(x) is a Jacobi polynomial [8]. The Fourier cosine transform for equation (10) is [9]
A(s) = ~(s,O) = Ea~G.(s)B.J~÷,[s(
)]s-'.
(1 1)
n-O
I
(-1) 2 cos(s where G , ( s ) = / n.t .+ [ ( - I ) 2 sin(s~---~),n---1,3,5,7 .... ....
Bn =2x/-~
l-(n+ 1 + -) n!
where F(x) and Jn (x) are the Gamma and Bessel functions, respectively. Substitution equation (11) into equations (7) and (8), respectively, the equation (7) has been automatically satisfied by using the Fourier transform. Then the remaining equation (8) reduces to the form after integration with respect to x for a < x -< 1
~,anBn ~=0
G,(s)Jn.l(
s)s-'[sin(sx)-sin(as)]ds=-~B ( x - a ) +
186
Z.-G. ZHOU, B. WANG and S.-Y. DU ®
..o~a"B"f£
®2[s;ph(sr) - sinh(aS)]dse~G j 1 - a sin(r/h) ~rsinn~sh) J0 "(r/) " " ( ~ - - - r / ) - ~ - ~ s 2drl
(12)
For a large s, the integrands of the double semi-infinite integral in the equation (12) almost all exponentially, so the double semi-infinite integral in equation (12) can be evaluated numerically by Filon's method[9]. The first semi-infinite integral in equation (12) can be easily obtained by using the integral formula (see [8]). Thus equation (12) can be solved for coefficients a. by the Schmidt's method [6,7]. For brevity, we have rewritten equation (12) as oC
~a.E.(x)
= U(x).
a
(13)
n~0
where E.(x)
and U(x) are known functions and coefficients a. are unknown and to be
determined. A set of functions P. (x) which satisfy the orthogonality condition et
J l~ (x)P. (x)dx = ,¥.d . . . . . .
I
V = f~ P" (x)dx
(14)
can be constructed from the function, E. (x), such that n
P.(x) = ~-~ M " E (x) ,=0
Mnn
(15)
'
where M,. is the cofactor of the element d,. of D., which is defined as d l 0 , d t t d°~.d:2' , d r 2 d°2 ...... . . . . do. d°°d2o,d:~' dz.d°"- ,
D. =
d,. = ~jE,(x)E.(x)dx
.........................
d.o,d.~,d
(16)
i
: ..... d
j
Using equations (13-16). we obtain a. = ~--~qj M.j with q, = ]=n
l~'(x)Pj(x)dx
(17)
MM
4. Stress Intensity Factor and Numerical Results
When coefficients a, are known, the entire stress field is obtainable. However, in fracture mechanics, it is of importance to determine stress r~ in the vicinity of the crack's tips. r~.. at
y= 0 is given by r~ -
a.B. {~iG. ( s)J.., ( •"*/" n = 0
s) cos( sx)ds
COLLINEAR CRACKS UNDER SHEAR
187
2seosh(sx) _ r* . . . . . 1 - a , sin(r/h) j , ~ "-":-TT~, , , , aS l. ~ , l.rl)J,+, l . r l - - Z - ) ~ r l ~ , (18) trsmn~sn) -. z r1 + s The singular portion o f the stress field results can be obtained by using the integral formulas (see [8]). At the left end o f the right crack, we obtain the stress intensity factor K L as K L = lim x / 2 x ( a - x) - o = / l
a.B.(-1)"
(19)
x--~a
At the right end o f the right crack, we obtain the stress intensity factor K R as K R = lira 2 x / ~ - 1 ) . o " x-w
.[
2
=/~ - - ~ - ' , . B .
-
(20)
V x ( l - a ) ..0 10
For a check of the accuracy, the values of ~ ' a , E , ( x )
and U ( x ) a r e given in table 1 for
n=O
a = 05,h = 1.8. Hence, it is clearly that the Schrnidt's method is carried out satisfactorily. The
variation with a o f the stress intensity factors K L and K R are given in table 2---4. For the case of a --->0, the present problem reduces to that for a single crack [2]. For the case o f h > 3, the results in this paper show a similar tendency the results of two Griffith cracks in an infinite medium. The results o f the stress intensity factors K L and K R become higher with the increase of the crack length. The stress intensity factors K L and Ku become lower with the increase of the strip width. For the case h = i.2,a > 0.3, the stress intensity factor K L is bigger than the stress intensity factor K R . For the case a --~ 1, K L and K R approach 0. For the case a ~ 0, K L increases more
quickly than K~.
~"a.E.(x).Eo 10
T a b l e 1. v a l u e s o f and
/(-~-~)
U(x)/(~)
for a = 0.5,h = 1.8
10
0.5 0.6 0.7 0.8 0.9
0.0 0.1 0.2 0.3 0.4
0.0 0.1 0.2 0.3 0.4
T a b l e 2. v a r i a t i o n w i t h a o f the intensity factor
K L and
K R for h = l . 2
a
KLIro
KRIr o
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
2.66708 2.15620 1.87970 1.65365 1.45074 1.25911 1.06747 0.856926 0.59819
2.49919 2.15586 1.92477 1.69832 1.48102 1.27729 1.07703 0.860411 0.598695 (Table continued)
188
Z.-G. ZHOU, B. WANG and S.-Y. DU
Table 2. variation with a of the intensity i'at:iur K L and K~ for h=l.8
Table 3. variation with a of the intensity factor A L and K R for h=3.5
a
KL / r0
KR / r0
a
KL / r0
KR / r0
0.005 0.02 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.93 0.97
5.29970 3.60718 2.35343 1.95718 1.72716 1.54558 !.38023 1.21621 1.04247 0.845288 0.595302 0.497724 0.325661
2.56963 2.41415 2.10011 1.86601 1.69710 1.53477 1.37663 1.21531 1.04238 0.845310 0.595310 0.497726 0.325661
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.99
2.21146 1.86532 1.66466 1.50385 1.35356 1.20049 1.03445 0.842234 0.594728 0.187998
1.96223 1.77971 1.62986 1.48927 1.34767 !.19834 1.03380 0.842101 0.594718 0.187998
Acknowledgment--This work was supported by the National Foundation for Excellent Young Investigators. References:
[1] [2]
[3] [4]
[5]
[6] [7] [8] [9]
Irwin, G.R., "Analysis of stresses and strains near the end of a crack transversing a plate," Journal of Applied Mechanics, vol. 24, 1957, pp361. Sneddon, I.N. and Srivastav, R.P., "'The stress field in the vicinity of a Griffith crack in a strip of finite width", International Journal of Engineering Science. Vol. 9, 1971, p 479488. Gupta, G.D., "'An integral equation approach to the semi-infinite strip problem", Journal of Applied Mechanics. December, 1973, p948. Bethem, J.P. and Koiter, W.T., "Asymptotic approximations to crack problems," Methods of Analysis and Solutions of Crack problems, Mechanics of Fracture I,(edited by Sih, G.C), 1973, Noordhoff, Holland. Yamamoto, Y. and Tokuda, N., "Stress Intensity Factors in Plate Structures Calculated by the Finite Element Method," Journal Society of Naval Arch, Japan, Vol. 130, 1971. pp 219233. Morse,P.M., and Feshbach,H., "Methods of Theoretical Physics," vol.l, McGraw-Hill, New York, 1958. ltou,S., "Transient response of a finite crack in a strip with stress free edges," Journal of Applied Mechanics, vol. 47, 1980, p801-805. Gradshteyn,I.S., and Ryzhik,I.M., "Table of integral , series and products," Academic Press, New York, 1980. Amemiya, A., and Taguchi,T., Numerical Analysis and Fortran, Maruzen, Tokyo, 1969.