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Analysis of Mode-III fracture problem with multiple cracks by boundary element alternating method
ht. Printed ELSEVIER
0308-0161(94)00018-2
J. Pm.
Vex & Piprng 62 (1995) 259-261 0 1995 Elsevier Science Limited in Northern Ireland. All rights reserved 0308~0161/95/$09.50
Analysis of Mode-III fracture problem with multiple cracks by boundary element alternating method Kuen Ting Department of Nuclear Regulation,Atomic Energy Council, 67, Lane 144, Keelung Road, Section4, Taipei, Taiwan
Kwei-Kuo
Chang & Ming-Fang
Yang
Institute of Svstetn Engineering, Chung ChengInstitute of Technology, Ta-Hsi, Tao-Yuan. Taiwan (Received 20 January 1994; accepted29 March 1994)
An efficient boundary ellement alternating method was developed in this present study for the analysisof a Mode-III fracture problem with single or multiple cracksin a finite sheet.Firstly, an analytical solution for a crack in an infinite domain, subjectedto an arbitrary longitudinal shearloading acrossthe crack surface,is developed herein. The solution can then be obtained through the iterative superpositionof this analytical solution and the boundary element of a finite untracked shelet.Several Mode-III fracture problems in a finite sheet,with singleand multiple cracks under various boundary conditions, are discussedfor confirming the validity of this work. Excellent agreementcan be observed.The interaction effects amongcracksand influenceof the boundaries are alsoconsidered.
1 INTRODUCTION
Mode-III in a rectangular sheet, where its boundary is free or constrained. Recently, Ma and Zhang6 also proposed a new solution for an eccentric crack off the center line of a rectangular sheet for Mode-III and clearly pointed out that the assumptions of Zhang’s paper’ were not suitable for an eccentric crack problem. As mentioned above, the studies were restricted to a central crack or an off-central crack in a finite sheet. However, the multiple cracks for the Mode-III fracture problem in a finite domain have received little attention in this regard. Thus, this paper will propose an efficient numerical procedure to solve the Mode-III fracture problems with multiple cracks. Some powerful means among those numerical techniques are developed, such as the finite element method and the boundary element method. Although both methods can be applied to deal with the complexities of the geometries,
The piping components containing thle crack along the generator under the twist moment can be revealed in the Mode-III fracture problem. The boundary of the cross-section is taken to be a sheet with the crack in order to make the mathematics tractable. In past years, the analysis of a sheet containing cracks subjected to longitudinal loadings had been studied1 extensively. Sih’ proposed the analytical solution of stress intensity factors for several Mode-III fracture problems in an infinite domain. Zhang2,3 applied Fourier transform and Fourier series to obtain a series of general solutions of a tearing mode crack in a rectangular sheet, where a central crack surface off the center line or a crack located anywhere were discussed in these papers. Ma4T5provided the analytical solution of the stress intensity factor for a central crack of 259
260
Kuen Ting et al.
boundary conditions and multicracks, the rigorous discretization of the mesh makes these problems complicated. Hence, an efficient, accurate and versatile method that combines the advantages of the boundary (or finite) element method and the Schwarz alternating method was developed so as to overcome these shortcomings. As is known, the finite element alternating method has been successfully applied in solving three-dimensional fracture problems for an elliptical crack,7,8 and multiple cracks.’ In two-dimensional elasticity problems, a mixed mode fracture problem were solved by Chen and Chang. lo As for analyzing the K-factor and weight function for two-dimensional mixed mode problems, a finite difference alternating method has been developed by Chen and Atluri” and a boundary element alternating method by Rajiyah and Atluri.12 Nevertheless, those researches were confined to three-dimensional Mode-I and two-dimensional mixed mode problems. As far as the authors’ knowledge is concerned, the boundary element alternating method has not been studied for purely Mode-III fracture problems with multiple cracks. The objective of this work is to apply the boundary element alternating method in dealing with the Mode-III problems of a finite sheet with single and multiple cracks. The analytical solution of a Mode-III problem in an infinite domain and the boundary element method are needed in the procedure of the boundary element alternating method. Referring to the solution obtained by Sneddon and Elliott13 for the problem with the crack surface under arbitrary normal and shear loading, the general solution for an infinite plate including a crack, for which its surface is subjected to arbitrary longitudinal shear loadings represented by a polynomial of any order, is rederived. Based on the analytical solution and boundary element method, the iterative superposition process is then repeatedly applied to satisfy the prescribed boundary condition for each crack in the problem. The iterative solution process would terminate until the increments of stress intensity factor and stress on each crack converge to a small value. Several Mode-III fracture problems with single or multiple cracks subjected to various types of boundary condition are analyzed to verify the versatility and validity of the present work. The results are compared with those of other
reference solutions and the method put forward in this work is shown to be quite efficient and accurate. 2 ANALYTICAL SOLUTION FOR A CRACK SUBJECTED TO ARBITRARY LONGITUDINAL SHEAR LOADING
To establish the boundary element alternating procedure, an analytical solution for an infinite sheet subjected to arbitrary longitudinal shear loading on crack surfaces is needed. Referring to the solution obtained by Sneddon and Elliott13 for a crack under symmetric normal pressure, the analytical solution for an infinite sheet subjected to arbitrary longitudinal shear force is derived here. Consider an infinite domain with a crack at y = 0 and -a I x I a (where a is half of the crack length), shown in Fig. 1. The crack surfaces are subjected to arbitrary longitudinal shear loading, which can be represented by a polynomial of any order. The governing equation is given as follows: v2w=o
(1)
where w is a displacement of the z-direction. The boundary condition for this problem can be prescribed as below: (i) Stresses rY,, z,, and displacement w approach zero at the infinite boundaries, i.e. x2+y2+q (ii) Stress rYZis represented as ryi = -f(x)
aty=Oand IxIsa. (iii) Displacement
$ Fig. 1. An
= 2 A, . (I)’ n=O
w = 0 at y = 0 and Ix/> a,
I+-2a-I
i
infinite plate containing a crack subjected arbitrarily longitudinal shear loading.
to
Analysis
of Mode-III
fracture
where N is a positive integer and A,, are polynomial coefficients, It follows from the symmetry and antisymmetry loading case with respect to the x-axis that only the semiinfinite elastic medium for y ~0 needs to be considered. One can employ the Fourier transform technique to solve this problem; the analytical solutions are derived as +,
Y> = -
problem
with multiple
even and odd cases are found from eqns (5)-(10) as follows:
;. J=m.cos[xd{=f,(x) 0
Olxra
(11)
x>a
(12)
01x la
(13)
J0“3x8 5’ ~
and
cos&d[=O
1 . sin tx d[=f,(x)
2%
. rm Ii- . e+“.(cos[x-i.sin&)d< J-r.
261
cracks
J0%tJ 5’
(2)
sin 5x d[ = 0 x > a
151
(14)
The variables are next defined as
. -lf(~ . e-l*‘y. (cosex- i . sin&x>idt (3) J z,,(x,y)
p=ija;
q=X
=2 27L
Since the function f(x) describing the prescribed longitudinal shear force zYz can be separated into two parts, i.e. even parts h(x) and odd parts h(x), eqns (2)-(4) can be set apart into two forms as follows. (a) For even function h(x):
cos(p. q> =
( )
7c. p . ‘I 1’2 2 .
J-lR(P.
7)
n . p . ‘I u2
sin(p, ‘I) = 2 (
* UP.
1
‘I)
where Lli2(p. 7) and J&p. 7) are Bessel functions of the first kind. Two sets of dual integrals are obtained by substituting eqn (1.5) into eqns (ll)-(14) as follows:
OL J cc J @G J cc J
p.F,(~).J-,,,(~.rl)d~=g,(rl)
O-=r)ll
0
F,(p). Jmm(p.
(b) for odd function fO(x): w(x,y)=L
(8)
7’ 1
(17)
and
P . F,(P).
j’?,(t).
q) dp = 0
0
mfd5) e--5.y, sin lx d5 f&o J 6’
z,,(x, y) = G.
(16)
e-*.’ . sin 6x Id[
(9)
0
.L(P.
77) dp
= gob)
0 5 rls
1
0
F,(P)
(18)
. Jm(p.
rl)
dp
= 0
rl ’
1
(19)
0
since w (x, y), ~,,=(x,y) and zX,(x, y) are determined by f&i> and .fX). Hence, f&Cl alnd .ko(S> must be decided. Two sets of dual inte.grals of
The solution of dual integrals was derived by Sneddon.14 In this derived procedure, the solutions F,(p) and F,(p) of the above dual
Kuen Ting et al.
262
integrals for the cases of even and odd are obtained as
few=($)‘;‘. [Jo(P)
s 1
x
x’/2
. (1
-
xy
. g,(x)
I
J&(x.
u)
(24)
No,
Al, Aa . . . , AIT
[S] is a relation function between {T} and {A}; [S] is a function of crack length and space. When the coefficient {A} is decided the traction {T} at any location of the infinite domain can be computed from eqn (24).
u”~. (1 - u2)l” du
1
s0
6’7 = [Sk4 where {T} and {A} are defined as
{Al = 1
0
X
= 0, 1,2 * * *) in the matrix form as
and
dx
0
+ /I.
A,(n
. .P2.
and
F,(p) =(y2. [UP)
3 COMPUTATION OF MODE-III INTENSITY FACTOR
STRESS
1
x
I
. (1 - X2)“2 . g&x) d.X
x3’2
0
A crack in a solid can be expressed in three different modes, namely, opening mode (ModeI), sliding mode (Mode-II) and tearing mode (Mode-III). The stability of a crack is described by three important parameters (K,, Ku, K,,,), called the stress intensity factors. From the definition by Erdogan,15 K,,, can be expressed as
1
+ p.
u3’*. (1 - u2)‘” du
I
0
1
X
I0
&(X
. u)
. x5”.
J2CP
* 4
h]
(21)
Substituting eqn (15) into eqns (20) and (21) and through suitable rearrangement, the functions fe([) and fO(t) are determined as
u3
n l/2 -.a2.[. =4
X
I
.A
i n=0,2.
Jo(a . 0 + a.
-.’ I- ; + 2 i 1
K,,, = lim vC27c.(x - a) . Tvz(x, 0) .x-Cl
(25)
Substituting the aforesaid analytical solution of z7 into eqn (25), and performing careful manipulation, K,,, is obtained as: (i) for even order ~1,stress intensity factor Kf,, can then be expressed as
5. i’xn+2.Jl(a.t.x)~] 0 (22)
. A, . (n + 2)
(26)
(ii) for odd order ~1,stress intensity factor Kg, can then be expressed as 1
X
Jl(a.
e)+a.
5.
~“+‘..&(a.
c.x)dx
1 (23)
where I(n) is a Gamma function. Substituting fe([) and fO([) into eqns (5)-(10) the closed form solutions of o, r,, and r,, are thus found. They can be expressed in terms of the coefficient
n-1 *
c n=1,3,..
r i ;+I 1 A,
. (n + 3)
(27)
Analysis
of Mode-III
fracture
The complete solution of stress intensity factor K,,, is thus Kill = G
+G
VC-= [WA)
(29)
where (K} is defined as
and subscripts A and B represent the craclk tips A and B; [R] is a function of crack length. Once the coefficient {A} is determined the stress intensity factor (K) can be evaluated from eqn (29).
INTEGRAL
EQUATION
The conventional boundary element method was applied to calculate the stress at the location of a fictitious crack in this work. Based on consideration of the weight residual, Betti’s reciprocal theorem and Green’s third identity, etc., the boundary integral equation for a potential problem (Laplace equation) was derived aP C’w’:lqo*dI-Lwq*dI
(30)
where C’ = $ on the smooth boundary, and o* and q* are the weight functions of displacement and traction for the potential problems. The weight functions of o* and q* are given in Ref. 16. According to eqn (30) the value of displacement and tractions on the boundary are solved. Once this is done, the displacement and the stresses at any internal point in the. domain can be computed by the following equations:
w’=Lqm*dI-LWq*dI-
(31)
and ~=i,q(!$$)idl.-~u(~)idl~
263
with multiple cracks
The detailed numerical procedure of the boundary element method can be obtained in Ref. 16.
(28)
As a result, K,,, can be indicated as
4 BOUNDARY
problem
5 BOUNDARY METHOD
ELEMENT
ALTERNATING
The procedure of the boundary element alternating method for solving a multiple crack problem in this work can be described as follows: (1) The relation between geometric arrangement and boundary condition of a bounded crack is input. (2) The conventional boundary element method is applied to compute the residual stress (rVz)$ at the location of a fictitious crack k (k = 1, 2, . . . , N), where N is the total number of cracks existing in the plate. The coordinate of the fictitious crack k can be presented by some internal point in a finite sheet. (3) As the crack surface is stress free, reverse the sign of the residual stress (z,-)fk and fit it with a polynomial of appropriate order. The fitted residual stress (zVz)“, can be expressed as (z,J~ = [L]l{A}‘j!, where
and {A}“, = {A”, A: A; A: a. ~A:}
n(= 0, 1, 2, . . .) is a polynomial order, x,, is the coordinate (internal point) of crack k, uk is one half of the crack length of crack k, and {A}: is a polynomial coefficient for crack k. For better expediency, {A}: can be determined by a least-squares method. (4) The interactive effect among cracks is induced by the longitudinal shear load zYzonly. Substitute {A}“, into the analytical solution which is shown by eqn (24). Therefore, the interactive residual stress (rYz) induced by crack k at a fictitious crack I (1 = 1, 2, . . . , N; 1# k) is computed. Simultaneously, the interactive residual coefficient {A}lk can be evaluated by the least-squares method. If I > k, add {A}, to {A}? and compute the updated {A}: as k-l
(32) {Ali’
=
{A):
-
c I=1
{&-.u(k
=
I,‘&
. . . , K)
Thereupon the residual stress of crack k induced by the rest of the crack is to be released for the
Kuen Ting et al.
264
next iteration process and can be found as {A}: = XEktl {A}kl. All the steps are repeated until the cumulative residual stress on each crack can be neglected. The resulting coefficient {A}k = {A}: + {A}: + {Afi + * * * for current iteration is obtained. (5) Substituting the resulting coefficient {A}k into eqns (24) and (29), the stress, displacement and stress intensity factor (K,,,) can be computed and updated. (6) When the variation of stress intensity factor for all crack tips is smaller than a small value y( = O.OOS),i.e. IK,,,(current step) - K1rl(previous step)/
where N is the number of cracks existing in the finite sheet. (8) To match the boundary condition, reverse the sign of the resultant residual nodal force or the resultant nodal displacement {Q}z calculated in procedure (7) and consider them as new applied load and displacement acting on the external boundaries of the bounded uncrack sheet. Repeat all the iteration steps (2) to (7) until the stress intensity factor KIII on each crack tip is negligible.
quadratic element is adopted and each crack surface is simulated by seven or eleven internal points in all analyses. Example 1: Single central crack in a rectangular sheet subject to uniform antiplane shear loading with various boundary conditions
A rectangular sheet of edge length 2b x 2h contains a single central crack of length 2~2.The crack surface is subjected to antiplane shear stress r as shown in Fig. 2. Boundary element meshes with fewer than 40 quadratic elements are employed. As seen in Fig. 3, four different boundary conditions, i.e. (a) four free edges, (b) fixed edge perpendicular to central crack, (c) fixed edge parallel to central crack, and (d) four fixed edges, are solved, respectively. The variation of normalized stress intensity factor F,,, = KIII/rV’& with different ratios of b/a versus the various ratios of h/a are presented in Fig. 4(a)-(d). The results are compared with the solution obtained by Ma.4,5 Excellent correlations between the computed results and the reference solutions are observed. When the value b/a is equal to 4.0, the values of F,,, are equivalent to the solutions for b/a = m obtained by Ma. These phenomena are also shown in Fig. 4(a)-(d). In Fig. 4(d), if the mark of 0(&r = O-915) relative to h/a = 4 were to be corrected to the mark of A(&, = O-975), it would be consistent with the present results. Example 2: Straight crack located at an arbitrary position in a square sheet subjected uniform antiplane shear stress with various boundary conditions
A straight crack of length 2a is located at an Quadratic
6 RESULTS
AND
to
element
DISCUSSION
Several Mode-III fracture problems in a finite sheet with single or multiple cracks under different boundary conditions are considered for verifying the validity of the technique developed in this work. The interactive coefficient among cracks and the influence of the boundary on the calculation of stress intensity factors are also considered. In the boundary element method, the
I k- 2a-I I
2b
II
___)I
condition and boundary a center-cracked plate.
element
mesh for
Analysis
of Mode-III
fracture
problem
265
with multiple cracks Quadratic
element
-x
(b)
2b
2b I< >I Fig. 5. Geometry condition and boundary element meshfor an eccentric cracked plate.
(4
Fig. 3. Four different boundary conditions: (a) four free edges; (b) fixed edge perpendicular to central crack; (c) fixed edgeparallel to central crack; (d) four fixeid edges.
intensity factor firI = KIIIlr& (where F,,, = max{F,,,,, F,,,,}) with various e, and e, versus the various ratios of a/b are presented in Fig. 6. For comparison purposes, the analytical solution of an eccentric crack off the center line, derived by Ma and Zhang,6 is also shown. Good agreement is presented in Fig. 6. The other solutions with various eccentric distances e, and e, are also presented.
arbitrary position in a square sheet with edges of length 2b and eccentric distances e, and e,,. A boundary element mesh of not more than 40 quadratic elements is adopted, as seen im Fig. 5. The crack surfaces are subjected to antiplane shear stress r. The variation of normalized stress
a ,.,(I
)
0l
2.2 2.0 i
:Present (b,a=m)
[:
1.2,1.4,1.6 b'a:
Ma4
i-l
2.0,4.0
/;I>yz3$iq 0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
0.5
h/a
1.0
1.5
2.0
1
2.5
3.0
3.5
4.0
4.5
4.0
4.5
h/a
(4 1.3r 1.2--
-
:Present
1.2,1.4,1.6 b/a :
l.f--
2.0.4.0
1.0~FIrI 0.9-0.6 -0.7 -0.6 -3:55
0.5-i
0.0
0.5
1.0
1.5
2.0
2.5
3.0
0.54
I
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
h/a h/a Fig. 4. Variation of normalized stressintensity lEactor(F,,,) with different ratios of b/a vs. the various ratios of h/a under various boundary conditions (shown in Fig. 3).
266
Kuen Ting et al.
0.0 0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.6
CI.9
Example 3: Two collinear cracks in a square sheet subjected to uniform antiplane shear stress To show the interaction effect among the cracks and boundaries, the problem of a square sheet containing two collinear cracks subjected to uniform antiplane stress z at the crack surface is considered and is shown in Fig. 7. The sheet has edges of length 2b, and a is one half of the crack length. A boundary element mesh with 40 quadratic elements was adopted for this case. The variation of fi,, = Krrr/&& versus the ratio 2ald is shown in Fig. 8, where d is the distance between the midpoint of the cracks. As seen in Fig. 8, three different ratios of b/a are considered; i.e. b/a = 6, 24 and 48. The strong interaction effect between cracks can be seen for the ratio 2a/d = O-9 (the larger the ratio 2a/d, the closer the two cracks). The value of fi,, for crack tip A(&,,) is larger than that for crack element
m7
I+2a-4 ! I<
0.9
1.0
Fig. 8. Variation of normalized stressintensity factor (&,) with different b/a vs. the ratio 2a/d.
tip B(&,). When the ratio 2a/d gradually decreases, the interaction effect between cracks also decreases. As 0.22 5 2a/d I 0.33 relative to b/a = 6 or as O-04 r2a/d ~0.11 relative to b/a = 24, the value of F,,, increases again (F’,,, is slightly larger than F,,,,) due to the influence of the boundary. When the ratio b/a is equal to 48, as would be expected, the presented value of FIIIn is similar to that of the solution for an infinite sheet displayed in the handbook by Sih.’ 7 CONCLUSIONS An efficient and accurate boundary element method for the stress analysis of a linearly elastic, homogeneous and isotropic solid, with one or multiple cracks under various boundary conditions subjected to antiplane shear, has been developed in this paper. Using only a simple conventional boundary element mesh, an excellent solution for the Mode-III stress intensity factor can be calculated. The interaction effect among cracks and the influence of boundaries can be evaluated efficiently. For the derivation of the potential problem, this work can also be extended to the study of thermal problems with one or multiple cracks in a finite (or infinite) domain in the near future.
I+ 2a4 !I
2b
0.6
2a/d
a/b
Fig. 6. Variation of normalized stress intensity factor (&) with various e, and e,,vs. the different ratios of a/b.
Quadratic
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
>i
Fig. 7. Geometry condition and boundary element meshfor a squareplate containing two collinear cracks.
REFERENCES 1. Sih, G. C., Handbook of Stress Intensity Factors, Noordhoff, Leyden, The Netherlands,1973. 2. Zhang, X. S., The general solution of a central crack off the center line of a rectangular sheet for Mode-III. Erzgng Fract. Me&., 28 (1987) 147-55.
Analysis
of Mode-III
fracture
3. Zhang, X. S., A tear mode crack located anywhere in a finite rectangular sheet. Engng Fract. Mech., 383(1989) 509-x.
4. Ma, S. W., A central crack in a rectangular sheNet where its boundary is subjectedto an arbitrary anti-plane load. Engng Fract. Mech., 30 (1988)435-43. 5. Ma, S. W., A central crack of Mode-III in a rectangular sheetwith fixed edges.Int. J. of Fract., 39 (1989) 323-9. 6. Ma, S. W. & Zhang, L. X., A new solution of an eccentric crack off the center line of a rectangular sheet for Mode-III. Engng Fract. Mech., 40 (1991) l--7. 7. Nishioka, T. & Atluri, S. N., Analytical solution for embeddedelliptical cracks, and finite element alternating method for elliptical surface cracks, subjected to arbitrary loadings. Engng Fract. Mech., 17 (1983) 247-68. 8. Rajiyah, H. & Atluri, S. N., Analysis of embeddedand surfaceelliptical flaws in transverselyisotropic bodiesby the finite element alternating method. J. ofApp/. Mech., 58 (1991) 435-43.
9. O’Donoghue, P. E., Nishioka, T. & Atluri, S. N., Multiple surfacecracksin pressurevessels.Engng Fract. Me&.,
20 (1984) 545-60.
problem
with multiple cracks
267
10. Chen, W. H. & Chang, C. S., Analysis of two dimension fracture problems with multiple cracks under mixed boundary conditions. Engng Fract. Mech., 34 (1989) 921-34. 11. Chen, K. L. & Atluri,
S. N., A finite difference alternating method for a cost effective determination of weight-functions for orthotropic materials in mixedmode fracture. Engng Fract. Mech., 36 (1990) 327-40. 12. Rajiyah, H. & Atluri, S. N., Evaluation of K-factors and weight functions for 2-D mixed-modemultiple cracksby the boundary elementalternating method. Engng. Fract.
Mech., 32 (1989) 911-22. 13. Sneddon, I. N. & Elliott, H. A., The opening of a Griffth crack under internal pressure.Q. Appl. Math., 4
(1946) 262-7. 14. Sneddon, I. N., Fourier Transform, McGraw-Hill, New York, 1951. 15. Erdogan, F., Stressintensity factors. Trans. ASME, J. Appl. Mech., 50 (1983) 992-1002. 16. Brebbia, C. A. & Dominguez, J., Boundary Elements; An introductory Course, McGraw-Hill, New York, 1989.
0308-0161(94)00018-2
J. Pm.
Vex & Piprng 62 (1995) 259-261 0 1995 Elsevier Science Limited in Northern Ireland. All rights reserved 0308~0161/95/$09.50
Analysis of Mode-III fracture problem with multiple cracks by boundary element alternating method Kuen Ting Department of Nuclear Regulation,Atomic Energy Council, 67, Lane 144, Keelung Road, Section4, Taipei, Taiwan
Kwei-Kuo
Chang & Ming-Fang
Yang
Institute of Svstetn Engineering, Chung ChengInstitute of Technology, Ta-Hsi, Tao-Yuan. Taiwan (Received 20 January 1994; accepted29 March 1994)
An efficient boundary ellement alternating method was developed in this present study for the analysisof a Mode-III fracture problem with single or multiple cracksin a finite sheet.Firstly, an analytical solution for a crack in an infinite domain, subjectedto an arbitrary longitudinal shearloading acrossthe crack surface,is developed herein. The solution can then be obtained through the iterative superpositionof this analytical solution and the boundary element of a finite untracked shelet.Several Mode-III fracture problems in a finite sheet,with singleand multiple cracks under various boundary conditions, are discussedfor confirming the validity of this work. Excellent agreementcan be observed.The interaction effects amongcracksand influenceof the boundaries are alsoconsidered.
1 INTRODUCTION
Mode-III in a rectangular sheet, where its boundary is free or constrained. Recently, Ma and Zhang6 also proposed a new solution for an eccentric crack off the center line of a rectangular sheet for Mode-III and clearly pointed out that the assumptions of Zhang’s paper’ were not suitable for an eccentric crack problem. As mentioned above, the studies were restricted to a central crack or an off-central crack in a finite sheet. However, the multiple cracks for the Mode-III fracture problem in a finite domain have received little attention in this regard. Thus, this paper will propose an efficient numerical procedure to solve the Mode-III fracture problems with multiple cracks. Some powerful means among those numerical techniques are developed, such as the finite element method and the boundary element method. Although both methods can be applied to deal with the complexities of the geometries,
The piping components containing thle crack along the generator under the twist moment can be revealed in the Mode-III fracture problem. The boundary of the cross-section is taken to be a sheet with the crack in order to make the mathematics tractable. In past years, the analysis of a sheet containing cracks subjected to longitudinal loadings had been studied1 extensively. Sih’ proposed the analytical solution of stress intensity factors for several Mode-III fracture problems in an infinite domain. Zhang2,3 applied Fourier transform and Fourier series to obtain a series of general solutions of a tearing mode crack in a rectangular sheet, where a central crack surface off the center line or a crack located anywhere were discussed in these papers. Ma4T5provided the analytical solution of the stress intensity factor for a central crack of 259
260
Kuen Ting et al.
boundary conditions and multicracks, the rigorous discretization of the mesh makes these problems complicated. Hence, an efficient, accurate and versatile method that combines the advantages of the boundary (or finite) element method and the Schwarz alternating method was developed so as to overcome these shortcomings. As is known, the finite element alternating method has been successfully applied in solving three-dimensional fracture problems for an elliptical crack,7,8 and multiple cracks.’ In two-dimensional elasticity problems, a mixed mode fracture problem were solved by Chen and Chang. lo As for analyzing the K-factor and weight function for two-dimensional mixed mode problems, a finite difference alternating method has been developed by Chen and Atluri” and a boundary element alternating method by Rajiyah and Atluri.12 Nevertheless, those researches were confined to three-dimensional Mode-I and two-dimensional mixed mode problems. As far as the authors’ knowledge is concerned, the boundary element alternating method has not been studied for purely Mode-III fracture problems with multiple cracks. The objective of this work is to apply the boundary element alternating method in dealing with the Mode-III problems of a finite sheet with single and multiple cracks. The analytical solution of a Mode-III problem in an infinite domain and the boundary element method are needed in the procedure of the boundary element alternating method. Referring to the solution obtained by Sneddon and Elliott13 for the problem with the crack surface under arbitrary normal and shear loading, the general solution for an infinite plate including a crack, for which its surface is subjected to arbitrary longitudinal shear loadings represented by a polynomial of any order, is rederived. Based on the analytical solution and boundary element method, the iterative superposition process is then repeatedly applied to satisfy the prescribed boundary condition for each crack in the problem. The iterative solution process would terminate until the increments of stress intensity factor and stress on each crack converge to a small value. Several Mode-III fracture problems with single or multiple cracks subjected to various types of boundary condition are analyzed to verify the versatility and validity of the present work. The results are compared with those of other
reference solutions and the method put forward in this work is shown to be quite efficient and accurate. 2 ANALYTICAL SOLUTION FOR A CRACK SUBJECTED TO ARBITRARY LONGITUDINAL SHEAR LOADING
To establish the boundary element alternating procedure, an analytical solution for an infinite sheet subjected to arbitrary longitudinal shear loading on crack surfaces is needed. Referring to the solution obtained by Sneddon and Elliott13 for a crack under symmetric normal pressure, the analytical solution for an infinite sheet subjected to arbitrary longitudinal shear force is derived here. Consider an infinite domain with a crack at y = 0 and -a I x I a (where a is half of the crack length), shown in Fig. 1. The crack surfaces are subjected to arbitrary longitudinal shear loading, which can be represented by a polynomial of any order. The governing equation is given as follows: v2w=o
(1)
where w is a displacement of the z-direction. The boundary condition for this problem can be prescribed as below: (i) Stresses rY,, z,, and displacement w approach zero at the infinite boundaries, i.e. x2+y2+q (ii) Stress rYZis represented as ryi = -f(x)
aty=Oand IxIsa. (iii) Displacement
$ Fig. 1. An
= 2 A, . (I)’ n=O
w = 0 at y = 0 and Ix/> a,
I+-2a-I
i
infinite plate containing a crack subjected arbitrarily longitudinal shear loading.
to
Analysis
of Mode-III
fracture
where N is a positive integer and A,, are polynomial coefficients, It follows from the symmetry and antisymmetry loading case with respect to the x-axis that only the semiinfinite elastic medium for y ~0 needs to be considered. One can employ the Fourier transform technique to solve this problem; the analytical solutions are derived as +,
Y> = -
problem
with multiple
even and odd cases are found from eqns (5)-(10) as follows:
;. J=m.cos[xd{=f,(x) 0
Olxra
(11)
x>a
(12)
01x la
(13)
J0“3x8 5’ ~
and
cos&d[=O
1 . sin tx d[=f,(x)
2%
. rm Ii- . e+“.(cos[x-i.sin&)d< J-r.
261
cracks
J0%tJ 5’
(2)
sin 5x d[ = 0 x > a
151
(14)
The variables are next defined as
. -lf(~ . e-l*‘y. (cosex- i . sin&x>idt (3) J z,,(x,y)
p=ija;
q=X
=2 27L
Since the function f(x) describing the prescribed longitudinal shear force zYz can be separated into two parts, i.e. even parts h(x) and odd parts h(x), eqns (2)-(4) can be set apart into two forms as follows. (a) For even function h(x):
cos(p. q> =
( )
7c. p . ‘I 1’2 2 .
J-lR(P.
7)
n . p . ‘I u2
sin(p, ‘I) = 2 (
* UP.
1
‘I)
where Lli2(p. 7) and J&p. 7) are Bessel functions of the first kind. Two sets of dual integrals are obtained by substituting eqn (1.5) into eqns (ll)-(14) as follows:
OL J cc J @G J cc J
p.F,(~).J-,,,(~.rl)d~=g,(rl)
O-=r)ll
0
F,(p). Jmm(p.
(b) for odd function fO(x): w(x,y)=L
(8)
7’ 1
(17)
and
P . F,(P).
j’?,(t).
q) dp = 0
0
mfd5) e--5.y, sin lx d5 f&o J 6’
z,,(x, y) = G.
(16)
e-*.’ . sin 6x Id[
(9)
0
.L(P.
77) dp
= gob)
0 5 rls
1
0
F,(P)
(18)
. Jm(p.
rl)
dp
= 0
rl ’
1
(19)
0
since w (x, y), ~,,=(x,y) and zX,(x, y) are determined by f&i> and .fX). Hence, f&Cl alnd .ko(S> must be decided. Two sets of dual inte.grals of
The solution of dual integrals was derived by Sneddon.14 In this derived procedure, the solutions F,(p) and F,(p) of the above dual
Kuen Ting et al.
262
integrals for the cases of even and odd are obtained as
few=($)‘;‘. [Jo(P)
s 1
x
x’/2
. (1
-
xy
. g,(x)
I
J&(x.
u)
(24)
No,
Al, Aa . . . , AIT
[S] is a relation function between {T} and {A}; [S] is a function of crack length and space. When the coefficient {A} is decided the traction {T} at any location of the infinite domain can be computed from eqn (24).
u”~. (1 - u2)l” du
1
s0
6’7 = [Sk4 where {T} and {A} are defined as
{Al = 1
0
X
= 0, 1,2 * * *) in the matrix form as
and
dx
0
+ /I.
A,(n
. .P2.
and
F,(p) =(y2. [UP)
3 COMPUTATION OF MODE-III INTENSITY FACTOR
STRESS
1
x
I
. (1 - X2)“2 . g&x) d.X
x3’2
0
A crack in a solid can be expressed in three different modes, namely, opening mode (ModeI), sliding mode (Mode-II) and tearing mode (Mode-III). The stability of a crack is described by three important parameters (K,, Ku, K,,,), called the stress intensity factors. From the definition by Erdogan,15 K,,, can be expressed as
1
+ p.
u3’*. (1 - u2)‘” du
I
0
1
X
I0
&(X
. u)
. x5”.
J2CP
* 4
h]
(21)
Substituting eqn (15) into eqns (20) and (21) and through suitable rearrangement, the functions fe([) and fO(t) are determined as
u3
n l/2 -.a2.[. =4
X
I
.A
i n=0,2.
Jo(a . 0 + a.
-.’ I- ; + 2 i 1
K,,, = lim vC27c.(x - a) . Tvz(x, 0) .x-Cl
(25)
Substituting the aforesaid analytical solution of z7 into eqn (25), and performing careful manipulation, K,,, is obtained as: (i) for even order ~1,stress intensity factor Kf,, can then be expressed as
5. i’xn+2.Jl(a.t.x)~] 0 (22)
. A, . (n + 2)
(26)
(ii) for odd order ~1,stress intensity factor Kg, can then be expressed as 1
X
Jl(a.
e)+a.
5.
~“+‘..&(a.
c.x)dx
1 (23)
where I(n) is a Gamma function. Substituting fe([) and fO([) into eqns (5)-(10) the closed form solutions of o, r,, and r,, are thus found. They can be expressed in terms of the coefficient
n-1 *
c n=1,3,..
r i ;+I 1 A,
. (n + 3)
(27)
Analysis
of Mode-III
fracture
The complete solution of stress intensity factor K,,, is thus Kill = G
+G
VC-= [WA)
(29)
where (K} is defined as
and subscripts A and B represent the craclk tips A and B; [R] is a function of crack length. Once the coefficient {A} is determined the stress intensity factor (K) can be evaluated from eqn (29).
INTEGRAL
EQUATION
The conventional boundary element method was applied to calculate the stress at the location of a fictitious crack in this work. Based on consideration of the weight residual, Betti’s reciprocal theorem and Green’s third identity, etc., the boundary integral equation for a potential problem (Laplace equation) was derived aP C’w’:lqo*dI-Lwq*dI
(30)
where C’ = $ on the smooth boundary, and o* and q* are the weight functions of displacement and traction for the potential problems. The weight functions of o* and q* are given in Ref. 16. According to eqn (30) the value of displacement and tractions on the boundary are solved. Once this is done, the displacement and the stresses at any internal point in the. domain can be computed by the following equations:
w’=Lqm*dI-LWq*dI-
(31)
and ~=i,q(!$$)idl.-~u(~)idl~
263
with multiple cracks
The detailed numerical procedure of the boundary element method can be obtained in Ref. 16.
(28)
As a result, K,,, can be indicated as
4 BOUNDARY
problem
5 BOUNDARY METHOD
ELEMENT
ALTERNATING
The procedure of the boundary element alternating method for solving a multiple crack problem in this work can be described as follows: (1) The relation between geometric arrangement and boundary condition of a bounded crack is input. (2) The conventional boundary element method is applied to compute the residual stress (rVz)$ at the location of a fictitious crack k (k = 1, 2, . . . , N), where N is the total number of cracks existing in the plate. The coordinate of the fictitious crack k can be presented by some internal point in a finite sheet. (3) As the crack surface is stress free, reverse the sign of the residual stress (z,-)fk and fit it with a polynomial of appropriate order. The fitted residual stress (zVz)“, can be expressed as (z,J~ = [L]l{A}‘j!, where
and {A}“, = {A”, A: A; A: a. ~A:}
n(= 0, 1, 2, . . .) is a polynomial order, x,, is the coordinate (internal point) of crack k, uk is one half of the crack length of crack k, and {A}: is a polynomial coefficient for crack k. For better expediency, {A}: can be determined by a least-squares method. (4) The interactive effect among cracks is induced by the longitudinal shear load zYzonly. Substitute {A}“, into the analytical solution which is shown by eqn (24). Therefore, the interactive residual stress (rYz) induced by crack k at a fictitious crack I (1 = 1, 2, . . . , N; 1# k) is computed. Simultaneously, the interactive residual coefficient {A}lk can be evaluated by the least-squares method. If I > k, add {A}, to {A}? and compute the updated {A}: as k-l
(32) {Ali’
=
{A):
-
c I=1
{&-.u(k
=
I,‘&
. . . , K)
Thereupon the residual stress of crack k induced by the rest of the crack is to be released for the
Kuen Ting et al.
264
next iteration process and can be found as {A}: = XEktl {A}kl. All the steps are repeated until the cumulative residual stress on each crack can be neglected. The resulting coefficient {A}k = {A}: + {A}: + {Afi + * * * for current iteration is obtained. (5) Substituting the resulting coefficient {A}k into eqns (24) and (29), the stress, displacement and stress intensity factor (K,,,) can be computed and updated. (6) When the variation of stress intensity factor for all crack tips is smaller than a small value y( = O.OOS),i.e. IK,,,(current step) - K1rl(previous step)/
where N is the number of cracks existing in the finite sheet. (8) To match the boundary condition, reverse the sign of the resultant residual nodal force or the resultant nodal displacement {Q}z calculated in procedure (7) and consider them as new applied load and displacement acting on the external boundaries of the bounded uncrack sheet. Repeat all the iteration steps (2) to (7) until the stress intensity factor KIII on each crack tip is negligible.
quadratic element is adopted and each crack surface is simulated by seven or eleven internal points in all analyses. Example 1: Single central crack in a rectangular sheet subject to uniform antiplane shear loading with various boundary conditions
A rectangular sheet of edge length 2b x 2h contains a single central crack of length 2~2.The crack surface is subjected to antiplane shear stress r as shown in Fig. 2. Boundary element meshes with fewer than 40 quadratic elements are employed. As seen in Fig. 3, four different boundary conditions, i.e. (a) four free edges, (b) fixed edge perpendicular to central crack, (c) fixed edge parallel to central crack, and (d) four fixed edges, are solved, respectively. The variation of normalized stress intensity factor F,,, = KIII/rV’& with different ratios of b/a versus the various ratios of h/a are presented in Fig. 4(a)-(d). The results are compared with the solution obtained by Ma.4,5 Excellent correlations between the computed results and the reference solutions are observed. When the value b/a is equal to 4.0, the values of F,,, are equivalent to the solutions for b/a = m obtained by Ma. These phenomena are also shown in Fig. 4(a)-(d). In Fig. 4(d), if the mark of 0(&r = O-915) relative to h/a = 4 were to be corrected to the mark of A(&, = O-975), it would be consistent with the present results. Example 2: Straight crack located at an arbitrary position in a square sheet subjected uniform antiplane shear stress with various boundary conditions
A straight crack of length 2a is located at an Quadratic
6 RESULTS
AND
to
element
DISCUSSION
Several Mode-III fracture problems in a finite sheet with single or multiple cracks under different boundary conditions are considered for verifying the validity of the technique developed in this work. The interactive coefficient among cracks and the influence of the boundary on the calculation of stress intensity factors are also considered. In the boundary element method, the
I k- 2a-I I
2b
II
___)I
condition and boundary a center-cracked plate.
element
mesh for
Analysis
of Mode-III
fracture
problem
265
with multiple cracks Quadratic
element
-x
(b)
2b
2b I< >I Fig. 5. Geometry condition and boundary element meshfor an eccentric cracked plate.
(4
Fig. 3. Four different boundary conditions: (a) four free edges; (b) fixed edge perpendicular to central crack; (c) fixed edgeparallel to central crack; (d) four fixeid edges.
intensity factor firI = KIIIlr& (where F,,, = max{F,,,,, F,,,,}) with various e, and e, versus the various ratios of a/b are presented in Fig. 6. For comparison purposes, the analytical solution of an eccentric crack off the center line, derived by Ma and Zhang,6 is also shown. Good agreement is presented in Fig. 6. The other solutions with various eccentric distances e, and e, are also presented.
arbitrary position in a square sheet with edges of length 2b and eccentric distances e, and e,,. A boundary element mesh of not more than 40 quadratic elements is adopted, as seen im Fig. 5. The crack surfaces are subjected to antiplane shear stress r. The variation of normalized stress
a ,.,(I
)
0l
2.2 2.0 i
:Present (b,a=m)
[:
1.2,1.4,1.6 b'a:
Ma4
i-l
2.0,4.0
/;I>yz3$iq 0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
0.5
h/a
1.0
1.5
2.0
1
2.5
3.0
3.5
4.0
4.5
4.0
4.5
h/a
(4 1.3r 1.2--
-
:Present
1.2,1.4,1.6 b/a :
l.f--
2.0.4.0
1.0~FIrI 0.9-0.6 -0.7 -0.6 -3:55
0.5-i
0.0
0.5
1.0
1.5
2.0
2.5
3.0
0.54
I
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
h/a h/a Fig. 4. Variation of normalized stressintensity lEactor(F,,,) with different ratios of b/a vs. the various ratios of h/a under various boundary conditions (shown in Fig. 3).
266
Kuen Ting et al.
0.0 0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.6
CI.9
Example 3: Two collinear cracks in a square sheet subjected to uniform antiplane shear stress To show the interaction effect among the cracks and boundaries, the problem of a square sheet containing two collinear cracks subjected to uniform antiplane stress z at the crack surface is considered and is shown in Fig. 7. The sheet has edges of length 2b, and a is one half of the crack length. A boundary element mesh with 40 quadratic elements was adopted for this case. The variation of fi,, = Krrr/&& versus the ratio 2ald is shown in Fig. 8, where d is the distance between the midpoint of the cracks. As seen in Fig. 8, three different ratios of b/a are considered; i.e. b/a = 6, 24 and 48. The strong interaction effect between cracks can be seen for the ratio 2a/d = O-9 (the larger the ratio 2a/d, the closer the two cracks). The value of fi,, for crack tip A(&,,) is larger than that for crack element
m7
I+2a-4 ! I<
0.9
1.0
Fig. 8. Variation of normalized stressintensity factor (&,) with different b/a vs. the ratio 2a/d.
tip B(&,). When the ratio 2a/d gradually decreases, the interaction effect between cracks also decreases. As 0.22 5 2a/d I 0.33 relative to b/a = 6 or as O-04 r2a/d ~0.11 relative to b/a = 24, the value of F,,, increases again (F’,,, is slightly larger than F,,,,) due to the influence of the boundary. When the ratio b/a is equal to 48, as would be expected, the presented value of FIIIn is similar to that of the solution for an infinite sheet displayed in the handbook by Sih.’ 7 CONCLUSIONS An efficient and accurate boundary element method for the stress analysis of a linearly elastic, homogeneous and isotropic solid, with one or multiple cracks under various boundary conditions subjected to antiplane shear, has been developed in this paper. Using only a simple conventional boundary element mesh, an excellent solution for the Mode-III stress intensity factor can be calculated. The interaction effect among cracks and the influence of boundaries can be evaluated efficiently. For the derivation of the potential problem, this work can also be extended to the study of thermal problems with one or multiple cracks in a finite (or infinite) domain in the near future.
I+ 2a4 !I
2b
0.6
2a/d
a/b
Fig. 6. Variation of normalized stress intensity factor (&) with various e, and e,,vs. the different ratios of a/b.
Quadratic
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
>i
Fig. 7. Geometry condition and boundary element meshfor a squareplate containing two collinear cracks.
REFERENCES 1. Sih, G. C., Handbook of Stress Intensity Factors, Noordhoff, Leyden, The Netherlands,1973. 2. Zhang, X. S., The general solution of a central crack off the center line of a rectangular sheet for Mode-III. Erzgng Fract. Me&., 28 (1987) 147-55.
Analysis
of Mode-III
fracture
3. Zhang, X. S., A tear mode crack located anywhere in a finite rectangular sheet. Engng Fract. Mech., 383(1989) 509-x.
4. Ma, S. W., A central crack in a rectangular sheNet where its boundary is subjectedto an arbitrary anti-plane load. Engng Fract. Mech., 30 (1988)435-43. 5. Ma, S. W., A central crack of Mode-III in a rectangular sheetwith fixed edges.Int. J. of Fract., 39 (1989) 323-9. 6. Ma, S. W. & Zhang, L. X., A new solution of an eccentric crack off the center line of a rectangular sheet for Mode-III. Engng Fract. Mech., 40 (1991) l--7. 7. Nishioka, T. & Atluri, S. N., Analytical solution for embeddedelliptical cracks, and finite element alternating method for elliptical surface cracks, subjected to arbitrary loadings. Engng Fract. Mech., 17 (1983) 247-68. 8. Rajiyah, H. & Atluri, S. N., Analysis of embeddedand surfaceelliptical flaws in transverselyisotropic bodiesby the finite element alternating method. J. ofApp/. Mech., 58 (1991) 435-43.
9. O’Donoghue, P. E., Nishioka, T. & Atluri, S. N., Multiple surfacecracksin pressurevessels.Engng Fract. Me&.,
20 (1984) 545-60.
problem
with multiple cracks
267
10. Chen, W. H. & Chang, C. S., Analysis of two dimension fracture problems with multiple cracks under mixed boundary conditions. Engng Fract. Mech., 34 (1989) 921-34. 11. Chen, K. L. & Atluri,
S. N., A finite difference alternating method for a cost effective determination of weight-functions for orthotropic materials in mixedmode fracture. Engng Fract. Mech., 36 (1990) 327-40. 12. Rajiyah, H. & Atluri, S. N., Evaluation of K-factors and weight functions for 2-D mixed-modemultiple cracksby the boundary elementalternating method. Engng. Fract.
Mech., 32 (1989) 911-22. 13. Sneddon, I. N. & Elliott, H. A., The opening of a Griffth crack under internal pressure.Q. Appl. Math., 4
(1946) 262-7. 14. Sneddon, I. N., Fourier Transform, McGraw-Hill, New York, 1951. 15. Erdogan, F., Stressintensity factors. Trans. ASME, J. Appl. Mech., 50 (1983) 992-1002. 16. Brebbia, C. A. & Dominguez, J., Boundary Elements; An introductory Course, McGraw-Hill, New York, 1989.