Multiple cracks inside and outside circular regions

Theoretical and Applied Fracture Mechanics 11 (1989) 199-208

199

MULTIPLE CRACKS INSIDE AND OUTSIDE CIRCULAR REGIONS W.Z. L I N and Y.Z. C H E N

Division of Engineering Mechanics, Jiangsu Institute of Technology, Zhenjian~ People's Repubfic of China

Analyzed in this work are the problems of multiple cracks inside and outside of a circular region, the boundary of which is fixed. The former represents a finite circular plate with internal cracks "while the latter refers to cracks exterior to a rigid circular inclusion. The method of superposition is applied such that the problem may be divided into two parts. The first part involves cracks each subjected to a pair of concentrated normal and tangential forces. The second corresponds to the same cracks on which distributed normal and tangential tractions are applied. Superposition of two gives rise to the desired boundary conditions on the cracks. A system of Fredholm integral equations is obtained that can be solved numerically for the stress intensity factors at the crack tips.

1. Introduction Elastostatic problems involving holes, notches, cracks and inclusions have attracted the attention of m a n y previous investigators in the past because their solutions can be applied to derive certain parameters for predicting the remaining strength of defective solids. The stress intensity factor quantity has been widely used in fracture mechanics for solids weakened by crack-like imperfections. To this end, a variety of crack problems have been solved b y application of the singular integral equation representation [1-9]. The majority of these works, however, did not consider cracks whose locations can be arbitrary. In contrast to the singular integral approach, the F r e d h o l m integral equation can be equally, if not more effective, in solving crack problems [10-12]. The work in [10] will be extended to solve multiple crack problems involving a fixed circular b o u n d a r y where the locations of the crack can be arbitrary. The m e t h o d involves using the concentrated load solutions in [13] as the Green's functions that can adjust for the necessary crack surface b o u n d a r y conditions by means of superposition. The stress intensity factor can be isolated from the singular portion of the stress field and solved numerically f r o m a system of F r e d h o l m integral equation. Such a procedure has been used extensively throughout in [12].

2. Analytical consideration Consider the potential functions ~ ( z ) and ~b(z) of the complex variable z = x + iy. According to the plane theory of elasticity, the displacement and stress c o m p o n e n t s can be obtained from the expressions

2 G ( u + iv) = r e ( z ) - z ~ - r ' ~ - ~ - - ( ~

(1)

and

oxx + Oyy = 4 Re ~ ( z ), Oey - i O x y = ~ ( z ) + ~----('~ + z~---r-~ + ~----(-~,

(2)

where ~¢ equals to (3 - p ) / ( 1 + p) for plane stress and 3 - 41, for plane strain with ~, being the Poisson's ratio. In eqs. (1) and (2), ~(z) = ~'(z),

~(z) = ~'(z),

where prime denotes derivative with respect to z and the bar denotes complex conjugate. 0167-8442/89/$3.50 © 1989, Elsevier Science Publishers B.V.

(3)

200

W.Z. Lin, Y.Z. Chen / Multiplecracksinsideand outsidecircularregions i

C,~

i

t

/

.R

~

%1

fl

r

-~x2a

-I

Fig. 1. Crack outside a rigid circular inclusion in an infinite medium.

2.1. Rigid inclusion." Single crack Let the xy-plane o c c u p y the infinite medium that contains a rigid circular inclusion of radius R, the exterior region of which contains a crack of length 2 a as shown in Fig. 1. The circular b o u n d a r y ./ being fixed cannot displace and hence eq. (1) gives

~,(t)- t,-~-~-~=

0 on./,

(4)

with t = R exp(i0) where t denotes z on ./. Making use of the relations

-t=R2/t,

- ( R / t ) 2 dt,

dt=

eq. (4) m a y be written as

x~-~--~-~(t)+

[

,2]

t~'(t)+--~(t)=0

on'/.

(5)

In the absence of the inclusion, the expressions of #p(Z) and qp(Z) for a pair of concentrated forces P and Q at x = s to the surface of a single crack in an infinite m e d i u m is given by [14]

%(z)

P - iQ

X(s)

2~i

X(z-zo)(Z-Zo-S)'

~/'p(z) = _ P + i Q 2~ri

X(s) _~p(Z)_(Z_Zo+~o)Cbp(z) ' X(z-zo)(Z-Z o-s)

(6)

where

(7) has a branch for which lim 1X(z_z [z[~oo z

o)=1.

A n additional pair of potentials ~ c ( z ) and q c ( z ) are introduced for the purpose of eliminating the displacements on y such that the solution of the original problem in Fig. 1 can be obtained by superposition:

) ( z ) = %(z) + )o(z),

~ ( z ) = %(z) + ~o(z).

(8)

201

W.Z. Lin, Y.Z. Chen / Multiple cracks inside and outside circular regions

Substituting eq. (8) into (5), the result is

[

][

,

K~p---~--~Pc(t) + tdPc(t) + - - ~ c ( t ) + K~c---~--~--~p(t)+ t~bp(t) + - ~ p ( t

)

] =0

on y,

(9) which leads to the following expressions

q ' c ( z ) = - x - ~R2-p(-~)

-1

RE[[ -- [|R._~_2t

R2~,( R 2] + R_._~4-~,,[R2 t

+3--v'/" k-z

+

'

+B-~

,

Izl>R.

(10) The constant fl in eq. (10) may be found from the behavior of ~c(z) and ~c(z) for large ]z [. Expanding eqs. (10) for large ]z ] render g2

g3

• o(~) -- [,3~03- ~ 1 / . + 7 + V + ....

h2 % ( ~ ) --

7

h3 +

7

(11) + ....

where (12) Since no stresses prevail at the infinity, it follows that Re[*¢(oo)]=0

or

Re[/3l=Re[(I)p(O)].

(13)

In addition, the unbalanced moment on the inclusion is proportional to Ira[h2] which is absent in this problem and hence Im[h2l = 0.

(14)

Inserting eq. (12) into (14), the result I m [ / 3 ] - - ( x - 1) Im[~p(0)]

(15)

can be used to yield 13= Re[ #p(0)] + i ( x - 1) Im[ ep(0)].

(16)

Equations (6) and (10) can thus be substituted into eq. (8) to obtain the complete potentials for the problem. The displacements and stresses follow directly from eqs. (1) and (2). For instance the components Oy,y, and O,,y, along CD in Fig. 1 can be computed as o~,~, + io~,~, = 0 ( ~ ) + 0 - - ~ +

e~°[~*'(~) + ~(z)].

(17)

202

Lin, Y.Z. Chen / Multiple cracks inside and outside circular regions

W..Z

Yk

P (Sk)

Y

k~

pj(sj)

~

I~

~Sk.~...#/qk(Sk)

~

2oj

~

xk

Zko

qj(sj) Fig. 2. Multiple cracksoutsidea fi#d circular inclusionin an infirutc medium. 2.2. Rigid inclusion: Multiple cracks Figure 2 shows the case of N cracks subjected to normal and tangential tractions that are exterior to the rigid circular inclusion of radius R. It suffices to illustrate the method of solution by considering the k th crack of length 2a k subjected to tractions Pk and qk since superposition may be applied to obtain the solution for N single crack problem. As before, a pair of concentrated forces Pk and Qk are applied at s k and used as Green' functions for assuming the appropriate crack surface tractions. The final results are given by the following system of Fredholm integral equations [11]. N

N

+ E f°' j=l

[sk[
e~(s~)+ E j=l

Is k [ < ak,

--a:

k=1,2,3

a)

+ E f Q:(s:)/,.,:(s:, s,,)dsj=Pk(Sk), j=l

--aj

. . . . . N; N

f°' Pj(sj)f.t,jk(sj, Sk)dS:+ Efa, Q./(sj)ftt.jk(Sj'sk)dsj=qk(sk)' --aj j=l

k=1,2,3

--a)

..... N.

(18)

Note that f.tjk(Sj,Sk) may be interpreted as an influence function for forces acting on the j t h crack at (sj, 0) and on the k t h crack at (s k, 0). The system of eqs. (18) can be solved numerically, the solution of which yields the crack tip stress intensity factors: KA, j

1 = K 1 A , j -- i K 2 A , j

KB, j = K]B,j -- iK2B,j = _ _ _

aj

~ aj -- Sj

~j

~aj [Pj(sj) - iaj(sj)l

-aj -+ s j

dsj,

1

+sj £ a~j[ Pj(sj ) - iQj(sbl ~ a ya:--sj

dsj.

(19)

The form of equation (19) is the same as that for a single crack which was first derived in [14]. The subscripts A and B denote, respectively, the left and right crack tip.

2. 3. Finite circular plate: Multiple cracks The problem of a finite circular plate of radius R can be solved in the same way as that of an infinite medium containing a rigid circular inclusion. The general expressions for the complex potentials may be

W.Z. Lin, Y.Z. Chen / Multiplecracksinsideand outsidecircularregions

s R

z

203

P

y

:B

f,

t

jB

!~

2a

= x

A r!

J Fig. 3. Fixed boundary finite circular plate with one crack.

Fig. 4. Two unparallel cracks outside a rigid circular inclusion.

obtained by inverting the complex variable z from I z I > R to I z I < R in equations (10). The function Oc(0 ) and ~/'c(0) must have definite limits and fl now becomes p~/a 2 - s 2 /3 = ¢t(1 - K)R 2'

(20)

where s is the location for a pair of concentrated forces measured from the midpoint of the crack. Configuration for the case of a single crack subjected to concentrated forces is shown in Fig. 3.

3. Numerical examples and results

The numerical procedures [12,15] for solving Fredholm integral equations are well-known. They usually involve reduction of the integral equations to algebraic expressions by application of some quadrature rule. The Chebyshev quadrature will be used [15]:

f"oc(t) -

dt = ~

M G ( t , . ) sin (2m - 1)0 ~_, 2M

m = l

(21) '

where t,, -- a cos[(2m - 1)~r/2M]. Equations (18) can thus be converted to a system of algebraic equations such that the values of Pj(sj) and Q j ( s j ) ( j = 1, 2 . . . . . N ) at M discrete points can be found. Applying eq. (21), the following can be obtained: (a J

F ( t ) dt ~r M ~ - - 7 = "-~ E F(tm),

(22)

where t,, = a cos[(2m - 1)0r/2M]. Equations (19) can thus be applied to yield the corresponding stress intensity factors. 3.1. Two unparallel cracks outside rigid inclusion

Let an infinite domain contain a rigid inclusion. Two cracks AB and CD are situated at the outside having length 2a and 2b, respectively with CD inclined at angle 8 with reference to AB as shown in Fig. 4.

W.Z. Lin, Y.Z. Chen / Multiple cracks inside and outside circular regions

204

Table 1 L o a d i n g s at i n f i n i t y f o r t w o u n p a r a l l e l c r a c k s o u t s i d e rigid i n c l u s i o n s C a s e NO.

a~

ov~

°~v

I

p

0

0

II I11

0 0

p 0

0 q

The three different loadings referred to as Case I, II and III in Table 1 will be considered. T h e stress intensity factors at the four crack tips in Fig. 4 for Case I and Case II are given by

Kij = ~J

(_~

f2 ' R'

a b 8)p~ R' R'

(23)

with i = 1, 2 and j = A, B, C, D such that a A = a B = a and a c = a D = b. T h e quantities ft and f2 are the respective distances of the mid-point of crack AB and C D to the inclusion center. F o r Case III, the results are (_~ Kij = aij

f2 ' R'

a b 8)q~j. R' R'

(24)

T h e subscripts i and j have the same interpretation as those in eq. (23). N u m e r i c a l values of the normalized stress intensity factors or functions F1A, F m . . . . . FED in eqs. (23) are plotted in Figs. 5 and 6 as

0.9 I

0.7 ~

0.5 ¸

/

0.9

FIC

FI B 0.3

h

0.7

IA

Z

o pL~

z

0.1

h

u_ 0 . 5

Z

-- -- , L ' F 2 B

"

'

•' / F2C

o I.J

~F2A

-0.1

~FIB

u-

/ /

0.5

I

/ / ,@.

~'-~,.~ F2D

-0.5

O. F2C

-0.5

i

15 °

I

30 °

,

I

i

I

60 °

45 °

i

,~ ~ ' -

I 75 °

I

-0. 90 °

8

~ ~

15 o

~F2B I

30 °

~

I

,

45 °

I

60 °

,

i

75 °

,

90 °

8

Fig. 5. N o r m a l i z e d stress intensity f a c t o r s for C a s e I in Table 1

Fig. 6. N o r m a l i z e d stress intensity factors f o r Case I ] in Table

f o r f l / R = 2, f 2 / R = 1.6, a / R = 0.8, b / R = 0.4 a n d x = 1.8.

] f o r f ] / R = 2, f2/R = 1 . 6 , a/R = 0.8, b/R = 0.4 and ~ = ] . 8 .

14<.Z. Lin, Y.Z. C h e n / Multiple cracks inside and outside circular regions

o.~L

_

~-~-

205

_

G2A 0.6

",.',,,

0.4

,./G2o

0.2

J', G2 %%

,%

y

%%~%/.' GIA 5

_

L...f

w

f

/~,

I-

-0.4 ~ ~/GID

%\%~%/~" ' / ', '

%. -0.8

i

15*

I

s

30"

I

,

45*

I

60*

,

I

90*

I Fig. 8. A schematic of rigid inclusion sandwiched by two equal and parallel cracks.

stress intensity factors for Case III in Table

1 f o r f i / R = 2, f 2 / R

= 1.6,

a/R

= 0.8,

b/R

| i

~ TM

75*

8 F i g . 7. N o r m a l i z e d

G

= 0 . 4 a n d ~ = 1.8.

a function of the angle 8. Figure 7 gives graphical representation of eq. (24) for Case III. For Case II, if K1A and K1B for a single crack are given by

f l / R = f 2 / R = 3.0, a / R = b / R = 1.0, R = 1 and 8 = ~r, the K1A = 0.9407p ~vr~-a,

Kin = 0.9816pyre-a,

(25)

which is in close agreement with the results in [9] for the same problem: K1A = 0.9266pC~-a,

K m = 0.9711pvr~-a.

(26)

Two equal cracks with radial symmetry. Another special case of Case II in Table 1 involves two equal radial cracks outside the circular inclusion which can be obtained by letting AB = C D in Fig. 4. With symmetry fl =f2, it can be shown from eq. (23) that Fig = F l c and FIB = F1D. Table 2 gives the values of F1A and Fla for a / R = b / R = 1, f l / R = f 2 / R and 8 = ~ r and two other special cases: one with a traction-free condition [11] and one with R - - 0 containing only cracks [10]. As it is well known, a fixed boundary tends to lower the neighboring crack tip stress intensity while the opposite prevails for the traction-free case. Two equal parallel cracks around inclusion. The situation of an inclusion surrounded by two equal and parallel cracks may also be regarded as a special case of the configuration in Fig. 4. This is shown in Fig. 8 such that K,A = K,B = K1¢ = KiD = F,,,

a)

, ~

p~vr~-.

(27)

W . Z Lin, Y.Z. Chen / Multiple cracks inside and outside circular regions

206

Table 2 Normalized stress intensity factors for two radial cracks outside rigid inclusion with Case II loading for a / R = b / R =1, f l / R = f 2 / R , and 8 = ~r, and two special cases [10,11]

~/R

2.1

2.3

2.5

2.7

2.9

3.1

0.4824 0.9154

0.7355 0.9455

0.8433 0.9624

0.8985 0.9726

0.9299 0.9792

0.9493 0.9836

Traction free [11] £'1A 2.4695 F~n 1.3285

1.6941 1.2059

1.4294 1.1495

1.2956 1.1160

1.2172 1.0934

1.1669 1.0778

Cracked plate [10] 1.0422 F] a 1.0253

1.0334 1.0211

1.0272 1.0179

1.0226 1.0154

1.0191 1.0133

1.0163 1.0117

Fixed F]A Fm

"y

FIA

For Case II loading and a/R = 1, Table 3 gives the numerical values of eq. (27). When the crack plane instead of the tips is facing the rigid inclusion, a rise in the stress intensity factor occurs. Hence, the influence of rigid inclusion on the local crack tip stress intensity depends on the relative position and orientation of the crack for a given loading.

I.I

i.o r

H

~

0.9 I2A y

HI

O.6

~

.

12D

a_z

_o 0.5

'

0

0.4

p

X

0,3

Z "

B

0.2

~,~/ ,,.

~

H 2B

0

0. I

IH2A ~¢

-O.I IIB

H2D I

~ ,~ ..,. _ - - ~ .~ ~ H 2 c

O.2

i

I O~ ~

0*

20*

40*

60*

80 °

Fig. 9. Normalized stress intensity factors for two unequal and unparallel cracks under normal pressure in circular plate for a / R = 0.2, b / R = 0.2 and h / R = 0.25.

O*

I

20 °

I

I

40*

I

I

60*

i

I

80 °

Fig. 10. Normalized stress intensity factors for two unequal and unparallel cracks under shear in circular plate for a / R = 0.2, b / R = 0.2 and h / R = 0.25.

W.Z. Lin, Y.Z. Chen / Multiple cracks inside and outside circular regions

207

Table 3 Normalized stress intensity factors for rigid inclusion sandwiched by two equal and parallel cracks for Case II loading with a / R = 1 fiR

1.1

1.3

1.5

1.7

1.9

2.1

0.8960

0.9386

0.9546

0.9697

0.9842

0.9973

Traction free V [11] FIA 0.7719

0.6970

0.7145

0.7317

0.7476

0.7625

Cracked plate [10] FIA 0.8553

0.8774

0.8961

0.9116

0.9243

0.9348

Fixed y F1A

3.2. Two unparallel cracks in circular plate: Normal pressure The problem of two unparallel cracks of unequal length 2a and 2b in a circular plate is shown in Fig. 9. These cracks are opened by uniform surface pressure p. Plotted in Fig. 9 are also the functions Hij in Kij=Hij

(a

b

h,8)p

R' R' R

~j.

(28)

The values of H m and H1D are seen to increase with the orientation of the inclined crack CD. The function H1A increases at first and then decreases while Hlc has the opposite behavior.

3. 3. Two unparallel cracks in circular plate: Shear tractions Suppose that two unparallel and unequal cracks in a circular plate are subjected to surface shear tractions of magnitude q as shown in Fig. 10. The expression for K~j becomes K~j=I~j

R' R' R

Refer to Fig. 10 for a display of the numerical results. The stress intensity factors at the cracks tips remained virtually constant as 8 varied from 0 ° to 90 o.

4. Conclusions The problem of multiple cracks in a two-dimensional domain bounded by a circular boundary or containing a circular inclusion is reduced to the evaluation of Fredholm integral equations, a procedure that has been used extensively in the past for solving crack problems. Boundary conditions on the circular boundary and those on the cracks can be adjusted to the desired values by application of the concentrated load solution used as a Green's function in conjunction with the use of auxiliary solution by means of superposition. The Chebyshev quadrature is used in the numerical computation for M up to 15 whereas changes in the stress intensity factors become negligible for M >t 9. Sufficiently accurate numerical results have been obtained for a variety of examples involving the interaction of cracks with a fixed circular boundary. In the special cases, the results agreed well with those published previously.

References [1] V.V. Panasyuk, M.P. Savruk and A.P. Datsyshyn, "A general method of solution of two-dimensionalproblems in the theory of cracks", Engrg. Fracture Mech. 9, 481-497 (1977). [2] P.S. Theocaris and N.I. Ioakimidis, "Numerical integral method for the solution of singular integral equations", Quarterly of Appl. Math. 35, 173-183 (1977).

208

W.Z Lin, Y.Z. Chen / Multiple cracks inside and outside circular regions

[3] F. Erdogan, G.D. Gupta and T.S. Cook, "Numerical solution of singular integral equations", in: Mechanics of Fracture VoL 1, ed. by G.C. Sih, Noordhoff, Leiden, 368-425 (1973). [4] P.S. Theocaris and D. Sardzokas, "The influence of a finite stringer on the stress intensities around cracks in plates", Engrg. Fracture Mech. 14, 493-506 (1981). [5] F. Erdogan, "Mixed boundary value problems in mechanics", in: Mechanics Today VoL 4, ed. by S. Nemat-Nasser, Pergamon Press, New York, 1-86 (1978). [6] D.P. Rooke and J. Tweed, "The stress intensity factors of a radial crack in a point loaded disc", lnternat, d. of Fracture 11, 285-290 (1973). [7] N.I. Ioakimidis, "Two methods for the numerical solution of Bueckner's singular integral equation for plane elasticity crack problems", Comput. Meths. AppL Mech. Engrg. 31, 167-177 (1982). [8] W. Gunther, "Asymptotic elastic solution for two straight cracks of arbitrary length and location", Theoret. and AppL Fracture Mech. 3, 247-255 (1985). [9] G.T. Zhorzholiani and A.I. Kalandiil, "Influence of a rigid inclusion on the stress intensity near the tips of a crack", PMM 38, 719-727 (1974). [10] Y.Z. Chen, "A Fredholm integral equation approach for multiple crack problems in an infinite plate", Engrg. Fracture Mech. 20, 767-776 (1984). [11] Y.Z. Chen, "Solution of multiple crack problems of a circular plate or an infinite plate containing a circular hole by using Fredholm integral equation approach", lnternat. J. of Fracture 25, 155-168 (1984). [12] Mechanics of Fracture, Vols. H - I V , editor: G.C. Sih, Noordhoff Publishing (Now Martinus Nijhoff Publishers), The Netherlands (1972-80). [13] N.I. Muskhelishvili, Some Basic Problems of the Mathematical Theory of Elasticity, Noordhoff, Groningen, Holland (1953). [14] G.C. Sih, P.C. Paris and F. Erdogan, "Crack-tip stress intensity factors for plane extension and plate bending problems", J. of AppL Mechanics 29, 306-312 (1962). [15] F.B. Hildebrand, Introduction to Numerical Analysis, McGraw-Hill, New York (1974).