Uncoupled characteristics of three-dimensional planar cracks

Pergamon

UNCOUPLED

Int. J. Engng Sci. Vol.36, No. 1, pp. 33--48,1998 t~) 1997ElsevierScienceLimited.All rightsreserved Printed in Great Britain PI~ 80020-7225(97)00059.1 0020-7225/98 $19.00+0.00

CHARACTERISTICS OF THREE-DIMENSIONAL PLANAR CRACKS

Y O N G L I N XU* and B R I A N M O R A N and T E D B E L Y T S C H K O Department of Civil Engineering,Northwestern University,Evanston, IL 60208, USA Abstract The uncoupled characteristicsof three-dimensionalplanar cracks under arbitrary loading is

investigated by boundary integral equation method. From theoretical analysis, the crack displacements under normal loading and shear loading on the crack surface are uncoupled. For a crack under shear loading (mode II and III), the special case of the crack with uncoupled displacements is investigated, whereby the :self-similarcrack expansion (SSCE) method, proposed in a previous paper by Xu et al. (1996, Journal of Applied Mechanics) is used for the analysis of three-dimensionalcracks under shear loading. A number of cracks with arbitrary geometry and mixed mode loading conditions are calculated in 1:hispaper. For cracks with uncoupled displacements under various loading conditions,the numerical re,;ults by the SSCE method show good accuracy, with errors in stress intensity factors around the penny-shaped crack edge less than 1%. Results of a penny-shaped crack under torsional loading show good applications of uncoupled characteristics to the stress analysis for threedimensional crack~ © 1997 Elsevier ScienceLtd.

1. I N T R O D U C T I O N As a crack propagates, the energy release rate of the crack generally consists of three parts which correspond to three fracture modes (mode I, II and III). The understanding of the uncoupled characteristics of a three-dimensional crack for these three modes is useful for crack growth analysis and fatigue life prediction of cast materials. The boundary integral equation for three-dimensional cracks has been presented in the literature (see [1-6]). For complex loading conditions, Weaver [7] presented expressions of the boundary integral equation with lower singularities for a crack under tensile loading and shear loading. In Weaver's equations for a crack under shear loading, however, the uncoupled characteristic of a planar crack is not explicit. In this paper, an explicit form of boundary integral equations for the uncoupled crack under shear loading is given. The equation indicates that three-dimensional planar cracks with tensile loading in the z-direction and shear loading in the x-direction or y-direction are uncoupled, and that crack-opening displacements in the two tangential directions (x and y) under shear loading are uncoupled as the Poisson's ratio of the material approaches zero. This special case is very useful for the stress distribution analysis of a crack under shear loading. The effort in crack analysis for the uncoupled crack can be greatly reduced since the SIFs can be calculated separately. In order to improve the accuracy of stress intensity factor evaluation for three-dimensional cracks, Xu et al. [8] proposed the self-similar crack expansion (SSCE) method, in which the SIFs of a crack are evaluated by the crack surface displacement, not solely estimated by the tip displacement. The singular and regular integrals are evaluated by the analytical form. Thus, the accuracy of the stress intensity factors are significantly improved. In this paper, the SSCE method, associated with the analytically numerical evaluation of element integrals, is extended to the analysis of uncoupled three-dimensional cracks with arbitrary geometry under shear loading. The appJtication of the uncoupled analysis to an interesting physical problem, cracks under torsional loading, is given, and stress intensity factor with error less than 1% is obtained for a penny-shaped crack under torsional loading. * Corresponding author. 33

34

YONGLIN XU et al. 2. B A S I C F O R M U L A T I O N

FOR MIXED

MODE

LOADING

CONDITIONS

The general boundary integral equation can be derived either by using reciprocity [1] or the body force method [4]. The boundary integral equations for a three-dimensional crack are

o'ii(x) = fta Sijk(X, ~) AUk(~) df~ (~), = 1,2,3

(2.1)

where AUk(~) represents the displacement jump across the crack surface. The functions Sok for isotropic materials are /*

Siik- 4¢r(1 --v)r 3 {3[(1 -- 2V)6ijr,k + v(6ikr,j + 6jkr,i) -- 5r,irqr,k]r,,n, + ni[(1 -- 2V)6jk + 3vr,krq] + nj[(1 -- 2V)6ik + 3 Vr,kr,i] + nk[3(1 -- 2v)r,irq -- (1 -- 4v)6j]}

(2.2)

For the typical case of a three-dimensional planar crack (Fig. 1), equation (2.1) can be written as (2.3)

tr3i(x) = fu S3jk(X, () A Uk(~) d a ({), j = 1,2,3 where _

$333

/x

/z

4~-(1 - v)r 3 ' S3jk -- 47r(1 -- v)r 3 [(1 -- 2V)6jkr,k + 3vr,jr,k],j,k = 1,2

(2.4)

It is obvious from equation (2.3) that the normal displacement and tangential displacement of a crack are uncoupled since the functions S3jk(X, () in the crack plane are /x 4¢r(1 - v)r 3

5313 = 0, S323 -~- 0, 5333 -

_

$321

u

{

47r(1-- v)r 3

$311 _

(2.5a)

( e - x) (1 - 2v) + 3v

~ t" 4¢r(1 --- v) r3 I. j

((_x)(Tl_

3

v rZ

X3

/

r2

j,

(2.5b)

y) ~ s331=O

,

q3

-/

ql • s

S

S

S

1En~

4 I I

! !

X2 S

S

S~

s S

X1

Fig. 1. A three-dimensional planar crack under a mixed mode loading.

Uncoupled characteristics of three-dimensional planar cracks

35

z q3

.. ~ - ~

\\

Pq2

Fig. 2. Displacements and tractions on the crack surface.

_

S312

Ix

(

47r(1 --- v)r 3 3v

(~-x)(~l-y)} r2

$322_ 4zr(1 /x- v)r 3 { ( 1 - 2 v ) + 3 v

. ,

( , / _r y2 ) 2 ~j ' $332= 0

(2.5c)

Equations (2.5a), (2.5b) and (2.5c) imply that the normal displacement on the crack surface only gives rise to the normal stress in the crack plane of the material, and the tangential displacements produce the shear stresses in the crack plane (Fig. 2). Therefore, a planar crack under tensile loadting (mode I problem) can be calculated directly by using equation (2.3) with equation (2.5a). For a crack under shear loading, the integral can be written as

,~3j( x ) = f, F3jkt( x, ~ ) A Uk.,( ~ ) df~ ( ~ ), j = 1,2

(2.6)

where the functions F3jkt(X, ~) given by Weaver [7] are

_ Faro

IX { (~:-x) 3 ) 8 rr(1 S_ v)r 3 3

_ F3112

F3121

/~ 8

-

rr(1

- v)r 3 X

r (~:_ x)2(~7 _ y) + (1 - 2 v ) ( n - Y) } ~3 r2

Iz { (~-x)2(rl-Y) 8 rr(1 " v)r 3 3 r2

!z ( (~-x)(rl-Y) F3122= 8~(1 - v ) r 3 3 r2 /.~ F3221 -

8 7/'(1 --- v)r3

F3222 =

8

rr(1

- v)r 3

--

(1

--

2v)(rl

--

y)}

2) .

f (~: - x)(~/- y)2 - (1 - 2v)(~: - x) ) ~3 r2

3

( '17 -- y)3 .) F3211 _

Iz ( (g-x)2(~-y)} 8'rr(1 ~- v)r 3 3 r2

tz { ( ~ - x ) ( r l - y)2 (1 2v)(~ x) } F3212 - 8,rr(1 - v)r 3 3 r2 -

(2.7)

36

YONGLIN XU et al.

It can be seen that the tangential displacement discontinuity in the Xl-direction produces the shear stresses in both of x~- and x2-directions. Note that the functions F3jk, are not unique. Following the method by Xu et al. [8], we can find a function f3jkt which satisfies V2f3jk= -S3jk and thus, the corresponding functions to F3jk, in equation (2.6) can be written as

--

f3111

I~

4~(1"-- v ) r 3

{ 3 v ((--X)3 r2

--

(1+ 2V)(~ X) }

_ /z { (sC-x)2(r/-y) f3112 47r(1--- V)F 3 3V r2

} -- (71 -- y)

_ /x { (~:- x)2(~7- Y) f3121 4~'(1--- v)r 3 3 v r2 - v(rl = /x { ((-x)(rl-Y) f3122 41r(1 - v)r 3 3 v r2

_

f3221

4zr(1 "- v)r 3

fa222- 47r(1 --- v)r 3 =

{ 3V (~: 3v

} -

y)

2

} - v((-

x)

x)(~ - y)2 r2

(~7 - y)3 r2

} - (~: - x)

(gr _ x) ] ,

/

(2.8)

=

It can be verified that equation (2.6) with functions f3jkt in equation (2.8) is equivalent to equation (2.6) with functions Fajk, in equation (2.7) through integration by parts. The advantage of using equation (2.8) is that the uncoupled characteristics for a planar crack become explicit. If the Poisson's ratio v approaches zero, the integral equations of a planar crack under shear loading are

O'3y(x'Y) -

8 7 r ( / E- - V 2) fD{('~3X--""~)Auj, e + (7 r 3 Y) Auj, n}df~(~:,~7),j = 1,2 -

(2.9)

Equations (2.9) show the uncoupled characteristics of a planar crack under shear loading as Poisson's ratio approaches zero. It is fortunate that equation (2.9) have the same form as the crack subject to tensile loading. Equation (2.6) with equation (2.8) can be used for general problem of cracks under shear loading. It is clear that as Poisson's ratio v = 0, the non-zero functions in equation (2.8) are only f3111, f3112, f3221 and f3222. Therefore, the displacements in the xl-direction and x2-direction are uncoupled, and the calculation can be greatly simplified and computer time consumption is reduced. In this paper, the uncoupled crack problem will be mainly discussed. In the case of an uncoupled crack under a complex loading condition, it can be independently dealt with as three problems of the crack subject to loading in the xi (i= 1,2,3) direction. The numerical implementation of equation (2.9) has been discussed by Xu et al. [8], where an analytically numerical method has been proposed. With this method, the regular and singular integrals can be evaluated by using the explicit expression of close form based on path integrals. An efficient technique in their work is using the SSCE method for precisely evaluating the stress intensity factors around the crack edge under tensile loading. In this paper, the SSCE method is extended to cracks subject to shear loading for more complicated problems.

Uncoupled characteristics of three-dimensionalplanar cracks

37

3. S E I , F - S I M I L A R C R A C K E X P A N S I O N M E T H O D F O R CRACKS UNDER SHEAR LOADING The self-similar crack expansion method stems from a method based on crack extension (crack extension method) which was originally proposed by Cruse and Besunner [9] and Cruse and Meyers [10]. It is well known that the stress intensity factors (/(ix and KIII) can be determined by the asymptotic field of crack tip displacement as

un -

1-v/2r I-~

1 - - KH, ut = - zr tx

/2r -- Km ~r

(3.1)

where un is the crack tip displacement in the crack edge normal direction and ut the crack edge tangential direction. Instead of using local displacement at the crack tip, the SSCE method uses the displacement field over the entire crack surface to estimate the stress intensity factors. For this purpose the relationship between the energy release rate and the displacement on the crack surface (for mode II and III) is employed. For a crack in the Xl-X2 plane under a uniform shear load qx and q2, this gives

6 2

(qlAul + qzAuz)dtl = f ~ 1 - v z KIZI+ 1 +_._____vK~II v 6a(s)dF

3rt

E

E

(3.2)

where 5a(s) is the crack advance as a function of a crack-edge contour parameter S, and F is the boundary of the crack. For details of the relationship between strain energy and the energy release rate one can refer to Rice [11]. If a crack expands self-similarly (Fig. 3), the change in crack size can be written as 8a(s) = aa(s)

(3.3)

where a is a parameter, and a(s) is the distance from the crack edge to the similarity center. The stress intensity factors around the crack edge can be expressed as Kii(s) = fit(s) Kiio, KHI(S) = fm(s)Kiio

(3.4)

where fxi(s) and fiii(s) are distribution functions of the stress intensity factors, which can be determined from the displacement at the crack tip, and Kito is a reference stress intensity factor of mode II at the reference point. The distance between the crack edge and the similarity center at the reference point is denoted by ao. For a crack under remote shear loading ql and q2, the left side of equation (3.2) can be written as 1/2(qld;V1 + q26V2), where 111 and V 2 is the so-called crack-opening volume in the xtand XE-direction respectively. Here 111 and V2 are induced by shear loading, while V3 is induced

,

X2

Fig. 3. Crack propagating self similarly.

YONGLIN XU et al.

38

by normal loading. The crack-opening volume can be expressed V = Aa 3 (where A is a constant since Au3/ao is constant) as long as the three-dimensional crack expands self-similarly as described by equation (3.3). Thus, the variation of the crack volume can be written as

8V=

/3V ao

6aoordV =/3Va

(3.5)

where /3 is taken as 3 for three-dimensional cracks and 2 for two-dimension cracks under uniform loading. Substituting equations (3.3) and (3.4) into the right side of (3.2) and using equation (3.5) on the left side of equation (3.2) yields

T

~

K2I°

1

}

f~i(s) "k- V(1- "--" ' ~ f2II(S) a(s)dF

(3.6)

Hence, the reference stress intensity factors can be written as

~ E(qlV1 + q2V2) KI1°=

f { (1 --1 ) (1 -- V2) Jr f2I(S) + V - " " ' ~ f2II(s) a(s)dF

(3.7)

To obtain fn(s) and fin(S), the crack tip displacements Au, and hu, as local quantities of COD in the crack edge's normal and tangential directions are used. Therefore, the stress intensity factors can be obtained by using (3.4).

4.

NUMERICAL IMPLEMENTATION F O R S T R E S S INTENSITY FACTOR CALCULATION

The boundary integrals equations, equation (2.8), for uncoupled planar cracks are E v2) f u ( (' ~ x r)~ Or3j(x'Y) -- 87;'(1---

) l ,Auj,,}df~(g, 2r----3~ Auj,~ + r( l~)-,Yj =

(4.1)

The kernel function in equation (4.1) has a lower order singularity than equation (2.1). To evaluate the above integral, the crack is discretized as N=m ×n elements, where m is the number of segments in the circumferential direction and n is the number of segments in the radial direction. Except for m triangular elements around the crack center, the other m × (n-l) elements are trapezoidal elements as shown in Fig. 4. In discretizing equation (4.1), we may put the source point (x,y) at the center of one element. The displacement in the elements is approximated by 4 A uj =--A uj = ~ Nr(x,y) A Ujr (4.2) K=I To simplify the evaluation of the integrals, the basis (1,x,y,xy), NK = D r is used, where D is a D ' determinant of nodal coordinates D=

i xl yl xlyl x2 Y2 xzy2/ X3 Y3 x3Y3] X4 Y4 x4Y4/

(4.3)

Uncoupled characteristics of three-dimensional planar cracks

A

39

a

AKI AI

" K+I Fig. 4. Boundaryelements(mx n) on the crack surface. and D r is the matrix where the kth row of D is replaced by (1 x y xy). The discrete form of the integral equation (4.1) can be written as N /z °'3J- 4rr(1 - v) ~ J'j

(4.4)

n=l

where Jnj = fn rl--T[(s~ - X)Nlc.e + ( ~ 7 - y)NK,,7]Au/Kd ~

(4.5)

n

and Au#< (K= 1, 2, 3, 4) represent the crack-opening displacements (j-direction) at the element nodes, and the derivatives of the shape functions may be written as NlC,e = AK + B I ~ , Nr,,7 = Cr + DK~

cq+B1 11

2

-7 H

II

4

crack edge

J

?,,

1

0~2~+ ~ 2 Fig. 5. Singular elements near the crack edge.

(4.6)

40

Y O N G L I N X U et aL

Fig. 6. Boundary elements (25 × 13) for a penny-shaped crack.

(a)

2.5

•••

2.0

1.5,

0

SSCEsolution

--

A n a l y t i c solution (Sih)

1.0'

0.5'

0.0

.

0

(b)

.

10

.

20

.

.

30

.

40

.

50

.

,

60

70

80 0

900

2.5

2.0,

O

SSCE solution

--

A n a l y t i c s o l u t i o n (Sih)

1.5-

_

J

Y

ff

0 0 " v

o

~

~

~

~

~

lO

20

ao

40

50

60

~

=

70

80

9'00

Fig. 7. (a) SIFs of a penny-shaped crack u n d e r shear loading (25 × 13 elements. (b) SIFs of a penny shaped crack under shear loading (25 × 13 elements).

41

Uncoupled characteristics of three-dimensionalplanar cracks Table 1. SIFs, KII/(¢V~a), for a penny-shaped crack under shear loading Element

O = 0°

O = 300

O = 450

O = 600

27 x 13 35 x 17 45 x 30 81 × 31 Analytical solution

2.0376 2.0263 2.0215 2.0118 2.0000

1.7646 1.7548 1.7506 1.7423 1.7321

1.4408 1.4328 1.4294 1.4225 1.4142

1.0188 1.0132 1.0107 1.0059 1.0000

where the constants AK, BI¢, CK and D r can be directly obtained from equation (4.2) and equation (4.3). Therefore, equation (4.4) becomes

°'3i= ~:

4¢r(1-v)

--~-[(AK+BKy)(~--X)+(CK+DKX)(71--y)

n=l

n

+ (BK + O K ) ( ~ -- x)(~l -- Y)] A ujkd~

(4.7)

In the above, three integrals need to be evaluated for each element which is not near the crack edge.

ii= fo e- x

df~ , 12 =

e

fo ,7-~'~ y

df~ , 13 =

e

r3 fo (e- x)(nY) da

(4.8)

e

These three integrals, for either regular elements or singular elements, have been precisely evaluated with explicit expression of closed form by Xu et al. [8]. More difficult integrals are the singular integrals on the element near the crack edge, where double singularities of the integral occur. One is the kernel singularity at the source point (x,y), and the other is the stress singularity at the crack tip. The displacement in the local coordinate system (s¢,~/) may be written as (see Fig. 5) X/L - ~: [ __~- , / 4 Auj3 +

~

AUa] --

k

"173 - -

7/74

~73-~7 Auj4] T]3 --

(4.9)

T]4

where L = ~2-~3, and the nodal displacements Aujl and Auj~ vanish because of the zero displacement cortdition at the crack edge. Equation (4.12) indicates the displacement has a linear variation in the tangential 07) direction and varies with the square root of the distance from the crack edge in the normal (~) direction. However, these singular integrals on the element near the crack edge are still possible to transform these two-dimensional singular integrals to one-dimensional regular integrals which are easier to evaluate precisely. For the detailed calculation of these integrals, one can refer to Xu et al. [8].

4.0

o

SSCE Solution

LLJ -.~ 3 . 0

Analytic solution

Q..

>< 2 . 0

1.0

0.0

0.0

0.4

0.8

1.2

1.6

D/a

2.0

Fig. 8. Normalized crack-openingdisplacement of a penny-shaped crack.

42

Y O N G L I N X U et al.

(a)

2.0

,,-,

1.5

CD

1.0

0.5 ¸

0.0 0 i4

0.0

0 i8

' 1.2

' 1.6

2.0 (x+r)/2

(b)

2.5 ""

/

t 2.01

O --

SSCE solution Analytic solution

1.5 ~.

1.0 0.5 0.0

•

0

I

.

.

15

.

30

.

.

45

60

75

0

900

Fig. 9. (a) Displacement along x-axis for a penny-shaped crack under shear loading r = aror/a (b) Stress intensity factors of a penny-shaped crack under shear loading r = a~'or/a.

C

D \ . %

, \

~

, I

, I

',

t

i

,

.

I

/

. /

~ /.I~l l .,'ll.,r'il

1 I I i / /1L, t " I l l t

/

/

/I,VIII~

I I I i/.~ L~III \ ~ 1 I t//,q,.rlJ,'fl I IJ

A "//

/ I

', \ \ ",~de

Fig. 10. Square crack with 37 × 13 elements.

Uncoupled characteristics of three-dimensional planar cracks

43

The crack surface discretization is shown in Fig. 4. There are rn × n elements and the displacement unknowns Aul and Au2 are m(n-1)+ 1, respectively. The least square method may be used for detelmining the 2[m(n-1) + 1] node displacement on the crack surface in terms of 2 m x n equations. The crack-opening volume 1/1 and V2 and their derivatives can be determined in terms of the analysis mentioned above, and the stress intensity factors around the crack are estimated by the SSCE method.

5. S T R E S S I N T E N S I T Y

FACTORS

OF CRACKS UNDER

SHEAR

LOADING

A penny-shaped crack with radius a under shear loading ~- in the x-direction is discretized as shown in Fig. 6 (25 x 13 = 325 elements). The stress intensity factor distributions around the

(a)

1.0 1;J~ 0"8 t

IXffO ,

B

o

~Y

Jll o

.

C

Ix

0"6t 0.4

0.2 0.0 0.0

(b)

I

0.2

•

0.4

0.6

0.8

1.0 y/a

.2

1.0

0.8 ¸

IX = 900

0.6-

6

~

5

0

•

,L

.

001

0.4 0.2 0.0 -0.2

0.0

0.2

0.4

0.6

0.8 ~/a

1.0

Fig. 11. (a) Stress intensity factors along side CD for a square crack under shear loading with angle a. (b) Stress intensity factors along side CD for a square crack under shear loading with angle a.

44

YONGLIN XU et al. 1.0

0 8_ •

o

SSCE solution

Y

--

Solution (Weaver)

~

m

X

|

0.6.

%r "~

0.4.

0.2.

0.0

0

0'.2

.

.

.

.

x/a

1

Fig. 12. Energy release rate along the edge of a square crack (81 x 13 elements) under shear loading.

crack edge are shown in Figs 7(a) and (b). The results are in good agreement with the analytical solutions by Kassir and Sih [12], where the analytical solutions for SIFs are Kn -

4

trV'-aacos0, KII I -

(2 - v) ~

4(1 - v)

trX/-aasin0

(5.1)

(2 - v)

Stress intensity factors of a penny-shaped crack with different elements are shown in Table 1. The accuracy of the calculation can be so high that errors of the SIFs are less than 1%, provided that the elements on the crack surface are greater than 81 x 31. The crack displacement on the crack surface AUl is shown in Fig. 8. It can be seen that, for uncoupled crack problems, the displacement in the shear direction is the same as the displacement due to tensile loading. Thus, the maximum shear displacement is also located at the center of the penny-shaped crack. For uncoupled cracks, the energy release rate on the crack edge is independent of the shear loading direction• The difference is the ratio of the SIFs between mode II and III for various shear loading directions. The SSCE method can also be applied to cracks with other loading conditions. For example, a penny-shaped crack under shear loading of ~-= a~'or/a, where the tangential traction r in the

Y

B J'

"C

a

A

x

I

Fig. 13. Triangular crack under shear loading.

Uncoupled characteristics of three-dimensional planar cracks

45

Fig. 14. Triangular crack with 61 × 13 elements.

x-direction is proportional to the distance to the center of the crack. In contrast to the case of uniform loading, the maximum displacement of the crack under such loading is not located at the center of the crack (Fig. 9(a)). This phenomenon may be explained by the fact that the loading at the crack center is the minimum, zero. The stress intensity factors around the crack edge is shown in Fig. 9(b). Results are in good agreement with the analytical solutions by Kassir and Sih [12]. For a square crack under shear loading r in various directions, the crack surface discretization (37 × 13 element:~) is shown in Fig. 10. The stress intensity factors KH and Km along the crack edge CD are shown in Figs ll(a) and (b), respectively. As the angle a increases, the stress intensity factor KH decreases, while Km (mode III) increases. For the case of an uncoupled crack, the energy release is independent of the direction of the shear loading. On the edge CD of the square crack, the energy release rate is independent of the loading angle. The energy release rate computed by the SSCE method is very close to the that calculated by Weaver [7] as shown in Fig. 12, where the quantity K = ~¢/K2I + K2H is used for the comparison. 3.0

2st 2.0

1.5

1.0

0.5 !

°°o.o

04

1o y/a

Fig. 15. SIFs along side AB for a triangular crack under shear loading (61 x 13 elements).

46

YONGLIN XU et al. 3

0.0

I

I

I

l

0.2

0.4

0.6

0.8

x/h

1.0

Fig. 16. Crack-opening displacement along CD for a triangular crack.

A triangular crack with three equivalent sides of length 2a, under shear loading ~"is shown in Fig. 13. The triangular is discretized as 61 x 13 elements (see Fig. 14). The stress intensity factors along side AB are calculated by using the SSCE method, and the distribution of SIFs is shown in Fig. 15. The crack opening displacement along the x-axis is shown in Fig. 16. As compared to the square crack, the crack opening displacement of the triangular crack is relatively small. However, the relative value of the stress intensity factor of the triangle crack to its area is comparable to the square crack. Figure 16 shows that the maximum displacement is also at the center of the triangular crack. One of the applications of cracks under shear loading is the analysis of stress intensity factors of a three-dimensional crack under torsional loading. As shown in Fig. 17, a penny-shaped crack with radius a is subjected to torsional loading (torque T). On the crack surface the shear stress ~'0~ perpendicular to the radial direction is

Fig. 17. A penny-shaped crack under torsional loading.

Uncoupled characteristics of three-dimensional planar cracks

47

1.0

L~

0.8-

B

Analyticalsolution

o

Uncoupled analysis

Y " ~

S

~

~ x

l.a

~

'l:x=(-yla, 'I:OL~'~J

0.6-

--

0.4Q 0

O

0.2-

O.O

0

i

i

i

i

i

30

60

90

120

150

0

180 0

Fig. 18. SIFs of a penny-shaped crack under shear loading in the x-direction.

where r0 is the shear stress near the crack edge. Thus, the torque T can be written as (5.3)

T = --~ a2ro

The shear loading on the crack surface can be decomposed into two components, respectively, in the x- and y-directions r. = - (y/a)ro (5.4a) (5.4b)

r r = (x/a) ro

Taking advantage of the independence of the torsional solution to the Poisson's ratio, we can select the Poisson's ratio v = 0 and thus, the displacements ux induced by shear stress rx and Uy induced by ry are uncoupled. Consider the crack under the shear loading Zx in the x-direction as equation (5.4a). The displacements of the crack under this kind of linear shear loading can be solved using equation (2.9). The maximum stress intensity factor of the crack under shear loading equation (5.4a) is located at the crack edge with 0=90 ° or 270 ° (Fig. 18), where Km =0.4238 r0-

~

using 25 x 15 elements. The stress intensity factor distribution of the

crack under shear loading in the y-direction is similar to that of the crack under shear loading in the x-direction, but the angle should shift 90 °. The stress intensity factor on the crack edge at 0 = 90 ° or 270 ° has a minimum value (zero). Thus, the stress intensity factor of the pennyshaped crack under torsional loading is K m = 0.4238ro

m

71" "

Using equation (5.2) gives the solution of the crack under torsional loading T, Kni= T 2.6628 - V'~a. As compared with the analytical solution by Kassir et aL [12] 7r2a 2

8

r

K i n - 3 ~2a-------~

(5.5)

the numerical results by uncoupled analysis show good accuracy, with the stress intensity factor error of less than 1%. Since the torsional problem is independent of the Poisson's ratio, the uncoupled analysis provides a simple way to estimate stress intensity factors of cracks under

48

YONGLIN XU et al.

torsional loading. This analysis can be applied to the planar cracks with m o r e complicated geometry under torsional loading.

6. C O N C L U S I O N It has been shown that stress intensity factors for three-dimensional cracks under shear loading can be accurately calculated by using the self-similar crack expansion method with the boundary integral equation technique. The numerical results show that the accuracy of the stress intensity factors for three-dimensional cracks can be improved by accurate evaluation of the element integrals (regular and singular). The numerical m e t h o d shows good accuracy for three-dimensional crack analysis, with errors in stress intensity factors around the penny-shaped crack edge less than 1%. For a planar crack, the normal displacement in the crack plane is induced by the normal traction on the crack surface; the tangential displacements in the crack plane are induced by the shear tractions on the crack surface. Generally, the tangential displacements and shear tractions in the two directions (x and y) are coupled. They are uncoupled as long as the Poisson's ratio v of the material approaches zero. For the uncoupled crack under shear loading, the distribution of the displacement amplitude is the same as that of the crack under tensile loading, while the displacement direction is the same as the shear loading direction. The stress intensity factors on the crack edge depends on the loading direction in general, while the energy release rate of an uncoupled crack under shear loading is independent of the loading direction. The calculation of cracks under torsional loading can be simplified using the uncouple characteristics of the threedimensional cracks under shear loading. Acknowledgements The support of the National Institute of Standards and Technology and the Army Office of

Research are gratefully acknowledged. A Grant from Cray Research, Inc. for access to the Pittsburgh Supercomputer Center is also gratefully acknowledged.

REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12.

Cruse, T. A., Boundary element analysis in computationalfracture mechanics. Kluwer Academic, 1988. Cruse, T. A. and Novati, (3. Fracture Mechanics: 22nd Symposium,II.ASTM STP 1992, 1131, 314 Polch, E. Z., Cruse, T. A. and Huang, C.-J. Computational Mechanics 1987, 2, 253 Murakami, Y. and Nemat-Nasser, Engineering Fracture Mechanics 1982, 16, 373 Lee, J. C. and Keer, L. M. ASME, Journal of Applied Mechanics 1986, 53, 311 Guo, Q., Wang, J. J., Clifton, J. J. and Mertaugh, L. J. ASME Journal of Applied Mechanics 1995, 62, 108 Weaver, J. International Journal of Solids and Structures 1977, 13, 21 Xu, Y., Moran, B. and Belytschko, T., ASME Journal of Appled Mechanics, in press. Cruse, T. A. and Besunner, E M. Journal of Aircraft 1975, 12, 369 Cruse, T. A. and Meyers, G. J. J. Struct. Divn., ASCE 1977, 103, 309 Rice, J. R. ASME Journal of Applied Mechanics 1985, 52, 71 Kassir, M. K. and Sih, G. C., Mechanics of Fracture, Vol. 2, Three-Dimensional Crack Problems, Noordhoff International, 1975. (Received 2 December 1996; accepted 25 April 1997)