Weight Function

vol.8 No.2 .,~ 20ol J. CENT. SOUTH UNIV. TECHNOL. Article ID: 1005-9784(2001)02-0099-06

Semi-analytical weight function method for center-cracked circular disk subjected to diametral compression

CHEN Feng, SUN Zong-qi, XU Ji-cheng

(The Key Laboratory of Nonferrous Metal Materials Science and Engineering, Central South University, Changsha 410083, China)

Abstract: Btieckner-Rice weight function method was used to analyse mixed-mode fracture of center-cracked circular disk subjected to uniaxial compression. Based on Wu-Carlsson procedure semi-analytical modes ~[ and [[ weight functions were derived from corresponding reference displacement fields and stress intensity factors calculated by finite element method. Normalized mode ~ and mode ]] stress intensity factors, f ] , fn , were derived from the obtained semi-analytical weight functions. The results were then fitted into polynomials, the precision is within 0.5 %. It is interesting to note that when the inclined angle 0 of a crack is less than 15 °, the f~ values are positive, when (9 = 15 °, the f~ values are positive for the crack length a varying from 0.1 to 0.7, but when a = 0.8, the f~ takes the negative value - 0.51. When O > 15 °, all the f1 values become negative, which denotes that the compression-shear mode is achieved at crack tips. These results are very useful in the investigation of mixed-mode fracture of brittle materials. Key words: weight function; stress intensity factor; compression; center-cracked circular disk Document code: A

Since the center-cracked circular disk specimen can be available directly from rock core, and mixed-mode fracture ranging from pure mode I to any K I / K I ratio can be easily achieved using

this geometry, it is widely used in the mixed-mode fracture investigation of brittle materials. The problem concerning stress intensity factors (SIFs) in center-cracked disk subjected to diametral compression has been tackled by many authors, but most solutions were obtained by numerical meth- ods. Atkinson h~ solved the problem using bound- my integral equation method and then solved it nu- merically. Awaiji et al. [2] studied mixed-mode fracture problems using complex analysis method, and Fett [3] obtained mode 11 SIFs for this geome-

try using weight function method. In this paper semi-analytical formula for both

mode T and mode 1] SIFs in center -cracked circular disk due to uniaxial compression were de-

rived using Wu-Carlsson ~4] approximate weight function method. Firstly, constant normal and shear stresses are chosen as mode I and mode 1] reference loads; secondly the corresponding reference displacements of crack face and related refer-

ence SIFs computed by finite element method were used to derive semi-analytical weight functions; finally, normalized SIFs were derived by combin- ing the obtained weight functions and Edlac crack line stresses [ iI .

1 Outline of weight function method

Based on Btieckner [5] weight function meth-

od, the SIFs due to practical load case can be cal-

culated by integrating the product of weight func-

tion M i (X, A ) and crack line stress O'li (X) as

follows :

Foundation item: Doctorate Program Found of China(No.98053318) Biography of the first author: CHEN Feng, professor, born in 1947, majoring in fracture mechanics. Received date: June 2, 2000

100 Journal CSUT Vol.8

E' 3Ui( ( x a) m ~ ( x , a ) - re) , ( i = 1 , 2 ) A(re f ) 3 a

(2) where E' is Young's modulus, E' = E for plain

stress and E' = E/( 1 - v ) for plain strain, v is

Poisson's ratio, and a = A/L , x = X/L are non-

dimensional crack length and coordinate, respec-

tively, L is the characteristic length parameter of

a cracked body, mi ( x, a ) ( i = 1, 2) are mode

I and mode II weight functions. Therefore, the

normalized SIFs f / ( i = 1, 2) in the practical load

case becomes

f/= ol,(x), m'(x'a)dx o ~ ( i = 1 , 2 )

(3)

In the present study, ali ( x ) in Eq. (3) is the

normalized stresses induced on the prospective

crack line of a uncracked circular disk subjected

to diametral compression. Atkinson [~3 developed

this solution into power series, which reads:

P ~ , A, (O)(~, /R) 2'-2 (4) zoo - ~ R i=1'

2Psin20 ~ Bi(O)(p/R)2i -2 (5)

where P is the concentrate force on per unit thickness of a disk, p is the polar coordinate with ori- gin at disk center, R is disk radius and 0 is the angle between load line and crack orientation, (see Fig. 1). A~(O) and Bi(O) are angle consta-

nts, the first five terms Ill are given in Table 1 for

later use. The comparison of two-term and five-term se-

ries approximation of stresses given by Atkinson [1] shows that five term solution is very close to analytical solution, therefore, we use the five-term approximation of stress in the following derivation.

P

P

Fig. 1 Geometry of circular disk

Table 1 The first five A~ (0) and B i ( 0 ) in Eqs. (4) and (5)

A1 1 - 4 s - " B1

A2 852 (1 - 4 c 2 ) B2

A3 - 4 s 2 ( 3 - 3 6 c 2 + 4 8 c 4 ) B3

A 4 - 16s z ( - 1 + 2 4 c 2 + 8 0 c 4 + 6 4 c 6 ) B4

A5 - 2 0 s 2 ( 1 - 4 0 c 2 + 2 4 0 c 4 - 4 4 8 c 6 + 2 5 6 c s ) B5

1

- 5 + 8 c 2

- 3 + 8 ( 1 - 2c z)(2- 3 c 2)

3 + 16 (1 - 2 c 2 ) - 1 2 ( l - 2 ¢ 2 )2 _ 3 2 ( 1 - 2 c 2 )3

5 - 1 6 ( 1 - 2 c z ) - 6 0 ( 1 - 2 c 2 ) z + 3 2 ( 1 - 2 c 2 ) 2 + 8 0 ( 1 - 2 c 2 ) 4

note : s = s i n 0 , c = c o s 0 , c o s 2 0 = 2 c 2 - 1

2 Derivation of weight functions for center- cracked circular disk and mixed-mode SIF formula

2.1 Mode I weight function and corresponding SIF formula

Consider a center-cracked disk with crack length 2A, radius R and coordinate X along crack line(see Fig. 2) . The disk radius is chosen as a characteristic dimension, the normalized quantities (see Fig. 2c) can then be written as:

a= A / R , x - X /R The constant normal and shear stresses shown

in Fig. 2a and 2b, respectively, were chosen as reference loads. The corresponding reference displacement fields and SIFs calculated by FEM were used to derive modes I and 11 weight functions. Fig. 3 shows a FEM mesh of a disk with the crack length a = 0 . 5 , six-node isoparametric triangular elements were used in FEM analysis, the areas near crack tips are refined to simulate the stress singularity.

First , the calculated normalized mode I

No.2 CHEN Feng, et al: Semi-analytical WFM for center-cracked circular disk 101

(a) (b) (~)

x

(a)--mode I reference toad; (b)--mode H reference load; (c)--non-dimensional quantities in disk specimen

Fig. 2 Geometry of disk specimen and reference loads

Fig. 3 The FEM mesh of a disk with a = 0.5

SIFs due to reference load a , with the limitation

for a = 1: 2A/(rr 2 - 4 ) ( 1 - a) = 0 . 8 2 6 / , f l - a are fitted into the following polynomial I4] :

fl(ref) ( a ) = K i (r,f) ( ct ) /a 8

= ~ a i a i - ' / , / 1 - a (6) i= l

The a~ ( i = 1- 8) coefficients are listed in Table 2.

With the same procedure as Wu-Carlsson I43 , the related mode I reference displacement field caused by a can be expressed as:

1

, =~Tj_~_~le) ( ,a ) 1 - - --Lt

(O<<. x / a <~ l ) (7) where Fj ( j = 1,2) can be determined by crack tip opening displacement and self-similar conditions [4J which reads: 5

Ft ( a ) = 2f~(,,y) ( a )

F 2 ( a ) = 3 [ cI)(a) -f,(~,¢)( a) /2]

(TI)(ct) - - 4 fa ~S/(ref)(s)]d8 {1+ o N N

= - -7 ~ ala+ ln(1 - a ) O~ i=1 "= k=2

Substituting Eqs. (6) and (7) into (2) yields the mode [ weight function for center cracked circular disk:

3 i - 3/2

/ n l ( X , ( ~ ) - l i~=lPi(a)[1--(~)2 1

(8) where

i l l ( a ) =2 , ~ 2 ( a ) = I 2 a / i ( r e f ) ( a ) + 3F2( a) ]/f~(ref)( a)

f13( a) = [ aF'2( a) - 2 F 2 ( a ) ]/f~(r¢) ( a ) Introducing the non-dimensional quantity p / R = X / R = x into Eqs. (4) and (5) gives the normalized tensile and shear stresses along prospective crack line (using five term approximation):

5 a i ~ ( x ) - ace ~a , (O)x2~_2 (9)

P/r~R - i=,

a~2(x) = a~o p/r~R = Zsin2O Z Bi( O) x (10)

Substituting Eqs. (8) and (9) into ( 3 ) yields the normalized mode ]~ SIFs:

f I ( a ,O) = K1/ao

= f2 a l l ( X ) m i ( x , a ) d x

5 3 _ 1 E E A , ( O ) g ( a ) "

7['a i= lj= 1

f~x2i -2[1- (~)2] j -312dx ( 1 1 )

where ao = PlrcaR, fl coefficients in Eq. ( 11 ) are

tabulated in Table 3(see Wu-Carlsson[4J ) . 2.2 Mode ]I weight function and corre-

sponding SIF formula For reference load case shown in Fig. 2 ( b ) ,

102 Journal C,$UT Vol.8

normalized mode 1I SIFs for different crack length

calculated by FEM are fitted into the following

polynomial: 5

i - 1 f u ( r ~ f ) ( a ) = ~ a , a / ' , / 1 - a (12)

' = 1

where the coetticients a i ( i = 1-5) are 1,

- 0.5 , 0. 9273, - 0. 8841, 0. 2823, respective-

ly, and mode II displacements of crack face are

fitted into following polynomial: 1

2 v a ~ __ 2]~- (O<~xla<~l) (13)

where Gj ( a ) can be determined with the same

procedure as used for Fj. ( a ) through fIl (r~f) ( a ) .

Substituting Eqs. (12) and ( 13 ) into (2)

gives the mode 1I weight function:

3 1 3 x 2 i - ~

m2(x, a) - 1- (7) ]

(14) The coefficients ?', for different crack length are

tabulated in Table 4. Substituting Eqs. (10) and (14) into ( 3 )

yields the normalized mode 1I SIF as follows:

f n ( a, O) = KH lao ~/rraR

= a l E ( x ) m 2 ( x , a ) d x o

5 3

_ 2 ~ ~ s in (20)B, (O)Yi (a) . ~ a i=1 j= l

• 3

The f I , fn values at different angles of crack inclination were oomputed vs erack length a and listed in Tables 5 and 6, respectively.

Table 2 O/i coefficients in Eq. ( 6 )

0;1 ~2 0;3 ~4 0;5 0;6 0;7 ~8

1.000 0 -0.4963 1.5581 -3.1816 10 .0961 -20.7782 20.7781 -7.5066

Table 3 fli coefficients for different crack length a a

8, 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

fll 2.000 0 2.000 0 2.000 0 2.000 0 2.000 0 2.000 0 2.000 0 2.000 0

f12 0.062 7 0.238 9 0.526 7 0.921 8 1.431 9 2. 105 0 3.062 2 4.619 1

f13 0.0002 0.0027 0.0186 0.0680 0.1587 0.7450 0.3906 0.453 1

Table 4 Y~ coefficients in Eq. (14) for different crack length a a

7~ 0.05 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

7~ 2.000 0 2.000 0 2.000 0 2.000 0 2.000 0 2.000 0 2.000 0 2.000 0 2.1300 0

Y~ 0.0103 0.0405 0.1568 0.3476 0.6229 1.0135 1.5905 2.5196 4.2527

Y3 - 0 . 0 0 4 9 - 0 . 0 1 9 9 - 0 . 0 8 1 6 - 0 . 1 9 4 3 -0 .3 7 7 7 - 0 . 6 7 1 0 - 1 . 1 5 0 9 -1 .9 8 1 8 - 3 . 5 8 9 6

Table 5 f i values at different angles of crack inclination a

0/(o) 0.01 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

15 0.74 0.75 0.77 0.75 0.74 0.72 0.70 0.42 -0.51

30 0.00 -0.01 -0.12 -0.24 -0.25 -0.63 -0.90 -1.48 -1.72

45 -1.00 -I .08 -1.13 -1.25 -1.50 -1.75 -1.90 -2.25 -2.75

60 -2.00 -2.02 -2.09 -2.11 -2.25 -2.50 -2.74 -2.90 -3.61

75 -2.70 -2.72 -2.75 -2.80 -2.91 -3.02 -3.11 -3.50 -4.01

No.2 CHEN Feng, eta/: Semi-analytical WFM for center-cracked circular disk 103

Table 6 fu values at different angles of crack inclination

0/(o ) a 0.01 0.1 0 .2 0 .3 0 .4 0 .5 0 .6 0 .7 0 .8

15 1.00 1.02 1.09 1.23 1.40 1.47 2.05 2.58 3.26

30 1.73 1.75 1.84 1.96 2.12 2.30 2.49 2.70 3.01

45 2.00 2.01 2.04 2.07 2.10 2.12 2.15 2.22 2.00

60 1.73 1.72 1.70 1.66 1.60 1.55 1.53 1.56 1.72

75 1.00 0.99 0.96 0.91 0.86 0.81 0.79 0.81 0.89

Table 7 Ci coefficients in Eq. (16)

0/(°) C0 C1 C2 Cs C4 C5 C6

15 0 . 7 3 0 0 - 0 . 0 1 1 3 2 . 1 3 5 9 - 1 3 . 4 4 3 6 35 .8979 - 4 4 . 4 8 7 7 16.6669

30 - 0 . 0 0 0 3 0 . 0 6 7 2 - 2 . 6 7 2 5 2 . 8 2 2 0 - 8 . 3 3 3 4 5 . 4 4 8 7 0.000 0

45 - 1.000 1 - 0 . 1 5 0 0 - 1 . 8 2 1 4 - 7 . 5 1 8 8 15.5301 - 9 . 2 9 4 8 0.000 0

60 - 2 . 0 0 0 0 0 . 1 2 9 9 - 5 . 0 2 2 8 8 . 7 6 5 9 - 1 1 . 6 4 4 1 10.1268 - 5 . 5 5 8 0

75 - 2 . 7 2 9 8 - 0 . 0 4 3 9 - 2 . 2 2 8 8 8 .9743 - 2 7 . 6 1 6 7 36 .7299 - 1 9 . 4 4 4 1

Table 8 D~ coefficients in Eq. (17)

0/(°) Do D1 D2 D3 04 D5 D6

15 1 .0004 - 0 . 0 9 8 1 3 . 5 8 4 2 - 5 . 8 8 9 8 11.3433 - 4 . 8 3 9 8 0.000 0

30 1 .7300 - 0 . 0 1 2 3 3 . 8 4 0 0 - 9 . 3 3 2 7 28 .3478 - 4 3 . 6 7 5 4 24.074 1

45 2 . 0 0 0 0 - 0 . 1 3 3 4 3 . 4 1 9 8 - 1 3 . 4 8 1 7 30 .1999 - 3 7 . 9 9 2 0 20 .3706

60 1.730 1 - 0. 330 1 4. 363 4 - 27.289 5 63. 372 5 - 67.286 4 28.703 7

75 1.0(300 0 . 0 2 3 7 - 1 . 2 5 5 0 0 .5261 - 0 . 3 5 6 3 3 .1625 - 1 . 8 5 1 9

These data are fitted into six order polyno-

mials: 6

fI (a) = ~ C i ai (i = 0 - 6 ) (16) i=0 6

fu (a) = ~ O i a i (i = 0 - 6 ) (17) i=0

where the coefficients C/and Di ( i = 0-6) for dif-

ferent angles of crack inclination are listed in Ta-

bles 7 and 8.

Graphical representations of Eqs. (16) and

(17) are illustrated in Fig. 4 (a ) and 4 ( b ) , re-

spectively. Results from Atkinson ~1] and Fett E3] are

also given in Fig. 4 for comparison. It can be seen

that results calculated by Eqs. (16) and (17) are

in good agreement with the numerical solutions

from literatures E1'33 .

The SIF values versus the angles of crack in-

clination are also given in the following polynomials

for a fixed crack length a = 0 .5 .

5

fI(O)= ~ mff (18) i=0

5

i n ( 0 ) = ~ n,O' (19) i=0

where, 0 in radian is the inelined angle of a

crack, the coefficients mi, ni ( i = 0-5) are listed

in Table 9.

The eomparison of SIF values obtained by

Eqs. (18) and (19) and numerieal solutions from

literatures [1'3] is shown in Fig.5.

It can be seen from Fig. 5 that Eqs. (18)

and (19) present almost the same estimation of

normalized SIFs as those given by Atkinson [1] and

Fett[3], fI varies from the maximum value at 0 =

13' to its minimum negative value at 0 = 9i f , while

at the angle at which f1 is near zero, the fu

reaches its maximum value, which is consistent

with the results reported by Chen and Sun E63 .

104 Joumal CSUT VoI. 8

"~- 2 . 0 ~ . , , -"---,a...~......, --.-e30o

3 5 - ~ ~ ' ~ - - e 60 - . ~ .Atkinson ~..?.~l_.ol | oFett 13 I

- 5 . 0 | . . . . I 0.0 0 .2 0 .4 0 .6 0 .8

normalized crack length a

Fig. 4

3.0 3

| ~ , ~ 4 5

1.5 ' r ~ ' ~ ~ - ' - '~ '~60°

5 ° • Atkinson o Fett

0 .0 0 :2 014 026 0:8 normalized crack length a

(,a)--mode I ; (b)--mode n

Normalized SIFs for different angles of crack inclination

1.5' m,~ • Atkinsin (a) " N O Fett 0 .5 X ~

~" - 0 . 5

< - 1.5

- 2.5 t~tq.eo ~

- 3 . 5 . . . . . , 0 15 30 45 60 75 90

0/(°)

Fig. 5

3.0 (b) • At kinsin o Fett

1.0

0 . 0 ~ , , , , , X X ~ 0 15 30 45 60 75 90

0/(°)

(a)--mode I ; (b)--mode [I

Normalized SIFs vs the angle of crack inclination(for fixed crack length a = 0 . 5 )

Table 9 m~, n~ coefficients in Eqs. (18) and (19)

i

0 1 2 3 4 5

mi 1.3368 0.0466 -0 .0098 0.0003 - 2 . 8 0 x 10 -6 1.09x 10 -s

n, -0 .0127 0.1417 -0.001 7 - 3 . 1 7 x 10 -5 6.34x 10 -7 -2.98 x 10 -9

4 Conclusions

Approximate weight function method was ap- plied to the analysis of mixed-mode fracture of center-cracked circular disk loaded by diametral compression. The proposed semi-analytical formula for normalized stress intensity factors is quite consistent with the numerical solutions from literatures. The obtained f~ , f~ results are very useful and

convenient in practical SIF calculations, since the only information needed is the concentrate force P on per unit thickness of disk, and combined-mode fracture with any K 1 / K I ratio can be easily

achieved by adjusting the angle of crack inclination with respect to load line.

References

[ 1 ] Atldnson C, Smelser R E, Sanchez J. Combined mode fracture via the cracked Brazilian disk test [J] . Int J of Fract, 1982, 18(4) : 279-291.

[2] Awaiji H, Sato S. Combined mode fracture toughness mea- surement by the disk test [ J ] . J Engng Mat Tech, 1978, 100: 175-182.

[ 3 ] Fett T. Mode U weight function for circular disk with internal radial crack and application to the Brazilian disk test [J]. Int J of Fract, 1988, 89:L9-L13.

[4] WU X R, Carlsson A J. Weight function and stress intensity factor solutions [M]. New York: Pergamon Press, 1991.

[5] BOeckner H. A novel principle for the computation of stress intensity factors [J]. Z Angew Math Mech, 1970, 50: 529- 546.

[6] CHEN Feng, SUN Zong-qi, XU Ji-cheog. A crack closure model subjected to compression-shear loading [ J ]. Trans Nonfenous Met Soc China, 1999, 12(1) : 427-432.

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Weight Function