InternationalJournal of Fracture 45: R37-R41, 1990. © 1990KluwerAcademic Publishers.Printedin the Netherlands.

R37

A METHOD FOR CALCULATING STRESS INTENSITY FACTORS OF CHEVRON-NOTCHED THREE-POINT BEND ROUND BARS Wang Qizhi and Xian Xuefu Department o f Resources and Environmental Engineering Chongqing University, Chongqing 630044 People's Republic o f China • tel: 6234

In fracture testing, a 3-point bend (3PB) round bar specimen (Fig. 1) can be made with a straight crack (Fig. 2) or with a chevron notch (Fig. 3). A crack can initiate and proceed through the chevron notch during loading (Fig. 4). This unique feature of chevron-notched specimens attracted much interest in recent years, see for example [1], but investigation of chevron-notched 3PB round bars is relatively scarce. In the present report, an approximate method is used to study their stress intensity factors (SIFs). Shown in Figs, 1,3,4 is a specimen recently suggested by the International Society for Rock Mechanics [2] for fracture toughness testing of rock; 0 = 90, the width of the formed crack b is

(i)

b = 2(a - ao) = 2D (o~ - o~o) where D is the diameter, a is the crack length, a is the initial crack length, and o~, O~oare the dimensionless forms respectively. For a straight cracked round bar, the crack width B is 1

B = 2D

1

1-

(2)

= 2D ~ ( 1 - o0?

Using the Irwin-Kies formula, the energy release rate for a chevron-notched specimen is

p2dCv(a) GI- 2

(3)

dA

where P is load, and Cv is the compliance of a chevron-notched specimen. Since the area of crack extension is AA = bAa, (3) can be written as

P2 dCv(a) G I - 2b

da

(4) Int Journ of Fracture 45 (1990)

R38

The average SIF K~can be obtained using K I = E/-E~, where E'=E/(1-x~9 for plain strain condition, with E as the elastic modulus and x) as Poisson's ratio, then for a chevron-notched specimen 1

K*=2D~ ot2o~o ~

(5)

J

where E'QD is the dimensionless compliance of a chevron-notched specimen. We can define the dimensionless SIF of a chevron-notched specimen as

1

r'=K,/

1

d(E'C,,D)

i

(6)

For a straight cracked specimen (Fig. 2), the dimensionless SIF can be obtained in a way similar to

Y =

KJ

[ /1 11 ,T

= 2[ W(I i o0½ d---~ ]

(7)

where E'CD is the dimensionless compliance of a straight cracked specimen. If we assume that the change in compliance with crack extension for the chevron-notched specimen is the same as that for a straight cracked specimen [3], i.e.

d (E'C,,D) d (E'CD) d~

(8)

dc~

then 1 1

1

-

51

L

(9)

]

Equation (9) is the formula for calculating dimensionless SIF of a chevronnotched 3PB round bar from dimensionless SIF of a straight cracked specimen; for comparison, (6) is the compliance method, which is sophisticated. Bush [4] performed experimental calibrations using the compliance method to obtain SIFs for straight cracked 3PB round bars. Bush's results should be multiplied by 0.835 to obtain Y of this paper, then using (9) to obtain Y*, the results are as in Table 1. Int Journ of Fracture 45 (1990)

R39

Daoud et al. [5] performed FEM analysis for SIFs of s~_.aightcracked 3PB round bars. Daoud's results should be multiplied by 15.03/0~ to obtain Y of this paper and then using (9) to calculate Y*, the results are listed in Table 2. Figure 5 shows 11 pairs of Y-(x data and 11 pairs of Y'-(x data of Table 1 and Table 2 plotted and connected into smooth curves respectively. It can be seen that Y* has a minimum value of Y*=, which corresponds to maximum load P in fracture toughness testing if K,o is considered constant, so that PITI&X *

(lO)

D~ In order to obtain Y'=, Y*-(x data are fitted into a 3rd order polynomial using the least square method, that is Y* = 38.358-184.926(x + 376.46hx2-202.329(x3

(11)

In order to find two real roots of the equation (Y*)'=0, the smaller root is the critical crack length (xo,which is found to be 0.337, substitute (xointo (11) to obtain Y ' = = I 1.0485. According to [2], A~=10.42, which corresponds to Y ' ~ of this paper, the difference ( Y = - A = ) / A ~ is 6 percent; this illustrates the reliability of (9), (10), and (11) of the present paper.

REFERENCES [1] J.H. Underwood, S.W. Freiman and F.I. Baratta (eds.), ASTM STP 855, Philadelphia (1984). [2] ISRM (F. Ouchterlony-co-coordinator), International Journal of Rock Mechanics and Mining Sciences and Geomechanics Abstracts 25 (1988) 781-96. [3] D. Munz, R.T. Bubsey and J.E. Srawley, International Journal of Fracture 16 (1980) 359-374. [4] A.J. Bush, Experimental Mechanics 16 (1976) 249-257. [5] O.E.K. Daoud and D.J. Cartwright, Engineering Fracture Mechanics 19 (1984) 701-71.

1 June 1990

Int Journ of Fracture 45 (1990)

R40

~k t~

r~ J~

q~

c~

R41

c-A

%

I

® ] -,

Q . . . . .

f=y.9~

4

I -I

L.- A

Figure I.,A 3PB round bar specimen.

i

J

Section A-A Section A-A Section A-A Figure 2. Straight crack Figure 3. Chevron-notch Figure 4. Chevron notch and its formed crack

%

>..

.~.

I

I

I

i

I

I

I

Figure 5. Dimensionless SIF vs. dimensionless crack length for chevronnotched and straight cracked 3 PB round bars.

Int Journ

of Fracture

45 (1990)