Rotational factor using bending moment approach under elasto-plastic situation

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~ Pergamon

ROTATIONAL FACTOR USING BENDING MOMENT APPROACH U N D E R ELASTO-PLASTIC SITUATION--II.

CRACK 3PB GEOMETRY

A. N. KUMARt and SOVA BHATTACHARYA

Applied Mechanics Department, Indian Institute of Technology, New Delhi ll0016, India Abstract--Part I dealt with the evaluation of rotational factor considering the bending moment approach arising out of the stress distributions in the plane of fracture in a notched 3PB specimen. Part II is an extension of a similar approach for the situation with a sharp crack. The rotational factor is found to be around 0.41-0.42 in elastic situation (without considering plastic zone) and varies between 0.42 and 0.47 in elastic-plastic situation (considering plastic zone). Effects of other variables like initial crack size (ao), load (P), elastic stress distribution range, state of stress, etc. have also been investigated. A mild dependency of ao/w on elastic rotational factor (re) is observed in the range 0.15 ~< ao/w <~ 0.70, while the same is found to be independent of loading.

1. I N T R O D U C T I O N

PLASTIC ROTATIONAL FACTOR (PRF) allows the determination o f crack tip opening displacement ( C T O D ) by extrapolating the displacement measured at a crack m o u t h in three point SEN bend specimens. Although m a n y studies have been made on this aspect, it is still a highly controversial issue [1-12]. Several parameters like occurrence of slow crack growth [l, 2], state of stress, crack depth to width ratio [3], crack tip plasticity [4], specimen geometry [5], etc. are known to influence the magnitude o f P R F significantly. A large variation in the value of P R F from 0.10 to as large as 0.96 has been reported in the literature [6, 7]. Various standards recommend different values of P R F [13-17]. In view o f such ambiguities, an unequivocal value is very much needed which justifies further work in this direction.

The authors have recently studied the subject both theoretically and experimentally to determine the value o f rotational factor in notched geometry under elastic as well as elastic-plastic situation [18, 19]. An analytical relation between the bend angle, plastic zone size and tip opening displacement is established [18]. The experimental investigation of rotational factor is based on the displacement measurements at several points along the notch length [19]. The range of rotational factor in elastic and elastic-plastic situations is found to be around 0.26--0.28 and 0.50-0.56, respectively.

A new a p p r o a c h for the evolution of rotational factor considering the resultant bending m o m e n t o f all stresses around the notched region was proposed recently for notched SENB geometry [20]. The significance of such studies lies in the fact that m a n y structural components contain functional holes, blunt notches, etc. Part II is an extension o f a similar a p p r o a c h for the situation with a sharp crack. The study has been carried out with the general formulation as discussed in Part I and is confined to two situations; namely elastic and elastic-plastic. The results obtained are c o m p a r e d with our values and also with the reported data.

1.1. Considering no plastic zone (elastic situation)

The elastic stress distribution (ay) at any element ahead of the crack tip is given by [21]

K . cos_02 [ 1 + sin_02, sin ~ ] +...,

⁽¹⁾

tAuthor to whom correspondence should be addressed.

507

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508 A . N . KUMAR and S. BHATI'ACHARYA

where K is the stress intensity factor and (x,O) denotes the co-ordinates of the element. Any element in the plane of interest PQRS [20] ahead of the crack tip makes an angle of 0 ° with the abscissa (X direction). Putting 0 = 0, the stress distribution can be expressed as

K

(2)

The stress intensity factor 'K' in the mode I loading situation is given by

K = K , = o F -~ ao]

, (3)

where o is the stress, ao the crack size and F(ao/w) is a function which considers the effect of finite geometry. For SEN bend geometry, K~ may also be expressed in terms of applied load (P)[13].

P 1

K=~.f(ao/W)

for ~ = 4 . (4) T w o stress distributions, i.e. I/V/~ type non-linear, eq. (2), and a linear nominal stress distribution are considered for the elastic situation as shown in Fig. I. The distribution of linear stresses m a y be expressed as

o2=-)-- ~ - x for 0~<x~<~

and

)

tr3=- f - x for b/2<~x<~b. (5)

Firstly, the non linear stress distribution is assumed to be present over the entire ligament and according to the general equation (eq. (5) of ref. [20]), the resultant bending moment equation may be given by,

fo b Pf(ao/W) ~ b/2 M

or it can be represented as I~ + 12 + I3 = 0.

- - X (hi -- x)B dx

_

f, u_

(

x x)B

Jb/2

1 \ 2

/

^dx^{= 0} ⁽⁶⁾

o- I . K O < x ~<b

d2,~x

o. I ..[~ ,__TtT..,, b o~..<b/z

. . . . x ) b / 2 ...< x ~ b

~ . - - b [2 -.D

Fig. I. Stress pattern I (without considering plastic zone) in cracked 3PB specimen.

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Rotational factor--lI. Crack 3PB geometry 509

and Let

= ~ b p f ( a o / W ) B ( h , - x ) Pf(ao/W)[f[hnxl/2 Io ~ dx]

~'

jo n w ' , ' , f i ;

~, ,/;

d~ =

w,,,,/~ dx-

^xl,,

_ _Pf(ao/W)

[2h, x/~ - ~ b 3/2 ] ⁼

Pf(ao/W)

[6hl x//b - 2b 3/2 ]

Adding Is, 12 and 13 we get

or,

[5~ 3 b2hl]

I3 = -~- k 48 8

(7) (from eq. (17) of ref. [20])

(from eq. (18) of ref. [20]).

Pf(ao/W)

^[6htx/~

. 2b3/2]_1~ - [ ^b2h, b8 ].O

3 W I/2x/2-~ L 4

Pf(ao/W) [6hnv/-~_ 2b3/2]+24ffB[ hl b ~-g]=o

3 W I/2x/~-~

(putting M - - P L / 4 and I = 1/12

Bb 3)

w(1 -ao/W)-

(1

-ao/W)

hl [3hn~/W(1

- ao/W) -

^W3/2(I

- aolW)3/2]f (ao/W)

3 x / ~ W i/2 ~- 6 = 0

or,

(substituting W = 2B and

b = W - ao =

W(1 -

ao/W))

3h,(1

-aolW)l/2f(ao/W)

W(I

-aolW)3/2f(ao/W )

3,/~ 3,/~

6kin 6 W ( I -

ao/W)

= 0

4(1

^--

ao/W)

8(1 -

ao/W)

hi [ 3(1 -

aolW)l/2f(ao/W)

₊

3 ] 3w

2(1 --

ao/W) = "-~'t

W(I -

aolW)3/2f (ao/W )

3,/~

h i =

3w W(l-ao/W)3/2f(ao/W) - T + 3,/~

3 3(1 --

ao/W)n/2f(ao/W )

2(1

^-

aolW) 3 , / ~

r,W(1 -ao/W) =

WI.~ (1-ao/W)3/2f(ao/W)3~/~ ]

(I -aolW)'/~f(aolW)

2(I

-.o/W) ~/~

~+

3

r e = 3

~+

(I

-aolW)~/~f(a.lW)

(1 - a o l W ) 3 / ~ f ( a o l W ) "

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510 A.N. KUMAR and S. BHATTACHARYA

Table I. Effect of initial crack size on elastic ro- Table 2. Variation of elastic rotational factor with elastic stress

tational factor distribution range

ao/W 0,15 0.20 0.50 0.70 xl/b 10 -4 10 -3 10 -2 0.5 1.0

r,, 0.424 0.421 0.416 0.417 r,, 0.495 0.485 0.455 0.362 0.416

Equation (8) thus gives the elastic rotation factor as a function of ao/W. In the equation putting

ao/W

^{= 0,} r e becomes 0.50. This situation is obtained for the no crack situation, i.e. in the case o f plane bend geometry as already discussed in Part I [20]. However, assuming different ao / W ratios the value o f rotational factor has been calculated and is shown in Table 1.

1.2. Effect o f elastic stress distribution range on r e

In the above approach the non-linear stress is assumed to be present over the whole ligament, but the stress equation can only approximate the stress field fairly accurately within a small vicinity of the crack tip [21]. Since no information is available in the literature about the distance from the crack tip over which the stress distributions are valid, several limits for x are assumed for the stress eq. (2). The general formulation of the resultant moment equation may be given by

~xlef(a°/W)(hl-x)gdxfb/2M( b )

0 ,w'/2v/~ +J0

7- - x ( h , - x ) B d x

- - - x ( h l - x ) B d x = 0

o r

P f ( a o / W ) [6h~x//~t_ 2x~/2] 24PB [ h~ b )

3 x / ~ W 1/2 + ~ ~ ~- - = 0. (9)

Carrying out the integration over the different integral limits as before, and on further simplification one may find ht values which are used to obtain re values for different limits (Xl) of elastic stress distribution. The re values are shown in Table 2.

2. E S T I M A T I O N O F P L A S T I C Z O N E S I Z E

In the earlier section the theoretical formulation and determination o f rotational factor considering elastic stress distribution assumes no plastic deformation at the crack tip. The theoretical formulation considering the presence o f plastic zone at the crack tip necessitates the approximation o f plastic zone size. Estimation o f plastic zone size in plane stress as well as plane strain situation is dealt with in this section.

2.1. Plane stress

The most simplistic way to estimate the size o f plastic zone in cracked geometry is by finding out the point o f intersection of the plastic stress (try = ays) with the non-linear elastic stress {ay = K t / ~ } . The point of intersection (x = rx) may be obtained as follows [22, 23]

1 . ( K . ' ] 2

r~ = ~--~ \ -~y~ / . (10)

Replacing K~ by ( P / B W ~/2) f ( a o / W ) for three point bend geometry and a~,s by 6PGr/b 2 [24], the expression o f rx may be written as

1 P2{f(ao/W)}2b4 1 [ p2 W(1--:_~o/W)4 1

rx = 2---~" 3 6 W B 2 p ~ r = 2--~ P~---r" {f ( a ° / W ) } 2 --

- 1 8 r e k 4 { f ( a ° / W ) } 2 (11)

(putting b = W - ao = W(1 - a o / W ) = Wk).

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Rotational factor--II. Crack 3PB geometry 511

4 - -

"~ 2

v i X

0.~

0 -

_ W = 2 O m m

P l a n e st r a i n P l a n e s t r e s s

0.7

/ 0 . 3 / ^{/ 0 . 5}

/ / / / "

/ /

/ / /

/ / /"~ao/w = o ,7 / / " / /

, , / / / . /

~ - - - - # ~ g r - I I I I I

0.2 o.~ o 6 o. 8 1.o 1.2 1.4

P / % v

Fig. 2. Effect of loading on plastic zone size in 3PB cracked specimen.

2.2. Plane strain

When the plane strain situation prevails near the crack tip, the yield stress increases by x/~

times, giving rise to a smaller plastic zone. The size of the plastic zone is obtained by putting x / 3 ay, in place of cry, in eq. (10) as follows

1 gj z = k 4 { f ( a o / W ) } 2. (12)

rx=~

The variation of plastic zone size with loading for different ao/W ratios in both states of stress is shown in Fig. 2.

3. DETERMINATION OF ROTATIONAL FACTOR CONSIDERING PLASTIC ZONE (ELASTIC-PLASTIC SITUATION)

3.1. Plane stress

The present section is devoted to the theoretical formulation of the plastic rotational factor considering the presence of a plastic zone at the crack tip. The plastic zone developed at the crack tip modifies the non-linear elastic stress distribution introducing an additional plastic stress. The stress pattern considered here is shown in Fig. 3. The stress components and their limits are given below.

at=a~,~ for O<~x<~r.,. (13)

K,

for rx<.x<~b (14)

a 2 -- 2 N / / ~

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512 A . N . K U M A R and S. BHATTACHARYA

Y

\

--- bh-"-I " ~

/

-1

Fig. 3. Stress pattern II (considering plastic zone) in cracked 3PB specimen, 0"1 = O'y s 0 ~ x ~: r x

k

0"2 : , J 2 x ' K rx ~: x ~< b

~

=~_

( - - f - x ) ^b b / 2 ~ x ~< ^b

63=-)-- - x for O~<x~<~ (15)

0"4 = ~ - -- x for~ ~< x ~< b. (16)

Combining eqs (13)-(16) and following eq. 5 [20], the resultant bending moment equation may be written as

or,

at (hi - x)Bdx (r2(hl -- x)Bdx + a3(h I - x)Bdx - a4(h, - x)Bdx = 0

d 0 rx dO b/2 (17)

Let

I~ + I2+ I 3 - / 4 = 0.

I~= a l ( h t - x ) B d x = a y s B h l x - ~ J o = ~ t r y s B [ 2 h l r . ~ - r ~ ]

(18)

it, = fb P f ( a o / W ) ( h , - x)

12= arx a2(h, - x)Bdx J,, ~ w / - - ~ " Bdx

_ P f ( a o / W ) [ 2 h , ~ / - ~ 2 ] b _ p f ( a o / W )

2 ~ --'3 x3/z~r,

3 2 ~ [6ht'v/b --

2 b 3 / 2 - 6 h t ~ x -t- 2r 3/2]

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Table 3. Variation of plastic rotational factor with P/Por, ao/W and state of stress in 3PB cracked specimen

rp

Plane strain (Plane stress)

P/PGr ao/W =0,3 ao/W =0.5 ao/W =0.7

0.2 0.428 (0.429) 0.425 (0.427) 0,426 (0.428) 0.4 0.436 (0.440) 0.434 (0.439) 0,435 (0.443) 0.6 0.444 (0.449) 0.445 (0.449) 0,443 (0.450) 0.8 0.452 (0.460) 0.453 (0.459) 0.452 (0.460) 1.0 0.460 (0.469) 0.460 (0.470) 0,461 (0.470)

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4 8 "

Adding eqs (18)-(20) we get aysB (2hi rx -- r~)

2

Now putting

(20)

3PGy

b2 (2hlrx-r~)-~

On further simplification

[ 6PGvrx 4Px/~f(ao/W)

h,--U-+ (v/-~)w~/~

(from eq. (17) of ref. [20])

Pf(ao/W)3

2 ~ [6h'w/b - 2b3/2 - 6htw/~ + 2r3/2]

24PB[ b2h,

^{b 3 ]}

+ - - b -5-- 4 ~-8 = 0 .

Substituting

6PG r

°YS ----" b 2

2Pf(~ao/W).

[6h, w/b - 2b 3/z - 6h, ~ +

2rf 21

(3~/2rOW a/2

24P[h,

b 3 ] +--b-- 4 48 = 0 .

4P.~f(ao/W)

( / ~ ) W 3 / 2

[""~-]

= 3P 4Pr3/2f(a°/W) 4Pb3/Zf(ao/W) 3PGyr~ + b - -

( 3 w / ~ ) W 3/2 ( 3 V / ~ ) W 3/2

( Oo)

b = w 1 - - f f = ~

r, [ 6PGYrx 4Pk3/2f(a°/W) ~ 2 ~

^{+ 6 P ]}

I 4Pr3/2f(ao/W ) 4Pk3/2f(ao/W) 3PGyr2x -]

= 3P ( 3 ~ ) w 3j2 + 3 , / ~ + W2k23

b 2

0.5 Z,

u 0.4

-~ 0.a

c O :.7. 0-2

0

,.r 0,I

• I I

. . . I , , . ; , . , I . . . I . . . I

16/. 103 x 1 / b 1 6 2 1

Fig. 4. Effect of elastic stress distribution range on the rotational factor.

EFM 50/4~G

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514 A . N . K U M A R a n d S. B H A T r A C H A R Y A

o r

rp[ 6P°wk rx

_ 4 P f ( a o / W ) 1/2}._[_ 6P

= [3P 4Pf(ao/W) r~

₍₂₁₎

Equation (21) thus becomes a function of P, Pot, rx and

ao/W.

Substituting the size of plastic zone (plane stress) from eq. (11) into eq. (21) and simplifying one may find

4ek 3/2f (ao / W) 7 r F k3e2{f(a°/w)}2 2P2k3{f(a°/W)}Z ~ +6P

"L 3rePay x//~ I

3k6p4{f(ao/W)} 4 4k6p4{f(ao/W)} 4 4Pf(ao/W)k 3/2

=3P-~

(18~)2e~y 3x/~ (18~)3/2p ~ 3v/~

I 4k3/2f(ao/W) k3{f(ao/W)}2P]=

rp 6q x / ~ 3 - ~ a r J 3 +

4f(ao/W)k 3/z k6p3{f(ao/W)}4

3 x / ~ (18n):e3r or,

or,

e 3 '

3+4f(a°/W)k3/2-Ik6{f(a°/W)}4(-~Gr)

/(18n)2] (22)

rp= 6 -~ 4f(a°/W)k3/2~ 3nl k3{f(a°/W)}2(P-~r )

Equation (22) has been used to estimate rp values as a function of

P/PGY

and

ao/W.

The results are reported in Table 3.

3.2.

Plane strain

In plane strain condition, replacing

ffys

by x/~ try s in eq. (17), the resultant bending moment equation may be written as

fr"x/~ Crys(h,-x)Bdx + fb K' (h,-x)Bdx

0 r x - - ~

) )

+,J0

I \ 2 - x (h~-x)Bdx- b/2I- --X (h~-x)Bdx=O.

(23) Carrying out the integration as before

rp

may be evolved as

3 +0.532f(ao/W)k3/Z + l.36 x lO-4{f(ao/W)}4k6( ~-r)3

rp - - (24)

6 + l.596 k3/2f (ao/W) + O.O85{f (ao/W)}2k3( ~r )

Assuming different

ao/W

values as well as

P/PGY

ratios,

rp

has been calculated and is shown in Table 3.

4. DISCUSSION

In the cracked situation the rotational factor (re) without considering the plasticity effect (elastic situation) is found to be around 0.416 assuming the non-linear stress part is extended over the entire ligament length. The theoretical analysis is found to be independent of

PIPer

and this is valid for smaller

P/P~y

values when the effects of plasticity may be neglected. A marginal dependency of re values is seen on the

ao/W

ratio. A change of

ao/W

from 0.15 to 0.70 causes a change in r e values from 0.424 to 0.417, i.e. around 1.5%. Since the non-linear 1/x/~ type stress distribution is only valid in the close vicinity of the crack tip and the precise range of validity of the stress function is not known, this aspect has been studied. As shown in Table 2 the r e value

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0.470

0.46 5

u 0

c o 0

o

0.4 60

0.4 55

0.450

0 , 4 4 5

0.~40

0 ~ 3 5

_ _ _ _

= 0 . 5 . . . ~ / w = 0 . 7

0.1.3(

0.42~

0.420

o.~ISl i t I I J

0.2 0 . 4 0.6 0.8 1.0

P/%v

Fig. 5. Effect o f ao/W, P/PGr and state o f stress on the rotational factor.

tends to increase as the range of validity of the 1/x/~ type stress function decreases. This is obvious as the contribution from the tensile part decreases with the decrease in the validity range o f the stress distribution in question. The r e values increase beyond

xt/b

= 0 . 5 , because the net contribution o f the compressive part decreases with increase in

x~/b

beyond 0.5. The effect is shown in Fig. 4. In the limiting situation when

x,/b

tends to 0, i.e. xt ----, 0, r e tends to 0.5 as the situation may be described by plane bend geometry. An exact knowledge is required for the validity of the I/x/-~ type of stress function in order to determine r e values precisely.

In situations where the plastic zone is large enough and cannot be ignored the "rp" value may be considered under plane stress and plane strain situations. Since the plastic zone size in the plane strain condition is about one-third o f the same in plane stress, the rp values also may be expected to be smaller. Table 3 reports the values for two situations when

P/Par

varies from 0.2 to 1.0. As the

P/Par

ratio increases the size of the plastic zone increases and therefore the

rp

values considering the plastic zone also increase. However, the increase in rp values is not significant and mostly lie within 10%. The effects o f state of stress,

P/Par

ratio and

ao/W

ratio are depicted in Fig. 5.

The effect of

ao/W

on the value o f rp appears to be very negligible for all values of

ao/W

considered, while a change of around 1-2% in the value of

rp

may be seen for two extensive states o f stress situations at any load levels. Again for loading up to

P/Par

level o f around 0.4, the effect o f the state o f stress may even reduce to nil.

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516 A . N . KUMAR and S. BHATTACHARYA

2 - -

A A p p a r e n t r o t a t i o n a l f a c t o r [ 1,7,11 ]

• A c t u a l r o t a t i o n a l f a c t o r [ !,7,11 ]

x P l a s t i c r o t a t i o n a l f a c t o r (proposed model)

I i I I t I I i

0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

P / P G Y

Fig. 6. Comparison of plastic rotational factor obtained by proposed model and Kolednik [1, 7, 11].

T o substantiate the above results on r and effects o f several parameters like ao/W, P/Par, etc.

the reported works in literature may be referred. Matsoukas et al. [4] demonstrated a very insignificant effect o f ao/W on rp values as the rp values change from 0.439 to 0.455 for 0.2 ~< ao/W <<. 0.6. A range ofrp values between 0.46 to 0.48 was shown by Lin et al. [25] who also found a slight decreasing effect of rp with increase in ao/W. However, a strong dependency between the two parameters (rp and ao/W) is demonstrated by the equation [3]:

rp = 0.029 + 1.3(ao/W) - 0.9(ao/W) 2.

As ao/W decreases from 0.5 to 0.1, the rp values reduce from 0.455 to 0.15. Nevertheless, at ao/W = 0.5 the rp value o f 0.455 matches well with the r values found in the present investigation.

Shi et al. [10] again confirmed the insensitiveness of ao/W on rp. The findings from the present work are also confirmed as rp values reported earlier [26] are not dependent on ao/W noticeably in the range 0.35 to 0.60. The same author reported a value o f 0.46 for the range P/Par (0.6 < P/Par < 1.5). As may be seen from Table 3, the values ofrp for ao/W = 0.5 lie in the ranges 0.449-0.47 and 0.445-0.46, respectively for the plane stress and plane strain situation over 0.6 < P/PGr < 1.0.

Kolednik [1, 7, 11] recently studied the effect o f P/PaY on the value o f rp and found an infinite value at the start o f loading when P/PaY---' O. However, with continued loading r~ sharply falls and reaches a saturated value of about 0.4 as P/PGY ~ 1 (general yielding). The variations between the two are shown in Fig. 6 along with the results o f the present work. At the start o f loading the r value is found to be 0.416 which slowly increases to 0.46 or 0.47, respectively for plane strain and plane stress conditions. The hinge axis lying outside the specimen ligament is hypothetical and

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Rotational factor--II. Crack 3PB geometry 517 meaningless. Therefore a value o f rp > 1.0 should n o t be acceptable. In view o f the a b o v e consideration, the rp values as reported in this study a p p e a r to be highly logical. It m a y also be m e n t i o n e d here that effects o f state o f stress at the crack tip have n o t been reported elsewhere.

5. C O N C L U S I O N S

T h e following conclusions on the theoretical determination o f rotational factor using resultant b e n d i n g m o m e n t can be drawn.

(1) In the case o f cracked b e n d g e o m e t r y the rotational factor assuming no plastic zone (linear elastic situation) is f o u n d to be a function o f the distance (Xl) over which the try distribution is present. T h e re value tends to decrease f r o m 0.5 for x~/b = 0 to 0.416 for x~/b = 1. A very mild d e p e n d e n c y is observed for re o n a o / W as re is f o u n d to increase f r o m 0.417 to 0.424 with the c h a n g e in the a o / W ratio f r o m 0.15 to 0.70. In this situation the theoretical f o r m u l a t i o n reveals that the value o f re is independent o f loading.

(2) C o n s i d e r a t i o n o f plastic zone size and the elastic-plastic stress pattern ( a r o u n d the general yield situation) in cracked b e n d g e o m e t r y reveals rp values to fall in the range o f 0 . 4 2 5 - 0 . 4 6 for P / P o t varying between 0.2 a n d 1.0 ( a o / W = 0.5). A negligible influence o f a o / W on rp values is h o w e v e r observed. F o r a given a o / W a n d load level P / P ~ r a change f r o m plane stress to plane strain c o n d i t i o n causes a difference o f a r o u n d 2% in the value o f rotational factor. T h e present study confirms the results on r o t a t i o n a l factor m a d e by several researchers earlier.

R E F E R E N C E S [1] O. Kolednik, Engng Fracture Mech. 33, 813-826 (1989),

[2] C. R. Pratap, R. K. Pandey and R. Chinadurai, Engng Fracture Mech. 31, 105-118 (1988).

[3] D. Z. Zhang and S. F. Zhu, Engng Fracture Mech. 31, 917-921 (1988).

[4] G. Matsoukas, B. Cotterell and Y. W. Mai, Int. J. Fracture 26, R49-R53 (1984).

[5] T. Hollstein and J. G. Blauel, Int. J. Fracture 13, 385-390 (1977).

[6] A. Bhattacharya, Engng Fracture Mech. 40, 187-200 (1991).

[7] O. Kolednik, Int. J. Fracture 39, 269-286 (1989).

[8] S. A. Paranjpe, Ph.D. Thesis, I.I.T., Bombay (1976).

[9] W. N. Sharpe, Jr and S. Paleebut, Proc. ICF 5, France, Vol. 5, p. 2517 (1981).

[I0] Y. W. Shi, Z. X. Han, N. N. Zhou and J. Li, Engng Fracture Mech. 41, 143-151 (1992).

[11] O. Kolednik, Engng Fracture Mech. 29, 173-188 (1988).

[12] C. L. Veerman and T. Muller, Engng Fracture Mech. 4, 25 (1972).

[13] British Standard Institution, BS 5762 (1979).

[14] Standard test method for COD fracture toughness measurement, ASTM Standards E-1290 (1989).

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[26] W. Shang Xian, Engng Fracture Mech. 18, 83-95 (1983).

(Received 24 December 1993)

Rotational factor using bending moment approach under elasto-plastic situation

~ Pergamon