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A model for crack growth behaviour of macroscopically deflected fatigue cracks
Scripta M E T A L L U R G I C A et M A T E R I A L I A
Vol.
26, pp. 1713-1718, 1992 Printed in the U.S.A.
Pergamon Press Ltd. All rights reserved
A MODEL FOR CRACK GROWTH B E H A V l O U R OF H A C R O S C O P I C A L L Y D E F L E C T E D F A T I G O E CRACKS S.V.Ea~at and N.Eswara P~asad Defence ~ t s l l u r g l c a l Research Laboratory, Hyderabad, India 500 258 (Received February 27, 1992) (Revised March 31, 1992)
Fatigue c r a c k growth c h a r a c t e r i z a t i o n under mode I loading c o n d i t i o n s is b a s e d on the premise that the fatigue crack path is linear and its plane of g r o w t h is normal to the loading axis. It is, however, well known that on a m i c r o s c o p i c level and in some cases even on a m a c r o s c o p i c level, cracks seldom p r o p a g a t e in a linear fashion. P r o n o u n c e d d e f l e c t i o n and b r a n c h i n g of the crack can occur due to factors such as stress state, environment, load e x c u r s i o n s and local m i c r o s t r u c t u r a l d i s c o n t i n u i t i e s [1-8]. N o n - l i n e a r i t i e s in crack path are g e n e r a l l y ignored in the c h a r a c t e r i z a t i o n of fatigue b e h a v i o u r b e c a u s e of the d i f f i c u l t i e s in i n c o r p o r a t i n g crack m e a n d e r i n g effects in the e s t i m a t e s of stress i n t e n s i t y factor range. Suresh [9] has p r o v i d e d a d e t a i l e d t h e o r e t i c a l a n a l y s i s for the case of d e f l e c t e d fatigue c r a c k s w h i c h m e a n d e r (illustrated s c h e m a t i c a l l y in Fig.l). However, in c e r t a i n cases such as 2090 AI-Li alloys in u n d e r a g e d c o n d i t i o n s [I0], the fatigue c r a c k s g r o w in a m a c r o s c o p l c a l l y d e f l e c t e d m a n n e r ( i l l u s t r a t e d s c h e m a t i c a l l y in Fig.2a). The o b j e c t i v e of this p a p e r is to p r o v i d e a simple elastic d e f l e c t i o n model for a m a o r o s c o p i c a l l y d e f l e c t e d fatigue c r a c k by i n c o r p o r a t i n g c h a n g e s in the e f f e c t i v e d r i v i n g force and in the a p p a r e n t p r o p a g a t i o n rate. T h e o r e t i c a l Analysis Stress i n t e n s i t y factors for d e f l e c t e d cracks are a v a i l a b l e for cases where the d e f l e c t e d kink is i n f i n i t e s i m a l l y small [11,12] as well as for cases w h e r e the kink is of finite size [18]. In fatigue crack growth tests, the crack g r o w t h after d e f l e c t i o n is quite a p p r e c i a b l e and it is more a p p r o p r i a t e to use the stress intensity factors for kinks of finite size. A schematic r e p r e s e n t a t i o n of a m a c r o s c o p i c a l l y d e f l e c t e d crack is shown in Fig.2a. For a finite size kink, the k i n k e d crack can be r e p l a c e d by a straight crack w i t h the same i n c l i n a t i o n angle, 8, and w i t h the same p r o j e c t e d crack length, a, as the d e f l e c t e d crack, as shown in Fig.2b [13]. The stress i n t e n s i t y factors at the d e f l e c t e d c r a c k tip are then given by the f o l l o w i n g e q u a t i o n s [13] :
kl = Kicos3nO
(1)
k2
(2)
=
KlCOSU20sin O
w h e r e El is the far field a p p l i e d mode I stress intensity factor and k, and are the m e s o l v e d mode I and mode II stress intensity factors at the tip of the d e f l e c t e d crack, respectively. A simple c o p l a n a r strain e n e r g y release rate c r i t e r i o n [14] can be used for c a l c u l a t i n g the e f f e c t i v e stress i n t e n s i t y factorat the tip of the d e f l e c t e d crack. The e f f e c t i v e stress i n t e n s i t y factor, ~ff is then g i v e n by the f o l l o w i n g :
Keff = (k~+ k~) v' 1713 0036-9748/92 $5.00 + .00 Copyright (c) 1992 Pergamon Press Ltd.
(3)
1714
DEFLECTED FATIGUE CRACKS
substituting equations
Keff or
=
(i) and ( 2 )
KI
Vol.
26,
No.
in ( 3 ) , one obtains:
C0$1/2B
(4)
in case of fatigue crack g r o w t h
AKeff = AKicosV20 For an u n d e f l e c t e d crack w h e r e O
AKef f
(5) = O"
= AK I
(6)
=
(7)
This implies,
( K)uo
(AK). cosV2o
where (aK)D and (AK)~ are the d r i v i n g forces for crack g r o w t h of d e f l e c t e d and u n d e f l e c t e d cracks, respectively. E q u a t i o n (7) can be r e w r i t t e n as,
(~K)O
=
(~K)u o see1/2 @
(8)
In other words, a larger e x t e r n a l l y a p p l i e d d r i v i n g force is r e q u i r e d to p r o p a g a t e a d e f l e c t e d crack at the same rate. The a p p l i e d driving force r e q u i r e d to p r o p a g a t e a d e f l e c t e d c r a c k i n c r e a s e s w i t h i n c r e a s i n g d e f l e c t i o n angle. This is i l l u s t r a t e d in Fig.3 w h i c h shows the v a r i a t i o n of (AK),/(AK)~ w i t h d e f l e c t i o n angle. The nominal mode I crack p r o p a g a t i o n rate for a d e f l e c t e d c r a c k can be w r i t t e n by simple g e o m e t r y as:
= ( /aN) ocosO
(9)
w h e r e (da/dN)~ and ( d a / d N ) ~ are the m e a s u r e d g r o w t h rates of d e f l e c t e d and u n d e f l e c t e d cracks, respectively. E q u a t i o n (9) shows that the g r o w t h rates of a d e f l e c t e d crack are always a p p a r e n t l y lower than those for an u n d e f l e c t e d c r a c k s u b j e c t e d to the same v a l u e of aK~ff, if the d e f l e c t i o n in the c r a c k path is not taken into account. The higher the d e f l e c t i o n a n g l e , t h e lower is the c r a c k g r o w t h rate. This is i l l u s t r a t e d in Fig.3, w h i c h shows the v a r i a t i o n of (da/dN),/(da/dN)n w i t h the d e f l e c t i o n angle. The above p r o c e d u r e for e s t i m a t i n g the fatigue c r a c k g r o w t h b e h a v i o u r of a d e f l e c t e d c r a c k is i l l u s t r a t e d for a c o n d i t i o n w h e r e the fatigue c r a c k grows at an angle of 45" (0=45 °) to the loading axis. S u b s t i t u t i o n of @=45 ° in e q u a t i o n (8) yields that an a p p a r e n t d r i v i n g force 1.19 times larger than the d r i v i n g force for an u n d e f l e c t e d crack is n e c e s s a r y to p r o p a g a t e the d e f l e c t e d crack at the same rate. S i m i l a r l y j e q u a t i o n (9) provides,
(da/dN)o
=
0.707 ( da/dN)u D
(10)
In other words, the d e f l e c t e d crack p r o p a g a t e s at a p p a r e n t l y 29 percent slower rate in mode I g r o w t h d i r e c t i o n for the same d r i v i n g force. The p r e d i c t e d c h a n g e s in the fatigue crack g r o w t h b e h a v i o u r for v a r i o u s d e g r e e s of d e f l e c t i o n are i l l u s t r a t e d in Fig.4. The h i g h e r the d e f l e c t i o n angle, the g r e a t e r is the improvement in fatigue crack g r o w t h r e s i s t a n c e of the material. ~
i
~
n
with K~De~Imant
The p r e d i c t e d v a r i a t i o n s in fatigue b e h a v i o u r w i t h d e f l e c t i o n are c o m p a r e d here w i % h the e x p e r i m e n t a l results of V e n k a t e s h w a r a R a o and R i t o h i e for a n A i - L i a l l o y 2090 (10). T h i s a l l o y in the o v e r a g e d (0/%) c o n d i t i o n e x h i b i t s a n o m i n a l l y u n d e f l e c t e d c r a c k path, w h e r e a s the same a l l o y in the lightly o v e r a g e d (LOA) c o n d i t i o n shows a m a c r o s c o p i c a l l y d e f l e c t e d crack path w i t h an average d e f l e c t i o n angle of about 27 ° . The fatigue data in the o v e r a g e d c o n d i t i o n was u s e d as a r e f e r e n c e to o b t a i n the p r e d i c t e d fatigue data for the lightly o v e r a g s d c o n d i t i o n u s i n g the e a r l i e r d e s c r i b e d d e f l e c t i o n model. In other w o r d s
"
Vol.
26, No.
ll
DEFLECTED FATIGUE CRACKS
1715
( d a / d N ) ~ = 0.89 (da/dN)~ and (AK)~a = 1.06 (AK)oa. (These calculations were done only up to a stress intensity factor of 8MPalm as the crack path m o r p h o l o ~ at higher aK in LOA condition is not shown in reference 10). The experimental fatigue crack growth data in OA and LOA conditions as well as the theoretically predicted LOA fatigue behaviour is illustrated in Fig.5. Figure 5 indicates that there is fairly reasonable agreement between the theoretically predicted and experimentally observed fatigue data for AI-Li alloy 2090 in LOA condition. The match is extremely good in the near threshold region whereas it is only reasonable in the Paris regime. This is expected because deflection will primarily affect the roughness induced closure which is predominant in the near threshold regime. At higher stress intensities, plasticity induced closure also begins to play a significant role and has to be taken into account. The implications of the above model are that the factors which promote macroscopic deflection can significantly improve the fatigue crack growth resistance of a material. For example, in 2090 AI-Li alloys a significant increase in fatigue crack growth resistance can he achieved by tailoring the microstructure with a suitable selection of the aging treatment, (LOA) which leads to appreciable macroscopic crack deflection.
Ao][nowle.~lsmnt~ The authors would like to acknowledge Shri S.L.N.Acharyulu, Director, DMRL for his permission to publish the paper. The authors would also llke to thank Dr.G.Malakondaiah for his encouragement.
i. 2. 3. 4. 5. 6. 7. 8. 9. 10.
F.Erdogan and G.C.Sih, J.Basic Eng., 85, 519 (1963). W.W.Gerberich and K.A.Peterson, In Micro and Macro Mechanism of Crack Growth, K.Sadananda, B.B.Rath, and D.J.Michel eds., TMS-AIME, Warrendale PA, p.1 (1982). J.Lankford and D.L.Davidson, in Advances in Fracture Research, D.Francois ed, Pergaman Press, Oxford, V.4, p.899 (1981). S.Suresh, Scripts Hetall., 16, 495 (1982). A.K.Vasudevan, P.E.Bretz, A.C.Miller, and S.Suresh, Mater. Sci. Eng., (1983). J.Lankford, Fat. Eng. Mater. Struct. ~, 233 (1982). J.Lankford and D.L.Davidson, Acts Metall., 31 1273 (1983). D.L.Davidson, Fat. Eng. Mater. Struct., ~ 229 (1981). S.Suresh, Metall. Trans. A, 14A 2375 (1983) K.T.Venkateshwara Rao and R.O.Ritchle, Mater. Sci. and Engg. 100, 23,
(1988). 11. 12. 13. 14.
B.A.Bilby, G.C.Cardew, and I.C.Howard, in Fracture 1977, D.M.R.Taplin, ed., University of Waterloo Press, Z, 197 (1977). B.Cotterell and J.R.Rice, Int. J.Fract., 16 155 (1980). H.Kitagawa, R.Yuukl, and T.Ohira, Eng. Fract. Mech., Z, 515 (1975). David Broek, in Elementary Engineering Fracture Mechanics, Martlnus Nijhoff Publlshers, 374 (1986).
1716
DEFLECTED FATIGUE CRACKS
Vol.
I - - s --I
Fig.l
Schematic illustration crack which meanders
of a (Ref.9).
segment
of
a
deflected
b a
_L U~ b cose~
IL. --
i_
a
Fig.2
_l
Schematic deflected
illustration crack and b)
o£ a) a an equivalent
macroscopioall¥ inclined crack.
26,
No.
11
Vol.
26, No.
Ii
DEFLECTED FATIGUE CRACKS
2A
1717
l
I I
2.2
I I I I
2.0 L't
1.8
1.0 0.8 Z "o
\,/
~ 1.6 A~ v
rib 0.6
i////~
1.2!
c~ OJ, z -1D 02
1.0
¢o
-o
0 30
60
90
DEFLECTION ANGLE (B), DEG. FIB.3
Variation of the normalized stress intensity factor range (dashed line) and the crack growth rate (solid line) for a macrosoopically deflsoted crack with ar~lle of deflection
-3 w
~
R llX~6.)
6O
E E
Z "~ \ ¢o
75
1~5 ;
,.,
1~6
-r
1~7
o ~
1~|
<
*
9! 1
I
2
STRESS INTENSITY FACTORRANGE(AK) MPaV-~ Fig.4
Predicted fatigue crack uacroe~optcally deflected angle of deflection
Er~wth cracks as
behaviour a function
for of
1718
DEFLECTED FATIGUE CRACKS
Vol.
26, No.
-4
10 .-I
l(~S -
Z
m
"C3
m'P
10 ----A---
LI.J 0
I--
. . . .
EXPT. OA EXPT. LOA THEORETICAL LOA
¢Y
~-~ 1~v
¢-y
~'J
-9
10
I
3
I
J
I
J
5
6
7
8
STRESS INTENSITY FACTORRANGE(AK), MPa~l-~Fig.5
Experimental fatigue crack growth behaviour in overfed (OA) a n d l i g h t l y o v e r a g o d (LOA) c o n d i t i o n B (from Rof.10) and theo~tlcally predicted fatigue c r a c k R r o w t h b e h a v i o u r i n LOA c o n d i t i o n u e i n a f a t i g u e c r a c k g r o w t h d a t a i n OA c o n d i t i o n and the meamared deflection a n g l e i n LOA c o n d i t i o n .
ii
Vol.
26, pp. 1713-1718, 1992 Printed in the U.S.A.
Pergamon Press Ltd. All rights reserved
A MODEL FOR CRACK GROWTH B E H A V l O U R OF H A C R O S C O P I C A L L Y D E F L E C T E D F A T I G O E CRACKS S.V.Ea~at and N.Eswara P~asad Defence ~ t s l l u r g l c a l Research Laboratory, Hyderabad, India 500 258 (Received February 27, 1992) (Revised March 31, 1992)
Fatigue c r a c k growth c h a r a c t e r i z a t i o n under mode I loading c o n d i t i o n s is b a s e d on the premise that the fatigue crack path is linear and its plane of g r o w t h is normal to the loading axis. It is, however, well known that on a m i c r o s c o p i c level and in some cases even on a m a c r o s c o p i c level, cracks seldom p r o p a g a t e in a linear fashion. P r o n o u n c e d d e f l e c t i o n and b r a n c h i n g of the crack can occur due to factors such as stress state, environment, load e x c u r s i o n s and local m i c r o s t r u c t u r a l d i s c o n t i n u i t i e s [1-8]. N o n - l i n e a r i t i e s in crack path are g e n e r a l l y ignored in the c h a r a c t e r i z a t i o n of fatigue b e h a v i o u r b e c a u s e of the d i f f i c u l t i e s in i n c o r p o r a t i n g crack m e a n d e r i n g effects in the e s t i m a t e s of stress i n t e n s i t y factor range. Suresh [9] has p r o v i d e d a d e t a i l e d t h e o r e t i c a l a n a l y s i s for the case of d e f l e c t e d fatigue c r a c k s w h i c h m e a n d e r (illustrated s c h e m a t i c a l l y in Fig.l). However, in c e r t a i n cases such as 2090 AI-Li alloys in u n d e r a g e d c o n d i t i o n s [I0], the fatigue c r a c k s g r o w in a m a c r o s c o p l c a l l y d e f l e c t e d m a n n e r ( i l l u s t r a t e d s c h e m a t i c a l l y in Fig.2a). The o b j e c t i v e of this p a p e r is to p r o v i d e a simple elastic d e f l e c t i o n model for a m a o r o s c o p i c a l l y d e f l e c t e d fatigue c r a c k by i n c o r p o r a t i n g c h a n g e s in the e f f e c t i v e d r i v i n g force and in the a p p a r e n t p r o p a g a t i o n rate. T h e o r e t i c a l Analysis Stress i n t e n s i t y factors for d e f l e c t e d cracks are a v a i l a b l e for cases where the d e f l e c t e d kink is i n f i n i t e s i m a l l y small [11,12] as well as for cases w h e r e the kink is of finite size [18]. In fatigue crack growth tests, the crack g r o w t h after d e f l e c t i o n is quite a p p r e c i a b l e and it is more a p p r o p r i a t e to use the stress intensity factors for kinks of finite size. A schematic r e p r e s e n t a t i o n of a m a c r o s c o p i c a l l y d e f l e c t e d crack is shown in Fig.2a. For a finite size kink, the k i n k e d crack can be r e p l a c e d by a straight crack w i t h the same i n c l i n a t i o n angle, 8, and w i t h the same p r o j e c t e d crack length, a, as the d e f l e c t e d crack, as shown in Fig.2b [13]. The stress i n t e n s i t y factors at the d e f l e c t e d c r a c k tip are then given by the f o l l o w i n g e q u a t i o n s [13] :
kl = Kicos3nO
(1)
k2
(2)
=
KlCOSU20sin O
w h e r e El is the far field a p p l i e d mode I stress intensity factor and k, and are the m e s o l v e d mode I and mode II stress intensity factors at the tip of the d e f l e c t e d crack, respectively. A simple c o p l a n a r strain e n e r g y release rate c r i t e r i o n [14] can be used for c a l c u l a t i n g the e f f e c t i v e stress i n t e n s i t y factorat the tip of the d e f l e c t e d crack. The e f f e c t i v e stress i n t e n s i t y factor, ~ff is then g i v e n by the f o l l o w i n g :
Keff = (k~+ k~) v' 1713 0036-9748/92 $5.00 + .00 Copyright (c) 1992 Pergamon Press Ltd.
(3)
1714
DEFLECTED FATIGUE CRACKS
substituting equations
Keff or
=
(i) and ( 2 )
KI
Vol.
26,
No.
in ( 3 ) , one obtains:
C0$1/2B
(4)
in case of fatigue crack g r o w t h
AKeff = AKicosV20 For an u n d e f l e c t e d crack w h e r e O
AKef f
(5) = O"
= AK I
(6)
=
(7)
This implies,
( K)uo
(AK). cosV2o
where (aK)D and (AK)~ are the d r i v i n g forces for crack g r o w t h of d e f l e c t e d and u n d e f l e c t e d cracks, respectively. E q u a t i o n (7) can be r e w r i t t e n as,
(~K)O
=
(~K)u o see1/2 @
(8)
In other words, a larger e x t e r n a l l y a p p l i e d d r i v i n g force is r e q u i r e d to p r o p a g a t e a d e f l e c t e d crack at the same rate. The a p p l i e d driving force r e q u i r e d to p r o p a g a t e a d e f l e c t e d c r a c k i n c r e a s e s w i t h i n c r e a s i n g d e f l e c t i o n angle. This is i l l u s t r a t e d in Fig.3 w h i c h shows the v a r i a t i o n of (AK),/(AK)~ w i t h d e f l e c t i o n angle. The nominal mode I crack p r o p a g a t i o n rate for a d e f l e c t e d c r a c k can be w r i t t e n by simple g e o m e t r y as:
= ( /aN) ocosO
(9)
w h e r e (da/dN)~ and ( d a / d N ) ~ are the m e a s u r e d g r o w t h rates of d e f l e c t e d and u n d e f l e c t e d cracks, respectively. E q u a t i o n (9) shows that the g r o w t h rates of a d e f l e c t e d crack are always a p p a r e n t l y lower than those for an u n d e f l e c t e d c r a c k s u b j e c t e d to the same v a l u e of aK~ff, if the d e f l e c t i o n in the c r a c k path is not taken into account. The higher the d e f l e c t i o n a n g l e , t h e lower is the c r a c k g r o w t h rate. This is i l l u s t r a t e d in Fig.3, w h i c h shows the v a r i a t i o n of (da/dN),/(da/dN)n w i t h the d e f l e c t i o n angle. The above p r o c e d u r e for e s t i m a t i n g the fatigue c r a c k g r o w t h b e h a v i o u r of a d e f l e c t e d c r a c k is i l l u s t r a t e d for a c o n d i t i o n w h e r e the fatigue c r a c k grows at an angle of 45" (0=45 °) to the loading axis. S u b s t i t u t i o n of @=45 ° in e q u a t i o n (8) yields that an a p p a r e n t d r i v i n g force 1.19 times larger than the d r i v i n g force for an u n d e f l e c t e d crack is n e c e s s a r y to p r o p a g a t e the d e f l e c t e d crack at the same rate. S i m i l a r l y j e q u a t i o n (9) provides,
(da/dN)o
=
0.707 ( da/dN)u D
(10)
In other words, the d e f l e c t e d crack p r o p a g a t e s at a p p a r e n t l y 29 percent slower rate in mode I g r o w t h d i r e c t i o n for the same d r i v i n g force. The p r e d i c t e d c h a n g e s in the fatigue crack g r o w t h b e h a v i o u r for v a r i o u s d e g r e e s of d e f l e c t i o n are i l l u s t r a t e d in Fig.4. The h i g h e r the d e f l e c t i o n angle, the g r e a t e r is the improvement in fatigue crack g r o w t h r e s i s t a n c e of the material. ~
i
~
n
with K~De~Imant
The p r e d i c t e d v a r i a t i o n s in fatigue b e h a v i o u r w i t h d e f l e c t i o n are c o m p a r e d here w i % h the e x p e r i m e n t a l results of V e n k a t e s h w a r a R a o and R i t o h i e for a n A i - L i a l l o y 2090 (10). T h i s a l l o y in the o v e r a g e d (0/%) c o n d i t i o n e x h i b i t s a n o m i n a l l y u n d e f l e c t e d c r a c k path, w h e r e a s the same a l l o y in the lightly o v e r a g e d (LOA) c o n d i t i o n shows a m a c r o s c o p i c a l l y d e f l e c t e d crack path w i t h an average d e f l e c t i o n angle of about 27 ° . The fatigue data in the o v e r a g e d c o n d i t i o n was u s e d as a r e f e r e n c e to o b t a i n the p r e d i c t e d fatigue data for the lightly o v e r a g s d c o n d i t i o n u s i n g the e a r l i e r d e s c r i b e d d e f l e c t i o n model. In other w o r d s
"
Vol.
26, No.
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DEFLECTED FATIGUE CRACKS
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( d a / d N ) ~ = 0.89 (da/dN)~ and (AK)~a = 1.06 (AK)oa. (These calculations were done only up to a stress intensity factor of 8MPalm as the crack path m o r p h o l o ~ at higher aK in LOA condition is not shown in reference 10). The experimental fatigue crack growth data in OA and LOA conditions as well as the theoretically predicted LOA fatigue behaviour is illustrated in Fig.5. Figure 5 indicates that there is fairly reasonable agreement between the theoretically predicted and experimentally observed fatigue data for AI-Li alloy 2090 in LOA condition. The match is extremely good in the near threshold region whereas it is only reasonable in the Paris regime. This is expected because deflection will primarily affect the roughness induced closure which is predominant in the near threshold regime. At higher stress intensities, plasticity induced closure also begins to play a significant role and has to be taken into account. The implications of the above model are that the factors which promote macroscopic deflection can significantly improve the fatigue crack growth resistance of a material. For example, in 2090 AI-Li alloys a significant increase in fatigue crack growth resistance can he achieved by tailoring the microstructure with a suitable selection of the aging treatment, (LOA) which leads to appreciable macroscopic crack deflection.
Ao][nowle.~lsmnt~ The authors would like to acknowledge Shri S.L.N.Acharyulu, Director, DMRL for his permission to publish the paper. The authors would also llke to thank Dr.G.Malakondaiah for his encouragement.
i. 2. 3. 4. 5. 6. 7. 8. 9. 10.
F.Erdogan and G.C.Sih, J.Basic Eng., 85, 519 (1963). W.W.Gerberich and K.A.Peterson, In Micro and Macro Mechanism of Crack Growth, K.Sadananda, B.B.Rath, and D.J.Michel eds., TMS-AIME, Warrendale PA, p.1 (1982). J.Lankford and D.L.Davidson, in Advances in Fracture Research, D.Francois ed, Pergaman Press, Oxford, V.4, p.899 (1981). S.Suresh, Scripts Hetall., 16, 495 (1982). A.K.Vasudevan, P.E.Bretz, A.C.Miller, and S.Suresh, Mater. Sci. Eng., (1983). J.Lankford, Fat. Eng. Mater. Struct. ~, 233 (1982). J.Lankford and D.L.Davidson, Acts Metall., 31 1273 (1983). D.L.Davidson, Fat. Eng. Mater. Struct., ~ 229 (1981). S.Suresh, Metall. Trans. A, 14A 2375 (1983) K.T.Venkateshwara Rao and R.O.Ritchle, Mater. Sci. and Engg. 100, 23,
(1988). 11. 12. 13. 14.
B.A.Bilby, G.C.Cardew, and I.C.Howard, in Fracture 1977, D.M.R.Taplin, ed., University of Waterloo Press, Z, 197 (1977). B.Cotterell and J.R.Rice, Int. J.Fract., 16 155 (1980). H.Kitagawa, R.Yuukl, and T.Ohira, Eng. Fract. Mech., Z, 515 (1975). David Broek, in Elementary Engineering Fracture Mechanics, Martlnus Nijhoff Publlshers, 374 (1986).
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DEFLECTED FATIGUE CRACKS
Vol.
I - - s --I
Fig.l
Schematic illustration crack which meanders
of a (Ref.9).
segment
of
a
deflected
b a
_L U~ b cose~
IL. --
i_
a
Fig.2
_l
Schematic deflected
illustration crack and b)
o£ a) a an equivalent
macroscopioall¥ inclined crack.
26,
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11
Vol.
26, No.
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DEFLECTED FATIGUE CRACKS
2A
1717
l
I I
2.2
I I I I
2.0 L't
1.8
1.0 0.8 Z "o
\,/
~ 1.6 A~ v
rib 0.6
i////~
1.2!
c~ OJ, z -1D 02
1.0
¢o
-o
0 30
60
90
DEFLECTION ANGLE (B), DEG. FIB.3
Variation of the normalized stress intensity factor range (dashed line) and the crack growth rate (solid line) for a macrosoopically deflsoted crack with ar~lle of deflection
-3 w
~
R llX~6.)
6O
E E
Z "~ \ ¢o
75
1~5 ;
,.,
1~6
-r
1~7
o ~
1~|
<
*
9! 1
I
2
STRESS INTENSITY FACTORRANGE(AK) MPaV-~ Fig.4
Predicted fatigue crack uacroe~optcally deflected angle of deflection
Er~wth cracks as
behaviour a function
for of
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DEFLECTED FATIGUE CRACKS
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-4
10 .-I
l(~S -
Z
m
"C3
m'P
10 ----A---
LI.J 0
I--
. . . .
EXPT. OA EXPT. LOA THEORETICAL LOA
¢Y
~-~ 1~v
¢-y
~'J
-9
10
I
3
I
J
I
J
5
6
7
8
STRESS INTENSITY FACTORRANGE(AK), MPa~l-~Fig.5
Experimental fatigue crack growth behaviour in overfed (OA) a n d l i g h t l y o v e r a g o d (LOA) c o n d i t i o n B (from Rof.10) and theo~tlcally predicted fatigue c r a c k R r o w t h b e h a v i o u r i n LOA c o n d i t i o n u e i n a f a t i g u e c r a c k g r o w t h d a t a i n OA c o n d i t i o n and the meamared deflection a n g l e i n LOA c o n d i t i o n .
ii