Fatigue crack propagation in aluminum-base copper alloys
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The c r a c k p r o p a g a t i o n r a t e a c c e l e r a t e s m o r e r a p idly than p r e d i c t e d b y the P a r i s r e l a t i o n with i n c r e a s e in AK a s Kma x a p p r o a c h e s the l i m i t i n g v a l u e of the s t a t i c c r i t i c a l s t r e s s i n t e n s i t y f a c t o r , K c. The M c E v i l y - F o r m a n s equation e m p i r i c a l l y t a k e s account of t h i s i n c r e a s e a s well a s the concept of a t h r e s h o l d v a l u e of M(, L~r(th. F o r i n e r t a t m o s p h e r e s M c E v i l y 3 proposed d a / d N = ( 8 / n E z) (Ms z - hK~h) (1 + A K / ( K c - Kmax)).
[1] Since K c is a function of a l l o y c o m p o s i t i o n , m e t a l l u r g i c a l s t r u c t u r e , and m o d e of f r a c t u r e 4 a d e p e n d e n c e Id a / d N [ A K on m e t a l l u r g i c a l v a r i a b l e s i s i m p l i e d in Eq. [1]. The r e l a t i o n of M~Tth to m e t a l l u r g i c a l s t r u c t u r e i s not known, n o r is it known that ~ K t h i s a r e a l m a t e r i a l p r o p e r t y . S The d e p e n d e n c e of d a / d N on M42, that is 5aZa, was d e r i v e d b y M c C l i n t o e k 6 a s s u m i n g that d a / d N is p r o p o r t i o n a l to the c r a c k tip d i s p l a c e ment. In a p r e v i o u s s t u d y of 4140 s t e e l in t h i s l a b o r a t o r y , ~ Eq. [1] fit the d a t a for s a m p l e s quenched and t e m p e r e d at 25, 200, 400, 550, and 650~ within a f a c t o r of t h r e e , t a k i n g ~ t h = 6.8 M N / m 3/~ for R = K m i n / K m a x = 0.05, the v a l u e r e p o r t e d for a n u m b e r of d i f f e r e n t s t e e l s , s'9 The p r e d i c t e d r a t e s w e r e m o s t l y too s m a l l , h o w e v e r , and the t r e a t m e n t with the h i g h e s t K c did not give the l o w e s t v a l u e s of I ~a/~NIzxK. S e v e r a l t h e o r e t i c a l t r e a t m e n t s have r e l a t e d d a / d N v s a p p l i e d s t r e s s to o t h e r m e c h a n i c a l p r o p e r t i e s . M c C l i n t o c k 6 t r e a t e d the c a s e f o r p r o p a g a t i o n by c o n tinued r e n u c l e a t i o n of the c r a c k ahead of the m a i n c r a c k front u s i n g the C o f f i n - M a n s o n f a i l u r e c r i t e r i o n and d e r i v e d da dN
AK a E2(y~(r )2~t
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VOLUME 7A, APRIL 1976-583
w h e r e a u is the u l t i m a t e t e n s i l e s t r e n g t h , e~ is the t r u e s t r a i n at f r a c t u r e in monotonic t e n s i o n , and X is an a p p r o p r i a t e s t r u c t u r a l p a r a m e t e r such as the i n c l u s i o n spacing. W e e r t m a n l~ d e r i v e d a fatigue c r a c k p r o p a g a t i o n r a t e equation b a s e d on continuous d i s l o c a tion t h e o r y by adopting the c r i t e r i o n that a s m a l l e l e m e n t of m a t e r i a l within the p l a s t i c zone f r a c t u r e s when its c u m u l a t i v e d i s p l a c e m e n t r e a c h e s a c r i t i c a l value, D*. S u m m i n g the a b s o l u t e v a l u e s of the d i s p l a c e m e n t s for each e l e m e n t in the p l a
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