A Criterion for ductile fracture in sheets under biaxial loading
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v a r i o u s c r i t e r i a which have been p r o p o s e d to e x p l a i n the f r a c t u r e of m e t a l s g e n e r a l l y a l l o w the d e r i v a t i o n of f r a c t u r e s t r a i n s u n d e r known c o n d i t i o n s of loading f r o m known c o n s t i t u t i v e b e h a v i o r of m a t e r i a l s . C r i t e r i a l i k e , " M a x i m u m T e n s i l e S t r e s s " ,~ " M a x i m u m S h e a r S t r e s s " ,2,3 , , M a x i m u m V o l u m e S t r a i n " ,4 , , M a x i m u m T e n s i l e W o r k " , s e t c , have b e e n a d v a n c e d to e x p l a i n the f r a c t u r e b e h a v i o r of v a r i o u s m a t e r i a l s . M o s t of t h e s e p r o v i d e a s i m p l i f i e d end condition f o r f r a c t u r e without t a k i n g into c o n s i d e r a t i o n the g r a d u a l i n c r e a s e of i n t e r n a l d a m a g e d u r i n g d e f o r m a t i o n of d u c t i l e m a t e r i a l s . C o c k c r o f t , 5 in his " M a x imum Tensile Work" criterion, accounted for material d a m a g e by i n t e g r a t i n g the p l a s t i c w o r k done by the m a x i m u m t e n s i l e s t r e s s . T h i s , h o w e v e r , l a c k e d any r e a l m i c r o s t r u c t u r a l input c o n c e r n i n g the p r o c e s s of d a m a g e a s e x p e r i m e n t a l l y o b s e r v e d by many i n v e s t i gators.6-9 The m o s t r e a l i s t i c f r a c t u r e c r i t e r i o n f o r d u c t i l e m e t a l s was p r o p o s e d by M c C l i n t o c k , ~~ who m o d e l e d the p r o c e s s of i n i t i a t i o n , growth and c o a l e s c e n c e of v o i d s . Although v o i d s can i n i t i a t e f r o m i n t e r s e c t i n g s h e a r bands in a p u r e m e t a l , in m o s t d u c t i l e m e t a l s void i n i t i a t i o n t a k e s p l a c e e i t h e r by s e p a r a t i o n of i n c l u d e d p a r t i c l e s f r o m the m a t r i x o r by c r a c k i n g of i n c l u s i o n s . T h i s happens r a t h e r e a r l y d u r i n g d e f o r m a t i o n and cons i d e r a b l e a d d i t i o n a l p l a s t i c flow t a k e s p l a c e b e f o r e f r a c t u r e . M c C l i n t o c k 1~ f u r t h e r pointed out that, s i n c e the i n c l u s i o n s in d u c t i l e m e t a l s a r e r a t h e r l a r g e c o m p a r e d to the d i s l o c a t i o n s p a c i n g s , it might not be n e c e s s a r y to c o n s i d e r i n d i v i d u a l d i s l o c a t i o n s (to d e a l with void growth and c o a l e s c e n c e ) but a continuum a p p r o a c h should be sufficient. D u c t i l e f r a c t u r e in s h e e t m e t a l d e f o r m a t i o n i s found to t a k e p l a c e a l o n g a m a c r o s h e a r p l a n e at 45 d e g to the p l a n e of the s h e e t . ~2-13 In t h e s e r e f e r e n c e s , a s w e l l a s o t h e r s 8'~4 on d u c t i l e f r a c t u r e , the void j o i n ing p r o c e s s i s not by d i r e c t i m p i n g e m e n t [Fig. l(a)] which i s u s u a l l y a c c o m p a n i e d by t r i a x i a l s t r e s s e s . R a t h e r the p r o c
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