The deformation and failure of a biaxially stretched sheet
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o c c u r r e d m o s t l y by s t r a i g h t t e a r i n g a f t e r c o n s i d e r a b l e local thinning. The q u e s t i o n is how this m i g h t develop on a p r o p o r t i o n a l s t r a i n i n g path of, i n i t i a l l y at least, p > 0. T h e r e a r e at l e a s t two p o s s i b i l i t i e s : f r a c t u r e , o r a g r a d u a l r e d i r e c t i o n f r o m along the o r i g i n a l l o a d ing path to one for p l a n e s t r a i n on which a local neck is allowed. F r a c t u r e was e m p h a s i z e d in the punch s t r e t c h i n g work, 3 the a l t e r n a t i v e has b e e n the focus of a r e c e n t a n a l y s i s by M a r c i n i a k and Kuczyfiski s ( r e f e r r e d to h e r e a f t e r as M-K). If the loading path is s t r a i g h t it can only be deflected in the e v e n t u a l f a i l u r e - r e g i o n by a local w e a k n e s s . M - K i m a g i n e an i n c i p i e n t trough (or s o m e s t r u c t u r a l e q u i v a l e n t a c t i n g to lower load c a r r y i n g a b i l i t y ) a l r e a d y n o r m a l to the d i r e c t i o n of e 1, as shown s c h e m a t i c a l l y in Fig. 1. The i n t e r m e d i a t e s t r a i n , ez, along the trough axis is r e q u i r e d to be the s a m e both in and out of the trough. T h e r e f o r e , t h i n n i n g (ca) and e x t e n s i o n (el) a r e n e c e s s a r i l y m o r e r a p i d in the t r o u g h than outside. With 5ezB = 5~zA , while 5E~B grows s t e a d i l y l a r g e r t h a n 5exA as en4 i n c r e a s e s , the i n - t r o u g h s t r a i n i n c r e m e n t r a t i o , 5EzB/SexB, a p p r o a c h e s z e r o . Thus p l a n e s t r a i n is e v e n t u a l l y produced in the trough, and the exA outside (the i m p o s e d - s t r a i n c o m p o n e n t ) at which the approach b e c o m e s r a p i d is the l i m i t i n g strain, el. The M - K idea is new, i m p o r t a n t , and b a s i c to an i n t e r p r e t a t i o n of the findings in this work. Unfortunately, however, in its only p r e s e n t a t i o n s to date, 5"7 a s s u m p tions a r e not e v a l u a t e d , steps a r e m i s s i n g , t h e r e is no d i s c u s s i o n of the c r i t i c a l p r o b l e m of i n i t i a l b o u n d a r y conditions, and t h e r e a r e t y p o g r a p h i c a l e r r o r s . The c e n t r a l idea is not d i m i n i s h e d , but it is e x t r e m e l y difficult to use. Much m u s t be done s i m p l y to get s t a r t e d --so much in fact that a n o t h e r p a p e r s e e m s n e c e s s a r y for showing how to handle these c o m p l i c a t i o n s f l The p r i n c i p a l c o n c e r n of M - K is with the l o a d i n g path m o v e m e n t , at the f a i l u r e site, t o w a r d s one r e p VOLUME 1, OCTOBER 1970-2857
tB f=
i
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' 8EtAj F i g . 1 - - A s c h e m a t i c v i e w of h o w t h e s t r a i n s t a t e i n a n i n c i p i e n t trough (B region) can approach plane strain while propor-
tional straining is maintained outside (A region).
r e s e n t i n g plane s t r a i n . They d e s c
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