Formability of sandwich sheet materials in plane strain compression and rolling
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I. INTRODUCTION W HE l a r g e d e f o r m a t i o n s i m p o s e d d u r i n g the p r i m a r y w o r k i n g of m e t a l s f r e q u e n t l y o c c u r in a s t a t e of p l a n e s t r a i n c o m p r e s s i o n . A m o n g the p r i m a r y m e t a l w o r k ing o p e r a t i o n s is that of p l a n e - s t r a i n cold r o l l i n g , which is used to p r o d u c e thin sheet m e t a l to close t h i c k n e s s and s u r f a c e f i n i s h t o l e r a n c e s . In view of the i m p o r t a n c e of this p r o c e s s , its t h e o r e t i c a l as well as p r a c t i c a l a s p e c t s , e s p e c i a l l y as they a r e conc e r n e d with the p r o d u c t i o n of m o n o l i t h i c sheet m a t e r i a l s , have b e e n thoroughly i n v e s t i g a t e d . A complex s t a t e of s t r e s s e x i s t s in the r o l l gap b e cause of g e o m e t r i c and f r i c t i o n a l e f f e c t s . W o r k i n g s t r e s s e s can be e s t i m a t e d , however, by a n a l y z i n g the plane s t r a i n d e f o r m a t i o n of a p l a s t i c m a s s comp r e s s e d b e t w e e n flat p a r a l l e l p l a t e s . T h e p r e s s u r e d i s t r i b u t i o n along the c o m p r e s s i o n p l a t e s is c o m posed e s s e n t i a l l y of two p a r t s . ~'2 The f i r s t c o n s i s t s of the p r e s s u r e to c o m p r e s s the p l a s t i c m a s s without f r i c t i o n . When the length of the c o m p r e s s i o n p l a t e s , L, is much g r e a t e r than the sheet t h i c k n e s s , h, this p r e s s u r e is v e r y close to the c o n s t r a i n e d flow s t r e n g t h of the m a t e r i a l , s,4 With f r i c t i o n , however, l a t e r a l c o m p r e s s i v e s t r e s s e s a r e induced in the p l a s t i c m a s s , and an i n c r e m e n t in p r e s s u r e m u s t be applied to b r i n g about y i e l d i n g . An i n c r e a s e in p r e s s u r e , whose d i s t r i b u t i o n along the length of the p l a t e s r e s e m b l e s that of a hill, r e s u l t s for the types of f r i c t i o n u s u a l l y e n c o u n t e r e d . ThuS, the d e s c r i p t i v e t e r m " f r i c t i o n h i l l " has come into use. ~ S. L. SEMIATIN,formerly Graduate Student, Department of Metallurgy and Materials Science, Carnegie-MellonUniversity, is Research Scientist, MetalworkingSection, Battelle Columbus Laboratories, Columbus, OH 43201. H. R. PIEHLER is Professor, Department of Metallurgy and Materials Science, Carnegie-MellonUniversity, Pittsburgh, PA 15213. Manuscript submitted June 28, 1978. METALLURGICALTRANSACTIONSA
The d e t a i l e d effect of f r i c t i o n on the p r e s s u r e d i s t r i b u t i o n and s t r e s s field in the c o m p r e s s e d m a s s may be a n a l y z e d by a v a r i e t y of t e c h n i q u e s . The two most often used a r e the s l a b technique and s l i p - l i n e field a n a l y s i s . In applying the s l a b technique, one n e g l e c t s the v a r i a t i o n of s t r e s s t h r o u g h the t h i c k n e s s , and the s o l u t i
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