Lateral constraints in plane strain compression of single crystals
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s t u d i e s have shown that m a n y f e a t u r e s of c o n s t r a i n e d d e f o r m a t i o n of c r y s t a l s can be p r e d i c t e d by a c r y s t a l p l a s t i c i t y a n a l y s i s , z In p a r t i c u l a r , good a g r e e m e n t with theory has b e e n obtained for a v a r i e t y of fcc and bcc m e t a l s with r e g a r d to s t r e s s s t r a i n c u r v e s , active slip and twin s y s t e m s , and lattice r o t a t i o n J -6 Even c o m p l i c a t e d a s p e c t s such as inhomogeneous d e f o r m a t i o n 7 and d e f o r m a t i o n bands a have been explained. A s a t i s f a c t o r y theory of d e f o r m a t i o n text u r e s has also been evolved. 9 To date, however, the s t r e s s e s n e c e s s a r y to cons t r a i n the d e f o r m a t i o n have not been m e a s u r e d although they can be c a l c u l a t e d t h e o r e t i c a l l y . In this study we decided to examine the l a t e r a l c o n s t r a i n i n g s t r e s s a r i s i n g in plane s t r a i n c o m p r e s s i o n , a s t r a i n state which a p p r o x i m a t e s d e f o r m a t i o n p r o c e s s i n g o p e r a t i o n s such as r o l l i n g . The p u r p o s e of the study is two-fold. F i r s t , i n f o r m a tion c o n c e r n i n g the c o n s t r a i n t s t r e s s b r o a d e n s the field of t e s t i n g the t h e o r e t i c a l a n a l y s i s . Secondly, s e v e r a l a u t h o r s have a s s u m e d a fixed c o n s t r a i n t s t r e s s ( r e l a t i v e to c o m p r e s s i o n s t r e s s ) in c h a r a c t e r i z i n g the r o l l i n g of single c r y s t a l s of v a r i o u s o r i e n t a t i o n s , z~ z (~xx - ~ which is The a s s u m p t i o n is that CXyy = ~g e n e r a l l y applicable only for plane s t r a i n d e f o r m a t i o n of an isotropic p l a s t i c solid. The s y m b o l s x, y , and z denote the sheet n o r m a l , t r a n s v e r s e , and r o i l i n g d i r e c tions, r e s p e c t i v e l y . T h i s fixed s t r e s s s t a t e a s s u m p t i o n was u s e d to c a l c u l a t e the a c t i v e slip s y s t e m s during r o l l i n g , and is c l e a r l y inadequate in view of the a n i s o tropic yield b e h a v i o r of s i n g l e c r y s t a l s .
shaded planes ( i , 2, 4, and 5) a r e activated equallly. The r e s u l t i s thinning in [110] a c c o m p a n i e d by extension in [001] with no s t r a i n in [110]. Thus plane s t r a i n in the (110)[001] o r i e n t a t i o n is s a t i s f i e d without the need for l a t e r a l c o n s t r a i n t . If the o r i e n t a t i o n is changed to (110)[i10], four new s y s t e m s (7, 8, 10, i i ) from the unshaded planes must be a c t i v a t e d to c a n c e l the expansion along [001]. This requires a lateral constraint stress. An i n t e r e s t i n g s e r i e s of e x p e r i m e n t s can be b a s e d on this s i m p l e model by fixing (110) as the c o m p r e s s i o n p l a n e and enforcing plane s t r a i n by varying the extension d i r e c t i o n in this plane. A n oft
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