Inhomogeneity of Plastic Flow in Constrained Deformation B. C. WONSIEWICZ AND G. Y. CHIN C r y s t a l s of c o p p e r , Cu + 6 wt pct A1, and Ag + 4 wt pct Sn w e r e c o m p r e s s e d along [111] with flow r e s t r i c t e d to [1i2]. A f t e r d e f o r m a t i o n , four d i f f e r e n t l y o r i e n t e d r e g i o n s w e r e o b s e r v e d . T h e i r o r i g i n is e x p l a i n e d by the i n s t a b i l i t y of the (111)[112] o r i e n t a t i o n which can r o t a t e to e i t h e r (112)[ i i 1] o r (110)[001] during the i m p o s e d shape change. The d i r e c t i o n of r o t a t i o n i s d e t e r m i n e d by which of two i n i t i a l l y e q u a l l y f a v o r e d p a i r s of s l i p s y s t e m s o p e r a t e . Surface f r i c t i o n p r o d u c e s s h e a r s t r e s s e s which f a v o r one p a i r o v e r the o t h e r (depending on the sign of the s h e a r s t r e s s ) and thus one of the final o r i e n t a t i o n s . Since the sign of the f r i c t i o n a l s t r e s s v a r i e s s y s t e m a t i c a l l y with p o s i t i o n in the d e f o r m i n g c r y s t a l , a s y s t e m a t i c v a r i a t i o n of o r i e n t a tion r e s u l t s . A n o t h e r o r i e n t a t i o n (001)[110] h a s a l s o been o b s e r v e d to b e h a v e s i m i l a r l y . D u r i n g r o l l i n g , the f r i c t i o n a l f o r c e s d r a w i n g the c r y s t a l into the r o l l gap a r e a l s o e x p e c t e d to l e a d to the d i v i s i o n of the c r y s t a l into two m i s o r i e n t e d r e g i o n s . The p r e d i c t i o n s a r e g e n e r a l i z e d to include b c c m e t a l s of (1i2)[111] and (110)[001] o r i e n t a t i o n s . P r e v i o u s l y r e p o r t e d o b s e r v a t i o n s of r o l l e d c r y s t a l s of FeNia, F e - 3 . 5 pct Si, and F e - 2 pct A1 a r e in a c c o r d with the p r e s e n t analysis. A s p a r t of a l a r g e r p r o g r a m to t e s t the a p p l i c a b i l i t y of t h e o r i e s of p o l y c r y s t a l l i n e d e f o r m a t i o n , a n u m b e r of fcc c r y s t a l s were s u b j e c t e d to plane s t r a i n c o m p r e s s i o n - - a c o m p r e s s i o n t e s t in which the e x p a n s i o n of the c r y s t a l i s l i m i t e d to one d i r e c t i o n , in o u r c a s e by the w a l l s of a channel. C r y s t a l s with (111) a s the c o m p r e s sion plane and [1i2] a s the e x t e n s i o n d i r e c t i o n b e h a v e d in an unusual m a n n e r . A f t e r d e f o r m a t i o n , two d i s t i n c t r e g i o n s could be seen on the c o m p r e s s i o n p l a n e . A p h o t o g r a p h of a Cu-6 wt pct A1 c r y s t a l i s shown in F i g . 1. U n d e r a m i c r o s c o p e , d i f f e r e n t s l i p m a r k i n g s w e r e o b s e r v e d in each r e g i o n . On the r i g h t hand side of F i g . 1, Region A, the c h e v r o n t r a c e s of the i n t e r s e c t i n g p l a n e s w e r e e v i d e n t , F i g . 2(a), while the left s i d e , Region B, contained p a r a l l e l s l i p t r a c e s , F i g . 2 (b). M o s t s u r p r i s i n g l y , X - r a y pole f i g u r e s , F i g . 3, r e v e a l e d that Region A had r o t a t e d n e a r l y to (112)[111], while Region B a p p r o a c h e d (110)[001] in the opposite d i r e c t i o n . The r e s u l t i n g m i s o r i e n t a t i o n b e t w e e n the two r e g i o n s was thus about 55 deg. S i m i l a r o b s e r v a t i o n s w e r e m a d e on (111)[112] c r y s t a l s of c o p p e r and Ag4 wt pct Sn. An e x p l a n a t i o n i s o f f e r e d through an a n a l y s i s of the s l i p s y s t e m s b a s e d on the t h e o r i e s of T a y l o r 1 and of Bishop and Hill. 2 F o r a f r i c t i o n l e s s d e f o r m a t i o n , two d i s t i n c t c o m b i n a t i o n s of s l i p s y s t e m s a r e e q u a l l y f a v o r e d at the i n i t i a l o r i e n t a t i o n . The p r e s e n c e of f r i c t i o n in the e x p e r i m e n t , h o w e v e r , s e t s up s h e a r s t r e s s e s that b i a s one of the c o m b i n a t i o n s depending on the sign of the s h e a r . T h i s sign c h a n g e s f r o m one p o r t i o n of the c r y s t a l to a n o t h e r and, a s a r e s u l t , the c r y s t a l s e p a r a t e s into m a c r o s c o p i c a l l y d i s t i n c t r e g i o n s . l e a s t amount of s l i p d T , a s s u m i n g that the c r i t i c a l s h e a r s t r e s s T is the s a m e f o r a l l s y s t e m s . F o r plane s t r a i n c o m p r e s s i o n of a (111)[1i2] c r y s t a l , s l i p i s e x p e c t e d on the ( i i l ) plane in both the [011] and [101] d i r e c t i o n s , r e f e r r e d to a s the c o p l a n a r p a i r , c o m b i n e d with s l i p in the [110] d i r e c t i o n on both the ( l i i ) and (111) p l a n e s , the c o l i n e a r p a i r , F i g . 4. T h e s e r e s u l t s w e r e o b t a i n e d p r e v i o u s l y 3'4 b y the m e t h o d of Bishop and Hill, which has r e c e n t l y b e e n shown s to be c o m p l e t e l y e q u i v a l e n t to the T a y l o r t h e o r y . T h i s o r i e n t a t i o n has b e e n a n a l y z e d in m o r e d e t a i l by D i l l a m o r e , B u t l e r and G r e e n 6 and by us 7 f o r p l a n e s t r a i n d e f o r m a t i o n with no s h e a r s t r a i n s p e r m i t t e d , i.e., exx =-ezz, Cyy = e y z = ~ z x = e x y = 0, such a s m a y be e n c o u n t e r e d in a p o l y c r y s t a l l i n e a g g r e g a t e . The (x, y , z) d i r e c t i o n s a r e taken a s the c o m p r e s s i o n plane n o r m a l [ 111], the c o n s t r a i n t d i r e c t i o n [ i10] and the e x t e n s i o n d i r e c t i o n [112], r e s p e c t i v e l y . In t h i s ANALYSIS F r o m the T a y l o r a n a l y s i s , s l i p i s e x p e c t e d on s y s t e m s which will p r o d u c e the d e s i r e d shape change with the l e a s t a m o u n t of w o r k ~-d7 and, t h e r e f o r e , the B. C. WONSIEWICZand G. Y. CHIN are Members of Technical Staff, Bell Telephone Laboratories, Murray Hill, N. J. Manuscript submitted June 30, 1969. METALLURGICAL TRANSACTIONS Fig. 1--View of the compression plane of a Cu-6 wt pct A1 crystal initially compressed on (111) with expansion limited to [112]. Regions A and B are the result of inhomogeneous deformation. Cx x = -0.48. VOLUME 1, JANUARY 1970-57 c a s e , if t h e r e w e r e a c o n s t r a i n t ezx = 0, it c o u l d be s a t i s f i e d only if b o t h the c o l i n e a r and c o p l a n a r p a i r s o p e r a t e t o g e t h e r by d e f i n i t e a m o u n t s of s h e a r . A s the c r y s t a l d e f o r m e d h o m o g e n e o u s l y , it w o u l d r o t a t e toward and past (112)[1il], achieving a stable orientat i o n about 8 d e g f r o m (112). H o w e v e r , in the p r e s e n t e x p e r i m e n t a l s e t u p , t h e r e i s no c o n s t r a i n t o n ezx o r eyz. The c r y s t a l i s t h e n p e r m i t t e d to d e f o r m by e i t h e r p a i r of s l i p s y s t e m s a l o n e . As the f o l l o w i n g a n a l y s i s s h o w s , t h e s e two p a i r s a r e m u t u a l l y e x c l u s i v e in that the o p e r a t i o n of one r a i s e s i t s r e s o l v e d s h e a r s t r e s s r e l a t i v e to the o t h e r . With the (x, y, z) d i r e c t i o n s d e f i n e d a s a b o v e , the r e s o l v e d s h e a r s t r e s s on e a c h of the f o u r s y s t e m s can be c a l c u l a t e d a s f o l l o w s : sion stress axx, plus possible constraining stresses Oyy and axy to i n s u r e eyy = 0 and exy = O. The l a t t e r c o n s t r a i n t r e q u i r e s e q u a l o p e r a t i o n of both m e m b e r s of e a c h p a i r of s y s t e m s . F r o m E q s . [1] the r e s o l v e d s h e a r s t r e s s on b o t h m e m b e r s i s e q u a l only if axy = O. With azz and the s h e a r s t r e s s e s e q u a l to z e r o , the c r i t i c a l r e s o l v e d s h e a r s t r e s s "rc i s r e a c h e d s i m u l t a n e o u s l y on a l l f o u r s y s t e m s w h e n (Xxx =-1.54-6Zc, Z \ T(~il)[011] = - 0 . 2 7 2 a x x + 0.272azz + 0.470~y z Coplanar pair + 0.674azx - 0.166axy T (111)[101] = - 0 . 2 7 2 v x x + 0 . 2 7 2 a z z - 0 . 4 7 0 a y z + 0"674~zx + 0"166axy [1] T ( l i i ) [ l l 0 ] = - 0 . 2 7 2 ~ x x + 0 . 2 7 2 ~ z z + 0.470Oyz Cotinear pair I - 0 . 1 9 4 a z x - 0.666axy / T ( ~ l i ) [ l l 0 ] = _ 0 . 2 7 2 ~ x x + 0.272az z _ 0.470~y z - 0.194(Xzx + 0.666Oxy The i n f l u e n c e of e a c h s t r e s s c a n be s e e n f r o m the e q u a t i o n . F i r s t , if f r i c t i o n i s n e g l e c t e d , the only s t r e s s e s a c t i n g on the c r y s t a l a r e the a p p l i e d c o m p r e s - (a) (a) 5 Io 2o 40 0v,) 8o (b) (b) Fig. 2--Photomicrographs of Regions A and B of Fig. 1. Magnification 390 times. (a) Region A. The chevron t r a c e s c o r respond to slip on intersecting planes of the colinear pair, see Fig. 4. (b) Region B. P a r a l l e l slip traces correspond to slip on the coplanar pair. 58-VOLUME 1, JANUARY 1970 Fig. 3--{111} X - r a y pole figures of the Regions A an d_B shown in Fig. 1. Circles indicate the original (l11)[112]_orientation. (a) Crystal has rotated about 20 deg to (112)[111] orientation, filled triangles. Note the presence of the twin orientation, open triangles. (b) Crystal has rotated about 35 deg to (110)[001]. No twinning is evident. METALLURGICAL TRANSACTIONS tt2 i n d e p e n d e n t of Cryy w i t h i n the r a n g e -x/6-~-c < Oyy < 0.* *It can be shown that at Oyy = 0, yieldingoccurson the (11 l) [ 101] and (11 l) [011] systemsin addition to the four in Eqs. [1]. At ayy = -V"6 "rc,yielding occurs on the (l~l) [01~] and (ll]) [10]'] systemsin addition to the four. T h u s , the c o n s t r a i n i n g s t r e s s g y y c a n n o t be l a r g e r t h a n ~ of the a p p l i e d s t r e s s . U n d e r t h e s e s t r e s s e s , the c r y s t a l is l i k e l y to d e f o r m i n h o m o g e n e o u s l y . If only the c o l i n e a r p a i r o p e r a t e s l o c a l l y , the c o m p r e s s i o n a x i s w i l l b e r o t a t e d a w a y f r o m [111] t o w a r d [001], F i g . 4. T h e r e s o l v e d s h e a r s t r e s s w o u l d t h e n i n c r e a s e on the a c t i v e s y s t e m s w h i l e d e c r e a s i n g on the c o p l a n a r p a i r . H e n c e , o n l y the c o l i n e a r p a i r w o u l d c o n t i n u e to o p e r a t e and the r o t a t i o n w o u l d p e r s i s t until [112] i s r e a c h e d . At t h a t o r i e n t a t i o n , two a d d i t i o n a l s l i p s y s t e m s w o u l d o p e r a t e , n a m e l y ( i ] i ) [ 0 ] 1 ] + (111)[101]. The o p e r a t i o n of the f o u r s y s t e m s would r e s u l t in a s t a b l e o r i e n t a t i o n at (112)[1]1], w h i c h i s 19.5 d e g f r o m the s t a r t i n g o r i e n t a t i o n . N o t e the d i f f e r e n c e f r o m the f u l l y c o n s t r a i n e d c a s e . 6 L i k e w i s e , if only the c o p l a n a r p a i r s h o u l d o p e r a t e at the i n i t i a l o r i e n t a t i o n , the c o m p r e s s i o n a x i s w i l l be r o t a t e d in the o p p o s i t e d i r e c t i o n t o w a r d [110], F i g . 4. The s h e a r s t r e s s would t h e n i n c r e a s e on the a c t i v e c o p l a n a r p a i r w h i l e d e c r e a s i n g on the i n a c t i v e c o l i n e a r p a i r . H e n c e , the c o p l a n a r p a i r w i l l c o n t i n u e to o p e r a t e , l e a d i n g to a f i n a l o r i e n t a t i o n (110)[001], 35 d e g f r o m the s t a r t i n g o r i e n t a t i o n . T h e f i n a l o r i e n t a t i o n is s t a b l e a s a r e s u l t of s l i p on the c o p l a n a r p a i r and two a d d i t i o n a l s y s t e m s , (]11)[01i] + ( ] i 1 ) [ 1 0 i ] . * Fig. ff--Stereographic projection for the (111}[112] orientation. Slip is favored on the (111) plane (its negative (111) is illustrated) in both the [101] and [011] directions (coplanar pair) and on both the ( l i i ) and ( i l i ) planes in the [110] direction (colinear pair). *The (110) [001] orientationis stable with respect to rotation about the [310] transverse axis. It has been observedthat at large deformation, it becomesunstable with respect to rotation around [ 110], leadingtoward a final (11 O) [112] + (11O) [1T2] orientation. In the a b s e n c e of f r i c t i o n , t h e n , the (111)[112] o r i e n t a t i o n i s i n h e r e n t l y u n s t a b l e , with the p o s s i b i l i t y of s e p a r a t i o n into l o c a l r e g i o n s with f i n a l o r i e n t a t i o n s ( 1 1 2 ) [ 1 i l ] a n d (110)[001]. T h u s , the g r o s s f e a t u r e s of the o b s e r v a t i o n s of F i g s . 1 t h r o u g h 3 a r e e x p l a i n e d . On the o t h e r hand, the s e p a r a t i o n of the c r y s t a l into the two m a c r o s c o p i c r e g i o n s can be t r a c e d to the p e r t u r b ing i n f l u e n c e of f r i c t i o n . A l t h o u g h f r i c t i o n w a s m i n i m i z e d by c o a t i n g the s a m p l e and d i e s with a t e f l o n s p r a y , it was s u f f i c i e n t to a f f e c t the o r i e n t a t i o n i n s t a b i l i t y . Since Crxx i s the l a r g e r s t r e s s , the f r i c t i o n f o r c e s a r e e x p e c t e d to be g r e a t e s t on the c o m p r e s s i o n p l a n e , g e n e r a t i n g a (rzx t e r m . C o n s u l t i n g F i g . 5, the r i g h t - h a n d p o r t i o n of the c r y s tal i s d i s p l a c e d to the r i g h t d u r i n g c o m p r e s s i o n . T h e f r i c t i o n a l f o r c e s which r e s i s t the d i s p l a c e m e n t a r e d i r e c t e d to the l e f t a l o n g the - z d i r e c t i o n . T h e r e s u l t i n g s h e a r s t r e s s ~z~ i s n e g a t i v e on the u p p e r f a c e , p o s i t i o n 1, a n d p o s i t i v e on the l o w e r f a c e , p o s i tion 4. S i n c e the l e f t p o r t i o n i s d i s p l a c e d in the oppos i t e d i r e c t i o n , s h e a r s t r e s s e s d e v e l o p e d on the two f a c e s a r e the r e v e r s e of the r i g h t p o r t i o n . R e g i o n A [lII] [] x J[--]~. ~ < ~I[-]~ [] -7 ~__ [ ] iD - Qy >z [TT2] qE3~ Fig. 5--Schematic of the constrained side, y face. The f r i c tional forces on the x face, represented by the half arrows, produce a s t r e s s crz x that changes sign with position. At (1) and (3) azx is negative while at (2) and (4) Crzx is positive. Regions A and B of Fig. 1 through 3 correspond to the portions of compression (x) face, located directly above positions (1) and (2), respectively. Fig. 6--Composite photomicrograph of the y face of (111)[112] Ag-4 wt pct Sn crystal. Note the division into four regions s i m i lar to Fig. 5. Similar observations were made on copper and Cu-6 wt pct A1 crystals, exx = -0.55. Magnification 6 times. METALLURGICAL TRANSACTIONS VOLUME 1, JANUARY 1970-59 of the c r y s t a l , Fig. 1, c o r r e s p o n d s to the c o m p r e s s i o n plane d i r e c t l y above position 1 of Fig. 5 where n e g a tive azx p r e v a i l s , and Region B c o r r e s p o n d s to the c o m p r e s s i o n plane d i r e c t l y above position 2 where positive c~zx p r e v a i l s . A c c o r d i n g to Eqs. [1], the negative azx of Region A would lower the s h e a r s t r e s s on the c o p l a n a r p a i r while i n c r e a s i n g that on the c o l i n e a r p a i r . T h u s , slip on the c o l i n e a r s y s t e m s would be favored and it i s the t r a c e of these p l a n e s which form the c h e v r o n m a r k i n g typical of Region A. A s i m i l a r a r g u m e n t a p p l i e s for Region B where cop l a n a r slip is favored and o b s e r v e d in the p a r a l l e l t r a c e s of Fig. 2(b). The a r g u m e n t does not depend on the magnitude of ~zx but only on its d i f f e r e n t i a l effect in Eqs. [1]. As seen f r o m Fig. 5, the c r y s t a l should divide into four s y m m e t r i c a l blocks with r e gions 1 and 3 at (112) and 2 and 4 at (110). A c o m posite photograph of the y face, Fig. 6~ i l l u s t r a t e s the four r e g i o n s . The f r i c t i o n on the side face l e a d s to some m i n o r c o m p l i c a t i o n s . As pointed out p r e v i o u s l y , the lower n o r m a l s t r e s s ayy is expected to r e s u l t in lower f r i c t i o n a l s t r e s s e s Oy z and Crxy on this plane. Since the c r y s t a l s were thin s l a b s , i n i t i a l d i m e n s i o n s about 3 by 6.5 by 8 m m , the d i s p l a c e m e n t s o c c u r m a i n l y in the z d i r e c t i o n and so I(ryzl >>lC~xyL Cryz tends to inc r e a s e the s h e a r s t r e s s on one s y s t e m of each p a i r , see Eqs. [1]. F o r e x a m p l e , if the plane of Fig. 5 is taken a s (110), ay z is negative at location 1, thus f a v o r i n g slip on (i11)[110] over (1ii)[110]. Location 3 with positive Oyz, would f a v o r the other c o l i n e a r 40 - (11o) [ool] 30 20 5: 0 o ill:: LI- @ - Cu + 6 w / o ~-Aq+4 10 A,( w/oSn (D LIJ IJ.I rr (,.9 W Q I -.25 X I -.50 I -.75 The l a t t i c e rotation due to slip can be c a l c u l a t e d dir e c t l y f r o m a knowledge of the a m o u n t of s h e a r on each s y s t e m . As stated p r e v i o u s l y , slip on the c o l i n e a r p a i r , favored in Region A, r e s u l t s in a r o t a t i o n toward (112)[1il] while the c o p l a n a r p a i r f a v o r e d in Region B r o t a t e s the c r y s t a l to (110)[001]. The c a l c u l a t e d r o t a t i o n s for the two r e g i o n s a r e shown in Fig. 7, together with the e x p e r i m e n t a l v a l u e s obtained f r o m pole f i g u r e s . The a g r e e m e n t i s quite good. The rotation is a c c o m p a n i e d by a l a r g e change in the yield s t r e s s . Region B which r o t a t e s toward (110)[001] b e c o m e s m o r e f a v o r a b l y o r i e n t e d f o r slip and its yield s t r e n g t h t h e o r e t i c a l l y d r o p s s t e a d i l y to x/6z c at (110). The yield s t r e s s for Region A is expected to r e m a i n r e l a t i v e l y c o n s t a n t at about 1.5xF6-7c. The o b s e r v e d s t r e n g t h of the composite c r y s t a l i s roughly the a v e r a g e of the s t r e n g t h s of the two o r i e n t a t i o n s . A m o r e detailed a n a l y s i s of the s t r e s s e s will be p r e s e n t e d in a s e p a r a t e p a p e r . ~ An i n t e r e s t i n g d e v e l o p m e n t o c c u r r e d in the low s t a c k i n g fault e n e r g y m a t e r i a l s , Cu + 6 wt pct A1 and Ag + 4 wt pct Sn), as the o r i e n t a t i o n of Region A app r o a c h e d (112). A twin o r i e n t a t i o n b a s e d on (111) twinning was detected in.pole figure o b s e r v a t i o n s , Fig. 3, and t w i n - l i k e b a n d s along the (111) t r a c e could be seen m e t a l l o g r a p h i c a l l y . No such twins were detected in Region B. The o b s e r v a t i o n s a r e c o n s i s t e n t with the modified T a y l o r a n a l y s i s of Chin, Hosford, and Mendorf, 9 who pointed out that t w i n n i n g i s not expected in the o r i e n t a t i o n range (111)[]]2] to (110)[001]. Howe v e r , a s the o r i e n t a t i o n r o t a t e s from (111)[[i2] toward ( 1 1 2 ) [ [ i l ] , twinning on the (111)[ii2] twin s y s t e m b e c o m e s favored in a c c o r d a n c e with o b s e r v a t i o n s of Region A. THE (001)[il0] ORIENTATION - [~;z] Oi;) Fig. 7--Rotation of the compression Rlane normal x vs compressive strain exx. Rotation about [110] axis toward [110] is taken as positive, see Fig. 4. Solid line indicates the rotation expected from theory. Symbols indicate the orientation estimated from the center intensity maxima of pole figure taken after deformation to the strain exx. 60-VOLUME l, JANUARY 1970 LATTICE ROTATION AND ITS CONSEQUENCE ON MECHANICAL STRENGTH AND TWINNING -(111) [liZ] -10 -20 s y s t e m , (111)[ 110]. O b s e r v a t i o n s n e a r the c o n t r a i n e d face of the c r y s t a l c o n f i r m e d the p r e d i c t i o n s . On the top s u r f a c e away from this face, e y z is r e d u c e d and the c h e v r o n m a r k i n g of both t r a c e s b e c a m e e q u a l l y v i s i b l e . A s i m i l a r a r g u m e n t can be made for Region B, l o c a t i o n s 2 and 4, where one c o p l a n a r s y s t e m i s favored. However, since both s y s t e m s s h a r e the same plane, the slip t r a c e s would be i d e n t i c a l in e i t h e r c a s e . T h u s , the role of f r i c t i o n is to provide a v e h i c l e for the m a s s i v e s e p a r a t i o n of two a l r e a d y " u n s t a b l e " r e gions. The f r i c t i o n - r e l a t e d p h e n o m e n o n i s g e n e r a l a n d other o r i e n t a t i o n s a r e expected to be s i m i l a r l y affected. An e x a m p l e is the (001)[1i0] o r i e n t a t i o n , r e lated to (111)[ii2] by a 54.7-deg r o t a t i o n about the [i10] t r a n s v e r s e d i r e c t i o n . H e r e , i n i t i a l l y two cop l a n a r slip s y s t e m p a i r s a r e equally favored: (iii)[0il] + (iii)[i01] and (11i)[101] + (11i)[011]. The f o r m e r i s b i a s e d by a positive ezx and the l a t t e r by a negative ezx. T h u s , the c r y s t a l i s expected to b r e a k up into two r e g i o n s with four p o s i t i o n s s i m i l a r to those of Fig. 6. A n a l y s i s shows that the end o r i e n t a t i o n s as a r e s u l t of slip on the two p a i r s a r e (112)[i1~] and (]12)[111], r e s p e c t i v e l y . The p r e d i c t e d f e a t u r e s have b e e n o b s e r v e d by us. METALLURGICALTRANSACTIONS DISC USSION Sheet r o l l i n g offers an i n t e r e s t i n g e x t e n s i o n of the p r e s e n t a n a l y s i s . The l i t e r a t u r e c o n t a i n s s e v e r a l obs e r v a t i o n s of lattice r o t a t i o n s d u r i n g r o l l i n g of single c r y s t a l s that a r e r e m a r k a b l y s i m i l a r to the plane s t r a i n c o m p r e s s i o n r e s u l t s r e p o r t e d h e r e . T h i s is not unexpected since r o l l i n g and plane s t r a i n c o m p r e s s i o n a r e b a s i c a l l y s i m i l a r , although r o l l i n g is much m o r e complex. The d e t a i l s of the d e f o r m a t i o n depend upon s e v e r a l v a r i a b l e s such as the size of the r o l l s r e l a t i v e to c r y s t a l t h i c k n e s s , r e d u c t i o n p e r p a s s , and so forth. As a s i m p l i f i c a t i o n , the f r i c t i o n a l f o r c e s drawing the c r y s t a l into the r o l l gap produce s h e a r s t r e s s e s of opposite sign on the top and bottom s u r face of the c r y s t a l . If the i n i t i a l c r y s t a l o r i e n t a t i o n were unstable, such a s (111)[112], the f r i c t i o n can r e sult in the c r y s t a l s e p a r a t i n g about the m i d - t h i c k n e s s plane into r e g i o n s of differing o r i e n t a t i o n , (110)[001] and (112)[111]. Hatherly and Brown 1~have rolled a (111)[112] copper c r y s t a l and o b s e r v e d l a y e r s e p a r a tions with o r i e n t a t i o n s close to the p r e d i c t e d . T h e s e a u t h o r s a l s o suggest s u r f a c e f r i c t i o n as a cause of the separation. Likewise, for a c r y s t a l r o l l e d in the (001)[110] o r i e n t a t i o n , it is expected to b r e a k up into two l a y e r s ; one l a y e r d e f o r m i n g by ( i i i ) [ 0 i t ] + (iii)[i01] s l i p and the other by (111)[101] + (111)[011] s l i p . T h i s b e h a v i o r was o b s e r v e d by Chikazumi, Suzuki, and Iwata n in a r o l l e d FeNi3 c r y s t a l . The o r i e n t a t i o n s c o n s i d e r e d h e r e , with [110] a s the t r a n s v e r s e d i r e c t i o n , have t h e i r i n t e r e s t i n g c o u n t e r p a r t s in c r y s t a l s of bcc s t r u c t u r e . If {110}(111) slip is c o n s i d e r e d , the i n d i c e s of the slip plane and slip d i r e c t i o n a r e m e r e l y i n t e r c h a n g e d f r o m the fcc case. The same slip s y s t e m s a r e thus expected. H o w e v e r , the s e n s e of lattice rotation is r e v e r s e d . It may be a n a l y z e d that for the bcc c a s e , the (1i2)[ 111] o r i e n t a tion is unstable and is expected to rotate to e i t h e r (001)[110] or (1il)[112] depending on the sign of f r i c tion. These end o r i e n t a t i o n s were o b s e r v e d by Dunn and Koh ~2 on a F e - 3 . 5 pct Si c r y s t a l , by Hu and Cline 13 on a F e - 2 pct A1 c r y s t a l , and by Taoka, F u r u b a y a s h i , and Takeuchi ~4 on a Fe-3 pct Si c r y s t a l . The l a t t e r a u t h o r s a l s o advanced a f r i c t i o n a l a r g u m e n t s i m i l a r to ours, In e a r l i e r work, Dunn ~5 r e p o r t e d that an (011)[ 100] F e - 3 . 5 pct Si c r y s t a l s p l i t s up into two l a y e r s of diff e r e n t o r i e n t a t i o n s . By analogy with the case of (100)[011] fcc r o l l i n g , we would expect this o r i e n t a t i o n METALLURGICAL TRANSACTIONS to rotate toward (111)[211] and (i11)[211] in s e p a r a t e r e g i o n s . T h e s e a r e the r o t a t i o n s o b s e r v e d by Dunn. It should be pointed out that f r i c t i o n can help to s e p a r a t e m i s o r i e n t e d m a t e r i a l into m a c r o s c o p i c r e gions only if an o r i e n t a t i o n is i n h e r e n t l y u n s t a b l e . If the o r i e n t a t i o n i s s t a b l e , the effect of f r i c t i o n is m e r e l y to displace the o r i e n t a t i o n somewhat away f r o m its stable point, but not to b r e a k up the c r y s t a l into g r o s s l y misoriented regions. F i n a l l y , although the T a y l o r - B i s h o p - H i l l a n a l y s e s a r e b a s e d on homogeneous d e f o r m a t i o n , they r e m a i n e s s e n t i a l l y v a l i d for i n h o m o g e n e o u s d e f o r m a t i o n . As d e f o r m a t i o n p r o c e e d s and lattice rotation o c c u r s , however, the p o s s i b i l i t y of i n s t a b i l i t y m u s t be cons i d e r e d . If the c r y s t a l does s e p a r a t e into r e g i o n s of differing o r i e n t a t i o n , the a n a l y s i s m u s t obviously be b a s e d on the c u r r e n t r a t h e r than the i n i t i a l o r i e n t a t i o n , with each r e g i o n t r e a t e d as a s e p a r a t e , h o m o g e n e o u s l y d e f o r m i n g body. 8 ACKNOWLEDGMENTS It is a p l e a s u r e to acknowledge the a s s i s t a n c e of Mr. R. R. Hart in these e x p e r i m e n t s . Helpful d i s c u s s i o n s were held with T. D. D u d d e r a r . T. D. Schlabach and J. H. W e r n i c k kindly r e v i e w e d the m a n u s c r i p t . RE FERENCES 1. G. I. Taylor: J. Inst. Metals, 1938, vol. 62, pp. 307-24; Stephen Timoshenko, 69th Anniversary Volume, pp. 218-24, The Macmillan Co., New York, 1938. 2. J. F. W. Bishop and R. Hill: Phil. Mag., 1951, vol. 42, pp. 414-27, 1298-1307. 3. G. Y. Chin, E. A. Nesbitt, and A. J. Williams: Acta Met., 1966, vol. 14, pp. 467-76. 4. W. F. Hosford: Acta Met., 1966, vol. 14, pp. 1085-94. 5. G. Y. Chin and W. L. Mammel: Trans. TMS-AIME, 1969, vol. 245, pp. 121114. 6. I. L. Di/lamore, E. Butler, and D. Green: Metal Sci. J., 1968, vol. 2, pp. 16167. 7. B. C. Wonsiewicz and G. Y. Chin: Bell Telephone Laboratories, Murray Hi/l, N.J., 1969. 8. B. C. Wonsiewicz and G. Y. Chin: Bell Telephone Laboratories, Murray Hill, N.J., 1969. 9. G. Y. Chin, W. F. Hosford, and D. R. Mendorf: Proe. Roy. Soc., 1969, vol. A309, pp. 433-56. 10. M. Hatherly and K. Brown: University of New South Wales, Kensington, N.S.W., Australia, 1969. 11. S. Cnikazumi, K. Suzuki, and H. Iwata: J. Phys. Soc., Japan, 1957, vol. 12, pp. 1259-76. 12. P. K. Koh and C. G. Dunn: AIME Trans., 1955, vol. 203, pp. 401-6. 13. H. Hu and R. S. Cline: Trans. TMS-AIME, 1962, vol. 224, pp. 784-97. 14. T. Taoka, E. Furubayashi, and S. Takeuchi: Trans. Nat. Res. Inst. Metals (Japan), 1967, vol. 9, pp. 158-85. 15. C G. Dunn: AIME Trans., 1946, vol. 167, pp. 373-94. 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