Calculation of the elastic anisotropy of Ti:6Al-4V alloy sheet from pole figure data
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T H E concept of t e x t u r e s t r e n g t h e n i n g of t i t a n i u m w a s i n i t i a l l y put f o r t h by B a c k o f e n , et al.,* who s u g g e s t e d that d i r e c t i o n a l i t y of m e c h a n i c a l p r o p e r t i e s could be c o n t r o l l e d t h r o u g h c o n t r o l of c r y s t a l l o g r a p h i c t e x t u r e . The f e a s i b i l i t y of t e x t u r e s t r e n g t h e n i n g has s i n c e b e e n c o n f i r m e d and the a n i s o t r o p i c c h a r a c t e r of the m e c h a n i c a l p r o p e r t i e s d e m o n s t r a t e d by v a r i o u s workers,2-4 and the c o r r e l a t i o n of a n i s o t r o p i c p r o p e r t i e s to pole f i g u r e s has a l s o b e e n d e m o n s t r a t e d , for e x a m p l e , by Z a r k a d e s and L a r s o n . 5 H o w e v e r , the d i r e c t c a l c u l a t i o n of m e c h a n i c a l p r o p e r t i e s of t i t a n i u m f r o m c r y s t a l l o g r a p h i c d a t a , i.e., pole f i g u r e s , has not b e e n r e p o r t e d . T h i s p a p e r r e p o r t s on the c a l c u l a t i o n of e l a s t i c m o d u l u s of Ti:6A1-4V a l l o y s h e e t f r o m pole f i g u r e d a t a and s i n g l e c r y s t a l e l a s t i c c o n s t a n t s . BACKGROUND AND THEORY In o r d e r to c a l c u l a t e e l a s t i c p r o p e r t i e s of a m e t a l l i c s p e c i m e n , the e l a s t i c c o n s t a n t s of s i n g l e c r y s t a l s a r e a v e r a g e d a c c o r d i n g to the t e x t u r e c r y s t a l l i t e d i s t r i bution function a s r e p r e s e n t e d b y the p o l e f i g u r e data. S e v e r a l d i f f e r e n t a p p r o x i m a t i o n s have b e e n p r o p o s e d to f a c i l i t a t e this a v e r a g i n g . Voigt 6 a s s u m e d c o n s t a n t s t r a i n in a l l the c r y s t a l l i t e s . R e u s s 7 a s s u m e d c o n s t a n t s t r e s s . H i l l 8 s u g g e s t e d the m e a n v a l u e of the Voigt and R e u s s a p p r o x i m a t i o n a s a b e t t e r a p p r o x i m a t i o n than e i t h e r . K r o n e r ' s t h e o r y ~ a c c o u n t s f o r the b o u n d a r y c o n d i t i o n s f o r s t r e s s and s t r a i n at the i n t e r f a c e s b e tween g r a i n s . Bunge and R o b e r t s *~ have c o m p a r e d the r e s u l t s of m o d u l u s m e a s u r e m e n t s with the p r e d i c t i o n s of the V o i g t - R e u s s - H i l l a p p r o x i m a t i o n s for low c a r b o n s t e e l s h e e t . T h e i r r e s u l t s have shown that a l l t h r e e m e t h o d s account f o r the a n g u l a r v a r i a t i o n of the m o d u l u s within the p l a n e of the s h e e t and that m a g n i t u d e of the m o d u l u s was b e t w e e n that p r e d i c t e d by the Voigt and H i l l a p p r o x i m a t i o n s . The a c c u r a c y of e i t h e r m e t h o d was within the c e r t a i n t y of s i n g l e c r y s t a l d a t a . M o r r i s *x has c o m p a r e d the r e s u l t s of c a l c u l a t i o n s b a s e d on K r o n e r ' s t h e o r y w i t h th
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