Resolved shear stress in elastically anisotropic polycrystals
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t r e a t m e n t s by Sachs 1 and T a y l o r , 2 which s e e k to r e l a t e yield in single and p o l y c r y s t a l s , n e g l e c t i n t e r a c t i o n s between a c r y s t a l l i t e and its e n v i r o n m e n t . F a i l u r e to c o n s i d e r these i n t e r a c t i o n s can lead to e r r o r s in e s t i m a t i n g r e s o l v e d s h e a r s t r e s s . T h e o r e t i c a l d e v e l o p m e n t s by Eshelby, s'4 K r o n e r , 5 Kneer, 6'7 Bunge, 8 Roe, 9''~ and the w r i t e r , 1~ p e r m i t c o n s i d e r a t i o n , in a s t a t i s t i c a l s e n s e , of such i n t e r a c t i o n s . Eshelby, by an ingenious s e q u e n c e of hypothetical cutting, s t r a i n i n g , and r e j o i n i n g o p e r a t i o n s , was able to satisfy s t r e s s e q u i l i b r i u m and c o m p a t i b i l i t y equations along the boundary between a homogeneous i s o tropic ellipsoidal i n c l u s i o n and a s u r r o u n d i n g h o m o geneous isotropic m a t r i x with e l a s t i c c o n s t a n t s diff e r e n t from those of the inclusion. K r o n e r extended E s h e l b y ' s a n a l y s i s to a homogeneous a n i s o t r o p i c ellipsoid of revolution imbedded in a homogeneous isotropic m a t r i x . He f u r t h e r a s s u m e d that, in a s t a t i s t i c a l s e n s e , the nonhomogeneous e n v i r o n m e n t of a c r y s t a l l i t e (inclusion) in a p o l y c r y s t a l can be r e p l a c e d by a homogeneous m a t r i x with e l a s t i c p r o p e r t i e s equal to the m a c r o s c o p i c p r o p e r t i e s of the p o l y c r y s t a l . In this way, he calculated bounds for the e l a s t i c constants of a p o l y c r y s t a l composed of r a n domly oriented a n i s o t r o p i c c r y s t a l l i t e s . K n e e r extended K r o n e r ' s a n a l y s i s to a homogeneous a n i s o t r o p i c m a t r i x . He r e c o m m e n d e d expansion of the c r y s t a l l i t e o r i e n t a t i o n d i s t r i b u t i o n in a s e r i e s of gene r a l i z e d s p h e r i c a l h a r m o n i c s . Since, at that time, the coefficients of such s e r i e s could only be obtained in s p e c i a l cases, K n e e r r e s t r i c t e d his a n a l y s i s to fiber physical symmetry~ Bunge and Roe have s i n c e shown how the coefficients can be obtained in the g e n e r a l case, and the w r i t e r has extended K n e e r ' s a n a l y s i s to orthotropic (rolled sheet) p h y s i c a l s y m m e t r y , so that it is p r e s e n t l y p o s s i b l e to c o n s i d e r i n t e r a c t i o n s between c r y s t a l l i t e s of a r b i t r a r y s y m m e t r y and m a t r i c e s of orthotropic and higher s y m metries. PETER R. MORRIS is Senior Research Spectrochemist, with Research and Technology, Armco Steel Corp., Middletown, Ohio. Manuscript submitted December 3, 1970. METALLURGICALTRANSACTIONS
THEORY A complete development of the theory r e l e v a n t to the c a l c u l a t i o n of the e l a s t i c c o n s t a n t s of p o l y c r y s t a l s for o r t h o t r o
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