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i) THE

(HALL-PETCH) STRESS-GRAIN RE LATIONS

SIZE

There are quite general grounds for expecting that the overall strength properties of polycrystalline materials should depend on their grain size. For example, the mechanical properties of these materials are often correlated with their stress-strain behavior, and it is known to depend on grain size. That is, the flow stress, a~, measured at constant values of strain, E, for polycrystals depends at relatively low temperatures on the average grain diameter, l, according to the relation a E

=a%

+kel -u2

[1]

where a% and k e a r e e x p e r i m e n t a l c o n s t a n t s . 1-5 Hall 1 and Petch 2 d e t e r m i n e d in detail that the t e n s i l e y i e l d and b r i t t l e f r a c t u r e s t r e s s e s of m i l d s t e e l followed Eq. [1]. A r m s t r o n g , Codd, Douthwaite, and P e t c h s showed that Eq. [1] applied for the c o m p l e t e s t r e s s - s t r a i n b e h a v i o r of the s a m e m a t e r i a l . Fig. 1 c o n t a i n s t h e i r r e s u l t s for the flow s t r e s s at the lower y i e l d point (1.y.p.) and for v a r i o u s s t r a i n v a l u e s past it, in c o m p a r i s o n with r e s u l t s e a r l i e r r e p o r t e d by P e t c h 4 on the g r a i n size dependence of the t r u e ductile f r a c t u r e s t r e s s . A r m s t r o n g et al. found, a s now is u s u a l l y a s s u m e d for Eq. [1], that the flow s t r e s s of v a r i o u s m e t a l s and t h e i r a l l o y s , with o r without a y i e l d point, and at v a r i o u s s t r a i n v a l u e s , followed the Hall-Petch relation. One of the r e a s o n s for thinking that Eq. [1] i t s e l f should be applied b r o a d l y to the d e f o r m a t i o n b e h a v i o r of p o l y c r y s t a l s is b e c a u s e the equation m a y be d e r i v e d a s a d i r e c t c o n s e q u e n c e of the g e n e r a l p o s t u l a t e s that p l a s t i c flow o c c u r s in l o c a l i z e d zones of s h e a r , as in s l i p b a n d s , and these s h e a r b a n d s m u s t i n d i v i d u a l l y g e n e r a t e an exceptional c o n c e n t r a t i o n of s t r e s s at the g r a i n b o u n d a r i e s to propagate p l a s t i c flow throughout the m a t e r i a l . 5 In this way, the yield and f r a c t u r e s t r e s s e s m e a s u r e d for a m a t e r i a l m i g h t be thought to R. W. ARMSTRONGis Professor, EngineeringMaterialsGroup, and Department of MechanicalEngineering,University of Maryland, College Park, Md. This manuscript is based on an invited talk presented at the symposium on Deformation and Strength of Polycrystals, sponsored by the IMD Physical Metallurgy Committee, Detroit, Mich., October 1415, 1968. METALLURGICALTRANSACTIONS

follow Eq. [1] as l i m i t i n g f o r m s of the g e n e r a l r e l a t i o n s h i p which a p p l i e s for the total d e f o r m a t i o n b e h a v i o r . A c o n s i d e r a b l e n u m b e r of the e x p e r i m e n t a l and t h e o r e t i c a l i n v e s t i g a t i o n s u n d e r l y i n g this e x p l a n a