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ATTEMPTS

t o predict polycrystalline stress-strain behavior have taken two routes. The first, given a strong impetus by Taylor, 1 l a t e r taken up by Bishop and Hill%3 and generalized by Chin and M a m m e l4 t o include twinning, considers the necessary slip systems to minimize the total internal work necessary t o produce the imposed external change of shape. No consideration is given t o effects of g r a i n size or differences in contribution t o strength by g r a i n bounda r i e s and g r a i n interiors. The second path, begun by Chalmers and his coworkers, has attempted to approach polycrystalline behavior by studying bicrystals %~ multicrystals 7 and completely surrounded bicrystalsfl Hauser and Chalmers s implied that the r e g i o n of complex slip in the vicinity of a bicrystal boundary can c a u s e inc r e a s e d strengthening in a macroscopically i n c o m patible bicrystal. Separate contributions to strength by dislocations geometrically necessary to maintain continuity at g r a i n boundaries and g r a i n interiors have been proposed by A s h b y ,9 and Thompson, Baskes and Flanagan ~° have been able t o plot s t r e s s s t r a i n curves, in agreement with experimental data, based on the separate contributions of g r a i n boundary and g r a i n interior regions. T h e i r Eq. [10] simplifies t o the following f o r m : a T = a b Vb + -6gbVg b = a b + V g b ( ~ gb - ab)

[1]

where aT = total applied stress, -dgb = a v e r a g e s t r e s s in the g r a i n boundary de-

formation region,

TAEK DONG LEE is VisitingResearch Associate, Metallurgical Engineering Department, OhioState University, Columbus, OH 43270. HAROLD MARGOLIN is Professor, Department of Physical and Engineering Metallurgy, Polytechnic Instituteof New York, Brooklyn, NY 11201,where Tack DongLee was formerly a GraduateStudent. This paperis basedon a thesis to be submittedby Tack DongLee in partialfulfillment of the requirements for the degree of Doctor of Philosophy at Polytechnic Instituteof New York. Manuscript submitted October20, 1975. METALLURGICAL TRANSACTIONS A

a b = a v e r a g e s t r e s s in the body of the g r a i n away from the g r a i n boundary, V g b , Vb = the volume fractions of g r a i n boundary and g r a i n interior deformation respectively. Using Eq. [1], Chuang and Margolinxx were able t o determine -6gb as a function of strain for a beta b r a s s bicrystal incompatible in s h e a r at the boundary. They found that ~ g b > ab and that ~ g b increased at about twice the rate of ab. Subsequent t o this work Margolin and Stanescu12 proposed an equation for polycrystalline strengthening which incorporated Eq. [1] and observations on slip line spacing. In the polycrystalline case the evidence indicated that ab increased more rapidly than-#gb with increasing strain. The present work on beta b r a s s tricrystals was undertaken to develop some understanding of how increasing constraints to plastic flow develop into polycrystalline behavior. Continuity requirements for a bicrystal boundary parallel t o the