Elastic interaction stresses: parti. the influence of bicrystal size on stresses in [213] Iso-Axial 70-30 alpha-brass bi
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I.
INTRODUCTION
E V I D E N C E has been accumulating that elastic interactions play an important role in the initiation of slip ~I-73and, recently, a model incorporating elastic interactions was proposed for yielding in anisotropic metals. Is~ This model was based on the extent of compatibility slip emanating from the boundary of an iso-axial shear incompatible bicrystal of beta brass. [9"~~ It was proposed ts~ that the extent of the compatibility slip depended on two factors: (1) the size of the bicrystal normal to the boundary, which size determined the magnitude of the torque produced by the elastic shearing of the bicrystal components in opposite directions parallel to the boundary; (2) the m~gnitude and inward extent of the enhanced grain boundary stress region arising from this torque. The enhanced stress region governed the extent of compatibility slip, and it was proposed that this enhanced stress region would effectively disappear at large and small grain sizes. The yield stress, o-r, was proposed to consist of two parts: (1) a resisting stress, oR, and (2) an assisting stress, o-a . Thus, o-y = o"R - o"a
9 d -1/2
[2]
with o"0 and K~ having values different from the HallPetch values. o'a was expressed as
TZI-KANG CHEN, Graduate Student, and HAROLD MARGOLIN, Professor, are with the Department of Metallurgy and Materials Science, Polytechnic University, Brooklyn, NY 11201. Manuscript submitted August 7, 1987.
METALLURGICALTRANSACTIONS A
Vgb=C[1--(-'~-)m]'[(-~ln--1].
[4]
[1]
o'R was expressed in a Hall-Petch form
O"R = o"o + K l
where K2 was expressed as a constant, in units of stress and depended on the average difference in the matrix of elastic constants between adjacent grains. D L is the effective grain size at which the elastic interaction goes to zero at large grain sizes. Ds is the effective grain size at which the elastic interaction goes to zero at small grain sizes. The exponent n governs the magnitude of the torque and recognizes that not all of the grain volume can contribute to torquing a neighboring grain. The exponent m provides a power-law decay to the induced, enhanced grain boundary resolved shear stress. tra was shown to pass through a maximum as the grain size, d, changed Is] and to become zero at a large and a small grain size. Because slip, emanating from the grain boundary of a shear incompatible bicrystal, stopped when the enhanced stress (which includes the applied stress) fell below the critical resolved shear stress, t9:~ the inward extent of o-a and the volume fraction of grain boundary deformation zone were taken to have essentially the same formJ sl Thus
Here c is a proportionality constant. Vgb, having the same form as o-A, should also pass through a maximum as the shear incompatible bicrystal size normal to the boundary varied. This study was undertaken in part in an attempt to examine whether the volume fraction of grain boundary deformation, Vgb, did in fact behave according to observed behavior of Eq. [3]. As will be seen, Vgb does behave as predicted. Howeve
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