Diffusion-accommodated sliding of irregularly shaped grain boundaries
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(Received 6 October 1989; accepted 13 November 1989) The diffusion-accommodated sliding of irregularly shaped grain boundaries in twodimensional bicrystals is considered. The following assumptions are made: the grains adjoining the boundaries are rigid, the boundaries do not support any shear stresses, sliding displacements are infinitesimal, and sliding is accommodated only by grain boundary diffusion. The solution to this problem is illustrated for a bicrystal with a grain boundary consisting of three segments. The results of calculations involving up to 35 segments agree with Raj and Ashby's theory for the sliding of periodic boundaries. The influence of boundary conditions on the normal stress distributions along grain boundaries is examined. Zero-flux conditions at the intersection of a grain boundary with a free surface, which correspond to low surface diffusivities, can lead to high normal grain boundary stresses. The stress distributions and sliding rates of boundaries containing randomly spaced equisized bumps or equispaced bumps of random size are compared to the periodic case (i.e., equispaced equisized bumps). Substantial normal stresses can build up at such nonperiodic grain boundaries.
I. INTRODUCTION
Raj and Ashby,1 in 1971, gave a complete description of the diffusion-accommodated sliding of grain boundaries for the cases of boundary as well as bulk diffusion. They assumed full relaxation of the shear stresses at periodically serrated boundaries. In the present paper we extend their work, for the special case of grain boundary diffusion, to irregularly shaped grain boundaries in a finite bicrystal. As in Raj and Ashby's work, sliding is controlled by the rate with which matter is plated out at, or removed from, the grain boundary. We assume that the plating and sliding rates at the grain boundary, as well as the rotation rate between the adjacent rigid crystals, are infinitesimal. As will be seen, the stress distributions, plating rates, sliding rate, etc. of a grain boundary consisting of n straight segments can be obtained by generating a system of An linear equations and solving for the An constants required for a complete description of the problem. II. FORMULATION AND SOLUTION OF THE THREE-SEGMENT CASE
If bulk diffusion is ignored, the stress-induced atomic flux; along a grain boundary is given by Db da (1) J = kTds'
a)Permanent
address: Department of Metallurgy, Parks Road, Oxford OX1 3PH, United Kingdom.
where Db is the grain boundary self-diffusivity, k is Boltzmann's constant, T the absolute temperature, and da/ds the gradient of the normal stress acting on the grain boundary. The rate with which atoms are plated out at, or removed from, the grain boundary is given by 8bDbQ. d2a v = —-
kT
ds2
where Sb is the diffusional grain boundary thickness and XI the atomic volume. Since the plating rate may vary along a grain boundary, wedge-shaped layers may be plated out or removed. This results in a rotation rate 8bDbQ, d3a dv (3) kT ds3 Since we assume the grains to be rigid, the rotation
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