Elastic interaction of two precipitates subjected to an applied stress field
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I.
INTRODUCTION
THE elastic
interaction of two or more precipitates can often affect critically material microstructures. For example, precipitate interactions are considered to be an important factor in the formation of periodic precipitate arrays in two phase materials ~-5 and void lattices in irradiated materials. 6 These elastic interactions also influence the stabilization of precipitates against coarsening] 's autocatalytic and sympathetic nucleation, 9 the bulk elastic properties of two phase materials, ~~ and the enhancement of fracture due to stress concentrations. Precipitate interactions are attributable to two sources. The first is the misfit or stress-free transformation strain associated with the precipitate. This misfit arises from differences in the volume and shape of the precipitate (manifested during nucleation and growth) as measured with respect to the matrix phase. The second source of precipitate interaction results from application of an external stress field. When the precipitate possesses elastic constants different from those of the matrix (due to differences in concentration or crystal structure), it perturbs the applied stress field. This perturbation leads to system energy changes that depend on the relative position of the precipitates and degree of elastic inhomogeneity. Under the assumption of elastic homogeneity (precipitate and matrix possess the same elastic constants), several workers have investigated the interaction of two or more precipitates possessing stress-free transformation strains. H.~2 Great progress can be made with the homogeneity assumption toward understanding precipitate alignment2 5 and in the prediction of precipitate habit planes. However, under certain conditions, elastic inhomogeneity can result in qualitative and quantitative differences in both the strain field and interaction energies of two misfitting particles as compared with the homogeneous case. 4 Elastic inhomogeneity has also been shown to be a critical factor in predicting morphology changes in misfitting particles. ~3'14 Another serious disadvantage of assuming elastic homogeneity is the inability of predicting morphology changes or interaction energies in the presence of an applied stress field. This is because an external load does not interact with a source of internal stress. ~5 (Changes in the potential energy of the loading
WILLIAM C. JOHNSON is Assistant Professor, Department of Metallurgical Engineering and Materials Science, Carnegie-Mellon University, Pittsburgh, PA 15213. Manuscript submitted June 15, 1983. METALLURGICALTRANSACTIONS A
mechanism are possible, however.) Thus, effects of the applied stress field on material microstructure are manifested only in elastically inhomogeneous systems. Although changes in material microstructure under applied load are well documented experimentally, ~6'~7'~8very little effort has been extended in determining changes in strain field and interaction energies between two or more precipitates subjected to an applied stress field. Sternberg and
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