Dynamic interaction between a coherent precipitate and an edge dislocation
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I.
INTRODUCTION
DYNAMIC interactions between coherent precipitates and dislocations play a crucial role in determining the mechanical properties of alloys. Indeed, the shapes, sizes, and distribution of second-phase precipitates are the principal factors determining the mechanical, magnetic, and optical properties of a wide variety of high technology alloys. In the initial processing, and also in high-temperature applications, such as jet engine superalloys, the precipitate morphologies evolve in time and the properties change. An understanding of the elastic interactions between matrix and precipitates, among the precipitates, and between precipitates and dislocations is crucial for predicting and manipulating the properties of the alloys. Therefore, there has been a need for a computational technique, through which one can analyze the elastic state associated with arbitrarily shaped precipitates whose elastic constants are different from those of the matrix phase. Eshelby[1] was the pioneer in the field of coherency strain who devised the seminal equivalency method and thus brought much understanding to the coherency strain problem; however, the method is limited to a single ellipsoidal precipitate.[2,3,4] Since his work, several numerical techniques have been developed, but most involve either computations of an elastically homogeneous state, or approximate solutions for integrodifJONG K. LEE, Professor, is with the Department of Metallurgical and Materials Engineering, Michigan Technological University, Houghton, MI 49931. This article is based on a presentation made in the symposium ‘‘Kinetically Determined Particle Shapes and the Dynamics of Solid:Solid Interfaces,’’ presented at the October 1996 Fall meeting of TMS/ASM in Cincinnati, Ohio, under the auspices of the ASM Phase Transformations Committee. METALLURGICAL AND MATERIALS TRANSACTIONS A
ferential equations when faced with an inhomogeneous system.[5–14] As a consequence, earlier theoretical treatments of the interaction between precipitates and dislocations have had to resort either to a homogeneous system or to a specific precipitate shape.[15–20] A new technique, called the discrete atom method (DAM), was recently developed to analyze a general coherency strain problem.[21,22] In this work, the DAM is applied to examine dynamic interactions between a coherent particle and an edge dislocation under a plane strain condition. Based on both classical statistical mechanics and linear elasticity, the DAM has not only eliminated shape restrictions and allowed interactions among many precipitates, but has also allowed computation of many relevant details of the evolution process.[21–28] Several major DAM findings which bear direct implications to this work are the following. (a) With a purely dilatational misfit strain, a ‘‘soft’’ particle tends to have a platelike shape, whereas a ‘‘hard’’ particle takes on a shape of high symmetry. If the stiffness is comparable, the shape depends sensitively on elastic anisotropy. (b) The orientation relationship betwe
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