A study on coherency strain and precipitate morphology via a discrete atom method
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INTRODUCTION
IN a precipitation-hardening process, the matrix phase is designed to bear finely dispersed, coherent precipitates, and the precipitates in turn interfere with dislocation motions. There arises a distortion of the lattice structure around and within the vicinity of coherent precipitates. During plastic deformation, dislocation motions are effectively impeded as a result of these distortions, and, consequently, the alloy becomes harder and stronger. As an example, nickel-base superalloys are widely used for aircraft and industrial gas turbines as they render high strengths and superior resistance to creep and rupture at elevated temperatures. The microstructural development of such elastically stressed two-phase systems has been a subject of great interest, as it is closely tied to the alloy performance during application and is also fundamentally different from that of unstressed systems. For unstressed two-phase systems, the equilibrium shape of a precipitate is established solely by the interfacial free energy and its dependence on crystallographic orientation, and the solution is provided by the well-known Wulff constructionY 1 On the other hand, the equilibrium morphology of a misfitting coherent precipitate is dictated by both the interracial free energy and the elastic strain energy. Consequently, 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.
JONG 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 during TMS/ASM Materials Week in the symposium entitled "Atomistic Mechanisms of Nucleation and Growth in Solids," organized in honor ofH.I. Aaronson's 70th Anniversary and given October 3-5, 1994, in Rosemont, Illinois. METALLURGICAL AND MATERIALS TRANSACTIONS A
Eshelbyf21was 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, t3,4,51Since his work, several techniquest6-14] have been developed. The works of the Khachaturyan schoolt5-81 are most noteworthy in that their generalized stochastic field model can predict the features of microstructural evolution for various transformations including diffusional precipitation, diffusionless transformations, and ordering phenomenon. Thompson et al.[9l used a variational principle and the thermodynamics of stressed solids and showed an enlightening work that a particle with a dilatational misfit strain undergoes a series of shape bifurcations from a radial to a fourfold then to a twofold symmetric morphology. The major limitation of these models, however, is that the system should be elastically homogeneous, i.e., the elastic constants of particles must be equal to those of the matr
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