On the Shape Evolution of Coherent Precipitates: Discrete Atom Method
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JONG K. LEE AND JUN H. CHOY Department of Metallurgical & Materials Engineering Michigan Technological University, Houghton, MI 49931
ABSTRACT Morphological evolution of coherent precipitates with arbitrary transformation strain is studied via a discrete atom method. With a purely dilatational misfit strain, a soft precipitate tends to have a plate-like equilibrium shape, whereas a hard precipitate takes up a shape of high symmetry such as a circle. If the stiffness is comparable between the matrix and precipitate, however, the equilibrium shape depends on the degree of anisotropy, misfit strain, size, and interfacial energy. Shape bifurcation phenomena from circular to rectangular shapes are also revealed during clustering. With either a tetragonal misfit strain of mixed signs or a pure shear misfit strain, a precipitate takes up a plate-like shape whose major axis lies along the direction containing an invariant line in accordance with the continuum elasticity prediction. Both elastic anisotropy and elastic inhomogeneity exert little influence on the preferred orientation relationship. With a misfit strain of combined shear and dilatation, a precipitate follows an orientation and shape dictated by both components of the misfit strain.
INTRODUCTION The microstructural evolution of an elastically stressed two-phase system is of great importance as it is intimately linked to alloy performance. An elastically-stressed, coherent precipitate undergoes shape evolution fundamentally different from that of an unstressed one, as its morphology is dictated by both the interfacial free energy and the elastic strain energy. Since Eshelby formulated the seminal inclusion method to evaluate the stress field associated with a coherent ellipsoidal inclusion [1], several numerical techniques have been developed [2-5]. Most of these treatments, however, involve either computations of an elastically homogeneous state, or approximate solutions for integro-differential equations when faced with an inhomogeneous system. In an effort to develop a tool to study general shape evolution of a coherent precipitate, a Discrete Atom method (DAM) was recently developed on the basis of a statistical approach [6,7]. The method has been applied to a number of coherency strain problems [8,9], demonstrating that an elastically inhomogeneous, multi-particle system can be readily analyzed. In this work, the DAM is reviewed to examine morphological evolution of coherent particles with generalstress-free transformation strains.
DICRETE ATOM METHOD In the DAM, a triangular lattice is constructed following the work of Hoover et al [10]. Atomic interactions are then mimicked through a parabolic potential function, k(r - a) 2/2, where k is spring constant, r is the interatomic distance, and a is the lattice parameter at stress-free state. A precipitate phase is represented with different spring constant, k*, and lattice parameter, a*, from 439 Mat. Res. Soc. Symp. Proc. Vol. 398 01996 Materials Research Society
those of the matrix phase, k and a. The l
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