The influence of applied stress on precipitate shape and stability
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I. INTRODUCTION The equilibrium shape of an isolated precipitate in a two-phase alloy is determined by minimizing the sum of the elastic strain energy and the interfacial energy. The shape that minimizes the elastic strain energy is not necessarily equivalent to the one that minimizes the interfacial energy. Because the elastic strain energy scales with the volume of the precipitate, V, and the interfacial energy scales as V213, it is not surprising that in a number of alloy systems precipitate shape transitions are observed as a function of precipitate size.' Our understanding of precipitate shape transitions can be greatly facilitated by using the properties of bifurcation theory. Bifurcation theory is a study of the branching of solutions of nonlinear equations.2'3 The singularity in the behavior of the precipitate shape transition versus size can be compared to well-known properties of bifurcation theory. The manner in which the precipitate shape transition is altered (broken) in an external field can also be likened to the manner in which bifurcations are perturbed (broken) by imperfections. Furthermore, the generic nature of the analysis avoids the necessity of numerous system-specific energy calculations that may often obfuscate the physics of the transition. However, effective use of bifurcation theory is predicted upon a complete understanding of the symmetry of the crystals, the external influences, and the parameters that describe the precipitate shape. Recently, Johnson and Cahn 4 used the properties of bifurcations and symmetry predictions to show how the self-energy extremizing shapes of a precipitate should change with precipitate size. Size-induced shape transitions are predicted to occur as a function of precipitate size when the precipitates are elastically softer than the matrix under the assumption of system isotropy and in the absence of an externally imposed field. For a twodimensional system under plane strain conditions, these transitions occur primarily in a continuous fashion analJ. Mater. Res. 1 (5), Sep/Oct 1986 http://journals.cambridge.org
ogous to a second- or higher-order phase transition (supercritical bifurcation). For a three-dimensional system, the shape transition is discontinuous (transcritical bifurcation) and is analogous to a first-order phase transition. The size-induced shape transition is predicted to occur at a distinct volume or cross-sectional area, Ac, determined by the system material parameters, and is a result of the increasing importance of minimizing the elastic strain energy at larger precipitate sizes. The equilibrium precipitate shape can change when subjected to directional external influences such as a mechanical stress, an electric field, or a magnetic field. Physically, the external influence (stress) alters the energy of the system, thereby affecting any size-induced shape transitions predicted or observed in the absence of the external influence. Many instances of precipitate shape changes resulting from an external influence have been observed experime
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