Criterion for predicting the morphology of crystalline cubic precipitates in a cubic matrix
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INTRODUCTION
I T is generally accepted that the shape and the orientation of precipitates in a solid matrix are determined by strain and surface energy effects. In many cases strain energy dominates the surface energy contribution, and the latter can be neglected. The calculation of the strain energy dependence on the shape and orientation of a crystalline precipitate is a complex task, and results have been obtained only under a number of simplifying assumptions. Khachaturyan Ldetermined the energy of a crystalline precipitate of arbitrary elastic anisotropy in a homogeneous system, namely, a system consisting of a matrix and a precipitate possessing the same elastic constants. Khachaturyan showed that a minimum of the strain energy is associated with plate-shaped precipitates. Khachaturyan's approach has also been employed to calculate the dependence of the strain energy of a plate-shaped precipitate on its orientation, 2'3 and to predict additional shapes associated with minimum strain energy for precipitates having a hexagonal crystal symmetry. 4 Eshelby 5 developed a method for calculating the strain energy associated with an isotropic precipitate having an ellipsoidal shape. Barnett and A s a r o 6 extended this method to anisotropic systems. The extension7 to an anisotropic inhomogeneous system is based on Eshelby's model of the "equivalent precipitate", a model which is strictly applicable to an ellipsoidal precipitate only, in which the stresses are uniform. The ellipsoidal shape, however, is convenient for approximating a variety of precipitate shapes, extending from a disk through a sphere to needle-like shapes. R. SCHNECK, Graduate Student, is with the Department of Materials Engineering, Ben-Gurion University of the Negev, P.O. Box 653, BeerSheva, Israel. S. I. ROKHLIN, Associate Professor, Department of Materials Engineering, Ben-Gurion University of the Negev, P.O. Box 653, Beer-Sheva, Israel, is now on sabbatical leave with the Department of Welding Engineering, The Ohio State University, 190 West 19th Avenue, Columbus, OH 43210. M. P. DARIEL, Professor, is with the Department of Materials Engineering, Ben-Gurion University of the Negev and Nuclear Research Centre-Negev, P.O. Box 9001, Beer-Sheva, Israel. Manuscript submitted July 5, 1984. METALLURGICALTRANSACTIONS A
Lee, Barnett, and Aaronson 7 carried out numerical calculations of the dependence of the strain energy on the shape and orientation of crystalline precipitates in various matrices, following a dilatational transformation. These authors concluded that for cubic precipitates in a cubic matrix, only plate-shaped or spherical precipitates are associated with a minimum strain energy. The objective of the present work was to determine an analytic criterion which would allow the prediction of the particular shape and orientation of a cubic precipitate in a cubic matrix, following a dilatational transformation. Formulation of such a criterion based on strain energy considerations was expected to allow identification of the elastic paramet
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