Phase-Field Modeling of Equilibrium Precipitate Shapes Under the Influence of Coherency Stresses
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PRECIPITATE strengthened alloys are one of the most commonly used materials for high-temperature applications, whereby, the strengthening is achieved through precipitate–dislocation interactions. The mechanical properties of these alloys predominantly depend on precipitate size, morphologies, and their distribution. Thus, there are several experimental studies carried out which focus on the precipitate morphology,[1,2] their growth and coarsening,[3–6] strengthening.[7,8] In this regard, there are two mechanisms, one in which the precipitates are large enough such that there is no coherency between the precipitate and the matrix from which it is formed, and the second in which the precipitates are small such that there is still substantial coherency between the matrix and the precipitate. While
BHALCHANDRA BHADAK and ABHIK CHOUDHURY are with the Department of Materials Engineering, Indian Institute of Science, Bangalore 560012, India. Contact e-mail: [email protected] R. SANKARASUBRAMANIAN is with the Defence Metallurgical Research Laboratory, Hyderabad 500058, India. Manuscript submitted April 4, 2018.
METALLURGICAL AND MATERIALS TRANSACTIONS A
in the former, the interaction of the dislocation with the precipitate is purely physical, in the latter, the coherency stresses around the precipitate also influence the interaction with the precipitate. In both cases, the shape of the precipitate plays an important role in the interaction with the dislocation. Our investigation in this paper will be related to the investigation of shapes of coherent precipitates, more particularly the understanding of the equilibrium morphology of precipitates as a function of the misfits, elastic, and interfacial energy anisotropies. The first theoretical efforts are from Johnson and Cahn[9] who predict an equilibrium shape transition of an elastically isotropic misfitting precipitate in a stiffer matrix. The equilibrium shape of a precipitate is determined by minimization under the constraint of constant volume of the precipitate, of the total energy, constituting of the sum of elastic and interfacial energies. The theory proposes the shape transition with size, akin to a second-order phase transition with the shape of the precipitate as an order parameter. The theory analytically predicts the equilibrium shape order parameters as a function of precipitate size whereby for certain conditions, below a critical size there is an unique order parameter describing the shape of the particle and a bifurcation into two or more variants beyond it. In their
work, one of the transitions the authors discuss is the case where, beyond a particular size of a precipitate, a purely circular cross section of a cylindrical precipitate elongates along either of two-orthogonal directions, thereby retaining only a twofold symmetry. This is therefore an example of a symmetry breaking transition. Voorhees et al.[10] and Thompson et al.[11] give numerical predictions for the equilibrium morphologies of a precipitate in a system with dilatational and tetrago
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