In this paper, we formulate a phase-field model for the computation of equilibrium configurations of multiple phases that arise out of a solid-state precipitate reaction, in the presence of coherency stresses. Here, we utilize the phase-field framework to minimize the sum of the elastic and the interfacial energies for a given volume of the precipitates using an extension of the volume-preserved Allen–Cahn algorithm (Garcke et al. in Math Models Methods Appl Sci 18(08):1347–1381, 2008; Bhadak et al. in Metall Mater Trans A 49A(11):5705–5726, 2018). Using this technique, we investigate the precipitate organization for three solid-state reactions. The first is the classical two-phase precipitate reaction that leads to the formation of core–shell microstructures, where we clarify the influence of elasticity on the formation of such clusters. Following this, we investigate two symmetry-breaking transitions (cubic to tetragonal) and (hexagonal to orthorhombic), that lead to the formation of multi-variant clusters where we study the organization of the precipitates as a function of the elastic properties of the precipitate and the matrix.

在本文中，我们建立了一个相场模型，用于计算在相干应力存在下固态沉淀反应引起的多相平衡构型。在这里，我们利用相体积框架扩展了体积保留的Allen-Cahn算法（Garcke等人在Math Models Methods Appl Sci中的扩展），以使给定体积的沉淀物的弹性能和界面能之和最小。 18（08）：1347–1381，2008; Bhadak等人在Metall Mater Trans A 49A（11）：5705-5726，2018）中。使用这种技术，我们调查了三个固态反应的沉淀物组织。第一个是经典的两相沉淀反应，导致形成核-壳微结构，在此我们阐明了弹性对此类团簇形成的影响。