A two-dimensional analysis of the evolution of coherent precipitates in elastic media

The morphological evolution of coherent inclusions in elastic media is studied in two-dimensions. The inclusions are simple dilations with isotropic surface energy in a system with homogeneous elastic constants of negative anisotropy. The equilibrium sizes at which a circular inclusion transforms to a rectangle or square, and at which a square splits into a doublet or quartet of separated inclusions are computed analytically. A finite-element model is then constructed to simulate the evolution of an arbitrary distribution of inclusions along the minimum-energy path. In the model, the circle evolves into a square, which splits into a doublet by hollowing from its center, or, if this is forbidden, by drawing in a perturbation on its surface. The sizes at which shapes spontaneously transform are compared to the equilibrium values. Finally, the simulation is used to study the evolution of a random distribution of inclusions. The first metastable state assumed by the distribution depends on the elastic interaction, surface energy and areal fraction of the inclusion phase through a single dimensionless parameter that groups these three effects. The results are compared to prior theoretical and experimental work on coarsening patterns in three dimensions.