Finite element analysis for surface diffusion-controlled shape instabilities of plate-like grains

Based on classical theory of surface diffusion and evaporation–condensation, a finite element program is developed to simulate the unstable shape evolution of plate-like grains. The program is used to analyze thermal grooving on a polycrystalline surface and compared with a non-linear solution and finite difference analysis. It shows that the finite element method used is robust, accurate and efficient. Then, the shape evolution kinetics of the plate-like grains are simulated as a function of the thermal grooving angle θ at the grain boundary–surface junctions and the initial aspect ratio of the plate β (plate width to thickness). When θ=0 (without internal boundary), the plate-like grain will evolve into cylinders directly. When an internal boundary exists, there is a critical thermal grooving angle θmin for given β. If θ<θmin, the plate cannot split, otherwise, the plate will split along the internal boundary of the plate-like grain. An approximate formulation of θmin as a function of β is given based on a number of finite element analyses. The effect of initial termination shape of the plate on θmin is also examined, and a weak effect was found. When β>10, its effect can be neglected.