A variational formulation and a double-grid method for meso-scale modeling of stressed grain growth in polycrystalline materials

摘要

A deterministic approach for meso-scale modeling of grain growth in stressed polycrystalline materials based on the principle of virtual power is presented. The variational equation is formulated based on the power balance of the system associated with grain boundary surface tension and curvature, rate of strain energy stored in each grain, strain energy density jump across the grain boundaries, and external work rate. The numerical solution of stressed grain growth variational equation requires discretization of grain interiors and grain boundaries. This cannot be effectively modeled by Lagrangian, Eulerian, or Arbitrary Lagrangian Eulerian finite element method if grain boundary migration (moving interfaces) and topological changes of grain boundary geometry are considered. This paper presents a double-grid method to resolve the above mentioned difficulty. In this approach, the material grid points carry material kinematic variables, whereas the grain boundary grid points carry grain boundary kinematic variables. The material domain is discretized by a moving least squares reproducing kernel approximation with strain discontinuity enrichment across the grain boundaries. The grain boundaries, on the other hand, are discretized by the standard finite elements. An interface enrichment function to accurately capture strain jump conditions across the grain boundaries is introduced. A reproducing kernel approximation that includes the periodicity of the unit cell in the construction of reproducing kernel shape function for material velocity is also presented. This proposed double-grid method allows modeling of arbitrary evolution of grain boundaries without remeshing.