A three dimensional adaptive multiscale method for crack growth in Silicon

A three dimensional concurrently coupled adaptive multiscale method is introduced here to simulate complex crack growth patterns in Silicon, by combining several numerical techniques across the length scales. The coarse scale material is modeled using the virtual atom cluster approach. The strong kinematic discontinuities in the bulk are simulated based on a three dimensional version of the phantom node method. A molecular statics model placed around the crack tip is concurrently coupled with the phantom-based discontinuous formulation, where the coupling between the fine and coarse scales is realized through the use of ghost atoms, whose positions are interpolated based on the coarse scale solution. The boundary conditions to the fine scale model, at the coupling region, are assigned by enforcing the interpolated displacements of ghost atoms. In order to optimize the computation costs, adaptivity schemes for adjustment of the fine scale region as the crack propagates, and coarse graining of the region behind the crack tip, are proposed. The crack tip location is detected based on an energy criterion. All the molecular simulations in the pure atomistic as well as the multiscale model are carried out using the LAMMPS software, triggered through the system command in MATLAB. The performance of the developed framework in terms of computation cost, robustness and versatility, is assessed through several numerical examples concerning crack growth in Silicon. Therefore, the diamond cubic lattice structure of Silicon is used at the fine scale, where the atom-atom interactions are modeled based on the Tersoff potential function. According to the numerical examples presented in this study, savings in computational time using the present multiscale method are observed to be up to 87%, as compared to the pure atomistic model.