A Multi-Scale Computational Method Integrating Finite Element Method with Atomic Interactions of Materials

Bridging the atomic and continuous analyses is an important aspect in multi-scale mechanics. This paper develops a computational method to integrate the atomic potential of a material with the finite element method. The novelty of this method is that strain energy is calculated from the atomic potential without the assumption in the Cauchy-Born rule that deformation in a virtual atomic cell is homogeneous. In this new method, the virtual atomic cell deformation is interpolated according to the continuum displacements associated with the shape functions. The applications of the method to single crystal Si and Ge bars under uniaxial tension and compression show that with a proper construction of the virtual atomic cell, the Young's modulii in the < 100 >, < 110 > and < 111 > directions obtained are in good agreement with the experimental measurements and MD simulations in the literature. Moreover, the simulated material's response to tension and compression is consistent with the interatomic interactions.