Computational method for atomistic homogenization of nanopatterned point defect structures

The development of an approximation method that rigorously averages small-scale atomistic physics and embeds them in large-scale mechanics is the principal aim of this work. This paper presents a general computational procedure based on homogenization to average frozen nanoscale atomistics and couple them to the equations of continuum hyperelasticity. The proposed application is to nanopatterned systems in which complex atomic configurations are organized in a repeating periodic array. The finite element method is used to solve the equations at the large scale, but the small-scale equation is representative of lattice-statics. The method is predicated on a quasistatic zero-temperature assumption and, through homogenization, leads to a coupled set of variational equations. The numerical procedure is presented in detail, and 2-D examples of ultra thin film layers of carbon one atom thick are shown to illustrate its applicability. Homogenization naturally gives rise to an inner displacement term with which point defects are explicitly modelled and their non-linear interactions with global states of multiaxial strain are studied