A continuum-atomistic multi-scale technique for nonlinear behavior of nano-materials.

Highlights • A hierarchical RVE-based continuum-atomistic multi-scale procedure is developed. • The inter-scale kinematic and energetic consistency principals are exploited. • The kinematic compatibility is applied by the atomistic periodic boundary conditions. • The energetic consistency is satisfied by the Hill-Mandel periodic boundary conditions. • Coarse-scale is modeled using the stress tensor and tangent modulus computed from atomistic RVE. Abstract In this paper, a hierarchical RVE-based continuum-atomistic multi-scale procedure is developed to model the nonlinear behavior of nano-materials. The atomistic RVE is accomplished in consonance with the underlying atomistic structure, and the inter-scale consistency principals, i.e. kinematic and energetic consistency principals, are exploited. To ensure the kinematic compatibility between the fine- and coarse-scales, the implementation of periodic boundary conditions is elucidated for the fully atomistic method. The material properties of coarse-scale are modeled with the nonlinear finite element method, in which the stress tensor and tangent modulus are computed using the Hill-Mandel principal through the atomistic RVE. In order to clearly represent the mechanical behavior of the fine-scale, the stress-strain curves of the atomistic RVE undergoing distinct type of deformation modes are delineated. These results are then assessed to obtain the proper fine-scale parameters for the multi-scale analysis. Finally, several numerical examples are solved to illustrate the capability of the proposed computational algorithm. Graphical abstract Schematically representation of the hierarchical atomistic-continuum multi-scale procedure. Image, graphical abstract [ABSTRACT FROM AUTHOR]