Coarse-Grained Modeling and Simulation of Nanoscale Systems Based on Discrete Hyper-Elastic Model

Abstract

The subject of developing equivalent continuum models from the atomistic models has attracted significant attention in recent years. An outstanding issue in extending the continuum model to smaller scales is the size effect. Such a size effect is intimately related to the discrete nature of the atomic structure and nonlocal interaction among the atoms. In many of the existing continuum approaches, discrete variables are introduced in the constitutive model to account for these non-continuum effects. In this paper, we present a discrete hyperelasticity model as an alternative. Our approach, however, is fundamentally different from the conventional approach in that it treats the concept of deformation mapping in the discrete sense. The resulting deformation measure is referred to as spatial secant and the corresponding material model is called the spatial secant model. We then formulate the potential energy density functional and derive stress-like measure based on the spatial secant. After a brief description on the formulation and its comparison with the classical hyper-elastic model, we show the application of this model to both low-dimensional carbon nanostructures and general three-dimensional nanostructures. The concept of geometric-exact mapping is discussed through the examples. Comparisons with full-scale molecular mechanics simulations are made to illustrate the robustness of this approach.