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The main objective of this paper is to present a coarse-grained material model for the simulation of threedimensional nanostructures. The developed model is motivated by the recent progress in establishing continuum models for nanomaterials and nanostructures. As there are conceptual differences between the continuum field defined in the classical sense and the nanomaterials consisting of discrete, space-filling atoms, existing continuum measures cannot be directly applied for mapping the nanostructures due to the discreteness at small length scale. In view of the fundamental difficulties associated with the direct application of the continuum approach, we introduce a unique discrete deformation measure called spatial secant and have developed a new hyperelastic model based on this measure. We show that the spatial secant-based model is consistently linked to the underlying atomistic model and provides a geometric exact mapping in the discrete sense. In addition, we outline the corresponding computational framework using the finite element and/or meshfree method. The implementation is within the context of finite deformation. Finally we illustrate the application of the model in studying the mechanics of low-dimensional carbon nanostructures such as carbon nanotubes (CNT). By comparing with full-scale molecular mechanics simulations, we show that the proposed coarsegrained model is robust in that it accurately captures the nonlinear mechanical responses of the CNT structures.

SUMMARYA discrete hyperelastic model was developed in this paper for a single atomic layer of graphene structure that was originally planar. This model can be viewed as an extension to the well-known continuum hyperelastic model. Based on the discrete nature of the atomic structure, the notion of discrete mapping and the concept of spatial secant were introduced. The spatial secant served as a deformation measure that provided a geometric exact mapping in the discrete sense between the atomistic and continuum representations. By incorporating a physics-based interatomic potential, the corresponding discrete hyperelastic model was then established. After an introduction of the model, the computational implementation using the Galerkin finite element and/or meshfree method was outlined. The computational framework was then applied to study of the mechanics of graphene sheets. Extensive comparisons with full-scale molecular mechanics simulations and experimental measurement were made to illustrate the robustness of this approach.

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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.