Multiscale simulation of nanostructures based on spatial secant model: a discrete hyperelastic approach

Abstract

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.

Coarse‐grained modeling and simulation of graphene sheets based on a discrete hyperelastic approach

“…The second approach is to develop a new measure to map the bonds directly. This method was proposed in [37], and the robustness of the model were demonstrated for 2D atomic rings [37] and carbon nanotube structures [39]. The newly developed measure is called spatial secant (SS) N F, and given as N F D S W S 0 !…”

“…It can be seen that F is not directly applicable in describing the deformation of the bond vector Δ X , i.e., Δ x ≠ F ·Δ X because of the fundamental difference between d X and Δ X . As a result, the assumption of homogeneous deformation does not hold in general and corrections must be made to remove the inconsistency in the mapping of the bond using F directly. There are generally two approaches to make this correction: the first is to modify F by introducing parameters that feature the length scale of the bond.…”

“…The second approach is to develop a new measure to map the bonds directly. This method was proposed in , and the robustness of the model were demonstrated for 2D atomic rings and carbon nanotube structures . The newly developed measure is called spatial secant (SS) , and given as , in which is a two‐point tensor that associates the bond vectors Δ X in Ω 0 to Δ x in Ω through .…”

“…Alternatively, Qian et al ., developed a discrete hyperelastic model called spatial secant (SS) model by establishing deformation mapping measures in the discrete sense. The resulting model has been implemented successfully on 2D generic nanostructures such as atomic rings and chains and also CNTs . The main objective of this paper is to extend the application of the SS model to study the mechanics of graphene sheets.…”