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International Journal for Multiscale Computational Engineering
IF: 1.016 5-Year IF: 1.194 SJR: 0.554 SNIP: 0.68 CiteScore™: 1.18

ISSN Print: 1543-1649
ISSN Online: 1940-4352

International Journal for Multiscale Computational Engineering

DOI: 10.1615/IntJMultCompEng.v4.i1.60
pages 71-94

A Constitutive Model for Nanomaterials Based on Spatial Secant

Dong Qian
Department of Mechanical, Industrial and Nuclear Engineering University of Cincinnati, Cincinnati, OH 45221-0072
Rohit H. Gondhalekar
Department of Mechanical, Industrial and Nuclear Engineering University of Cincinnati, Cincinnati, OH 45221-0072

ABSTRACT

In this paper a finite deformation hierarchical model based on the concept of spatial secant is presented. Motivated by the fundamental differences between the definitions of continuum in the classical sense and discrete atomic structure studied in this paper, this model proposes a systematic description for the mechanics of nanomaterials with regular lattice structures. Although the proposed model is similar in its form to the crystal elasticity model, it distinguishes itself from the classical continuum model in that it directly considers the finite size effect of the atomic bond by introducing the concept of spatial secant. As a result, the proposed model is consistently linked to the mechanics of the underlying atomic structure by deriving directly from the interatomic potential with the proposed deformation measure. In contrast, it is shown that the crystal elasticity model based on the deformation gradient (or equivalently, the spatial secant) is not suitable for describing the mechanics of nanostructures due to its inability to account for the finite size effect of the interatomic bond. Following the presentation of the model, the numerical procedure to solve the constitutive model based on spatial secant is described. The implementation of the model in a typical mesh-free Galerkin formulation is illustrated in a set of benchmark problems involving two-dimensional nanostructures. The robustness and accuracy of the proposed model are shown by directly comparing the results to those obtained from full-scale atomistic simulations for the same benchmark problems. The erroneous results obtained from the spatial tangent-based crystal elasticity model highlight the importance of the proposed model for resolving the mechanics of nanostructures.


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