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The main objective of this paper is to develop a multiscale method for the analysis of carbon nanotube systems. The multiscale coupled governing equations are first derived based on a multiscale partition, which consists of a coarse scale component represented by meshfree approximation, and a fine scale component to be resolved from molecular dynamics. It is noted that the coarse scale representation is present in the entire domain of the problem, and coexists with the fine scale representation in the enriched region. Using the projection properties of the partition, the decoupled non-linear set of equations are obtained and solved iteratively using Newton''s method. In this multiscale analysis, a virtual atom cluster (VAC) model is also proposed in the coarse scale treatment. This model does not involve a stress update scheme employing the Cauchy–Born hypothesis, which has been widely used in the crystal elasticity theory. Finally, to approximate a curved surface at the nanoscale, meshfree approximations are introduced to interpolate a single layer of atoms. Unlike traditional shell or continuum elements, the geometric constraint is only imposed in two dimensions. The high order continuity property of the meshfree shape functions guarantees an accurate description of the geometry and thus the energy of the atomic bond. The accuracy and efficiency of the method are illustrated in the post-buckling analysis of carbon nanotube structures. To our knowledge, this is the first multiscale post-buckling analysis presented for such a nanoscale system. [Copyright &y& Elsevier]

Abstract: Inspired by the pioneering work of Professor T.J.R. Hughes on the variational multi-scale method, this document summarizes recent developments in multiple-scale modeling using a newly developed technique called the bridging scale. The bridging scale consists of a two-scale decomposition in which the coarse scale is simulated using continuum methods, while the fine scale is simulated using atomistic approaches. The bridging scale offers unique advantages in that the coarse and fine scales evolve on separate time scales, while the high frequency waves emitted from the fine scale are eliminated using lattice impedance techniques. Recent advances in extending the bridging scale to quantum mechanical/continuum coupling are briefly described. The method capabilities are demonstrated via quasistatic nanotube bending, dynamic crack propagation and dynamic shear banding. [Copyright &y& Elsevier]