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A parametric variational principle for van der Waals force simulation between any two adjacent nonbonded atoms and the corresponding improved quadratic programming method for numerical simulation of mechanical behaviors of carbon nanotubes are developed. Carbon nanotubes are modeled and computed based on molecular structural mechanics model. van der Waals force is simulated by the network of bars (called bar network) with a special nonlinear mechanical constitutive law (called generalized parametric constitutive law) in the finite element analysis. Compared with conventional numerical methods, the proposed method does not depend on displacement and stress iteration, but on the base exchanges in the solution of a standard quadratic programming problem. Thus, the model and method developed present very good convergence behavior in computation and provide accurate predictions of the mechanical behaviors and displacement distributions in the nanotubes. Numerical results demonstrate the validity and the efficiency of the proposed method.

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Based on the higher order Cauchy-Born rule, a nanoscale finite deformation continuum theory, which links interatomic potentials and atomic microstructure of carbon nanotubes to a constitutive model, is presented for analysis of the mechanics of carbon nanotubes. By using of Tersoff-Brenner potential with two sets of parameters, the energy and Young’s modulus of graphite sheet and single-walled carbon nanotubes are studied based on the theory presented. The findings are in good agreement with the existing experimental and theoretical results.