On the modelling of carbon nano tubes as generalized continua

A 1D continuum endowed with internal structure, previously introduced by the authors in order to describe some nonlinear behaviours of Carbon Nano Tubes (CNTs), is extended and generalised by giving a procedure for constructing the constitutive functions. Starting from the reference configuration of a Carbon Nano Sheet (CNS), a Representative Elementary Volume (REV) is chosen. The deformation measures of the REV are identified with the change of the length of the Carbon-Carbon (C-C) bonds and the angle variation between each pair of adjacent bonds. The strain energy density of the REV is given as a standard function of the microscopic strain measures. A relationship between the micro and the continuum strain measures is then put down, this leading to an expression of the strain energy density of the REV in terms of the latter strains. Making the derivative of this energy with respect to its argument the constitutive functions for the 1D continuum are obtained. The geometric and mechanical properties of a graphene nano sheet are used to construct its equivalent continuum and some numerical comparisons are discussed. Although the procedure is set up for a CNS, its extension to Carbon Nano Tubes (CNTs) involves only simple geometric computations.