On the Continuum Modelling of Carbon Nano Tubes

A one-dimensional continuum endowed with internal structure, previously introduced by the authors, is extended and generalized 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 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 established, 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 arguments the constitutive functions for the one-dimensional continuum are derived. The geometric and mechanical properties of a graphene nano sheet are used to con- struct its equivalent continuum and some numerical comparisons discussed. Although the procedure is established for a CNS, its extension to carbon nano tubes (CNTs) involves only simple geometric computations.