| A method for the systematic reduction of degrees of freedom in the static analysis of lattice systems of reduced dimensionality is presented. The traditional methods of crystal elasticity, valid for space-filling crystals, are extended to deal with crystalline films in three dimensions, and chains in two or three dimensions. A generalization of the Cauchy-Born rule, the exponential Cauchy-Born rule, is key to these developments. This methodology allows us to formulate hyperelastic constitutive relations for continua of reduced dimensionality (lines, surfaces) exclusively in terms of the underlying lattice model, and written in closed-form, i.e. they do not involve local or constrained atomistic calculations. These models are shown to very accurately mimic the parent discrete model in the full nonlinear regime. This theory is applied to the mechanics of carbon nanotubes. The continuum model is discretized with finite elements, providing a computationally advantageous alternative to atomistic calculations. Large multi-walled nanotubes containing millions of atoms are efficiently handled in this manner, and unusual experimental observations are reproduced. The symmetry of several deformation modes can be treated analytically, and reduced two and one-dimensional models which encapsulate interesting mechanics of nanotubes are formulated. The linear response of nanotubes is characterized by elastic moduli which are written explicitly in terms of the interatomic potential. |
