A first order FEM-based formulation for the analysis of molecular structures with bonded interactions

In this paper we present a formulation, denoted as the Molecular Element Method, that allows, given a certain force field, the possibility of performing a first order analysis as is common in structural mechanics. The stiffness matrices have been obtained in an exact way, without the need to resort to any assumptions other than those used to define the force field. Obtaining these matrices analytically allows the exploitation of algorithms and techniques typical of finite elements, being able to perform both static and vibration mode analysis independently of the molecular geometry. In this aspect, the formulation has been validated by studying the frequencies and vibration modes of several molecules, comparing with both theoretical and experimental results. Values of the modulus of elasticity of both graphene sheets and carbon nanotubes have also been obtained in accordance with the values present in the literature. Finally, the formulation shows its potential allowing to obtain the flexural wave dispersion in nanotubes compared to results using Tersoff-Brenner potentials and non-local elasticity models.