H-FEM AND FSDT TO APPROACH THE FREE AND FORCED VIBRATION OF A GRAPHENE SHEET USING A NON-LOCAL CONTINUUM MECHANICS THEORY

Resumo

The allotropic forms of carbon as nano tubes and more recently graphene sheet has gained significant
notoriety in the last two decades where its use has been spread in the most diverse areas such as composite
materials, electronics, medicine, fine chemistry, among others. Within this context, the mechanical behavior of
wave propagation at the nanoscale level has received special attention for its relevance in the application of
transport problems of molecules, sensors for detecting gas atoms, resonators for high frequencies, among
others. Recently, classical approaches to nanoscale problems with atomic and hybrid models, more accurate but
with high computational cost, have left some room for approaches that use the principles of continuum mechanics
with the classic (local) and non-local versions. The results observed in the literature on free vibration problems,
obtained with non-local continuous mechanics, have been closer to the results of molecular dynamics than those
obtained with classical or local continuous mechanics. In this work, the authors propose an approach using the
non-local continuum mechanics for the free and forced vibrations problems in graphene sheets using the
approximation spaces obtained with Hermite finite elements (H-FEM) with regularity

Ω, 1,2 together
with the first order plate model “First Shear Deformation Theory” (FSDT). This approach intends to investigate
some aspects of the analyzed problem to improve the proximity of the response to that obtained using molecular
dynamics: provide the regularity requirements of the non-local model equilibrium equations; improve accuracy in
natural modes and frequencies using highly regular approximation spaces; improve the accuracy of the response
of natural modes and frequencies by incorporating the rotational inertia introduced by the kinematic FSDT model.