Uniform FEM B-Splines in the analysis of free vibration problem in the axisymmetric nano shells in elastic medium using a non-local elasticity theory

Resumo

The search for mathematical and numerical models to address problems of nanostructures has gained
centrality in the last two decades. This importance stems from the increasing applications of allotropic forms of
carbon as nanotubes and more recently graphene in high-performance composite materials, resonators for high
frequency, biosensors, gas sensors, among others. In fullerene nanotubes and graphene sheets, the natural
frequencies are of the order of THz, a fact that enabled the application of these materials in ultra-frequency
resonators and sensors based on the dispersion of mechanical waves in solid media. In the simulation of the
dynamic behavior of nanostructures, the molecular dynamics (MD) method is frequently used, however, with a
high computational cost. Recently, low-cost computational approaches based on the principles of non-local
continuum mechanics have been used to include neighborhood effects of fundamental importance in the treatment
of nanoscale problems. In this work, the authors propose the use of non-local continuous mechanics in the approach
of free vibrations of axisymmetric nano-shells on Winkler's foundation modeled by first-order kinematic theories
considering shear deformation (FSDT) and by cubic kinematic theories (TSDT). To increase the accuracy of the
relatively high frequencies (above ten percent of the eigenvalues approximated by the numerical model) the
approximation space will be built according to Uniform FEM B-Spline (U- FEM B-Spline) technique, with high
order and high regularity. The results will be analyzed under three aspects: the sensitivity of the first natural
frequency to variations in dimensions and the nanoscale coefficient; the relative error with respect to a target
frequency; the relative error for a pre-stipulated frequency range.