Conforming and nonconforming FEMs for the free vibration problem of a CNT microbeam

Abstract

This work investigates the eigenfrequencies of a carbon nanotube (CNT), simulated as a gradient Rayleigh beam, to explore the effect of the macro-inertia dynamical terms in the response. Analytical and numerical methods are employed to deduce credible results for three types of boundary conditions (BCs), i.e. a cantilever, a simply supported and a clamped–clamped beam, respectively. In particular, the analytical eigenfrequencies are calculated by means of modal analysis and respectively the numerical ones by both the conforming C2-continuous finite element method (C2CFEM) and interior penalty discontinuous Galerkin finite element methods (IPDGFEMs), for the first time. The above numerical methods are used, since the coefficient of the sixth-order differential term (as a perturbation term) takes very small values, and that triggers stability and numerical issues. A thorough investigation concerned with each method’s suitability is conducted. The conclusions drawn are very encouraging for the study of higher-dimensional problems with a complicated domain.