Discontinuous Galerkin FEMs for a gradient beam in static carbon nanotube applications

Abstract

The engineering applications of the innovative materials, such as carbon nanotubes (CNTs) and graphene sheets, constitute a developing branch of the modern science. CNTs, in particular, exhibit extraordinarily mechanical properties capable of making them a reliable material for the above applications. The simulation of the mechanical response of a CNT is effectively conducted by a generalized beam model. This work investigates the response of a CNT, subjected to a static loading, by means of a gradient beam model. Given that the existence of a boundary layer is demonstrated by the analytical solutions of the aforementioned problem, the interior penalty discontinuous Galerkin finite element methods (IPDGFEMs) are crucial to its solution. The hp‐version IPDGFEMs are developed in this study for the solution of a static gradient elastic beam in bending, derived from two different equilibrium formulations, for the first time. An a priori error analysis is also performed for the above method, and numerical simulations are then carried out. A comparison is finally drawn between the exact deflection and those of the IPDGFEM and the conforming C2‐continuous finite element method (C2CFEM) for a number of discretizations and each length scale that is investigated. The deduced results highlight the suitability, the efficiency, and the accuracy of the investigated models over the already existing ones, and they have considerable significance for the engineering design in the range of micro and nanodimensions.