An exact stiffness matrix method for nanoscale beams

Conventional continuum theories are inapplicable to nanoscale structures due to their high surface-to-volume ratios and the effects of surface energy and inter-atomic forces. Although atomistic simulations are more realistic and accurate for nanostructures, their use in practical situations is constrained by the high computational cost. Modified continuum methods accounting for the surface energy are therefore considered computationally efficient engineering approximations for nanostructures. The modified continuum theory of Gurtin and Murdoch accounting for the surface energy effects has received considerable attention in the literature. This paper focuses on developing an exact stiffness matrix method for nanoscale beams based on the Gurtin-Murdoch theory. Past research has presented a classical finite element formulation to analyze nanoscale beams using the Galerkin weighted residual method. The proposed approach is based on the analytical solutions to the governing partial differential equations of nanobeams. These governing equations are established by using the Gurtin-Murdoch continuum theory. The general analytical solutions are used to derive the exact stiffness matrix and mass matrix of a beam finite element in closed form. The study examines the static and time-harmonic dynamic response of thin nanoscale beams. Normalized deflections and bending moments under concentrated and distributed loads are obtained for aluminum and silicon thin beams subjected to simply supported, cantilevered and clamped-clamped edges. Our results were compared with the available solutions in the literature, and close agreement was observed. Therefore, the method presented in this study serves as an efficient and accurate scheme to analyze nanobeams under static and dynamic loading compared to the conventional finite element schemes.