An exact stiffness matrix method for nanoscale beams

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Conventional continuum theories are inapplicable to nanoscale structures due to their high surface-to-volume ratios and the effects of inter-atomic forces. Although atomistic simulations are more accurate for nanostructures, their use in practical situations is often constrained by the high computational cost. Therefore, simplified continuum methods accounting for the surface energy are 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 for nanoscale beams based on the Gurtin-Murdoch theory. The proposed approach is based on the analytical solutions to the governing partial differential equations established in the Gurtin-Murdoch continuum theory for defining nanoscale beams. For the first time, an exact stiffness matrix and a mass matrix were derived based on the general analytical solution of a nanobeam. The study examines the response of thin nanoscale beams under both static and dynamic loading conditions. Normalized displacements are obtained from the finite element analysis for thin nanoscale beams under point and distributed loading for different boundary conditions. Natural frequencies and mode shapes are also computed. The final results agree with the solutions in the literature, demonstrating the accuracy and efficiency of the presented method.