Vibrations of Lattice Nanobeams in Strain Gradient Elasticity

This study investigates vibrations of nanoscale structures using a novel finite element method based on {the theory of the strain gradient elasticity}. {The governing equations are derived from the Euler–Bernoulli beam theory enhanced with Eringen’s strain gradient formulation to capture the axial and bending behavior with nonlocal effects. The weak form enables efficient implementation of finite element,} introducing additional degrees of freedom and using Hermite shape functions for both axial and bending vibrations. Stiffness and mass matrices explicitly include nonlocality and are transformed into global coordinates. Numerical validation shows excellent agreement with analytical solutions under various boundary conditions. The comparative analysis of L-shaped frames highlights the advantages over classical FEM. The influence of nonlocal and geometric parameters on lattice structure frequencies is also examined. The method offers an efficient and accurate tool for nanoscale beam analysis and optimization in NEMS applications.