Impact of fracture and nonlocal elasticity on the free vibration behavior of nanobeams: A bi-Helmholtz finite element study

This study presents a nonlocal bi-Helmholtz finite element formulation for the free vibration analysis of fractured nanobeams. Based on Eringen’s nonlocal elasticity theory with bi-Helmholtz kernels, the governing equations of Euler–Bernoulli beams are derived. Fractures are modeled by dividing the beam into segments connected through rotational springs, capturing discontinuities in rotational displacement proportional to the transmitted bending moment. A higher-order four-node finite element method is employed to compute the stiffness and mass matrices using the weighted residual approach, enabling accurate evaluation of natural frequencies under elastic boundary conditions. The proposed formulation allows calculation of nonlocal nondimensional frequency parameters for clamped nanobeams with axial springs. A comprehensive parametric study is conducted to evaluate the effects of fracture location, fracture severity, and the nonlocal length-scale parameter on the vibration characteristics. The results demonstrate that the methodology effectively captures the combined effects of fractures and nonlocal elasticity, providing a reliable framework for structural health monitoring and design of nanoscale beam-like structures.