Random Vibration and Dynamic Analysis of Fractured Nanobeams Using a Stress-Driven Bi-Helmholtz Nonlocal Finite Element Framework

This study examines the dynamic behaviour of nanobeams with multiple transverse cracks under random excitation using a stress-driven Bi-Helmholtz nonlocal elasticity framework. In contrast to classical formulations based on Eringen-type nonlocal elasticity and conventional strain-driven models, the proposed approach employs a higher-order differential stress operator that captures both short- and long-range interactions while avoiding boundary-condition paradoxes. The classical Euler–Bernoulli beam theory is extended by incorporating nonlocal stress gradients to represent size-dependent stiffness softening and hardening effects. Multiple cracks are modeled using equivalent rotational springs introducing slope discontinuities. A nonlocal finite element formulation is developed to ensure mathematical well-posedness and accurate spatial interaction modelling. The stochastic response under Gaussian white-noise loading is evaluated using a linear modal-based spectral approach to compute displacement variance and stress measures. Parametric studies demonstrate that the proposed framework provides physically consistent and boundary-compatible predictions, particularly for nanobeams with multiple cracks subjected to random excitation.