Vibration-based non-destructive evaluation of dual cracks in bidirectionally functionally graded nanobeams: a nonlocal elasticity–metaheuristic optimisation framework for structural health monitoring

Cracks in functionally graded nanobeams used in MEMS/NEMS devices can significantly reduce stiffness and reliability, and identifying multiple cracks at the nanoscale is challenging because nonlocal effects and bidirectional material gradation must be accounted for. To address this problem, we present a vibration-based non-destructive evaluation (NDE) framework for dual-crack identification in bidirectionally functionally graded (BDFG) nanobeams governed by Eringen’s nonlocal elasticity. Cracks are modelled as physically consistent rotational-spring interfaces, and the forward problem is discretised by the Differential Quadrature Method (DQM) to obtain a generalised eigenproblem. The inverse problem is formulated as a frequency-based optimisation problem, in which crack severity is expressed via a dimensionless parameter. A Modified Cuckoo Optimization Algorithm (MCOA) with adaptive egg-laying radius and mode weighting is benchmarked against the Genetic Algorithm (GA) and the standard COA over several graded and nonlocal scenarios. Results show that MCOA attains faster and more robust convergence, accurately recovering crack locations and dimensionless crack-intensity parameters from a small set of natural frequencies. The main contributions are a physically consistent crack interface with a dimensionless damage index, an explicit nonlocal-to-eigenproblem mapping within the DQM framework, and an adaptive metaheuristic strategy that enhances convergence for vibration-based SHM of graded MEMS/NEMS components.