Free vibration of multi-cracked functionally graded nanobeams using nonlocal elasticity theory

Abstract This paper presents an analytical study on the free transverse vibration behavior of functionally graded (FG) nanobeams containing multiple cracks, modeled using rotational springs. The nanobeam’s material properties are assumed to vary through the thickness according to a power-law gradation, while the size-dependent effects are captured using Eringen’s nonlocal elasticity theory. A segment-wise closed-form formulation is developed by enforcing nonlocal continuity and jump conditions at cracked sections, leading to an explicit characteristic determinant for natural frequencies. Unlike many existing studies limited to single-crack cases or numerical schemes, the proposed approach efficiently handles an arbitrary number of cracks and provides analytical mode shapes. Parametric investigations highlight the coupled effects of crack parameters, material gradation, and nonlocality on vibration characteristics, demonstrating the effectiveness of the model for nano-scale structural analysis and damage assessment.