Investigation of free vibration of multi-cracked nonlocal nanobeams

Abstract

This paper investigates the free transverse vibration characteristics of nanobeams containing multiple cracks using the Euler–Bernoulli beam framework augmented by Eringen’s nonlocal elasticity theory. A size-dependent model is formulated by introducing discontinuities in slope and transverse displacement at cracked sections, which are represented by rotational springs. The nonlocal elasticity is incorporated via an integral constitutive relation, leading to an enriched governing differential equation. Analytical expressions for natural frequencies and mode shapes are derived by enforcing kinematic and continuity conditions across crack locations and beam boundaries. The analysis reveals that the presence of multiple cracks significantly alters dynamic behavior, notably lowering natural frequencies compared to intact and local-behavior predictions. The effect is further amplified by increasing nonlocal parameter and crack severity. The proposed method is applicable for arbitrary crack configurations and boundary conditions, providing a versatile tool for assessing vibrational responses in nanobeams. Results are validated against existing solutions and serve as benchmarks for future theoretical and computational studies.