Softening and Stiffening Size Effects in Free Flexural Vibration of Small-scale Cracked Beams

Studying the dynamics of small-scale beams with cracks is crucial for damage detection and maintenance in micro- and nano-electromechanical systems. This paper studies the size-dependent free transverse vibration of miniaturized cracked beams using a local/nonlocal stress-driven gradient elasticity model. A key novelty of this model is its ability to simulate both softening and stiffening behaviors at small scales, unlike other stress-driven nonlocal models in the literature.  The effect of cracks on the vibration of beams is modeled by introducing rotational and translational springs at the cracked cross-sections. The analysis considers a Timoshenko small-scale beam with an arbitrary number of cracks. To solve the problem, the beam is divided into sub-beams at the cracked cross-sections, and the higher-order equations of motion are solved for each sub-beam individually. The frequencies and associated mode shapes are derived by solving the eigenvalue problem constructed by the imposition of the standard variationally consistent and non-standard constitutive boundary and continuity conditions. The accuracy of the model is verified against experimental, molecular dynamics, and analytical results from the literature. The formulated model is further used to investigate the frequencies of intact and cracked beams with different boundary conditions. The effects of shear rigidity, boundary conditions, nonlocal parameters, crack length and location, and multiple cracks on frequencies and mode shapes are systematically analyzed.