Nonlinear Instability and Bifurcation Analysis of Breathing-Cracked Cantilever Beams Using a Polynomial Stiffness Representation

Breathing cracks in beam-like structures introduce stiffness degradation and displacementdependent nonlinearity owing to periodic opening and closure, leading to amplitude-dependent resonance shifts, jump phenomena, and bistability. This study presents a reduced-order nonlinear vibration framework for the dominant instability characterization of breathingcracked beams, in which crack breathing, geometric nonlinearity, and crack-induced damping are represented by a unified smooth polynomial stiffness formulation. Unlike discontinuous piecewise representations that complicate perturbation-based analyses, the proposed formulation provides a continuous approximation of the stiffness variation while retaining the essential nonlinear features of breathing-crack dynamics. The governing system is analyzed using the Method of Multiple Scales to characterize the nonlinear amplitude–frequency response, saddle-node bifurcation, and associated jump behaviour. Comparisons with bilinear representations and experimental observations on aluminium and mild steel cantilever beams showed that the formulation captured key nonlinear characteristics, including resonance shifts, multivalued response behaviour, and instability boundaries. Parametric analysis further demonstrates the coupled influence of stiffness degradation and crack-induced damping on the nonlinear instability regions and bistability. This study establishes a physically consistent and analytically tractable reduced-order framework for the nonlinear instability characterization of breathing-cracked beams.