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Abstract

The current work presents a thorough nonlinear framework, for initial crack detection and characterization in an Euler–Bernoulli beam, with breathing cracks, using a hybrid of the Mathieu–Duffing formalism and Recurrence Quantification Analysis (RQA). Taking into consideration the periodic modulation of stiffness in the breathing crack mechanism, a model, with such dynamics, is able to provide complicated dynamical signatures that often escape analysis by linear systems. The results show a transition where with growing relative crack depth, η, the structure transits from being a damped dissipative system to being self-sustained, and then weakly chaotic. This transition in dynamics occurs due to an energetic interaction whereby the breathing phenomenon counters the damping inherent in the structure, with its appearance indicated by the breakdown of regularity in the recurrence plots. Robust diagnostic capabilities were also developed with a statistical analysis performed on the responsiveness of several RQA measures to damage. These results show that determinism (DET), entropy (ENTR), and TREND were all highly responsive to changes in the topology of the phase space, with damage detected at a much earlier stage than through traditional spectral methods. Statistical tests (ANOVA) showed DET, ENTR, and TREND to be of a very high level of significance (p < 0.0001) for detecting the initial stage of structural degradation (η ≤ 0.2), with the ROC curve for DET giving a value of 0.999 for the Area Under the Curve (AUC) and thus classifying this measure as one of the best possible. TREND in fact showed perfect correlation (ρ = 1.0) with the level of damage and can be considered a superb quantitative measure for progression of the damage. A sophisticated multi-scale tool for structural health monitoring has been thus presented, which integrates contact mechanics and the topology of the signal, with the capability of detecting the initiation of the system instability.