Flexural mode NRUS: Theory

Nonlinear resonant ultrasound spectroscopy (NRUS) has been widely utilized for detecting substructural damage and characterizing strain-dependent properties of materials. While the longitudinal resonance of well-prepared specimens has been predominantly used, flexural and torsional resonances have also been explored recently. The objective of this paper is to provide a formal analysis of the flexural vibration of a slender beam with nonlinear hysteresis, serving as a model for a flexural mode NRUS experiment. The Davidenkov hysteresis function is employed to describe hysteretic motions, enabling the explanation of the experimentally observed nonlinear dependence of resonance frequency shifts on strain. Classical nonlinearities are also considered to obtain more general formulae for the resonance frequency shift and the increase in damping capacity with strain, which can be used to experimentally determine the hysteresis parameters. Additionally, it is shown that the classical quadratic and cubic nonlinearity parameters can be derived from NRUS experiments. The origins of strain-dependent behaviors of hysteretic nonlinear materials are clarified through the derivation of quasi-static backbone curves. Finally, the processing of transient vibration signals from impact-based nonlinear resonance experimental techniques is discussed. It is emphasized that calibrating the measured transducer signals into physical quantities (strain or acceleration signals) is crucial for obtaining absolute hysteresis parameters. It is because the signals from a nonlinear system are not scale-invariant, unlike those from a linear system.