Modeling the Nonlinear Hysteretic Response of Distributed Damage in a 1D Resonance Experiment

The resonant response of alloys and geo-materials to a harmonic excitation reveals the existence of hysteretic mechanisms but provides little information, if any, about their nature. A dynamic version of the classical elasto-dynamic technique, on the other hand, seems to be able to discriminate a variety of nonlinear responses, offering a potentially viable avenue to investigate these mechanisms. To this date, however, even disregarding the relative complexity of its setup, lack of theoretical models supporting the interpretation of these results appears to hinder further application of this technique. In this work, the nonlinear hysteretic response of a 1D bar subjected to a longitudinal excitation is investigated theoretically with the aim of advancing the use of resonance-based techniques for characterization purposes. Three types of distributed damage are considered: dislocations, micro-cracks with and without adhesion, and defects leading to hysteretic quadratic nonlinearity at the macroscopic level. Each type of damage is represented by a constitutive relation that captures the essence of the mechanism responsible for the hysteresis of the material. Arbitrary distributions of damage along the bar are allowed. Spectral features characteristic of these distinct forms of damage are predicted, and their use for characterization purposes is discussed.