LAUSR

Resonant bar experiments have revealed that dynamic deformation induces nonlinearity in rocks. These experiments produce resonance curves that represent the response amplitude as a function of the driving frequency. We propose a model to reproduce the resonance curves with observed features that include (a) the log-time recovery of the resonant frequency after the deformation ends (slow dynamics), (b) the asymmetry in the direction of the driving frequency, (c) the di↵erence between resonance curves with the driving frequency that is swept upward and downward, and (d) the presence of a “cli↵” segment to the left of the resonant peak under the condition of strong nonlinearity. The model is based on a feedback cycle where the e↵ect of softening (nonlinearity) feeds back to the deformation. This model provides a unified interpretation of both the nonlinearity and slow dynamics in resonance experiments. We further show that the asymmetry of the resonance curve is caused by the softening which is documented by the decrease of the resonant frequency during the deformation; the cli↵ segment of the resonance curve is linked to a bifurcation that involves a steep change of the response amplitude when the driving frequency is changed. With weak nonlinearity, the di↵erence between the upwardand downward-sweeping curves depends on slow dynamics; a su ciently slow frequency sweep eliminates this up-down di↵erence. With strong nonlinearity, the up-down di↵erence results from both the slow dynamics and bifurcation; however, the presence of the bifurcation maintains the respective part of the up-down di↵erence, regardless of the sweep rate.