Slow elastic dynamics in a resonant bar of rock

Recent resonant bar experiments on Berea sandstone show that nonlinear excitation of the sample excites a slow dynamics with a time scale many orders of magnitude longer than the excitation period, 2 pi/omega. That is, a nonlinear resonant frequency decays to the linear resonant frequency long after the high amplitude drive has been turned off. We postulate a phenomenological theory of slow nonlinear dynamics in the context of a resonant bar experiment. The normalized elastic modulus of the resonant bar is allowed to be nonlinear and time dependent. The nonlinear terms are derived from a model of elasticity in rocks that includes anharmonic and hysteretic contributions. We use this theory to explain the experimental results. We find an explanation for the slow relaxation of the experimental resonant frequency using an anharmonic contribution to the modulus that responds instantaneously to a disturbance, and a contribution derived from elastic hysteresis that displays slow dynamics. We suggest an acoustic NMR-type experiment to explore slow nonlinear dynamics.