Unified interpretation of nonlinear elasticity in granular solids, A

Abstract

Granular solids such as natural rocks and concrete show nonlinear elasticity in response to dynamic deformation with a large strain amplitude. Resonance experiments can measure the nonlinear elasticity using resonance curves which show response amplitudes as a function of driving (oscillation) frequencies. To analyze the nonlinear elasticity observed in resonance experiments, I first simulate a nonlinear oscillation system (i.e., Duffing equation) that involves a cubic term in the equation of state. The simulation illustrates three critical factors, i.e., driving frequency, driving amplitude, and the initial condition of the deformation; these factors control the stable solution that is the sustained amplitude of the Duffing oscillation. I propose a thermodynamics-based model to reproduce the nonlinear resonance features observed in laboratory experiments of rocks and concrete including (a) the log-time recovery of the resonant frequency after the deformation ends (slow dynamics), (b) asymmetric resonance curves in the direction of the driving frequency, (c) the difference between resonance curves when the driving frequency is swept upward and downward, and (d) the presence of a cliff segment to the left of the resonant peak under the condition that the nonlinearity in the oscillation system is strong. This model provides a unified interpretation of nonlinear elasticity. The asymmetry of the resonance curve is caused by softening, which is documented by a reduction of the resonant frequency during the deformation; the cliff segment of the resonance curve is linked to a bifurcation that involves a steep change in the response amplitude when the driving frequency is changed. The simulated Duffing oscillation system shows similar behavior as the resonance simulations. The bifurcation originates from the strong nonlinearity in the oscillation system and is present in both simulations. Extensions of the thermodynamics-based model could include temperature, moisture (pore pressure), and confining pressure. This thesis could contribute to geophysical applications such as monitoring of fracture healing after hydraulic fracturing in unconventional oil and gas reservoirs as well as in enhanced geothermal systems.