Slow dynamic nonlinear elasticity during and after conditioning, a unified theory and a lock-in probe

Of the non-classical nonlinear elastic phenomena, slow dynamics (SD) has received particular attention due to recent modeling efforts and experiments in new systems. SD is characterized by a loss of stiffness after a minor conditioning strain, followed by a slow recovery back towards the original stiffness. It is observed in many imperfectly consolidated granular materials (e.g., rocks and concrete) and unconsolidated systems (e.g., bead packs). Here we posit a simple unified phenomenological model capable of seamlessly describing modulus evolution for SD materials during steady-state conditioning, during conditioning ringdown, and during recovery. It envisions a distribution of breaking and healing bonds, with healing rates governed by the usual spectrum of relaxation times. Well after the end of conditioning, the model recovers the characteristic logarithmic-in-time relaxation. For times during conditioning ringdown, when recovery has initiated but conditioning has not fully ceased, the model predicts deviations from log(t) and a dependence on the ringdown rate. We compare these model predictions with SD measurements on four different systems. To perform the measurements, an ultrasonic digital lock-in (DLI) probe is developed. The advantages of DLI over other techniques to measure SD are a sufficiently high time resolution and an insensitivity to noise from conditioning. We find good agreement between theory and experiment. The model in conjunction with DLI also allows for estimates of the minimum relaxation time. Our measurements indicate that the minimum relaxation time is material dependent.