Simple models for intermittent deformation and slip avalanches: from crystals to granular materials and earthquakes

Slowly sheared solid or densely packed granular materials often deform in an intermittent way with slip avalanches. The distribution of sizes follows often a power law over a broad range of sizes. In these cases, universal (i.e. detail-independent) scaling behavior governs the statistics of the slip-avalanches. Under some conditions, there are also "characteristic" statistics associated with enhanced occurrence of system-size events, and long-term mode switching between power law and characteristic behavior. These dynamic regimes can be understood with basic micromechanical model for deformation of solids with only two tuning parameter: weakening and dissipation of elastic stress transfer. For granular materials the packing fraction plays the role of the dissipation parameter and it sets the size of the largest slip avalanche. The model can reproduce observed stress-strain curves, power spectra of acoustic emissions, statistics of slip avalanches, and geometrical properties of slip, with a continuous phase transition from brittle to ductile behavior. Exact universal predictions for the power law exponents of the avalanche size distributions, durations, power spectra of acoustic emissions, and scaling functions are extracted using an analytical mean field theory and renormalization group tools. For granular materials a dynamic phase diagram with solid-like behavior and large slip avalanches at large packing fractions, and fluid-like behavior at lower packing fractions is obtained. The results agree with recent experimental observations and simulations of the statistics of dislocation dynamics in sheared crystals such as ice [1], slip avalanches in sheared granular materials [2], and avalanches in magnetic and fault systems [3,4]. [1] K. A. Dahmen, Y. Ben-Zion, and J.T. Uhl, "A micromechanical model for deformation in solids with universal predictions for stress strain curves and slip avalanches", Physical Review Letters 102, 175501/1-4 (2009). [2] K. A. Dahmen, Y. Ben-Zion, and J.T. Uhl, "A simple analytic theory for the statistics of avalanches in sheared granular materials" Nature Physics 7, 554-557 (2011) [3] K.A. Dahmen and Y. Ben-Zion, "The physics of jerky motion in slowly driven magnetic and earthquake fault systems", Encyclopedia of Complexity and Systems Science Springer. (Eds.: M.C. Marchetti and R. Meyers), Vol. 5, 5021-5037 Springer, (2009). [4] Ben-Zion, Y., K. A. Dahmen and J. T. Uhl, A unifying phase diagram for the dynamics of sheared solids and granular materials , Pure Appl. Geophys., DOI: 10.1007/s00024-011-0273-7, 2011.