A basic micromechanical model for deformation of solids with only one tuning parameter (weakening epsilon) is introduced. The model can reproduce observed stress-strain curves, acoustic emissions and related power spectra, event statistics, and geometrical properties of slip, with a continuous phase transition from brittle to ductile behavior. Exact universal predictions are extracted using mean field theory and renormalization group tools. The results agree with recent experimental observations and simulations of related models for dislocation dynamics, material damage, and earthquake statistics.

Slowly sheared granular materials at high packing fractions deform via slip avalanches with a broad range of sizes. Conventional continuum descriptions 1 are not expected to apply to such highly inhomogeneous, intermittent deformations. Here, we show that it is possible to analytically compute the dynamics using a simple model that is inherently discrete. This model predicts quantities such as the avalanche size distribution, power spectra and temporal avalanche profiles as functions of the grain number fraction v and the frictional weakening ε. A dynamical phase diagram emerges with quasi-static avalanches at high number fractions, and more regular, fluidlike flow at lower number fractions. The predictions agree with experiments and simulations for different granular materials, motivate future experiments and provide a fresh approach to data analysis. The simplicity of the model reveals quantitative connections to plasticity and earthquake statistics. Slip avalanches in slowly sheared granular materials, such as sand and powders, are important for many industrial, engineering, and geophysical processes. Understanding and predicting the dependence on packing fraction, shear rate, and frictional properties are the questions addressed here. In contrast to traditional models based on continuum mechanics 1 or on simulations of each individual grain, we use an analytical, discrete, coarse-grained approach. We analytically derive predictions for the statistical properties of slip avalanches at slow shear rates, where grain inertia is negligible (the 'quasi-static' regime). (We do not consider the regimes where grain inertia is non-negligible, such as granular gases 2,3). Previous studies focused on jamming 4,5 , force chains 3,5-9 , stress drops during avalanches (refs 10-12; P. Yu, T. Shannon, B. Utter & P. R. Behringer, unpublished data and R. P. Behringer, private communication), and shear localization in shear bands 1,2,13,14. Here, we consider a simple model for slip avalanche statistics. We model the simplified system on a coarse-grained scale (larger than the grain diameter) with a lattice of sites that can either stick or slip under shear. The lattice is either two-or three-dimensional. It has linear extent L and N = L d sites, where d is the dimension of the lattice. N occ sites are occupied by grains and N − N occ sites are empty (voids). The 'grain number fraction', v ≡ N occ /N is proportional to the rescaled packing fraction / max , with v = 1 for the densest possible packing = max. Initially all sites are stuck at random initial stresses. We apply a slow shear strain rate by moving one boundary of the lattice at a very slow parallel velocity V (see Fig. 1). This leads to a slow increase of shear stress at each lattice point. A site i slips in the shear direction when its local shear stress τ i exceeds a random static 'frictional' failure stress τ s,i (i = 1,...,N). (The shape of the narrow distribution of the τ s,i does not affect the behaviour on long length scales 15 .) A failing

We show that slowly sheared metallic nanocrystals deform via discrete strain bursts (slips), whose size distributions follow power laws with stress-dependent cutoffs. We show for the first time that plasticity reflects tuned criticality, by collapsing the stress-dependent slip-size distributions onto a predicted scaling function. Both power-law exponents and scaling function agree with mean-field theory predictions. Our study of 7 materials and 2 crystal structures, at various deformation rates, stresses, and crystal sizes down to 75 nm, attests to the universal characteristics of plasticity.