A Simple Model for Stress Fluctuations in Plasticity with Application to Granular Materials

When granular material is modeled as a continuum, plastic constitutive behavior is often assumed. The use of plasticity amounts to replacing a complicated micromechanical system by its average behavior. Recent experiments have shown that, at least for small-scale systems, stress fluctuations may be of the same order, or even much larger, than average stresses. In this paper a first generation of discrete models for stress fluctuations is discussed. These models consist of many spring-slider elements in parallel. The sliders all obey the same law for frictional resistance, and this resistance varies with the position, but not the velocity, of the slider. The initial positions (and hence the initial frictional resistances) of the sliders are taken to be random. The usual elastoplastic response emerges as the ensemble average over all possible initial positions of the sliders. The stress response resulting from any particular choice of initial conditions exhibits fluctuations similar to those in the experiments. It is shown that the magnitude of fluctuations is governed by two parameters, namely, the system size and the roughness, the latter defined as the ratio of particle contact length to particle size. In numerical simulations, it is observed that the roughness parameter controls the shape of the stress response as a function of applied strain.