Disorder effects in dynamic fragmentation of brittle materials

While fragmentation of brittle materials is a phenomenon that occurs commonly, a consistent theoretical description of this process is not yet available. In this paper, we introduce a one-dimensional micromechanics-based statistical model for fragmentation in a brittle material and analyze the model using two different approaches, namely, an exact numerical method based on the method of characteristics and a dynamic mean-field theory. The mean-field theory is derived by averaging the equations that govern the dynamics of flaws in a self-consistent manner and explicitly accounts for both the statistical nature of flaw distributions and the stress-wave-mediated communication between flaws. It is shown that the mean-field theory provides an accurate description of the damage evolution process that takes place during fragmentation and predicts fragment sizes that are in close quantitative agreement with the numerical calculations.