Power Law of Crack Length Distribution in the Multiple Damage Process

Multiple fatigue damage, which is characterized by crack initiation and propagation processes, is considered. We proposed two models of multiple damage, which imply random crack initiation and further propagation, with the exponential dependence between their length on the number of loading cycles. Crack initiation is modeled by the stationary Poisson flow with a constant intensity, while crack propagation is characterized by the rate parameter controlling the dependence of crack propagation rate and its length. The first model describes the deterministic case of multiple crack propagation at a fixed value of the above rate parameter, while the second one predicts their propagation by random trajectories according to distribution of the rate parameter. In the former case, crack length distribution is shown to be the Pareto power law with the exponent, which is defined by the ratio of kinetic parameters of initiation and propagation of defects. In the latter case, the rate parameter is uniformly distributed, in accordance with experimental data, so that the power-law distribution of crack length is close to the Pareto distribution. The respective distribution exponent also depends on the ratio of kinetic parameters of multiple damages and tends to drop during damage accumulation to the threshold level (namely, reaches the value of 2). This finding complies with experimental data on multiple damages of various classes of materials, including metals and rock masses. We also substantiated the range of ratios of kinetic parameters of defect initiation and propagation, which ensure the Pareto law of cracks length distribution and can be used to estimate the critical state of cracked bodies.