Defect Population Statistics near and Far from a Critical Event

We consider the defect size distributions at different stages of damage evolution, from the initial stage of defect nucleation and accumulation, through the intermediate stage of defect propagation and into the final stage leading to the appearance of the major crack. Critical events considered in this context are related to (a) the formation of microcracks exceeding the structural threshold associated with the grain size, and (b) the formation of a major crack that corresponds to the attainment of the maximum load. The significance of these two critical events is that they are often used to define the boundary between microand macro-mechanical analyses, and used in order to establish scale and size effects on material strength. We consider the statistical distribution functions which are in widespread use for the description of cumulative defect size distributions, namely, the exponential, power law and the exponential-power law (also referred to as the Rosin-Rammler or Weibull) distributions. We note that the defect population statistics at different stages of evolution are best described by different statistical functions. We discuss of the relationship between defect size distributions and the statistics of material strength. We point out that Weibull strength statistics implies power law defect size distribution, and note the direct correspondence between the power law defect size distribution exponent and the Weibull modulus in the statistics of strength, giving the relationship between these parameters. We propose to use the transition between different distribution statistics as an indicator of the approach of a critical event. The effect of the structural size barrier associated with the grain size is to retard temporarily the crack growth beyond this critical size. This results in increasingly steep distribution curves, reflected in the increase of the Rosin-Rammler exponent parameter. Once crack growth proceeds beyond the structural barrier, a power law ‘tail’ of the crack size distribution appears. Microstructurally large defects evolve from the exponential towards power law cumulative size distributions. The appearance of the major crack is preceded by the decrease in the exponent of the power law ‘tail’. Observations confirming these conclusions have been reported in materials science and seismology.