Further Features of the Mott Statistical Theory

Abstract

In this chapter some of the theoretical issues introduced by Mott are expanded on and discussed in further detail. Additionally, several issues inspired by Mott’s ideas are introduced. First, the statistical development of Mott uses facets of classical survival (or hazard) statistics. Survival statistics methods are pursued here in an alternative way to directly develop statistical fragment size distributions. Next, the statistical treatment of the interaction of multiple fractures was not treated in the original work of Mott (1947). This part of the theory was applied to both the one- and two-dimensional fragmentation problem and is considered in further detail in this chapter. Further, the analytic determination of statistical fragment size distributions from the statistical theory has been performed in the previous chapter for specific Mott fracture activation and growth laws. This development involves analytic details, which are not readily transparent. Here, a more general development of the size distribution relations is developed, which provides a clearer outline of the procedures. After completion of fracture, continued expansion of the fragments also results in a statistical distribution of opening cracks and the associated crack-opening displacement. An analysis is presented which provides an analytic statistical description of the crack-opening displacement for the one-dimensional expanding Mott cylinder. The solution provided earlier for Mott fracture including fracture resistance was restricted to a linear decreasing fracture resistance with crack opening displacement. Here the solution is extended to a power law crack-opening resistance which provides for discussion of fracture resistance ranging from brittle to ductile in character. Lastly, the Mott γ parameter integral to the Mott fragment size prediction is examined further.