A statistical theory of fragmentation processes

Abstract

The goal of the work reviewed here is a theory of material behavior accounting for the average deformation that results from the opening, shear, growth and coalescence of an ensemble of microcracks. A concomitant is the calculation of permeability from crack structure. The first part of this paper summarizes previous developments. In particular, the initial work on this problem made use of a linear Liouville equation to characterize the change in crack distribution resulting from crack growth and coalescence. Straightforward analytic solutions to this equation were possible because the mean free path of cracks was assumed constant. Though this assumption is useful for the early stages of crack growth, increasing crack size reduces the mean free path in the later stages of fragmentation. This problem is addressed in the second part of this paper. The governing (nonlinear) Liouville equation is derived therein, and it is shown that it can be reduced to an ordinary differential equation of third order involving only a single free parameter, denoted by β. This equation has now been solved numerically to determine the limiting value of the mean free parth as a function of β, and the results are presented in graphical form. In the third part of this paper prospects for further developments are briefly discussed.