Fracture of Brittle Solids. IV. Two-Dimensional Distribution Function for Fragment Size in Single Fracture (Theoretical)

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Abstract
Prior work in paper I on the size distribution of the fragments in single fracture of a three-dimensional solid is extended to the two-dimensional case (that of a thin plate). The underlying physical assumptions are the two-dimensional analogs of those made in I. These assumptions yield directly the probability dπ (c,a) of formation of a fragment with perimeter and area in the ranges c to c+dc and a to a+da, respectively, as e−RdR in the general case, with R linear in c and a. The derivation yielding this Poisson form requires no assumption on the shape of a fragment. The number dv (c,a) of fragments with perimeter and area in the ranges c to c+dc and a to a+da, respectively, is evaluated as the product of dπ (c,a) by the a priori number of particles with these values of c and a. The distribution function dv (c,a) meets the necessary physical requirement that the fracture process conserve surface area independently of particle shape. By assuming that all fragments are geometrically similar, one can replace dπ (c,a) and dv (c,a) by forms, π(x)dx and v(x)dx, respectively, which depend only on a mean linear dimension x of a fragment. The moments of the distribution corresponding to the total number and total perimeter of the fragments are divergent; this anomaly is explained as the result of neglect of depletion of Griffith flaws.