Fragment Size Distributions in Dynamic Fragmentation

Up to this point we have addressed primarily the flaw structure and energy concepts in stress-wave loaded solids governing the creation of new fracture surface area (or the mean fragment size) in catastrophic fragmentation events. It remains to consider a concept which is frequently the end concern in impulsive fracture applications, namely, the distribution in sizes of the particles produced in the dynamic fragmentation event. 

Theoretical efforts a step beyond simply fitting standard statistical curves to fragment size distribution data have involved applications of geometric statistical concepts, i.e., the random partitioning of lines, areas, or volumes into the most probable distribution of sizes. The one-dimensional problem is reasonably straightforward and has been discussed by numerous authors 

Consequently, consider an infinite one-dimensional line or rod along which fractures occur randomly with an average frequency of Nq per unit length as illustrated in Fig. 8.19. Randomly distributed points on an infinite line obey Poisson statistics and the probability of finding n fractures in a length, /, is given by 

To determine the probability distribution in fragment lengths, first determine the probability of finding no fractures in length, /, 

the probability of finding one fracture whose length is within a range of I to I + dl is given by 

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