Model Inspired by Population Genetics to Study Fragmentation of Brittle Plates

Abstract

We use a model whose rules were inspired by population genetics and language competition, the random capability growth model, to describe the statistical details observed in experiments of fragmentation of brittle plate-like objects, and in particular the existence of (i) composite scaling laws, (ii) small critical exponents τ associated with the power-law fragment-size distribution, and (iii) the typical pattern of cracks. The proposed computer simulations do not require numerical solutions of Newton's equations of motion, nor several additional assumptions normally used in discrete element models. The model is also able to predict some physical aspects which could be tested in new experiments of fragmentation of brittle systems.

Minimal fragmentation of regular polygonal plates

“…Therefore, one could call it a "proximity" crossover. The appearance of composite power laws has been considered one of the most interesting features in the fragmentation of brittle solids and has been reported in experiments involving long thin rods [14] and, more explicitly, in fragmentation of plates [9,10,15] as well as in computer simulations [16].…”

Fragmentation of brittle plates by localized impact

“…Part of the trouble in dealing with fragmentation problems is the variety of possible scenarios. Measurable quantities, notably, the mass distribution of the fragments, are sensitive to the mechanical character of the material, e. g., brittle or ductile [5], to its effective dimensionality (aspect ratios) [6][7][8][9][10], to its intrinsic geometry, e. g., a flat plate [11][12][13] or a spherical shell [14,15], to the magnitude [16,17] and spacio-temporal distribution of the energy input (uniform compression [18], explosion [14,15], projectile impact [18,19] , etc). There is also a difficulty of distinct nature, namely, a considerable gap between geometric statistical models and what one could refer to as first principle fracture theories [14].…”

“…In fragmentation processes in general, and specifically in the situations we are interested in, it is common to have for small fragments, although the distribution as a whole is often better described by a power law with an exponential cutoff F (m) ∼ m −α exp(−m/m 0 ). More intricate behaviors are also observed as, for example, composite power laws [7,[11][12][13].…”