Universal Dynamic Fragmentation in $D$ Dimensions

A generic model is introduced for brittle fragmentation in $D$ dimensions, and this model is shown to lead to a fragment-size distribution with two distinct components. In the small fragment-size limit a scale-invariant size distribution results from a crack branching-merging process. At larger sizes the distribution becomes exponential as a result of a Poisson process, which introduces a large-scale cutoff. Numerical simulations are used to demonstrate the validity of the distribution for $D=2$. Data from laboratory-scale experiments and large-scale quarry blastings of granitic gneiss confirm its validity for $D=3$. In the experiments the nonzero grain size of rock causes deviation from the ideal model distribution in the small-size limit. The size of the cutoff seems to diverge at the minimum energy sufficient for fragmentation to occur, but the scaling exponent is not universal.

Figure 1

(a) Snapshot of a small brittle membrane at an early stage of fragmentation (broken bonds are removed so they appear white). (b) The final fragment-size distribution averaged over $\sim 30$ configurations for $\sigma =1.2$. The curve is a fit to the data by Eq. (5): $1/{\gamma }_v={\lambda }^D=22.5$ (in lattice units), and $M_f=0.007$. (c) Fragment-size distribution in the case of minimal external strain ($\sigma =1.0$) required for fragmentation. The line is a fit by Eq. (4).

Figure 2

Experimental fragment-size distributions from quarry blastings (a) and laboratory experiments on cylinders (b). The quarry data are fitted by $d{n}_2(v)$ (broken line), and by $\widehat{d{n}_2}(v)$ (full line), with $1/{\gamma }_v,\lambda \to \infty$, and $c=14.0$. The laboratory data are fitted likewise, except that $c=2.6$.

Figure 3

(a) The large fragment-size part of the data from the laboratory experiments fitted by $dn(v)$, with ${\gamma }_v$ as an adjustable parameter. (b) Scaling of the correlation length $\sqrt{1/{\gamma }_v}$ near the transition point $E=E_c$: $\sqrt{E}-\sqrt{E_c}$ is plotted as a function of $(1/{\gamma }_v{)}^{1/3}$, and fitted by a power law with the exponent $-0.9$. The resulting correlation-length exponent is $\nu \approx 1.1$.