Universality of fragment shapes

The shape of fragments generated by the breakup of solids is central to a wide variety of problems ranging from the geomorphic evolution of boulders to the accumulation of space debris orbiting Earth. Although the statistics of the mass of fragments has been found to show a universal scaling behavior, the comprehensive characterization of fragment shapes still remained a fundamental challenge. We performed a thorough experimental study of the problem fragmenting various types of materials by slowly proceeding weathering and by rapid breakup due to explosion and hammering. We demonstrate that the shape of fragments obeys an astonishing universality having the same generic evolution with the fragment size irrespective of materials details and loading conditions. There exists a cutoff size below which fragments have an isotropic shape, however, as the size increases an exponential convergence is obtained to a unique elongated form. We show that a discrete stochastic model of fragmentation reproduces both the size and shape of fragments tuning only a single parameter which strengthens the general validity of the scaling laws. The dependence of the probability of the crack plan orientation on the linear extension of fragments proved to be essential for the shape selection mechanism.

: Definition of equilibrium points, illustrated on a cube. (a) Stable/unstable equilibrium points are the local minima/maxima of the distance function R measured from the center of gravity G. (b) On a cube, stable equilibrium points (red) are located at the middle of the 6 faces (n S = 6), while unstable equilibrium points (green) are located at the 8 vertices (n S = 8).
Supplementary information on the probability distribution of the number of stable and unstable equilibrium points Although, the mass distribution of fragments is well understood, to the best of our knowledge, nothing has been published about the distributions of the number of stable n S and unstable n U equilibrium points of fragments. One can, however, make some simple observations based on earlier mathematical results [S1,S2]: E(n U ) ≤ 8 both densities p(n S ) and p(n U ) decay to zero at n S , n U → ∞.
Here E denotes the expected value of the stochastic variable. While there are infinitely many distributions fitting the above observations, the lognormal seems to be the first obvious choice. We also mention that there are partial results for the distribution of n S and n U on mature (well abraded) pebble populations: a Markov model suggests geometric distributions [S3]. However, the two cases (fragments and abraded particles) are fundamentally different. In Figure 5 of the manuscript we used the lognormal distribution only as a heuristic approximation of the data.

Supplementary information for Model-Rect
The sequential breakup model Model-Rect can be solved analytically for the mass distribution of fragments [S4]. Power law distribution is obtained with an exponent τ depending on the value of the breaking probability p and on the number of fragments n created by the breakup of a single piece τ = ln n ln (1/p) .
In our model n = 2 daughter pieces are created, hence, the breaking probability p ≈ 0.665 results in the exponent τ ≈ 1.7 obtained in the experiments. However, one has to note that in the analytic derivation an infinite breakup sequence is assumed without a lower cutoff. In our measurements and modeling a finite observation window of fragment sizes is considered which implies an upper and lower cutoff. This is the reason why the measured exponent is reproduced by a higher value p = 0.8 of the breaking probability. The cutoffs, i.e. the size range of fragments are summarized in Table 1 of the manuscript.
Supplementary Figure S2 to illustrate the definition of the lists of length l (r) Relative frequency

Relative frequency
Relative frequency  l (v) and l (r) . The histograms for l (f ) and l (v) are based on the hypothesis that we sample uniformly from the lists l (f ),i (i = 1, 2, . . . F ) and l (v),i (i = 1, 2, . . . V ), respectively. In the current example F = V = 4, so the relative frequency to pick any listelement is 0.25. The histogram for l (r) is based on randomly picking N = 1000 orientations n i (i = 1, 2, . . . N ) from a uniform distribution on the unit circle and calculating the distances to the corresponding N tangents. Bold black dashed vertical lines correspond to the mean values of mean(l (f ) )=0.6297, mean(l (v) ) = 0.9207 and mean(l (r) ) = 0.8261, respectively.