Solution for the fragment-size distribution in a crack-branching model of fragmentation

Particle fracture regimes from impact simulations

We introduce an approach to particle breakage, wherein the particle is modeled as an aggregate of polyhedral cells with their common surfaces governed by the Griffith criterion of fracture. This model is implemented within a discrete element code to simulate and analyze the breakage behavior of a single particle impacting a rigid plane. We find that fracture dynamics involves three distinct regimes as a function of the normalized impact energy ω. At low values of ω, the particle undergoes elastic rebound and no cracks occur inside the particle. In the intermediate range, the particle is damaged by nucleation and propagation of cracks, and the effective restitution coefficient declines without breakup of the particle. Finally, for values of ω beyond a well-defined threshold, the particle breaks into fragments and the restitution coefficient increases with ω due to kinetic energy carried away by the fragments. We show that particle damage, restitution coefficient, and fracture efficiency (the amount of energy input consumed for particle fracture) collapse well as a function of dimensionless scaling parameters. Our data are also sufficiently accurate to scale fragment size and shape distributions. It is found that fragment masses (volumes) follow a power-law distribution with an exponent decreasing with fracture energy. Interestingly, the average elongation and flatness of fragments are very close to those observed in experiments and lunar samples at the optimal fracture efficiency.

Prince Rupert's Drops: An analysis of fragmentation by thermal stresses and quench granulation of glass and bubbly glass

When volcanic eruptions involve interaction with external water (hydrovolcanism), the result is an ash-rich and energetic volcanic plume, as illustrated dramatically by the January 2022 Tonga eruption. The origin of the high explosive energy of these events remains an important question. We investigate this question by studying Prince Rupert's Drops (PRDs)-tadpole-shaped glass beads formed by dripping molten glass into water-which have long fascinated materials scientists because the great strength of the head contrasts with the explosivity of the metastable interior when the tail is broken. We show that the fragment size distribution (FSD) produced by explosive fragmentation changes systematically with PRD fragmentation in air, water, and syrup. Most FSDs are fractal over much of the size range, scaling that can be explained by the repeated fracture bifurcation observed in three-dimensional images from microcomputed tomography. The shapes of constituent fragments are determined by their position within the original PRD, with platey fragments formed from the outer (compressive) shell and blocky fragments formed by fractures perpendicular to interior voids. When molten drops fail to form PRDs, the glass disintegrates by quench granulation, a process that produces fractal FSDs but with a larger median size than explosively generated fragments. Critically, adding bubbles to the molten glass prevents PRD formation and promotes quench granulation, suggesting that granulation is modulated by heterogeneous stress fields formed around the bubbles during sudden cooling and contraction. Together, these observations provide insight into glass fragmentation and potentially, processes operating during hydrovolcanism.

Plato's cube and the natural geometry of fragmentation

Plato envisioned Earth's building blocks as cubes, a shape rarely found in nature. The solar system is littered, however, with distorted polyhedra-shards of rock and ice produced by ubiquitous fragmentation. We apply the theory of convex mosaics to show that the average geometry of natural two-dimensional (2D) fragments, from mud cracks to Earth's tectonic plates, has two attractors: "Platonic" quadrangles and "Voronoi" hexagons. In three dimensions (3D), the Platonic attractor is dominant: Remarkably, the average shape of natural rock fragments is cuboid. When viewed through the lens of convex mosaics, natural fragments are indeed geometric shadows of Plato's forms. Simulations show that generic binary breakup drives all mosaics toward the Platonic attractor, explaining the ubiquity of cuboid averages. Deviations from binary fracture produce more exotic patterns that are genetically linked to the formative stress field. We compute the universal pattern generator establishing this link, for 2D and 3D fragmentation.

Generation of fine fragments during dynamic propagation of pressurized cracks

High-resolution numerical simulations of cracks driven by an internal pressure in a heterogeneous and brittle granular medium produce fragment-size distributions with the same characteristics as experiments on blasted cylinders of mortar and rock in both the fine- and the intermediate-size-fragment regions. To mimic full-scale blasts used, e.g., within the mining industry, the cracks propagate in a medium that is under compression, neutral, or under tension. In a compressive environment, shear fracture produces a large volume of fines, whereas in a neutral or tensile environment, unstable crack branching is responsible for a much smaller volume of fines. The boundary between the fine- and the intermediate-size fragments scales as the average grain size of the material. The ultimate goal is to develop a blasting process that minimizes the fines, which, in mining, are both an environmental hazard and useless for further processing.

Using Small-scale Blast Tests and Numerical Modelling to Trace the Origin of Fines Generated in Blasting

Waste fines from rock breakage often negatively influence economics and environment. The Austrian Science Fund (FWF) sponsors a project to investigate the cause of the fines by studying blast fragmentation throughout small-scale blast tests and numerical simulations. The tests include blast-loading confined granite and mortar cylinders by detonating cord with 6, 12, and 20 g/m of PETN. The blast-driven dynamic cracking at the end face of the cylinder opposite to the initiation point is filmed with a high-speed camera. The filming is followed up by an analysis of surface and internal crack systems and sieving of the blasted cylinders to quantify the amount of fine material created. The numerical simulations cover the blast fragmentation of a mortar cylinder. These simulations use Finite and Discrete Element Methods (FEM, DEM) with explicit time integration. The model cylinders are loaded by a pressure evolution acting on the borehole wall. Both methods produce realistic crack patterns, consisting of through-going radial cracks with crack intersections around a crushed zone at the borehole. Furthermore, the DEM models have also yielded realistic fragment size distributions (FSD). The paper covers the present progress of the ongoing project and related future work.

Effect of disorder on shrinkage-induced fragmentation of a thin brittle layer

We investigate the effect of the amount of disorder on the shrinkage-induced cracking of a thin brittle layer attached to a substrate. Based on a discrete element model we study how the dynamics of cracking and the size of fragments evolve when the amount of disorder is varied. In the model a thin layer is discretized on a random lattice of Voronoi polygons attached to a substrate. Two sources of disorder are considered: structural disorder captured by the local variation of the stiffness and strength disorder represented by the random strength of cohesive elements between polygons. Increasing the amount of strength disorder, our calculations reveal a transition from a cellular crack pattern, generated by the sequential branching and merging of cracks, to a disordered ensemble of cracks where the merging of randomly nucleated microcracks dominate. In the limit of low disorder, the statistics of fragment size is described by a log-normal distribution; however, in the limit of high disorder, a power-law distribution is obtained.

Characterizing the size and shape of sea ice floes

Monitoring drift ice in the Arctic and Antarctic regions directly and by remote sensing is important for the study of climate, but a unified modeling framework is lacking. Hence, interpretation of the data, as well as the decision of what to measure, represent a challenge for different fields of science. To address this point, we analyzed, using statistical physics tools, satellite images of sea ice from four different locations in both the northern and southern hemispheres, and measured the size and the elongation of ice floes (floating pieces of ice). We find that (i) floe size follows a distribution that can be characterized with good approximation by a single length scale , which we discuss in the framework of stochastic fragmentation models, and (ii) the deviation of their shape from circularity is reproduced with remarkable precision by a geometric model of coalescence by freezing, based on random Voronoi tessellations, with a single free parameter expressing the shape disorder. Although the physical interpretations remain open, this advocates the parameters and as two independent indicators of the environment in the polar regions, which are easily accessible by remote sensing.

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.

Emergence of energy dependence in the fragmentation of heterogeneous materials

The most important characteristics of the fragmentation of heterogeneous solids is that the mass (size) distribution of pieces is described by a power law functional form. The exponent of the distribution displays a high degree of universality depending mainly on the dimensionality and on the brittle-ductile mechanical response of the system. Recently, experiments and computer simulations have reported an energy dependence of the exponent increasing with the imparted energy. These novel findings question the phase transition picture of fragmentation phenomena, and have also practical importance for industrial applications. Based on large scale computer simulations here we uncover a robust mechanism which leads to the emergence of energy dependence in fragmentation processes resolving controversial issues on the problem: studying the impact induced breakup of platelike objects with varying thickness in three dimensions we show that energy dependence occurs when a lower dimensional fragmenting object is embedded into a higher dimensional space. The reason is an underlying transition between two distinct fragmentation mechanisms controlled by the impact velocity at low plate thicknesses, while it is hindered for three-dimensional bulk systems. The mass distributions of the subsets of fragments dominated by the two cracking mechanisms proved to have an astonishing robustness at all plate thicknesses, which implies that the nonuniversality of the complete mass distribution is the consequence of blending the contributions of universal partial processes.

Scaling laws for impact fragmentation of spherical solids

We investigate the impact fragmentation of spherical solid bodies made of heterogeneous brittle materials by means of a discrete element model. Computer simulations are carried out for four different system sizes varying the impact velocity in a broad range. We perform a finite size scaling analysis to determine the critical exponents of the damage-fragmentation phase transition and deduce scaling relations in terms of radius R and impact velocity v(0). The scaling analysis demonstrates that the exponent of the power law distributed fragment mass does not depend on the impact velocity; the apparent change of the exponent predicted by recent simulations can be attributed to the shifting cutoff and to the existence of unbreakable discrete units. Our calculations reveal that the characteristic time scale of the breakup process has a power law dependence on the impact speed and on the distance from the critical speed in the damaged and fragmented states, respectively. The total amount of damage is found to have a similar behavior, which is substantially different from the logarithmic dependence on the impact velocity observed in two dimensions.

New universality class for the fragmentation of plastic materials

We present an experimental and theoretical study of the fragmentation of polymeric materials by impacting polypropylene particles of spherical shape against a hard wall. Experiments reveal a power law mass distribution of fragments with an exponent close to 1.2, which is significantly different from the known exponents of three-dimensional bulk materials. A 3D discrete element model is introduced which reproduces both the large permanent deformation of the polymer during impact and the novel value of the mass distribution exponent. We demonstrate that the dominance of shear in the crack formation and the plastic response of the material are the key features which give rise to the emergence of the novel universality class of fragmentation phenomena.

Difference between fracture of thin brittle sheets and two-dimensional fracture

Recently there has been some suggestions that fragmentation of thin brittle sheets is qualitatively different from pure two-dimensional fragmentation. The obvious reason for such a discrepancy is the possibility of the sheet to deform out of plane. There is a generic crack-branching mechanism that creates power-law fragment size distribution in the small fragment range for two-dimensional (2D) and three-dimensional bulk fragmentation with the power exponent (2D-1)/D. For thin sheets, the power exponent seems to be close to 1.2 which differs from the D=2 exponent 1.5. In order to make a distinct separation between sheet and 2D fragmentation, high-resolution fragment size distributions are required for fragmentation models with minimal differencies other than dimensionality. Here a very efficient numerical model which can be switched from 2D fragmentation to out-of-plane sheet fragmentation with minimal changes is used to produce high-resolution fragment size distribution for the two cases. The model results cast some doubt on the existence of separate universality classes for sheet and 2D fragmentation.