Universality of fragment shapes

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.

Characterization of Clasts in the Glen Torridon Region of Gale Crater Observed by the Mars Science Laboratory Curiosity Rover

The morphology and composition of clasts have the potential to reveal the nature and extent of erosional processes acting in a region. Dense accumulations of granule- to pebble-sized clasts covering the ground throughout the Glen Torridon region of Gale crater on Mars were studied using data acquired by the Mars Science Laboratory Curiosity rover between sols 2300 and 2593. In this study, measurements of shape, size, texture, and elemental abundance of unconsolidated granules and pebbles within northern Glen Torridon were compiled. Nine primary clast types were identified through stepwise hierarchical clustering, all of which are sedimentary and can be compositionally linked to local bedrock, suggesting relatively short transport distances. Several clast types display features associated with fragmentation along bedding planes and existing cracks in bedrock. These results indicate that Glen Torridon clasts are primarily the product of in-situ physical weathering of local bedrock.

Cracking of Colloidal Films to Generate Rectangular Fragments

Cracks are common in nature. Cracking is known as an irreversible and uncontrollable process. To control the cracking patterns, many researchers have proposed methods to prepare notches for stress localization on films. In this work, we investigate a method of controlling cracks by making microscale pyramid patterns that have notches between the pyramids. After preparing pyramid patterns consisting of colloidal particles with organic residue, we annealed them to induce volume shrinkage and cracking between the pyramids. We studied the effect of film thickness on cracking and the generation of rectangular fragments consisting of multiple pyramids. The area of rectangular fragments was in good agreement with the results of scaling analysis. The concept of controlling cracks by imprinting notches on a film and the relationship with the film thickness can guide the study of cracking phenomena.

Pebbles and sand on asteroid (162173) Ryugu: In situ observation and particles returned to Earth

The Hayabusa2 spacecraft investigated the C-type (carbonaceous) asteroid (162173) Ryugu. The mission performed two landing operations to collect samples of surface and subsurface material, the latter exposed by an artificial impact. We present images of the second touchdown site, finding that ejecta from the impact crater was present at the sample location. Surface pebbles at both landing sites show morphological variations ranging from rugged to smooth, similar to Ryugu's boulders, and shapes from quasi-spherical to flattened. The samples were returned to Earth on 6 December 2020. We describe the morphology of >5 grams of returned pebbles and sand. Their diverse color, shape, and structure are consistent with the observed materials of Ryugu; we conclude that they are a representative sample of the asteroid.

The mechanics of brittle granular materials with coevolving grain size and shape

The influence of particle shape on the mechanics of sand is widely recognized, especially in mineral processing and geomechanics. However, most existing continuum theories for engineering applications do not encompass the morphology of the grains and its evolution during comminution. Similarly, the relatively few engineering models accounting for grain-scale processes tend to idealize particles as spheres, with their diameters considered as the primary and sole geometric descriptor. This paper inspires a new generation of constitutive laws for crushable granular continua with arbitrary, yet evolving, particle morphology. We explore the idea of introducing multiple grain shape descriptors into Continuum Breakage Mechanics (CBM), a theory originally designed to track changes in particle size distributions during confined comminution. We incorporate the influence of these descriptors on the elastic strain energy potential and treat them as dissipative state variables. In analogy with the original CBM, and in light of evidence from extreme fragmentation in nature, the evolution of the additional shape descriptors is postulated to converge towards an attractor. Comparisons with laboratory experiments, discrete element analyses and particle-scale fracture models illustrate the encouraging performance of the theory. The theory provides insights into the feedback among particle shape, compressive yielding and inelastic deformation in crushable granular continua. These results inspire new questions that should guide future research into crushable granular systems using particle-scale imaging and computations.

Evolution of anisotropic crack patterns in shrinking material layers

Anisotropic crack patterns emerging in desiccating layers of pastes on a substrate can be exploited for controlled cracking with potential applications in microelectronic manufacturing. We investigate such possibilities of crack patterning in the framework of a discrete element model focusing on the temporal and spatial evolution of anisotropic crack patterns as a thin material layer gradually shrinks. In the model a homogeneous material is considered with an inherent structural disorder where anisotropy is captured by the directional dependence of the local cohesive strength. We demonstrate that there exists a threshold anisotropy below which crack initiation and propagation is determined by the disordered micro-structure, giving rise to cellular crack patterns. When the strength of anisotropy is sufficiently high, cracking is found to evolve through three distinct phases of aligned cracking which slices the sample, secondary cracking in the perpendicular direction, and finally binary fragmentation following the formation of a connected crack network. The anisotropic crack pattern results in fragments with a shape anisotropy which gradually gets reduced as binary fragmentation proceeds. The statistics of fragment masses exhibits a high degree of robustness described by a log-normal functional form at all anisotropies.

Curvature flows, scaling laws and the geometry of attrition under impacts

Impact induced attrition processes are, beyond being essential models of industrial ore processing, broadly regarded as the key to decipher the provenance of sedimentary particles. Here we establish the first link between microscopic, particle-based models and the mean field theory for these processes. Based on realistic computer simulations of particle-wall collision sequences we first identify the well-known damage and fragmentation energy phases, then we show that the former is split into the abrasion phase with infinite sample lifetime (analogous to Sternberg's Law) at finite asymptotic mass and the cleavage phase with finite sample lifetime, decreasing as a power law of the impact velocity (analogous to Basquin's Law). This splitting establishes the link between mean field models (curvature-driven partial differential equations) and particle-based models: only in the abrasion phase does shape evolution emerging in the latter reproduce with startling accuracy the spatio-temporal patterns (two geometric phases) predicted by the former.

Bedload transport in rivers, size matters but so does shape

Bedload transport modelling in rivers takes into account the size and density of pebbles to estimate particle mobility, but does not formally consider particle shape. To address this issue and to compare the relative roles of the density and shape of particles, we performed original sediment transport experiments in an annular flume using molded artificial pebbles equipped with a radio frequency identification tracking system. The particles were designed with four distinct shapes and four different densities while having the same volume, and their speeds and distances traveled under constant hydraulic conditions were analyzed. The results show that particle shape has more influence than particle density on the resting time between particle displacement and the mean traveling distance. For all densities investigated, the particle shape systematically induced differences in travel distance that were strongly correlated (R² = 0.94) with the Sneed and Folks shape index. Such shape influences, although often mentioned, are here quantified for the first time, demonstrating why and how they can be included in bedload transport models.

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.

Dynamics and fragmentation of small inextensible fibres in turbulence

The fragmentation of small, brittle, flexible, inextensible fibres is investigated in a fully developed, homogeneous, isotropic turbulent flow. Such small fibres spend most of their time fully stretched and their dynamics follows that of stiff rods. They can then break through tensile failure, i.e. when the tension is higher than a given threshold. Fibres bend when experiencing a strong compression. During these rare and intermittent buckling events, they can break under flexural failure, i.e. when the curvature exceeds a threshold. Fine-scale massive simulations of both the fluid flow and the fibre dynamics are performed to provide statistics on these two fragmentation processes. This gives ingredients for the development of accurate macroscopic models, namely the fragmentation rate and daughter-size distributions, which can be used to predict the time evolution of the fibre size distribution. Evidence is provided for the generic nature of turbulent fragmentation and of the resulting population dynamics. It is indeed shown that the statistics of break-up is fully determined by the probability distribution of Lagrangian fluid velocity gradients. This approach singles out that the only relevant dimensionless parameter is a local flexibility which balances flow stretching to the fibre elastic forces. This article is part of the theme issue 'Fluid dynamics, soft matter and complex systems: recent results and new methods'.

Fine Comminution of Pine Bark: How Does Mechanical Loading Influence Particles Properties and Milling Efficiency?

The use of lignocellulosic plant biomass as an alternative to fossil feedstocks for chemistry, energy and materials often involves an intense dry comminution step, for which the energy consumed can vary significantly according to the process parameters, the particle size targeted, and the properties of the biomass. Here we studied the fine milling of maritime pine bark in an impact-mill configuration and in an attrition-mill configuration. The properties of the resulting powders (particle size distribution, particle shape, specific surface area, agglomeration level) obtained in each configuration were compared in relation to process energy consumption. Results evidenced that the agglomeration phenomena drive milling efficiency and limit the possibilities for reaching ultrafine particles. Interestingly, impact loading proved more effective at breaking down coarse particles but tended to generate high agglomeration levels, whereas attrition milling led to less agglomeration and thus to finer particles.

Effect of disorder on the spatial structure of damage in slowly compressed porous rocks

Faults and damage zone properties control a range of important phenomena, from the hydraulic properties of underground reservoirs to the physics of earthquakes on a larger scale. Here, we investigate the effect of disorder of porous rocks on the spatial structure of damage emerging under compression. Model rock samples are numerically generated by sedimenting particles where the amount of disorder is controlled by the particle size distribution. To obtain damage bands with a sufficiently large length along axis, we performed simulations of 'Brazilian'-type compression tests of cylindrical samples. As failure is approached, damage localization leads to the formation of two conjugate shear bands. The orientation angle of bands to the loading direction increases with disorder, implying a decrease in the internal coefficient of friction. The width of the damage band scales as a power law of the degree of disorder. Inside the damage band, the sample is crushed into a large number of pieces with a power law mass distribution. The shape of fragments undergoes a crossover at a disorder-dependent size from the isotropy of small pieces to the anisotropic flattened form of the large ones. The results provide important constraints in understanding the role of disorder in geological fractures.This article is part of the theme issue 'Statistical physics of fracture and earthquakes'.

Comminution of Dry Lignocellulosic Biomass, a Review: Part I. From Fundamental Mechanisms to Milling Behaviour

The comminution of lignocellulosic biomass is a key operation for many applications as bio-based materials, bio-energy or green chemistry. The grinder used can have a significant impact on the properties of the ground powders, of those of the end-products and on the energy consumption. Since several years, the milling of lignocellulosic biomass has been the subject of numerous studies most often focused on specific materials and/or applications but there is still a lack of generic knowledge about the relation between the histological structure of the raw materials, the milling technologies and the physical and chemical properties of the powders. This review aims to point out the main process parameters and plant raw material properties that influence the milling operation and their consequences on the properties of ground powders and on the energy consumption during the comminution.

Universal characteristics of particle shape evolution by bed-load chipping

River currents, wind, and waves drive bed-load transport, in which sediment particles collide with each other and Earth's surface. A generic consequence is impact attrition and rounding of particles as a result of chipping, often referred to in geological literature as abrasion. Recent studies have shown that the rounding of river pebbles can be modeled as diffusion of surface curvature, indicating that geometric aspects of impact attrition are insensitive to details of collisions and material properties. We present data from fluvial, aeolian, and coastal environments and laboratory experiments that suggest a common relation between circularity and mass attrition for particles transported as bed load. Theory and simulations demonstrate that universal characteristics of shape evolution arise because of three constraints: (i) Initial particles are mildly elongated fragments, (ii) particles collide with similarly-sized particles or the bed, and (iii) collision energy is small enough that chipping dominates over fragmentation but large enough that sliding friction is negligible. We show that bed-load transport selects these constraints, providing the foundation to estimate a particle's attrition rate from its shape alone in most sedimentary environments. These findings may be used to determine the contribution of attrition to downstream fining in rivers and deserts and to infer transport conditions using only images of sediment grains.

Fragmentation and shear band formation by slow compression of brittle porous media

Localized fragmentation is an important phenomenon associated with the formation of shear bands and faults in granular media. It can be studied by empirical observation, by laboratory experiment, or by numerical simulation. Here we investigate the spatial structure and statistics of fragmentation using discrete element simulations of the strain-controlled uniaxial compression of cylindrical samples of different finite size. As the system approaches failure, damage localizes in a narrow shear band or synthetic fault "gouge" containing a large number of poorly sorted noncohesive fragments on a broad bandwidth of scales, with properties similar to those of natural and experimental faults. We determine the position and orientation of the central fault plane, the width of the shear band, and the spatial and mass distribution of fragments. The relative width of the shear band decreases as a power law of the system size, and the probability distribution of the angle of the central fault plane converges to around 30 degrees, representing an internal coefficient of friction of 0.7 or so. The mass of fragments is power law distributed, with an exponent that does not depend on scale, and is near that inferred for experimental and natural fault gouges. The fragments are in general angular, with a clear self-affine geometry. The consistency of this model with experimental and field results confirms the critical roles of preexisting heterogeneity, elastic interactions, and finite system size to grain size ratio on the development of shear bands and faults in porous media.

Ordered fragmentation of oxide thin films at submicron scale

Crack formation is typically undesirable as it represents mechanical failure that compromises strength and integrity. Recently, there have also been numerous attempts to control crack formation in materials with the aim to prevent or isolate crack propagation. In this work, we utilize fragmentation, at submicron and nanometre scales, to create ordered metal oxide film coatings. We introduce a simple method to create modified films using electroplating on a prepatterned substrate. The modified films undergo preferential fragmentation at locations defined by the initial structures on the substrate, yielding ordered structures. In thicker films, some randomness in the characteristic sizes of the fragments is introduced due to competition between crack propagation and crack creation. The method presented allows patterning of metal oxide films over relatively large areas by controlling the fragmentation process. We demonstrate use of the method to fabricate high-performance electrochromic structures, yielding good coloration contrast and high coloration efficiency.

Fracture and related processes are typically considered detrimental, but have also attracted interest in more constructive roles. Here authors demonstrate ordered fragmentation at submicron scales of a metal oxide/hydroxide thin film by introducing preferential sites for fracture on the underlying substrate.

Reconstructing the transport history of pebbles on Mars

The discovery of remarkably rounded pebbles by the rover Curiosity, within an exhumed alluvial fan complex in Gale Crater, presents some of the most compelling evidence yet for sustained fluvial activity on Mars. While rounding is known to result from abrasion by inter-particle collisions, geologic interpretations of sediment shape have been qualitative. Here we show how quantitative information on the transport distance of river pebbles can be extracted from their shape alone, using a combination of theory, laboratory experiments and terrestrial field data. We determine that the Martian basalt pebbles have been carried tens of kilometres from their source, by bed-load transport on an alluvial fan. In contrast, angular clasts strewn about the surface of the Curiosity traverse are indicative of later emplacement by rock fragmentation processes. The proposed method for decoding transport history from particle shape provides a new tool for terrestrial and planetary sedimentology.