Fragmentation of shells

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.

Petal formation law in a cellophane diaphragm subjected to a pressure difference

In this study, a layer of cellophane, subjected to an air-pressure difference, was ruptured using a piercing needle. Accordingly, petal-like fragmentation was observed in the layer via high-speed imaging. Two types of crack-propagation regimes were subsequently observed experimentally. If a tensile stress lower than 20.6 MPa acted on the cellophane diaphragm, a single crack was generated, whose propagation speed was lower than that under higher-stress conditions. For tensile stresses greater than 23.7 MPa, the crack-propagation speed remained constant at approximately 0.86 km/s, even after altering the device size, pressure, and humidity on the low-pressure side. The number of cracks equidistant from the piercing point was expressed as a linear function of the tensile stress acting on the diaphragm.

Critical Scaling of Solid Fragmentation at Quasistatic and Finite Strain Rates

Using two-dimensional simulations of sheared, brittle solids, we characterize the resulting fragmentation and explore its underlying critical nature. Under quasistatic loading, a power-law distribution of fragment masses emerges after fracture which grows with increasing strain. With increasing strain rate, the maximum size of a grain decreases and a shallower distribution is produced. We propose a scaling theory for distributions based on a fractal scaling of the largest mass with system size in the quasistatic limit or with a correlation length that diverges as a power of rate in the finite-rate limit. Critical exponents are measured using finite-size scaling techniques.

Universal fluctuation of polygonal crack geometry in solidified lava

Outcrops of columnar joints made of solidified lava flows are often covered by semiordered polygonal cracks. The polygon diameters are fairly uniform at each outcrop, but their shapes largely vary in the number of sides and internal angles. Herein, we unveil that the statistical variation in the polygon shape follows an extreme value distribution class: the Gumbel distribution. The Gumbel law was found to hold for different columnar joints, regardless of the locality, lithologic composition, and typical diameter. A common distribution for columnar joints implies a universal class that may integrate the polygonal crack networks observed on the surface of various fractured brittle materials.

Explosive fragmentation of Prince Rupert's drops leads to well-defined fragment sizes

Anyone who has ever broken a dish or a glass knows that the resulting fragments range from roughly the size of the object all the way down to indiscernibly small pieces: typical fragment size distributions of broken brittle materials follow a power law, and therefore lack a characteristic length scale. The origin of this power-law behavior is still unclear, especially why it is such an universal feature. Here we study the explosive fragmentation of glass Prince Rupert’s drops, and uncover a fundamentally different breakup mechanism. The Prince Rupert’s drops explode due to their large internal stresses resulting in an exponential fragment size distribution with a well-defined fragment size. We demonstrate that generically two distinct breakup processes exist, random and hierarchical, that allows us to fully explain why fragment size distributions are power-law in most cases but exponential in others. We show experimentally that one can even break the same material in different ways to obtain either random or hierarchical breakup, giving exponential and power-law distributed fragment sizes respectively. That a random breakup process leads to well-defined fragment sizes is surprising and is potentially useful to control fragmentation of brittle solids.

Fragmentation of breaking glass as a brittle solid is a problem of equal practical and theoretical importance. Kooij et al. demonstrate that the fragment size distribution can surprisingly be both, either power-law or exponential, depending on how a particular specimen is broken.

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.

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'.

Controlling fracture cascades through twisting and quenching

Fracture fundamentally limits the structural stability of macroscopic and microscopic matter, from beams and bones to microtubules and nanotubes. Despite substantial recent experimental and theoretical progress, fracture control continues to present profound practical and theoretical challenges. While bending-induced fracture of elongated rod-like objects has been intensely studied, the effects of twist and quench dynamics have yet to be explored systematically. Here, we show how twist and quench protocols may be used to control such fracture processes, by revisiting Feynman's observation that dry spaghetti typically breaks into three or more pieces when exposed to large pure bending stresses. Combining theory and experiment, we demonstrate controlled binary fracture of brittle elastic rods for two distinct protocols based on twisting and nonadiabatic quenching. Our experimental data for twist-controlled fracture agree quantitatively with a theoretically predicted phase diagram, and we establish asymptotic scaling relations for quenched fracture. Due to their general character, these results are expected to apply to torsional and kinetic fracture processes in a wide range of systems.

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.

Discrete Particle Method for Simulating Hypervelocity Impact Phenomena

In this paper, we introduce a computational model for the simulation of hypervelocity impact (HVI) phenomena which is based on the Discrete Element Method (DEM). Our paper constitutes the first application of DEM to the modeling and simulating of impact events for velocities beyond 5 kms^-1. We present here the results of a systematic numerical study on HVI of solids. For modeling the solids, we use discrete spherical particles that interact with each other via potentials. In our numerical investigations we are particularly interested in the dynamics of material fragmentation upon impact. We model a typical HVI experiment configuration where a sphere strikes a thin plate and investigate the properties of the resulting debris cloud. We provide a quantitative computational analysis of the resulting debris cloud caused by impact and a comprehensive parameter study by varying key parameters of our model. We compare our findings from the simulations with recent HVI experiments performed at our institute. Our findings are that the DEM method leads to very stable, energy-conserving simulations of HVI scenarios that map the experimental setup where a sphere strikes a thin plate at hypervelocity speed. Our chosen interaction model works particularly well in the velocity range where the local stresses caused by impact shock waves markedly exceed the ultimate material strength.

Frozen Impacted Drop: From Fragmentation to Hierarchical Crack Patterns

We investigate experimentally the quenching of a liquid pancake, obtained through the impact of a water drop on a cold solid substrate (0 °C to -60 °C). We show that, below a certain substrate temperature, fractures appear on the frozen pancake and the crack patterns change from a 2D fragmentation regime to a hierarchical fracture regime as the thermal shock increases. The different regimes are discussed and the transition temperatures are estimated through classical fracture scaling arguments. Finally, a phase diagram presents how these regimes can be controlled by the drop impact parameters.

Popping Balloons: A Case Study of Dynamical Fragmentation

Understanding the physics of fragmentation is important in a wide range of industrial and geophysical applications. Fragmentation processes involve large strain rates and short time scales that take place during crack nucleation and propagation. Using rubber membranes, we develop an experimental analysis that enables us to track the fragmentation process in situ in both time and space. We find that bursting a highly stretched membrane yields a treelike fragmentation network that originates at a single seed crack, followed by successive crack tip-splitting events. We show that a dynamic instability drives this branching mechanism. Fragmentation occurs when the crack tip speed attains a critical velocity for which tip splitting becomes the sole available mechanism of releasing the stored elastic energy. Given the general character of the fragmentation processes, this framework should be applicable to other crack networks in brittle materials.

Fragmentation of fractal random structures

We analyze the fragmentation behavior of random clusters on the lattice under a process where bonds between neighboring sites are successively broken. Modeling such structures by configurations of a generalized Potts or random-cluster model allows us to discuss a wide range of systems with fractal properties including trees as well as dense clusters. We present exact results for the densities of fragmenting edges and the distribution of fragment sizes for critical clusters in two dimensions. Dynamical fragmentation with a size cutoff leads to broad distributions of fragment sizes. The resulting power laws are shown to encode characteristic fingerprints of the fragmented objects.

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.

Impact fragmentation of model flocks

Predicting the bulk material properties of active matter is challenging since these materials are far from equilibrium and standard statistical-mechanics approaches may fail. We report a computational study of the surface properties of a well known active matter system: aggregations of self-propelled particles that are coupled via an orientational interaction and that resemble bird flocks. By simulating the impact of these models flocks on an impermeable surface, we find that they fragment into subflocks with power-law mass distributions, similar to shattering brittle solids but not to splashing liquid drops. Thus, we find that despite the interparticle interactions, these model flocks do not possess an emergent surface tension.

Fracturing highly disordered materials

We investigate the role of disorder on the fracturing process of heterogeneous materials by means of a two-dimensional fuse network model. Our results in the extreme disorder limit reveal that the backbone of the fracture at collapse, namely, the subset of the largest fracture that effectively halts the global current, has a fractal dimension of 1.22 ± 0.01. This exponent value is compatible with the universality class of several other physical models, including optimal paths under strong disorder, disordered polymers, watersheds and optimal path cracks on uncorrelated substrates, hulls of explosive percolation clusters, and strands of invasion percolation fronts. Moreover, we find that the fractal dimension of the largest fracture under extreme disorder, d(f) = 1.86 ± 0.01, is outside the statistical error bar of standard percolation. This discrepancy is due to the appearance of trapped regions or cavities of all sizes that remain intact till the entire collapse of the fuse network, but are always accessible in the case of standard percolation. Finally, we quantify the role of disorder on the structure of the largest cluster, as well as on the backbone of the fracture, in terms of a distinctive transition from weak to strong disorder characterized by a new crossover exponent.

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.

Power-law behavior in a cascade process with stopping events: a solvable model

The present paper proposes a stochastic model to be solved analytically, and a power-law-like distribution is derived. This model is formulated based on a cascade fracture with the additional effect that each fragment at each stage of a cascade ceases fracture with a certain probability. When the probability is constant, the exponent of the power-law cumulative distribution lies between -1 and 0, depending not only on the probability but the distribution of fracture points. Whereas, when the probability depends on the size of a fragment, the exponent is less than -1, irrespective of the distribution of fracture points. The applicability of our model is also discussed.

Effect of disorder on temporal fluctuations in drying-induced cracking

We investigate by means of computer simulations the effect of structural disorder on the statistics of cracking for a thin layer of material under uniform and isotropic drying. For this purpose, the layer is discretized into a triangular lattice of springs with a slightly randomized arrangement. The drying process is captured by reducing the natural length of all springs by the same factor, and the amount of quenched disorder is controlled by varying the width ξ of the distribution of the random breaking thresholds for the springs. Once a spring breaks, the redistribution of the load may trigger an avalanche of breaks, not necessarily as part of the same crack. Our computer simulations revealed that the system exhibits a phase transition with the amount of disorder as control parameter: at low disorders, the breaking process is dominated by a macroscopic crack at the beginning, and the size distribution of the subsequent breaking avalanches shows an exponential form. At high disorders, the fracturing proceeds in small-sized avalanches with an exponential distribution, generating a large number of microcracks, which eventually merge and break the layer. Between both phases, a sharp transition occurs at a critical amount of disorder ξ(c)=0.40±0.01, where the avalanche size distribution becomes a power law with exponent τ=2.6±0.08, in agreement with the mean-field value τ=5/2 of the fiber bundle model. Moreover, good quality data collapses from the finite-size scaling analysis show that the average value of the largest burst ⟨Δ(max)⟩ can be identified as the order parameter, with β/ν=1.4 and 1/ν≃1.0, and that the average ratio ⟨m(2)/m(1)⟩ of the second m(2) and first moments m(1) of the avalanche size distribution shows similar behavior to the susceptibility of a continuous transition, with γ/ν=1, 1/ν≃0.9. These results suggest that the disorder-induced transition of the breakup of thin layers is analogous to a continuous phase transition.

Experimental analysis of lateral impact on planar brittle material: spatial properties of cracks

The breakup of alkaline glass and alumina plates due to planar impacts on one of their lateral sides is studied. Particular attention is given to investigating the spatial location of the cracks within the plates. Analysis based on a phenomenological model suggests that bifurcations along the cracks' paths are more likely to take place closer to the impact region than far away from it, i.e., the bifurcation probability seems to lower as the perpendicular distance from the impacted lateral increases. It is also found that many observables are not sensitive to the plate material used in this work, as long as the fragment multiplicities corresponding to the fragmentation of the plates are similar. This gives support to the universal properties of the fragmentation process reported in previous experiments. However, even under the just mentioned circumstances, some spatial observables are capable of distinguishing the material of which the plates are made, which therefore suggests that this universality should be carefully investigated.

Dynamic fragmentation of a ring: predictable fragment mass distributions

We employ a finite element framework, coupled to cohesive elements, to model material decohesion of a uniformly expanding ring. Our study focuses on the average fragment mass, the distribution of fragment masses, and the heaviest fragments. The computed fragment mass distributions are best captured by generalized gamma distributions, regardless of the model parameters. However, the distribution of the heaviest fragments depends on toughness, specimen size, and loading rate.

Universal shapes formed by two interacting cracks

We investigate the origins of the widely observed "en passant" crack pattern, which forms through interactions between two approaching cracks. A rectangular elastic plate is notched on each long side and then subjected to quasistatic uniaxial strain from the short side. The two cracks propagate along approximately straight paths until they pass each other, after which they curve and release a lenticular fragment. We find that, for materials with diverse mechanical properties, the shape of this fragment has an aspect ratio of 2:1, with the length scale set by the initial crack offset s and the time scale set by the ratio of s to the pulling velocity. The cracks have a universal square root shape, which we understand by using a simple geometric model of the crack-crack interaction.

Experimental analysis of lateral impact on planar brittle material

The fragmentation of alumina and glass plates due to lateral impact is studied. A few hundred plates have been fragmented at different impact velocities and the produced fragments are analyzed. The method employed in this work allows one to investigate some geometrical properties of the fragments, besides the traditional size distribution usually studied in former experiments. We found that, although both materials exhibit qualitative similar fragment size distribution function, their geometrical properties appear to be quite different. A schematic model for two-dimensional fragmentation is also presented and its predictions are compared to our experimental results. The comparison suggests that the analysis of the fragments' geometrical properties constitutes a more stringent test of the theoretical models' assumptions than the size distribution.

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.

Role of particle shape on the stress propagation in granular packings

We present an experimental and numerical study on the influence that particle aspect ratio has on the mechanical and structural properties of granular packings. For grains with maximal symmetry (squares), the stress propagation in the packing localizes forming chainlike forces analogous to the ones observed for spherical grains. This scenario can be understood in terms of stochastic models of aggregation and random multiplicative processes. As the grains elongate, the stress propagation is strongly affected. The interparticle normal force distribution tends toward a Gaussian, and, correspondingly, the force chains spread leading to a more uniform stress distribution reminiscent of the hydrostatic profiles known for standard liquids.

Fragmentation processes in impact of spheres

We study the brittle fragmentation of spheres by using a three-dimensional discrete element model. Large scale computer simulations are performed with a model that consists of agglomerates of many particles, interconnected by beam-truss elements. We focus on the detailed development of the fragmentation process and study several fragmentation mechanisms. The evolution of meridional cracks is studied in detail. These cracks are found to initiate in the inside of the specimen with quasiperiodic angular distribution. The fragments that are formed when these cracks penetrate the specimen surface give a broad peak in the fragment mass distribution for large fragments that can be fitted by a two-parameter Weibull distribution. This mechanism can only be observed in three-dimensional models or experiments. The results prove to be independent of the degree of disorder in the model. Our results significantly improve the understanding of the fragmentation process for impact fracture since besides reproducing the experimental observations of fragment shapes, impact energy dependence, and mass distribution, we also have full access to the failure conditions and evolution.

Elastocapillary coalescence: aggregation and fragmentation with a maximal size

Aggregation processes generally lead to broad distributions of sizes involving exponential tails. Here, experiments on the capillary-driven coalescence of regularly spaced flexible structures yields a self-similar distribution of sizes with no tail. At a given step, the physical process imposes a maximal size for the aggregates, which appears as the relevant scale for the distribution. A simple toy model involving the aggregation of nearest neighbors exhibits the same statistics. A mean-field theory accounting for a maximal size is in agreement with both experiments and numerics. This approach is extended to iterative fragmentation processes where the largest object is broken at each step.

Phase transition in liquid drop fragmentation

A liquid droplet is fragmented by a sudden pressurized-gas blow, and the resulting droplets, adhered to the window of a flatbed scanner, are counted and sized by computerized means. The use of a scanner plus image recognition software enables us to automatically count and size up to tens of thousands of tiny droplets with a smallest detectable volume of approximately 0.02 nl . Upon varying the gas pressure, a critical value is found where the size distribution becomes a pure power law, a fact that is indicative of a phase transition. Away from this transition, the resulting size distributions are well described by Fisher's model at coexistence. It is found that the sign of the surface correction term changes sign, and the apparent power-law exponent tau has a steep minimum, at criticality, as previously reported in nuclear multifragmentation studies. We argue that the observed transition is not percolative, and introduce the concept of dominance in order to characterize it. The dominance probability is found to go to zero sharply at the transition. Simple arguments suggest that the correlation length exponent is nu=1/2 . The sizes of the largest and average fragments, on the other hand, do not go to zero abruptly but behave in a way that appears to be consistent with recent predictions of Ashurst and Holian.

Mechanisms for fragment formation in brittle solids

A model for mode I fracture in brittle materials is used to elucidate the relationship between characteristics of the fracture process, such as crack roughness, fractal dimension, and fragment size distributions. It is shown that different roughness in local regions of the crack path leads to different mechanisms for the subsequent fracture of those regions. Formation of two robust power laws for the distribution of formed fragments is observed, governing the size distribution of smaller and larger fragments. We connect measurements in fragment size distribution with the local roughness of cracks in the region of fragment formation.

Scaling behavior of fragment shapes

We present an experimental and theoretical study of the shape of fragments generated by explosive and impact loading of closed shells. Based on high speed imaging, we have determined the fragmentation mechanism of shells. Experiments have shown that the fragments vary from completely isotropic to highly anisotropic elongated shapes, depending on the microscopic cracking mechanism of the shell. Anisotropic fragments proved to have a self-affine character described by a scaling exponent. The distribution of fragment shapes exhibits a power-law decay. The robustness of the scaling laws is illustrated by a stochastic hierarchical model of fragmentation. Our results provide a possible improvement of the representation of fragment shapes in models of space debris.

Fragment velocity distribution in the impact disruption of thin glass plates

We present the experimental results of the measurement of fragment velocity in an impact disruption. Cylindrical projectiles impact on a side (edge) of thin glass plates, and the dispersed fragments were observed using a high-speed camera. The fragment velocity did not depend on the mass but rather on the initial position of the fragment; the velocity component parallel to the projectile direction increased with the distance from the impacted side, while the component perpendicular to the projectile direction increased with the distance from the central axis parallel to the projectile direction. It appears that there are two mechanisms for fragment ejection: one is "spallation," where the fragment velocities depend on the particle velocity induced by shock waves, and the other is "elastic ejection," where the velocities are controlled by the strain energy stored in targets and are at most a few tens of meters per second. We performed a one-dimensional numerical simulation of elastic ejection with a discrete element method and obtained the velocity distribution as a function of the initial position. The numerical results are qualitatively consistent with the experimental ones.

Explosive fragmentation of a thin ceramic tube using pulsed power

This study experimentally examined the explosive fragmentation of thin ceramic tubes using pulsed power. A thin ceramic tube was threaded on a thin copper wire, and high voltage was applied to the wire using a pulsed power generator. This melted the wire and the resulting vapor put pressure on the ceramic tube, causing it to fragment. We examined the statistical properties of the fragment mass distribution. The cumulative fragment mass distribution obeyed the double exponential or power law with exponential decay. Both distributions agreed well with the experimental data. Finally, we obtained universal scaling for fragmentation, which is applicable to both impact and explosive fragmentation.

Fragmentation of rods by cascading cracks: why spaghetti does not break in half

When thin brittle rods such as dry spaghetti pasta are bent beyond their limit curvature, they often break into more than two pieces, typically three or four. With the aim of understanding these multiple breakings, we study the dynamics of a bent rod that is suddenly released at one end. We find that the sudden relaxation of the curvature at this end leads to a burst of flexural waves, whose dynamics are described by a self-similar solution with no adjustable parameters. These flexural waves locally increase the curvature in the rod, and we argue that this counterintuitive mechanism is responsible for the fragmentation of brittle rods under bending. A simple experiment supporting the claim is presented.

Dimensional effects in dynamic fragmentation of brittle materials

It has been shown previously that dynamic fragmentation of brittle D -dimensional objects in a D -dimensional space gives rise to a power-law contribution to the fragment-size distribution with a universal scaling exponent 2-1/D . We demonstrate that in fragmentation of two-dimensional brittle objects in three-dimensional space, an additional fragmentation mechanism appears, which causes scale-invariant secondary breaking of existing fragments. Due to this mechanism, the power law in the fragment-size distribution has now a scaling exponent of approximately 1.17 .

Memory effects on the statistics of fragmentation

We investigate through extensive molecular dynamics simulations the fragmentation process of two-dimensional Lennard-Jones systems. After thermalization, the fragmentation is initiated by a sudden increment to the radial component of the particles' velocities. We study the effect of temperature of the thermalized system as well as the influence of the impact energy of the "explosion" event on the statistics of mass fragments. Our results indicate that the cumulative distribution of fragments follows the scaling ansatz F(m) proportional to m(1-alpha)exp-(m/m(0))(gamma), where m is the mass, m(0) and gamma are cutoff parameters, and alpha is a scaling exponent that is dependent on the temperature. More precisely, we show clear evidence that there is a characteristic scaling exponent alpha for each macroscopic phase of the thermalized system, i.e., that the nonuniversal behavior of the fragmentation process is dictated by the state of the system before it breaks down.

Breakup of shells under explosion and impact

A theoretical and experimental study of the fragmentation of closed thin shells made of a disordered brittle material is presented. Experiments were performed on eggshells under two different loading conditions: fragmentation due to an impact with a hard wall and explosion by a combustion mixture giving rise to power law fragment size distributions. For the theoretical investigations a three-dimensional discrete element model of shells is constructed. Molecular dynamics simulations of the two loading cases resulted in power law fragment mass distributions in satisfactory agreement with experiments. Based on large scale simulations we give evidence that power law distributions arise due to an underlying phase transition which proved to be abrupt and continuous for explosion and impact, respectively. Our results demonstrate that the fragmentation of closed shells defines a universality class, different from that of two- and three-dimensional bulk systems.

Crossover of weighted mean fragment mass scaling in two-dimensional brittle fragmentation

We performed vertical and horizontal sandwich two-dimensional brittle fragmentation experiments. The weighted mean fragment mass was scaled using the multiplicity mu. The scaling exponent crossed over at log(10) mu(c) approximately equal to -1.4 . In the small mu (<< mu(c) ) regime, the binomial multiplicative (BM) model was suitable and the fragment mass distribution obeyed log-normal form. However, in the large mu (>> mu(c) ) regime, in which a clear power-law cumulative fragment mass distribution was observed, it was impossible to describe the scaling exponent using the BM model. We also found that the scaling exponent of the cumulative fragment mass distribution depended on the manner of impact (loading conditions): it was 0.5 in the vertical sandwich experiment and approximately 1.0 in the horizontal sandwich experiment.