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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 2D fragments, from mud cracks to Earth's tectonic plates, has two attractors: "Platonic" quadrangles and "Voronoi" hexagons. In 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., main: 6 pages, 6 figures, supplementary: 18 pages, 12 figures

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.

Natural catastrophes like landslides are often caused by the nucleation and propagation of fractures in heterogeneous materials. Landslides are typically initiated by heavy raining events when water penetrates the pores and reduces the cohesion of soils leading to instability and cracking. When it happens on a steep slope, the moving mass could break up into pieces and the landslide gives rise to a debris flow composed of rapidly traveling fragments of soil and rocks. Such devastating catastrophes endanger the infrastructure and take thousands of lives every year.In order to understand the emergence of landslides and debris flows we investigated the collapse of a granular column under the action of gravity by means of discrete element simulations. In the model, a cylindrical sample of soil is represented as a random packing of spherical particles. Cohesion is introduced by connecting the particles with non-linear spring elements along the edges of Delaunay triangles determined in the initial configuration of the particles. The constitutive law of springs captures the elastic behavior of particle contacts at small deformations, the plasticity beyond a yield threshold, and the gradual softening and final breaking at large separation distances. A very important feature of the interaction is that particle contacts can be healed, i.e. if two particles approach each other within a capture distance, a new cohesive contact is established between them. Computer simulations were performed varying the strength of cohesion in a broad range.Our calculations revealed that at high cohesion the granular column sinks in, i.e. its height gradually decreases while it undergoes restructuring and flattening, however, in the final state the system keeps its integrity. When the cohesion is sufficiently week, the process of collapse cannot stop: the system breaks up into a large number of fragments which run out at a high speed. The two phases of high and low cohesion represent the mass movement and the debris flow states of real landslides, respectively. We demonstrate that the transition occurs at a critical cohesion showing analogies to continuous phase transitions.

Layers of dense pastes, colloids attached to a substrate often undergo sequential cracking due to shrinkage stresses caused by desiccation. From the spectacular crack patterns of dried out lake beds through the polygonal ground patterns of permafrost regions to the formation of columnar joints in cooling volcanic lava, shrinkage induced cracking is responsible for a large variety of complex crack structures in nature. Under laboratory conditions this phenomenon is usually investigated by desiccating thin layers of dense colloidal suspensions in a container, which typically leads to polygonal crack patterns with a high degree of isotropy.It is of great interest how to control the structure of shrinkage induced two-dimensional crack patterns also due to its high importance for technological applications. Recently, it has been demonstrated experimentally for dense calcium carbonate and magnesium carbonate hydroxide pastes that applying mechanical excitation by means of vibration or flow of the paste the emerging desiccation crack pattern remembers the direction of excitation, i.e. main cracks get aligned and their orientation can be tuned by the direction of mechanical excitation.In order to understand the mechanism of this memory effect, we investigate a fragmentation process of a brittle, cylindrical sample, where the driving force of the cracking coming from a continous shrinkage, which sooner or later destroys the cohesive forces between the structure’s building blocks. Our study is based on a two dimensional discrete element model, where the material is discretised via a special form of the Voronoi-tesselation, with the so-called randomised vector lattice which allows to fine-tune the initial disorder of the system. We assume that the initial mechanical vibration imprints plastic deformation into the paste, which is captured in the model by assuming that the local cohesive strength of the layer has a directional dependence: the layer is stronger along the direction of vibration. We demonstrate that - based on this simple assumption - the model well reproduces the qualitative features of the anisotropic crack patterns observed in experiments. Gradually increasing the degree of anisotropy the system exhibits a crossover from an isotropic cellular structure to an anisotropic one.

We investigate how the interplay of the topology of the network of load transmitting connections and the amount of disorder of the strength of the connected elements determines the temporal evolution of failure cascades driven by the redistribution of load following local failure events. We use the fiber bundle model of materials’ breakdown assigning fibers to the sites of a square lattice, which is then randomly rewired using the Watts–Strogatz technique. Gradually increasing the rewiring probability, we demonstrate that the bundle undergoes a transition from the localized to the mean field universality class of breakdown phenomena. Computer simulations revealed that both the size and the duration of failure cascades are power law distributed on all network topologies with a crossover between two regimes of different exponents. The temporal evolution of cascades is described by a parabolic profile with a right handed asymmetry, which implies that cascades start slowly, then accelerate, and eventually stop suddenly. The degree of asymmetry proved to be characteristic of the network topology gradually decreasing with increasing rewiring probability. Reducing the variance of fibers’ strength, the exponents of the size and the duration distribution of cascades increase in the localized regime of the failure process, while the localized to mean field transition becomes more abrupt. The consistency of the results is supported by a scaling analysis relating the characteristic exponents of the statistics and dynamics of cascades.

We investigate the impact induced damage and fracture of a bar shaped specimen of heterogeneous materials focusing on how the system approaches perforation as the impact energy is gradually increased. A simple model is constructed which represents the bar as two rigid blocks coupled by a breakable interface with disordered local strength. The bar is clamped at the two ends and the fracture process is initiated by an impactor hitting the bar in the middle. Our calculations revealed that depending on the imparted energy, the system has two phases: at low impact energies the bar suffers damage but keeps its integrity, while at sufficiently high energies, complete perforation occurs. We demonstrate that the transition from damage to perforation occurs analogous to continuous phase transitions. Approaching the critical point from below, the intact fraction of the interface goes to zero, while the deformation rate of the bar diverges according to power laws as function of the distance from the critical energy. As the degree of disorder increases, further from the transition point the critical exponents agree with their zero disorder counterparts, however, close to the critical point a crossover occurs to a higher exponent., Comment: 10 pages, 11 figures

We investigate the effect of the amount of disorder on the statistics of breaking bursts during the quasi-static fracture of heterogeneous materials. We consider a fiber bundle model where the strength of single fibers is sampled from a power law distribution over a finite range, so that the amount of materials' disorder can be controlled by varying the power law exponent and the upper cutoff of fibers' strength. Analytical calculations and computer simulations, performed in the limit of equal load sharing, revealed that depending on the disorder parameters the mechanical response of the bundle is either perfectly brittle where the first fiber breaking triggers a catastrophic avalanche, or it is quasi-brittle where macroscopic failure is preceded by a sequence of bursts. In the quasi-brittle phase, the statistics of avalanche sizes is found to show a high degree of complexity. In particular, we demonstrate that the functional form of the size distribution of bursts depends on the system size: for large upper cutoffs of fibers' strength, in small systems the sequence of bursts has a high degree of stationarity characterized by a power law size distribution with a universal exponent. However, for sufficiently large bundles the breaking process accelerates towards the critical point of failure which gives rise to a crossover between two power laws. The transition between the two regimes occurs at a characteristic system size which depends on the disorder parameters., Comment: 10 pages, 12 figures, revtex

We investigate how the dimensionality of the embedding space affects the microscopic crackling dynamics and the macroscopic response of heterogeneous materials. Using a fiber bundle model with localized load sharing computer simulations are performed from 1 to 8 dimensions slowly increasing the external load up to failure. Analyzing the constitutive curve, fracture strength and avalanche statistics of bundles we demonstrate that a gradual crossover emerges from the universality class of localized behavior to the mean field class of fracture as the embedding dimension increases. The evolution between the two universality classes is described by an exponential functional form. Simulations revealed that the average temporal profile of crackling avalanches evolves with the dimensionality of the system from a strongly asymmetric shape to a symmetric parabola characteristic for localized stresses and homogeneous stress fields, respectively., Comment: 9 pages, 12 figures, revtex

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’.

We investigate the fracture of heterogeneous materials occurring under unloading from an initial load. Based on a fiber bundle model of time dependent fracture, we show that depending on the unloading rate the system has two phases: for rapid unloading the system suffers only partial failure and it has an infinite lifetime, while at slow unloading macroscopic failure occurs in a finite time. The transition between the two phases proved to be analogous to continuous phase transitions. Computer simulations revealed that during unloading the fracture proceeds in bursts of local breakings triggered by slowly accumulating damage. In both phases the time evolution starts with a relaxation of the bursting activity characterized by a universal power law decay of the burst rate. In the phase of finite lifetime the initial slowdown is followed by an acceleration towards macroscopic failure where the increasing rate of bursts obeys the (inverse) Omori law of earthquakes. We pointed out a strong correlation between the time where the event rate reaches a minimum value and of the lifetime of the system which allows for forecasting of the imminent catastrophic failure., 10 pages, 10 figures, revtex

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.

We study the impact fragmentation of two-and three-dimensional disordered solids in a discrete element model of heterogeneous brittle materials focusing on the spatial distribution and mass-velocity correlation of fragments. Our calculations revealed that depending on the energy of impact the breakup process can have two different outcomes: at low impact energy the sample gets damaged, however, to achieve fragmentation the imparted energy has to surpass a critical value. Based on large scale computer simulations we show that the position of fragments inside the original body with respect to the impact site determines their mass and velocity in the final state. A novel relation of the mass and velocity of fragments is revealed: In the damage phase the mass and velocity of fragments are strongly correlated, however, in the fragmented phase correlation emerge solely for large fragments. The correlation function decays as a power law with a universal exponent in an excellent agreement with recent experimental findings.

We present a theoretical study of the fracture of two-dimensional disc-shaped samples due to projectile penetration focusing on the geometrical structure of the crack pattern. The penetration of a cone is simulated into a plate of circular shape using a discrete element model of heterogeneous brittle materials varying the speed of penetration in a broad range. As the cone penetrates a destroyed zone is created from which cracks run to the external boundary of the plate. Computer simulations revealed that in the low speed limit of loading two cracks are generated with nearly straight shape. Increasing the penetration speed the crack pattern remains regular, however, both the number of cracks and their fractal dimension increases. High speed penetration gives rise to a crack network such that the sample gets fragmented into a large number of pieces. We give a quantitative analysis of the evolution of the system from simple cracking through fractal cracks to fragmentation with a connected crack network.

We present a theoretical study of the creep rupture of heterogeneous materials based on a fiber bundle model which provides a direct connection between the microscopic fracture mechanisms and the macroscopic time evolution. In the model, material elements fail either due to immediate breaking or undergo a damage accumulating ageing process. We found that on the micro-level the competition of the two failure modes gives rise to bursts of breakings with power law distributed size and waiting time between events. We demonstrate that approaching macroscopic failure the system accelerates which can be fully described as a non-homogeneous Poissonian process for long range load sharing, however, when localization occurs breaking events get clustered. Bursts are composed of sub-avalanches which lead to a non-trivial temporal shape comparable to measurements. The pulse shape proved to be sensitive to the range of load sharing.

We investigate the size scaling of the macroscopic fracture strength of heterogeneous materials when microscopic disorder is controlled by fat-tailed distributions. We consider a fiber bundle model where the strength of single fibers is described by a power law distribution over a finite range. Tuning the amount of disorder by varying the power law exponent and the upper cutoff of fibers' strength, in the limit of equal load sharing an astonishing size effect is revealed: For small system sizes the bundle strength increases with the number of fibers and the usual decreasing size effect of heterogeneous materials is only restored beyond a characteristic size. We show analytically that the extreme order statistics of fibers' strength is responsible for this peculiar behavior. Analyzing the results of computer simulations we deduce a scaling form which describes the dependence of the macroscopic strength of fiber bundles on the parameters of microscopic disorder over the entire range of system sizes., Comment: 6 pages, 4 figures, revtex

Localized fragmentation is an important phenomenon associated with the formation of shear bands and faults ingranular 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 thestrain-controlled uniaxial compression of cylindrical samples of different finite size. As the system approachesfailure, damage localizes in a narrow shear band or synthetic fault “gouge” containing a large number of poorlysorted noncohesive fragments on a broad bandwidth of scales, with properties similar to those of natural andexperimental faults. We determine the position and orientation of the central fault plane, the width of the shearband, and the spatial and mass distribution of fragments. The relative width of the shear band decreases as apower law of the system size, and the probability distribution of the angle of the central fault plane converges toaround 30 degrees, representing an internal coefficient of friction of 0.7 or so. The mass of fragments is powerlaw distributed, with an exponent that does not depend on scale, and is near that inferred for experimental andnatural fault gouges. The fragments are in general angular, with a clear self-affine geometry. The consistencyof this model with experimental and field results confirms the critical roles of preexisting heterogeneity, elasticinteractions, and finite system size to grain size ratio on the development of shear bands and faults in porousmedia.

‘Crackling noise’ occurs in a wide variety of systems that respond to steady-state external forcing in an intermittent way, leading to sudden bursts of energy release similar to those heard when crumpling a piece of paper or listening to a fire. In rock physics sudden changes in internal stress associated with microscopically-brittle rupture events lead to acoustic emissions that can be recorded on the sample boundary, and used to infer the state of internal damage. Crackling noise is inherently stochastic, but the population of events often exhibits remarkably robust scaling properties, in terms of the source area, duration, energy, and in the waiting time between events. Here we describe how these scaling properties emerge and evolve spontaneously in a fully-dynamic discrete element model of sedimentary rocks subject to uniaxial compression applied at a constant strain rate. The discrete elements have structural disorder similar to that of a real rock, and this is the only source of heterogeneity. Despite the stationary strain rate applied and the lack of any time-dependent weakening processes, the results are all characterized by emergent power law distributions over a broad range of scales, in agreement with experimental observation. As deformation evolves, the scaling exponents change systematically in a way that is similar to the evolution of damage in experiments on real sedimentary rocks . The potential for real-time forecasting of catastrophic failure obeying such scaling rules is then examined by using synthetic and real data from laboratory tests and prior to volcanic eruptions. The combination of non-linearity in the constitutive rules and an irreducible stochastic component governed by the material heterogeneity and finite sampling of AE data leads to significant variations in the precision and accuracy of the forecast failure time. This leads to significant proportion of ‘false alarms’ (forecast too early) and ‘missed events’ (forecast too late), as well as an over-optimistic assessments of forecasting power and quality when the failure time is known (the ‘benefit of hindsight’). The evolution becomes progressively more complex, and the forecasting power diminishes, in going from ideal synthetics to controlled laboratory tests to open natural systems at larger scales in space and time.

Quasi-brittle behavior where macroscopic failure is preceded by stable damaging and intensive cracking activity is a desired feature of materials because it makes fracture predictable. Based on a fiber bundle model with global load sharing we show that blending strength and stiffness disorder of material elements leads to the stabilization of fracture, i.e. samples which are brittle when one source of disorder is present, become quasi-brittle as a consequence of blending. We derive a condition of quasi-brittle behavior in terms of the joint distribution of the two sources of disorder. Breaking bursts have a power law size distribution of exponent 5/2 without any crossover to a lower exponent when the amount of disorder is gradually decreased. The results have practical relevance for the design of materials to increase the safety of constructions., Comment: 9 pages, 6 figures, revtex

We investigate the effect of the amount of disorder on the fracture process of heterogeneous materials in the framework of a fiber bundle model. The limit of high disorder is realized by introducing a power law distribution of fiber strength over an infinite range. We show that on decreasing the amount of disorder by controlling the exponent of the power law the system undergoes a transition from the quasi-brittle phase where fracture proceeds in bursts to the phase of perfectly brittle failure where the first fiber breaking triggers a catastrophic collapse. For equal load sharing in the quasi-brittle phase the fat tailed disorder distribution gives rise to a homogeneous fracture process where the sequence of breaking bursts does not show any acceleration as the load increases quasi-statically. The size of bursts is power law distributed with an exponent smaller than the usual mean field exponent of fiber bundles. We demonstrate by means of analytical and numerical calculations that the quasi-brittle to brittle transition is analogous to continuous phase transitions and determine the corresponding critical exponents. When the load sharing is localized to nearest neighbor intact fibers the overall characteristics of the failure process prove to be the same, however, with different critical exponents. We show that in the limit of the highest disorder considered the spatial structure of damage is identical with site percolation—however, approaching the critical point of perfect brittleness spatial correlations play an increasing role, which results in a different cluster structure of failed elements.

We investigate the geometrical structure of breaking bursts generated during the creep rupture of heterogeneous materials. Based on a fiber bundle model with localized load sharing we show that bursts are compact geometrical objects, however, their external frontier has a fractal structure which reflects their growth dynamics. The perimeter fractal dimension of bursts proved to have the universal value 1.25 independent of the external load and of the amount of disorder in the system. We conjecture that according to their geometrical features breaking bursts fall in the universality class of loop-erased self-avoiding random walks with perimeter fractal dimension 5/4 similar to the avalanches of Abelian sand pile models. The fractal dimension of the growing crack front along which bursts occur proved to increase from 1 to 1.25 as bursts gradually cover the entire front., 6 pages, 6 figures, revtex

Discrete element modelling (DEM) is one of the most efficient computational approaches to the fracture processes of heterogeneous materials on mesoscopic scales. From the dynamics of single crack propagation through the statistics of crack ensembles to the rapid fragmentation of materials DEM had a substantial contribution to our understanding over the past decades. Recently, the combination of DEM with other simulation techniques like Finite Element Modelling further extended the field of applicability. In this paper we briefly review the motivations and basic idea behind the DEM approach to cohesive particulate matter and then we give an overview of on-going developments and applications of the method focusing on two fields where recent success has been achieved. We discuss current challenges of this rapidly evolving field and outline possible future perspectives and debates.

An accurate understanding of the interplay between random and deterministic processes in generating extreme events is of critical importance in many fields, from forecasting extreme meteorological events to the catastrophic failure of materials and in the Earth. Here we investigate the statistics of record-breaking events in the time series of crackling noise generated by local rupture events during the compressive failure of porous materials. The events are generated by computer simulations of the uni-axial compression of cylindrical samples in a discrete element model of sedimentary rocks that closely resemble those of real experiments. The number of records grows initially as a decelerating power law of the number of events, followed by an acceleration immediately prior to failure. We demonstrate the existence of a characteristic record rank k^* which separates the two regimes of the time evolution. Up to this rank deceleration occurs due to the effect of random disorder. Record breaking then accelerates towards macroscopic failure, when physical interactions leading to spatial and temporal correlations dominate the location and timing of local ruptures. Sub-sequences of bursts between consecutive records are characterized by a power law size distribution with an exponent which decreases as failure is approached. High rank records are preceded by bursts of increasing size and waiting time between consecutive events and they are followed by a relaxation process. As a reference, surrogate time series are generated by reshuffling the crackling bursts. The record statistics of the uncorrelated surrogates agrees very well with the corresponding predictions of independent identically distributed random variables, which confirms that the temporal and spatial correlation of cracking bursts are responsible for the observed unique behaviour., 11 pages, 13 figures, revtex

We present two extensions of the classical fibre bundle model to study the creep rupture of heterogeneous materials and the shear failure of glued interfaces of solid blocks. To model creep rupture, we assume that the fibres of a parallel bundle present time dependent behaviour under an external load and fail when the deformation exceeds their local breaking threshold. Assuming global load sharing among fibres, analytical and numerical calculations showed that there exists a critical load below which only partial failure occurs while above which the system fails globally after a finite time. Approaching the critical point from both sides the system exhibits scaling behaviour which implies that creep rupture is analogous to continuous phase transitions. To describe interfacial failure, we model the interface as an array of elastic beams which experience stretching and bending under shear load and break if the two deformation modes exceed randomly distributed breaking thresholds. The two breaking modes can be independent or combined in the form of a von Mises type breaking criterion. In the framework of global load sharing, we obtain analytically the macroscopic constitutive behaviour of the system and describe the microscopic process of the progressive failure of the interface.

Fragmentation can be observed in nature and in everyday life on a wide range of length scales and for all kinds of technical applications. Most studies on dynamic failure focus on the behaviour of bulk systems in one, two and three dimensions under impact and explosive loading, showing universal power law behaviour of fragment size distribution. However, hardly any studies have been devoted to fragmentation of shells. We present a detailed experimental and theoretical study on the fragmentation of closed thin shells of various materials, due to an excess load inside the system and impact with a hard wall. Characteristic fragmentation mechanisms are identified by means of a high speed camera and fragment shapes and mass distributions are evaluated. Theoretical rationalisation is given by means of stochastic break-up models and large-scale discrete element simulations with spherical shell systems under different extreme loading situations. By this we ex-plain fragment shapes and distributions and prove a power law for the fragment mass distribution. Satisfactory agreement between experimental findings and nu-merical predictions of the exponents of the power laws for the fragment shapes is obtained.

An experimental and theoretical study of the formation of structures in a binary monolayer of dipolar particles is presented. We construct an experimental technique for an easy to control realization of a binary monolayer where the two components have oppositely oriented dipole moments constrained perpendicular to the plane of motion. The experimental set-up ensures that hydrodynamic effects do not play a crucial role in the structure formation, the particles move deterministically due to the dipole–dipole interaction. At low concentrations cluster–cluster aggregation occurs with chain-like morphologies, while at high concentration the particles self-assemble into various types of binary crystal lattices. We propose a simple theoretical framework to describe the observed structure formation. Based on molecular dynamics simulations and analytic calculations we show that the total concentration of the particles, the relative concentration and the relative dipole moment of the components determine the struct...

The break-up of a two-dimensional circular disc by normal and oblique impact on a hard frictionless plate is investigated by molecular dynamics simulations. The disc is composed of numerous unbreakable randomly shaped convex polygons connected together by simple elastic beams that break when bent or stretched beyond a certain limit. It is found that for both normal and oblique impacts the crack patterns are the same and depend solely on the normal component of the impact velocity. Analysing the pattern of breakage, amount of damage, fragment masses and velocities, we show the existence of a critical velocity which separates two regimes of the impact process: below the critical point only a damage cone is formed at the impact site (damage), cleaving of the particle occurs at the critical point, while above the critical velocity the disc breaks into several pieces (fragmentation). In the limit of very high impact velocities the disc suffers complete disintegration (shattering) into many small fragments. In agreement with experimental results, fragment masses are found to follow the Gates-Gaudin-Schuhmann distribution (power law) with an exponent independent of the velocity and angle of impact. The velocity distribution of fragments exhibit an interesting anomalous scaling behavior when changing the impact velocity and the size of the disc., submitted to J. Phys: Condensed Matter special issue on Granular Media

In the present work a fiber bundle model with slowly relaxing fibers and a continuous range of interaction rule is presented. The elements of the model are linearly elastic until they break, but after breaking they undergo a slow relaxation process, giving rise to time-dependent local load on intact fibers. The load sharing among the fibers is described by a power law as a function of the distance which allows to recover the global and local load sharing approaches as limit cases. Our results demonstrate that the two universality classes of creep rupture are robust with respect to the details of the relaxation mechanisms of the fibers.

Objects subject to a steady load will often resist it for a long time before weakening and suddenly failing. This process can be studied by fiber bundle models, in which fibers fail in a random fashion that depends both upon the integrity of their neighbors and the load to which they are subjected. In this letter, we introduce a new fiber bundle model that allows us to examine how the behavior of such a system depends upon the range over which each fiber interacts with its neighbors. Using analytical and numerical arguments, we show that for all systems there is a critical load below which the system reaches equilibrium, and above which it fails in finite time. We consider how the time to failure depends upon how much the applied load exceeds the critical load, and find two universality classes. For short-range interactions, failure is abrupt, and time to failure is independent of load. For long-range interactions, the time to failure depends upon the excess load as a power law. The transition between these two universality classes is sharp as a function of the range of interaction. In both universality classes, the distribution of times between breaking events obeys a power law as the system creeps towards failure.

This paper reports on a combined experimental and theoretical study on the size dependency of tension strength of clear wood at loading parallel to %ber direction. The fracture behavior of the testedsoftwoodspecimens was foundto be rather brittle with low precursory activity anda statistical variation of the strength. The distribution of the strength values can be well %tted with a Weibull distribution distinguished by a shape parameter � ∼ 8 − 10. A signi%cant dependency of the mean strength of the material on the cross-sectional size of the specimens was obtained. The range of loadred istribution in clear woodsubjectedto tension parallel to %ber was assessed by the theoretical concept of %ber bundle models for %ber composites. Hereby the macroscopic behavior was modelled in terms of the microscopic damage process. c � 2003 Elsevier Science B.V. All rights reserved.

We investigate the statistics of record breaking events in the time series of crackling bursts in a fiber bundle model of the creep rupture of heterogeneous materials. In the model fibers break due to two mechanisms: slowly accumulating damage triggers bursts of immediate breakings analogous to acoustic emissions in experiments. The rupture process accelerates such that the size of breaking avalanches increases while the waiting time between consecutive events decreases towards failure. Record events are defined as bursts which have a larger size than all previous events in the time series. We analyze the statistics of records focusing on the limit of equal load sharing (ELS) of the model and compare the results to the record statistics of sequences of independent identically distributed random variables. Computer simulations revealed that the number of records grows with the logarithm of the event number except for the close vicinity of macroscopic failure where an exponential dependence is evidenced. The two regimes can be attributed to the dominance of disorder with small burst sizes and to stress enhancements giving rise efficient triggering of extended bursts, respectively. Both the size of records and the increments between consecutive record events are characterized by power law distributions with a common exponent 1.33 significantly different from the usual ELS burst size exponents of fiber bundles. The distribution of waiting times follows the same behavior, however, with two distinct exponents for low and high loads. Studying the evolution of records we identify a load dependent characteristic scale of the system which separates slow down and acceleration of record breaking as failure is approached., 12 pages, 6 figures

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 plate-like 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 non-universality of the complete mass distribution is the consequence of blending the contributions of universal partial processes., 9 pages, 8 figures

International Journal of Fracture, 154 (1-2), ISSN:0376-9429, ISSN:1573-2673

We study the statistical features of magnetic noise accompanying the dynamic fracture of steel samples during mixed type fracture, where the overall ductile character of crack propagation is interrupted by a sudden brittle jump. The structure of the voltage time series is investigated by identifying discrete peaks which correspond to elementary steps of the jerky cracking process. We show that the height, duration, area, and energy of peaks have power law distributions with exponents falling close to the corresponding values of pure ductile fracture. For single peaks a power law correlation of the height and area with the width is evidenced, however, the mixed nature of fracture gives rise to a crossover between two regimes of different exponents. The average pulse shape of micro-cracking events has a parabolic form with a right handed asymmetry similarly to quasi-static fracture propagation. The asymmetry emerges due to stress localization at the advancing crack front., Comment: 4 pages, 5 figures

We present a two-dimensional model of heterogeneous cohesive frictional solids where the material structure is idealized by a discrete granular particle assembly. Our discrete model is composed of convex polygons which are linked together by simple beams accounting for cohesive effects. Varying its parameters the model naturally interpolates between a continuous solid state and a discontinuous dry granular state of the material. We performed simulations of the quasi-static uniaxial loading with two different boundary conditions, and shearing of a solid. To provide a geometrical description of the damage state, and of the microscopic fracture events, different characteristic quantities will be introduced and their relevance will be examined. The results obtained from simulations are in satisfactory agreement with experimental observations.

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.

Fiber bundle models are one of the most fundamental modeling approaches for the investigation of the fracture of heterogeneous materials being able to capture a broad spectrum of damage mechanisms, loading conditions, and types of load sharing. In the framework of the fiber bundle model we introduce a kinetic Monte Carlo algorithm to investigate the thermally induced creep rupture of materials occurring under a constant external load. We demonstrate that the method overcomes several limitations of previous techniques and provides an efficient numerical framework at any load and temperature values. We show for both equal and localized load sharing that the computational time does not depend on the temperature; it is solely determined by the external load and the system size. In the limit of low load where the lifetime of the system diverges, the computational time saturates to a constant value. The method takes into account the secondary failures induced by subsequent load redistributions after breaking events, with the additional advantage that breaking avalanches always start from a single broken fiber.

An experimental and theoretical study of the deformation and break-up process of rings, formed by magnetic microspheres, under the application of an external magnetic field is reported in this Letter. When the external magnetic field is applied parallel to the plane of the rings, we found that the break-up process has three different outcomes depending on the way of application and time history of the external field: (a) deformation into a compact set of dipoles with a triangular lattice structure, (b) opening into a single chain, and (c) break-up into two chains with various relative sizes. A thorough theoretical investigation of the break-up process has been carried out based on computer simulations, taking into account solely the dipole–dipole and dipole–external field interactions, without thermal noise. The experimental results and the simulations are in good agreement.

We present a theoretical investigation of damage development in Ceramic Matrix Composites (CMC's) before catastrophic failure under gradual loading conditions. The aim of the work is to give a theoretical interpretation of the experimental results on the development of damage in the C/C-SiC composite obtained by cyclically loading, unloading, and reloading a planar specimen. Our approach is based on statistical, micromechanical models of the failure of CMC's in the framework of global load sharing, where the failure mechanism includes quasi-periodic matrix cracking, gradual breaking of fibers, and sliding of broken fibers with respect to the matrix. It is demonstrated that the non-linear characteristics of the constitutive behavior and the potential drop measured under uniaxial loading of a planar C/C-SiC specimen can be described with reasonable accuracy in the framework of statistical, micromechanical models.