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.

Temporal evolution of failure avalanches of the fiber bundle model on complex networks.

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.

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.

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'.

Rupture cascades in a discrete element model of a porous sedimentary rock.

We investigate the scaling properties of the sources of crackling noise in a fully dynamic numerical model of sedimentary rocks subject to uniaxial compression. The model is initiated by filling a cylindrical container with randomly sized spherical particles that are then connected by breakable beams. Loading at a constant strain rate the cohesive elements fail, and the resulting stress transfer produces sudden bursts of correlated failures, directly analogous to the sources of acoustic emissions in real experiments. The source size, energy, and duration can all be quantified for an individual event, and the population can be analyzed for its scaling properties, including the distribution of waiting times between consecutive events. Despite the nonstationary loading, the results are all characterized by power-law distributions over a broad range of scales in agreement with experiments. As failure is approached, temporal correlation of events emerges accompanied by spatial clustering.

Scale-free bursting activity in shrinkage induced cracking.

Based on computer simulations of a realistic discrete element model we demonstrate that shrinkage induced cracking of thin layers of heterogeneous materials, generating spectacular crack patterns, proceeds in bursts. These crackling pulses are characterized by scale free distributions of size and duration, however, with non-universal exponents depending on the system size and shrinking rate. On the contrary, local avalanches composed of micro-cracking events with temporal and spatial correlation are found to obey a universal power law statistics. Most notably, we demonstrate that the observed non-universality of the integrated signal is the consequence of the temporal superposition of the underlying local avalanches, which pop up in an uncorrelated way in homogeneous systems. Our results provide an explanation of recent acoustic emission measurements on drying induced shrinkage cracking and may have relevance for the acoustic monitoring of the electro-mechanical degradation of battery electrodes.

Temporal and spacial evolution of bursts in creep rupture.

We investigate the temporal and spacial evolution of single bursts and their statistics emerging in heterogeneous materials under a constant external load. Based on a fiber bundle model we demonstrate that when the load redistribution is localized along a propagating crack front, the average temporal shape of pulses has a right-handed asymmetry; however, for long range interaction a symmetric shape with parabolic functional form is obtained. The pulse shape and spatial evolution of bursts proved to be correlated, which can be exploited in materials' testing. The probability distribution of the size and duration of bursts have power law behavior with a crossover to higher exponents as the load is lowered. The crossover emerges due to the competition of the slow and fast modes of local breaking being dominant at low and high loads, respectively.

Effect of the loading condition on the statistics of crackling noise accompanying the failure of porous rocks.

We test the hypothesis that loading conditions affect the statistical features of crackling noise accompanying the failure of porous rocks by performing discrete element simulations of the tensile failure of model rocks and comparing the results to those of compressive simulations of the same samples. Cylindrical samples are constructed by sedimenting randomly sized spherical particles connected by beam elements representing the cementation of granules. Under a slowly increasing external tensile load, the cohesive contacts between particles break in bursts whose size fluctuates over a broad range. Close to failure breaking avalanches are found to localize on a highly stressed region where the catastrophic avalanche is triggered and the specimen breaks apart along a spanning crack. The fracture plane has a random position and orientation falling most likely close to the centre of the specimen perpendicular to the load direction. In spite of the strongly different strengths, degrees of 'brittleness' and spatial structure of damage of tensile and compressive failure of model rocks, our calculations revealed that the size, energy and duration of avalanches, and the waiting time between consecutive events all obey scale-free statistics with power law exponents which agree within their error bars in the two loading cases.

Scaling laws of failure dynamics on complex networks.

The topology of the network of load transmitting connections plays an essential role in the cascading failure dynamics of complex systems driven by the redistribution of load after local breakdown events. In particular, as the network structure is gradually tuned from regular to completely random a transition occurs from the localized to mean field behavior of failure spreading. Based on finite size scaling in the fiber bundle model of failure phenomena, here we demonstrate that outside the localized regime, the load bearing capacity and damage tolerance on the macro-scale, and the statistics of clusters of failed nodes on the micro-scale obey scaling laws with exponents which depend on the topology of the load transmission network and on the degree of disorder of the strength of nodes. Most notably, we show that the spatial structure of damage governs the emergence of the localized to mean field transition: as the network gets gradually randomized failed clusters formed on locally regular patches merge through long range links generating a percolation like transition which reduces the load concentration on the network. The results may help to design network structures with an improved robustness against cascading failure.

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.

Impact-induced transition from damage to perforation.

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, farther 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.

Record statistics of bursts signals the onset of acceleration towards failure.

Forecasting the imminent catastrophic failure has a high importance for a large variety of systems from the collapse of engineering constructions, through the emergence of landslides and earthquakes, to volcanic eruptions. Failure forecast methods predict the lifetime of the system based on the time-to-failure power law of observables describing the final acceleration towards failure. We show that the statistics of records of the event series of breaking bursts, accompanying the failure process, provides a powerful tool to detect the onset of acceleration, as an early warning of the impending catastrophe. We focus on the fracture of heterogeneous materials using a fiber bundle model, which exhibits transitions between perfectly brittle, quasi-brittle, and ductile behaviors as the amount of disorder is increased. Analyzing the lifetime of record size bursts, we demonstrate that the acceleration starts at a characteristic record rank, below which record breaking slows down due to the dominance of disorder in fracturing, while above it stress redistribution gives rise to an enhanced triggering of bursts and acceleration of the dynamics. The emergence of this signal depends on the degree of disorder making both highly brittle fracture of low disorder materials, and ductile fracture of strongly disordered ones, unpredictable.

Fiber bundle model with stick-slip dynamics.

We propose a generic model to describe the mechanical response and failure of systems which undergo a series of stick-slip events when subjected to an external load. We model the system as a bundle of fibers, where single fibers can gradually increase their relaxed length with a stick-slip mechanism activated by the increasing load. We determine the constitutive equation of the system and show by analytical calculations that on the macroscale a plastic response emerges followed by a hardening or softening regime. Releasing the load, an irreversible permanent deformation occurs which depends on the properties of sliding events. For quenched and annealed disorder of the failure thresholds the same qualitative behavior is found, however, in the annealed case the plastic regime is more pronounced.

Cluster-cluster aggregation of Ising dipolar particles under thermal noise.

The cluster-cluster aggregation processes of Ising dipolar particles under thermal noise are investigated in the dilute condition. As the temperature increases, changes in the typical structures of clusters are observed from chainlike (D approximately 1) to crystalline (D approximately 2) through fractal structures (D approximately 1.45), where D is the fractal dimension. By calculating the bending energy of the chainlike structure, it is found that the transition temperature is associated with the energy gap between the chainlike and crystalline configurations. The aggregation dynamics changes from being dominated by attraction to diffusion involving changes in the dynamic exponent z=0.2 to 0.5. In the region of temperature where the fractal clusters grow, different growth rates are observed between charged and neutral clusters. Using the Smoluchowski equation with a twofold kernel, this hetero-aggregation process is found to result from two types of dynamics: the diffusive motion of neutral clusters and the weak attractive motion between charged clusters. The fact that changes in structures and dynamics take place at the same time suggests that transitions in the structure of clusters involve marked changes in the dynamics of the aggregation processes.

System-size-dependent avalanche statistics in the limit of high disorder.

We investigate the effect of the amount of disorder on the statistics of breaking bursts during the quasistatic 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 quasibrittle where macroscopic failure is preceded by a sequence of bursts. In the quasibrittle 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.

Damage process of a fiber bundle with a strain gradient.

We study the damage process of fiber bundles in a wedge-shape geometry which ensures a constant strain gradient. To obtain the wedge geometry we consider the three-point bending of a bar, which is modeled as two rigid blocks glued together by a thin elastic interface. The interface is discretized by parallel fibers with random failure thresholds, which become elongated when the bar is bent. Analyzing the progressive damage of the system we show that the strain gradient results in a rich spectrum of novel behavior of fiber bundles. We find that for weak disorder an interface crack is formed as a continuous region of failed fibers. Ahead of the crack a process zone develops which proved to shrink with increasing deformation, making the crack tip sharper as the crack advances. For strong disorder, failure of the system occurs as a spatially random sequence of breakings. Damage of the fiber bundle proceeds in bursts whose size distribution shows a power law behavior with a crossover from an exponent 2.5 to 2.0 as the disorder is weakened. The size of the largest burst increases as a power law of the strength of disorder with an exponent 23 and saturates for strongly disordered bundles.

Continuous damage fiber bundle model for strongly disordered materials.

We present an extension of the continuous damage fiber bundle model to describe the gradual degradation of highly heterogeneous materials under an increasing external load. The breaking of a fiber in the model is preceded by a sequence of partial failure events occurring at random threshold values. In order to capture the subsequent propagation and arrest of cracks, furthermore, the disorder of the number of degradation steps of material constituents, the failure thresholds of single fibers, are sorted into ascending order and their total number is a Poissonian distributed random variable over the fibers. Analytical and numerical calculations showed that the failure process of the system is governed by extreme value statistics, which has a substantial effect on the macroscopic constitutive behavior and on the microscopic bursting activity as well.

Thermodynamics of a binary monolayer of Ising dipolar particles. II. Effect of relative moment.

Thermodynamic behaviors of a binary monolayer of Ising dipolar particles are studied using particle dynamics simulation, varying the relative intensity between the upward and downward dipole moments. The orientational order of the solid phase changes from tetragonal to hexagonal as the moment ratio increases. On the basis of the arguments of the candidates for ground state structures, the energy of the structures are well estimated. The transition point is also determined theoretically, which is consistent with the value obtained from the simulation results. Critical condensation is also studied. While the system whose moment ratio is unity does not exhibit the gas-liquid critical condensation, the transition appears as the moment ratio changes. The local structure of the liquid phase is found to be characterized by the ground state of the tetramer. The above-mentioned results imply that the gas-liquid critical point comes close to the melting transition point as the local structure of the liquid phase becomes closer to the structure of the solid phase, and therefore, the critical condensation is vanished.

Time-dependent fracture under unloading in a fiber bundle model.

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.

Thermodynamics of a binary monolayer of Ising dipolar particles.

Thermodynamic behavior of a binary monolayer of Ising dipolar particles is studied using numerical simulation. The thermal equilibrium states of the system under the canonical ensemble are observed. The boundary of gas-liquid phase is determined from the power law growth of critical clusters. But that temperature is slightly lower than that of solidification. So it means that this system does not have a liquid phase. This system is also studied theoretically using virial expansion. The coefficients obtained from standard virial expansion, however, do not show any phase transition. An improvement of the virial expansion is also given by taking tetragonal local structures of alternate types of particles into consideration. Such structures are observed within critical clusters of our simulation. The thermodynamic state equation obtained from the improved virial expansion agrees well with the simulation result, and this expansion also shows that the critical point is almost at the same temperature with solidification. These results suggest that not simply the strength of attractive force by the dipole interaction but a typical configuration caused by the binary condition of this system plays a dominant role in phase transitions.