The explosive breakup of rings exhibits a transition from segmentation to fragmentation and ultimately to shattering as the loading strain rate increases. In this work, we investigate how this transition is governed by the evolving structure of the underlying crack pattern. Using a two-dimensional discrete element model of ring-shaped shells, we uncover a rich structural evolution of the crack ensemble characterized by increasing complexity and connectivity. At low strain rates, segmentation arises from straight, radially oriented cracks that span the ring. As the strain rate increases, these cracks develop branching structures with a growing fractal dimension, leading to a connected crack network. This network eventually spans the entire ring and induces fragmentation. Quantitative analysis reveals that the fractal dimension of spanning cracks increases logarithmically with strain rate, approaching the embedding dimension at the onset of fragmentation. The orientation of cracks becomes increasingly isotropic, reflecting the loss of memory of the radial loading direction. The transition to fragmentation is signaled by the emergence of a dominant, system-spanning crack resembling percolation-like critical phenomena. Our results demonstrate that the structural transition of crack patterns provides a fundamental mechanism driving the breakup of shells under dynamic loading.

We investigate the fragmentation of ring-like brittle structures under explosive loading using a discrete element model. By systematically varying ring thickness and strain rate, we uncover a transition from one-dimensional (1D) segmentation to two-dimensional (2D) planar fragmentation and, ultimately, to complete shattering. This transition is driven by the effective dimensionality of the crack pattern, which evolves with increasing strain rate. We identify a critical ring thickness beyond which segmentation ceases, and fragmentation directly follows a power-law mass distribution characteristic of 2D systems. In the crossover regime, spanning and non-spanning fragments coexist, enabling control over the power-law exponent of the mass distribution. At very high strain rates, we observe a transition to complete shattering, where the system follows a novel scaling law relating the shattering strain rate to ring thickness. Our results provide fundamental insights into fragmentation universality classes and offer potential applications in space debris prediction, controlled detonation technologies, and materials engineering.

We investigate the shrinkage induced breakup of thin layers of heterogeneous materials attached to a substrate, a ubiquitous natural phenomenon with a wide range of potential applications. Focusing on the evolution of the fragment ensemble, we demonstrate that the system has two distinct phases: damage phase, where the layer is cracked, however, a dominant piece persists retaining the structural integrity of the layer, and a fragmentation phase, where the layer disintegrates into numerous small pieces. Based on finite size scaling we show that the transition between the two phases occurs at a critical damage analogous to continuous phase transitions. At the critical point a fully connected crack network emerges whose structure is controlled by the strength of adhesion to the substrate. In the strong adhesion limit, damage arises from random microcrack nucleation, resembling bond percolation, while weak adhesion facilitates stress concentration and the growth of cracks to large extensions. The critical exponents of the damage to fragmentation transition agree to a reasonable accuracy with those of two-dimensional bond percolation. Our findings provide a novel insights into the mechanism of shrinkage-induced cracking revealing generic scaling laws of the phenomenon.

We investigate the failure process of fiber bundles with structural disorder represented by the random misalignment of fibers. The strength of fibers is assumed to be constant so that misalignment is the only source of disorder, which results in a heterogeneous load distribution over fibers. We show by analytical calculations and computer simulations that increasing the amount of structural disorder a transition occurs from a perfectly brittle behavior with abrupt global failure to a quasibrittle phase where failure is preceded by breaking avalanches. The size distribution of avalanches follows a power-law functional form with a complex dependence of the exponent on the amount of disorder. In the vicinity of the critical point the avalanche exponent is 3/2; however, with increasing disorder a crossover emerges to a higher exponent 5/2. We show analytically that the mechanical behavior of the bundle of misaligned fiber with no strength disorder can be mapped to an equal load sharing fiber bundle of perfectly aligned fibers with properly selected strength disorder.

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.

Mechanical perturbation of thin layers of pastes on a substrate results in anisotropic cracking during desiccation which allows for a controlled generation of crack patterns with a high relevance for industrial applications. In the present paper we propose a modelling framework to capture the anisotropy induced by initial perturbations which imprint directional dependent material features into the layer. Our approach is based on a random tessellation of the layer using convex polygons which are made anisotropic by elongating or compressing them along the direction of perturbation. Physical properties of the particles’ cohesive contacts are determined by the geometrical characteristics of polygons giving rise to a well controlled anisotropy in the mechanical response of the material and in the structure of the fracture pattern emerging during shrinking. The main advantage of the approach is that both the structural disorder of the layer and the strength of anisotropy are controlled by a single parameter. Based on computer simulations we demonstrate that varying its two parameters the model reproduces the qualitative features of experimental findings and provides advantages over existing modelling approaches.

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.

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.

We present a two-dimensional discrete model of solids that allows us to follow the behavior of the solid body and of the fragments well beyond the formation of simple cracks. The model, consisting of polygonal cells connected via beams, is an extension of discrete models used to study granular flows. This modeling is particularly suited for the simulation of fracture and fragmentation processes. After calculating the macroscopic elastic moduli from the cell and beam parameters, we present a detailed study of an uniaxial compression test of a rectangular block, and of the dynamic fragmentation processes of solids in various experimental situations. The model proved to be successful in reproducing the experimentally observed subtleties of fragmenting solids.

We investigate how the macroscopic response and the size scaling of the ultimate strength of materials change when their local strength is sampled from a fat-tailed distribution and the degree of disorder is varied in a broad range. Using equal and localized load sharing in a fiber bundle model, we demonstrate that a transition occurs from a perfectly brittle to a quasi-brittle behaviour as the amount of disorder is gradually increased. When the load sharing is localized the high load concentration around failed regions make the system more prone to failure so that a higher degree of disorder is required for stabilization. Increasing the system size at a fixed degree of disorder an astonishing size effect is obtained: at small sizes the ultimate strength of the system increases with its size, the usual decreasing behaviour sets on only beyond a characteristic system size. The increasing regime of the size effect prevails even for localized load sharing, however, above the characteristic system size the load concentration results in a substantial strength reduction compared to equal load sharing. We show that an adequate explanation of the results can be obtained based on the extreme order statistics of fibers' strength.

We study how the competition of the disordered local strength and of the evolving inhomogeneous stress field affects the evolution of the series of breaking avalanches accompanying the fracture of heterogeneous materials. To generate fracture processes, we use a fiber bundle model of localized load sharing where the degree of strength disorder is controlled by varying two parameters of the distribution of the breaking threshold of fibers. Analyzing the record statistics of avalanches of breaking fibers, we demonstrate that both for low and high disorders the series of crackling events remains stationary until global failure making the collapse of the system unpredictable. Based on computer simulations, we determine a region of the parameter plane of strength disorder where global failure is preceded by an accelerating breaking activity. We show that the record avalanche with the longest lifetime can be used to identify the onset of acceleration of the fracture process towards the catastrophic failure. Comparison of the results to their equal load sharing counterparts reveals that the accelerating regime is shorter than in case of a homogeneous stress field due to the higher degree of brittleness of the system caused by stress localization.

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 study the failure process of fiber bundles on complex networks focusing on the effect of the degree of disorder of fibers' strength on the transition from localized to mean field behaviour. Starting from a regular square lattice we apply the Watts-Strogatz rewiring technique to introduce long range random connections in the load transmission network and analyze how the ultimate strength of the bundle and the statistics of the size of failure cascades change when the rewiring probability is gradually increased. Our calculations revealed that the degree of strength disorder of nodes of the network has a substantial effect on the localized to mean field transition. In particular, we show that the transition sets on at a finite value of the rewiring probability, which shifts to higher values as the degree of disorder is reduced. The transition is limited to a well defined range of disorder, so that there exists a threshold disorder of nodes' strength below which the randomization of the network structure does not provide any improvement neither of the overall load bearing capacity nor of the cascade tolerance of the system. At low strength disorder the fully random network is the most stable one, while at high disorder best cascade tolerance is obtained at a lower structural randomness. Based on the interplay of the network structure and strength disorder we construct an analytical argument which provides a reasonable description of the numerical findings.

EDITORIAL article Front. Phys., 18 November 2021Sec. Interdisciplinary Physics https://doi.org/10.3389/fphy.2021.795803

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.

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.

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.

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 present an extension of fiber bundle models to describe the mechanical response of systems which undergo a sequence of stick-slip cycles taking into account the changing stiffness and the fluctuating number of slip events of local material elements. After completing all stick-slip cycles allowed, fibers can either ultimately break or can keep their final stiffness leading to softening or hardening of the bundle, respectively. Under the assumption of global load sharing we derive analytic expressions for the constitutive response of the bundle with both quenched and annealed disorder of the failure thresholds where consecutive slips occur. Our calculations revealed that on the macro-scale the bundle exhibits a plastic behavior, which gets more pronounced when fibers undergo a higher number of stick-slip cycles with a gradually degrading stiffness. Releasing the load a permanent deformation remains, which increases monotonically for hardening bundles with the maximum deformation reached before unloading starts, however, in the softening case a non-monotonous behavior is obtained. We found that the macroscopic response of hardening bundles is more sensitive to fluctuations of the number of stick-slip cycles allowed than of the softening ones. The quenched and annealed disorder of failure thresholds gives rise to the same qualitative macro-scale behavior, however, the plastic response is found to be stronger in the annealed case.