Crack roughness and avalanche precursors in the random fuse model

Fluctuation-based fracture mechanics of heterogeneous materials

We present results of a hybrid analytical-simulation investigation of the fracture resistance of heterogeneous materials. We show that bond-energy fluctuations sampled by Monte Carlo simulations in the semigrand canonical ensemble provide a means to rationalize the complexity of heterogeneous fracture processes, encompassing probability and percolation theories of fracture. For a number of random and textured model materials, we derive upper and lower bounds of fracture resistance and link bond fracture fluctuations to statistical descriptors of heterogeneity, such as two-point correlation functions, to identify the origin of toughening mechanisms. This includes a shift from short- to long-range interactions of bond fracture processes in random systems to the transition from critical to subcritical bond fracture percolation in textured materials and the activation of toughness reserves at compliant interfaces. Induced by elastic mismatch, they connect to a number of disparate experimental observations, including toughening of brittle solids by deformable polymers or organics in, e.g., gas shale, nacre; stress-induced transformational toughening in ceramics; and toughening of sparse elastic networks in hydrogels, to name a few.

Unified scenario for the morphology of crack paths in two-dimensional disordered solids

A combined experimental and numerical investigation of the roughness of intergranular cracks in two-dimensional disordered solids is presented. We focus on brittle materials for which the characteristic length scale of damage is much smaller than the grain size. Surprisingly, brittle cracks do not follow a persistent path with a roughness exponent ζ≈0.6-0.7 as reported for a large range of materials. Instead, we show that they exhibit monoaffine scaling properties characterized by a roughness exponent ζ=0.50±0.05, which we explain theoretically from linear elastic fracture mechanics. Our findings support the description of the roughening process in two-dimensional brittle disordered solids by a random walk. Furthermore, they shed light on the failure mechanism at the origin of the persistent behavior with ζ≈0.6-0.7 observed for fractures in other materials, suggesting a unified scenario for the geometry of crack paths in two-dimensional disordered solids.

Transient flow modeling in fractured media using graphs

We describe a method to simulate transient fluid flows in fractured media using an approach based on graph theory. Our approach builds on past work where the graph-based approach was successfully used to simulate steady-state fluid flows in fractured media. We find a mean computational speedup of the order of 1400 from an ensemble of a 100 discrete fracture networks in contrast to the O(10^{4}) speedup that was obtained for steady-state flows earlier. However, the transient flows considered here involve an additional degree of complexity that was not present in the steady-state flows considered previously with a graph-based approach, that of time marching and solution of the flow equations within a time-stepping scheme. We verify our method with an analytical test case and demonstrate its use on a practical problem related to fluid flows in hydraulically fractured reservoirs. By enabling the study of transient flows, we create an opportunity for a wide set of possibilities where a steady-state approximation is not sufficient, such as the example motivated by hydraulic fracturing that we present here. This work validates the concept that graphs are able to reliably capture the topological properties of the fracture network and serve as effective surrogates in an uncertainty-quantification framework.

Fracture dynamics of correlated percolation on ionomer networks

This article presents a random network model to the study fracture dynamics on a scaffold of charged and elastic ionomer bundles that constitute the stable skeleton of a polymer electrolyte membrane. The swelling pressure upon water uptake by this system creates the internal stress under which ionomer bundles undergo breakage. Depending on the local stress and the strength of bundle-to-bundle correlations, different fracture regimes can be observed. We use kinetic Monte Carlo simulations to study these dynamics. The breakage of individual bundles is described with an exponential breakdown rule and the stress transfer from failed to intact bundles is assumed to exhibit a power-law-type spatial decay. A central property considered in the analysis is the frequency distribution of percolation thresholds, which is employed to analyze fracture regimes as a function of the stress and the effective range of stress transfer. Based on this distribution, we introduce an order parameter for the transition from random breakage to crack growth regimes. Moreover, as a practically important outcome, the time to fracture is analyzed as a descriptor for the lifetime of polymer electrolyte membranes.

Microscopic precursors of failure in soft matter

The mechanical properties of soft matter are of great importance in countless applications, in addition of being an active field of academic research. Given the relative ease with which soft materials can be deformed, their non-linear behavior is of particular relevance. Large loads eventually result in material failure. In this Perspective article, we discuss recent work aiming at detecting precursors of failure by scrutinizing the microscopic structure and dynamics of soft systems under various conditions of loading. In particular, we show that the microscopic dynamics is a powerful indicator of the ultimate fate of soft materials, capable of unveiling precursors of failure up to thousands of seconds before any macroscopic sign of weakening.

Avalanche precursors of failure in hierarchical fuse networks

We study precursors of failure in hierarchical random fuse network models which can be considered as idealizations of hierarchical (bio)materials where fibrous assemblies are held together by multi-level (hierarchical) cross-links. When such structures are loaded towards failure, the patterns of precursory avalanche activity exhibit generic scale invariance: irrespective of load, precursor activity is characterized by power-law avalanche size distributions without apparent cut-off, with power-law exponents that decrease continuously with increasing load. This failure behavior and the ensuing super-rough crack morphology differ significantly from the findings in non-hierarchical structures.

From damage percolation to crack nucleation through finite size criticality

We present a unified theory of fracture in disordered brittle media that reconciles apparently conflicting results reported in the literature. Our renormalization group based approach yields a phase diagram in which the percolation fixed point, expected for infinite disorder, is unstable for finite disorder and flows to a zero-disorder nucleation-type fixed point, thus showing that fracture has a mixed first order and continuous character. In a region of intermediate disorder and finite system sizes, we predict a crossover with mean-field avalanche scaling. We discuss intriguing connections to other phenomena where critical scaling is only observed in finite size systems and disappears in the thermodynamic limit.

Experimental study of the effect of disorder on subcritical crack growth dynamics

The growth dynamics of a single crack in a heterogeneous material under subcritical loading is an intermittent process, and many features of this dynamics have been shown to agree with simple models of thermally activated rupture. In order to better understand the role of material heterogeneities in this process, we study the subcritical propagation of a crack in a sheet of paper in the presence of a distribution of small defects such as holes. The experimental data obtained for two different distributions of holes are discussed in the light of models that predict the slowing down of crack growth when the disorder in the material is increased; however, in contradiction with these theoretical predictions, the experiments result in longer lasting cracks in a more ordered scenario. We argue that this effect is specific to subcritical crack dynamics and that the weakest zones between holes at close distance to each other are responsible for both the acceleration of the crack dynamics and the slightly different roughness of the crack path.

Damage-cluster distributions and size effect on strength in compressive failure

We investigate compressive failure of heterogeneous materials on the basis of a continuous progressive-damage model. The model explicitly accounts for tensile and shear local damage and reproduces the main features of compressive failure of brittle materials like rocks or ice. We show that the size distribution of damage clusters, as well as the evolution of an order parameter--the size of the largest damage cluster--argue for a critical interpretation of fracture. The compressive failure strength follows a normal distribution with a very small size effect on the mean strength, in good agreement with experiments.

Crack roughness in the two-dimensional random threshold beam model

We study the scaling of two-dimensional crack roughness using large scale beam lattice systems. Our results indicate that the crack roughness obtained using beam lattice systems does not exhibit anomalous scaling in sharp contrast to the simulation results obtained using scalar fuse lattices. The local and global roughness exponents (zetaloc and zeta, respectively) are equal to each other, and the two-dimensional crack roughness exponent is estimated to be zetaloc = zeta = 0.64+/-0.02 . Removal of overhangs (jumps) in the crack profiles eliminates even the minute differences between the local and global roughness exponents. Furthermore, removing these jumps in the crack profile completely eliminates the multiscaling observed in other studies. We find that the probability density distribution p[Deltah(l)] of the height differences Deltah(l)=[h(x+l)-h(x)] of the crack profile obtained after removing the jumps in the profiles follows a Gaussian distribution even for small window sizes (l) .

Crackling dynamics in material failure as the signature of a self-organized dynamic phase transition

We derive here a linear elastic stochastic description for slow crack growth in heterogeneous materials. This approach succeeds in reproducing quantitatively the intermittent crackling dynamics observed recently during the slow propagation of a crack along a weak heterogeneous plane of a transparent Plexiglas block [K. J. Måløy et al., Phys. Rev. Lett. 96, 045501 (2006)10.1103/PhysRevLett.96.045501]. In this description, the quasistatic failure of heterogeneous media appears as a self-organized critical phase transition. As such, it exhibits universal and to some extent predictable scaling laws, analogous to that of other systems such as, for example, magnetization noise in ferromagnets.

Energy dissipation statistics in the random fuse model

We study the statistics of the dissipated energy in the two-dimensional random fuse model for fracture under different imposed strain conditions. By means of extensive numerical simulations we compare different ways to compute the dissipated energy. In the case of an infinitely slow driving rate (quasistatic model), we find that the probability distribution of the released energy shows two different scaling regions separated by a sharp energy crossover. At low energies, the probability of having an event of energy E decays as approximately E(-1/2), which is robust and independent of the energy quantifier used (or lattice type). At high energies, fluctuations dominate the energy distribution, leading to a crossover to a different scaling regime, approximately E(-2.75), whenever the released energy is computed over the whole system. On the contrary, strong finite-size effects are observed if we consider only the energy dissipated at microfractures. In a different numerical experiment, the quasistatic dynamics condition is relaxed, so that the system is driven at finite strain load rates, and we find that the energy distribution decays as P(E) approximately E(-1) for all the energy range.

Mapping of the roughness exponent for the fuse model for fracture

The roughness exponent for fracture surfaces in the fuse model has been thought to be universal for narrow threshold distributions and has been important in the numerical studies of fracture roughness. We show that the fuse model gives a disorder dependent roughness exponent for narrow disorders when the lattice is influencing the fracture growth. When the influence of the lattice disappears, the local roughness exponent approaches zeta(local)=0.65+/-0.03 for distribution with a tail toward small thresholds, but with large jumps in the profiles giving corrections to scaling on small scales. For very broad disorders the distribution of jumps becomes a Lévy distribution and the Lévy characteristics contribute to the local roughness exponent.

Effect of disorder and notches on crack roughness

We analyze the effect of disorder and notches on crack roughness in two dimensions. Our simulation results based on large system sizes and extensive statistical sampling indicate that the crack surface exhibits a universal local roughness of zeta(loc)=0.71 and is independent of the initial notch size and disorder in breaking thresholds. The global roughness exponent scales as zeta=0.87 and is also independent of material disorder. Furthermore, we note that the statistical distribution of crack profile height fluctuations is also independent of material disorder and is described by a Gaussian distribution, albeit deviations are observed in the tails.

Crack surface roughness in three-dimensional random fuse networks

Using large system sizes with extensive statistical sampling, we analyze the scaling properties of crack roughness and damage profiles in the three-dimensional random fuse model. The analysis of damage profiles indicates that damage accumulates in a diffusive manner up to the peak load, and localization sets in abruptly at the peak load, starting from a uniform damage landscape. The global crack width scales as W approximately L(0.5) and is consistent with the scaling of localization length xi approximately L(0.5) used in the data collapse of damage profiles in the postpeak regime. This consistency between the global crack roughness exponent and the postpeak damage profile localization length supports the idea that the postpeak damage profile is predominantly due to the localization produced by the catastrophic failure, which at the same time results in the formation of the final crack. Finally, the crack width distributions can be collapsed for different system sizes and follow a log-normal distribution.

Crossover behavior in failure avalanches

Composite materials, with statistically distributed thresholds for breakdown of individual elements, are considered. During the failure process of such materials under external stress (load or voltage), avalanches consisting of simultaneous rupture of several elements occur, with a distribution D(Delta) of the magnitude Delta of such avalanches. The distribution is typically a power law D(Delta) proportional to Delta (-xi). For the systems we study here, a crossover behavior is seen between two power laws, with a small exponent xi in the vicinity of complete breakdown and a larger exponent xi for failures away from the breakdown point. We demonstrate this analytically for bundles of many fibers where the load is uniformly distributed among the surviving fibers. In this case xi=3/2 near the breakdown point and xi=5/2 away from it. The latter is known to be the generic behavior. This crossover is a signal of imminent catastrophic failure of the material. Near the breakdown point, avalanche statistics show nontrivial finite size scaling. We observe similar crossover behavior in a network of electric fuses, and find xi=2 near the catastrophic failure and xi=3 away from it. For this fuse model power dissipation avalanches show a similar crossover near breakdown.

Crossover behavior in burst avalanches: signature of imminent failure

The statistics of damage avalanches during a failure process typically follows a power law. When these avalanches are recorded only near the point at which the system fails catastrophically, one finds that the power law has an exponent which is different from that one finds if the recording of events starts away from the vicinity of catastrophic failure. We demonstrate this analytically for bundles of many fibers, with statistically distributed breakdown thresholds for the individual fibers and where the load is uniformly distributed among the surviving fibers. In this case the distribution D(Delta) of the avalanches (Delta) follows the power law Delta-xi with xi=3/2 near catastrophic failure and xi=5/2 away from it. We also study numerically square networks of electrical fuses and find xi=2.0 near catastrophic failure and xi=3.0 away from it. We propose that this crossover in xi may be used as a signal of imminent failure.

Statistical properties of fracture in a random spring model

Using large-scale numerical simulations, we analyze the statistical properties of fracture in the two-dimensional random spring model and compare it with its scalar counterpart: the random fuse model. We first consider the process of crack localization measuring the evolution of damage as the external load is raised. We find that, as in the fuse model, damage is initially uniform and localizes at peak load. Scaling laws for the damage density, fracture strength, and avalanche distributions follow with slight variations the behavior observed in the random fuse model. We thus conclude that scalar models provide a faithful representation of the fracture properties of disordered systems.