We study anomalous scaling and multiscaling of two-dimensional crack profiles in the random fuse model using both periodic and open boundary conditions. Our large scale and extensively sampled numerical results reveal the importance of crack branching and coalescence of microcracks, which induce jumps in the solid-on-solid crack profiles. Removal of overhangs (jumps) in the crack profiles eliminates the multiscaling observed in earlier studies and reduces anomalous scaling. We find that the probability density distribution $p(Δh(\ell))$ of the height differences $Δh(\ell) = [h(x+\ell) - h(x)]$ of the crack profile obtained after removing the jumps in the profiles has the scaling form $p(Δh(\ell)) = <Δh^2(\ell)>^{-1/2} ~f(\frac{Δh(\ell)}{<Δh^2(\ell)>^{1/2}})$, and follows a Gaussian distribution even for small bin sizes $\ell$. The anomalous scaling can be summarized with the scaling relation $[\frac{<Δh^2(\ell)>^{1/2}}{<Δh^2(L/2)>^{1/2}}]^{1/ζ_{loc}} + \frac{(\ell-L/2)^2}{(L/2)^2} = 1$, where $<Δh^2(L/2)>^{1/2} \sim L^ζ$.
We measure the roughness exponent and the correlation length exponent of a stress-weighted percolation process in the central force model in 2D. The roughness exponent is found to be zeta = 0.75 \pm 0.03 and the correlation length exponent is found to be nu = 1.7 \pm 0.3. This result supports a conjecture that the fracture roughness for large scales is controlled by a stress weighted percolation process, and the fracture roughness can by calculated from the correlation length exponent by zeta = 2*nu/(1+2*nu). We also compare global and local measurements of the fracture roughness and do not fi…
We study the fracture surface of three dimensional samples through a model for quasi-static fractures known as Born Model. We find for the roughness exponent a value of 0.5 expected for ``small length scales'' in microfracturing experiments. Our simulations confirm that at small length scales the fracture can be considered as quasi-static. The isotropy of the roughness exponent on the crack surface is also shown. Finally, considering the crack front, we compute the roughness exponents for longitudinal and transverse fluctuations of the crack line (both 0.5). They result in agreement with exper…
Using a multi-resolution technique, we analyze large in-plane fracture fronts moving slowly between two sintered Plexiglas plates. We find that the roughness of the front exhibits two distinct regimes separated by a crossover length scale $δ^*$. Below $δ^*$, we observe a multi-affine regime and the measured roughness exponent $ζ_{\parallel}^{-} = 0.60\pm 0.05$ is in agreement with the coalescence model. Above $δ^*$, the fronts are mono-affine, characterized by a roughness exponent $ζ_{\parallel}^{+} = 0.35\pm0.05$, consistent with the fluctuating line model. We relate the crossover length scal…
Roughness of i-AlPdMn cleaved surfaces are presently analysed. From the atomic scale to 2-3 nm, they are shown to exhibit scaling properties hiding the cluster (0.45 nm) aperiodic structure. These properties are quantitatively similar to those observed on various disordered materials, albeit on other ranges of length scales. These properties are interpreted as the signature of damage mechanisms occurring within a 2-3 nm wide zone at the crack tip. The size of this process zone finds its origin in the local temperature elevation at the crack tip. For the very first time, this effect is reported…
We report results on the interrelation between driving force, roughness exponent, branching and crack speed in a finite element model. We show that for low applied loadings the crack speed reaches the values measured in the experiments, and the crack surface roughness is compatible with logarithmic scaling. At higher loadings, the crack speed increases, and the crack roughness exponent approaches the value measured at short length scales in experiments. In the case of high anisotropy, the crack speed is fully compatible with the values measured in experiments on anisotropic materials, and we a…
We analyze the scaling of the crack roughness and of avalanche precursors in the two dimensional random fuse model by numerical simulations, employing large system sizes and extensive sample averaging. We find that the crack roughness exhibits anomalous scaling, as recently observed in experiments. The roughness exponents ($ζ$, $ζ_{loc}$) and the global width distributions are found to be universal with respect to the lattice geometry. Failure is preceded by avalanche precursors whose distribution follows a power law up to a cutoff size. While the characteristic avalanche size scales as $s_0 \…
In an effort to investigate the link between failure mechanisms and the geometry of fractures of compacted grains materials, a detailed statistical analysis of the surfaces of fractured Fontainebleau sandstones has been achieved. The roughness of samples of different widths W is shown to be self affine with an exponent zeta=0.46 +- 0.05 over a range of length scales ranging from the grain size d up to an upper cut-off length ξ= 0.15 W. This low zeta value is in agreement with measurements on other sandstones and on sintered materials. The probability distributions P(delta z,delta h) of the var…
In this paper, a computational model in (2+1)-dimensions which simulates the rupture process of a fibrous material submitted to a constant force $F$, is analyzed. The roughness exponent $ζ$ at the boundary that separates two failure regimes, catastrophic and slowly shredding, is evaluated. In the catastrophic (dynamic) regime the initial strain creates a crack which percolates rapidly through the material. In the slowly shredding (quasi-static) regime several cracks of small size appear in all parts of the material, the rupture process is slow and any single crack percolates the sample. At the…
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 …
We investigate propagating fronts in disordered media that belong to the universality class of wetting contact lines and planar tensile crack fronts. We derive from first principles their nonlinear equations of motion, using the generalized Griffith criterion for crack fronts and three standard mobility laws for contact lines. Then we study their roughness using the self-consistent expansion. When neglecting the irreversibility of fracture and wetting processes, we find a possible dynamic rough phase with a roughness exponent of $ζ=1/2$ and a dynamic exponent of z=2. When including the irrever…
{\it Critical slowing down} associated with the iterative solvers close to the critical point often hinders large-scale numerical simulation of fracture using discrete lattice networks. This paper presents a block circlant preconditioner for iterative solvers for the simulation of progressive fracture in disordered, quasi-brittle materials using large discrete lattice networks. The average computational cost of the present alorithm per iteration is $O(rs log s) + delops$, where the stiffness matrix ${\bf A}$ is partioned into $r$-by-$r$ blocks such that each block is an $s$-by-$s$ matrix, and …
In this paper, we introduce a model for fracture in fibrous materials that takes into account the rupture height of the fibers, in contrast with previous models. Thus, we obtain the profile of the fracture and calculate its roughness, defined as the variance around the mean height. We investigate the relationship between the fracture roughness and the fracture toughness.
The correlations among elements that break in random fuse network fracture are studied, for disorder strong enough to allow for volume damage before final failure. The growth of microfractures is found to be uncorrelated above a lengthscale, that increases as the the final breakdown is approached. Since the fuse network strength decreases with sample size, asymptotically the process resembles more and more mean-field-like (``democratic fiber bundle'') fracture. This is found from the microscopic dynamics of avalanches or microfractures, from a study of damage localization via entropy, and from…
The permeability of two-dimensional fractures with self-affine fractal roughness is studied via analytic arguments and numerical simulations. The limit where the roughness amplitude is small compared with average fracture aperture is analyzed by a perturbation method, while in the opposite case of narrow aperture, we use heuristic arguments based on lubrication theory. Numerical simulations, using the lattice Boltzmann method, are used to examine the complete range of aperture sizes, and confirm the analytic arguments.
We simulate the propagation of a planar crack in a quasi-two dimensional fuse model, confining the crack between two horizontal plates. We investigate the effect on the roughness of microcrack nucleation ahead of the main crack and study the structure of the damage zone. The two dimensional geometry introduces a characteristic length in the problem, limiting the crack roughness. The damage ahead of the crack does not appear to change the scaling properties of the model, which are well described by gradient percolation.
The roughness properties of two-dimensional fracture surfaces as created by the slow failure of random fuse networks are considered and compared to yield surfaces of perfect plasticity with similar disorder. By studying systems up to a linear size L=350 it is found that in the cases studied the fracture surfaces exhibit self-affine scaling with a roughness exponent close to 2/3, which is asymptotically exactly true for plasticity though finite-size effects are evident for both. The overlap of yield or minimum energy and fracture surfaces with exactly the same disorder configuration is shown to…
The high computational cost involved in modeling of the progressive fracture simulations using large discrete lattice networks stems from the requirement to solve {\it a new large set of linear equations} every time a new lattice bond is broken. To address this problem, we propose an algorithm that combines the multiple-rank sparse Cholesky downdating algorithm with the rank-p inverse updating algorithm based on the Sherman-Morrison-Woodbury formula for the simulation of progressive fracture in disordered quasi-brittle materials using discrete lattice networks. Using the present algorithm, the…
We study the roughness of a crack interface in a sheet of paper. We distinguish between slow (sub-critical) and fast crack growth regimes. We show that the fracture roughness is different in the two regimes using a new method based on a multifractal formalism recently developed in the turbulence literature. Deviations from monofractality also appear to be different in both regimes.
We argue that fracture surfaces may exhibit anomalous dynamic scaling properties akin to what occurs in some models of kinetic roughening. We determine the complete scaling behavior of the local fluctuations of a brittle fracture in a granite block from experimental data. We obtain a global roughness exponent $χ= 1.2$ which differs from the local one, $χ_{loc} = 0.79$. Implications on fracture physics are discussed.
We analyse the statistical distribution function for the height fluctuations of brittle fracture surfaces using extensive experimental data sampled on widely different materials and geometries. We compare a direct measurement of the distribution to a new analysis based on the structure functions. For length scales $δ$ larger than a characteristic scale $δ^*$, we find that the distribution of the height increments $Δh = h(x+ δ) -h(x)$ is Gaussian. Self-affinity enters through the scaling of the standard deviation $σ$, which is proportional to $δ^ζ$ with a unique roughness exponent. Below the sc…
The roughness of fracture surfaces has been shown to exhibit self-affne scale invariance for a wide variety of materials and loading conditions. The range of scales over which this regime extends remains a matter of debate, together with the universality of the associated exponents. The topography of these surfaces is however often investigated with a contact probe that is larger than the micro-structure. In this case, we show that the correlation function of the roughness and the corresponding Hurst exponent $ζ$ can only be measured down to a length scale $Δxc$ which depends on the probe size…
We address the role of the nature of material disorder in determining the roughness of cracks which grow by damage nucleation and coalescence ahead of the crack tip. We highlight the role of quenched and annealed disorders in relation to the length scales $d$ and $ξ_c$ associated with the disorder and the damage nucleation respectively. In two related models, one with quenched disorder in which $d \simeq ξ_c $, the other with annealed disorder in which $d \ll ξ_c$, we find qualitatively different roughening properties for the resulting cracks in 2-dimensions. The first model results in random …
We study the acoustic emission produced by micro-cracks using a two-dimensional disordered lattice model of dynamic fracture, which allows to relate the acoustic response to the internal damage of the sample. We find that the distributions of acoustic energy bursts decays as a power law in agreement with experimental observations. The scaling exponents measured in the present dynamic model can related to those obtained in the quasi-static random fuse 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 ($ζ_{loc}$ and $ζ$, respectively) are equal to each other, and the two-dimensional crack roughness exponent is estimated to be $ζ_{loc} = ζ= 0.64 \pm 0.02$. Removal of overhangs (jumps) in the crack profiles eliminates even the minute differences between the local …
Fractures are a critical process in how materials wear, weaken, and fail whose unpredictable behavior can have dire consequences. While the behavior of smooth cracks in ideal materials is well understood, it is assumed that for real, heterogeneous systems, fracture propagation is complex, generating rough fracture surfaces that are highly sensitive to specific details of the medium. Here we show how fracture roughness and material heterogeneity are inextricably connected via a simple framework. Studying hydraulic fractures in brittle hydrogels that have been supplemented with microbeads or gly…
We consider the morphology of two dimensional cracks observed in experimental results obtained from paper samples and compare these results with the numerical simulations of the random fuse model (RFM). We demonstrate that the data obey multiscaling at small scales but cross over to self-affine scaling at larger scales. Next, we show that the roughness exponent of the random fuse model is recovered by a simpler model that produces a connected crack, while a directed crack yields a different result, close to a random walk. We discuss the multiscaling behavior of all these models.
The geometry of post mortem rough fracture surfaces of porous glass ceramics made of sintered glass beads is shown experimentally to be self-affine with an exponent zeta=0.40 (0.04) remarkably lower than the 'universal' value zeta=0.8 frequently measured for many materials. This low value of zeta is similar to that found for sandstone samples of similar micro structure and is also practically independent on the porosity phi in the range investigated (3% < phi < 26%) as well as on the bead diameter d and of the crack growth velocity. In contrast, the roughness amplitude normalized by d increase…
We investigate the scaling properties of post-mortem fracture surfaces in silica glass and glassy ceramics. In both cases, the 2D height-height correlation function is found to obey Family-Viseck scaling properties, but with two sets of critical exponents, in particular a roughness exponent $ζ\simeq 0.75$ in homogeneous glass and $ζ\simeq 0.4$ in glassy ceramics. The ranges of length-scales over which these two scalings are observed are shown to be below and above the size of process zone respectively. A model derived from Linear Elastic Fracture Mechanics (LEFM) in the quasistatic approximati…
The scaling properties of post-mortem fracture surfaces of brittle (silica glass), ductile (aluminum alloy) and quasi-brittle (mortar and wood) materials have been investigated. These surfaces, studied far from the initiation, were shown to be self-affine. However, the Hurst exponent measured along the crack direction is found to be different from the one measured along the propagation direction. More generally, a complete description of the scaling properties of these surfaces call for the use of the 2D height-height correlation function that involves three exponents zeta = 0.75, beta = 0.6 a…
Tracer diffusion and hydrodynamic dispersion in two-dimensional fractures with self-affine roughness is studied by analytic and numerical methods. Numerical simulations were performed via the lattice-Boltzmann approach, using a new boundary condition for tracer particles that improves the accuracy of the method. The reduction in the diffusive transport, due to the fractal geometry of the fracture surfaces, is analyzed for different fracture apertures. In the limit of small aperture fluctuations we derive the correction to the diffusive coefficient in terms of the tortuosity, which accounts for…
Transport properties of three-dimensional self-affine rough fractures are studied by means of an effective-medium analysis and numerical simulations using the Lattice-Boltzmann method. The numerical results show that the effective-medium approximation predicts the right scaling behavior of the permeability and of the velocity fluctuations, in terms of the aperture of the fracture, the roughness exponent and the characteristic length of the fracture surfaces, in the limit of small separation between surfaces. The permeability of the fractures is also investigated as a function of the normal and…
We study the stability and roughness of propagating cracks in heterogeneous brittle two-dimensional elastic materials. We begin by deriving an equation of motion describing the dynamics of such a crack in the framework of Linear Elastic Fracture Mechanics, based on the Griffith criterion and the Principle of Local Symmetry. This result allows us to extend the stability analysis of Cotterell and Rice to disordered materials. In the stable regime we find stochastic crack paths. Using tools of statistical physics we obtain the power spectrum of these paths and their probability distribution funct…
The fracture strength distribution of materials is often described in terms of the Weibull law which can be derived by using extreme value statistics if elastic interactions are ignored. Here, we consider explicitly the interplay between elasticity and disorder and test the asymptotic validity of the Weibull distribution through numerical simulations of the two-dimensional random fuse model. Even when the local fracture strength follows the Weibull distribution, the global failure distribution is dictated by stress enhancement at the tip of the cracks and sometimes deviates from the Weibull la…
This article reports results concerning the fracture of a 2d triangular lattice of atoms linked by springs. The lattice is submitted to controlled strain tests and the influence of both porosity and temperature on failure is investigated. The porosity is found on one hand to decrease the stiffness of the material but on the other hand it increases the deformation sustained prior to failure. Temperature is shown to control the ductility due to the presence of cavities that grow and merge. The rough surfaces resulting from the propagation of the crack exhibit self-affine properties with a roughn…
State Key Laboratory of Solid Lubrication, Lanzhou Institute of Chemical Physics, Chinese Academy of Sciences（中国科学院兰州化学物理研究所固体润滑国家重点实验室）；Peter Grünberg Institute (PGI-1), Forschungszentrum Jülich（于利希研究中心彼得·格林伯格研究所）；MultiscaleConsulting；INSA Lyon, CNRS, LaMCoS（里昂国立应用科学学院法国国家科学研究中心多尺度接触力学实验室）
We investigate the role of disorder on the fracturing process of heterogeneous materials by means of a two-dimensional fuse network model. Our results in the extreme disorder limit reveal that the backbone of the fracture at collapse, namely the subset of the largest fracture that effectively halts the global current, has a fractal dimension of $1.22 \pm 0.01$. This exponent value is compatible with the universality class of several other physical models, including optimal paths under strong disorder, disordered polymers, watersheds and optimal path cracks on uncorrelated substrates, hulls of …
In this work, we present an experimental investigation of the fuse model. Our main goal was to study the influence of the disorder on the fracture process. The experimental apparatus used consisted of an $L\times L$ square lattice with fuses placed on each bond of the lattice. Two types of materials were used as fuses: copper and steel wool wires. The lattice composed only by copper wires varied from a weakly disordered system to a strongly disordered one. The lattice formed only by steel wool wires corresponded to a strongly disordered one. The experimental procedure consisted of applying a p…
Stress enhancement in the vicinity of brittle cracks makes the macro-scale failure properties extremely sensitive to the micro-scale material disorder. Therefore: (i) Fracturing systems often display a jerky dynamics, so-called crackling noise, with seemingly random sudden energy release spanning over a broad range of scales, reminiscent of earthquakes; (ii) Fracture surfaces exhibit roughness at scales much larger than that of material micro-structure. Here, I provide a critical review of experiments and simulations performed in this context, highlighting the existence of universal scaling fe…
The blasting of brittle materials with sharp particles is an important fabrication technology in many industrial processes. In particular, for micro-systems, it allows the production of devices with feature sizes down to few tens of microns. An important parameter of this process is the surface roughness of post-blasted surfaces. In this work the scaling properties of Pyrex glass and silicon surfaces after bombardment with alumina particles is investigated. The targets were bombarded at normal incidence using alumina particles with two different average sizes, $29μm$ and $9μm$, respectively. T…
