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 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 $p(Δh(\ell))$ of the height differences $Δh(\ell) = [h(x+\ell) - h(x)]$ of the crack profile obtained after removing the jumps in the profiles follows a Gaussian distribution even for small window sizes ($\ell$).
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 $ζ_{loc} = 0.71$ and is independent of the initial notch size and disorder in breaking thresholds. The global roughness exponent scales as $ζ= 0.87$ 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…
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 \…
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 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…
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.
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…
We consider numerically the roughness of a planar crack front within the long-range elastic string model, with a tunable disorder correlation length $ξ$. The problem is shown to have two important length scales, $ξ$ and the Larkin length $L_c$. Multiscaling of the crack front is observed for scales below $ξ$, provided that the disorder is strong enough. The asymptotic scaling with a roughness exponent $ζ\approx 0.39$ is recovered for scales larger than both $ξ$ and $L_c$. If $L_c > ξ$, these regimes are separated by a third regime characterized by the Larkin exponent $ζ_L \approx 0.5$. We disc…
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 $Δh(\ell) = [h(x+\ell) - h(x)]$…
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…
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…
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 study avalanches in a model for a planar crack propagating in a disordered medium. Due to long-range interactions, avalanches are formed by a set of spatially disconnected local clusters, the sizes of which are distributed according to a power law with an exponent $τ_{a}=1.5$. We derive a scaling relation $τ_a=2τ-1$ between the local cluster exponent $τ_a$ and the global avalanche exponent $τ$. For length scales longer than a cross-over length proportional to the Larkin length, the aspect ratio of the local clusters scales with the roughness exponent of the line model. Our analysis provides…
We propose a model for the formation of drying cracks in a viscoplastic material. In this model, we observe that when an external force is applied to a viscoplastic material before drying, the material memorizes the effect of the force as a plastic deformation. The formation of the drying cracks is influenced by this plastic deformation. This outcome clarifies the result of a recent experiments which demonstrated that a drying fracture pattern on a powder-water mixture depends on the manner in which an external force is applied before drying. We analytically express the position of the first c…
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 …
Roughness of driven elastic interfaces in random media is typically understood to be characterized by a single roughness exponent $ζ$. We show that at the depinning threshold, due to symmetry breaking caused by the direction of the driving force, elastic interfaces with local, long-range and mean-field elasticity exhibit asymmetric roughness. It is manifested as a skewed distribution of the local interface heights, and can be quantified by using detrended fluctuation analysis to compute a spectrum of local, segment-level scaling exponents. The asymmetry is observed as approximately linear depe…
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.
We investigate the origin of Paris's law, which states that the velocity of a crack at subcritical load grows like a power law, $da/dt \sim (ΔK)^{m}$, where $ΔK$ is the stress intensity factor amplitude. Starting from a damage accumulation function proportional to $(Δσ)^γ$, $Δσ$ being the stress amplitude, we show analytically that the asymptotic exponent $m$ can be expressed as a piecewise-linear function of the %damage accumulation exponent $γ$, namely, $m=6-2γ$ for $γ< γ_{c}$, and $m=γ$ for $γ\ge γ_{c}$, reflecting the existence of a critical value $γ_{c}=2$. %In this way, here we discover …
We present a detailed study of a two-dimensional minimal lattice model for the description of mud cracking in the limit of extremely thin layers. In this model each bond of the lattice is assigned to a (quenched) breaking threshold. Fractures proceed through the selection of the part of the material with the smallest breaking threshold. A local damaging rule is also implemented, by using two different types of weakening of the neighboring sites, corresponding to different physical situations. Some analytical results are derived through a probabilistic approach known as Run Time Statistics. In …
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…
We study numerically the depinning transition of driven elastic interfaces in a random-periodic medium with localized periodic-correlation peaks in the direction of motion. The analysis of the moving interface geometry reveals the existence of several characteristic lengths separating different length-scale regimes of roughness. We determine the scaling behavior of these lengths as a function of the velocity, temperature, driving force, and transverse periodicity. A dynamical roughness diagram is thus obtained which contains, at small length scales, the critical and fast-flow regimes typical o…
