Origin of the universal roughness exponent of brittle fracture surfaces:stress-weighted percolation in the damage zone

Inequality of avalanche sizes in models of fracture

Prediction of an imminent catastrophic event in a driven disordered system is of paramount importance-from the laboratory scale controlled fracture experiment to the largest scale of mechanical failure, i.e., earthquakes. It has long been conjectured that the statistical regularities in the energy emission time series mirror the "health" of such driven systems and hence have the potential for forecasting imminent catastrophe. Among other statistical regularities, a measure of how unequal avalanche sizes are is potentially a crucial indicator of imminent failure. The inequalities of avalanche sizes are quantified using inequality indices traditionally used in socioeconomic systems: the Gini index g, the Hirsch index h, and the Kolkata index k. It is shown analytically (for the mean-field case) and numerically (for the non-mean-field case) with models of quasi-brittle materials that the indices show universal behavior near the breaking points in such models and hence could serve as indicators of imminent breakdown of stressed disordered systems.

Combinatorial topology and geometry of fracture networks

A map is proposed from the space of planar surface fracture networks to a four-parameter mathematical space, summarizing the average topological connectivity and geometrical properties of a network idealized as a convex polygonal mesh. The four parameters are identified as the average number of nodes and edges, the angular defect with respect to regular polygons, and the isoperimetric ratio. The map serves as a low-dimensional signature of the fracture network and is visually presented as a pair of three-dimensional graphs. A systematic study is made of a wide collection of real crack networks for various materials, collected from different sources. To identify the characteristics of the real materials, several well-known mathematical models of convex polygonal networks are presented and worked out. These geometric models may correspond to different physical fracturing processes. The proposed map is shown to be discriminative, and the points corresponding to materials of similar properties are found to form closely spaced groups in the parameter space. Results for the real and simulated systems are compared in an attempt to identify crack networks of unknown materials.

Rigidity-Controlled Crossover: From Spinodal to Critical Failure

Failure in disordered solids is accompanied by intermittent fluctuations extending over a broad range of scales. The implied scaling has been previously associated with either spinodal or critical points. We use an analytically transparent mean-field model to show that both analogies are relevant near the brittle-to-ductile transition. Our study indicates that in addition to the strength of quenched disorder, an appropriately chosen global measure of rigidity (connectivity) can be also used to tune the system to criticality. By interpreting rigidity as a timelike variable we reveal an intriguing parallel between earthquake-type critical failure and Burgers turbulence.

Statistical physics perspective of fracture in brittle and quasi-brittle materials

We discuss the physics of fracture in terms of the statistical physics associated with the failure of elastic media under applied stresses in presence of quenched disorder. We show that the development and the propagation of fracture are largely determined by the strength of the disorder and the stress field around them. Disorder acts as nucleation centres for fracture. We discuss Griffith's law for a single crack-like defect as a source for fracture nucleation and subsequently consider two situations: (i) low disorder concentration of the defects, where the failure is determined by the extreme value statistics of the most vulnerable defect (nucleation regime) and (ii) high disorder concentration of the defects, where the scaling theory near percolation transition is applicable. In this regime, the development of fracture takes place through avalanches of a large number of tiny microfractures with universal statistical features. We discuss the transition from brittle to quasi-brittle behaviour of fracture with the strength of disorder in the mean-field fibre bundle model. We also discuss how the nucleation or percolation mode of growth of fracture depends on the stress distribution range around a defect. We discuss the corresponding numerical simulation results on random resistor and spring networks.This article is part of the theme issue 'Statistical physics of fracture and earthquakes'.

Force percolation of contractile active gels

Living systems provide a paradigmatic example of active soft matter. Cells and tissues comprise viscoelastic materials that exert forces and can actively change shape. This strikingly autonomous behavior is powered by the cytoskeleton, an active gel of semiflexible filaments, crosslinks, and molecular motors inside cells. Although individual motors are only a few nm in size and exert minute forces of a few pN, cells spatially integrate the activity of an ensemble of motors to produce larger contractile forces (∼nN and greater) on cellular, tissue, and organismal length scales. Here we review experimental and theoretical studies on contractile active gels composed of actin filaments and myosin motors. Unlike other active soft matter systems, which tend to form ordered patterns, actin-myosin systems exhibit a generic tendency to contract. Experimental studies of reconstituted actin-myosin model systems have long suggested that a mechanical interplay between motor activity and the network's connectivity governs this contractile behavior. Recent theoretical models indicate that this interplay can be understood in terms of percolation models, extended to include effects of motor activity on the network connectivity. Based on concepts from percolation theory, we propose a state diagram that unites a large body of experimental observations. This framework provides valuable insights into the mechanisms that drive cellular shape changes and also provides design principles for synthetic active materials.

Turbulent fracture surfaces: a footprint of damage percolation?

We show that a length scale ξ can be extracted from the spatial correlations of the "steep cliffs" that appear on a fracture surface. Above ξ, the slope amplitudes are uncorrelated and the fracture surface is monoaffine. Below ξ, long-range spatial correlations lead to a multifractal behavior of the surface, reminiscent of turbulent flows. Our results support a unifying conjecture for the geometry of fracture surfaces: for scales larger than ξ, the surface is the trace left by an elastic line propagating in a random medium, while for scales smaller than ξ, the highly correlated patterns on the surface result from the merging of interacting damage cavities.

Effect of the porosity on the fracture surface roughness of sintered materials: from anisotropic to isotropic self-affine scaling

To unravel how the microstructure affects the fracture surface roughness in heterogeneous brittle solids like rocks or ceramics, we characterized the roughness statistics of postmortem fracture surfaces in homemade materials of adjustable microstructure length scale and porosity, obtained by sintering monodisperse polystyrene beads. Beyond the characteristic size of disorder, the roughness profiles are found to exhibit self-affine scaling features evolving with porosity. Starting from a null value and increasing the porosity, we quantitatively modify the self-affine scaling properties from anisotropic (at low porosity) to isotropic (for porosity >10%).

High frequency monitoring reveals aftershocks in subcritical crack growth

By combining direct imaging and acoustic emission measurements, the subcritical propagation of a crack in a heterogeneous material is analyzed. Both methods show that the fracture proceeds through a succession of discrete events. However, the macroscopic opening of the fracture captured by the images results from the accumulation of more-elementary events detected by the acoustics. When the acoustic energy is cumulated over large time scales corresponding to the image acquisition rate, a similar statistics is recovered. High frequency acoustic monitoring reveals aftershocks responsible for a time scale dependent exponent of the power law energy distributions. On the contrary, direct imaging, which is unable to resolve these aftershocks, delivers a misleading exponent value.

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.

Nonuniversality of roughness exponent of quasistatic fracture surfaces

Numerous experiments have indicated that the fracture front (in three dimensions) and crack lines (in two dimensions) in disordered solids and rocklike materials is rough. It has been argued that the roughness exponent ζ is universal. Using extensive simulations of a two-dimensional model, we provide strong evidence that if extended correlations and anisotropy-two features that are prevalent in many materials-are incorporated in the models that are used in the numerical simulation of crack propagation, then ζ will vary considerably with the extent of the correlations and anisotropy. The results are consistent with recent experiments that also indicate deviations of ζ from its supposedly universal value, as well as with the data from rock samples.

Local dynamics of a randomly pinned crack front during creep and forced propagation: an experimental study

We have studied the propagation of a crack front along the heterogeneous weak plane of a transparent poly(methyl methacrylate) (PMMA) block using two different loading conditions: imposed constant velocity and creep relaxation. We have focused on the intermittent local dynamics of the fracture front for a wide range of average crack front propagation velocities spanning over four decades. We computed the local velocity fluctuations along the fracture front. Two regimes are emphasized: a depinning regime of high velocity clusters defined as avalanches and a pinning regime of very low-velocity creeping lines. The scaling properties of the avalanches and pinning lines (size and spatial extent) are found to be independent of the loading conditions and of the average crack front velocity. The distribution of local fluctuations of the crack front velocity are related to the observed avalanche size distribution. Space-time correlations of the local velocities show a simple diffusion growth behavior.

Stress concentration and size effect in fracture of notched heterogeneous material

We study theoretically and experimentally the effect of long-range correlations in the material microstructure on the stress concentration in the vicinity of the notch tip. We find that while in a fractal continuum the notch-tip displacements obey the classic asymptotic for a linear elastic continuum, the power-law decay of notch-tip stresses is controlled by the long-range density correlations. The corresponding notch-size effect on fracture strength is in good agreement with the experimental tests performed on notched sheets of different kinds of paper. In particular, we find that there is no stress concentration if the fractal dimension of the fiber network is D≤d-0.5, where d is the topological dimension of the paper sheet.

Fracture roughness in three-dimensional beam lattice systems

We study the scaling of three-dimensional crack roughness using large-scale beam lattice systems. Our results for prenotched samples indicate that the crack surface is statistically isotropic, with the implication that experimental findings of anisotropy of fracture surface roughness in directions parallel and perpendicular to crack propagation is not due to the scalar or vectorial elasticity of the model. In contrast to scalar fuse lattices, beam lattice systems do not exhibit anomalous scaling or an extra dependence of roughness on system size. The local and global roughness exponents (ζ(loc) and ζ, respectively) are equal to each other, and the three-dimensional crack roughness exponent is estimated to be ζ(loc)=ζ=0.48±0.03 . This closely matches the roughness exponent observed outside the fracture process zone. The probability density distribution p[Δh(ℓ)] of the height differences Δh(ℓ)=[h(x+ℓ)-h(x)] of the crack profile follows a Gaussian distribution, in agreement with experimental results.

Effects of finite probe size on self-affine roughness measurements

The roughness of fracture surfaces exhibits self-affinity for a wide variety of materials and loading conditions. The universality and the range of scales over which this regime extends are still debated. The topography of these surfaces is however often investigated with a finite contact probe. In this case, we show that the correlation function of the roughness can only be measured down to a length scale Deltax{c} which depends on the probe size R, the Hurst exponent zeta of the surface and its topothesy l, and exhibits spurious behavior at smaller scales. First, we derive the dependence of Deltax{c} on these parameters from a simple scaling argument. Then, we verify this dependence numerically. Finally, we establish the relevance of this analysis from AFM measurements on an experimental glass fracture surface and provide a metrological procedure for roughness measurements.

Crack propagation through phase-separated glasses: effect of the characteristic size of disorder

We perform fracture experiments on nanoscale phase-separated glasses and measure crack surface roughness by atomic force microscopy. The ability of tuning the phase domain size by thermal treatment allows us to test thoroughly the predictions of crack front depinning models about the scaling properties of crack surface roughness. It appears that, in the range of validity of these depinning models developed for the fracture of brittle materials, our experimental results show a quantitative agreement with theoretical predictions: Beyond the characteristic size of disorder, the roughness of crack surfaces obeys the logarithmic scaling early predicted by Ramanathan, Ertaş, and Fisher [Phys. Rev. Lett. 79, 873 (1997)].

Transient damage spreading and anomalous scaling in mortar crack surfaces

The scaling properties of a post-mortem mortar crack surface are investigated. The root mean square of the height fluctuations is found to obey anomalous scaling properties, but with three exponents, two of them characterizing the local roughness ( zeta approximately 0.79 and zetae approximately 0.41 ) and the third one driving the global roughness (zetag approximately 1.60) . The critical exponent zeta approximately 0.79 is conjectured to reflect damage screening occurring for length scales smaller than the process zone size, while the exponent zetae approximately 0.41 characterizes roughness at larger length scales, i.e., at length scales where the material can be considered as linear elastic. Finally, we argue that the global roughness exponent could be material dependent contrary to both local roughness exponents ( zeta approximately 0.8 and zetae approximately 0.4 ) which can be considered as universal.

Self-affinity in the gradient percolation problem

We study the scaling properties of the solid-on-solid front of the infinite cluster in two-dimensional gradient percolation. We show that such an object is self-affine with a Hurst exponent equal to 23 up to a cutoff length approximately g{-4/7}, where g is the gradient. Beyond this length scale, the front position has the character of uncorrelated noise. Importantly, the self-affine behavior is robust even after removing local jumps of the front. The previously observed multiaffinity is due to the dominance of overhangs at small distances in the structure function. This is a crossover effect.

Scaling exponents for fracture surfaces in homogeneous glass and glassy ceramics

We investigate the scaling properties of postmortem 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 zeta approximately 0.75 in homogeneous glass and zeta approximately 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 the process zone, respectively. A model derived from linear elastic fracture mechanics in the quasistatic approximation succeeds to reproduce the scaling exponents observed in glassy ceramics. The critical exponents observed in homogeneous glass are conjectured to reflect the damage screening occurring for length scales below the size of the process zone.

Low self-affine exponents of fractured glass ceramics surfaces

The geometry of postmortem 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 microstructure and is also practically independent on the porosity phi in the range investigated (3%< or =phi< or =26%) as well as on the bead diameter d and of the crack growth velocity. In contrast, the roughness amplitude normalized by d increases linearly with phi while it is still independent, within experimental error, of d and of the crack propagation velocity. An interpretation of this variation is suggested in terms of a transition from transgranular to intergranular fracture propagation with no influence, however, on the exponent zeta.

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.

Mean-field theory of localization in a fuse model

We propose a mean-field theory for the localization of damage in a quasistatic fuse model on a cylinder. Depending on the quenched disorder distribution of the fuse thresholds, we show analytically that the system can either stay in a percolation regime up to breakdown, or start at some imposed current, to localize starting from the smallest scale (lattice spacing), or instead go to a diffuse localization regime where damage starts to concentrate in bands of width scaling as the width of the system, but remains diffuse at smaller scales. Depending on the nature of the quenched disorder on the fuse thresholds, we derive analytically the phase diagram of the system separating these regimes and the current levels for the onset of these possible localizations. We compare these predictions to numerical results.

Local waiting time fluctuations along a randomly pinned crack front

The propagation of an interfacial crack along a heterogeneous weak plane of a transparent Plexiglas block is followed using a high resolution fast camera. We show that the fracture front dynamics is governed by local and irregular avalanches with very large size and velocity fluctuations. We characterize the intermittent dynamics observed, i.e., the local pinnings and depinnings of the crack front by measuring the local waiting time fluctuations along the crack front during its propagation. The deduced local front line velocity distribution exhibits a power law behavior, P(v) alpha v-eta with eta=2.55+/-0.15, for velocities v larger than the average front speed <v>. The burst size distribution is also a power law, P(S) alpha S-gamma with gamma=1.7+/-0.1. Above a characteristic length scale of disorder Ld approximately 15 microm, the avalanche clusters become anisotropic providing an estimate of the roughness exponent of the crack front line, H=0.66.

Two-dimensional scaling properties of experimental fracture surfaces

The self-affine properties of postmortem fracture surfaces in silica glass and aluminum alloy were investigated through the 2D height-height correlation function. They are observed to exhibit anisotropy. The roughness, dynamic, and growth exponents are determined and shown to be the same for the two materials, irrespective of the crack velocity. These exponents are conjectured to be universal.

Void formation and roughening in slow fracture

Slow crack propagation in ductile, and in certain brittle materials, appears to take place via the nucleation of voids ahead of the crack tip due to plastic yields, followed by the coalescence of these voids. Postmortem analysis of the resulting fracture surfaces of ductile and brittle materials on the microm-mm and the nm scales, respectively, reveals self-affine cracks with anomalous scaling exponent zeta approximately = 0.8 in 3 dimensions and zeta approximately = 0.65 in 2 dimensions. In this paper we present an analytic theory based on the method of iterated conformal maps aimed at modelling the void formation and the fracture growth, culminating in estimates of the roughening exponents in 2 dimensions. In the simplest realization of the model we allow one void ahead of the crack, and address the robustness of the roughening exponent. Next we develop the theory further, to include two voids ahead of the crack. This development necessitates generalizing the method of iterated conformal maps to include doubly connected regions (maps from the annulus rather than the unit circle). While mathematically and numerically feasible, we find that the employment of the stress field as computed from elasticity theory becomes questionable when more than one void is explicitly inserted into the material. Thus further progress in this line of research calls for improved treatment of the plastic dynamics.

Interacting damage models mapped onto Ising and percolation models

We introduce a class of damage models on regular lattices with isotropic interactions between the broken cells of the lattice. Quasi-static fiber bundles are an example. The interactions are assumed to be weak, in the sense that the stress perturbation from a broken cell is much smaller than the mean stress in the system. The system starts intact with a surface-energy threshold required to break any cell sampled from an uncorrelated quenched-disorder distribution. The evolution of this heterogeneous system is ruled by Griffith's principle which states that a cell breaks when the release in potential (elastic) energy in the system exceeds the surface-energy barrier necessary to break the cell. By direct integration over all possible realizations of the quenched disorder, we obtain the probability distribution of each damage configuration at any level of the imposed external deformation. We demonstrate an isomorphism between the distributions so obtained and standard generalized Ising models, in which the coupling constants and effective temperature in the Ising model are functions of the nature of the quenched-disorder distribution and the extent of accumulated damage. In particular, we show that damage models with global load sharing are isomorphic to standard percolation theory and that damage models with a local load sharing rule are isomorphic to the standard Ising model, and draw consequences thereof for the universality class and behavior of the autocorrelation length of the breakdown transitions corresponding to these models. We also treat damage models having more general power-law interactions, and classify the breakdown process as a function of the power-law interaction exponent. Last, we also show that the probability distribution over configurations is a maximum of Shannon's entropy under some specific constraints related to the energetic balance of the fracture process, which firmly relates this type of quenched-disorder based damage model to standard statistical mechanics.

Correlation length exponent in the three-dimensional fuse network

We present numerical measurements of the critical correlation length exponent nu in the three-dimensional fuse model. Using sufficiently broad threshold distributions to ensure that the system is the strong-disorder regime, we determine nu to be nu=0.83+/-0.04 based on analyzing the fluctuations of the survival probability. This value is different from that of ordinary percolation, which is 0.88.

Roughening of fracture surfaces: the role of plastic deformation

Post mortem analysis of fracture surfaces of ductile and brittle materials on the microm-mm and the nm scales, respectively, reveal self-affine cracks with anomalous scaling exponent zeta approximately 0.8 in three dimensions and zeta approximately 0.65 in two dimensions. Attempts to use elasticity theory to explain this result failed, yielding exponent zeta approximately 0.5 up to logarithms. We show that when the cracks propagate via plastic void formations in front of the tip, followed by void coalescence, the void positions are positively correlated to yield exponents higher than 0.5.