Depinning transition in the failure of inhomogeneous brittle materials

A model for micro-front dynamics using a ϕ4 equation

In this article, we propose a numerical model based on the ϕ4 equation to simulate the dynamics of a front inside a microchannel that features an imperfection at a sidewall to different flow rates. The micro-front displays pinning-depinning phenomena without damped oscillations in the aftermath. To model this behavior, we propose a ϕ4 model with a localized external force and a damping coefficient. Numerical simulations with a constant damping coefficient show that the front displays pinning-depinning phenomena showing damped oscillations once the imperfection is overcome. Replacing the constant damping coefficient with a parabolic spatial function, we reproduce accurately the experimental front-defect interaction.

Theory and experiments for disordered elastic manifolds, depinning, avalanches, and sandpiles

Domain walls in magnets, vortex lattices in superconductors, contact lines at depinning, and many other systems can be modeled as an elastic system subject to quenched disorder. The ensuing field theory possesses a well-controlled perturbative expansion around its upper critical dimension. Contrary to standard field theory, the renormalization group (RG) flow involves a function, the disorder correlator Δ(w), and is therefore termed the functional RG. Δ(w) is a physical observable, the auto-correlation function of the center of mass of the elastic manifold. In this review, we give a pedagogical introduction into its phenomenology and techniques. This allows us to treat both equilibrium (statics), and depinning (dynamics). Building on these techniques, avalanche observables are accessible: distributions of size, duration, and velocity, as well as the spatial and temporal shape. Various equivalences between disordered elastic manifolds, and sandpile models exist: an elastic string driven at a point and the Oslo model; disordered elastic manifolds and Manna sandpiles; charge density waves and Abelian sandpiles or loop-erased random walks. Each of the mappings between these systems requires specific techniques, which we develop, including modeling of discrete stochastic systems via coarse-grained stochastic equations of motion, super-symmetry techniques, and cellular automata. Stronger than quadratic nearest-neighbor interactions lead to directed percolation, and non-linear surface growth with additional Kardar-Parisi-Zhang (KPZ) terms. On the other hand, KPZ without disorder can be mapped back to disordered elastic manifolds, either on the directed polymer for its steady state, or a single particle for its decay. Other topics covered are the relation between functional RG and replica symmetry breaking, and random-field magnets. Emphasis is given to numerical and experimental tests of the theory.

Effective Toughness of Heterogeneous Materials with Rate-Dependent Fracture Energy

We investigate the dynamic fracture of heterogeneous materials experimentally by measuring displacement fields as a rupture propagates through a periodic array of obstacles of controlled fracture energy. Our measurements demonstrate the applicability of the classical equation of motion of cracks at a discontinuity of fracture energy: the crack speed jumps at the entrance and exit of an obstacle, as predicted by the crack-tip energy balance within the brittle fracture framework. The speed jump amplitude is governed by the fracture energy contrast and by the combination of the rate dependency of the fracture energy and the inertia of the medium, which allows the crack to cross a fracture energy discontinuity at a constant energy release rate. This discontinuous dynamics and the rate dependence cause higher effective toughness, which governs the coarse-grained behavior of these cracks.

Dynamic crack growth along heterogeneous planar interfaces: Interaction with unidimensional strips

We examine theoretically and numerically fast propagation of a tensile crack along unidimensional strips with periodically evolving toughness. In such dynamic fracture regimes, crack front waves form and transport front disturbances along the crack edge at speed less than the Rayleigh wave speed and depending on the crack speed. In this configuration, standing front waves dictate the spatiotemporal evolution of the local crack front speed, which takes a specific scaling form. Analytical examination of both the short-time and long-time limits of the problem reveals the parameter dependency with strip wavelength, toughness contrast and overall fracture speed. Implications and generalization to unidimensional strips of arbitrary shape are lastly discussed.

Thermally rounded depinning of an elastic interface on a washboard potential

The thermal rounding of the depinning transition of an elastic interface sliding on a washboard potential is studied through analytic arguments and very accurate numerical simulations. We confirm the standard view that well below the depinning threshold the average velocity can be calculated considering thermally activated nucleation of defects. However, we find that the straightforward extension of this analysis to near or above the depinning threshold does not fully describe the physics of the thermally assisted motion. In particular, we find that exactly at the depinning point the average velocity does not follow a pure power law of the temperature as naively expected by the analogy with standard phase transitions but presents subtle logarithmic corrections. We explain the physical mechanisms behind these corrections and argue that they are nonpeculiar collective effects which may also apply to the case of interfaces sliding on uncorrelated disordered landscapes.

Correlations between avalanches in the depinning dynamics of elastic interfaces

We study the correlations between avalanches in the depinning dynamics of elastic interfaces driven on a random substrate. In the mean-field theory (the Brownian force model), it is known that the avalanches are uncorrelated. Here we obtain a simple field theory which describes the first deviations from this uncorrelated behavior in a ε=d_{c}-d expansion below the upper critical dimension d_{c} of the model. We apply it to calculate the correlations between (i) avalanche sizes (ii) avalanche dynamics in two successive avalanches, or more generally, in two avalanches separated by a uniform displacement W of the interface. For (i) we obtain the correlations of the total sizes, of the local sizes, and of the total sizes with given seeds (starting points). For (ii) we obtain the correlations of the velocities, of the durations, and of the avalanche shapes. In general we find that the avalanches are anticorrelated, the occurrence of a larger avalanche making more likely the occurrence of a smaller one, and vice versa. Examining the universality of our results leads us to conjecture several exact scaling relations for the critical exponents that characterize the different distributions of correlations. The avalanche size predictions are confronted to numerical simulations for a d=1 interface with short range elasticity. They are also compared to our recent related work on static avalanches (shocks). Finally we show that the naive extrapolation of our result into the thermally activated creep regime at finite temperature predicts strong positive correlations between the forward motion events, as recently observed in numerical simulations.

Depinning Dynamics of Crack Fronts

We investigate experimentally and theoretically the dynamics of a crack front during the microinstabilities taking place in heterogeneous materials between two successive equilibrium positions. We focus specifically on the spatiotemporal evolution of the front, as it relaxes to a straight configuration, after depinning from a single obstacle of controlled strength and size. We show that this depinning dynamics is not controlled by inertia, but instead by the rate dependency of the dissipative mechanisms taking place within the fracture process zone. This implies that the crack speed fluctuations around its average value v_{m} can be predicted from an overdamped equation of motion (v-v_{m})/v_{0}=[G-G_{c}(v_{m})]/G_{c}(v_{m}) involving the characteristic material speed v_{0}=G_{c}(v_{m})/G_{c}^{'}(v_{m}) that emerges from the variation of fracture energy with crack speed. Our findings pave the way to a quantitative description of the critical depinning dynamics of cracks in disordered solids and open up new perspectives for the prediction of the effective failure properties of heterogeneous materials.

Crack growth and energy dissipation in paper

Here, we follow the stable propagation of a roughening crack using simultaneously Digital Image Correlation and Infra-Red imaging. In a quasi-two-dimensional paper sample, the crack tip and ahead of that the fracture process zone follow the slowly, diffusively moving “hot spot” ahead of the tip. This also holds when the crack starts to roughen during propagation. The well-established intermittency of the crack advancement and the roughening of the crack in paper are thus subject to the dissipation and decohesion in the hot spot zone. They are therefore not only a result of the depinning of the crack in a heterogeneous material.

Glassy dynamics of landscape evolution

Soil is apparently solid as it moves downhill at glacial speeds, but can also liquefy from rain or earthquakes. This behavior is actually similar to that of glass, which creeps very slowly at low temperatures but becomes a liquid at higher temperatures. We develop a discrete granular-physics hillslope model, which shows that the similarities between soil and glass are more than skin deep. Despite the geologic and climatic complexity of natural environments, the shapes and erosion rates of hillsides over geologic timescales appear to be governed by generic dynamics characteristic of disordered and amorphous materials.

Soil creeps imperceptibly downhill, but also fails catastrophically to create landslides. Despite the importance of these processes as hazards and in sculpting landscapes, there is no agreed-upon model that captures the full range of behavior. Here we examine the granular origins of hillslope soil transport by discrete element method simulations and reanalysis of measurements in natural landscapes. We find creep for slopes below a critical gradient, where average particle velocity (sediment flux) increases exponentially with friction coefficient (gradient). At critical gradient there is a continuous transition to a dense-granular flow rheology. Slow earthflows and landslides thus exhibit glassy dynamics characteristic of a wide range of disordered materials; they are described by a two-phase flux equation that emerges from grain-scale friction alone. This glassy model reproduces topographic profiles of natural hillslopes, showing its promise for predicting hillslope evolution over geologic timescales.

Statistics of zero crossings in rough interfaces with fractional elasticity

We study numerically the distribution of zero crossings in one-dimensional elastic interfaces described by an overdamped Langevin dynamics with periodic boundary conditions. We model the elastic forces with a Riesz-Feller fractional Laplacian of order z=1+2ζ, such that the interfaces spontaneously relax, with a dynamical exponent z, to a self-affine geometry with roughness exponent ζ. By continuously increasing from ζ=-1/2 (macroscopically flat interface described by independent Ornstein-Uhlenbeck processes [Phys. Rev. 36, 823 (1930)PHRVAO0031-899X10.1103/PhysRev.36.823]) to ζ=3/2 (super-rough Mullins-Herring interface), three different regimes are identified: (I) -1/2<ζ<0, (II) 0<ζ<1, and (III) 1<ζ<3/2. Starting from a flat initial condition, the mean number of zeros of the discretized interface (I) decays exponentially in time and reaches an extensive value in the system size, or decays as a power-law towards (II) a subextensive or (III) an intensive value. In the steady state, the distribution of intervals between zeros changes from an exponential decay in (I) to a power-law decay P(ℓ)∼ℓ^{-γ} in (II) and (III). While in (II) γ=1-θ with θ=1-ζ the steady-state persistence exponent, in (III) we obtain γ=3-2ζ, different from the exponent γ=1 expected from the prediction θ=0 for infinite super-rough interfaces with ζ>1. The effect on P(ℓ) of short-scale smoothening is also analyzed numerically and analytically. A tight relation between the mean interval, the mean width of the interface, and the density of zeros is also reported. The results drawn from our analysis of rough interfaces subject to particular boundary conditions or constraints, along with discretization effects, are relevant for the practical analysis of zeros in interface imaging experiments or in numerical analysis.

Crack propagation through disordered materials as a depinning transition: A critical test of the theory

The dynamics of a planar crack propagating within a brittle disordered material is investigated numerically. The fracture front evolution is described as the depinning of an elastic line in a random field of toughness. The relevance of this approach is critically tested through the comparison of the roughness front properties, the statistics of avalanches, and the local crack velocity distribution with experimental results. Our simulations capture the main features of the fracture front evolution as measured experimentally. However, some experimental observations such as the velocity distribution are not consistent with the behavior of an elastic line close to the depinning transition. This discrepancy suggests the presence of another failure mechanism not included in our model of brittle failure.

Universal correlations between shocks in the ground state of elastic interfaces in disordered media

The ground state of an elastic interface in a disordered medium undergoes collective jumps upon variation of external parameters. These mesoscopic jumps are called shocks, or static avalanches. Submitting the interface to a parabolic potential centered at w, we study the avalanches which occur as w is varied. We are interested in the correlations between the avalanche sizes S_{1} and S_{2} occurring at positions w_{1} and w_{2}. Using the functional renormalization group (FRG), we show that correlations exist for realistic interface models below their upper critical dimension. Notably, the connected moment 〈S_{1}S_{2}〉^{c} is up to a prefactor exactly the renormalized disorder correlator, itself a function of |w_{2}-w_{1}|. The latter is the universal function at the center of the FRG; hence, correlations between shocks are universal as well. All moments and the full joint probability distribution are computed to first nontrivial order in an ε expansion below the upper critical dimension. To quantify the local nature of the coupling between avalanches, we calculate the correlations of their local jumps. We finally test our predictions against simulations of a particle in random-bond and random-force disorder, with surprisingly good agreement.

Length scales and pinning of interfaces

The pinning of interfaces and free discontinuities by defects and heterogeneities plays an important role in a variety of phenomena, including grain growth, martensitic phase transitions, ferroelectricity, dislocations and fracture. We explore the role of length scale on the pinning of interfaces and show that the width of the interface relative to the length scale of the heterogeneity can have a profound effect on the pinning behaviour, and ultimately on hysteresis. When the heterogeneity is large, the pinning is strong and can lead to stick-slip behaviour as predicted by various models in the literature. However, when the heterogeneity is small, we find that the interface may not be pinned in a significant manner. This shows that a potential route to making materials with low hysteresis is to introduce heterogeneities at a length scale that is small compared with the width of the phase boundary. Finally, the intermediate setting where the length scale of the heterogeneity is comparable to that of the interface width is characterized by complex interactions, thereby giving rise to a non-monotone relationship between the relative heterogeneity size and the critical depinning stress.

Crackling versus continuumlike dynamics in brittle failure

We study how the loading rate, specimen geometry, and microstructural texture select the dynamics of a crack moving through an heterogeneous elastic material in the quasistatic approximation. We find a transition, fully controlled by two dimensionless variables, between dynamics ruled by continuum fracture mechanics and crackling dynamics. Selection of the latter by the loading, microstructure, and specimen parameters is formulated in terms of scaling laws on the power spectrum of crack velocity. This analysis defines the experimental conditions required to observe crackling in fracture. Beyond failure problems, the results extend to a variety of situations described by models of the same universality class, e.g., the dynamics in wetting or of domain walls in amorphous ferromagnets.

Avalanche dynamics of elastic interfaces

Slowly driven elastic interfaces, such as domain walls in dirty magnets, contact lines wetting a nonhomogeneous substrate, or cracks in brittle disordered material proceed via intermittent motion, called avalanches. Here we develop a field-theoretic treatment to calculate, from first principles, the space-time statistics of instantaneous velocities within an avalanche. For elastic interfaces at (or above) their (internal) upper critical dimension d≥d(uc) (d(uc)=2,4 respectively for long-ranged and short-ranged elasticity) we show that the field theory for the center of mass reduces to the motion of a point particle in a random-force landscape, which is itself a random walk [Alessandro, Beatrice, Bertotti, and Montorsi (ABBM) model]. Furthermore, the full spatial dependence of the velocity correlations is described by the Brownian-force model (BFM) where each point of the interface sees an independent Brownian-force landscape. Both ABBM and BFM can be solved exactly in any dimension d (for monotonous driving) by summing tree graphs, equivalent to solving a (nonlinear) instanton equation. We focus on the limit of slow uniform driving. This tree approximation is the mean-field theory (MFT) for realistic interfaces in short-ranged disorder, up to the renormalization of two parameters at d=d(uc). We calculate a number of observables of direct experimental interest: Both for the center of mass, and for a given Fourier mode q, we obtain various correlations and probability distribution functions (PDF's) of the velocity inside an avalanche, as well as the avalanche shape and its fluctuations (second shape). Within MFT we find that velocity correlations at nonzero q are asymmetric under time reversal. Next we calculate, beyond MFT, i.e., including loop corrections, the one-time PDF of the center-of-mass velocity u[over ·] for dimension d<d(uc). The singularity at small velocity P(u[over ·])~1/u[over ·](a) is substantially reduced from a=1 (MFT) to a=1-2/9(4-d)+... (short-ranged elasticity) and a=1-4/9(2-d)+... (long-ranged elasticity). We show how the dynamical theory recovers the avalanche-size distribution, and how the instanton relates to the response to an infinitesimal step in the force.

Toughening and asymmetry in peeling of heterogeneous adhesives

The effective adhesive properties of heterogeneous thin films are characterized through a combined experimental and theoretical investigation. By bridging scales, we show how variations of elastic or adhesive properties at the microscale can significantly affect the effective peeling behavior of the adhesive at the macroscale. Our study reveals three elementary mechanisms in heterogeneous systems involving front propagation: (i) patterning the elastic bending stiffness of the film produces fluctuations of the driving force resulting in dramatically enhanced resistance to peeling; (ii) optimized arrangements of pinning sites with large adhesion energy are shown to control the effective system resistance, allowing the design of highly anisotropic and asymmetric adhesives; (iii) heterogeneities of both types result in front motion instabilities producing sudden energy releases that increase the overall adhesion energy. These findings open potentially new avenues for the design of thin films with improved adhesion properties, and motivate new investigations of other phenomena involving front propagation.

Average crack-front velocity during subcritical fracture propagation in a heterogeneous medium

We study the average velocity of crack fronts during stable interfacial fracture experiments in a heterogeneous quasibrittle material under constant loading rates and during long relaxation tests. The transparency of the material (polymethylmethacrylate) allows continuous tracking of the front position and relation of its evolution to the energy release rate. Despite significant velocity fluctuations at local scales, we show that a model of independent thermally activated sites successfully reproduces the large-scale behavior of the crack front for several loading conditions.

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.