Avalanches and extreme value statistics in interfacial crackling dynamics

Microscopic interplay of temperature and disorder of a one-dimensional elastic interface

Elastic interfaces display scale-invariant geometrical fluctuations at sufficiently large lengthscales. Their asymptotic static roughness then follows a power-law behavior, whose associated exponent provides a robust signature of the universality class to which they belong. The associated prefactor has instead a nonuniversal amplitude fixed by the microscopic interplay between thermal fluctuations and disorder, usually hidden below experimental resolution. Here we compute numerically the roughness of a one-dimensional elastic interface subject to both thermal fluctuations and a quenched disorder with a finite correlation length. We evidence the existence of a power-law regime at short lengthscales. We determine the corresponding exponent ζ_{dis} and find compelling numerical evidence that, contrarily to available analytic predictions, one has ζ_{dis}<1. We discuss the consequences on the temperature dependence of the roughness and the connection with the asymptotic random-manifold regime at large lengthscales. We also discuss the implications of our findings for other systems such as the Kardar-Parisi-Zhang equation and the Burgers turbulence.

Statistical properties of avalanches via the c-record process

We study the statistics of avalanches, as a response to an applied force, undergone by a particle hopping on a one-dimensional lattice where the pinning forces at each site are independent and identically distributed (i.i.d.), each drawn from a continuous f(x). The avalanches in this model correspond to the interrecord intervals in a modified record process of i.i.d. variables, defined by a single parameter c>0. This parameter characterizes the record formation via the recursive process R_{k}>R_{k-1}-c, where R_{k} denotes the value of the kth record. We show that for c>0, if f(x) decays slower than an exponential for large x, the record process is nonstationary as in the standard c=0 case. In contrast, if f(x) has a faster than exponential tail, the record process becomes stationary and the avalanche size distribution π(n) has a decay faster than 1/n^{2} for large n. The marginal case where f(x) decays exponentially for large x exhibits a phase transition from a nonstationary phase to a stationary phase as c increases through a critical value c_{crit}. Focusing on f(x)=e^{-x} (with x≥0), we show that c_{crit}=1 and for c<1, the record statistics is nonstationary. However, for c>1, the record statistics is stationary with avalanche size distribution π(n)∼n^{-1-λ(c)} for large n. Consequently, for c>1, the mean number of records up to N steps grows algebraically ∼N^{λ(c)} for large N. Remarkably, the exponent λ(c) depends continuously on c for c>1 and is given by the unique positive root of c=-ln(1-λ)/λ. We also unveil the presence of nontrivial correlations between avalanches in the stationary phase that resemble earthquake sequences.

Thermally activated intermittent dynamics of creeping crack fronts along disordered interfaces

We present a subcritical fracture growth model, coupled with the elastic redistribution of the acting mechanical stress along rugous rupture fronts. We show the ability of this model to quantitatively reproduce the intermittent dynamics of cracks propagating along weak disordered interfaces. To this end, we assume that the fracture energy of such interfaces (in the sense of a critical energy release rate) follows a spatially correlated normal distribution. We compare various statistical features from the obtained fracture dynamics to that from cracks propagating in sintered polymethylmethacrylate (PMMA) interfaces. In previous works, it has been demonstrated that such an approach could reproduce the mean advance of fractures and their local front velocity distribution. Here, we go further by showing that the proposed model also quantitatively accounts for the complex self-affine scaling morphology of crack fronts and their temporal evolution, for the spatial and temporal correlations of the local velocity fields and for the avalanches size distribution of the intermittent growth dynamics. We thus provide new evidence that an Arrhenius-like subcritical growth is particularly suitable for the description of creeping cracks.

Effects of external noise on threshold-induced correlations in ferromagnetic systems

In the present paper we investigate the impact of the external noise and detection threshold level on the simulation data for the systems that evolve through metastable states. As a representative model of such systems we chose the nonequilibrium athermal random-field Ising model with two types of the external noise, uniform white noise and Gaussian white noise with various different standard deviations, imposed on the original response signal obtained in model simulations. We applied a wide range of detection threshold levels in analysis of the signal and show how these quantities affect the values of exponent γ_{S/T} (describing the scaling of the average avalanche size with duration), the shift of waiting time between the avalanches, and finally the collapses of the waiting time distributions. The results are obtained via extensive numerical simulations on the equilateral three-dimensional cubic lattices of various sizes and disorders.

State and Rate Dependent Contact Line Dynamics over an Aging Soft Surface

We report direct atomic-force-microscope measurements of capillary force hysteresis (CFH) of a circular contact line (CL) formed on a long glass fiber, which is coated with a thin layer of soft polymer film and intersects a water-air interface. The measured CFH shows a distinct overshoot for the depinning of a static CL, and the overshoot amplitude grows logarithmically with both the hold time τ and fiber speed V. A unified model based on the slow growth of a wetting ridge and force-assisted barrier crossing is developed to explain the observed time (or state) and speed (or rate) dependent CL depinning dynamics over an aging soft surface. The experimental findings have important implications to a common class of problems involving depinning dynamics in a defect or roughness landscape, such as friction of solid interfaces.

Statistical physics of fracture and earthquakes

Manifestations of emergent properties in stressed disordered materials are often the result of an interplay between strong perturbations in the stress field around defects. The collective response of a long-ranged correlated multi-component system is an ideal playing field for statistical physics. Hence, many aspects of such collective responses in widely spread length and energy scales can be addressed by the tools of statistical physics. In this theme issue, some of these aspects are treated from various angles of experiments, simulations and analytical methods, and connected together by their common base of complex-system dynamics.This article is part of the theme issue 'Statistical physics of fracture and earthquakes' .

Thermally activated crack fronts propagating in pinning disorder: simultaneous brittle/creep behaviour depending on scale

We study theoretically the propagation of a crack front in mode I along an interface in a disordered elastic medium, with a numerical model considering a thermally activated rheology, toughness disorder and long-range elastic interactions. This model reproduces not only the large-scale dynamics of the crack front position in fast or creep loading regimes, but also the small-scale self-affine behaviour of the front. Two different scaling laws are predicted for the front morphology, with a Hurst exponent of 0.5 at small scales and a logarithmic scaling law at large scales, consistently with experiments. The prefactor of these scaling laws is expressed as a function of the temperature, and of the quenched disorder characteristics. The cross-over between these regimes is expressed as a function of the quenched disorder amplitude, and is proportional to the average energy release rate, and to the inverse of temperature. This model captures as well the experimentally observed local velocity fluctuation probability distribution, with a high-velocity tail P(v)∼v -2.6 This feature is shown to arise when the quenched disorder is sufficiently large, whereas smaller toughness fluctuations lead to a lognormal-like velocity distribution. Overall, the system is shown to obey a scaling determined by two distinct mechanisms as a function of scale: namely, the large scales display fluctuations similar to an elastic line in an annealed noise excited as the average front travels through the pinning landscape, while small scales display a balance between thresholds in possible elastic forces and quenched disorder.This article is part of the theme issue 'Statistical physics of fracture and earthquakes'.