Growing interfaces uncover universal fluctuations behind scale invariance

Stochastic motion of a point -- known as Brownian motion -- has many successful applications in science, thanks to its scale invariance and consequent universal features such as Gaussian fluctuations. In contrast, the stochastic motion of a line, though it is also scale-invariant and arises in nature as various types of interface growth, is far less understood. The two major missing ingredients are: an experiment that allows a quantitative comparison with theory and an analytic solution of the Kardar-Parisi-Zhang (KPZ) equation, a prototypical equation for describing growing interfaces. Here we solve both problems, showing unprecedented universality beyond the scaling laws. We investigate growing interfaces of liquid-crystal turbulence and find not only universal scaling, but universal distributions of interface positions. They obey the largest-eigenvalue distributions of random matrices and depend on whether the interface is curved or flat, albeit universal in each case. Our exact solution of the KPZ equation provides theoretical explanations.Comment: 5 pages, 3 figures, supplementary information available on Journal pag

Fracture in Three-Dimensional Fuse Networks

We report on large scale numerical simulations of fracture surfaces using random fuse networks for two very different disorders. There are some properties and exponents that are different for the two distributions, but others, notably the roughness exponents, seem universal. For the universal roughness exponent we found a value of zeta = 0.62 +/- 0.05. In contrast to what is observed in two dimensions, this value is lower than that reported in experimental studies of brittle fractures, and rules out the minimal energy surface exponent, 0.41 +/- 0.01.Comment: 4 pages, RevTeX, 5 figures, Postscrip

Roughness of Crack Interfaces in Two-Dimensional Beam Lattices

The roughness of crack interfaces is reported in quasistatic fracture, using an elastic network of beams with random breaking thresholds. For strong disorders we obtain 0.86(3) for the roughness exponent, a result which is very different from the minimum energy surface exponent, i.e., the value 2/3. A cross-over to lower values is observed as the disorder is reduced, the exponent in these cases being strongly dependent on the disorder.Comment: 9 pages, RevTeX, 3 figure

Anomalous roughening of wood fractured surfaces

Scaling properties of wood fractured surfaces are obtained from samples of three different sizes. Two different woods are studied: Norway spruce and Maritime pine. Fracture surfaces are shown to display an anomalous dynamic scaling of the crack roughness. This anomalous scaling behavior involves the existence of two different and independent roughness exponents. We determine the local roughness exponents ${\zeta}_{loc}$ to be 0.87 for spruce and 0.88 for pine. These results are consistent with the conjecture of a universal local roughness exponent. The global roughness exponent is different for both woods, $\zeta$ = 1.60 for spruce and $\zeta$ = 1.35 for pine. We argue that the global roughness exponent $\zeta$ is a good index for material characterization.Comment: 7 two columns pages plus 8 ps figures, uses psfig. To appear in Physical Review

Anomalous Roughening in Experiments of Interfaces in Hele-Shaw Flows with Strong Quenched Disorder

We report experimental evidences of anomalous kinetic roughening in the stable displacement of an oil-air interface in a Hele-Shaw cell with strong quenched disorder. The disorder consists on a random modulation of the gap spacing transverse to the growth direction (tracks). We have performed experiments varying average interface velocity and gap spacing, and measured the scaling exponents. We have obtained beta=0.50, beta*=0.25, alpha=1.0, alpha_l=0.5, and z=2. When there is no fluid injection, the interface is driven solely by capillary forces, and a higher value of beta around beta=0.65 is measured. The presence of multiscaling and the particular morphology of the interfaces, characterized by high slopes that follow a L\'evy distribution, confirms the existence of anomalous scaling. From a detailed study of the motion of the oil--air interface we show that the anomaly is a consequence of different local velocities over tracks plus the coupling in the motion between neighboring tracks. The anomaly disappears at high interface velocities, weak capillary forces, or when the disorder is not sufficiently persistent in the growth direction. We have also observed the absence of scaling when the disorder is very strong or when a regular modulation of the gap spacing is introduced.Comment: 14 pages, 17 figure

Size Effect in Fracture: Roughening of Crack Surfaces and Asymptotic Analysis

Recently the scaling laws describing the roughness development of fracture surfaces was proposed to be related to the macroscopic elastic energy released during crack propagation [Mor00]. On this basis, an energy-based asymptotic analysis allows to extend the link to the nominal strength of structures. We show that a Family-Vicsek scaling leads to the classical size effect of linear elastic fracture mechanics. On the contrary, in the case of an anomalous scaling, there is a smooth transition from the case of no size effect, for small structure sizes, to a power law size effect which appears weaker than the linear elastic fracture mechanics one, in the case of large sizes. This prediction is confirmed by fracture experiments on wood.Comment: 9 pages, 6 figures, accepted for publication in Physical Review

Statistical Physics of Fracture Surfaces Morphology

Experiments on fracture surface morphologies offer increasing amounts of data that can be analyzed using methods of statistical physics. One finds scaling exponents associated with correlation and structure functions, indicating a rich phenomenology of anomalous scaling. We argue that traditional models of fracture fail to reproduce this rich phenomenology and new ideas and concepts are called for. We present some recent models that introduce the effects of deviations from homogeneous linear elasticity theory on the morphology of fracture surfaces, succeeding to reproduce the multiscaling phenomenology at least in 1+1 dimensions. For surfaces in 2+1 dimensions we introduce novel methods of analysis based on projecting the data on the irreducible representations of the SO(2) symmetry group. It appears that this approach organizes effectively the rich scaling properties. We end up with the proposition of new experiments in which the rotational symmetry is not broken, such that the scaling properties should be particularly simple.Comment: A review paper submitted to J. Stat. Phy