Spatial scaling in fracture propagation in dilute systems

The geometry of fracture patterns in a dilute elastic network is explored using molecular dynamics simulation. The network in two dimensions is subjected to a uniform strain which drives the fracture to develop by the growth and coalescence of the vacancy clusters in the network. For strong dilution, it has been shown earlier that there exists a characteristic time $t_c$ at which a dynamical transition occurs with a power law divergence (with the exponent $z$) of the average cluster size. Close to $t_c$, the growth of the clusters is scale-invariant in time and satisfies a dynamical scaling law. This paper shows that the cluster growth near $t_c$ also exhibits spatial scaling in addition to the temporal scaling. As fracture develops with time, the connectivity length $\xi$ of the clusters increses and diverges at $t_c$ as $\xi \sim (t_c-t)^{-\nu}$, with $\nu = 0.83 \pm 0.06$. As a result of the scale-invariant growth, the vacancy clusters attain a fractal structure at $t_c$ with an effective dimensionality $d_f \sim 1.85 \pm 0.05$. These values are independent (within the limit of statistical error) of the concentration (provided it is sufficiently high) with which the network is diluted to begin with. Moreover, the values are very different from the corresponding values in qualitatively similar phenomena suggesting a different universality class of the problem. The values of $\nu$ and $d_f$ supports the scaling relation $z=\nu d_f$ with the value of $z$ obtained before.Comment: A single ps file (6 figures included), 12 pages, to appear in Physica

Self-Similar Law of Energy Release before Materials Fracture

A general law of energy release is derived for stressed heterogeneous materials, being valid from the starting moment of loading till the moment of materials fracture. This law is obtained by employing the extrapolation technique of the self-similar approximation theory. Experiments are accomplished measuring the energy release for industrial composite samples. The derived analytical law is confronted with these experimental data as well as with the known experimental data for other materials.Comment: Latex, 15 pages, no figure

New model for surface fracture induced by dynamical stress

We introduce a model where an isotropic, dynamically-imposed stress induces fracture in a thin film. Using molecular dynamics simulations, we study how the integrated fragment distribution function depends on the rate of change and magnitude of the imposed stress, as well as on temperature. A mean-field argument shows that the system becomes unstable for a critical value of the stress. We find a striking invariance of the distribution of fragments for fixed ratio of temperature and rate of change of the stress; the interval over which this invariance holds is determined by the force fluctuations at the critical value of the stress.Comment: Revtex, 4 pages, 4 figures available upon reques

Slider-Block Friction Model for Landslides: Application to Vaiont and La Clapiere Landslides

Accelerating displacements preceding some catastrophic landslides have been found empirically to follow a time-to-failure power law, corresponding to a finite-time singularity of the velocity $v \sim 1/(t_c-t)$ [{\it Voight}, 1988]. Here, we provide a physical basis for this phenomenological law based on a slider-block model using a state and velocity dependent friction law established in the laboratory and used to model earthquake friction. This physical model accounts for and generalizes Voight's observation: depending on the ratio $B/A$ of two parameters of the rate and state friction law and on the initial frictional state of the sliding surfaces characterized by a reduced parameter $x_i$, four possible regimes are found. Two regimes can account for an acceleration of the displacement. We use the slider-block friction model to analyze quantitatively the displacement and velocity data preceding two landslides, Vaiont and La Clapi\`ere. The Vaiont landslide was the catastrophic culmination of an accelerated slope velocity. La Clapi\`ere landslide was characterized by a peak of slope acceleration that followed decades of ongoing accelerating displacements, succeeded by a restabilizing phase. Our inversion of the slider-block model on these data sets shows good fits and suggest to classify the Vaiont (respectively La Clapi\`ere) landslide as belonging to the velocity weakening unstable (respectively strengthening stable) sliding regime.Comment: shortened by focusing of the frictional model, Latex document with AGU style file of 14 pages + 11 figures (1 jpeg photo of figure 6 given separately) + 1 tabl

Oscillatory Finite-Time Singularities in Finance, Population and Rupture

We present a simple two-dimensional dynamical system where two nonlinear terms, exerting respectively positive feedback and reversal, compete to create a singularity in finite time decorated by accelerating oscillations. The power law singularity results from the increasing growth rate. The oscillations result from the restoring mechanism. As a function of the order of the nonlinearity of the growth rate and of the restoring term, a rich variety of behavior is documented analytically and numerically. The dynamical behavior is traced back fundamentally to the self-similar spiral structure of trajectories in phase space unfolding around an unstable spiral point at the origin. The interplay between the restoring mechanism and the nonlinear growth rate leads to approximately log-periodic oscillations with remarkable scaling properties. Three domains of applications are discussed: (1) the stock market with a competition between nonlinear trend-followers and nonlinear value investors; (2) the world human population with a competition between a population-dependent growth rate and a nonlinear dependence on a finite carrying capacity; (3) the failure of a material subjected to a time-varying stress with a competition between positive geometrical feedback on the damage variable and nonlinear healing.Comment: Latex document of 59 pages including 20 eps figure