Weakest-link scaling and extreme events in finite-sized systems

Weakest-link scaling is used in the reliability analysis of complex systems. It is characterized by the extensivity of the hazard function instead of the entropy. The Weibull distribution is the archetypical example of weakest-link scaling, and it describes variables such as the fracture strength of brittle materials, maximal annual rainfall, wind speed and earthquake return times. We investigate two new distributions that exhibit weakest-link scaling, i.e., a Weibull generalization known as the κ-Weibull and a modified gamma probability function that we propose herein. We show that in contrast with the Weibull and the modified gamma, the hazard function of the κ -Weibull is non-extensive, which is a signature of inter-dependence between the links. We also investigate the impact of heterogeneous links, modeled by means of a stochastic Weibull scale parameter, on the observed probability distribution

Scaling strength distributions in quasi-brittle materials from micro- to macro-scales: A computational approach to modeling Nature-inspired structural ceramics

International audienceThis paper presents an approach to predict the strength distribution of quasi-brittle materials across multiple length-scales, with emphasis on Nature-inspired ceramic structures. It permits the computation of the failure probability of any structure under any mechanical load, solely based on considerations of the microstructure and its failure properties by naturally incorporating the statistical and size-dependent aspects of failure. We overcome the intrinsic limitations of single periodic unit-based approaches by computing the successive failures of the material components and associated stress redistributions on arbitrary numbers of periodic units. For large size samples, the microscopic cells are replaced by a homogenized continuum with equivalent stochastic and damaged constitutive behavior. After establishing the predictive capabilities of the method, and illustrating its potential relevance to several engineering problems, we employ it in the study of the shape and scaling of strength distributions across differing length-scales for a particular quasi-brittle system. We find that the strength distributions display a Weibull form for samples of size approaching the periodic unit; however, these distributions become closer to normal with further increase in sample size before finally reverting to a Weibull form for macroscopic sized samples. In terms of scaling, we find that the weakest link scaling applies only to microscopic, and not macroscopic scale, samples. These findings are discussed in relation to failure patterns computed at different size-scales

Dynamic Length Scale and Weakest Link Behavior in Crystal Plasticity

Plastic deformation of heterogeneous solid structures is often characterized by random intermittent local plastic events. On the mesoscale this feature can be represented by a spatially fluctuating local yield threshold. Here we study the validity of such an approach and the ideal choice for the size of the representative volume element for crystal plasticity in terms of a discrete dislocation model. We find that the number of links representing possible sources of plastic activity exhibits anomalous (super-extensive) scaling which tends to extensive scaling (often assumed in weakest-link models) if quenched short-range interactions are introduced. The reason is that the interplay between long-range dislocation interactions and short-range quenched disorder destroys scale-free dynamical correlations leading to event localization with a characteristic length-scale. Several methods are presented to determine the dynamic length-scale that can be generalized to other types of heterogeneous materials

Universality-class dependence of energy distributions in spin glasses

We study the probability distribution function of the ground-state energies of the disordered one-dimensional Ising spin chain with power-law interactions using a combination of parallel tempering Monte Carlo and branch, cut, and price algorithms. By tuning the exponent of the power-law interactions we are able to scan several universality classes. Our results suggest that mean-field models have a non-Gaussian limiting distribution of the ground-state energies, whereas non-mean-field models have a Gaussian limiting distribution. We compare the results of the disordered one-dimensional Ising chain to results for a disordered two-leg ladder, for which large system sizes can be studied, and find a qualitative agreement between the disordered one-dimensional Ising chain in the short-range universality class and the disordered two-leg ladder. We show that the mean and the standard deviation of the ground-state energy distributions scale with a power of the system size. In the mean-field universality class the skewness does not follow a power-law behavior and converges to a nonzero constant value. The data for the Sherrington-Kirkpatrick model seem to be acceptably well fitted by a modified Gumbel distribution. Finally, we discuss the distribution of the internal energy of the Sherrington-Kirkpatrick model at finite temperatures and show that it behaves similar to the ground-state energy of the system if the temperature is smaller than the critical temperature.Comment: 15 pages, 20 figures, 1 tabl

Using the κ-exponential to deform Gumbel and Gompertz probability distributions

Here we propose the use of the κ-exponential to deform Gumbel and the Gompertz distributions. The κ-exponential is a function of κ-statistics, a statistics which has been developed by G. Kaniadakis, Politecnico di Torino, in the framework of special relativity, and used for statistical analyses involving power law tailed distributions. The exponentiated κ-Gumbel function is also considered, to have a further deformation of the probability distribution