The fiber bundle model describes a collection of elastic fibers under load. The fibers fail successively and, for each failure, the load distribution among the surviving fibers changes. Even though very simple, this model captures the essentials of failure processes in a large number of materials and settings. A review of the fiber bundle model is presented with different load redistribution mechanisms from the point of view of statistics and statistical physics rather than materials science, with a focus on concepts such as criticality, universality, and fluctuations. The fiber bundle model is discussed as a tool for understanding phenomena such as creep and fatigue and how it is used to describe the behavior of fiber-reinforced composites as well as modeling, e.g., network failure, traffic jams, and earthquake dynamics.

We study via simulation how a lattice breaks if each bond is an elastic beam having longitudinal and flexural breaking thresholds randomly selected according to various probability distributions. We observe scaling of force, displacement, and number of broken beams in the controlled regime. The distribution of local forces just before breaking is characterized by a multifractal spectrum f(α).

We record the height of the crack surface as a function of position along one-dimensional cuts orthogonal to the crack for six different brittle materials. We find that the width w of this one-dimensional trace as a function of its length L behaves as w∼${\mathit{L}}^{\mathrm{\zeta }}$, where ζ=0.87(7). This result is in agreement with recent conjectures of a universal roughness exponent ζ for these materials.

We propose a method for (global) Hurst exponent determination based on wavelets. Using this method, we analyze synthetic data with predefined Hurst exponents, fracture surfaces, and data from economy. The results are compared to those obtained with Fourier spectral analysis. When many samples are available, the wavelet and Fourier methods are comparable in accuracy. However, when one or only a few samples are available, the wavelet method outperforms the Fourier method by a large margin.

We present a simple criterion based on the Einstein relation for determining whether diffusion in systems governed by a generalized Langevin equation with long-range memory is normal, superdiffusive, or subdiffusive. We support our analysis with numerical simulations.