Scaling of fracture strength in disordered quasi-brittle materials

The European Physical Journal B - This paper presents two main results. The first result indicates that in materials with broadly distributed microscopic heterogeneities, the fracture strength...     

Scaling behaviour in the fracture of fibrous materials

We study the existence of distinct failure regimes in a model for fracture in fibrous materials. We simulate a bundle of parallel fibers under uniaxial static load and observe two different failure regimes: a catastrophic and a slowly shredding. In the catastrophic regime the initial deformation produces a crack which percolates through the bundle. In the slowly shredding regime the initial deformations will produce small cracks which gradually weaken the bundle. The boundary between the catastrophic and the shredding regimes is studied by means of percolation theory and of finite-size scaling theory. In this boundary, the percolation density scales with the system size L, which implies the existence of a second-order phase transition with the same critical exponents as those of usual percolation. Received 24 June 1999     

Statistical kinetics of quasi-brittle fracture

The statistical laws of fracture under conditions of static fatigue and loading at a constant rate are discussed. Phenomenological crack-growth models are used to theoretically analyze the statistics of quasi-brittle fracture of solids that undergo static fatigue and whose experimentally measured strength is controlled by kinetic factors. The strength and longevity distributions that correctly reflect the main empirical dependences are constructed. It is shown that the often detected specific features of the statistics of quasi-brittle fracture are likely to be caused by the qualitative similarity between the asymptotic behaviors of the statistics of thermal fluctuations and “dangerous” structural defects, namely, by the exponential distribution of the strong-thermal-fluctuation expectation time, a large number of statistically equivalent dangerous defects in a sample, and a sharp decrease in the probability of defects whose sizes are larger than a certain limiting size. The results obtained suggest that the generally accepted theoretical interpretation of the Weibull distribution is likely to require revision, since the assumption regarding the power asymptotics of the size distribution of cracklike defects during quasi-brittle fracture is in conflict with the empirical data on static fatigue.     

From brittle to ductile fracture in disordered materials

We introduce a lattice model able to describe damage and yielding in heterogeneous materials ranging from brittle to ductile ones. Ductile fracture surfaces, obtained when the system breaks once the strain is completely localized, are shown to correspond to minimum energy surfaces. The similarity of the resulting fracture paths to the limits of brittle fracture or minimum energy surfaces is quantified. The model exhibits a smooth transition from brittleness to ductility. The dynamics of yielding exhibits avalanches with a power-law distribution.     

Scaling effect on the mixed-mode fracture path of rock materials

In this paper, a new approach is presented to predict the crack growth path in the rock materials by taking into account the size effect. The proposed approach is an incremental method in which the crack initiation angle for each step is determined from the modified forms of the maximum tangential stress criterion. These modified maximum tangential stress criteria take into account the influence of the higher order terms of the stress series at the crack tip in addition to the singular terms. As an important parameter in the proposed method, the critical distance r _c is also assumed to be size dependent. Finally the incremental method is evaluated by experimental results obtained from Guiting limestone and CJhorveh marble specimens reported in the previous studies. It is shown that the proposed approach can predict the fracture trajectory of cracked specimens with different sizes in good agreement with the experimental results when three terms of Williams series expansion are considered for characterizing the stress field around the crack tip.     

Space-time scale invariance in dynamically fragmented quasi-brittle materials

The paper reports on a study to investigate the influence of load intensity and material structure on statistical regularities of fragmentation in ZrO₂-based ceramics differing in porosity. The study was performed on a modified split Hopkinson pressure bar system and allowed a comparative analysis of dynamic stress-strain curves and statistical characteristics of fragmentation such as distributions of emitted light pulses (fractoluminescence) and fragment sizes. It is shown that increasing the ceramic porosity changes both the form of stress-strain curves and the pulse distribution. In ceramic specimens with up to 45% porosity, the pulse distribution is described by a bimodal power law; in ceramic specimens with 60% porosity, by a power law. The fragment size distribution in the material corresponds to a power law with the exponent dependent on porosity and load intensity.     

Some aspects of nucleation and evolution of microscopic and mesoscopic cracks and quasi-brittle fracture of homogeneous materials

The dynamics of nucleation of microcracks in the region of a developing macroscopic crack has been studied using scanning electron microscopy during in situ experiment and the acoustic emission method. An explosive-like nucleation of microcracks and the influence of dissipative properties of a material on the size and the rate of redistribution of local stresses of the microcracks have been established. Each of nucleations of microscopic and mesoscopic defects is considered as an act of local testing of the dissipative ability of the system, and the interaction between microcracks is determined by relaxation processes when the ability is sufficient. With deteriorating dissipative properties (with increasing residence time under loading, deformation rate, etc.), an elastic linear interaction becomes possible, and the macroscopic fracture occurs. In the microcrack dynamics, a specific ductile-brittle transition is determined by a time deterioration of the dissipative properties of the material under the action of a mechanical load. Generally, as a large latent energy is stored during deformation, a macroscopic fracture can be caused by any little discrete structural transformation.     

Scaling in the disordered quantum antiferromagnetic chain

Numerical calculations on the disordered quantum Heisenberg antiferromagnetic chain yield low-temperature behavior independent of the detailed form of the randomness. A simple scaling interpretation, which makes contact with earlier theoretical work, of these complicated calculations is presented. An analogy with a one-parameter scaling theory of localization is explored.     

Scaling behavior of one-dimensional weakly disordered models

The density of states and various characteristic lengths of one-dimensional tight-binding models and disordered harmonic chains are calculated in the limit of weak disorder at the band edge of the ordered system. The density of states and a localization length of the one-dimensional Anderson model were already calculated by Derrida and Gardner; we recover their results. For the tight-binding models with off-diagonal disorder our results are in agreement with numerical calculations of Krey.     

Scaling behavior of disordered lattice fermions in two dimensions

We propose a lattice model for Dirac fermions which allows us to break the degeneracy of the node structure. In the presence of a random gap we analyze the scaling behavior of the localization length as a function of the system width within a numerical transfer-matrix approach. Depending on the strength of the randomness, there are different scaling regimes for weak, intermediate and strong disorder. These regimes are separated by transitions that are characterized by one-parameter scaling.     

Scaling theory of quantum resistance distributions in disordered systems

We have derived explicitly, the large scale distribution of quantum Ohmic resistance of a disordered one-dimensional conductor. We show that in the thermodynamic limit this distribution is characterized by two independent parameters for strong disorder, leading to a two-parameter scaling theory of localization. Only in the limit of weak disorder we recover single parameter scaling, consistent with existing theoretical treatments.     

Optical phonons in disordered materials

A quantized Hamiltonian, including both the short- and long-range forces, for optical phonons in disordered materials is presented in this paper. The spectra of the optical phonons and the LO---TO splitting are calculated under the coherent potential approximation. The competed effects of the long-range correlation and the local structural fluctuation on the optical phonons are also discussed.     

Scaling and universality in the optics of disordered exciton chains

The joint probability distribution of exciton energies and transition dipole moments determines a variety of optical observables in disordered exciton systems. We demonstrate numerically that this distribution obeys a one-parameter scaling, originating from the fact that both the energy and the dipole moment are determined by the number of coherently bound molecules. A universal underlying distribution is found, which is identical for uncorrelated Gaussian disorder in the molecular transition energies or in the intermolecular transfer interactions. The universality breaks down for disorder in the transfer interactions resulting from variations in the molecular positions. We suggest the possibility to probe the joint distribution by means of single-molecule spectroscopy.     

Scaling and universality in rock fracture

We present a detailed statistical analysis of acoustic emission time series from laboratory rock fracture obtained from different experiments on different materials including acoustic emission controlled triaxial fracture and punch-through tests. In all considered cases, the waiting time distribution can be described by a unique scaling function indicating its universality. This scaling function is even indistinguishable from that for earthquakes suggesting its general validity for fracture processes independent of time, space, and magnitude scales.     

Scaling and localization lengths of a topologically disordered system

We consider a noninteracting disordered system designed to model particle diffusion, relaxation in glasses, and impurity bands of semiconductors. Disorder originates in the random spatial distribution of sites. We find strong numerical evidence that this model displays the same universal behavior as the standard Anderson model. We use finite-size scaling to find the localization length as a function of energy and density, including localized states away from the delocalization transition. Results at many energies all fit onto the same universal scaling curve.     

Scaling theory for percolative charge transport in disordered molecular semiconductors

We present a scaling theory for charge transport in disordered molecular semiconductors that extends percolation theory by including bonds with conductances close to the percolating one in the random-resistor network representing charge hopping. A general and compact expression is given for the charge mobility for Miller-Abrahams and Marcus hopping on different lattices with Gaussian energy disorder, with parameters determined from numerically exact results. The charge-concentration dependence is universal. The model-specific temperature dependence can be used to distinguish between the hopping models.     

Electrons in disordered systems. Scaling near the mobility edge

Renormalization group arguments are applied to an ensemble of disordered electronic systems (without electron-electron interaction). The renormalization group procedure consists of a sequence of transformations of the length and the energy scales, and of orthogonal transformations of the electronic states. Homogeneity and power laws are obtained for various one and two-particle correlations and for the low-temperature conductivity in the vicinity of the mobility edge. Two types of fixed point ensembles are proposed, a homogeneous ensemble which is roughly approximated by a cell model, and an inhomogeneous ensemble.