B. Kunin

A probabilistic model of brittle crack formation

Probability of a brittle crack formation in an elastic solid with fluctuating strength is considered. A set Ω of all possible crack trajectories reflecting the fluctuation of the strength field is introduced. The probability P(X) that crack penetration depth exceeds X is expressed as a functional integral over Ω of a conditional probability of the same event taking place along a particular path. Various techniques are considered to evaluate the integral. Under rather nonrestrictive assumptions we reduce the integral to solving a diffusion‐type equation. A new characteristic of fracture process, ‘‘crack diffusion coefficient,’’ is introduced. An illustrative example is then considered where the integration is reduced to solving an ordinary differential equation. The effect of the crack diffusion coefficient and of the magnitude of strength fluctuations (ratio of minimal and mean values of the strength field) on probability density of crack penetration depth is presented. Practical implications of the propo...

Kinematics of damage zone accompanying curved crack

This paper presents a methodology of a quantitative characterization of the kinematics of evolution of a damage zone surrounding the tip of a slowly propagating crack. On the basis of the Crack Layer Theory, the evolution of the damage zone is modelled as a combination of a few elementary motions: translation, rotation, isotropic expansion and distortion. A procedure for evaluating the rates of the elementary motions on the basis of direct measurements is developed. The procedure is illustrated for curved crack layer growth in the vicinity of a hole in commercial polystyrene. The important role of the damage zone in determining the main crack trajectory and speed is clearly demonstrated.

Modeling of brittle fracture based on the concept of crack trajectory ensemble

Abstract The objective of this paper is to present a comprehensive review of an approach which stands aside from the mainstream of statistical modeling of fracture. The approach is essentially based on the concept of an ensemble of macroscopically identical fracture specimens and on averaging over it. Equivalently, an ensemble Ω of virtual crack trajectories is associated with a single specimen; the averaging is then expressed in the form of functional integration over Ω. The approach combines the concepts of weakest link theories with fracture mechanics formalism and models crack propagation through a brittle microheterogeneous solid. The statistics of microheterogeneity, e.g. the population of pre-existing defects, is reflected in a random field of specific fracture energy γ and in the statistical features of Ω. The fracture parameters employed in the approach are: parameters of the pointwise distribution of the γ-field; its correlation distance; and the characteristics of roughness of the fracture surfaces, including their fractal dimension. The probability of crack formation between any two points in a two-dimensional solid (referred to as “crack propagator”) is introduced as the main building block of the approach. It is expressed as a functional integral (over the set Ω) of the probability of crack formation along a particular path. The probability distributions of critical loads, critical crack lengths, G 1c , crack arrest locations, etc., are derived in terms of crack propagator. The dependence of the distributions on the statistical characteristics of the material as well as on the roughness of the crack trajectories is analyzed by both analytical and numerical means.

Statistical fracture mechanics

We offer this lecture on fracture of solids to the school on “... complex hydrodynamic phenomena” to attract your attention to a vast field which has many similarities with turbulence and other hydrodynamic phenomena, while remaining a virgin territory. Fracture resembles turbulence through the presence of the scaling hierarchy of the defects involved and the complexity of their interaction. We believe that many concepts and formalism developed in hydrodynamics could be useful if applied to fracture.

A new type of extreme value distributions

Abstract The three classical pairs of extreme value distributions correspond to random variables with ranges of values unbounded from either one or both sides. Some applications of statistics of extremes, however, deal with variables, which are bounded on both sides (local values of specific fracture energy in crack diffusion theory is one such example). In this paper, we derive a fourth pair of extreme value distributions, which are supported on a finite segment (one for maxima and one for minima). First, we propose a derivation of the three known maximal value distributions, which lends itself to a generalization (everything is done for maxima, since transition to minima is standard). The derivation is then extended to a slightly more general setting, and the fourth distribution is obtained. It is explained that certain fact concerning groups of transformations of the real line prevents any further generalization, i.e. the extended list of extreme value distributions is complete. The three classical maximal value distributions can be obtained as limits of the new one. A possible criterion of when one may expect the new distribution to be more adequate than the Weibull distribution is offered. An illustrative numerical example is considered, in which the scatter of sample minima is modeled by both Weibull and the new distribution. Another example shows that when the modeling of data requires very high values of the shape parameter of the Weibull distribution, the new distribution may be expected to have much smaller “shape parameter” values. The modeling of experimentally observed scatter of crack arrest length, using the Weibull distribution, is compared to that using the new distribution.

Lorenz-type controlled pendulum

It is known that the popular Lorenz system admits an equivalent representation as a controlled Duffing system. It is also known that the Duffing system is an approximation to the simple pendulum. We introduce a new class of controlled pendulum systems that may also be interpreted as Pendulum-Lorenz systems. These systems contain the Lorenz system as an approximation and have a wide range of potential applications. Examples of chaotic attractors associated with a controlled pendulum system are presented.

Probabilistic fracture mechanics of 2D carbon-carbon composites

The fracture toughness of two types of carbon fabric reinforced carbon composite (KKARB®, Types A and C) is evaluated, the mechanisms of crack propagation resistance are identified and both are related to microstructural differences.The two composites have the same constituents, i.e. fibers, yarns, fabric weaving and matrix precursor. However, different processing cycles result in apparent differences in microstructure (e.g. different number and length distribution of microcracks, crimp angle) and toughness.The crack diffusion model (CDM) is invoked to parameterize the fluctuating strength field of the composite in terms of an average fracture energy 〈γ〉, a minimum fracture energy 179-1 and a shape parameter α. The values of 179-1 are in direct correlation with the average size of microcracks in each composite, and α is found to correlate with the scatter in fracture toughness.

Evaluation of statistical fracture toughness parameters on the basis of crack arrest experiment

The crack diffusion theory addresses the origins of the observed scatter, in brittle fracture, of macroscopic fracture parameters under seemingly identical test conditions. An essential question of experimental evaluation of the nonconventional fracture parameters introduced there appears to be buried in the formalism. In this paper we report a pilot crack arrest experiment, in which a methodology of evaluating some of the new fracture parameters is demonstrated. Experimental observations are found to not disprove the theory.

A Stochastic Model for Controlled Discontinuous Crack Growth in Brittle Materials

Statistical Fracture Mechanics aims at describing fracture of brittle solids with complex microstructure and pronounced combination of scatter and’ scale effect’ for conventional fracture parameters such as fracture toughness, critical energy release rate, etc. It achieved significant progress in prediction of distributions of critical loads, crack lengths, displacements, etc., for a stressed structural element under a variety of conditions. The present paper addresses another important question: predicting life-time scatter for a stressed brittle structural element. We model an observed mode of slow crack growth in brittle materials, namely a Markovian stochastic pattern of a microscopic random jump, followed by a random waiting time, followed by a random jump, and so on. The waiting times are related, on physical grounds, to random energy barriers at the arrest points, whereas random magnitudes of the jumps are treated within the existing framework of Crack Diffusion Theory. The transition probability density for the resulting random process of crack growth is shown to satisfy a differential equation which admits a solution in quadratures. A simple illustrative example is considered.

A Probabilistic Approach to Crack Instability

A wide scatter of critical crack lengths, critical loads, critical energy release rates, etc., is well known in brittle fracture. It is also recognized that the scatter of such macroparameters is caused by microstructural fluctuations of material morphology. This is typical for critical phenomena, where microscopical fluctuations produce a macroscopical effect. In fracture, microstructural fluctuations are reflected most explicitly in the morphology of fracture surfaces.