Quasi-static hydraulic crack growth driven by Darcy’s law

In the framework of rate independent processes, we present a variational model of quasi-static crack growth in hydraulic fracture. We first introduce the energy functional and study the equilibrium conditions of an unbounded linearly elastic body subject to a remote strain {\epsilon\in\mathbb{R}} and with a sufficiently regular crack Γ filled by a volume V of incompressible fluid. In particular, we are able to find the pressure p of the fluid inside the crack as a function of Γ, V, and ϵ. Then we study the problem of quasi-static evolution for our model, imposing that the fluid volume V and the fluid pressure p are related by Darcy’s law. We show the existence of such an evolution, and we prove that it satisfies a weak notion of the so-called Griffith’s criterion.

Convergence of alternate minimization schemes for phase-field fracture and damage

We consider time-discrete evolutions for a phase-field model (for fracture and damage) obtained by alternate minimization schemes. First, we characterize their time-continuous limit in terms of parametrized [Formula: see text]-evolutions, introducing a suitable family of “intrinsic energy norms”. Further, we show that the limit evolution satisfies Griffith’s criterion, for a phase-field energy release, and that the irreversibility constraint is thermodynamically consistent.

A Generalized Model for Adhesive Contact Between a Rigid Cylinder and a Transversely Isotropic Substrate

In this paper, a solution to the quasi-static adhesive contact problem between a rigid cylinder and a transversely isotropic substrate is extended to the most general case by taking adhesion hysteresis into account. An analytical solution to the contact stress is obtained by solving the integral equations established on the basis of the Green's function for the two-dimensional transversely isotropic half-space problem. By using equilibrium conditions and Griffith's criterion, the adhesion force and resistant moment to rolling are determined as functions of contact geometries and material properties of the contacting solids. Detailed discussions on the adhesion force and resistant moment are presented for some specific cases, revealing adhesion behaviors that have not been predicted by previous models. As the most generalized solution to the discussed problem, our results would have extensive applications in predicting the adhesion behavior between solids undergoing sophisticated mechanical loadings.

QUASI-STATIC CRACK PROPAGATION BY GRIFFITH'S CRITERION

We consider the propagation of a crack in a brittle material along a prescribed crack path and define a quasi-static evolution by means of stationary points of the free energy. We show that this evolution satisfies Griffith's criterion in a suitable form which takes into account both stable and unstable propagations, as well as an energy balance formula which accounts for dissipation in the unstable regime. If the load is monotonically increasing, this solution is explicit and almost everywhere unique. For more general loads we construct a solution via time discretization. Finally, we consider a finite element discretization of the problem and prove convergence of the discrete solutions.

Possible deviations from Griffith’s criterion in shallow slabs, and consequences on slab avalanche release

Possible reasons for deviations from Griffith’s criterion in slab avalanche triggerings are examined. In the case of a major basal crack, we show (i) that the usual form of Griffith’s criterion is valid if elastic energy is stored in a shallow and hard slab only, and (ii) that rapid healing of broken ice bonds may lead to shear toughnesses larger than expected from tensile toughness experiments. In the case of avalanches resulting from failure of multi-cracked weak layers, where a simple Griffith’s criterion cannot be applied, frequency/size plots obtained from discrete elements and cellular automata simulations are shown to obey scale invariant power law distributions. These findings are confirmed by both frequency/acoustic emission duration and frequency/size plots obtained from field data, suggesting that avalanche triggerings may be described using the formalism of critical phenomena.

A MODEL FOR THE QUASI-STATIC GROWTH OF BRITTLE FRACTURES BASED ON LOCAL MINIMIZATION

We study a variant of the variational model for the quasi-static growth of brittle fractures proposed by Francfort and Marigo.9 The main feature of our model is that, in the discrete-time formulation, in each step we do not consider absolute minimizers of the energy, but, in a sense, we look for local minimizers which are sufficiently close to the approximate solution obtained in the previous step. This is done by introducing in the variational problem an additional term which penalizes the L2-distance between the approximate solutions at two consecutive times. We study the continuous-time version of this model, obtained by passing to the limit as the time step tends to zero, and show that it satisfies (for almost every time) some minimality conditions which are slightly different from those considered in Refs. 9 and 8, but are still enough to prove (under suitable regularity assumptions on the crack path) that the classical Griffith's criterion holds at the crack tips. We also prove that, if no initial crack is present and if the data of the problem are sufficiently smooth, no crack will develop in this model, provided the penalization term is large enough.

Mechanics of Contact Fatigue

Contact fatigue between typical machine elements such as gears, roller followers, bearings and other components involves many complex interacting features. There are the effects of geometry, mechanical properties, material properties and surface chemistry. The present discussion will center around analytical prediction techniques that are concerned only with the mechanical aspects of contact fatigue between two elements. Aspects related to the initiation of a crack under repeated loading will be discussed. The application of an approach developed by Mura, analogous to Griffith’s criterion for fracture, will be used to show how estimates of initiation life can be made and how these estimates are related to currently used ones. Once a crack has been initiated, then issues related to crack propagation become important. Some fracture mechanics based methods developed to calculate crack growth will be described along with estimates of crack propagation life.

Structural Designing With Ceramic Materials

A central problem in attempting to use ceramic materials in demanding structural applications is uncertainty about the stresses to which they can be safely subjected. A ceramic rarely, if ever, exhibits a characteristic failure stress. This stress depends on the nature and distribution of microscopic flaws that intensify stress locally, and fracture initiates at a single “worst” flaw when Griffith’s criterion for crack instability is met. Within the basic framework, theories are available for treating effects of time, size, and stress distribution on failure stress. This paper reviews these theories, and discusses their use in specifying limiting stresses in designing structural members.