Blaise Bourdin

The Variational Approach to Fracture

1 Introduction 2 Going variational 2.1 Griffiths theory 2.2 The 1-homogeneous case - A variational equivalence 2.3 Smoothness - The soft belly of Griffiths formulation 2.4 The non 1-homogeneous case - A discrete variational evolution 2.5 Functional framework - A weak variational evolution 2.6 Cohesiveness and the variational evolution 3 Stationarity versus local or global minimality - A comparison 3.1 1d traction 3.1.1 The Griffith case - Soft device 3.1.2 The Griffith case - Hard device 3.1.3 Cohesive case - Soft device 3.1.4 Cohesive case - Hard device 3.2 A tearing experiment 4 Initiation 4.1 Initiation - The Griffith case 4.1.1 Initiation - The Griffith case - Global minimality 4.1.2 Initiation - The Griffith case - Local minimality 4.2 Initiation - The cohesive case 4.2.1 Initiation - The cohesive 1d case - Stationarity 4.2.2 Initiation - The cohesive 3d case - Stationarity 4.2.3 Initiation - The cohesive case - Global minimality 5 Irreversibility 5.1 Irreversibility - The Griffith case - Well-posedness of the variational evolution 5.1.1 Irreversibility - The Griffith case - Discrete evolution 5.1.2 Irreversibility - The Griffith case - Global minimality in the limit 5.1.3 Irreversibility - The Griffith case - Energy balance in the limit 5.1.4 Irreversibility - The Griffith case - The time-continuous evolution 5.2 Irreversibility - The cohesive case 6 Path 7 Griffith vs. Barenblatt 8 Numerics and Griffith 8.1 Numerical approximation of the energy 8.1.1 The first time step 8.1.2 Quasi-static evolution 8.2 Minimization algorithm 8.2.1 The alternate minimization algorithm 8.2.2 The backtracking algorithm 8.3 Numerical experiments 8.3.1 The 1D traction (hard device) 8.3.2 The Tearing experiment 8.3.3 Revisiting the 2D traction experiment on a fiber reinforced matrix 9 Fatigue 9.1 Peeling Evolution 9.2 The limitfatigue law when d tends to 0 9.3 A variational formulation for fatigue 9.3.1 Peeling revisited 9.3.2 Generalization Appendix Glossary References.

Numerical experiments in revisited brittle fracture

Abstract The numerical implementation of the model of brittle fracture developed in Francfort and Marigo (1998. J. Mech. Phys. Solids 46 (8), 1319–1342) is presented. Various computational methods based on variational approximations of the original functional are proposed. They are tested on several antiplanar and planar examples that are beyond the reach of the classical computational tools of fracture mechanics.

Numerical implementation of the variational formulation for quasi-static brittle fracture

This paper presents the analysis and implementation of the variational formulation of quasi-static brittle fracture mechanics proposed by G. A. Francfort and J.-J. Marigo in 1998. We briefly present the model itself, and its variational approximation in the sense of -convergence. We propose a numerical algorithm based on Alternate Minimizations and prove its convergence under restrictive assumptions. We establish a new necessary condition for optimality for the entire time evolution from which we derive the Backtracking algorithm. We give some elements of analysis of the Backtracking algorithm on a simple problem. We present realistic numerical simulations of a traction experiment on a fiber-reinforced matrix, and of the propagation of cracks in a perforated sample under mode-I loading.

Implementation of an adaptive finite-element approximation of the Mumford-Shah functional

Summary. We present and detail a method for the numerical solving of the Mumford-Shah problem, based on a finite element method and on adaptive meshes. We start with the formulation introduced in [13], detail its numerical implementation and then propose a variant which is proved to converge to the Mumford-Shah problem. A few experiments are illustrated.

Optimal Partitions for Eigenvalues

We introduce a new numerical method for approximating partitions of a domain minimizing the sum of Dirichlet-Laplacian eigenvalues of any order. First we prove the equivalence of the original problem and a relaxed formulation based on measures. Using this result, we build a numerical algorithm to approximate optimal configurations. We describe numerical experiments aimed at studying the asymptotic behavior of optimal partitions with large numbers of cells.

A Variational Approach to the Numerical Simulation of Hydraulic Fracturing

One of the most critical capabilities of realistic hydraulic fracture simulation is the prediction of complex (turning, bifurcating, or merging) fracture paths. In most classical models, complex fracture simulation is difficult due to the need for a priori knowledge of propagation path and initiation points and the complexity associated with stress singularities at fracture tips. In this study, we follow Francfort and Marigo’s variational approach to fracture, which we extend to account for hydraulic stimulation. We recast Griffith’s criteria into a global minimization principle, while preserving its essence, the concept of energy restitution between surface and bulk terms. More precisely, to any admissible crack geometry and kinematically admissible displacement field, we associate a total energy given as the sum of the elastic and surface energies. In a quasistatic setting, the reservoir state is then given as the solution of a sequence of unilateral minimizations of this total energy with respect to any admissible crack path and displacement field. The strength of this approach is to provide a rigorous and unified framework accounting for new cracks nucleation, existing cracks activation, and full crack path determination (including complex behavior such as crack branching, kinking, and interaction between multiple cracks) without any a priori knowledge or hypothesis. Of course, the lack of a priori hypothesis on cracks geometry is at the cost of numerical complexity. We present a regularized phase field approach where fractures are represented by a smooth function. This approach makes handling large and complex fracture networks very simple yet discrete fracture properties such as crack aperture can be recovered from the phase field. We compare variational fracture simulation results against several analytical solutions and also demonstrate the approach’s ability to predict complex fracture systems with example of multiple interacting fractures. Introduction Conventionally, in most numerical modeling strategy of hydraulic fracturing, fracture propagation is assumed to be planar and perpendicular to the minimum reservoir stress (Adachi et al. 2007), which simplifies fracture propagation criteria to mode-I and aligns the propagation plane to the simulation grid. Restricting propagation mode search into one direction and prescribing fracture growth plane can greatly reduce computation overhead and make practical numerical modeling tractable. However, recent observations suggest creation of nonplanar complex fracture system during reservoir stimulation (Mayerhofer et al. 2010) or waste injection (Moschovidis et al. 2000). To address predictive capabilities of complex fracture propagation, several different approaches, namely, mixed-mode fracture growth criterion with a single fracture (Rungamornrat, Wheeler, and Mear 2005), multiple discrete fractures that grow with empirical correlations (Gu et al. 2012), and implicit fracture treatment with the idea of stimulated reservoir volume where averaged properties are estimated over an effective volume (Hossain, Rahman, and Rahman 2000) have been proposed. In this study, we propose to apply the variational approach to fracture (Francfort and Marigo 1998; Bourdin, Francfort, and Marigo 2008) to hydraulic fracturing. One of the strengths of this approach is to account for arbitrary numbers of pre-existing or propagating cracks in terms of energy minimization, without any a priori assumption on their geometry, and without restricting their growth to specific grid directions. The goal of this paper is to present early results obtained with this method. At this stage, we are not trying to account for all physical, chemical, thermal, and mechanical phenomena involved in the hydraulic fracture process. Instead, we propose a mechanistically sound yet mathematically rigorous model in an ideal albeit not unrealistic situation, for which we can perform rigorous analysis and quantitative comparison with analytical solutions. In particular, we neglect all thermal and chemical effects, we assume that the injection rate is slow enough that all inertial effects can be neglected, and place ourself in the quasi-static setting. Furthermore, we consider a reservoir made of an idealized impermeable perfectly brittle linear material with no porosity and assume that the injected fluid is incompressible. These assumptions imply that no leak-off can take place and that the fluid pressure is constant throughout the fracture system (infinite fracture conductivity), depending only on total injected fluid volume and total cracks opening respectively. Also, throughout the analyses presented, we only deal with dimensionless parameters, which are normalized by the Young’s moduli for mechanical parameters and by the total domain volume for volumetric parameters. The variational approach to hydraulic fracturing In classical approaches to quasi-static brittle fracture, the elastic energy restitution rate, G, induced by the infinitesimal growth of a single crack along an a priori known path (derived from the stress intensity factors) is compared to a critical energy rate Gc and propagation occurs when G = Gc, the celebrated Griffith criterion. The premise of the variational approach to fracture is to recast Griffith’s criterion in a variational setting, i.e. as the minimization over any crack set (any set of curves in 2D or of surface in 3D, in the reference configuration) and any kinematically admissible displacement field u, of a total energy consisting of the sum of the stored potential elastic energy and a surface energy proportional to the length of the cracks in 2D or their area in 3D. More specifically, consider a domain Ω in 2 or 3 space dimension, occupied by a perfectly brittle linear material with Hooke’s law A and critical energy release rate (also often referred to as fracture toughness) Gc. Let f(t, x) denote a time-dependent body force applied to Ω, τ(t, x), the surface force applied to a part ∂NΩ of its boundary, and g(t, x) a prescribed boundary displacement on the remaining part ∂DΩ. To any arbitrary crack set Γ and any kinematically admissible displacement set u, we associate the the total energy

The variational formulation of brittle fracture: numerical implementation and extensions

This paper presents the implementation of a variational formulation of brittle fracture mechanics proposed by G.A. Francfort and J.-J. Marigo in 1998. The essence of the model relies on successive global minimizations of an energy with respect to any crack set and any kinematically admissible displacement field. We briefly present the model itself, and its variational approximation in the sense of Gamma—convergence. We propose a globally convergent and monotonically decreasing numerical algorithm. We introduce a backtracking algorithm whose solution satisfy a global optimality criterion with respect to the time evolution. We illustrate this algorithm with three dimensional numerical experiments. Then we present an extension of the model to crack propagation under thermal load and its numerical application to the quenching of glass.

Crack patterns obtained by unidirectional drying of a colloidal suspension in a capillary tube: experiments and numerical simulations using a two-dimensional variational approach

Basalt columns, septarias, and mud cracks possess beautiful and intriguing crack patterns that are hard to predict because of the presence of cracks intersections and branches. The variational approach to brittle fracture provides a mathematically sound model based on minimization of the sum of bulk and fracture energies. It does not require any a priori assumption on fracture patterns and can therefore deal naturally with complex geometries. Here, we consider shrinkage cracks obtained during unidirectional drying of a colloidal suspension confined in a capillary tube. We focus on a portion of the tube where the cross-sectional shape cracks does not change as they propagate. We apply the variational approach to fracture to a tube cross-section and look for two-dimensional crack configurations minimizing the energy for a given loading level. We achieve qualitative and quantitative agreement between experiments and numerical simulations using a regularized energy (without any assumption on the cracks shape) or solutions obtained with traditional techniques (fixing the overall crack shape a priori). The results prove the efficiency of the variational approach when dealing with crack intersections and its ability to predict complex crack morphologies without any a priori assumption on their shape.

The Phase-Field Method in Optimal Design

We describe the phase-field method, a new approach to optimal design originally introduced in Bourdin and Chambolle (2000, 2003). It is based on the penalization of the variation of the properties of the designs, and its variational approximation (in the sense of Г-convergence. It uses a smooth function, the phase-field, to represent all materials involved.

Numerical Implementation of Overlapping Balancing Domain Decomposition Methods on Unstructured Meshes

The Overlapping Balancing Domain Decomposition (OBDD) methods can be considered as an extension of the Balancing Domain Decomposition (BDD) methods to the case of overlapping subdomains. This new approach, has been proposed and studied in [4, 3]. In this paper, we will discuss its practical parallel implementation and present numerical experiments on large unstructured meshes.