A variational model of quasistatic crack evolution is proposed. Although close in spirit to Griffith’s theory of brittle fracture, the proposed model however frees itself of the usual constraints of that theory : a preexisting crack and a well-defined crack path. In contrast, crack initiation as well as crack path can be quantified, as demonstrated on explicitly computable examples. Furthermore the model lends itself to numerical implementation in more complex settings.

Revisiting brittle fracture as an energy minimization problem

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 experiments in revisited brittle fracture

1 Introduction 2 Going variational 2.1 Griffith's theory 2.2 The 1-homogeneous case - A variational equivalence 2.3 Smoothness - The soft belly of Griffith's 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.

The Variational Approach to Fracture

A phase-field description of dynamic brittle fracture

This paper presents a modified regularized formulation of the Ambrosio–Tortorelli type to introduce the crack non-interpenetration condition in the variational approach to fracture mechanics proposed by Francfort and Marigo [1998. Revisiting brittle fracture as an energy minimization problem. J. Mech. Phys. Solids 46 (8), 1319–1342]. We focus on the linear elastic case where the contact condition appears as a local unilateral constraint on the displacement jump at the crack surfaces. The regularized model is obtained by splitting the strain energy in a spherical and a deviatoric parts and accounting for the sign of the local volume change. The numerical implementation is based on a standard finite element discretization and on the adaptation of an alternate minimization algorithm used in previous works. The new regularization avoids crack interpenetration and predicts asymmetric results in traction and in compression. Even though we do not exhibit any gamma-convergence proof toward the desired limit behavior, we illustrate through several numerical case studies the pertinence of the new model in comparison to other approaches.

Regularized formulation of the variational brittle fracture with unilateral contact: Numerical experiments

The paper is concerned with the fine properties of functions in , the space of functions with bounded deformation. We analyse the set of Lebesgue points and the set where these functions have one-sided approximate limits. Moreover, following the analogy with , we decompose the symmetric distributional derivative into an absolutely continuous part , a jump part , and a Cantor part . The main result of the paper is a structure theorem for functions, showing that these parts of the derivative can be recovered from the corresponding ones of the one-dimensional sections. Moreover, we prove that functions are approximately differentiable in almost every point of their domain.

Fine Properties of Functions with Bounded Deformation

We prove that local minimizers u : R n → R N of the functional ∫ ∣D u ( x )∣ p(r) d x are of class C 1, α for some α > 0, provided p(x) > 1 is Holder continuous.

Hölder continuity of the gradient of p(x)-harmonic mappings

Compactness and lower semicontinuity properties in SBD(Omega)

We approximate by discrete Г-convergence a functional proposed by Mumford-Shah for a variational approach to image segmentation. Such a functional is first relaxed with a sequence of nonconvex functionals, which in turn, are dis-cretized by piecewise linear finite elements. Under a suitable relation between the relaxation parameter ∊and the meshsize h, the convergence of the discrete functionals and the compactness of any sequence of discrete minimizers are proved. The proof relies on the techniques of Г-convergence and on the properties of the Lagrange interpolation and Clement operators.

Discrete approximation of a free discontinuity problem

Background: Despite the increasing prevalence of food allergy, few studies have assessed the prevalence of perceived food-induced symptoms among school-aged children. There is also a paucity of data on how children with food reactions are managed. We investigated the frequency and characteristics of perceived food reactions in school-aged children.



Methods:  Children aged 5–14 years were included in this cross-sectional study. A standardized self-administered questionnaire on food reactions was handed out to 900 parents.



Results:  We achieved a response rate of 69%. The lifetime prevalence of parental perceived allergic reactions to food was 10.5%; the point prevalence was 1.6%. Medical care included a call to a general practitioner in 54% of cases, self-management in 37%, an emergency call in 6%, and hospitalization in 3%. Antihistamines were administered in 45% of food reactions, topical steroids in 24%, oral or parenteral steroids in 16%, and epinephrine in 1.5%. In children who reported food reactions, skin prick tests for foods were performed in 54% of cases; the oral food challenge test was performed in 7.5%.



Conclusion:  Parent perception of food allergic disorders is common in school-aged children. Few children have undergone diagnostic tests to ascertain clinical food hypersensitivity. This is warranted to avoid unnecessarily restricted diets. Efforts should be made to train primary care physicians to manage food-allergic children.

Alessandra Coscia

Papers

Fine Properties of Functions with Bounded Deformation

Hölder continuity of the gradient of p(x)-harmonic mappings

Compactness and lower semicontinuity properties in SBD(Omega)

Discrete approximation of a free discontinuity problem

Parents' estimate of food allergy prevalence and management in Italian school-aged children.