Issue 19

K. Pham et alii, Frattura ed Integrità Strutturale, 19 (2012) 5-19; DOI: 10.3221/IGF-ESIS.19.01 5 Damage localization and rupture with gradient damage models K. Pham Université Pierre et Marie Curie, Institut Jean le Rond d'Alembert, F75005 Paris pham@lmm.jussieu.fr J.-J. Marigo Ecole Polytechnique, Laboratoire de Mécanique des Solides, F91128 Palaiseau Cedex marigo@lms.polytechnique.fr A BSTRACT . We propose a method of construction of non homogeneous solutions to the problem of traction of a bar made of an elastic-damaging material whose softening behavior is regularized by a gradient damage model. We show that, for sufficiently long bars, localization arises on sets whose length is proportional to the material internal length and with a profile which is also characteristic of the material. The rupture of the bar occurs at the center of the localization zone when the damage reaches there the critical value corresponding to the loss of rigidity of the material. The dissipated energy during all the damage process up to rupture is a quantity c G which can be expressed in terms of the material parameters. Accordingly, c G can be considered as the usual surface energy density appearing in the Griffith theory of brittle fracture. All these theoretical considerations are illustrated by numerical examples. K EYWORDS . Damage Mechanics; Gradient Damage Models; Variational Methods; Crack Initiation. I NTRODUCTION t is possible to give an account of rupture of materials with damage models by the means of the localization of the damage on zones of small thickness where the strains are large and the stresses small. However the choice of the type of damage model is essential to obtain valuable results. Thus, local models of damage are suited for hardening behavior but cease to give meaningful responses for softening behavior. Indeed, in this latter case the boundary-value problem is mathematically ill-posed (Benallal et al. [1], Lasry and Belytschko, [5]) in the sense that it admits an infinite number of linearly independent solutions. In particular damage can concentrate on arbitrarily small zones and thus failure arises in the material without dissipation energy. Furthermore, numerical simulation with local models via Finite Element Method are strongly mesh sensitive. Two main regularization techniques exist to avoid these pathological localizations, namely the integral (Pijaudier-Cabot and Bažant [13]) or the gradient (Pham and Marigo[10, 11]) damage approaches, see also [6] for an overview. Both consist in introducing non local terms in the model and hence a characteristic length. We will use gradient models and derive the damage evolution problem from a variational approach based on an energetic formulation. The energetic formulations, first introduced by Nguyen [9] and then justified by Marigo [7, 8] by thermodynamical arguments for a large class of rate independent behavior, constitute a very promising way to treat in a unified framework the questions of bifurcation and stability of solutions to quasi-static evolution problems. Francfort and Marigo [4] and Bourdin, Francfort and Marigo [3] have extended this approach to Damage and Fracture Mechanics. Considering the one-dimensional problem of a bar under traction with a particular gradient damage model, Benallal and Marigo [2] apply the variational formulation and emphasize the scale effects in the bifurcation and stability analysis: the instability of the homogeneous response and the localization of damage strongly depend on the ratio between the size of I