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. | Vietnam Journal of Mechanics, VAST, Vol. 31, No. 3 &4 (2009), pp. 233 – 246 CONSTRUCTION AND ANALYSIS OF LOCALIZED RESPONSES FOR GRADIENT DAMAGE MODELS IN A 1D SETTING K. Pham and . Marigo Université Paris 6, Institut Jean le Rond d’Alembert, 4 Place Jussieu 75005 Paris Abstract. 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. We point out the very sensitivity of the responses to the parameters of the damage law. All these theoretical considerations are illustrated by numerical examples. 1. INTRODUCTION It 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 Baˇzant [10]) or the gradient (Triantafyllidis and Aifantis [11]) damage approaches, see also [6] for an overview. Both consist in .
