Résumé Nous présentons un modèle original pour décrire le comportement des matériaux fragiles ou semi-fragiles. Les expériences menées sur ces matériaux mettant en évidence dispersion et effet d'échelle, le souci du modèle est de rendre compte de façon correcte de ces phénomènes. L'écriture du modèle repose sur des considérations faites à deux niveaux. A un premier niveau, caractérisé de ‘micro-’, le comportement est guidé par la présence et l'évolution de défauts (pores, fissures, hétérogénéités…) et l'outil choisi pour la description est celui de la rupture fragile probabiliste. Au niveau qualifié de ‘macro-’, le matériau peut être considéré comme continu et la mécanique de l'endommagement est l'outil descriptif. Le modèle couple les deux descriptions en assurant la continuité de la description entre les deux niveaux. Il conduit à une formulation probabiliste de la loi d'évolution de l'endommagement. Ce modèle a été implanté dans un code de calcul par éléments-finis et les résultats de deux simulations sont présentés. Ces simulations mettent en évidence le rôle joué par les hétérogénéités dans le comportement macroscopique de la structure. Le matériau étant donné, ce rôle dépend fortement de la configuration étudiée: géométrie extérieure de la structure, conditions aux limites en effort et en déplacement.

Summary A model is presented for the description of brittle or semi-brittle materials (concrete, rocks, glass, etc.). The macroscopic behaviour of such materials depends deeply on the presence and the evolution of micro-defects (voids, micro-cracks, inhomogeneities, etc.). For instance, the progressive loss of stiffness of concrete is due to the initiation, growth and coalescence of many micro-cracks. Many authors have used probabilistic brittle fracture for the description of the failure of fibres and fibrous composites. On the other hand, continuous damage mechanics has proved to be an attractive tool for a correct description of the progressive failure in concrete structures. The idea developed in this paper is to mix these two kinds of methods using a two-level approach: (i) The ‘micro-level’ is that of the micro-defects. At this scale, the behaviour is assumed to be elastic-brittle, the value of the strength being distributed according to a given probabilistic density function. Then the defect has only two states: broken or unbroken. (ii) The ‘macro-level’ is that where the material can be considered as continuous, and damage continuous mechanics is then used. The state of the macro-volume results as a sum of many micro-volumes; the damage then appears as related to the local distribution of failure. The model assumes a few simple hypotheses on the relation between the micro- and macro-level, and it results into a probabilistic formulation of the damage evolution law of the material. In the present stage of development of the model, we assume a Weibull distribution function (with two parameters) at the micro-level and a parallel loose bundle for the micro-macro transition (each bar in the bundle being a micro-defect and the whole bundle being the macro-volume considered). The model has been implemented with the finite-element program CESAR and a few results are shown and commented on. Due to the occurrence of different random events, the path followed during two numerical experiments on initially identical structures are different and it exhibits scattering, just as do the real experiments (where the distribution of inhomogeneities differs from one specimen to the other). We show that, for the same material, the sensitivity of the structure to the presence of defects is a function of its geometry and of the boundary conditions (loading and displacements). The explanation is that the evolution of the damage is guided by two competitive factors. While the ‘macro’ field of strains, given by the geometry of the problem, tends to average and regulate the evolution, the local field of strains, deeply influenced by the defects, tends to make it diverge from an averaged behaviour.