XVII Convegno IGF –Bologna 16-18 Giugno 2004 An implicit gradient stress failure condition R. Tovo, P. Livieri, E. Benvenuti Department of Engineering, University of Ferrara, Via Saragat 1, 44100 Ferrara (Italy), plivieri@ing.unife.it ABSTRACT. This contribution focuses on gradient formulations for the prediction of the static failure load of V-notched and cracked components made of brittle materials. A weighted average of the local equivalent stress, called non-local equivalent stress, is first considered and subsequently approximated by a gradient expansion up to the spatial second-order derivative. A distinguishing characteristic of the present approach is that the non-local equivalent stress is calculated by solving a differential equation of implicit type. The numerical solution for a V-notch in the presence of Neumann’s boundary conditions is presented. Moreover, the analytical solutions of an one-dimensional case is proposed. Finally, the static failure loads predicted by to the present formulation through the finite element technique are compared with the experimentally determined ones. 1. INTRODUCTION Accurate prediction of the static failure initiation of brittle components in the presence of high stress concentrations is of great interest in mechanics. In this case, classical continuum models cannot be used. For instance, volume averages of the local stress scalar representative of the adopted failure criterion have been using in fracture and fatigue mechanics (e.g. Seweryn and Mroz, 1995 1 ; Taylor, 1999 2 ). However, these averages-based approaches possess an empirical origin, while lacking of an explicit theoretical basis. In particular, the interaction of points placed within the volume of integration is independent of the distance between the points. Moreover, the choice of the volume of integration has to be stated a priori, as a function of the mode of loading. On the other hand, these averages- based approaches can be seen as simple versions of the so-called non-local approaches developed in elasticity starting from the 60’s by Kröner 3 , Edelen 4 , Eringen 5 and based on the assumption that the principle of local action does not hold in the presence of stress- or strain- singularities. Later, Bazant, 6 de Borst 7 and several others 8 ,developed a simpler class of non- local models for the finite element analysis of strain localization exhibited by structural elements made of softening materials. The purpose was to overcome pathological mesh- dependency of finite element models for softening materials. Classical finite element techniques converge indeed to a physically meaningless solution, where the strain field tends to localize into the smallest finite element. That is in contrast with the experimental observations that failure is reached by strain localization in a process zone with finite size. Moreover, the computed load-displacement response in the post-elastic phase tends to the elastic unloading as the mesh size decreases, leading to mesh-dependency. That motivated the adoption of enhanced formulations such as, for instance, formulations of non-local type. Basically, a non-local field is defined as the volume average of the local field weighted by a suitable weight function, ( 29 y x, α . The weight function reaches its maximum at the actual point x , and decays to zero at increasing distances. Such non-local models are usually called of integral type. After Taylor series expansions of the local field and its subsequent weighted integration, gradient-approximations of the non-local field can be obtained, of both explicit and implicit gradient types 9 . In this contribution, an implicit gradient type stress-failure criterion is proposed for the analysis of the static failure of V-shaped notches under mode I loading assuming a linear elastic material. A stress scalar of non-local type, called the non-local equivalent stress, is