Continuum to discrete transition in non local damage models
Résumé
Most failure models for strain softening materials involve non locality. Non locality finds its origin in the interaction between growing defects in the course of failure. Nearby the boundary of the solid, interactions between defects are expected to be different compared to those observed in the bulk material. On the boundary of a solid, the material response should be local according to [1]. Cracks should also modify the non local interactions and should induce some shielding effect: the interaction between two points located apart from a crack should not exist. This means also that non locality, to some extend, should depend on the state of damage in the material and, for instance, that the internal length entering in non local models should evolve in the course of failure. Capturing existing and evolving boundaries in non local model is a pending issue with various approaches. The purpose of this paper is to discuss how boundary, and emerging boundary effects can be folded into a new non local damage formulation based on integral models [2] that provides a consistent transition towards discrete cracking. The goals of the presented modified non local formulation are twofold: (1) at the boundary of the solid, non locality should vanish; (2) when a crack has formed - i.e. when a band of damage close to 1 has been created, the constitutive relations should again be local in this band. One dimensional examples illustrate the capabilities of the new model to describe progressive failure and also complete failure with a view to the estimate of crack opening [3]. [1] Krayani et al., Eng. Fract. Mech., 2009. [2] Pijaudier-Cabot and Bazant, J. Eng. Mech., 1987. [3] Dufour et al., Comput. Concr., 2008.