A multi-scale numerical method for the study of size-scale effects in ductile fracture

The use of a stress-strain constitutive relation for the undamaged material and a traction-separation cohesive crack model with softening for cracking has been demonstrated to be an effective strategy to predict and explain the size-scale effects on the mechanical response of quasi-brittle materials. In metals, where ductile fracture takes place, the situation is more complex due to the interplay between plasticity and fracture. In the present study, we propose a multi-scale numerical method where the shape of a global constitutive relation used at the macro-scale, the so-called hardening cohesive zone model, can be deduced from meso-scale numerical simulations of polycrystalline metals in tension. The shape of this constitutive relation, characterized by an almost linear initial branch followed by a plastic plateau with hardening and finally by softening, is in fact the result of the interplay between two basic forms of nonlinearities: elasto-plasticity inside the grains and classic cohesive cracking for the grain boundaries.