Issue 49

M. Hadj Miloud et alii, Frattura ed Integrità Strutturale, 49 (2019) 630-642; DOI: 10.3221/IGF-ESIS.49.57 631 subsequently [2-4]. The modified Gurson-Tvergaard-Needeleman (GTN) model is widely used in the modeling of ductile fracture. Metallographic studies [5-6] demonstrate that the ductile fracture process of a metal is basically characterized by three mechanisms of void evolution as represented in the Fig. 1. a) Nucleation of voids due to the debonding of particle–matrix interface, fracture of the particle or micro-cracking of the matrix surrounding the inclusion (Part III); b) Growth of voids leading to an enlargement of existing cavities (Part IV); c) Finally, coalescence of micro-cracks initiated from voids leading to the drop of the load-carrying capacity of the material, when the void volume fraction (VVF) reaches to its final value (Part V). From Fig. 1, we note also that Part I represents the elastic zone and Part II represents the Lüder bands. Figure 1 : Schematic representation of different stages of ductile fracture. Gurson [1] developed a constitutive model for porous metal plasticity. This model was derived from an approximated one through an upper bound approach limit-analysis of a hollow sphere made of ideal plastic Mises material. Tvergaard et al. [4] perceived that the Gurson model give adequate results for high triaxiality rates of the stresses but overestimate the fracture strains (ductility) for low triaxialities. Therefore, they introduced the constitutive parameters: q 1 , q 2 and q 3 in the Gurson model as: 2 eq 2 m eq y 1 2 3 2 y y σ σ (σ , σ ,f) q f q ( q f ) 0 σ σ 3 Φ 2 cosh 1 2              (1) where: σ eq Von Mises stress, σ y matrix yield limit, σ m hydrostatic stress, f the porosity For the previous yielding criteria and to take into account the fast softening of the material during the coalescence stage (Fig. 1-stage V), Tvergaard and Needleman [2-4] introduced the f*(f) function. The yield surface of the Gurson-Tvergaard- Needleman (GTN) model is written with the following form: 2 eq * 2 *2 m eq 1 2 1 2 σ σ 3 Φ(σ ,σ,f) 2q f cosh q (1 q f ) 2 σ σ 0            (2) where: σ is an equivalent tensile flow stress representing the actual microscopic stress-state in the matrix material, [4]. f* is the modified porosity follows the law below: