Issue 37

R. Sepe et alii, Frattura ed Integrità Strutturale, 37 (2016) 369-381; DOI: 10.3221/IGF-ESIS.37.48 370 propagation of a crack to the nucleation and growth of micro-voids in the material, and then it is able to connect the micromechanical characteristics of the component under examination to crack initiation and propagation up to a macroscopic scale. The three stages of nucleation, growth and coalescence of micro-voids are well established results of metallographic observation for polycrystalline metals at ductile failure. The simulation of these microstructural damage processes has been considered in various micromechanical and macromechanical model approaches in the literature. A macromechanical model can be exactly obtained by the statistical averaging of microscopic quantities in a homogenization process. The Gurson model [8], which derived a macroscopic yield function and an associated constitutive flow law for an ideally plastic matrix containing a certain volume fraction of spherical voids, is a well-known analytical approach to this problem. Empirical modifications of this approach have been proposed to improve the prediction at low void volume fractions [9] and to provide a better representation of final void coalescence [10]. It must be pointed out that even if the statistical character of some of the physical parameters involved in this model has been put in evidence, no effective attempt has been made insofar to relate the corresponding statistic to the experimental results, as for example the R-Curve [1,11]. In the first part of the work, after briefly recalling the ductile fracture mechanism and the classical approach to determine the parameters of the GT model, we pointed out the main aspects of the Stochastic Design Improvement (SDI) technique [12], implemented in the commercial code ST-ORM [13], which can be coupled with specialized FEM codes as LS- DYNA [14] or WARP 3D [15] for what concerns the deterministic structural part of the process; we then illustrated the use of the SDI technique to perform a robust numerical calibration of the parameters of the GT model. In the second part of the work, some numerical results regarding a simple structural component are shown, pointing out the influence of the statistical distribution of the physical parameters of the GT model on the toughness of the considered material. Material toughness was numerically evaluated both in terms of the critical value of the J-integral, by using the Equivalent Domain Integral method [16] implemented in the finite element code WARP 3D, and in terms of critical applied load, by using the explicit finite element code LS Dyna. In this case, as the load is monotonically increased by applying increments that are considered to be constant in time, the critical load value is univocally linked to a correspondingly crack propagation initiation time. For the validation of the numerical results, experimental data from literature are used [17-18]. D UCTILE FRACTURE MECHANISMS : OVERVIEW ON THE G URSON -T VERGAARD MODEL s it is well known, a typical process of ductile fracture develops through the nucleation and the following growth and coalescence of cavities nucleated at crack tip. The best known nucleation mechanisms of such cavities are those linked to the presence of hard particles (inclusions or precipitates) in a ductile matrix (base material). In such cases, the brittle failure of a particle or its decohesion from the matrix, because of the interface breaking, leads to the nucleation of small voids, whose growth causes real cavities in the material. It is clear that the toughness of a material increases for smaller dimensions of such particles and for increasing mean distance and homogeneity of the distribution of such inclusions. From an analytical point of view, we can observe that a void nucleates when the elastic potential energy associated with the material surrounding the particles reaches a particular value, which can be determined by following one of the two approaches, which, even if rather different from each other, lead to similar or equivalent conclusions. The first one is based on the existence of a critical value of the strain in the direction of the applied load, and the second one on the existence of a critical value of the stress in the same direction [19]; once such critical values are overcome, the amount of elastic potential energy necessary for the voids nucleation is considered as reached. As those critical values for both stress and strain are usually low, the nucleation process develops rather easily; therefore, the ductile failure of metallic materials depends substantially on the way the cavities grow and coalesce, rather than on their nucleation. As a rule, a growing cavity is modelled as an ellipsoid of revolution, whose major axis (2 b ) is parallel to the direction of the load. As the strain increases, the cavity grows in volume while keeping its minor axis equal to the initial mean dimension R 0 of the enclosed particle considered to be non-deformable, and b varying along with the deformation in the load direction, A