Issue 37

R. Sepe et alii, Frattura ed Integrità Strutturale, 37 (2016) 369-381; DOI: 10.3221/IGF-ESIS.37.48 373 of the induced softening of the surrounding material the critical value of cavity volume fraction, f c , can be reached and, in this case, the coalescence starts. From a numerical point of view, when the coalescence of the cavities starts, the cell elements reduce their ability to equilibrate the applied load, according to a given parameter,  , which can be expressed as follows:   0 0 1.0 0 1.0 D D D         (10) where D is the effective cell dimension, 0 D is the cell dimensions at the time at which f = f c and  is a numerical parameter. From the considerations above, it follows that the GT model must be calibrated for every chosen material if it wants to be considered as a valid tool for fracture mechanics analysis within the non linear field. C ALIBRATION OF THE GT MODEL PARAMETERS . he numerical calibration of the GT model parameters can be carried out with a two-step procedure [23-24]. The first stage involves, on a micro-mechanical scale, the correction parameters, q 1 , q 2 ( q 3 = q 1 2 [15]), the fixing of the physical parameters (  , f c ) and of the geometric ones ( D o , f o ) on the basis of experimental considerations. The second stage involves, on a macro-mechanical scale, the geometric parameters ( D o , f o ). For what concerns the parameters governing the nucleation mechanism of cavities, as already noted, they can be initially roughly determined by experimental data related to the initial distribution of the particles in a material sample, while their final refined values can be determined in the second stage of the calibration process. Beside these parameters, it is obviously necessary to know the material properties of the base material ( E     , n ), or the relationship      from a standard tensile test. The validity and the reliability of the GT model are strongly dependent on the correct determination of all the aforesaid parameters and material properties, which can be summarized as follows: - elastic-plastic material model parameters: n ,    ; - Tvergaard correction parameters: q 1 , q 2 , q 3 ; - nucleation process parameters: f N ,  N , S N ; - initial dimension of the elementary cell: D 0 ; - initial and critical cavities volume fraction: f 0 , f c ; - parameter  . In the present work, the first stage of the calibration was performed by comparing two different numerical models: the first one (reference model) was built using ANSYS code. The model (Fig. 1) was a cubic three-dimensional cell, with an initial characteristic dimension D o of the same order of magnitude of the distance between the largest inclusions (100  m), having the same material properties as the base material with a spherical cavity in the centre. A second FEM model (developed with WARP 3D v.14.2 code) consists of a unique 8-noded cubic finite element, whose material shows a constitutive law based upon the GT model, with parameters relative to the base material and to the cavity volume fraction equal to those adopted in the previous reference model. In the first stage of the calibration the stress-strain curve, obtained from the reference model under given boundary conditions, was compared with the curve obtained in the same conditions by the second model, on the basis of the maximum stress and of the work needed to deform the cell [23-24]. All the combinations of q 1 , q 2 e q 3 that give the smallest difference of the said quantities in the second model, with respect to those obtained from the reference model, can be used as parameters of the GT model for the considered material. The second stage of the calibration process enabled to evaluate the parameters D 0 and f 0 , as well as the nucleation parameters f N ,  N and S N , by fitting the numerical R-curves to the experimental one. Anyway, metallographic analyses and numerical calculations suggest that an indicative value of D 0 can be given by the Crack Tip Opening Displacement ( CTOD i ) at the beginning of the propagation process [23-24], which depends on the average distance between the largest inclusions in the material. This value of the CTOD can be measured with standard static propagation tests carried out on a CT specimen [25]. T