Issue 49

M. Hadj Miloud et alii, Frattura ed Integrità Strutturale, 49 (2019) 630-642; DOI: 10.3221/IGF-ESIS.49.57 632     C * C C C F * U F f if f f f f f δ f f if f f f f if f f             (3) with: * U C F C f f f f     , called the coalescence acceleration f C is the critical porosity corresponding to the beginning of the coalescence f F designate the porosity corresponding to the final fracture of the material. The ultimo value * U f is reached when the macroscopic fracture occurs by loss of the bearing load and is calculated by the following equation * 1 1 f q U  when 2 3 1 q q  . During the plastic flow, the porosity evolution is due to both the voids growth and the voids nucleation: croissance nucléation f f f      (4) Assuming the matrix incompressibility, the term due to the voids growth is given by the following equation: p croissance kk f (1 f )ε     (5) p kk   is the trace of the macroscopic strain rates tensor. When the nucleation is controlled by the plastic strain, its contribution is as follow: p nucléation f Aε    (6) Where p  represents the equivalent plastic strain. Chu and Needleman [7] assumed that the priming of the voids follows a normal distribution with a mean strain N  and a standard deviation S N : 2 p N N N N f ε ε 1 A exp 2 S S 2π                  (7) The GTN model parameters can be subdivided in three subsets as: - Constitutive parameters: q 1 , q 2 and q 3 . Commonly, these parameters are fixed as: q 1 =1.5, q 2 =1 and q 3 = ( q 1 ) 2 [2, 8]. - Nucleation parameters: ε N , S N and f N . Generally, the ε N and S N values are 0.3 and 0.1 respectively for most materials. Thus, they are considered as constants of the statistical distribution law and not as intrinsic material characteristics. The void volume fraction f N is the volume fraction of particles available for void nucleation, while the initial void volume fraction f 0 concerns all the inclusions [9, 10]. - The porosities f 0 , f C and f F are considered as material parameters. The initial VVF parameter f 0 characterizes the initial state of the material. Generally, this parameter is evaluated by microscopic analysis of the undamaged material. For the studied material, the initial void volume fraction f 0 is small ( f 0 =10 -5 ) [9]. There are several methods to determine the critical void volume fraction f C . From a physical point of view, there is a threshold of porosity from which the specimen rigidity drops suddenly. But its determination is very difficult. Sun et al. [11-13] have suggested that f C can be numerically obtained by fitting the numerical curve with the experimental one. The final void volume fraction f F describes the state of the material at the fracture phase. This parameter is considered as constant and can be determined experimentally [14]. It has been considered as an unimportant parameter, it is interesting to know whether it is a constant Zhang et al. [15]. In this work, the final failure void volume fraction is preset to f F =0.1.