13th International Conference on Fracture June 16–21, 2013, Beijing, China -5- volumetric strain. Compared to the Gurson model, no modifications are needed to describe the evolution of porosity. The rate of f consists of two terms describing growth and nucleation of voids. nuc growth f f f &= & + & (6) Void growth is governed by the rate of volumetric plastic strain, whereas void nucleation is assumed to be controlled by the equivalent plastic strain pl M, ε of the matrix material. Under the assumption that void nucleation follows a normal distribution, its rate is given by Eq. (8). pl kk growth f f ε& & (1 ) = − (7) pl M N pl M N N n nuc S S f f , 2 , 2 1 exp 2 ε ε ε π & & ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ − − = (8) fn denotes the overall volume fraction of void nucleating inclusions, εN is the mean value of the equivalent plastic strain, at which inclusions nucleate voids and SN is the standard deviation of the normal distribution. Together with the Gologanu model the fracture criteria of Thomason [14], in a modified version of Pardoen and Hutchinson [15], and Brown and Embury [16] are used. The Thomason model assumes that void coalescence begins once the plastic limit load on the ligament between the voids is reached. The limit load is derived using an axisymmetric unit-cell containing one ellipsoidal void of a regular array shown in Fig. (5). Figure 5. Geometrical quantities of the void distribution used for the Thomason model The coalescence criterion is then expressed by Eq. (9), where α and β are material parameters. Unit cell analyses suggest to choose α = 0.1 and β = 1.2 [15]. Based on the geometry the ligament size ratio can be written as Eq. (10). 0 / 1 1 1/2 2 2 22 Thomason = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ⎟ + ⎠ ⎞ ⎜ ⎝ ⎛ − ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − − Σ = b R w R b R b C M β α σ (9) ( ) 1/3 ' 2 22 3 0 0 exp R H w f 2 3 R b ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ = ε (10) The Gologanu model calculates f, w, Σ22 and σM in Eq. (9) and (10). The initial aspect ratio H0/R0 can be considered, besides α and β, as a third material parameter of the failure criterion. To model material failure at low stress triaxilities, the criterion of Brown and Embury [16] is used here. This model is based on the assumption that the formation of shear bands takes place in the ligament between two voids and leads to ductile fracture, if the mean radius of a void equals half the distance between the centers of the two voids. Using the void aspect ratio w, calculated by the
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