13th International Conference on Fracture June 16–21, 2013, Beijing, China -3- and plastic strain to the finite-element program. The macroscopic stress and strain rate tensors are denoted as, Σij, and Dij, respectively. The porosity (void volume fraction), spheroidal aspect ratio and void spacing ratio are denoted as f, W, and χ, respectively. A Gurson-based yield criterion [1,7] using the modification of Ragab [8] for the qi parameters is employed to account for material softening within the percolation element and is expressed as 2 eq hyd 2 2 , , 1 hyd eq 2 1 hyd eq 3 2 ( , )cosh ( , ) 1 0 2 d d f q n q q n f σ σ Σ Σ ⎛ ⎞ ⎛ ⎞ Φ= + Σ Σ − Σ Σ − = ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ (1) where the global porosity and average q2 value of the voids and cracks are defined as 1 1 v c n n v c d i i i i f f f = = = + ∑ ∑ 2 2 2 1 1 1 1 v c n n v v c c i i j j i j v c q f q f q f f = = = + ∑ ∑ (2, 3) with the subscripts v and c denote quantities for the voids and cracks, respectively, and an overbar symbol denotes a global quantity. The q1 parameter does not require an averaging procedure since it is a function of the stress triaxiality and hardening exponent and these quantities are assumed to be homogeneous in the element. Alternatively, the q2 parameter is a function of the void shape and stress state and will typically be different for each void and crack. The relations for the qi parameters and the triaxiality, T, are 2 3 1 0 1 2 3 q A AT AT AT = + + + ( ) , 2 T n q Wη = hyd eq = / T Σ Σ (4-6) where Ai, and η are coefficients found in Ragab [8]. The associated flow rule of the GT model yields the void growth relation that is applied for each void and crack as hyd 1 2 2 p growth 1 1 hyd hyd 1 2 2 3 3 (1 ) sinh 2 3 3 sinh 2 f f qq q f D fqq q σ σ σ ⎛ Σ ⎞ − ⎜ ⎟ ⎝ ⎠ = Σ −Σ Σ ⎛ ⎞ ⎛ ⎞ + ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ & (7) where Σ1 and D1 p are the stress and plastic strain rate in the principal loading direction. Void coalescence is modeled using the criterion of Pardoen and Hutchinson [9] for internal necking coalescence. The onset of coalescence occurs when the following constraint is satisfied: ( ) ( ) 2 2 1 2 1 1 1.24 0.1 0.22 4.8 uc n n W κ χ χ σ χ χ − ⎛ ⎞ Σ ⎛ − ⎞ ≥ + + + ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ (8) where / 4 ucκ π = for a cubic cell. The void spacing ratio and the cell aspect ratio, λ, are defined as 1 3 W f λ χ γ = ⎛ ⎞ ⎜ ⎟ ⎝ ⎠ y x z L L L λ= (9a,b) where /6 γ π = for a cubic unit cell and Li are its side-lengths that evolve with the applied strain. The criterion in Eq. (8) is also used to identify the onset of profuse void coalescence and failure of the percolation element. In this case, the global void aspect ratio and spacing ratios are:
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