13th International Conference on Fracture June 16–21, 2013, Beijing, China -5- Using explicit void representation, the void growth and coalescence mechanisms and the effects of the initial relative void spacing, void pattern, void shape and void volume fraction on ductile fracture toughness can also be studied directly [16]. 2.2. Porous Plasticity Models Various forms of porous plasticity models have been developed to describe void growth in ductile solids and the associated macroscopic softening, among which the most famous model is due to Gurson [4] with the modification by Tvergaard and Needleman [5, 6]. The yield function of the GTN model has the form 0 1 2 3 2 cosh Φ 2 2 2 2 − − = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = + q f σ q q f σ 1 h 2 1 e Σ Σ (3) where Σe denotes the macroscopic Mises effective stress, Σh represents the macroscopic hydrostatic stress, σ is the current flow stress of the matrix material, and f defines the current void volume fraction. The evolution law for void volume fraction is determined by requiring the matrix material to be plastically incompressible ( ) p kk f -f E & & 1 = (4) where p kk E & is the trace of the macroscopic plastic strain rate tensor. The GTN model was derived for growth of spherical voids, but voids are often non-spherical in actual materials. The GLD model [7, 8], with the yield function given by Eq. (5), was derived to describe the macroscopic plastic response of ductile solids containing spheroidal voids ( )( ) ( ) ( ) 0 1 cosh 2 1 2 2 2 ' 2 ⎟− + − + = ⎠ ⎞ ⎜ ⎝ ⎛ + + + + = q g f g Σ q g g f Σ C Φ h h σ κ η σ X Σ (5) where S is the shape parameter, denotes the von Mises norm, ' Σ is the deviatoric stress tensor, hΣ is the generalized hydrostatic stress defined by ( ) ( ) yy zz xx h Σ Σ Σ Σ 2 2 1 α α + + − = , X is a tensor defined as ( ) ( ) ( ) z z x x y y e e e e e e = ⊗ − ⊗ − ⊗ 1/3 1/3 2/3 X , and (ex, ey, ez) is an orthogonal basis with ey parallel to the axisymmetric axis of the void, and⊗denotes tensor product. The evolution equation for f is the same as Eq. (4) and derivations of the evolution equation for S can be found in Gologanu et al. [7, 8]. In order to simulate ductile fracture process, these porous plasticity models must be calibrated such that the material behavior in the fracture process zone is accurately captured. Calibration of these models requires the predicted macroscopic stress-strain response and void growth behavior of the representative material volume to match the results obtained from detailed finite element models with explicit void representation obtained from the unit cell analysis outlined in Section 2.1. Faleskog et al. [13], Kim et al. [9] and Pardoen and Hutchinson [17] describe the procedures to
RkJQdWJsaXNoZXIy MjM0NDE=