13th International Conference on Fracture June 16–21, 2013, Beijing, China -2- form ( ) , cE T θ . In this paper, two different void coalescence criteria are combined with the original GT model to simulate the whole process of voids nucleation, growth and coalescence. Axisymmetric round tensile bars with different notch root radii are simulated using the extended damage models to investigate the variation of critical damage at coalescence as a function of stress triaxiality. 2. Extended Damage Models 2.1. Modeling the void growth process The growth of a void and the associated macroscopic softening is adequately captured by GT constitutive relationship. The most widely used form, which applies to strain hardening materials under the assumption of isotropic hardening, has the shape 2 2 2 1 3 3 2 cosh 1 0 2 q q p q f q f σ σ ⎛ ⎞ ⎛ ⎞ Φ= + − − − = ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ , (1) where prepresents the macroscopic hydrostatic pressure 1 : 3 p I σ =− , (2) qdenotes the macroscopic Mises equivalent stress 3 : 2 q s s = , (3) σis stress tensor; s is stress deviator; I is the second order identity tensor; ( ) p m σ ε is the current flow stress of the fully dense matrix material as a function of p mε , the equivalent plastic strain in the matrix; and f is the current void volume fraction in the material. Tvergaard (1980, 1981) [3] introduced the constants 1q , 2q and 2 3 1 q q= to account for void interaction effects due to multiple-void arrays and to give a better agreement with experimental data. The hardening of the matrix material is described through ( ) p m σ σ ε = . The evolution of p mε is assumed to be governed by the equivalent plastic work expression: ( ) 1 : p p m f d d σ ε σ ε − = , (4) where p d ε is the macroscopic plastic strain rate tensor; p m d ε is equivalent plastic strain rate of the matrix material. The change in volume fraction of the voids is due partly to the growth of existing voids and partly to the nucleation of voids. It can be expressed as [12]: growth nucleation df df df = + , (5) with ( ) 1 : p growth df f d I ε = − , (6) Nucleation of voids can occur as a result of micro-cracking and/or decohesion of the particle-matrix interface. It can be assumed to be strain controlled, so that the rate of increase of void volume fraction due to nucleation of new voids is given by p nucleation m df Ad ε = , (7)
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