13th International Conference on Fracture June 16–21, 2013, Beijing, China -2- microcracks where intense shear bands are present, may result in a void sheeting effect further contributing to void coalescence [3, 4]. The metallurgical characteristics of the microstructure, including the size and distribution of the initiating particles which can often concentrate close to or on the grain boundaries will contribute to the nucleation and coalescence process. The distribution of these particles may also be uneven within the material with banding regions of greater concentration of particles or varying grain sizes [5]. There exists a range of mechanistically based models that have been developed to describe the ductile fracture process. One of these is the Gurson-Tvergaard-Needleman (GTN) model [6] which characterises failure by defining a material yield function which depends greatly on the stress states and on material specific characteristics. These characteristics need to be calibrated to enable a simulation of ductile crack growth. 1.1 The Gurson Tvergaard Needleman The GTN model assumes the material is homogeneous and behaves as a continuum with an idealised void volume fraction distribution. Crucially, the model takes into consideration both the strain softening effects of void nucleation, growth and coalescence as well as the competing effect of the matrix hardening behaviour to define a material yielding function Φ, defined as (Eq.1): Φ σ e,σ m,σ, f * = σ e σ 2 +2q 1 f * cosh 3q 2 σ m 2σ - 1+q 3 f * 2 =0 (1) Where: σ e = macroscopic Von Mises Stress σ m = macroscopic mean stress σˉ = flow stress for the matrix material f∗ = current void fraction The values for q1, q2 and q3 were introduced by Tvergaard and Needleman to better simulate the experimental observations. These are often taken as q1 = 1.5, q2 = 1.0 and q3 = q1 2. The rate of void growth is related to the plastic part of the strain rate tensor ε k p k and the void nucleation rate is related to the equivalent plastic strain rate, p eq ε& in (Eq. 2): f*= fgrowth+ fnucleation= !1- f$ε k p k+Λε e p q (2) The first term expresses the growth rate of existing voids assuming the matrix material is incompressible and the second term defines the quantity of new voids that have nucleated as a result of the increasing plastic strain. The scaling coefficient, is characterised by (Eq. 3):
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