13th International Conference on Fracture June 16–21, 2013, Beijing, China -4- 1 1 1 1 v c n n v v c c i i j j i j v c W f W f W f f = = = + ∑ ∑ 1 3 d e f W λ χ γ ⎛ ⎞ = ⎜ ⎟ ⎝ ⎠ (10,11) where eλ is the aspect ratio of the finite-element with respect to the principal loading direction. 2.1 Calibration of the void evolution and coalescence models To ensure that the extended GT constitutive model described in the previous section is accurate for a single void, an extensive study of axisymmetric unit cells containing a dilute concentration of voids was performed with initial void aspect ratios ranging from 0.001 to 6. Each unit cell was subjected to constant triaxial loadings ranging from, T = 1/3 - 3, for a material with a hardening exponent of 0.1. Following the method of Ragab [8], the q2 coefficient in Eq. (7) was calibrated for each void shape to ensure accurate predictions of the porosity. Second-order polynomials were used to describe the evolution of the void aspect ratios to high accuracy. See Butcher [2] for the complete results of this numerical study along with the calibrated coefficients for void evolution. When void growth and shape evolution are properly modeled, the coalescence strains predicted using Eq. (8) were in very good agreement with the numerical results. An example of the accuracy of the calibrated porosity trends for penny-shaped voids is shown in Figure 3a when the unit cell data is used to evaluate Eq. (7). Similarly, the predicted porosity and coalescence strains are shown in Figure 3b when the GT model in Eq. (1) is used to integrate the stress state. The good agreement of the calibrated GT model in Fig. 3b demonstrates that the predictions for void evolution and coalescence within the particle field are well represented for isolated voids. Figure 3: (a) Comparison of the porosity evolution in the unit cell and with Eq. (7) when it is evaluated using the stress state from the unit cell. (b) Comparison of the predicted void evolution and coalescence when using the GT yield criterion in Eq. (1) to integrate the stress state for a penny-shaped void in uniaxial tension. 2.2 Void nucleation The secant-based particle homogenization scheme of Tandon and Weng [10] was implemented using the procedure of Butcher et al. [11]. Using this method, the stress state within the particles can be estimated based upon the global stress state, particle shape, volume fraction and its mechanical properties. For AA5182, the iron-rich intermetallic particles are assumed to remain elastic during deformation and nucleate voids via a penny-shaped crack in a brittle-type fracture.
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