13th International Conference on Fracture June 16–21, 2013, Beijing, China -4- In the later stage of the material failure process, secondary voids often nucleate in the ligament between enlarged primary voids and rapid growth and coalescence of these secondary voids accelerates the final ligament separation. In our analyses, it is assumed that void nucleation is plastic strain controlled and follows a normal distribution proposed by Chu and Needleman [15]. The nucleated voids are regarded to be smeared in the material and the material behavior is governed by the GTN model. Figure 3(a) compares the macroscopic effective stress versus effective strain curves between models including and not including the secondary voids. Here several values of stress triaxiality ratio, T = 1/3, 2/3, 1, 1.5 and 2, are considered. The open circles denote the onset of coalescence for models where secondary voids are not taken into account. The filled circles represent the onset of coalescence for models where nucleation, growth and coalescence of secondary voids are accounted for. It is clear that secondary voids significantly accelerate the void coalescence process. To demonstrate the Lode angle effect on ductile failure, let the stress triaxiality ratio T be fixed and consider a series of stress states corresponding to different θ-values. Figure 3(b) shows the variation of Ec with θ as T taking a fixed value of 2/3. Clearly the Lode angle has an important effect on Ec. Ee 0.0 0.5 1.0 1.5 Σe / σo 0.0 0.5 1.0 1.5 2.0 2.5 T=1/3 T=2/3 T=1.0 T=1.5 T=2.0 f0=0.02 Now consider an array of T and θ values and perform unit cell analysis for each case. The variation of Ec with T and θ can be expressed by a function Ec(T,θ). Therefore, a ductile failure criterion for a given material can be established as ( , )θ E E T c e = (2) where Ee denote the macroscopic effective strain of the RMV. The RMV fails when Ee reaches a critical value dependent of its stress state characterized by T and θ. Figure 3. (a) Comparison of the macroscopic effective stress versus effective strain curves between models including and not including secondary voids. The parameters for nucleation of secondary voids are fN = 0.04, εN = 0.1 and sN = 0.05 [15]. (b) Variation of Ec with θ as T taking a fixed value of 2/3. 0 0.4 0.8 1.2 1.6 -30 -20 -10 0 10 20 30 no secondary voids with secondary voids Θ(degree) Ec T = 2/3 (a) (b)
RkJQdWJsaXNoZXIy MjM0NDE=