13th International Conference on Fracture June 16–21, 2013, Beijing, China -3- using the finite element program ABAQUS [12], which employs a finite strain, J2 plasticity theory within an updated Lagrangian formulation. The displacement boundary conditions on the outer surfaces of the RMV are prescribed such that the macroscopic parameters T and θ are kept constant during the entire deformation history. Faleskog et al. [13] and Kim et al. [9] provide the details of how to prescribe the boundary conditions. A case of axisymmetric loading is considered first, where 3 1 2Σ ≥Σ =Σ ( o 30 =− θ ). Figure 2(a) shows the variation of X with the macroscopic effective strain (Ee) of the RMV. As loading continues, X gradually decreases. But when the deformation reaches a certain level, X stops decreasing and remains at a constant value. This implies that further deformation takes place in a uniaxial straining mode, which corresponds to flow localization in the ligament between adjacent voids. The shift to a macroscopic uniaxial strain state indicates the onset of void coalescence. Detailed explanation of the uniaxial straining mode can be found in references Koplik and Needleman [14] and Kim et al. [9]. Here we use Ec to denote the macroscopic effective strain at the onset of void coalescence. Ee 0.0 0.2 0.4 0.6 0.8 1.0 X / X0 0.6 0.7 0.8 0.9 1.0 T=1 T=2/3 T=2 Ee 0.0 0.2 0.4 0.6 0.8 1.0 Σe / σ0 0.0 0.5 1.0 1.5 2.0 T=1 T=2/3 T=2 Figure 2. (a) Variation of the deformed cell width in x-direction with the macroscopic true effective strain of the cell revealing the shift to uniaxial straining. (b) Macroscopic true effective stress versus true effective strain of the void-containing RMV displaying the macroscopic softening. The macroscopic effective stress versus effective strain curve, Figure 2(b), provides an overview of the competition between matrix material strain hardening and porosity induced softening. As deformation progresses, a maximum effective stress is reached (indicated by the filled circle), and then Σe decreases as strain-hardening of matrix material is insufficient to compensate for the reduction in ligament area caused by void growth. As the macroscopic effective strain reaches Ec (indicated by the open circle), a rapid drop in macroscopic effective stress occurs. As expected, both the peak stress value and the value of Ec decrease with the stress triaxiality ratio T, reflecting the decease of ductility. (a) (b)
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