# Convergence Check Phase-Field Scheme for Modelling of Brittle and Ductile Fractures
## 5 Numerical Examples
### Ductile Fractures  
The benchmark problem related to verification of the proposed CCPF algorithm in modelling of ductile fractures involves an asymmetrically notched specimen loaded onto tension. For the specific problem, experimental testing was conducted in [29] on the aluminium alloy Al-5005. Geometry and boundary conditions of the numerical model are displayed in Figure 14. Thickness of the specimen was 3 mm. Vertical displacement loading \(v=3\) mm was imposed in 100 increments. In the numerical model, the symmetry boundary condition was imposed on the \(xy\) plane (\(w=0\)). Figure 12: Crack pattern of an asymmetric three-point bending test: (**a**) whole model, (**b**) crack detail. Figure 13: Number of iterations utilised through loading increments for an asymmetric three-point bending test. Elastic material properties of Al-5005 were [PERSON]'s modulus \(E=70.9\) GPa and [PERSON]'s ratio \(\
u=0.34\). The isotropic hardening law was employed, according to the relation:  
\[\sigma_{y}=\sigma_{y}^{0}+Q_{\infty}\Big{(}1-e^{-be_{\text{adv}}^{\text{P}}} \Big{)} \tag{33}\]  
In Equation (33), initial yield stress is \(\sigma_{y}^{0}=113\) MPa, saturation coefficient \(Q_{\infty}=22\) MPa, and saturation exponent \(b=24.5\). \(e_{\text{adv}}^{\text{P}}\) represents equivalent plastic strain. The material parameters are defined according to [29]. It is important to mention that in [29], large-strain elastoplasticity theory was used. Furthermore, in this example a 3D setting was adopted. Hence, the model is discretised by 16,395 hexahedral finite elements, with refinement in the expected FPZ. Regarding phase-field modelling, the TH model was utilised in this example. The critical fracture energy density \(G_{\text{c}}=254.66\) N/mm was taken from [29], while length scale parameter used in the example is \(l=1\) mm, different from the literature [29], where the length scale was set to \(l=0.3\) mm. Furthermore, to improve the convergence rate, the line search option available in ABAQUS was activated. All simulations were finished in approximately one hour. The results obtained are displayed in Figure 15.  
As can be observed from Figure 15, small-strain elastoplasticity theory is inadequate for this problem. Therefore, large-strain theory was activated in ABAQUS using the NL-GEOM option, where the logarithmic strain is computed without any intervention from UMAT. The force-displacement dependency was in better agreement with the experimental investigation compared to the small-strain setting, as can be seen. However, due to differences in tensorial variables used in the constitutive law used in [29] and this paper, discrepancy still exists. To improve agreement of the results, after several attempts, modifications to initial yield stress \(\sigma_{y}^{0}=118\) MPa, saturation coefficient \(Q_{\infty}=32\) MPa, and length scale \(l=0.9\) mm were made. With modified parameters, a significant improvement in numerical results was obtained, as demonstrated by Figure 15. Additional changes should be made to result in better agreement, but this is out of the scope of this paper. A contour plot of equivalent plastic strain and phase-field parameters for a loading displacement of 3 mm is displayed in Figure 16, for the case with modified parameters. Figure 14: Geometry and boundary conditions of an asymmetrically notched specimen. Figure 16 demonstrates the physical consistency of the numerical results, which is in good agreement with [29] although with different values. Hence, according to the results in the example, it can be concluded that the CCPF framework is applicable to the problem of ductile damage, not only for small-strain assumptions but also for large-strain elastoplasticity. To switch between different theories, it is only necessary to activate the NLGEOM option in ABAQUS. Figure 16: Distribution of contours on deformed shape of an asymmetrically notched specimen: (**a**) equivalent plastic strain, (**b**) phase-field variable.