# Convergence Check Phase-Field Scheme for Modelling of Brittle and Ductile Fractures
## 5 Numerical Examples
### Low Cyclic Fatigue  
The next example demonstrates the ability of the proposed CCPF framework to model LCF phenomena. The problem considered is a CT specimen loaded by force. An analogous benchmark example was taken from [48] for verification of the phase-field method in modelling of material fatigue. A phase-field extension to fatigue derived from [48] was adopted, as explained in Section 2.1. Geometry, material properties, and loading were also taken from [48]. The dimensions of the CT specimen along with boundary conditions are shown in Figure 17.  
The parameter \(w\) was set to \(w=30\) mm, thickness of the specimen was 15 mm, radius of the notch tip was \(r=0.08\) mm, and parameter \(a_{0}\) was \(a_{0}=7.5\) mm. Boundary conditions were imposed via kinematic constraints, mimicking pins. The material of the specimen was nodular cast iron, according to [64]. Elastic properties were [PERSON]'s modulus \(E=140\) GPa and [PERSON]'s ratio \(v=0.3\). Isotropic hardening is described by the saturation law (33). Initial yield stress equals \(\sigma_{y}^{0}=123\) MPa, saturation coefficient (\(Q_{\infty}=95\) MPa, and saturation exponent \(b=18\). Aside from isotropic hardening, due to cyclic loading in the plastic regime, kinematic hardening [65] was also exhibited, according to the relation:  
\[\alpha_{k}=\frac{C_{k}(\sigma-\alpha)c_{\mathrm{ekv}}^{\mathrm{P}}}{\sigma_{y} \Big{(}\epsilon_{\mathrm{ekv}}^{\mathrm{P}}\Big{)}}-\gamma_{k}\alpha_{k}c_{ \mathrm{ekv}}^{\mathrm{P}}. \tag{34}\]  
In Equation (34), \(\alpha_{k}\) represents back stress tensor and \(C_{k}\) and \(\gamma_{k}\) are the kinematic hardening modulus and decrease rate of kinematic hardening, respectively. The total value of back stress tensor can be calculated by summation of \(k\) back stress tensors. In this example, \(k=2\), with parameters \(C_{1}=22,734\) MPa, \(\gamma_{1}=261.8\), \(C_{2}=136,029\) MPa, and \(\gamma_{2}=2113.5\). The specimen was loaded at the reference point of the upper pin by the concentrated force of amplitude \(F=6000\) N and load ratio \(R=0.1\). The numerical model was discretised by 4224 finite elements in a single layer. Mesh discretisation was densified in the expected FPZ. Since the simulation was conducted for approximately half of the possible crack propagation length, mesh refinement was applied only in this part of the CT specimen. The critical strain energy density parameter used in simulation was \(G_{\mathrm{c}}=740\) N/mm, while length scale was \(l=0.1\) mm. For consideration of fatigue behaviour, the fatigue parameter \(\psi_{\infty}\) had a value of \(\psi_{\infty}=5000\) MPa. The fatigue degradation function \(F_{\mathrm{LCF}}\) was used for the degradation of \(G_{\mathrm{c}}\) according to Equation (19). Since this example deals with LCF phenomena and material elastoplasticity, cycle skipping was omitted. Furthermore, the line search option was activated for easier convergence of the solver. For better visualisation and numerical efficiency, the element deletion option available in ABAQUS was also utilised  
Figure 17: Geometry and boundary conditions of CT specimen. in this problem. Hence, all finite elements which exhibit values of phase-field variables \(\phi\geq 0.95\) in all integration points were deleted. The fatigue crack propagation through loading cycles is displayed in Figure 18.  
The crack path obtained through cycles is correct, and further verification of the crack propagation can be achieved by comparison with [48], where crack propagation after the same number of loading cycles is displayed. In addition, Figure 19 represents lower and upper displacement amplitude of the reference point of the top pin through loading cycles. From Figure 19, it can be concluded that after 13,000 loading cycles accelerated crack propagation is expected, leading to the complete fracture of the specimen based on the  
Figure 19: Displacement amplitude of the top pin through cycles in a low cyclic fatigue regime.