# Convergence Check Phase-Field Scheme for Modelling of Brittle and Ductile Fractures
## 5 Numerical Examples
### High Cyclic Fatigue  
The final example demonstrates the ability of the CCPF framework to model HCF phenomena. The CT specimen from the LCF problem presented in Figure 17 is used herein, similar to [48]. The material of the specimen was steel, where only elastic properties were accounted for. [PERSON]'s modulus was \(E=210\) GPa and [PERSON]'s ratio \(\
u=0.3\). For this problem, the specimen was loaded at the reference point of the upper pin by the concentrated force of amplitude \(F=400\) N and load ratio \(R=0\). The specimen was discretised by 29,112 finite elements in a single layer, which is a considerably greater number of finite elements than in the LCF problem. However, in this example, mesh discretisation was densified in the expected FPZ throughout the entire length of the specimen. The critical strain energy density parameter used in the simulation was \(G_{\mathrm{c}}=5000\) N\(/\)mm while the length scale was \(l=0.1\) mm. The fatigue parameter was set to \(\varPhi_{\infty}=50\) MPa. The logarithmic fatigue degradation function \(\hat{F}_{\mathrm{HCF}}\) was used for degradation of \(G_{\mathrm{c}}\) according to Equation (19). To speed up computation time, the two-step cycle-skipping technique was tested in this example. In the first stage of cycle skipping, 2504 cycles were jumped prior to any degradation of material, according to Relation (30). In the second stage of cycle skipping, the possibility of the extrapolation procedure occurring was checked every 10 th loading cycle by setting the parameter nFE = 20 (10 loading and 10 unloading increments). Two values of relative error parameters, \(q=5\) and \(q=25\), were tested. Furthermore, depending on the number of skipped cycles in the first stage, the maximum allowed number of extrapolated cycles was determined. This is a security mechanism because in the early stage of damage development, an unphysically large number of extrapolated cycles can be obtained via Equation (31). Hence, for the example considered, the maximum allowed number of extrapolated cycles was 1000. This is the author-defined value based on experience with the cycle-skipping technique (it was chosen as a round number in the range of 30-50% of stage-one skipped cycles). As in the previous problem, the element deletion option was utilised with a threshold limit of \(\phi\geq 0.95\) in all integration points for deletion of the element. The displacement of the top pin through loading cycles with insight into extrapolated cycles is displayed in Figure 20.  
As shown in Figure 20, extrapolation of the fatigue variable \(\overline{\psi}\) is at the highest degree at the beginning of the simulation for both values of relative error parameter \(q\) when the initial fatigue crack develops. The first 20,000 loading cycles are extrapolated by the  
Figure 20: Displacement amplitude of the top pin through loading cycles: (**a**) \(q=5\), (**b**) \(q=25\). maximum allowed number. Afterwards, a sudden decrease is observable. Setting a greater limit on the maximum allowable extrapolated cycle number could increase computational speed, but as can be seen from Figure 20, the maximum number of extrapolated cycles is exhibited only at the very beginning of the simulation. In the continuation of the simulation, extrapolation is driven by the value of the relative error. In that sense, for \(q=5\), after approximately 30,000 loading cycles there is no extrapolation at all. Therefore, the results obtained herein are considered as relevant since fatigue crack propagation is computed cycle-by-cycle, except the early initiation. With the increase in the relative error parameter to \(q=25\), an increase in extrapolated cycles is exhibited, as expected. However, after reaching approximately 50,000 loading cycles, number of extrapolated cycles drops to \(\approx 20\) extrapolated cycles and preserves a steady value until the final fatigue fracture.