## 4 Results and Discussion
### Prediction of the Fatigue Crack Growth Rate  
To carry out a quantitative investigation of the AE parameters in the crack propagation process, this section begins with a finite element analysis of the prefabricated fracture rod, as shown in Figure 17a. The prefabricated fracture rod, consisting of Q235 material with a density of 7850 kg/m\({}^{3}\), a Young's modulus of 210 GPa, and a Poisson's ratio of 0.28, is simulated. A solid support constraint is applied to one end of the rod, while the rod load calculated using Equation (1) is applied to the opposite side. The model is discretised using a tetrahedral mesh, with the mesh refined to 0.5 mm at the thin rod diameter and transition fillet, and to 2 mm in other regions. Several simulations with different mesh densities are performed to evaluate the mesh independence. Following the mesh-refinement rule, denser meshes are generated, demonstrating consistency and convergence in the simulation results. The average stress change in the centre section of the thin section of the prefabricated fracture rod in models with mesh sizes of 0.6 mm, 0.5 mm, and 0.4 mm at the transition fillet and thin rod diameter is less than 5%. The final model is constructed with 103,568 meshes using a mesh scheme of 103,568. When the rod was loaded, the stress intensity factor at the crack tip was determined using the M-integral method. The crack propagation equation, represented by the Paris equation (Equation (12)), was used for the piston rod material, specifically the Q235 with  
Figure 16: Dye-penetrant testing of crack on the prefabricated fracture rod. Figure 17: Finite element analysis of the crack propagation of prefabricated fracture rods. Finite element model (**a**); crack propagation analysis process (**b**). the coefficients \(C\) set at \(3.16\times 10^{-11}\) and \(m\) set at 2.83. The fatigue threshold, denoted \(\Delta K_{th}\), was set at 6.35 and the fracture toughness was characterised by \(K_{IC}=110.7\) MPa\(\cdot\sqrt{\text{m}}\)[60]. \[\frac{da}{dN}=C(\Delta K)^{m}=C\big{(}\Delta\sigma\sqrt{\pi a}\big{)}^{m}\,, \tag{12}\]  
where \(\frac{da}{dN}\) is the crack propagation rate, mm/cycle; \(\Delta K\) is the stress intensity factor of the corresponding section, MPa\(\cdot\sqrt{\text{m}}\); \(\Delta\sigma\) is the stress amplitude of the corresponding nominal section, MPa; \(a\) is the crack length, mm; \(C\) and \(m\) are the material correlation constants, respectively. Following the crack tip mesh-refinement and propagation calculations, the submodel mesh is updated and integrated into the full model for further iterations in subsequent finite element analyses. The calculation results use the midpoint of the crack tip length as the reference point for calculating the crack path length. The AE counts and energy data are extracted for 11,586 cycles during the period from thew mutation onset to the prefracture and correlated with the crack length data, as shown in Figure 18a,b. The observed correspondence between the crack propagation cycle and the actual fracture cycle period indicates that the simulation results can serve as a basis for the crack propagation rate analysis. Polynomial fitting of the variation curves for the AE counts and AE energy are performed to identify the trend of the parameter changes. Notably, the trends of the AE counts and AE energy show good agreement with the increment trend of the crack length, especially in approximately the first 8000 crack propagation cycles. To investigate the ability of the AE parameters to quantitatively describe the crack growth rate, the logarithmic relationship between the rate of change in the AE parameters and the crack growth rate is plotted in Figure 18c. It can be seen that the relationship between the rate of change in the AE energy and the crack growth rate shows a more favourable linear relationship with a smaller R\({}^{2}\). The slightly inferior performance of the AE counts can be attributed to the strong influence of the chosen threshold value in the calculation of the AE counts.