## 4 Applications
### Concrete Fatigue in Axial Compression
#### 4.1.2 Finite Element Simulation  
The finite element model of the cylindrical fatigue specimen is shown in Figure 21. Rigid elements are provided at both ends of the cylinder to simulate the loading device. Only tangential and normal behavior contact properties are set when defining the interaction between the specimen and rigid elements. Reference points RP-1 and RP-2 are placed on the top and bottom surfaces of the model, respectively, and are connected to the rigid element surfaces by coupling constraint. The hinge constraint is applied on RP-2 to simulate the boundary conditions during the fatigue test, while the cyclic load is defined on RP-1 as a concentrated force. For specimen F1, the fatigue-loaded stress levels are \(S_{\text{max}}=0.76\) and \(S_{\text{min}}=0.05\); for specimen F3, \(S_{\text{max}}=0.84\) and \(S_{\text{min}}=0.05\). In the finite element model, concrete is modeled as C3D8R (an 8-node linear hexahedral element with reduced integration and hourglass control) elements. A total of 1536 elements are used in the model with a standard mesh size of 20 mm along the periphery and height. The number of elements is chosen to strike a balance between computational efficiency and model accuracy, ensuring reliable results without excessive computational cost. The embedded CDPM is used to simulate material properties, where the plasticity parameters are defined as listed in Table 7[46]. A suggested value between \(10^{-4}\) and \(10^{-3}\) of the viscosity parameter is utilized in Abaqus to improve the convergence rate in strain softening and stiffness degradation [9]. The compressive behavior of the CDPM is calculated from converted parameters in Table 6 according to Equations (15)-(18). \begin{table}
\begin{tabular}{c c c c c c} \hline \multirow{2}{*}{**Specimen**} & \(f_{c}^{\prime}\) & **Converted**\(f_{cu}\) & **Converted**\(f_{c}\) & **Converted**\(\varepsilon_{\text{p,c}}\) & **Converted**\(E_{0}\) \\  & **(MPa)** & **(MPa)** & **(MPa)** & \(\varepsilon_{\text{p,c}}\) & **(MPa)** \\ \hline F1 & 46.2 & 54 & 48.6 & 0.001899 & 35,573 \\ F3 & 43.9 & 51 & 45.9 & 0.001865 & 35,124 \\ \hline \end{tabular}
\end{table} 
Table 6: Converted material parameters in the monotonic constitutive curve (Equation (22)) for fatigue tests under axial compression from the literature [42]. \(f_{c}\), \(\varepsilon_{\text{p,c}}\), and \(E_{0}\) are the prismatic compressive strength, peak strain, and initial modulus of elasticity, respectively. Figure 21: Finite element model of the cylindrical specimen in fatigue tests under axial compression from the literature [42]. The stress-strain simulation curves of specimens F1 and F3 under monotonic axial compression loading are shown in Figure 22.  
Figure 22 shows that the compressive strength of F1 is 44.4 MPa, and the associated peak strain is 0.00174 from simulation; the simulated strength of F3 is 42.5 MPa, and the peak strain is 0.001698. The simulated compressive strength is slightly smaller than the experiment value (shown in Table 6). However, the relative errors are within 5%. The literature does not specify the peak strain from monotonic loading [42]. The ratio of the maximum fatigue strains to the peak strain \(\epsilon_{\text{max}}^{\text{n}}/\epsilon_{\text{p,c}}\) for fatigue loading is reported in the literature [42]. Maximum fatigue strains under axial compression fatigue loading were obtained from simulation by incorporating the equivalent fatigue damage constitutive model into Abaqus. These maximum fatigue strains at specified cycle ratios are then divided by the simulated peak strain in Figure 22. The comparison between the numerical simulation results and experimental results of \(\epsilon_{\text{max}}^{\text{n}}/\epsilon_{\text{p,c}}\) is presented in Figure 23.  
Figure 23 shows satisfactory simulation results for both specimens F1 and F3, and the relative errors between the simulation and experiment values of the two specimens do not exceed 7%. The RMSE of simulation and test values in Figure 23a is 0.04, and 0.02 in Figure 23b.