# A Review of Prediction Methods for Ultra-Low Cycle Fatigue Damage of Steel Piers Under Earthquakes
## 2 Research Status and Development Trend at Home and Abroad
### Cyclic Void Growth Model
The calculation formula of \(\eta\) is given as  
\[\eta=\int\limits_{0}^{\epsilon_{p}}\exp(1.5T)\times d\epsilon_{p}. \tag{6}\]  
Finally, the calculation Formula (7) of the fracture index \(FI_{CVGM}\) in the cyclic cavity expansion model (i.e., CVGM) can be obtained using Formulas (3) and (5). \[FI_{CVGM}=VGI_{cyclic}/VG\epsilon_{cyclic}^{critical}\geq 1 \tag{7}\]  
When \(FI_{CVGM}\geq 1\), the material point is determined to exhibit ultra-low cycle fatigue ductile cracking, and the fracture criterion must be satisfied on characteristic length \(l^{*}\) to predict the fracture [35]. From the preceding formulas, the CVGM accounts for changes in stress triaxiality during loading, enhancing prediction accuracy for fatigue life in steel structure joints. In the ultra-low cycle fatigue prediction process of the CVGM, when the \(FI_{CVGM}\) value of the unit satisfies Formula (7), the finite element is deleted, and then the post-cracking path is predicted. The CVGM accounts for changes in stress triaxiality during loading, enhancing prediction accuracy for fatigue life in steel structure joints. When stress triaxiality remains relatively constant during loading, the CVGM can be simplified to the DSPS, although this may reduce calculation accuracy. Building on the CVGM, researchers have studied methods for predicting ultra-low cycle fatigue damage in steel structures [35; 59; 60; 61; 62; 63]. In 2012, [PERSON] et al. [35] calibrated the parameter \(\eta\) of the CVGM for Q345 steel and the damage degradation parameter \(\lambda_{CVGM}\) under cyclic loading. Their findings indicated a need for numerous calibration tests, which often produced widely varying results. In 2022, [PERSON] [64] calibrated the fracture toughness parameters of S220503 duplex stainless steel under the CVGM. At the component level, [PERSON] et al. [62] established a mixed-element model in 2018 to predict ultra-low cycle fatigue damage in the high-strain areas of steel arch bridges, based on the CVGM fracture criterion. However, the CVGM assumes that the expansion and contraction rates of microvoids are equal, which does not accurately reflect the actual stresses in steel. The simulation agreed well with the test results, which verified the applicability of the CVGM to the numerical simulation of the ULCF fracture of structural steels [65]. [PERSON] [66] carried out experimental research on specimens made of high-strength steel at different positions (base metal, welding material, heat-affected zone). After calibrating the parameters of the kinematic-isotropic hybrid strengthening model, the finite element software was used for calculation. The results were substituted into the CVGM for fracture prediction. By comparing the experimental data, it was found that the CVGM can accurately predict the ultra-low cycle fatigue life. Based on the CVGM, [PERSON] [67] proposed a new stress-weighted ductile fracture model (SWDFM) to simulate ductile fracture induced by ultra-low cyclic fatigue (ULCF). The SWDFM can accurately predict the cycle life of ULCF under a complex stress state and loading system. The research indicates that although the CVGM effectively accounts for the influences of stress triaxiality and cumulative equivalent plastic strain on cavity expansion ratio, leading to a more accurate prediction of ultra-low cycle fatigue life in steel [59; 60; 61; 62; 63], it also has several shortcomings. The CVGM is both semi-empirical and semi-theoretical, relying on numerous parameter calibration tests. Additionally, variations in the materials of welds and heat-affected zones result in significant discrepancies in damage degradation parameters for welded specimens. Moreover, the model assumes that the expansion and contraction rates of micro-pores are identical, which does not reflect actual conditions.