# 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  
Recent micro-damage mechanism fracture models offer advantages over empirical formula methods. These models not only predict the fatigue life of steel structures but also describe how stress-strain fields affect the inherent microstructural characteristics of materials. As a result, they provide clear physical criteria for more accurate predictions of ductile cracks in steel structure nodes [52; 53; 54]. Key fracture models for predicting ultra-low cycle fatigue damage based on the micro-damage mechanism include the CVGM) [30] and degraded significant plastic strain (DSPS) [55]. The following discussion describes the development process of the cyclic void growth model (CVGM). In 1969, [PERSON] and [PERSON] [56] believed that when microvoids exceeded the critical void size, unstable necking would occur between voids and lead to void aggregation and crack initiation. Therefore, a void growth model (VGM) that was suitable for ductile cracking prediction under monotonic loading was proposed. In 1976, [PERSON] et al. [57] proposed a stress-corrected critical strain (SMCS) model based on VGM for predicting ductile cracking under monotonic loading, assuming that stress triaxiality remains basically unchanged during loading. Based on the VGM and the SMCS model, scholars have conducted research on the prediction of steel fracture under monotonic tensile load. In 2006, [PERSON] et al. [58] corrected the SMCS model parameters of ASTM A572 Grade 50 steel through the tensile test and finite element analysis of notched steel bar specimens. They then verified the prediction accuracy of the SMCS model for the ductile failure of large-scale joints through beam-column joint tests. Researchers believe that SMCS can predict the ductile failure of metal materials without initial cracks. In 2007, [PERSON] [17] verified the effectiveness of VGM and the SMCS model in predicting the ductile cracking of steel through the tensile tests of 12 specimens. Based on the preceding analysis, VGM and the SMCS model are suitable for the ductile cracking of steel specimens under monotonic loading. Moreover, VGM can consider the change in stress triaxiality, and its prediction accuracy is higher than that of the SMCS model. Based on VGM, [PERSON] et al. [17] proposed Formula (3) for calculating the cavity expansion index \(VGI_{cyclic}\) under tension and compression cyclic loading based on the cavity radius growth formula of Rice-Tracey in 2007. \[VGI_{cyclic}=\sum_{tensile}\int_{\epsilon_{1}}^{\epsilon_{2}}\exp(|1.5T|)d \epsilon_{p}-\sum_{compressive}\int_{\epsilon_{1}}^{\epsilon_{2}}\exp(|1.5T| )d\epsilon_{p}, \tag{3}\]  
Figure 4: Cross-sections of test specimens [38]. where \(\epsilon_{1}\) and \(\epsilon_{2}\) are the equivalent plastic strain at the beginning and end of the tension or compression cycle, respectively; \(\epsilon_{p}\) is the equivalent plastic strain; and \(T\) is the stress triaxiality. The calculation formula of \(T\) is  
\[T=\frac{1}{3}(\sigma_{1}+\sigma_{2}+\sigma_{3})/\sqrt{\frac{1}{2}\Big{[}( \sigma_{1}-\sigma_{2})^{2}+(\sigma_{2}-\sigma_{3})^{2}+(\sigma_{3}-\sigma_{1}) ^{2}\Big{]}}, \tag{4}\]  
where \(\sigma_{1}\), \(\sigma_{2}\), and \(\sigma_{3}\) represent principal stress in the fracture zone. The calculation formula of the critical hole expansion index \(VGI_{cyclic}^{critical}\), which is used to compare with the hole expansion index \(VGI_{cyclic}\), is given as  
\[VGI_{cyclic}^{critical}=\eta\exp\Big{(}-\lambda_{CVGM}\epsilon_{p}^{accumalat }\Big{)}, \tag{5}\]  
where \(\lambda_{CVGM}\) is the damage degradation parameter of the material under cyclic loading, \(\epsilon_{p}^{accumalat}\) is the cumulative equivalent plastic strain, and \(\eta\) is the material toughness parameter under monotonic tension.