# Study on the Prediction Method of the Ultra-Low-Cycle Fatigue Damage of Steel
## 1 Introduction  
In the past earthquakes, the seismic damage of steel piers has led to ultra-low cycle fatigue (ULCF) damage at the weld joints in addition to the residual displacement caused by excessive plastic deformation and local buckling of steel piers [1]. For example, during the Kobe earthquake in 1995, ULCF damage was observed at the joints of the steel structures [1; 2]. Since then, domestic and foreign researchers have conducted numerous studies on the local buckling of steel plate and the ULCF damage at weld joints. Local buckling failure has been solved [3; 4], but ULCF damage is still being explored. In recent years, researchers have made a series of studies on the mechanism of low cycle fatigue (LCF) and ULCF damages of steel [5; 6]. The LCF life ranges from \(10^{2}\) to \(10^{4}\) cycles, and the fracture mechanism is brittle failure [7]. ULCF failure is caused by the ductile failure of steel under high strain cyclic load after a cyclic plastic deformation of less than 100 cycles, and the fracture mechanism is ductile failure [8; 9; 10; 11; 12]. The fracture mechanism of steel with ULCF is different from that of LCF [13; 14]. Thus, the damage prediction method suitable for LCF cannot be directly applied to ULCF. Therefore, research on the prediction method of ULCF must be urgently conducted. At present, several methods can be used to predict fatigue life, such as traditional fracture mechanics, Coffin-Manson formula, micromechanics model and continuum damage mechanics (CDM) model. Early researchers have mainly used fracture mechanics to study the fracture properties of steel and its weld joints. The traditional fracture mechanics method assumes that the crack alreadyexists and a high strain constraint exists at the crack tip; this method is mainly suitable for the study of brittle or pseudo-brittle fracture [15]. However, the prediction method is unsuitable for the ductile fracture problem [16; 17]. In 1954, [PERSON] [18] and [PERSON] [19] proposed the Coffin-Manson formula for predicting the LCF life of materials on the basis of the relationship between fatigue life and plastic strain amplitude. The Coffin-Manson formula is the main method of predicting the LCF life of steel and its accuracy has been verified by many test results; however, its prediction of the ULCF life of steel is biased [20; 21]. The above-mentioned descriptions indicate that the traditional fracture mechanics method and the Coffin-Manson formula are unsuitable for the damage prediction of ULCF. Micromechanical models have been proposed in recent years to accurately predict the ULCF life [22]. These models mainly include the cyclic void growth model (CVGM) [23] and the degraded significant plastic strain (DSPS) model [24]. The two models attribute ductile cracking to the process of micro-hole nucleation, expansion and aggregation, which can reflect the triaxial stress state and plastic strain characteristics of the structure [25]. CVGM assumes that stress triaxiality is a variable during loading, whilst DSPS model assumes otherwise. The prediction results of CVGM are more accurate because CVGM can simultaneously consider the effects of stress triaxial and cumulative plastic strain on damage accumulation [26]. CDM model is another approach to predict the damage of ULCF method. By introducing appropriate damage variables, CDM model can directly describe the macroscopic mechanical behaviour and process of damage evolution of materials by considering the effect of damage on the material constitutions [27; 28; 29]. CDM model is applicable not only to brittle fracture but also to ductile fracture [30]. According to the abovementioned analysis, the cyclic void growth model (CVGM) and continuum damage mechanics (CDM) are both suitable for predicting the damage of ULCF theoretically. However, the study on the prediction of ULCF damage is actually lacking.