# 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
### Continuum Damage Mechanics (CDM) Model
#### 2.4.1 The Principle and Development of Traditional CDM Model  
In 1963, [PERSON] [70] proposed the concepts of damage factor and effective stress while formulating the creep constitutive equation for metals, laying the groundwork for the CDM model. In 1997, building on [PERSON]'s research [71], [PERSON] et al. [31; 72] assumed that the damage strain threshold under uniaxial stress corresponds to that under multiaxial stress. They derived a CDM model suitable for predicting ductile fracture under monotonic loading, represented in Formula (8):  
\[dD=c\frac{\left(D_{cr}-D_{0}\right)^{1/c}}{\ln(\varepsilon_{f}/\varepsilon_{th })}f(T)(D_{cr}-D)^{\left(1-1/c\right)}\frac{d\overline{\varepsilon}^{p}}{ \overline{\varepsilon}^{p}} \tag{8}\]  
where \(D\), \(D_{0}\), and \(D_{cr}\) are the damage, initial damage, and critical damage variables, respectively. \(c\) is the damage exponent characteristic of the material. \(\varepsilon_{th}\) represents the plastic strain threshold. \(\varepsilon_{f}\) represents the fracture-accumulated plastic strain under uniaxial stress. \(T\) and \(f(T)\) represent the stress triaxiality and stress triaxial function, respectively. \(\overline{\varepsilon}^{p}\) is the accumulated plastic strain. The uniaxial damage strain threshold \(\epsilon_{th}\) in Formula (8) is calibrated by Formula (9):  
\[\bar{\epsilon}_{f}^{p}=\epsilon_{th}(\frac{\epsilon_{f}}{\epsilon_{th}})^{1/f(T)} \tag{9}\]  
\(\bar{\epsilon}_{f}^{p}\) represents the fracture-accumulated plastic strain under multiaxial stress. It can be seen from the formulas above that the CDM model, suitable for the monotonic tensile loading of metals, requires only the calibration of the uniaxial damage strain threshold value \(\epsilon_{th}\) and the uniaxial fracture strain to effectively predict the material's damage process. The damage variable accounts for the effects of cumulative plastic strain and stress triaxiality, resulting in high prediction accuracy. In 2016, [PERSON] et al. [73] suggested that the damage to steel under ultra-low cycle fatigue load results solely from tensile strain. They determined the tensile and compressive strains during loading based on the positive and negative signs of stress triaxiality. They extended Formula (8) to apply to ultra-low cycle fatigue fracture prediction, as shown in Formulas (10)-(12):  
\[dD=c\frac{\left(D_{cr}-D_{0}\right)^{1/c}}{\ln(\epsilon_{f}/\epsilon_{th})}f(T )(D_{cr}-D)^{\left(1-1/c\right)}\frac{d\bar{\epsilon}^{p+}}{\bar{\epsilon}^{p}} \tag{10}\]  
\[d\bar{\epsilon}^{p+}=d\bar{\epsilon}^{p}\times H(T) \tag{11}\]  
\[H(T)=\left\{ \begin{array}{cc}0&T<0\\ 1&T\geq 0\end{array} \right\} \tag{12}\]  
\(d\bar{\epsilon}^{p+}\) is the cumulative plastic tensile strain increment, and \(H(T)\) is the function describing the damage state. By modifying the elastic modulus of steel, Formula (13) realizes the influence of the damage variable \(D\) on the macroscopic mechanics of steel. \[E=E_{0}[1-D\times H(T)] \tag{13}\]  
In the formula, \(E_{0}\), \(E\) are the initial elastic modulus and the elastic modulus after damage, respectively. Building on the traditional CDM model, [PERSON] et al. [73] calibrated the parameters solely through monotonic tensile tests. They predicted the crack initiation life and fracture life of six beam-column joints, validating the formula through experiments. However, this prediction approach does not account for the influence of equivalent plastic strain on the damage increment, contributing to discrepancies between the predicted and test lives. The CDM model treats the entire system as solid elements, which affects calculation efficiency. There is also a lack of refined modeling methods that balance efficiency and accuracy. In 2020, [PERSON] et al. [37] investigated ultra-low cycle fatigue damage prediction in Q345 qC steel-welded specimens using the solid element approach of the traditional CDM model.