# 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
#### 2.4.2 The Method of Testing the Critical Value of Damage Variable  
All of the aforementioned studies on the prediction of the ultra-low cycle fatigue damage of steel using the traditional continuum damage mechanics (CDM) model have assumed that the critical value of the damage and fracture of materials is constant at 1. That is, the complete fracture state of a specimen is set as the criterion for the critical value of damage and fracture, which is inaccurate. The reason is as follows: regardless of whether the monotonic tensile test of steel or the load-displacement curve measured by the ultra-low cycle fatigue test is used, the load drops sharply when the bearing capacity passes through a certain point, and the steel sample can be regarded as a failure, but the steel sample is not completely broken at this moment. The method of determining the critical value of damage and fracture in the traditional CDM model must be further studied [74]. At present, two major methods are used to measure damage variables. The first one directly measures various microscopic defects in the process of material damage from a microscopic point of view [75]. However, this method cannot reflect the relationship between microscopic damage and macroscopic mechanics. It also cannot predict the macroscopic mechanical damage characteristics of steel structures, and thus, it is difficult to apply. The other one is an indirect measurement method that can consider the influence of micro-damage on the constitutive relationship of materials. It can directly establish a constitutive relationship with macro-mechanical quantities, and thus, it is widely used. In [PERSON] et al. [75], an elastic modulus method for testing metal damage based on the coupling law of elasticity and damage of thermodynamic potential was proposed, as shown in Figure 5. This method measures the elastic modulus under different strains through repeated loading and unloading, and it calculates the development process of the damage variable \(D\). Although the elastic modulus method is widely used, its measurement accuracy of strain is affected by the early microplasticity of the metal and the reversible movement and texture development of the dislocation, resulting in a considerable difference in the test results. Simultaneously, repeated loading and unloading lead to the complexity of the test process. To simplify the test process and improve the test accuracy of the damage variable, [PERSON] et al. [76] defined the effective cross-sectional area as the damage variable. Based on the relationship between the effective cross-sectional area of a material and the potential difference, the change in the effective cross-sectional area of a material is indirectly reflected by measuring the change in the potential difference between the two sides of the break point of a notched round bar sample. The basic principle and formula of test are as follows. In accordance with [PERSON]'s law, the two-point potential difference \(V\) without damage and the potential difference \(\overline{V}\) in a damaged state can be computed using Formula (14). \[\left\{ \begin{array}{l}V=r\frac{L}{M}I\\ \overline{V}=r\frac{L}{M_{0}}I\end{array} \right. \tag{14}\]  
where \(r\) is the resistivity of the specimen, \(L\) is the initial distance between the measuring points, and \(I\) is the applied current. Figure 5: Development process of the damage variable of copper [75]. The damage variable \(D\) can be expressed by the potential difference through Formula (15). \[D=1-\frac{V}{\overline{V}} \tag{15}\]  
The alternating current potential damage measurement method is not required to consider the insulation between the specimen and the testing machine and the extensometer and the specimen during the test. The test results are not affected, and test accuracy is high. This method is only required to perform a monotonic tensile test to measure the potential difference between the measuring points when the steel sample is undamaged or damaged. Then, the evolution law of the damage variable can be determined.