# Notch Fatigue Life Research Based on Critical Distance Theory
## 2 Critical Distance Theory  
The critical distance theory belongs to the combined field of continuum media mechanics and linear elastic fracture mechanics, and its fracture mechanics is based on the assumption that crack extension does not occur smoothly and continuously as described in the traditional linear elastic fracture mechanics theory, and that crack extension is usually discontinuous, and occurs in a discontinuous manner through a series of jumps (see Figure 1), the step size of which is determined by the microstructure and deformation behavior of the material [35]. On the other hand, critical distance theory has many features in common with linear elastic fracture mechanics in terms of the mechanism. For example, the application of a small-scale yield criterion is expected in both critical distance theory and linear elastic fracture mechanics, requiring that the plastic zone at the root of the notch falls within a very small region of the specimen size in order to effectively predict fatigue fracture. According to the different methods of calculating stress, the critical distance theory is divided into the point method, line method, area method, and volume method. Among them, the point method and the line method are the two most important calculation methods of the critical distance theory, where the point method considers the characteristic stress as the maximum principal stress at \(L/2\) from the tip of the notch, and the line method considers the characteristic stress as the average value of the maximum principal stress within a line along the hot spot path of \(2L\), as shown in Figure 2.  
The characteristic stress in the line method can be calculated as  
\[\sigma_{\text{ave}}=\frac{1}{2L}\int_{0}^{2L}\sigma(x)\text{d}x \tag{1}\]  
where \(\sigma(x)\) denotes the maximum principal stress at distance x from the defect along the hot spot path direction. Figure 1: Crack expansion-time relationship diagram. Figure 2: Schematic diagram of critical distance theory point method and line method. There are two important parameters in the critical distance theory: material strength \(\sigma_{0}\) and critical distance \(L\). Generally speaking, the values of both are only related to the material properties themselves. When the radius of the stress concentration region at the root of the notch in a structure or member is infinitely close to 0, the notch can be approximated as a sharp crack, and fatigue damage occurs in the notched member when the stress intensity factor of the member is equal to the fracture toughness of the material in this scenario. The fracture toughness is a constant for a given material, independent of the crack itself and the applied load, which is an inherent property of the material and is only related to the material itself. Certain parameters in the critical distance theory point method can be expressed in the form of fracture toughness correlation. [PERSON], in his book _Critical Distance Theory: Engineering Fracture Mechanics_, states that [36], the following expressions exist for fracture toughness \(K_{\text{c}}\), brittle fracture strength \(\sigma_{f}\), and crack length \(a\):  
\[\sigma_{f}=\frac{K_{\text{c}}}{\sqrt{\pi a}} \tag{2}\]  
The notch is approximately equivalent to a sharp crack condition; the relationship between the stress at the root of the notch and the distance can be expressed by Equation (3):  
\[\sigma_{(r)}=\frac{\sigma}{\left[1-\left(\frac{a}{a+r}\right)^{2}\right]^{1/2}} \tag{3}\]  
\[\sigma_{(r)}=\sigma\sqrt{\frac{a}{2r}} \tag{4}\]  
Relation (4) is only applicable in the case of \(r<<a\), that is, the length of the sharp crack must be much larger than the critical distance \(L\) of the material. Relationships (2) and (4) can be combined to obtain the following relationship for calculating the critical distance:  
\[L=\frac{1}{\pi}\left(\frac{K_{\text{c}}}{\sigma_{0}}\right)^{2} \tag{5}\]  
where \(K_{\text{c}}\) is the material fracture toughness; \(\sigma_{\text{o}}\) is the characteristic stress.