## 2 Damage Accumulation Theory  
The most widely used linear damage accumulation theory is [PERSON]'s theory [16]. The theory defines the fatigue damage \(D\) as the ratio of the number of cycles \(n\) under a certain stress to the fatigue life \(N_{f}\) of the material under the stress:  
\[D=\frac{n}{N_{f}} \tag{1}\]  
[PERSON]'s theory believes that under the action of multiple levels of different stress amplitudes, fatigue failure occurs  
\[\sum\frac{n_{i}}{N_{fi}}=1 \tag{2}\]  
where \(n_{i}\) is the number of cycles under the \(i\)th stress level; \(N_{fi}\) is the fatigue life under the \(i\)th stress. Taking into account the decrease in the material's bearing capacity under cyclic loading, nonlinear fatigue damage accumulation theory introduces the concept of the material's physical property degradation into damage accumulation [17], a typical tough dissipation model proposed by [PERSON] [18]. According to the [PERSON] fracture criterion [19], [PERSON] [20] established the empirical relationship between material fatigue toughness and static toughness, which was verified by a large number of experimental results [21; 22]:  
\[\frac{U}{W_{f}}=\left(\frac{e_{a}}{e_{f}}\right)^{4} \tag{3}\]  
\(U\) is the initial toughness without damage, \(W_{f}\) is fatigue toughness, \(e_{a}\) is the applied stress amplitude, \(e_{f}\) is the breaking strength of the material. For metal materials with certain damage, Formula (3) can be rewritten as:  
\[\frac{U_{N}}{W_{fN}}=\left(\frac{e_{a}}{e_{fN}}\right)^{4} \tag{4}\]  
\(U_{N}\) is the toughness after \(N\) cycles of loading, \(W_{fN}\) is the remaining fatigue toughness, \(e_{fN}\) is the residual breaking strength of the damaged material. From the energy consumption process of fatigue damage, the fatigue damage variable is defined as [20]:  
\[D_{N}=\frac{\sum\limits_{i=1}^{N}\Delta W_{i}}{W_{f}}=\frac{W_{f}-\sum\limits _{i=N+1}^{N_{f}}\Delta W_{i}}{W_{f}}=1-\frac{W_{fN}}{W_{f}} \tag{5}\]  
\(\sum\limits_{i=1}^{N}\Delta W_{i}\) represents the plastic hysteresis energy accumulated and dissipated under a certain damage state, \(N\) is the load cycles experienced, \(N_{f}\) is the cycle of fatigue fracture. Substitute Formulas (3) and (4) into Formula (5):  
\[D_{N}=1-\left(\frac{e_{fN}}{e_{f}}\right)^{4}\frac{U_{N}}{U} \tag{6}\]  
For materials with a power-hardening law, according to the experimental results [18], the tensile strength of the material does not show a sharp decline until it is close to fracture. The Formula (6) is further simplified as:  
\[D_{N}=1-\frac{U_{N}}{U} \tag{7}\]  
This is the calculation formula of the damage variable defined by the material's toughness dissipation. Its physical meaning is that the degree of fatigue damage of the material can be measured by the amount of change in the metal's energy or the ability to absorb deformation and fracture during the fatigue process. In order to obtain the damage evolution law under the ductile dissipation model, the damage variable calculation, Formula (7), and the ductile dissipation model, Formula (8) [18], are combined together as  
\[U_{N}=U+\frac{\left(U-U_{N_{f}-1}\right)}{\ln N_{f}}\ln(1-\frac{N}{N_{f}}) \tag{8}\]  
where \(U_{N_{f}-1}\) is the residual toughness of the material after \(N_{f}-1\) cycles of loading, that is, the energy absorbed by the material under the tensile load before fatigue fracture. For most fatigue problems, because the macroscopic fracture presents brittle fracture characteristics,there is no obvious necking phenomenon; therefore. \(D_{N_{f}-1}\approx 1\)[18], the fatigue damage evolution law with toughness as a parameter is obtained as:  
\[ \begin{split} D_{N}&=-\frac{\left(1-U_{N_{f}-1}/U \right)}{\ln N_{f}}\ln\left(1-\frac{N}{N_{f}}\right)\\ &=-\frac{D_{N_{f}-1}}{\ln N_{f}}\ln(1-\frac{N}{N_{f}})\approx- \frac{\ln\left(1-\frac{N}{N_{f}}\right)}{\ln N_{f}}\end{split}