# A Combined High and Low Cycle Fatigue Life Prediction Model for Wind Turbine Blades
## 2 Miner's Rule and Fatigue Damage Accumulation Under Variable Amplitude Cyclic Load
### Nonlinear Damage Accumulation Considering Load Interactions  
The H-L sequence was analyzed to explore how load sequences and interactions influence fatigue damage accumulation, as shown in Figure 4.  
The damage evolution route under the first-level load \(\sigma_{1}\) is represented by the red solid line in Figure 4. The damage evolution route under secondary loads \(\sigma_{2}\) is represented by the blue solid line if the interaction between loads is ignored. However, the first-level load \(\sigma_{1}\) influences the second-level load \(\sigma_{2}\) damage development due to the interaction  
Figure 4: Cumulative damage in H-L loading sequence. between loads, which accelerates fatigue damage accumulation in materials. Consequently, the damage evolution path's slope increases when secondary loads \(\sigma_{2}\) are applied [32; 33]. The orange solid line in Figure 4 illustrates how the fatigue damage route appears as a red, green, and orange solid line when both load sequence and interaction are considered. The relationship between applied loads under variable amplitude loading is modeled using their load ratio. Fatigue life prediction accuracy decreases with increasing load differences. Based on equivalent damage theory and load interaction considerations, Equation (5) is modified to Equation (6) to enhance fatigue life prediction accuracy under variable amplitude loading. \[D\bigg{(}\frac{n_{2}}{N_{2}}\bigg{)}=\bigg{[}D\bigg{(}1-\frac{n^{\prime\prime} }{N_{2}}\bigg{)}\bigg{]}^{\frac{\sigma_{2}}{\sigma_{1}}} \tag{6}\]  
where \(n^{\prime\prime}\) represents the number of loading cycles under the interaction of stress levels \(\sigma_{1}\) and \(\sigma_{2}\). The modified Equation (6) improves fatigue life prediction accuracy by incorporating load interaction effects and nonlinear damage accumulation, which are critical for more accurate life prediction under variable amplitude loading. The modified Equation (6) considers the relationship between the applied loads under variable amplitude loading by using their load ratio, which accounts for the influence of load sequence (H-L or L-H) on the damage accumulation process. This allows the model to reflect the impact of load history, which is often neglected in simpler models like Miner's rule. The term \(\left(\frac{n^{\prime\prime}}{N_{2}}\right)\) in Equation (6) introduces this load interaction effect, which adjusts the damage accumulation based on the varying loads. The original Equation (5) assumes linear damage accumulation, which can lead to overestimation or underestimation of fatigue life, especially when there are significant load variations. Equation (6) modifies the damage calculation by including a nonlinear factor \(\left(1-\frac{n^{\prime\prime}}{N_{2}}\right)\) raised to a power, which better captures the material's response to cyclic loading. The addition of the stress ratio \(\left(\frac{\sigma_{2}}{\sigma_{1}}\right)\) further enhances the model's ability to predict fatigue life by accounting for the differing effects of high and low stress levels on damage accumulation. By integrating these factors, the modified Equation (6) enhances the model's ability to predict fatigue life more accurately, especially under CCF loading conditions. This results in a more accurate prediction of material behavior, aligning better with experimental data and reducing errors associated with traditional models. Based on the damage curves, for composites, the nonlinear cumulative damage model was presented [34; 35]:  
\[D(n)=1-\bigg{[}1-\Big{(}\frac{n}{N}\Big{)}^{B}\bigg{]}^{A} \tag{7}\]  
where material properties \(A\) and \(B\) are used. Fatigue life \(N\) and applied load relate to parameter \(B\). The expression is given in Equation (8) [33]. \[B=\frac{K\times\text{lg}N}{(1-R)\bigg{(}\frac{\sigma_{\text{max}}}{\sigma_{all }}\bigg{)}} \tag{8}\]  
where \(K\), which stands for the scaling parameter, is set at 0.06. \(\sigma_{\text{max}}\) represents the maximum stress, \(\sigma_{all}\) denotes the original strength of the material.