FIGURE 1 Growth law of fatigue driving stress under two-stage cyclic loading
According to Figure 1, under a high-low loading sequence, the member is first loaded under stress for cycles, and the fatigue driving stress reaches point E. Then we have
Assuming that the member is loaded under stress for cycles to point F such that points E and F have the same fatigue driving stress and is the equivalent number of cycles under stress loading, the fatigue driving stress at point F can be expressed as
Because the E and F points have the same fatigue driving stress, equations (4) and (5) are required to be equivalent. Then equating the fatigue driving stress reached by cycles of loading at stress to the fatigue driving stress reached by cycles of loading at stress, according to equation (2), the equivalence equation can be expressed as
The equivalent driving stress at points E and F can be expressed as:
Equation (7) shows that the fatigue driving stress achieved by loading stressat life-fraction of is equal to the fatigue driving stress achieved by loading stress at life-fraction of .
Subsequently, if the loading is continued for cycles under stress to achieve the equivalent driving stress, we have
If fatigue failure occurs under the second stage of loading, then the fatigue driving force will follow the path FG from until the critical fatigue driving stress is reached. From equations (7) and (8) we can get
Taking equation into the equation, the remaining life fraction of the second stage of loading is obtained as
Similarly, if there is a third stage of loading, then the fatigue driving stress reached by cycles of loading stress is equal to the fatigue driving stress reached by ()cycles of loading stress , then the equivalent equation can be expressed as
The equivalent driving stress is expressed as
Then if the loading is continued for cycles under stress to reach the equivalent driving stress, we have
Combining equation(7), equation (8), and equation (12), equation (13) can be transformed as
Thus, for a multi-stage cyclic loading with stresses of , loading cycles of , and fatigue failure lives of , the equivalent fatigue driving stress takes the form of
At fracture, the critical fatigue driving stress will be equal to the fatigue strength constant
Taking the equation into the equation, a new damage model is defined in the form of equation (17)
Therefore, each item on the left-hand side of the equation defines the damagedue to applied loading stressas
where is the cycle life fraction under loading stress, is the fatigue failure life of loading , and is the life of the initially applied loading . Thus, the cumulative damage can be expressed as
When fatigue failure occurs at =1, the remaining life fraction of the ith load is
2.2.2. Modified K-R model
Zhu model
In order to consider the load interaction effects, Zhu15 et al. proposed a modified K-R model by imposing a load ratio between two stages to obtain a multi-stage residual life prediction model as
Li model
Since the K-R model only considers fatigue damage at a constant temperature, Li17 et al proposed a modified K-R model at variable temperatures. Fatigue driving stress was initially normalized to
where is the temperature-dependent fatigue strength coefficient.
The remaining life fraction for the second-stage loading at a temperature different from the first stage can be obtained based on the equivalence of fatigue driving stress.
where and are the temperature-dependent fatigue strength coefficients for the first stage and the second stage respectively.
In order to consider the effect of the first-stage loading on the remaining life fraction of the second stage, Li et al obtained the following expression by proposing to use as a pre-cycling factor.
Extending the model to multi-stage loading as
Li and Zhu give a modified form of the K-R model by adding a power exponential term to the remaining life model respectively, but do not give a specific explanation of the reason for the correction.
3. A new residual life prediction model