FIGURE 4 Fatigue driving stress evolution under two-stage high-low loading
After translation and extension of the fatigue driving stress curve EI, a fatigue driving stress curve F′G is finally obtained with a low loading applied alone. In contrast to EI, the curve F′G accounts for the loading interaction effects. Compared with EI, it can be expressed as:
The fatigue driving stress curve F′G is built based on curve FG, and the following conditions should be satisfied:
Under low-high loading
According to the equivalence of fatigue driving stress, the fatigue driving stress generated by cycles under stress is equal to the fatigue driving stress generated by cycles under stress according to curve F′G, as shown in Figure 3. Therefore, considering the loading interaction effects, the fatigue driving path is E F′→F′G, and the corresponding equivalent state can be described as:
According to the studies20-23, the greater the difference between loading levels, the more significant the interaction effects are. The loading interaction effects can be described by the ratio of loading levels. Therefore, the interaction factor is defined in this paper as the ratio of loading levels of two adjacent stages,.
In order to satisfy conditions (29) - (33), referring to the Marco-Starkey model6 or Manson model25, we can obtain by adding the power exponent to :
Then the equivalent driving stress can be expressed as:
By taking logarithms of both sides of equation (36), the equivalent cyclic fraction under stress loading can be obtained:
If the member fails by fatigue under a secondary loading, the fatigue driving stress will follow the path from until the critical driving stress is reached.
By substituting the equation into , the remaining life fraction of the second stage of loading is obtained as
By fatigue driving stress equivalence, extending (39) to the multi-stage loading case, the remaining life fraction at the ith stage is
Where the equivalent life fraction is expressed as
Where is the loading interaction factor, expressed as the ratio of two levels of stress:
In summary, equation (40) is the general form of the residual life prediction model when considering the loading interaction effects. The unknown parameters involved in the model can be determined only by the S-N curve.