Fig 5 low amplitude load region in low-high sequence. (A), The definition of load region. (B), Influence of preload on the residual fatigue life
As shown in Fig 5(A), σH and σL are the maximum and minimum strengthening stress, respectively; σ is the stress level of preload. Its influence on residual fatigue life can be omitted for loads lower than the minimum strengthening stress. In the strengthening region, the residual fatigue life can be improved when the number of preload cycles is within the appropriate range. Several investigations29,38,55 confirmed this phenomenon and suggested that σH is slightly below the fatigue limit.
For loads over the σH, the residual fatigue life firstly increases and then decreases with the number of preload cycles. This opinion is supported by the test results exhibiting the strengthening effect in Fig 2. In fact, for loads over the fatigue limit, the finite fatigue life makes it inevitable that the damaging effect is dominated when the number of cycles exceeds a specific value. The strengthening effect is also affected by the stress level of the preload. Two-level tests on various materials indicated that the strengthening effect decreases with increasing stress level of the preload. Ishihara et al.45 conducted two-level load tests on S45C steel with preload stresses of 400 MPa and 500 MPa, respectively. The former loading condition corresponds to a significant strengthening effect, while the latter one can be ignored. The test results for DTD683 steel17 and Maraging steel49 also show this trend. In Li et al.’s study44, two-level load tests with different stress levels were conducted on K417 alloy. It was found that the residual fatigue life fraction decreased from 5.37 to 2.33 with increasing preload stress. Only the damaging effect can be observed when the preload stress σa1 is equal to or higher than the subsequent stress σa2. The relationship betweenn 1/Nf 1 andn 2/Nf 2must follow Miner’s rule when σa1 equals σa2.
In the Chaboche model, the damage evolution in the low-high loading path is shown in Fig 6. For materials exhibiting the strengthening effect, it can be assumed that the fatigue resistance is improved after low amplitude loading, i.e., the damage growth rate is reduced. The expression of the damage rate can be modified from Equation to:
where q is a factor characterizing the strengthening effect and can be referred to as the “strengthening factor.”