Fig 6 Schematic of the influence of the strengthing effect on the evolution of damage in the low-high loading path
The strengthening factor q is a function of the preload stress level and the number of preload cycles. It has been deduced earlier that the strengthening effect decreases as the first level of stress approach the second level of stress. Therefore the ratio of two stress levels σa1a2 can be used to characterize the influence of the preload stress. Since the fatigue life of virgin materials can implicitly indicate that the load is higher than the fatigue limit, σa1a2 was replaced by the ratio of fatigue life corresponding to the first and second levels of stress, i.e.,Nf 1/Nf 2.
In general, the influence of the stress level and the number of cycles of the preload on the strengthening effect can be summarized in the following four features:
(a) The strengthening effect can be ignored when the preload cycle life fractionn 1/Nf 1=0 or 1.
(b) The strengthening effect first increases and then decreases with the preload cycle life fractionn 1/Nf 1, and the location of the ”turning point” is related to the material.
(c) The strengthening effect increases with the life ratioNf 1/Nf 2between the two stress levels.
(d) The strengthening effect is only observed under low-high sequence loading, i.e.,Nf 1/Nf 2>1.
From Equation , it is clear that q is always greater than or equal to unity, and q equals unity means that the strengthening effect can be ignored. Thus the exponential function was used as the basic form of the expression. Combining features (a), (b), (c), and (d), the general form of q was developed as:
where A , B , and m are material parameters, all are greater than zero. (n 1/Nf 1)A*(1-n 1/Nf 1)Bwas used to represent features (a) and (b). Parameters A andB affect the “turning point” in the curve of q . The preloading fatigue life fraction corresponding to the maximum strengthening factor qmax isA /(A +B ). It still has the same function ifA +B is set as a constant. The factor q can be determined only by the parameters A and m . It should be noted that the influence of this constant on theqmax can be offset by taking a suitable value for the parameter m . Therefore, this constant can take an arbitrary value when fitting the test results. For simplicity, the value ofA +B was taken as 2 here. Features (c) and (d) were represented by in Equation . The parameter m characterizes the sensitivity of the strengthening factor q to the life ratioNf 1/Nf 2.
Derived from Equation, the cumulative damage can be expressed as:
Then the residual life fraction is modified from Equation as follows:
By substituting Equation into Equation, the expression for the residual cycle life fraction considering the strengthening effect can be written as: