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
σa1/σa2 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, σa1/σa2 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: