Some mathematical models have been developed to account for
retardation in crack-growth-integration procedures. All models are based on simple assumptions, but within certain
limitations and when used with experience, each model will produce results that
can be used with reasonable confidence.
The two yield zone models by Wheeler [1972] and by Willenborg, et al.,
[1971], and a crack-closure model by Bell & Creager [1975] will be briefly
discussed. Detailed information and
applications of closure models can be found in Bell & Creager [1975], Rice
& Paris [1976], Chang & Hudson [1981], and Wei & Stephens [1976].

Wheeler Model

Wheeler defines a crack-growth reduction factor, *C*_{p}:

where *f(**DK*) is the usual crack-growth function, and (*da/dN*) is the retarded crack-growth
rate. The retardation factor, *C*_{p} is given as

where (see Figure 5.2.6):

**Figure
5.2.6.** Yield Zone Due to Overload (*r*_{poL}), Current Crack Size (*a*_{i}), and Current Yield Zone (*r*_{pi})

There is retardation as long as the current plastic zone (*r*_{pi}) is contained within a
previously generated plastic zone (*r*_{poL})
; this is the fundamental assumption of yield zone models.

Some examples of crack-growth predictions made by means of the
Wheeler model are shown in Figure 5.2.7. Selection of the proper value for the
exponent *m* will yield adequate
crack-growth predictions. In fact, one
of the earlier advantages of the Wheeler model was that exponent *m *could be tailored to allow for
reasonably accurate life predictions of spectrum test results. Through the course of time, it has become
recognized, however, that the exponent m was dependent on material, crack size,
and stress-intensity factor level as well as spectrum. The reader is cautioned against using the
Wheeler model for service life predictions based on limited amounts of
supporting test data and more specifically against estimating the service life of
structures with spectra radically different from those for which the exponent *m* was derived. Estimates made without the supporting data required to tailor the
exponent *m* can lead to inaccurate and
unconservative results.

**Figure 5.2.7.** Crack Growth Predictions by Wheeler Model
Using Different Retardation Exponents [Wood, et al. 1971]

Willenborg Model

The Willenborg model also relates the magnitude and extent of
the retardation factor to the overload plastic zone. The extent of the retardation is handled exactly the same as that
of the Wheeler model. The magnitude of
the retardation factor is established through the use of an effective
stress-intensity factor that senses the differences in compressive residual stress
state caused by differences in load levels.
The effective stress-intensity factor (*K*^{eff}_{i}) is equal to the typical remote
stress-intensity factor (*K*_{i})
for the i^{th} cycle minus the residual stress-intensity factor (*K*_{R}):

where in the original formulation [Willenborg, et al., 1971;
Gallagher, 1974; Gallagher & Hughes, 1974; Wood, 1974]

in which (see Figure 5.2.6):

*a*_{i}
– current crack size

*a*_{oL}
–- crack size at the occurrence of
the overload

*r*_{poL}
– yield zone produced by the
overload

*K*^{oL}_{max}
– maximum stress intensity of the
overload

*K*_{max,i}
– maximum stress intensity for the
current cycle.

The equations show that retardation will occur until the crack
has generated a plastic zone size that reaches the boundary of the overload
yield zone. At that time, *a*_{i}-a_{oL}= r_{poL}
and the reduction becomes zero.

Equation 5.2.3 indicates that the complete stress-intensity
factor cycle, and therefore, its maximum and minimum levels (*K*_{max, i} and *K*_{min, i}), are reduced by the
same amount (*K*_{R}). Thus, the retardation effect is sensed by
the change in the effective stress ratio calculated by

since the range in stress-intensity factor is unchanged by the
uniform reduction. Thus, for the i^{th}
load cycle, the crack growth increment *(**Da*_{i}) is:

For many of the early calculations with the Willenborg model,
it was assumed that *R*_{eff}
was never less than zero and that _{} when *R*_{eff} was calculated to be
less than zero. Recent evidence,
however, supports the calculations of *R*_{eff}
as given by Equation 5.2.5 and the use of a negative stress ratio cut-off in
the crack growth rate calculation (Equation 5.2.6) for more accurate modeling
of crack growth behavior.

Another problem that was identified with the original
Willenborg model was that it was always assigned the same level of residual
stress effect independent of the type of loading. In particular, it can be noted (through the use of Equation 5.2.3
and 5.2.4) that the model predicts that _{}, and therefore crack arrest, immediately after overload if _{}. That is, if the
overload is twice as large as (or larger than) the following loads, the crack
arrests. To account for the
observations of continuing crack propagation after overloads larger than a
factor of two or more, Gallagher & Hughes [1974] introduced an empirical
(spectra/material) constant into the calculations. Specifically, they suggested that

where f is given by

There are two empirical constants in Equation 5.2.7a: *K*_{max, th} is the threshold
stress-intensity factor level associated with zero fatigue crack growth rates
(see Section 5.1.2), and *S** *^{oL} is the overload (shut-off) ratio required to
cause crack arrest for the given material.
The type of underload/overload cycle, as well as the frequency of
overload cycle occurrence, affects this ratio.
Results of some life predictions made using what has become to be called
the “Generalized” Willenborg model are presented in Figure
5.2.8 [Engle & Rudd, 1974].
Compressive stress levels were ignored in this analysis.

**Figure 5.2.8.** Predictions of Crack Growth Lives with the
Generalized Willenborg Model Compared to Test Data [Engle & Rudd, 1974]

Closure Models

One of the earliest crack-closure models developed for aircraft
structural applications is attributed to Bell & Creager [1975]. The closure model makes use of a
crack-growth-rate equation based on an effective stress-intensity range *DK*_{eff}.
The effective stress intensity is the difference between the applied
stress intensity and the stress intensity for crack closure. Some examples of predictions made with the
model are presented in Figure 5.2.9. The final equations contain many
experimental constants, which reduces the versatility of the model and make it
difficult to apply. Recent work by Dill
& Saff [1977] shows that the closure model can be simplified to the point
of practicality while retaining a high level of accuracy in life prediction.

**Figure 5.2.9. **Predictions by Crack Growth Closure
Model as Compared with Data Resulting From Constant-Amplitude Tests with
Overload Cycles [Bell & Creager, 1975]

Crack-growth calculations are the most useful for comparative
studies, where variations of only a few parameters are considered (i.e.,
trade-off studies to determine design details, design stress levels, material
selection, etc.). The predictions must
be verified by experiments. (See
Analysis Substantiation Tests in Section 7.3).
Example calculations of crack-growth curves will be given in Section
5.5.

Other factors contributing to uncertainties in crack-growth
predictions are:

·
Scatter in baseline *da/dN*
data,

·
Unknowns in the effects of service environment,

·
Necessary assumptions on flaw shape development,

·
Deficiencies in *K*
calculation,

·
Assumptions on interaction of cracks,

·
Assumptions on service stress history.

In view of these additional shortcomings of crack-growth
predictions, the shortcomings of a retardation model become less pronounced;
therefore, no particular retardation model has preference over the others. From a practical point of view, the
Generalized Willenborg model is easier to use since it contains a minimum
number of empirical constants.