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:

(5.2.1)

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

(5.2.2)

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):

(5.2.3)

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

(5.2.4)

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

(5.2.5)

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:

(5.2.6)

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

(5.2.7)

where f is given by

(5.2.7a)

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.