An Analytical Model for Fatigue Crack Propagation Prediction with Overload Effect

In this paper a theoretical model was developed to predict the fatigue crack growth behavior under the constant amplitude loading with single overload. In the proposed model, crack growth retardation was accounted for by using crack closure and plastic zone. The virtual crack annealing model modified by Bauschinger effect was used to calculate the crack closure level in the outside of retardation effect region. And the Dugdale plastic zone model was employed to estimate the size of retardation effect region. A sophisticated equation was developed to calculate the crack closure variation during the retardation area. Model validation was performed in D16 aluminum alloy and 350WT steel specimens subjected to constant amplitude load with single or multiple overloads. The predictions of the proposed model were contrasted with experimental data, and fairly good agreements were observed.


Introduction
The damage tolerance concept is widely used in modern aircraft design to ensure flight safety, which has made the prediction of fatigue crack propagation lives of aircraft components under service loading necessary [1,2].In 1860s, Paris and Erdogan [3] proposed a fracture mechanics based method for fatigue life prediction, which correlated the fatigue crack growth rate to the applied stress intensity factor range.One issue of Paris' model is that the stress ratio effect is not considered.Many modifications of Paris' law have been proposed in the literatures [4][5][6], one of the most important modifications being the inclusion of the crack closure concept.Crack closure was first introduced into the fatigue crack growth analysis by Wolf [4].And then, plenty of research has been done concerning the crack closure using experimental investigation, numerical analysis, and theoretical studies [7][8][9][10][11][12][13][14][15][16].Zhang and Liu [7,8] performed a state-of-the-art in situ SEM testing to investigate fatigue crack growth behavior continuously within a load cycle.In their study, crack closure's existence and its influence on crack propagation was directly observed.Newman [9,10] used a strip yield model to analyze the crack closure problems.Budiansky and Hutchinson [13] proposed a method to estimate the crack closure caused by plasticity in plane stress cases under constant cyclic loading.The crack closure models reviewed above can eliminate stress ratio  effect and describe fatigue crack behavior under constant amplitude loading well.However, since crack closure calculation involves the highly nonlinear analysis of cyclic plasticity and contact analysis, direct tracking of crack closure under a variable amplitude loading is extremely difficult and of high computational cost.So, the plastic zone concept is also employed to describe the crack growth behavior under general loading conditions.Willenborg et al. and Wheeler [17,18] proposed a series of models to analyze the fatigue crack growth under variable loading cases.In their papers, the plastic zone concept was used to explain the overload retardation effect.Plentiful literatures were presented to investigate fatigue crack growth behavior with the single overload [19][20][21][22], and many modifications have been proposed to correlate experimental observations.However, in most of these models, empirical coefficients are added to match the testing data under the influence of retardation, instead of modeling directly based on the mechanisms, such as the crack closure.Moreover, the recent in situ SEM experimental observations have shown that single overload has a significant impact on the crack closure within the large plastic zone [23].Therefore, crack closure variation is probably a good explanation of the retardation phenomenon.In this paper, a mechanism-based method will be developed to predict fatigue crack growth behavior under the single overload, in which crack closure and plastic zone concept are considered.The paper is organized as follows.First, the fatigue crack growth model and the modified virtual crack annealing crack closure model are briefly reviewed.Second, our proposed model for crack growth retardation behavior estimation is discussed in detail.The model is derived based on plastic zone and crack closure variation.And then model validation is performed using the experimental data in D16 aluminum alloy and 350WT steel from the literature.Finally, some conclusions and future work are given based on the current study.

Methodology Development
A general fatigue crack growth model can be expressed as (1) by Wolf [4].The fatigue crack growth rate is determined by the effective stress intensity factor range where / is the crack growth rate, Δ eff is the effective stress intensity factor,  max is the stress intensity factor of the peak load, and  op is the crack closure level; ,  are calibration parameters.
In this paper the virtual crack annealing model is employed to avoid considering the complex contact of crack closure [7].This analytical closure model is based on plasticity ahead of crack tip, but it does not consider the Bauschinger effect.So a brief derivation of the modified closure model is discussed in the following part.
As shown in Figure 1, in the process of unloading, when the stress level decreases from  max to  op , a reversed plastic zone with diameter   appears ahead of the real crack tip "".And as the stress level continues reducing from  op to  min , the crack tip will fully close and the reversed plastic zone remains constant due to the crack closure.According to the assumption of crack annealing, the closed part that has the length of "" can be considered as nonfractured material.So, there is an equivalent reversed plastic zone with diameter   , and the virtual crack tip position is   .In the following loading process, when the stress level increases from  min to  op , the superposition of the stress within the forward plastic zone with diameter   and the residual stress within the distance "" ahead of the virtual crack is zero and the crack tip opens.So, the equation   =   −   can be established.Additionally, for anisotropic materials, the Bauschinger effect has a significant impact on the cyclic plastic deformation ahead of crack tip.Thus, equation can be expressed as where   is tensile yield strength,  max is maximum stress level,  min is minimum stress level,  op is stress level of crack opening,  is the Bauschinger effect factor in loading process, and  is the Bauschinger effect in unloading process.Since the crack overlapping length is very small compared with the true crack length (i.e.,  ≪ ), its effect on the stress intensity factor calculation can be ignored.Equation ( 2) can be rewritten as where  is equal to /.Equation ( 3) is the general solution using the virtual crack annealing model for the crack opening stress calculation under constant amplitude loadings.Equation (3) can be further simplified as There are two possible solutions for the opening stress level under the proposed virtual crack annealing model.One solution is  op =  min , which indicates that there is no crack closure and the overlapped length is zero.The other solution is  op / max = (1/(1 + 4 2 ))[2 + (4 2 − 1)].This observation indicates that either there is no crack closure or there is a unique crack closure level under constant amplitude loadings.In this paper, the modified virtual crack annealing model is used to make predictions of the fatigue crack growth behavior in the outside of retardation effect region.
Retardation caused by overloads is a typical phenomenon of loading interaction effect.A great number of papers were presented to investigate the single overload [24][25][26][27][28]. Generally, overloads will induce large plastic deformation ahead of the crack tip and retard the crack growth rate in the subsequence loading of a certain range.As the developing of the crack, the retardation effect gradually recedes.When the crack grows beyond the large plastic zone, the retardation effect vanishes completely.Before the crack growth rate decreases, there is an accelerated crack growing stage, which has been termed delayed retardation.That transient acceleration right after overload is small enough to be neglected in this investigation.As it is shown in Figure 2, the crack growth length is "Δ" under the influence of a single overload.
The crack growth behavior is shown in Figure 3.After the overload, the crack growth rate increases transitorily and then decreases sharply.Once the / value reaches to minimum, crack growth rate increases gradually and finally recovers to the equilibrium constant amplitude loading growth rate.
The crack tip plastic zone associated with the application of a single overload is shown in Figure 4(a).A new approach to estimation of crack opening stress under the influence of single overload is proposed, as shown in Figure 4(b).When the crack grows to point  1 , a single overload stress is applied and a large plastic zone appears ahead of the crack tip.As the crack penetrates into this plastic zone and grows to point  3 , the retardation effect gradually recedes until it vanishes.
A function of  op / max and the crack length is established to calculate the crack closure variation during the retardation region, which is shown in Figure 4(b).The points  and  in this figure, respectively, represent the situation of crack growing to points  1 and  3 in Figure 4(a).Equation ( 5) can be obtained based on the modified virtual crack annealing model above: where  is Bauschinger effect factor,  is the load ratio, and  ol is the overload ratio.The left side of ( 5) is the ordinate of point .Similarly, the ordinate of point  can be illustrated by the following: The horizontal ordinate of point  is where  0 is the horizontal ordinate of point , which is the crack length when single overload is applied.  is the diameter of plastic zone created by that single overload.If point  is considered to be the ordinate origin, the equation of the curve between points  and  is where   and   are, respectively, the ordinate of points  and  and   is the horizontal ordinate of point .When the value of  can be 2, 1, and 0.5, the curve is, respectively, shown as (A), (B), and (C) in Figure 4. Based on the current study,  = 1 is a good approximation for aluminum alloy.If the  is considered to be the ordinate origin, (7) can be rewritten as Based on the discussion above, the new crack closure model is established as (10), which is used to describe fatigue crack propagation behavior under single overload interspersed in a constant amplitude loading spectrum.Consider after influence of single overload. ( The cycle by cycle algorithm is used to implement the above model.The flow chart for fatigue crack growth prediction based on the modified crack closure model is shown in Figure 5. Before any predictions can be made, there are several unknown parameters (see (1) and (10)) that need to be calibrated.Three sets of / ∼  testing data are used to evaluate these parameters (Figure 6), which are assumed to depend on the material only [27].The calibrated parameters are  = 2.4534 and  = 1.2331E-9, and Bauschinger effect factor  is 0.97.
Then additional two sets of - data are used to validate the proposed model.The specimen configuration for these tests is [27] width = 100 mm, length = 500 mm, and thickness = 4 mm.Initial half crack length is  = 5.0 mm.The specimen was subjected to a constant amplitude load sequence with  max = 64 MPa,  = 0.And the overload ratio is 2.  The model predictions are compared to the experimental data in Figure 7.The -axis is the number of cycles, and the -axis is crack length.The red circlets represent the fatigue crack growth under the constant amplitude loading; and the small blue squares are the counterpart under the constant amplitude load with single overload.It is observed that the predictions by proposed model match the testing data well.
More detailed information can be obtained in the /- curve in Figure 8.Right after the single overload, the fatigue crack growth rate increases in a short time and then decreases sharply.After a certain crack length, the retardation phenomenon vanishes and the crack growth rate turns back to the original trend.Note that very good agreement is observed between the model prediction and experimental data.
Additional set of testing data is used for the model validation in the same material and similar specimen configuration [28].Specimen dimensions are as follows: length = 180 mm, width = 45 mm, and thickness = 1.5 mm.Initial crack length is  = 4.0 mm.The specimens are subjected to two different loading spectra: (1) constant amplitude loading with no overloads ( max = 60MPa,  = 0.1) and (2) constant amplitude load with single overload applied at the certain crack length (overload ratio is 2) [28].Following the same procedure, the comparisons between predictions and experimental data are shown in Figure 9  former loading spectrum, both of our proposed model and Manjunatha's model can give very good predictions [28].In the latter loading case, the single overloads are applied at the crack length of 5.5, 9, 12, and 16 mm, respectively.The prediction of our proposed model is just slightly slower than the experimental data, while Manjunatha's prediction is much faster than that of the experimental data.The fatigue life ratios expressed as  expt / pred are, respectively, calculated to be 0.89 by using the proposed model and 1.44 by using   Manjunatha's model.Obviously, the proposed closure based model can give the better predictions.

Crack Growth Prediction in 350WT under Constant
Loading and Single Overload.In this section, the model validation will extend to 350WT steel.The experimental data are given by Taheri et al. [29].The specimen dimensions are as follows: length = 300 mm, width = 100 mm, and thickness = 5 mm.Initial center crack length is 2 = 20.0 mm.Recording test data was started at crack length of 11.17 mm.In addition, Mathematical Problems in Engineering  the mechanical properties of 350WT in longitudinal (LT) orientation are as follows:   = 350 MPa,  UTS = 524 MPa, Modulus of elasticity = 205 GPa, and Poisson's ration = 0.30 [30].
Similarly, the parameters are calibrated by the / ∼ Δ data in the literature [19], as shown in Figure 10.The fitting coefficients are  = 4.3136 − 11 and  = 2.9245, and the Bauschinger effect factor  is 1.05.
Figure 11 shows the comparison between the predicted - curve and the experimental data in 350WT steel specimens.The single overloads are applied at the crack length of 15 and 25 mm.It can be seen that there are two obvious retarded regions right after the overload in the experimental data.The prediction follows almost the same trend as the experimental observation.Predicted total fatigue lives are slightly lower than the experimental values.In Figure 12, crack growth rate variation caused by overloads is also compared with our model prediction.It is clear that the prediction matches the experimental data very well, especially in the retardation region.Hence, it is proven that the method proposed in this paper is applicable to predict the fatigue crack growth behavior under the constant loading with single overloads in 350WT steel.

Conclusion
In this investigation, a theoretical model is developed to predict the fatigue crack growth behavior under the constant amplitude loading with single overload.The crack growth retardation was accounted for by using crack closure concept and plastic zone.Model was validated in D16 aluminum alloy and 350WT steel subjected to several different loading spectra, and the predictions matched experimental data well.The following conclusions can be drawn based on the current investigation.
(1) Fatigue crack growth is slowed down by application of single overload cycle.A convincing reason for this retardation phenomenon is that after the overload a large plastic zone will form ahead of the crack tip, which can increase the crack closure level within this region.And as the crack grows through the large plastic zone, the crack closure level will gradually decrease which can be described as a linear function.
The retardation effects disappear after a certain characteristic crack length extension from the overload position.This extension is approximately equal to monotonic plastic zone size caused by the overload.
(2) The proposed model is derived from fatigue crack growth mechanisms (such as crack closure, plastic zone, and Bauschinger effect), and it does not require any additional parameters which has no physical meaning.
(3) The above statement is only valid under the current investigated loading spectrums and materials.In the future, the whole frame work should be extended to other materials.Additionally, branching and bifurcation caused by overload can also retard the crack growth rate, which should be investigated in the future.Bauschinger effect factor in loading process :

Nomenclature
Bauschinger effect factor in unloading process : Bauschinger effect factor.

2 MathematicalFigure 1 :
Figure 1: Schematic illustration of real crack and virtual crack model.

Figure 2 :
Figure 2: Schematic illustration of CA load interspersed with singlecycle tensile overload.

Figure 3 :
Figure 3: Schematic of delayed retardation due to single tensile overload.

Figure 4 :
Figure 4: (a) Crack-tip plastic zone due to single tensile overload; (b) the mathematical derivation of  op based on crack closure model.

Figure 5 :
Figure 5: Flow chart for fatigue crack growth calculation.

Figure 7 :
Figure 7: - curve for constant amplitude and single overload cases.

Figure 8 :
Figure 8: /- curve for constant amplitude and single overload cases.

Figure 11 :
Figure 11: - curve for constant amplitude and single overload cases.
Fatigue crack growth rate per cycle  min ,  max : Minimum and maximum stress in one loading cycle  op : Stress level at which the crack begins to grow  ol : Stress level of single overload : S t r e s sr a t i o  ol : O v e r l o a dr a t i o  max ,  min : Maximum/minimum stress intensity factor Δ: