Experimental Verification and Estimation of Fatigue Crack Growth Delay due to Single Overload

A novel crack driving force parameter based on the plastic zone size for overload/constant amplitude loading is proposed to estimate the fatigue crack growth (FCG) and the benefcial fatigue life due to overload. By considering the stress distribution in the plastic zone, the efective stress intensity factor (SIF) integrated from equivalent residual stress is presented. It is illustrated that the FCG rate decreases rapidly after overload and then gradually returns to the baseline FCG rate. Besides, the overload ratio R pic has a signifcant infuence on the afected length a d and the benefcial fatigue life N d .


Introduction
Te fatigue life prediction is required in such applications as aerospace, automobiles, and pressure vessel industries for damage-tolerant design. As a common consequence of fatigue loading, fatigue cracks can be generated in the geometrical stress concentration areas of structures. When an overload is applied to a crack subjected to constant amplitude loading, the FCG rate could be signifcantly delayed, largely improving the fatigue life. Numerous works have been devoted to the study of crack propagation behavior under variable amplitude loading [1][2][3][4][5]. However, the estimation of the FCG rate is not a trivial issue in fatigue crack life prediction, even though overloads are well known to retard crack growth. Indeed, there is no widely accepted method, which can model all infuential parameters correctly.
Generally, the fatigue life can be divided into three stages: (1) the crack nucleation, (2) the crack growth, and (3) the rupture of structure. Paris' model is one of the most frequently employed methods to predict the macro FCG rate. Tis empirical model postulates that the FCG rate, da/ dN, is dominated by the SIF range ΔK � K max -K min , which depends on the loads and crack length. However, this model is limited by its initial hypothesis. In recent years, it is recognized that the Paris formula based on linear elastic fracture mechanics (LEFM) cannot exactly describe the overload delay efect, where crack fatigue propagation behavior is related to the plastic zone size in front of the crack tip [6][7][8].
Based on the experimental results, it shows that the FCG rate is not proportional to the range of the applied stress intensity factor ∆K but rather to the efective stress intensity factor ∆K ef . Indeed, only a portion of the fatigue loading cycle is efective for crack propagation, and the crack may remain closed for the other part due to elastic constraints acting on the plastically stretched material in the crack wake, according to Elber [9]. Te crack closure is caused by plasticity, oxidation, roughness of the crack faces, or a phase transformation induced by mechanical loading. Te plasticity-induced closure, which is particularly sensitive to loading history efects, derives from two diferent sources: the frst is the plastic wake created by the plastic zone along the crack path, and the second is the occurrence of residual compressive stresses in front of the crack. In reality, these efects are not independent, leading to the same delay phenomenon. Terefore, the relationship between the FCG rate and the plastic stress distribution in front of the crack tip is extremely signifcant for exploring crack propagation mechanisms and improving the accuracy of fatigue life prediction. In this procedure, the plasticity afected region, i.e., the monotonous plastic zone, is the origin of residual compressive stresses in front of the crack tip, which makes the subsequent cycles less efective. When the crack grows, the residual stresses are then exerted on the crack face and increase its opening level.
To refect delay efects due to overload, Willenborg et al. [10] proposed a delay coefcient C p associated with the plastic zone size in front of the crack tip to predict the FCG after overload. Based on the model, Bacila et al. modifed the FCG model [11] using a piecewise linear function to ft the growth rate curve. Moreover, the delay efect on FCG has also been confrmed through EBSD, FIB-DIC, and FEM methods by Salvati et al. [12]. However, these empirical models need additional parameters ftted through experimental data. In addition, Wu and Carlsson [13] used a weighted residual stress intensity factor (RSIF) to quantify the crack closure efect due to the residual stress, as described in (5). And several analytical solutions for standard specimens are provided in [14][15][16].
In the past decades, considerable attention has been paid to the plastic zone in front of the crack tip because of its close correlation with the delay of FCG after overloading [17]. Dai et al. proposed a method that defnes a plasticity-corrected SIF as a new mechanical driving force for predicting the FCG rate [18].
Fatigue crack delay caused by overload is a general phenomenon found in most engineering applications. Even though the traditional rainfow counting techniques are widely used for variable amplitude loading events, it is impossible to decide a priori, which points should be considered load reversions, decreasing the crack life prediction accuracy. Accordingly, many crack delay models are proposed based on diferent mechanisms. In this feld, the crack closure theory has drawn many attentions, which purports that a crack will not propagate unless the crack driving force is over a specifc point. However, the closure efect is still highly controversial since this efect cannot explain many of its peculiarities, such as high loading ratio conditions or high-strength steels [19][20][21][22][23]. Besides, many phenomenological delay models have also been proposed based on the following main hypotheses: (1) the crack length afected by overload is estimated by the plastic zone sizes, which are infuenced by loading conditions; (2) the crack length corresponding to the value of the FCG rate is relative to the plastic zone size. Tese empirical models confrm that the plastic zone plays an important role in the quantifcation of the variation of the FCG rate due to overload. However, the stress distribution in the plastic zone is rarely considered in classical models. Moreover, most of the models contain complex ftting parameters, e.g., the severity coefcient in Bacila's model, both increasing the possible accumulative errors.
In this paper, with the introduction of equivalent residual stress σ res and equivalent RSIF ΔK pl , a novel crack driving force model without ftting parameters with clear physical meaning is proposed, by considering the material's hardening efect in the plastic zone near the crack tip. Tis model can be further incorporated with the rainfow algorithm to estimate the FCG rate under variable amplitude loading, without losing prediction accuracy. Moreover, the detailed experimental data of steels 1045 and 1080 have also been presented. A brief description of residual stress distribution ahead of the crack tip under cyclic loading is presented in Section 2. Section 3 develops a novel efective stress intensity factor, which incorporates essential delay parameters to capture the infuence of a single overload. Section 4 presents the experimental setup and diferent applied overloads to quantify how the infuential parameters afect the FCG rate and fatigue life. Proposed driving force evaluations are carried out through experimental data. Finally, conclusions are drawn in Section 5.

Residual Stress Distribution
LEFM presents a characteristic distribution of the tensile stress ahead of a crack tip with the following expression: (1) Generally, as shown in Figure 1, in order to meet the static equilibrium conditions, the stress reduced by the material yielding motivates the stress augmentation in the nearby elastic materials, bringing more materials to yield. Terefore, the stress distribution (dotted line) based on LEFM is transformed into the elastoplastic stress distribution (solid line). In the plane stress state, Irwin calculates the plastic zone size as follows: Te residual stress can then be calculated through the plastic superposition method proposed by Rice. Te unloading process is regarded as Δσ � 2σ 0 with a reverse plastic fow. Te schematic superposed residual stress is shown in Figure 2.
Although Rice's method is widely accepted, the stress gradient in the plastic zone has not yet been considered. Actually, during cyclic loading events, the material near the crack tip plastically deforms, and the stress increases with the decrease in the distance from the crack tip. As a result, the local stress near the crack tip must be higher than the original yield stress. In other words, the neglect of material elastoplastic hardening in front of the crack tip may lead to a deviation of estimation on the fatigue crack driving force. Moreover, it can be proved that the stress near the crack tip varies from the yield stress to the strength limit according to the FEM results and physical observations [24]. Analogously, in the proposed method, the stress near the crack tip σ 1 is assumed to be equal to the average of the yield stress σ 0 and the strength limit σ b , as follows: According to equations (1)-(3), we have 2 Advances in Materials Science and Engineering Correspondingly, due to the increased hardened stresses within the plastic zone, the yielded material causes a larger plastic zone than that estimated by Rice's method. During the unloading process, the radius of the yield plane is assumed to be fxed due to the Bauschinger efect with stress range Δσ � 2σ 0 , and the plastic zone sizes can then be estimated by the following: where the equivalent reverse radius of unloading plastic zone r t and the equivalent radius of loading plastic zone r s are as follows: By substituting r t and r s , the stress distribution in front of the crack tip after cyclic loading can then be expressed as follows: where σ is the stress distribution of loading and σ′ is the stress distribution of unloading.
Te residual stress can then be superposed by equation (8), and the results are shown in Figure 3.

Proposed Effective SIF
Based on [13], the expression of equivalent residual stress σ res considering elastoplastic hardening in the plastic zone is correspondingly presented in (9).
where ϕ(x/W) is a weight function. According to Qylafku's results [25], the weight function ϕ(x/W) is a monotonically decreasing function and relates to the geometry of specimens, having the following characteristics: A power exponential function is proposed as follows: where D has the expression Δσ -σ0

Advances in Materials Science and Engineering
with W 0 � 2.5 mm, which is the baseline width of the specimen. In addition, f is a dimensionless function with the following equation for CT specimens [26]: Terefore, equivalent RSIF ΔK res based on Δσ res is given by the following: Obviously, ΔK pl is mainly dependent on load amplitude, which can refect the diference between constant and overloaded loading events. ΔK ef is derived from ΔK res , as described in equations (15) Finally, ΔK ef can be obtained by substituting Eq. (7), (8), and (14) into (15), given by the following: Moreover, the crack length afected by delay a d can be calculated by (17).
One advantage of this method is that it can reach rather quickly the calculation of the afected delay cycle number N d . Indeed, it is very simple to get the characteristic and essential delay parameters by the integral method. In summary, to quantify the infuence of the plastic zone on FCG life after a single overloading, the proposed efective SIF can be used in the prediction of FCG rate based on the traditional form of the Paris formula, as described in (18).
Te delay cycles number N d is given by the following expression: Te integration starts at the position a 0 where the overload occurs.

Experimental Investigation and Discussions
To verify the prediction capabilities of the presented estimated framework, compact tension (CT) specimens made of 1045 and 1080 steel are employed for the fatigue crack propagation study, see Figure 4. Te recommendation for the dimensions of specimens in the ASTM standard [22] is adopted with a thickness B � 15 mm and a width W � 80 mm. In order to characterize the mechanical properties, tensile tests are performed on cylindrical specimens (diameter d � 11 mm). Te results illustrated in Table 1 reveal two yield stress families: low (1045) and high (1080). Te Fatigue tests are performed under constant ΔK with a stress ratio of R � 0.1 in order to maintain the same baseline plastic zone sizes with a loading frequency of 15 Hz. During each test, a single overload cycle, as shown in Figure 5 is applied with a loading frequency of 0.1 Hz. Table 2 specifes the parameters for two materials with fve diferent overload ratios R pic .
Te curve of ΔK ef versus crack length under the overload ratio R pic � 1.5, 2, and 2.5 is presented in Figure 6. It can be found that after overloading, ΔK ef is reduced to a minimum value and gradually recovers to the baseline SIF value. Moreover, the minimum value of ΔK ef decreases with the increase of the overload ratio R pic . And the result can well interpret the delay hysteresis of the overloading, which means the FCG rate is not immediately reduced to the lowest point when the overloading occurs but gradually reduced to the lowest rate, which depends on the plastic zone size in front of the crack tip.
To validate the proposed empirical driving force parameter in fatigue crack life prediction, the estimated overloaded FCG rate is compared with the experimental data for two materials with the same overload ratio R pic � 2.5, as shown in Figure 7. It is illustrated that this proposed method leads to an acceptable description of the two stages of the FCG rate after a single overloading because the position of the lowest point, the afected delay length, and the curve tendency are all within a reasonable error range with respect to the experimental results.
Moreover, for the material with low yield strengths such as 1045, the a d is relatively longer and the recovery rate is much higher than that of materials with high yield strength such as 1080. Similar phenomena are also found in Louah's results [23], which can prove that the crack closure efect is weakly infuenced by the overloaded cycle in the case of high-strength steels.
Te estimated FCG rates with diferent overload ratios R pic for two materials are presented in Figure 8. According to the tendency of the curves, it can be concluded that the delay efect of the FCG rate is mainly dominated by R pic , which manifests in the form of a convex function. It means that the curves drop rapidly to the lowest point immediately after overloading and are followed by gradual returns to the  Advances in Materials Science and Engineering baseline rate, which is infuenced by the stress distribution in the plastic zone. Te estimated and observed fatigue lives for materials 1045 and 1080 are listed in Table 3. Moreover, the comparison of the estimated and experimental results is plotted in Figure 9. It can be found that the prediction results correlated well with the experimental data, as all points fall within a factor of 2 time scatter band. Compared to the traditional empirical models, the stress hardening in front of the crack tip is considered in the proposed method, which can more reasonably quantify the overload delay efect. It is necessary to point out that for the proposed method, which has no ftting parameter in the calculation, the agreement of the predictions at diverse levels of the FCG rate, overload level, afected length, and fatigue life is remarkable.
Besides, two empirical FCG delay models are also used to compare the fatigue life prediction capability of the proposed    method under the same conditions. Te frst one is based on the weight function solutions proposed by Wu [13], and the second one is based on the modifed equivalent inclusion theory proposed by Li [17,28]. Te life prediction results of these two methods are given in Table 4. Figures 10 and 11 illustrate a comparison of fatigue life predictions with the experimental results based on Wu's and Li's methods. Te correlation coefcient R c and mean square error (MSE) defned by Mohanty [29] as shown in (21) are adopted to describe the deviation between the predicted and experimental results, detailed in Table 5. It can be found that, for overload events, these three methods have strong applicability in fatigue life prediction. However, according to R c and MSE, the proposed method in this paper has much higher accuracy and better stability.      Advances in Materials Science and Engineering

Conclusions
Te delay efect of overloading on the FCG rate for diferent materials has been studied in the paper. By taking into account of the residual stress in the plastic zone in front of the crack tip, a novel empirical crack driving force parameter with no need for ftting parameters is presented, and the infuenced FCG rate is used to estimate the benefcial fatigue life. A series of fatigue experiments with fve overload ratios R pic for two materials are performed. Te conclusions below can be drawn: (1) Te residual compressive stress in the plastic zone in front of the crack tip due to overloading is the main reason for FCG delay; (2) After a single overloading, the FCG rate curve will gradually decrease to the lowest point, and then slowly return to the baseline rate in the form of a convex function; (3) Te higher the overload ratio R pic value is, the less is the minimum FCG rate after overloading; (4) Te proposed crack driving force parameter, taking into account of the elastoplastic hardening in the plastic zone, can be used to efectively predict the benefcial fatigue life due to overloading, which is validated by comparing the predicted and experimental results.
Abbreviations σ 0 : Yield strength σ 0 : Stress close to crack tip σ res : Equivalent residual stress σ res : Residual stress distribution in front of the crack tip a 0 : Crack length where overload occurs a: Crack length a d : Crack length afected by delay K c : Fracture toughness ΔK ef : Efective stress intensity factor K res : Equivalent residual stress intensity factor N d : Crack initiation life R pic : Overload ratio r s : Equivalent radius of loading plastic zone r t : Equivalent reverse radius of unloading plastic zone r p : Irwin radius of plastic zone r pl : Radius of the plastic zone in front of the crack tip RSIF: Residual stress intensity factor SIF: Stress intensity factor W: Specimen width.

Data Availability
Te data that support the fndings of this study are available from the corresponding author, Hao Wu, upon reasonable request.

Conflicts of Interest
Te authors declare that they have no conficts of interest.