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Article

The Overload-Induced Delay Model of 7055 Aluminum Alloy Under Periodic Overloading

State Key Laboratory of Mechanics and Control for Aerospace Structures, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China
*
Author to whom correspondence should be addressed.
Current Address: College of General Aviation and Flight, Nanjing University of Aeronautics and Astronautics, Liyang 213300, China.
Metals 2025, 15(6), 644; https://doi.org/10.3390/met15060644
Submission received: 7 May 2025 / Revised: 2 June 2025 / Accepted: 6 June 2025 / Published: 9 June 2025
(This article belongs to the Section Metal Failure Analysis)

Abstract

Aluminum alloys, serving as critical structural materials in the aviation and aerospace industry, frequently endure variable amplitude loading under complex service conditions. The resulting non-steady-state crack propagation behavior directly impacts structural safety. This study considers the engineering application requirements of the 7055-T7751 aluminum alloy and conducts fatigue crack growth experiments on compact tensile specimens subjected to constant amplitude loading and periodic variable amplitude overloading conditions. The findings indicate that the 7055 aluminum alloy exhibits an instantaneous acceleration period under tensile overload, which is important in the comprehensive analysis of crack growth life. The experimental findings show no significant correlation between post-overload minimum crack growth rate deviation and thickness or crack size at overload, where the values are 50.3% and 94.8% at 1.4 and 1.7 ROL, respectively. An analytical model for the crack growth increment aii during this period was developed. Additionally, the delay distance influenced by overloads ad and the number of delay cycles Nd are identified as effective parameters for evaluating the retardation effects induced by overloading. Our comparative analysis of crack growth experimental data under varying overload ratios ROL and specimen thicknesses B revealed that existing plastic zone models inadequately assess ad, prompting the establishment of a corresponding evaluation model. By incorporating the parameters aii and ad into the Wheeler model, a method for calculating the delay cycles Nd was constructed, which effectively captured the variation trend. Finally, an analysis of fractography revealed numerous secondary cracks within the overload damage zone, and the ductile fracture characteristics in this region were significantly weaker compared to areas subjected to fatigue loading.

1. Introduction

In the aviation industry, aluminum (Al) alloys have been successfully employed for over 90 years as the primary materials for safety-critical structures, including aircraft fuselages, wings, tails, and wing spars [1,2]. This extensive application is attributed to advantageous properties, such as low density and high specific strength. The operational environment of aircraft is highly complex and variable, with conditions such as gusts, turbulence, and hard landings during severe weather causing structures to endure variable amplitude (VA) loading [3]. This makes crack propagation under VA loading a critical factor influencing structural integrity and reliability. Therefore, an in-depth investigation of fatigue crack growth (FCG) behavior in Al alloy materials under VA loading holds significant theoretical and practical importance for ensuring the safety and stability of engineering structures.
VA loading significantly differs from the constant amplitude (CA) loading commonly employed in laboratory settings. Parameters such as the overload ratio ROL [4], overload interval Nb [5,6], number of overloads Ns [6], and loading sequence [7] all influence crack propagation behavior under VA loading, thereby complicating the prediction of FCG life. According to the type of load spectrum, VA loading can be categorized into simple VA fatigue loading and complex VA fatigue loading [8]. The former refers to scenarios where single tensile overload or consecutive multiple tensile overloads are superimposed on the CA load. The latter can be further subdivided into stable VA fatigue loading and unstable VA fatigue loading. Stable VA fatigue loading comprises several identical program load cycle blocks. Within each program load cycle block, loading parameters—such as the stress amplitude σa, loading frequency f, loading waveform, and stress ratio R—are typically maintained constant. Examples include periodic single overloads and repetitive loading of identical program load cycle blocks. In contrast, the sequence of program load blocks in unstable VA load spectra lacks repeatability.
In recent decades, scholars both domestically and internationally have conducted extensive research on FCG under VA loading [9]. It is well-known that applying a single tensile overload under CA loading tends to induce crack growth retardation in ductile alloys. However, the study of FCG mechanisms under VA loading has continuously developed amidst ongoing debates and controversies [10,11,12,13]. Currently, it is generally acknowledged that the complex phenomena of crack acceleration and retardation during VA loading cannot be explained by a single mechanism. Instead, they are governed by the simultaneous presence and potential competition of multiple mechanisms, including crack closure [14], crack tip blunting [15], residual stresses [12] at the crack tip, crack deflection and branching [3], and strain hardening [16]. Identifying the true dominant mechanisms and their interactions during crack propagation under VA loading remains a significant challenge. Moreover, although crack size is a critical factor in evaluating structural integrity [17], numerous uncertainties pose difficulties and challenges for the accurate prediction of FCG life. These uncertainties include not only the inherent variability of material properties and manufacturing processes but also the unpredictability of actual loading and service environments in aircraft operation. Transitioning from the FCG behavior under stable VA fatigue loading to the accurate prediction of FCG life under unstable VA fatigue loading is far from straightforward. Despite the widespread proposal and application of plastic zone models, crack closure models, strip yield models [18], and their improved versions [19,20,21], challenges persist regarding their predictive accuracy and applicability to different materials. In addition, real service environments, such as coastal salt spray and saltwater conditions, extremely low temperatures at high altitudes, and aviation fuel inside wing tanks, also influence the FCG behavior of metallic materials and affect the accuracy of predictions [22]. Analytical methods that neglect these factors in FCG analysis often yield overly conservative or unsafe predictions, leading to a series of adverse consequences, such as unreasonable structural weight, inspection thresholds, repetitive inspection intervals, and the use of non-destructive testing techniques. These issues ultimately result in a dramatic increase in maintenance costs.
The delay distance ad and delay cycles Nd affected by crack retardation are common quantitative parameters used to measure the degree of retardation following an overload. Typically, as the overload ratio ROL increases, both ad and Nd also increase [4]. Vecchio et al. [23] were the first to define Nd as the difference between the number of cycles required to traverse the overload-affected zone after an overload and the number of cycles required to traverse the same distance under normal CA loading. Furthermore, they discovered that the relationship between Nd and the ratio of the overload plastic zone size to the specimen thickness exhibits a U-shaped trend. Shin and Hsu [24] investigated the overload retardation phenomena in AISI 304 stainless steel under different baseline stress intensity factor (SIF) ΔKBL, overload ratio ROL, and stress ratio R. They obtained an inverted U-shaped relationship curve, providing the explanation that the maximum retardation is associated with an optimal ΔKBL. Under the applied values of 16, 19, and 22 MPa m , crack closure did not undergo significant changes. Consequently, the effective FCG driving force ΔK and the delay cycles Nd compete with each other. The study also found that the degree of overload retardation increases with increasing ROL and decreases with increasing R, while its relationship with the baseline SIF ΔKBL is more complex. McEvily and Yang [25] investigated the FCG behavior in Al alloys 6061-T6, 7090-T6, and titanium alloy IMI829 following a single tensile overload. They proposed a semi-empirical method for calculating the number of delay cycles Nd after an overload. Subsequently, McEvily et al. [26] analyzed Skorupa’s [27] overload experiment data and discovered that as the specimen thickness B increases, the post-overload crack opening level decreases and Nd reduces, indicating a transition from plane stress-dominated to plane strain-dominated retardation behavior. Li et al. [28] investigated the FCG behavior of DP780 dual-phase steel under a single tensile overload and found that increases in the overload ratio ROL and the crack length at overload application aOL enhance the retardation effect. They introduced the factor (1 − n)/(1 + n) into the Irwin model, thereby reducing the error in characterizing the overload-induced delay distance ad by approximately 30% when using the monotonic plastic zone size. Additionally, they incorporated the aforementioned factor into the Wheeler model to describe the FCG rate during the retardation process following an overload. Bahram et al. [29] studied the crack propagation characteristics of 2024-T351 aluminum alloy under a single tensile overload using the AFGROW code and the Willenborg model. Their research demonstrated that the plastic zone size rp and Nd increase with ROL. Zhou et al. [30] designed four overload modes to investigate the effects of overload ratio, overload type, and specimen thickness on the FCG behavior of 300 steel. They found that the FCG rate during the retardation process decreases with increasing overload ratio, multiple peak overload types influence the retardation effect, and the residual stress at the crack tip is greater under continuous multiple overloads compared to a single overload.
Currently, research on FCG behavior following an overload primarily focuses on the impact of a single overload on the subsequent FCG rate, as this forms the foundation for studying FCG behavior under VA loading. However, the quantitative parameters, including delay cycles Nd and distance ad, which measure the degree of overload-induced retardation, are evidently related to loading parameters throughout the entire lifespan of FCG. Furthermore, current studies lack sufficient investigation into these two parameters under periodic overload conditions or across the full lifespan of the cycle. Therefore, the main objective of this study is to investigate the variation trends of Nd and ad with cyclic loading SIF ΔK, overload ratio ROL, and specimen thickness B through FCG experiments on aluminum alloy 7055-T7751 under periodic overload conditions. The experiments reveal that the often-overlooked instantaneous acceleration period following an overload is significant and cannot be ignored in the analysis of FCG life. Based on the Wheeler model, a characterization model for Nd is established. Different methods are employed to calculate the plastic zone size, and the effects of these methods on the characterization of ad are evaluated. Additionally, fatigue failure fracture surfaces are examined using a VHX-1000 ultra-depth-of-field digital microscope.

2. Materials and Test Conditions

2.1. Specimens

For the experimental research, 7055-T7751 aviation Al alloy sheets (Southwest Aluminum (Group) Co., Ltd., Chongqing, China) were selected. The chemical composition of the Al alloy used in this study was determined through Agilent 7800 Inductively Coupled Plasma—Mass Spectrometry (ICP-MS, Agilent Technologies, Inc., Santa Clara, CA, USA) analysis, and is as follows: Zinc (Zn) at 7.714%, Copper (Cu) at 2.297%, Magnesium (Mg) at 1.838%, Manganese (Mn) at 0.626%, Iron (Fe) at 0.039%, Titanium (Ti) at 0.024%, Silicon (Si) at 0.013%, Gallium (Ga) at 0.010%, and the balance is Al. According to ASTM E8M-22 [31], flat specimens were used, and the tensile speed was set at 1 mm/min. The tensile stress–strain curve of the material at room temperature was obtained, as shown in Figure 1. The resulting mechanical properties are as follows: a tensile yield stress σy of 419 MPa, an ultimate tensile stress σb of 522 MPa, an elasticity modulus E of 73 GPa, and an elongation of 10%.
The preparation and testing procedures for compact tension C(T) specimens adhere to the requirements specified in the ASTM E647-15e1 [32] test standard. Figure 2a illustrates the dimensions of the C(T) specimens, while Figure 2b presents images of the Al alloy 7055-T7751 specimens. Three specimen thicknesses are employed, specifically B = 5 mm, 10 mm, and 15 mm. All specimens are oriented longitudinal-transverse, L-T. The initial machined notch of each specimen is produced using electrical discharge machining (EDM), with an initial notch length of an = 20 mm. Prior to testing, the surface of each specimen is polished using conventional metallographic preparation methods to ensure clear observation of the crack. The similarity principle in fracture mechanics indicates that for metallic structures with long cracks under cyclic loading, similar crack growth behavior will be exhibited as long as they experience the same ΔK [33]. Therefore, the crack growth behavior observed in C(T) specimens can serve as a reference for the design of actual aerospace structures.

2.2. FCG Experiments

FCG experiments were conducted at room temperature using the electro-hydraulic servo fatigue testing system MTS 370.10 (MTS Systems Corporation, Eden Prairie, MN, USA). The uniaxial tensile testing machine demonstrates excellent alignment, with a bending strain ratio below 2.8%. It has a load range of ±100 kN, and the relative error of the indicated value within this range is less than 0.3%. The COD gauge used to measure the crack tip opening displacement is the MTS 632.02F-20 model, which has a displacement travel range of +3/−1 mm, and the relative error of the indicated value within this range is less than 0.2%. A MV-CH120-10UM CMOS industrial camera(Hikvision Digital Technology Co., Ltd., Hangzhou, China) is simultaneously used to monitor the real-time crack propagation length, with a resolution of 4096 × 3000 and a pixel size of 3.45 µm, meeting the requirements for high-precision observation. The experimental setup and specimen clamping configuration are shown in Figure 3. Prior to performing the formal FCG tests, fatigue cracks were pre-cracked in accordance with the requirements of ASTM E647 [32] to create crack tips sharper than those produced by machining the notch root. Fatigue cracks were pre-grown through a five-level K-decreasing process with an R ratio of 0.1. During this process, the maximum cyclic load at stage i, Pmax,i, did not decrease by more than 20% relative to the maximum cyclic load of the preceding stage Pmax,i−1. The fifth-level load was maintained consistent with the load used in the FCG tests, i.e., Pmax,5 = Pmax. Additionally, the crack extension increment Δai at each level satisfies [32],
Δ a i     3 π K max ,   i - 1 σ y 2
where Kmax,i−1 represents the maximum SIF at the crack tip during the (i − 1)th stage of the pre-cracking process. The crack pre-cracking process is completed when the fatigue crack reaches a = 32.7 mm (i.e., extends 12.7 mm from the root of the machined notch in the direction of crack propagation). During the test, the crack length a is measured using the compliance method and visually verified with the assistance of a CMOS camera. In the visual inspection method, the current crack size is determined as the arithmetic mean of the crack lengths on both sides of the C(T) specimen. The compliance method calculates the crack size using measurement data from the COD gauge(MTS Systems Corporation, Eden Prairie, MN, USA), with the specific procedure detailed in ASTM E 647 [32]. Based on the results from the visual inspection method, the effective elastic modulus E′ used in the compliance method is corrected to ensure that the difference between E′ and the typical elastic modulus E does not exceed 10%. Additionally, E′ must satisfy the condition EE′ ≤ E/(1 − ν2), where ν is Poisson’s ratio.
According to linear elastic fracture mechanics, for finite-sized components in engineering, the stress intensity factor can be expressed as
K = σ β ( a ) π a
where σ is the magnitude of the applied stress, a is the crack size, and β(a) is the geometry-dependent shape factor related to the crack length and specimen configuration. For C(T) specimens, the calculation formula for β(a) is [32],
  β a = 2 + α 4 π × 0.886 + 4.64 α 13.32 α 2 + 14.72 α 3 5.6 α 4 α 1 / 2 1 α 3 / 2
where α = a/W is the normalized crack size, and W and P are the dimensions relative to the width of the specimen and the magnitude of the applied load, respectively. For the specimens in this study, W = 100 mm. Equation (3) is valid for 0.2 ≤ α ≤ 1.0.
In the FCG tests, the stress ratio R of the applied load on the specimens was maintained at 0.1. The loading waveform was a sine wave with a frequency f of 15 Hz. The overload ratio ROL = POL/Pmax is defined as the ratio between the applied overload POL and the maximum baseline cyclic load Pmax. The values of Pmax for specimens with thicknesses of 5 mm, 10 mm, and 15 mm are 2.9 kN, 5.8 kN, and 7.7 kN, respectively. Three test conditions (TC) were designed for each thickness: TC1 involved CA loading, while TC2 and TC3 involved periodical VA loading with overload ratios ROL of 1.4 and 1.7, respectively, as these represent characteristic overload ratios found in civil aircraft wings’ load spectra. The initial overload was applied starting from the first cycle immediately after the completion of the pre-cracking stage. Moreover, the number of overload interval cycles Nb was set to 10,000 cycles, that is, NOL = 1 + 10,000i, where i = 1, 2, 3…. CA and VA loading modes are illustrated in Figure 4. In variable amplitude loading, the FCG life was accumulated from the first cycle after the end of the pre-cracking stage until the specimen fractures. Upon completion of the crack propagation tests, the fatigue failure surfaces of the specimens were examined using a VHX-1000 ultra-depth-of-field optical microscope(Keyence Corporation, Osaka, Japan). The microscope lenses covered magnifications from 20× to 5000×, with a resolution of up to 0.1 µm and a minimum imaging area of 61 µm × 46 µm.

3. Experiment Results and Discussion

3.1. FCG Behavior Affected by Overload

Table 1 presents the results of the fatigue tests, where the values represent the average outcomes of three repeated experiments for each test condition (with specimen numbers #1, #2, and #3 for each group). CV denotes the coefficient of variation (CV = sN/ n - ), and sN represents the standard deviation. The CV for all test groups remains below 10%, indicating good data consistency and reproducibility. It can be observed that FCG life increases for all three specimen thicknesses following the application of periodic overloads. Furthermore, for the 5 mm and 15 mm thick 7055 Al alloy specimens, the FCG life under an overload ratio ROL of 1.7 is higher than that under 1.4 ROL. Notably, for the 10 mm thick specimens, the FCG life under an overload ratio of 1.4 is slightly higher than that under 1.7. The reason for this phenomenon will be further discussed in Section 3.3.
Since the crack length a is measured using the compliance method, it is also essential to observe the variations of the load–displacement (P-V) curves and crack opening displacement (COD, V) with respect to the number of cycles N. Figure 5a illustrates the variation of the P-V curves with N for a specimen with thickness B = 10 mm under TC2. It can be observed that both Vmax and Vmin exhibit an increasing trend as the number of loading cycles N increases. This trend closely follows the variation observed in the a-N curves, consistent with the trends observed under TC1 and TC3. Additionally, it is evident that the application of cyclic loading remains consistent throughout the entire FCG test duration. Figure 5b presents the trend of COD values V corresponding to 95% of the maximum baseline load Pmax with respect to N for a specimen with B = 15 mm under TC3. A detailed plot for N = 1 to 40,000 cycles is shown in Figure 5c. It is apparent that even during the initial stages of crack growth, the COD values clearly reflect the effects of the applied overloads.
Figure 6 presents the relationship curves between a and N for specimen #1 under each test condition across three different B. It is evident that smooth V-N and corresponding a-N curves are obtained under CA loading. In contrast, under VA loading, crack propagation is significantly influenced by overloads, resulting in retardation. This leads to noticeable abrupt changes in both the V-N and a-N curves. Consequently, the FCG rate da/dN data can be obtained using the incremental polynomial method. It is important to note that for calculating the FCG rate under CA loading, the a-N curve is sufficiently smooth, allowing the window center of the incremental polynomial to increase continuously from Nj = 4 to Nj = Nf − 4 before failure. However, for VA loading, calculating the FCG rate is not feasible due to the evident discontinuities in the a-N curve at overload positions. Therefore, the secant method is employed to determine da/dN at the overload point NOL, while the window centers for data points within individual fitting windows before and after NOL are selected as Nj = NOL − 4 and Nj = NOL + 3, respectively. In fact, the window size of the incremental polynomial also affects the calculation results; in this study, a window size of 7, which is the most commonly used, was selected. The resulting da/dNK curves are shown in Figure 7. However, the use of the 7-point incremental polynomial method prevents the effective utilization of the last three data points in the a-N data cluster (corresponding to the final approximately 300 load cycles before specimen failure in this study). This is also why the da/dNK curve under CA loading in Figure 7a does not exhibit a significant unstable crack growth stage. If the final three data points in the a-N data cluster were processed using successively reduced window sizes or the secant method, the unstable crack growth stage would become clearly observable.
Moreover, by comparing the da/dNK curves for the three thicknesses under CA loading in Figure 7, it can also be observed that as the specimen thickness increases, the unstable crack growth stage is reached at lower ΔK values. In reality, crack propagation is influenced by various factors, including microstructural effects and crack deflection, which may accelerate or decelerate the local crack growth rate. Even if the a-N curve appears smooth, such effects can still be reflected in the da/dNK curve. However, the overall crack growth rate during the stable propagation stage still conforms to the Paris law, as shown in Figure 7a. In comparison, da/dNK curves under VA loading exhibit noticeable discontinuities, with a significant overload retardation effect, and the overall crack growth rate is lower than the FCG rate under CA loading.
Figure 8 further illustrates the variation of the FCG rate da/dN with a during the final overload event for specimen #1 under TC3 with B = 10 mm. For the 7055-T7751 Al alloy, five typical stages can be observed under overload conditions [26,28], with schematic annotations of the corresponding stages shown in Figure 8:
(i)
Stable period before overload: In this stage, the FCG behavior is consistent with that under CA loading. The FCG rate da/dN can still be calculated and predicted by the Paris equation.
(ii)
Instantaneous acceleration period: Following the application of overload, the crack tip undergoes blunting, and the crack closure level established under the baseline loading immediately decreases. Concurrently, the applied overload POL, which exceeds the maximum baseline cyclic load Pmax, results in an effective SIF range ΔKeff higher than that before the overload (ΔKeff = KmaxKop, where Kop is the SIF at crack opening). Consequently, the crack tip propagates at an increased FCG rate. It is worth noting that the observable portion of this stage, as indicated by the area enclosed by the yellow dashed lines in Figure 8, was generated during a single overload cycle. This is because no fatigue striations were observed in the localized examination of this region, and its characteristics are consistent with those of static fracture. In other words, under a single tensile overload during periodic loading, this stage corresponds to only one overload cycle, after which the subsequent loading transitions directly into Stage iii.
(iii)
Delayed retardation period: After the overload is applied, the crack tip enters the overload plastic zone. The compressive residual stresses generated in the crack tip region during the overload process appear in the wake of the crack tip, leading to a rapid increase in the crack closure level. This causes ΔKeff to decrease swiftly, reaching the minimum FCG rate.
(iv)
Retardation period: As the crack continues to propagate further within the overload plastic zone, Kop decreases, and ΔKeff gradually increases, resulting in a gradual rise in da/dN. Due to the inherently three-dimensional nature of the crack opening, Kop in the plane strain region at the crack front is lower than that in the plane stress region on the specimen surface [34]. Therefore, the crack initially begins to open at the mid-thickness of the specimen and then extends towards the specimen surface, concurrently propagating forward.
(v)
Stable period after overload: Once the crack propagation extends beyond the influence range of the overload, the FCG rate returns to that under CA loading conditions. The hysteretic process induced by the single overload event concludes.
Figure 8. Localized da/dN-a curve of 10 mm thickness specimen at TC3 with 5 stages of overload.
Figure 8. Localized da/dN-a curve of 10 mm thickness specimen at TC3 with 5 stages of overload.
Metals 15 00644 g008
Previous studies [28] have indicated that the minimum FCG rate (da/dN)min after an overload decreases with increasing ROL and the crack length at the application of overload aOL. Figure 9 presents the relative deviation (RD) between (da/dN)min and the baseline FCG rate before overload (da/dN)BL. According to the experimental results obtained in this study, for specimens with all three thicknesses, the (da/dN)min after the application of overload indeed decreases as ROL increases. This is reflected in the RD, where the deviation of (da/dN)min from (da/dN)BL under 1.7 ROL is significantly higher than under 1.4 ROL. Specifically, the average RD for each overload event under 1.7 ROL is 94.8%, compared to 50.3% under 1.4 ROL. However, (da/dN)min does not exhibit a straightforward relationship with aOL. In contrast, based solely on the values of da/dN, (da/dN)min tends to increase with increasing aOL. Except for the 5 mm thick specimens under near-plane stress conditions and low overload ratios, there is no significant correlation between RD and aOL, and no apparent relationship with specimen thickness B. Therefore, (da/dN)min should be closely related to ΔKBL and ROL and show no significant correlation with aOL, at least under plane strain conditions.

3.2. Delay Distance ad

From Figure 8, it is evident that the crack length a calculated using the compliance method represents an average level of crack size describing the morphology of the crack front along the entire thickness direction. We define the delay distance ad length as the sum of the crack growth sizes during stages ii, iii, and iv in crack propagation after overload, that is,
a d = a ii + a iii + a iv
where aii, aiii, and aiv are obtained through experimental data in this study. aii represents the crack growth increment from the crack size right before overloading to the crack size corresponding to (da/dN)max after overloading. aiii represents the crack growth increment from the crack size corresponding to the (da/dN)max after overloading to the crack size corresponding to the strongest retardation effect ((da/dN)min). aiv represents the crack growth increment from the crack size corresponding to (da/dN)min to the crack size where the FCG rate gradually recovers to the pre-overloading FCG rate.
Overload-induced plastic zone size rp,OL at the crack tip is crucial for evaluating the FCG behavior influenced by overloads [20,26,28,35]. Based on the von Mises yield criterion and the principle of stress relaxation, Irwin [36] provided a two-dimensional analytical method for calculating the monotonic plastic zone size in a finite-width center-cracked plate. The calculation of the overload plastic zone size rpOL,I is presented in Equation (5),
r pOL , I = 1 α 0 π K OL σ y 2
Here, the value of α0 depends on the stress state, with α0 = 1 under plane stress and α0 = 2 2 under plane strain conditions. KOL represents the SIF at the crack tip under overload, and σy is the material’s yield strength.
Dugdale [37], based on fracture mechanics theory and by considering the plastic yielding characteristics at the crack tip, also provided the size of the plastic zone. The corresponding overload plastic zone size rpOL,D can be expressed as
r pOL , D = a sec π σ OL 2 σ y 1
In this study, under all test conditions, we have σOL/σy < 0.095. For small-scale yielding conditions, the overload plastic zone size rpOL,D can be further expressed as
r pOL , D = π 8 α 0 K OL σ y 2
Shin [38] considered strain hardening in a manner consistent with the small-scale yielding anti-plane shear solution by introducing the coefficient (1 − n)/(1 + n). Li et al. [22] utilized this coefficient to adjust the plastic zone size. The corresponding overload plastic zone size rpOL,S can be expressed as
r pOL , S   = 1   -   n 1   +   n 1 π K OL σ y 2
where the strain hardening index n is calculated using the Ramberg–Osgood [39] constitutive relation,
ε ture   = σ ture   E + σ ture   K n 1 / n
where E is the elastic modulus, Kn is the strength coefficient, and σtrue and εtrue represent true stress and true strain, respectively. Figure 1 illustrates the true stress–strain curve of 7055 Al alloy. Based on data from the uniform plastic deformation stage between the yield point and the necking point, the strain hardening index n = 0.101, Kn = 731.496, and the coefficient of determination R2 = 0.996 were calculated.
Since the overload plastic zone is formed during the monotonic tensile process of the overload load and affects subsequent fatigue crack propagation, Equations (5), (7) and (8) provide the expressions for the monotonic plastic zone. Under cyclic loading, the fatigue process zone is formed within the monotonic and cyclic plastic zones. Therefore, when studying the stress–strain state and fracture conditions near stress concentrators or crack tips, many researchers utilize concepts such as material structural parameters [40], critical distance [41], or the fatigue process zone [42] to characterize and control the material’s resistance to fatigue crack initiation and propagation in structural materials. However, regardless of whether it is the monotonic or cyclic plastic zone size, it can be observed that the calculation methods consider the plastic zone size as a function of KOLy. Therefore, based on the experimental results and Equations (2) and (4), we can obtain the ad-KOLy curves, as shown in Figure 10. The fitted curves for each dataset, along with the calculated results from the Irwin, Dugdale, and Shin models, are also presented in the figure.
It can be observed that the relationship between ad and KOLy still adheres to a power–law relationship, meaning that it exhibits a linear relationship on a double logarithmic scale, and can be expressed as
a d = C d   K OL / σ y m d
where Cd and md are the coefficients and exponents in the power–law relationship, respectively, with the corresponding parameter fitting results listed in Table 2. However, unlike the exponent value of 2 in the aforementioned three formulas describing the rp at the crack tip, the exponent in the power–law relationship between the delay distance ad and KOLy has average values of 4.14 and 3.70 for three different thicknesses when ROL = 1.4 and 1.7, respectively, with CV of 0.68% and 2.04%. This indicates that variations in ROL result in a more significant change in the average exponent md, whereas under the same ROL value, the exponent data for specimens with different B are closely clustered around their respective means with low dispersion. This suggests that the thickness B has a less significant influence on the exponent compared to the overload ratio ROL. The Mean Absolute Percentage Error (MAPE) between rp predicted by the Irwin, Dugdale, and Shin models and the experimental results as a function of KOLy is illustrated in Figure 11. This indicates that when describing ad using the Irwin and Dugdale models, the MAPE initially decreases and then increases with KOLy, with errors within 50% for ranging from 0.055 to 0.076. In contrast, for the Shin model, the error decreases to within 50% when KOLy exceeds 0.073. For all three plastic zone calculation models, the MAPE values at 1.7 ROL are generally lower than those at 1.4 ROL.
We hypothesize that this discrepancy arises from two factors. Firstly, the presence of second phases, substructures, and grains [43] in 7055 Al alloy causes anisotropy at the crack tip on a microscopic scale. This differs from the closed-form approximate solutions for the plastic zone size ahead of long cracks obtained under monotonic loading in uniform and isotropic non-hardening metals. Secondly, current plastic zone models typically represent the plastic zone size as the distance from the crack tip to the plastic boundary along the crack propagation direction [44]. However, the morphology of the crack front and the shape of the plastic zone are irregular both along the thickness direction and the crack propagation direction. Therefore, the delay distance ad influenced by overload should be more closely related to the plastic zone area, which is a possible reason why the exponent md exceeds 2.
Assuming that the parameters Cd and md are related to the power function of ROL and B, respectively, i.e.,
C d = c 0 R OL c 1 B c 2 m d = 2 + b 0 R OL b 1 B b 2
where c0, c1, c2, b0, b1, and b2 are the parameters to be solved. Taking logarithms on both sides of Equation (11) simultaneously, the equation can be further transformed into a linear form shaped like Y = C0 + C1X1 + C2X2 and expressed as a matrix form Y = XC. The expressions for the parameters Cd and md are then obtained by the least-squares method C = (XTX)−1XTY as follows:
C d =   e 11.051 R OL 4.822 B 0.622 m d = 2 + e 1.073 R OL 1.198 B 0.041
For the parameters Cd and md, the coefficients of determination R2 are 0.963 and 0.965, respectively. Three-dimensional scatter plots and the fitted surfaces are given in Figure 12, which indicate a high agreement between the test data points and the fitted surfaces, further verifying the validity of the high R2. The relationship in Equation (12) shows that the correlations of the parameters Cd and md with the overload ratio ROL are significantly higher than that for the thickness B, which is consistent with the observation of the experimental results.

3.3. Instantaneous Acceleration Period

When an overload is applied to the specimen, the overload not only induces crack tip blunting but also causes the crack to propagate rapidly under a higher crack tip SIF, as illustrated in Figure 8. We compiled the crack propagation increments aii of specimen #1 after each overload application under all test conditions and posit that they satisfy the following relationship with the SIF range ΔK:
a ii = exp λ 1 +   λ 2 Δ K   +   λ 3 Δ K   2
where λ1, λ2, and λ3 are material parameters determined from the tests. The experimental data and their fitting results are shown in Figure 13. The values of λ1, λ2, and λ3 are −5.973, 0.149, and 0.004 for operating condition TC2 (ROL = 1.4), while the values of the three parameters are −6.099, 0.236, and 0.004 for TC3 (ROL = 1.7), and the parameter correlations are higher than 0.99. The values of the parameters are −6.099, 0.236, and 0.004, and the parameter correlation is higher than 0.99. The values of the parameters are −6.099, 0.236, and 0.004, and the parameter correlations are higher than 0.99.
In most studies analyzing FCG behavior under tensile overloads, the analysis of the instantaneous acceleration phase is often overlooked. However, in reality, the crack propagation increments during this period inevitably have a significant impact on the overall FCG life of the specimen. For instance, in this study, the crack growth rate corresponding to the overload is calculated using the secant method, where the denominator is the number of load cycles between data sampling points (100 cycles). As a result, the instantaneous crack growth rate after overloading, (da/dN)OL, on the da/dNK curve differs numerically from the experimentally measured instantaneous crack growth increment, aii, by approximately a factor of 100. Thus, by calculating the ratio of the FCG rates before and after each overload point, the normalized instantaneous crack growth increment can be obtained. For a given specimen, this ratio can be calculated for each overload throughout the entire crack growth lifetime, and the average value can be determined. This average value approximately reflects the average level of aii caused by each overload under the given conditions for specimens of this thickness. We determined the multiplication factors by which this average level under TC3 increases relative to TC2 for specimens with thicknesses of 5 mm, 10 mm, and 15 mm based on experimental results, which are 2.72, 7.09, and 4.08, respectively. In other words, when subjected to an overload ratio of 1.7 ROL, the crack propagation increment aii of the 10 mm thick specimen is significantly higher than those of the other two thickness specimens. This results in a shorter FCG life under conditions where the deviation degree RD between (da/dN)min and (da/dN) BL is relatively consistent.

3.4. Delay Cycles Nd

McEvily and Ishihara [45] provided a definition for the number of delay cycles Nd after an overload, which is the total number of cycles required for the FCG rate to regain its initial value minus the number of cycles needed to traverse the distance over which retarded growth occurred at the baseline rate. Li et al. [28] determined Nd by measuring the horizontal displacement between the two a-N curves under VA and CA loading. This approach is essentially consistent with the definition provided by McEvily.
To further illustrate the definition of Nd, the local a-N curve of a 10 mm thick specimen under TC3 is presented in Figure 14. When the slope of the a-N curve after the overload matches the da/dN before the overload, it can be considered that the crack propagation rate has returned to its pre-overload level. At this point, the experimental value of Nd can be determined.
Wheeler [46] describes the retardation behavior of crack extension after overloading by introducing the retardation coefficient Cp as shown in Equation (14):
  d a d N   VA = C p C Δ K m
where based on the crack extension test data under CA loading in Figure 5, the material-dependent constants C and m in the Paris model can be obtained as shown in Table 3. The retardation coefficient Cp is a function related to the size of the plastic zone as follows:
C p = r p a OL + r p , OL a m ,     a + r p <     a OL + r p , OL 1 ,     a + r p     a OL + r p , OL
where a is the current crack length, aOL is the crack length at the overload applied, rp is the plastic zone size under the current crack length a, rp,OL is the plastic size when the overload is applied, and m′ is a dimensionless shape parameter obtained by fitting the test data.
Based on the discussion in Section 3.2 and Section 3.3, the retardation coefficient Cp is corrected to account for the influence of aii and ad in the FCG behavior after overloading.
C p = r p a OL + a ii + r p , OL a m ,     a   <     a d + a OL     1 ,     a       a d + a OL
Therefore, after applying overloads, when a = aOL + aii, that is, after the instantaneous acceleration period, Cp′ obtains the minimum value, which corresponds to the (da/dN)min. When aaOL + ad, the crack expands beyond the range of overload influence, there is Cp′ = 1 and the retardation effect disappears. Then, according to the definition, the number of delay cycles Nd can be expressed as
N d = a OL a OL + a d 1   C p C p C Δ K m   d a
The calculation methods for aii and ad are provided in Equations (10) and (13), respectively. The model was validated using FCG data from specimen #3 under VA loading conditions, with experimental results and average fitted results presented in Figure 15a. Nd varies between 32 and 494 cycles under 1.4 ROL, whereas under 1.7 ROL ranges from 599 to 3700 cycles. This indicates that the number of Nd, which characterizes the degree of overload-induced retardation, increases with both ROL and ΔK. Additionally, Nd shows a certain upward trend with increasing B, which is more pronounced at 1.7 ROL compared to 1.4. Furthermore, additional crack propagation experiments were conducted on 5 mm thick C(T) specimens of 7055 Al alloy with overload interval cycles Nb of 20,000 and 50,000 cycles, respectively. The selection of overload cycles ensured the integrity of the crack retardation process, with 1.7 ROL, and the average FCG lives were 77,891 and 97,496 cycles, respectively. At this point, different overload interval cycles Nb correspond to different ΔK or aOL values at the time the overload is applied. The experimental and fitted results are shown in Figure 15b. The results indicate that for 7055 aluminum alloy specimens with thicknesses of 5 mm, 10 mm, and 15 mm under periodic overload conditions, the proposed model can effectively capture the trend of the delayed cycles Nd affected by the overload as a function of ΔK.

3.5. Characteristics of the Fracture Surface

By observing the fracture morphology of C(T) specimens using the VHX-1000 high-depth-of-field three-dimensional microscopy system, we can discern the variation patterns of the crack front during crack propagation under cyclic overloads, as illustrated by the macroscopic failure surface of a 15 mm thick specimen under 1.7 ROL in Figure 16. After the crack extends to 51.2 mm (ΔK = 14.7 MPa·m1/2), the influence of the overload becomes clearly observable on the fracture surface. Prior to this extension, higher-magnification microscopy is required to effectively observe the effects of the overload. Additionally, as shown in the figure, unlike specimens of the other two thicknesses, the crack front of the 15 mm thick specimen after overload does not exhibit a crescent-shaped arc but instead displays a double-peaked morphology. Based on the stop line morphology between the pre-cracking and crack propagation stages, it can be inferred that this crack front morphology has persisted since the first overload application and becomes more pronounced during the last three overload cycles. This may be related to the microstructure of the material, which leads to a redistribution of stress at the crack tip along the thickness direction of the crack front under overload conditions. Moreover, the larger the ΔK, the greater the curvature of the crack front, meaning the radius of curvature is smaller. This is because the specimen surface is under plane stress conditions, while the interior is under plane strain conditions. Under plane strain, the crack tip experiences a triaxial stress state, making the material less prone to plastic deformation and facilitating easier crack propagation [30].
Furthermore, as can be clearly observed from Figure 16, with an increase in aOL, that is, ΔK, the distance of the crack front in the direction of crack propagation after the application of an overload also increases. In other words, the crack propagation increment aii during the instantaneous acceleration period exhibits an upward trend, which is consistent with the trend of data variation in Equation (13). Similarly, the area of the damage zone induced by the overload also increases significantly. This further indicates that characterizing the affected delay length ad using only the closed-form approximations based on the size of the plastic zone ahead of a long crack as shown in Equations (5) and (7) is limited. Instead, it should be related to the area, as expressed by the exponential term md in Equation (10).
Under CA loading, the failure surface of the specimen is smoother and more uniform. This is because, under CA loading, the crack faces repeatedly open and close, and the friction between the faces leads to a more planar, smooth, and glossy fracture morphology during the stable crack propagation phase. As shown in Figure 17a, the fracture surface clearly exhibits ductile fracture features such as wavy patterns and dimples, as well as arcuate fatigue striations perpendicular to the crack propagation direction. Ductile fracture occurs through the interactions of void nucleation, void growth, and deformation processes between adjacent voids [47]. Fatigue striations are traces of plastic deformation at the instantaneous crack front during FCG. Under VA loading, fatigue striations can also be observed on the failure surfaces before and after the application of an overload, as shown in Figure 17b. The spacing between the striations gradually increases with an increase in ΔK, and this effect is more pronounced under overload conditions. In the damage zone formed during the application of an overload, fatigue striations are not observable, and the characteristics of ductile fracture are diminished. Instead, quasi-cleavage faceting and voids are more frequently observed, and the microscopic surface becomes flatter. This is because the local stress exceeds the fracture resistance of the Al alloy after the application of an overload, causing the local material to be rapidly pulled apart and accelerating crack propagation. Additionally, as seen in the figure, the application of an overload significantly increases the probability of secondary crack formation. The fatigue striations in Figure 17b are further given in Figure 17c. Figure 17d presents the fracture morphology of a 10 mm thick specimen under the final overload phase at 1.7 ROL after the application of the overload. A clear boundary can be observed between the damage zone formed by the overload and the subsequent crack propagation phase influenced by retardation. Beyond this boundary, the features of ductile fracture become more pronounced.

4. Conclusions

This study conducted FCG tests on 7055-T7751 Al alloy under constant amplitude and periodic tensile overload conditions, considering different overload ratios and specimen thicknesses. The post-test data and fatigue fracture surfaces were analyzed, and the main conclusions are as follows:
1.
All specimens of the studied material exhibited an increased FCG life under periodic overload conditions compared to their CA counterparts. The larger the ROL, the greater the relative deviation degree RD of (da/dN)min from the FCG rate before overload. For 7055 aluminum alloy, the average RD values at 1.4 and 1.7 ROL are 50.3% and 94.8%, respectively, while the experimental data reveal that there is no significant correlation between this value and either the specimen thickness B or the crack length at overload aOL.
2.
The experimental results confirm that the crack growth increment aii during the instantaneous acceleration period plays a significant role in the assessment of FCG life under overload conditions. Periodic overload makes the cumulative impact of this period on FCG life more pronounced. An empirical model for the crack growth increment in this FCG was established and successfully fitted the experimental data. Furthermore, it was found that aii increases with the overload ratio ROL and the SIF range ΔK, while there is no significant correlation with specimen thickness B.
3.
In this research, the MAPE between the plastic zone size calculation models proposed by Irwin, Dugdale, Shin and the experimental data for ad was assessed as a function of KOLy. The results indicate that ad follows a power–law relationship with KOLy, but the exponent bd exceeds 2. The established evaluation model for ad, which accounts for specimen thickness B and overload ratio ROL, demonstrates greater predictive accuracy for ad compared to the plastic zone models. Finally, by incorporating ad and aii into the Wheeler model, a calculation method for Nd was developed. The calculated Nd values under 1.4 and 1.7 ROL conditions show good agreement with the experimental results. Under periodic overload conditions, both ad and Nd exhibit an increasing trend with the elevation of ROL and B.
4.
Specimens with thickness B = 15 mm exhibit a double-peaked crack front morphology under overload conditions, which is inconsistent with the morphology observed in specimens with thicknesses of B = 5 mm and 10 mm. In all three thicknesses, numerous secondary cracks were observed in the fracture morphologies before and after the application of overload at 1.4 and 1.7 ROL. The ductile fracture characteristics within the overload damage zone are significantly weaker than those in regions subjected to fatigue loading, thereby providing a means to distinguish the boundaries of overload initiation and termination.

Author Contributions

Conceptualization, Z.L. and W.Y.; methodology, Z.L.; software, Z.L. and J.C.; validation, Y.Y., S.L. and W.Y.; formal analysis, Z.L.; investigation, S.L.; resources, J.C.; data curation, J.C. and S.L.; writing—original draft preparation, Z.L.; writing—review and editing, W.Y.; visualization, Y.Y.; supervision, W.Y.; project administration, W.Y.; funding acquisition, W.Y. All authors have read and agreed to the published version of the manuscript.

Funding

This research was funded by the National Natural Science Foundation of China, grant numbers 52235003 and 52075244.

Data Availability Statement

The original contributions presented in this study are included in the article. Further inquiries can be directed to the corresponding author.

Acknowledgments

The financial support by the National Natural Science Foundation of China is gratefully acknowledged. Furthermore, the authors would like acknowledge the industrial partners Commercial Aircraft Corporation of China Ltd. (Shanghai, China) and Southwest Aluminum (Group) Co., Ltd. (Chongqing, China), for the excellent mutual scientific collaboration, as well as eceshi (Zhengzhou, China, www.eceshi.com) for the ICP-MS testing.

Conflicts of Interest

The authors declare no conflicts of interest. The funding sponsors had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results.

Abbreviations

The following abbreviations are used in this manuscript:
acrack length, mm
addelay distance, mm
anmachined notch length, mm
aOLcrack length at overload application, mm
aiithe amount of crack propagation during instantaneous acceleration, mm
aiiithe amount of crack propagation during the delayed retardation period, mm
aivthe amount of crack propagation during the retardation period, mm
Bspecimen thickness, mm
Ccoefficient in the Paris formula
Cdcoefficient in the ad expression
Cpretardation parameter in the Wheeler formula
CODddelay crack opening distance, mm
da/dNcrack growth rate, mm/cycle
(da/dN)minminimum crack growth rate after overload application, mm/cycle
(da/dN)BLcrack growth rate before overload application, mm/cycle
(da/dN)OLinstantaneous accelerated crack growth rate after overload, mm/cycle
fload frequency, Hz
Kstress-intensity factor, MPa·m1/2
KOLoverload stress intensity factor, MPa·m1/2
Kopcrack opening stress intensity factor, MPa·m1/2
mexponent in the Paris formula
mexponent in the Cp expression
mdexponent in the ad expression
Nnumber of loading cycles
Nbnumber of overload interval cycles
Ndnumber of delay cycles
Nfnumber of cycles to failure
Njnumber of cycles of the increment polynomial’s window center
NOLnumber of cycles elapsed when overload applied
Nsnumber of overload cycles applied
Papplied load, kN
Pmaxapplied baseline maximum load, kN
Pmax,imaximum value of cyclic load at stage i during pre-cracking, kN
Pminapplied baseline minimum load, kN
POLapplied overload, kN
rpplastic zone dimensions, mm
rp,OLplastic zone dimensions under the applied overload, mm
Rstress ratio (σmin/σmax)
R2coefficient of determination
ROLoverload ratio (σOL/σmax)
sNstandard deviation of experimental FCG life data, cycles
Vcrack opening displacement, mm
Wdimension relative to the width of the specimen, mm
Δaiincrement of crack propagation in stage i of pre-cracking, mm
ΔKrange of stress-intensity factor (KmaxKmin), MPa·m1/2
ΔKBLrange of baseline stress-intensity factor, MPa·m1/2
αnormalize crack length (a/W)
α0coefficient related to the stress state in rp calculations
β(a)shape factor related to the specimen’s geometric characteristics
εtruetrue strain, mm/mm
σstress applied, MPa
σastress amplitude, MPa
σbultimate tensile stress, MPa
σytensile yield stress, MPa
σmaxstress under applied baseline maximum load, MPa
σminstress under applied baseline minimum load, MPa
σOLstress under applied overload, MPa
σtruetrue stress, MPa
Alaluminum
ASTMAmerican society for testing and materials
CAconstant amplitude
CVcoefficient of variation
C(T)Compact-tension specimen
EDMelectrical discharge machining
FCGfatigue crack growth
ICP-MSinductively coupled plasma mass spectrometry
L-Tlongitudinal-transverse
MAPEmean absolute percentage error
RDrelative deviation between (da/dN)min and (da/dN)BL
VAvariable amplitude

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Figure 1. Stress–strain curves of 7055-T7751 Al alloy.
Figure 1. Stress–strain curves of 7055-T7751 Al alloy.
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Figure 2. (a) Geometry dimensions of C(T) specimen sample, and (b) C(T) specimen of 7055-T7751 Al alloy.
Figure 2. (a) Geometry dimensions of C(T) specimen sample, and (b) C(T) specimen of 7055-T7751 Al alloy.
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Figure 3. Servo hydraulic fatigue testing apparatus with C(T) specimen fixture and specimen.
Figure 3. Servo hydraulic fatigue testing apparatus with C(T) specimen fixture and specimen.
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Figure 4. Loading sequence: (a) CA loading; (b) VA loading.
Figure 4. Loading sequence: (a) CA loading; (b) VA loading.
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Figure 5. (a) P-V curves of 10 mm thickness specimen at TC2, (b) V-N curve of 15 mm thickness specimen at TC3, and (c) its localized V-N curve.
Figure 5. (a) P-V curves of 10 mm thickness specimen at TC2, (b) V-N curve of 15 mm thickness specimen at TC3, and (c) its localized V-N curve.
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Figure 6. a-N curves of (a) B = 5 mm, (b) B = 10 mm, (c) B = 15 mm specimen thickness, for specimen #1.
Figure 6. a-N curves of (a) B = 5 mm, (b) B = 10 mm, (c) B = 15 mm specimen thickness, for specimen #1.
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Figure 7. da/dNK curves of (a) B = 5 mm, (b) B = 10 mm, (c) B = 15 mm specimen thickness, for specimen #1.
Figure 7. da/dNK curves of (a) B = 5 mm, (b) B = 10 mm, (c) B = 15 mm specimen thickness, for specimen #1.
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Figure 9. The variation of RD between (da/dN)min and (da/dN)BL with aOL for 3 thicknesses of #1 specimens.
Figure 9. The variation of RD between (da/dN)min and (da/dN)BL with aOL for 3 thicknesses of #1 specimens.
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Figure 10. Comparison of ad-KOLy curves for specimens under VA loads: experimental data, fitted curves, and predictions from Irwin, Dugdale, and Shin models.
Figure 10. Comparison of ad-KOLy curves for specimens under VA loads: experimental data, fitted curves, and predictions from Irwin, Dugdale, and Shin models.
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Figure 11. Variation of MAPE with KOLy for Irwin, Dugdale, and Shin Models under TC2 and TC3.
Figure 11. Variation of MAPE with KOLy for Irwin, Dugdale, and Shin Models under TC2 and TC3.
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Figure 12. Three-dimensional fitted surface for (a) parameter Cd and (b) parameter md.
Figure 12. Three-dimensional fitted surface for (a) parameter Cd and (b) parameter md.
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Figure 13. Correlation between aii and ΔK: experimental results and fitted curves.
Figure 13. Correlation between aii and ΔK: experimental results and fitted curves.
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Figure 14. Definition of the number of delay cycles Nd and delay distance ad.
Figure 14. Definition of the number of delay cycles Nd and delay distance ad.
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Figure 15. The relationship between Nd and ΔK for (a) specimen #3 under TC2 and TC3 with different thicknesses and (b) 5 mm specimens under 1.7 ROL with different Nb.
Figure 15. The relationship between Nd and ΔK for (a) specimen #3 under TC2 and TC3 with different thicknesses and (b) 5 mm specimens under 1.7 ROL with different Nb.
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Figure 16. Macroscopic failure surface of 15 mm thickness specimen at TC3.
Figure 16. Macroscopic failure surface of 15 mm thickness specimen at TC3.
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Figure 17. Fractography of 10 mm thickness specimens at (a) TC1, (b) TC2 with its (c) localized fatigue striations, and (d) TC3.
Figure 17. Fractography of 10 mm thickness specimens at (a) TC1, (b) TC2 with its (c) localized fatigue striations, and (d) TC3.
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Table 1. FCG test results of specimens by the no. of cycles to failure.
Table 1. FCG test results of specimens by the no. of cycles to failure.
TC IdOverload Ratio, ROLInterval Cycles, Nb (Cycles)B = 5 mmB = 10 mmB = 15 mm
FCG   Life ,   N ¯ f (Cycles)CV FCG   Life ,   N ¯ f (Cycles)CV FCG   Life ,   N ¯ f (Cycles)CV
TC1N/AN/A71,6107.23%83,6546.07%97,0579.25%
TC21.410,00075,1500.11%128,6336.07%134,9033.03%
TC31.710,000101,4212.48%112,6917.99%170,0008.32%
Table 2. Parameter fitting results between ad and KOLy.
Table 2. Parameter fitting results between ad and KOLy.
Thickness BCd × 103/mm·m−b/2mdR2
TC2TC3TC2TC3TC2TC3
5 mm36.82513.7714.10323.66250.98420.9637
10 mm48.85614.0954.15323.62670.92000.9369
15 mm67.95230.8474.16943.80140.98790.9716
Table 3. Parameter fitting results for the Paris formula.
Table 3. Parameter fitting results for the Paris formula.
Thickness BlnCmR2
5 mm−14.5362.5530.977
10 mm−14.1062.3250.948
15 mm−12.8331.8850.947
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Liu, Z.; Cao, J.; Liu, S.; Yang, Y.; Yao, W. The Overload-Induced Delay Model of 7055 Aluminum Alloy Under Periodic Overloading. Metals 2025, 15, 644. https://doi.org/10.3390/met15060644

AMA Style

Liu Z, Cao J, Liu S, Yang Y, Yao W. The Overload-Induced Delay Model of 7055 Aluminum Alloy Under Periodic Overloading. Metals. 2025; 15(6):644. https://doi.org/10.3390/met15060644

Chicago/Turabian Style

Liu, Zuoting, Jing Cao, Shilong Liu, Yuqi Yang, and Weixing Yao. 2025. "The Overload-Induced Delay Model of 7055 Aluminum Alloy Under Periodic Overloading" Metals 15, no. 6: 644. https://doi.org/10.3390/met15060644

APA Style

Liu, Z., Cao, J., Liu, S., Yang, Y., & Yao, W. (2025). The Overload-Induced Delay Model of 7055 Aluminum Alloy Under Periodic Overloading. Metals, 15(6), 644. https://doi.org/10.3390/met15060644

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