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

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 R_OL, respectively. An analytical model for the crack growth increment a_ii during this period was developed. Additionally, the delay distance influenced by overloads a_d and the number of delay cycles N_d 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 R_OL and specimen thicknesses B revealed that existing plastic zone models inadequately assess a_d, prompting the establishment of a corresponding evaluation model. By incorporating the parameters a_ii and a_d into the Wheeler model, a method for calculating the delay cycles N_d 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 R_OL [4], overload interval N_b [5,6], number of overloads N_s [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 σ_a, 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 a_d and delay cycles N_d affected by crack retardation are common quantitative parameters used to measure the degree of retardation following an overload. Typically, as the overload ratio R_OL increases, both a_d and N_d also increase [4]. Vecchio et al. [23] were the first to define N_d 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 N_d 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) ΔK_BL, overload ratio R_OL, and stress ratio R. They obtained an inverted U-shaped relationship curve, providing the explanation that the maximum retardation is associated with an optimal ΔK_BL. Under the applied values of 16, 19, and 22 $\mathrm{MPa}\sqrt{\mathrm{m}}$, crack closure did not undergo significant changes. Consequently, the effective FCG driving force ΔK and the delay cycles N_d compete with each other. The study also found that the degree of overload retardation increases with increasing R_OL and decreases with increasing R, while its relationship with the baseline SIF ΔK_BL 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 N_d 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 N_d 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 R_OL and the crack length at overload application a_OL 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 N_d increase with R_OL. 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 N_d and distance a_d, 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 N_d and a_d with cyclic loading SIF ΔK, overload ratio R_OL, 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 N_d is established. Different methods are employed to calculate the plastic zone size, and the effects of these methods on the characterization of a_d 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 σ_y of 419 MPa, an ultimate tensile stress σ_b 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 a_n = 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, P_max,i, did not decrease by more than 20% relative to the maximum cyclic load of the preceding stage P_max,i−1. The fifth-level load was maintained consistent with the load used in the FCG tests, i.e., P_max,5 = P_max. Additionally, the crack extension increment Δa_i at each level satisfies [32],

$$\Delta a_{\mathrm{i}} \ge {\displaystyle \frac{3}{\pi }}{\left({\displaystyle \frac{K_{\max , \mathrm{i}-1}}{{\sigma }_{\mathrm{y}}}}\right)}^2$$

(1)

where K_max,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 E ≤ E′ ≤ 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

$$K=\sigma \cdot \mathit{\beta }\left(a\right)\sqrt{\pi a}$$

(2)

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],

$$\mathit{\beta }\left(a\right)={\displaystyle \frac{\left(2+\alpha \right)}{4\sqrt{\pi }}}\times {\displaystyle \frac{\left(0.886+4.64\alpha -13.32{\alpha }^2+14.72{\alpha }^3-5.6{\alpha }^4\right)}{{\alpha }^{1/2}{\left(1-\alpha \right)}^{3/2}}}$$

(3)

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 R_OL = P_OL/P_max is defined as the ratio between the applied overload P_OL and the maximum baseline cyclic load P_max. The values of P_max 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 R_OL 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 N_b was set to 10,000 cycles, that is, N_OL = 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. FCG test results of specimens by the no. of cycles to failure.

TC Id	Overload Ratio, ROL	Interval Cycles, Nb (Cycles)	B = 5 mm		B = 10 mm		B = 15 mm
			$\mathbf{FCG} \mathbf{Life}, {\overline{\mathit{N}}}_{\mathbf{f}}$ (Cycles)	CV	$\mathbf{FCG} \mathbf{Life}, {\overline{\mathit{N}}}_{\mathbf{f}}$ (Cycles)	CV	$\mathbf{FCG} \mathbf{Life}, {\overline{\mathit{N}}}_{\mathbf{f}}$ (Cycles)	CV
TC1	N/A	N/A	71,610	7.23%	83,654	6.07%	97,057	9.25%
TC2	1.4	10,000	75,150	0.11%	128,633	6.07%	134,903	3.03%
TC3	1.7	10,000	101,421	2.48%	112,691	7.99%	170,000	8.32%

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 = s_N/$\stackrel{\mathrm{-}}{\mathrm{n}}$), and s_N 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 R_OL of 1.7 is higher than that under 1.4 R_OL. 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 V_max and V_min 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 P_max 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 N_j = 4 to N_j = N_f − 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 N_OL, while the window centers for data points within individual fitting windows before and after N_OL are selected as N_j = N_OL − 4 and N_j = N_OL + 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/dN-ΔK 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/dN-ΔK 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/dN-ΔK 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/dN-ΔK 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/dN-ΔK 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 P_OL, which exceeds the maximum baseline cyclic load P_max, results in an effective SIF range ΔK_eff higher than that before the overload (ΔK_eff = K_max − K_op, where K_op 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 ΔK_eff to decrease swiftly, reaching the minimum FCG rate.

(iv)

Retardation period: As the crack continues to propagate further within the overload plastic zone, K_op decreases, and ΔK_eff gradually increases, resulting in a gradual rise in da/dN. Due to the inherently three-dimensional nature of the crack opening, K_op 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.

Previous studies [28] have indicated that the minimum FCG rate (da/dN)_min after an overload decreases with increasing R_OL and the crack length at the application of overload a_OL. 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 R_OL increases. This is reflected in the RD, where the deviation of (da/dN)_min from (da/dN)_BL under 1.7 R_OL is significantly higher than under 1.4 R_OL. Specifically, the average RD for each overload event under 1.7 R_OL is 94.8%, compared to 50.3% under 1.4 R_OL. However, (da/dN)_min does not exhibit a straightforward relationship with a_OL. In contrast, based solely on the values of da/dN, (da/dN)_min tends to increase with increasing a_OL. Except for the 5 mm thick specimens under near-plane stress conditions and low overload ratios, there is no significant correlation between RD and a_OL, and no apparent relationship with specimen thickness B. Therefore, (da/dN)_min should be closely related to ΔK_BL and R_OL and show no significant correlation with a_OL, at least under plane strain conditions.

3.2. Delay Distance a_d

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 a_d length as the sum of the crack growth sizes during stages ii, iii, and iv in crack propagation after overload, that is,

$$a_{\mathrm{d}}=a_{\mathrm{ii}}+a_{\mathrm{iii}}+a_{\mathrm{iv}}$$

(4)

where a_ii, a_iii, and a_iv are obtained through experimental data in this study. a_ii represents the crack growth increment from the crack size right before overloading to the crack size corresponding to (da/dN)_max after overloading. a_iii 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). a_iv 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 r_p,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 r_pOL,I is presented in Equation (5),

$$r_{\mathrm{pOL},\mathrm{I}}={\displaystyle \frac{1}{{\alpha }_0\pi }}{\left({\displaystyle \frac{K_{\mathrm{OL}}}{{\sigma }_{\mathrm{y}}}}\right)}^2$$

(5)

Here, the value of α₀ depends on the stress state, with α₀ = 1 under plane stress and α₀ = $2\sqrt{2}$ under plane strain conditions. K_OL represents the SIF at the crack tip under overload, and σ_y 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 r_pOL,D can be expressed as

$$r_{\mathrm{pOL},\mathrm{D}}=a\left[\sec \left({\displaystyle \frac{\pi {\sigma }_{\mathrm{OL}}}{2{\sigma }_{\mathrm{y}}}}\right)-1\right]$$

(6)

In this study, under all test conditions, we have σ_OL/σ_y < 0.095. For small-scale yielding conditions, the overload plastic zone size r_pOL,D can be further expressed as

$$r_{\mathrm{pOL},\mathrm{D}}={\displaystyle \frac{\pi }{8{\alpha }_0}}{\left({\displaystyle \frac{K_{\mathrm{OL}}}{{\sigma }_{\mathrm{y}}}}\right)}^2$$

(7)

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 r_pOL,S can be expressed as

$$r_{\mathrm{pOL},\mathrm{S} }={\displaystyle \frac{1 - n}{1 + n}}\cdot {\displaystyle \frac{1}{\pi }}{\left({\displaystyle \frac{K_{\mathrm{OL}}}{{\sigma }_{\mathrm{y}}}}\right)}^2$$

(8)

where the strain hardening index n is calculated using the Ramberg–Osgood [39] constitutive relation,

$${\epsilon }_{\mathrm{ture} }={\displaystyle \frac{{\sigma }_{\mathrm{ture} }}{E}}+{\left({\displaystyle \frac{{\sigma }_{\mathrm{ture} }}{K_{\mathrm{n}}}}\right)}^{1/n}$$

(9)

where E is the elastic modulus, K_n is the strength coefficient, and σ_true and ε_true 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, K_n = 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 K_OL/σ_y. Therefore, based on the experimental results and Equations (2) and (4), we can obtain the a_d-K_OL/σ_y 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 a_d and K_OL/σ_y still adheres to a power–law relationship, meaning that it exhibits a linear relationship on a double logarithmic scale, and can be expressed as

$$a_{\mathrm{d}}=C_{\mathrm{d}} {\left(K_{\mathrm{OL}}/{\sigma }_{\mathrm{y}}\right)}^{m_{\mathrm{d}}}$$

(10)

where C_d and m_d are the coefficients and exponents in the power–law relationship, respectively, with the corresponding parameter fitting results listed in

Table 2. Parameter fitting results between a_d and K_OL/σ_y.

Thickness B	Cd × 103/mm·m−b/2		md		R2
	TC2	TC3	TC2	TC3	TC2	TC3
5 mm	36.825	13.771	4.1032	3.6625	0.9842	0.9637
10 mm	48.856	14.095	4.1532	3.6267	0.9200	0.9369
15 mm	67.952	30.847	4.1694	3.8014	0.9879	0.9716

. However, unlike the exponent value of 2 in the aforementioned three formulas describing the r_p at the crack tip, the exponent in the power–law relationship between the delay distance a_d and K_OL/σ_y has average values of 4.14 and 3.70 for three different thicknesses when R_OL = 1.4 and 1.7, respectively, with CV of 0.68% and 2.04%. This indicates that variations in R_OL result in a more significant change in the average exponent m_d, whereas under the same R_OL 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 R_OL. The Mean Absolute Percentage Error (MAPE) between r_p predicted by the Irwin, Dugdale, and Shin models and the experimental results as a function of K_OL/σ_y is illustrated in Figure 11. This indicates that when describing a_d using the Irwin and Dugdale models, the MAPE initially decreases and then increases with K_OL/σ_y, 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 K_OL/σ_y exceeds 0.073. For all three plastic zone calculation models, the MAPE values at 1.7 R_OL are generally lower than those at 1.4 R_OL.

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 a_d influenced by overload should be more closely related to the plastic zone area, which is a possible reason why the exponent m_d exceeds 2.

Assuming that the parameters C_d and m_d are related to the power function of R_OL and B, respectively, i.e.,

$$\begin{gathered} C_{\mathrm{d}}=c_0\cdot R_{\mathrm{OL}}^{c_1}\cdot B^{c_2} \\ {m}_{\mathrm{d}}=2+b_0\cdot R_{\mathrm{OL}}^{b_1}\cdot B^{b_2} \end{gathered}$$

(11)

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 C_d and m_d are then obtained by the least-squares method C = (X^TX)⁻¹X^TY as follows:

$$\begin{gathered} C_{\mathrm{d}}={ e}^{11.051}\cdot R_{\mathrm{OL}}^{4.822}\cdot B^{0.622} \\ {m}_{\mathrm{d}}=2+e^{1.073}\cdot R_{\mathrm{OL}}^{-1.198}\cdot B^{0.041} \end{gathered}$$

(12)

For the parameters C_d and m_d, 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 C_d and m_d with the overload ratio R_OL 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 a_ii of specimen #1 after each overload application under all test conditions and posit that they satisfy the following relationship with the SIF range ΔK:

$$a_{\mathrm{ii}}=\exp \left({\lambda }_1+{ \lambda }_2\cdot \Delta K +{ \lambda }_3\cdot \Delta K^{ 2}\right)$$

(13)

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 (R_OL = 1.4), while the values of the three parameters are −6.099, 0.236, and 0.004 for TC3 (R_OL = 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/dN-ΔK curve differs numerically from the experimentally measured instantaneous crack growth increment, a_ii, 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 a_ii 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 R_OL, the crack propagation increment a_ii 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 N_d

McEvily and Ishihara [45] provided a definition for the number of delay cycles N_d 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 N_d 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 N_d, 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 N_d can be determined.

Wheeler [46] describes the retardation behavior of crack extension after overloading by introducing the retardation coefficient C_p as shown in Equation (14):

$${\left( {\displaystyle \frac{\mathrm{d}a}{\mathrm{d}N}} \right)}_{\mathrm{VA}}=C_{\mathrm{p}}\left[C{\left(\Delta K\right)}^m\right]$$

(14)

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. Parameter fitting results for the Paris formula.

Thickness B	lnC	m	R2
5 mm	−14.536	2.553	0.977
10 mm	−14.106	2.325	0.948
15 mm	−12.833	1.885	0.947

. The retardation coefficient C_p is a function related to the size of the plastic zone as follows:

$$C_{\mathrm{p}}=\left\{\begin{array}{cc}\hfill {\left({\displaystyle \frac{r_{\mathrm{p}}}{a_{\mathrm{OL}}+r_{\mathrm{p},\mathrm{OL}}-a}}\right)}^{m\prime },& a+r_{\mathrm{p}}< a_{\mathrm{OL}}+r_{\mathrm{p},\mathrm{OL}}\\ \hfill 1,& a+r_{\mathrm{p}} \ge a_{\mathrm{OL}}+r_{\mathrm{p},\mathrm{OL}}\end{array}\right.$$

(15)

where a is the current crack length, a_OL is the crack length at the overload applied, r_p is the plastic zone size under the current crack length a, r_p,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 C_p is corrected to account for the influence of a_ii and a_d in the FCG behavior after overloading.

$${C\prime }_{\mathrm{p}}=\left\{\begin{array}{cc}\hfill {\left({\displaystyle \frac{r_{\mathrm{p}}}{a_{\mathrm{OL}}+a_{\mathrm{ii}}+r_{\mathrm{p},\mathrm{OL}}-a}}\right)}^{m\prime },& a < a_{\mathrm{d}}+a_{\mathrm{OL}}\\ \hfill 1,& a \ge a_{\mathrm{d}}+a_{\mathrm{OL}}\end{array}\right.$$

(16)

Therefore, after applying overloads, when a = a_OL + a_ii, that is, after the instantaneous acceleration period, C_p′ obtains the minimum value, which corresponds to the (da/dN)_min. When a ≥ a_OL + a_d, the crack expands beyond the range of overload influence, there is C_p′ = 1 and the retardation effect disappears. Then, according to the definition, the number of delay cycles N_d can be expressed as

$$N_{\mathrm{d}}=\underset{a_{\mathrm{OL}}}{\overset{a_{\mathrm{OL}}{+a}_{\mathrm{d}}}{\int }}\left({\displaystyle \frac{1 -{C\prime }_{\mathrm{p}}}{{C\prime }_{\mathrm{p}}\left[{C\left(\Delta K\right)}^m\right]}}\right) \mathrm{d}a$$

(17)

The calculation methods for a_ii and a_d 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. N_d varies between 32 and 494 cycles under 1.4 R_OL, whereas under 1.7 R_OL ranges from 599 to 3700 cycles. This indicates that the number of N_d, which characterizes the degree of overload-induced retardation, increases with both R_OL and ΔK. Additionally, N_d shows a certain upward trend with increasing B, which is more pronounced at 1.7 R_OL 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 N_b of 20,000 and 50,000 cycles, respectively. The selection of overload cycles ensured the integrity of the crack retardation process, with 1.7 R_OL, and the average FCG lives were 77,891 and 97,496 cycles, respectively. At this point, different overload interval cycles N_b correspond to different ΔK or a_OL 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 N_d 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 R_OL in Figure 16. After the crack extends to 51.2 mm (ΔK = 14.7 MPa·m^1/2), 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 a_OL, 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 a_ii 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 a_d 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 m_d 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 R_OL 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 R_OL, 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 R_OL 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 a_OL.

2.

The experimental results confirm that the crack growth increment a_ii 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 a_ii increases with the overload ratio R_OL 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 a_d was assessed as a function of K_OL/σ_y. The results indicate that a_d follows a power–law relationship with K_OL/σ_y, but the exponent b_d exceeds 2. The established evaluation model for a_d, which accounts for specimen thickness B and overload ratio R_OL, demonstrates greater predictive accuracy for a_d compared to the plastic zone models. Finally, by incorporating a_d and a_ii into the Wheeler model, a calculation method for N_d was developed. The calculated N_d values under 1.4 and 1.7 R_OL conditions show good agreement with the experimental results. Under periodic overload conditions, both a_d and N_d exhibit an increasing trend with the elevation of R_OL 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 R_OL. 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.