Numerical Simulation of the Effect of Pre-Strain on Fatigue Crack Growth in AA2024-T351

Abstract

The objective here is to study the effect of pre-strain on fatigue crack growth (FCG) in 2024-T351 aluminum alloy. Three pre-strain conditions were considered: without pre-strain, compressive and tensile permanent pre-strains of 4%. A numerical approach based on cumulative plastic strain at the crack tip was followed to predict FCG rate. The compressive pre-strain increased FCG rate, while the tensile pre-strain reduced the da/dN relative to the situation without pre-strain. The influence of pre-strain was linked with plasticity-induced crack closure. In fact, a linear trend was obtained between da/dN and ΔK_eff for three crack lengths (a = 16.184; a = 15.048 mm and a = 15.152 mm) and three pre-strain conditions. The increase in the stress ratio from R = 0.1 to R = 0.5 and the elimination of the contact of crack flanks significantly reduced the effect of pre-strain, also pointing to the huge relevance of crack closure in this context. Finally, the effect of pre-strain on da/dN after an overload was also explained by crack closure variations.

1. Introduction

Real structural components are submitted to plastic deformation during fabrication. Operations like blanking, trimming, stretching, bending and deep drawing are involved in structure fabrication. Sheet metal parts for automotive industry are produced by stamping [1], involving large plastic deformation. Material deformation can also be performed to improve mechanical properties, namely tensile strength and yield stress [2,3]. Plastic deformation resulting from fabrication affects material behavior [4,5], so it must be included in the design of components and structures.

In laboratory testing or numerical simulations, the effect of pre-strain is studied by loading a material up to the desired amount so that it can experience deformation in the plastic strain regime and then unloading the material. Pre-strain increases the yield stress and tensile strength while decreasing the ductility. In other words, the material becomes less ductile with pre-strain. This was observed by Ghosal et al. in dual-phase (DP590) steel [6,7], by Le et al. [8] in DP600 steel, and by Fredriksson et al. [9] in DP400, DP600, HSLA steel (HSLA500) and a deep-drawing quality steel. Similar trends were also found by Wang et al. [10] for TWIP steel, Walker et al. [11] for DP and FB steels, Ji et al. [12] for different grades of DP steels, and Wu et al. [13] for stainless steel (SUH660). These effects of pre-strain are explained by a substantial increase in dislocations [6], leading to localized alterations in lattice orientations. The increased dislocation density of the pre-strained material leads to significant work hardening.

In the high-cycle fatigue regime (HCF), fatigue strength can be defined as the specific stress amplitude below which a material can withstand an infinite number of cycles without failure. Ghosal et al. [6] and Paul et al. [7] observed an increase in the endurance limit with pre-strain in dual-phase steel. Wang et al. [10] studied AA2024-T3 and found a decrease in fatigue strength. Considering these opposite trends, the influence of tensile pre-strain on high-cycle fatigue life is expected to be material-dependent. In fact, Froustey et al. [14] studied AA5454 and AA2017. The AA2017 showed a considerable reduction in fatigue life, but the AA5454 revealed unaffected fatigue properties with increased tensile pre-straining magnitudes.

The effect of pre-strain on low-cycle fatigue (LCF) has also been widely studied. Le Roux et al. [15] studied 304L stainless steel subjected to uniaxial pre-strain. They found a decrease in fatigue life compared to the as-received condition under strain-controlled cycling, which was attributed to the reduction in ductility. Gustavsson et al. [16] studied dual-phase (DP) steel subjected to uniaxial pre-strain. They also observed a reduction in the fatigue life under fully reversed strain-controlled fatigue loading. Parker et al. [17] investigated the effect of 40% balanced biaxial stretching on hot rolled low-carbon steel SAE1008. They reported that fatigue resistance starts to degrade in the LCF region compared to the as-received condition due to the cyclic softening nature of the stretched material. The pre-straining decreases ductility, which can cause poor fatigue performance in the LCF region. In contrast, it was found that fatigue resistance improves in the HCF region. The same trends were observed in DP600 steel submitted to 12.5% uniaxial pre-strain [18]. Rex et al. [19] studied the 2024-T4 aluminum alloy. A considerable increase in fatigue life was found for uniaxial tensile pre-strained in comparison to the 0% pre-strained condition at a higher strain amplitude. Furthermore, equi-biaxial tensile pre-strained specimens showed a quite small increment in fatigue life compared to 0% pre-strained material. As-received specimens exhibit noticeable cyclic hardening at all the strain amplitude values. The pre-strained specimen shows an almost stable cyclic stress–strain response for each strain amplitude. Branco et al. [20] studied the effect of 4 and 8% tensile pre-strains on the LCF in AA7050-T6. They observed a reduction in fatigue life with pre-strain. Without pre-strain, the dislocations tended to have a more homogeneous distribution, while under higher tensile pre-strains, the dislocations were mainly observed at the slip bands.

The effect on fatigue crack growth (FCG) has also been studied. Schijve [21] studied aluminum alloys and observed a substantial rise in the FCG rate of 2024-T3 alloy after applying a 3% tensile pre-strain. He suggested that the elevated yield strength due to pre-straining leads to a smaller plastic zone size, intensifying tensile stresses and diminishing crack closure within the plastic zone near the crack tip, thereby accelerating crack propagation. Al-Rubaie et al. [22] studied AA7475-T7351 and found a limited effect of pre-straining in regimes I and II, but also an effect in regime III associated with a loss of fracture toughness. Anandavijayan et al. [23] also found a limited effect in S355 steel submitted to pre-strains of 0, 5 and 10%. On the contrary, they found a great influence on high-cycle fatigue life. Tai Shan and Liu [24] observed an increase in the FCG rate in AA2024-T351. Wasen et al. [25] found that pre-straining reduces the FCG rate in dual-phase steel. Arora et al. [26] studied mild steel and observed a decrease in the FCG rate in regime II. Radhakrishnan et al. [27] investigated the influence of pre-straining and stress ratio on FCG in mild and stainless steel. They observed that the FCG rate decreased by factors of 2.4 and 3.7 for 6% and 9% pre-strains and 1.4 and 2.4 for 9% and 14% pre-strains in mild and stainless steel, respectively. In contrast to earlier findings, Kim et al. [28] showed that the FCG rate increases with higher pre-strain levels in the lower and intermediate ΔK regimes in PH steel. Leitner et al. [29] also found that increased plastic pre-deformation increases the FCG rate and ΔKth in pearlitic steel. Wang et al. [30] investigated the influence of different pre-deformation conditions on the microstructure evolution and fatigue performance of Al-Cu-Li alloy sheets. As the pre-stretching level increased from 1% to 4.5%, the FCG rate initially decreased and then increased at a stress intensity factor range (ΔK) of 20 MPa⋅m^1/2. In contrast, pre-rolled samples exhibited a distinctly different trend at the same ΔK. Specifically, the FCG rate of pre-rolled samples increased with higher pre-rolling levels, reaching up to 1.23 × 10⁻³ mm/cycle for the 12.4% pre-rolled sample at ΔK = 20 MPa⋅m^1/2. The study revealed that pre-deformation increased dislocation density, enhanced precipitation, and inhibited the growth of precipitates. The pre-strain may be accomplished by residual stresses. Gao et al. [31] studied shot-peened specimens of Q235B steel, with and without relaxation of residual stresses. They found that the effect of compressive residual stresses on FCG mainly depended on the residual stresses. The conclusions from the above literature demonstrate that the influence of plastic pre-strain on FCG response is greatly dependent on the material. However, there is a lack of understanding of the mechanisms explaining the effect of pre-strain. In addition, the effect of compressive pre-strain on FCG has not been reported.

The objective of this research is to study the effect of pre-strain on FCG in 2024-T351 aluminum alloy. A numerical model was built to simulate the application of pre-strain and the subsequent propagation of a fatigue crack by node release. Tensile and compressive plastic pre-strains of 4% were studied. These initial plastic deformation values were defined empirically. The results obtained showed an effect on da/dN, which indicates that the initial pre-strain was high enough to have an effect on fatigue life.

2. Numerical Model

Figure 1a shows the CT specimens considered in the numerical study. These specimens had a width W = 36 mm and an initial crack length a₀ = 15 mm. Only 1/4 of CT specimen was modeled, as indicated in Figure 1b,c, considering adequate boundary conditions. The symmetry conditions are dy = 0 at plane xz, and dz = 0 at plane xy, which define the two planes of symmetry. The cyclic load was applied in the upper point of the specimen hole, considering a nodal force. Rigid body movement is inhibited fixing the lower right corner, as indicated in Figure 1b. The contact of crack flanks was simulated using a rigid plane placed at the crack symmetry plane. A small thickness of 0.1 mm was considered for the specimen, in order to model a plane stress state. Simulations without contact between crack flanks were also run, to understand the effect of crack closure.

The material studied was the 2024-T351 aluminium alloy. The 2XXX series of aluminum alloys are widely used in aerospace applications owing to their high strength, good fracture toughness, and low density [30]. As a result of its high strength-to-weight ratio, the Al–Cu–Mg–Mn-based AA2024 aluminum alloy is widely used for various engineering applications, including aircraft and transportation applications [32,33,34]. Typically, this material is used in the design of fuselages, wings, and lightweight components in the aerospace industry. This material was also selected because it was characterized in previous works in terms of static behavior, fatigue behavior, and elastic-plastic modelling [35,36].

Table 1. Chemical composition in weight % of 2024-T351 aluminum alloy. The balance is Al. Reprinted from Ref. [35].

Si	Fe	Cu	Mn	Mg	Cr	Zn	Ti
0.50	0.50	3.8–4.9	0.3–0.9	1.2–1.8	0.10	0.25	0.15

shows the chemical composition of the 2024-T351 aluminum alloy used in this research. The material properties of the 2024-T351 aluminum alloy without pre-strain, determined with a tensile test, are as follows: Young’s modulus = 73 GPa, yield stress = 325 MPa, ultimate tensile strength (UTS) = 470 MPa, elongation at break = 20% and Brinell hardness = 137.

The accurate modeling of material elastic-plastic behavior is fundamental to obtaining good-quality numerical predictions of crack tip plastic deformation. Elastic behavior is assumed to be isotropic and described by Hooke’s law. On the other hand, plastic behavior is characterized by the von Mises yield criterion and the Swift isotropic hardening law coupled with the Lemaître–Chaboche kinematic hardening law under an associated flow rule. The Swift hardening law [37] is described by the following equation:

$$\mathrm{Y}=\mathrm{C}{\left[{\left({\displaystyle \frac{{\mathrm{Y}}_0}{\mathrm{C}}}\right)}^{{\displaystyle \frac{1}{\mathrm{n}}}}+{\stackrel{-}{\epsilon }}^p\right]}^{\mathrm{n}}$$

(1)

where Y₀, C and n are the material parameters and ${\stackrel{-}{\epsilon }}^p$ denotes the equivalent plastic strain. The Lemaître–Chaboche kinematic hardening law [38] is as follows:

$$\dot{\mathit{X}}=C_x\left[{\mathit{X}}_{\mathit{Sat}}{\displaystyle \frac{\sigma \prime -\mathit{X}}{\overline{\sigma }}}-\mathit{X} \right]{\dot{\overline{\epsilon }}}^p$$

(2)

where C_x and X_Sat are the material parameters of the Lemaître–Chaboche law, $\overline{\sigma }$ is the equivalent stress and ${\dot{\overline{\epsilon }}}^p$ is the equivalent plastic strain rate. The material parameters, presented in

Table 2. List of material parameters involved in the Swift and Lemaître–Chaboche laws.

Material	Y0 [MPa]	C [N]	n	Cx [mm]	XSat [MPa]
2024-T351	288.96	389.00	0.056	138.80	111.84

, were identified using LCF fatigue curves obtained for R_ε = −1 and Δε/2 = 1.5% in a previous study [36].

The numerical model was implemented using the in-house software DD3IMP (Three-Dimensional Elasto-Plastic Finite Element Code), originally developed for the simulation of deep-drawing [39]. An updated Lagrangian approach is used to describe the evolution of the deformation process, assuming a hypoelastic-plastic model. The numerical model considers large elastoplastic strains and rotations, while the elastic strains are assumed to be negligibly small. Figure 2 shows the finite element mesh considered, where the specimen was discretized with linear eight-node hexahedral finite elements, using a selective reduced integration technique to avoid volumetric locking. It was refined in the crack growth region, with elements of 8 × 8 μm². In regions relatively distant from the propagation zone, finite elements have larger dimensions in order to reduce the total number of elements and thus the numerical effort. Only one layer of elements was considered along the thickness direction. The model comprises 14,625 finite elements and 29,756 nodes.

Crack propagation was numerically modeled by successive debonding of both current crack front nodes. This crack propagation is fundamental to generate a residual plastic and therefore to stabilize the crack closure level. The release occurs at minimum load to avoid convergence problems that may arise due to crack propagation at maximum load. The size of each crack increment corresponded to the finite element size in the refined region. The two crack front nodes were released when the accumulation of plastic strain reached the critical value of strain. This critical value was calibrated using one experimental value of da/dN, and a value of 110% was obtained in a previous study of the authors [40]. The values of FCG rate are obtained using the following equation:

$${\displaystyle \frac{da}{dN}}={\displaystyle \frac{∆a}{∆N}}$$

(3)

where Δa is the individual crack increment (which is equal to the size of finite elements, as already said) and ΔN is the number of load cycles required to reach the critical cumulative plastic strain at the crack tip. The size of finite elements is usually a major parameter of FEM studies. However, in this case, the mesh refinement reduces both Δa and ΔN, thus having a limited effect on the FCG rate.

The pre-strain was applied by displacing a range of nodes along the entire width of the specimen. This displacement simulated what would experimentally be a distributed tensile or compressive load on the specimen. This loading produces elastic and plastic deformation, which is the initial pre-strain. The region where plastic deformation was induced is indicated by the red rectangle in Figure 3. The region above it is free from pre-deformation. The displacements were adjusted to obtain plastic deformations of 4% in tension and compression (ε_ps = −4, +4%). After the application of pre-deformation, a crack with a length a₀ = 15 mm was simulated and submitted to cyclic loading and crack propagation by node release.

The CT specimen was submitted to constant amplitude loading with stress ratios R (=F_min/F_max) of 0.1 and 0.5. In the first case, the maximum and minimum forces were 41.67 N e 4.17 N, being 75 N and 37.5 N for a stress ratio R = 0.5. Simulations of overloads were also considered. In the overload cycle the maximum and minimum loads were 79.17 N and 4.17, respectively, giving an overload ratio OLR = 1.5, calculated according to the following equation:

$$OLR={\displaystyle \frac{F_{OL}-F_{min}}{F_{max}-F_{min}}}$$

(4)

The baseline loading had F_max = 41.67 N and F_min= 4.17 N, respectively.

Table 3. Loading conditions (a₀ = 15 mm; plane stress).

Test	Loading	R	εps [%]	Contact
1	CA	0.1	0	Yes
2	CA	0.1	+4	Yes
3	CA	0.1	−4	Yes
4	CA	0.1	0	No
5	CA	0.1	+4	No
6	CA	0.1	−4	No
7	CA	0.5	0	Yes
8	CA	0.5	+4	Yes
9	CA	0.5	−4	Yes
10	OL	0.1 (BL) 1	0	Yes
11	OL	0.1 (BL)	+4	Yes

¹ BL, Baseline loading.

presents all the load cases considered. Simulations without contact of crack flanks were also considered, as already mentioned.

3. Results

3.1. Effect of Pre-Strain on da/dN

Figure 4 shows the variation in FCG rate versus crack length, a, with and without pre-strain. The crack propagation started at a₀ = 15 mm and ended at a = 16.2 mm. This propagation is enough to stabilize the level of plasticity-induced crack closure.

The specimen without pre-strain presents a small initial increase in da/dN, followed by a progressive decrease to a minimum value, which occurs at about a = 15.5 mm. This decrease is explained by the progressive increase in crack closure level with the formation of residual plastic wake. Above this transient regime, there is a stable regime characterized by a progressive increase in da/dN, which is explained by the increase in crack tip fields with crack length.

The specimen with a tensile pre-strain of 4% presents a similar behavior, but the initial variations are much smaller. There is a small increase in da/dN, followed by a smooth decrease to a minimum value which occurs at about a = 15.2 mm. Finally, there is a progressive increase with crack propagation as a result of the increase in crack length.

The specimen submitted to a compressive pre-strain of 4% presents a small initial increase in da/dN, but without subsequent decrease. The slope of increase in da/dN with crack length is similar to that observed for the specimen without pre-strain. On the other hand, the specimen with tensile pre-strain has a lower rate of variation in da/dN in the stable regime. The FCG rate in the stable regime is higher for the compressive pre-strain, intermediate for the specimen without pre-strain, and lower for the tensile pre-strain. For a crack length of 16.184 mm, indicated by the vertical dashed line, da/dN has values of 0.95, 0.71 and 0.39 μm/cycle for ε_ps = −4, 0 and +4%, respectively. Finally, both the tensile and compressive pre-strains eliminated the initial transient effect associated with crack closure.

Figure 5 presents the CTOD versus load plots, which are particularly interesting to understand what is happening at the crack tip. The CTOD curves plotted were measured at the first node behind crack tip for a crack length a = 16.184 mm, indicated by the vertical dashed line on the right-hand side of Figure 4. The load ranged between F_min = 4.17 N and F_max = 41.67 N, as can be seen in the horizontal axis. The crack is closed at the lower part of the load cycle, having a linear elastic behavior after opening. At relatively high loads, there is a departure from linearity, resulting from crack tip plastic deformation. The elastic and plastic CTOD ranges (δ_e and δ_p, respectively) are indicated for the compressive pre-strain.

The crack opening level is lower for the compressive pre-strain (F_open ≈ 10 N) and higher for the tensile pre-strain (F_open ≈ 20 N), with an intermediate value for the specimen without pre-strain (F_open ≈ 15 N). The crack opening level was quantified using the following equation:

$$U^{*}={\displaystyle \frac{F_{max}-F_{open}}{F_{max}-F_{min}}}$$

(5)

The compressive pre-strain has U* = 18.23%, the situation without pre-strain has U* = 27.15%, and the tensile pre-strain has U* = 42.53%. The increase in the crack opening level decreases the effective load range, which reduces the maximum CTOD, the elastic and plastic deformation ranges, and the area of the loop (which is proportional to the dissipated energy). The sizes of monotonic plastic and reversed plastic zones are also expected to decrease. Therefore, a relation may be established between these non-linear parameters and the da/dN values presented in Figure 4.

The relation between the FCG rate and the effective stress intensity factor range, ΔK_eff, was studied for three crack lengths (a = 16.184; a = 15.048 mm and a = 15.152 mm) and three pre-strain conditions (ε_ps = −4, 0, +4%). Figure 6 shows the results obtained, which indicate a well-defined correlation between da/dN and ΔK_eff, although some scatter can be observed. The increase in ΔK_eff increases da/dN, and a linear trend is obtained. This correlation clearly indicates that plasticity-induced crack closure (PICC) is behind the variations in da/dN observed in Figure 4 and therefore behind the effect of pre-strain on FCG.

3.2. Influence of Stress Ratio on Pre-Strain Effect

Figure 7 presents results of da/dN versus crack length for a stress ratio R = 0.5 and different pre-strain values (ε_ps = −4, 0, +4%). The dashed lines indicate results obtained for R = 0.1. The increase in R increased significantly da/dN, which means that the stress ratio has a major impact. The effect of R is usually associated with the crack closure phenomenon; therefore, the results in Figure 7 indicate that crack closure is affecting the results. The increase in the FCG rate with the stress ratio has been observed by different authors, including Mehrzadi e Taheri [41] in a magnesium alloy and Seifi and Hosseini [42] in pure copper.

On the other hand, the effect of pre-strain is less evident for R = 0.5 than it was for R = 0.1. Note that linear scales are being used instead of logarithmic scales. Nevertheless, the tensile pre-strain produces a slightly higher da/dN for the compressive pre-strain, while the pre-strain ε_ps = 0 has intermediate values. The slope, i.e., the rate of variation in da/dN with crack length, is similar for the different pre-strains. Thus, there is an inversion of the effects of tensile and compressive pre-strains relative to the results obtained for R = 0.1.

Some oscillations were observed in all curves. In fact, the node release for the simulation of crack growth occurs at minimum load, when the accumulated plastic strain reaches the critical value (110%). Normally, this critical value is reached in the middle of a load cycle. Therefore, there is some inaccuracy in the predicted value of N, i.e., the number of load cycles between node releases used to calculate the da/dN. The problem becomes more relevant when N is relatively small, i.e., when the crack growth rate is relatively high. In this case, although the error is just a fraction of unit, it has some impact on the da/dN. These oscillations are a numerical issue, which does not affect the global trend; therefore, a smoothing strategy was used to remove them in Figure 7.

Figure 8 plots the variation in CTOD with the applied load for a stress ratio R = 0.5 and different pre-strain values (ε_ps = −4, 0, +4%). Both the maximum and minimum loads increased relative to the situation with R = 0.1, which is represented by the dashed lines. The curves for R = 0.5 are very similar, showing a limited effect of pre-strain. Compressive pre-strain presents a slightly lower value of opening load. The comparison with the dashed lines indicates a great decrease in the crack opening level, which has values of U* in the range 6–11%, while for R = 0.1, it ranged from 18 to 40%. This reduction in U* increases the effective load range, which significantly increases the total CTOD range and the elastic and plastic components, δ_e e δ_p, respectively. The variation in U* also explains the increase in da/dN produced by the increase in R and the reduced effect of pre-strain at R = 0.5. In fact, since the effect of pre-strain is associated with crack closure, and there is almost no crack closure at R = 0.5, a minor effect of pre-strain on FCG may be expected.

3.3. Effect of Pre-Strain on Variable Amplitude Loading

The influence of pre-strain on the effect of an overload was also studied. The overload was applied at the load cycle number 1500, at a baseline stress ratio R = 0.1 and considering an overload ratio of 1.5. Figure 9 compares the results of da/dN obtained without pre-strain and with a tensile pre-strain of 4%. These tests took a long time because the FCG rate drops significantly after the overload.

The variation in da/dN is in line with the typical behavior observed after an overload [43,44]. Immediately after the overload, there was a sudden increase in da/dN, followed by a fast decrease to a minimum value. Some crack extension is required to reach this minimum, which is named delayed retardation. After the minimum, there was a progressive increase in the FCG rate to the stable value, corresponding to constant amplitude loading. The increase in da/dN is due to crack tip blunting, which eliminates previous crack closure. The variations after blunting were due to the formation of a new residual plastic wake, stronger than that corresponding to constant amplitude loading. In fact, overload increases the monotonic plastic zone, and this has an impact on crack closure, which is felt after crack propagation. The two main mechanisms of plasticity-induced crack closure, i.e., the crack tip blunting and the effect of residual plastic wake, are clearly visible in the FCG observed after an overload.

When comparing the curves for the two pre-strain values (ε_ps = 0, +4%), significant differences are evident. Overload is applied at a shorter crack length for ε_ps = +4% because the crack growth rate is lower while the number of load cycles is the same (1500). In addition, the variations in da/dN are much more pronounced for the case without pre-strain. The extent of propagation affected by overload is also much higher for ε_ps = 0.

For a better understanding of the effect of pre-strain, Figure 10 plots CTOD curves. Figure 10a presents CTOD variation exactly in the overload cycle. For relatively low loads, the crack is closed, i.e., the CTOD is zero. The crack opening occurs first for ε_ps = 0, as already observed in Figure 5. After crack opening, the CTOD increases—first linearly and then non-linearly—with the load (region II). The linear variation is a result of elastic deformation, while the non-linear variation is due to plastic deformation. The increase in the load above the baseline values, i.e., above F_max,BL = 41.67 N, produces a sudden change in the slope of the curve (region III). The material has not experienced such high loads before; therefore, there is a strong plastic deformation—significantly higher than that observed in region II. Consequently, CTOD reaches relatively high values. The material with ε_ps = +4% has less plastic deformation because it had already experienced plastic deformation, i.e., it was hardened previously. The subsequent decrease in load from F_OL to F_min reduces the blunting (region IV), initially linearly and then non-linearly. The CTOD at the minimum load is the crack tip blunting produced by overload. The dashed line indicates the behavior without overload.

Figure 10b shows the CTOD curves at the minimum values of da/dN, which occur at crack lengths of 16.336 mm and 15.448 mm for ε_ps = 0 and +4%, respectively. The dashed line indicates the behavior without overload and without pre-strain. The crack closure is quite relevant, particularly for the tensile pre-strain, being much more than observed under constant amplitude loading (Figure 5). U* = 53.1% for the specimen without pre-strain and U* = 71.0% for the tensile pre-strain, at the minimum da/dN after the overload. These high values of U* reduce the effective load range, and therefore the crack tip plastic deformation. This is particularly dramatic for ε_ps = +4%, because there is no evidence of plastic CTOD, δ_p. This explains the very small value of da/dN in Figure 9. The slopes of the elastic regimes are very similar, which means that the elastic behavior is not affected by the overload, as could be expected. Therefore, the minimum da/dN is linked to a peak of crack closure, as was observed by Baptista et al. [45].

4. Discussion

The first issue is the validity of the conclusions obtained using only numerical simulation. Naturally, it would be interesting to have experimental results to validate the numerical predictions. However, this is expensive, since it requires the preparation of specimens with pre-strain and their testing, and time-consuming. On the other hand, the present numerical model has already proved to be competent and robust in the simulation of FCG. In fact, the numerical model based on cumulative plastic strain was able to predict the effect of ΔK [40], stress ratio [46], overloads [47] and load blocks [48], among other parameters. These successive validations are also a clear indication that cyclic plastic deformation is the main mechanism responsible for FCG. In addition, numerical approaches are very interesting to develop parametric studies and to analyze the fundamental mechanisms. This is particularly important in the study of the effect of pre-strain, where the mechanisms are not completely understood.

The present results clearly indicate that plasticity-induced crack closure is behind the effect of pre-strain. The decrease in ductility is expected to decrease the size of monotonic and reversed plastic zones. This reduces residual plastic deformations and, therefore, crack closure. On the other hand, there is a decrease in crack blunting, which potentiates the effect of residual plastic wake. These mechanisms of opposite effect may explain the different trends observed for different materials, or, in other words, the dependence on material. Schijve [21] studied AA2024-T3 submitted to a pre-strain of 3%. He also linked the increase in da/dN with the crack closure phenomenon.

The material hardening resulting from plastic deformation may also be expected to have a direct effect on crack tip damage. In fact, the cumulative damage will be lower in the hardened material. Therefore, a numerical study was developed without contact between crack flanks, which eliminates the crack closure phenomenon from the results. Figure 11a presents da/dN versus crack length. The dashed lines indicate the predictions obtained with contact between crack flanks also for R = 0.1. As can be seen, there is some effect of pre-strain, but significantly less than observed in the presence of contact between crack flanks. Tensile pre-strain now has a damaging effect, relative to the situation without pre-strain. On the other hand, compressive pre-strain reduces da/dN. The numerical oscillations, which were removed from Figure 7, were not eliminated here and are now clearly visible.

Figure 11b presents the CTOD curves versus load for a crack length a = 16.048 mm. Negative values of CTOD can be observed, which means that the crack flanks are overlapping. The elimination of contact is not physically correct, but is particularly interesting to isolate crack tip phenomena from the influence of crack closure. The applied load range is now totally effective. In other words, the crack tip feels all load cycle without any protection from the crack closure phenomenon. This produces an increase in elastic, plastic and total CTOD ranges.

Table 4. Fatigue crack growth rate at a = 16 mm (a₀ = 15 mm; plane stress).

Test	R	εps [%]	Contact	da/dN [μm/cycle]	%Variation
1	0.1	0	Yes	0.67	-
2	0.1	+4	Yes	0.36	−46.3
3	0.1	−4	Yes	0.89	+32.8
4	0.1	0	No	1.61	-
5	0.1	+4	No	1.81	+12.4
6	0.1	−4	No	1.48	−8.0
7	0.5	0	Yes	1.54	-
8	0.5	+4	Yes	1.49	−3.2
9	0.5	−4	Yes	1.68	+9.1

summarizes the crack propagation rates obtained for a crack length a = 16 mm, i.e., after 1 mm of crack propagation. The cases without pre-strain are used as a reference. As already mentioned, there is a great influence of pre-strain for R = 0.1, and a lower influence for R = 0.5 and for the situations without contact between crack flanks. Concerning the relative importance of the three parameters studied (R, ε_ps and crack length), the least influential parameter is crack length. The comparison of results for R = 0.1 and R = 0.5 in Table 4 indicates that this parameter has the greatest influence.

The inversion of the effect of tensile and compressive pre-strain relative to the situation with contact (Figure 4) is curious and warrants closer investigation. Figure 12 shows the variation in accumulated plastic strain at the crack tip with time (i.e., with load cycling). There are two aspects that deserve special attention: the slope of the curves between node releases and the initial value of plastic strain after node release. The slope represents the rate of accumulation of damage, while the initial value is a consequence of the accumulation of damage in previous positions of the crack tip. Figure 12a presents the results obtained for five load increments starting at a = 15.952 mm, with contact of crack flanks activated. The slopes are clearly different, being higher for the compressive pre-strain. The initial value of accumulated strain is also higher for ε_ps = −4%, which indicates larger monotonic and reversed plastic zones. Consequently, assuming 70 load cycles, there are five node releases for the compressive pre-strain, while the tensile pre-strain only suffers two node releases. These trends are in line with the results of Figure 4, i.e., they explain the higher da/dN obtained for compressive strain. Figure 12b presents similar results, but without contact between crack flanks. Now, the tensile pre-strain has slightly higher values of initial plastic strain and rate of accumulation of plastic strain. This explains the slightly higher value of da/dN obtained for ε_ps = +4% in Figure 11a.

5. Conclusions

The main objective here is to study the effect of pre-strain on FCG in AA2024-T351. A numerical model was created to predict FCG assuming that cyclic plastic deformation is the main damage mechanism and using the cumulative plastic strain at the crack tip as the driving parameter. Two values of pre-strain were considered—a compressive pre-strain of 4% and a tensile pre-strain of 4%—which were compared with the situation without pre-strain (i.e., ε_ps = −4, 0, +4%). After stabilization of the FCG rate, the da/dN is higher for compressive pre-strain, intermediate for the specimen without pre-strain, and lower for tensile pre-strain. Values of U* of 18.23, 27.15 and 42.53% were obtained for ε_ps = −4, 0 and +4%, respectively.

The main conclusion of this paper is that the effects of pre-strain in AA2024-T351 are linked to crack closure. In fact, a well-defined correlation was obtained between da/dN and ΔK_eff for three crack lengths (a = 16.184; a = 15.048 mm and a = 15.152 mm) and the three pre-strain conditions (ε_ps = −4, 0, +4%). In addition, the increase in the stress ratio from R = 0.1 to R = 0.5 significantly reduced the relevance of pre-strain, which is also explained by the reduction in the crack closure phenomenon. The effect of pre-strain on da/dN after an overload was also explained by crack closure variations. Interestingly, both the tensile and compressive pre-strains eliminated the initial transient effect associated with crack closure.

Concerning the relative importance of the three parameters studied (R, ε_ps and crack length), the stress ratio has the greatest influence, followed by the pre-strain. The least influential parameter is crack length.