Effect of Specimen Thickness and Stress Intensity Factor Range on Plasticity-Induced Fatigue Crack Closure in A7075-T6 Alloy

materials

Article Effect of Specimen Thickness and Stress Intensity Factor Range on Plasticity-Induced Fatigue Crack Closure in A7075-T6 Alloy

Kenichi Masuda 1,* , Sotomi Ishihara 1,2 and Noriyasu Oguma 1

1 Department of Mechanical Engineering, University of Toyama, Gofuku 3190, Toyama 930-8555, Japan; [email protected] (S.I.); [email protected] (N.O.) 2 National Institute of Technology, Toyama College, Toyama 939-8630, Japan * Correspondence: [email protected]; Tel.: +81-76-445-6772

Abstract: Fatigue crack growth experiments are performed using A7075-T6 compact tension (CT) specimens with various thicknesses t (1–21 mm). The stress intensity factor at the crack opening level Kop is measured, and the effects of t and the stress intensity factor range ∆K on Kop are investigated. In addition, the change in Kop value due to specimen surface removal is investigated. Furthermore, we clarify that the radius of curvature of the leading edge of the fatigue crack decreases as t becomes thinner. Using the three-dimensional elastoplastic finite element method, the amount of plastic lateral contraction (depression depth d) at the crack tip after fatigue loading is calculated quantitatively. The following main experimental results are obtained: In the region where ∆K is 5 MPam1/2 or higher, the rate of fatigue crack growth da/dN at a constant ∆K value increases as t increases from 1 to 11 mm. The da/dN between t = 11 and 21 mm is the same. Meanwhile, in the region where ∆K is less than 5 MPam1/2, the effect of t on da/dN is not observed. The effects of t and ∆K on the da/dN–∆K relationship are considered physically and quantitatively based on d. Keywords: fatigue crack growth behavior; aluminum alloy; CT specimen; plasticity-induced fatigue Citation: Masuda, K.; Ishihara, S.; crack closure; specimen thickness; plane stress and plane strain; 3D elastoplastic finite element Oguma, N. Effect of Specimen method; plastic lateral contraction at the fatigue crack tip Thickness and Stress Intensity Factor Range on Plasticity-Induced Fatigue Crack Closure in A7075-T6 Alloy. Materials 2021, 14, 664. 1. Introduction https://doi.org/10.3390/ma14030664 The importance of fatigue crack closure (FCC) on fatigue crack growth (FCG) behavior is acknowledged by many researchers. In many materials, even if the minimum stress Academic Editor: Tomasz Tański intensity factor (K ) is on the tension side, a crack will not open unless the K value Received: 28 December 2020 min reaches the opening stress intensity factor K of the crack (K > K ). Therefore, K Accepted: 26 January 2021 op op min op Published: 31 January 2021 values must be evaluated well to accurately predict FCG behavior. Elber introduced the important concept of FCC [1,2] in the 1970s. He conducted FCG experiments using alu-

Publisher’s Note: MDPI stays neutral minum alloy 2024-T3 under a constant stress ratio—R value and demonstrated that Kop − with regard to jurisdictional claims in increased with the stress intensity factor width ∆K (= Kmax Kmin). Here, Kmax is the max- published maps and institutional afﬁl- imum stress intensity factor. He proposed that the plastic stretch (plastic wake) occurring iations. behind the crack tip contributed to the increase in Kop. Similar results were observed in other low- and medium-strength aluminum alloys [3], and this type of FCC behavior is called plasticity-induced fatigue crack closure (PIFCC). After the initial study by Elber, other types of FCCs were discovered, such as roughness-induced fatigue crack closure (hereinafter RIFCC) [4–6], oxide-induced crack closure, and corrosion product-induced Copyright: © 2021 by the authors. Licensee MDPI, Basel, Switzerland. crack closure [7–10]. In RIFCC, unlike PIFCC, Kop levels do not change signiﬁcantly or This article is an open access article remain constant with increasing ∆K. distributed under the terms and In a review paper regarding the PIFCC phenomenon [11], Schivje indicated that the conditions of the Creative Commons Kop level in the plane-stress region (specimen surface region) was higher than that in the Attribution (CC BY) license (https:// plane-strain region (inside the specimen). Ishihara et al. [12] conducted FCG experiments creativecommons.org/licenses/by/ using A6061 and carbon steel S25C. They reported that A6061 exhibited PIFCC behavior, 4.0/). and the slope of Kop–∆K was almost 0.5, whereas carbon steel S25C exhibited RIFCC

Materials 2021, 14, 664. https://doi.org/10.3390/ma14030664 https://www.mdpi.com/journal/materials Materials 2021, 14, 664 2 of 16

behavior and the slope of Kop–∆K was almost 0. In addition, they measured the fracture surface roughness (Ra) near the specimen surface and specimen inside as a function of ∆K to specifically observe the interaction between the specimen surface and the specimen inside. Their results showed that the Ra near the specimen surface increased with ∆K, but the Ra inside the specimen decreased. Newman et al. reported that FCG behavior in aluminum alloy 7075-T651 (t = 5.7 mm) showed PIFCC behavior [13]. Furthermore, Matos et al. [14] investigated the effect of specimen thickness t on Kop using aluminum alloy 6082-T6. They observed that the Kop for thin specimens was higher than that for thick specimens. In addition, Camas et al. [15] reported that the leading edge of the fatigue crack had a curvature, and that this curvature was affected by the specimen thickness. The conclusion of a review that we conducted on previous studies focusing on PIFCC for aluminum alloys indicates that knowledge regarding the effects of t and ∆K on FCG and FCC behaviors of aluminum alloys is limited to qualitative understanding. In particular, few studies have been carried out on the effect of ∆K on PIFCC. In this study, FCG and Kop behaviors of A7075-T6 were investigated experimentally using CT specimens with various thicknesses. Subsequently, the effect of t on Kop and the range of ∆K at which the effect of t occurs were clarified. The effect of t on the leading edge shape of the crack was experimentally investigated; furthermore, the effect of the leading edge shape of the crack on Kop is discussed herein. In addition, the PIFCC behavior involving plastic deformation was significantly affected by the ∆K level. It is assumed that the effect of RIFCC appeared in the low ∆K region. Therefore, we analyzed the ∆K value at which the transition from PIFCC to RIFCC occurred. The effects of t and ∆K on Kop were quantitatively and physically considered by analyzing the plastic lateral contraction (depression depth d) of the crack tip via the three-dimensional (3D) elastoplastic finite element method (FEM).

2. Materials, Specimens, and Experimental Methods 2.1. Material/Specimen The test material was aluminum alloy 7075-T6 (hereinafter A7075-T6). Its chemical composition and mechanical properties are shown in Tables1 and2, respectively. A tensile test was performed according to JIS standards, and two specimens were used. The difference between the results of the two tests was not large; thus, the average value of these is presented in Table2. The stress–strain curve obtained from the tensile test is shown in Figure1. The stress–strain curve was approximated by Ramberg–Osgood power law, and a strain-hardening exponent of 0.036 was obtained.

Table 1. Chemical composition of the test material (wt%).

Si Fe Cu Mn Mg Cr Zn Ti Al 0.40 0.50 1.60 0.30 2.50 0.24 5.50 0.20 Bal.

Table 2. Mechanical properties of the test material.

Yield Tensile Young’s Poisson’s Strain-Hardening Elongation Strength Strength Modulus Ratio Exponent 510 MPa 577 MPa 70 GPa 0.3 12% 0.036 Materials 2021, 14, x FOR PEER REVIEW 3 of 17

Materials 2021,, 14,, 664x FOR PEER REVIEW 33 of of 1617

Figure 1. Stress−strain curve for the present material A7075−T6. Figure 1. Stress− strain curve for the present material A7075−−T6. FigureFCG 1. Stress experimentsstrain curve were for performed the present using material American A7075 T6. society for testing and materials (ASTM) standard compact tension (CT) specimens; their shapes and dimensions are FCG experiments were performed using using American American society for testing and materials shown in Figure 2. FCG experiments and Kop measurements were performed using four (ASTM) standard compact tension (CT) specimens; their shapes and dimensions are shown (ASTM) standard compact tension (CT) specimens; their shapes and dimensions are CTin Figurespecimens2. FCG with experiments t = 1, 6, 11, andand K21op mmmeasurements, and with specime were performedn width (W using) = 57.2 four mm CT fixed. speci- shown in Figure 2. FCG experiments and Kop measurements were performed using four FCG experimentt was performed using one or two CT specimensW for one specimen CTmens specimens with = with 1, 6, 11,t = 1, and 6, 11, 21 mm,and 21 and mm with, and specimen with specime widthn width ( ) = 57.2(W) = mm 57.2 ﬁxed. mm fixed. FCG thickness,experiment t. Such was performeda method is using employed one or in two a usual CT specimensFCG experiment for one using specimen CT specimen thickness,s. FCG experiment was performed using one or two CT specimens for one specimen Then,t. Such the a methodrelationship is employed between in the a usual rate FCG of fatigue experiment crack using growth CT da/dN specimens. and Then,ΔK was the thickness, t. Such a method is employed in a usual FCG experiment using CT specimens. measuredrelationship by betweenperforming the a rate constant of fatigue ΔK experiment, crack growth a da/dNΔK increasing and ∆K experiment, was measured and bya Then, the relationship between the rate of fatigue crack growth da/dN and ΔK was ΔKperforming decreasing a constantexperiment.∆K experiment,As for the measurement a ∆K increasing of K experiment,op, during a constant and a ∆K ΔK decreasing test, the measured by performing a constant ΔK experiment, a ΔK increasing experiment, and a valuesexperiment. were measured As for the measurementmultiple times of (up Kop to, during 4 times) a constant at an arbitrary∆K test, crack the values length. were In mea-the ΔK decreasing experiment. As for the measurement of Kop, during a constant ΔK test, the FCGsured experiments, multiple times experiments (up to 4 times) using at three an arbitrary-point bending crack length. [16] have In thebeen FCG also experiments, used rela- values were measured multiple times (up to 4 times) at an arbitrary crack length. In the tivelyexperiments often. using three-point bending [16] have been also used relatively often. FCG experiments, experiments using three-point bending [16] have been also used relatively often. P 2 14.3 t P 2 14.3 t

33 1.4 31.5 °

33 30

1.4 68.6

31.5 14.3 W=57.2 30°

14.3 W=57.2 68.6

P 71.5

P 71.5 FigureFigure 2. 2. ShapeShape an andd dimensions dimensions of of ASTM ASTM standard standard compact compact tension tension ( (CT)CT) specimens specimens ( (WW == 57.2 57.2 mm; mm; thickness t = 1, 6, 11, and 21 mm). Figurethickness 2. Shapet = 1, 6,an 11,d dimensions and 21 mm). of ASTM standard compact tension (CT) specimens (W = 57.2 mm; thickness t = 1, 6, 11, and 21 mm). 2.2.2.2. FCG FCG E Experimentxperiment and and K Kopop MMeasurementeasurement In a laboratory environment, an FCG experiment was performed using a hydraulic 2.2. FCGIn a laboratoryExperiment environment,and Kop Measurement an FCG experiment was performed using a hydraulic servoservo fatigue fatigue-testing-testing machine machine under under a a stress stress ratio ratio R R of of 0.1 0.1 and and a a frequency frequency of of 15 15 Hz. Hz. The The In a laboratory environment, an FCG experiment was performed using a hydraulic replicareplica method method [17] [17 ]was was used used to to measure measure the the crack crack length. length. The The FCG FCG experiment experiment was wasin- servointerrupted fatigue after-testing every machine predetermined under a number stress ratio of load R of cycles. 0.1 and Subsequently, a frequency methylof 15 Hz. acetate The replica method [17] was used to measure the crack length. The FCG experiment was in-

Materials 2021, 14, 664 4 of 16

was dropped on the specimen surface, and an acetyl cellulose ﬁlm was quickly attached to the specimen surface to obtain a replica of the specimen surface. The crack length was measured by observing the replica at a 100× to 200× magniﬁcation using an optical microscope. Equation (1) [18] was used to calculate the stress intensity factor K, where P is the applied load, and t is the specimen thickness.

K = √P 2+α 0.886 + 4.64α − 13.32α2 + 14.72α3 − 5.6α4, t W (1−α)3/2 (1) α = α/W, α ≥ 0.2,

The elastic compliance method [19] was used to measure Kop. To improve the measurement accuracy, a strain gage was attached in front of the crack tip, and the strain in the direction perpendicular to the crack propagation direction was measured. To increase the measurement sensitivity of Kop, the signal of a load cell (PCD-320A; Kyowa Electronic Instruments Co., LTD, Tokyo, Japan) and the signal of the strain gage (PCD-300B; Kyowa Electronic Instruments Co., LTD, Tokyo Japan) were input to the subtraction circuit. The opening load of the crack was measured using the elastic compliance method, and the value was used to calculate Kop. For details regarding the experimental method, please refer to a previous report [12].

2.3. Specimen Surface Removal Experiment A CT specimen of t = 6 mm was used for the specimen surface removal experiment. First, a fatigue crack was propagated under a constant ∆K value (8.0 MPam1/2), and the FCG velocity and Kop value were measured each time. When the Kop value stabilized, the FCG experiment was stopped, and the specimen was removed from the fatigue testing machine. Subsequently, 0.5 mm on one side of the specimen (1 mm on both sides) was removed using an electric discharge machine (Tape Cut Model; FANUC Corporation, Chicago, IL, USA). At that time, sufﬁcient care was taken such that processing strain was not introduced in the specimen. Subsequently, the specimen was attached to the testing machine again, and the FCG experiment was restarted. Kop was measured for each constant crack growth amount in this manner.

2.4. Measurement of the Curvature on the Front Edge of the Crack To observe the shape of the leading edge of the fatigue crack inside the specimen, the specimen was removed from the testing machine when the ∆K value reached 12.0 MPa·m1/2. A red penetrant ﬂaw-detection dye was inﬁltrated into the crack surface. Subsequently, the specimen was attached to the testing machine again, and the specimen was broken by applying a larger fatigue load. The fracture surface was observed using a scanning electron microscope to examine the leading edge shape of the crack. The observations above were performed on specimens with different t.

3. 3D Elastoplastic FEM

To quantitatively consider the effects of t and ∆K on Kop, the amount of the plastic lateral contraction (concavity depth d) of the specimen was analyzed using 3D elastoplastic FEM. After the maximum load was applied to the CT specimen, the load was removed. The concavity depths, d, generated at the crack tip at that time were obtained using the FEM for various ∆K values with t values as a parameter. In the analysis, we assumed that the crack length a was constant at 15 mm (α = 0.26) and that the leading edge of the crack was straight (radius of curvature = ∞). In addition, in some of the calculations, an analysis was performed when the leading edge of the crack had a radius of curvature; subsequently, it was compared with a straight leading edge on the crack. Commercially available software [20] was used for the FEM analysis. Examples of element division for the entire specimen and the vicinity of the crack tip are shown in Figure3a,b, respectively, where t is 6 mm. A quarter of the CT specimen was analyzed, considering the symmetry of the specimen. Isoparametric elements were Materials 2021, 14, x FOR PEER REVIEW 5 of 17

curvature; subsequently, it was compared with a straight leading edge on the crack. Commercially available software [20] was used for the FEM analysis. Materials 2021, 14, 664 Examples of element division for the entire specimen and the vicinity of the crack tip5 of 16 are shown in Figure 3a,b, respectively, where t is 6 mm. A quarter of the CT specimen was analyzed, considering the symmetry of the specimen. Isoparametric elements were usedused for for the the FEM FEM analysis; analysis; the the total total numbers numbers of ofele elementsments and and nodes nodes were were 11, 11,200200 and and 46,46,275,275, respectively. respectively. In addition, In addition, the minimum the minimum element element size sizenear nearthe crack the crack tip was tip set was toset approximatelyto approximately 1/75 of 1/75 the ofplastic the plastic zone size zone at size the atcrack the cracktip [15]. tip For [15]. element For element division division in thein specimen the specimen thickness thickness direction, direction, the half the thickness half thickness of the CT of thespecimen CT specimen was divided was divided into 40 intoparts 40 (minimum parts (minimum element elementsize = 20 size m) = in 20 theµm) vicinity in the of vicinity the crack of thetip crackto ensure tip to calcu- ensure lationcalculation accuracy. accuracy. With an Withincrease an increasein the distance in the distancefrom the fromcrack thetip, crackthe numbers tip, the numbersof ele- mentof elementdivision divisionin the specimen in the specimen thickness thickness direction direction were 20 were (element 20 (element size = size150 =m), 150 10µm), (element10 (element size = size300 =m), 300 andµm), 5 and(element 5 (element size = size600  =m); 600 theyµm); were they gradually were gradually reduced. reduced.

FigureFigure 3. (a 3.) Element(a) Element division division for a for quarter a quarter of CT of specimen CT specimen;; (b) enlarged (b) enlarged view viewof the of crack the cracktip in tip in FigureFigure 3 (a3)(. a).

TheThe material material properties properties used used in the in the analysis analysis were were a Young’s a Young’s modulus modulus of 70 of 70GPa, GPa, a a Poisson’s ratio of 0.3, and a yield strength of 510 MPa. Three-dimensional elastoplastic Poisson’s ratio of 0.3, and a yield strength of 510 MPa. Three-dimensional elastoplastic FEM analysis was performed on a material with a strain-hardening exponent of 0.036, and FEM analysis was performed on a material with a strain-hardening exponent of 0.036, the plastic lateral depression depth d at the crack tip was calculated. and the plastic lateral depression depth d at the crack tip was calculated. 4. Experimental Results and Discussion 4. Experimental Results and Discussion 4.1. Relationships between FCG Rate da/dN and ∆K and between da/dN and ∆Keff 4.1. Relationships between FCG Rate da/dN and ΔK and between da/dN and ΔKeff Figure4 shows the relationship between da/dN and ∆K of A7075-T6 alloy on a log–log graph, where a is the crack length, and N is the number of stress cycles. In the ﬁgure, the experimental results for t = 1, 6, 11, 21 mm are shown. For comparison, the results for t = 5.7 mm obtained by Newman [13] and that for t = 10 mm by Jono et al. [21] are provided. As shown in the ﬁgure, the da/dN values for t = 6 and 11 mm in this study almost agreed with the values of t = 5.7 mm measured by Newman and t = 10 mm measured by Jono et al., respectively. Materials 2021, 14, x FOR PEER REVIEW 6 of 17

Figure 4 shows the relationship between da/dN and ΔK of A7075-T6 alloy on a log– log graph, where a is the crack length, and N is the number of stress cycles. In the figure, the experimental results for t = 1, 6, 11, 21 mm are shown. For comparison, the results for t = 5.7 mm obtained by Newman [13] and that for t = 10 mm by Jono et al. [21] are provided. As shown in the figure, the da/dN values for t = 6 and 11 mm in this study almost agreed with the values of t = 5.7 mm measured by Newman and t = 10 mm measured by Jono et al., respectively. In the region where ΔK was 5 MPa∙m1/2 or higher, da/dN increased as t increased from 1 to 21 mm at a constant ΔK value. Meanwhile, in the region where ΔK was 5 MPa∙m1/2 or less, the da/dN value at each ΔK value did not depend on t; specifically, for ΔK between 3.5 and 5 MPa∙m1/2, the da/dN values were the same for t = 1, 6, and 11 mm. Materials 2021, 14, 664 6 of 16 For ΔK between 2.5 and 3.5 MPa∙m1/2, the da/dN values for t = 5.7 mm (Newman) and t =10 mm (Jono et al.) were equal.

10-5 A7075-T6, R = 0.1, CT 10-6 t = 1 mm t = 6 mm

-7 t = 11 mm 10 t = 21 mm ③ -8 10 ② Newman t = 5.7 mm ① da/dN, m/cycle 10-9 Jono et al CCP, t = 10 mm 10-10 2 3 4 5 6 7 8 910 20 K, MPam1/2

FigureFigure 4. 4.FatigueFatigue crack crack growth growth–stress–stress intensity intensity factor factor range range (da/dN (da/dN––ΔK∆) K)relationship relationship of A7075 of A7075-T6-T6 andand the the effect effect of oft ont onthe the relationship relationship,, ① a1 lowa low-rate-rate region; region; ② 2a mediuma medium-rate-rate region; region; ③ 3a a high- highrate-rate region. region.

FigInure the 5 region shows where the relationship∆K was 5 MPa between·m1/2 or da/dN higher, and da/dN the effective increased stress as t intensityincreased factorfrom range1 to 21 ΔK mmeff at(= a K constantmax-Kop). ∆ Furthermore,K value. Meanwhile, the figure in shows the region the where result ∆ ofK the was 1/2 A70755 MPa-T651·m CTor specimen less, the studied da/dN by value Newman at each [13]∆K and value that did of A7075 not depend-T6 alloy on reportedt; specifi- 1/2 bycally, Tokaji for et∆ Kal. between [22]. As 3.5 shown and 5 in MPa the· mfigure,, the the da/dN results values of this were study the almost same for correspond-t = 1, 6, and 1/2 ed11 to mm. the Forresults∆K betweenof Newman 2.5 and[13]3.5 and MPa Tokaji·m et, theal. [21]. da/dN Moreover, values for thet =effect 5.7 mm of t (Newman) was not observedand t =10 in mm the (Jonoda/dN et–ΔK al.)eff were relationship. equal. Similar results were obtained by Ishihara et al. [12] usingFigure A60615 shows CT the specimens relationship and between by Matos da/dN et al. and [14] the using effective A6082 stress CT intensity specimens. factor − Therefore,range ∆K effΔK(=eff Kwasmax effectiveKop). Furthermore,as the FCG driving the figure force shows even thewhen result t changed of the A7075-T651 [12,14]. CT specimen studied by Newman [13] and that of A7075-T6 alloy reported by Tokaji et al. [22]. As shown in the figure, the results of this study almost corresponded to the results of Newman [13] and Tokaji et al. [21]. Moreover, the effect of t was not observed in the Materials 2021, 14, x FOR PEER REVIEW 7 of 17 da/dN–∆Keff relationship. Similar results were obtained by Ishihara et al. [12] using A6061 CT specimens and by Matos et al. [14] using A6082 CT specimens. Therefore, ∆Keff was effective as the FCG driving force even when t changed [12,14].

10-5 A7075-T6, R = 0.1, CT 10-6 Tokaji CT, t = 4 mm 10-7 t = 1 mm 10-8 t = 6 mm t = 11 mm

t = 21 mm da/dN,m/cycle 10-9 Newman, CT t = 5.7 mm 10-10 1 5 10 50 K MPam 1/2 eff

FigureFigure 5. Relationship 5. Relationship between between da/dN da/dN and and effective effective stress stress intensity intensity factor factor range range ΔK∆effK. eff.

WhenWhen the the da/dN da/dN––△K∆ andK and da/dN da/dN––△K∆effK relationshipseff relationships are are plotted plotted on ona log a log–log–log graph, graph, anan inverted inverted S S-shape-shape is shown shown [ 23 [23].]. In In steel steel materials, materials, excluding excluding the vicinity the vicinity of the threshold of the thresholdof the FCG of the (∆ KFCGth) and (ΔK unstableth) and unstable crack growth crack regions,growth regions, the da/dN– the ∆da/dNK relationship–ΔK relation- in the ship in the intermediate FCG region is shown as a straight line; hence, the Paris law applies. However, in aluminum alloys, as shown in Figures 4 5, the intermediate FCG region can be further classified into three regions [20]; ① a low-rate region: da/dN = 10−9 to 108 m/c; ② a medium-rate region: da/dN = 10−8 to 10−7 m/c; ③ a high-rate region: da/dN = 10−7 to 2×106 m/c. Jono et al. [20] reported the fracture surface of the ZK141-T7 aluminum alloy as follows: in the low velocity region of ➀, the fracture surface was featureless and comprised flat and uneven fracture surfaces; in the medium velocity region of ②, it was primarily a flat area (plateau area) surrounded by steps; in the high velocity region of ③, striations appeared. The authors conducted an FCG experiment using an A7075-T6 CT specimen with a thickness of 1 mm and observed striations on the fracture surface at ΔK = 10 MPam1/2 and a da/dN of approximately 107 m/c. Furthermore, it was observed that the striation interval near the specimen surface was narrower than that inside the specimen [24]. In the high-velocity region of ③ (ΔK is 5 MPam1/2 or higher), where the effect of t was clearly observed on da/dN (as evident from the ductile striation on the fracture surface), plastic deformation at the crack tip was considered to be an important factor for FCG.

4.2. Specimen Surface Removal Experiment and ΔKeff To investigate the interaction between the surface and the inside of the specimen affecting the FCG behavior, a surface removal experiment was performed when the fatigue crack was growing. Before and after removing the specimen surface, the change in Kop with the crack growth amount Δa was measured. Figure 6 shows the measurement results. As shown in the figure, the Kop value decreased from 3 to 1.75 MPam1/2 due to the removal of the specimen surface, and the Kop value gradually returned to the original value as the crack grew. The Kop value on the specimen surface was higher than that inside the specimen by approximately 1 MPam1/2. Hence, the effect of the specimen surface on the Kop value was significant. A similar finding has been reported by the authors of [12].

Materials 2021, 14, 664 7 of 16

intermediate FCG region is shown as a straight line; hence, the Paris law applies. However, in aluminum alloys, as shown in Figures4 and5, the intermediate FCG region can be − further classified into three regions [20]; 1 a low-rate region: da/dN = 10 9 to 108 m/c; 2 − − − a medium-rate region: da/dN = 10 8 to 10 7 m/c; 3 a high-rate region: da/dN = 10 7 to 2 × 106 m/c. Jono et al. [20] reported the fracture surface of the ZK141-T7 aluminum alloy as follows: in the low velocity region of 1 , the fracture surface was featureless and comprised flat and uneven fracture surfaces; in the medium velocity region of 2 , it was primarily a flat area (plateau area) surrounded by steps; in the high velocity region of 3 , striations appeared. The authors conducted an FCG experiment using an A7075-T6 CT specimen with a thickness of 1 mm and observed striations on the fracture surface at ∆K = 10 MPam1/2 and a da/dN of approximately 107 m/c. Furthermore, it was observed that the striation interval near the specimen surface was narrower than that inside the specimen [24]. In the high-velocity region of 3 (∆K is 5 MPam1/2 or higher), where the effect of t was clearly observed on da/dN (as evident from the ductile striation on the fracture surface), plastic deformation at the crack tip was considered to be an important factor for FCG.

4.2. Specimen Surface Removal Experiment and ∆Keff To investigate the interaction between the surface and the inside of the specimen affecting the FCG behavior, a surface removal experiment was performed when the fatigue crack was growing. Before and after removing the specimen surface, the change in Kop with the crack growth amount ∆a was measured. Figure6 shows the measurement results. 1/2 As shown in the figure, the Kop value decreased from 3 to 1.75 MPam due to the removal of the specimen surface, and the Kop value gradually returned to the original value as the crack grew. The Kop value on the specimen surface was higher than that inside the Materials 2021, 14, x FOR PEER REVIEW 8 of 17 specimen by approximately 1 MPam1/2. Hence, the effect of the specimen surface on the Kop value was significant. A similar finding has been reported by the authors of [12].

5 A7075-T6 R = 0.1

4 K = 8 MPam1/2 1/2 3

, MPam , 2 op

K Surface removal 1 Before, t = 6 mm After, t = 5 mm 0 0.5 1 1.5 2 a, mm

FigureFigure 6. Specimen 6. Specimen surface surface removal removal experiment experiment (A7075 (A7075-T6,-T6, ΔK= 8 ∆MPa∙mK = 8 MPa1/2, R ·m＝1/2 0.1,, R t = = 0.1,6 mm).t = 6 mm).

4.3. Effects of t and ∆K on Kop 4.3. Effects of t and ΔK on Kop 4.3.1. Kop–∆K Relationship 4.3.1. Kop–ΔK Relationship Figure7a shows the K op–∆K relationship for the A7075-T6 alloy used in this study. In Figure 7a shows the Kop–ΔK relationship for the A7075-T6 alloy used in this study. the ﬁgure, the experimental results for various t from 1 to 21 mm are plotted. Additionally, In the figure, the experimental results for various t from 1 to 21 mm are plotted. Addi- the ﬁgure shows Newman’s result [13] for CT specimens with a thickness of 5.7 mm, which tionally, the figure shows Newman’s result [13] for CT specimens with a thickness of 5.7 correspond to the thickness of 6 mm in this experiment. When ∆K was 5 MPam1/2 or mm, which correspond to the thickness of 6 mm in this experiment. When ΔK was 5 higher, Kop increased with ∆K, and the degree of increase was affected by t. The Kop–∆K MPam1/2 or higher, Kop increased with ΔK, and the degree of increase was affected by t. relationships for each t were approximated by the straight lines; the slopes of the straight The Kop–ΔK relationships for each t were approximated by the straight lines; the slopes of lines were 0.58, 0.32, and 0.2 for t of 1, 6, and 11 and 21 mm, respectively, and hence the straight lines were 0.58, 0.32, and 0.2 for t of 1, 6, and 11 and 21 mm, respectively, and hence decreased with increasing t. Meanwhile, when ΔK was less than 5 MPam1/2, the Kop value was not affected by t. More specifically, the Kop values for t of 5.7, 11, and 21 mm were the same, and the slope of the straight line was not affected by t. The Kop value of t = 6 mm at ΔK = 3.5 MPam1/2 in the present study was larger by approximately 1 MPam1/2 than that of t = 5.7 mm. As this value was larger than the experimental results of other thickness (t = 1, 5.7, 11, and 21 mm) in the low ΔK region, it was assumed that the value contained an error. Hence, in this study, a Kop value of 5.7 mm [13] was used. In order to confirm the certainty and generality of the experimental results for A7075-T6 in this study, this experimental result (Figure 7a) is compared with the experimental results of Bao et al. [3] and Matos et al. [14]. The Kop–ΔK relationships shown below were newly created for this study using the relationships da/dN–ΔK, and da/dN– ΔKeff obtained by Bao and Matos [3,14]. Figure 7b shows the Kop–ΔK relationship measured by Bao using A6061 CT specimens with a thicknesses of 0.35 and 6.35 mm [3]. From the figure, similar to A7075-T6, when ΔK was 5 MPam1/2 or higher, the Kop value of the specimen with a t of 0.35 mm was larger than that with t = 6.35 mm. Furthermore, the slope of the straight line for t = 0.35 mm was approximately 0.5, which was larger than 0.33 for the specimen with t = 6.35 mm. The values of these slopes were almost identical to those of A7075-T6. Meanwhile, when ΔK was 5 MPam1/2 or less, the Kop values for the specimens with t = 0.35 and 6.35 mm were almost equal. Figure 7c shows the Kop–ΔK relationship (R = 0.1) of the A6082-T6 alloy measured by Matos et al. [14]. Kop was measured using a backside strain gage. This figure was recreated by the author based on data obtained directly from the original paper. For comparison, the results (Figure 7a) for A7075-T6 of the present study are shown by broken lines in the figure. From the figure, for a constant ΔK value, Kop decreased with increasing t from 3 to 10 mm; however, the Kop values for the 10 and 25 mm specimens differed only

Materials 2021, 14, 664 8 of 16

1/2 decreased with increasing t. Meanwhile, when ∆K was less than 5 MPam , the Kop value Materials 2021, 14, x FOR PEER REVIEWwas not affected by t. More speciﬁcally, the K values for t of 5.7, 11, and 2110 of mm 17 were the op same, and the slope of the straight line was not affected by t.

Figure 7. Effect of ∆K on the stress intensity factor at the crack opening level Kop for (a) A7075-T6; Figure(b) A6061(Bao); 7. Effect of ΔK (c) A6082-T6on the stress (Matos intensity et al.).factor at the crack opening level Kop for (a) A7075-T6; (b) A6061(Bao); (c) A6082-T6 (Matos et al.).

Materials 2021, 14, 664 9 of 16

1/2 The Kop value of t = 6 mm at ∆K = 3.5 MPam in the present study was larger by approximately 1 MPam1/2 than that of t = 5.7 mm. As this value was larger than the experimental results of other thickness (t = 1, 5.7, 11, and 21 mm) in the low ∆K region, it was assumed that the value contained an error. Hence, in this study, a Kop value of 5.7 mm [13] was used. In order to confirm the certainty and generality of the experimental results for A7075- T6 in this study, this experimental result (Figure7a) is compared with the experimental results of Bao et al. [3] and Matos et al. [14]. The Kop–∆K relationships shown below were newly created for this study using the relationships da/dN–∆K, and da/dN–∆Keff obtained by Bao and Matos [3,14]. Figure7b shows the K op–∆K relationship measured by Bao using A6061 CT specimens with a thicknesses of 0.35 and 6.35 mm [3]. From the figure, similar to A7075-T6, when ∆K 1/2 was 5 MPam or higher, the Kop value of the specimen with a t of 0.35 mm was larger than that with t = 6.35 mm. Furthermore, the slope of the straight line for t = 0.35 mm was approximately 0.5, which was larger than 0.33 for the specimen with t = 6.35 mm. The values of these slopes were almost identical to those of A7075-T6. Meanwhile, when ∆K 1/2 was 5 MPam or less, the Kop values for the specimens with t = 0.35 and 6.35 mm were almost equal. Figure7c shows the K op–∆K relationship (R = 0.1) of the A6082-T6 alloy measured by Matos et al. [14]. Kop was measured using a backside strain gage. This figure was recreated by the author based on data obtained directly from the original paper. For comparison, the results (Figure7a) for A7075-T6 of the present study are shown by broken lines in the figure. From the figure, for a constant ∆K value, Kop decreased with increasing t from 3 to 10 mm; however, the Kop values for the 10 and 25 mm specimens differed only slightly. The slopes of the straight lines shown in the figure showed values similar to the result (broken lines) of A7075-T6 in the figure. As discussed above, although the materials differed, the Kop–∆K relationships of A7075-T6 (this study), A6061 [3], and A6082-T6 [14] exhibited similar characteristics. Elber [1] conducted FCG experiments using A2023-T3 at various R ratios and discovered an empirical formula of ∆Keff/∆K = 0.5 + 0.4R. Considering the relation of ∆Keff = Kmax−Kop,Kop is expressed as the following equation (Equation (2)) [25]:

Kop = {1/(1 − R) − 0.5 − 0.4R}∆K, (2)

Substituting R = 0.1 of this study into Equation (2) yields a Kop/∆K value of 0.57. This value corresponds to a value of 0.58 when t is thin (plane-stress PIFCC) in the present study; however, its value is 0.2 when t is thick (plane-strain PIFCC). PIFCC depends largely on the ∆K level as it is largely dominated by plastic deformation in the specimen surface region. Therefore, strictly speaking, the condition without the inﬂuence of PIFCC is considered to be represented by Kop/∆K = 0 [12]. The physical meanings of the values of 0.57 and 0.2 for Kop/∆K, obtained in the present study, are currently unknown. Further research is needed in the future on this point.

4.3.2. Kop–t Relationship

Figure8 shows the K op–t relationship with the ∆K value as a parameter. Experimental 1/2 data are not available for the values of Kop at ∆K = 4.5 and 5 MPa m in the ﬁgure. Hence, these Kop values were estimated by extrapolating the straight lines in Figure7a to the low ∆K region. In the ﬁgure, the result at ∆K = 12 MPam1/2 (R = 0.1) obtained by Matos [14] and that for t = 5.7 mm obtained by Newman [13] are shown. Materials 2021, 14, x FOR PEER REVIEW 11 of 17

4.3.2. Kop–t Relationship

Figure 8 shows the Kop–t relationship with the ΔK value as a parameter. Experi- mental data are not available for the values of Kop at ΔK = 4.5 and 5 MPa m1/2 in the figure. Hence, these Kop values were estimated by extrapolating the straight lines in Figure 7a to the low ΔK region. In the figure, the result at ΔK = 12 MPam1/2 (R = 0.1) obtained by Matos [14] and that for t = 5.7 mm obtained by Newman [13] are shown. As shown in the figure, at ΔK = 12 MPam1/2, the Kop value decreased from 6 to 2.5 MPam1/2 as t increased from 1 to 15 mm. As the ΔK value decreased to less than 12 MPam1/2, the effect of t on the decreasing rate of Kop became less. At ΔK = 4.5 and 5 MPam1/2, the t dependence of Kop almost disappeared. Meanwhile, when t was 11 to 15 mm or more, Kop was constant without depending on t. This was because the plane-strain PIFCC [26] dominated when t became thicker than a certain value, and Kop was not affected by t. This result is consistent with the fact that the 21 mm slope of Kop–ΔK with t ＝ 11 is of the same value, 0.2 (Figure 7a), which also corresponds with the result [14] of Materials 2021, 14, 664 10 of 16 Matos et al. in Figure 7c.

8 A7075-T6 A6082-T6, 1/2 7 K = 12 MPam1/2 K = 12 MPam 1/2 6 K = 8 MPam

K = 6 MPam1/2 1/2 5 K = 5 MPam1/2 1/2 4 K = 4.5 MPam

MPam 3 op

K 2 1 0 0 5 10 15 20 25 30 Specimen thickness, t mm

FigureFigure 8. 8. EffectEffect of of tt onon K Kopop (R(R = = 0.1). 0.1).

1/2 TheAs shownresults inof the FCC ﬁgure, experiments at ∆K = 12 can MPam be summarized, the Kop asvalue shown decreased in Table 3. from 6 to 2.5 MPam1/2 as t increased from 1 to 15 mm. As the ∆K value decreased to less than 1/2 Table12 MPam 3. Effects, theof t and effect ΔK of ont Konop. the decreasing rate of Kop became less. At ∆K = 4.5 and 1/2 5 MPam , the t dependence of Kop almost disappeared. Meanwhile, when t was 11 to ΔK > 5 MPa∙m1/2 15Classification mm or more, by K op was constant without depending on t. This was because the plane- ③ΔK < 5 MPa∙m1/2 ➀ ② strain PIFCCΔK [26] dominated when t became thicker than a certain value, and Kop was not affected by t. This result is consistent with the factt < that11–15 the mm 21 mm slopet of > 11 Kop–15–∆ mmK with t = 11 is of the same value, 0.2 (Figure7a), whichPIFCC also under corresponds interaction with the result [ 14] of Plane strain MatosClassification et al. in Figure of 7c. RIFCC between plane stress and PIFCC TheFCC results of the FCCKop/ΔK experiments = 0.2 can be summarizedplane strain as shown in Table3. Kop/ΔK = 0.2 Kop/ΔK = 0.2–0.58 TableEffect 3. Effects of t of t and ∆K onNo K opeffect. Affected No effect

➀ When ΔK was 5 MPam1/2 or higher and t was 11 to∆ K15 > mm 5 MPa or· mless,1/2 PIFCC under 1/2 theClassiﬁcation interaction by of∆ K plane 3 stress∆K < 5 and MPa · planem strain occurred. 1 As t became thinner, 2 the plane-stress PIFCC dominated, the Kop value increased,t < 11–15 mm and the slopet > of 11–15 Kop mm–ΔK approached 0.58. On the contrary, as t became thicker,PIFCC under the plane-strain PIFCC became more significant, and the slope of Kop–ΔK approachedinteraction 0.2. between ② When ΔKPlane was strain5 MPam1/2 RIFCC orClassiﬁcation higher and oft was FCC 11 to 15 mm or more, theplane plane stress-strain and planePIFCC dominated,PIFCC the t de- Kop/∆K = 0.2 strain Kop/∆K = 0.2 Kop/∆K = 0.2–0.58 Effect of t No effect Affected No effect

1 When ∆K was 5 MPam1/2 or higher and t was 11 to 15 mm or less, PIFCC under the interaction of plane stress and plane strain occurred. As t became thinner, the plane-stress PIFCC dominated, the Kop value increased, and the slope of Kop–∆K approached 0.58. On the contrary, as t became thicker, the plane-strain PIFCC became more signiﬁcant, 1/2 and the slope of Kop–∆K approached 0.2. 2 When ∆K was 5 MPam or higher and t was 11 to 15 mm or more, the plane-strain PIFCC dominated, the t dependence of Kop 1/2 disappeared, and the slope of Kop–∆K approached 0.2. 3 When ∆K was 5 MPam or less, it was assumed that RIFCC occurred instead of PIFCC because the plastic deformation was small. As the deformation was small, no interaction occurred between the plane-stress and plane-strain regions, and the slope of Kop–∆K was 0.2 regardless of t. Materials 2021, 14, 664 11 of 16

In the ASTM standard [27], the following equation is recommended as a condition for t to obtain a plane-strain fracture toughness value KIC using a CT specimen. Here, σY is the yield strength of the material:

t ≥ 2.5(KIC/σY), (3)

1/2 Osaki et al. [28] reported that the KIC and σY values of A7075-T6 were 31.2 MPam and 486 MPa for SL (Short transverse–Longitudinal) specimens, respectively, and 43.6 MPam1/2 and 548 MPa for LS (Longitudinal–Short transverse) specimens, respectively. By substituting these values into Equation (3), the minimum values of t for obtaining KIC were calculated; the values of t = 10.3 and 15.8 mm were obtained for SL and LS specimens, respectively. Compared with these values, the minimum value of t (11 to 15 mm) for the plane-strain PIFCC obtained in this study is considered a reasonable value.

4.4. Shape of the Front Edge of the Crack Figure9 shows the fracture surfaces of specimens with various t (1, 6, 11, and 21 mm). The leading edge shape of the crack was revealed by dyeing it red with a penetrant ﬂaw dye. As shown in the ﬁgure, the leading edge shape of the crack was not straight but curved, i.e., the crack grew faster in the specimen interior than on the specimen surface. Based on the specimen surface removal experiment shown in Figure6, the K op near the specimen surface was higher than that at the specimen interior. The FCG driving force ∆Keff (= Kmax − Kop) near the specimen surface was lower than that at the specimen interior. Therefore, the FCG velocity near the specimen surface area became lower than that at the specimen interior, and the leading edge of the crack exhibited an arcuate shape. Next, the radius of curvature Materials 2021, 14, x FOR PEER REVIEW 13 of 17 r was measured by approximating the leading edge of the crack with an arc; the circular arc passed through both the center and surface of the specimen.

Figure 9. Leading edge shapes of the cracks of specimens with various thicknesses. Figure 9. Leading edge shapes of the cracks of specimens with various thicknesses. ΔK=12.0 1/2 MPam∆K =1/2 12.0, (a) MPam t = 1 mm,(; (ab)) tt == 16 mm;mm; ((bc) t = 11 6 mm; mm; ( c(d) )t t= = 11 21 mm; mm. ( d) t = 21 mm.

Figure 10 is a log–log graph showing the relationship between the radii of curvature r and t. As shown, for a specimen with a small t, r was small and increased with t.

100

A7075-T6 CT specimen

Curvature Curvature radius, r mm 0.1 0.5 1 5 10 50 Specimen thickness, t mm

Figure 10. Relationship between curvature radius r and t. ΔK＝12 MPa∙m1/2.

5. Depression Depth d Generated at the Crack Tip

In order to quantitatively consider the effect of t on the Kop–ΔK relationship (Figure 7) and the transition conditions for the ΔK value from RIFCC to PIFCC (Table 3), the depression depth d generated at the crack tip of the CT specimen was analyzed using 3D FEM. Analysis was performed for various ΔK values with t as a parameter.

5.1. Effect of d on Kop

Materials 2021, 14, x FOR PEER REVIEW 13 of 17

Materials 2021, 14, 664 Figure 9. Leading edge shapes of the cracks of specimens with various thicknesses. ΔK=12.0 12 of 16 MPam1/2, (a) t = 1 mm; (b) t = 6 mm; (c) t = 11 mm; (d) t = 21 mm.

Figure 10 is a log–log graph showing the relationship between the radii of curva- Figure 10 is a log–log graph showing the relationship between the radii of curvature r ture r and t. As shown, for a specimen with a small t, r was small and increased with t. and t. As shown, for a specimen with a small t, r was small and increased with t.

100

Materials 2021, 14, x FOR PEER REVIEW 14 of 17

Figure 11 shows the relationship between d and ΔK obtained using the 3D elasto- plastic1 FEM for various t values. The curves in the figure were obtained by approximating the data with quadratic curves using the least-squares method. As shown in the figure, atA7075-T6 the region where ΔK was 5 MPam1/2 or higher (regions CT specimen ➀ ➁ Curvature radius, r mm and0.1 in Table 3), d increased as t became thinner and ΔK increased; d can be con- sidered0.5 as a 1physical quantity5 10 that comprehensively50 represents the interaction between the surface Specimen of the specimen thickness, (plane t mm-stress deformation) and the specimen interior (plane-strain deformation). Therefore, when t was small, the constraint on the lateral Figure 10. Relationship between curvature radius r and t. ∆K = 12 MPa·m1/2. Figurecontraction 10. Relationshi due top between the specimen curvature interior radius wasr and smallt. ΔK＝, and12 MPa∙m, hence1/2., d increased. Further- more,5. Depression as ΔK increase Depth d,d Generatedthe out-of- atplane the Crackplastic Tip deformation at the crack tip increased, 5. andDepression, hence, Dd epthincreased. d Generated Meanwhile, at the C atrack the T regionip where ΔK was less than 5 MPa In order to quantitatively consider the effect of t on the K –∆K relationship (Figure7 ) m1/2(region ➂ in Table 3), the d value was small, i.e., approximatelyop 0.05 µm, regardless andInthe order transition to quantitatively conditions consider for the ∆ theK value effect from of t RIFCCon the K toop PIFCC–ΔK relationship (Table3), the (Fig depres-ure of t. Hence, it was believed that RIFCC occurred instead of PIFCC because the plastic 7) sionand the depth transitiond generated conditions at the for crack the tip ΔK of value the CT from specimen RIFCC wasto PIFCC analyzed (Table using 3), the 3D de- FEM. deformation necessary to cause PIFCC did not occur in this region. In a PIFCC, the pressionAnalysis depth was d performed generated for at variousthe crack∆K tip values of the with CT tspecimenas a parameter. was analyzed using 3D FEM.amount Analysis of plastic was performed stretch ε (plastic for various wedge) ΔK valuesthat occurs with behindt as a parameter. the crack tip and that at the5.1. crack Effect-tip of d ope on Kningop displacement are considered to be important factors [12]. It can be assumed that the material at the lateral contraction portion is supplied as the material 5.1. EffeFigurect of d on11 Kshowsop the relationship between d and ∆K obtained using the 3D elastoplastic forFEM forming for various the plastict values. stretch The ε curves behind in the the crack ﬁgure tip. were In that obtained case, it by isapproximating assumed that as the d becomesdata with deeper, quadratic both curves ε and using Kop increase. the least-squares method.

Figure 11. Relationship between depth d and ΔK∆K..

1/2 5.2. EffectAs shown of the L ineading the ﬁgure, Edge S athape the of region the Crack where on ∆theK d was Value 5 MPam or higher (regions 1 and 2 in Table3), d increased as t became thinner and ∆K increased; d can be considered as a Figure 12 shows the relationship between d and ΔK obtained using FEM for r = 4.5 physical quantity that comprehensively represents the interaction between the surface of the mm, the radius of curvature of the leading edge of the crack, and r = ∞ (straight edge). specimen (plane-stress deformation) and the specimen interior (plane-strain deformation). The FEM analysis conditions were t = 6 mm and a = 15 mm. The curves in the figure Therefore, when t was small, the constraint on the lateral contraction due to the specimen were obtained by approximating the analysis result with a quadratic curve. As shown in interior was small, and, hence, d increased. Furthermore, as ∆K increased, the out-of-plane the figure, at a constant ΔK value, the value of d at r = 4.5 mm was larger than that at r = plastic deformation at the crack tip increased, and, hence, d increased. Meanwhile, at the ∞. Moreover, this difference increased with ΔK. region where ∆K was less than 5 MPa m1/2 (region 3 in Table3), the d value was small, Because Kop increases with d, the value of Kop for a curved edge with a radius of i.e., approximately 0.05 µm, regardless of t. Hence, it was believed that RIFCC occurred curvature of r = 4.5 mm, increases to larger than that for a straight edge. Camas et al. [15] instead of PIFCC because the plastic deformation necessary to cause PIFCC did not occur investigated the effect of r on the plastic zone size in front of a crack using a 3D elastoplastic FEM and obtained results similar to those in this study. The leading edge of the crack immediately after the start of the FCG experiment was a straight edge (r = ∞); however, as the crack grew, the leading edge shape of the crack gradually exhibited a curve (Figure 9). When the leading edge of the crack became curved, the value of Kop on the specimen surface increased to larger than that when it was straight; therefore, the change in the leading edge of the crack into an arc shape accelerated. Recently, FCG and Kop has been evaluated using FEM simulation [29,30]. To

Materials 2021, 14, 664 13 of 16

in this region. In a PIFCC, the amount of plastic stretch ε (plastic wedge) that occurs behind the crack tip and that at the crack-tip opening displacement are considered to be important factors [12]. It can be assumed that the material at the lateral contraction portion is supplied as the material for forming the plastic stretch ε behind the crack tip. In that case, it is assumed that as d becomes deeper, both ε and Kop increase.

5.2. Effect of the Leading Edge Shape of the Crack on the d Value

Materials 2021, 14, x FOR PEER REVIEW Figure 12 shows the relationship between d and ∆K obtained using FEM for r =15 4.5 of mm 17 , the radius of curvature of the leading edge of the crack, and r = ∞ (straight edge). The FEM analysis conditions were t = 6 mm and a = 15 mm. The curves in the ﬁgure were obtained by approximating the analysis result with a quadratic curve. As shown in the ﬁgure, at a predictconstant more∆K value, accurate the FCG value and of d Katopr =behavior, 4.5 mm was an FCG larger simulation than that at incorporatingr = ∞. Moreover, the shapethis difference change of increased the leading with edge∆K. of the crack is necessary.

Figure 12. Effect of the radius of curvature r of the leading edge of the crack on the d value. t = 6 mm, Figure 12. Effect of the radius of curvature r of the leading edge of the crack on the d value. t ＝ 6 mm,crack crack length lengtha = 15a = mm. 15 mm. Because K increases with d, the value of K for a curved edge with a radius of 6. Conclusions op op curvature of r = 4.5 mm, increases to larger than that for a straight edge. Camas et al. [15] investigatedIn this study, the effect FCG of r experimentson the plastic zoneand Ksizeop inmeasurements front of a crack were using performed a 3D elastoplastic using A7075FEM and-T6 CT obtained specimens results with similar different to those thicknesses in this study. t. In addition, an FEM analysis was performedThe leading to supplement edge of the and crack quantitatively immediately consider after the the start exp oferimental the FCG experimentresults. The wasef- fectsa straight of t and edge ΔK ( ron= FCG∞); however, and Kop were as the studied, crack grew, and the following leading edge conclusions shape of were the crack ob- tainedgradually: exhibited a curve (Figure9). When the leading edge of the crack became curved, 1.the valueIn the of da/dN Kop on–ΔK the relationship, specimen surface in the increased region towhere larger ΔK than was that 5 whenMPam it1/2 was or straight;higher, therefore,da/dN the at changea constant in the ΔK leading value increased edge of the as crack t increased into an arcfrom shape 1 to accelerated. 11 mm. The Recently, da/dN FCGbetween and Kop thas = 11 been and evaluated21 mm was using the FEMsame. simulation Meanwhile, [29 ,in30 ].the To region predict where more ΔK accurate was FCGless and than Kop 5behavior, MPam1/2 an, the FCG effect simulation of t on da/dN incorporating was not theobserved. shape changeFurthermore, of the leadingit was edgediscovered of the crack that is necessary.the relationship of da/dN–ΔKeff was not affected by t and that ΔKeff was an effective FCG driving force. 6. Conclusions 2. When ΔK was 5 MPam1/2 or higher and t was 11 to 15 mm or less (region ➀), the Kop In this study, FCG experiments and Kop measurements were performed using A7075- value increased as t decreased. The slope of Kop–ΔK was asymptotic to 0.58 as t be- T6 CT specimens with different thicknesses t. In addition, an FEM analysis was performed came thinner. Conversely, as t thickened, the slope of Kop–ΔK approached 0.2. When to supplement and quantitatively consider the experimental results. The effects of t and ΔK was 5 MPam1/2 or higher and t was 11 to 15 mm or more (region ②), plane-strain ∆K on FCG and Kop were studied, and the following conclusions were obtained: PIFCC dominated, and the slope of Kop–ΔK approached 0.2, which did not depend 1/2 1. onIn t. the Meanwhile da/dN–,∆ inK the relationship, region where in the ΔKregion was 5 MPam where1/2∆ Kor waslower 5 MPam(region ③or), RIFCC higher, occurredda/dN atinstead a constant of PIFCC∆K value because increased the plastic as t increased deformation from was 1 to small, 11 mm. and The the da/dN slope between t = 11 and 21 mm was the same. Meanwhile, in the region where ∆K was of Kop–ΔK became 0.2. Hence, ΔK = 5 MPam1/2 was the transition stress intensity 1/2 factorless than from 5 RIFCC MPam to ,PIFCC. the effect of t on da/dN was not observed. Furthermore, it was discovered that the relationship of da/dN–∆Keff was not affected by t and that ∆Keff 3. When the specimen surface was removed during the FCG, the Kop value decreased; was an effective FCG driving force. subsequently, the FCG velocity returned gradually to the original value. The experimental results showed that the interaction between plane stress and plane strain was an important factor for FCG and Kop. 4. The leading edge shape of the fatigue crack showed an arc shape rather than a straight line. The radius of curvature r increased with t. Due to the interaction between the specimen surface and the specimen interior, ΔKeff near the specimen surface became lower than that inside the specimen. Therefore, the FCG velocity near the specimen surface became lower than that inside the specimen, and the leading edge of the crack arcuated. 5. According to the FEM analysis of the depression depth d in the region where ΔK was 5 MPam 1/2 or higher, the value of d increased as t decreased, and ΔK increased.

Materials 2021, 14, 664 14 of 16

2. When ∆K was 5 MPam1/2 or higher and t was 11 to 15 mm or less (region 1 ), the Kop value increased as t decreased. The slope of Kop–∆K was asymptotic to 0.58 as t became thinner. Conversely, as t thickened, the slope of Kop–∆K approached 0.2. When ∆K was 5 MPam1/2 or higher and t was 11 to 15 mm or more (region 2 ), plane-strain PIFCC dominated, and the slope of Kop–∆K approached 0.2, which did not depend on t. Meanwhile, in the region where ∆K was 5 MPam1/2 or lower (region 3 ), RIFCC occurred instead of PIFCC because the plastic deformation was small, and 1/2 the slope of Kop–∆K became 0.2. Hence, ∆K = 5 MPam was the transition stress intensity factor from RIFCC to PIFCC. 3. When the specimen surface was removed during the FCG, the Kop value decreased; subsequently, the FCG velocity returned gradually to the original value. The experimental results showed that the interaction between plane stress and plane strain was an important factor for FCG and Kop. 4. The leading edge shape of the fatigue crack showed an arc shape rather than a straight line. The radius of curvature r increased with t. Due to the interaction between the specimen surface and the specimen interior, ∆Keff near the specimen surface became lower than that inside the specimen. Therefore, the FCG velocity near the specimen surface became lower than that inside the specimen, and the leading edge of the crack arcuated. 5. According to the FEM analysis of the depression depth d in the region where ∆K was 5 MPam 1/2 or higher, the value of d increased as t decreased, and ∆K increased. When t was thin, the resistance against lateral contraction by the specimen inside was small, and, thus, d was large. When ∆K was large, d increased because the out-of-plane plastic deformation at the crack tip increased. Meanwhile, when ∆K was 5 MPa m1/2 or lower, the d value was small, i.e., approximately 0.05 µm, regardless of t; hence, RIFCC occurred instead of PIFCC. Therefore, the effects of t and ∆K on Kop (conclusion (2)) can be quantitatively and physically explained using d.

Author Contributions: Conceptualization, S.I.; Data curation, K.M.; Formal analysis, K.M.; Investi- gation, K.M.; Supervision, S.I. and N.O.; Writing—original draft, K.M. and S.I.; Writing—review & editing, S.I. All authors have read and agreed to the published version of the manuscript. Funding: This research received no external funding. Institutional Review Board Statement: Not applicable. Informed Consent Statement: Not applicable. Data Availability Statement: The data presented in this study are available on request from the corresponding author. Acknowledgments: We would like to express our deep gratitude to Arthur J. McEvily (University of Connecticut) for his guidance in performing this research. Conﬂicts of Interest: The authors declare no conﬂict of interest.

Abbreviations

FCC Fatigue crack closure FCG Fatigue crack growth 1/2 Kmin Minimum stress intensity factor (MPam ) 1/2 Kmax Maximum stress intensity factor (MPam ) 1/2 ∆K Stress intensity factor range, ∆K = KmaxKmin (MPam ) 1/2 Kop Crack opening stress intensity factor (MPam ) 1/2 ∆Keff Effective stress intensity factor range, ∆Keff = Kmax − Kop (MPam ) d Plastic lateral contraction (depression depth; m) t Specimen thickness (m) Materials 2021, 14, 664 15 of 16

PIFCC Plasticity-induced fatigue crack closure RIFCC Roughness-induced fatigue crack closure R Stress ratio Ra Fracture surface roughness CT Compact tension FEM Finite element method P Applied load (N) a Crack length (m) N Number of stress cycles (cycles) da/dN FCG rate (m/cycle) α Non-dimensional quantity (= a/W, W = 57.2 × 10−3 (m)) r Radius of curvature of the leading edge of the crack (m)

References 1. Elber, W. Fatigue crack closure under cyclic tension. Eng. Fract. Mech. 1970, 2, 37–45. 2. Elber, W. The Signiﬁcance of Fatigue Crack Closure. In Damage Tolerance in Aircraft Structures; ASTM International: West Conshohocken, PA, USA, 2009; p. 230. 3. Bao, H. Fatigue Crack Growth under Plane Stress Conditions. Ph. D. Dissertation, University of Connecticut, Storrs, CT, USA, 1994. 4. Minakawa, K.; McEvily, A. On crack closure in the near-threshold region. Scr. Met. 1981, 15, 633–636. [CrossRef] 5. Suresh, S.; Ritchie, R.O. A geometric model for fracture surface roughness-induced crack closure during fatigue crack-growth. J. Met. 1982, 34, 71. 6. Morris, W.; James, M.; Buck, O. A simple model of stress intensity range threshold and crack closure stress. Eng. Fract. Mech. 1983, 18, 871–877. [CrossRef] 7. Paris, P.; Bucci, R.; Wessel, E.; Clark, W.; Mager, T. Extensive Study of Low Fatigue Crack Growth Rates in A533 and A508 Steels. In Stress Analysis and Growth of Cracks: Proceedings of the 1971 National Symposium on Fracture Mechanics: Part 1; ASTM International: West Conshohocken, PA, USA, 2009; p. 141. 8. Skelton, R.P.; Haigh, J.R. Fatigue crack growth rates and thresholds in steels under oxidizing conditions. Mater. Sci. Eng. 1978, 36, 17–25. [CrossRef] 9. Suresh, S.; Zamiski, G.F.; Ritchie, D.R.O. Oxide-Induced Crack Closure: An Explanation for Near-Threshold Corrosion Fatigue Crack Growth Behavior. Met. Mater. Trans. A 1981, 12, 1435–1443. [CrossRef] 10. Minakawa, K.; LeVan, G.; McEvily, A.J. The inﬂuence of load ratio on fatigue crack growth in 7090-t6 and in9021-t4 p/m aluminum alloys. Met. Mater. Trans. A 1986, 17, 1787–1795. [CrossRef] 11. Schivje, J. Mechanics of Fatigue Crack Closure; ASTM: West Conshohocken, PA, USA, 1988; pp. 5–34. 12. Ishihara, S.; Sugai, Y.; McEvily, A.J. On the Distinction Between Plasticity- and Roughness-Induced Fatigue Crack Closure. Met. Mater. Trans. A 2012, 9, 3086–3096. [CrossRef] 13. Newman, J.C., Jr.; Anagnostou, E.L.; Rusk, D. Fatigue and crack-growth analyses on 7075-T651 aluminum alloy coupons under constant- and variable-amplitude loading. Int. J. Fatigue 2014, 62, 133–143. [CrossRef] 14. De Matos, P.F.P.; Nowell, D. Experimental and numerical investigation of thickness effects in plasticity-induced fatigue crack closure. Int. J. Fatigue 2009, 31, 1795–1804. [CrossRef] 15. Camas, D.; Lopez-Crespo, P.; Gonzalez-Herrera, A.; Moreno, B. Numerical and experimental study of the plastic zone in cracked specimens. Eng. Fract. Mech. 2017, 185, 20–32. [CrossRef] 16. Rozumek, D.; Faszynka, S. Fatigue crack growth in 2017A-T4 alloy subjected to proportional bending with torsion. Frattura ed Integrità Strutturale 2017, 11, 23–29. [CrossRef] 17. Ishihara, S.; McEvily, A.J. On the early initiation of fatigue cracks in the high cycle regime. In Proceedings of the 12th International Conference on Fracture, Ottawa, ON, Canada, 12–17 July 2009; pp. 12–17. 18. Murakami, Y. Stress Intensity Factors; U.S. Department of Energy: Washington, DC, USA, 1987. 19. Kikukawa, M.; Jono, M.; Tanaka, K. Fatigue crack closure at low stress intensity level. In Proceedings of the Second International Conference on Mechanical Behavior of Materials, Boston, MA, USA, 16–20 August 1976; pp. 716–720. 20. Marc, MSC Software Corporation—MSC SimCompanion. Available online: https://www.mscsoftware.com/(accessed on 30 January 2021). 21. Jono, M.; Song, J.; Sugeta, A. Fatigue crack growth and crack closure under variable loadings on aluminum alloys. (Effect of load variation on characteristic form of crack growth rate curve in region II). J. Soc. Mater. Sci. Jpn. 1985, 34, 1193–1199. [CrossRef] 22. Tokaji, K.; Ogawa, T.; Kameyama, Y. The effect of stress ratio on growth behaviour of small fatigue cracks in an aluminum alloy 7075-T6 (With special interest in stage I crack growth). J. Soc. Mater. Sci. Jpn. 1989, 38, 1019–1025. [CrossRef] 23. Suresh, S. Second edition, Fatigue of Materials. Aeronaut. J. 1999, 1026, 398. 24. Masuda, K.; Ishihara, S.; McEvily, A.J.; Okane, M. Experimental investigation of thickness effects on fatigue crack closure behavior in Al7075-T6 alloy. In Proceedings of the 7th Engineering Integrity Society International Conference on Durability & Fatigue, Cambridge, UK, 3–5 July 2017; pp. 456–465. Materials 2021, 14, 664 16 of 16

25. Anderson, T.L. Fracture Mechanics: Fundamentals and Applications, 2nd ed.; CRC Press, Inc.: Boca Raton, FL, USA, 1995. 26. Pippan, R.; Hohenwarter, A. Fatigue crack closure: A review of the physical phenomena. Fatigue Fract. Eng. Mater. Struct. 2017, 40, 471–495. [CrossRef][PubMed] 27. ASTM E 399-90, Standard Test Method for Plane-Strain Fracture Toughness of Metallic Materials. America Society for Testing Materials, Philadelphia. 1990. Available online: https://www.astm.org/DATABASE.CART/HISTORICAL/E399-90R97.htm (accessed on 26 January 2021). 28. Ohsaki, S.; Utsue, M.; Yokofujita, H.; Iino, M. Fracture Toughness and SCC Behavior of 7075 Aluminum Alloy under Mixed-Mode I-II Loading. Zairyo 1996, 45, 661–666. [CrossRef] 29. Roychowdhury, S.; Dodds, R.H., Jr. A numerical investigation of 3-D small-scale yielding fatigue crack growth. Eng. Fract. Mech. 2003, 70, 2363–2383. [CrossRef] 30. Camas, D.; Garcia-Manrique, J.; Perez-Garcia, F.; Gonzalez-Herrera, A. Numerical modelling of three-dimensional fatigue crack closure: Plastic wake simulation. Int. J. Fatigue 2020, 131, 105344. [CrossRef]