Fatigue Life Appraisal and Its Corrected Stress Intensity Factor for Repaired Oﬀ-Centrallycracked Aluminum Plates

materials

Article Fatigue Life Appraisal and Its Corrected Stress Intensity Factor for Repaired Oﬀ-CentrallyCracked Aluminum Plates

Xiang You 1,2, Zhiyu Wang 1,2,*, Xiafang Zhou 1, Zifeng Liu 1, Ruijuan Jiang 3,* and Weiming Gai 3

1 Key Laboratory of Deep Underground Science and Engineering (Ministry of Education), School of Architecture and Environment, Sichuan University, Chengdu 610065, China; you_ﬂ[email protected] (X.Y.); [email protected] (X.Z.); [email protected] (Z.L.) 2 Sichuan Provincial Key Laboratory of Failure Mechanics and Engineering Disaster Prevention & Mitigation, Sichuan University, Chengdu 610065, China 3 Innovation Design Department, Shenzhen Municipal Engineering Design and Research Institute Co. Ltd., Shenzhen 518029, China; [email protected] * Correspondence: [email protected] (Z.W.); [email protected] (R.J.); Tel.: +86-28-8540-6919 (Z.W.)

Received: 27 July 2020; Accepted: 31 August 2020; Published: 10 September 2020

Abstract: This paper presents an experimental study on the fatigue life estimation of off-centrally cracked aluminum plates. Typical theoretical equations for off-central, central and edge cracks were reviewed and compared in terms of their sensitive parameters and applicability. A finite element model has been validated in its capacity in modelling the influences of eccentricity and crack size on the boundary correction coefficients. The Forman equation has been employed along with numerical results for the prediction of fatigue lives. Based on the test data, the fatigue life results of aluminum plates with and without patched laminate repair have been compared with codified fatigue classes. It is demonstrated that the repair at the crack tip close to the plate edge is effective in the fatigue life improvement for off-centrally crackedaluminum plates.

Keywords: oﬀ-central crack; fatigue life; stress intensity factor; fracture mechanics

1. Introduction Tension structural components, such as tension clamps made up of clevis cover plates in tension, play an important role in many railway [1,2] or aircraft [3,4] mechanical systems. Using codified variables, it is relatively easy to find a premade tension structural component to suit the specifications of a specific project. Fatigue cracks in a tension structural component usually initiate at notches as a result of stress concentration. Since the fatigue cracks can start very early within the fatigue endurance, the fatigue life estimation becomes a major question for the fatigue crack propagation analysis. For such a purpose, the stress intensity factors for a central crack or an edge crack as typical crack formations have been intensively investigated by contemporary researchers and documented in design manuals, such as Reference [5]. During services, however, mechanical failure can take place, due to the formation of cracks at the initial imperfections or flaws away from the center or the edge. An example in Figure1 indicates an offset imperfection-related fatigue crack initiated between the edge and the center line of the clevis cover plate subject to tension.

Materials 2020, 13, 4014; doi:10.3390/ma13184014 www.mdpi.com/journal/materials Materials 2020, 13, 4014 2 of 13 Materials 2020, 13, x FOR PEER REVIEW 2 of 13

Tension

Eccentric crack Clevis cover plate

Tension

Figure 1. Off-centralOﬀ-central crack in a typical tension clamp.

The linear elasticelastic fracturefracture mechanicsmechanics method method has has gained gained a a good good application application for for the the prediction prediction of theof the strength strength and and life oflife cracked of cracked structures, structures, when thewhen understanding the understanding of the crackof the tip crack stress tip intensity stress factorintensity as afactor function as a offunction applied of load applied and structuralload and structural geometry geometry is known. is The known. crack The tip cancrack be tip assumed can be asassumed an energy as an sink, energy around sink, which around dissipation which takesdissipation place for takes energy place propagation. for energy Thepropagation. research work The reportedresearch work by Isida reported [6] seems by Isida to give [6] the seems first to theoretical give the first equation theoretical in the equation prediction in ofthe stress prediction intensity of factorsstress intensity for the tension factors of for an the eccentrically tension of cracked an eccentri strip.cally The cracked Airy stress strip. function The Airy for astress state offunction generalized for a planestate of stress generalized can be formulated plane stress in can terms be offormulated complex potentials, in terms of which complex can bepotentials, expanded which in Laurent’s can be series,expanded convergent in Laurent’s in a regionseries, boundedconvergent by in two a regi certainon bounded concentric by circles. two certain The resultant concentric stress circles. intensity The factorsresultant are stress given inintensity power series,factors and are their given related in coepowerfficients series, are tabulated.and their Furtherrelated understandingcoefficients are of thetabulated. stress intensity Further factorunderstanding of eccentrically of the cracked stress plateshasintensity deepenedfactor of ineccentrically theory in the cracked past two plateshas decades. Fordeepened example, in theory Wang in [7 ]the proposed past two a decades. quite useful, For example, simple theoretical Wang [7] proposed expression a quite with theuseful, balance simple of forcetheoretical and moment expression for the with above-mentioned the balance of stress force intensity and moment factor, which for the results above-mentioned in a good correlation stress withintensity Isida’s factor, report which [6] withresults less in than a good 6% error.correlation Following with Isida’s Isida’s report report [6] [4], with Wang less et than al. [8 ]6% recently error. developedFollowing Isida’s means basedreport on[4], the Wang Westergaard et al. [8] stressrecently function, developed which means is simplified based on to providethe Westergaard solutions forstress the function, crack opening which displacement is simplified and to stress-strainprovide solutions field ahead for the of ancrack eccentric opening crack. displacement The length fromand thestress-strain maximum field crack ahead opening of an site eccentric to the crackcrack. tipTh wase length suggested from the as themaximum length ofcrack the crackopening when site the to Westergaardthe crack tip stresswas suggested distribution as is the concerned. length of To the validate crack when the theoretical the Westergaard findings, stress numerical distribution modelling is hasconcerned. also been To performed validate the by researcherstheoretical findings, in the study numerical of stress modelling intensity factors has also of eccentricallybeen performed cracked by plates.researchers For example,in the study Wang of etstress al. [ 9intensity] adopted fact a macroors of elementeccentrically method cracked in the plates. calculation For example, of stress intensityWang et factorsal. [9] adopted and analysis a macro of cracked element plate. method It was in shownthe calculation that the crack of stress propagation intensity may factors become and stableanalysis when of cracked the eccentricity plate. It is was considered. shown that Later, the the crack detrimental propagation effect may of the become presence stable of off whenset cracks the oneccentricity the tension is considered. plate with Later, eccentric the cracksdetrimental embedded effect of in the bi-materials presence of was offset observed cracks byon Ismailthe tension [10], sinceplate thewith formation eccentric of cracks mixed modeembedded stress in intensity bi-mater factorsials was at the observed crack tip by lead Ismail to premature [10], since failure. the Subsequently,formation of Hanmixed et al.mode [11] comparedstress intensity and discussed factors theat ethefficiencies crack oftip the lead displacement to premature extrapolation failure. method,Subsequently, stress extrapolationHan et al. [11] method, compared node displacementand discussed method, the efficiencies and J-integral of the method displacement based on finiteextrapolation element analysesmethod, ofstress stress extrapolation intensity factors method of eccentrically, node displacement cracked plates. method, Triangular and J-integral singular elements,method based and anon element finite element size of 1analyses/10 of the of crack stress length, intensity are suggestedfactors of foreccentrically increased accuracycracked plates. in the predictionTriangular ofsingular stress intensityelements, factors. and an Recently, element an size equivalent of 1/10 of model the crack was developed length, are by suggested He et al. [12for] forincreased an eccentrically accuracy in cracked the prediction plate with of clamped stress intensity ends. Experimental factors. Recently, data an waste equivalent was avoided model using was approximatedeveloped by solutions, He et al. [12] since for invalid an eccentrically crack growth cracked data, failingplate with to meet clamped the crack ends. symmetry Experimental condition, data couldwaste bewas utilized avoided in using the data approximate processing. solutions, since invalid crack growth data, failing to meet the crackWhere symmetry cracking condition, occurs could in aluminum be utilized plates, in the adata satisfactory processing. repair may be made by composite patchesWhere and cracking adhesive occurs bonding in aluminum in the reinforcement. plates, a satisfactory Their attractive repair may merits be made are good by composite flexibility, highpatches strength-to-weight and adhesive bonding ratio, and in the good reinforcement. capacity in resisting Their attractive environment merits deterioration are good flexibility, or corrosion. high Ayatollahistrength-to-weight and Hashemi ratio, [13 and] conducted good capacity finite element in resisting modelling environment in the study deterioration of the effect or of compositecorrosion. patching,Ayatollahi such and as Hashemi composite [13] laminate conducted configuration, finite el andement adhesive modelling properties in the on thestudy crack of tip the parameters. effect of Relatedcomposite stress patching, intensity such factors as alongcomposite with thelaminate fracture configuration, strength of the and repaired adhesive specimens properties under on mixed the modecrack tip loading parameters. conditions Related were stress analyzed. intensity Khan factors et al. [14 along] performed with the fatigue fracture tests strength on V-notched of the repaired repaired andspecimens unrepaired under pre-cracked mixed mode specimens loading underconditions fatigue were loading. analyzed. It was Khan found et al. that [14] very performed high fatigue fatigue life extensiontests on V-notched can be achieved repaired when and the unrepaired patch precedes pre-cracked the overload, specimens which under can be fatigue owed toloading. a cumulative It was retardationfound that very effect high of patching fatigue andlife overload.extension Recently,can be achieved the authors when have the investigated patch precedes the etheffect overload, of CFRP (carbonwhich can fiber be reinforced owed to polymer)a cumulative reinforcement retardation on effect the fatigue of patching strength and of corrugatedoverload. Recently, plates [15 ,16the], authors have investigated the effect of CFRP (carbon fiber reinforced polymer) reinforcement on the fatigue strength of corrugated plates [15,16], steel plates [17,18], and aluminum alloy plates [19–25]

Materials 2020, 13, 4014 3 of 13 Materials 2020, 13, x FOR PEER REVIEW 3 of 13 withsteel central plates [17fastener,18], and holes. aluminum Based on alloy the platestest and [19 modelling–25] with central data, a fastener new model holes. has Based been onproposed the test forand the modelling contributions data, aof new the modelplate and has carbon been proposed fiber in tension, for thecontributions the adhesive ofinterface the plate in andshear, carbon and additionalfiber in tension, secondary the adhesive bending interfaceeffect when in shear,the improvement and additional fatigue secondary endurance bending is concerned. effect when the improvementDespite the fatigue above-reviewed endurance isresearch concerned. work, there are no unanimous conclusions for the stress intensityDespite factors the above-reviewedof off-centrally researchcracked aluminum work, there plates, are no with unanimous and without conclusions patched for thelaminate stress repairintensity and factors subject of otoff -centrallyfatigue loading. cracked In aluminum this paper, plates, the theoretical with and without equations patched in the laminate calculation repair of andthe stresssubject intensity to fatigue factors loading. for In thisoff-centrally paper, the cracked theoretical aluminum equations plates in the calculationare compared. of the A stress counterpart intensity three-dimensionalfactors for off-centrally finite crackedelement aluminummodel is developed plates are for compared. the stress A intensity counterpart factor three-dimensional calculation and validatedfinite element with modelthe calculation is developed results for of the cracked stress aluminum intensity factor plates calculation without strengthening. and validated Based with theon thecalculation fatigueresults test results, of cracked the aluminum improvement plates withoutof fatigue strengthening. life of the Based patched, on the laminate-repaired, fatigue test results, off-centrallythe improvement cracked of aluminum fatigue life plate of the is patched,discussed laminate-repaired, with codified curves. off-centrally cracked aluminum plate is discussed with codified curves. 2. Experimental Procedure and Results 2. Experimental Procedure and Results Test specimens were machined from aluminum sheet (Al 2024T3) (Alcoa Korea, Ltd., Seoul, Korea),Test satisfying specimens the were American machined specification from aluminum AMS-QQ-A-250/5A. sheet (Al 2024T3) (AlcoaThe specimen Korea, Ltd., plates Seoul, were Korea), 300 mmsatisfying long by the b American= 30 mm wide specification by t = 2 AMS-QQ-A-250mm thick. The off-central/5A. Thespecimen crack was plates considered were 300between mm long the centerlineby b = 30 mmand widethe tip by oft =the2 mmplate, thick. which The can off be-central regarded crack at was an intermediate considered between location the between centerline the centraland the crack tip of and the plate,edge crack, which as can shown be regarded in Figure at an 2. intermediateBefore patched location laminate between repair, the the central specimen crack platesand edge were crack, notched as shownthrough in the Figure thickness2. Before of the patched crack laminatestarted on repair, both sides the specimen (tip A and plates tip B), were for whichnotched the through total initial the thickness crack length of the and crack the started eccentricity on both are sides denoted (tip A andas c tip and B), s for, respectively, which the total as showninitial crackin Figure length 3. andThe thepatched eccentricity laminates are denotedwere made as c usingand s ,unidirectional respectively, as CFRP shown (carbon in Figure fiber3. reinforcedThe patched polymer) laminates (Toray were madeGroup using Co., unidirectionalLtd., Tokyo, Japan) CFRP (carbonapplied fiber at one reinforced side of polymer)test aluminum (Toray alloyGroup plate Co., in Ltd., alignment Tokyo, with Japan) the applied longitudinal at one direct sideion oftest (x direction) aluminum of alloythe test plate specimen. in alignment The CFRP with wasthe longitudinalcured and bonded direction to the (x direction)pre-cracked of specimen the test specimen. using epoxy The resin CFRP matrix was curedSikadur and 330CN bonded (Sika to Corporation,the pre-cracked Lyndhurst, specimen NJ, using USA). epoxy As listed resin in matrix Table Sikadur 1, the mechanical 330CN (Sika property Corporation, of the Lyndhurst,aluminum sheetNJ, USA). was Asobtained listed infrom Table monotonic1, the mechanical tensile test propertys of six of coupons(Toray the aluminum sheetGroup was Co., obtained Ltd., Tokyo, from Japan)monotonic under tensile the loading tests of rate six coupons(Toray of 0.02 mm per Group second, Co., while Ltd., that Tokyo, of the Japan) carbon under fiber the and loading epoxy rate resin of matrix0.02 mm were per obtained second, while according that of to the GB50367 carbon fiberand DI andN epoxy53,455,respectively, resin matrix were referring obtained to the according testing reportsto GB50367 from and the DINmanufacturers. 53,455,respectively, In total, referring 21 specimens to the testinggrouped reports in the from four theseries manufacturers. (“centrally crackedIn total, 21+ unrepaired”, specimens grouped “centrally in the cracked four series + repa (“centrallyired”, “off-centrally cracked + unrepaired”, cracked + “centrallyunrepaired” cracked and +“off-centrallyrepaired”, “o crackedff-centrally + repaired”) cracked +wereunrepaired” tested with and a “ovaryingff-centrally eccentricity cracked ratio+ repaired”) of 2s/b and were the testedcrack sizewith ratio a varying of c/b eccentricityas listed in ratioTable of 2. 2Fatigues/b and thetests crack were size conducted ratio of usingc/b as listeda servo-hydraulic in Table2. Fatigue testing machinetests were (Shimadzu, conducted Kyoto, using aJapan), servo-hydraulic as shown in testing Figure machine 4, with (Shimadzu,the test frequency Kyoto, Japan),of 8 Hz asuntil shown the fracturein Figure of4, withthe test the specimen. test frequency As tress of 8 Hzratio until of theR = fracture 0.1 and of maximum the test specimen. stress range As tressof 170 ratio MPa of sinusoidalR = 0.1 and waveform maximum was stress used range to fatigue of 170 specimens. MPa sinusoidal Repetitive waveform load signals was used were to fatiguemeasured specimens. using a PC-basedRepetitive automatic load signals data were acquisition measured system. using a PC-based automatic data acquisition system.

Edge crack Eccentric crack

Central crack

FigureFigure 2. CrackCrack modes modes in in a a tension tension strip. strip.

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B 0.5 - 0.5 0.5 0.5 A

y 0.5 x

Figure 3. Analytical illustration of an oﬀ-centrally cracked strip. Figure 3. Analytical illustration of an off-centrally cracked strip. Table 1. Mechanical properties of test materials.

Critical Stress Yield Strength Ultimate Strength Elastic Modulus Material Elongation (%) Intensity Factor (MPa) (MPa) (MPa) Materials 2020, 13, x FOR PEER REVIEW (MPa mm0.5) 4 of 13 · Aluminum 307 445 7.2 105 15 1181 × Carbon Fiber - 4216 2.52 105 1.76 - × Adhesive - 30 4.5 103 0.9 - B ×0.5 - Unrepaired Case 0.5 0.5 0.5 A Table 2. List of test specimens.

Specimen Series Test Number Crack Mode Repairy 2s/b c/b 0.5 T1 9 Centrally x 0 0.15–0.45 Repaired× Case T2 4 Centrally √ 0 0.3

T3 4 Oﬀ-Centrally 0.5 0.3 Figure 4. Test set-up and fracture mode.× T4 4 Oﬀ-Centrally √ 0.5 0.3 Figure 3. Analytical illustration of an off-centrally cracked strip. Table 1. Mechanical properties of test materials.

Yield Strength Ultimate Strength Elastic Modulus Elongation Critical Stress Intensity Factor Material (MPa) (MPa) (MPa) (%) (MPa·mm0.5) Aluminum 307 445 7.2 × 105 15 1181 Carbon - 4216 2.52 × 105 1.76 - Fiber Adhesive - 30 4.5 × 103 Unrepaired0.9 Case -

Table 2. List of test specimens.

Specimen Series Test Number Crack Mode Repair 2s/b c/b

T1 9 Centrally Repaired× 0 Case 0.15–0.45 T2 4 Centrally √ 0 0.3 T3 FigureFigure 4.4.4 Test set-upset-upOff-Centrally andand fracturefracture mode.mode.× 0.5 0.3 T4 4 Off-Centrally √ 0.5 0.3 3. Finite Element Analysis Table 1. Mechanical properties of test materials.

3. FiniteThe Element commercialYield Strength Analysis finite Ultimate element Strength modelling Elastic package Modulus ANSYSElongation software Critical (ANSYS Stress Inc., Intensity Canonsburg, Factor Material PA, USA) was(MPa) employed to numerically(MPa) simulate(MPa) the stress distribution(%) in front of(MPa·mm both crack0.5) tips, and The commercial finite element modelling package5 ANSYS software (ANSYS Inc., Canonsburg, theAluminum stress intensity 307 factors for the 445 off-central cracks. 7.2 × 10 The modelling15 results were adopted 1181 to validate PA,Carbon USA) was employed to numerically simulate the stress distribution in front of both crack tips, the results calculated- from referred 4216 theoretical 2.52 equations, × 105 which1.76 will be presented in - the following andFiber the stress intensity factors for the off-central cracks. The modelling results were adopted to section.Adhesive The material - mechanical 30 properties in 4.5 the × 10 finite3 element0.9 models are the same - as those in the validate the results calculated from referred theoretical equations, which will be presented in the experimental tests, as listed in Table1. following section. The material mechanical properties in the finite element models are the same as Since the test specimens are shownTable in2. List Figure of test4 to specimens. rupture perpendicular to the longitudinal those in the experimental tests, as listed in Table 1. direction of the test plate with slight necking on the plate fringe, only the upper half of the test specimen Since the testSpecimen specimens Series are Test shown Number in Figure Crack 4 to Mode rupture Repair perpendicular 2s/b c /tob the longitudinal was modelled due to the symmetry condition before and after the rupture of the test specimen, as direction of the test plateT1 with slight necking9 on Centrallythe plate fringe,× only 0the 0.15–0.45 upper half of the test shown in Figure5. As such, the bottom two edges except the crack line between two crack tips are specimen was modelledT2 due to the symmetry4 conditCentrallyion before √and after0 the0.3 rupture of the test symmetrically restrained, while the longitudinal tensile stresses were applied at the tip edge. specimen, as shown inT3 Figure 5. As such,4 the boOff-Centrallyttom two edges except× 0.5 the crack0.3 line between two crack tips are symmetricallyT4 restrained, while4 theOff-Centrally longitudinal tensile√ stresses0.5 were0.3 applied at the tip edge. 3. Finite Element Analysis The commercial finite element modelling package ANSYS software (ANSYS Inc., Canonsburg,

PA, USA) was employed to numerically simulate the stress distribution in front of both crack tips, and the stress intensity factors for the off-central cracks. The modelling results were adopted to validate the results calculated from referred theoretical equations, which will be presented in the following section. The material mechanical properties in the finite element models are the same as those in the experimental tests, as listed in Table 1. Since the test specimens are shown in Figure 4 to rupture perpendicular to the longitudinal direction of the test plate with slight necking on the plate fringe, only the upper half of the test specimen was modelled due to the symmetry condition before and after the rupture of the test specimen, as shown in Figure 5. As such, the bottom two edges except the crack line between two crack tips are symmetrically restrained, while the longitudinal tensile stresses were applied at the tip edge.

Materials 2020, 13, 4014 5 of 13 Materials 2020, 13, x FOR PEER REVIEW 5 of 13 Materials 2020, 13, x FOR PEER REVIEW 5 of 13

Figure 5. Typical FE model. Figure 5. TypicalTypical FE model.

This failure is expected, asas thethe maximummaximum tensiletensile stressstressin in the the vicinity vicinity of of the the crack crack tip tip B isB is close close to tothe the edge edge of the of plate,the plate, which which agrees agrees well withwell thewith modelling the modelling results asresults shown as inshown Figure in6. Figure The J-integral 6. The J-integralcalculation calculation technique withtechnique a square with root a square singularity root atsingularity the crack tipat the was crack applied tipfor was the applied analysis for of the analysisstress intensity of the stress factor intensity at the crack factor tip. at The the quadratic crack tip. solid The elements quadratic SOLID95 solid elements were used SOLID95 in the meshwere usedconstruction in the mesh of the construction crack tip zone of the with crack a swept tip zo meshingne with method.a swept meshing Only the method. meshes nearOnly the the crack meshes tip nearzone the were crack intensiﬁed tip zone with were the intensified element size with of the 0.5 mmelement due tosize the of consideration 0.5 mm due to of the computational consideration costs. of computationalFor the geometry costs. of the For ﬁnite the geometry element model, of the the fini eccentricityte element ratiomodel, of the 2s/b eccentricityand the crack ratio size of ratio 2s/b of andc/b theare variedcrack size as 0.17~0.7 ratio of and c/b 0.05~0.5,are varied respectively, as 0.17~0.7 so and as not0.05~0.5, to surpass respectively, the plate so width as not and to contradict surpass the the platecases width related and to centralcontradict or edge the cases cracks. related to central or edge cracks.

Figure 6. Stress concentration at crack tip. Figure 6. StressStress concentration concentration at at crack crack tip. 4. Results and Discussion 4. Results and Discussion 4.1. Summary of K and F Theoretical Equations for O -Central, Central and Edge Cracks 4.1. Summary of KII and FwwTheoretical Equations for Off-Central,ff Central and Edge Cracks 4.1. Summary of KI and FwTheoretical Equations for Off-Central, Central and Edge Cracks Based on the theory of fracture mechanics, the mode I elastic stress intensity factor can be assessed Based on the theory of fracture mechanics, the mode I elastic stress intensity factor can be by the following criteria: assessed by the following criteria: KI = FWσ √0.5πc (1) =πσ KFIW=πσ 0.5 c (1) which is determined by the nominal tensionKFIW stress, σ, and0.5 the c boundary correction coefficient, Fw.(1) For whichthe surface is determined crack scenario by the studied nominal herein, tension suitable stress, parameters σ, and the are neededboundary for correction the rational coefficient, assessment F ofw. which is determined by the nominal tension stress, σ, and the boundary correction coefficient, Fw. F . On this basis, two analytical equations are compared as below. Forw the surface crack scenario studied herein, suitable parameters are needed for the rational assessment of Fw. On this basis, two analytical equations are compared as below. assessment4.1.1. Crack of Line Fw. StressOn this Field basis, Method two analytical for Off-Centrally equations Cracked are compared Case as below.

4.1.1.Based Crack on Line the Stress model Field proposed Method by for Wang Off-Centrally [5], a crack Cracked extended Case line subject to uniaxial tensile stress σ consists of KI dominated plastic segments at the crack tips (A and B) and outside plastic segments. Based on the model proposed by Wang [5], a crack extended line subject to uniaxial tensile stress σ consists of KI dominated plastic segments at the crack tips (A and B) and outside plastic stress σ consists of KI dominated plastic segments at the crack tips (A and B) and outside plastic segments. Based on the basic theory for a crack opening symmetrically with respect to the undeformed crack plane, the corresponding normal stress σx along the x axis is written as: undeformed crack plane, the corresponding normal stress σx along the x axis is written as:

Materials 2020, 13, 4014 6 of 13

Based on the basic theory for a crack opening symmetrically with respect to the undeformed crack plane, the corresponding normal stress σx along the x axis is written as:

 √  FWAσ 0.5πc FWAσ √c  = (0 < 0.5d 0.5d0A)  √2π0.5d √2d σx = ≤ (2)  FWBσ √c  = (0 < 0.5d 0.5d0B) √2d ≤ where, FWA and FWB are the correction factors, d is the diameter of pseudo plastic zone at the crack front. When the distances from the centers of the pseudoplastic zone are greater than 0.5d0A or 0.5d0B, the square FWA or FWB is chosen due to an allowance of ﬁnite width and the corresponding normal stress σx is expressed as: ( F2 σ (0.5d 0.5d 0.5b + s 0.5c) σ = WA 0A ≤ ≤ − (3) x F2 σ (0.5d 0.5d 0.5b s 0.5c) WB 0B ≤ ≤ − − Given the stress continuity at the boundary of the plastic zone around the crack origin, i.e., 2 Equation (2) equal to Equation (3) when d = d0A and d = d02B, it can be obtained as d0A = 0.5c/FWA 2 and d0B = 0.5c/FWB . Equating the tensile stress along the crack extended line to the remote tension stress, σ, along the y axis yields:

R 0.5d0A FWAσ √c R 0.5d0B FWBσ √c 2 2 ∂d + ∂d + F σ(0.5b + s 0.5d0A 0.5c) + F σ(0.5b + s 0.5d0B 0.5c) = bσ (4) 0 √2d 0 √2d WA − − WB − − In equilibrium along the crack extended line, the sum of bending moments about the centerline of the tubular ﬂange is zero, and thus:

R 0.5d √ h i 0A FWAσ c d(0.5d + 0.5c s) + 0.5F2 0.25b2 (0.5d + 0.5c s) 2 0 √ ∂ 0A WAσ 0A 2d − h − − i (5) R 0.5d0B FWBσ √c 2 2 2 = ∂d(0.5d0B + 0.5c + s) + 0.5F σ 0.25b (0.5d0B + 0.5c + s) 0 √2d WB −

2 2 Given the (1/FWA -1/FWB ) approaching zero when s = 0 (no eccentricity), solving Equations (4) and (5) gives: r 2b + 4s c F = − (6) WA 2b + 4s 2c − r 2b 4s c F = − − (7) WB 2b 4s 2c − − 4.1.2. The Westergaard Function Based Method for Oﬀ-Centrally Cracked Case The stress distribution in front of the crack tip along crack line can be expressed using the Westergaard function as: FW σx = (8) q 2 1 0.5c − x where, x is the distance from the crack center in the crack line. At two sides of the crack (close to crack tip A and B), the corresponding loads carried by the stress distributions at uncracked parts are given by: Z 0.5b +s 0.5c 0 − PA = σxt∂y (9) 0.5c Z 0.5b s 0− PB = σxt∂y (10) 0.5c Materials 2020, 13, x FOR PEER REVIEW 7 of 13

Materials 2020, 13, 4014 =+−σ 7 of 13 PtbscA0(0.5 0.5 ) (11)

=−σ The stress distributions far away fromPB0 the crackedtbs(0.5 section ) are given by: (12)

Implementing the load equilibrium Equation (9) = Equation (11), and Equation (10) = Equation PA = σt(0.5b0 + s 0.5c) (11) (12) at the cracked section, the boundary correction coefficient− developed by Wang et al. [6] is written as: P = σt(0.5b s) (12) B 0 − 1 Implementing the load equilibriumF Equation= (9) = Equation (11), and Equation (10) = Equation WA 2 (12) at the cracked section, the boundary correction coefficcient developed by Wang et al. [6] is written(13) as: 1−  bsc+−2 1 FWA = q (13) 2 = 1 1 c FWB b+2s c − 2 c − (14) 1−  bs1− 2 FWB = (14) q 2 1 c − b 2s 4.1.3. Comparison of Referred Calculations] − 4.1.3. Comparison of Referred Calculations] In Figure 7, the boundary correction coefficients at the crack tips for off-centrally cracked platesIn calculated Figure7, the using boundary referred correction methods coeare fficomparedcients at theagainst crack the tips crack for osizeff-centrally ratio ofc cracked/b. The results plates calculatedobtained usingfrom referredthe crack methods line stress are compared field meth againstod theare crackslightly size ratiohigher of cthan/b. The the results Westergaard obtained fromfunction-based the crack line method, stress fieldespecially method when are slightly c/b is higherranging than between the Westergaard 0.2 and 0.35. function-based However, method, such a especiallydifference whenis graduallyc/b is ranging reduced between as c/ 0.2b is and greater 0.35. However,than 0.4. suchWith athe diff erenceincreasein is gradually the eccentricity, reduced ascalculatedc/b is greater Fw is than increased 0.4. With which the increaseinindicates thea much eccentricity, higher calculatedstress concentrationFw is increased of the which stress indicates field at athe much crack higher tip. stressCalculated concentration Fw for crack of the tip stress A fieldis consistently at the crack lower tip. Calculated than that Fforw for crack crack tipB tip Aand is consistentlyseemingly less lower influenced than that by for the crack eccentricity tipB and seeminglyof the crack. less Such influenced an underestimation by the eccentricity can ofbe theowed crack. to Suchthe different an underestimation plastic zones can at bethe owed crack to fronts the di offferent tip A plastic and B, zones which at thewas crack simplified fronts as of tipthe Asame and in B, whichreferred was methods. simplified For as design the same purpose, in referred however, methods. only Forthe designlarger purpose,Fw, i.e., for however, crack tip only B, related the larger to Fgreaterw, i.e., forstress crack concentration tip B, related is toconsidered greater stress in the concentration following analysis. is considered in the following analysis.

2s/b=0.3: B: Referred calculation [7] Referred calculation [8]

A: Referred calculation [7] Referred calculation [8] 2s/b=0.5: B: Referred calculation [7] Referred calculation [8] A: Referred calculation [7] Referred calculation [8] 2.5

1.5 w F 1

0.5 0 0.1 0.2 0.3 0.4 0.5 c/b

Figure 7. Comparison of referred calculations of stressstress intensity factorsfactors (SIF)(SIF) atat AA andand B.B.

4.2. Finite Element Analysis Based Evaluation of F for Off-Centrally Cracked Plates 4.2. Finite Element Analysis Based Evaluation of Fwfor Off-Centrally Cracked Plates Given that the aforementioned theoretical methods are limited by simplified assumptions, F for Given that the aforementioned theoretical methods are limited by simplified assumptions,w Fw oforff-centrally off-centrally cracked cracked plates plates is evaluated is evaluated based on ba ased finite on element a finite parametric element study.parametric One hundred study. One and twelvehundred finite and element twelve models—whichfinite element aremodels—which permutations are and permutations combinations and of seven combinations sets of 2s/ bofranging seven fromsets of 0.17 2s/ tob ranging 0.7, and from fourteen 0.17 setsto 0.7, of cand/b ranging fourteen from sets 0.1 of toc/b 0.5—are ranging built from for 0.1 the to stress0.5—are intensity built for factor the

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stress intensity factor analysis. Based on the finite element parametric study, the calculation for Fwcan be developed as the following poly-fit equation, including the ratios of 2s/b and c/b.

32 2 22s s 5.5 0.12  222sssc   bb  c  = −+  FWB 212 -173  +53  -3.7 0.05ee  bbbb b (15)     

As a validation example for the cases of 2s/b = 0.3 and 0.5 shown in Figure 8, Fw predicted from modelling agrees well with those from referred calculation. The cases with central cracks and edge cracks are also referred to further the comparison with analytical results. For the centrically cracked plate of finite width, the stress intensity factor at the crack tip was originally modified from that for an infinite sheet with a periodic array of through-thickness cracks under uniformly distributed Materials 2020, 13, 4014 8 of 13 stress. The formation is deemed accurate for up to c/b = 0.5, and the tangent correction as documented in Reference [3] is given by: analysis. Based on the finite element parametric study, the calculation for Fw can be developed as the 2bcπ following poly-fit equation, including theF ratios= of 2s/tanb and c/ b. (16) W π cb2 " 3 2 # 2 2s 2s 2s c 5.5( 2s ) c 0.12( 2s ) An alternativeFWB = solution212 was173 later presented+53 3.7by Feddersen0.05e [26],b and+ knowne b as the secant(15) b − b b − b − b correction, as: As a validation example for the cases of 2s/b = 0.3π and 0.5 shown in Figure8, Fw predicted from modelling agrees well with those from referred= calculation.c The cases with central cracks and edge FW sec (17) cracks are also referred to further the comparison with2 analyticalb results. For the centrically cracked plate of finite width, the stress intensity factor at the crack tip was originally modified from that for an The stress intensity factor at the crack tip for the edge cracked plate was originally modified infinite sheet with a periodic array of through-thickness cracks under uniformly distributed stress. from the long surface crack solution when in-plane bending was not accounted. The corresponding The formation is deemed accurate for up to c/b = 0.5, and the tangent correction as documented in boundary correction coefficient documented in BS 7910 [27] is: Reference [3] is given by: r 234 cc   c  c F =−1.12 0.23 + 10.62b − 21.7πc + 30.4 W F= tan   (16)(18) Wbb πc 2b  b  b

2 2s/b=0.3: Referred calculation [7] Referred calculation [8] 1.8 Modelling 2s/b=0.5: Referred calculation [7] 1.6 Referred calculation [8] Modelling wB F 1.4

1.2

1 0 0.1 0.2 0.3 0.4 0.5 c/b

Figure 8. OOff-centralﬀ-central crack in tension clamp.

AnAs observed alternative from solution Figure was9, the later curves presented predicted by from Feddersen the above [26 poly-fit], and equation known asare the governed secant correction, as: by off-central cracks (i.e., within the range betweenr those for the edge crack and those for the central crack) when 2s/b is increased from 0.17 to 0.55. πConversely,c Fw is more or less involved with FW = sec (17) the edge cracks or the central cracks when the 2s/b is greater2b than 0.55 or less than 0.17. The stress intensity factor at the crack tip for the edge cracked plate was originally modiﬁed from the long surface crack solution when in-plane bending was not accounted. The corresponding boundary correction coeﬃcient documented in BS 7910 [27] is:

c c 2 c 3 c 4 F = 1.12 0.23 + 10.6 21.7 + 30.4 (18) W − b b − b b As observed from Figure9, the curves predicted from the above poly-ﬁt equation are governed by oﬀ-central cracks (i.e., within the range between those for the edge crack and those for the central crack) when 2s/b is increased from 0.17 to 0.55. Conversely, Fw is more or less involved with the edge cracks or the central cracks when the 2s/b is greater than 0.55 or less than 0.17. Materials 2020, 13, 4014 9 of 13 Materials 2020, 13, x FOR PEER REVIEW 9 of 13

1.8

2s/b: 1.6 Edge crack 0.40 wB F 1.4 0.30 0.20 1.2 0.17 Central crack 1 0 0.1 0.2 0.3 0.4 0.5 c/b

Figure 9. CrackCrack modes modes in in a a tension tension strip. strip.

4.3. Fatigue Fatigue Life Life Evaluation Evaluation Using the above-discussed boundary correction coefficient, F , and the mode I elastic stress Using the above-discussed boundary correction coefficient, FW, and the mode I elastic stress intensityintensity factor, factor, the the fatigue fatigue life life of of the the test test specim specimensens can can be be evaluated. evaluated. As Asper per the thesuggestion suggestion by Su by etSu al. et [28], al. [28 the], thealuminum aluminum material material was was defined defined following following a arate-independent rate-independent isotropic isotropic elastic-plastic elastic-plastic relation using the classical J -flow theory of the plasticity. It fits the experimentally measured relation using the classical J2-flow theory of the plasticity. It fits the experimentally measured stress-strainstress-strain curvecurve with with the the extraction extraction of strain-hardening of strain-hardening features. features. When aboveWhen discussed above discussed boundary correction coefficient, F , is determined, the mode I elastic stress intensity factor can be calculated using boundary correction coefficient,W FW, is determined, the mode I elastic stress intensity factor can be calculatedEquation (1),using and Equation then the fatigue(1), and life then of testthe specimensfatigue life can of betest evaluated. specimens It iscan generally be evaluated. recognized It is generallythat three recognized regions exist that for three the regions relationship exist betweenfor the relationship the fatigue between crack growth the fatigue rate andcrack the growth stress rateintensity and factor;the stress threshold, intensity linear factor; growth, threshold, and accelerated linear growth, growth. and To accelerated calculate the growth. fatigue life,To calculate the crack thepropagation fatigue life, during the thecrack latter propagation two regions during can be analyzedthe latter using two theregions concept can of be fracture analyzed mechanics using [the29] conceptand the well-knownof fracture mechanics Forman equation [29] and [3 ],the including well-known the stress Forman ratio equation effect and [3],F Wincludingaforementioned the stress in the Section 4.2, as: ratio effect and FW aforementioned in the Sectionm 4.2, as: m 0.5m dc CF(∆KI) CF(FW∆σ) (0.5πc) = = 0.5m (19) Δ m Δ m ()π dNdc (1CKR())KC ∆KI CF(FW1()0.5R)Kσ F ∆σ √ c0.5πc ==−FI− − C − W (19) dN(1−−Δ R ) K K −−Δσ π 3 0.5 where, Kc is the fracture toughness of theCI test material,(1R which )KFCW is equal to 0.5 1.181 c10 MPa mm . CF is × 10· the material constant converted from its counterpart in Paris equation as 0.174 10− . Assuming where, Kc is the fracture toughness of the test material, which is equal to 1.181 × 10×3 MPa·mm0.5. CF is m = 3, the integration of Equation (19) results in the fatigue life, N, of the cracked aluminum plate as: the material constant converted from its counterpart in Paris equation as 0.174 × 10−10. Assuming m = 3, the integration of Equation (19) results in the fatigue life, N, of the cracked aluminum plate as: 2(1 R)KC( √c0 √c1) 1 c N = − − ln 1 (20) 3 2 −−( σ) − ( σ) c0 2(1CRKF )FWC0 (√π c∆ c 1 ) CF FW1√π∆ c N =−ln 1 (20) CF()()ππΔΔσσ32 CF c where, c0 and c1 are the initial andFW final crack lengths, FW respectively. As0 the magnitude of ∆KI is determined by c0 and c1, N can be obtained via the numerical analysis and corresponding data for S-N where, c0 and c1 are the initial and final crack lengths, respectively. As the magnitude of ΔKI is relation can be deduced. Figure 10 shows a good correlation of fatigue lives between experimental determined by c0 and c1, N can be obtained via the numerical analysis and corresponding data for Stest-N resultsrelation and can finite be elementdeduced. modelling Figure based10 shows analytical a good results correlation using the Formanof fatigue equation lives whenbetween the experimentalcentrally cracked test andresults off-centrally and finite cracked element aluminum modelling plates based are analytical concerned. results This using also indicates the Forman that F equationw defined when by the the proposed centrally poly-fitcracked equationand off-centrally would be cracked a good aluminum choice in representing plates are concerned. stress intensity This factors, and thus the fatigue life. also indicates that Fw defined by the proposed poly-fit equation would be a good choice in representing stress intensity factors, and thus the fatigue life.

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400400 Test data: Central crack Eccentric crack: 2s/b=0.52ss//b Prediction: Central crack Eccentric crack: 2s/b=0.22ss//b Eccentric crack: 2s/b=0.352ss//b 2s/b=0.522s/b=0.5ss//b (MPa) (MPa) σ σ Δ Δ

6060 5050 Stressrange, Stressrange, 4242 3535 2929 4040 10001,0001,000 10,000 10,000 100,000 100,000 1,000,000 1,000,000 Number of cycles, N

FigureFigure 10. ComparisonComparison of of fatigue fatigue live live results results between between centrally and off-centrally oﬀ-centrally cracked plates.plates.

Based on the fatigue criterion recommended by the the BS8118 BS8118 standard standard for for structural structural use use of of aluminum [30], [30], the the fatigue fatigue test test data data and and prediction prediction results results are are compared compared against against detail detail classes. classes. As shownAs shown in Figure in Figure 10, 10 codified, codified S--NS- Ncurvescurvescurves areare are representedrepresented represented asas as thethe the fatiguefatigue fatigue strengthstrength strength levelslevels levels atat at 22 million million cycles. The The data data of of log logΔ∆σσ-log-log-logNN calculatedcalculatedcalculated forfor for off-centrallyoff-centrally off-centrally crackedcracked cracked platesplates withwith 22 ss//b = 0.20.2 areare slightly slightly lowerlower than than thosethose forfor centrallycentrally crackedcracked platesplates overoverover thethethe classesclassesclasses 505050 andandand 60.60.60. InInIn contrast,contrast,contrast, thethethe testtest datadata for for off-centrallyoff-centrally crackedcracked platesplates withwith 22sss///bb == 0.50.5 areare shownshown toto fallfall aboveabove thethe classclass ofof 35.35. Despite Despite the the scatter scatter of the predictions, a consistent trend can be observ observeded for the off-centrally off-centrally cracked plate with a listed variation of 2ss//b.b. The comparisoncomparison ofofS S-N--Ncurves curvescurves for forfor test testtest aluminum aluminumaluminum plates platesplates with withwith and withoutandand withoutwithout patched patchedpatched laminate laminatelaminate repair repairisrepair shown isis shownshown in Figure inin Figure11Figure. The 11.11. test TheThe fatigue testtest fatiguefatigue lives under livelives stress under ranges stress of ranges 100, 120, of 100, and 120, 140 MPaand are140 extractedMPa are extractedfor the comparison for the comparison in Figure 12in. Figure After repair, 12. After it can repair be seen,, itit cancan that bebe the seenseen fatigue thatthat thethe lives fatiguefatigue of the liveslives test specimens ofof thethe testtest specimenswith central with cracks central are showncracks toare increase shown byto increase 30–60%. byby This 30–60%.30–60%. can be ThisThis due tocancan the bebe beneficial duedue toto thethe eff beneficialbeneficialect of the effectcarbon of fiber the andcarbon interface fiber and strengthening, interface strengthen and the decreaseing,ing, andand in thethe stress decreasedecrease concentration inin stressstress at concentrationconcentration thecentral crack atat thecentralhole.thecentral In contrast, crackcrack hole.hole. such InIn abeneficial contrast,contrast, esuchsuchffect aa becomes beneficibenefici moreal effect effective becomes for omoreff-centrally effective cracked for off-centrally plates with cracked2s/b = 0.5, plates for which with 2 thess//b fatigue= 0.5,0.5, forfor lives whichwhich areincreased thethe fatiguefatigue by liveslives 90–120%, areare inin ascreased a result by of 90–120%, repair especially as a result at the of repaircrackrepair tipB, especiallyespecially close toatat thethethe platecrackcrack edge tipB,tipB, in clclose Figure to the3. plate edge in Figure 3.

400 No repair Central crack Eccentric crack: 2s/b=0.52ss//b After repair: Central crack Eccentric crack: 2s/b=0.52ss//b (MPa) (MPa) σ σ Δ Δ

60 50 Stress range, Stress Stress range, Stress 42 35 29 40 1,0001000 10,000 100,000 1,000,000 Number of cycles, N

FigureFigure 11. ComparisonComparison of of fatigue fatigue live live results results between between unrepaired and repaired plates.plates.

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0.5 Central crack: No repair After repair 0.4 Eccentric crack, 2s/b=0.5: No repair After repair N 0.3

0.2 (million cycles) (million N

Number of cycles, cycles, Number of 0.1

0 100 MPa 120 MPa 140 MPa Stress range, Δσ (MPa)

Figure 12. Comparison of fatigue lives of cracked plates under three typical stress ranges. 5. Concluding Remarks 5. Concluding Remarks The stress intensity factors for off-centrally cracked aluminum plates have been presented in The stress intensity factors for off-centrally cracked aluminum plates have been presented in this paper. Typical theoretical equations for off-central, central and edge cracks were reviewed and this paper. Typical theoretical equations for off-central, central and edge cracks were reviewed and compared in terms of their sensitive parameters and applicability. A finite element model has been compared in terms of their sensitive parameters and applicability. A finite element model has been developed to explore a wide range of parameters related to the eccentricity and crack size influencing developed to explore a wide range of parameters related to the eccentricity and crack size the boundary correction coefficients. The Forman equation has been employed, along with numerical influencing the boundary correction coefficients. The Forman equation has been employed, along results for the prediction of fatigue lives. Based on the test data, the fatigue lives result of aluminum with numerical results for the prediction of fatigue lives. Based on the test data, the fatigue lives plates with and without patched laminate repair have been compared with codified fatigue classes. result of aluminum plates with and without patched laminate repair have been compared with The efficiency of repair for the improvement of fatigue lives is compared and discussed for central and codified fatigue classes. The efficiency of repair for the improvement of fatigue lives is compared off-central cracked conditions. The following conclusions can be drawn: and discussed for central and off-central cracked conditions. The following conclusions can be drawn:The developed finite element parametric study-based poly-fit equation for the boundary correction • coefficient incorporating the eccentricity ratio and the crack size ratio is demonstrated to agree • The developed finite element parametric study-based poly-fit equation for the boundary well with referred calculations. Moreover, it fits in well between centrally cracked cases and edge correction coefficient incorporating the eccentricity ratio and the crack size ratio is crack casesfor the off-centrally cracked aluminum plates. demonstrated to agree well with referred calculations. Moreover, it fits in well between The fatigue life prediction on the basis ofthe Forman equation accounting for the crack in linear • centrally cracked cases and edge crack casesfor the off-centrally cracked aluminum plates. • growthThe fatigue and acceleratedlife prediction growth on the is shownbasis ofthe to correlate Forman well equation with the accounting test results for of the aluminum crack in plateslinear withgrowth central and cracksaccelerated and o ffgrowth-central is cracks. shown The to corre correspondinglate well withS-N curvesthe test are results also comparableof aluminum to thoseplates suggestedwith central by codified cracks curves.and off-central cracks. The corresponding S-N curves are also The beneficial effect of patched laminate repair can be identified from the increase in test fatigue • comparable to those suggested by codified curves. • livesThe beneficial by 30–60% effect and 90–120%of patched for centrallylaminate crackedrepair can and be off identified-centrally crackedfrom the aluminum increase in plates, test respectively.fatigue lives by Thus, 30–60% the repairand 90–120% at the crack for centrally tip close cracked to the plateand off-centrally edge is deemed cracked to be aluminum effective inplates, the fatigue respectively. life improvement Thus, the repair for off -centrallyat the crac crackk tip aluminum close to the plates.The plate edge strengthening is deemed etoff ectbe ofeffective patched in laminate the fatigue repair, life however, improvement was not modelledfor off-centrally in detail, crack and thereforealuminum requires plates.The more experimentalstrengthening work effect in furtheringof patched the laminate correction repa ofir, defined however, parameters was not in themodelled fatigue in life detail, prediction. and therefore requires more experimental work in furthering the correction of defined parameters Authorin Contributions:the fatigue lifeX.Y. prediction. conducted engineering trial tests and analysis of the test data. Z.W. supervised the projects and the students in the preparation of this manuscript. X.Z. and Z.L. assisted in the preparation of the test samples.Author Contributions: R.J. and W.G. oX.Y.ffered conducted helpful suggestionsengineering fortrial the tests numerical and anal modellingysis of the andtest engineeringdata. Z.W. supervised practices. Allthe authors have read and agreed to the published version of the manuscript. projects and the students in the preparation of this manuscript. X.Z. and Z.L. assisted in the preparation of the Funding:test samples.The R.J. research and W.G. presented offered was helpful sponsored suggestions by the Sichuan for the Science numerical and Technologymodelling and Program engineering (No. 2020YJ0307), practices. theAll Scienceauthors andhave Technology read and agreed Project to of Ministrythe published of Housing version and of Urban-Ruralthe manuscript. Construction (Grant No. 2018-K9-004). Conflicts of Interest: The authors declare no conflict of interest. Funding: The research presented was sponsored by the Sichuan Science and Technology Program (No. 2020YJ0307), the Science and Technology Project of Ministry of Housing and Urban-Rural Construction (Grant No. 2018-K9-004).

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