A Crack Growth Rate Conversion Module: Theory,

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A Crack Growth Rate Conversion Module: Theory,Development, User Guide and ExamplesYu Chee Tong, Weiping Hu and David MongruAir Vehicles DivisionDefence Science and Technology OrganisationDSTO-TR-2050ABSTRACTThe use of crack growth analysis tools based on plasiticty-induced crack closure model,such as FASTRAN, CGAP and AFGROW, requires the conversion of crack growth rateversus the nominal stress intensity range curves to a "single" curve of crack growth rateversus the effective stress intensity range. In order to minimise the error arising from crackgrowth rate conversion and judicially utilise these software tools, a user-friendly tool wasintegrated into CGAP. This report documents the theory, implementation, the user guideand examples of the crack growth rate conversion software module.RELEASE LIMITATIONApproved for public release

Published byAir Vehicles DivisionDSTO Defence Science and Technology Organisation506 Lorimer StFishermans Bend, Victoria 3207 AustraliaTelephone: (03) 9626 7000Fax: (03) 9626 7999 Commonwealth of Australia 2007AR-014-020September 2007APPROVED FOR PUBLIC RELEASE

A Crack Growth Rate Conversion Module: Theory,Development, User Guide and ExamplesExecutive SummaryCrack growth assessment is an essential element of the aircraft certification procedure foraddressing structural durability and fatigue concerns on all Australian Defence Force(ADF) air platforms.The current modelling tools for fatigue crack growth all involve numerous assumptionsand extrapolation methodologies, in order to estimate the life of a real structure from thedata obtained from simple coupon tests. These assumptions introduce uncertainties notonly in the results themselves but also in the procedures used to obtain the results. Thereis, therefore, a need for DSTO to explicitly define the procedures in each stage of crackgrowth analyses, including data conversion.As an example of the efforts to codify the knowledge in this area, this report develops anddocuments procedures for the intuitive and routine processing of crack growth rate data,in order to allow fatigue life prediction tools to be applied appropriately, and anycorrelation and comparison with experimental data to be made rigorously. The use ofcrack growth assessment tools based on the plasticity-induced crack closure model, suchas FASTRAN, CGAP and AFGROW, requires a single curve of crack growth rate versuseffective stress intensity range, where the effective stress intensity range is dependent onthe crack opening stress. However, the available experimental growth rates are routinelydefined against the nominal stress intensity range, with the stress ratio as a parameter.Therefore, the crack growth rate versus the nominal stress intensity range needs to beconverted to a "single" curve of growth rate versus the effective stress intensity range. Thisconversion is by no means straightforward. In order to minimise the error arising from theconversion and judicious use of software, a user-friendly tool for conversion of crackgrowth rate has been developed, implemented and integrated into CGAP.This report presents the theory and the algorithms involved in the conversionmethodology. It discusses, in detail, the concept of plasticity-induced crack closure, crackopening stress, the constraint factor and the plasticity-corrected stress intensity factor. Auser manual and examples are included to assist the use of this software module in CGAP.

AuthorsYu Chee TongAir Vehicles DivisionDr Chee Tong is presently a research scientist in the Air Vehiclesdivision of DSTO. He joined AVD-DSTO in 1999 after graduatingfrom the Royal Melbourne Institute of Technology with a BachelorDegree in Aerospace Engineering with Honours. In 2006, hecompleted his Ph.D. at the University of Sydney, supported by AVDDSTO, on probabilistic fatigue life analysis methods for aerospacevehicles. Since joining DSTO in 1999, Dr Tong has worked in thefields of structural risk and reliability for airframe, engine andhelicopter components, propulsion systems life management, fracturemechanics research, aircraft structural lifing standards, and structurallifing methodologies and tools. Currently, he is working in the areasair vehicle risk and reliability assessment, and structural lifingmethodologies and tools.Weiping HuAir Vehicles DivisionDr Weiping Hu joined DSTO in 1998 as a research scientist. He iscurrently a senior research scientist leading the development ofmodelling capabilities for the analysis of structural integrity of aircraftstructures.After obtaining his PhD degree in 1993 at Dublin City Univeristy,Ireland, he held various academic positions at Dublin City University,Monash University and Deakin University. His research interestsinclude fatigue and fracture of engineering materials and structures,fatigue crack growth in aircraft structures, constitutive models andplasticity, and numerical methods in engineering.

David MongruAir Vehicles DivisionMr Mongru graduated from RMIT in 1990 with a Bachelor ofAerospace Engineering (Honours). He commenced work at DSTO in1994 working on loads development and fatigue interpretation for thePC9 fatigue test. He joined IFOSTP in 1996 performing fatigueanalysis of critical components on the aft fuselage of the F/A-18. In2001 he commenced work on the P-3 SLAP. His primary functionsincluded fatigue test interpretation and provision of technical supportto the P-3 empennage test. He is currently working on the standardsand lifing methodologies task.

Contents1. INTRODUCTION . 12. FUNDAMENTALS OF FATIGUE CRACK GROWTH MODELLING . 22.1 Linear Elastic Fracture Mechanics and Paris Law. 22.2 Mechanical Loading Effects. 42.2.1Stress Ratio Effect . 42.2.2Load Interaction Effect. 52.3 Plasticity-Induced Crack Closure . 52.4 Fatigue Crack Growth Rate Modelling. 72.5 Elastic-Plastic Effective Stress Intensity Factors. 83. NEWMAN’S CRACK OPENING STRESS EQUATIONS . 93.1 Centre-Crack Tension (CCT) Specimen . 103.2 Compact Tension (CT) Specimen . 113.3 Limitations . 124. DKEFF PROGRAM AND CGAP CRACK GROWTH RATE CONVERSIONMODULE. 144.1 Effective Stress Intensity Factor Calculation. 154.2 Constant Constraint Factor. 154.3 Variable Constraint Factor . 165. EXAMPLES . 185.1 7050-T7451 Aluminium Alloy Example . 185.1.1Constant Constraint Factor . 195.1.2Variable Constraint Factors. 225.2 2219-T851 Aluminium Alloy Example . 235.2.1Constant Constraint Factor . 255.2.2Variable Constraint Factor . 256. DISCUSSION. 266.1 Finding the Optimum Constraint Factor . 266.2 Limitations . 267. SUMMARY AND FUTURE WORK. 27REFERENCE. 28APPENDIX A– CGAP CRACK GROWTH RATE CONVERTER: USER GUIDE 31A.1. Introduction. 31A.2. FCGR Program Flowchart. 31A.3. Running FCGR . 31

A.4. Example Problem Using 7075-T651. 44APPENDIX B- COEFFICIENT OF DETERMINATION . 47APPENDIX CINPUT FILES FOR THE EXAMPLES . 49

NomenclatureaCC5FGHmNKKCK maxΔKΔK 0ΔK effCrack length, or half length for a symmetric crack.Crack growth rate coefficient in Paris law.Cyclic fracture toughness.Geometry correction factor.Threshold function, G 1 (ΔK 0 / ΔK eff ) p .Fracture function, H 1 (K max / C5 ) .Crack growth rate exponent in Paris law.Number of cycles.Stress intensity factor.Fracture toughness.Maximum stress intensity factor.Stress intensity factor range.Long crack threshold. It is considered to be a material constant, and inparticular, independent of crack length.Effective stress intensity range.qS maxLoad.Stress ratio of a load cycle, R S S min / S max .Applied remote stress.Effective stress range, ΔSeff S max SoThe maximum stress in a load cycle.S minThe minimum stress in a load cycle.PRSΔSeffSoΔStUWγλρσσ0Δσ eσ maxσ minσuσyωCrack opening stress.Applied stress range. ΔS S max S min .Thickness of the specimen.ΔK effS So1 γ max ΔKS max S min 1 RSpecimen width, or half width for symmetric crack.γ So / S maxλ a /W .Plastic zone size.Local stress.Averaged flow stress, σ 0 (σ y σ u ) / 2 .Effecitve stress intensity factor ratio, U Endurance limit.Local maximum stress.Local minimum stress.Ultimate stress of material.Uniaxial yield stress of material.Cyclic plastic zone size.

DSTO-TR-20501. IntroductionCrack growth assessment is an essential element of the aircraft certification procedure foraddressing structural durability and fatigue concerns of all Australian Defence Force(ADF) air platforms. It provides a means for estimating and assessing the growth offatigue cracks in structures from flaws either pre-existing at the time of manufacture orgenerated under in-service conditions. The output of the assessment provides guidance forthe development of inspection programs to ensure the timely detection of fatigue cracks incomponents or to allow repair or replacement of the components, wherever feasible, to becarried out.The current modelling methodologies for fatigue crack growth are, to varying degrees,empirical in nature. In classical linear elastic fracture mechanics (LEFM), the quantitativeprediction of fatigue crack growth in structures is obtained by extrapolating thecharacteristic fatigue crack growth rate (FCGR) data derived from simple crack growthtests using an appropriate crack growth model. As a result, the accuracy of fatigue crackgrowth life prediction can be significantly affected by the uncertainty in the source dataand the uncertainty in the procedures used to process the data. For the judicious use ofcrack growth analysis tools, it is highly desirable to codify the procedures for dataprocessing, to ensure that the data are processed consistently, and any correlation andcomparison of analytical results with experimental data be made meaningfully andrigorously.The plasticity-induced crack closure model has been one of the most widely used modelsfor characterising FCGR under constant amplitude loading and predicting fatigue crackgrowth under variable amplitude loading in recent decades. Some of the computerprograms used in DSTO for crack growth analysis are fully or partially based on thismodel, such as FASTRAN [1], AFGROW 1 [2] and CGAP [3]. These programs require theFCGR da dN be defined by a "single" curve against the effective stress intensity factor(SIF) range ΔK eff , but FCGR data are traditionally plotted against the nominal SIF rangeΔK for different stress ratios R . Due to the complexity in the calculation of the crackopening stress, the conversion to a single da dN versus ΔK eff relation is neither simple norstraightforward. Therefore, it has been recognised [4] that there is an immediate need todevelop a user-friendly tool to codify this conversion procedure, in order to minimise theuncertainties arising from this FCGR conversion process, and better utilise these softwaretools.This report details the principle and the implementation of the FCGR conversionprocedure. The FCGR conversion module, based on Newman’s work [1], has beenintegrated into CGAP to take advantage of its graphical user interface. This FCGRconversion module enables the conversion of the nominal FCGR data to a da dN versus1Crack closure is one of the options available.1

DSTO-TR-2050ΔK eff curve, allowing the rate generation and fatigue crack growth analysis to be carriedout within a single software environment. This report discusses the concept of crackclosure, the relationship between the nominal SIF and the effective SIF, the determinationof crack opening stress, and the procedures involved in the FCGR conversion. It alsodocuments the FCGR conversion module under the CGAP graphical user interface. Someexamples are also included to demonstrate the use of this software within the CGAPenvironment.2. Fundamentals of Fatigue Crack Growth ModellingThis section provides a brief introduction to the crack growth models used by FASTRAN,CGAP and AFGROW.2.1 Linear Elastic Fracture Mechanicsand Paris LawIn 1957, Irwin [5] derived the linear elasticstress solutions for an isolated flaw inside aninfinite plate that formed the foundation oflinear elastic fracture mechanics (LEFM). Irwinidentified three basic modes [6, 7] of fracture,(I) the opening mode, (II) the shear mode and(III) the anti-plane shear mode, andsubsequently derived the linear elasticsolutions of stresses and displacements forthese three modes of fracture. As an example,for the opening mode (mode-I) with an isolatedflaw of length 2a subjected to a uniform farfield stress S inside an infinite plate, as shownin Figure 1, the stress components are given as,S 2aS Figure 1A central mode I crack in aninfinite plate subjected touniform far-field stress 3θ θ θ cos 2 1 sin 2 sin 2 S xx KI θ 3θ θ cos 1 sin sin S yy 2 22 2πr τ xy 3θ θθ cos sin cos 2 22 (1)where r is the distance from the crack tip, and θ is the angle between the crack planeand r .The parameter K in these stress equations has been termed the SIF, and the subscript I isused to indicate the mode-I fracture. Similar solutions of stresses for the other modes offracture were also derived, with K II and K III , respectively, replacing K I in the aboveequation. Depending on the load and geometry, all three modes of crack opening or2

DSTO-TR-2050fracture may co-exist during crack extension, but under uniaxial loading and when thecrack is long, the tensile opening mode K I is dominant.The significant outcome from Irwin’s stress equations is that it shows the stress field insidea linear elastic body containing an isolated flaw and subjected to a far field applied stressis uniquely characterized by a single parameter, K . This means that provided K for anycombination of crack length, geometry and applied stress is the same, the stress, strain anddeformation would also be the same. This principle, known as the principle of similitude, isapplicable in the slow stable growth stage. The significance of the principle is that itprovides a theoretical basis for allowing the material response of simple laboratoryspecimens to be extrapolated to that of real engineering structures, and vice versa.However, the principle of similitude breaks down in the short crack regime [8] where thecrack length is either comparable to the size of the microstructures, or it is comparable tothe crack tip plastic zone size. Furthermore, these linear elastic stress solutions predict aninfinite stress at the crack tip ( r 0 ) (meaning a sharp crack has a stress concentrationfactor of infinity), a situation that cannot exist in real materials. This anomaly implies thata region of plastically-deformed material may exist at the crack tip. This plasticallydeformed region in the vicinity of the crack tip has a significant influence on the FCGR.Under cyclic fatigue loading, Paris et al [9] related the FCGR, da dN , to the SIF range, ΔK ,to giveda CΔK m .dN(2)This is now well-known as the Paris law. Here C and m are regression parameters butare also known as the FCGR coefficient and exponent, respectively. This empiricalrelationship was based on the principle of similitude and experimental evidence obtainedfor long cracks and constant amplitude testing. An example of this is shown in Figure 2. Itis customary to obtain such long crack da dN versus ΔK data for a material from constantamplitude loading crack growth tests on either compact tension (CT) or centre-cracktension (CCT) specimens.3

DSTO-TR-2050da/dNParis RegimeRegion IrisPaRegion IIIlawΔKthFigure 2:Region IIΔKΔKcrFCGR behaviour for long crack under constant amplitude loadingAs shown in Figure 2, the Paris law only applies in region II or the Paris regime.Graphically the parameters C and m are simply the parameters of linear regression ofda dN versus ΔK data within this region plotted on a double logarithm scale. Region Ishows that there is a stress intensity range threshold, ΔK 0 , below which long cracks willnot grow. Region III, right of the Paris regime, shows increasingly higher FCGR, signallingthe final fracture of the component as K max approaches KC , the fracture toughness of thematerial. Note that at K max KC , ΔK C K C (1 R ) , since the effect of crack closure is notrelevant at this point of loading. Then, for long cracks the fatigue crack propagation life,N P , can be obtained by numerically summing Equation (2) cycle by cycle from the initialcrack length until the critical SIF is reached, or until any other failure criterion is met, suchas gross section yield.2.2 Mechanical Loading EffectsThe rate of fatigue crack growth is influenced by various mechanical loading andenvironmental effects, which are not taken into account by the Paris law. Numerousmodifications and corrections have been made to the Paris law, in an attempt to bettercharacterise fatigue crack growth behaviour. In this investigation however, onlymechanical loading effects, in particular the effects of stress ratio and load sequence, atroom temperature are considered.2.2.1 Stress Ratio EffectUnder the same cyclic SIF range, FCGRs vary with the stress ratio, R S min S max . Figure 3shows the effect of stress ratio on FCGR for the 7050-T7451 aluminium alloy. These data4

DSTO-TR-2050were obtained from Sharp et al [10]. The effect of R -ratio on FCGR is not surprisingbecause a change in the R -ratio for the same ΔK means a change in the mean stress,which in turn, affects the plastic deformation at the crack tip. The obvious explanation forthe R -ratio effect on FCGR is that the cyclic plastic

Crack growth assessment is an essential element of the aircraft certification procedure for addressing structural durability and fatigue concerns of all Australian Defence Force (ADF) air platforms. It provides a means for estimating and assessing the growth of

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