Prepared by: R. Jones and S. Pitt COURSE NOTES

Prepared by: R. Jones and S. Pitt COURSE NOTES - Damage Tolerance, Microstructure and Airworthiness Recommended Reading: Jones R., Baker A., Matthews N., Champagne V, Aircraft Sustainment and Repair, ButterworthHeinemann Press, 2018, ISBN 9780081005408. (Book). https://www.elsevier.com/books/aircraft-sustainmentand-repair/jones/978-0-08-100540-8 Prac’s:

Both the Prac Classes and associated notes are on the class web site.

Class Exercises:

On the class web site

Resources:

On the class web site

MILSPEC 5: USAF METALLIC MATERIAL HANDBOOK USAF DAMAGE TOLERANT DESIGN HANDBOOK FAA crack growth data base FAA LOV ruling and relevant FAA-ac’s CASA – “How old is too old” and much more NOTE: All material covered in the Lectures and the Prac Classes is examinable.

© S. Pitt and R. Jones – S. Pitt, January 2018 © Susan Pitt and Rhys Jones

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TOPICS COVERED AND LOCATION IN NOTES 1 The Structure Of The Course ...................................................9 2 The Comet Accidents ..............................................................12 3 The Aloha Accident ................................................................15 3.1 Brief summary of the effect of the Alohoa accident on airworthiness regulations .........................................................21 4 The Role Of Full Scale Fatigue Tests in Establishing Life of Type .............................................................................................23 5 Fracture Mechanics: Background & Theory ........................34 5.1 The no growth criteria .....................................................45 5.1 Crack growth from etch pits and manufacturing defects at a fastener hole under an operational F/A-18 flight load spectrum ...................................................................................53 5.2 Fracture and yielding sites do not coincide ....................57 5.3 The Fatigue Threshold .....................................................58 5.4 The crack length dependency of the fatigue threshold ..59 6 Terminology And Definitions In Linear Elastic Fracture Mechanics ...................................................................................61 7 Derivation Of The Stress Intensity Factor .............................63 7.1 Crack tip stress and displacement fields ........................70 7.2 View of the crack tip stress field .....................................73 8 Common Solutions .................................................................80 9 The General Form Of The Kujawski Approximation ............92 10 Derivation of the stress field associated with a circular hole in a large body .................................................................................94 10.1 The stress field at a hole .................................................102 10.2 The stress concentration factor at a fastener hole in a civil aircraft fuselage .............................................................105 11 Weight Function Methods ....................................................108 11.1 Application to cracks at notches ...................................109 11.2 Derivation of weight functions ......................................122 12 Crack Tip Plastic Zone Size..................................................129 13 ASTM E399 Fracture Toughness Tests ...............................131

© Susan Pitt and Rhys Jones

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13.1 ASTM E399 Compact Tension Fracture Toughness Test 132 13.2 The effect of orientation .................................................136 14 The Energy Release Rate: G ................................................137 14.1 Definition of the Energy Release Rate ..........................137 14.2 Compliance techniques for measuring the Energy Release Rate: Slightly more general approach ...................138 14.3 An Alternative approach ...............................................144 15 The J Integral .......................................................................147 15.1 Limitations ......................................................................153 16 Failure by Fracture Of Primary Structure ..........................154 16.1 The Newman and Raju Formulae for 3D Cracks ........155 16.2 Practical 3D Fracture Mechanics: Real Cracks ...........160 17 Life of Type, Remaining Life, Through Life Support & Inspection Intervals...................................................................166 18 Modeling Crack Growth .......................................................171 19 Paris Crack Growth Equation:.............................................177 19.1 The similitude hypothesis ..............................................177 19.2 The Paris crack growth equation ..................................179 19.3 The importance of initial flaws and the local stress states .........................................................................................180 20 The Role of Initial Flaws in Design .....................................184 21 Crack Growth Models ...........................................................189 21.1 Paris equation: ................................................................192 21.2 Forman’s equation: ........................................................192 21.3 Modified Forman’s equation: ........................................193 21.4 Walker’s equation: .........................................................195 21.5 The Nasgro equation: .....................................................196 21.6 The Hartman-Schijve variant of the Nasgro equation: ..... .........................................................................................197 21.7 The Wheeler retardation model ....................................199 22 Crack Closure Based Equations...........................................201 22.1 FASTRAN: The Analytical Crack-Closure Model ......202 22.2 The Hartman-Schijve-McEvily equation .....................212


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23 Hartman-Schijve-McEvily (Nasgro) Representation Of A Number Of Common Aerospace Structural Materials: ...........218 24 An Obvious Question: ..........................................................223 25 Which Crack Growth Curve ? ..............................................228 26 Composite Repairs To Cracked Wing Skins. .......................232 27 The Difference Between Design And Aircraft Sustainment (Operational Considerations). ..................................................240 27.1 The generic shapes of crack growth curves associated with ASTM E647 tests and crack growth in operational aircraft ....................................................................................246 27.2 The role of crack closure in crack growth in operational aircraft ....................................................................................249 27.3 Crack growth in panel reinforced with a supersonic particle deposition (SPD) doubler ........................................259 27.4 Crack growth in 1 mm thick 2024-T3 specimens .........263 28 The Keith Donald ACR Test Method ...................................266 29 The Paris Crack Growth Equation Revisited:......................267 30 Variability In Crack Growth.................................................270 31 Equivalent Block Methods For Predicting Fatigue Crack Growth .......................................................................................278 31.1 Crack growth from etch pits and manufacturing defects at a fastener hole under an operational F/A-18 flight load spectrum .................................................................................284 31.2 Crack growth at a fastener hole in a RAAF P3C (Orion) aircraft containing intergranular cracking..........................287 31.3 Blocks to Failure .............................................................302 32 Other Alternative Approaches ..............................................305 33 Fractals & The DSTO-DGTA Lead Crack Approach - Also Built Into The USAF Risk of Failure Assessment Computer Program (PROF) ......................................................................307 33.1 Real life damage growth- Exponential crack growth: .308 33.2 Cubic Stress Equation: ..................................................312 33.3 Application of the Cubic Rule: Combat Aircraft Spectra: ...................................................................................314


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33.4 Application of the Cubic Rule: Maritime Aircraft Spectra: ...................................................................................317 33.5 Application of the Cubic Rule: USAF Military Transport Aircraft Spectra:..................................................319 33.6 Scaling from one fatigue critical area (FCA) to a second FCA with a different spectrum and a different peak stress. .... .........................................................................................322 33.7 Cubic Rule: Conclusion .................................................329 33.8 Generalised Frost-Dugdale Equation: ..........................330 33.9 Fractals and The Cubic Rule .........................................333 33.9.1 Nature of the crack tip stress field for a fractal crack .. .....................................................................................335 33.10 Relationship been Fractal, Frost-Dugale and Hartman-Schijve crack growth equations ...........................337 33.10.1 ... The Equivalance of the Fractal, Generalised FrostDugdale and the Nasgro Crack Growth equations ...........339 33.11 Implications for the effect of micro-structure on crack growth .....................................................................................345 33.12 Cubic Stress Equation: Composite Repairs ............351 34 Example Of The Role Of Initial Flaws Using The FrostDugdale (Exponential) Equation: Lead Cracks .......................361 34.1 Application of the Lead Crack/Frost-Dugdale Approach to Predicting the Growth of Cracks In a Repaired Structure .. .........................................................................................364 35 Palmgren-Miner damage accumulation equation and its relationship to exponential crack growth (Lead Crack) ..........368 36 The Equivalent Initial Flaw Size (EIFS) .............................372 37 Inspection Intervals ..............................................................384 38 The Interaction of Micro-Structure and Crack Growth ......385 38.1 Microstructure and long cracks ....................................390 38.2 Characterising microstructural effects using the Hartman-Schijve equation ....................................................393 38.3 Implications For Additively Manufactured Components . .........................................................................................395 38.4 Atomistic models and the Hartman-Schijve equation .396


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39 Fractography: A (Very) Brief Introduction .........................399 39.1 Beachmarks and striations ............................................399 39.2 Definitions: ......................................................................399 40 Quantitative Fractography (QF) ..........................................404 40.1 Determining the point of crack initiation .....................406 41 Mandated Initial Flaw Sizes .................................................409 42 The Role of Proof Tests in Establishing Continuing Operational Safety ....................................................................411 42.1 Description of the Proof Test Method...........................414 43 Why Not Use S-N Curves In Sustainment? .........................424 44 Your Legal Responsibility: The Concorde Example ...........425 44.1 Remarks ..........................................................................437 45 Putting It All In Context .......................................................438 45.1 The aging aircraft challenge ..........................................438 46 An Example Of Australia’s Indigenous Capabilities...........447 47 Additive Metal Technology ...................................................448 48 CLASS ASSESSMENT TASKS ...........................................449

** Note: Page Numbers are out by 1 Note: Section 28 Refers to Notes prepared by Mr. K. Donald (FTA, US) Sections 41 and 42 are taken directly from the USAF Damage Tolerance Design Handbook. Section 22.1 is (essentially) a précis of that given by J. Newman in the FASTRAN user and theory manual. NOTE: The weeks associated with each Section are clearly marked. NON EXAMINABLE subsections are clearly marked in the text as being NOT EXAMINABLE.


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NOTE: The certification requirements for military aircraft are delineated in MIL-STD-1530 and the associated guidelines are contained in JSSG2006, also see: R. Wanhill, Chapter 2 - Fatigue Requirements for Aircraft Structures, Jones R., Baker A., Matthews N., Champagne V, Aircraft Sustainment and Repair, Butterworth-Heinemann Press, 2018, ISBN 9780081005408. (Book). https://www.elsevier.com/books/aircraft-sustainment-andrepair/jones/978-0-08-100540-8 Guidelines for Composite and Adhesively Bonded structures are given in CMH-17-3G.

The US FAA requirements for composite and bonded civil aircraft are contained in FAA Advisory Circular No: 20107B. Also see: Jones R., Baker A., Matthews N., Champagne V, Aircraft Sustainment and Repair, ButterworthHeinemann Press, 2018, ISBN 9780081005408. (Book). https://www.elsevier.com/books/aircraft-sustainmentand-repair/jones/978-0-08-100540-8 References 1. MIL-STD-1530D, Department Of Defense Standard Practice: Aircraft Structural Integrity Program (ASIP) (31-Aug-2016)


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2. Department of Defense Joint Service Specification Guide, Aircraft Structures, JSSG-2006, October 1998. 3. Composite Materials Handbook, Volume 3: Polymer Matrix Composites Materials Useage, Design and Analysis, March 2012, Published by SAE International. 4. Federal Aviation Authority, (2009) Airworthiness Advisory Circular No: 20-107B. Composite Aircraft Structure, 09/08/2009. 5. R. Wanhill, Fatigue Requirements for Aircraft Structures, Chapter 2, Jones R., Baker A., Matthews N., Champagne V, Aircraft Sustainment and Repair, Butterworth-Heinemann Press, 2018, ISBN 9780081005408. (Book).


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WEEK 2 LECTURES START HERE Note: The First weeks Introductory lectures, Practical Classes and Class Assignments are on the Moodle we site. 1 The Structure Of The Course The introductory lecture spelt out the fact that in accordance with the damage tolerance design philosophy as spelt out in JSSG2006 we require:  An estimate of the residual strength of a component at any point during its operational life, Topics: Fracture and stress corrosion cracking (SCC).  An estimate of whether a defect will grow during service, Fatigue The first two points are covered in the 1st half of the course, i.e. Sections 2-16.


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We also require:  An estimate of the structural life (flight hours) for a crack to grow an initial (damage) size to its critical size and the associated crack length history, Fatigue.  An estimate of the inspection intervals needed to insure continued airworthiness, if it transpires that a defect will grow, Fatigue. The last two points are covered in the 2nd half of the course, i.e. Sections 17-46.  The link between fractals, exponential crack growth, the Cubic Rule and the HartmanSchijve crack growth equation is covered in Section 33.  The Interaction of MicroStructure and Crack Growth and implications for Additive Manufacturing are covered in Section 38.  A brief discussion on Atomistic Modelling is also given in Section 38 

Your Legal Responsibility is briefly discussed in Section 44. This Section is NOT EXAMINABLE. © Susan Pitt and Rhys Jones

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A key point to be noted when evaluating crack growth in operational aircraft

When discussing cracking in operational aircraft key points that are highlighted in this course are: That, as a result of the programs performed in support of F/A-18, F-15 and F-16 aircraft it is now known that the life of operational aircraft is a strong function of the time taken for a crack to grow from a small naturally occurring material discontinuity to a size that is readily detectable, i.e. 2-5 mm. In this case, in operational aircraft, the fatigue crack growth rate can often be described by a simple “Paris” like equation. This phenomenon is discussed in detail in Sections 25, 27 and 29. A large number of examples of how to compute the growth of small naturally occurring cracks under operational flight loads are presented in the text: Jones R., Baker A., Matthews N., Champagne V, Aircraft Sustainment and Repair, Butterworth-Heinemann Press, 2018, ISBN 9780081005408. (Book). https://www.elsevier.com/books/aircraft-sustainment-andrepair/jones/978-0-08-100540-8


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2 The Comet Accidents Recall the Comet failure discussed in the Introductory Lecture • There were two catastrophic in flight failures:  1/10/54 - G-ALYP, @30,000ft., 3680 hrs., 1286 cycles – 35 fatalities  4/8/54 - G-ALYY, @35,000ft., 2702 hrs., 903 cycles – 21 fatalities • A full scale fatigue test was then undertaken at RAE Farnborough (UK) on the fuselage from a/c G-ALYU. • Prior to fatigue testing the section was subjected to proof load = 2 x nominal pressurization. • This prior proof loading introduced residual stresses that retarded crack initiation. • Failure occurred at 1830 test cycles giving a total of 3060 cycles. • Examination of G-ALYU & G-ALYP showed evidence of fatigue at corner of passenger window and automatic direction finding windows • Failure initiated from a manufacturing defect.


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Figure is from: T. Swift, 1987, Damage tolerance in pressurized fuselages, 11th Plantema Memorial Lecture, New Materials and Fatigue Resistant Aircraft Design (ed. D L Simpson), pp. 1-77, Engineering Materials Advisory Services Ltd., Warley, UK. Also see: R. J. H. Wanhill, Milestone Case Histories in Aircraft Structural integrity, NLR-TP-2002-521, October 2002.


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This led the FAA to: Establish rules for Fatigue Management Programs (FMP)  1956 for transport airplanes  1957 for small airplane pressurized structure  Safe-life (safety-by-retirement), or  Fail-safe (safety-by-design)

More details about the Comet accident are available at http://lessonslearned.faa.gov/ Also see: R. Wanhill, Fatigue Requirements for Aircraft Structures, Chapter 2, Jones R., Baker A., Matthews N., Champagne V, Aircraft Sustainment and Repair, Butterworth-Heinemann Press, 2018, ISBN 9780081005408.


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3 The Aloha Accident Public awareness of the problem of aging structures first arose via the Aloha Airlines accident in April 1989.

Despite the fact that there were few casualties involved in the Aloha Airlines accident, it continues to be described as the event which first brought the aging aircraft issue to the attention of the general public and to those who have responsibility to ensure aircraft safety.


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On April 28th, 1988 Aloha Airlines Flight 243, a Boeing 737-200 was undertaking a regularly scheduled passenger flight from the island of Hilo to Honolulu, Hawaii. After levelling the aircraft at flight level 240 the crew heard a loud “clap” or “whooshing” sound followed by rushing air behind them. They observed that the cockpit door was gone and noticed “blue sky” where the first class cabin ceiling had been. The crew also noted that they had lost power on the No. 1 engine. By using the brakes and the No. 2 engine thrust reverser to stop on the runway, a normal touchdown and landing was achieved. The entire episode from the time of the structural failure in level flight to touchdown on Maui lasted only approximately 11 minutes.


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Figure 1 – Aircraft on ground immediately after landing, from [1].


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Figure 2 - Relative positions of body stations 360 through to 540.

Figure 3 - Cross-sectional view of the Boeing 737 showing damage done during explosive decompression.


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Figure 4 - Catastrophic failure of the Aloha Airlines aircraft.


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There were 95 people on board this flight - 89 passengers, 2 flight crew, 3 flight attendants, and an FAA air traffic controller in the jump seat. The senior flight attendant in the first class cabin section was sucked out of the aircraft during the explosive decompression. Fifty nine passengers were taken to the hospital with a range of injuries. It was in fact a lucky escape for all but one person on board the flight and is indeed testimony to the skill of the flight crew and to the residual strength of the B-737 [1].

Reference 1. NTSB/AAR-89/03, Aircraft Accident Report--Aloha Airlines, Flight 243


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3.1 Brief summary of the effect of the Alohoa accident on airworthiness regulations • Aging Airplane Safety Act 1991; • Change to §25.571 to require full scale fatigue test to validate no WFD up to design life goal (1998); • AC 120-73 “Damage tolerance assessment of repairs to pressurized fuselages” (2000); • Aging Airplane Safety Final Rule (2005); • AC No: 25.1529-1A “Instructions for continued airworthiness of structural repairs on transport airplanes” (2007); • AC 120-93 “Damage tolerance inspections for repairs and alterations” (2007); • AC 91-82 “Fatigue Management Programs for Airplanes with Demonstrated Risk of Catastrophic Failure Due to Fatigue” (2008); • AC 43-13-2b “Acceptable Methods, Techniques, and Practices – Aircraft Alterations” (2008); • AC 91-56B Continuing structural integrity program for airplanes (2008); • AD 2011–01–15 “Airworthiness Directives; The Boeing Company Model 757–200, –200CB, and 300 Series Airplanes” (2011); • AC 120-104, “Establishing and implementing limit of validity to prevent widespread fatigue damage” (10/1/2011). Introduces LOV established via full scale fatigue testing. More details about the Comet and Aloha accidents are available at http://lessonslearned.faa.gov/ © Susan Pitt and Rhys Jones

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A repair to a fuselage lap joint

Picture showing a repair to a Boeing aircraft fuselage lap joint. From the Boeing Structural Repair Training Course Description, available at https://www.myboeingtraining.com/

The effect of the environment on aircraft structural integrity is covered in the companion course notes available via Research Gate. © Susan Pitt and Rhys Jones

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4 The Role Of Full Scale Fatigue Tests in Establishing Life of Type

A typical “wiffle tree” arrangement (from SWRI, Static and Fatigue Testing of Full-Scale Aircraft Structures, from: www.aerospacestructures.swri.org)


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Contrast this with the DSTG F/A-18 full scale fatigue test. This involved the simultaneous application of manoeuvre and dynamic buffet using pneumatic loading The load spectra was obtained from operational flight loads

From Photograph courtesy of L. Molent (DSTG).


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Airbus A380 MegaLiner Barrel (MLB) full-scale fatigue test ‘specimen’, the general loading conditions, and the number of simulated flights applied during testing, from Wanhill, R.J.H., Platenkamp, D.J., Hattenberg, T., Bosch, A.F. and De Haan, P.H., 2009, “GLARE teardowns from the MegaLiner Barrel (MLB) fatigue test”, in ‘ICAF 2009: Bridging the Gap between Theory and Operational Practice’, Ed. Bos, M.J. (Springer Netherlands, Dordrecht, the Netherlands), pp. 145-167. Figure is courtesy of Dr Russell Wanhill


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HISTORY: THIS SECTION IS LARGELY TAKEN FROM TIFFANY, GALLAGHER AND BABISH [1]. In the late 1950s, the solution to preventing catastrophic fatigue failures of commercial aircraft (and large military transport and bomber aircraft) was to ensure that the design resulted in a structure which had sufficient structural redundancy to maintain adequate residual strength after any structural element failed. Aircraft manufactured using the structural redundancy concept became known as fail-safe structure. The capability of fail-safe structure to carry the required residual strength was initially demonstrated via full scale experimental testing. Appendix K, in [1], describes the early development of fail-safe testing by US aircraft manufacturers. Reference 1. C. F. Tiffany, J. P. Gallagher, and C. A. Babish, IV, Threats To Aircraft Structural Safety, Including A Compendium Of Selected Structural Accidents/Incidents, ASC-TR-2010-5002. March 2010. © Susan Pitt and Rhys Jones

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Operational lives were established (calculated) using fatigue tests and design analyses utilized safe-life fatigue analysis methods based on elastic stress analysis, stressfatigue life (S-N) curves, and Miner’s Rule.

The stress-fatigue life curves used to calculated the fatigue life the aircraft were based on smooth bar fatigue tests and notched coupon fatigue tests, see ASTM Standard E466 on Moodle

Since this surface condition DOES NOT represent/reflect that seen in operational aircraft it means that it is difficult to relate the calculated fatigue life to the actual operation life of an aircraft.


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In the later part of the 1950’s and beyond, part of the fatigue design verification usually included at a full-scale fatigue test.

One of the main functions of the full-scale fatigue test is to identify fatigue critical locations and local high stress locations that were not identified by the structural analysis.

Many of the aircraft that were operating in the late 1950s and early 1960s did not have a full-scale fatigue test until fatigue failures occurred in operational aircraft.


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In the late 1950’s the USAF realised that operational loads monitoring was a key element that was required to extend operational lives beyond the initial full-scale fatigue test.

It was also realised that the safe-life fatigue analysis approach did not accurately capture the rates at which fatigue damage was accumulating in operational aircraft.


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For those aircraft that had design operational lives validated by full-scale fatigue tests (FSFT) there still were early fatigue failures in service; these in-service failures were found to start at manufacturing and material defects. For example, the Northrop F-5 (Appendix A.6 in Reference [1]) was fatigue tested to 16,000 cyclic test hours without failure, but failed in 1900 hours due to a tool mark in a highly stressed radius at the speed brake cut-out in the lower wing surface.

The KC-135 (Appendix B.2 in [1]) was fatigue tested to 55,000 cyclic test hours without failure, but experienced 28 instances of unstable cracking (at 1,800 to 17,000 flight hours) in the wings due to poor quality fastener holes in a very low toughness material.

The F-111 (Appendix B.5 in [1]) was eventually cycled to 40,000 cyclic test hours (10 lifetimes) but failed in 105 hours due to a relatively large forging flaw in a low toughness material.


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The lack of correlation between operational lives validated during the design phase and those experienced during operational experience resulted in significant concern to the aircraft community. The above description is largely taken from Tiffanny, Gallagher and Babish [1]. It was not until the F-15 program that it was realised that in order to life an aircraft so that the crack cycles versus flight hours curves agreed with fractographic data the associated da/dN versus ∆K curve to be used should be an amalgam of the ASTM Long crack and the Short crack curves for that material and thickness.

This topic is discussed in more detail in Sections 24, 25, 27 and 29.


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As such the role of the full scale fatigue tests is primarily to establish where fatigue cracking may occur in the fleet. Other aims include: Determine the life (and for civil transport aircraft the LOV) of the airframe; Validate modifications and repairs; Obtain strain measurements for subsequent use in through life support of the fleet; Obtain crack growth data to support damage tolerance analysis/assessment of the airframe and subsequent inspection programs. Provide the basis for certifying the airframe (LOV) and, if required, for extending the LOV. See L. Molent, Analysis and Prevention of Aircraft Structural Failures, 2011. available at: demar.eel.usp.br/~baptista/arquivos/BAP01


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The operational life of the airframe is then calculated as: Operational Life = FSF Test Life/Scatter Factor The Scatter Factor (SF) is typically in the range 2-4 Thus if the test gave a life of 120,000 flight hours the various scatter factors would yield the following operation lives.

In recent years it is common to use a SF =2 Probability of Survival (%)

No of Flight Operational Life (Years) Hours obtained assuming (3,000 flight hrs/year)

2.0

94.0

60,000

20

2.5

97.5

48,000

16

3.0

98.8

40,000

13.3

3.5

99.3

34,300

11.4

4.0

99.54

30,000

10.0

SF

A more detailed discussion on the timeline and the historical development of the Damage Tolerant Design Philosophy as well as SafeLife and Failsafe is given in [3]. References 1. C. F. Tiffany, J. P. Gallagher, and C. A. Babish, IV, Threats To Aircraft Structural Safety, Including A Compendium Of Selected Structural Accidents/Incidents, ASC-TR-2010-5002. March 2010. 2. Lincoln JW., Melliere RA., Economic Life Determination for a Military Aircraft, AIAA Journal of Aircraft, 36,5, 1999. 3. R. Wanhill, Fatigue Requirements for Aircraft Structures, Chapter 2, Jones R., Baker A., Matthews N., Champagne V, Aircraft Sustainment and Repair, Butterworth-Heinemann Press, 2018, ISBN 9780081005408.


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5 Fracture Mechanics: Background & Theory All new aircraft design and repairs made to cracked aircraft structure, and all structure and structural changes made to in-service aircraft now require some form of damage tolerance analysis, see the FAA-ac’s and the USAF Damage Tolerant Design Handbook (Assisst02) on Moodle. Also see MIL-STD-1530 and JSSG2006

The discipline of Fracture Mechanics underpins this (damage tolerance) analysis. The nature of each analysis needed depends on the consequences of failure in the repaired or modified structure.


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For example, the damage tolerance analysis associated with a repair to a compressively loaded fuselage member that has corrosion damage would be (relatively) small.

In contrast a fleet wide modification to the lower wing skin of a fighter aircraft requires a detailed evaluation of the effect on both residual strength and on durability as outlined in the US DoD Joint Services Structural Guidelines, JSSG-2006.

At this stage it should be stressed that when assessing the damage tolerance of repairs or structural modifications it is important to note that this assessment should be consistent with the methodology used to certify the aircraft.


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This specification (JSSG2006) requires that all safetyof-flight critical structure be designed using a damage tolerance analysis.

The purpose of the analysis is to ensure that cracks potentially present in the structure will not cause loss of the aircraft during flight for some predetermined period of in-service operation.

Thus all structural repairs and modifications to safety-offlight critical structure, must ensure that, for the operational period contemplated, the safety of flight is not compromised as a result of the repair (or modification), see FAA-ac 120-93 on Moodle.


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This raises the question as what constitutes a safety-offlight critical structure and their locations within the airframe.

To support the force management operations MILHDBK-1530 requires that a critical parts list is prepared by the airframe contractor and appended to the Force Structural Maintenance (FSM) plan supplied to the (US) Air Force.

Any potential problems associated with the repair or modification of safety-of-flight structure needs to be identified and detailed in the repair manual.


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The complexity and level of the analysis required is variable. If the repair or modification can be incorporated into any given aircraft or will be applied to all aircraft in the force, then a more careful analysis of the impact of a crack potentially existing in the structure should be conducted. For one off repairs performed to return the aircraft to a depot for more extensive repair, the analysis need only address residual strength considerations This requirement is now mandated and is built into FAA- ac 120-93 which is on the class website. This a/c deals with the requirement for all repairs/mods to be designed in accordance with damage tolerance principles. This also implies that the effect of corrosion and any corrosion reworks should also be assessed for its/their effect on structural integrity. This requirement is implied in US Congress Public Law 107-314 Sec: 1067, “Prevention and mitigation of corrosion of military equipment and infrastructure”. Ask students to look up the relevant FAA/ac on Moodle.. © Susan Pitt and Rhys Jones

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This leads us to the question of: How do we estimate the residual strength of a damaged structure? NOTE: As stated in MIL-STD-1530 - The role of testing is merely to validate and correct analysis MIL-STD-1530D, Department Of Defense Standard Practice: Aircraft Structural Integrity Program (ASIP) (31-Aug-2016)

Before addressing this it is important to note that there are three basic elements in any damage tolerance analysis: 1) A residual strength analysis

2) An estimate of whether a flaw will grow in service.

3) A (sub critical) crack growth analysis.


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Residual strength, crack size and time-in-service as per the USAF Damage Tolerance Design Handbook. © Susan Pitt and Rhys Jones

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It should be stressed that a structure should always be designed such that: At no time in the operational life of the structure will the residual strength of the structure fall beneath limit load.

Limit load = max load seen in service.

Proof load = 1.5 * Limit load


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Once a crack has been detected/initiated the discipline of fracture mechanics can be used to address the following questions:  Is the residual strength of a cracked component beneath its design (limit) load?  What is the critical crack size for a given load case?  For a given load spectra will a crack grow? These 3 questions are addressed in the 1st half of the course.  Will crack growth occur in a fast unstable, or a slow stable manner?  What is the fatigue life of a cracked structure?  What are the appropriate inspection intervals for a cracked structure? These last 3 questions are addressed in the 2nd half of the course. © Susan Pitt and Rhys Jones

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NOTE The stresses at a crack are infinite and take the form: yy  KI / (2r)

where K is termed the stress intensity factor and r is the radial distance from the crack tip. (The precise equations will be presented later in the course.) Thus stress can’t be used as a design parameter for a body containing a flaw. This is discussed in the Tutorials (Prac Classes). The stress intensity factor K is now used in design to assess both residual strength and fatigue life.


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The stress intensity factor KI at a crack tip is defined as KI = limit r 0 (2r) yy y

r



x

Figure – Local “crack-tip axis system(s)


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5.1

The no growth criteria (For metallic airframes)

As the load is cycled the value of K = Kmax –Kmin in a cycle may reach a certain critical value Kth, the fatigue threshold for the material, at which stage the crack will grow. (The units of K are usually expressed as: MPa m or k.s.i in.)

Thus the no growth criteria is K < Kth If the crack grows then as the load increases K (the crack tip stress intensity factor) will eventually reaches a certain critical value Kc (the critical fracture toughness for that thickness of material). (The units of K are usually expressed as MPa m or k.s.i in. Note: 1 k.s.i in = 1.0987 MPa m ) © Susan Pitt and Rhys Jones

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RESIDUAL STRENGTH CRITERIA Accordingly the crack will suddenly propagate (usually in an unstable manner) and failure wi1l occur when:

KI = Kc (the Mode 1 fracture toughness)

although stable crack growth is also possible and cracks can even arrest (i.e. stop). y

r



x

Figure – Local “crack-tip axis system(s)

At this point the airframe will fail !! © Susan Pitt and Rhys Jones

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46

Table 1a.Plane Strain Fracture Toughness Of Selected Alloys, from the USAF Damage Tolerance Handbook (on Moodle) Material Yield TypicalFracture Stress, MPa Toughness, MPa m Aluminiums 2014-T651 2024-T3 2024-T851 7075-T651 7178-T651

455 345 455 495 570

25 44. 26 24 23.

7178-T7651 490

33.

Titanium Alloys Ti-6At-4V Ti-6At-4V Steels 4340 4340 4335 + V 17-7 PH 15-7 Mo 350 Maraging

910

115

1035

150

860 1515 1340 1435 1415 1550

99 60. 73 77 50 55.


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47

Tough materials have high Kc values, e.g. > 100 MPa m for high alloy steels, and low values, i.e. 10 MPa m, for some cast alloys.

For long cracks in thick (plane strain) aerospace aluminium alloys the values of KIc values typically lie in the range 25 – 35 MPa m.

For long cracks, i.e. greater than say 10 mm, in aerospace aluminium alloys Kth values typically lie in the range 1 – 6 MPa m. For short cracks representative of the initial flaw sizes found in combat aircraft, viz: 3 – 30 microns, Kth is very small – typically 0.1 MPa √m. Representative values for the long crack plane strain fracture toughness KIC are given in Table 1.


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48

The fracture toughness of a material can be measured experimentally and is dependent on both temperature and specimen thickness (constraint).

It can also be dependent on the yield strength, crack length and strain rate. As such it is not a true material constant.

The plane strain value is termed the plane strain fracture toughness and is denoted by the symbol KIC

For non plane strain conditions the symbol KC is commonly used.


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49

Table 1b.Plane Strain Fracture Toughness of Selected Steels, from USAF Dam Tol Handbook, on Moodle Alloy

Product Orientation form

Heat treatment

D6ac

Plate

L-T

D6AC

Plate

L-T

D6AC

Forging L-T

D6AC

Plate

1650F, AusBay Quench 975F, SQ 375F, 1000F 2 + 2 1650 F, AusBay Quench 975F, SQ 400F, 1000F 2 1650F, AusBay Quench 975F, SQ 400F, 1000F 2 + 2 1700 F, AusBay Quench 975F, OQ 140F, 1000F 2 + 2 1700F, AusBay Quench 975F, OQ 140F, 1000F 2 + 2

D6AC

L-T

Forging L-T

L-T

Yield Sample Spec stress size Thick k.s.i (inches) 217 19 0.6

Kmax

Kmin Kav

88

40

62

217

103

0.6-0.8

92

44

64

214

53

0.6-0.8

96

39

66

217

30

0.6-0.8

101

64

92

214

34

0.7

109

81

95

Quench and Temper

185192

27

1.0-2.0

147

107

129

9Ni-4Co-.20C

Hand forging

9Ni-4Co-.20C

Forging L-T

1650F, 1-2 Hr, AC, 1525F, 1-2 Hr, OQ, -100F, Temp

186192

17

1.5-2.0

147

120

134

PH13-8Mo

Forging L-T

H1000

205212

12

0.7-2.0

104

49

90


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50

Plane Strain Fracture Toughness Of selected Aluminium alloys, from USAF Dam Tol Handbook, on Moodle. Alloy

Product form

Orientation

2014-T651 2014-T651 2014-T652

Plate Plate Hand Forging Hand Forging Plate Plate Plate Plate Forging Hand Forging Hand Forging Plate Plate Plate Plate Plate Plate Forging Extrusion Forging Hand Forging Hand Forging Plate Plate Die Forging Die Forging Hand Forging Hand Forging Hand Forging Plate Plate Plate Die Forging

2014-T652 2024-T351 2024-T851 2024-T851 2024-T851 2024-T852 2024-T852 2024-T852 2124-T851 2124-T851 2124-T851 2219-T851 2219-T851 2219-T851 2219-T851 2219-T8511 2219-T852 2219-T852 2219-T852 2219-T87 2219-T87 7049-T73 7049-T73 7049-T73 7049-T73 7049-T73 7050-T7351 7050-T7351 7050-T7351 7050-T74

Kmax

Kmin

Kav

L-T T-L L-T

Sample Spec size Thick (inches) 24 0.5-1.0 34 0.5-1.0 15 0.8-2.0

25 23 48

19 18 24

22 21 31

T-L

15

0.8-2.0

30

18

21

L-T L-S L-T T-L T-L L-T

11 11 102 80 20 35

0.8-2.0 0.5-0.8 0.4-1.4 0.4-1.4 0.7-2.0 0.8-2.0

43 32 32 25 25 38

27 20 15 18 15 19

31 25 23 20 19 28

T-L

17

0.7-2.0

22

14

18

L-T T-L S-L L-T T-L S-L S-L T-L S-L L-T

497 509 489 67 108 24 85 19 60 32

0.5-2.5 0.5-2.0 0.3-1.5 1.0-2.5 0.8-2.5 0.5-1.5 1.0-1.5 1.8-2.0 0.8-2.0 1.5-2.5

38 32 27 38 37 26 34 34 35 46

18 19 16 30 20 20 19 23 20 30

29 25 21 33 29 22 25 29 25 38

T-L

28

1.5-2.5

30

22

27

L-T T-L L-T

11 11 21

0.8-2.0 1.0 0.5-1.0

34 22 34

25 19 27

27 22 30

S-L

46

0.5-1.0

26

18

22

L-T

28

0.5-1.0

37

23

30

T-L

27

1.0

28

18

22

S-L

24

0.8-1.0

22

14

19

L-T T-L S-L S-L

31 29 30 12

1.0-2.0 1.5-2.0 0.8-1.5 0.6-2.0

43 35 30 27

28 25 25 21

35 30 28 24


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51

The thickness effect is associated with whether the fracture takes place under plane stress or plane strain conditions. Summary of Kapp data for 7075-T6 sheet at Room Temperature, from USAF Dam Tol Handbook, on Moodle. Kapp Thickness Inches Ksi in

Direction

Specimen Width (inches)

Buckling Restraint

0.039 0.063 0.090

49.4 51.1 64.6

L-T L-T L-T

6 15.8 12

NO NO YES

0.039 0.063 0.063 0.064 0.061 0.061 0.062 0.062 0.064 0.063

45.2 53.3 54.3 56.7 59.2 62.9 65.0 61.2 65.7 69.5

T-L T-L T-L T-L T-L T-L T-L T-L T-L T-L

6 3.03 4.5 6 7 8 10 12 18 24

NO YES YES YES YES YES YES YES YES YES

0.061 0.063 0.064 0.063 0.063 0.063 0.063 0.081

50.1 56.2 58.4 54.4 55.0 54.8 46.0 56.0

T-L T-L T-L T-L T-L T-L T-L T-L

3 6 10 12 15 15.8 24 30

NO NO NO NO NO NO NO NO


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52

5.1 Crack growth from etch pits and manufacturing defects at a fastener hole under an operational F/A18 flight load spectrum From the Figure below it can be seen that fracture in thick plates (primary structure), where plane strain conditions apply, has the lowest value of KC.

Consequently, it is common, if at times conservative, to design on the basis of the plane strain fracture toughness KIC.


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53

Effect of thickness (constraint) on fracture toughness

From http://www.ndted.org/EducationResources/CommunityCollege/Materials/ Mechanical/FractureToughness.htm

We see that as the thickness decreases, i.e. as the level of constraint seen by the material at the crack tip decreases, the apparent fracture toughness Kc increases.


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54

That is to say that the amount of work per unit area of crack surface created is higher when there is a larger plastic zones surrounding the crack tip.

From http://www.ndted.org/EducationResources/CommunityCollege/Materials /Mechanical/FractureToughness.htm © Susan Pitt and Rhys Jones

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55

The temperature effect shown below is associated with the existence of a transition temperature where as the temperature is lowered the fracture toughness undergoes a rapid drop. Some steels can have a transition temperature close to 0C. For aerospace quality steels it is often closer to -40C. This effect is used in Cold Proof Load Tests (CPLT) to ensure continued airworthiness.


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56

5.2

Fracture and yielding sites do not coincide

Iso von Mises stress e contour

crack


Direction of yielding

Crack path, i.e. direction of fracture

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57

Week 3 Starts Here 5.3

The Fatigue Threshold

For long cracks in aerospace aluminium alloys values of Kth typically lie in the range 1 – 6 MPa m

Kth is a function of both the R ratio and the crack length and asymptotes to a constant value, at typically 5 mm or so, as the crack length increases.

For short (lead) cracks, i.e. the fastest growing cracks in the fleet, that are representative of the initial flaw sizes found in combat aircraft*, viz: 3 – 30 microns, Kth is very small, typically 0.1 MPa √m. However, there can be a significant variation dependent on the local microstructure.

* See Section 36 on equivalent initial flaw sizes (EIFS) © Susan Pitt and Rhys Jones Page 58

5.4 The crack length dependency of the fatigue threshold

It is common to express the fatigue threshold Kth as Kth = Kth0 √(a/(a+a0)) where Kth0 is the asymptotic value seen by long cracks and a0 is a constant Kth0 is a function of the R ratio and the expressions

Kth0 = Kth (1- λ R) or Kth0 = Kth √[(1-R)/(1+R)]

are sometimes used to represent this R ratio dependency λ is typically 0.8. Kth is the long crack threshold value associated with R = 0.


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59

Whilst there are several ways to measure Kth it now appears that the ASTM load decreasing approach can yield erroneous values.

Dr. Forth was, at the time, the Chairman of the NASA Johnson Fracture Control Board. This report is on Moodle

This has been the topic of a past Class Exercise, see Moodle.


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60

6 Terminology And Definitions In Linear Elastic Fracture Mechanics A crack present in a structure which is subjected to a far field load can have three modes of extension. Irwin suggested three basic types of kinematic movements of the upper and lower crack surfaces with respect to each other. These are illustrated below.

(a)

(b)

(c)

The three basic modes of crack surface displacement: (a) opening Mode I; (b) sliding Mode II; (c) tearing Mode III


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61

These three modes can be categorised as follows :(i) The opening mode (Mode I) is encountered in symmetrical extension and bending of cracked structures where displacement discontinuity is perpendicular to the plane of the crack. (ii) The sliding, or shearing, mode (Mode II) occurs in skew-symmetric plane loading of cracked structures with displacement discontinuity of the leading edge of the crack occurring in the plane and parallel to the direction of the crack. (iii) The tearing mode (Mode III) occurs in skewsymmetric bending (twisting) or loading of cracked structures by forces perpendicular to the crackplane and displacement discontinuity is perpendicular to the plane of the material and in the plane of the crack.


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62

7 Derivation Of The Stress Intensity Factor

? What is the stress intensity factor (K) ? ? How is it related to the stress field ?

For any 2D problem the stress field can be obtained via the Airy’s stress function  that satisfies the biharmonic equation, viz:

4  = 0


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63

4  = 0 For a Mode I crack under symmetric loading we can express  in the form:  = rλ+1 (C1 cos((λ + 1) ) + C4 cos((λ - 1) )) where r,  is a local crack centred coordinate system, see below.

=

r =0

 = -

Crack tip, r =0

Local crack tip coordinate system


Page

64

In this case the faces of the crack are stress free and the boundary conditions on the crack faces are  = r = 0, on  = , viz:  = 2/2 r = 0, on  =  which implies that C1 cos( (λ + 1) ) + C4 cos( (λ - 1) ) = 0

i.e. λ = 1/2, 3/2, 5/2, etc, and

The requirement that: r = - /r (1/r/) = 0, on  =  means that / = 0, on  =  © Susan Pitt and Rhys Jones

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65

From this we find that

C1 (λ + 1) sin( (λ + 1) ) + C4 (λ - 1) sin( (λ - 1) ) = 0 or C4 = -C1 (λ + 1) sin( (λ + 1) )/ [(λ - 1) sin( (λ - 1) ) ] From this it follows that  = 2 /2 r = (λ + 1) λ rλ-1 (C1 cos((λ + 1) ) + C4 cos((λ - 1) ))

We obtain similar expressions for rr = 1/r  /r +1/r2 2 /2 , and r. Thus the strain energy density (W) which is ½ ijij is proportional to r2λ-2. © Susan Pitt and Rhys Jones

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66

The total strain energy U in a volume of material ahead of the crack tip is given by U = 1/2  ij ij dV

Since we can write dV = r dr d we see that U   r2λ-2 r dr d =  r2λ-1 dr d For this to be finite for a vanishingly small element in front of the crack we require λ 1/2


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67

Thus from energy considerations the first term (λ = 1/2) in the series for  can be written in the form  = C4 r3/2(1/3 cos(3/2) + cos(/2)) and  = 2 /2 r = - C4 r-1/2( 3/4 cos(3/2) + 1/4 cos(/2)) Defining K1 = -(2 ) C4 where K1 is the mode I stress intensity factor we see that the stresses have a r singularity at the crack tip and  = K1 ( 3/4 cos(3/2) + 1/4 cos(/2))/(2 r) rr = K1 ( 5/4 cos(/2) - 1/4 cos(3/2))/(2 r) Note that along the line  = 0, directly in front of the crack, we have an equal bi-axial stress state, viz:

rr =  =  (say) rr = = K1/ (2 r) and r= 0 © Susan Pitt and Rhys Jones

Page

68

Thus ahead of a Mode I crack we have

equal biaxial stresses, a state of hydrostatic tension minimum distortion (yield)

rr = = K1/ (2 r) and r= 0 We thus see that yield and fracture events do not coincide, see diagram below. Iso von Mises stress e contour

crack

Direction of yielding

Crack path, i.e. direction of fracture

A specimen will be passed around in class to reinforce this observation.


Page

69

7.1 Crack tip stress and displacement fields For a through thickness crack in an elastic body the elastic stress distribution in the immediate vicinity of the crack tip can be expressed in the form:

x 

K II   3    3  cos  1  sin sin   sin  2  cos cos  2 2 2 2 2 2 2r 2r

KI

y 

 xy 

  3 cos 1  sin sin 2 2 2 2r

KI

  3  K II sin cos cos  2 2 2 2r 

KI   3 K II   3  sin cos cos  cos 1  sin sin  2 2 2 2 2 2  2r 2r

Here KI, KII , KIII, are the Modes I, II, and III stress intensity factors respectively


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70

The displacement field in the immediate vicinity of a crack-tip can be expressed as :

u

KI  r   3  K   (2  1) cos  cos   II 4   2   2 2  4

v

KI  r  4  2

 r   2

 3  K   (2  1) sin  sin   II 2 2  4 

 3    (2  3) sin  sin  2 2 

 r   2

 3    (2  3) cos  cos  2 2 

where  (=G) is the shear modulus of the material, and

 3  4   (3  ) /(1  )

Plane strain Plane stress

and where  is Poisson’s ratio for the material. These equations are commonly used when using finite element analysis to compute K solutions.


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71

Here the Mode I and II stress intensity factors KI and KII are defined as, viz:

K I  lim ( yy 2r ) r 0

K II  lim ( xy 2r ) r 0


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72

7.2 View of the crack tip stress field SECTION 7.2 WILL NOT BE EXAMINED The high stress field associated with a crack tip can be seen experimentally. To illustrate this consider the “double ear” specimen shown in Figure 1 which has cracks growing out of each side of a central “rabbit ear” like hole. This specimen geometry was developed as part of the Lockheed P3C (Orion) service life extension program (SLEP) [1] and used to study crack growth under a range of measure flight load spectra [2].

Figure 1 Test specimen with cracks emanating from central holes. These cracks have grown naturally from small material discontinuities

As can be seen in the picture of the specimen real cracks are invariably not straight. (Fractal? See Section 33.9) © Susan Pitt and Rhys Jones

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73

A picture of the bulk stress field B (= (x + y + z)) in the structure obtained using lock-in infra-red thermography is given in Figure 2 where the high stresses at the crack tip are clearly visible. 45 20 mm

40 35 30 25 20 15 10 5 0 -5 -10 -15 -20 -25 -30

Figure 2. The bulk stress field (∆(1 + 2 + 3)), in MPa, around the crack, note in the region just behind the crack tip the faces of the crack are OPEN AND DO NOT CLOSE DURING FATIGUE


Page

74

The second example is a crack between stiffeners in an Airbus upper fuselage rib stiffened upper fuselage panel.

Figure 3 Schematic of an Airbus upper fuselage rib stiffened panel with two SPD strips © Susan Pitt and Rhys Jones

Page

75

A picture of the bulk stress field B (= (x + y + z)) near the crack in the plate obtained using lock-in infra-red thermography is given in Figure 4 where the high stresses at the crack tip are clearly visible.

Figure 4 Stress distribution associated with a crack in an Airbus upper fuselage panel


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76

The third example is two 3.18 mm thick 2024-T3 plates each of which contains a 32 mm long edge crack. Each crack repaired with a seven ply unidirectional boron epoxy patch that is bonded over the crack. To stop secondary bending the two plates are bonded back to back and are separated by an aluminium honeycomb core, see Figure 5.

2024-T3 plate

160 75 2024-T3 t = 3.18 mm

460

280 boron/epoxy patch 7 layer boron/epoxy patch (inside) honeycomb core grips

Figure 5 Specimen geometry. © Susan Pitt and Rhys Jones

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77

A picture of the bulk stress field B near the crack in the plate obtained using lock-in infra-red thermography for a cyclic remote stress  = 100 MPa and R= 0.1, is given in below where the high stresses at the crack tip are clearly visible.

Figure The stress field around the crack, from [3] This figure further illustrates the fact that real cracks have a complex shape and are not straight, see Sections 33.9 and 33.10. © Susan Pitt and Rhys Jones

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78

The topology of such cracks can often be described as a fractal [4] with a fractal box dimension of approximately 1.2 [5, 6] see Sections 33.9 and 33.10.

REFERENCES 1. 2. 3. 4. 5. 6.

Lincoln JW., Melliere RA., Economic Life Determination for a Military Aircraft, AIAA Journal of Aircraft, 36,5, 1999. Jones R. and Tamboli D., Implications of the lead crack philosophy and the role of short cracks in combat aircraft, Engineering Failure Analysis, 29, 2013, pp.149-166. Jones R., A scientific evaluation of the approximate 2D theories for composite repairs to cracked metallic components, Composite Structures, 87, , 2009, pp. 151-160. Mandelbrot BB, Passoja DE, Paullay AJ. Fractal character of fracture surfaces of metals. Nature 1984;6:721–2. Jones R., Chen F., Pitt S., Paggi M., Carpinteri A., From NASGRO to fractals: Representing crack growth in metals, International Journal of Fatigue 82 (2016) 540–549. Molent L., Spagnoli A., Carpinteri An., Jones R., Using the lead crack concept and fractal geometry for fatigue lifing of metallic structural components, International Journal of Fatigue, (2017), 102, pp. 2014-2020.

END OF SECTION THAT WILL NOT BE EXAMINED


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79

Week 4 Starts Here 8 Common Solutions This section presents a range of solutions for commonly encountered geometries and loads. One of the most useful solutions is a central crack of length 2a in an infinite sheet, of thickness t, under a remote uniform stress σ, see Figure. σ 2W

2a

σ

Schematic of a centre-cracked plate.

In this case the stress intensity factor can be expressed as K1 = σ(a) © Susan Pitt and Rhys Jones

Page

80

If the plate is 2W wide we obtain K1 = σ(a) [sec (a/2W)] = (a,w) σ(a) Here (a,w), which is termed the beta factor, accounts for finite geometry effect. An alternative commonly used expression for  is  = [2W/(a) tan (a)/2W]1/2


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81

σ

W

a

σ

Figure Schematic of a centre-cracked plate.

For an edge crack in an semi-infinite panel the solution is K1 = (a,w) σ(a) where  = 1.12 + 0.41a/(W() )+18.7 (a/(()W)2 38.48 (a/(W())3 + 53.85(a/(W())4 © Susan Pitt and Rhys Jones

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82

For small a/W the first term 1.12 is the so called free surface correction factor. This equation can also be approximated by K1 = 1.12 σ(a) [sec (a/2W)]

This highlights the term 1.12 as being the factor applied to the centre crack problem to allow for the free surface.


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83

Example: A large square flat panel in a 1 m by 1m section of a 3 mm thick 7075-T6 aluminium alloy wing skin contains a mid bay crack that lies at 90 degrees to the maximum principal stress. The apparent critical fracture toughness for this thickness of material has been measured to be Kc = 50 MPa √m. If (at limit load) the panel sees a stress of 150 MPa determine the largest crack length that the wing can withstand. Answer: At fracture K = Kc = 50 MPa √m Let us first assume that finite width effects can be ignored. This assumption gives K = 150 √(π acr) where acr is the critical crack length. Thus acr = (50/150)2/ π = 0.03536 m Hence the largest tip to tip crack length that the wing skin can withstand is 2a = 0.07072 m or 70.72 mm. This length is less than 1/10th of the width of the panel thereby justifying the approximation that the width effect on K is negligible. © Susan Pitt and Rhys Jones

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84

Example: If in the wing skin discussed above a crack with a tip to tip length of 20 mm was found in the wing skin during a routine inspection then if the long crack threshold ∆Kth is 5 MPa √m will the crack: a) Fail and hence should the aircraft be grounded ? b) Grow ? Answer: The peak stress intensity factor experienced by the crack during flight will be K = 150 √(π 10/1000) =18.9 MPa √m So the answer to part a) is: No the wing will not fail. The answer to part b) requires you to calculate ∆K but you are not told the R ratio. To be conservative assume R = 0 so that ∆K = K = 18.9 MPa √m. Since ∆K > ∆Kth then yes the crack will grow. A crack growth rate analysis must now be performed to determine how long it will take for the crack to grow to its critical length (as determined above) before any subsequent decision can be made about when and how to repair the crack and whether to allow the aircraft back into service. © Susan Pitt and Rhys Jones

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85

K = P[0.92 + 6.12 +0168 (1)2 + 1.32 2 (1-)2]/{ t(W) (1-)3/2} where  =a/W Figure 32 Schematic of point loads acting on an edge cracked plate

Concentrated forces at the centre of an embedded crack in a finite width panel. K = P[1.297 -0.297cos(/2)] /{ t(W sin(/2))}

Figure 33 Schematic of point loads acting in the middle of an embedded crack.

K = Sg(  ) [1.99 -(1)(2.15-3.93+2.72] /{ (1+2)(1-)3/2} where Sg = 6 M/(b2t) and M = Ph/2 Figure 34 Edge crack under bending moments


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86

b

P

K = P/[2t( a )](a+b/a-b)1/2 P

2a a

Figure 35 Embedded crack with a crack face force

b

P

K = P/[t( a )] (a+b/a-b)1/2 P

2a a

Figure 36 Embedded crack with opposing crack face forces

b

b

P

P

P

2a P

K = 2P/[t(  )] (a+b/a-b)1/2 P

a Figure 37 Embedded crack, symmetric opposing crack face forces


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87

K =  ( l ) (l /r)





r

l



 Figure 38 Cracking at a hole under remote uniform stress The Bowie approximation for a single crack can be written as:

β(l,r) = 0.6762 + [0.8733/(0.3245 +( l /r))]

Tabulated values for (l /r) =0 l /r One Two crack cracks 0.0 3.39 3.39 0.1 2.73 2.73 0.2 2.30 2.41 0.3 2.04 2.15 0.4 1.86 1.96 0.5 1.73 1.83 0.6 1.64 1.71 0.8 1.47 1.58 1.0 1.37 1.45 1.5 1.18 1.29 2.0 1.06 1.21 3.0 0.94 1.14 5.0 0.81 1.07 10.0 0.75 1.03 0.707 1.0 



KI =  ( a ) sin2()

2a

KII =  ( a ) cos2() 

 Figure 39 Uniform stress acting on an inclined crack


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88

KI = h ( a ) β() where the hoop stress h is 2a

h=PRr/t  = a/(Rt)

Uniform internal pressure P

β() = [ 1 +0.52 +1.2920.0743 ] Here R and t are the radius and the thickness of the pipe.

Figure 40 Crack in a pressurised pipe containing a axial running through crack

The solution for a crack emanating from a semicircular edge notch of radius r is given by Newman, Wu, et. al. [1] and will be presented later in this course. This solution is needed in the Prac Classes (Tutorials). Reference 1. Newman, J.C., Wu, X.R., Venneri, S.L., and Li, C.G., Small-Crack Effects in High-Strength Aluminium Alloys, NASA, Editor. 1994, NASA.


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89

For the problem of a crack at a circular hole in an infinite sheet the value of KI may be approximated as 

R a



Schematic of a crack at a circular hole in an infinite sheet. a

KII = F 0 2(a/) n(x) /(a2-x2) 1/2dx where

F = 1/2 + (1.12 - 1/2) exp(- a/)

and  is the local radius of curvature 1. Mode I;  = 0.8 (1.0 – 0.3 a/ + 0.13 ( a/.)2) 2. Mode II;  = 0.32 if a/ < 1.3, and  = 0.32 ( 1.- 0.5 (a/ -1.3)/a/ ) if a/ > 1.3. and n(l) = / 2 [2+ (R/ l)2 + 3(R/ l)4] where x = 0 is the edged of the hole, and where l = R +a.


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An alternative approximate expression for KI is KI = 1.12 (/ 2) [2+ (R/l)2 + 3(R/l)4] (a) This is based on the so called Kujawski approximation.


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9 The General Form Approximation

Of

The

Kujawski

y axis Stress field = (n, ) at the notch

Notch

 is the local radius of curvature

x=0



Crack

l

in the uncracked body

x=l

Schematic diagram showing notch, crack and the stress field at the notch. For the notch problem given above the Kujawski approximation can be used to determine K

K = 1.12 * σy (evaluated at x = l ) √(πl) where σy is the stress in the uncracked body acting in the direction normal to the crack at the location of the crack tip (i.e. evaluated at x = l in the uncracked body)

This approximation is reasonably accurate for crack lengths l < ρ (the local radius of curvature of the notch). © Susan Pitt and Rhys Jones

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Thus if l > a then since stress is a second order tensor it follows that: rr (evaluated at r = b) = σ cos2(θ) = σ (1+ cos(2θ))/2 and r (evaluated at r = b) = σ cos2(θ) = -σ sin(2θ)/2 © Susan Pitt and Rhys Jones

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The solution for the stresses at any point in the structure can be obtained via the Airy’s stress function  that satisfies the biharmonic equation, viz:

4  = 0 where 4 = 2(2) and (2) = 2/2r + 1/r /r +1/r2 2/2  So that 4 = (2 /2r + 1/r  /r +1/r2 2/2 ) (2/2r + 1/r /r +1/r2 2/2 ) = 0


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95

Let us seek a solution for  in the form  =  r2/4 – ( a2ln(r)/2) + f(r) cos(2) Substituting for  we find that f(r) must satisfy the following differential equation: (2 /2r + 1/r  /r -4/r2) (2f/2r + 1/r f/r -4f/r2) = 0 The general solution for f is thus f = A r2 + B r4+ C/r2+ D The stress function thus takes the form  = r2/4 – ( a2ln(r)/2) + (A r2 + B r4+ C/r2+ D) cos(2)


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Consider the normal and shear stresses on a circular curve radius b such that b >>> a. The expression for the shear and normal stresses on the curve r =b are directly related to the Airy’s stress function , viz: r = - /r (1/r/) = (2A + 6B r2 -6C/r4- 2D/r2) sin(2) and rr = 1/r /r +1/r2 2/2  = (1-a2/r2)/2 - (2A+6C/ r4+ 4D/r2) cos(2)  = 2/2 r = (1+a2/r2)/2 (2A+ 12B r2 + 6C/r4) cos(2)


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The constants can now be determined from the boundary conditions on the hole, r = a, and on the curve r = b, viz: r = 0 and rr = 0 at r =a and rr (evaluated at r = b) = σ cos2(θ) = σ (1+ cos(2θ))/2 r (evaluated at r = b) = -σ sin(2θ)/2 Taking b to be large this gives 2A = -/2 B=0 2A + 6C/ a4+ 4D/a2 = 0 2A - 6C/ a4 - 2D/a2 = 0 The solution to these equations is: A = -/4, B = 0, C= - a4/4 and D = a2/2


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Substituting these values gives the stresses to be: r = - /r (1/r/) = - (1 -3a4/ r4 + 2a2/r2) sin(2)/2 and rr = 1/r /r +1/r2 2/2  =  (1 - a2/r2)/2 + 0.5  (1 + 3a4/r4 - 4a2/r2) cos(2)  = 2/2 r =  (1 + a2/r2)/2 - 0.5  (1 +3a4/r4) cos(2) At the hole, r = a, the variation of hoop stress can be expressed as  =  -2 cos(2) The normal (rr) and shear stresses (r) at the hole are zero, viz: rr = r = 0 © Susan Pitt and Rhys Jones

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Examining the stress field we see that the max value of the hoop stress () at the hole at an angles of θ = π/2 or 3π/2, i.e. at points n and m, and has the value  = Kt  = 3 . Thus the stress concentration factor (Kt) for a circular hole under remote uniform tension is 3. On the other hand at θ = 0 and π, i.e. points p and q, we see that  = -  Along the line n-n1 the stress field is given by: r =0


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100

Stress solution for an arbitrary biaxial stress field:  = 1 (2 + a2/r2+3a4/ r4)/2 + 2 (a2/r2-3a4/ r4)/2

Thus the Kujawski solution for a crack of length l that emanates from a hole under an arbitrary biaxial stress field becomes: K = 1.12 x [1 (2 + a2/r2+3a4/ r4)/2 + 2 (a2/r2-3a4/ r4)/2] x sqrt(3.1415* l)

l = length of crack from hole, r = a+l and a is the radius of the hole.

This solution enables you to determine the stress field due to any arbitrary combination of biaxial stresses acting on a plate with a small hole.


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101

10.1

The stress field at a hole

In class we will talk about how the load flow, for this problem, results in re-circulation directly above the hole. (To see this plot the work vector ij uj where ij is the stress tensor and uj are the components of the displacement, i,j = 1,3.) The stress field associated with a hole in a structure can be easily seen using infra-red thermography. Consider the panel geometry shown below which has a large central hole with two smaller holes on either side.


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102

Figure: Picture of the specimen and a close up of the three holes in the specimen Removing those parts of your structure where there is stress recirculation will result in a reduction in the stresses and structures which are both stronger and more durable.


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103

Figure: The bulk stress field (∆(1 + 2 + 3)) and stress concentrations associated with the three holes as measured along the line shown in the picture

Note the dead zone above the central hole. This is where recirculation occurs. Removing those parts of your structure where there is stress recirculation will result in a reduction in the stresses and structures which are both stronger and more durable.

This specimen represents a dome nut hole with its satellite attachment holes in a P3C (Orion) wing.


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104

10.2 The stress concentration factor at a fastener hole in a civil aircraft fuselage The stress concentration at a fastener hole is not 3 and is significantly reduced by the presence of the fastener. As such when calculating the stress intensity factors at a fastener hole it is important to use the actual stress field and NOT the stress field associated with an open hole. To illustrate this consider the case of a fuselage lap joint in a Boeing 737 aircraft. A picture of the local detail is shown below

Figure Photograph showing the rivet numbers pasted over the first row of rivets in a Boeing 737 lap joint specimen © Susan Pitt and Rhys Jones

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105

A close up of the bulk stress field (∆(1 + 2 + 3)) in the critical first row of fasteners is shown below. The remote stress was approximately 100 MPa.

Figure Close up of the (∆(1 + 2 + 3)) field, in MPa, in the critical rows of fasteners.

Note the stress concentrations at each of the fasteners Note the reduction in the Kt (< 3) due to the fasteners. Also note the dead zone behind the fastener. This is where the load is recirculating. © Susan Pitt and Rhys Jones

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106

CLASS EXERCISE

Use the solution above, for the case of a hole in a plate subjected to a uniform remote stress, to determine the stress distribution along line n-n1 for the case of a uniform remote stress 1 acting in the x direction and a uniform remote stress 2 acting in the y direction.


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107

11 Weight Function Methods If the stresses on the crack face are not constant we can use the weight function approach to determine the stress intensity factors. For two dimensional cracks the value of K can be written in the following form:

KI =

 mI(x, l) n(x) dx,

KII =



mII(x, l) (x) dx

The integration is over the length of the crack.

Here mI and mII are termed the Mode I and Mode II weight functions.


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108

11.1 Application to cracks at notches First perform a finite element analysis of the uncracked problem to determine the stresses in the uncracked body that are acting normal, n, and tangential, , to the location where crack would lie, see Figure 15.

Stress field = (n, ) at the notch

Notch x=0

 is the local radius of curvature



Crack

l

x=l

Figure 15. Schematic diagram showing notch, crack and the stress field at the notch.


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109

Once these stresses have been computed the values of KI and KII can then be computed using the following formulae, viz: l

KI = 0 mI(x, l) n(x) dx

(14)

KII = 0 mII(x, l)  (x) dx

(15)

l

where mI(x, l) and mII(x, l) are the Mode I and II weight functions and n(x) and are  (x) are the normal and shear stresses acting along the line where the crack face would be in the uncracked problem, see Figure 15.


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110

For this particular problem the 2D weight functions mI(x, l) and mII(x, l) can be expressed as: mI(x, l) = 2(l/) FI /(l2-x2) 1/2

(16)

mII(x, l) = 2(l/) FII /(l2-x2) 1/2

(17)

The (Mode I) values of FI and the associated (Mode II) values for FII can be expressed in the form: FI, FII = 1/2 + (1.12 - 1/2) exp(- l/)

(18)

where for i) Mode I;  = 0.8 (1.0 – 0.3 l/ + 0.13 ( l/.)2)

(19)

ii) Mode II;  = 0.32 if l/ < 1.3, and  = 0.32 ( 1.- 0.5 (l/ -1.3)/l/ ) if l/ > 1.3


Page

(20)

111

The solution for a centre notched panel under arbitrary loads is contained, as a special case, of this formulation. You make the radius of the hole very much smaller than the crack length. Note: When performing the integration to obtain K the terms FI and FII come outside the integral. Three dimensional weight function techniques are given in Chapter 4 of the text: Jones R., Baker A., Matthews N., Champagne V, Aircraft Sustainment and Repair, ButterworthHeinemann Press, 2018, ISBN 9780081005408. (Book). https://www.elsevier.com/books/aircraft-sustainmentand-repair/jones/978-0-08-100540-8 © Susan Pitt and Rhys Jones

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Exact Solution: For a centre crack (in an infinite sheet) if we move coordinate system such that x =0 is the location of the left hand tip and x = 2a is the location of the right hand tip x=0 K(x = 0)

x = 2a K(x = 2a)

Then

K ( x  2a ) 

1 ( a )

1/ 2

 ( x) 

2a

x   2a  x  dx

 0

The second expression i.e. K(x =0) is obtained from the first by changing the direction of integration from x to –x.

K ( x  0) 

1

2a

( a ) 

 ( x )  2a  x 

0




x

1/ 2

 dx

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113

For edge cracks we write KI = KII =



a

0



a

0

mI (x, l) n(x) dx, mII(x, l) (x) dx

Here mI(x, l) and mII(x, l) are the Mode I and II weight functions for the associated fracture problem and n(x) and are (x) are the normal and shear stresses acting on the crack face.


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For edge crack problems [1, 2] mI can be approximated by mI(x, l) = (2/(2(a-x))) {(1+M1(1-x/a) 1/2 + M2(1-x/a) + M3(1-x/a) 3/2} where

Here w is the width of the section which contains the crack.


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115

This approximation also works reasonably well up to value of a/r ~ 1. For edge cracks from a hole in a wide panel, where as a first approximation you can assume a/w = 0, the above expressions reduce to: M1 = 0.0719768, M2 = 0.246984 and M3 = 0.529659 References 1. Moftakhar A., Glinka G., 1992, Calculation of Stress Intensity Factors by Efficient Integration of Weight Functions, Engng. Fract. Mech., 43, 5, pp. 749-756. 2. Glinka G., Shen G., 1991, Universal Features of Weight Functions for Cracks in Mode I, Engng. Fract. Mech., 40, 6, pp. 11 © Susan Pitt and Rhys Jones Page 116

FOR Aircraft Sustainment Most of the life of AN OPERATIONAL AIRCRAFT IS CONSUMED GROWING FROM A SMALL CRACK TO ~ 1-2 mm. Weight Function Techniques are useful when the crack length is small typically less than 1mm. Why: Because finite element (f.e.) analysis requires element sizes < 1/15th crack length. This results in a mismatched mesh with a very fine mesh around the crack. Since f.e. is based on variational calculus this can result in the crack tip region being too stiff and can cause PARASITIC STIFFENING. This can yield erroneous results. Extract stresses at the location of the crack from your uncracked fe model. Use WFT in conjunction with these stresses to get K. © Susan Pitt and Rhys Jones

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A more detailed discussion of the use of finite element analysis and weight function techniques for aircraft design and sustainment can be found in Chapter 4 of the text:

Jones R., Baker A., Matthews N., Champagne V, Aircraft Sustainment and Repair, Butterworth-Heinemann Press, 2018, ISBN 9780081005408. (Book). https://www.elsevier.com/books/aircraft-sustainment-andrepair/jones/978-0-08-100540-8


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Class Exercise Use weight function solutions to determine K for a centre cracked panel. We will go through this in class/Prac Classes using an Excel spread sheet on Moodle.


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119

Class Assignment Number 3: As discussed above for edge crack problems mI can be approximated by mI(x, l) = (2/(2(a-x))) {(1+M1(1-x/a) 1/2 + M2(1-x/a) + M3(1-x/a) 3/2} (A1) where

Here w is the width of the section which contains the crack. However, the weight function solutions are dependent on the geometry.


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To illustrate this use the (above) edge crack weight function to obtain the stress intensity factors for 0.5, 1, 5 and 10 mm through thickness edge cracks, i.e. 2D edge cracks, that emanate from a 10 mm diameter hole in a 2 mm thick and 500 mm by 500 mm square panel wing (skin) panel. The principle stresses in the panel are approximately 200 and 0 MPa and the crack(s) lies at 90 degrees to the maximum principle stress (200 MPa). Compare your answers to the solution given in the lecture notes* and also to the solution obtained using the Kujawski approximation and the Bowie solution for this problem. *The

Bowie beta factor solution for the stress intensity factor for a through crack of length “a” at a hole of radius r in a wide plate is: K1 = β(a,r) σ(a) β(a,r) = 0.6762 + [0.8733/(0.3245 +(a/r))]

End of Class Assignment Number 3 © Susan Pitt and Rhys Jones

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Week 5 Starts Here Section 11.2 will NOT BE PRESENTED and is NOT EXAMINABLE 11.2

Derivation of weight functions

If for a given geometry, boundary conditions and load we know both the stress intensity factor Kr(a) and the crack displacement field ur(x,a). Here the subscript r denotes that these solutions are reference solutions.

Then the associated weight function mI(x, a) for any arbitrary load can be determined using the formulae: mI(x, a) = (E’ ∂ur(x,a)/ ∂a)/ Kr Here E’ = E for plane stress E’ = E/(1-ν2) for plane strain


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122

Schematic diagram showing the definition of u(x, a)

END OF SECTION THAT WILL NOT BE

PRESENTED

AND

IS

NOT

EXAMINABLE


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123

Practical Class Problems a) Use weight function solutions to determine K for: 1, 3, 6, 10 and 20 mm cracks emanating from a 6 mm diameter fastener hole in a 300 mm wide plate under a remote uniform stress of 150 MPa. b) Repeat the above for the case when the local stress field at the hole, normal to the crack, n can be expressed as: n = (1+0.8 (R/(R+x))2 + 1.2 (R/(R+x))4) Here R = the radius of the hole = 3 mm and x is the distance from the edge of the hole.


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Class Exercise A wing panel near where the wing meets the fuselage contains a 20 mm by 50 mm region that is badly corroded. Removal of the corrosion leaves a 40 mm by 60 mm elliptical hole in the wing, see the Figure below. The panel is in a state of near plane shear with the peak shear stress seen at limit load of 80 MPa. The process of removing the corrosion can be assumed to leave a small 1mm initial crack at the most highly stressed point around the hole, i.e. at point A. If ΔKth for this material is 6 MPa √m will the flaw grow during operational service ?

20 mm 30 mm

A

If following corrosion removal an equal thickness doubler is fastened over the hole will the flaw grow.


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Answer: The principle stresses in the shear panel are σ1 = 80 MPa and σ2 = -80 MPa. Assume a worst case scenario that the cut out is aligned such that the major axis is perpendicular to the maximum principle stress and that the crack lies at point A. Then the peak stress for this problem occurs at point A where the hoop stress is given by

Hoop stress = Kt *80 = (1+2* 30/20) *80 – (-80) = 400 MPa Since the associated value of the minimum stress is not mentioned we need to assume a minimum stress of 0 MPa to be conservative. This yields:

ΔK = 1.12*400 √ (π 0.001) = 25.1 MPa √m > ΔKth (= 6 MPa √m) So yes the crack will grow.


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Question: Since 80 MPa is a realistic, in reality it is a bit on the mid to low range side, wing skin stress for the region where the wing meets the fuselage and other regions where there is significant load transfer, what are the implications of this for managing corrosion in fleet aircraft?


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Part 2: Assume that the mechanical doubler reduces the stress by a factor of 2 since it has an equal thickness. In that case

ΔK = 200 √ (π 0.001) = 12.6 MPa √m >ΔKth So yes the crack will still grow.

Question: Is the assumption that the repair will reduce the stress by a factor of two valid?


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12 Crack Tip Plastic Zone Size Plastic deformation occurs at the crack tip as a result of the high stresses that are generated by a sharp stress concentration. Irwin found that the extent of plastic deformation could be approximated by: ry =1/2(K/ σy) 2 plane stress =1/6 (K/ σy) 2 plane strain

The Irwin plastic zone approximation


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129

In the Irwin concept the effective stress intensity factor is obtained by replacing the crack length a by the value a+ ry. For example for the simple embedded crack in an infinite panel under remote uniform stress the linear elastic stress intensity factor K is given by K σ[ a] so that we can write

Keff  σ[ (a+ ry)]. As mentioned previously the stress intensity factor is a measure of the stress and strain fields at the crack tip. The stress intensity factor is only meaningful if the plastic zone at the crack tip is small and is contained.

Question: What does JSSG2006 say about the allowable extent of plasticity in an operational aircraft? Answer-- GO AND FIND OUT! © Susan Pitt and Rhys Jones

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13 ASTM E399 Fracture Toughness Tests There are a number of specimen geometries that are used to determine fracture toughness, viz:

M(T)

CT

SENT

SMT

A few common specimen geometries, from Forth 2008 M(T)

=

Middle crack tension, also termed centre cracked panel

CT

=

Compact tension

SENT

=

Single edge notch tension

SM(T) =

Short middle tension panel


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131

13.1 ASTM E399 Compact Tension Fracture Toughness Test The ASTM E399 standard geometry for determining the plane-strain fracture toughness KIc specimens is as shown below

Applied Load

Pin diameter is 0.25 W

Sharpened crack produced by fatigue cycling specimen 1.2 W

Notch

0.55 W

Applied Load

B

W 1.25 W

The ASTM E399 plane strain fracture toughness compact tension test specimen geometry


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132

For a valid compact tension plane strain fracture toughness KIc test the crack size (a), thickness (B), and remaining ligament size (W-a) should all be greater 2.5 (KIc/ys)2, viz: a, B, (W-a) > 2.5 (KIc/ys)2 3

Load-Displacement curves for plane strain fracture toughness test, from ASTM E399.


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133

The effective crack length was then calculated as aeff = a+ry This new crack length aeff is then used to calculate the effective or apparent fracture toughness Kapp using the formulae Kapp = (PQ / BW½) ƒ(aeff /W) where: (aeff /W) =

2  a

eff



/ W  0.886  4.64aeff / W  13.32a 2 / W 2  14.72aeff / W 3  5.6a 4 / W 4 2

1  a

/W 



3/ 2

eff

Here: PQ B W aeff

= = = =

load as defined in E-399 specimen thickness. specimen width effective crack length.


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134

The other widely used geometry is the Single Edge Notch Bending (SENB)

From http://www.ndted.org/EducationResources/CommunityCollege/Materi als/Mechanical/FractureToughness.htm


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135

13.2 The effect of orientation The fracture toughness and the fatigue behaviour of a material generally varies with the grain direction. In the ASTM tests it is usual to specify orientations by an ordered pair of symbols. In this notation the first letter designates the grain direction normal to the crack plane. The second letter designates the grain direction parallel to the fracture plane. For flat sections of various products, e.g., plate, extrusions, forgings, etc., in which the three grain directions are designated (L) longitudinal, (T) transverse, and (S) short transverse, the six principal fracture path directions are: L-T, L-S, T-L, T-S, S-L and S-T.


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14 The Energy Release Rate: G 14.1 Definition of the Energy Release Rate Definition: G is the energy required to create a unit area of new crack The concept of an energy release rate is only valid in linear elastic fracture mechanics. For elastic-plastic fracture G is identically zero. In such cases the term G needs to be replaced by the quantity T* which is briefly discussed in Section 15.


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14.2 Compliance techniques for measuring the Energy Release Rate: Slightly more general approach Consider the centre cracked panel (MT), with a tip to tip crack length of 2a, shown below. The panel is subjected to a remote load P and the load point moves a distance δ.

P, 

2W

2a

P, 


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138

The Load displacement diagrams, shown below, present the load displacement relationships that arise for two different crack length, viz: a and a + ∆a. The diagrams also present the changes that occur in the elastic strain energy as a crack grows under the two defined conditions. G = P2/2B C/a where C = ΔL/P is the compliance ΔL is the change in the

movement of the load point, P is the load B is the specimen thickness, and a is the crack length.


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Example Consider three fracture toughness samples, of an aluminium alloy that have identical external dimensions and a thickness of 100 mm. During these three tests the following information was obtained: Test 1 2 3

Sample crack length (mm) 20 19.5 20.5

Applied load (kN)

Sample elongation (mm)

180 120 120

Sample fractured 0.365 0.370

The Young’s modulus and Poisson’s ratio values for the aluminium are 70,000 MN/m2 (MPa) and 0.3 respectively.


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Answer: The Relationship between G, Compliance, and Elastic Strain Energy G = P2/(2 B) C/a Here Compliance C = ΔL/P ΔL is the change in the movement of the load point, P is the load B is the specimen thickness, and a is the crack length. Now

K2  G (plane stress) E





K 2 1 2  G (plane strain) E

so that in plane strain K = (E/(1-2) P2/(2 B) C/a)1/2


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In above P =120 kN Thus ΔC = 0.005 mm/120 kN and Δa = 1 mm. Hence C/a = 4.17 10 -8/N Thus G = (180 103)2/(200) 4.17 10 -8 and thus K = (70000 (180 103)2/(200) 4.17 10 -8 )1/2

or

K ~ 23 MPa m


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14.3 An Alternative approach

Section 14.3 will NOT be presented in class and is NOT EXAMINABLE

Load-displacement (P - u) records up to the point of crack growth for different crack lengths, from E. Gdoutos, Fracture Mechanics: and Introduction, Springer, ISBN 14020-2863-6, 2005.

In such cases we can use the formulae: Gc = (Piuj-Pjui)/[2B(aj-ai)] An example using this approach is given in the Practical Classes and is repeated below. © Susan Pitt and Rhys Jones

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Prac Class Question: (From E. Gdoutos, Fracture Mechanics: and Introduction, Springer, ISBN 1-4020-2863-6, 2005.)

The following data were obtained from a series of tests conducted on pre-cracked aluminium alloy specimens with an E = 70, 000 MPa, υ = 0.3 and a thickness of 1 mm. Crack length Critical load Critical displacement P(kN) u(mm) a(mm) 30.0 40.0 50.5 61.6 71.7 79.0

4.00 3.50 3.12 2.80 2.62 2.56

0.40 0.50 0.63 0.78 0.94 1.09

where P and u are the critical load and displacement at crack growth. The load-displacement record for all crack lengths is linearly elastic up to the critical point. Determine the critical value of the strain energy release rate Gc from the load-displacement records. From the notes we have Gc = (Piuj-Pjui)/[2B(aj-ai)] Work through the calculations to obtain Gc from this test data and then determine the associated values of K.


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Answer: Area OA1A2 OA2A3 OA3A4 OA4A5 OA5A6 Gc 30.0 30.7 30.2 29.1 29.8 (kJ/m) Mean values is 29.96 kJ/m. This gives K = 45.8 MPa √m.

End of Section that is NOT EXAMINABLE


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15 The J Integral Section 15 will NOT be presented in class and is NOT EXAMINABLE The J-Integral: Consider a body containing a crack as shown in Figure 14.1.

Figure 14.1 Evaluation of the J-Integral In 1968, Rice [2] proposed the evaluation of a line integral taken anti clockwise along any contour i enclosing the crack tip.   ui  J   Wdx2  Ti ds    x i  


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  ui  J   Wdx2  Ti ds   xi  

where W is the strain energy density per unit volume. 

1

W    ij d ij   ij  ij for linear elastic materials  2

Here Ti   ij n j is the surface traction on the contour element d in the direction of the outward normal ni and ui, i = 1- 3, are the components of the displacement field. Eshelby [1] was the first to define a similar integral in the 1950’s. The significance of this integral lies in the fact that it can be shown that when evaluated for a given set of load conditions and crack geometries, the value of J is independent of the integration path  and therefore may represent a useful material parameter for monotonic loading only. © Susan Pitt and Rhys Jones

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Note: The analysis contained in Rice’s derivation does not extend to cyclic plasticity, highly nonproportional paths or unloading paths. To illustrate this path independence consider the

T * integral developed by Atluri et al [3]  ui  * dS T  lim  Wn1  Ti x1    0 e 

Here  is a closed path of vanishingly small distance  from a blunted (non-singular) crack tip.


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Next consider the integral  ui   dS J   Wn1  Ti  xi  

where  is another path surrounding the tip. Using Green’s theorem it follow that

T*  J  

V V

W   u  dV '1

ij i ,1 , j

where the integral is over the volume V  V . The term W,1 ( the partial derivative wrt x, can now be written as: 

W'1  W'ij  ij ,1   ij  ij ,1 Noting that in the absence of body forces, i.e. when  ij , j  0 where the comma subscript means differentiation,

 ij ij ,1   ij ui ,1 , j © Susan Pitt and Rhys Jones

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it follows that T *  J . Hence in two dimensional Mode I linear-elastic isothermal fracture, in the absence of body forces and alternate load paths over/around the crack, the J integral is path independent. Substituting the asymptotic expressions for the elastic displacement and stress fields into the * expression for T and performing the integration around a vanishingly small circular path we find that K2 J T   G (plane stress) E K 2 1   2    G (plane strain) E *

Hence for 2-D linear and non-linear elastic problems J (=T*) is equal to the energy release rate G, i.e. the change in potential energy  per unit of crack area. That is:J © Susan Pitt and Rhys Jones

 A Page

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It should be noted however, that in threedimensional linear elasticity J is no longer path independent and is not related to the energy release rate and T* should be calculated on a path normal to the crack front [1]. For linear and non-linear iso-thermal elasticity J is equal to the rate of change of potential energy per unit of crack area. For non iso-thermal elasticity J is path dependent and is NOT equal to G. Similarly under fatigue loading J is path dependent [1]. Reference 1. Wong AK. and Jones R., A numerical study of two integral type elasto-plastic fracture parameters under cyclic loading, Engng. Fracture Mech., 26, pp .741-752, (1987).


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15.1 Limitations

As a result J integral calculations do not return G (or K) for aircraft structures with cracks in rib stiffened components or for cracks under mechanical or composite repairs. This is because in each case there are secondary load paths around/over the crack. As a result the f.e. section of this course will explain how to calculate K for these situations. References 1. Eshelby J. D, “The continuum theory of lattice defects”, in Solid State Physics, Vol. III, pp. 79-144, Academic Press, New York, 1956. 2. Rice J. R, “A path independent integral and the approximate analysis of strain concentration by notches and cracks”, Journal of Applied Mechanics, Vol. 35, pp. 379-386, 1968. 3. Satya N. Atluri, “Energetic approaches and path independent integrals in Fracture mechanics”, Computational Methods in the Mechanics of Fracture, Edited by Satya N. Atluri, Elsevier Science Publishers B. V., 1986, pp 121-165.

END OF SECTION EXAMINABLE © Susan Pitt and Rhys Jones

THAT

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Week 6 Starts Here 16 Failure by Fracture Of Primary Structure In reality the flight critical components often tend to be thick and highly loaded, i.e. spars, carry through boxes, wing pivot fittings, etc. In these cases the critical crack size, i.e. the size at failure, is generally very small, a few mm, and the critical crack is generally 3D in shape and not a through crack. Thus it is essential to have a knowledge of the stress intensity factors associated with a complex 3D crack.

The Macchi and the F111 failures


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16.1 The Newman and Raju Formulae for 3D Cracks

To this end let us consider an embedded elliptical flaw 

Z

Y

h

a

X

2c

h

2w



t

Only ½ of crack is shown, i.e. an embedded crack. K varies around crack front, viz: K() = (a/Q) (sin()2 +(a/c)2 cos()2) ¼


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K is also commonly written as K = (a) (sin()2 +(a/c)2 cos()2) ¼/  ≈ (a) (sin()2 +(a/c)2 cos()2) 1/4/(3/8 + (a/c)2/8) where Q is the flaw shape factor Here:  = 0 is the visible surface in picture, &  = /2 is the deepest point


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For a semi elliptical surface flaw we can thus approximate K as K() = fs fd fw (a) (sin()2 +(a/c)2 cos()2) ¼/(3/8 + (a/c)2/8) where fs , fd , fw are the free surface and back surface and finite width correction factors respectively.

   1  0.1  0.35c / a a / t  1  sin  

f s  1  0.1  0.35a / t  1  sin   for a / c  1 2

fs

2

2

2

for

a/c 1

It is common to use fs = 1.12 at the free surface.   c a   f w  sec    2w t  .

fd is a complex expression and as a first approximation we can use fd ~ [sec (a/2t)] which will overestimate the back face correction factor. © Susan Pitt and Rhys Jones

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For a quarter elliptical flaw at a fastener hole we can approximate we can thus approximate K as K() = fs fw Kt (a) (sin()2 +(a/c)2 cos()2) ¼/(3/8 + (a/c)2/8)

For small cracks only!!!!! Note: There are two fs terms since there are two free surfaces. where Kt  is the local stress concentration, this thus allows for biaxial loading, and we often set fs = 1.12. This method for obtain approximate values for K is termed the method of compounding stress intensity factor solutions.


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Example: The accident report on an aluminium pressure vessel stated that at an internal pressure of 19 MN/m2 the vessel failed by brittle fracture. A post failure investigation of the failure found that the vessel had a longitudinal surface crack 8mm long and 3.2 mm deep, i.e. c = 4 mm and a= 3.2 mm. Subsequent material tests on a sample of the aluminium showed that it had a KIC value of 65MNm-3/2. If the vessel diameter was 1m and its wall thickness was 10mm, determine whether the data reported are consistent with the observed failure. Answer Hoop stress = PR/t = 19 500/10 = 950 MPa K max ≈ 1.12 (a) /(3/8 + (a/c)2/8) = 1.12 950 (3.1415 * 0.0032) / /8( 3+ 0.64) = 73 MPa m Thus is consistent with observation


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16.2 Practical 3D Fracture Mechanics: Real Cracks Many real life problems involve fatigue cracks with complex three dimensional shapes that initiate on the surface or at a corner or a notch. Appendix X3 in the ASTM fatigue test standard E647-13a, USAF approach to assessing the “Risk of Failure” [1], [2] and [3] explain that the majority of the life of an aircraft component is consumed in growing from a small initial sub mm crack, which as noted in Chapter 3 of [3] typically has dimensions of the order of 0.01 mm, to an observable length. These small initial cracks often first grow as a complex 3D shape and then transition into a through-the-thickness flaws with an oblique (part) elliptical crack front. As a result stress-intensity factor (SIF) solutions are required in order to assess fracture strength and residual fatigue life, i.e. for a damage tolerance analysis.


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Whilst finite element analysis, with the nodes moved to the quarter points, is the most common way to analyse the growth of long cracks, see Chapter 4 in [3] and [4, 5], using finite element analysis to obtain the stress intensity factors associated with small sub mm three dimensional cracks is: a) time consuming and b) due to the size of the cracks, requires extensive mesh refinement which results in a fine mesh with a large number of degrees of freedom. Since the problem has to be re-meshed as the crack shape grows it is very man power intensive.


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Furthermore, new crack shapes can’t be determined until after an incremental fatigue crack growth analysis has been performed. For the large complex problems that are generally associated with aircraft sustainment even without this level of mesh refinement the uncracked finite element problem often involves many hundreds of thousands of degrees of freedom. Since the crack growth analysis will often involve several hundred steps and since the next shape is unknown prior to the analysis this means that several hundred different shapes would have to be modelled and analysed. This (in-turn) means that traditional finite element based approaches for assessing the growth of such small sub mm cracks are to a large extent impractical.


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A practical means for overcoming these shortcomings is presented in Chapter 4 of [3]. The advantage of this approach is that it negates the need to explicitly model cracks. A crack of any size and shape can be analysed using the original (un-cracked) finite element model. As cracks are not modelled explicitly, a coarser mesh can be used to minimize the number of degrees of freedom, thereby reducing the analysis time. Solutions for the stress-intensity factors can then be obtained for a variety of cracks using the original finite element analysis quickly and easily. For example the lifing of a structure with a finite element mesh containing approximately a million nodes will take under ten minutes elapse time on a modern PC.


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Chapter 3 [3] focuses on tools for determining the growth of small sub mm three dimensional cracks that first grow as a complex 3D shape and then transition into a throughthe-thickness flaws with an oblique (part) elliptical crack front sustainment problems associated with thin skins often involve the growth of small through the thickness cracks. Whilst the computational and meshing problems associated with such 2D cracks are much less [4], problems still arise when attempting to use finite element analysis to determine the stress intensity factor (K) associated with small sub mm 2D cracks. In such cases the use of weight function solutions is recommended [3].

Numerous examples that illustrate how to determine the stress intensity factors associated with complex 3D cracks in real aircraft structures under arbitrary loads and how to use these solutions to life full scale aircraft are given in Chapters 4, 5 and 8 in [3] and also in [5].


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References 1. Berens AP., Hovey PW., Skinn DA., Risk analysis for aging aircraft fleets - Volume 1: Analysis, WL-TR-913066, Flight Dynamics Directorate, Wright Laboratory, Air Force Systems Command, Wright-Patterson Air Force Base, October 1991. 2. Schijve J., Fatigue of structures and materials, Kluwer Academic Publishers, ISBN: 0-7923-7013-9, 2001. 3. Jones R., Baker A., Matthews N., Champagne V, Aircraft Sustainment and Repair, Butterworth-Heinemann Press, 2018, ISBN 9780081005408. 4. Williams JF., Jones R. and Goldsmith N., An introduction to fracture mechanics theory and case studies, Transactions of the Institution of Engineers, Australia Mechanical Engineering, 14(4), 185-223 (1989). 5. Jones R., Fatigue Crack Growth and Damage Tolerance, Fatigue and Fracture of Engineering Materials and Structures, (2014), Volume 37, Issue 5, pages 463-483.

Mid Semester Test At This Point Covers All Of The Material Presented Above, Except Section 16.2


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17 Life of Type, Remaining Life, Through Life Support & Inspection Intervals We have now dealt with the first two requirements outlined at the beginning of this course, viz: 1.

Estimating the effect of a crack on the residual strength of the structure

2.

Estimating whether a crack in a structure will grow during service.

Note: It has long been known [1, 2] and ASTM E64713a that for operational aircraft the majority of the fatigue life is consumed in growing from a small initial material discontinuity to a length of the order of 2 mm. 1. Schijve J., Differences between the growth of small and large fatigue cracks. The relation to threshold K values, Proceedings International Symposium on Fatigue thresholds, Stockholm, Sweden, 1-3 June, 1981. Editors J. Backlund, A. F. Blom and C. J. Beevers, p 881-891, 1982. 36. 2. Iyyer NS. and Dowling NE., Fatigue growth and closure of short cracks, AFWAL-TR-89-3008 (1989). © Susan Pitt and Rhys Jones

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CLASS EXERCISE

The FAA, USAF and US Navy approaches to the certification of both metallic and composite airframes requires a building block approach.

What is meant by this?

Hint, see: R. Wanhill, Fatigue Requirements for Aircraft Structures, in Jones R., Baker A., Matthews N., Champagne V, Aircraft Sustainment and Repair, Butterworth-Heinemann Press, 2018, ISBN 9780081005408.


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Thus in aerospace applications (sustainment) the equations used need to be able to represent crack growth from small naturally occurring defects.

That aside let us first focus on estimating the fatigue life and the associated inspection intervals. In the late 1950’s Paul Paris revealed that for long cracks increment in the crack length per cycle da/dN, which is usually (incorrectly) referred to as the crack growth rate, can be related to the increment in the stress intensity factor KI, = Kmax – Kmin where Kmax is the maximum value of K in the cycle and Kmin is the maximum value of K in the cycle, see the Figure below.


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KC

Increasing R ratio

Crack growth da/dN I

II

III

Log K For a given K increasing the R ratio (R = Kmin/Kmax)

often leads, for long cracks, to an increase in da/dN. The ASTM standards for performing tests to determine the da/dN versus K curves for long cracks are given in ASTM E647-13a. An extension to this procedure, termed the ACR method, to represent the growth of physically short cracks is also given in E647 and is briefly discussed in Section 28. NOTE: Section 28 and the ACR method are NOT EXAMINABLE.


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A schematic diagram (a), and (b) a typical crack growth v K relationship for various R ratio’s, from Mil HDBK 5. Note 1 ksi in ½ 1.089 MPa m1/2, whilst 1 in = 25.4 mm

In the next section we will build on this to address questions related to remaining life, inspection intervals, etc


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18 Modeling Crack Growth In previous Sections we have mentioned that in accordance with the damage tolerance design philosophy we require: 

An estimate of the residual strength of a component at any point during its operational life, see Figure 1;



An estimate of the structural life to grow from an initial (damage) size to its critical (crack) size, see Figure 1.

A detailed discussion on fatigue crack growth is given in Chapter 5 of the text: Jones R., Baker A., Matthews N., Champagne V, Aircraft Sustainment and Repair, Butterworth-Heinemann Press, 2018, ISBN 9780081005408. (Book). https://www.elsevier.com/books/aircraft-sustainment-andrepair/jones/978-0-08-100540-8


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Fig 1. Relationship between residual strength, crack and time-in-service as per the USAF Dam Tol Design Hdbk © Susan Pitt and Rhys Jones Page 172

We have dealt with the first requirement, which involves estimating the residual strength, In this section we will focus on the second requirement, i.e. estimating the fatigue life and the associated inspection intervals. Crack growth can be related to the increment in the stress intensity factor KI, see Figures 2 and 3. In general there are three distinct regions: Region III being associated with rapid crack growth.

Region II we will term the “mid growth” range. (Paris Region)

Regions I is a region where crack growth is slow and where several authors have introduced the concept of a fatigue threshold stress intensity factor range KIth beneath which cracks will not grow.


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KC

Increasing R ratio

Crack growth da/dN I

II

III

Log K Figure 2 Schematic of da/dN versus ΔK curves

? What happens for small naturally occuring defects in operational aircraft ? More On This Later © Susan Pitt and Rhys Jones

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Figure 3 A schematic diagram (a), and (b) a typical crack growth v K relationship for various R ratio’s, from Mil Hdbk 5. Note 1 ksi √in 1.089 MPa √m, whilst 1 in = 25.4 mm


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What is metal fatigue? 

In materials science, fatigue is progressive and localized structural damage that occurs when a material is subjected to cyclic loading#. Spectrum loading

Over a period of time

Over a period of time

BROKEN

Fatigue crack # Source: Wikipedia


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Week 7 Starts Here 19 Paris Crack Growth Equation: 19.1 The similitude hypothesis The Paris crack growth equation and all “long crack” growth equations are based on the similitude hypothesis, also referred to in [1] as the similarity principle. This hypothesis plays a central role in both design and aircraft sustainment. For metals the similitude hypothesis can be expressed [2] as: “Two different cracks growing in identical materials with the same thickness and the same crack driving force (CDF) and the same Kmax, will grow at the same rate.” For metals the crack driving force, which is often written as is generally taken to be the increment per cycle in the stress intensity factor: Kmax –Kmin,


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where Kmax is the maximum value of K in the cycle and Kmin is the minimum value of K in the cycle. There is a corollary to this hypothesis, viz:

“In two different tests on identical materials with the same thickness for cracks growing with the same crack driving force (CDF) the test with the highest value of Kmax will have a higher value of da/dN.” This will be discussed in more detail in class. A detailed discussion on fatigue crack growth is given in Chapter 5 of the text: Jones R., Baker A., Matthews N., Champagne V, Aircraft Sustainment and Repair, Butterworth-Heinemann Press, 2018, ISBN 9780081005408. (Book). https://www.elsevier.com/books/aircraft-sustainment-andrepair/jones/978-0-08-100540-8 References 1. Schijve J., Fatigue of structures and materials, Springer, 2008, ISBN-13:978-1-4020-6807-2. © Susan Pitt and Rhys Jones

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2. Jones R., Molent L., and Pitt S., Crack growth from physically small flaws, International Journal of Fatigue, Volume 29, (2007), pp 1658-1667. 19.2

The Paris crack growth equation

In the late 1950’s and early 1960’s it was observed that, for long cracks, in Region II the relationship between crack growth and K could be expressed in the form: da / dN = C (K)n

This is called the Paris crack growth law and is one of the earliest, circa 1958, and most widely used crack growth laws. It is generally used over a wide range of Mode-I crack growth rates, typically between 10-9 and 10-5 m/cycle.

The material parameters C and n are experimentally determined.


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19.3 The importance of initial flaws and the local stress states

Let us first use this simple model to examine the effects on crack growth.

To this end consider a (2D) edge crack in a semi-infinte plate under remote uniform stress  =  sin(t) .

In this case K1 = 1.12 (a)


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If the initial crack size is ai and the size at failure is af the number of cycles to failure Nf can be written as af

Nf = 1/C ai

af

= 1/C ai

(K)-nda 1/(1.12 (a))nda

= 2/{C (n-2)(1.12 )n }(1/ ai(n-2)/2 - 1/ af(n-2)/2) (For values of n NOT equal to 2) The final crack length af can be calculated by equating K to the apparent fracture toughness Kc for the material, viz:

af = 1/ (Kc /[1.12 ]) 2


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Nf =2/{C (n-2)(1.12 ()n } (1/ ai(n-2)/2 - 1/ af(n-2)/2)

For most metals for long cracks the value of n lies in the range 2.5 < n < 4. In many instances we have ai 0.05 inch), the assumed initial flaw is a 0.05 inch (1.27 mm) radius corner flaw.


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At locations other than holes, the assumed initial flaw is a semi-circular surface flaw with a length of 0.25 inch and depth of 0.125 inch, or, for material thickness less than 0.125 inch, a through thickness flaw of 0.25 inch length, see Figure 3. Initial Flaw Assumptions for Metallic Structure, from JSSG-2006 Category Slow crack growth and Fail Safe primary element

Critical Initial Flaw Assumption* Detail Hole, Cutouts, For t  1.27 mm, a 1.27 mm etc. through thickness flaw. For t > 1.27 mm, a 1.27 mm radial corner flaw Other For t  3.175 mm, a 6.35 mm through thickness flaw For t > 3.175 mm, a 3.175 mm deep x 6.35 long surface flaw Welds, TBD embedded defects

 - Flaw is orientated in the most critical direction


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Figure 3. Summary of Initial-Flaw Assumption for Intact Structure, from the USF Damage Tolerance Design Handbook.


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21 Crack Growth Models The Paris law discussed previously was used as a starting point for discussing the effect of load and initial flaws on operational life. However, when modelling the growth of long cracks it suffers from a number of shortcomings, viz:  It does not reflect the effect of R ratio on the growth rate.  It does not account for rapid increase in growth rate when Kmax approaches KC.  It does not reflect a fatigue threshold  It does not allow for overload or load interaction effects.

A range of crack growth models have been developed in an attempt to allow for some, or all, of these effects.


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The most commonly used crack growth laws are: the Paris, Wheeler, Forman, modified Forman, Wheeler, Willenborg, Crack Closure, Hartman-Schijve-McEvily models that relate: The crack growth rate da/dN to the increment in the stress intensity factor

K = Kmax - Kmin. or an effective crack driving force Keff = K - Kth or = Kmax – Kop. or = K(1-p)Kmaxp

The most commonly used (commercially available) crack growth computer codes are: AFGROW, NASGRO and FASTRAN All generally work well for long cracks, i.e. in an initial design scenario.


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Here Kmax is the maximum stress intensity factor experienced during a particular load cycle, Kmin

is the corresponding minimum value of the stress intensity factor

Kth

is the fatigue threshold beneath which a crack will not grow. This is a function of the R ratio, thickness and crack length.

Kop

is the value of the stress intensity factor at which the crack opens during a given load cycle

p

is a constant that accounts for R ratio dependency


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These models can be summarised as follows, viz: 21.1 Paris equation: da / dN = C (K)n 21.2 Forman’s equation: The Paris equation does not account for either the rapid rate of crack growth seen when K approaches the fracture toughness, or for mean stress effects. To overcome this Forman suggested the following crack growth relationship:

da C (K ) n  dN (1  R) K c  K where Kc is the critical stress intensity factor and stress ratio R = min/max = Kmin / Kmax.


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21.3 Modified Forman’s equation:

Forman’s equation can be modified to produce slightly more accurate results, viz: da C (K  ( PR  Q))K n  dN (1  BR ) K c  K

The 5 material parameters in this formulation are C, n, P, Q, and B.


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Kth

K th  P  R  Q

R

Relationship between the stress intensity factor threshold and the R ratio

KIC

K IC effective  1  B  R K IC

R

Relationship between the effective fracture toughness K1C and the R ratio

Forman’s equation works very well for long cracks. © Susan Pitt and Rhys Jones

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21.4 Walker’s equation: Walker's equation is an alternative extension of the Paris equation to include the influence of the stress ratio (R) effects, viz: da/dN = C [(K) 1-M (Kmax)M] n where M is a constant. Note: that the Paris law is contained as a special case of the Walker equation, i.e. setting M = 0 returns the Paris equation. K is the +ve part only K = Kmax - Kmin K = Kmax

if Kmin > 0 if Kmin < 0

Walker’s equation works well for long cracks.


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21.5 The Nasgro equation: The general form of the Nasgro equation, which forms the basis of the NASA crack growth program NASGRO, AFGROW and FASTRAN and which is mandatory for use in the design and assessment of space vehicles, can be expressed in the form da/dN = D ∆K(m-p) (∆K – ∆Kthr) p/(1-K max/A))q where m, α and q are constants. Reference Forman, RG., and Mettu, SR. (1992) Behavior of Surface and Corner Cracks Subjected to Tensile and Bending Loads in Ti-6Al-4V Alloy, Fracture Mechanics 22nd Symposium, Vol. 1, ASTM STP 1131, H.A. Ernst, A. Saxena and D.L. McDowell, eds., American Society for Testing and Materials, Philadelphia, 1992.


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WEEK 8 Starts Here 21.6 The Hartman-Schijve variant of the Nasgro equation: The Hartman-Schijve-McEvily equation is a subset of the Nasgro equation. It is obtained by setting m = p and q = p/2, see Jones (2014). It is generally written in the form da/dN = D [(∆K - ∆Kthr) /(1-K max/A) ]p where D and A are constants and p is approximately 2. The constant A reflects the apparent cyclic fracture toughness value Kcy which can differ from the fracture toughness (Kc) obtained from static testing. The role of the denominator is to match the predicted and measured crack growth rates for large values of Kmax. In general the exponent p is approximately 2, see Jones Fatigue and Fracture of Engineering Materials and Structures, (2014). © Susan Pitt and Rhys Jones

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There are many materials for which the Hartman-Schijve equation works for both long and short cracks, see Jones (2014). Note: For cracks that grow from small naturally occurring material discontinuities the value of ΔKth is typically in the range 0.10 to 0.30 (i.e. it is very small.) The formulation is available in NASGRO, FASTRAN and AFGROW. For more details on the NASGRO equation and its use in both design and sustainment of aircraft see the review paper Jones (2014) listed below: Reference [1] Jones R. (2014) Fatigue crack growth and damage tolerance, Fatigue and Fracture of Engineering Materials and Structures, 37, 463-483.

Tabulated data: In many instances the constant

amplitude data for da/dN versus K relationship can be input via tabulated values. © Susan Pitt and Rhys Jones

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21.7 The Wheeler retardation model Retardation models have been developed to account for the reduction in crack growth rate observed after the application of a high load (overload). Three retardation models are presented, viz: Wheeler and two crack closure retardation models. The Wheeler Model: This model introduces a retardation factor CP to reduce the growth rate, viz:

 da   da   C Pi   C Pi f K i      dN  i retarded  dN  i linear where  Ri C Pi    a 0  R0  a i

  

m

as long as

ai  Ri  a 0  R0

~ (i/overload) 2p

C Pi  1

if ai  Ri  a 0  R0 ie: no retardation


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Here R0 is the size of the crack tip plastic zone due to the overload, and a0 is the crack length at the instance when the overload occurred. Ri is size of the plastic zone when the crack has propagated to a length ai , see Figure 4. The power m, p are determined from experimental data.

Figure 4. Crack retardation due to overload, from USAF Damage Tol Design Handbook.


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22 Crack Closure Based Equations There are several “crack closure” based crack growth equations. We will only discuss one, viz: The closure variant of the Nasgro equation, which is available in both of the industry standard crack growth codes NASGRO, FASTRAN and AFGROW;

Note: There are minimal R ratio effects and hence little crack closure associated with small naturally occurring cracks in operational aircraft, see the ASTM 647 test standard and the three references below for more details Jones R. (2014) Fatigue crack growth and damage tolerance. Fatigue and Fracture of Engineering Materials and Structures, 37, 463-483. Jones R. and Tamboli D., Implications of the lead crack philosophy and the role of short cracks in combat aircraft, Engineering Failure Analysis, 29, 2013, pp.149-166. Chapter 5, Jones R., Baker A., Matthews N., Champagne V, Aircraft Sustainment and Repair, Butterworth-Heinemann Press, 2018, ISBN 9780081005408. (Book). https://www.elsevier.com/books/aircraft-sustainment-andrepair/jones/978-0-08-100540-8 © Susan Pitt and Rhys Jones

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22.1 FASTRAN: The Analytical Crack-Closure Model When calculating the crack closure and the crack opening stresses at any instance of crack growth, we need the elastic-plastic solutions for the stress and the displacements. The crack face displacements, used to determine the contact (closure) stresses during unloading, are a function of crack tip plasticity and the residual deformations left in the wake of an advancing crack. On reloading, the level of the applied load at which the crack again fully opens can be directly related to there contact stresses. We will call the load/stress level at which the crack becomes fully open the "crack-opening stress" σo. This model was based on the Dugdale model, modified account for the plastically deformed material left in the wake of the crack.


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Break up of the stress field needed to compute K for the crack closure model, from the FASTRAN users manual by J. Newman


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Crack face displacements and stress distributions, from the FASTRAN manual by J. Newman. Picture modified by Dr. Chris Wallbrink. The shaded regions in the above Figure indicate that the material is behaving plastically. At any applied stress level, the bar elements are either intact (in the plastic zone), or broken (residual plastic deformation).


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The broken elements can only carry compressive loads and then only if they are in contact. The contact elements yield when the contact (compressive) stress reaches -σ0. The elements that are not in contact do not effect crack face displacement. To account for the different triaxiality states on plastic-zone a constraint factor (α) is used. The role of the constraint factor was to elevate the tensile flow stress. Thus the effective flow stress was taken to be ασ0 . For plane stress conditions  =1, and for plane strain  =3.

The Plastic-Zone Length The plastic-zone size was estimated by using the smallscale yielding solution

  K max    8   0 © Susan Pitt and Rhys Jones

  

2

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The Closure Based Fatigue Crack Growth Rate Equation Most closure based laws now use the Nasgro crack growth equation proposed by Forman et al, viz: da/dN = D ∆Keff(m-p) (∆Keff – ∆Keff, thr) p/(1-K max/A))q where for uniaxial loading we have K max  Smax c 

and

 K eff  ( S max  S 0 )  c M

f

 K eff  ( K max  K 0 P ) Here  is the stress intensity beta factor. Kop is the value of K at which the crack opens.

In Fastran the crack opening stress (S0) is calculated using the analytical closure model described above. For constant amplitude loading we have


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S 0 /S max  A 0  A 1 R  A 2 R

and

2

A 3 R

3

S 0 /S max  A 0  A 1 R

for R > 0 (16) for R < 0 (17)

where So= Smin if So/Smin is less than R, and So= 0.0 if So= Smax is negative. The Aj coefficients are functions of α and S max /  0 and are given by:

A O  (0.825 - 0.34  0.05 2 ) [COS(S max F/2 0 ]1/ A1  (0.415 - 0.07l ) S max F/ 0 A 2 1 - A 0- A 1 - A

3

A 3  2A 0  A 1 - 1 for α = 1 to 3. The boundary correction factor, F, accounts for the influence of finite width on the stresses required to propagate the crack.


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For arbitrary loading the term S0/Smax is replaced by K0/Kmax

NOTE: The precise form of the closure based equations used in FASTRAN/Nasgro and AFGROW are not examinable.


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A number of investigators have reported a fatigue limit KIth beneath which there is no crack growth. In general this limit, the existence of which is debated in the literature, is a function of the R ratio, viz: KIth = KIth0 (1- R) where  is a constant and KIth0 is the threshold corresponding to R = 0.0.


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The crack growth rate is a strong function of the constraint seen at the crack tip. For simple crack growth specimens this translates to a dependency on the thickness of the test specimen. This is shown in Figure 6 where we compare cracking in 0.25 mm and 0.5 in thick 7050 T3517 aluminium alloy specimens. Note: The FASTRAN formulation generally works quite well for long cracks.


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However, there is minimal crack closure associated with small naturally occurring cracks, see Annex X3 of ASTM E647-13a. As such it needs to be modified to work for small cracks.

Recall that, as explained by ASTM E647 Appendix X3, Schijve and also by Iyyer and Dowling, for aerospace applications the majority of the life is spent in growing from a small crack to a size of (say) 5 mm.

In such cases, i.e. for short cracks, Keff is approximately equal to K.


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22.2 The Hartman-Schijve-McEvily equation The Hartman-Schijve-McEvily variant is a special case of the Nasgro equation which has m = p and q = p/2 da/dN = D ((∆Keff - ∆Keff, thr)/(1-K max/A)0.5)p

where

ΔKeff = ΔK- ΔKop

In this formulation we express ΔKop in the form suggested by McEvily and co-workers, viz: ΔKop = Kop – Kmin = (1-e-λa) ΔKopl where ΔKopl is the long crack value of ΔKop and λ is a material dependent constant (a typical value is 10000 m-1).


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Thus for cracks that initiate from small naturally occurring discontinuities ΔKop asymptotes to zero. Hence for cracks that initiate from small naturally occurring discontinuities in operational aircraft ΔKeff asymptotes to ΔK. As such the Hartman-Schijve-McEvily equation overcomes the limitation inherent in the use of FASTRAN to model the growth of cracks from small naturally occurring material discontinuities in operational aircraft, see [1-9]. References 1. Jones R. (2014) Fatigue crack growth and damage tolerance, Fatigue and Fracture of Engineering Materials and Structures, 37, 463-483. 2. Lo M., Jones R., Bowler A., Dorman M., and Edwards D., Crack growth at fastener holes containing intergranular cracking, Fatigue and Fracture of Engineering Materials and Structures, (2017) doi: 10.1111/ffe.12597. 3. Jones R., Huang P. and Peng D., (2016) Crack growth from naturally occurring material discontinuities under constant amplitude and operational loads, International Journal of Fatigue, 91 pp. 434–444. 4. Molent L., Jones R (2016) The influence of cyclic stress intensity threshold on fatigue life scatter, International Journal of Fatigue, 82, pp. 748-756.


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5. Jones R., Peng D., Singh RRK., Pu Huang, Tamboli D., Matthews N., (2015) On the growth of fatigue cracks from corrosion pits and manufacturing defects under variable amplitude loading, JOM, Vol. 67, No. 6, pp. 1385-139 6. Jones R, Molent L., Barter S., Calculating crack growth from small discontinuities in 7050-T7451 under combat aircraft spectra, International Journal of Fatigue, 55 (2013), pp. 178182. 7. Barter S., Tamboli D. and Jones R., On the growth of cracks from small naturally occurring material discontinuities (etch pits) under a Mini-TWIST spectrum, Proceedings 2015 Aircraft Structural Integrity Program (ASIP), At The Hyatt Regency, San Antonio, Texas, USA, 1st-3rd December, 2015. 8. Tamboli DZ., Jones R., Barter S, Decoupling of Fatigue and Corrosion in AA7050-T7451, Proceedings ICAF2017 29th Symposium, Nagoya, Japan, 5th-6th June, 2017. 9. Chapter 5, Jones R., Baker A., Matthews N., Champagne V, Aircraft Sustainment and Repair, Butterworth-Heinemann Press, 2018, ISBN 9780081005408. (Book). https://www.elsevier.com/books/aircraft-sustainment-andrepair/jones/978-0-08-100540-8


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FASTRAN, NASGRO, AFGROW generally work well for long cracks in aluminium alloys. In this formulation the crack opening stress is a function both of the R ratio and also the crack length and both effects become small as the crack length becomes very small. This is important when assessing crack growth in operational aircraft.

The Nasgo equation as used in the NASA crack growth code NASGRO, is mandatory for all space vehicles.


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(a)


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(b) Figure 6 Crack growth curves for (a) 0.25 in thick (6.4 mm), and (b) 0.5 in (12.7 mm) thick 7050-T3571 Aluminium alloy in 50% RH, from Mil Handbook 5.


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23 Hartman-Schijve-McEvily (Nasgro) Representation Of A Number Of Common Aerospace Structural Materials: D6AC C3-5-lt D6ac Ct3-12-lt D6ac Ct3-27-lt D6ac Ct3-25-lt D6ac Ct3-47-lt D6ac Ct3-46-lt D6ac Ct3-29-lt D6ac Ct3-10b-lt 4340 R = 0.1 4340 R = 0.7 10-Ni R = 0.1 10-Ni R = 0.8

1.0E-05

da/dN (m/cycle)

1.0E-06

1.0E-07

1.0E-08 1.95

y = 2.36E-10 x 2

R = 9.88E-01

1.0E-09

1.0E-10 0.1

1

10

100

1000

(-th)/(1-Kmax/A)1/2 (MPa m)

The Hartman-Schijve-McEvily (Nasgro) representation for D6ac, 4340 and 10Ni-8Co-1Mo steels.


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1.0E-05

2024-T351 R = 0.1 [36] 2219-T851 R = 0.1 [15] 2219-T851 R = 0.3 [15]

da/dN (m/cycle)

1.0E-06

1.93

y = 2.64E-09 x 2

R = 0.998

2219-T851 R = 0.8 [15] 2025-T6 R = 0.5 [28] 2025-T6 R -= 0.7 [28]

1.0E-07

2024-T3 R = 0.06 [37] 2024-T3 R = 0.3 [37]

1.0E-08

2024-T3 R = 0.64 [37] 2.13

y = 1.26E-09 x R2 = 0.985

1.0E-09

1.0E-10 0.1

1

10

100

(-th)/(1-Kmax/A)1/2 (MPa √m)

Hartman-Schijve-McEvily (Nasgro) representations for various 2000 series aluminium alloys.


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1.0E-04 7050-T7451 R = 0.7 [40] 7075-T6 R = 0.7 [31]

da/dN (m/cycle)

1.0E-05

7075-T6 R = 0.1 [37] 1.0E-06

7050-T76511 [39]

1.0E-07 1.97

y = 7.05E-10 x 2 R = 0.989

z

1.0E-08

1.0E-09 1.97

y = 1.20E-09 x 2

R = 0.981 1.0E-10 0.1

1 (-th)/(1-Kmax/A)

10 1/2

100

(MPa √m)

Hartman-Schijve-McEvily (Nasgro) representation for several 7000 series aluminium alloys.


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da/dN (mm/cycle)

Although the Hartman-Schijve-McEvily equation and the Frost-Dugdale equation look very different they are often operationally equivalent

1.0E-02

1.0E-03

y = 0.6 x -2.0 e-7

1.0E-04

1.0E-05

1.0E-06

1.0E-07 y = 0.507 x - 5.49E-06

D6AC C3-5-lt D6ac Ct3-12-lt D6ac Ct3-27-lt D6ac Ct3-25-lt D6ac Ct3-47-lt D6ac Ct3-46-lt D6ac Ct3-29-lt D6ac Ct3-10b-lt Hudson Ti-6AL-4V R =0.85 Hudson Ti-6AL-4V R =0.66 Hudson Ti-6AL-4V R =0.43 Hudson Ti-6AL-4V R =0.25 Hudson Ti-6AL-AV R = 0.0 Porter Ti-6AL-4V R=0.03 Porter Ti-6Al-4V R = 0.05 7050-T7451 R = 0.1 7050-T7541 R = 0.8 7050-T7541 R = 0.5 Generalised Frost-Dugdale Grade C wheel steel

1.0E-08 1.E-07

1.E-06

1.E-05 1.E-04

1.E-03

1.E-02 1.E-01 1.E+00 1.E+01

g

((K 1-p Kmax p)/y) a1-g/2 /(1-Kmax/Kc) (metres)

Comparison of crack growth, ala the Generalised Frost-Dugdale equation, in several disparate materials, log scales, see Section 33.8.


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The work on this page will not be discussed in class and IS NOT EXAMINABLE The crack closure variant can be directly linked to the Hartman-Schijve equations since as shown by Hudak S.J and Davidson DL, The dependence of crack closure on fatigue loading variables, Mechanics of crack closure, ASTM STP 982, Edited by J. C. Newman and W Elber, American Society for Testing of Materials, Philadelphia, USA, pp 121-158, 1988. ΔKeff – (ΔKeff )th = ΔK – ΔKth where

ΔKeff = ΔK- ΔKop

and

ΔKop = = Kop- Kmin

and (ΔKeff )th is the associated threshold value. As such the Hartman-Schijve-McEvily crack closure equation can be thought of as equivalent to the original Hartman-Schijve hypothesis.

Note: The above page will not be discussed in class and IS NOT EXAMINABLE © Susan Pitt and Rhys Jones

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24 An Obvious Question: Given that the above equations were derived to model crack growth under constant amplitude loading and that the service load spectra seen by aircraft is complex then when can you use these formulae ? Answer: A very good question and one that raises the topic of load interaction. In general if you have a large crack growing under have constant amplitude loading interspersed with an occasional tensile overload then the overload will produce a residual compressive stress in front of the crack which will reduce crack growth. (This sort of thing is termed load interaction.) Thus there are numerous cases when these equations need to be modified to account for load interaction. © Susan Pitt and Rhys Jones

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However, there are many practical cases when you can use these simple equations as is, viz: Crack growth in pressurised fuselage skins. In this case the dominant load cycle is due to the pressurisation and depressurisation that results on take off and landing which is (of course) a constant amplitude load cycle. Crack growth from small naturally occurring defects in fighter wings and bulkheads. Here the industry standard spectra is termed FALSTAFF and has very little load interaction. In the Australian context the RAAF F/A-18 wing load spectra also has very little load interaction. Crack growth from small naturally occurring defects in civil aircraft (transport) wings and bulkheads. Here the industry standard spectra is termed TWIST and for such small naturally occurring cracks there is very little load interaction. © Susan Pitt and Rhys Jones

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Crack growth in the lift frames of helicopters. In this instance it has been shown that crack growth under the ASTERIX load spectra has little load interaction.

In each of the cases you can use equations developed to represent crack growth under constant amplitude loadings and use linear damage accumulation.


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You can also often use a variant of these equations when the load spectra consists of a number of repeated load blocks. This is the case for many parts the aircraft. (More on this later in the course.) Warning: For combat aircraft the initial flaw size is generally very small, typically of the order of 3-50 microns. In such cases when using FASTRAN crack closure growth law or variants thereof you need to use different da/dN versus ΔKeff relationships than you would use for long cracks and the constraint factor needs to changed as the crack length increases. A more detailed discussion is given in Chapter 5 of the text: Jones R., Baker A., Matthews N., Champagne V, Aircraft Sustainment and Repair, ButterworthHeinemann Press, 2018, ISBN 9780081005408. (Book). https://www.elsevier.com/books/aircraft-sustainmentand-repair/jones/978-0-08-100540-8 © Susan Pitt and Rhys Jones Page 226

This shortcoming is removed if you use equations based on the Nasgro (HartmanSchijve-McEvily variant) crack closure equation. However, you still need to allow for the crack length dependency of ΔKeff th It will later be shown that for cracks in operational aircraft conservative estimates for crack growth can be obtained by using a simple Paris law with an exponent of 2. This means that AFGROW can also be used to estimate the life of operational aircraft. Note: The crack closure based growth equations do not appear to be very accurate for a number of high strength aerospace steels, i.e. 4340 and D6ac, or for several Ti-6Al-4V alloys which show only a small effect of the R ratio on crack growth. © Susan Pitt and Rhys Jones

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25 Which Crack Growth Curve ? The importance of using a crack growth curve that is an amalgam of the short and long crack growth curves was raised in Section 4 and was first highlighted by Lincoln [1] in the USAF report on cracking in F-15 aircraft. A detailed discussion on this is given in Chapter 5 of the text: Jones R., Baker A., Matthews N., Champagne V, Aircraft Sustainment and Repair, ButterworthHeinemann Press, 2018, ISBN 9780081005408. (Book). https://www.elsevier.com/books/aircraft-sustainmentand-repair/jones/978-0-08-100540-8 This topic is also discussed in [2-4]. 1. Lincoln JW., Melliere RA., Economic Life Determination for a Military Aircraft, AIAA Journal of Aircraft, 36,5, 1999. 2. Jones R. and Tamboli D., Implications of the lead crack philosophy and the role of short cracks in combat aircraft, Engineering Failure Analysis, 29, 2013, pp.149166. © Susan Pitt and Rhys Jones

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3. Jones R. (2014) Fatigue crack growth and damage tolerance, Fatigue and Fracture of Engineering Materials and Structures, 37, 463-483. 4. Lo M., Jones R., Bowler A., Dorman M., and Edwards D., Crack growth at fastener holes containing intergranular cracking, Fatigue and Fracture of Engineering Materials and Structures, (2017) doi: 10.1111/ffe.12597.


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In this context the USAF and Boeing evaluation of crack growth in F-15 aircraft, Lincoln [1] stated that: “Figure 1 shows the short-crack correction to the longcrack data for both aluminium and titanium that McDonnell used for this study. The inclusion of the short-crack effect was key to the success of the program. The inclusion of the short-crack effect enabled McDonnell to pool the coupon test EIFSs derived from constant-amplitude tests with the random flight-by-flight loaded test aircraft EIFSs. This is a significant finding in that earlier use of the long-crack threshold for crack growth made it appear that the equivalent initial flaws in the structure were spectrum dependent.”

Figure 1 The short crack modification used in the F-15 study, from Lincoln [1]. © Susan Pitt and Rhys Jones

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As such any sustainment analysis of the fatigue behaviour of cracks in metal airframe structures in operational aircraft generally requires an assessment of short crack behaviour. For more details see Jones [3] and Chapter 5 of the text; Jones R., Baker A., Matthews N., Champagne V, Aircraft Sustainment and Repair, ButterworthHeinemann Press, 2018, ISBN 9780081005408. (Book). https://www.elsevier.com/books/aircraft-sustainmentand-repair/jones/978-0-08-100540-8

However, as mentioned above this behaviour can, as a first approximation, be determined from long crack data via the Nasgro (Hartman-Schijve-McEvily) equation with the variability in the crack length versus cycles history being controlled by the parameter ∆Kthr. da/dN = D ((∆K - ∆Kthr) /(1-Kmax/Kcy)0.5)α where D and Kcy (also referred to as A) are constants and the exponent α is approximately 2. © Susan Pitt and Rhys Jones

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26 Composite Repairs To Cracked Wing Skins.

Section 26 is NOT EXAMINABLE This approach can be easily used to predict the effect of composite repairs to cracked wing skins [1].

Composite repairs are now widely used to ensure the continued airworthiness of military aircraft. It involves bonding high stiffness composite patches over the cracked region so that, see [1]: a) The fibres provide an alternative load path over the crack. b) The fibres help restrain the crack from opening. A detailed discussion on the use of Composite repair technology is given in Chapters 6-13 of the text: Jones R., Baker A., Matthews N., Champagne V, Aircraft Sustainment and Repair, ButterworthHeinemann Press, 2018, ISBN 9780081005408. (Book). https://www.elsevier.com/books/aircraft-sustainmentand-repair/jones/978-0-08-100540-8 © Susan Pitt and Rhys Jones

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A schematic diagram of a typical repair is given below.

2024-T3 plate

160 75 2024-T3 t = 3.18 mm

460

280 boron/epoxy patch 7 layer boron/epoxy patch (inside) honeycomb core grips

Figure 1 Specimen geometry.

This specimen will be passed around in class.


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Example 1: Baker [2], Figure 6.23 page 155, presented crack growth history for a 3.14 mm thick 2024-T3 edge cracked 2024-T3 specimen with an initial 5.0 mm crack, as shown in Figure 1, subjected to constant amplitude fatigue testing with a remote uniaxial stress with σmax = 138 MPa and R= 0.1.

The specimen which had a Youngs modulus of 70,000 MPa and a Poison’s ratio of 0.3 was repaired with a unidirectional 0.889 mm thick boron epoxy patch with a primary modulus of 209,000 MPa, see the Figure below. (The fibres were perpendicular to the crack and as such lay in the direction of the load.)

Bending, due to neutral axis offset effects, was eliminated by testing two specimens joined back to back. The resultant crack growth data from 5 to 12 mm’s are shown in below. Use the Hartman-Schijve equations with the value of D=1.26 10-9, which is taken from the data given earlier in this course for 2024-T3, a threshold of ΔK = 0.2 MPa √m and A = 40 MPa √m to predict growth up to 11 mm. © Susan Pitt and Rhys Jones

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Answer: Jones [3] used 3D finite element analysis to determine the solution to this particular problem for a remote stress of 229 MPa. This paper gave a K of 15.8 MPa √m. Correcting for a stress of 138 MPa give a K of 9.5 MPa √m. Note: For composite repairs to cracked metal skins the stress intensity factor asymptotes to a constant value as the crack length increases [1, 2]. Jones [3] revealed that for the particular geometry under consideration this asymptote was (essentially) reached at approximately 5 mm. Note: The high residual tensile stresses induced in the aluminium have resulted in a low value for the threshold, i.e. 0.2 MPa √m, as a result there is little R ratio effects seen in panels repaired using bonded composite patches [5].


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Crack length a (mm)

100

y = 4.80 e1.03E-05x R2 = 0.995

10

Experimental data

Computed crack length

1 8000

58000

108000

158000

Cycles


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236

Example 2: Repeat the analysis for a remote peak stress of 80 MPa and compare the computed [4] crack length history with that given in [4]. In this instance the value of K for this load level was calculated to be: K = 15.8 * (80/229) = 5.52 MPa √m so that we obtain a ∆K = 4.97 MPa √m.


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Crack length a (mm)

100

y = 4.80 e1.03E-05x R2 = 0.995

10

Experimental data Computed crack length 1 100000

300000

500000

Cycles

700000

A large number of additional examples are given in Chapters 8-11 of the text [6], viz: Jones R., Baker A., Matthews N., Champagne V, Aircraft Sustainment and Repair, ButterworthHeinemann Press, 2018, ISBN 9780081005408. (Book). https://www.elsevier.com/books/aircraft-sustainmentand-repair/jones/978-0-08-100540-8


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References 1. Bonded Repair of Aircraft Structure, A. Baker and R. Jones, Martinus Nijhoff Publishers, The Hague, 1988. (Book), pp 107- 173. 2. Baker AA., “Crack patching: experimental studies, practical applications”, Chapter 6, Bonded Repair of Aircraft Structure, A. Baker and R. Jones, Martinus Nijhoff Publishers, The Hague, 1988. (Book), pp 107173. 3. Jones R., Chiu WK., and Marshall IH., Weight functions for composite repairs to rib stiffened panels , Engineering Failure Analysis, 11(1), 49-78, 2004. 5. Baker AA., “Boron epoxy patching efficiency studies”, Chapter13, “Advances in the Bonded Composite Repair of Metallic Aircraft Structure”, A. Baker, L. R. F. Rose and R. Jones, Elsevier Applied Science Publishers, 2002. ISBN 0-08-042699-9. 6. Jones R., Baker A., Matthews N., Champagne V, Aircraft Sustainment and Repair, ButterworthHeinemann Press, 2018, ISBN 9780081005408. (Book). https://www.elsevier.com/books/aircraft-sustainmentand-repair/jones/978-0-08-100540-8

END OF Section THAT is NOT EXAMINABLE © Susan Pitt and Rhys Jones

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27 The Difference Between Aircraft Sustainment Considerations).

Design And (Operational

The 2014 Fatigue and Fracture of Engineering Materials and Structures review paper [1] and the Chapter 5 in the text Aircraft Sustainment and Repair [2] highlight the difference between the damage tolerance analysis tools needed for design and for aircraft sustainment related problems.

For more details see Chapters 5 and 8 of the text: Jones R., Baker A., Matthews N., Champagne V, Aircraft Sustainment and Repair, Butterworth-Heinemann Press, 2018, ISBN 9780081005408. (Book). https://www.elsevier.com/books/aircraft-sustainment-andrepair/jones/978-0-08-100540-8


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In this context the USAF F-15 program, see the paper by Lincoln [1] mentioned above, was the first to reveal the difference between the approach used in design to that needed to assess actual crack growth in aircraft structures. Here Lincoln [1] stated: “Figure 1 shows the short-crack correction to the longcrack data for both aluminum and titanium that McDonnell used for this study. The inclusion of the short-crack effect was key to the success of the program. The inclusion of the short-crack effect enabled McDonnell to pool the coupon test EIFSs derived from constant-amplitude tests with the random flight-by-flight loaded test aircraft EIFSs. This is a significant finding in that earlier use of the longcrack threshold for crack growth made it appear that the equivalent initial flaws in the structure were spectrum dependent.”

Thus the use of the long crack da/dN versus ∆K curves that are used in in the design of the aircraft needs to be abandoned when assessing cracking in operational aircraft. © Susan Pitt and Rhys Jones

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The Royal Australian Air Force (RAAF) approach to the management of fatigue cracking in combat and trainer aircraft is similar in that it makes use of the “lead crack” concept [2]. This “lead crack” approach is also built into the USAF approach for assessing the Risk of Failure [3], see [2] for more details. In this approach, the life of the fleet is determined by lead fatigue cracks which in [2] were defined to have the following features: a) Crack growth initiates from small naturally occurring material discontinuities, such as inclusions and pits, which have dimensions that are equivalent to a fatigue crack-like size typically of about 10 μm in depth [2, 4, 6]. b) Crack growth essentially starts from the day that the aircraft enters service [1, 2, 4-6]. REFERENCES 1. Jones R. (2014) Fatigue crack growth and damage tolerance. Fatigue and Fracture of Engineering Materials and Structures, 37, 463-483. 2. Chapters 3 and 5 in Jones R., Baker A., Matthews N., Champagne V, Aircraft Sustainment and Repair, Butterworth-Heinemann Press, 2018, ISBN 9780081005408. (Book). © Susan Pitt and Rhys Jones

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https://www.elsevier.com/books/aircraft-sustainmentand-repair/jones/978-0-08-100540-8 3. Lincoln JW., Melliere RA., Economic Life Determination for a Military Aircraft, AIAA Journal of Aircraft, 36,5, 1999. 4. Molent L., Barter S.A., Wanhill R.J.H., The lead crack fatigue lifing framework, International Journal of Fatigue, 33 (2011) 323–331. 5. Berens AP., Hovey PW., Skinn DA., Risk analysis for aging aircraft fleets - Volume 1: Analysis, WL-TR-913066, Flight Dynamics Directorate, Wright Laboratory, Air Force Systems Command, Wright-Patterson Air Force Base, October 1991. 6. Barter SA., Molent L. and Wanhill RH. (2012) Typical fatigue-initiating discontinuities in metallic aircraft structures, International Journal of Fatigue, 41. 1-198. 7. Jones, R., Barter, S., and Chen, F., 2011, “Experimental studies into short crack growth”, Engineering Failure Analysis, 18, 7, pp.1711-1722.

Therefore understanding the growth of fatigue cracks from small naturally occurring defects is of fundamental importance to managing the Australian fleet (airworthiness) [1, 2, 4-6]. © Susan Pitt and Rhys Jones

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This means that it is important to account for the so called “short crack anomaly” which is one of the basic problems in materials science and particularly in fatigue crack growth prediction.

The short crack anomaly arises as experimental studies have shown that for a given ΔK, R ratio and specimen thickness the increment in the crack length (or depth) per cycle (da/dN) seen in tests on short cracks is significantly greater than that seen in tests on long cracks.


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An example of this “anomaly” is shown in the Figure below, which presents the da/dN versus ΔK relationships obtained for 7050-T7451, which is used in both the F/A18 Hornet, the F/A-18 Super Hornet and the Joint Strike Fighter (JSF), for constant amplitude short crack tests [1, 2, 7] for R = -0.3, 0.1, 0.5 and 0.7, and the long crack growth data presented in [1, 2, 7].

da/dN (m/cycle)

-5 100 -6

100 -7

10 0 -8

100

short crack effect

-9 100

Long crack thresholds

-10 10 0

-11 10 0

0.1

1

10

100

 K  MPa  m NASA Kmax

DSTO CT R = 0.1

DSTO CT R = 0.1

NASA R = 0.7

NASA R = 0.1

DSTO CT R = 0.5

Short crack Test 2 R = 0.1


DSTO MT R = 0.2



Comparison of the short crack and long crack da/dN versus ΔK test data for 7050-T7451 from [1, 2, 7] © Susan Pitt and Rhys Jones Page 245

In the short crack tests detailed above the cracks had initial sizes of approximately 7 μm. This represents typical in-service initial defect sizes, see Section 36 on EIFS and EPS. 27.1 The generic shapes of crack growth curves associated with ASTM E647 tests and crack growth in operational aircraft

da/dN (m/cycle)

1.0E-05

D

Short crack R = 0.7

1.0E-06 x 1.0E-07

B

Short crack data

1.0E-08

Long crack: ASTM curve

F x

1.0E-09

x

1.0E-10

E

A

C

G KB

1.0E-11 0.1

1

10

100 K MPa m

Generic representation of long and short crack growth curves, from [2] © Susan Pitt and Rhys Jones

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This begs the question: Given that the short and long crack da/dN versus ∆K relationships are very different which curve should be used in any crack growth analysis ?

Given that the short and long crack da/dN versus ∆K relationships are very different which curve should be used in any crack growth analysis ? Answer: (a) JSSG2006 implies that for design the ASTM long crack test data, i.e. curve CBD (which will vary with the R ratio), is to be used, together with an initial flaw size(s) as mandated in JSSG2006 and outlined previously in the course. (It does not specifically state this, but it is implied.)


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(b) To assess the life, or the effect of a structural modification/repair, of an operational aircraft then since cracking in service aircraft generally grows from small naturally occurring defects then the curve ABD must be used since using the ASTM curve CBD will NOT be conservative. Recall Lincoln’s (1999) statement on the F-15 program.

For more details see Chapters 5 and 8 of the text: Jones R., Baker A., Matthews N., Champagne V, Aircraft Sustainment and Repair, Butterworth-Heinemann Press, 2018, ISBN 9780081005408. (Book). https://www.elsevier.com/books/aircraftsustainment-and-repair/jones/978-0-08100540-8


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27.2 The role of crack closure in crack growth in operational aircraft

Section 27.2 is NOT EXAMINABLE To illustrate the lack of crack closure effect in the growth of cracks in operational aircraft consider the “double ear” specimen shown in Figure 3 which has cracks growing out of each side of a central “rabbit ear” like hole. This specimen geometry was developed as part of the Lockheed P3C (Orion) service life extension program (SLEP) [9] and used to study crack growth under a range of measure flight load spectra [8].


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As can be seen in Figure 3 the specimen real cracks are invariably not straight. The stress field associated with the double ear specimen is shown in Figure 4,

Figure 3 Double ear specimen with cracks emanating from central holes. The cracks have grown naturally from small material discontinuities


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Region just behind the crack tips are open NOT CLOSED

Figure 4. The bulk stress field (∆(1 + 2 + 3)), in MPa, around the crack, note in the region just behind the crack tip the faces of the crack are OPEN AND DO NOT CLOSE DURING FATIGUE


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As a result of the complex topology of the crack the opposing faces of the crack will often rub and in the process dissipate energy. This phenomena is in Figure 5. Figure 5 reveals that the region directly behind the crack remains OPEN during fatigue loading, i.e. IT DOES NOT CLOSE during fatigue loads. 0.0125 0.0115 0.0105 0.0095 0.0085 0.0075 0.0065 0.0055 0.0045 0.0035 0.0025 0.0015 0.0005 0.000

Figure 5 The energy dissipation field, in degrees Celsius. © Susan Pitt and Rhys Jones

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To further illustrate this phenomena consider the problem of crack growth between stiffeners in an Airbus upper fuselage rib stiffened panel, see Figure 6.

Figure 6 Schematic of an Airbus upper fuselage rib stiffened panel with two SPD strips © Susan Pitt and Rhys Jones

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The stress distribution in the panel is given in Figure 7 where the stress concentration at the crack tips is clearly evident.

Figure 7 Stress distribution associated with a crack in an Airbus upper fuselage panel


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The energy dissipation field is given in the Figure 8.. Here we again see two regions that are dissipating energy, viz:: i) The region directly in front of the crack ii) A region back from the crack tip where the faces of the crack are rubbing and hence dissipating energy.

Figure 8. Energy dissipation associated with a crack in an Airbus upper fuselage panel © Susan Pitt and Rhys Jones

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REMARKS: These two examples both reveal energy dissipation due to rubbing of the crack face away from the crack tip (due to it closing during loading). The region in front of the crack tip also dissipates energy. The region behind the crack that rubs (closes) and thereby dissipates energy is NOT associated with crack tip plasticity. REFERENCES 7. Lincoln JW., Melliere RA., Economic Life Determination for a Military Aircraft, AIAA Journal of Aircraft, 36,5, 1999. 8. Jones R. and Tamboli D., Implications of the lead crack philosophy and the role of short cracks in combat aircraft, Engineering Failure Analysis, 29, 2013, pp.149-166. © Susan Pitt and Rhys Jones

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9. Jones R. (2014) Fatigue crack growth and damage tolerance, Fatigue and Fracture of Engineering Materials and Structures, 37, 463-483. 10. Jones, R., Barter, S., and Chen, F., 2011, “Experimental studies into short crack growth”, Engineering Failure Analysis, 18, 7, pp.1711-1722. 11. Jones R., Peng D., Pu Huang, Singh RRK.,(2015) Crack growth from naturally occurring material discontinuities in operational aircraft, Proceedings 3rd International Conference on Material and Component Performance under Variable Amplitude Loading, VAL2015, Prague, Czech Republic, 23rd – 26th March, 2015. Procedia Engineering, 101, pp. 227 – 234. 12. Barter S., Tamboli D. and Jones R., On the growth of cracks from small naturally occurring material discontinuities (etch pits) under a mini-twist spectrum, Proceedings 2015 Aircraft Structural Integrity Program (ASIP), Hyatt Regency San Antonio, 1st-3rd December 2015. 13. Molent L. and Jones R., The influence of cyclic stress intensity threshold on fatigue life scatter, International Journal of Fatigue 82 (2016) 748–756. 14. P. Jackson, C. Wallbrink, K. Walker, D. Mongru and W. Hu, Exploration of Questions Regarding Modelling of Crack Growth Behaviour under Practical Combinations © Susan Pitt and Rhys Jones

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of Aircraft Spectra, Stress Levels and Materials, DSTORR-0368, Melbourne, Australia, July 2011. 15. Veul, R.P.G., and Ubels, L.C., Results of the FMS Spectra Coupon Test Prorgam Performed within the Framework of the P-3C Service Life Assessment Program, NLR-CR-2003-488, December 2003.

END OF SECTION THAT IS NOT EXAMINABLE


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27.3 Crack growth in panel reinforced with a supersonic particle deposition (SPD) doubler To illustrate this let us consider the case of a 3 mm thick SENT (single edge notch tension) specimen, see Figure 1, with a 0.5 mm thick 7075 aluminium alloy SPD patch on one side, see Figure 2. The 7050-T7451 panel was 350 mm long by 42 mm wide and 3 mm thick with a 0.69 mm radius semi-circular edge notch on one side. The specimen was tested at a peak (remote) load of 17.64 kN with R = 0.1. This corresponds to a ∆σ under the patch of 108 MPa.


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100 mm

50mm

125mm

42 mm

350 mm

125mm

50mm

Rivets

Figure 1 Geometry of the 7050-T7451 SENT test specimen.


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Crack Length (mm)

6 Measured crack length data Computed 5

4

3 20000

30000

40000

Cycles (Total)

Figure 2 Comparison of computed and measured crack lengths


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Note: The stress intensity factor for a though the thickness crack of length c emanating from the centre of the notch of radius r is given by Newman, Wu et. al in [1] as: K = f1 x g4 x fw x √( c)

(1)

where c is the length of the crack emanating from the notch and  is the stress in the 7050-T7451 underneath the SPD. The values of f1, g4 and fw were taken from [1] to be: f1 = 1+0.358  +1.425 2 -1.578 3 + 2.156 4  = 1/(1+c/r) g4 = Kt (0.36 – 0.32/(1+c/r)1/2) fw = 1+ 2.7 2- 3.5 4 + 3.8 6 Kt = 3.17

(2) (3) (4) (5) (6)

Since the specimen was tested using hydraulic grips the formulae used for fw was the fixed displacement expression given in [1]. This solution is to be used in Tutorial 9. In this example use r = 0.69 mm 1. Newman, J.C., Wu, X.R., Venneri, S.L., and Li, C.G., Small-Crack Effects in High-Strength Aluminium Alloys, NASA, Editor. 1994, NASA. © Susan Pitt and Rhys Jones

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27.4 Crack growth in 1 mm thick 2024-T3 specimens

This is Tutorial 9 SENT (single edge notch tension) tests were performed on 1mm thick 2024-T3 specimens. The specimens had a small 0.5 mm semi-circular notch from which the cracks grew. In the first test we had a maximum stress of 160 MPa and R = 0.1. In the second test we had a maximum stress of 107 MPa and R = 0.1. In the first test the cracking was initially 3D and then later became 2D whilst in the 2nd test cracking was essentially 2D. Thus the smallest through-the – thickness crack analysed in this study was approximately 0.29 mm. The resultant crack growth histories are shown in Figure 3. Compute the crack length versus cycles history using the Hartman-Schijve equation: da/dN

= D (ΔK – ΔKth)2/√(1-Kmax/A)

(2)

where D and A are constants with the value of D taken from data presented in earlier in the course, viz: D and A =1.2 10-9, and A = 50 MPa √m (as previously) and to simulate the growth of cracks from small naturally occurring defects set ΔKth =0.

Use the same equations as before to compute K this time use r = 0.5 mm © Susan Pitt and Rhys Jones

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Crack Length (mm)

10.00

1.00 Test 1 Side A Test 1 Side B Test 2 side A Test 2 side B Hartman-Schijve 107 MPa Hartman-Schijve 160 MPA

0.10 35000

55000

75000

95000

115000

Figure 3 Measured and predicted crack length histories for a 1 mm thick 2024-T3 skin


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SUMMARY We thus see that for short cracks in both 7050-T7451 and thin sheet 2024-T3 crack growth conservative estimates can often be obtained by using a simple Paris like equation, viz: da/dN

= D ΔK2/(1-Kmax/A)


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28 The Keith Donald ACR Test Method Section 28 will not be discussed in Class and IS NOT EXAMINABLE Donald and Paris subsequent developed a method, termed the Adjusted Compliance Ratio (ACR) technique, that can be used to determine the da/dN versus ∆K curve for physically short cracks. This approach is NOW adopted as part of E647. A detailed outline of this approach is given in the file “ACR Summary for FCG Manual” which is located on Moodle. This file was written by and is provided courtesy of Mr. Donald. END OF SECTION THAT IS NOT EXAMINABLE


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WEEK 9 Starts Here

29 The Paris Revisited:

Crack

Growth

Equation

Since for cracks that grow from small naturally occurring defects i) The da/dN curve is not very R ratio sensitive ii) The da/dN versus ΔK curve exhibits a near power law relationship This means that many of the shortcomings of the Paris crack growth equation vanish for growth from small naturally occurring defects.


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This is aptly illustrated in the paper by Gallagher* and Molent for crack growth in 7050-T7451 under 6 different combat aircraft load spectra.

Crack growth per flight hours plotted against a reference value of K in a load block for tests under an operational F/A-18 flight load spectra *Gallagher is the author of the USAF Damage Tolerance Design Handbook © Susan Pitt and Rhys Jones

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Hence the Paris crack growth equation, which is available in AFGROW, is useful for obtaining conservative estimates for crack growth in operational aircraft.

Reference 1. Gallagher JP., Molent L., The equivalence of EPS and EIFS based on the same crack growth life data, International Journal of Fatigue, 80, 162–170, 2015.


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30 Variability In Crack Growth One of the major challenges in any fleet management process is how to account for the variability in crack growth that can be expected to occur in operational aircraft. This variability is dominantly due to: 1. The nature of the initial discontinuity that leads to fatigue cracking; 2. Variations in the local fatigue thresholds; 3. Variation in the stress concentration factors due to inter-aircraft variations; 4. Fit-up or residual stresses; 5. Variations in the fracture toughness; This effect is usually small. 6. The initiation period; 7. Variations in the crack growth rate. Points 1, 3 and 4 are considered to define the aircraft’s build quality from a fatigue perspective. Points 2, 5, 6 and 7 define the metal’s fatigue variability. © Susan Pitt and Rhys Jones

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With respect to point 2 it is now known [1- 4] that for cracks that arise and grow from small naturally occurring material discontinuities, for a given initial size, the variability in crack growth could be accounted using the Nasgro (Hartman-Schijve-McEvily) crack growth equation and accounting for the variability in the apparent threshold. However, since for operational aircraft you are interested in the fastest growing crack, i.e. the lead crack [5, 6], conservative estimates for operational life and the associated inspection intervals can also be obtained using the Paris crack growth law option in AFGROW.


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Let us now address point 7. The paper by Virkler, Hillberry and Goel [7] is the definitive study that illustrated the variability in crack growth. This paper presented the results of sixty eight tests on 2024-T3 panels where the initial crack length was 9 mm. As such the effect of variations in the initial crack length was removed.

Crack Length (mm)

100

Virkler-Hillberry Data

10

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

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22

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28

29

30

31

33

34

35

36

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44

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61

62

63

64

65

66

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68

1 0

50000

100000

150000

200000

250000

300000

350000

Cycles

Figure 1 Variability in the measured curves, R = 0.2 [2].


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References [1-4] revealed that Nasgro (Hartman-SchijveMcEvily) can be used to estimate the variability in fatigue life we used Equation (1) with the constant of proportionality as given in [8], viz: da/dN

= D (K - Kthr)2/(1-Kmax/A)

(1)

with D = 1.2 10-9 and then merely allowed for small changes in the threshold term Kthr. The result is follows. (This will be a tutorial/Prac class question).

100 Threshold = 3.6 Threshold = 3.8 Threshold = 3.2

Crack Length (mm)

Threshold = 2.9

Threshold = 4 Threshold = 4.2

10

1 0

50000

100000

150000

200000

250000

300000

2

3

6

7

12

15

19

24

25

26

36

37

38

39

45

46

49

51

57

60

61

68

Threshold 2.9

Threshold 3.2

Threshold 3.6

Threshold 3.8

Threshold 4

Threshold 4.2

350000

400000

Cycles

Figure 2 Computed and measured crack length curves, from [4]. © Susan Pitt and Rhys Jones

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Figure 2 reveals that the variability in the data is captured reasonably well by merely allowing for small changes in Kthr, i.e. using vales of 2.9, 3.2, 3.4, 3.6, 3.8, 4 and 4.2, see [1, 3] for more details. As such this approach can be used to help determine the natural variability that will be seen in operational vehicles. However, it should be noted that for operational aircraft the variability in initial defect size also needs to be accounted for, see [9].


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Recall that for operational aircraft the life of the fleet is determined by the fastest growing crack from small naturally occurring material defects. (Such cracks are termed “Lead Cracks”, see Section 33.) In such cases the threshold effect can be neglected so that the variability in life is determined by the the variability in equivalent initial pre-crack size (EPS), which is the size of the initiating material discontinuity that is determined via quantitative fractography. This is discussed in more detail in Section 35. Gallagher and Molent [10] have shown that cracks in operational aircraft follow a simple Paris like equation and that if this approach is adopted then for operational aircraft the EPS and the EIFS coincide. EPS = Equivalent Precrack Size, EIFS = Equivalent Initial Flaw Size. © Susan Pitt and Rhys Jones

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For more details see Chapter 5 of the text Jones R., Baker A., Matthews N., Champagne V, Aircraft Sustainment and Repair, ButterworthHeinemann Press, 2018, ISBN 9780081005408. (Book). https://www.elsevier.com/books/aircraft-sustainmentand-repair/jones/978-0-08-100540-8 References 1. Molent L., Jones R (2016) The influence of cyclic stress intensity threshold on fatigue life scatter, International Journal of Fatigue, 82, pp. 748-756. 2. Jones R, Molent L., Barter S., Calculating crack growth from small discontinuities in 7050-T7451 under combat aircraft spectra, International Journal of Fatigue, International Journal of Fatigue, 55 (2013), pp. 178-182. 3. Jones R., Fatigue Crack Growth and Damage Tolerance, Invited Review Paper, Fatigue and Fracture of Engineering Materials and Structures, (2014), Volume 37, Issue 5, pp. 463–483. 4. Jones R., Huang P. and Peng D., (2016) Crack growth from naturally occurring material discontinuities under constant amplitude and operational loads, International Journal of Fatigue, 91 pp. 434–444. 5. Berens AP., Hovey PW., Skinn DA., Risk analysis for aging aircraft fleets - Volume 1: Analysis, WL-TR-913066, Flight Dynamics Directorate, Wright Laboratory, Air Force Systems Command, Wright-Patterson Air Force Base, October 1991. © Susan Pitt and Rhys Jones

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6. Molent L., Barter S.A., Wanhill R.J.H., The lead crack fatigue lifing framework, International Journal of Fatigue, 33. pp.323-331. 2011 7. Virkler DA, Hillberry BM, Goel PK. The statistical nature of fatigue crack propagation, Trans ASME 1979;101:148–53. 8. Jones R, Molent L., Walker K., Fatigue crack growth in a diverse range of materials, Int. J. Fatigue, (2012), Volume 40, July 2012, pp 43-50. 9. Adriano Francisco Siqueira, Carlos Antonio Reis Pereira Baptista, Loris Molent, On the Determination of a Scatter Factor for Fatigue Lives Based on the Lead Crack Concept, Journal of Aerospace Technology Management, Vol.5, No 2, pp.223-230, Apr.-Jun., 2013 10. Gallagher JP., Molent L., The equivalence of EPS and EIFS based on the same crack growth life data, International Journal of Fatigue, 80, 162–170, 2015.


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31 Equivalent Block Methods For Predicting Fatigue Crack Growth Many practical engineering problems, i.e. cracking in rail and aircraft structures, involve complex load spectra that can be approximated by a number of repeating load blocks. As a result Schijve [1], Gallagher and Stalnaker [2], Miedlar, Berens, Gunderson, and Gallagher [3, 6], and Barsom and Rolfe [4] developed a simple equivalent block approach in where each load bock was taken to be equivalent to a single cycle. Whilst this approach was developed for long cracks Gallagher and Molent [6] and Huang, Peng and Jones [7] have shown that it also holds for crack growth where the cracks grow from small naturally occurring material discontinuities under a range of operational flight load spectra (civil and military aircraft).


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References 1. Schijve J., Fatigue crack growth under variableamplitude loading, Engineering Fracture Mechanics, Engineering Fracture Mechanics, 1979, pp 207-221. 2. Gallagher JP. and Stalnaker HD., Developing normalised crack growth curves for tracking damage in aircraft, American Institute of Aeronautics and Astronautics, Journal of Aircraft, 15, 2, (1978), pp 114120. 3. Miedlar PC., Berens AP., Gunderson A., and Gallagher JP., Analysis and support initiative for structural technology (ASIST), AFRL-VA-WP-TR-2003-3002. 4. Barsom JM and Rolfe ST, Fracture and fatigue control in structures: Applications of fracture Mechanics, Butterworth-Heinemann Press, November 1999. 5. Gallagher JP., Estimating fatigue crack lives for aircraft: Techniques, Experimental Mechanics, pp. 425-433, 1976. 6. Gallagher JP., Molent L., The equivalence of EPS and EIFS based on the same crack growth life data, International Journal of Fatigue, 80, 162–170, 2015. 7. Huang, P., Peng D., Jones R., The USAF characteristic K approach for cracks growing from small material discontinuities under combat aircraft and civil aircraft load spectra, Engineering Failure Analysis, 2017, http://dx.doi.org/10.1016/j.engfailanal.2017.03.008


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This approach requires that: i)

The slope of the a versus block curve has a minimal number of discontinuities.

ii) There are a large number of blocks before failure It is written into the USAF Damage Tolerant Design Handbook as an accepted method of analysis. Termed the characteristic K (or Mini-Block) approach This method uses a characteristic stress intensity factor in the load block, Kmax or ∆Krms, see Chapter 5 of Miedlar PC., Berens AP., Gunderson A., and Gallagher JP., Analysis and support initiative for structural technology (ASIST), AFRL-VA-WP-TR-2003-3002. For more details see Chapter 5 of the text Jones R., Baker A., Matthews N., Champagne V, Aircraft Sustainment and Repair, ButterworthHeinemann Press, 2018, ISBN 9780081005408. (Book). https://www.elsevier.com/books/aircraft-sustainmentand-repair/jones/978-0-08-100540-8


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Example To illustrate this consider the USAF test program on centre cracked panel tested under a B1-B bomber spectrum. Three tests were performed with maximum stresses in the spectrum of 36.31, 31.12 and 24.17 k..s.i. (Note 1 ksi = 6.894 MPa, please change the units to grasp how to do this yourself.)

From Gallagher JP. and Stalnaker HD., Developing normalised crack growth curves for tracking damage in aircraft, American Institute of Aeronautics and Astronautics, Journal of Aircraft, 15, 2, (1978), pp 114-120. An average block variant of the Hartman-Schijve equation was used to predict the crack length histories, viz:

da D K max  K th   K max dN (1  ) A

2

where D and A are constants and Kmax is the maximum value of K in a load block.


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The predicted behaviour with D = 1.44 10-11 A = 77 MPa √m and

Kth = 4.08 MPa √m

is shown below together with the experimental data from Gallagher

a (mm)

100

10

36.31 ksi 31.12 ksi 24.17 Hartman-Schijve 24.17 Hartman-Schijve 31.12 Hartman-Schijve 36.31 1 0

200

400

600

800

1000

1200

1400

1600

1800

Flights

Here you see that the predictions are in good agreement with the test data. © Susan Pitt and Rhys Jones

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(We will click on this spreadsheet and work our way through these predictions in class.) Other examples are presented in Chapters 5 and 8 of the text Jones R., Baker A., Matthews N., Champagne V, Aircraft Sustainment and Repair, ButterworthHeinemann Press, 2018, ISBN 9780081005408. (Book). https://www.elsevier.com/books/aircraft-sustainmentand-repair/jones/978-0-08-100540-8


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31.1 Crack growth from etch pits and manufacturing defects at a fastener hole under an operational F/A-18 flight load spectrum Crack growth from etch pits and manufacturing defects at a fastener hole under an operational F/A-18 flight load spectrum with a maximum remote stress in the spectrum of 155 MPa. This example is taken from Tamboli, Huang and Jones [1] and Huang et al [2]. The geometry of the test specimen is given below

Schematic of the specimen studied in [1, 2] The “Characteristic” crack growth equation used in [1, 2] was © Susan Pitt and Rhys Jones

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The value of ΔKrms of the spectrum was calculated using the formula

The values of D = 7.0 x 10-10, A = 47 MPa√m and p= 2 were taken from [3] which presented data on constant amplitude tests for Al 7050-T7451 with small naturally occurring cracks.


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Resultant Computed versus Measured Crack Growth History

Measured vs computed crack growth histories for “etched” coupons under a representative F/A-18 Spectra, from [1]


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31.2 Crack growth at a fastener hole in a RAAF P3C (Orion) aircraft containing intergranular cracking

Let us next consider crack growth at a fastener hole in a RAAF P3C (Orion) aircraft that contained intergranular cracking, resulting from an aggressive environment, under representative maritime aircraft flight load spectra is given in [6].

Location (arrowed) of the problem areas in the inner engine nacelles at the lower wing panels on the P-3C Orion.


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Schematic of the DNH coupon used in analysis of the interaction between IGC and fatigue crack growth, from [6]. Note that 2.032 mm is a nominal thickness however the thickness of the test coupons varied because of the taper in the thickness of the AP-3C wing panels.

Typical finite element model of RAAF P3C dome nut hole test coupon with a 1.5 mm deep IGC representing the IGC found in DST Group coupon P3-IG-33, from [6]. © Susan Pitt and Rhys Jones

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Typical agreement between measured and computed, using the Hartman-Schijve equation, crack growth histories, from Chapter 5 in the text by Jones, Baker, Matthews and Champagne.

Additional examples of how to combine finite element analysis with crack growth equations so as to life/assess cracking in operational aircraft are given in Chapters 5 and 8 of the text: Jones R., Baker A., Matthews N., Champagne V, Aircraft Sustainment and Repair, ButterworthHeinemann Press, 2018, ISBN 9780081005408. (Book). https://www.elsevier.com/books/aircraft-sustainmentand-repair/jones/978-0-08-100540-8 © Susan Pitt and Rhys Jones

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CLASS EXERCISE Repeat the analysis for the case when the hole contains a single small 0.05 mm deep through the thickness cracks on either side of the hole. In this analysis you can assume that ∆σrms = 0.3 σmax, where is the maximum remote stress in the spectrum.

To do this you will need to use the Bowie beta factor solution, given in Gallagher 1976 [4], for the stress intensity factor for a through crack of length “a” at a hole of radius r in a wide plate, viz: K1 = β(a,r) σ(a) β(a,r) = 0.6762 + [0.8733/(0.3245 +(a/r))]


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From Proceedings USAF 1989 ASIP Conference


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FxF = flight by flight


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Other examples are given in [6].


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References 1. Tamboli D, Pu Huang and Jones R., Validation Of The USAF Characteristic K Approach For Long Cracks Under Variable Amplitude Loading And Naturally Occurring Small Cracks Under A Combat Aircraft Load, Proceedings 28th International Conference on Aeronatical Fatigue and Structural Integrity, 28th ICAF Symposium – Helsinki, 3–5 June 2015. 2. Huang, P., Peng D., Jones R., The USAF characteristic K approach for cracks growing from small material discontinuities under combat aircraft and civil aircraft load spectra, Engineering Failure Analysis, 2017, http://dx.doi.org/10.1016/j.engfailanal.2017.03.008 3. Jones R, Molent L and Barter S., (2013). Calculating crack growth from small discontinuities in 7050-T7451 under combat aircraft spectra, International Journal of Fatigue, vol 55, pp. 178-182. 4. Jones R, Molent L and Walker K (2012). Fatigue crack growth in a diverse range of materials, International Journal of Fatigue, vol 40, pp 43-50. 5. Gallagher JP., Estimating fatigue crack lives for aircraft: Techniques, Experimental Mechanics, pp. 425-433, 1976. © Susan Pitt and Rhys Jones

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6.

Lo M., Jones R., Bowler A., Dorman M., and Edwards D., Crack growth at fastener holes containing intergranular cracking, Fatigue and Fracture of Engineering Materials and Structures, (2017) doi: 10.1111/ffe.12597.


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There are other alternative (equivalent block) approaches that have been developed for long cracks Example A crack is growing from the edge of a wing skin on the lower tension surface. The wing skin is subjected to block loading. The block starts with a stress  = 0 MPa, and has turning points at 125 MPa, -125 MPa, 75 MPa, -50 MPa, 25 MPa, and -100 MPa. Use Wheeler’s crack growth law with C = 10 -8, the exponent of K, i.e. m, equal to 3, p = 2, and Kc = 50 MPa m , and the yield stress y = 500 MPa to determine the number of cycles to failure from initial crack lengths of 0.1 mm and 0.5 mm.


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WEEK 10 Starts Here 31.3 Blocks to Failure Assume that the crack length is constant in a given block. Then in a single block da/db

=  (max i/overload) 2p C (K)n =C  (i/overload) 2p (K)n = C (1.12 (a))n  in (max i/overload)2p

So the blocks to failure Bf = af

Bf =1/([1.12] i (maxi/overload) ) ai 1/(a)nda = 2/{(n-2)* ([1.12 ]n  i n (maxi/overload) 2p C)} (1/ ai(n-2)/2 - 1/ af(n-2)/2) n

n

2p

The final crack length af can be calculated by equating K to the apparent fracture toughness Kc for the material, viz: af = 1/ (Kc /1.12  overload) 2


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We now merely substitute the values of C, n, p, I and overload in the block, and ai to first determine af and the Bf. By using two different initial crack sizes we show the importance, i.e. dominant role, of initial flaws. In this example: The block starts with a stress  = 0 MPa, and has turning points at 125 MPa, -125 MPa, 75 MPa, -50 MPa, 25 MPa, and -100 MPa. C = 10 -8. m = 3, p = 2, and Kc = 50 MPa max 1 = 125 MPa, max 2 = 75 MPa, max 3 = 25 MPa and overload = 125 MPa.

,

Thus (max1/ overload )4 = 1 MPa, (max 2/ overload )4 = 0.13 MPa, (max 3/ overload )4 = 0.00016 MPa and ai = 0.1 mm.


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Thus Bf ~ 2/ ([1.12 ]3 1253 10 -8. m) (1/ ai1/2 - 1/ af1/2) where af = 1/ (50 /1.12 125) 2 = 40.6 mm, 1/ af1/2 = 0.157, 1/ ai1/2 = 0.09 mm -1= 3.16 mm -1 1/ ai1/2 -1/ af1/2 = 3.0, Thus it is sufficient to dealt with the first term only., viz: Thus Bf ~ 2/ ([1.12 ]3 1253 10 -8.) 3.0 *√(1000) = 7.2 *31* 10 -8/ 247 ~ 128 blocks. When using an initial flaw 0.5 mm long this number reduces by the ratio 5 to ~ 54 blocks.


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32 Other Alternative Approaches Section 32 will NOT be presented in Class and IS NOT Examinable Boeing (Seattle) [1] developed a related (non-similitude) approach which was used in the design of the 757 and 767. In the Boeing approach da/dB was expressed as: da/dB = C (Kmax/g(a/t))m where the function g(a/t), which is a function of ratio of the crack length (a) to the thickness (t) of the specimen, was experimentally determined. This formulation was necessary to enable the predictions to match the measured crack length histories. 1.

Miller M., Luthra VK., and Goranson UG., Fatigue crack growth characterization of jet transport structures, Proc. of 14th Symposium of the International Conference on Aeronautical Fatigue (ICAF), Ottawa, Canada, June 10-12, 1987.


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An alternative approach is to use the equivalent block variant of the generalized Frost-Dugdale law as developed by the DSTO Combat Aircraft Group.

This will be discussed later in the course.

NOTE: Section 32 will NOT be presented in Class and IS NOT Examinable


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33 Fractals & The DSTO-DGTA Lead Crack Approach - Also Built Into The USAF Risk of Failure Assessment Computer Program (PROF) The DSTO Combat Aircraft Group in conjunction with the RAAF Directorate General Technical Airworthiness (DGTA, now DASA) Air Structural Integrity Section have developed an alternative approach which is based on observations associated with fleet cracking and can be related to the fractal nature of crack growth [12-14], see Section 33.8. This will be discussed in the next few slides.


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33.1 Real life damage growth- Exponential crack growth: As first explained by the USAF [1] the (long) crack growth equations that until recently were used to describe crack growth in operational aircraft often fail in real life. The USAF [1] then noted that lead cracks in fleet aircraft exhibit exponential crack growth, see [1-4].

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The DST Group combat aircraft group subsequently developed a methodology that is based on an extension of the Frost-Dugdale crack growth equation and is related to the fractal nature of crack growth, see Section 33.8. In its simplest form this states that crack growth in operational aircraft is often exponential, viz: ln(a) = N + ln(a0) OR a = a0 eN where N is the number of cycles, flights or repeated load blocks, (also contained in USAF approach to assessing the Risk of Failure [1])  is a parameter that is geometry, material and load dependent (not to be confused with the beta factor), a is the crack depth and time n, a0 is the initial crack-like flaw size and  = f (σ)


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33.2

Cubic Stress Equation:

For aluminium alloys there is often a near cubic stress dependency, see [2-6, 15-17] above:  = λ (∆σ)3 where λ is a constant. (The USAF approach did not realise this.) The cubic variation of crack growth rate with stress is known as the “Cubic Rule” and is now used in the management of cracking in RAAF F/A-18 (Classic Hornet) [7] and AP3C (Orion) [8] aircraft as well as in the assessment of cracking in the RAAF PC-9 (basic trainer) fleet. The cubic rule is also consistent with Def Standard 970 [9] which states “damage is proportional to the cube of the (stress) amplitude”.


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In the non Aerospace field one of the most-widely used standards, that of the American Association of State Highway and Transportation Officials (AASHTO)[10], also stipules that crack growth shall be assumed to depend on the cube of the stress amplitude.

However, this approximation, i.e. that the crack growth rate is proportional to the cube of the stress amplitude, is not correct if the applied stress exceeds the yield stress of the material.

Recall JSSG2006 requires NO YIELDING at 115% DLL.

The mathematical basis of the Cubic Rule can be linked to the fact that for much of its life cracks in operational structures can be represented as a fractal with a fractal box dimension of approximately 1.2, see Section 33.8.


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33.3 Application of the Cubic Rule: Combat Aircraft Spectra: The experimental data presented above for crack growth at a fastener hole under an operational F/A-18 flight load spectrum can be used to aptly illustrate the application of the cubic equation to predict crack growth. For example in the Figure below we see that when using this approach the fastest growing crack associated with the 155 MPa tests, i.e. specimen KK1H353, has value of  = 0.123. 10

y = 0.0173e0.1113x R² = 0.9949 y = 0.0247e0.123x R² = 0.9959

1

Crack size (mm)

y = 0.0097e0.1059x R² = 0.9907

0.1 155_Etched_KK1H353 155_Etched_KK1H348

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155_Etched_KK1H193 155_Etched_KK1H339 155_Etched_KK1H343

0.001 0

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20

30

40

50

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70

80

90

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Load Blocks

Figure 9 Crack growth histories associated with the 155 MPa etched tests, from [5, 15].


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Thus the cubic law as outlined in the F/A-18 Structural Assessment Manual (SAM) would give, using the value of  from the fastest 155 MPa test (KK1H343), a value of  for the test with a peak stress of 250 MPa to be:  (for fastest crack in the 250 MPa test) = 0.123 x (250/155)3 = 0.516


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As can be seen in the Figure below the fastest growing crack in the 250 MPa tests is somewhat greater than that predicted using a value of  = 0.516. Nevertheless using the cubic equation yields a predicted crack depth history that is in reasonably good agreement with the measured crack depth history.

10

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actual crack growth

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250_Etched_KK1H373

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Comparison between measured and predicted crack growth history, from [5, 15].


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33.4 Application of the Cubic Rule: Maritime Aircraft Spectra: Crack growth data were provided for centre-notched AA7075-T6 (unclad) coupons (Matricciani, Duthie and Walker, 2015 [18]). The coupons were 2.032 mm thick, 96.52 mm wide and 279.4 mm long. The coupon notch feature consisted of a 2.032mm open hole with two 0.508mm slits either side of the hole. The slits were introduced manually using a jeweller’s saw during coupon manufacture. Two coupons were tested under a spectrum associated with fatigue critical location FCA 361 (see Iyyer et. al, 2007 [19]) at four stress scales, namely: baseline, +5%, +10% and +20%. The baseline spectrum was tested at a peak stress of 124.8 MPa. As seen in Figure below the +20% spectrum results were predicted using the Cubic Rule [20] well using the baseline results.


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Figure: Measured and Cubic predictions of crack growth histories for AA7075-T6 tested under a RAAF P3C spectrum, from [5].


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33.5 Application of the Cubic Rule: USAF Military Transport Aircraft Spectra: In the mid-1981’s, the USAF contracted Rockwell International, North American Aircraft Division to evaluate crack growth under a typical USAF military transport aircraft spectra, Chang [1]. The test specimens used ASTM standard 6.35 mm thick centre cracked tension (CCT) panels machined from AA2219-T851 plates. The centre notches were installed through the electrical discharge machining process. Test specimens were pre-cracked before applying the spectra loading. The baseline spectrum represented a typical four engine military transport aircraft. Test TB-1 had a maximum tensile stress of 14 ksi (96.5 MPa) and maximum compressive stress of 11.5 ksi (79.3 MPa). A test (TB-V-2) was also performed with stress increased by 160%.


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The resultant crack length versus cycles curves were essentially linear, i.e. crack growth was exponential, see Figure. The value of  for the base line specimen T-B-1 was 5.14 10-7. This value of  was then used together with the cubic rule to predict the crack growth history for specimen T-B-2. The measured [20] and predicted [5] crack growth histories are shown below. In this example we note that the pre-crack was in the physically large crack regime. A large number of additional examples of this approach are given in Chapters 5 and 8 of the text: Jones R., Baker A., Matthews N., Champagne V, Aircraft Sustainment and Repair, ButterworthHeinemann Press, 2018, ISBN 9780081005408. (Book). https://www.elsevier.com/books/aircraft-sustainmentand-repair/jones/978-0-08-100540-8


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T-B-V-2 Test Data

Crack length (mm)

T-B-V-2 Prediction

10 0.0E+00

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Cycles

Figure: Measured and Cubic predictions of crack growth history of AA2219-T851 tested under a military transport spectrum, from [5].


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33.6 Scaling from one fatigue critical area (FCA) to a second FCA with a different spectrum and a different peak stress. You sometimes may have data associated with cracking, that has arisen and grown from a small naturally occurring material discontinuity, at one location (location 1) and cracking arises at a different location (location 2) which sees a different spectrum and a different peak stress. The Cubic Rule can be used to estimate the growth rate associated with cracking at this 2nd location provided that: i) The crack growth history associated with location 1 is exponential. ii) The ratio σmax/Δσrms is the same at both locations.


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EXAMPLE: Crack growth in P3C dome nut hole (DNHS) specimens under two different RAAF P3C (Maritime) flight load spectra, viz: FCA352 and FCA361. Location FCA352-PDN-1 correspond to “the fairing dome nut holes in the lower surface panels at the inboard nacelle”; location FCA361-PDN-3 correspond to “the outer wing lower panels 1 to panel 3 – inboard nacelle – outboard fillet fairing dome nut holes”.

Figure is courtesy of Adam Bowler RAAF MPSPO


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Figure is from [21]


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• Typical IGC at the central hole

Figure is from: [21].


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Stress = 133 MPa

Crack growth in DNH specimens with IGC under the FCA352 (clipped) spectrum is essentially exponential with a slope ψ = 4.78 x 10-4, from [18].


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• Since for FCA352 and FCA361 the ratio ∆σrms/σmax in the spectra are essentially the same • We can use the cubic rule together with the value of ψ for the FCA352 tests to estimate the crack growth history in the FCA361 tests. Stress was 124 MPa. • This yields a predicted slope of Ψ(for FCA361) = (4.78 x 10-4 ) x (124/133)3 = 3.87 x 10-4. Ratio of the remote stresses in the two tests


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The resultant predicted crack growth history in DNH specimens with IGC (for the FCA361 spectrum) is in good agreement with the experimental data.

This illustrates the potential for the RAAF-DST Group approach to use data at one location to predict crack growth at a different location providing that the differences the ratio ∆σrms/σmax in the spectra are essentially the same.


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33.7 Cubic Rule: Conclusion

These three disparate examples illustrate how the Cubic Rule works well for Combat, Military Transport and Maritime Aircraft Flight Load Spectra.

Additional examples of this approach are given in Chapters 5 and 8 of the text: Jones R., Baker A., Matthews N., Champagne V, Aircraft Sustainment and Repair, ButterworthHeinemann Press, 2018, ISBN 9780081005408. (Book). https://www.elsevier.com/books/aircraft-sustainmentand-repair/jones/978-0-08-100540-8


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33.8 Generalised Frost-Dugdale Equation:

This Subsection, i.e. 33.8, WILL NOT be presented AND IS NOT EXAMINABLE Exponential crack growth and the cubic stress dependency follows directly from the “Generalised Frost-Dugdale” growth equation [35] which can be expressed in the form:

da / dN = Ca(1-m/2) (K)m where m =3 gives the Frost Dugdale law and also yields both near exponential crack growth and the cubic stress equation. If a is in mm and K in MPa mm then C is typically of the order of 10-12


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Taking m =3 K = a da / dN = C (K)3/ a = a C ()3

N  {1/ C ()3 }[ ln(a) –ln(ao)] Provided that the  is a weak function of the crack length a. Otherwise a

N = 1/ {C ()3 } a0 da/( a)


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Crack growth in Mirage 1110 full-scale fatigue test wing Predictions were made using a Paris crack growth equation with m = 3, i.e. a cubic stress dependency.

END OF SECTION EXAMINABLE © Susan Pitt and Rhys Jones

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33.9 Fractals and The Cubic Rule As we have seen in Section 7.2 cracks DO NOT grow as idealized straight lines, or as a plane in Euclidean space as shown in Figure 1.

Figure 1. A classical Mode I fatigue crack growth. It is now known that fracture surfaces can be considered as an invasive fractal set, see Mandelbrot et al 1984 [11].

Figure 2. Plan and surface view of a fractal like cracks, from [23]. © Susan Pitt and Rhys Jones Page 333

In this work Mandelbrot et al [11] wrote: “When a piece of metal is fractured either by tensile or impact loading the facture surface that is formed is rough and irregular. Its shape is affected by the metal’s microstructure (such as grains, inclusions, and precipitates where characteristic length is large relative to the atomic scale), as well as by ‘macrostructural’ influences (such as the size, the shape of the specimen, and the notch from which the fracture begins). However, repeated observation at various magnifications also reveal a variety of additional structures that fall between ‘micro’ and ‘macro’ and have not yet been described satisfactorily in a systematic manner. The experiments reported here reveal the existence of broad and clearly distinct zone of intermediate scales in which the fracture is modelled very well by a fractal surface.”

Bouchard [12] subsequently revealed that fatigue crack growth surfaces have a fractal dimension of approximately 2.2, i.e., their profiles have D = 1.2.

This conclusion Mandelbrot [33].

was

subsequent


validated

by

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33.9.1

Nature of the crack tip stress field for a fractal crack

If the crack has a fractal dimension D (1 50 years old!

10%

0%

Over 20

Over 25

Over 30

Over 35

Over 40

Over 45

Over 50

Average Age (Years) Source: GCSS/AF Data Services, as of FY09 SLIDE 1


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46 An Example Of Australia’s Indigenous Capabilities Hardy Hornets save $400m on upgrade By John Kerin, Sep 03, 2008 (The Australian Financial Review - ABIX via COMTEX) -“The Australian Air Force's upgrade of its Hornet fighter aircraft will not be as expensive as thought. Fatigue testing at DSTO revealed that nearly half the fleet will not need upgrading, reducing the $A1 billion cost by $A400 million. The aircraft will also be able to fly for two years longer than thought, if the delivery of new fighter aircraft is delayed.”


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47 Additive Metal Technology Although the application of additive metal technology to metallic airframes is outside the scope of this course it should be noted that a detailed treatise on additive metal technology is contained in Chapters 14-18 of the text: Jones R., Baker A., Matthews N., Champagne V, Aircraft Sustainment and Repair, ButterworthHeinemann Press, 2018, ISBN 9780081005408. (Book). https://www.elsevier.com/books/aircraft-sustainmentand-repair/jones/978-0-08-100540-8


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48 CLASS ASSESSMENT TASKS Assessment Tasks Participation Assessment Task 1: Mid Semester Test Due Date: Week 6, 10:00 – 11:00 am Details of task: Mid Semester Test Release dates (where applicable): Word limit (where applicable): Value: 10 % Presentation requirements: Criteria for Marking: As per the final exam. Assessment Task 2: Discuss in detail the FAA LOV its origin and impact on the Civil Aviation Industry Due Date: Friday Week 7 Details of task: On Moodle The period 2010-2012 saw the introduction by the US Federal Aviation Administration of a LOV for civil transport aircraft. Explain in detail: i) ii) iii) iv) v) vi) vii) viii) ix)

What is meant by the term LOV ? Why was it introduced and how is it expected to help continued airworthiness ? How was it introduced and is it mandatory ? Are there any exceptions to this rule ? How is the LOV of an aircraft to be determined ? How, if at all, is the LOV linked to the operational environment in which the aircraft operates ? Can the LOV be extended and if so how ? Is it limited to civil transport aircraft ? The FAA documentation in which LOV is discussed uses the terms WFD, MSD and MED. What do these terms mean. Give examples.

Please answer the question in about 1,500 words essay format with subject headings. Release dates (where applicable): Word limit (where applicable): Value: 15% Assessment Task 3: As discussed in class weight functions are dependent on geometry. This Assignment requires you to use the 2D edge crack weight function to obtain solutions for a crack at a hole in a wide plate with the geometry as described below and compare it to the Bowie solution and the solution obtained using the Kujawski approximation solution. Due Date: Friday week 10


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Details of task: On Moodle and given below As discussed in class weight functions are dependent on geometry. This Assignment requires you to use the 2D edge crack weight function to obtain solutions for a crack at a hole in a wide plate with the geometry as described below and compare it to the Bowie solution and the solution obtained using the Kujawski approximation solution. Due Date: Friday week 10 Details of task: On Moodle and given below As discussed in class weight functions are dependent on geometry. This Assignment requires you to use the 2D edge crack weight function to obtain solutions for a crack at a hole in a wide plate with the geometry as described below and compare then to the ASTM solution. For edge crack problems mI can be approximated by: mI(x, l) = (2/(2(a-x))) {(1+M1(1-x/a) 1/2 + M2(1-x/a) + M3(1-x/a) 3/2}

(1)

where

Here w is the width of the section which contains the crack. Use the (above) edge crack weight function to obtain the stress intensity factors for 0.5, 1, 5 and 10 mm through thickness edge cracks, i.e. 2D edge cracks, that emanate from a centrally located 10 mm diameter hole in a 2 mm thick and 500 mm by 500 mm square panel wing (skin) panel. The principle stresses in the panel are approximately 200 and 0 MPa and the crack(s) lies at 90 degrees to the maximum principle stress (200 MPa). Compare your answers to the solution given in the lecture notes to the Bowie solution and also to the solution obtained using the Kujawski approximation for this problem. The Bowie beta factor solution for the stress intensity factor for a through crack of length “a” at a hole of radius r in a wide plate which can be approximated by:


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K = β(a,r) σ(a) β(a,r) = 0.6762 + [0.8733/(0.3245 +(a/r))] Release dates (where applicable): Word limit (where applicable): Value: 10% Presentation requirements: Assessment Task 4: Summarise the engineering problems that led to Ansett Australia grounding their Boeing 767 fleet in 2000 and 2001 and to the subsequent demise of Ansett Australia. Due Date: Friday week 12. Details of task: On Moodle Assessment Four is a written assignment worth 15%. The Assessment is due for submission as a hard copy in the MAE 4408 assessment box located in the foyer, Building 31 no later than Friday of Week 12. Results will be published along with the results for MAE 4408. Please answer the question in about 1,500 words essay format with subject headings. Release dates (where applicable): Word limit (where applicable): Value: 15%


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