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MECHANISM-BASED MODELS OF FATIGUE CRACK GROWTH

DISSERTATION

Presented in Partial Fulfillment of the Requirements For the Degree Doctor of Philosophy in the Graduate School of The Ohio State University

By Sassan Steven Shademan, B.S, M.S

*****

The Ohio State University 2000

Dissertation Committee: Professor W.O. Soboyejo, Advisor Approved by Professor A.B.O. Soboyejo, Co-Advisor Professor Hamish Fraser Professor Glenn Daehn Advisor Professor Susan Olisek (Graduate School Rep) Department of Material Science and Engineering UMI Number: 9994937

Copyright 2001 by Shademan, Sassan Steven

All rights reserved.

UMi

UMI Microform 9994937 Copyright 2001 by Bell & Howell Information and Learning Company. All rights reserved. This microform edition is protected against unauthorized copying under Title 17, United States Code.

Bell & Howell Information and Leaming Company 300 North Zeeb Road P.O. Box 1346 Ann Arbor, Ml 48106-1346 ABSTRACT

In recent years, investigators in the area of fatigue have realized the importance of alternative empirical equations to the widely accepted Paris equation. However, most of the existing crack growth equations are still restricted to cases where fatigue crack growth is dependent on only one or two variables. As a result, most of the efforts to develop life prediction methodologies have been limited by the lack of fatigue laws that can account for more than one or two variables. In this work, a single multiparameter equation is introduced that can asses the combined effects of mechanical, physical and microstructural variabilities on fatigue crack growth rate of TI-6AI-4V alloys. The multiparameter equation showed that, for mill annealed Ti-6A1-4V microstructures, specimen thickness and test frequency have minimal effects on fatigue crack growth rate, while AK and Kmax are the most important contributors to the crack growth rate. For Widmanstatten microstructures, the model shows that the coarser the colony size, the lower the fatigue crack growth rate.

Also, the existing empirical fatigue crack growth equations are only applicable over parametric ranges of crack driving force that are within the bounds where the expressions provide good fits to measured data. However, It is well known that the mechanisms of fatigue crack growth may vary significantly over the parametric ranges of crack driving force. There is, therefore, a need for a simple physically-based model that considers the essential features of fatigue crack growth. In this work, A physically-based model is presented for the prediction of fatigue crack growth in Ti-6AI-4V alloys. The model assumes that the crack extension per cycle is directly proportional to the change in the crack-tip opening displacement, CTOD, during cyclic loading between the maximum and minimum stress intensity factor. A simple power law equation is then derived for the prediction of fatigue crack growth as a function of the stress intensity factor range. The model is validated for fatigue crack growth in the near­ threshold, Paris and high-AK regimes, for both mill annealed and Widmanstatten Ti-6A1-4V microstructures.

Anomalous short crack growth behavior is a problem in Ti-6AI-4V. However, the basic causes of anomalous short crack growth in Ti-6AI-4V are not well understood. This lack of understanding may lead to a non-conservative estimates of the total fatigue lives of engineering structures and components fabricated from Ti-6AI-4V. It may also lead to unpredicted or unpredictable failures. In this work, microstructurally short fatigue cracks are shown to grow at stress intensity factors below the long crack fatigue threshold in Widmanstatten Ti-6AI-4V microstructures. Anomalously high fatigue crack growth rates and crack growth retardation is also shown to occur in the short crack regime at stress ratio of 0.1. The fast crack growth rates are associated with partially developed crack wakes that result in lower levels of plasticity-induced closure. These result in higher effective stress intensity factor ranges and faster fatigue growth rates in the short crack regime. The work also shows that cracks are retarded when they interact with prior p grain boundaries in fine and intermediate Widmanstatten microstructures. Similar crack growth retardation phenomena are observed at colony boundaries in coarse Widmanstatten microstructure. The slower crack growth rates are attributed to the retarding effects of the boundaries and the crack deflection that is generally associated with crack growth beyond the retarding boundaries.

Ill Dedicated to my wife

IV ACKNOWLEDGMENTS

I would like to thank my, Dr. Wole Soboyejo and my co-advisor, Dr. Alfred Soboyejo for intellectual support, encouragement, and enthusiasm throughout the course of this research program. Without the efforts on their part, this dissertation would not have been completed.

I would also like to thank my dissertation committee members. Dr. Hamish Fraser and Dr. Glenn Daehn. The courses taught by them during my graduate study at Ohio State University were also useful in this research.

I thank my colleagues. Dr. Chris Mercer and Dr. Vikas Sinha for helping me get started with my experiments. Useful technical discussions with them are also acknowledged.

I also with to thank the technical staff members, Mr. Lloyd Barnhart, Mr. Ken Kushner, Mr. Steve Bright, Mr. Gary Dodge and Mr. Cameron Begg at Ohio State for providing assistance with the experimental work. VITA

October 18, 1959 Bom-Shahpoor, Iran

1988 B.S. Welding Engineering The Ohio State University, Columbus, OH

1991 M.S. Welding Engineering The Ohio State University, Columbus, OH

1994 M.S. Materials Science and Engineering The Ohio State University, Columbus, OH

1998-present Graduate Research Associate The Ohio State University, Columbus, OH

PUBLICATIONS

1. A. B. 0. Soboyejo, S. Shademan, M. Foster, N. Katsube and W. O. Soboyejo, “A multiparameter Approach to the Prediction of Fatigue Crack Growth in Metallic Materials”, submitted.

2. S. Shademan, A. B. O. Soboyejo, J. F. Knott and W. O. Soboyejo, “A Physically-Based Model for the Prediction of Long Fatigue Crack Growth in Ti-6AI-4V”, submitted.

3. V. Sinha, S. Shademan and W. O. Soboyejo, “Effect of Colony Size on Long Fatigue Crack Growth Behavior in Ti-6AI-4V”, To be submitted.

4. A. B. 0. Soboyejo, S. Shademan, V. Sinha and W. O. Soboyejo, “Statistical Modeling of Microstructural Effects on Fatigue Behavior of cx/p Titanium Alloys”, To be submitted.

5. J. C. Lipplod, S. Shadem an and W. A. Baeslack in, “The Effect of Specimen Strength and Thickness on Cracking Susceptibility During the Sigmajig Weldability Test", The Welding Journal, March 1996, pp. 81-92.

VI FIELD OF STUDY

Major Field: Materials Science and Engineering

vu TABLE OF CONTENTS

Abstract.

Dedication.

Acknowledgments.

Vita...... vi

List of Tables ...... xiii

List of Figures ...... xv

Chapters:

1. introduction ...... 1 1.1 Background ...... 1 1.2 R eferences...... 4

2. Literature Review...... 6

2.1 Fundamentals of Fracture Mechanics ...... 6 2.1.1 Introduction ...... 6 2.1.2 Griffith Fracture Theory ...... 8 2.1.3 Strain Energy Release Rate (G) ...... 8 2.1.4 Linear Elastic Fracture Mechanics and the Stress Intensity Factor (K) ...... 8 2.1.5 Plastic Zone Size Under Monotonie and Cyclic Loading Conditions ...... 10 viii 2.2 Fundamentals of Fatigue ...... 11 2.2.1 introduction ...... 11 2.2.2 Micromechanisms of Fatigue Crack Initiation and Propagation ...... 12 2.2.3 Fatigue Crack Propagation ...... 12 2.2.4 Crack Closure in Fatigue ...... 14 2.2.5 Crack Closure Measurement ...... 15 2.2.5.1 Sources of Crack Closure ...... 16 2.2.6 Short Crack Anomalies ...... 17 2.2.7 Multiparameter Approach to Fatigue Crack Growth ...... 19 2.2.8 Physically-Based Approach to Fatigue Crack Growth ...... 20

2.3 Titanium Alloys and Their Fatigue Behavior ...... 22 2.3.1 Physical Metallurgy of Titanium Alloys ...... 22 2.3.1.1 Introduction ...... 22 2.3.1.2 Origin and Basic Applications of Titanium ...... 22 2.3.1.3 Classification of Titanium Alloys ...... 23 2.3.1.4 Phase Transformations in (a+P) Titanium Alloys ...... 23 2.3.2 Fatigue and Fracture Behavior of (a+P) Titanium Alloys ...... 24 2.3.2.1 Introduction ...... 24 2.3.2.2 Effects of Microstructure ...... 25 2.3.2.3 Effects of Stress Ratio, R ...... 28 2.3.2.4 Effect of the Ratio of Specimen Thickness to Colony Size ...... 28 2.3.2.5 Mechanisms of Fatigue Crack Growth ...... 29 2 3.2.6 Crack Closure Studies in (a+p) Titanium Alloys ...... 31 2.2.21 Short Crack Studies in (a+p) Titanium Alloys ...... 32

2.4 References...... 34

3. Material and Experimental Procedures ...... 61 3.1 Material...... 61

ix 3.2 Experimental Procedures ...... 61 3.2.1 Long Crack Fatigue Experiments ...... 61 3.2.2 Short Fatigue Crack Growth Experiments ...... 65 3.2.3 Fractography ...... 66 3.2.4 Statistical Analysis ...... 66 3.3 References...... 67

4. Fatigue Crack Growth Behavior of Mill-Annealed Ti-6AI-4V ...... 70 4.1 Introduction ...... 70 4.2 Effects of Stress Ratio and Specimen Thickness on Fatigue Crack Growth R ates...... 70 4.3 Fatigue Fracture Modes ...... 72 4.4 Multiparameter Modeling of Fatigue Crack Growth ...... 73 4.4.1 Evolution of Multiparameter Model ...... 73 4.4.2 Results and Discussion of the Multiparameter Modeling ...... 78 4.4.3 Fatigue Life Prediction Using Multiparameter Approach...... 82 4.5 Conclusions ...... 83 4.6 References...... 85

5. Fatigue Crack Growth Behavior in p-Annealed Widmanstatten Microstructures ...... 100 5.1 Introduction ...... 100 5.2 Heat Treatments and Phase Transformations ...... 100 5.3 Fatigue Crack Growth Rate Behavior of the Three Widmanstatten Microstructures ...... 101 5.3.1 Comparisons of the Fatigue Crack Growth Rates of the Three Widmanstatten Microstructures (Containing Almost the Same Volume % of P Phase) and Mill-Annealed Microstructure ...... 103 5.3.2 Comparisons of the Fatigue Crack Growth Rates of the Widmanstatten Microstructures (Containing Different Volume % of p Phase) and Mill-Annealed Microstructures ...... 104 5.4 Crack/Microstructure Interactions ...... 104

X 5.5 Fatigue Fracture Modes ...... 105 5.6 Modeling of Roughness-Induced Crack Closure ...... 107 5.7 Multiparameter Model Containing both mechanical and Microstructural Variables...... I l l 5.8 Conclusions ...... 114

5.9 References...... 116

6. A Physically-Based Model For the Prediction ofLong Fatigue Crack Growth ...... 149 6.1 Introduction ...... 149 6.2 Derivation of the Physically-Based Model ...... 150 6.3 Application of the Physically-Based Model to Mill-Annealed and Widmanstatten Microstructures ...... 152 6.4 Fatigue Fracture Mechanism Maps ...... 154 6.5 Conclusions ...... 156 6.6 References...... 158

7. Short Crack Behavior in Widmanstatten T1-6AI-4V Microstructures ...... 171 7.1 Introductions ...... 171 7.2 Comparison of Short and Long Crack Fatigue Behavior ...... 171 7.3 Fatigue Fracture Modes ...... 172 7.4 Conclusions ...... 174 7.5 References ...... 175

8. Suggestions for Future Work ...... 184 8.1 R eferences ...... 186

BIBLIOGRAPHY...... 187

XI LIST OF TABLES

Table Page

1. Classification of titanium-base alloys [38,40] ...... 44

2. Compositions of the a and (3 phases in a Widmanstatten Ti-6242 alloy [38]. . .. 46

3. Mechanical properties of mill-annealed Ti-6AI-4V ...... 68

4. Summary of exponents and correlation coefficients for mill-annealed TI-6AI-4V...... 98

5. Summary of the correlation coefficients for mill-annealed Ti-6AI-4V ...... 98

6. Summary of multiparameter exponents and correlation coefficients for HY 80 steel and Inconel 718 ...... 99

7. Variations in the microstructural parameters and the mechanical properties as a function of heat treatment [8] ...... 120

8. Average calculated dimensions of the deflected segments on the fracture surface of the three Widmanstatten microstructures ...... 146

9. Summary of the cross correlation between the microstructural variables for the Widmanstatten Ti-6AI-4V microstructures ...... 148

10. Summary of the coefficients and correlation coefficients for the physically-based fatigue crack growth rate model for the mill-annealed TÎ-6AI-4V microstructures ...... 163

11. Summary of the coefficients and correlation coefficients for the

xii physically-based fatigue crack growth rate model for the mill-annealed Ti-6A1-4V microstructures ...... 163

12. Summary of the coefficients and correlation coefficients for the physically-based fatigue crack growth rate model using (a) AK and (b) AKgff for the Widmanstatten Ti-6A1-4V microstructure A (fine microstructure) ...... 164

13. Summary of the coefficients and correlation coefficients for the physically-based fatigue crack growth rate model using (a) AK and (b) AKgf, for the Widmanstatten Ti-6A1-4V microstructure B (intermediate microstructure) ...... 165

14. Summary of the coefficients and correlation coefficients for the

physically-based fatigue crack growth rate model using (a) AK and (b) AKsb for the Widmanstatten Ti-6A1-4V microstructure C (coarse microstructure) 166

15. The stress intensity factor range, AK, and the corresponding crack length at which transition occurs between the short and long crack regime for the three Widmanstatten microstructures ...... 183

xui LIST OF FIGURES

Figure Page

1. A large plate ef an elastic material of a thickness, B, containing a center crack of length, 2a, subjected to an applied stress a ...... 39

2. The three modes of fracture, (a) Mode I (crack opening mode), (b) Mode II (sliding mode) and (c) Mode III (tearing mode) ...... 39

3. Schematic illustration of the different regimes of stable fatigue crack propagation ...... 40

4. Schematic illustration of the plastic blunting model, (a) zero load, (b) small tensile load, (c) peak tensile load, (d) onset of load reversal, (e) peak compressive load, (f) small tensile load in the subsequent tensile cycle [18] ...... 40

5. A schematic illustration of the relationship between the applied stress and the measured displacement [19]...... 41

6. The offset procedure for closure m easurem ents ...... 41

7. Schematic illustration of (a) plasticity induced closure, (b) oxide induced closure, (c) roughness induced closure and (d) phase transformation induced closure ...... 42

8. A schematic representation of the typical fatigue crack growth behavior of long and short cracks [25] ...... 42

9. A schematic of sub-threshold growth and transient retardation characteristics

of a microstructurally small fatigue crack (dashed line): dg 2>dg, [25] ...... 43

xiv 10. Hard sphere models of basic crystal structures of (a) hep and (b) bcc ...... 43

11. Typical microstructures of a, a+(3 and (3 titanium alloys, (a) Equiaxed a in unalloyed titanium, (b) Equiaxed a+p, (c) Acicular a+p and (d) Equiaxed p [38] ...... 44

12. Typical microstructure of mill annealed a+p titanium alloy (Ti-6AI-4V)[39j ...... 45

13. X-ray analysis of the compositional distribution across the (a+P) Widmanstatten structure in Ti-6AI-2Sn-4Zr-2Mo-0.1Si [42] ...... 45

14. Fatigue crack growth rates for the mill annealed (MA) and p annealed (BA) microstructures [45] ...... 47

15. Fatigue crack growth behavior of the (a+P) Widmanstatten microstructures with different packet sizes [47] ...... 48

16. Schematic comparison of the computed cyclic plastic zone size to the average Widmanstatten packet size [47] ...... 49

17. Influence of clustering on the sharpness of the transition observed in the fatigue crack growth rate curves for the Widmanstatten microstructure.

mB/mA is the ratio of the slopes for a K>a Kt and AK

18. Fatigue crack path profiles at low AK levels for Widmanstatten Ti-6AI-4V alloy heat treated a s (a) Argon quenched, (b) cooled at 30®C/min, (c) cooled at 6°C/min, and (d) cooled at 2®C/min from the p solutionizing temperature [46]...... 51

19. Fracture morphology at low AK levels for Widmanstatten Ti-6AI-4V alloy heat treated as (a) Argon quenched, (b) cooled at 30°C/min, (c) cooled

XV at 6°C/min, and (d) cooled at 2°C/min from tfie (3 solutionizing temperature temperature [46] ...... 52

20. Fatigue crack growtfi rate curves obtained at (a) R-ratio = 0.35 and (b) R-ratio = 0.12 for Ti-6AI-4V alloy in martensitic, mill annealed (as received), transformed p, and p annealed (Widmanstatten microstructure) conditions [43] ...... 53

21. Influence of colony sizes on fatigue crack growtti rate, at constant level of AK, a scfiematic representation [55] ...... 54

22. Contrast in (a) structure sensitive and (b) structure-insensitive modes of crack growtti in a Widmanstatten (a+P) titanium alloy [47]...... 55

23. TEM micrograph) stiowing a portion of the crack tip unzipping along the intersecting slip bands in mill annealed Ti-6AI-4V alloy [51] ...... 56

24. (a) Fatigue crack growth rate for mill annealed Ti-6AI-4V alloy obtained at different stress ratios, and (b) Closure corrected fatigue crack growth rate for the same alloy at different stress ratios [51] ...... 57

25. The variation of the crack tip opening displacement, 5t, as a function of crack length in a Ti-6AI-4Zr-2Sn-6Mo alloy [58] ...... 58

26. Comparison of long and short fatigue crack growth rates at stress ratio of R = 0.1 [62]...... 59

27. Comparison of short and long fatigue crack growth rates with the long crack growth rates and also with the closure-corrected long crack growth rates (i.e. da/dN versus AKetr plots)[62] ...... 60

28. Triplicate optical micrograph of the mill annealed T1-6A1-4V forging ...... 68

xvi 29. Schematic illustration showing the location of the current and potential drop wires spot welded to the single edge notched (SEN) specimens ...... 69

30. (a) Procedure for precraking of single edge notched (SEN) specimens: (b) the starter notch is removed using electrical discharge machine (EDM), and the surface of the sample is ground using fine emery paper so as to obtain a precrack length comparable to the grain size for this microstructure 69

31. Fatigue crack growth rates versus stress intensity factor range, AK, for specimens with thicknesses of: (a) 12.7 mm, (b) 6.4 mm, (c) 3.2 mm and (d) 1.6 mm...... 88

32. Fatigue crack growth rate versus stress intensity factor range, AK, for stress ratios, R. of: (a) 0.02, (b) 0.25, (c) 0.50 and (d) 0.80 ...... 90

33. Fatigue crack growth rate versus (a) stress intensity factor range, AK, and (b) effective stress intensity factor range, AKg^ ...... 92

34. Plots of Elber closure ratios for the 3.2 mm and 6.4 mm thick specimens tested at stress ratios of 0.02 and 0.25 ...... 94

35. Crack growth morphology of the 3.2 mm thick specimen tested at stress ratio of 0.02 at low AK. intermediate AK and high AK. Striations and ductile dimples are denoted by 8 and D, respectively ...... 95

36. Crack growth morphology of the 12.7 mm thick specimen tested at stress ratio of 0.50 at low AK. intermediate AK and high AK. Striations and ductile dimples are denoted by S and D. respectively ...... 96

37. Goodness of fit between ln(da/dN)-predicted and ln(da/dN)-measured using the multi-variate model ...... 97

X V ll 38. Goodness of fit between (da/dN)-predicted and {da/dN)-measured using the multi-variate model ...... 97

39. Relevant portion of Ti-6A1-4V phase diagram [7] ...... 118

40. Microstructural modifications after heat treatments, (a) 1070 °C for 30 mln. cool @ 25 °C/min. to 400 ®C/Ar quench to RT, (b) 1070 °C for 30 mln. cool @ 3 °C/min. to 400 °C/Ar quench to RT, and (c) 1070 °C for 30 min. cool @ 3 °C/min. to 400 °C/Ar quench to RT, and (c) 1070 °C for 30 min. cool @ 1 °C/min. to 400 °C/Ar quench to RT ...... 119

41. Fatigue crack growth rate versus stress intensity factor, AK, at stress ratio of 0.1, 0.25, 0.50, and 0.80 for (a) Widmanstatten Ti-6AI-4V microstructure A, (b) W idmanstatten Ti-6Ai-4V microstructure B, and (c) Widmanstatten Ti-6AI-4V microstructure C ...... 121

42. Closure-corrected fatigue crack growth rate at stress ratios of 0.1 and o.8 for (a) Widmanstatten microstructure A, (b) Widmanstatten microstructure B, and (c) Widmanstatten microstructure 0 ...... 123

43. Effects of microstructural variables on Fatigue Crack Growth Rates in W idmanstatten Ti-6Ai-4V. (a) FCGR as function of AK at R = 0.1, (b) FCGR as function of AK at R = 0.25, (c) FCGR as function of AK at R = 0.50,and (d) FCGR as function of AK at R = 0.80 ...... 125

44. Effects of microstructural variables on Fatigue Crack Growth Rates in Widmanstatten Ti-6AI-4V. (a) FCGR as function of AKgff at R = 0.1, (b) FCGR as function of AKeff at R = 0.80 ...... 127

45. Comparison of Fatigue Crack Growth Rates in Widmanstatten Ti-6AI-4V (P - 22 Vol%) and mill annealed microstructures, (a) FCGR as function of AK at R = 0.1, (b) FCGR as function of AK at R = 0.25, (c) FCGR as xviii function of AK at R = 0.50, and (d) FCGR as function of AK at R = 0.80 ...... 128

46. Comparison of Fatigue Crack Growth Rates in Widmanstatten Ti-6A1-4V (P - 22 Vo!%) and mill annealed microstructures, (a) FCGR as function of AKeff at R = 0.1, and (b) FCGR as function of AKeff at R = 0.80 ...... 130

47. Comparison of Fatigue Crack Growth Rates in Widmanstatten Ti-6AI-4V and mill annealed microstructures, (a) FCGR as function of AK at R = 0.1, (b) FCGR as function of AK at R = 0.25, (c) FCGR as function of AK at R = 0.50, and (d) FCGR as function of AK at R = 0.80 ...... 131

48. Comparison of Fatigue Crack Growth Rates in Widmanstatten Ti-6AI-4V and mill annealed microstructures, (a) FCGR as function of AKetf at R = 0.1, and (b) FCGR as function of AKeff at R = 0.80 ...... 133

49. Fatigue crack interactions at (a) prior p grain boundaries and (b) colony boundaries for Widmanstatten microstructure A, tested at R = 0.1 ...... 134

50. Fatigue crack deflections in mill annealed microstructures ...... 135

51. Morphology of the fatigue fracture surfaces of the different microstructures observed at low magnification under stereo optical microscope. All specimens were tested at R = 0.1. (a) microstructure A (fine Widmanstatten), (b) microstructure B (intermediate Widmanstatten), (c) microstructure C (coarse Widmanstatten), and (d) microstructure D (mill annealed) [8] ...... 136

52. Fatigue fracture modes observed in the different crack growth regime of the Widmanstatten microstructure A (fine) at R = 0.1. (a) near threshold (AK - 11 Mpa v'm), (b) intermediate AK (AK - 18 Mpa Vm), and (c) high AK (AK ~ 27 Mpa Vm) ...... 137

XIX 53. Fatigue fracture modes observed in ttie different crack growth regime of the Widmanstatten microstructure A (fine) at R = 0.8. (a) near threshold

(AK - 5 Mpa Vm), (b) intermediate AK (AK ~ 10 Mpa Vm). and (c) high

AK (AK ' 14 Mpa Vm )...... 138

54. Fatigue fracture modes observed in the different crack growth regime of the Widmanstatten microstructure B (intermediate) at R = 0.1. (a) near threshold (AK - 11 Mpa Vm), (b) intermediate AK (AK ~ 15 Mpa Vm),

and (c) high AK (AK ~ 25 Mpa Vm)...... 139

55. Fatigue fracture modes observed in the different crack growth regime of the Widmanstatten microstructure B (intermediate) at R = 0.8. (a) near threshold (AK ~ 5 Mpa Vm), (b) intermediate AK (AK - 9 Mpa Vm),

and (c) high AK (AK ~ 13 Mpa Vm)...... 140

56. Fatigue fracture modes observed in the different crack growth regime of the Widmanstatten microstructure C (coarse) at R = 0.1. (a) near

threshold (AK ~ 12 Mpa Vm), (b) intermediate AK (AK - 18 Mpa Vm),

and (c) high AK (AK ~ 25 Mpa Vm)...... 141

57. Fatigue fracture modes observed in the different crack growth regime of the Widmanstatten microstructure C (coarse) at R = 0.8. (a) near

threshold (AK ~ 6 Mpa Vm), (b) intermediate AK (AK ~ 10 Mpa Vm),

and (c) high AK (AK - 15 Mpa Vm)...... 142

58. Idealization of a small segm ent of a crack with periodic tilts [14] ...... 143

59. Schematic representation of a deflected crack in fully opened condition at the peak load of fatigue cycle (on the left) and relative mismatch between the fracture surfaces at the point of first contact during unloading (on the right) [14] ...... 143

XX 60. Lateral profile of the deflected segments on the fracture surface of (a) Widmanstatten microstructure A, (b) Widmanstatten microstructure B and (c) Widmanstatten microstructure C ...... 144

61. Comparison of the originai closure-corrected fatigue crack growth rate data using a clip gage, and the growth rate data using the roughness- induced closure model at mismatch values of 0.25 and 0.50 for (a) Widmanstatten microstructure A, (b) Widmanstatten microstructure B and (c) Widmanstatten microstructure 0 ...... 146

62. Schematic representation of the extent of irreversibility, p, during cycling between Kmax and Kmm...... 161

63. Schematic representation of closure stress intensity factor, K^, and effective stress intensity factor, Kgm ...... 161

64. Schematic representation of the near threshold (regime I), Paris (regime II) and high AK (regime III) regions on plot of da/dN versus AK ...... 162

65. Fatigue mechanism maps for (a) mill annealed, (b) Widmanstatten microstructure A (fine), (c) Widmanstatten microstructure B (intermediate) and (c) Widmanstatten microstructure C (coarse) ...... 167

66. Typical crack/microstructure interactions in the short and long crack regimes for (a) Widmanstatten microstructure A, (b) Widmanstatten microstructure B and (c) Widmanstatten microstructure C ...... 176

67. Comparison of long and short fatigue crack growth rates at stress ratio of 0.1 for (a) Widmanstatten microstructure A, (b) Widmanstatten microstructure B and (c) Widmanstatten microstructure 0 ...... 178

68. Dependence of fatigue fracture modes on crack length, a, and stress

xxi intensity factor range, AK, at stress ratio of 0.1 for W idmanstatten microstructure A. (a) a ~ 75 pm, AK ~ 7.5 MPaVm, (b) a - 200 pm, AK ~ 9.15 MPaVm and (c) a - 350 pm, AK ~ 12.5 MPaVm...... 180

69. Dependence of fatigue fracture modes on crack length, a, and stress intensity factor range, AK, at stress ratio of 0.1 for Widmanstatten microstructure B. (a) a ~ 80 pm, AK - 4.30 MPaVm, (b) a - 300 pm. AK ~ 7 MPaVm and (c) a - 500 pm, AK ~ 10.5 MPaVm ...... 181

70. Dependence of fatigue fracture modes on crack length, a, and stress intensity factor range, AK, at stress ratio of 0.1 for Widmanstatten

microstructure C. (a) a - 100 pm, a K - 5 MPaVm, (b) a - 450 pm, AK - 10 MPaVm and (c) a ~ 900 pm, AK ~ 14 MPaVm...... 182

X X ll CHAPTER 1

INTRODUCTION

1.1 Background:

Since the pioneering work of Paris and co-workers [1] about forty years ago, the use of Paris equation has gained widespread acceptability. Their early work was particularly important because it was the first to recognize the direct relationship between the fatigue crack growth rate and linear elastic fracture mechanics parameters based on stress intensity factor.

Recently investigators have realized the importance of alternative empirical equations, which combines the effect of multiple variables on fatigue crack growth rate. One proposed equation [2] assesses the combined effects of stress intensity factor range, AK, and stress ratio, R, on fatigue crack growth rate, while another equation [3], estimates the fatigue crack growth rate as a function of AK and the maximum stress intensity factor, Kma%- However, most of the existing crack growth equations are still restricted to cases where fatigue crack growth is dependent on only one or two variables. As a result, there is still a need for a single multiparameter equation that can asses the combined effects of mechanical, physical and microstructural variabilities on fatigue crack growth rate.

Furthermore, fatigue crack growth rate equations are typically empirical in nature [4], like Newton’s laws of motion. They are based on mathematical expressions that provide good fits to experimental data. Therefore, fatigue crack growth equations are only applicable over parametric ranges of crack driving force that are within the bounds where the expressions provide good fits to measured data. However, it is well known that the mechanisms of fatigue crack growth may vary significantly over the parametric ranges of crack driving force [5-6]. Also, the crack-tip deformation [7-9] and chemisorption [10-12] processes are unlikely to be fully described by empirical expressions that do not assess the chemical and physical

I processes involved in the extension of a crack during a single fatigue cycle. However, modeling of all the physical/chemical processes that occur during fatigue crack growth is not possible at the current time. Furthermore, it is recognized that our existing understanding of fatigue crack growth processes is still insufficient for the development of models that will include the full details of all the relevant physical/chemical processes. There is, therefore, a need for simpler physically-based models that consider the essential features of fatigue crack growth.

The material used in this work is the Ti-6AI-4V alloy. This alloy has been extensively used in the aerospace industry (in both air frame and engine components) and in the chemical and biochemical industries where the required high strength to weight ratios and corrosion resistance make it the material of choice. However, although the Ti-6AI-4V alloys was developed more than fifty years ago, the mechanisms of fatigue failure in this alloy are not fully understood.

The need for the improved understanding of the fatigue of Ti-6AI-4V has been highlighted in recent years by recent high-cycle fatigue problems in the U.S. air force fleet [13] and the Sioux City, Iowa, crash that was attributed to crack nucléation from hard a phase in cast Ti- 6AI-4V [14]. T hese two examples highlight the limited extent of our current understanding of the fatigue behavior of Ti-6AI-4V. In the case of the high-cycle fatigue problem, it has resulted in a multi-million dollar U.S. air force research program designed to develop new tools for the modeling of high-cycle fatigue of Ti-6AI-4V. Similarly, in the case of the hard a problem, the Federal Aviation Authority has invested heavily in research programs designed to evaluate the role of hard a phases in the nucléation and growth of fatigue cracks in Ti-6AI-4V.

Nevertheless, our current understanding of fatigue damage processes in Ti-6AI-4V is still limited. Furthermore, most of the efforts to develop life prediction methodologies for Ti-6AI-4V have been partly limited by the lack of fatigue laws that can account for more than one or two variables. Since fatigue processes are often controlled by more than two variables [2-3], there is a need to develop multiparameter fatigue crack gro’jvth laws for the modeling of fatigue damage. There is also a need to explore the effects of microstructural variables on fatigue crack growth in Ti-6AI-4V. This dissertation presents the results of a systematic study of the mechanisms and mechanics of fatigue crack growth in Ti-6A!-4V. The dissertation is divided into seven chapters. Following the introduction, a detailed literature review on ct/p titanium alloys, fracture mechanics and fatigue is presented in chapter 2. The material and microstructures are then described in chapter 3 before exploring the effects of multiple mechanical variables on fatigue crack growth in mill annealed Ti-6A1-4V in chapter 4. The effects of Widmanstatten microstructural parameters on fatigue crack growth are then considered in chapter 5. A physically-based model for fatigue crack growth is presented in chapter 6, along with fatigue maps. The anomalous growth of short fatigue cracks is considered in chapter 7 before summarizing the suggestions for future work in chapter 8. 1.2 References:

1. P.C. Paris, M. Gomez and W. E. Anderson (1961) A rational anaiytic theory of fatigue. Trend Engng, Vol. 13, p. 9-14.

2. R. G. Forman, V. E. Kearney and R. M. Engle (1967) Numerical analysis of crack propagation in cyclic loaded structures. J. Basic Engng, Vol. 89, p. 459-464.

3. N. Walker, 1970, The effect of stress ratio during crack propagation and fatigue for 2024- T3 and 7075-T6 aluminum. In Effects of Environment and Complex Load History for Fatigue life, special technical publication 462, p. 1-14. Philadelphia: American Society for Testing and Materials.

4. P. C. Paris and F. Erdogan (1963) A critical analysis of crack propagation laws. J. Basic Engng, Vol. 85, p. 528-534.

5. S. Suresh, 1998, Fatigue of materials, Cambridge University Press, 2"" Edition.

6. S. Suresh and R. O. Ritchie, 1984a, Near-threshold fatigue propagation: a perspective on the role of crack closure. In Fatigue Crack Growth Threshold Concepts. Editors, D. L. Davidson and S. Suresh, p. 227-261. Warrendale: The Metallurgical Society of the American Institute of Mining, Mineral and Petroleum Engineers.

7. P. J. E. Forsyth, 1961, A two stage process of fatigue crack growth. In Crack Propagation: Proceeding of Cranfield Symposium, p. 76-94. London: Her Majesty’s Stationary Office.

8. C. Laird, 1967, The influence of metallurgical structure on the mechanisms of fatigue crack propagation. In Fatigue Crack Propagation, Special Technical Publication 415, p. 131-168. Phiiadelphia: The American Society for Testing and Materials.

9. P. Neuman, 1969, Coarse slip model of fatigue. Acta metallurgica. Vol. 17, p. 1219-1225. 10. R. P. Wei, 1970, Some aspects of environmental-enhanced fatigue-crack growth. Engineering Fracture Mechanics, Vol. 1, p. 633-651. 11. R. M. N. Pelloux, 1969, Mechanisms of formation of ductile fatigue striations. Transactions of the American Society for Metals, Vol. 62, p. 281-285.

12. D. A. Meyn, 1968, Observations of micromechanisms of fatigue crack propagation in 2024 aluminum. Transactions of the American Society for Metals, Vol. 61, p. 42-51.

13. T. Nicholas and J. R. Zuiker, International Journal of Fracture, Vol. 80, p. 219-235, 1996.

14. B. Fenton, Federal Aviation Authority, Atlantic City, New Jersey, Private Communication, 1998. CHAPTER 2

LITERATURE REVIEW

This literature review consists of two sections. The first section presents the fundamentals of fracture and fatigue behavior in metallic materials. This includes the basic concepts and terminology used in fracture, and a summary of empirical and mechanistic approaches to fatigue crack initiation and propagation. The concept of crack closure is reviewed along with experimental methods for the measurement of crack closure. Anomalous short crack behavior, and the major factors contributing to the short crack behavior are also discussed. Two newly proposed statistical and physically-based fatigue crack growth models are also reviewed. In the second section, the fatigue behavior of (a+P) titanium alloys is reviewed. The section focuses on the effects of microstructure on fatigue crack growth behavior. The results of prior studies of the effects of crack closure on fatigue crack growth behavior of the (a+P) titanium alloys are described.

2.1 Fundamentals of Fracture Mechanics:

2.1.1 Introduction:

In this section, the fundamentals of fracture and fatigue are reviewed. These include Griffith fracture theory, linear elastic fracture mechanics concepts, crack closure phenomena, short crack effects, and fracture mechanics approaches to the modeling of fatigue crack growth.

2.1.2 Griffith Fracture Theory:

Griffith [1] was the first person to propose a thermodynamic criterion for the prediction of unstable crack extension in a brittle solid. This was obtained in terms of a balance between potential and surface energies. He postulated that, for a unit crack extension to occur under influence of an applied stress, the decrease in potential energy of the body must be equal to the increase in surface energy due to crack extension (creation of two new surfaces with crack

6 extension). Using the stress anaiysis of Inglis [2] for an elliptical hole in an infinite body, Griffith deduced that the net change in potential energy for crack growth in a semi-infinitely large elastic plate (Figure 1 ) is given by;

where a, is the crack length, a is the applied stress, B is the plate thickness, and E' = E for plane stress, or E' = E /(1-v^) for plane strain conditions, and v is the Poisson’s ratio.

The surface energy of the crack in the plate in Figure 1 is given by:

fFj = (2) where Ys is the free surface energy per unit area. Griffith showed that if the total system energy is U = Wp+Ws, then the critical condition for the onset of crack extension is obtained when dU/da = 0. The critical stress for fracture is thus given by:

Griffith’s original formulation assum ed that the reduction in potential energy is balanced by the surface energy. It also laid the foundation for subsequent studies of the physics of fracture in brittle solids. However, in most engineering materials, plastic deformation occurs in the vicinity of the crack tip. To account for these non-linear processes, Orowan [3] introduced a plastic energy dissipation term, y,, to supplement the surface energy term in Equation 3. The resultant critical expression for fracture initiation is thus given by:

V m 2.1.3 Strain Energy R elease Rate (G):

The strain energy release rate, G, is defined as the rate of decrease of the total potential energy, PE, of the system with respect to crack length (per unit thickness of total crack front). This rate of reduction in PE is related to the strain energy release rate via:

where ‘a’ is the total crack length and B is the specimen thickness. Crack extension occurs when G is equal to the energy required for crack growth. In a brittle material, the energy required for crack growth is equal to the surface energy to form the new free surfaces plus the plastic energy dissipation term, yp. Hence, the Griffith fracture criterion can be expressed in terms of the strain energy release rate, G = (2ys+"/p). Thus, G is a m easure of the crack driving force, and the critical value of G is a measure of the inherent resistance of a material to crack propagation.

2.1.4. Linear Elastic Fracture Mechanics and The Stress Intensity Factor (K):

Irwin [4], in the mid 1950’s, quantified the near-tip stress fields for the linear elastic crack in terms of the stress intensity factor. Using analytical methods proposed originally by Westergaard [5], he showed that the near tip-stress fields can be expressed as:

a,: = ,— = fij (^ ) + higher terms in r (6)

In Equation 6, K is the stress intensity factor. It represents the magnitude of the crack-tip stress field under liner elastic conditions. The terms r and 0 represent the polar coordinates of points within the near-tip stress field, and fij(0) is a function of crack and specimen geometries. In an elastic body subjected to an applied stress, Oa, containing a through thickness crack, a, the stress intensity factor, K, can generally be expressed as:

8 K = Y a ^ y fm (7) where Y is a function of specimen and loading geometry. As shown in Figure 2, there are three modes of loading. Mode I (Figure 2a) is caiied the crack opening mode, it is generally induced by tensile stresses, (oa), that are normal to the crack faces. Mode II (Figure 2b) is called the sliding mode or the in-plane shear mode. The shear stress (xa) is applied in the plane of the advancing crack front. Mode III (Figure 2c) is called the tearing mode or out-of- plane shear mode. The shear stress (Qa) in this is applied to induce out-of-plane deformation in a direction that is parallel to the advancing crack front.

The stress field in Mode I is characterized by the Mode I stress intensity factor, K,, the Mode II field by K,,, and the Mode III field by Km. Therefore, depending upon the loading condition, the stress intensity factors K,, K„ and Km can be expressed as:

K, =Y,cr„Æ (8.1)

^11 " ^ ll^ a (8.2)

K-m = Y,„ yfm (8.3)

Using linear elastic fracture mechanics [6], it has been shown that both the strain energy release rate, G, and the stress intensity factor, K are measures of “crack driving force". They are related by:

K." G^ = (for plane stress conditions) and (9.1 )

G| (for plane strain conditions) (9.2) E

Similarly, it can be shown that for Mode n and Mode III: 9 . . y : Kr.ni G, = ( l - u ' ) —- and G,,, = ( l + u) (9.3)

The total energy release rate in combined mode cracking can easily be obtained by linear superposition of the energies associated with the different modes. This gives:

1-u- G = G, + G„ + G„, = (K;+K;+K.f„) (9.4)

Under plane stress loading conditions, the above expression for G reduces to:

G = (9.5)

2.1.5 Plastic Zone Size Under Monotonie and Cyclic Loading Conditions;

According to Equation 6. for r-»0 (at the very crack tip), the crack tip stresses approach infinity. In practice however, materials cannot support infinite stresses. In the c a se of metallic or ductile materials, a stress is reached at which plastic deformation occurs. This means that there is always a region around the crack tip in which plastic deformation occurs. For monotonie loading, under mode I conditions, the extent of the plastic zone, r,, may be estimated from:

(for plane strain conditions) and ( 10. 1)

(for plane stress conditions) ( 10.2)

The plastic zone size under cyclic loading is obtained by substituting the reversed yield stress, a = 2oy, into Equations 10.1 and 10.2. This gives: 10 1 (for plane strain conditions) and (10.3)

(for plane stress conditions) (10.4) ,

The cyclic plastic zone size Is, therefore, considerably smaller than the monotonie plastic zone size. Furthermore, the plastic zone size under plane stress conditions Is larger than the plastic zone size under plane strain conditions.

2.2 Fundamentals of Fatigue:

In this part of the literature review, the fundamentals of fatigue are reviewed. Fatigue crack Initiation and propagation are discussed. The concept of crack closure is discussed before describing the different types of short crack anomalies that can occur during the early stages of fatigue crack growth. This section presents the background Information required for the Interpretation of the literature review on the fatigue behavior of o/p titanium alloys In the next section.

2.2.1 Introduction:

In this section, the micromechanlsms of fatigue crack Initiation and propagation are reviewed. A framework Is also presented for the application of fracture mechanics to the prediction of fatigue crack growth and fatigue life. The effects of crack closure on fatigue crack growth are examined along with the different mechanisms of crack closure. Compliance-based crack closure measurement techniques are also described. Two recently proposed multiparameter and physically-based approaches to fatigue are then Introduced. Finally, the sources of short crack anomalies are described.

11 2.2.2 Micromechanisms of Fatigue Crack Initiation and Propagation:

During the 19“’ century, it was postulated that fatigue occurred due to the “crystallization” of materials under cyclic loading. The “crystallization" theory was laid to rest following the pioneering work of Ewing and Humfrey [7] during the early part of 20“’ century. Ewing and Humfrey [7] provided the first report of the microscopic changes associated with fatigue crack nucléation in metals. Using nominally smooth fatigue specimens, these investigators showed that fatigue deformation was associated with intense slip bands that were induced during cyclic deformation. These slip bands were later referred to as persistent slip bands (PSBs) by Thompson et.al. [8], who found that the slip bands persistently reappeared at the same sites in copper and nickel during continued cycling, even after a thin layer of the surface containing these bands was removed.

The origin of fatigue cracks in metals and alloys of high purity was first proposed by Wood [9]. He proposed that repeated cyclic straining lead to different amounts of net slip on different glide planes. This difference in net slip (due to slip irreversibility) along the slip planes results in roughening of the surface of the material. This surface roughening is manifested as microscopic PSB 'hills' (extrusions), ‘valleys' (intrusions) and protrusion at the free surface. Various investigators [10-13] have used optical interferometry to show that the strains within the PSB are highly inhomogeneous and localized at the PSB-matrix interface. As a result, these interfaces act as preferential sites for crack nucléation. Kinematic slip irreversibility, which is the strongly influenced by the test environment, is another dominant factor in promoting fatigue micro-crack formation in metals and their alloys. Investigators [14-15] have shown that fatigue microcrack formation occurs faster in air than in vacuum. This was attributed to the chemisorption of species such as oxygen and hydrogen (commonly found in air) to the freshly produced slip steps causing a greater extend of slip irreversibility. Forsyth [15] indicated that microcracks propagate predominantly by single shear, in the direction of the primary slip system termed stage I (mode II) crack growth. Continual cycling allows the microcracks to grow and eventually coalesce to form a dominant macrocrack.

12 2.2.3 Fatigue Crack Propagation:

When cyclic stresses are applied to a component, the linear elastic fracture mechanics characterization of the fatigue crack growth rate is based on the stress intensity factor range, AK, which is given by:

= ( 11)

where Kma% and Kmin are the maximum and minimum stress intensity factors in a fatigue cycle, respectively. Paris et al. [16] showed that the fatigue crack growth rate, da/dN, is related to the stress intensity factor range by the power law relationship:

^ = C(AK)- (12) aN

where C and m are material constants, and for metallic structural materials, the values of m are generally between 2 and 4 [16]. Fatigue crack growth rate data are most commonly represented on log-log plots of da/dN versus AK, as shown in Figure 3 in which three distinct regions can be identified. In regime A or the near-threshold regime (AK,h), the average growth increment per cycle is comparable to the lattice spacing (<10'® mm/cycle). Below AK^, cracks either remain dormant or grow at undetectable rates. In regime B (the Paris regime), a linear relationship is observed between log (da/dN) and log (AK). The Paris Equation (Equation 12) is only applicable in this regime. In regime C or the high AK regime, fatigue crack growth increases non-linearly with log AK.

As shown by Figure 3, fatigue crack growth rates in regimes A (threshold regime) are

lower than those predicted by Paris Equation, while in regime C (high a K regime) they are higher. The slower fatigue crack growth rates in the near-threshold regime are generally attributed to the effects of crack closure at low stress ratios. Also, in this region, the fracture

mode is predominancy cleavage-like type or “crystallographic" in morphology. In the high a K regime, the maximum stress intensity factor, Kmax, approaches the materials fracture

13 toughness, K,c. Hence it is now possible to induce “static" fracture modes in favorably oriented grains. Since an increasing umber of grains will become suitably oriented for “static" fracture as AK increases, an increasing incidence of static fracture modes is observed in this regime.

In addition to the fatigue fracture modes from regime B Forsyth et al. [17] has indicated that in some cases, the spacing between adjacent striations is correlated with the experimentally measured fatigue crack growth rate per cycle. One of the most acceptable models to rationalize the formation of fatigue striations is the plastic blunting model by Laird [18]. In this model, crack extension per fatigue cycle is envisioned as occurring due to the plastic blunting of the crack tip, as shown in Figure 4. During the loading portion of the fatigue cycle, the crack tip blunts and the crack extends due to duplex slip. During the load reversal portion of the fatigue cycle, the crack tip re-sharpens (Figure 4). Since the closure of the crack during the load reversal cannot fully reverse the blunting and the crack extension that occurred prior to load reversal, a net extension of the crack occurs during a fatigue cycle, leading ultimately to striations or surface markings. The process is repeated during the next cycle.

Since real components generally contain some inherent flaws, the Paris equation can be utilized to predict the fatigue life of the component. This can be achieved by separation of variables and integration to give:

j d a = Jc(AK)'".r/iV (13)

where a, and a, are the initial and final crack lengths, respectively, and N, is the number of cycles to failure. From Equation 13, the failure condition, which could be associated with either Nf or 3f can be predicted by integration.

2.2.4 Crack Closure In Fatigue:

The Paris equation (equation 12) shows that the stress intensity factor, AK, is the driving force for fatigue crack growth. The equation assumes that the fatigue crack remains fully open throughout the fatigue cycle. However, Elber [19] was first to recognize that fatigue cracks can 14 be closed for significant fractions of the fatigue cycle. Since the crack-tip stresses are reduced essentially to zero during crack face contact, Elber suggested that the closed portion of the fatigue cycle does not contribute to the effective crack driving force. As a result, he proposed that the crack driving force reduces from AK to AKg». Therefore, the Paris equation can now be expressed as:

In Equation 14, the effective stress intensity factor range, AKgm, is the difference between the maximum stress intensity factor, Kmax, and the stress intensity factor (in a fatigue cycle) at which the crack becomes fully open (Kop) during the fonward part of the fatigue cycle. Hence, AKetf can be written as:

(15)

2.2.5 Crack Closure Measurement:

Elber [19] measured the far field applied stress at which the crack remains fully open by monitoring the applied stress, a, and the displacement of the crack faces, 5, at some distance behind the crack tip, as shown in Figure 5. As the load is reduced from point A to point B, the slope of the curve remains constant. This slope is equal to the measured stiffness of an identical specimen having the same notch length as the fatigue crack. As the unloading continues from point B to points C, the curvature of the stress displacement curve changes, indicating that the crack length is changing, i.e. the crack is closing gradually. Beyond point C, the stress-displacement curve becomes linear and the slope of the line CD is equal to the stiffness of an identical notched specimen without the fatigue crack (line OE). This indicates that the fatigue crack is fully closed in this regime.

15 Figure 6 shows a typical compliance plot, which is presented in terms of the stress intensity factor, K, rather than the applied stress, cr. Upon loading in tension, the crack begins to open gradually until Kop is reached where the crack becomes fully open. During unloading, the first contact between the crack faces occurs at the closure stress intensity factor. Ko,. Below this point, the crack closes gradually. However, it is generally difficult to accurately determine Kop or Kd from the simple plots of load versus displacement. This is because of the difficulties associated with the determination of he exact point of transition from linearity. In order to accurately identify the Kc from the compliance curve, an offset displacement (6on) method is often used. The offset-displacement is defined as:

=G,(d'-orK) ( 16 )

where G, is the gain of the electronic circuit, 5 is the crack mouth opening displacement

usually measured by a clip gage, and a is the inverse of the slope of the linear portion of the

curve between Kma% and K^. For a fully open crack, = 0, and ô = or K. Also, when the crack begins to close, Ôp# takes on positive values.

2.2.5.1 Sources of Crack Closure:

Various investigators [19-23] have identified a range of fatigue crack closure mechanisms that are induced by a variety of mechanical, microstructural and environmental factors. These are illustrated by Figure 7. The most important sources of fatigue crack closure are:

fa) Plasticitv-lnduced Closure: In ductile alloys, as the fatigue crack propagates, residual tensile strains are left in the material behind the advancing crackfront, as only elastic recovery occurs after the creation of the fracture surfaces [19]. With an increase in the stress intensity factor and the size of plastic zone due to crack advance, the material which has previously been deformed permanently within the plastic zone now forms an envelope of plastic zones in the wake of the crackfront as shown in Figure 7. Closure occurs when the compressive stresses supplied by the underformed material surrounding the crack front causes premature contact of the crack faces.

16 (b) Oxide-Induced Closure: This source of closure is most dominant at elevated temperatures, aggressive environments and humid air. During propagation of fatigue crack at near-threshold AK levels and low stress ratios (R), the freshly formed fracture surface is oxidized due to presence of moist air. Also at these conditions, the possibility of repeated crack face contact during cycling is enhanced due to locally mixed-mode crack opening, microscopic roughness of the fracture surfaces and some plasticity induced closure. This results in continual breaking and reforming of the oxide scale, which promotes the build-up of a thick oxide layer behind the crackfront [20-21]. As a result, the possibility of crack face contact is increased since the maximum crack tip opening displacement at near-threshold (a small fraction of a um) is lower than the thickness of the oxide film.

(c) Roughness-Induced Closure: This type of closure is promoted due to discrete points of contact between fracture surface asperities [22-23]. The effect is dominant at low AK levels and low stress ratios [24]. At low AK levels, the plastic zone size is typically smaller than a characteristic microstructural dimension such as grain size. As a result crack propagation occurs by a single slip mechanism, which leads to a highly serrated (high volume of asperities) fracture morphology. At low stress ratios, the crack tip opening displacement is low which promotes a higher possibility of contact between the fracture surface asperities.

(d) Phase-Transformation Induced Closure: This type of closure is induced when phase transformations occur in the areas surrounding the crack tip, which leads to a net increase in the volume of the transforming region. As the crack advances, the enlarged material in the transformed zone is left behind the advancing fatigue crack tip, which promotes a net reduction in the crack tip opening displacement. This reduction in crack tip opening displacement promotes closure similar to plasticity induced closure.

2.2.6 Short Crack Anomalies:

The characterization of fatigue crack growth rate based on linear elastic fracture mechanics relies primarily on test specimens that contain long cracks that are between millimeter and tens of millimeters in length. However, in a number of fatigue-critical engineering components (turbine discs and blades), the fatigue cracks have significantly smaller dimensions (from tens

17 of micron? to fractions of a millimeter). Since between 80 and 90% of the fatigue lives may be spent in the so-called short crack regime, it is important to develop a basic understanding of the fatigue behavior if short cracks [25]. It has been shown that for the sam e crack driving force (AK). the growth rates of small cracks can be significantly higher than the corresponding growth rates of long cracks as illustrated in Figure 8. Hence, the use of long crack data in the short crack regime can lead to severe overestimation of the fatigue lives of engineering components and structures.

Short cracks have been classified by Suresh and Ritchie [26] as follows:

(a) Microstructurallv Small Cracks: These are fatigue cracks with sizes that are comparable to the characteristic microstructural dimensions such as grain size.

(b) Mechanicallv Small Cracks: These are cracks with lengths that are small compared to the size of the plastic zones at the crack tips. (c) Phvsicallv Small Cracks: These are small cracks that are larger than the microstructurally and mechanically small cracks, but smaller than a millimeter or two.

(d) Chemicallv Small Cracks: These cracks are amenable to linear elastic fracture mechanics analysis, but exhibit anomalies in grovrth rate below a certain size. These anomalies are due to environmentally-assisted corrosion-fatigue processes that are partly dependant on crack size.

One example of short fatigue crack growth behavior is presented in Figure 9. This shows a schematic of short crack growth in a peak-aged 7075-aluminum alloy [27]. The figure shows a marked reduction in the fatigue crack growth rate of the microstructurally short fatigue crack, with increasing crack length (increasing AK). The growth rate of the retarded short crack later increases with increasing AK, until it merges with the long fatigue crack growth data. Various investigators [27-31] have indicated that the retardation occurs when the crack tip reaches a grain boundary in the material. The consequences of crack tip-grain boundary interactions are as follows; (a) retardation of crack growth until a sizable plastic zone is established in the adjoining grain [28], (b) slip bands emanating from the crack tip are pinned at the grain

18 boundary [29], and (c) a change in the crack tip driving force (AK) occurs due to the crack deflection associated with crystallographic reorientation of the crack tip as it passes through the grain boundary into the adjacent grain [31]. The effects of crack tip-grain boundary interactions are diminished as the small fatigue crack grows to a size where its behavior can be characterized by linear elastic fracture mechanics. In that case, the crack tip samples many grains and the overall fatigue crack growth behavior is averaged over several individual grains.

Investigators [32-33] have also shown that crack closure effects are much less pronounced in short cracks than in long cracks. Using finite element simulations of plasticity- induced closure, Newman [32] has shown that the plastic wake behind an advancing short crack is much smaller than that of a corresponding long crack subjected to the same stress intensity factor range. As a result, the effective stress intensity factor range, AK@„, is greater for a short crack than in a long crack. This results ultimately in higher fatigue crack growth rates in the short crack.

2.2.7 Multiparameter Approach to Fatigue Crack Growth:

Since the pioneering work of Paul Paris and co-workers, the use of the Paris equation (Equation 12) has gained a widespread acceptability. Following the work of Paris et al.[16], Forman and co-workers [34] proposed alternative empirical crack growth equations for the assessment of the combined effects of stress intensity factor range, AK, and stress ratio, R. on fatigue crack growth rate, da/dN. Walker [35] also developed a simple two-parameter crack growth equation for the estimation of da/dN as a function of AK and the maximum stress intensity factor, Kmax- Most recently, a multiparameter linear regression model [36], which is a modified version of Equation 12 has been developed, based upon original work of Laplace [37].

The multiparameter model is based upon statistical and random variable analysis and assesses the combined effects of identifiable multiple variables. The multiple variables include stress intensity factor range, AK, stress ratio, R, nominal specimen thickness, t, maximum stress intensity factor, Kmax. and crack closure stress intensity factor, K^ which can contribute to fatigue crack growth rate. Based on this model, the fatigue crack growth rate is given by;

19 ^ =a„(AK)*'(K„)“’{R)“’(/)-(K„)''’ (17) aN

Equation 17 can be linearized to give:

da In = ln (a J + or, ln(A K )+a, ln(/?) + £Z3 \n{t) + a , l n ( K ^ J + a j tn (K :,J (18) dN

The unknown coefficients/exponents and the multiple correlation coefficients, r^ are calculated using multiple linear regression analysis. The correlation coefficient, r^, indicates that the closer its value to 1, the better the correlation between the measured (experimental) and predicted values of fatigue crack growth rate. Equation 17 suggests that the most important variable that contributes to fatigue crack growth rate will have the largest positive exponent, conversely, the lower the exponent values, the lower the contribution of that particular variable to fatigue crack growth rate.

2.2.8 Phvsicallv-Based Approach to Fatigue Crack Growth:

Since fatigue crack growth laws are typically empirical in nature, they are only applicable over parametric ranges of crack driving force that are within the bounds where the expressions provide a good fit of measured fatigue crack growth rate data, as a function of crack driving force. However, it is well known that the mechanisms of fatigue crack growth may very significantly over the parametric ranges of crack driving force in the near threshold, Paris and high AK regimes. Furthermore, the crack-tip deformation and chemisorption processes are unlikely to be fully described by empirical expressions that do not assess the chemical and physical processes involved in the extension of a crack during a single fatigue cycle.

A physically-based model has, therefor, been proposed for the analysis of fatigue crack growth. The model assumes that fatigue crack growth rate is directly proportional to the change in the crack-tip opening displacement during cycling between the maximum and minimum stress intensity factors, i.e., the physically-based model assumes that the fatigue

20 crack extension per cycle, 5a, is directly proportional to the change in the crack-tip opening displacement, CTOD. This gives:

Ôa oc ^C T O D (19)

Under cyclic loading conditions, the ACTOD is given by;

K-K; àCTOD = (cr0D)_ - (corD)_ = (20) 4E(T„ 4 E ct^,

where E is the Young’s modulus and 2 oys is assumed to be the cyclic yield stress. Substituting Equation 20 in to 19, and replacing the proportionality sign with equality sign gives;

' \ ^ R ' da K;.„ ' P = P[hCTOD) = p (AK): (2 1 ) I n _4Eo-,, 4Eo- „ _ 4£o-„_l-^_

where (3 is a measure of the extent of irreversibility. However, it is important to note here that the so-called constant, [3, is unlikely to remain constant as a function of AK during cycling between Kmax and Kmin- If it is now assumed that (3 exhibits a power law dependence on AK [(3 = ao(AK)", where Co and n are power law constants], Equation 21 can then be rewritten as:

\ + R (AK)" (22) d N \-R

Where C = Oo /(4Ecty 5) and m = n+2. it is important to note that crack closure phenomenon may significantly decrease the effective driving force, AK@m. especially at low stress ratios. Hence, in cases where closure levels are significant, AK^m may be substituted for AK in Equation 22. This gives;

21 Equation 23 is generally applicable to long fatigue cracks growth problems in the different regimes of crack growth. It is valid for stress ratios within the range where between -1

2.3 Titanium Alloys and Their Fatigue Behavior:

Titanium and its alloys have found widespread use in the aerospace industries (both air frame and aroengine components) and biomedical industries where their high strength to weight ratios and corrosion resistance make them the materials of choice. In this section, the physical metallurgy of titanium alloys is first described. Secondly, the influence of microstructure and testing variables on the fatigue crack growth behavior of (a+P) titanium alloys is discussed.

2.3.1 Physical Metallurgy of Titanium Alloys:

2.3.1.1 Introduction:

The origins of titanium and their basic applications are described in this section. The titanium alloys based on their important alloying elements are defined. Finally, the phase transformations in (a+P) titanium alloys are discussed.

2.3.1.2 Origin and Basic Applications of Titanium:

Titanium was first discovered in minerals now known as rutile. It is widely distributed on the surface of the earth. Its concentration within the earth’s crust of about 0.6% m akes it the fourth most abundant of the structural metal after Al, Fe and Mg [38]. Due to its moderate high temperature properties, high strength to weight ratio and good corrosion resistance their application in aerospace and chemical industries quickly became recognized. In the late 1960’s, the application of pure titanium and some of its alloys in orthopedic surgery to replace human bones and joints became very popular because of their biological compatibility with

22 human tissue. The density of titanium is approximately 4.51 gmcm'^ [38] (=60% of the density of steel) and its mechanical properties are greatly enhanced by alloying.

2.3.1.3 Classification of Titanium Alloys:

As shown in Figure 10, pure titanium undergoes an allotropie transformation. At room temperature, titanium has an hep (a) crystal structure. This transforms to a bcc (P) structure as the temperature is increased through 882.5°C (1621°F). Elements such as Al, 0, or N, which when dissolved in titanium produce little change or increases the transformation temperature, are known as “a stabilizers”. Alloying additions such as V, Nb, or Ta, which decrease the transformation temperature, are referred to as “P stabilizers” [38]. Other alloying elements such as Zr or Sn when added do not change the transformation temperature. They are, therefor, neither a or p stabilizers. These alloying elements are referred to as “neutral” elements. Hence, titanium alloys as classified as “a”, “a+p” and “P", as shown in Table 1. Typical microstructures of these alloys are shown in Figure 11.

2.3.1.4 Phase Transformations In (a+P) Titanium Alloys:

The (a+P) alloys (as their name indicates) contain both a and p stabilizers. As a result, their microstructure consists of both a and p phases. The microstructure of a mill-annealed (a+P) alloy such as Ti-6AI-4V is shown in Figure 12. The microstructure consists of equiaxed a plus small amounts of intergranular p. To produce this microstructure, heat treatment is done at a temperature low in the a+p phase field (for about 4 hours) followed by furnace cooling to room temperature. The mill-annealed structure is very ductile and relatively machinable.

The a+p alloys can be strengthened by solution treating followed by aging. Solution treating is usually done at a temperature high in the a+p phase field followed by rapid quenching. Due to quenching, the p phase present at the solution treating temperature may be retained or may be transformed during cooling. Aging is performed to precipitate out a phase, and to produce a fine mixture of a and p in the retained or transformed p phase.

23 When (a+P) alloys such as TI-6AI-4V are cooled slowly from the p region, a phase begins to form below the p transus. The a phase forms in plates or laths with a crystallographic relationship to the p in which it forms. This results in an alternating layered arrangement of a and p plates. This alternating layered structure of a and p phase is known as the Widmastatten microstructure. As a result of this type of transformation, each prior p grains is converted to a number of colonies. Within each colony, the crystallographic orientation of the a plates is fixed. In Widmanstatten microstructures, the Burgers orientation relationship [41] between the alternating layers of the a and p phase is given by: (101)^1 (OOOlX: [111] J [2llOL

A knowledge of the composition of the a laths and the p matrix of the Widmanstatten structure is necessary for a proper interpretation of the mechanical properties. Figure 13 and Table 2 present the measured compositions of a plates and the p matrix in a Widmanstatten microstructure of a Ti-6AI-2Sn-4Zr-2Mo-0.lSI alloy. These were obtained using X-ray fluorescence analyses [42]. Fig. 13 clearly shows the compositional uniformity of the neutral elements Zr and Sn across the sample. Also evident is the preference of Al for the a laths and Mo for the p matrix. This is consistent with the role of Al as an a stabilizer, and Mo as a p stabilizer.

2.3.2 Fatigue and Fracture Behavior of (a+61 Titanium Alloys:

2.3.2.1 Introduction:

In this section, the influence of microstructural differences on the fracture and fatigue crack growth behavior of (a+p) titanium alloys is reviewed. The effects of stress ratio, R, and specimens thickness on fatigue crack growth rate are also discussed. For different microstructures, the fatigue crack growth mechanisms and the crack closure studies are summarized.

24 2.3.2 2 Effects of Microstructure:

Several investigators [43-44] have shown that (a+P) titanium alloys exhibit higher fracture toughness plus reduced fatigue crack growth rates when the alloys are in the p annealed condition, rather than the mill annealed condition. The fracture toughness of alloy Ti-6AI-4V in the p annealed condition (K,c = 87 MPa Vm, [44]) has been measured to be twice the fracture toughness in the mill annealed condition (K,c = 42 MPa Vm [44]).

Fig. 14 compares the fatigue crack growth rate of the two microstructures [45]. Fatigue crack growth rates are substantially reduced as a result of the p anneal heat treatment. As shown by Figure 14, the data for the p anneal microstructure shows a transition point T ’ below which the most beneficial effect of this type of heat treatment is observed. Below the transition point, T , the fatigue crack growth rate is microstructure-sensitive, and the reduction in growth rate is attributed to crack tip bifurcation and crack deflection in the Widmanstatten microstructure [45]. Crack bifurcation and crack deflection cause reductions in the effective AK and consequently, da/dN. The occurrence of the transition point is not observed in the fatigue crack growth rate data for the mill-annealed microstructure.

It has also been shown that the fatigue crack growth behavior of p annealed microstructure is related to the Widmanstatten packet size (or colony size) and/or a lath size for (a+p) titanium alloys [46-47]. The fatigue crack growth behavior of five p anneal microstructures with different Widmanstatten packet sizes are shown in Figure 15 [47]. The different packet sizes are obtained by changing the cooling rate from the p anneal temperature. As illustrated by Figure 15, the growth rate plot for each condition exhibits a bilinear form, with

a transition at the stress intensity factor range, AKt.

In the hypo-transitional region (for AK < A K t) , fatigue crack growth rate, da/dN, varies

inversely with mean packet size, with a 20-25-fold reduction in da/dN for a 3.5-fold

increase in mean packet size. In the hyper-transitional region (for AK > AK;), the growth rates seem to converge, and they are independent of mean packet size.

25 Figure 16 compares the relative size of the computed cyclic plastic zone, r j , to the mean

Widmanstatten packet size, I ^p, in the different regions of the fatigue crack growth curve [47].

Below the transition point, where fy ; / „.p, a microstructuraüy sensitive mode of crack growth occurs which involves crystallographic bifurcation in Widmanstatten packets bordering the crack plane. Crack bifurcation reduces the effective AK and consequently da/dN. As a result, crack bifurcation can be considered as a crack-tip shielding process. Since bifurcated cracks in the crack tip region increase with increasing mean packet size [48], thereby decreasing the effective AK, da/dN thus decreases with increased mean packet size, as observed in Figure 15.

In contrast, above the transition where fy>î the packets within the larger cyclic plastic zone must deform as a continuum. Therefore, the overall crack growth behavior is the average of the differently oriented individual packets. This results in a microstructuraüy insensitive, non bifurcated mode of crack growth which as shown in Figure 15 causes the growth rate curves of the alloys with varying mean packet sizes to approach each other. Figure 16 shows that at

A K t, the mean packet size is equal to the computed cyclic plastic zone size.

Though mean packet size appears to have predominant influence on the fatigue crack growth behavior of Widmanstatten microstructures, the nature of packet size distribution appears to have an important effect on the sharpness of the transition observed at a K t . The degree of clustering, / , of Widmanstatten packet sizes around the mean value is calculated by [47]:

X = k r - L ) " ' (24)

where is the packet size from the 95“’ percentile of the packet size distribution. Figure 17 shows that, as the degree of clustering around the mean value is increased, the sharpness of transition increases. This effect is attributed to a residual microstructural sensitive crack growth mode that remains in the regions where crack growth mode is microstructuraüy 26 insensitive (AK > AKj). This is because as clustering increases, fewer packets meet the condition, lwp)K)^wp< necessary for any residual microstructural sensitive crack growth to occur in the hyper-transitional region.

Other investigators [46,49,50] have shown that besides Widmanstatten colony or packet size, other microstructural features, such as the thickness of the [3-layer and a-lath size in the (3 annealed microstructures have significant effects on the near-threshold fatigue crack growth behavior of (a+p) titanium alloys. It has also been shown that for fast cooled microstructures (fine colony and a-lath size with discontinuous P-layers), the controlling microstructural features are the colonies, whereas those are the a-laths in slow cooled microstructures (large colony and a-lath size with thick and continuous p-layers).

The above observations have been supported by fatigue crack path profiles and SEM fractographs of the fast and slow cooled alloys. Fatigue crack path profiles for the fast and slow cooled alloys at low AK values are shown in Figure 18 [46]. For fast cooled microstructures, the fatigue crack path profiles [18(a) and 18(b)], exhibit extensive crack branching and microstructure-induced crack deflection. The growth of main and secondary cracks occurs across the a-laths without any retardation, and crack deflection occurs mainly at colony boundaries and also to some extent at prior p grain boundaries.

In contrast, fatigue crack path profiles in the slow cooled microstructures [18(c) and 18(d)], exhibit a much smaller extent of crack branching, with crack deflection occurring primarily at thick p phases at a/p interfaces [46].

The SEM fractographs taken at low AK values for the fast and slow cooled alloys are shown in Figure 19. As indicated, fracture is strongly crystallographic in all the microstructures. For the fast cooled microstructures [19(a) and 19(b)], the size of fracture facets corresponds to the size of the colonies, whereas for the slow cooled microstructures [19(c) and 19(d)], the size of fracture facets corresponds to the a-lath size.

27 Also, the calculated plastic zone size (at AK = AKj) for the fast cooled alloys has been found to correspond to the packet or colony size, whereas for the slow cooled alloys, the calculated plastic zone size (at AK = AKj) corresponds to the a-lath size [46]. These observations clearly indicates that the fatigue crack growth behavior of Widmanstatten (a+0) titanium alloys is controlled by the colony size and a-lath size for the fast cooled and slow cooled microstructures, respectively.

2.5.2.3 Effects of Stress Ratio. R.:

The effects of stress ratio on fatigue crack growth rate behavior of mill annealed Ti-6AI-4V is well documented [51-54]. It has been shown that fatigue crack growth rate decreases with decreasing stress ratio for a given AK value. It will be shown later in this section that the decrease in fatigue crack growth rate is attributed to the increasing levels of crack closure as the stress ratio is decreased [51,52]. Fatigue crack growth rate curves obtained from (a+P) titanium alloys with different microstructures (at stress ratios of 0.35 and 0.12) are presented in Figures 20a and 20b, respectively [43]. The fatigue crack growth curves at a stress ratio of 0.35 show a transition between a microstructure sensitive (AK < 12 MN m'^^) and a microstructure insensitive region (AK > 12 MN m"^). The investigators were able to correlate the transition point with the attainment of a critical value of a cyclic plastic zone size that was equal to the a grain size of each microstructure. At low AK values and a stress ratio of 0.12, enhanced microstructure-induced crack growth rate differences are observed, as shown in Figure20b. The investigators in Ref 44 were not able to fully explain these important differences.

2.3.2.4 Effect of the Ratio of Specimen Thickness to Colonv Size:

Earlier on in section 2.2.2, it was mentioned that the fatigue crack growth rate decreases with increasing Widmanstatten microstructure colony size. However, Yoder and Eyion [55] have shown that, for Widmanstatten microstructures with a ratio of specimen thickness to colony

size of approximately unity (B:/^p=1), fatigue crack growth rate decreases with decreasing

colony size. Figure 21 shows a schematic illustration of the effect of colony size on fatigue

28 crack growth rate for B:/,^p=1 and B:/^.p=100. The specimens with were used to study the fatigue crack propagation within a single colony with a minimal influence of adjacent colonies.

In ref [55] investigators found that the actual transcolony growth was roughly equivalent to crack growth in equiaxed microstructures of the same alloy. However, fatigue cracks were arrested at the colony boundaries, while many cycles were consumed to reinitiate the crack into the next colony. As a result, this cycle consuming reinitiation process at the colony boundaries promotes a lower crack growth rate in Widmanstatten microstructure than in an equiaxed microstructure. Hence, it is suggested that a reduction in colony size will promote an increase in the number of cycles spent in reinitiating the fatigue crack at the adjacent colonies. This, therefor, reduces the overall fatigue crack growth rate (case A, Figure 21 ).

The specim ens with B:/,^p=iOO were used to capture the true polycrystalline fatigue crack growth behavior of the Widmanstatten microstructure [47,48]. As illustrated schematically in case B of Figure 21, the decrease in fatigue crack growth rate with an increase in colony size is explained in terms of a structure sensitive mode of crack growth that involves crystallographic bifurcation. The larger colonies permit larger bifurcations, which increase crack path tortuousity, reduce effective AK, decrease crack growth rate.

2.3.2.5 Mechanisms of Fatigue Crack Growth:

In the previous sections, it has been shown that fatigue crack growth rates in some microstructures are lower than in others. The lower growth rates in some structures suggests that fatigue cracking is intrinsically more difficult. The difficulty in fatigue cracking has generally been associated with crack path tortuosity, and crack branching or bifurcation. Crack path tortuosity occurs when the crack leaves the plane of mode I cracking due to interaction with obstacles (eg. Colony boundary, or continuous p-layer). This allows the actual crack to travel farther in the microstructure than the actual projected crack. This phenomenon promotes the consumption of more fatigue cycles, which in turn reduces the growth rate, da/dN. Crack branching or bifurcation occurs when a microstructure has tendency to crack

29 along multiple paths. As a result, crack bifurcation serves to reduce the effective AK (and thus da/dN) by dispersing the strain energy of the macroscopic crack along multiple crack tips.

Figure 22 shows the contrast in the microstructurally sensitive (AK < a K t ) and microstructurally insensitive (AK > A K t) modes of crack growth observed by crack path sectioning. The Widmanstatten (a+(3) titanium alloy contained colonies that were approximately 60 |im in size [47]. As shown in Figure 22a, for AK < AKj, the fracture mode is structure sensitive and crystallographically bifurcated. For AK > AKr, Figure 22b shows that the fracture mode is structure insensitive and nonbifurcated.

Using crack tip Transmission Electron Microscopy (TEM) to investigate the fatigue crack growth behavior of mill annealed (equiaxed microstructure) Ti-6AI-4V alloy, Dubey et.al [51] have shown that fatigue cracks propagate by unzipping along the intersecting slip bands (Figure 23). This mechanism of crack growth corresponds to the “alternating slip model" which was proposed by Neumann [56]. Figure 18d shows that fatigue crack growth in p-annealed Widmanstatten (a+P) titanium alloys containing large colonies and a-lath size with continuous interplatelet p (i.e. slow cooled microstructures) occurred preferentially along the continuous p- layers [46].

Other investigators have observed similar results in a p-annealed (Widmanstatten microstructure) and slow cooled TI-6AI-4V alloy [57]. By using electron diffraction and dark- field transition electron microscopy, these investigators were also able to show that the advancing crack (through the continuous p-layers) contained a thin layer of p on each side of the crack, thus indicating that crack path was through the p phase itself rather than along the o/p interface. These investigators have also been able to show that crack along the thick continuous p-layers continues until obstructed by a platelet. In these situations, the crack was observed to turn though angles of up to =80° in order to continue to extend until similarly obstructed. Each time the crack was constrained to tum, a period of arrest occurred during which the deformation spread primarily along the p layer, and little or no dislocation activity was noted in the obstructing a platelet [57].

30 2.3 2.6 Crack Closure Studies in (g+B) Titanium Alloys:

Figure 24a shows the fatigue crack growth rate behavior of a mill annealed TI-6AI-4V alloy tested at different stress ratios [51]. The curves clearly show an increase in fatigue growth rate with increasing stress ratio, which is consistent with the results of other investigators [52-54], Figure 24b shows the closure corrected data for the different stress ratios. The figure clearly shows that the data collapses onto a single curve [51]. Comparison of the two fatigue crack growth rate curves indicates that crack closure increases with decreasing stress ratio.

By recording the crack mouth opening during fatigue crack growth of a p-annealed (Widmanstatten microstructure) and a mill annealed TI-6AI-4V specimen, it was observed that for the p-annealed alloy, the rough fracture surfaces contact prematurely (during unloading) and there is a mismatch between the mating surfaces at zero load [22]. The roughness in fracture surfaces and their premature contact is attributed to crack path tortuousity and secondary cracking in p-annealed microstructure. For the p-annealed microstructure, it was also observed that at zero load the crack was open over a large part of the area behind the crack tip across which load bearing contact was being made. In discrete regions behind the crack tip, the crack was apparently held open due to local mismatch of the fracture faces. Hence, the stress intensity factor range, AK, at the crack tip is lower than that expected from the applied load range.

The mill-annealed material did not exhibit this load bearing contact behind the crack tip. However, as discussed earlier, the work of Dubey et.al [51] has shown extensive crack closure in mill annealed microstructures tested at low stress ratios (R = 0.02 and 0.25). This can be attributed to an improved sensitivity of the instrumentation to measure crack closure levels. However, the results presented in Ref. [22], clearly indicate that, due to crack path tortuousity and crack branching, closure levels are much higher in p-annealed microstructure than in mill annealed microstructure. The higher closure levels in the p-annealed microstructure reduce the effective stress intensity factor range at the crack tip, which in tum decreases da/dN.

The extent of crack closure is increased by the tortuousity of the fracture path and by crack branching. Sudden changes in the plane of the crack from one controlling

31 microstructural unit (e.g. colony size, a-lath) to the next result in different local mode I and mode n contributions to crack growth, and hence relative local displacements of the fracture faces in the plane of the crack. This difference in local displacements of the fracture faces causes mismatch and contact at discrete points. The load level at which contact across the fracture faces occurs is influenced by the coarseness of the fracture faces, which in turn is dependant upon the size of the microstructural units (colony size which could be as large as the prior p grain size). The coarseness of the microstructure can be significantly increased by P-annealing of Ti-6AI-4V alloy, as compared with mill annealed condition. This coarse and highly irregular fracture surface of the p-annealed alloy (Widmanstatten microstructure) is a prime contributor to roughness induced crack closure.

2.3.2.7 Short Crack Studies in (g+P) Titanium Allov:

As mentioned in section 2.2.4, crack closure mechanisms arise as a result of premature contact between the crack faces behind the advancing crack tip. Since a short crack has a limited crack wake behind the crack tip, crack closure effects are less pronounced in short cracks than in long cracks. This phenomenon is illustrated for an (a+P) titanium alloy in Figure

25. The figure shows the measured nil-load crack tip opening displacement, 6,, after complete unloading following a fatigue test, as a function of the crack size in a Ti-6AI-2Sn-42r-6Mo alloy [58]. With an increase in crack length from 50 pm to 150 pm, 5, is observed to increase and it is leveled off for crack lengths greater than 150 pm. This behavior suggests that for crack lengths from 50 pm to 150 pm, fracture surface roughness and hence, closure level gradually increases.

Short crack anomaly, which is defined as fatigue crack growth at stress intensity factors significantly below the long crack fatigue threshold has been observed in many commercial alloys such as, nickel-aluminum-bronze alloys [59], titanium alloy IMI 685 [60], and titanium- aluminum alloys [61]. Recently, short crack growth anomaly has been observed in mill- annealed TI-6AI-4V alloy [62]. Figure 26 clearly shows that short fatigue crack growth occurs at stress intensity factor levels that were significantly below the long crack fatigue threshold at stress ratio of R = 0.1. Figure 27 compares the short crack growth rate data (at R = 0.1) with the closure corrected (AK@„) long crack growth rate data (at R = 0.1). The figure shows that

3 2 there is a very good matching between the closure-corrected long crack data and the short crack data obtained at the beginning of the short crack test. The figure also shows that the short crack growth rate data merges with the long crack data (without closure-correction) as the test is continued (under constant load range conditions). This strongly suggests that the difference in the roughness induced crack closure levels are a major factor that contributes to the observed difference in the growth rates of the short and the long cracks in mill-annealed Ti- 6AI-4V alloys. The observed crack growth data also suggests that the closure levels increase with increasing crack length. As a result, the difference in short crack and long crack data can be rationalized largely by crack closure arguments.

This closure effect, in conjunction with other crack wake retardation mechanisms, provides a rationale for the higher fatigue crack growth rates found in the case of short cracks. However, further work is needed to develop an improved understanding of short crack behavior in a/p Ti- alloys.

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37 54. H. Doker, and V. Bachmann, (1988). Determination of crack opening load by use of threshold behavior, in mechanics of fatigue crack closure. ASTM STP 982. American Society of Testing and Materials, 247-259.

55. Yoder, G.R. and EyIon, D., (1979). On the effect of colony size on fatigue crack growth in vVidmansiatten structure a+p Titanium Aiioys, Metallurgical Transactions A, 1ÛA, p. 1808- 1810.

56. Neumann, P., (1974). Modeling of Changes in Dislocation Structure in Cyclically Deformed Crystals, Constitutive Equations in Plasticity (ed. Hassen, A.S.), p.251-326.

57. Hall, I.W., and Hammond, C., (1978). Fracture Toughness and Crack Propagation in Titanium Alloys, Material Science and Engineering, 32, p. 241-253.

58. Jam es, M.R. and Morris, W.L., Effect of Fracture Surface Roughness on Growth of Short Fatigue Cracks, Metallurgical Transactions, vol. 14A, p. 153-155, 1983.

59. Taylor, D., and Knott, J.F., (1981). Fatigue Crack Propagation Behavior of Short Cracks: The Effect of Microstructure, Fatigue of Engineering Materials and Structures, vol. 4, p. 147- 155.

60. Brown, C. W., and Hicks, M. A., (1983). A Study of Short Fatigue Crack Growth Behavior in Titanium Alloy IMI 685. Fatigue of Engineering Materials and Structures, vol. 6 , p. 67-76.

61. Larson, S. G., Nicholas, T., Thompson, A. W., and Williams, J. C., (1986). Small Crack Growth in Titanium-Aluminum Alloys. In Small Fatigue Cracks, (eds. R. O. Ritchie and J. Lankford), p. 499-512. Warrendale: The Metallurgical Society.

62. Sinha, V., C. Mercer., W. O. Soboyejo., (2000). An Investigation of Short and Long Fatigue Crack Growth Behavior o Ti-6AI-4V, Materials Science and Engineering, A287, p. 30-42.

38 Figure 1 : A large plate of an elastic material of a thickness, B, containing a center crack of length, 2 a, subjected to an applied stress a.

X

(a)

Figure 2: The three modes of fracture: (a) Mode I: (crack opening mode): (b) Mode n; (sliding mode); and (c) Mode HI: (tearing mode).

39 10-*

regim e A regim e B I m m , m m

10-'

regim e C I O'*

«-one laitice «pacing I m m /d a y

Figure 3: Schematic illustration of the different regimes of stable fatigue crack propagation.

// ia) \\

ib) (e)

(0

Figure 4: Schematic illustration of the plastic blunting model, (a) zero load, (b) small tensile load, (c) peak tensile load, (d) onset of load reversal, (e) peak compressive load, and (f) small tensile load in the subsequent tensile cycle [18].

40 s tr a in g a g e

O'.

fa tig u e c ra c k

0 (a)

Figure 5: A schematic illustration of the relationship between the applied stress and the m easured displacement [19].

K fully o p en c ra c k

p a rtia lly o p en crac k fully clo sed c ra c k (a)

Figure 6 ; The offset procedure for closure measurements.

41 plastic fracture surface asperity (c) wake fatigue crack zone (a)

oxide film (b) transformed zone (d)

Figure 7: Schem atic illustration of (a) plasticity induced closure, (b) oxide induced closure, (c) roughness induced closure and (d) phase transformation induced closure.

short crack F rom n o tc h

ea long crack (LEFM)

crack length, a (or) log A/T

Figure 8 : A schematic representation of the typical fatigue crack growth behavior of long and short cracks [ 2 ^ .

42 long crack (LEFM) short cracks -~

crack length, a (or) log A/T

Figure 9: A schematic of sub-threshold growth and transient retardation characteristics of a microstructurally small fatigue crack (dashed lines): dg 2>dg, [25].

(a) hexagonal, close-packed (b) cubic, body centered

Figure 10: Hard sphere models of basic crystal structures of (a) hep and (b) bcc [39].

43 Alloy Classification

Ti-5A!-2.5Sn a Ti-BAl-lMo-l V 1 . n e a r-a * Ti-6Al-2Sn-4Zr-2Mo j TÎ-6A 1-4V Ti-6Al-2Sn-6V " a + Ti-3A1-2.5V ^ Ti-6Al-2Sn-4Zr-6Mo Ti-5Al-2Sn-2Zr-4Cr-4Mo Y n e a r - â Ti-3Al-10V-2Fe

Ti-13V-llCr-3Al Ti-15V-3Cr-3.AI-3Sn Ti-4Mo-8V.6Cr-4Zr-3.AI >/3 Ti-8Mo-8V-2Fe-3Al** Ti-ll.5M o-6Zr-4.5Sn

•The terms “lean-d" and "super-o" may also be used. ••Obsolete alloy.

Table 1; Classification of titanium-base alloys [38,40]. s w w

J "1 >- -0.5 mm------«4 ■I(b) !-*-50nm-^

H-5c/xim-H Q.5 mm

Figure 11: Typical microstructures of a, a+p, and p titanium alloys, (a) Equiaxed a in unalloyed titanium, (b) Equiaxed a+p, (c) Acicular a+p and (d) Equiaxed p [38].

44 S?

% 500 X

Figure 12; Typical microstructure of a mill annealed a+(3 titanium alloy (Ti-6AI-4V) [39].

Mo

Plate Plate Plate

& I .9

js I A/

----1

Mo

îOOO 2000 3000 4000 Distance Across Sample, A

Figure 13: X-ray analysis of the compositional distribution across the (a+P) Widmanstatten structure in Ti-6AI-2Sn-4Zr-2Mo-0.1Si [42].

45 -Composition in wt.^o (ai.*a)- Componeni T: A! Sn Zr Mo

■Average* 86 (85) 6 (11) 2 (1) 4 (2) 2 (1) 3 platelet** 78.5 (87) 0 .5 (1) 2.0 (1) 4 .0 (2) 15.0 (8) a platelet** 88.5 (88) 5 .0 (8) 2.0 (1) 4 .0 (2) 0.5 (< 1 )

•Nominal composition. ••STEM/ED.AX analysis.

Table 2; Compositions of the a and p phases in a Widmanstatten Ti-6242 alloy [38].

46 TÎ-6AI-4V (0.20 % 0 )

MA

<

I g

su 3 SA O

RM TEM P A IR KAVERSINE WAVE ■

20 50 lOQ STRESS-tNTENSlTY FACTOR RANGE AK( MPC «1/21

Figure 14: Fatigue crack growth rates for the mill annealed (MA) and p annealed (BA) microstructures [45].

47 : WIDMANSTATTEN Ti ALLOYS

-4 X 10

o u

iNCREASeO PACKET SIZE

10 20 30 40 SO 100 STRESS-INTENSTTY RANGE. AK

Figure 15: Fatigue crack growth behavior of the (a+(3) Widmanstatten microstructure with different packet sizes [47].

48 WIDMANSTATTEN Ti ALLOYS

CYCLIC «.ASTIC lOWE trjl VS PACKET (7 „ ) OIMCNSIOMS fi

Î

* I

5 I Ï

4 K • a * T TRANSITION

AK, loe STRESS-IWTEWSITY KAMSC lAKI

Figure 16; Schematic comparison of the computed cyclic plastic zone size to the average Widmanstatten packet size [47].

49 WIDMANSTATTEN Ti-6Al-4V ‘ CLUSTERING-TRANSITION EFFECT

■ INCREASED CLUSTERING

■043 y 021 , 013

1 10-5 :

29 / 38 t #

f T t 54 i L. I 10 Î

i I . t I u. icr’ mg/m, «2.26 2.01 1.55 • SHARPER TRANSITION

10 20 30 50 10 20 30 50 10 20 SO 50 STRESS-INTENSITY RANGE. ûKÎMPo m '^ )

Figure 17: influence of clustering on the sharpness of the transition observed in the fatigue crack growth rate curves for the Widmanstatten microstructure. mB/mA is the ratio of the slopes for AK > a Kt and AK < AKt in the three curves [47).

50 40 a m (c ) Z O um

4 0 a m

Figure 18: Fatigue crack path profiles at low AK levels for Widmanstatten Ti-6AI-4V alloy heat treated as (a) Argon quenched, (b) cooled at 30° C/min, (c) cooled at 6 ° C/min, and (d) cooled at 2° C/min, from the P solutionising temperature [46].

51 i m

Ê

. 40pm

Figure 19: Fracture morphology at low AK levels for W idmanstatten Ti-6AI-4V alloy heat treated as (a) Argon quenched, (b) cooled at 30° C/min, (c) cooled at 6 ° C/min, and (d) cooled at 2° C/min, from the (3 solutionising temperature [46].

52 7 M*RTENSITJC i *S RECEIVED Î 1 TRANSPCflMEO 3 o3 annealed

Ï < I «5

AK KN in

(a)

o transformed f i

10-5 • AS RECEIVED Û MARTENSITIC . 5 u i • • • >«. o

: 1 0 - 6 •5 S 5 c o

ICT? 5 S 10 12 AK MN m'^f2

(b)

Figure 20: Fatigue crack growth rate curves obtained at (a) R-ratio = 0.35 and (b) R-ratio = 0.12 for Ti-6AI-4V alloy in martensitic, mill annealed (as received), transformed P, and p annealed (Widmanstatten microstructure) conditions [43].

53 CASE A CASE a SMALL a: F , RATIO LARGE a: in RATIO (ORDER OF 10°) (ORDER OF lO^l ui

u u U i

COLONY SIZE ( / . ) ' COLONY SIZE (£,)

Figure 21 : Influence of colony sizes on fatigue crack growth rate, at constant level of AK, a schem atic representation [55].

54 • ^ A > 'hv *\ vs A; ■■ ' V

\T \, 60*m ,

Figure 22: Contrast in (a) structure sensitive and (b) structure-insensitive modes of crack growth in a Widmanstatten (a+P) titanium alloy [47].

55 -> direction of crack growth

1/

\ % W

0.5|im

Figure 23: TEM micrograph showing a portion of the crack tip unzipping along the intersecting slip bands in mill annealed Ti-6AI-4V alloy [51].

56 u 3 u

Z - 4

■a1 SJ 3 5 ai g □ R=0.02 2 o 6 ♦ R=0.25 15 2 U u a U l n 100 O. Stress Intensity Factur Range, AK, MPaVm

•3

-4 Z 10

# lO 'S

R=0.02 2 -6 R=0.25 O 10 u R=0.5 R=0.8 U u -7 JL 3ca 10 1 10 100 Effective Stress Intensity factor Range, AKeff, (MPaVm)

Figure 24: (a) Fatigue crack growth rate for mill annealed T1-6AI-4V alloy obtained at different stress ratios, and (b) Closure corrected fatigue crack growth rate for the sam e alloy at different stress ratios [51].

57 0.3

0.2

0 100 200 300 400 500 Crack Length (um)

Figure 25: The variation of the crack tip opening displacement, &, as a function of crack length in a Ti-6AI-4Zr-2Sn-6Mo alloy [58].

58 Long Crack Short crack

u

z

u

2 Eh

10 100 Stress Intensity Factor Range, AK (MPaVm)

Figure 26; Comparison of long and short fatigue crack growth rates at stress ratio of R = 0.1 [62].

59 ° Short Crack) AK ° Long Crack, AK ^ Lons Crack- AK.

Z

•j

10 ICO Stress Intensity Factor Range. AK (MPa'imV or Effective Stress Intensity Factor Range, AK^,f (MPavm)

Figure 27; Comparison of short fatigue crack growth rates with the long crack growth rates and also with the closure-corrected long crack growth rates (i.e. da/dN versus AKe» plots) [62].

60 CHAPTER 3

MATERIAL AND EXPERIMENTAL PROCEDURES

3.1 Material:

The Ti-6Ai-4V alloy that w as used in this investigation w as supplied by Wyman Gordon Forgings, Houston, TX. The duplex cx/p microstructure was revealed by first poiishing the alloy to a 0.06 nm finish and then etching in KroH’s reagent (3% HP, 6 % HNO3 by volume in H 2O) for about 30 seconds. Figure 28 shows a triplicate optical photomicrograph of the forged alloy. The Figure shows the elongated a grains (light phase) in a continuous matrix of p (dark phase). The average width of the a grains along the longitudinal direction was -16.7 ± 1 um, while the corresponding size of the a grains along the long-transverse direction was -23.4 r 0.7 pm. The mechanical properties of the as-forged material in the longitudinal and transverse orientations are summarized in Table 3.

3.2 Experimental Procedures:

3.2.1 Long Crack Fatigue Experiments:

The long crack fatigue experiments were carried out on standard compact tension C(T) and three-point bend SE(B) specimens. The nominal thickness of the compact tension specimens ranged from 1.59 mm (1/16 in.) to 12.7 mm (1/2 in.). The fatigue tests were conducted under computer-control using a software package supplied by Fracture Technology Associates (FTA), Pleasant Valley, PA. The stress intensity factor, K, for compact tension specimens is expressed in ASTM E399 [1] by:

61 K = ( 1 ) BW TTt / where

a 0.886 + 4.64 1-13.32 ' l ''' + 14.32 -5.6 " a " w W / n 3/: \rrW j f 1 - W and B is the specimen thickness, P is the applied load, and a and W are the crack length and specimens width, respectively. For the computer to calculate K, a/W must be calculated. During the fatigue test, the computer acquired amplified crack mouth opening displacement signals. These were measured using a clip gage that was placed at the crack mouth. The measured crack mouth opening displacement, 5, was then used to calculate the compliance of the system. This is defined as:

EâB Compliance = (2 )

where E is the elastic modulus, and P and 8 are the same as defined above. The ratio, a/W, is then calculated as a function of this specimen compliance using the formula:

— = l-4.67(«)+l8.46 (h )' -236.83( h )' + 1214.88(h )'’ -2143.57( h )' IF

where H = (3) EâB + 1

Using this a/W and the applied load P, the computer is able to calculate the applied stress intensity factor, K.

62 After pre-cracking at a stress intensity factor range, A K , of -20 MPaVm at R = 0.1, the fatigue crack growth tests were conducted using a K-decreasing (K-shedding) procedure in accordance with ASTM E647 [2] specifications. A load-shedding exponent, C = 1/K«dK/da, of -0.08 mm"’ (-2 in. ') was used. Load shedding was continued until a fatigue threshold was reached. This was taken to correspond to a stress intensity factor range at which crack growth was not detected after - 10^ cycles. After reaching the fatigue threshold, the tests were

restarted under constant load ranges (i.e. increasing A K ) conditions. The applied load range

corresponded to the threshold A K values at the beginning of the constant load range test. This

provides an overlap of the data points obtained under the decreasing A K (load-shedding) and

increasing A K (constant load range) conditions. Very good agreement was observed between the load-shedding and constant load range data sets.

One of the advantages of using a crack mouth clip gage to measure crack mouth opening displacement (and hence crack length) is that it could also simultaneously measure crack closure throughout the fatigue test. Crack closure loads were determined from the plots of load versus offset-displacement using a technique described in detail in Ref [3]. The closure offsets, 5o(f, employed in the computation of effective stress intensity ranges are given by [4];

(4)

where Gi is the gain of the electronic circuit, 5 is the crack mouth opening displacement as

m easured by the clip gage, and a is the inverse of the slope of the linear portion of the curve

between K^ax and Kg,. For a fully open crack, 5on = 0, and 5 = a K and when the crack begin to close, 5off takes on positive values. In the calculation of K^, the stress intensity factor at which

crack closure is detected, a 1 % offset was used in the determination of the crack closure loads.

For the three-point bend specimens SE(B), fatigue tests were conducted using both computer control method, and conventional travelling microscope method. The stress intensity factor, K, for three-point bend specimens is expressed in ASTM E399 [1] by:

63 PS IC = (5) BW W j where S is the span of the test soecimen (Figure 29). and P. B. and W were defined earlier The function f(a/W) is defined by:

1 . 9 9 - I - — 2.15-3.93 — + 2 .7 -^ W W W w -

/' 3/2 W 2 a Y , a ^ 1 + W , W j in the case of computer-controlled fatigue tests, the direct current potential dorp method was used to calculate the crack length. Figure 29 schematically illustrates the locations where the current (I) and potential drop (V) were measured using wires that were spot welded to the specimens. Knowing the value of a/W during the tests, and the applied load P, the computer was able to calculate the applied stress intensity factor, K.

After pre-cracking at a stress intensity factor range, AK, of -12 MPaVm at R = 0.1, the fatigue crack growth tests were conducted using the same K-shedding procedure defined earlier. For all tests, the starting AK was 15 MPaVm. Load-shedding was continued until a fatigue threshold was reached. During this portion, the tests were frequently stopped and the actual crack length was physically measured using a light microscope. This enabled a useful comparison between the measured (obtained by DC potential drop method) and actual crack lengths. After the determination of the fatigue threshold, the test was resumed at the starting AK. The starting AK was recalculated using equation 5 for the current crack length. In this portion of the test, the load was gradually increased using a positive exponent of C = 1/K«dK/da, of 0.08 mm ' (2 in. ') until final fracture occurred. As mentioned earlier, fatigue tests were also performed on three-point bend specimens using the conventional travelling microscope method. After precracking the specimens (same procedure as above), fatigue tests were conducted at constant load range. For these tests, a low starting AK was selected, and the corresponding loads were calculated using equation 5.

64 The crack length w as monitored using a travelling microscope connected to a digital readout. If no crack growth was observed after ~ 100,000 cycles, the applied AK was increased in the steps of - 0.5 MPaVm. When the crack was observed to grow at a particular AK, the fatigue test was continued under constant load range conditions. The tests were stopped when a crack growth of ~ 0.25 mm (0.010 in.) was observed on the digital readout, and the fatigue cycles were recorded. This procedure continued until final fracture.

3.2.2 Short Fatigue Crack Growth Experiments:

Short crack fatigue tests were performed using single edge notched (SEN) bend specimens. The specimens were 32 mm (1.26 in.) long, and had square cross sections with dimensions of 8.3 mm (0.327 in.) x 8.3 mm (0.327 in.). The specimens also contained initial notches that were ~ 1.9 mm (0.075 in.) deep. The SEN specim ens were precracked under far-field compression loading at a stress ratio, R = Kmin/Kmax, of 0.1 (Figure 30), and a cyclic frequency of 10 Hz. Under applied cyclic compressive loads, precracks were initiated from the notches due to the influence of the residual tensile stresses at the notch tip [4]. The tops of the specimens were then electro-discharged machined (EDM) away to leave behind through­ thickness edge cracks that were ~ 125 to 150 pm deep (Figure 30). After EDM, both sides of the specimens were polished to 0.06 pm finish [5], prior to etching of the polished sides of the specimen in Kroll's solution for - 30 seconds to reveal the microstructure. In order to reduce the crack lengths to - 20-80 pm, the tops of the specimens were further ground with fine emery paper. The specimens were then tested under three-point bending loading in a servohydraulic testing machine.

The short crack fatigue tests were conducted at a stress ratio R = KmiVKmax. of 0.1. A cyclic frequency of 10 Hz was used, and all tests were carried out at room temperature in the laboratory air. The tests were started at AK = 1 MPaVm. For the appropriate initial crack lengths, the stress intensity factor (K) values were calculated using the stress intensity factor

solutions reported by Tada et. al. [ 6 ]. An optical microscope with an in-built millimeter scale was used to measure the crack length after regular intervals. If no crack growth was detected after - 50,000 cycles at a particular applied AK, the applied AK was increased in steps of - 0.5 MPaVm until the crack was observed to grow. At this point, the test was continued at constant 65 load range conditions. The optical microscope was used to measure crack length and to observe the crack/microstructure interactions at frequent intervals (between - 1 0 '* and 1 0 ® cycles per interval). The short crack fatigue tests were continued until the short crack data merged with the long crack growth rate data in the Paris regime.

3.2.3 Fractograohv:

Upon completion of the fatigue tests, the fracture surfaces of the failed long crack fatigue specimens (tested at a specific stress ratio, R) were examined using a scanning electron microscope. Fracture modes were examined from the start of the fatigue test (low AK regime) to the end of the fatigue test (high AK regime). Each time the fracture mode changed, the corresponding crack length was recorded and representative photomicrographs were taken. The crack lengths were then used to calculate the corresponding AK s using equations 1 or 5, depending upon the type of the test specimen. These corresponding AK s along with fatigue test stress ratio, R, were later used to produce fatigue mechanism maps. The fatigue mechanism maps will be explained in the future chapters.

3.2.4 Statistical Analysis:

The long crack growth rate data were statistically analyzed using a software program known as Minitab. This software program was used to calculate the coefficients/exponents and the multiple correlation coefficients of equations 17 and 23 in sections 2.2.7, and 2.2.8, respectively.

66 3.3 References:

1. Standard Test Method for Plane-Strain Fracture Toughness of Metallic Materials. Annual Book of ASTM Standards, 1992, section 3. Metals-Mechanical Testing; Elevated and Low- Temperature Tests; Metallography, vol. 03.01, p. 506-536.

2. Standard Test Method for Measurement of Fatigue Crack Growth Rates. Annual Book of ASTM Standards, 1992, section 3. Metals-Mechanical Testing; Elevated and Low- Temperature Tests; Metallography, vol. 03.01, p. 674-701.

3. J. K. Donald, “A Procedure for Standardizing Crack Closure Levels," Mechanics of Fatigue Crack Closure, ASTM STP 982, American Society for Testing and Materials, pp. 222-229, 1988.

4. Suresh, S., Christman, T., and Bull, C., “Crack Initiation and Growth under Far-Field Cyclic Compression: Theory, Experiments and Applications”, in “Small Fatigue Cracks", Edited by R. 0. Ritchie and J. Lankford, Proceedings of the Second Engineering Foundation International Conference/Workshop, Santa Barbara, California, pp. 513-540,1986.

5. Sinha, V., Ph.D. dissertation. Effects of Microstructure on Fatigue Behavior of o/p Titanium Alloys, 1999.

6 . Tada, H., Paris, P.C., and Irwin, G.R., “The Stress Analysis of Cracks Handbook”, Second edition, Paris Productions Incorporated and Del Research Corporation, pp.2.13-2.14, 1985.

67 20 |im I____ I

i

Figure 28; Triplicate optical micrograph of the mill-annealed Ti-6AI-4V forging.

Orientation 0.2% offset Yieid Stress Ultimate Tensile Stress % Elongation MPa (Ksi) MPa (Ksi) LT 881 (128) 887 (129) 7.5 TL 825 (120) 885 (128) 7.1

Table 3: Mechanical properties of mill-annealed Ti-6A1-4V

68 Figure 29; Schematic illustration showing the location of the current and potential dorp wires spot welded to the single edge notched (SEN) specimens.

(b) EDM slicing and fine grinding

(a) Precracking under cyclic compression

Figure 30: (a) Procedure for precracking of single edge notched (SEN) specimens; (b) the starter notch is removed using electrical discharge machine (EDM), and the surface of the sample is ground using fine emery paper so as to obtain a precrack length comparable to the grain size for this microstructure.

69 CHAPTER 4

FATIGUE CRACK GROWTH BEHAVIOR OF MILL-ANNELEALED TI-6AL-4V

4.1 Introduction:

In this chapter, the effect of variables such as stress ratio and specimen thickness on fatigue crack growth behavior of mill-annealed T1-6AI-4V alloy are discussed. It will be shown that the difference in fatigue crack growth rates, caused by changing stress ratio, is due primarily to the effects of crack closure. Closure-corrected fatigue crack growth rate data will be compared with non-closure-corrected growth rate data. The fatigue fracture modes in the near-threshold regime (low AK), Paris regime (intermediate AK), and high AK regime are also compared. Finally, a multiparameter model is used for the characterization of fatigue crack growth rate as a function of multiple variables. In order to confirm the general applicability of the multiparameter model, the predicted results are compared with experimental fatigue crack growth rate data.

4.2 Effects of stress ratio and specimen thickness on fatigue crack growth rates:

The effects of stress ratio on fatigue crack growth rates, da/dN, are shown in Figures 31a to 31 d. For each specimen thickness, data obtained from tests completed at different stress ratios are plotted together for comparison. The plots show that, for each specimen thickness, the crack growth rates for the stress ratio R = 0.02 were much slower than the growth rates at the higher stress ratios (R = 0.25, R = 0.50, and R = 0.80). Also, not much difference was observed between the fatigue crack growth rate data obtained at the stress ratios, R = 0.25 and R = 0.50, while the growth rates for the stress ratio R = 0.80 were somewhat faster. The trends in the data (increasing fatigue crack growth rates with increasing stress ratio for a given AK) are consistent with prior reports of the effects of stress ratio on fatigue crack growth in Ti- 6AI-4V[1-3].

70 Figures 32a to 32d show the effects of specimen thickness on fatigue crack growth rate. For each stress ratio, the effects of thickness on fatigue crack growth rate are minimal, for the range of specimen thickness that was studied (1.6-12.7 mm). Later on in this section, a multiparameter model will be introduced. This model will also confirm that specimen thickness does not have a significant effect on fatigue crack growth rates.

A summary of all the fatigue crack growth rate data, da/dN, is presented in Figure 33a, in which da/dN is plotted against the applied/remote stress intensity factor range, AK. Note that much of the spread in the data is due to the effects of stress ratio. However, when the da/dN is plotted against the effective stress intensity factor range, AKeff, the data collapses onto a narrow scatter band around a straight line (Figure 33b). The effective stress intensity factor range in Figure 33 and subsequent discussion is given by;

AKeff = Kmax - Kd (1) where Kmax is the maximum stress intensity factor and K^ is the closure stress intensity factor. The above results suggest that the obsen/ed effects of stress ratio may be attributed largely to the effects of different mechanisms of crack closure. Prior experimental work suggests that crack closure occurs primarily by roughness-induced crack closure in o/(3 titanium alloys [4-7]. Furthermore, the measurements of crack closure/opening in titanium alloys have also been

shown to be consistent with the recent results of Sehitoglu and Garcia [ 8 ], who carried out mechanics simulations of roughness-induced closure processes in which the deformation of asperities (due to contact forces) were assessed.

At large stress ratios, the crack mouth opening displacements are larger than for smaller stress ratios. This is because, at high stress ratios, the specimen encounters relatively large stresses throughout the entire fatigue cycle, while at low stress ratios, large stresses are encountered during only part of the fatigue cycle. As a result, at high stress ratios, there is not much contact between fracture surface asperities as there is at lower stress ratios. Consequently, the amount of roughness induced closure would be expected to decrease, as the stress ratio is increased. This results in higher fatigue crack growth rates with increasing stress ratio. The amount of roughness induced closure would not be expected to change

71 dramatically with specimen thickness, since the fracture surface roughness would not be expected to vary significantly with specimen thickness.

However, it is also possible that plasticity-induced crack closure [9-11] may also contribute to the overall levels of crack closure in Ti-6AI-4V, but its contribution is not as significant as roughness induced closure. This is because the thicker specimens contain a large constraint along the crack front (plain strain condition), and as a result, the plastic zone is relatively small and the contribution from plasticity induced closure should be reduced in thicker specimens. However, the plastic zone in thinner specimens is larger (plane stress condition) and contributions from plasticity induced closure could be much more significant under plane strain conditions. Also, the thinner specimens are more compliant, and as a result their crack opening displacements are much larger. This reduces the contributions from plasticity induced closure.

The closure data obtained from the current study at stress ratios of 0.02 and 0.25 are presented in Figure 34. No evidence of crack closure was detected at stress ratios of 0.50 and 0.80. Figure 34 shows a range of Elber closure ratios, AKen/AK, for various AK values and R- ratios. This shows quite clearly that the Elber closure ratio increases with increasing stress ratio (R=0.02 to 0.25 for both thicknesses). As in most metallic materials [12], U, decreases rapidly in the near-threshold regime. The decrease in the Elber closure ratio at low AK values is most likely due to contact of fracture surface asperities (roughness-induced closure). This closure effect is generally more pronounced at lower stress ratios due to the lower levels of crack opening displacement. This results in a higher levels of fracture surface/asperity contact at R = 0.02 than at R = 0.25. The “closure-corrected” low stress ratio data, also collapses onto the “closure free" fatigue crack growth rate data at high stress ratios, as shown in Figure 33b.

4.3 Fatigue fracture modes:

The effects of stress ratio and stress intensity factor range on the fatigue fracture modes are summarized in Figures 35 and 36 for the 3.2 mm (1/8 in.) and 12.7 mm (1/2 in.) thick

specimens. Since similar fracture modes features were obsen/ed for the remaining 1 .6 mm

72 and 6.4 mm thick specimens tested at different stress ratios, the fracture modes in these specimens are not presented here. Figures 35 and 36 show that fracture in the low AK regime occurred by a flat transgranular “cleavage-like" fracture mode at both stress ratios. Crack growth in the intermediate AK regime occurred by a classicai crack-tip blunting mechanism [13], giving rise to fatigue striations at both stress ratios. The spacing of the striations also increased with increasing AK. Fatigue crack growth in the high AK regime occurred by a combination of fatigue striations and ductile dimpled fracture at both stress ratios. As in prior studies on steels [14], the incidence of static ductile dimpled fracture modes increased with increasing AK and R, i.e., with increasing Kmax- However, specimen thickness did not have a significant effect on the fatigue fracture modes over the range of specimen thicknesses that was examined. Also, no shear lips were detected on the fracture surfaces of the fatigue specimens that were tested.

4.4 Multioarameter modeling of fatigue crack growth:

4.4.1 Evolution of multioarameter model:

As indicated in section 2.2.3, the Paris equation (shown below) is the most widely used and accepted expression for characterization of fatigue crack growth behavior. This is given by:

^ = C (A K )-

where C and m are constants, and their value is influenced by variables such as stress ratio, R, test temperature, T, cyclic frequency, type of waveform, W, material microstructure, M, and others. Since these variabies affect the fatigue crack growth rate, da/dN, the Paris equation can be thought of as being a function of more than one variable, AK. Hence, the Paris equation can be rewritten as:

^ = a , (AK)" {Rf' (T)"’ (n)" (Wf' {M f‘ (3) aJy 73 or

{y. )“' (K )"■ {y, T ' r c ) a N dN %,

In equation 4, variables V| will have an effect on fatigue crack growth rate, and a, are unknown parameters. Equation 4 is multiplicative in nature, and in order to solve for unknown parameters a,, 0 2 , ..., ok, it has to be linearized. Equation 4 can be linearized by taking the logarithms of both sides, which yield:

In = l n ( a J + ^ a ,l n ( K ,.) (5)

It should be indicated that the random variables V|, for i = 1,2 ..... K, do not have to be independent. It is also important to note that expression of the form shown in equation 5 are covered by the log-probability law, which was first proposed by Laplace [15]. For simplicity and ciarity in the up-coming analysis, the variables in equation 5 are renamed as follow:

da In

ink)=A , in(^^)=x, 0 = 1,2 ..../:)

As a result, equation 5 can be written as:

K

1*0

In equation 6 , XjS are the observable parameters [=ln(V|)], Xo is equal to unity, and r is a random variable with a mean value of zero [15]. The quantity y is considered to be a random variable since its exact value cannot be calculated for known values of the XjS. Equation 6 is the basis for the calculation of the parameters pi using a multi-variate analysis. The least square method can be used to calculate the parameters Po.Pi, P 2, .... Pk- The least squares 74 method fits a straight line through the experimental data points in such a way as to minimize the sum of the squared deviations of the data points from the straight line. This squared deviation can be expressed as:

= £ ( y , - > ' p ) (J) 1=1 where y, can be written as:

(8)

Substituting the expression for y, into equation 7 yields:

A’ = X U - (A. + A,X, + + K + A X , )] (9) (=1

To determine the p, values which minimize A^, A^ should be differentiated with respect to each Pi and each equation should be set equal to zero. This procedure produces the following K+1 equations:

5A- = 2 Z[>'.-(Ao + AX,+^,x , + k +^^xJK-i) =0 ( 10) dP . im\

dA- = 2 Î U -(^. + AX, +AX, +K +a x JK-x,)1=0 l=,|

dA- = 2 X U -(Ao + AX, + A.x, + K. + A X JK- X.) = 0 dP .

ÔA- = 2 £ k - (/9o + AX, + AX: + K+AX^ )X- XK ) = 0 1=1

75 After distributing ttie summation of eacti term, and dividing each equation by the number of data points, n, the coefficients Po,Pi, P 2 PK are determined. After the determination of these coefficients, one has to know how well the analysis predicts, or models, the actual experimental data. This is achieved by determining the correlation coefficient, r^. r^ is defined as;

r- = 1 - n — K —\ /si or

/

r = 1 - - ( 11] n — K — I I»! where n is the number of data, K is the number of variables in the multiparameter model, and

V is the mean of y. The correlation coefficient, r^, predicts how good of a linear fit the

multiparameter model is to the actual data. An r^ = 1 denotes a perfect linear correlation

between the predicted and the actual values of y, while a r^ = 0 denotes a no linear correlation. The spread, or scatter, of the experimental data about the predicted best-fit straight line can be characterize by a useful parameter know as variance. Variance is simply equal to the square of the standard deviation. The variance of the variable y, which is denoted by Var(y), can be expressed as follows:

K K r)=£A’Mx,)+X £a^/Cov(x„X,) (1 2 )

W here Cov(x,,Xy) is the covariance of X , and Xy , and can be expressed by;

76 C 0v(x, , Xy ) — /7,yCTx,Cryy — E (X ,X y ) E (X ; )E (X y ) (13)

In Equation 13, pij is the correlation coefficient between X, and X ,, and and are the standard deviations of the variables X,. and X y, respectively. The expressions E( ) denote the expected, or mean, value of the variables indicated in the parenthesis. When pi, = =

1 , variables X,. and Xy are perfectly linearly related, and when pij = ± 0 , there is no linear relationship between the two variables. Therefore, the magnitude of the correlation coefficient Pij is a measure of the degree of interrelationship between two variables [16]. Using the concept of the correlation coefficient pi,-, Equation 12 can be rewritten as:

K K

/«I ya

When all the variables are linearly independent of one another, equation 14 reduces to;

= (15) 1=1

In the evolution of the multiparameter model, the fatigue crack growth rate, da/dN was modeled using a multi-varlate expression which was multiplicative in nature (Equation 4). This can be linearized, as shown by equation 5. The central limit theorem of statistics deals with the limiting distributions of sum of random variables. According the central limit theorem, a physical process that is the result of a large number of individual effects tends toward a normal, Gaussian, distribution. Another result of the central limit theorem is the statement that when a quantity is equal to the product of a number of factors, then the distribution of that quantity will tend towards a lognormal distribution, for a large number of observation [16]. As a result, the multiparameter model introduced above is only appropriate if da/dN has a lognormal distribution. This has been shown to be the case for the sam e mill annealed Ti-6AI-4V alloy that was employed in this study [17].

77 As mentioned in section 3.2.4, a statistical software known as Minitab was used to calculate the coefficients/exponents (Equation 5), correlation coefficients (Equation 11). and the statistical distributions of fatigue crack growth rate of the multiparameter model introduced In this section.

4.4.2 Results and discussion of the multioarameter modeling:

Using the multiparameter model described above, a single multiparameter relationship was obtained between da/dN and the stress intensity factor range, AK, stress ratio, R, specimen thickness, t, maximum stress intensity factor, K^ax, and closure stress intensity factor, K^t. From Equation 4, this gives:

^ = a,(AK)"' {r Y'- (/)“’ (K„r (16)

As shown earlier in Equation 5, the linearized form of Equation 16 is given by ;

^ da ^ In = ln (a j+ a , ln(AK) + a , ln(/?) + aj ln(/) + or 4 In(K^) + a; ln(K,,) (17)

The unknown coefficients/exponents and the correlation coefficient, r^, were calculated during the statistical analysis. Substituting the results of the analyses into equations 16 and 17 gives:

— = 2.81 xIO "(AK)^°'(R)'"^(t)^°°'(Km«)"'(Kc,)'^'' (18) d N and

In = -24.30+3.01 [ In (AK)]+0.44[ In (R)]-0.002[ In (t)]+3.82[ In (Kma,)]-2.32[ In (K^)] (19)

78 The correlation coefficient ^ = 0.94 for Equation 19. and ^ = 0.89 for Equation 18. This indicates a very good correiation between the measured (experimental) and calculated values of ln(da/dN), as shown in Figure 37. Perfect correlation, r^ = 1, would correspond to all the data points being on the 45° iine where the value predicted from the muiti-variate anaiysis is exactiy equal to the measured value. It is also important to note here that the correlation coefficient between the calculated da/dN and measured da/dN is only r^ = 0.89, as shown in Figure 38. Since the r^ of 0.89 obtained for the da/dN data is iess than the r^ of 0.94 obtained for the in(da/dN) data, the iognormal distribution should give a better prediction of errors (in the fatigue crack growth data) than the normai distribution, as discussed earlier.

The exponents of the variables in Equation 18 can be used to denote the relative importance of the different variabies. Equation 18 suggests that the most important factor that contributes to fatigue crack growth in Ti-6/\l-4V is K^ax, which has a positive exponent of 3.82. The second most important variabie is the stress intensity factor range, AK, which has a positive exponent of 3.01. This is foilowed by the stress ratio, R, with the least positive exponent of 0.44. The closure stress intensity factor, Kc, is also significant since it has a relatively high negative exponent. -3.32. The specimen thickness, t, has the least effect on fatigue crack growth rate in Ti-6AI-4V, with a smali negative exponent of -0.002. This insignificant effect of thickness on fatigue crack growth rate was also observed in Figure 32.

Figure 33 shows that when aii of the experimental fatigue crack growth curves were piotted against the effective stress intensity factor, AK^ff, they aii coiiapsed onto the same line. As a resuit the fatigue crack growth rate can be expressed by:

^-C(aK,^)" (2 0 )

By linearizing Equation 20, and performing the statisticai analysis. Equation 20 can be rewritten as:

= 6.07E -1 o(aK^^ )'*'*' mm/cycle (2 1 )

79 where AKe» is expressed in MPa Vm. Equation 21 shows a very high correlation coefficient (r^ = 0.93). This suggests that the single variable, AKeff, characterizes the fatigue crack growth rates in Ti-6AI-4V very well. The current multiparameter results, along with results obtained from other previous tests from the same block of Ti-6AI-4V [18,19], are shown in Table 4. It is important to note here that the discussion above neglects the inter-dependence between the different variables. For example, R and Kmax are inter-related. Hence, the exponent of R of 0.44 is relatively small when both Kmax and R are included in the multiparameter formulation (Table 4). Similarly, the closure stress intensity factor, K^, and the stress ratio are inter-related. Hence the exponent of R is small when the multiparameter framework includes the effects of which account for most of the differences in fatigue crack growth rates over the range of positive stress ratios between 0.025 and 0.8. The thickness variable, t, which corresponds to the transition from nearly plane stress (at low thickness) to plane strain conditions (at high thickness) appears to have a relatively small effect on da/dN for values of t between 1.6 mm and 12.7 mm.

Similarly, frequencies between 10 and 100 Hz appear to have very little effect on fatigue crack growth rates, since the exponent of O in Table 4 is close to zero. Similar observations have been made by Ritchie et al. [20] in their studies of fatigue crack growth at -1000 Hz, i.e. the fatigue crack growth rates at -1000Hz and close to those at lower frequencies (-10-100 Hz). Tschegg et al. [21] have also reported almost no frequency effects at cycle frequencies up to 20 kHz. Studies at higher frequencies than those examined in this study are, therefore, consistent with the low value of the exponent of -0.175 (close to zero) obtained for Q (Table 4).

The results in Refs 20 and 21 suggest that crack growth predictions for aeroengines operating at cyclic frequencies up to -20 kHz may neglect the role of frequency as an additional variable. Similarly, section thickness, t, does not appear to have a significant effect on fatigue crack growth rates for the range of thickness (1.6 mm-12.7 mm) that was examined. The remaining variables in Table 4, however, do appear to have a significant effect on fatigue crack growth rates. Furthermore, the relative contributions from the different variables depend to some extent on the inter-relationships between variables. These inter-relationships can be quantified by considering the correlation coefficients between the variables.

80 The correlation coefficients that were obtained for Ti-6AI-4V are summarized in Table 5. Note that large positive correlation coefficients indicate that when large values of one variable are observed, large values of the other variable will occur. On the other hand, large negative correlation coefficients suggest that when large values of one variable are observed, small values of the other variable will occur. The correlation coefficients in Table 5 provide some important new insights. For example, the correlation coefficients between ln(t) and the other variables are very small. This implies that thickness is almost independent of the other variables. Also, the correlation coefficient between In(Kmax) and In(Kc) is very large. This implies that there is a strong linear relationship between Kmax and K^. For the most part, all of the variables, except for thickness, are linearly correlated to some degree.

Form the above discussion, it can be seen that for the wide range of data that were examined, the multiparameter fatigue crack growth equation was found to provide a good fit to the measured fatigue crack growth rate data reported for Ti-6AI-4V [18,19]. Typical correlation coefficients were between 0.93 and 0.99 in all the cases that were investigated (Table 4). However, the multiparameter constants were significantly different, presumably because of the

difference in the microstructures. Also, a K appears to be the variable with most contributions to fatigue crack growth rate, da/dN, in most data sets [18,19] that are presented in Table 4. The relative magnitude of the exponents also vary, depending on the number of variables considered, and the possible interdependence between variables. Nevertheless, it is clear that the multiparameter crack growth equation does provide an adequate fit for the parametric ranges of stress intensity factor range, maximum stress intensity factor, stress ratio, closure stress intensity factor, frequency and temperature that were considered in this investigation (Table 4).

It should also be apparent from the above discussion that the multiparameter constants are only applicable to particular alloy/microstructure combinations. No single multiparameter equation (of the form produced here) may, therefore, be used to fit all the available fatigue crack growth rate data for Ti-6AI-4V without further attempts to asse ss the effects of microstructure and slight variations in alloy chemistry. Nevertheless, the parameters presented in Table 4 are still very useful because they fit the fatigue crack growth behavior of particular alloy/chemistry/microstructure combinations [18,19]. They may, therefore, be used

81 to predict fatigue crack growth in specific structures and components that are fabricated from materials with similar alloy chemistry/microstructure combinations.

The multiparameter framework may also be extended to other materials apart from the Ti- 6AI-4V alloy considered in the above discussion. This has been demonstrated in the recent work on inconel 718 [22,23] and HY 80 steel [24]. The multiparameter constants obtained for

these alloys are summarized in Table 6 . As shown earlier for Ti-6AI-4V (Table 4), the correlation coefficients were very good for the parametric ranges of variables (AK, K^w, K^, and t) that were considered. The multiparameter framework, therefore, appears to provide a general framework for the combined assessment of the effects of multiple variables on the fatigue crack growth rates in metallic materials. It may also provide a general framework for the prediction of fatigue crack growth in practical cases that are likely to be controlled by the combined effects of multiple variables.

4.4.3 Fatigue life prediction using the multioarameter approach:

Finally, it is important to note here that the multiparameter approach may be used to simplify the prediction of fatigue crack growth and fatigue life, since the combined effects of multiple variables can be readily assessed by a single equation. Furthermore, the amount of fatigue crack growth, Aa, that occurs with an incremental number of cycles, AN, may be estimated by the separation of variables in Equation 4, and integration between the appropriate limits. This gives;

\da = aJ\^ X :'dN (22)

0 >=•

where a' = ^ A a = ^ Y(AA(), and N' = ^ (a ^ ). Equation 2 2 is, therefore, the basis for

the estimation of fatigue life N' . This is achieved readily by the numerical integration of

Equation 4, from assumed initial crack size at Y = 0, up to the value of crack size

82 a = a , when the number of cycles to failure is TV = N'. The determination of the total crack size, a = a ' , therefore, reduces to a simple step-by-step numerical integration process.

However, the required (fc + 1) constants ( , rz,, rr, ) in equation 22 must be determined from experimental data, using multiple linear regression methods described earlier. This should make it possible to incorporate the combined effects of stress intensity factor range, stress ratio, closure stress intensity factor, thickness, temperature, and frequency into numerical integration schemes for the prediction of fatigue crack growth or fatigue lives of engineering structures and components that operate over wide ranges of frequency, temperature and stress.

The implications of the above modeling framework are really significant. For example, considering the general case of an aeroengine with components rotating at high speeds and temperature. It should be possible to formulate multiparameter fatigue crack growth prediction schemes in which the combined effects of temperature, section thickness, stress intensity factor range and mean stress are all included within a single life prediction equation. The problem of fatigue crack growth prediction could thus be reduced to the simple numerical integration of multiparameter equations.

Finally, it is important to note that the above discussion applies largely to the prediction of long crack growth. Further work is clearly needed to extend the current framework to the prediction of short crack growth. There is also a need to extend the current deterministic multiparameter approach to a probabilistic approach that will allov/ for the prediction of material reliability as a function of multiple variables. Nevertheless, it is hoped that the multiparameter equation will provide engineers in industry with a robust tool for the prediction of long fatigue crack growth.

4.5 Conclusions:

1. Specimen thickness in the range between 1.59 mm and 12.7 mm (1/16 in. to 1/2 in.) does not significantly affect the fatigue crack growth rates and crack closure levels in T1-6AI-4V alloy. On the other had, positive stress ratios between 0.02 and 0.80 have a significant effect on 83 crack closure levels and fatigue crack growth rates. However, when the fatigue crack gro\wth rate data for all the thicknesses and stress ratio were plotted against the effective stress intensity factor, AKeff, the data collapsed into a single effective fatigue crack growth rate curve. This indicates that the effects of stress ratio and specimen thickness on the fatigue crack growth rates in Ti-6AI-4V may be attributed largely to shielding contributions from crack closure.

2. Fatigue crack growth in Ti-6A1-4V occurs by a static “cleavage-like" fracture mode in the low AK regime. Crack growth in the intermediate AK regime occurs by classical crack-tip blunting mechanism that give rise to fatigue striations. A combination of fatigue striations and static ductile dimpled fracture modes is observed in the high-AK regime. The incidence of static (ductile dimples and cleavage-like) fracture modes increases with increasing Kma* and AK. However, specimen thickness does not appear to have a significant effect on the fracture modes over the range of specimen thicknesses that were examined in this study.

3. A multiparameter framework is proposed for the assessment of the combined effects of multiple variables on fatigue crack growth in metallic materials. The framework provides a single equation that assesses the combined effects of stress intensity factor range, maximum stress intensity factor, closure stress intensity factor, frequency, test temperature and nominal specimen thickness. The multiparameter approach has also been validated for fatigue crack growth in T1-6AI-4V, inconel 718 and HY-80 pressure vessel steel. It may, therefore, be used as a general framework for the estimation of fatigue crack growth in metallic materials.

4. The multiparameter equation is a simple extension of the single parameter Paris equation. The combined effects of all the different variables may, therefore, be considered within the Paris coefficient, C, which is also a function of multiple variables.

5. The variabilities in da/dN due to mechanical fatigue variables such as AK, Kmax, R. K^, and t are described adequately with lognormal distributions (of In da/dN) for fatigue crack grovirth in TÎ-6A1-4V. furthermore, the lognormal dependence of the fatigue crack growth rate suggests that the effects of the above variables are additive.

84 4.6 References:

1. S. Dubey, A. B. O. Soboyejo and W. 0. Soboyejo, 1996 An investigation of the effects of stress ratio and crack closure on the micromechanism of fatigue crack growth in Ti-6A!-4V. Acta Mater, Vol. 45, 2777-2787.

2. M. Katcher, and M. Kaplan, 1974, Effects of R-factor and crack closure on fatigue crack growth in Aluminum and Titanium alloys, fracture toughness and slow-stable cracking, ASTM STP 559. American Society for Testing and Materials, 264-282.

3. H. Doker, and V. Bachmann, 1988, Determination of crack opening load by use of threshold behavior, in mechanics of fatigue crack closure, ASTM STP 982. American Society of Testing and Materials, 247-259.

4. A. J. McEvily, 1977, Current aspects of fatigue. Metal Science 11, 274-284.

5. S. Suresh, 1985, Fatigue crack deflection and fracture surface contact: micromechanical models. Metallurgical Transactions 16A, 249-260.

6 . 0. J. Beevers and M. D. Halliday, 1979, Non-closure of cracks and fatigue crack growth in P heat treated TI-6AI-4V. International Journal of Fracture, 15, R27-R30.

7. J. E. Allison, 1988, The measurem ent of crack closure during fatigue crack growth. Fracture Mechanics, 18*^ Symposium, ASTM STP 945, D. T. Read and R. P. Reed, Editors. American Society for Testing and Materials, Philadelphia, 913-933.

8 . H. Sehitoglu and A. M. Garcia, 1997, Contact of crack surfaces during fatigue. Metall.Mater Trans, 28A, 2263-2289.

9. B. Budiansky and J. W. Hutchinson, 1978, Analysis of closure in fatigue crack growth. Journal of Applied Mechanics 45, 267-276.

85 10. J. c.Newman, 1976, A finite element analysis of fatigue crack closure. Mechanics of Fatigue Crack Growth, Special technical publication 590, 281-301. Philadelphia: American Society of Testing and Materials.

11. N. A. Fleck, 1986, Finite element analysis of plastlclty-lnduced crack closure under plane strain conditions. Engineering Fracture Mechanics 25, 441-449.

12. W. O. Soboyejo and J. F. Knott, 1990, Effects of stress ratio on fatigue crack propagation in QIN (HY-80) pressure vessel steel. Intematlonal Journal of Fatigue, 12, 403-407.

13. S. Suresh, 1998, Fatigue of materials, Cambridge University Press, 2"" Edition.

14. R. O. Ritchie and J. F. Knott, 1973, Mechanisms of fatigue crack growth In a low alloy steel. Acta Metallurgica 21, 639-650.

15. P. S. Laplace, Theorle analytique des probabilités. Pahs, l". ed. 1812; 2"". Ed.,1814; and 3"*. ed., 1820.

16. A. H. Ang and W. H. Tang, Probability concepts In engineering planning and design, Volume I, basic concepts, John Wiley & Sons, 1975.

17. Shen. W. Shademan. S. Soboyejo. A. B. O, and Soboyejo. W, A probabilistic framework to the modeling of fatigue crack growth. International Journal of Fatigue. In preparation.

18. A. B. O. Soboyejo, S. Shademan, M. Foster, N. Katsube, and W. O. Soboyejo. To be published.

19. M. A. Foster, 1998, An Investigation of the fatigue crack growth and fracture behavior of Tl- 6AI-4V, M.S thesis. Department of Aerospace Engineering, Applied Mechanics and Aviation, The Ohio State University, Columbus, Ohio, USA.

86 20. R. O. Ritchie, High cycle fatigue and time-dependent failure in metallic alloys for propulsion systems, Proceedings of the Metallic Materials Contractors Meeting, San Diego, CA, March 1999, Air Force Office of Scientific Research, District of Columbia, 129-133.

21. S. E. Stanzl-Tschegg and E. K. Tschegg, 1998 unpublished research. Institute for solid state physics, Vienna University, Austria.

22. C. Mercer, A. B. 0. Soboyejo and W. 0. Soboyejo, 1997, An investigation of fatigue crack growth in Inconel 718, in High Cycle Fatigue of Structural Materials. In Honor of Professor Paul C. Paris, at Indianapolis, Indiana. Edited by W. O. Soboyejo, and I. S. Srivatsan, The Mineral, Metals and Materials Society, and ASM International, 269-285.

23. C. Mercer, W. O. Soboyejo, and A. B. 0. Soboyejo, 1999, Mechanisms of fatigue crack growth in a single crystal Inconel 718 nickel-base superalloy. Acta Mater, (submitted).

24. W. O. Soboyejo, and J. F. Knott, 1995, An investigation of crack closure and the propagation of semi-elliptical fatigue cracks in QIN pressure vessel steel. International Journal of Fatigue, 17, 577-581.

87 : R=0.25, t=12.7 m m 10" I I I I I I. • R=0.50, t=12.7 m m E R=0.80, t=12.7 m m

_© 0 ^ 10"^

E

1 10^ ■o

10^ ■ ■ ■ ■ I ■I I 1 I I I I 10 1 0 0 AK, MPa (m) 1/2 (a)

R=0.02, t=6.4 mm 1-3 R=0.25, t=6.4 mm R=0.50, t=6.4 mm R=0.80, t=6.4 mm

o

Z ? ■O

1 0 0 AK, MPa (b) ...continued on next page Figure 31. Fatigue crack growth rate versus stress intensity factor range, AK, for specimens with thicknesses of: (a) 12.7 mm; (b) 6.4; (c) 3.2 mm and (d) 1.6 mm.

88 da/dN, (mm/cycle) da/dN, (mm/cycle) CO c (D w I n I 11 n il] I I I m i l -I— I I 1 1 n i |------1— I I I i iT if 83 5- c a> Q.

> 7s o 00 KJ o VO s g in K» 0) O o cn to "ir ir ir TT TT TT WWW to to io n IX o ai at Ui

O o O o lllL ~ t=3.2 m m , R=0.02 • t=6.4 mm, R=0.02

Ü I E z

■o

1 0 0 AK. MPa (m)^^ (a)

- t=1.6mm. R=0.25 TT 10" • t=3.2 mm. R=0.25 Z t=6.4 mm. R=0.25 > t=12.7 mm. R=0.25 o 10" fE

1 10-® •D

10" - i l — t I t il I 11 I > I > l . i 1 10 1 0 0 AK, MPa

(b) ...continued on next page Figure 32. Fatigue crack growth rate versus stress intensity factor range, AK, for stress ratios, R, of: (a) 0.02; (b) 0.25; (c) 0.50 and (d) 0.80. 90 Figure 32 (continued)

t=1.6 mm, R=0.50 10 “ t=6.4 mm, R=0.50 t=12.7 mm, R=0.50 f u >> I & 10" Z % •o S 10" I I I I 1 I J. 10 1 0 0 AK, MPa (m) 1/2 (c)

t=1.6mm, R=0.80 1-4 t=3.2 mm, R=0.80 t=6.4 mm, R=0.80 t=12.7 mm, R=0.80

0 >. 1 E ,-5

% •o

1-6 1 1 0 0 AK, MPa (d) 91 2 R=0,25, #1.6 m m = R=0.2S, # 6 .4 m m R—0.50, t=1.6 m m R =0.50, # 6 .4 m m 5 R=0.80, # 1 .6 m m - R =0.80, # 6 .4 m m O R=0.02, #3.2 mm R=0.2S, #12.7 mm 10" 5 R=0.25, # 3 .2 m m R=0.50, #12.7 mm X R=0.80, #3.2 mm : R=0.80, #12.7 mm R=0.02, # 6 .4 m m o 0 1 1 0 ^

E _____ z % 10"

10" ± 7 8 9 10 20 AK, MPa (m) 1/2

(a)

Figure 33. Fatigue crack growth rate versus (a) stress intensity factor range. AK, and (b) effective stress intensity factor range, AKeff.

92 da/dN, (mm/cycle)

en ^ ° I I r n nq------1—i i 111ii|------1—i i 11m i w w 0 5 83 I A III O H II < ) D -vj C 8 00 ???????p p p p p p p o ô» k» o 00 ai k> CD joo ph» p o p

oTT pir pir p V ir p TT p TT ^k> loto a> b> b>

"0 iiiiiii 0> o I ■ I i ■ l I

?p op ? ? p ?p ?p ? k kk) k g k) O O en O ai '■'S N3 O -k -k 0 > o> o> lïiiii

I I i I l i i l - 1 i J.J ill I Ê i U i l i J t=3.2 mm, R=0.02 t=3.2 mm, R=0.25 t=6.4 mm, R=0.02 t=6.4 mm, R=0.2S

£ 3 0.7 (0 o Ü u . m 0.6 n ui

0.5 5 10 15 20 AK, MPa (m) 1/2

Figure 34. Plots of Eiber closure ratios for the 3.2 mm and 6.4 mm thick specimens tested at stress ratios of 0.02 and 0.25.

94 Direction of crack growth

_ Low a K Intermediate a K — 5 p m 5 p m

High a K

Figure 35. Crack growth morphology of the 3.2 mm thick specimen tested at stress ratio of 0.02 at low a K, intermediate a K and high a K. Striations and ductile dimples are denoted by 8 and D. respectively.

95 Direction of crack growth

Low AK Intermediate AK

m

High AK

Figure 36. Crack growth morphology of the 12.7 mm thick specimen tested at stress ratio of 0.50 at low AK, intermediate AK and high AK. Striations and ductile dimples are denoted by 8 and D, respectively.

96 T T I I I n ■! I I I I I I I I T I I I I I I I I T 'J 1^ I

R‘ = 0.9433

C -14 » ■ ■ ■ I ■ ‘ ■ I I ■ ‘ ■ I ■ ■ ■ ' I ■ ' ' ■ ' -14 -13 -12 -11 -10 -9 -8 In (da/dN), mm/cycle - measured

Figure 37. Goodness of fit between ln(da/dN)-predicted and ln(da/dN)-measured using the multi-variate model.

3 10

o 2 .5 1 0 — 0,8886

È 2 1 0

5 1 0 G - 1 10

5 1 0

0 10' 010' 5 1 0 110“* 1.510 2 1 0 2 .5 1 0 da/dN, mm/cycle - measured

Figure 38. Goodness of fit between (da/dN)-predicted and (da/dN)-measured using the multi-variate model.

97 Multi­ Multiparameter exponents Multiple parameter Correlation Coefficients coefficient References ai a? fl4

2.81 X 10'" 3.01 0.44 -0.002 3.82 -2.32 0.94 [18.19] (AK) (R) (t) (Kmax) (Kd) 6.087X10'’° 4.412 0.93 [18.19] (AKe«) 4.90X10'” 10.33 0.35 -0.497 - - 0.97 [18.19] (AK) (R) (t) 1.439 X 10'" 6.165 1.29 0.791 -0.63 - 0.99 [18-19] (AK) (R) (t) (Kd) 1.068 X 10'" 3.664 0.205 -0.205 -0.175 0.268 0.93 [19] (AK) (R) (t) (n) (T)

Q Frequency T Temperature a K Stress Intensity Factor Range Kmax Maximum Stress Intensity Factor Kd Closure Stress Intensity Factor R Stress Ratio t Specimen Thickness

Table 4. Summary of exponents and correlation coefficients for the mill-annealed Ti-6AI-4V microstructure.

Parameter ln(A/C) M«) In(r) In K ) ln(A/:) 1 -0.781 -0.102 -0.263 -0.545 ln(/î) -0.781 1 0.023 0.470 0.641 ln(() -0.102 0.023 1 -0.112 -0.058 0.263 0.470 -0.112 1 0.926 0.545 0.641 -0.058 0.926 1

Table 5. Summary of the correlation coefficients for the mill-annealed Ti-6AI-4V microstructure.

98 Multi­ Multiparameter exponents Multiple parameter Correlation Material Coefficients coefficient References Oo at 0 2 0 3 0 4 0 5

HY-80 Steel 2.9 X 10’® 3.24 0.357 --- 0.92 [12.24] (Independent (AK) (R) Variables) HY-80 Steel 3.0 X 10 ® 3.33 0.468 -0.124 -- 0.94 [12.24] (Independent (AK) (R) (Kd) Variables) Inconel 718 3.39 X 10'® 3.63 0.52 0.97 [22.23] Polycrystal (AK) (R) (Independent Variables) Inconel 718 1 X 10''® 4.25 0.71 0.94 [23] Single Crystal (AK) (Kmax) (Independent Variables)

Table 6. Summary of multiparameter exponents and correlation coefficients for HY 80 steei and inconel 718.

99 CHAPTER 5

FATIGUE CRACK GROWTH BEHAVIOR OF p-ANNEALED WIDMANSTATTEN MICROSTRUCTURE

5.1 introduction:

As discussed in chapter 2, p-annealed titanium alloys have been shown to exhibit superior fatigue crack growth resistance and fracture toughness, as compared to mill annealed and other a+p processed titanium alloys [1-6]. However, the reasons for the improved crack growth resistance are not fully understood. In this chapter, the effects of microstructure on the fatigue crack growth behavior of p-annealed Widmanstatten Ti-6AI-4V are investigated.

The chapter begins with a description of p heat treatments that were used to produce the well controlled Widmanstatten microstructures. The fatigue crack growth rates and crack growth/shielding mechanisms observed in the Widmanstatten microstructures are then described. The role of crack closure is examined using closure-corrected (using clip gage) and uncorrected data. A roughness-induced crack closure model is then used to estimate the shielding associated with actual lateral roughness profiles of the fracture surfaces. Closure- corrected data obtained from this model are compared with experimental crack closure- corrected data (obtained from clip gage measurements) for the different Widmanstatten microstructures. Finally, the multiparameter model Introduced in chapter 4, will be extended to the modeling of the effects of Widmanstatten microstructural parameters.

5.2 Heat Treatments and Phase Transformations:

The specimens machined from the T1-6AI-4V mill annealed alloy were heated to 1070 °C and held for 30 minutes. This temperature is well above the p-transus temperature (980°C) for this alloy, as shown in Figure 39. In order to produce the different Widmanstatten microstructures,

100 the specimens were cooled at three different cooling rates (25°C/min, 3°C/min, and 1°C/min) to 400 °C. These different cooling rates resulted in three different Widmanstatten microstructures of varying colony and lath sizes. Since the kinetics of phase transformation is very slow and sluggish at temperatures below 400“C, thermal exposures below 400°C do not strongly influence the final microstructure. The specimens were therefore, argon quenched below 400°C.

Figure 40 shows the effect of different cooling rates on the microstructure. Both the colony and lath sizes are observed to increase with decreasing cooling rate. This is due to the lower nucléation rate and higher growth rate associated with the slower cooling rates. Table 7 [8] shows that by changing the cooling rate from 25°C/min to 1°C/min, the a lath size changes from 3.4 pm to 14.1 pm, and the colony size changes from 118 pm to 312 pm, respectively. It is also observed that for the slowest (1°C/min) and the intermediate (3°C/min) cooling rates, the p volume % is roughly the same (~ 20 volume %) as in the mill annealed condition [8]. However, for the fastest cooling rate (25°C/min), the p volume % in the microstructure is about 50% higher than in the other microstructures.

5.3 Fatigue Crack Growth Rate Behavior of the Three Widmanstatten Microstructures:

Figures 41a to 41c show the fatigue crack grovirth rate data obtained for the three Widmanstatten microstructures at four different stress ratios. The crack growth rates for the lowest stress ratio (R = 0.1 ) is much slower than the growth rate at the higher stress ratios (R = 0.25, R = 0.50, and R = 0.80). The trends in these fatigue crack growth rate data (increasing fatigue crack growth rate with increasing stress ratio for a given AK) are consistent with the trends in the fatigue crack growth rate data obtained for the mill-annealed microstructure (chapter 4).

In chapters 2 and 4, it was shown that the differences in fatigue crack growth rates at different stress ratios were attributed largely to the effects of different mechanisms of crack closure, especially, roughness-induced crack closure. As a result, when growth rates were plotted against the effective stress intensity factor, AKetr, the data collapsed on to a narrow 101 scatter band around a straight line (Figures 24b and 33b). For each Widmanstatten microstructure, in order to find out the extent of which the differences in the growth rates (at different stress ratios) are caused by differences in crack closure levels, closure levels were measured at stress ratios of R = 0.1 and R = 0.8. The ciosure measurements were obtained using a compliance technique, as explained in chapter 3. This involved the measurement of crack mouth opening displacements with a clip gage [8].

Figures 42a to 42c show the fatigue crack growth rate curves plotted against the effective stress intensity factor range, AK@R, for Widmanstatten microstructures A, B, and C, respectively. The figures show that the data for each Widmanstatten microstructure collapse Into a single effective fatigue crack growth rate curve at both stress ratios. This again indicates that the effects of stress ratio on the fatigue crack growth rates in each Widmanstatten microstructure may be explained largely by different shielding contributions from crack closure.

Figures 43a to 43d compare the fatigue crack growth rates for the three Widmanstatten microstructures for similar stress ratios. Figures 43a to 43c show that for stress ratios of 0.1, 0.25, and 0.50, the coarsest microstructure (C) exhibits the slowest growth rates in the Paris regime. For microstructures (A), and (B) (finest and intermediate microstructures) the growth rates appear to be similar across the different regimes of crack growth. However, at stress ratio of 0.8 (Figure 43d), the fatigue crack growth rates appear to be similar for the three Widmanstatten microstructures, except in the upper Paris regime. In the upper Paris regime, the lowest growth rates are associated with coarsest microstructure (C).

In order to explore the extent to which the differences in the growth rates were caused by differences in crack closure levels, crack closure measurements were made at stress ratios, R = 0.1 and R = 0.8. Figures 44a and 44b show the fatigue crack growth rates plotted against the effective stress intensity factor, AKgm, for stress ratios of 0.1 and 0.80, respectively. Figure 44a shows that at R = 0.1, the closure-corrected data collapse closer together in the near­ threshold and Paris regimes. However, the closure-corrected data do not collapse in the high AK regime where static modes also contribute to crack growth. Also, the coarsest microstructure (C) still exhibits slower growth rates at the same applied AK. This data indicates that there are some intrinsic differences between the fatigue crack growth resistance

102 of the different microstructures. Hence, the differences in the fatigue crack growth rates can not explained solely by crack closure arguments. For R = 0.8, the corrected data did not show any evidence of closure. As a result, the applied stress intensity factor, AK, rather than can still be used to describe the effective crack driving force at this stress ratio. Therefore, Figure 44b is the same as Figure 43d.

5.3.1 Comparisons of the fatigue crack growth rates of the Widmanstatten microstructures (containing almost the same volume % of B phase) and mill annealed microstructure:

Figures 45a to 45d compare the fatigue crack growth rates in Widmanstatten microstructures (C) and (B) and the mill-annealed microstructure (D) at stress ratios of 0.1, 0.25, 0.50, and 0.8, respectively. As indicated in Table 7, for all the microstructures, the p volume % is approximately the same (~ 22%). Figures 45a to 45c show that at stress ratios of 0.1, 0.25, and 0.50, the growth rate for the mill annealed microstructure is faster. Also, for a fixed p volume %, the coarser Widmanstatten microstructure exhibits a slower growth rate. This slower growth rate can be attributed to the coarser colony size of microstructure C, which is consistent with the earlier studies on o/p titanium alloys by Yoder et. al. [9.10]. In contrast, Figure 45d shows that at stress ratio of 0.8, the three microstructures exhibit similar fatigue crack growth rates in the near-threshold and lower Paris regime. However, in the high a K regime, the two Widmanstatten microstructures exhibit slower growth rates than the mill- annealed microstructure. This is attributed to differences in the intrinsic resistance to fatigue crack growth rate in the different microstructures.

The closure-corrected data obtained at a stress ratio of 0.1 and 0.8 are shown in Figures 46a and 46b, respectively. Figure 46a shows that at R = 0.1, the closure-corrected data for the three microstructures collapse closer together in the near threshold and lower Paris regime. This strongly suggests that closure is a major contributor to the differences in the crack growth of the three microstructures in these regions. However, at high AK levels, even after closure- correction, the coarsest microstructure (C) still shows some mismatch in the growth rate data. This again can be attributed to the differences in intrinsic resistance of the three microstructures to fatigue crack growth. There was no closure detected at R = 0.8 for the three

103 microstructures. As a result, the closure-corrected plots (Figure 46b) are the same as the plots without closure-correction (Figure 45d).

5.3.2 Comparisons of the fatigue crack growth rates of the Widmanstatten microstructures (containing different volume % of B phase) and mill annealed microstructure:

Figures 47a to 47d compare the fatigue crack growth rates of Widmanstatten microstructure A (containing - 33 volume % P) with those of Widmanstatten microstructure B and mill annealed (both containing ~ 22 volume % P) at stress ratios of 0.1, 0.25, 0.50 and 0.8, respectively. Figures 47a and 47b show that at R = 0.1 and R = 0.25, the mill-annealed microstructure exhibits the fastest fatigue crack growth rate. At higher stress ratios (R = 0.50 and R = 0.80), Figures 47c and 47d show that growth rates in the near threshold and lower Paris regimes are generally similar for the different microstructures. However, at higher AK values (in the upper Paris regime), the mill-annealed data again exhibits faster growth rates. These results indicate that microstructural morphology (near-equiaxed or Widmanstatten) rather than p volume % has a greater effect on fatigue crack growth rates.

Figures 48a and 48b show the closer-corrected fatigue crack growth rate data obtained for the three microstructures at stress ratios of 0.1 and 0.8, respectively. At R = 0.1, Figure 48a shows that the closure-corrected crack growth rate data for the three microstructures collapse closer together. This again confirms that, at low stress ratios the differences in fatigue crack growth rates are due primarily to different levels of crack closure associated with the different microstructures. As in the last two cases, there was no crack closure detected at stress ratio of 0.8. Therefore, Figure 48b shows that the closure-corrected plot is the same as the growth rate plots without closure-correction (Figure 47d).

5.4 Crack/Microstructure interactions:

In the three Widmanstatten microstructures studied in this investigation, the fatigue crack path was frequently observed to deviate from the pure mode I direction. In all the three microstructures, cracks deviated from their original growth direction at prior p grain boundaries

104 and at colony boundaries, as shown in Figures 49. in contrast, the deviations in the crack growth directions for the miii-anneaied microstructure was relatively small (Figure 50).

Crack deviation or deflection result in different local mode I and mode II contributions to crack growth, and hence relative local displacements of the fracture faces in the plane of the crack. This difference in local displacements of the fracture faces causes mismatch and contact at discrete points. The load at which contact across the fracture faces occurs is influenced by the coarseness of the fracture faces, which in tum is dependent upon the size of the microstructural units (colony size).

Table 7 shows that the coarseness of the microstructure can be significantly increased by P-annealing of the Ti-6AI-4V alloy (Widmanstatten microstructure), as compared with mill- annealed condition. The coarseness of the Widmanstatten microstructure gives rise to higher levels of crack deflection and roughness-induced crack closure and hence, lower growth rates during the low stress ratio fatigue tests. This is consistent with the results of prior studies of fatigue crack growth in o/p titanium alloys [9-11]. Roughness-induced crack closure has recently been shown to be the dominant crack closure mechanism in a/p titanium alloys [12,13]. We will, therefore, explore the extent to which roughness-induced crack closure can be used to explain the obsen/ed differences in the fatigue crack growth behavior.

5.5 Fatigue fracture modes:

Figure 51 shows the macroscopic view of the fatigue fracture surfaces of the three Widmanstatten microstructures and the mill-annealed (near-equiaxed) microstructure, using a stereo optical microscope [8]. All the specimens were tested at a stress ratio of R = 0.1. By comparing these fractographs, it is clear that the fracture surfaces of the Widmanstatten microstructures are much more faceted than the mill-annealed microstructure. Also, the facet size for the Widmanstatten microstructure C (coarse) is much larger than the others.

The observed differences in the fracture modes, as illustrated in Figure 51, may explain the differences in the closure-corrected fatigue crack growth rates for the different

105 microstructures. As a result, these clearly observed differences in fracture modes could be related to differences in the intrinsic resistance of different microstructures to fatigue crack growth rate. The larger facet size in microstructure C, increases the fracture surface roughness, which may give rise to higher roughness-induced crack closure since the crack faces are more likely to come into contact when the heights of the asperities are comparable to crack-tip opening displacements. As a result, the closure-corrected intrinsic fatigue crack growth rate for the coarser microstructures should be slower, as was observed in Figures 44 and 45.

The fatigue fracture modes for the three Widmanstatten microstructures at stress ratios of 0.1, 0.25, 0.50, and 0.80 were very similar in the near threshold, intermediate AK, and high AK regimes. Only the relative proportions of each fracture modes observed on the entire fracture surface appeared to vary with stress ratio. As a result, the fracture modes in the three Widmanstatten microstructures at stress ratios of 0.25 and 0.50 are not presented here.

Figures 52 and 53 show the SEM fractographs of the fracture modes at near threshold, intermediate AK and high AK regimes for the Widmanstatten microstructure A (fine) tested at stress ratios of 0.1, and 0.80, respectively. In the near threshold regime (Figures 52a and 53a), the fracture modes exhibits cleavage-like features at both stress ratios. There was some evidence of closure debris on the fracture surfaces at R = 0.1 (Figure 53a). The debris are formed due to contact between the fracture surfaces in the near threshold regime. The debris are absent on the fracture surface at R = 0.80 (Figure 53a). This is attributed to the lack of contact between the fracture faces at this high stress ratio where the crack opening displacements were large, even in the near threshold regime. Crack growth in the intermediate AK regime occurred by a crystallographic “step-like” fracture mode at both stress ratios (Figures 52b and 53b). A crystallographic “step-like” fracture mode, and some evidence of secondary cracking were observed in the high AK regime at both stress ratios (Figures 52c and 53c).

Figures 54 and 55 show the SEM fractographs of the fracture modes at near threshold, intermediate AK, and high AK regimes for the Widmanstatten microstructure B (intermediate) tested at stress ratio of 0.1, and 0.80, respectively. In the near threshold (Figures 54a and

106 55a), and intermediate AK regimes (Figures 54b and 55b), the fracture modes at both stress ratios are similar to those in the Widmanstatten microstructure A. However, fracture mode in the high AK regime, at both stress ratios, exhibited cleavage facets plus secondary cracking (Figures 54c and 55c).

Figures 56 and 57 show the SEM fractographs of the fracture modes at near threshold, intermediate AK, and high AK regimes for the Widmanstatten microstructure C (coarse) tested at stress ratio of 0.1, and 0.80, respectively. Similar to the other Widmanstatten microstructures, in the near threshold regime, the fracture modes at both stress ratios exhibit “cleavage-like” features (Figures 56a and 57a). In the intermediate AK regime, the fracture modes were very tortuous and showed “cleavage-like” features plus large facets at both stress ratios (Figures 56b and 57b). In the high AK regime, the fracture mode, as illustrated by Figures 56c and 57c, exhibited small and large facets at both stress ratios.

5.6 Modeling of roughness-induced crack closure:

It is apparent form the above discussion that roughness-induced crack closure plays a significant role in the shielding of crack-tips in the Ti-6AI-4V alloy. It is, therefore, of interest to quantify the shielding contributions form roughness-induced crack closure. This may be attempted by considering first an idealized case of a fatigue crack which consists of segments of periodic deflections in its path, such as the one shown in Figure 58 [14]. In this segment, 6 is the kink angle, D the distance over which the tilted crack advances along the kink, and S the distance over which Mode I crack growth occurs in each segment.

In the tilted or deflected segment, D, the local tensile opening (Mode I) and sliding (Mode n) stress intensity factors are given by k, and ka, respectively. The elastic solutions for k, and ka are given by [15-17]:

K =aii(^) = cos^|^- K,

107 \ 1 ( e \ COS K, ( 1) ) UJ where K, is the instantaneous value of the nominal Mode I stress intensity factor for the fatigue cycle. The effective stress intensity factor, k, for coplanar growth along the deflected segment, D, is provided by the strain energy release rate criterion, which is given by;

k = ^jk; + k ; (2)

as a result, the effective stress intensity factor range along the deflected span, D, and the straight span, S, can be expressed, respectively by;

- ^,n.n ^^d M = AK, (3)

where k^æ, and k^m represent the values of k at maximum and minimum loads of the fatigue cycle, respectively, and AK, is the nominal far-field stress intensity factor range. By combining Equations 2 and 3, the weighted average of effective stress intensity factor range for the periodically deflected crack can be determined. This can be expressed by;

D c o s ' + S A k = *AK, = DAK, (4) D + S

The fatigue crack growth rate, da/dN, along the deflected segment, D, is given by;

d a D c o s d + S (—] (5 ) d N ~ D + S

where (da/dN)u is the growth rate along the Mode I direction.

108 Equations 4 and 5 describe the modifications to crack driving force and growth rates of deflected cracks without premature contact. However premature contact is promoted between the fracture surfaces because of net mismatch between them. The net mismatch arises from a variety of irreversible deformation mechanisms. This net mismatch causes roughness-induced crack closure and further reduces the effective driving force for fatigue crack growth in Equation 4.

Figure 59 schematically shows the opening profile of a deflected fatigue crack, located just behind the crack tip, at the maximum far-field tensile stresses. As shown, A6 denotes the opening displacement between the crack faces. As the crack unloads, a net mismatch, un, develops (due to irreversible deformation) above the for the fatigue cycle (note that U|, is not the Mode II displacement due to local k;). Due to this lateral shift u„, the cyclic opening displacement becomes, A5* = A6-u, = Uutan 9, where u, is the Mode I opening displacement form the peak of the fatigue cycle to the point of first contact. By representing the extent of mismatch by the nondimensional parameter, X = u»/ U|, the magnitude of the closure effect can be written as:

AÔ' f Àtanû V '" (6 ) AK, AS I -t- A tan ^ I

When only crack deflection is considered, the effective stress intensity range is give by Equation 4. This value of effective stress intensity factor range is further reduced by fracture surface mismatch. As a result, the total effective stress intensity factor range, Ak^p, due to both crack deflection and mismatch is:

A A ,^ = d (a K , - A K , ^ ) (7)

By substituting Equations 4 and 6 into Equation 7, this effective stress intensity factor range can be expressed in non-dimensional form by:

109 / a \ \ -I

Dcos' + S 1/2 AK 2 j Xiznd 1 - (8) àk‘ff D + S I + A tan ^

Figures 60a to 60c show the lateral views of the deflected segments on the fracture surface of the three Widmanstatten microstructures. Table 8 shows the average measured values of 0, D, S, and the calculated value of the extent of tilt, D/D+S. Note that the extent of tilt, and the kink angle, 0, are highest for the microstructure C (coarse). Using these calculated values for the deflected segments. Equation 8 was used to calculate the effective stress intensity factor range, Akg^, for arbitrary extents of mismatch values of A. = 0 (deflection only), X = 0.25 and X = 0.50, respectively. These arbitrary values for the extent of mismatch were chosen since their actual values were not measured before the test specimens were fractured. The values of X (between 0.25 and 0.50) also correspond to the range that was used by Suresh [14] in earlier work.

Figures 61a to 61c compare the original closure-corrected fatigue crack growth rate data obtained using the clip gage, and the growth rate data obtained using Equation 8 at mismatch values of A. = 0 (deflection only), A. = 0.25 and X = 0.50 (for the three Widmanstatten microstructures). The plots show that, for the extent of mismatch of A. = 0.25, the measured and modeled closure-corrected fatigue crack growth rate data match well, especially for microstructure B (Figure 61b). The plots also show clearly that deflection alone (A. = 0), and the extent of deflection (defined in Table 8) does not fully capture the effects of crack tip shielding on the fatigue crack growth rate. It is the extent of deformation irreversibility at the crack tip, which results in the fracture surface mismatch, that contributes most to the shielding contributions from roughness-induced crack closure.

It is important to note here that the roughness-induced crack closure model introduced in this section is two-dimensional. Hence, it does not include the crack tip twisting contributions through the thickness of the specimen. These could further reduce the total Ak«R in Equation 8. The modeling of twisting contributions is beyond the scope of the current work. However, the current model does seem to capture the trends in the closure-corrected growth rate data that 110 were obtained using the clip gage measurements. The two-dimensional roughness-induced closure model presented in this section, therefore, appears to be adequate.

5.7 Multiparameter model containing both mechanical and microstructural variables:

In chapter 4, a multiparameter model was introduced. The model was used to relate both mechanical and physical variables (AK, K^ax. R. W, B, T, and Q) to the measured fatigue crack growth rates. However, it did not include the effects of microstructural variables. In this section, the multiparameter model is extended to the modeling of the combined effects of mechanical and microstructural variables. The variables considered include the stress intensity factor range, AK, maximum stress intensity factor, K^a*. Widmanstatten colony size, C, a lath size. A, volume % a. Va, volume % p, Vo, and the cooling rate, CR. The multiparameter model can be written as:

^ = a.(A K )“' (K„„ )"■ ( c r (aY- (V, Y' (V, )“• (CRY' (9) clN

Before determining the coefficients/exponents, and the correlation coefficient, r^, for Equation 9, cross correlation between the microstructural variables (CR, C, A, Va, and V&) were determined. The results are shown in Table 9. As mentioned in section 4.4.2 in chapter 4, a large positive cross correlation coefficient indicates that, when large values of one variable are observed, large values of the other variable will occur. On the other hand, large negative correlation coefficients suggest that when large values of one variable is observed, small values of the other variable will occur.

As expected, the cross correlation coefficients between the microstructural variables are very high. This indicates that there is strong inter-relationship between these variables. Column 1 in Table 9 indicates that as the cooling rate, CR, increases, the colony size, C, a lath size, A, and % volume a. Va decreases, while % volume P, Vb, increases. From metallurgical point of view this is expected, as indicated in the measured Widmanstatten microstructural variations shown in Table 7. Some inter-relationship was also observed between the

111 mechanical variables in chapter 4 (Table 5). However, these were generally not as high as those between the microstructural variables.

Due to the high inter-relationship between the microstructural variables, the Minitab statistical software would only allow two microstructural variables, along with the mechanical variables to be analyzed. As a result. Equation 9 was rewritten as:

^ = a,(AK)"'{K„„r(C)-(rJ“- (10)

Substituting the results of the regression analysis into Equation 10 gives:

^ = 9.21x10-’ ( A K r (K )■ “ ( c r “ (K. (mm/cycle) ( 11) aN

The correlation coefficient obtained for Equation 11 is r^ = 0.923. This Indicates a very good correlation between the measured (experimental) and calculated values of fatigue crack growth rate, da/dN. The exponents of the variables in Equation 11 also suggest that the most important factor that contributes to fatigue crack growth in Widmanstatten Ti-6AI-4V is AK, which has a positive exponent of 2.05. The second most important variable is the maximum stress intensity factor, Kmax, which has a positive exponent of 1.60. The microstructural variables Vu (volume % P) and C (Widmanstatten colony size) have negative exponents of - 0.974 and -0.649, respectively. The negative exponents of the microstructural variables indicate that, +as the Widmanstatten colony size or volume % p increase, there would be a reductions in the fatigue crack growth rate, da/dN. These effects of colony size and volume % p on fatigue crack growth rate are observed in the experimental data, as shown in Figures 43 and 47, respectively.

Figures 44a, 44b, 46a, 46b, 48a, and 48b show that when the experimental fatigue crack growth curves for the Widmanstatten microstructures were plotted against the effective stress intensity factor range, AKetr, the data nearly collapsed onto the same line. As a result, the fatigue crack growth rate can be expressed by:

112 | t = c (a K „ )- (12,

By performing regression analysis on the experimental data, Equation 12 can be rewritten as;

^ = 5.07 X1 0 ' ^ y (mm/cycle) (13)

Equation 13 shows a very high correlation coefficient (r^ = 0.940). This suggests that the single variable, AKeff, characterizes the fatigue crack growth rates in Widmanstatten Ti-6AI-4V very well.

Finally, since the Widmanstatten microstructural variables are highly inter-related, the cooling rate. CR, can be expressed in terms of any other microstructural variables such as Widmanstatten colony size, C, and volume % p, Vt,. The relationship is given by:

CR=a,(Cf'(V,Y' (14)

By linearizing Equation 14, and performing the statistical analysis. Equation 14 can be rewritten as;

CR = 4.06(C )-' (r, (oQ/min) (15)

Equation 15 exhibits perfect correlation (r^ = 1). Hence, Equation 15 may be used to relate the cooling rate and microstructural variables such as Widmanstatten colony size and volume % p. Also, for any Widmanstatten Ti-6AI-4V microstructure, if the colony size and volume % p are metallographically determined, the corresponding cooling rate (from the same initial temperature) can be obtained from Equation 15.

113 5.8 Conclusions:

1. Positive stress ratios between 0.1 and 0.80 have a significant effect on crack closure levels and fatigue crack growth rates in Widmanstatten Ti-8AI-4V microstructures. When the fatigue crack growth rate data for stress ratios of 0.1 and 0.8 were plotted against the effective stress intensity factor, AKeff. the data for each Widmanstatten microstructure collapsed into a single effective fatigue crack growth rate curve. This indicates that the effects of stress ratio on the fatigue crack growth rates in Widmanstatten Ti-6AI-4V microstructure may be attributed largely to shielding contributions from crack closure.

2. The fracture surfaces of none of the three Widmanstatten microstructures exhibited any evidence of fatigue striations. Fatigue striations commonly occur in mill-annealed Ti-6AI-4V microstructures. For a fine Widmanstatten Ti-6AI-4V microstructure, fatigue crack growth rate in the near threshold regime occurs by “cleavage-like" fracture mode. Fatigue fracture in the Paris regime occurs by a combination of “cleavage-like” and step-like fracture mode. Crack growth in the high AK regime occurs by a combination of crystallographic fracture mode and secondary cracking. For an intermediate Widmanstatten Ti-6AI-4V microstructure, fatigue crack growth rate in the near threshold regime occurs by “cleavage-like" fracture mode. Fatigue fracture in the Paris regime occurs by a combination of “cleavage-like” and step-like fracture mode. Crack growth in the high AK regime occurs by a combination of cleavage facet fracture mode and secondary cracking. For a coarse Widmanstatten Ti-6AI-4V microstructure, fatigue crack growth rate in the near threshold regime occurs by “cleavage-like" fracture mode. Fatigue fracture in the Paris regime occurs by a combination of “cleavage-like" fracture mode and large facets. Crack growth in the high AK regime occurs by a combination of small and large facets fracture mode.

3. The coarser the Widmanstatten microstructure, the lower the fatigue crack growth rate. This is associated with a higher closure level in the coarser microstructure and more tortuous fracture modes. Fatigue crack growth rates in Widmanstatten microstructures are observed to be lower than in the miil-annealed microstructures. These differences are largely attributed to the higher closure levels associated with rougher fracture surface morphology in the Widmanstatten microstructures.

114 4. Crack deviations or deflections from initial Mode I growth direction are much greater for Widmanstatten microstructures compared to those in the mill-annealed microstructure. These deviations in Widmanstatten microstructure occur mostly at prior p grain boundaries and colony boundaries. They result in higher roughness-induced crack closure levels, especially at lower stress ratios. The roughness-induced crack closure levels are much more pronounced at the lowest stress ratio of 0.1, that was examined in this study.

5. A two-dimensional roughness-induced crack closure model can be used to adequately estimate the crack-tip shielding due to roughness-induced closure. This uses measurements of the periodic deflections in fatigue crack profiles in Widmanstatten Ti-6AI-4V microstructures. The model provides a single equation for the effective crack tip stress intensity factor range, Akefl, which includes the average dimensions of the periodic deflections and the extent of mismatch between fracture surfaces. Good agreement was observed between the roughness- induced closure-corrected fatigue crack growth rate data and the clip gage closure-corrected data when the extent of mismatch, X, was assumed to be 0.25 in the roughness-induced analysis.

6. The multiparameter framework proposed in chapter 4 is extended to capture the effects of both mechanical and microstructural variables in Widmanstatten Ti-6Ai-4V microstructures. The extended multiparameter framework provides a single equation that assesses the combined effects of stress intensity factor range, maximum stress intensity factor, Widmanstatten colony size, volume % p and cooling rate. A multiparameter expression has also been developed between cooling rate and Widmanstatten microstructural parameters.

115 5.9 References:

1. Harrigan, M. J., Kaplan, M. P., and Sommer, A. W., Effect of chemistry and heat treatment on the fracture properties of Ti-6A1-4V Alloy, in fracture prevention and control, edited by D. W. Hoepner, Vol 3, ASM Materials/Metal Working Technology Series, American Society for Metals, 1974, p. 22.

2. Yoder, G. R., Cooley, L. A., and Crooker, T. W., Enhancement of fatigue crack growth and fracture resistance in Ti-6AI-4V and Ti-6AI-6V-2Sn through microstructural modification. Journal of Engineering Materials and Technology, 1977, p. 313.

3. Thompson, A. W., Williams, J. C., Frandsen, J. D., and Chesnutt, J. C., The effect of microstructure on fatigue and crack propagation rate in Ti-6AI-4V, in Titanium and Titanium Alloys, Vol. 1, p. 691-704, 1982.

4. Amateau, M. P., Hanna, W. D., and Kendall, E. G., Mechanical Behavior, Proceedings of the international conference on mechanical behavior of materials. Vol. 2, p. 77-89, The Society of Material Science, Japan, 1972.

5. Chesnutt, J. C., Rhodes. C. G., and Williams, J. C., ASTM STP 600, Beacham, C. D., and Warke, W. R., editors, p. 99-138, ASTM, Philadelphia, PA, 1976.

6. Eyion, D., Hall, J. A., Peirce, C. M., and Ruckle, D. L, Metallurgical Transactions A, Vol. 7A, p. 1817-26, 1976.

7. Donachie, Jr., M. J., Titanium; A technical guide, ASM international, 1988.

8. Sinha, V., Ph.D. dissertation. Effects of microstructure on fatigue behavior of o/p titanium alloys, 1999.

9. Yoder, G. R., Cooley, L. A., and Crooker, T. W., Observations on microstructurally sensitive fatigue crack growth in a Widmanstatten TÎ-6AI-4V alloy. Metallurgical Transactions A, Vol. 8A, p. 1737, 1977.

116 10. Yoder, G. R., Cooley, L. A., and Crooker, T. W., Quantitative analysis if microstructural effects on fatigue crack growth rate in Widmanstatten Ti-6AI-4V and Ti-8AI-1Mo-1V, Engineering Fracture Mechanics, Vol. 11, p. 805-816,1979.

11. Ravichandaran, K. S., N ear threshold fatigue crack growth behavior of a titanium alloyiTi- 6A1-4V, Acta Metallurgica, Vol. 39, NO. 3, p. 401-410, 1991.

12. Sehitoglu, H., and Garcia, A. M., Contact of fracture faces during fatigue. Parti. Formulation of the model. Metallurgical and Materials Transactions A, Vol. 28A, p. 2263-2275.

13. Sehitoglu, H., and Garcia, A. M., Contact of fracture faces during fatigue, Parti. Simulations, Metallurgical and Materials Transactions A, Vol. 28A, p. 2277-2289.

14. Suresh, S., Fatigue crack deflection and fracture surface contact: Micromechanical models. Metallurgical Transactions 16A, p. 249-260,1985.

15. Bilby, B. A.., Cardew, G. E., and Howard, 1. C., Stress intensity factors at the tips of kinked and forked cracks. In fracture 1977 (ed. D. M. R. Taplin), Vol. 3, p. 197-200,1977. Pergamon Press.

16. Cotterell, B., and Rice, J. R., Slightly curved or kink cracks. International Journal of Fracture, Vol. 16, p. 155-169, 1980.

17. Faber, K. T., and Evans, A. G., Crack deflection processes: I. Theory. Acta Metallurgica, Vol. 31, p. 565-574, 1983.

117 p Transus ~ 980 °C 1100 c j 1000

% 03 900 ( a + P ) I “ 700

Weight Percent V

Figure 39: Relevant portion of Ti-AI-V phase diagram [7],

118 #////;T

/ , / / Vy Microstructure A Microstructure B (a) (b)

/ / / / / /

/ / V Z O u r n Microstructure C (C) Figure 40: Microstructural modifications after heat treatments, (a) 1070 °C for 30 min. cool @ 25 "C/min. to 400 “C/Ar quench to RT, (b) 1070 “C for 30 min. cool @ 3 "C/min. to 400 “C/Ar quench to RT, and (c) 1070 °C for 30 min. cool @ 1 “C/min. to 400 °C/Ar quench to RT.

119 Micro­ p layer Widmanstatten a lath Volume Volume UTS E| Fracture structure continuity Colony size size % a %P (MPa) (%) Toughness, (yes/no) (pm) (pm) Klc (MPaVm) A no 118.20 ±2.74 3.45 ± 0.61 66.65 33.35 912 7 121 B no 168.88 ±5.03 6.59 ± 0.21 78.56 21.44 878 11.9 109" C yes 312.50 ± 17.22 14.14 ±0.88 77.91 22.09 934 7.6 152" D yes 19.57 ±2.22" 77.26 22.74 922 15 57"

a grain size (|im) Not valid according to ASTM 399-90

K> O Table 7; Variations in the microstructurai parameters and the mechanical properties as a function of heat treatment [8],

IVIicrostructure A; 1070 °C/ 30 min./ cool @ 25 “C/min. to 400 “C / Ar quench to RT

Microstructure B: 1070 °C/ 30 min./ cool @ 3 °C/min. to 400 °C / Ar quench to RT

Microstructure C: 1070 °C/ 30 min./ cool @ 1 °C/min. to 400 °C / Ar quench to RT

Microstructure D; Mill Annealed (as received) R = 0.1 R = 0.25 R = 0.50 R = 0.80

1 100

(a)

1 0 ^ p- I 1 I I I n I I I I I I u 2 R = 0.1 e R = 0.25 R = 0.50 R = 0.80 t 10-" I E 1 •o

,-7 10 I -il .1 t ilt t.t 1 10 1 0 0 .1/2 AK, MPa(m)‘ (b) ...continued on next page

Figure 41: Fatigue crack growth rate versus stress intensity factor, a K, at stress ratios of 0.1, 0.25, 0.50, and 0.80 for (a) Widmanstatten Ti-6Ai-4V microstructure A, (b) Widmanstatten Ti-6A!-4V microstructure B, and (c) Widmanstatten Ti-6A1-4V microstructure C. 121 Figure 41 (continued)

1 0 R = 0.1 R = 0.25 R = 0.50 o R = 0.8 o r / ZI

1 10 1 0 0 AK, MPa(m) 1/2 (c)

122 1 0

R = 0.1 R = 0.8

S'

E z

■o

1 1 0 0

(a)

1 0

R = 0.1 R = 0.8

IE I

1 1 0 0

(b) ...continued on next page Figure 42: Closure-corrected fatigue crack growth rate at stress ratios of 0.1 and 0.8 for (a) Widmanstatten microstructure A, (b) Widmanstatten microstructure B, and (c) Widmanstatten microstructure C.

123 Figure 42 (continued)

1 0

R = 0.1 R = 0.8 C

1 ■O

101 1 0 0 MPa(m) 1/2

(c)

124 c Microstructure C 1 0 ' • Microstructure B s Microstructure A

I 10" E

10"

■ 10 100 AK, MPa(m) 1/2 (a)

- Microstructure C • Microstructure B s Microstructure A

I E zI

1 0 0

(b) ...continued on next page Figure 43: Effects of microstructurai variables on Fatigue Crack Growth Rates as a function of AK in Widmanstatten Ti-6Ai-4V. (a) FCGR at R = 0.1, (b) FCGR at R = 0.25, (c) FCGR at R = 0.50, and (d) FCGR at R = 0.8.

125 Figure 43 (continued)

Microstructure C Microstructure B 1111 Microstructure A 10“ u u I IQ-®

% •o / 10^

» i I I I t 1 10 100 AK, MPa(m) 1/2 (c)

Microstructure C Microstructure B Microstructure A

_o I E

1 ■o

1 10 100 1/2 AK, MPa(m) (d)

126 Microstructure C Microstructure B Microstructure A

Q) !

1 ■o

1 10 1 0 0 AK . MPa(m) 1/2

(a)

Microstructure C Microstructure B Microstructure A

0 1 E z f 1 ■o

1 10 1 0 0 AK^^ MPa(m)^'*

(b) Figure 44: Effects of microstructural variables on Fatigue Crack Growth Rates In W idmanstatten Ti-6AI-4V. (a) FCGR as a function of AKefr at R = 0.1, and (b) FCGR as a function of AKes at R = 0.8.

127 - Microstructure C • Microstructure B - Microstructure D ,-4 10 u I E z % ■o

1 10 100 AK, MPa(m) 1/2 (a)

: Microstructure C • Microstructure B - Microstructure D

u >« I E z 1 •o

1 10 0

(b) ...continued on next page Figure 45: Comparison of Fatigue Crack Growth Rates in Widmanstatten Ti-6Ai-4V (P ~ 22 Vol%) and mill annealed microstructures, (a) FCGR at R = 0.1, (b) FCGR at R = 0.25, (c) FCGR at R = 0.50, and (d) FCGR at R = 0.8.

128 Figure 45 (continued)

Microstructure C Microstructure B Microstructure D

1 0 0

(c)

- Microstructure C • Microstructure B 5: Microstructure D

u u

z 1

1 10 1 0 0 AK, MPa(m) (d)

129 - Microstructure C Microstructure B Microstructure 0

g. I z % ■o

1 10 1 0 0 MPa(m) 1/2 (a)

Microstructure C Microstructure B Microstructure D

0

1 E z § ■o

1 10 1 0 0 MPa(m)^®

(b) Figure 46: Comparison of Fatigue Crack Growth Rates in Widmanstatten Ti-6AI-4V (p - 22 Voi%) and mill annealed microstructures, (a) FCGR as a function of AKeff at R = 0.1, (b) FCGR as a function of at R = 0.8.

130 Microstructure A Microstructure B Microstructure D

0 >> 1 E 2 “

1 0 0

(a) Microstructure A Microstructure B 10" Microstructure D

£ ^ 10- I E 10"

■o1

lO""

X X 1 10 1 0 0 .1/2 AK, MPa(m)’ (b) ...continued on next page Figure 47: Comparison of Fatigue Crack Growth Rates in Widmanstatten Ti-6AI-4V and mil! annealed microstructures, (a) FCGR at R = 0.1, (b) FCGR at R = 0.25, (c) FCGR at R = 0.50, and (d) FCGR at R = 0.8.

131 Figure 47 (continued)

Microstructure A n I I Microstructure B Microstructure D 1 0 ^ _o I 1 - 1 0 ^ ? ■O

1 0 ' ' ■ 1 1 ■ ■ 11 t ■ 1 1 1 11-1 10 1 0 0 AK, MPa(m) 1/2 (c)

Microstructure A Microstructure B Microstructure D

% ■o

1 0 0

(d)

132 Microstaicture A Microstructure B Microstructure D

u >> I E

1 ■o

1 10 100 AK . MPa(m) 1/2 (a)

Microstructure A Microstructure B Microstructure D

U I E z % ■o

1 10 1 0 0 AK^^ MPa(m)^®

(b) Figure 48: Comparison of Fatigue Crack Growth Rates in Widmanstatten Ti-6A!-4V and mill annealed microstructures, (a) FCGR as a function of AKeff at R = 0.1, (b) FCGR as a function of AKeff at R = 0.8. 133 (a) 25 |im

(b) 25 nm

Figure 49: Fatigue crack interactions at (a) prior (3 grain boundaries, and (b) colony boundaries for Widmanstatten microstructure A, tested at R = 0.1.

134 ■ y t

Figure 50: Fatigue crack deflections in mill-annealed microstructures.

135 (a)

(b)

: - u .

(c)

(d)

Figure 51; Morphology of the fatigue fracture surfaces of the different microstructures observed at low magnification under stereo optical microscope. All specimens were tested at R = 0.1. (a) microstructure A (fine Widmanstatten), (b) microstructure B (intermediate Widmanstatten), (c) microstructure C (coarse Widmanstatten), and (d) microstructure D (mill-annealed) [8]. 136 Direction of crack growth

(a) — — (b) — 5 |im 5 |im

(c) 5 |im

Figure 52: Fatigue fracture modes observed in the different crack growth regime of the Widmanstatten microstructure A (fine) at R = 0.1. (a) near threshold (AK - 11 MPa Vm), (b) intermediate AK (AK ~ 18 MPa Vm), and (c) high AK (AK - 27 MPa Vm).

137 Direction of crack growth

(a) (b) 5 jim 5 pm

(c) 5 pm

Figure 53: Fatigue fracture modes observed in the different crack growth regime of the Widmanstatten microstructure A (fine) at R = 0.8. (a) near threshold (AK ~ 5 MPa Vm), (b) intermediate aK (AK ~ 10 MPa Vm), and (c) high aK (aK ~ 14 MPa Vm).

138 Direction of crack growth

(a) (b) 5 jim 5 |im

(c) 5 jim

Figure 54; Fatigue fracture modes observed in the different crack growth regime of the Widmanstatten microstructure B (intermediate) at R = 0.1. (a) near threshold (AK - 11 MPa Vm), (b) intermediate AK (AK ~ 15 MPa Vm), and (c) high AK (AK - 25 MPa Vm).

139 Direction of crack growth

(a) (b) 5 jim 5 |im

(c) 5 |im

Figure 55: Fatigue fracture modes observed in the different crack growth regime of the Widmanstatten microstructure B (intermediate) at R = 0.8. (a) near threshold (AK - 5 MPa Vm), (b) intermediate a K (AK - 9 MPa Vm), and (c) high AK (AK ~ 13 MPa Vm).

140 Direction of crack growth

(a) (b) ____ 5 n m 5 |im

(c) 5 |im

Figure 56: Fatigue fracture modes observed in the different crack growth regime of the Widmanstatten microstructure C (coarse) at R = 0.1. (a) near threshold (AK ~ 12 MPa Vm). (b) intermediate AK (AK ~ 18 MPa Vm), and (c) high aK (aK - 25 MPa Vm).

141 Direction of crack growth

(a) (b) __ 5 |im 5 |im

(c) 5 |im

Figure 57: Fatigue fracture modes observed in the different crack growth regime of the Widmanstatten microstructure C (coarse) at R = 0.8. (a) near threshold (AK - 6 MPa Vm), (b) intermediate AK (AK -1 0 MPa Vm). and (c) high AK (AK - 15 MPa Vm).

142 > t z i z d , D / 8 C e

U— s —J

0 kink angle D the distance over which the tilted crack advances along the kink S the distance over which Mode I crack growth occurs in each segment

Figure 58; idealization of a small segment of a crack with periodic tilts [14].

A

AS

k—S— J

Figure 59: Schematic representation of a deflected crack in fully opened condition at the peak load of fatigue cycle (on the left) and relative mismatch between the fracture surfaces at the point of first contact during unloading (on the right)[14].

143 100 urn

...continued on next page Figure 60: Lateral profile of the deflected segments on the fracture surface of (a) Widmanstatten microstructure A, (b) Widmanstatten microstructure B, and (c) Widmanstatten microstructure C.

144 Figure 60 (continued)

100 |im

145 Microstructure 0 D S Extent of tilt (Degrees) (pm) (pm) D/D+S A 42 150 175 0.462 B 44 230 235 0.495 C 48 969 894 0.520

Table 8: Average calculated dimensions of the deflected segments on the fracture surface of the three Widmanstatten microstructures.

Clip Gage Corrected Data X — 0 X = 0.25 ® 1 0 ^ k = 0.50 0 E Etr 1

■o

10"

■ 1 1 1 ' ■ ■ ■ ■ ■ 10 1 0 0 MPa(m) 1/2

(a) ...continued on next page Figure 61: Comparison of the original closure-corrected fatigue crack growth rate data using a clip gage, and the growth rate data using the roughness-induced closure model at mismatch values of 0.25 and 0.50 for (a) Widmanstatten microstructure A, (b) Widmanstatten microstructure B, and (c) Widmanstatten microstructure 0.

146 Figure 61 (continued)

Clip Gage Corrected Data TT 1 0 ^ r A. ~ 0 X = 0.25 X = 0.50 o > o 10"

% ■o ,T3 . < Z 10 Ë S - I I . — I L I II II ..1. I I I I I I I 10 1 0 0 MPa(m) vz

(b)

: Clip Gage Corrected Data 5 A. = 0 TTl : A = 0.25 s A = 0.50 10" u

Z

■o 10"

- t - i « t t t I I 10 100 AK^^ MPa(m) 1/2

(c)

147 parameter In(CR) ln(C) ln(A) ln(%Va) ln(%Vb)

In(CR) 1 -0.935 -0.969 -0.867 0.855

ln(C) -0.935 1 0.994 0.635 -0.616

ln(A) -0.969 0.994 1 0.717 -0.700

In(Va) -0.867 0.635 0.717 1 -1.00

In(Vb) 0.855 -0.616 -0.700 -1.00 1

Table 9: Summary of the cross correlations between the microstructural variables for the Widmanstatten microstructures.

148 CHAPTER 6

A PHYSICALLY-BASED MODEL FOR THE PREDICTION OF LONG FATIGUE CRACK GROWTH

6.1 Introduction:

Like Newton’s laws of motion, fatigue crack growth laws are typically empirical in nature [1-2]. They are based in mathematical expressions that provide good fits to experimental data. In the case of Newton's laws of motion, the fits to the experimental data are generally very good for rigid body motion at speeds well below the speed of light. Similarly, fatigue crack growth laws are only applicable over parametric ranges of crack driving force that are within the bounds where the expressions provide good fits to measured data. Once the accuracy of the empirical crack growth laws are established, they provide a useful basis for the estimation of fatigue life via analytical or numerical integration schemes [3],

However, it is well known that the mechanisms of fatigue crack growth may vary significantly over the parametric ranges of crack driving force [4-5]. Furthermore, the crack-tip deformation [6-8] and chemisorption [9-11] processes are unlikely to be fully described by empirical expressions that do not assess the chemical and physical processes involved in the extension of a crack during a single fatigue cycle. Two possible approaches may be explored in making such predictions. One involves the rigorous modeling of all the physical/chemical processes that occur during fatigue crack growth, while the other requires only the modeling of the essential features of fatigue crack growth. The latter approach is employed in this chapter partly because if its simplicity. Furthermore, it is recognized that our existing understanding of fatigue crack growth processes is still insufficient for the prediction of fatigue crack growth from detailed considerations of physical/chemical processes.

This chapter presents a simple fracture mechanics model for the prediction of fatigue crack growth. Like Newton’s second law of motion, which assumes that the force on a body is 149 directly proportional to its rate of change of linear momentum, the current model Is based on one fundamental assumption. It assumes that fatigue crack growth rate is directly proportional to the change in the crack-tip opening displacement during cycling between the maximum and minimum stress intensity factors. This assumption is used as a basis for the derivation of physically-based power law equation for the prediction of fatigue crack growth as a function of stress intensity factor range, AK, and stress ratio, R. The model is validated for long fatigue crack growth in mill annealed and Widmanstatten Ti-6AI-4V microstructures. The constants of the power law equation are then compared for fatigue crack growth in the near-threshold, Paris and high-AK regimes. Finally, The parametric ranges of AK and K^a* corresponding to the three regimes of crack growth are summarized in fatigue mechanism maps.

6.2 Derivation of the Phvsicallv-Based Model:

Following the example of Knott [12] and McEvily [13], it is assum ed that fatigue crack extension per cycle, 5a, is directly proportional to the change in the crack-tip opening displacement, CTOD or 6. This gives:

5 a oc ACTOD (1 )

under cyclic loading conditions, the CTOD is given by [14]:

CTOD = (2) 4E c7„

where E is the Young’s modulus and 2oyj is assumed to be the cyclic yield stress. Hence, ACTOD is given by:

ACTOD = {CTOD)^ - (CTOD)^ = (3) 4E(T_ 4Ec7„.

150 where subscripts max and min correspond to the maximum and minimum loads in a given fatigue cycle. Substituting Equation 3 into Equation 1 and replacing the proportionality sign with an equality sign gives:

d a k L = J3 {a C T O D ) = /3 (4) 7n 4Ecr,.. 4Ecr„

- [(K-nm -K-mm + K-mm )] 4Ecr„,

substituting Kma% = àKJ{^-R) and Km,n = RK^ax into Equation 4 gives:

d a \^ R (AK): (5) d N 4Ect„ \-R

Where p is a proportionality constant/factor and the other constants have their usual meaning. Equation 5 is similar to an expression derived earlier by Knott [12]. It suggests that the exponent of AK is 2, which is generally not true for structural metallic materials that typically have exponents of AK that are between 2 and 10 [4]. However, it is important to note that the so-called constant, P, is unlikely to remain constant as a function of AK, since it represents the extent of irreversibility during cycling between K^ax and K^n (Figure 62). Assuming that p exhibits a power law dependence on AK, we can write:

,8 = a„{AK)- (6 )

where oq and n are power law constants. Substituting Equation 6 into Equation 5 now gives:

d a l + R (AK)""' (7) d N 4E cr>■» 1 —i?

substituting C =

\ + R dN l-R

Equations 8 and 9 are generally applicable to long fatigue crack growth problems in the different regimes of crack growth. They are valid for stress ratios between R = -1 and R = 1. However, they are not valid for R = 1 for which the equations suggest that da/dN is not prescribed (equals to infinity), or R = -1, for which the expression suggests that da/dN is always equal to zero. This clearly is not the case, since stable crack growth can occur at R = - 1 and R = 1. Alternative approaches are, therefore, needed to develop appropriate functions for these stress ratios. These are beyond the scope of current work, which focuses on long fatigue crack growth at positive stress ratios between 0.02 and 0.80.

6.3 Application of the Physically-Based Model to Mill-Annealed and Widmanstatten Ti-6AI-4V Microstructures:

The fatigue crack growth rate data for the mill-annealed Ti-6AI-4V microstructure (Figures 31 and 32 in chapter 4) was analyzed using the Minitab statistical software. As discussed in section 6.2, the data was fitted to the expressions derived from the physically-based model (Equations 8 and 9). The fitting was done with data in regimes I, n, and III of fatigue crack growth (Figure 64). These correspond, respectively, to the near threshold, Paris and high AK regimes. The overall data obtained from the all three regimes (I, II, and III) were also 152 analyzed to identify differences between the model constants in individual regimes and the overall fatigue data.

The results of the analysis with Equations 8 and 9 are presented in Tables 10 and 11, respectively. Table 10 shows that the exponent of AK (a,) in the near threshold regime. Paris regime and the high AK regime are 3.43, 4.24 and 4.50, respectively. The exponents of AK do not appear to vary significantly in the three regimes (I, 11 and III) of fatigue crack growth. This is somewhat surprising since the slopes of the da/dN-AK curves vary significantly over the three regimes of crack growth (with slope being highest in the high AK regime). However, it also suggests that the (1+R)/(1-R) term in Equation 8 accounts for much of the increase in fatigue crack growth rates that is generally observed to occur with increasing stress ratio. In any case, the relatively high exponents of AK and correlation coefficients (r - 0.878-0.902) in the three regimes suggests that AK is the primary driving force for fatigue crack growth is these regimes of crack growth.

Table 11 shows that much better correlation coefficients are achieved in the three regimes of crack growth when the effective stress intensity factor range, AK@„, is used instead of AK. Table 11 also shows that, the exponents of AKefr increase gradually from a, = 3.74 in regime I (near threshold regime) to a, = 4.62 in regime III (high AK regime). However, the correlation coefficients reduce gradually from regime I (r = 0.967) to regime III (r = 0.949). This may indicate that AK@M characterizes the fatigue crack growth best in the near threshold regime. This is intuitively appealing, since closure mechanisms are much more dominant in the near threshold regime.

The results obtained for the Widmanstatten microstructures are presented in Tables 12 - 14, respectively. Similar to the results obtained for the mill-annealed microstructure, all three tables show that the exponents of AK do not appear to vary significantly in the three regimes (I, II and III) of fatigue crack growth. However, this could again be attributed to the (1+R)/(1-R) term in Equation 8, which accounts for much of the increase in fatigue crack growth rates with increasing stress ratio. The three tables also show that the correlation coefficients in regime I for all microstructures are low. This is due partly to the non-linear dependence of log (da/dN) on log (AK) in the near threshold regime. It is possible that the addition of other variables, 153 especially microstmctural variables such as colony size and volume % (3 to the model, could improve the correlation coefficients in regime I. However this could not be done, since these microstructural variables are constant for each Widmanstatten microstructure, and the number of microstructural conditions was not sufficient for the statistical software to consider them as variables. In any case, the relatively high exponents of AK in the three regimes suggests that AK is the primary driving force for fatigue crack growth in the three regimes of crack growth.

Good correlation coefficients are achieved in the three regimes of crack growth when the effective stress intensity factor range, AK«ff, is used. The tables (12b, 13b and 14b) show that for all three Widmanstatten microstructures, the greatest improvement in correlation coefficients was obtained in regime I, when AK,,, was used instead of AK. This indicates that AK«(f characterizes the fatigue crack growth best in the near threshold regime for the three Widmanstatten microstructures. This again appears to be intuitively correct, since closure mechanisms are much more dominant in the near threshold regime.

6.4 Fatigue Fracture Mechanism Maos:

Mercer et al. [23-24] have recently shown that the fracture mechanisms that occur in metallic materials during fatigue crack growth (especially in the high AK regimes) are not only controlled by the stress intensity factor range (AK), but aiso by the maximum stress intensity factor, Kmax- This important conciusion was based upon evidence of monotonie modes of failure during fatigue crack growth in the high AK regime. The observed proportions of the monotonie modes of failure increased with increases in stress ratio, R, or Kma%, especially in the high AK regime. Evidence of monotonie modes of failure in the high AK regime was also observed in the mill-annealed TI-6AI-4V microstructure (Figures 35 and 36 in chapter 4). This is because, for a constant stress ratio fatigue test, as AK increases, the maximum stress intensity factor, Kma%, of the fatigue cycle also increases, and approaches the materials fracture toughness. Therefore, as Kmax increases, the proportions of the monotonie modes of failure increases, which gives rise to faster fatigue crack growth rates in regime III (Figure 64).

In chapters 4 and 5, the fractographic results for both mill-annealed and Widmanstatten microstructures were discussed, respectively. The results indicated that; 154 The micromechanisms of fatigue crack growth in mill-annealed Ti-6AI-4V microstructure are strongly dependent on AK and R or Kmax- In the near-threshold regime, fatigue crack growth occurs by a “cleavage-like" fracture mode. Fatigue fracture in the Paris regime occurs by classical crack-tip blunting mechanism that gives rise to the formation of striations, while crack growth in the high AK regime occurs by a combination of fatigue striations and ductile dimpled fracture.

For a fine Widmanstatten Ti-6A1-4V microstructure, fatigue crack growth rate in the near threshold regime occurs by “cleavage-like” fracture mode. Fatigue fracture in the Paris regime occurs by a combination of “cleavage-like" and step-like fracture mode. Crack growth in the high AK regime occurs by a combination of crystallographic fracture mode and secondary cracking.

For an intermediate Widmanstatten Ti-6AI-4V microstructure, fatigue crack growth rate in the near threshold regime occurs by “cleavage-like" fracture mode. Fatigue fracture in the Paris regime occurs by a combination of “cleavage-like” and step-like fracture mode. Crack growth in the high AK regime occurs by a combination of cleavage facet fracture mode and secondary cracking.

For a coarse Widmanstatten Ti-6AI-4V microstructure, fatigue crack growth rate in the near threshold regime occurs by “cleavage-like” fracture mode. Fatigue fracture in the Paris regime occurs by a combination of “cleavage-like” fracture mode and large facets. Crack growth in the high AK regime occurs by a fracture mode that involves the formation of a combination of small and large facets.

The fractographic information obtained for the mill-annealed (Figures 35 and 36) and Widmanstatten Ti-6AI-4V microstructures (Figures 56 to 61) can also be summarized on fatigue mechanism maps, as suggested by the recent work of Mercer et al. [23-24]. The fatigue mechanism maps are shown in Figures 65a to 65d. The fatigue mechanism maps present the fracture modes on a plot of Kmax (ordinate) against AK (abscissa). Constant R-ratio tests are summarized on straight lines that radiate outwards from the origin, and the limiting value of Kmax, which is equal to the fracture toughness, K,c of the material. As shown in 155 Figures 65a to 65d, the maps also show a series of arcs (hyperbolae) which corresponds to transitions from one fracture mode to another. These arcs intersect at a common point located on the Kmax axis. This point again corresponds to the fracture toughness of the material. It is apparent that the transitions between each fracture mode are defined by the arcs for each Ti-

6AI-4V microstructure. They also depend upon both A K and Kmax-

It is also important to note here that the transitions from one mechanism to another do not occur suddenly at a given A K , Kmax combination. Instead the transitions appear to occur gradually over a range of A K and Kmax values. This gradual transition is shown by the dashed hyperbolae that are plotted next to the solid lines that describe the mean conditions for the transitions. Mixed fracture modes are commonly observed between the dashed lines within which fracture mode transitions occur. Also, since the fatigue mechanism maps describe the

parametric ranges of A K and Kmax associated with regimes I, II, and III, it is possible to link the

physically-based model constants ( a o and a , ) with the different domains of AK and Kmax associated with a particular mechanism of fatigue crack growth. Hence, the exponents of a

particular variable can be obtained for different domains of AK and Kmax in which the underlying fatigue crack growth mechanisms are the same, as mapped by the plots shown in Figures 65a- 65d.

6.5 Conclusions:

2. A physically-based fatigue crack growth equation is presented in this chapter for the prediction of long fatigue crack growth. The equation was derived from a fundamental assumption that the fatigue crack growth rate, da/dN, is proportional to the change in the CTOD between Kmax and Kmn, i.e. ACTOD. The fatigue crack growth rate, da/dN, is thus given by:

l + R 1 + /? l-R 1-/Î

156 Where oo and a, are constants. AK is the applied stress intensity factor range, a Kch is the effective stress intensity factor range and R is the stress ratio. The exponent, ai, did vary in the three regimes of fatigue crack growth in the mill-annealed and Widmanstatten microstructures that were examined. However, the variations in the a, exponents were not as large, presumably because the (1+R)/(1-R) term accounts for some of the stress ratio effects. The equations provided the best fit to the fatigue crack growth rate data in the near threshold regime when AKg„ was used in place of AK.

2. The fatigue fracture modes in mill-annealed and Widmanstatten Ti-6AI-4V microstructures can be summarized on fatigue fracture mechanism maps. These show the fatigue fracture mechanisms on plots of Kmax (ordinate) against AK (abscissa), in which all the possible loading conditions corresponding to positive stress ratios can be summarized within a triangle with radial lines corresponding to parametric ranges of AK and Kmax encountered in positive stress ratio tests. The fatigue maps show that the transitions from one mechanism to another are well described by curves (hyperbolae) that radiate from a point on the ordinate corresponding to the monotonie fracture toughness.

3. The linkage of the fracture mechanism maps and the physically derived fatigue crack growth law could serve as a basis for the future development of a mechanistically-based framework for the prediction of long fatigue crack growth. The current work has established the applicability of this approach to Ti-6AI-4V. However, future work is needed to demonstrate the general applicability of the proposed mechanistically-based approach to the prediction of long fatigue crack growth in a wide range of metallic materials.

157 6.6 References:

18. P. C. Paris, M. Gomez and W. E. Anderson A rational analytic theory of fatigue. Trend Engng, Vol. 13, p. 9-14, 1961.

19. P. C. Paris and F. Erdogan A critical analysis of crack propagation laws. J. Basic Engng, Vol. 85, p. 528-534, 1963.

20. W. O. Soboyejo, Y. Ni, Y. Li, A. B. 0. Soboyejo and J. F. Knott, 1998, A new multiparameter approach to the prediction of fatigue crack growth. Fatigue and Fracture of Engineering Materials and Structures, Vol. 21, p. 541-555.

21. S. Suresh, 1998, Fatigue of materials, Cambridge University Press, 2"“ Edition.

22. S. Suresh and R. O. Ritchie, 1984a, Near-threshold fatigue propagation: a perspective on the role of crack closure. In Fatigue Crack Growth Threshold Concepts. Editors, D. L. Davidson and S. Suresh, pp. 227-261. Warrendale: The Metallurgical Society of the American Institute of Mining, Mineral and Petroleum Engineers.

23. P. J. E. Forsyth, 1961, A two stage process of fatigue crack growth. In Crack Propagation: Proceeding of Cranfield Symposium, pp. 76-94. London: Her Majesty’s Stationary Office.

24. C. Laird, 1967, The influence of metallurgical structure on the mechanisms of fatigue crack propagation. In Fatigue Crack Propagation, Special Technical Publication Vol. 415, p. 131- 168. Philadelphia: The American Society for Testing and Materials.

25. P. Neuman, 1969, Coarse slip model of fatigue. Acta metallurgica, Vol. 17, 1219-1225.

26. R. P. Wei, 1970, Some aspects of environmental-enhanced fatigue-crack growth. Engineering Fracture Mechanics, Vol. 1, p. 633-651.

27. R. M. N. Pelloux, 1969, Mechanisms of formation of ductile fatigue striations. Transactions of the American Society for Metals, Vol. 62, p. 281-285. 158 28. D. A. Meyn, 1968, Observations of micromechanisms of fatigue crack propagation in 2024 aluminum. Transactions of the American Society for Metals, Vol. 61, p. 42-51. 29. J. F. Knott, Models of fatigue crack growth, 1984, R. A. Smith, Editor, Fatigue crack growth: 30 years of progress. Proceedings of a conference on fatigue crack growth Cambridge, UK, 1994.

30. A. J. McEvily, 1977, Current aspects of fatigue. Metal Science, Vol. 11, p. 274-284.

31. D. S. Dougdale, 1960, Yielding of steel sheets containing slits. Journal of the Mechanics and Physics of Solids, Vol. 8, p. 100-108.

32. R. O. Ritchie and J. F. Knott, 1973, Mechanisms of fatigue crack gro\Alh in low alloy steel, Acta Metallurgica, Vol. 21, p. 639-650.

33. S. Suresh, 1985, Fatigue crack deflection and fracture surface contact: micromechanical models. Metallurgical Transactions, Vol. 16A, p. 249-260.

34. C. J. Beevers and M. D. Halliday, 1979, Non-closure of cracks and fatigue crack growth in P heat treated Ti-6AI-4V. Intemational Journal of Fracture, Vol. 15, R27-R30.

35. J. E. Allison, 1988, The measurement of crack closure during fatigue crack growth. Fracture Mechanics, 18“' Symposium, ASTM STP 945, D. T. Read and R. P. Reed, Editors. American Society for Testing and Materials, Philadelphia, p. 913-933.

36. B. Budiansky and J. W. Hutchinson, 1978, Analysis of closure in fatigue crack growth. Journal of Applied Mechanics, Vol. 45, p. 267-276.

37. J. C. Newman, 1976, A finite element analysis of fatigue crack closure. Mechanics of Fatigue Crack Growth, Special technical publication 590, 281-301. Philadelphia: American Society of Testing and Materials.

159 38. N. A. Fleck, 1986, Finite element analysis of plasticity-induced crack closure under plane strain conditions. Engineering Fracture Mechanics, Vol. 25, p. 441-449.

39. H. Sehitoglu and A. M. Garcia, 1997, Contact of crack surfaces during fatigue. Metallurgical and Materials Transactions, Vol. 28A, p. 2263-2289.

40. C. Mercer, A. B. O. Soboyejo, and W. 0. Soboyejo, 1999, Micromechanisms of fatigue crack growth in forged Inconnel 718 nickel-base superalloy. Materials Science and Engineering.

41. 0. Mercer, A. B. O. Soboyejo, and W. 0. Soboyejo, 1999, Micromechanisms of fatigue crack growth in a single crystal Inconnel 718 nickel-base superalloy. Acta Materialia.

160 K

■max

CTODmax AK

mm

Time

Figure 62: Schematic representation of the extent of irreversibility, p, during cycling between Kma% and K^n- K

■max Fully open

Partially closed

■mm Fully closed

Time

Figure 63: Schematic representation of closure stress intensity factor, K^, and effective stress intensity factor, AKeff.

161 Regime Regime Regime

(Paris Regime) dN

da/dN = C(AK)

AK

Figure 64: Schematic representation of the near-threshold (regime I), Paris (regime n) and high AK (regime m) regions on plot of da/dN versus AK.

162 1 + /? \ + R (AK)- or ^=a.(AK)“' d N \-R dN l - R

Regime I

Material Constants I II III Overall Oo (mm/cycle) 1.13 X 10* 2.52X10'’° 5.57X10' 9.28X10'” 3.43 4.24 4.50 4.57 r^ 0.771 0.801 0.813 0.818 r 0.878 0.895 0.902 0.904

Table 10: Summary of the coefficients and correlation coefficient for the physically- based fatigue crack growth rate model for the mill-annealed Ti-6AI-4V microstructure.

d a 1 + /? d a l + R = C or d N l - R d N l - R

Regime

Material Constants I II III Overall Og (mm/cycle) 2.52X10'* 1.25X10* 3.41 X 10"’° 8 .3 8 X 1 0 ’° oti 3.74 4.15 4.62 4.51 r^ 0.935 0.913 0.902 0.934 r 0.967 0.956 0.949 0.966

Table 11: Summary of the coefficients and correlation coefficient for the physically- based fatigue crack growth rate model for the mill-annealed Ti-6AI-4V microstructure.

163 '\ + R '\ + R' (AK)" or — -ûf(,(AK)“' dN dN

Regime

Material Constants I II III Overall Oo (mm/cycle) 5.62 X 10'’° 8.38 X 10''° 2.08X 10''2 3.41 X 10''° ai 3.07 3.25 4.79 3.35 r" 0.341 0.915 0.960 0.925 r 0.640 0.957 0.980 0.962

(a)

' l + i? ' 'i + a ' i 2 . = c or ^ = ajAK^^)“' dN .1 - ^ . J-R.

Regime

Material Constants I II III Overall

« 0 (mm/cycle) 9.24 X 10'® 2.06 X 10'® 4.60 X 10’’° 8.35 X 10'® Ctl 3.18 3.46 5.10 3.55 r" 0.982 0.965 0.958 0.935 r 0.991 0.982 0.979 0.967

(b)

Table 12: Summary of the coefficients and correlation coefficient for the physically- based fatigue crack growth rate model using (a) AK and (b) AK«H for the Widmanstatten Ti-6AI-4V microstructure A (fine microstructure).

164 '\ + R 'l + R' (AK)'" or — -(2o(aK)“' dN .1 - ^ . dN

Regime

Material Constants I II III Overall Oo (mm/cycle) 5.68 X 10 " 1.69X10'" 1.26 X 10 ': 7.60 X 1 0 " ctl 3.94 4.80 5.33 4.39 r^ 0.467 0.886 0.971 0.923 r 0.683 0.941 0.985 0.961

(a)

'l + R' 'l + R' )" or — = a:g(AK^^)"' dN J-R.

Regime

Material Constants I II III Overall Oo (mm/cycle) 3.82 X 10 " 3.40 X 10'® 3.42 X 1 0 " 1.38 X 10'® ctl 3.35 3.55 5.24 3.90 1 r^ 0.971 0.899 0.860 0.929 r 0.985 0.948 0.927 0.964

(b)

Table 13: Summary of the coefficients and correlation coefficient for the physically- based fatigue crack growth rate model using (a) AK and (b) AK«ff for the Widmanstatten Ti-6AI-4V microstructure B (intermediate microstructure).

165 '! + /?■ 'l + ^ ‘ (AK)- or ^ = cr,(AK)"' dN .1 - ^ . dN .1 - ^ .

Regime

Material Constants I II III Overall Oo (mm/cycle) 7.65 X 10's 1.13X 10"’° 1.88 X 10"’° 3.42 X 10"” «1 4.20 4.60 6.01 4.49 r" 0.649 0.930 0.929 0.939 r 0.806 0.964 0.963 0.969

(a)

’\ + R 1 + ^ ' )” — = bTo (AK. ^ Y' dN .1 - ^ .

Regime

Material Constants I II III Overall Oo (mm/cycle) 9.26 X 10*’° 3.07 X 10"* 3.41 X 10"’° 3.09 X 10"* ai 4.30 4.55 5.33 4.40 r^ 0.956 0.925 0.920 0.946 r 0.978 0.962 0.959 0.973

(b)

Table 14: Summary of the coefficients and correlation coefficient for the physically- based fatigue crack growth rate model using (a) AK and (b) AKgft for the Widmanstatten Ti-6AI-4V microstructure C (coarse microstructure).

166 60

■R = 1 R = 0.80 R = 0.50 R = 0.25 R = 0.02 ■ 50

a

(0 Q. 30 Striations plus, ductile dimples*

20 Striations

10 Cleavage

0 10 20 30 40 50 60 AK, MPa (m) 1/2 (a)

...continued on next page Figure 65: Fatigue mechanism maps for (a) mill-annealed, (b) Widmanstatten microstructure A (fine), (c) Widmanstatten microstructure B (intermediate), and (d) Widmanstatten microstructure C (coarse).

167 Figure 65 (continued)

140

•R=1 R = 0.80 R = 0.50 R = 0.25 R = 0.1 1 2 0

100 a E (0 Q. S M I Crystallographic fracture and secondary cracks

Cleavage plus step-like features Cleavage

0 20 40 60 80 100 120 140 AK, MPa (m) 1/2

(b)

168 Figure 65 (continued)

120 R = 0.50 R = 0.25 R = 0.1■ R = 1 R = 0.80 R = 0.50 R = 0.25 R = 0.1■

100

Cleavage facets and secondary cracks

Cleavage and step-like features Cleavage

100 1 2 0 AK. MPa (c)

169 Figure 65 (continued)

1G0 7R=i R = 0.80 R = 0.50 R = 0.25, R = O.-Ç

140

120

Small and K large facets" i

Cleavage plus large facets Cleavage 0 20 40 60 80 100 120 140 AK, MPa (m) 1/2

(d)

170 CHAPTER 7

SHORT CRACK BEHAVIOR IN WIDMANSTATTEN TÎ-6AI-4V MICROSTRUCTURES

7.1 Introduction:

Anomalous short crack growth behavior, which is defined as fatigue crack growth at stress intensity factors significantly below the long crack fatigue threshold, has been observed ofp titanium alloys. These include titanium alloy IMI 685 [1], and titanium-aluminum alloys [2]. Recently, short crack growth anomalies have also been observed in mill-annealed TI-6AI-4V alloy [3] and the Ti-6AI-2Sn-4Zr-6Mo alloy [4],

As mentioned in section 2.2.4 (chapter 2), crack- closure mechanisms in long fatigue cracks arise as a result of premature contact between the crack faces behind the advancing crack tip. Since a short crack has a limited crack wake behind the crack tip, crack closure effects are less pronounced in short cracks than in long cracks. This results in a higher effective stress intensity factor range, which promotes higher fatigue crack growth rates.

In this chapter, short crack fatigue growth behavior is compared with long crack fatigue behavior for the three Widmanstatten microstructures. The higher crack growth rates in the short crack regime, at stress ratio of 0.1, are attributed to lower closure levels due to partially developed crack wake. Finally, the fatigue fracture modes in the short and long crack regimes are compared.

7.2 Comparison of short and long crack fatigue behavior:

The crack/microstructure interactions observed in the short crack specimens for the three Widmanstatten microstructures are shown in Figures 66a to 66c, respectively. The initial fatigue crack lengths correspond to position 1 on the optical micrographs. The figures show 171 that the initial stages of short crack growth are not associated with much crack path tortuously in the Widmanstatten microstructures examined in this study. Hence, their extrinsic resistance to crack growth is low in the short crack regime since the levels of roughness-induced closure and crack deflection mechanisms that contribute to crack tip shielding in a/p titanium alloys [5- 7] are relatively low. Plots of fatigue crack growth rate, da/dN, versus stress intensity factor range, AK, for the three Widmanstatten microstructures are presented in Figures 67a to 67c. The plots show clearly that short crack growth occurred at stress intensity factor levels that are significantly below the long crack fatigue threshold at stress ratio of 0.1. The figures also show that the subsequent crack growth exhibited a combination of acceleration and retardation phenomena. From Figures 67a to 67c, the AK levels at which crack growth retardation occurs were recorded and the corresponding crack lengths were determined. These crack lengths correspond to locations L* (- 275 pm from the edge) for Widmanstatten microstructure A, Lb (-400 nm from the edge) for Widmanstatten microstructure B and Lc (- 500 pm from the edge) for Widmanstatten microstructure C. At these locations, the cracks are retarded and crack deflection occurs from the prior p boundaries for Widmanstatten microstructures A and B and colony boundaries for Widmanstatten microstructure C. Crack deflection results in reduction in the crack driving force, which in turn reduces the crack growth rate, da/dN. It should be noted here that the crack path on the polished and etched surface of the specimens may also be affected by the crack/microstructure interactions below the surface. Hence, the surface observations alone may not be sufficient to account fully for the observed fatigue crack growth retardation.

7.3 Fatigue fracture modes:

The fracture surfaces of the short and long crack specimens were also examined in the short and long crack regimes for the three Widmanstatten microstructures. This was done in a scanning electron microscope. Typical fracture modes observed in the specimens are presented in Figures 68 to 70, respectively. Figures 68a, 69a and 70a show that the initial stages of short crack growth occur by a cleavage type fracture mode. These fracture modes are similar to the long crack fracture modes for the respective Widmanstatten microstructures observed in the near threshold regime at a stress ratio of 0.1 (Figures 52a, 54a and 56a).

172 However, unlike the long crack near threshold fracture modes, the fracture surfaces in the short crack regime do not exhibit any evidence of debris. This suggests that the evolving crack wake in short cracks are generally less well developed than that of long cracks [8]. As a result, the overall levels of plasticity-induced crack closure are, therefore, lower in short cracks. As a result, the effective stress intensity ranges for short cracks are greater than those for long cracks under the same applied far field stress intensity range. The occurrence of lower levels of crack closure in the short crack regime has also been demonstrated by James and Morris [4] in their studies of an (a+P) titanium alloy (Ti-6Al-2Sn-4Zr-6Mo).

Figures 68b, 69b and 70b show evidence of debris at stress intensity factors range values just prior to fatigue crack growth retardation (Figures 67a to 67c) for the three Widmanstatten microstructures. The figures indicate that, at these stress intensity factor ranges, the cracks are long enough to have fully developed a crack wake, which results in fracture surface contact.

Figures 68c, 69c and 70c show that at higher stress intensity factor range values (short crack/long crack transition regime), the fracture modes are similar to the long crack fracture modes for the respective Widmanstatten microstructures in the Paris regime at stress ratio of 0.1 (Figures 52b, 54b and 56b).

It is important to discuss the transition from short to long crack growth behavior. This has been observed to occur at different crack lengths in the current study, as shown in Figures 67a-67c, and Table 15 in which the microstructural parameters are also provided for each of the Widmanstatten microstructures. The AK at which the short crack data links to the long crack data increases with increasing colony size (Table 15). This is consistent with the results obtained from prior studies by Taylor and Knott [9]. The latter suggests that the transition form short to long crack growth behavior occurs at crack lengths between 3-5 grain sizes. Such a transition is associated presumably with the development of a steady-state crack wake as the crack length increases beyond a critical size range [9]. However, considerable modeling and short crack closure measurement are needed to establish the local crack-tip and crack wake conditions associated with the transitions from short to long crack behavior.

173 7.4 Conclusions:

1. The straight undeflected crack profiles observed during the initial stages of short crack growth at stress ratio of 0.1 suggest that the Widmanstatten microstructure does not provide significant extrinsic resistance to crack growth in the short crack regime. However, crack/microstructure interactions occur as the cracks extend to the prior (3 boundaries for Widmanstatten microstructure A (fine) and microstructure B (intermediate) and at colony boundaries for microstructure C (coarse).

2. Short crack anomalies are observed in Widmanstatten Ti-6AI-4V microstructures at stress ratio of 0.1. The anomalies include; fast fatigue crack growth rates and crack retardation phenomena at AK levels well below the long fatigue crack growth thresholds. The fast crack growth rates are associated with the short cracks undeveloped crack wake. The undeveloped crack wake results in lower plasticity-induced closure levels, which results in higher effective stress intensity factor and faster fatigue growth rates in the short crack regime.

3. The fatigue fracture modes in the short crack regime are similar to those observed in the near-threshold regions of the long crack fatigue specimens. This was true for all the three Widmanstatten microstructures that were examined. However, one important difference is the lack of debris on the fracture surface of short crack specimens. This suggests that significant crack face contact does not occur in the short crack regime. Hence, the fast fatigue crack growth rates observed in the short crack regime may be partly attributed to the absence or much lower levels of crack closure, compared to those in the near-threshold long crack regime.

4. Anomalous short crack growth phenomena are observed for crack lengths between approximately 3 and 4 times the colony sizes.

174 7.5 References:

1. Brown, C. W., and Hicks, M. A., (1983). A Study of Short Fatigue Crack Growth Behavior in Titanium Alloy IMI 685. Fatigue of Engineering Materials and Structures, vol. 6, p. 67-76.

2. Larson, S. G., Nicholas, T., Thompson, A. W., and Williams, J. C., (1986). Small Crack Growth in Titanium-Aluminum Alloys. In Small Fatigue Cracks, (eds. R. O. Ritchie and J. Lankford), p. 499-512. Warrendale: The Metallurgical Society, Warrendale, PA.

3. Sinha, V., C. Mercer., W. 0. Soboyejo., (2000). An Investigation of Short and Long Fatigue Crack Growth Behavior o Ti-6A1-4V, Materials Science and Engineering, A287, p. 30-42.

4. M. R. James and W. L. Morris, Effect of Fracture Surface Roughness on Growth of Short Fatigue Cracks, Metallurgical Transactions, Vol. 4, p. 303-320,1988.

5. Yoder, G.R., Cooley, L.A., and Crooker, T.W., (1979). Quantitative Analysis of Microstructural Effects on Fatigue Crack Growth in Widmanstatten Ti-6AI-4V and Ti-9AI-1Mo- IV, Engineering Fracture Mechanics, 11, p. 805-816.

6. Yoder, G.R., Cooley, L.A., and Crooker, T.W., (1978). Fatigue Crack Propagation Resistance of Beta-Annealed Ti-6AI-4V Alloys of Differing Interstitial Oxygen Contents. Met. Trans. A9A, p. 1413-1420.

7. Hall, I.W., and Hammond, C., (1978). Fracture Toughness and Crack Propagation in Titanium Alioys, Material Science and Engineering, 32, p. 241-253.

8. Newman, Jr., J. C., (1983). A Nonlinear Fracture Mechanics Approach to the Growth of Small Cracks, AGARD (Advisory Group for Aerospace Research and Development) Conference Proceedings no. 328.

9. Taylor, D. and Knott, J. F., (1981). Fatigue Crack Propagation Behavior of Short Cracks. Effect of Microstructure, Fatigue of Engineering Materials and Structures, Vol. 4, p. 147-155.

175 At location La , a ~ 275 pm (a)

'.-M À • prior P boundary . f '-

At location Lb, a ~ 400 pm (b) 50 p m

...continued on next page Figure 66: Typical crack/microstructure interactions in the short and long crack regimes for (a) Widmanstatten microstructure A, (b) Widmanstatten microstructure B and (c) Widmanstatten microstructure C.

176 Figure 66 (continued)

C;..»- / -,

6 .'R .4

W N’-Z'7? y

At location Le, a - 500 nm 50 |im

(c)

177 Long Crack Data Short Crack Data

I E z ■o

1 1 0 0

(a)

Long Crack Data Shon Crack Data

I E

I

100

(b) ...continued on next page Figure 67; Comparison of long and short fatigue crack growth rates at stress ratio of 0.1 for (a) Widmanstatten microstructure A, (b) Widmanstatten microstructure B and (c) Widmanstatten microstructure 0.

178 Figure 67 (continued)

Long Crack Data Short Crack Data

u I E z 1 •o

100

(c)

179 Direction of crack growth

Figure 68: Dependence of fatigue fracture modes on crack length, a, and stress intensity factor range, AK, at stress ratio of 0.1 for Widmanstatten microstructure A. (a) a ~ 75 pm, AK ~ 7.5 MPaVm, (b) a ~ 200 pm, AK - 9.15 MPaVm and (c) a - 350 pm, AK - 12.5 MPaVm.

180 Direction of crack growth

Figure 69: Dependence of fatigue fracture modes on crack length, a, and stress intensity factor range, AK, at stress ratio of 0.1 for Widmanstatten microstructure B. (a) a - 80 pm, AK ~ 4.30 MPaVm, (b) a ~ 300 pm, a K ~ 7 MPaVm and (c) a - 500 pm, AK ~ 10.5 MPaVm.

181 Direction of crack growth

(b) (b ) 5 | im 5 |p m

(c) 5 | im

Figure 70; Dependence of fatigue fracture modes on crack length, a, and stress intensity factor range, AK, at stress ratio of 0.1 for Widmanstatten microstructure C. (a) a -100 um, AK ~ 5 MPaVm, (b) a ~ 450 pm, AK ~ 10 MPaVm and (c) a - 900 pm, AK - 14 MPaVm.

182 Transition From Transition From Colony Size Short/Long Crack Short/Long Crack Microstructure (nm) (~AK, MPaVm) (~a, pm) A 118 ±2.74 12.20 325

B 169 ±5.03 12.35 575

C 312 ± 17.2 15.00 1000

Table 15: The stress intensity factor range, AK, and the corresponding crack length at which transition occurs between the short and long crack regime for the three Widmanstatten microstructures.

183 CHAPTER 8

SUGGESTIONS FOR FUTURE WORK

1. Extension of muitiparameter model to other alloys: The multiparameter model introduced in this work showed that the effects of physical, mechanical and microstructural variabilities on fatigue crack growth rate in Ti-6AI-4V can be predicted. The model shows that for mill annealed microstructures, specimen thickness and test frequency have minimal effects on fatigue crack growth rate, while AK and Kma* are the most important contributors to growth rate. For Widmanstatten microstructures, the model shows that the coarser the colony size, the lower the fatigue crack growth rate. A natural extension of this work will be to apply the multiparameter model to other alloys.

2. Fatigue maos for other alloys: In this work, it was shown that the fracture modes associated with the parametric ranges of AK and Kmax corresponding to the three regimes of crack growth can be summarized on fatigue mechanism maps. This was done for both mill annealed and Widmanstatten microstructures. One of the important outcomes of this work was that fatigue fracture mode in the Paris regime does not always include fatigue striations, as was the case for the Widmanstatten microstructures. Therefore, as creep maps are readily available for other metallic alloys, an extension of this work will be to produce fatigue mechanism maps for other alloys.

3. Fatigue crack initiation: Including this work, there is a significant body of information on fatigue crack growth behavior of both mill annealed and Widmanstatten Ti-6AI-4V alloys. Unfortunately, there is not much information available on fatigue crack initiation in these alloys. Detailed studies are, therefore needed to improve our understanding of the mechanisms of fatigue crack initiation in both mill annealed and Widmanstatten TÎ-6AI-4V alloys.

4. Mixed microstructures (equiaxed and lamellar): In this work, it was shown that the Ti-ôAI- 4V alloys exhibit higher fracture toughness and slower fatigue crack growth rates when the

184 alloys are in the p annealed condition (lamellar), rather than the mill annealed (equiaxed) condition. This reduction in fatigue crack growth rates was attributed to higher levels of roughness-induced closure levels in the p annealed (lamellar) material. Hence, a natural extension of this study will be to examine the effects of mixed microstructures (equiaxed and lamellar) on the fatigue crack growth behavior of Ti-6AI-4V alloys.

5. Creeo-fatique interactions: Room temperature creep has been observed to occur in Ti-6AI- 4V alloys at stresses that are considerably below the yield strength. Since fatigue crack growth behavior of a material is intimately related to the non-elastic deformation occurring at the crack tip, it is quite possible that the creep properties of the material will also affect its fatigue crack growth behavior. As a result, an extension of this work will be an in-depth study of creep-fatigue interactions in Ti-6AI-4V alloys.

6. Kitagawa diagrams to define short crack length thresholds: Based upon the short crack growth rate data for a variety of ductile material, Kitagawa and Takahashi [1]. have demonstrated that there exist a critical crack size below which the threshold stress intensity factor range, AK*, decreases with decreasing crack length. They also indicated that this threshold condition is characterized by a critical stress, Ao*, which approaches the smooth bar fatigue limit, Aog for vanishing small cracks. Thus, it will be interesting to determine the fatigue thresholds as functions of crack length in both mill annealed and Widmanstatten Ti-6AI-4V.

7. Three dimensional short crack fatigue studies of the different microstructures: There is a need to conduct three dimensional short crack fatigue studies on all the four microstructures studied in this research. It will be interesting to compare the short crack growth rates of the different microstructures. It is also important to develop mechanistic understanding that could lead to the development of physically-based mechanics models for the prediction of three- dimensional short crack growth.

185 8.1 References:

1. H. Kitagawa and S. Takahashi, (1976) Applicability of fracture mechanics to very small cracks or the cracks in the early stages. In proceedings of second international conference on mechanical behavior of materials, p. 627-631. Metals park, American society for metals.

186 BIBLIOGRAPHY

P. C. Paris, M. Gomez and W. E. Anderson (1961) A rational analytic theory of fatigue. Trend Engng. Vol. 13. p. 9-14.

R. G. Forman, V. E. Kearney and R. M. Engle (1967) Numerical analysis of crack propagation in cyclic loaded structures. J. Basic Engng, Vol. 89, p. 459-464.

N. Walker, 1970, The effect of stress ratio during crack propagation and fatigue for 2024-T3 and 7075-T6 aluminum. In Effects of Environment and Complex Load History for Fatigue life, special technical publication 462, p. 1-14. Philadelphia: American Society for Testing and Materials.

P. C. Paris and F. Erdogan (1963) A critical analysis of crack propagation laws. J. Basic Engng, Vol. 85, p. 528-534.

S. Suresh, 1998, Fatigue of materials, Cambridge University Press, 2"" Edition.

S. Suresh and R. O. Ritchie, 1984a, Near-threshold fatigue propagation: a perspective on the role of crack closure. In Fatigue Crack Growth Threshold Concepts. Editors, D. L. Davidson and S. Suresh, p. 227-261. Warrendale: The Metallurgical Society of the American Institute of Mining, Mineral and Petroleum Engineers.

P. J. E. Forsyth, 1961, A two stage process of fatigue crack growth. In Crack Propagation: Proceeding of Cranfield Symposium, p. 76-94. London: Her Majesty’s Stationary Office.

C. Laird, 1967, The influence of metallurgical structure on the mechanisms of fatigue crack propagation. In Fatigue Crack Propagation, Special Technical Publication 415, p. 131-168. Philadelphia: The American Society for Testing and Materials

P. Neuman, 1969, Coarse slip model of fatigue. Acta metallurgica. Vol. 17, p. 1219-1225.

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