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Numerical and Experimental Investigation of Plasticity (Slip) Evolution in Notched Single Crystal Superalloy Specimens


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NUMERICAL AND EXPERIMENTAL INVEST IGATION OF PLASTICITY (SLIP) EVOLUTION IN NOTCHED SINGLE CRYSTAL SUPERALLOY SPECIMENS By SHADAB SIDDIQUI A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLOR IDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2006

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Copyright 2006 by Shadab A Siddiqui

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To my parents and to my wife Rukshana

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iv ACKNOWLEDGMENTS I owe thanks and gratitude to many people in my life, whose help and support have led to the successful consolidation of this research. I will begin with thanking both my mentors, Dr. Nagaraj and Dr. Ebrahimi, who have relentlessly provided me with more than I could ask in terms of their time, a dvice, encouragement and support to facilitate this work. I thank Dr. Nagaraj for encouragi ng me to pursue doctoral research after the completion my masters degree under his ow n guidance, and for giving me the support and freedom to undertake exploratory paths in my research. I thank Dr. Ebrahimi, first of all for teaching a wonderful course on Adva nced Metallurgy from which we started a collaborative effort and journey towards th is research. Words are just not enough to express the many ways in which she has influe nced my life; I consider her more than being a mentor, and think of her as a clos e friend. I would also like to thank my supervisory committee members for their contri butions to the completion of this project. Next I would like to thank my friends and colleagues Eric, George, Jeff, Matt, Srikant, and TJ for not just the valuable disc ussions of research and course work, but also for patiently bearing with my grunts and nods during those hard moments in my research when I kept my attention on my screen monito r. I owe a lot to Srik ant who played a very important role in my success and was always there for me whenever I needed him. Special thanks also go to Jeff for helping me in my research. I would also like to thank my experimentalist colleagues Yanli, Luis, Mike, Shankara, Ian, Eboni and Krishna for their warm welcome and for all their help and suggestions.

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v Last but not the least, the one person who has been cons tantly by me throughout all the ups and downs of not just research but ever y aspect of my life, is my wife. I thank her not just for her endless love and patience, but for being my biggest critic and source of motivation, and also for her edit orial help in this thesis.

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vi TABLE OF CONTENTS page ACKNOWLEDGMENTS.................................................................................................iv LIST OF TABLES...........................................................................................................viii LIST OF FIGURES.............................................................................................................x ABSTRACT...................................................................................................................xviii CHAPTER 1 INTRODUCTION........................................................................................................1 1.1 Project Background................................................................................................1 1.2 Objectives...............................................................................................................8 2 LITERATURE REVIEW...........................................................................................10 2.1 Elastic Anisotropy................................................................................................10 2.2 Plastic (Slip) Deformation....................................................................................13 2.2.1 Slip Systems in FCC Single Crystals.........................................................19 2.3 Evaluation of Plasticity at Crack Tips..................................................................22 3 MATERIAL AND EXPERIMENTAL PROCEDURE..............................................37 3.1 Material.................................................................................................................37 3.2 Experimental Procedure........................................................................................43 4 THREE-DIMENSIONAL ELASTIC ANISOTROPIC FEA OF A NOTCHED SINGLE CRYSTAL SPECIMEN..............................................................................46 4.1 Development of a Three-Dimensional Linear Elastic FEA Model......................47 4.2 Geometries of the Specimen A, B and C..............................................................49 4.3 Numerical Model Characteristics.........................................................................51 4.3.1 Elements and Meshing...............................................................................51 4.3.2 Solution Location.......................................................................................52

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vii 5 RESULTS AND DISCUSSION.................................................................................54 5.1 Definition of Dominant Slip Systems...............................................................54 5.2 Development of the Polar Plots............................................................................58 5.3 Comparison of Experiment al and Numerical Results..........................................64 5.4 Evolution of Slip Sector s as a Function of Load..................................................77 5.5 Comparison of Anisotropic and Isotropic Results................................................79 5.6 Summary...............................................................................................................84 5.7 Effect of Secondary Orientation on S lip Systems of Single Crystal Nickel Base Superalloy....................................................................................................84 5.7.1 Zero Degree Secondary Orientation...........................................................86 5.7.2 Fifteen Degree Secondary Orientation.......................................................93 5.7.3 Thirty Degree Secondary Orientation........................................................99 5.7.4 Forty Five Degree S econdary Orientation................................................102 5.7.5 Calculation of Failure Life for Different Orientations.............................112 5.8 Effect of Temperature on Slip Syst ems of Single Crystal Nickel Base Superalloys.........................................................................................................117 5.8.1 Zero Degree Secondary Orientation.........................................................120 5.8.2 Forty Five Degree S econdary Orientation................................................127 5.8.3 Summary...................................................................................................131 5.9 Comparison of Experimental and Nu merical Results in a Copper Single Crystal subjected to Four Point Bending............................................................132 5.9.1 Experimental Procedure...........................................................................132 5.9.2 Numerical Analysis..................................................................................136 6 CONCLUSIONS......................................................................................................146 LIST OF REFERENCES.................................................................................................148 BIOGRAPHICAL SKETCH...........................................................................................154

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viii LIST OF TABLES Table page 2-1 Direction cosines......................................................................................................11 2-2 Slip planes and slip dire ctions in an FCC crystal.....................................................20 2-3 Comparison of sector boundary angles....................................................................28 2-4 Comparisons of experimental sect or boundary angle with numerical and analytical solutions of Orientation 2........................................................................29 2-5 Comparison among the plane strain, pl ane stress and experimental data................33 4-1 Material properties used in the analysis of the notched single crystal specimens...48 4-2 Actual and finite element specimen dimensions of specimens A, B and C in mm........................................................................................................................50 5-1 Comparison of numerical and experi mental results on the surface of the specimen A, at r = 5 ..............................................................................................68 5-2 Numerical prediction of do minant slip systems on the surface of specimen A, for varying radii, r, from the notch................................................................................68 5-3 Numerical predictions of dominant sl ip systems on the surface of specimen B for varying radii r, from the notch............................................................................70 5-4 Material properties used for the isotropic model.....................................................81 5-5 Summary of maximum RSS on the surface of the notch at several thicknesses, for 0 orientation.......................................................................................................92 5-6 Summary of maximum RSS on the surf ace of the notch, at several thicknesses for the 15 orientation...............................................................................................99 5-7: Summary of maximum RSS on the su rface of the notch at several thicknesses, for the 30 orientation............................................................................................101 5-8 Summary of maximum RSS on the surface of the notch at several thicknesses, for 45 orientation..................................................................................................104

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ix 5-9 Summary of the maximum RSS, and the predicted number of cycles for failure for 500N cyclic load...................................................................................116 5-10 Cube slip systems in FCC crystals.........................................................................118 5-11 Material proper ties used at 38 C and 927 C in this analysis.................................120 5-12 Summary of the maximum RSS and the corresponding load fo r slip initiation on the surface and midplane of the specimen..............................................................125 5-13 Summary of maximum RSS and corresponding loadsreq uired for slip initiation on the surface and midplane of the specimen.........................................................130 5-14 Material properties used in th e analysis of coppe r single crystal...........................136 5-15 Comparison between experimental, nume rical and analytical sector boundaries at r = 0.35mm for orientation I...............................................................................140 5-16 Comparison between experimental, nume rical and analytical sector boundaries at r = 0.7 mm for orientation I................................................................................141 5-17 Comparison between experimental, nume rical and analytical sector boundaries at r = 0.35mm for orientation II..............................................................................144 5-18 Comparison between experimental, nume rical and analytical sector boundaries at r = 0.7mm for orientation II................................................................................144

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x LIST OF FIGURES Figure page 1-1 Figure showing the microstructure of SCNBS...........................................................2 1-2 Different crystals used for th e manufacturing of turbine blades................................2 1-3 Blade leading edge crack location and orientation for the SSME AHPFTP 1st Stage Turbine Blade...................................................................................................3 1-4 Figure showing the {001} orientation of the turbine blade........................................4 1-5 Convention for defining single crys tal orientation in turbine blades.........................5 1-6 Slip systems predicted by Rice. Th e thick lines indicat e the sector boundary angles......................................................................................................................... .6 1-7 Sector boundaries at the crack tip. A) Slip sector bounda ry. B) Kink sector boundary.....................................................................................................................8 1-8 Figure showing dislocations A) shearing the precipitate, B) by-passing the precipitate...................................................................................................................9 2-1 Slip lines observed on the surface of copper crystal................................................15 2-2 Slip elements in a specimen subjected to uniaxial tension.......................................16 2-3 Yield behavior of anth racene single crystals............................................................17 2-4 Schematic diagram of the deformation ch aracteristics of a single-crystal in a tensile machine.........................................................................................................18 2-5 Shear stress/strain cu rves of metal crystals..............................................................19 2-6 Four octahedral slip planes of FCC crystal showing primary slip directions..........20 2-7 Effect of orientation on the number of active slip systems {111}<101> for the FCC crystal lattice structure.....................................................................................21 2-8 Effect of orientation on the shape of the flow curve for FCC single crystals..........22 2-9 Orientations of specimens used by Rice..................................................................23

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xi 2-10 Orientation of specimen used by Shield et al. (1993)..............................................25 2-11 The E22 strain components near a notch in an iron silicon single crystal..............26 2-12 Orientations of specimens used by Crone and Shield for their experiment.............29 2-13 Experimental slip sectors.........................................................................................31 2-14 Orientations of specimens used by Schulson et al. (1997).......................................32 2-15 Optical micrographs of slip sectors around a notch tip of Ni3Al.............................33 2-16 Figure shows the results for the specim en having (011) as the crack plane and [100] as the notch growth direction..........................................................................35 2-17 Figure shows the results for the specim en having its crack plane as (001) and the notch growth direction as [100]..........................................................................36 3-1 SEM pictures showing the tested supera lloys microstructure which consists of the precipitate and channels...............................................................................38 3-2 Figure shows the dog-bone specimen, A) schematic, B) actual and C) including double notch.............................................................................................................39 3-3 Figures showing orientations of specimens.............................................................39 3-4 Engineering stressstrain curves for specimen A and B..........................................40 3-5 Engineering stressstrain curves for specimen C.....................................................40 3-6 The evolution of slip lin es in the tensile specimen of nickel-base superalloy immediately, a) after yi elding and b) fracture..........................................................41 3-7 SEM pictures showing the shearing of precipitate in the SCNBS........................42 3-8 A 001 standard stereographic projecti on of cubic crystals showing different poles or directions....................................................................................................43 3-9 Fixture designed for notch cutting...........................................................................44 3-10 Figure showing the specimen grips a nd the dog-bone specimen used in the experiments..............................................................................................................44 4-1 Figure showing the material and specimen co-ordinate system...............................47 4-2 Flow chart for the analys is of the slip fields............................................................49 4-3 Dimension of the specimens...................................................................................51

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xii 4-4 PLANE2 2-D 6 NODE Tria ngular Structural Solid.................................................52 4-5 SOLID95 3-D 20Node Structural Solid...................................................................52 4-6 Close-up view of the notch.......................................................................................53 4-7 Figure shows the 3-D FEA model............................................................................53 5-1 Figure shows the RSS values at r=2 on the surface of specimen A at 1500 N load...........................................................................................................................55 5-2 Figure shows the RSS values at r=2 on the surface of specimen A at 2100 N load...........................................................................................................................56 5-3 Figure shows the RSS values at r=2 on the surface of specimen A at 4982 N load...........................................................................................................................56 5-4 Figure shows the RSS values at r = 1 on the surface of specimen A at 1600 N load...........................................................................................................................59 5-5 Figure shows the RSS values at r = 1.5 on the surface of specimen A at 1600 N load...........................................................................................................................59 5-6 Figure shows the RSS values at r = 4 on the surface of specimen A at 1600 N load...........................................................................................................................60 5-7 Polar plot shows the dominant slip systems around the notch on the surface of specimen A loaded to 1600 N..................................................................................60 5-8 Figure shows the RSS values at r = 1 on the surface of specimen A at 3200 N load...........................................................................................................................62 5-9 Figure shows the RSS values at r = 1.5 on the surface of specimen A at 3200 N load...........................................................................................................................62 5-10 Figure shows the RSS values at r = 4 on the surface of specimen A at 3200 N load...........................................................................................................................63 5-11 Polar plots shows the dominant slip systems around the notch on the surface of specimen A loaded to 3200 N..................................................................................63 5-12 Comparison between numerical and experimental results from r = 0.5 to 8 on surface of specimen A at load=4982 N (KI = 50MPam1/2)......................................67 5-13 Figure shows RSS values at r = 5 on the surface of specimen A at 4982 N load..........................................................................................................................67

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xiii 5-14 Comparison between numerical and e xperimental results on the surface of specimen B loaded to A) 1780 N (KI = 20MPam1/2) and B) 3456 N (KI = 40MPam1/2)...............................................................................................................71 5-15 Plot shows the front and back surface slip fields on the left and right notches of specimen C...............................................................................................................73 5-16 Stress distribution on the surface of sp ecimen C, at the left notch at r = 2 ...........74 5-17 Stress distribution on the surface of sp ecimen C, at the ri ght notch, at r = 2 .......75 5-18 Comparison between numerical and e xperimental results on the surface of specimen C loaded to 3500 N (KI = 30MPam1/2).....................................................76 5-19 Traces of different {111} slip planes, on the plane of observation (that is (130) plane)........................................................................................................................77 5-20 Polar plots showing the evolution of slip fields around the notch of the specimen A loaded to A) 1600 N (KI = 15MPam1/2), B) 3200 N (KI = 30MPam1/2) and C) 4982 N (KI = 50MPam1/2)........................................................78 5-21 Comparison between the anisotropic and isotropic slip fields of the specimen A...............................................................................................................81 5-22 Comparison between the anisotropic and isotropic slip fields of specimen B.........82 5-23 Stress distributions on the surface of specimen A (anisotropic case) at r = 5 at load = 4982 N...........................................................................................................83 5-24 Stress distributions on the surface of specimen A (isotropic case) at r = 5 at load = 4982 N...........................................................................................................83 5-25 Figure showing the variati on of secondary orientation from 0 to 45.................85 5-26 Figure showing the elements through th e thickness in a 3D finite element model........................................................................................................................87 5-27 Polar plots showing the evolution of s lip fields around the left notch of the 0 orientation specimen at various thicknesses at load = 4982N..................................88 5-28 Figure shows the nomenclature of th e finite element nodal points along the specimen notch.........................................................................................................90 5-29 Plot showing the maximum RSS on the surface of the notch, on the specimens front surface, for 4982 N load..................................................................................92 5-30 Plot showing the maximum RSS on the surface of the notch, on the specimens midplane, for 4982 N load........................................................................................93

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xiv 5-31 Plots show the front and back surface slip fields on the left and right notches of the15 orientation specimen.....................................................................................94 5-32 Polar plots showing the evolution of s lip fields around the left notch of the 15 orientation specimen at various thicknesses at load=4982N....................................96 5-33 Polar plots showing the evolution of s lip fields around the left notch of the 30 orientation specimen at various thicknesses at 4982N load...................................101 5-34 Polar plots showing the evolution of s lip fields around the left notch of the 45 orientation specimen at various thicknesses at 4982N load...................................103 5-35 Polar plots showing the evolution of s lip fields as a function of the secondary orientation at the front surface of the specimen around the left notch at 4982N load.........................................................................................................................105 5-36 Polar plots showing the evolution of s lip fields as a function of the secondary orientation at 0.3mm thickness of the sp ecimen around the left notch at 4982N load.........................................................................................................................106 5-37 Polar plots showing the evolution of s lip fields as a function of the secondary orientation at 0.6mm thickness of the sp ecimen around the left notch at 4982N load.........................................................................................................................107 5-38 Polar plots showing the evolution of s lip fields as a function of the secondary orientation at the midplane of the specimen around the left notch at 4982N load.........................................................................................................................108 5-39 Polar plots showing the evolution of s lip fields as a function of the secondary orientation at 1.2mm thickness of the sp ecimen around the left notch at 4982N load.........................................................................................................................109 5-40. Polar plots showing the evolution of s lip fields as a function of the secondary orientation at 1.5mm thickness of the sp ecimen around the left notch at 4982N load.........................................................................................................................110 5-41. Polar plots showing the evolution of s lip fields as a function of the secondary orientation at 1.8mm thickness of the sp ecimen around the left notch at 4982N load.........................................................................................................................111 5-42. Extrusion and intrusion on the su rface of a copper single crystal..........................113 5-43. Persistent Slip band obs erved in a SCNBS HCF specimen....................................114 5-44. Fatigue crack initiation at PSB in a copper single crystal at 20C.........................115 5-46. Three cube slip planes in FCC crystal....................................................................119

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xv 5-47. Slip fields on the surface of the specimen with zero degree secondary orientation...............................................................................................................121 5-48. Plot shows the maximum RSS at r = 1 on {111} slip plane, on the surface of the specimen at 38 C for 1600N load....................................................................123 5-49. Plot shows the maximum RSS at r = 1 on {111} slip plane, on the surface of the specimen at 927 C for 1600N load..................................................................124 5-50. Plot shows the maximum RSS on the surface of the notch on {111} slip plane on the specimen surface, at 927C..........................................................................126 5-51. Plot shows the maximum RSS on the surface of the notch on {100} slip plane at the specimen surface, at 927C..........................................................126 5-52. Slip field at the surf ace of the specimen with fo rty five degree secondary orientation...............................................................................................................127 5-53. Plot shows the maximum RSS at r=1 on {111} slip plane, at the surface of the specimen at 38 C, for 1600N load...................................................................129 5-54. Plot shows the maximum RSS at r=1 on {111} slip plane at the surface of the specimen at 927 C, for1600N load..................................................................129 5-55. Figure shows orientations I and II..........................................................................132 5-56. Figure showing the dimensions of the specimen....................................................133 5-57. Optical micrograph of the slip line fiel d around the notch tips in orientation I.....134 5-58. Slip trace around a notch tip in orientation I..........................................................135 5-59. Optical micrograph of the slip line fi eld around notch tips in orientation II..........135 5-60. Slip traces around a notch tip in orientation II.......................................................136 5-61. Radial and angular coordinates used for producing slip sector plots.....................137 5-62. Figure shows the 3-D FEA model of th e four point bending specimen with boundary conditions...............................................................................................138 5-63. Plot shows the stress di stribution for orientation I at r = 0.35 mm distance from the notch tip...................................................................................................139 5-64. Plot shows the stress distributi on for orientation I at r = 0.7mm...........................139

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xvi 5-65. Plot shows the stress distributi on for orientation II at r = 0.35mm........................143 5-66. Plot shows the stress distributi on for orientation II at r = 0.7mm..........................143

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xvii KEY TO ABBREVIATIONS SCNBS Single Crystal Nickel Base Superalloy HCF High Cycle Fatigue LCF Low Cycle Fatigue FCC Face Centered Cubic BCC Body Centered Cubic PSB Persistent Slip band SSME Space Shuttle Main Engine EDM Electron Discharge Machine CRSS Critical Resolved Shear Stress RSS Resolved Shear Stress

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xviii Abstract of Dissertation Pres ented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy NUMERICAL AND EXPERIMENTAL INVEST IGATION OF PLASTICITY (SLIP) EVOLUTION IN NOTCHED SINGLE CRYSTAL SUPERALLOY SPECIMENS By Shadab Siddiqui August 2006 Chair: Dr. Nagaraj Arakere Cochair: Dr. Fereshteh Ebrahimi Major Department: Mechanical and Aerospace Engineering Single crystal nickel base superalloys (S CNBS) are being used increasingly for high temperature turbine blade and vane applica tions in aircraft and rocket engines. As a first step toward developing a mechanistica lly based fatigue life prediction system for SCNBS components, an understanding of the evol ution of plasticity in regions of stress concentration, under the action of tr iaxial stresses, is necessary. A detailed numerical and expe rimental investigation of th e evolution of plasticity and slip sector boundaries near notches in SCNBS double-notched tensile specimens was conducted. The evolution of plastic ity in the vicinity of notches in three specimens with a <100> loading orientation, and having their notches parallel to one of the <010>, <110> and <310> directions (secondary orientation), were studie d. A three dimensional (3D) linear elastic anisotropic finite element m odel of the specimens was developed using ANSYS. Ni-base superalloys which deform by the shearing of the precipitate, were selected for the experimental study to insure that slip bands followed the slip planes,

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xix similar to single-phase materials. The tens ile testing of the notched specimens was carried out using a 1125 Instron system, and optical microscopy was utilized to observe the slip bands on the surface of the specimens near notches. The experimental tests were conducted at room temperature to limit the pl astic deformation to {111} planes, similar to FCC metals. In this study, we demonstrate that a 3D linear elastic anisotropic finite element model is able to predict the activated slip planes and sect or boundaries accurately on the surface of the specimens. The experimental and numerical results suggest that the dominant slip planes activated at low load le vels persist even at hi gh load levels, and the activation of other slip bands within a domain is initially inhibited. Results reveal that slip sector boundaries have complex curved sh apes, rather than stra ight sector boundaries as predicted previously. Moreover, both the experimental and numeri cal results indicate that sector boundaries change with increasing load. A comp arison between the isotropic and anisotropic results demonstrates that el astic anisotropy has a noticeable effect on the slip evolution near notches. The numerical model was further exploited to systematically evaluate the effects of crystallographic orientation, thickness and test temperature, on the evolution of plasticity in SCNBS. An analysis of the stresses as a function of thickne ss revealed that the activated slip systems and sector boundaries drastically change from the surface to the interior of the specimens. These numerical re sults suggest that e xperimental observation of slip lines on the surface is not representa tive of plasticity within the samples. Slip plane and sectors predicted near notches are found to be strong f unctions of the notch orientation, not only on the surface of the speci men but also at various thickness planes.

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xx Furthermore, results indicate that the slip fi elds are orientation de pendent not only at low temperature (38C), but also at high temperature (927C). Based on the dominant slip system c oncept developed here, good correlation between numerical and experimental results was also found in copper single crystals subjected to four point bending load. This finding confirms that the dominant slip system concept not only works for single crystal supera lloys, but is also applicable to other FCC single crystals, as well as for other loading modes.

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1 CHAPTER 1 INTRODUCTION 1.1 Project Background Fracture behavior of single crystalline materi als is an important area of research. In addition to the applicability of single crysta ls at micro scales, the fracture behavior in strong and relatively brittle po lycrystalline alloys is cont rolled by the response of the single grain encompassing the crack tip. Ther efore from a fundament al point of view, knowledge of fracture behavior of single cr ystals is important for understanding and predicting the toughness of polycrystalline alloys. Furthermore, the deformation mechanisms and failure modes of face cente red cubic (FCC) single crystal components subjected to triaxial st ates of static and fatigue stress are very complicated to predict. This is because plasticity precedes fracture in regions of stress concentration, and the evolution of plasticity on the surface and th rough the thickness is influenced by elastic and plastic anisotropy. Thus in order to design single crystal material against fatigue and fracture failure, more experimental and numerical testing will be necessary. This study deals with single crystal nickel base superalloy (SCNBS) in particular. SCNBS are precipitation st rengthened cast monograin a lloys based on the Ni-Cr-Al system. The microstructure of this alloy consists of approximately 60% by volume (depends on the superalloy generation) of ordered -precipitates (L12 structure) coherently set in a FCC nickel-base solid solution matrix (Figure 1-1).

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2 (A) (B) Figure 1-1. Figure showing the microstruc ture of SCNBS. A) Schematic of precipitate in a matrix (Arakere and Swanson, 2002). B) SEM micrograph showing precipitate in a matrix (Ebrahimi, et al. 2005). Due to their creep, thermal fatigue and corrosion resistance properties over polycrystalline and columnar crystal alloy, th ese alloys are widely used in the production of turbine blades and vanes, used in the aircraft and rocket engine (Figure 1-2) Polycrystal Columnar crystal Single crystal (A) (B) Figure 1-2. Different crystals used for the manufacturing of turbine blades. A) Polycrystal, columnar crystal and single crystal turbine blades. B) Comparative mechanical properties and surface stability of polycrystalline, columnar crystal and single cr ystal superalloys (Fermin, 1999). 10m

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3 The use of single crystal superalloys for the turbine blade materials has led to complex relationships between load, temperat ure and deformation. This is exacerbated by the complex geometry and anisotropy of blad e components, and the triaxial stress state they experience at the blade tip leading edge. Many of thes e blades have failed during operation due to the nucleation and propagation of fatigue cracks from an area of high concentrated stress at the blade tip leadi ng edge (Arakere and Swanson, 2002). Figure 1-3 shows the blade leading edge pr one to fatigue crack growth failure. In an attempt to study the state of stress in a component with comp lex geometry (e.g. turbine blades), notched specimens are often used to represent the areas of stress concentration or the theoretical fracture condition. As a first step towards unders tanding the effect of triaxial states of static and fatigue stress in complex stru ctural alloys, a complete numerical and experimental investigation of nickel-base si ngle crystal notched specimen is essential. Figure 1-3. Blade leading edge crack locati on and orientation for the SSME AHPFTP 1st Stage Turbine Blade (Arakere and Swanson, 2002). The growth direction of th e single crystal turbine blade is controlled in the preferred low modulus [001] cr ystallographic direction to enhance the thermal fatigue resistance and creep strength of the alloy (D reshfield and Parr; 1987). The [001] axis is

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4 parallel to the span of the blade, which is also in the direction of the centrifugal loading (Figure 1-4). Figure 1-4. Figure showing the {001} or ientation of the turbine blade. There is some variation between the [001] primary crystallographic direction and the airfoil stacking line of the si ngle crystal (commonly referred as the primary orientation angle) from one blade to anothe r due to the manufacturing process (Figure 15). However, current manufacturing capability permits control of to within 5 of the stacking line (Arakere an d Swanson, 2002). Past st udies indicate that the influence of the primary orientation angle, when constrained between 0 and 10 on the elastic stresses generated within the nickel-base single crys tal superalloy, is substa ntially lower (Deluca and Annis, 1995). Hence a variation of the prim ary orientation angle (if kept within the limit of 0-10 ) will have an insignificant impact on the life of the turbine blade. Due to this, the effect of primary orientation on the re solved shear stresses is not included in this study. In most turbine blade casti ngs, the secondary crystallogr aphic direction is neither specified nor controlled and is randomly oriented with respect to the fixed geometric axes

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5 in the turbine blade. The orientation of th e secondary direction may be controlled by using a seed crystal during its solidification (Kalluri et al., 1991). The control of the secondary orientation was not considered ne cessary, until recent re views of space shuttle main engine (SSME) turbine blade lifetime data, which indicate that the secondary orientation has a significant impact on the high -cycle fatigue resistan ce (Arakere et al., 2002). Furthermore, the creep properties of sing le crystals are greatly influenced by the secondary crystallographic orie ntation (Kakehi, 2004). Theref ore an optimization of the secondary orientation angl e (commonly referred as which is the angle between secondary crystallographic dir ection and airfoil mean chord line) has the potential to increase the fatigue life of the single crystal material, wit hout additional weight or cost. Although extensive research has been done on the effect of pr imary orientation, little is known about that of the secondary orientation. Figure 1-5. Convention for defini ng single crystal orientation in turbine blades (Arakere and Swanson, 2002).

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6 For single crystal materials, the analysis of the stress and deformation fields near a crack tip goes back to the wo rk of Rice (1987). Rice proposed th e first asymptotic (i.e., as the radius from the crack tip approaches zero) solution of the crack tip stress field in FCC and BCC single crystals for two specific crack orientations under pl ane strain conditions using a small strain plasticity framework. He predicted plastic deformation in the form of patchy fan shaped sectors around the crack tip of single crysta ls (Figure 1-6). Figure 1-6. Slip systems predicted by Rice. The thick lines indicat e the sector boundary angles (Crone et al. 2001). Later his work was investigated analy tically (Drugan, 2001), experimentally (Cho et al. 1991; Li et al. 1991; Shield et al. 1993; Shield, 1996; Schulson et al. 1997; Crone and Shield, 2001; Kysar et al. 2001; Crone and Shield, 2003; Crone et al. 2003) and numerically (Mohan et al. 1992; Cuitino and Ortiz, 1996; Forest et al., 2000; Flouriot et al. 2003). Some of the models (Rice, 1987; Mohan et al. 1992; Cuitino and Ortiz 1996) were able to predict certain features seen in the experi ments, however there existed significant areas of discrepancies. For ex ample, Rices analytic al solution does not distinguish between the two orientations sector boundaries an d also the sector boundaries between FCC and BCC crystals (Ric e, 1987). However, Shield et al., (1993, 1996) later discovered from their experimental an alysis that the sector boundaries of FCC crack

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7 and BCC crystals, and of two different orientations, are dissimilar. The non-hardening analytical solution provided by Rice (1987) a nd Drugan (2001) provides the closed form solution for the stress field around the crack, but are asymptotic in nature with an unspecified region of dominance. Moreover, Ri ces analytical solution predicts either kink bands (slip direction perpe ndicular to the band) or slip bands (slip direction along the band) on the sector boundaries and require s the presence of both the bands in the slip fields (Figure 1-7), whereas Drugan came up with an analytical solution that only involves localized slip bands. This indicates that the non-hardening plasticity solution is not unique. Besides, the majority of publishe d results use plane st ress or plane strain assumptions in their analytical and numeri cal models (Rice, 1 987; Mohan et al., 1992; Drugan, 2001; Crone et al., 2003). In fact, the main drawback of the 2D models is their inability to accommodate disp lacements along the burgers vect ors that do not lie in the plane of analysis. Moreover many experiment alists (Shield et al., 1993, Shield et al., 1996, Crone et al., 2003) have measured deforma tion fields using moire interferometry, which allows precise determination of in-plane normal and shear strains. However, since the technique is applied to a free surface, the resulting measurements are under conditions of neither plane strain nor plane stress, which introduces ambiguity when comparing experimental measurements with theoretical and numerical predictions under plane strain conditions. Therefore in order to evaluate the usual plane st rain condition at the specimen center or to determine the stressstrain fiel ds at the free surface, usage of a threedimensional numerical model will be require d instead of using two dimensional plane stress or plane strain models.

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8 Figure 1-7. Sector boundaries at the crack tip. A) Slip se ctor boundary. B) Kink sector boundary (Rice, 1987). 1.2 Objectives An accurate modeling of the anisotropic ma terial behavior and the fatigue damage process is fundamental to th e development of a mechanis tically based life-prediction system for single crystals. An ideal numerical test would incorporate the parameters for the specimen size, type of test, plastici ty, hardening, lattice rotation and so on, and eventually a fatigue crack rather than a notch. However before incorporating such complexity, a basic model must be developed. Keeping this in mind, the objectives for the study is motivated by the need to unders tand the deformation behavior of single crystals under a triaxial state of stress. The first step towards realizing this goal is to conduct a comprehensive experimental and numer ical investigation of the evolution of plasticity (slip) near notches in one of the SCNBS. A double edged notched rectangular specimen is chosen for the study. There are seve ral goals in this project. The first is to develop a 3-D linear elastic anisotropic fi nite element model (using ANSYS) which can predict slip activation near notches and the second is to va lidate the same by experimentally generating the slip ba nds. The next goal is to understand how crystallographic orientation, thickness, load level, and test temperature affect the evolution of plasticity in th is single crystal material.

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9 The reason for choosing SCNBS for our anal ysis is due to th eir application in turbine blades of commercial and military aircraft engines and also because of their high strength, which makes the handling of th e specimen little easier. Moreover their availability and cost also play a major role in their selection. As is already shown in the previous s ection, SCNBS is a two phase materials. Depending on the composition and heat treatmen t of SCNBS dislocation in some of the SCNBS shear (cut) through the precipitate (Figure 1-8A) or with the help of cross slip by-pass the precipitate (Figure 1-8B) at room te mperature. SCNBS in which a shearing of the precipitate takes places during yielding, be haves like a single phase FCC crystal. Since we model this SCNBS as a single phase crystal in ANSYS, we select the SCNBS which will deform by shearing of the precipitate so that the results between the FEA and experiments may be directly compared. (A) (B) Figure 1-8. Figure showing disl ocations A) shearing the precipitate, B) by-passing the precipitate. Dislocation

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10 CHAPTER 2 LITERATURE REVIEW This chapter will provide the background for understanding the deformation mechanism in cubic single crystal materials. A review of recent literature regarding the slip mechanism and plastic deformation near a notch tip in cubic sing le crystals will be provided. Elastic anisotropy found in cubic single crystals will be explained in a greater detail. In this chapter we will also discuss the different approaches (analytical, experimental and numerical) taken by rese archers in the past to understand the phenomena within the regi on close to the tip of a crack in single crystals. Advantages and limitation of their results and approaches will al so be highlighted to justify the need of a complete model which can explain the slip evol ution near stress concentrations in single crystals. 2.1 Elastic Anisotropy The generalized Hookes law relating stress and strain tensor s in linear elastic solids is shown in Equation (2-1). The Sij are compliance coefficients. For the general case of an elastic solid, there are 21 independent material constants. ijS (2-1) Since the elastic properties of FCC crystals exhibit cubic symmet ry also described as cubic syngony, only three independent elas tic compliance constants are required to describe the elastic properties of the crys tal. In material coordinate system, Sij can be expressed as shown in Equation (2-2).

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11 111212 121112 121211 ij 44 44 44SSS000 SSS000 SSS000 S 000S00 0000S0 00000S (2-2) The compliance constants of the single cr ystal vary with th e direction of the coordinate axis (that is the Sij matrix varies with the crys tal orientation). The elastic behavior in the specimen coordinate system can be described by a transformation of the compliance tensor from the material coordi nate system to the specimen coordinate system according to the law of transformati on of fourth rank tensors. Lieberman and Zirinsky developed a method for transforming th e compliance and stiffness constants of a single crystal between two coor dinate systems that are rela ted by a matrix of direction cosines as shown in Table 2-1(Lieberman and Zirinsky, 1956). Table 2-1. Direction cosines x y z x 1 2 3 y 1 2 3 z 1 2 3 Consider a Cartesian co ordinate system (x, y, z) th at has rotated about the origin O of (x, y, z). The elastic constant matrix [Sij] in the (x, y, z) coordinate system that relates { } and { }: [{ } = [Sij] { }] is given by the following transformation,. T ijijSS (2-3) The transformation matrix [ ] is a 6x6 matrix that is a function of the direction cosines between the (x, y, z) and (x, y, z) coordinate axes (Equation 2-4).

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12 222 123233121 222 123233121 222 123233121 112233233213312112 112233322331131221 112233233213312112222 222 222 (2-4) Knowing the state of stress at a given locati on in the material coordinate system (x, y, z), the resolved shear stresses on the 12 primary octahedral slip systems denoted by 1, 2,, 12 can be readily obtained using the fo llowing method. The stre ss vector S acting on some plane with an outward normal vector N is given as: 1 11213 221223 31323 3S l Sm n S (2-5) where N= li+mj+nk is a unit vector no rmal to the octahedral plane and ij are the stress components in the principal coordinate of th e material at the point of interest. The components of the stress in some direction p (which will be the slip directions), is then calculated using, 123123S S Sp p p (2-6) where p is also a unit vector. Equation 27 will resolve the com ponent stresses in the primary directions as listed in Table 2-2 from 1 to 12.

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13 1 2 3 4 5 6 7 8 9 10 11 12101101 011110 110011 101101 110011 011110 1 110011 6 011110 101101 011110 101101 110011 1 2 3 12 31 23 (2-7) To resolve the component stresses on cube planes, equation 2-8 will be used. 13 1 14 2 15 3 16 12 17 31 18 23000101 000101 000110 1 000110 2 000011 000011 (2-8) 2.2 Plastic (Slip) Deformation Having discussed the elastic properties of cubic single crystals, we now proceed towards the plastic deformation in single cr ystals. Slip, twinning and diffusion assisted plastic deformation are the three main mechan isms responsible for inelastic deformation in metals. When the temperature is less than approximately 0.5 Tm, where Tm is the absolute melting temperature, crystalline metals deform primarily by the propagation of dislocations through the lattice. The plastic de formation occurs in metals due to the glide of dislocations on the slip plane (the plane of high atomic density ) in the close packed direction (which represents the shortest distance between two atoms equilibrium

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14 positions). The type of slip systems varies wi th the materials crystal lattice. At higher temperatures, deformation occurs by diffusioncontrolled processes such as dislocation climb. Twinning, a rotation of at oms in the lattice structure is not as important, as strains are very small compared to slip and climb, however the process becomes important at very low temperatures. The resistance of a crystal to slip, as well as the level of resolved shear stress along the slip direction in the slip plane controls the activation of slip in single crystals. The resistance to slip depends on the atomic bondi ng and crystal structure, while the level of resolved shear stress (RSS) depends on the leve l of loads and their direction relative to the crystal axes. Slip begins when the resolved shear stress on the slip plane in the slip direction reaches a threshold value called the critical reso lved shear stress (CRSS) (Dieter, 1986). The value of the critical re solved shear stress depends chiefly on the material composition, strain rate, and temperature. During plastic deformation, slip bands can be observed on the polished surface of specimens. Each atom in the slipped part of the crystal moves forward the same integral number of lattice spacings and causes change s in the surface elevation, which at high magnification can be seen as slip lines. Figure 2-1 shows slip lines on the polished surface of copper single crystal (Dieter, 1986).

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15 Figure 2-1. Slip lines observed on the surface of copper crystal (Dieter, 1986). As dislocations can only glide under the eff ect of shear stresses, these shear stresses have to be determined. Figure 2-2 shows the orientation of the slip plane and the slip direction in the crystal relative to the loadi ng axis in a simple tensile test. According to Figure 2-2, we see that the cross secti onal area of the slip plane is given by: slip planeA A cos (2-9) where A cross-sectional area perpendicular to the loading axis, angle between the rod axis and the normal to the slip plane. Furthermore, the load on this plane resolv ed in the slip direct ion, is given by: resolvedF Fcos (2-10) where F applied axial force and angle between load axis and slip direction. Dividing equation 2-10 by 2-9, we get the resolved shear stre ss acting on the slip plane. resolved RSS slip planeF Fcoscos AA (2-11)

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16 Figure 2-2. Slip elements in a specimen subjected to uniaxial tension. The factor m = coscos represents an orientation factor which is known as the Schmid factor. From equation 2-11, we can s ee that slip will occur on the slip system possessing the greatest Schmid factor. Equati on 2-11 shows that in some circumstances RSSwill be zero (viz. if the tension ax is is normal to the slip plane (=90) or if it is parallel to the slip plane (=90)). Thus, slip deformation will not take place for these two extreme orientations, as there is no shear stress on the slip plane. On the other hand, RSSwill be maximum for = =45. Single crystals are elastically anisotropi c, whereby each crystallographic direction may respond differently to similar loadi ng conditions. Schmid and his coworkers (Schmid et al. 1935) experimentally confirme d that in hexagonal crystals (such as cadminum, zinc and magnesium), the tensile yield stress varies greatly with the F Cross-sectional area A Slip Plane Slip Direction Applied force F= A N ormal n Tensile Axis F

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17 crystallographic orientation. However, when the tensile yield stress is converted to resolved shear stress by using equation 2-11, it is found that the calculated resolved shear stress is constant for a particular material. This simple yield criterion for crystallographic slip is called Schmids Law. For example, it can be seen in Figure 2-3 that the axial stress necessary for yielding anthracene crystals vari es dramatically with crystal orientation, while the critical resolved shear stress is unchanged (Robinson and Scott 1967). From Figure 2-3A we can see that the elastic m odulus of the crystal also depends on the crystallographic orientation. It is also worth noting that th e yield stress is minimum for M=0.5. Consequently, it is very importan t to specify the orientation of the load. (A) (B) Figure 2-3. Yield behavior of anthracene single crystals. A) Axial stress-strain curves for crystals possessing different orientations relative to the loading axis. B) Axial stress for many crystals versus respectiv e Schmid factors (Robinson and Scott, 1967). It has been seen in the literature that the deformation mechanism of single crystals are mostly studied by loading them in simple uniaxial tension. During the experiment, the tensile specimen is fully constrained at both the ends as the specimen is in grips, which are attached to the crossheads. Therefore when the load is applied, the specimen is not

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18 allowed to deform freely by uniform glide on every slip plane along the gage length of the specimen (Figure 2-4). Due to this, the slip planes rotate towards the tensile axis of the specimen. For example, consider the deformation of the single crystal tensile specimen oriented in a single slip orientation. The grips hold the ends of the specimen in a fixed alignment. During application of the load, slip will initiate on a number of parallel slip planes located close to the maximum shear stress, and slip bands will be observed on the surface of the specimen. As deformation progresses, continued slip will not be possible without rotation of the slip planes to accommodate the change in length of the specimen. Bending will occur near the grips to maintain continuity and alignment of the specimen. During elongation, the slip planes ro tate towards the tensile axis and the angle between the slip plane and te nsile axis decreases. As ro tation continues a second slip plane will become oriented in a position to permit slip. When the stress on this plane reaches the CRSS, slip will occur in a sec ond slip system. Eventually, the slip on the primary and secondary planes will become equa l and duplex slip (identical slip in both systems) will occur. Figure 2-4. Schematic diagram of the deformati on characteristics of a single-crystal in a tensile machine (Dieter, 1986).

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19 Since slip in single crystals takes place in a specific pl ane in a specific direction, the increase in length of th e specimen for a given amount of slip will depend on the orientations of the slip plane and its direction with the specimen loading axis. In single crystals the fundamental measure of plastic strain is the shear strain or glide strain which is defined as the relative disp lacement of two slip planes that are unit distance apart. Typical resolved shear stress vs. resolved shear strain curve can be seen in Figure 2-5 for different metal cr ystals. It can be seen that the curves for FCC and HCP metals are different although in all the cas es the resolved shear stress increases as deformation proceeds. This phenomenon is known as strain hardening in metals. Also from the graph we infer that the rate of wo rk hardening in FCC single crystals is much greater than in HCP crystals, which is due to the difference in the number of slip systems that get activated during deformation in FCC and HCP crystals. Figure 2-5. Shear stress/strain curves of metal crystals (Dieter, 1986). 2.2.1 Slip Systems in FCC Single Crystals For FCC crystals, the high-density planes are the octahedral {111} planes and the close-packed directions ar e the primary <110> slip di rections. There are eight {111} planes in the FCC unit cell. Since the plan es at opposite faces of the octahedron are

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20 parallel to each other, ther e are only four sets of octa hedral planes. Each {111} plane contains three <110> directions. The slip pl ane with slip direction constitutes a slip system. Therefore the FCC single crystal cons ists of 12 slip systems. Table 2-2 and Figure 2-6 show the 12 slip directions for an FCC single crystal. Table 2-2. Slip planes and slip directions in an FCC crystal (Stouffer and Dame, 1996). Slip System Slip Plane <110>{111} Slip Direction 1 (111) [ 1 10] 2 (111) [ 1 1 0] 3 (111) [ 0 1 1] 4 ( 1 1 1) [ 1 10] 5 ( 1 1 1) [110] 6 ( 1 1 1) [011] 7 ( 1 1 1) [110] 8 ( 1 1 1) [ 1 1 0] 9 ( 1 1 1) [101] 10 ( 1 1 1) [011] 11 ( 1 1 1) [101] 12 ( 1 1 1) [ 0 1 1] Figure 2-6. Four octahedral s lip planes of FCC crystal show ing primary slip directions (Stouffer and Dame, 1996).

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21 The number of active slip systems in FCC single crystals depends on the orientation and magnitude of the applied lo ad. If the loading orientation of the FCC single crystals lies within the stereographic triangle, the deformation will take place on the slip system with highest Schmid factor of all the twelve {111}<101> slip systems. However if the loading axis of the crysta l lies on the boundaries of the stereographic triangle, then the CRSS values will be the same on more than one slip systems and plastic deformation will occur simultaneously on the slip systems with equivalent Schmid factors. It has been seen that for {111} <101> slip system, loading in the [001] direction will activate eight slip systems simultaneously if the stress is equivalent to the CRSS. Loading in the [110] will activate four slip systems simultaneously, whereas only one slip system will be activated for loading in th e [123] direction. Figure 2-7 summarizes the number of slip systems similar to the {111}< 101> slip system with equal stresses as a function of orientation. Figure 2-7. Effect of orient ation on the number of active slip systems {111}<101> for the FCC crystal lattice structur e (Stouffer and Dame, 1996). The effect of crystallographic orientation on the flow curve of FCC single crystals can also be explained by the number of activ ated slips for different loading directions. Figure 2-8 shows that the increase in shear st ress with shear strain is different for

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22 different loading orientations. When the speci men in loaded in the <011> direction, one slip system is experiencing appreciably more shear stress than any other and the flow curve shows a less strain hardening. When th e specimen is loaded in the <111> or <100> direction, the stress on severa l slip systems is not very different and the material experiences more strain hardening. Figure 2-8. Effect of orientation on the shap e of the flow curve for FCC single crystals (Dieter, 1986). 2.3 Evaluation of Plasticity at Crack Tips Rice in 1987, provided the foundation for much recent and current work in the area of crack/notch tip stress and strain analysis by examining the mechanics of both FCC and BCC notched specimens loaded in tension. Although the analysis techniques are applicable to other orientations, he paid atte ntion to two specific crack orientations in FCC and BCC crystals. The first orientation defined the notch plan e as (101), the notch growth direction as [ 010], and the notch ti p direction as [ 101]. The second orientation defined the notch plane as ( 010), the notch growth directi on as [101], and the notch tip direction as [ 101]. Due to symmetry of the sample, only half of the notched sample is shown in Figure 2-9.

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23 Figure 2-9. Orientations of specimens used by Rice. He presented an asymptotic an alysis of the plane stress field at a crack tip in an elastic-ideally plastic crystal, which predicts a stress field with st rong strain localization along certain radial directions around the crack tip. His work predicted that the cartesian components of stress remain constant within each sector, and ch ange discontinuously from sector to sector, that is sector bo undaries are necessarily stress and displacement discontinuities. Further he claimed that the or ientation of a sector boundary is constrained to lie either parallel or perpendicular to pl astic slip planes that intersect the crack tip. Accordingly, two different types of displacem ent discontinuities at the sector boundaries exist which Rice postulated, are the result of di fferent types of disloc ation structures, as illustrated in Figure 1-7. Those sector boundaries which are parallel to a slip plane are referred to as slip discontinuities and those se ctor boundaries perpendicular to a slip plane are referred to as kink discontinuities. He also predicted the angles at which these sectors meet, for two crack orientations, for both FCC and BCC single crystals. However, the solution does not distinguish between the tw o orientations sector boundaries or between

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24 FCC or BCC crystal structures. Both crys tal orientations and structures predict boundaries at 55 90 and 125 (Figure 1-6). Rice notes the weakness of this attribute, based on contradictory experiment al studies, related to the rotation of the crystal lattice. He acknowledges the simplification of the plane strain assumption and encourages incorporating anisotropy strain hardening, and 3D effect s into future models. In 1988, Saeedvafa and Rice extended this analysis and considering hardening effects, assumed that the crystals obeyed Tayl or power law hardening, and pr oposed HRRtype solutions for crack tip singular fields (Ri ce, 1987; Saeedvafa and Rice, 1988). On the footsteps of Rice, Drugan ( Druga n, 2001) also derived the asymptotic solutions for the near-tip stress fields for stat ionary plane strain tensile cracks in elasticideally plastic single crystals. He also analy zed the same orientations (Figure 2-9), which were studied by Rice (1987) for both FCC and BCC crystals. His analytical solution was different than Rices solution in regard to th e type of sector bounda ries around a crack tip of a symmetric orientation. Rice predicted ki nk and slip type of sector boundary for a symmetric orientation whereas Drugan predicte d only slip type of sector boundary for the symmetric orientation. Also Drugans solu tion had two interesti ng differences with Rices solution: first one is th at Rices solution predicted id entical sector boundaries for cracks having the same orientations in F CC and BCC crystals whereas Drugans solution for the FCC and BCC cases differed substantially ; second is that Rice s near-tip stress field solutions were identical for both th e orientations whereas Drugans solutions differed substantially from one orientati on to another. Drugan also compared his analytical solution with Crones experime ntal observations and measurements and showed that his asymptotic solutions ag reed quite well with the experiments.

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25 Following the theoretical work of Rice, Cho et al. in 1991 performed the experimental analysis of a notched copper sing le crystal. They did a 3-point bending test and examined the surfaces near the crack tip s by an optical microscope and a stylus profilometer. They verified the theoretical predictions of Rices by confirming that the plastic field around a crack tip (i n ductile single crystals) cons ists of fan-shaped sectors with each sector characterized by a family of dominant slip lines. From their experimental findings they also concluded that the crack tip slip field was different for the two orientation tested, contradi cting Rices theoretical prediction. Shield and Kim (1993) followed the work of Rice to correlate their experimental solution with Rices analytical solution. Re sults were presented for determining the plastic deformation fields near a crack tip (200m wide notch) in an iron 3% silicon single crystal (FE-11). The notch in Shields and Kims specimen was in a (011) plane with prospective crack growth in a [100] direction. Since this orientation is symmetric about the [100] axis, only the upper half-plane was considered. The specimen was loaded in four-point bending with measurements ma de at zero load after extensive plastic deformation had occurred. Figure 2-10. Orientation of specime n used by Shield et al. (1993).

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26 The specimen they considered had dimensions of 7.45 mm 6.00mm 26.05 mm. The bar was extended to a length of 51.95 mm by welding 12.95 mm long polycrystalline bars of the same cross-section to each end. They introduced a singl e-edge notch at the center of the crystal with a dept h of 2.05mm and a width of 200m. To verify that the surface strains reflect the behavior of the mate rial in the interior of the specimen, the specimen was sectioned and etched. They presen ted strains as a func tion of angle, since the strains do not vary much with radial distances from the notch tip. The angle was measured from the crack propagation dire ction and was taken as positive in the counterclockwise direction. Figure 2-11. The E22 strain components near a notch in an iron silicon single crystal (Shield et al. 1993). Shield et al. (1993) predicte d slip sectors similar to Rice based on the plastic strain field data. They assumed that the total strain was equal to the plastic strain neglecting the elastic strain. From the experiment, a pattern of four (eight symmetric) sectors was found. This pattern is shown in fi gure 2-11, which displays the strain components. Sectors 1 and

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27 2 had constant strains, although sector 2 had some small variations in strain. The third sector had the largest strain values which varied with radius in an approximately 1/r manner. The fourth sector had roughly constant strain, though the stra in levels were too low to make an absolute statement. These sectors were separate d by transition regions where the strains are changing very rapidl y. A good agreement between the interior dislocation pattern and surface strains was f ound. Thus, it was suggested that the surface measurements accurately reflect the deformati ons that occurred in the interior of the specimen and a comparison with the plane strain result of Rice was justified. However on the contrary with the help of some numerical and experimental analysis (Kysar et al. 2001; Cuitino and Ortiz, 1996; Flouriot et al. 200 3; Siddiqui et al. 2005), it was later discovered that the surface and interi or slip fields are not similar. Following his work on iron-silicon single cr ystals, Shield extended his work to copper single crystals (C1-B1). He chose to study the (011) as the crack plane with the prospective crack growth directi on in the [100] direction. This is the same orientation as the iron-silicon (FE-11) specimen. Because iron-silicon has a BCC structure and copper has a FCC structure, the slip systems in th e two materials are different. However since the orientations are the same, it was possi ble to directly compare on the basis of orientation. Also the effect of the different slip systems on the strain fields was assessed. Shield compared this work with his previ ous work and concluded that the discrete strain fields observed in FE11 were also present in this specimen. The sector boundary angles were similar to, but not exactly the same as, those observed in FE-11, which had the same orientation but differe nt slip systems (Table 2-3).

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28 Table 2-3. Comparison of sector boundary angles (Shield, 1996). FE-11 (BCC) C1-B1 (FCC) Rice (1987) Sector boundary angles (degrees) 1-2 boundary 35 43 55 2-3 boundary 65 62 90 3-4 boundary 110 100 125 The greatest difference in sector boundary angles occurred in the 1-2 sector boundary. The angle of the maximum strains (in s ector 3) was almost identical in both the specimens, suggesting that this angle may be related more to the notch tip geometry than the crystal structure. Shield also observed that load leve l had no effect on the sector boundary angles. However as the load increase d, the amount of plas ticity near the notch tips also increased. It was found that the results obtained for low loads showed similarities to Rices model but the experimental results did not correlate to Rices model at high plastic strain (Table 2-3). The boundary angles between the copper and ironsilicon samples were similar but not constant This disagreement was explained to be due to the material structure alone or due to fl aws that might be present in the material structure, regardless of a constant specimen orientation and test condition. This disagreement can also be due to the geomet ry of the notch, which is very difficult to duplicate accurately. The contradictory results of Shields experiment and Rices results provoke the need to replace the existing m odel, which can provide more accurate solutions. Crone and Shield continued experimental studies of notch tip deformation in two different orientations of single crystal copper and copper-beryllium tensile specimens. Two crystallographic orientations were consid ered in this research. Orientation 1 was defined as the orientation cont aining a crack or notch on the (101) plane and its tip along the [ 101] direction. This orientation was investig ated experimentally by Shield (1996) and

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29 Shield et al. (1993). Orientati on 2 was defined as the orient ation containing a crack or notch on the (010) plane with a tip along the [ 101] direction. Slip sector boundaries were determined experimentally, using Moir inte rferometry. The plane of observation was the same for these two orientations with the crack or notch being rotated by 90. Both of these orientations were also analytically investigated by Rice (Rice, 1987). Figure 2-12. Orientations of specimens used by Crone and Shield for their experiment (Crone et al. 2001) The visible slip patterns determine slip activ ity, but as the authors note, a lack of visible slip does not rule out any activity. Slip systems may be activated internally, rather than at the surface, or may show varying patterns on the surface as deformation continues. They compared their experimental results with Rices analytical solution, as well as numerical FEA solutions by Mohan et al. (1992) and Cuitino and Ortiz (1996). Table 2-4. Comparisons of experimental sector boundary angle with numerical and analytical solutions of Orie ntation 2 (Crone et al. 2001). Sector boundary Experimental AnalyticalNumerical In degrees Crone and ShieldRice Mohan, et al.Cuitino and Ortiz (2001) (1987) (1992) (1996) 1-2 50-54 55 40 45 2-3 65-68 90 70 60 3-4 83-89 125 112 100 4-5 105-110 130 135 5-6 150

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30 The numerical solutions were based on pl ane strain assumptions, however Cuitino and Ortiz later concluded that the problem under consideration is not a plane strain problem because of the differences between th e interior and surface fields. Even with the plane strain assumption, the results from Cuitino et als and Mohans numerical model and those from Rices analytical models do not match well with the experimental results (Table 2-4). The experimental results are somewhat am biguous due to the annulus of validity," where Crone and Shield (2001) take their measurements (Figure 2-13). This annulus corresponds to an area spanni ng the region between the radi al distances of 350 through 750 m from the notch tip. The notch width is between 100-200 m, making the notch radius to be between 50-100 m. Therefore the annulus, and the region where the sectors are measured by Crone et al. (2001), was anywhere between 3.5 7.0 and 7.5-15.0 times the notch radius from the tip. They chose th is annulus so that they could avoid the material close to the notch which is dominate d by the notch geometry and the material in the farfield, which is affected by the surf ace boundary. They note that the observed slip activity begins in a single sector and as the deformation proceeds, more slip lines become visible in the same sector at larger radial distances from the notch.

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31 Figure 2-13. Experimental slip sectors (Crone et al. 2001). They also clearly observed horizontal s lip traces directly ahead of the notch, however they discounted their observations and labeled the slip as elastic in order to compare their solution to other perfectly plastic sharp crack so lutions (Figure 2-13) (It is also interesting to note that perfectly plasti c models do not give st ress distributions, and yielding is judged based on elas tic stresses although the mode l is plastic in nature). Contrary to the equivalent se ctors predicted by Rice, Crone and Shields observed sectors show a marked difference with orientation, varying in bot h specific boundary angles and in the number of sectors. The FEA plane strain results from Cuition et al. (1996) appeared to correspond more closely to the expe rimental results of Crone et al. (2001). Schulson and Xu (1997) examined the state of stress at a notch tip for single crystal Ni3Al, the -component of single crystal supe ralloys using th ree-point bending Anulus of validity

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32 specimens. The orientation they analyzed has the notch plane as ( 101), the notch growth direction as [010], and the notch tip direction as [ 101]. Figure 2-14. Orientations of specime ns used by Schulson et al. (1997). An analytical model based on elastic isot ropic assumptions was used to calculate stress field around the notch, based on the equations for a sharp notch. Two solutions based on plane stress and plane strain assump tions were evaluated. The solution based on plane stress predicted four sectors to be activated around the notch however the plane strain solution predicted the activation of five sectors around the notch. They also found that within each sector, one slip system domi nates. Experimental re sults from Schulson et al (1997) show that the notch tip deformation field of Ni3Al is characterized by fan shaped sectors (Figure 2-15). Results also in dicated that the primary octahedral slip systems, {111}<110>, were activated in thes e sectors. Experimental results after significant plastic deformation reveal results that deviate from thos e predicted by either plane stress or plane strain assumption, but are closer to the plane stress assumption (Table 2-5). However they note that since the notch causes a triaxial state of stress, both plane stress and plane strain assumptions ignoring anisotropy are approximations.

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33 Figure 2-15. Optical micrographs of slip sectors around a notch tip of Ni3Al (Schulson et. al. 1997). Table 2-5. Comparison among the plane stra in, plane stress and experimental data (Schulson et al. 1997). Sector Plane Strain Plane Stress Experimental (deg) Slip systems (deg) Slip systems (deg) Slip systems I 0-23 ( 111) [ 110] 0-43 ( 111) [ 110] 0-38 ( 111) II 23-60 ( 111)[ 110] 43-60 (111)[ 110] 38-58 (111) III 60-107 ( 111)[ 110] 60-103 ( 111)[110] 58-100 ( 111) IV 107-133 ( 111)[101] 103-180 ( 111)[011] 100-max ( 111) V 133-180 ( 111)[ 011] -Kysar (2002) published his experimental results regardi ng the crack tip deformation fields in ductile aluminum single crystal. He applied Mode I loading to his specimen and sectioned it to map in plane rotation field using Electron Backscatter

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34 Diffraction. His observations pr ovided evidence of the main features of the deformation fields predicted by Rice (1987) using continuum single crysta l plasticity, es pecially the existence of kink shear sector boundaries wh ich had not been unambiguously identified in previous studies. Numerical investigations to study the evolut ion of slip systems from the surface to the midplane of notched FCC single crystals subj ected to triaxial stat es of stress have been limited. Experimental techniques ge nerally depend on surface observations for the determination of deformation field in the vici nity of notches and cracks. Experimentalists have often compared the surface slip results w ith stress states computed on the basis of plane strain assumptions (Shield et al. 1994; Shield, 1996; Crone et. al. 2001). There is ample evidence that (Arakere et al. 2005; Siddiqui et al. 2005; Ebrahimi et. al; 2005; Cuitino et al. 1996; Flouriot et al. 2003; Kim et al. 2003) the st ate of stress, changes very rapidly from the surface to interior and neith er plane stress nor plane strain isotropic assumptions adequately describe the surface or midplane stress states respectively. To address this problem Cuitino and Ortiz (Cuitino et al. 1996) developed a 3-D finite element analysis (FEA) model of a c opper single crystal specim en loaded in four point bending. Their results suggested that there is a large discrepancy between surface and interior fields. They also found that the plane strain condi tion is not fully attained at the specimen midplane. Forest et al. (2000) also i nvestigated the effect of a generalized continuum theory on the localized deformation patterns arising at the crack tip in elastoplastic FCC single crystals by using a plane strain finite element model. He us ed the material properties of single crystal nickel base s uperalloy SC 16 for his FEA mode l. He also included lattice

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35 rotation and hardening factor in his model a nd varied the hardening parameter to see the effect of lattice rotation on localization. He noticed that when he increased the hardening parameter in his model, the slip localizati on bands started disappear ing (Figure 2-16). He justified this behavior due to an increase in the local stresses, which is causing limited localization. Figure 2-16 show s the results for the specimen having (011) as the crack plane and [100] as the notch growth direction. (A) (B) Figure 2-16. Figure shows the results for the specimen having (011) as the crack plane and [100] as the notch growth direction. A) Strain localizatio n at the crack tip of the FEA model with lower hardening parameter. B) Strain localization for higher hardening parameter. Flouriot et al. (2003) presente d 3-D finite element simulations of mode I crack tip fields for elastic ideally plastic FCC single crystal tensile specimens. The analysis was specifically applied to the single crystal nick el-base superalloy AM1 at low temperatures. They discussed their FEA result on the su rface and the midplane of the specimen and found that the plastic strain fields predicte d by the 3-D FEA are high ly three-dimensional and strongly rely on crack orientations. They reported that ther e is strong disparity between the plastic strain field obtained at th e free surface and the one in the mid-section of the specimen, which indicated that the slip activity is different in the bulk as compared

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36 to the surface of the specimen (Figure 2-17). In addition to th is, they also discovered that the strain localization found in the plane strain model is different as compared to the midplane results of the 3-D FEA model, whic h again shows the necessity of having 3-D models. Figure 2-17 shows the results for th e specimen having its crack plane as (001) and the notch growth direction as [100]. (A) (B) Figure 2-17. Figure shows the results for the sp ecimen having its crack plane as (001) and the notch growth direction as [100]. A) Strain localization at the crack tip of the plane strain model. B) Strain loca lization at the free surface and the mid plane of the 3-D FEA model. The existing analytical and numerical works provide some insight into the deformation behavior of ductile single crystals in the presence of a crack or notch, but the results available do not completely predict the behavior seen in the experiments. Additionally, experimental and numerical work available in this area is limited and only begins to elucidate the complex behavior th at occurs at a notch in single crystals. A single-crystal model that inco rporates 3-D elastic anisotr opy and near-notch plasticity effects necessary to accurately pr edict the evolution of slip s ectors in 3-D stress fields, is far from complete.

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37 CHAPTER 3 MATERIAL AND EXPERIMENTAL PROCEDURE The literature review presented in chapter 2 indicates that a co mplete understanding of the mechanics of single crys tal materials, their failure m echanisms, and methods of life prediction have yet to be achieved. Requi rement of an accurate modeling of the deformation of single crystals, occurring under a triaxial state of stress has been highlighted for optimizing the efficiency in design of components with respect to both, preventing failure and avoiding over design. Keep ing this in mind, this research involves a comprehensive experimental and numerical investigation of SCNBS material. This chapter will serve to explain the expe rimental aspect of this research. 3.1 Material Pratt and Whitney provided single crystals of two nickel-base superalloys for this research. Figure 3-1 shows the microstructure of these alloys, which consists of cubical precipitates (L12 structure) coherently set in a FCC nickel-base solid solution matrix. The crystallographic orientations of the cr ystals were known when they were first received. Specimens with different orientat ions were cut using an electron discharge machine (EDM) and were tested experimentally. Two of the specimens (specimen A and specimen B) were machined from the same alloy and therefore have similar nominal composition and same CRSS while the third one (specimen C) was machined from the other alloy and therefore ha s a different nominal composition and CRSS than specimens A and B.

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38 Figure 3-1. SEM pictures show ing the tested superalloys mi crostructure which consists of the precipitate and channels. The specimens used in this study were de signed as flat dog-boned samples. Figure 3-2 shows the schematic and actual pictures of the dog-bone samples used in this study The experimental testing of two (specimens A and B) out of the three single crystal nickel base notched tensile specimens had b een carried out by Luis Forrero (a Ph.D. student from Dr. Ebrahimis Lab), in which he obtained the stress strain curves and the optical pictures of the slip bands of bot h the specimens, while the third specimen (specimen C) was analyzed in this thesis. The specimens defined as A/ B/ C have the notch planes as (001) / (001)/ (001), the notch gr owth directions as [ 110] / [010]/ [ 310] and the direction along the notch tip (i.e. dire ction normal to the pl ane of observation) as [110] / [100]/ [130] respectively (Figure 3-3A/3-3 B/3-3C). Only gauge length and half of the specimens are shown in the Figure 3-3. 2m

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39 (A) (B) (C) Figure 3-2. Figure shows the dog-bone speci men, A) schematic, B) actual and C) including double notch. (a) (b) (c) Figure 3-3. Figures showing or ientations of specimens. A) Specimen A, B) specimen B and C) specimen C. The un-notched dog bone tensile specimens we re loaded to evaluate the CRSS of the material at room temperature. All the test s were carried out at a strain rate of 3.33 x

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40 10-5s. The observation of the slip lines confir med that this Ni-base superalloy deformed on the octahedral slip planes. The CRSS wa s calculated based on the yield stress and schmid factor for each orientation. The average yield strength was found to be 794 MPa and the CRSS on the octahedral planes was cal culated to be 324 MPa for specimen A and specimen B (Figure 3-4) and the average yi eld strength and CRSS of specimen C was found to be 868 MPa and 378 MPa respectively (Figure 3-5). Figure 3-4. Engineering stressstrain curves for specimen A and B (Ebrahimi et al. 2002). Figure 3-5. Engineering stressstrain curves for specimen C (Westbrooke, 2005)

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41 From the stress-strain curve of specimens A and B (Figure 3-4) we note that the strain-hardening rate in this particular SCNB S was relatively small. Figure 3-6 shows the evolution of slip lines in the tensile sp ecimen of a SCNBS (Ebrahimi et al. 2005) one immediately after the yield point and the other after fracture. Initially when the load was applied, deformation occurred by the form ation of Lder band (Figure 3-6a) which eventually traversed the lengt h of the sample with increa sing load. Moreover at higher strain levels, traces of other slip systems we re detected, which were consistent with the deformation on the {111}<110> slip system s (Figure 3-6b). For the <001> loading orientation, multiple slip activation (8 slip systems) is expected based on the Schmids law. The observation of the Lder band which is consistent with the lack of strain hardening, suggests that localiz ation occurs easily in the su peralloy investigated. Detailed analysis of tensile specimens also revealed that the slip bands pr opagated by shearing of the precipitates (Figure 3-7). Figure 3-6. The evolution of s lip lines in the tensile specimen of nickel-base superalloy immediately, a) after yielding and b) fracture (Ebrahimi et al. 2005).

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42 Figure 3-7. SEM pictures showing the shearing of precipitate in the SCNBS (Ebrahimi et al. 2002). SCNBSs when grown in the [001] crys tallographic direction have good thermal fatigue resistance and creep strength; this is the reason why single cr ystal turbine blades are directionally so lidified in [001] di rection. So from applicati on point of view we chose [001] as our loading or primary orientation for all three specimen. Figure 3-8 shows a 001 standard stereographic projecti on of a cubic crystal. In this figure, different numbers represents different crystallographic dire ctions. The secondary directions of the specimens ([ 110] / [010]/ [ 310]) are shown in the red circles on the 001 stereographic projection. These secondary cr ystallographic directions of the specimens A, B and C were decided on the basis of their four fold, two fold and mirror symmetries respectively.

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43 Figure 3-8. A 001 standard ster eographic projection of cubic crystals showing different poles or directions (Honeycombe, 1984). 3.2 Experimental Procedure After calculating the yield properties of the specimens, two symmetric notches were created in the specimens to analyze the results near stress concentrations. A slow speed saw with a thin diamond blade was used to cut the notches in the samples. A special fixture (Figure 3-9) was created for notch cutting, which helped create straight and symmetric notches. Following the notch cut ting, both the sides of the specimens were mechanically polished. Initially the speci men was polished to remove at least 50 m (thickness of the damage layer) from the specimen surface during the EDM cutting of the specimen.

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44 Figure 3-9. Fixture desi gned for notch cutting. Figure 3-10. Figure showing the specimen gr ips and the dog-bone specimen used in the experiments. The tensile testing of the notched spec imens was carried out using a 1125 Instron system. Special specimen grips were create d to accommodate the design of the dog-bone sample during tensile testing (Figure 3-10). The design of the grips allo ws the head of the specimen to sit flat inside the grips so that it remains straight dur ing testing. Finally optical microscopy was utilized to observe the deformation or slip bands on the surface of the specimen near notches. In this analysis, the specimens are loaded to different load levels of apparent isotropic stress intensity factors calculated on the sharp crack assumption. Equations 3-1 and 3-2 are used to calculate the stress intensity factor, KI. IKYa 3-1

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45 23aaa Y1.990.768.4827.36 WWW 3-2 Where is the far field stress, Y is dimens ionless quantity (which depends on the geometry and type of loading) and a is the crack length.

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46 CHAPTER 4 THREE-DIMENSIONAL ELASTIC ANISOTR OPIC FEA OF A NOTCHED SINGLE CRYSTAL SPECIMEN In the field of linear elastic fracture mechanics, various approaches such as analytical, numerical and expe rimental approach have been taken in order to study the elastic response of isotropi c-notched specimens under the ac tion of a tensile load. These methods developed for isotropic specimens pos e many difficulties wh en applied to three dimensional anisotropic specimen models. For ex ample, in the case of isotropic analytical models, the current solution re lies on many simplifications/app roximations that lead to inaccurate results when compared with the experimental results. However, these limitations are overcome by the use of a thr ee-dimensional FEA approach, which yields solutions that correlate well with actual e xperimental results. Moreover, unlike the analytical solutions, both the numerical and experimental model specimens have the capability of introducing notches, which act as very simplified cracks to model the fracture behavior. This chapter serves to discuss the pro cedure to develop a 3-D linear anisotropic elastic numerical model, which has the ability to predict slip initiation in single crystal superalloys. We will also discuss the geometri es of the specimen used in the numerical and experimental analyses and the characteristics of the numerical model such as material properties, meshing and coordinate systems.

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47 4.1 Development of a Three-Dimensional Linear Elastic FEA Model In a polycrystalline material each grain has its own primary and secondary crystallographic orientation but because of the random orientation of grains, these materials are isotropic in nature (except some textured materials). Moreover in a polycrystalline material, two elastic constants govern the tr ansformation from stress to strain, unlike the case with single crysta l materials. A single crystal has specific crystallographic orientation in which each di rection may respond differently to similar loading conditions. These materials (with cu bic symmetry) require three independent elastic constants (elastic m odulus, shear modulus and poisso n ratio) for stress-strain transformation. Therefore while developing a finite element model of a single crystal, special care must be taken while defining the material properties of the 3D model. (A) (B) Figure 4-1. Figure showing the material a nd specimen co-ordinate system. A) The alignment of specimen (x, y, z) and mate rial coordinate system (x, y, z), B) Finite element model is created arou nd specimen coordinate system and the material coordinate system is specified later. The commercial software ANSYS (Finite Element Software Version 8.1) was employed to model the specific geometries a nd orientations of th e double-notched tensile

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48 test specimens. ANSYS has vari ous 3D elements available to account for the anisotropic material properties. These elements in conjunction with the three independent stress tensors (S11, S12, S44), or the three independent directional properties (G, E and ), can be used to model a single crystal material. The Sij values are always reported along the <100> direction, here defined as the material coordinate system. In experiments, specimens with specific crystallographic orientations are machined from a solid single crystal material. Theref ore, while modeling a single crystal specimen in finite element software, we defined two co-ordinate systems: the material and the specimen coordinate systems (Figure 4-1) The model was created around a specimen coordinate system in the finite element soft ware. Then the material coordinate system was aligned with the specimen coordinate sy stem using direction cosines (Figure 4-1B). Finite element software aligns the materi al properties with the element coordinate system; therefore the element coordinate system must be aligned with the material coordinate system in order that the directi onal material properties are suitably applied. The stress can now be transformed in any re quired coordinate system as the properties have been defined in the material coordinate system. The material properties used in the analysis of the single crystal material are gi ven in Table 4-1. These values are for the PWA 1480, a typical Ni-base superalloy used in the manufacturing of blades for turbine engines. Table 4-1. Material properties used in the an alysis of the notched single crystal specimens (Milligan et al., 1987). Elastic Modulus (Ex=Ey=Ez) 1.21 x 1011 Pa Shear Modulus (Gx=Gy=Gz) 1.29 x 1011 Pa Poissons Ratio ( x= y= z) 0.395

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49 Before creating the 3D notched finite el ement model, the stress transformation between the material and specimen coordina te system should be properly verified. A solid anisotropic rectangular specimen was fi rst created and the six component stresses ( xx, yy, zz, xy, zx, and yz) were calculated from the material coordinate system of the specimen. Subsequently, the numerically obtained component stresses were compared with the component stresses from an analytic al solution. After ensuring the correct stress transformation in the solid model, two notches were incorporated in the model. After creating the notched FEA model, component st resses was calculated from the material coordinate system of the finite element mode l at the desired location, and used in the transformation equations (Equation 2-7) to calc ulate the individual resolved shear stresses (RSS) on the octahedral planes. Data was an alyzed over a wide range of radial and angular distances around the notch tip to creat e a complete stress field, which eventually was used to draw conclusions on sectors a nd slip activation. Fi gure 4-2 describes the entire procedure for the analysis in terms of a flow chart. Figure 4-2. Flow chart for the analysis of the slip fields. 4.2 Geometries of the Specimen A, B and C The geometries of the numerical model of specimen A, B and C are based on their experimental counterparts. The specimens used in the experimental analysis possessed end shoulders for gripping the samples (Figur e 3-2). However, FEM will not take into account the entire geometry of the experimental specimen but will be limited to the gauge length of the specimen considering that the ob jective of the study, is to evaluate the evolution of slip near the notch rather than the shoulders of the samples. An elliptical

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50 notch tip is used in the finite element mode l of the specimen, which closely approximates the experimental notch geometry. We also performed a systematic investigation on the sensitivity of the computed notch stress fields for several notch tip geometries and found minimal variation in stress fields. Moreover the length of the notches in each sample were not exactly the same on both sides while for the FEA model both notch lengths and heights were set equal to t hose of the largest actual di mensions. Table 4-2 shows the actual geometry of the specimens, as well as the finite element specimen geometries. Also, Figure 4-3 shows the dimension of the specimens. Table 4-2. Actual and finite element specime n dimensions of specimens A, B and C in mm. Specimen A Specimen B Specimen C Dimensions Actual FEM Actual FEM Actual FEM Width 5.100 5.100 5 5 . 0 0 4 4 5 5 . 0 0 4 4 5 5 . 1 1 5 5 . 1 1 Height 19.00019.000 1 1 7 7 . 5 5 9 9 4 4 1 1 7 7 . 5 5 9 9 4 4 1 1 9 9 . 0 0 1 1 9 9 . 0 0 Thickness 1.800 1.800 1 1 . 8 8 2 2 1 1 . 8 8 2 2 1.77 1.77 Right Notch Length 1.300 1.550 1 1 . 3 3 9 9 9 9 1 1 . 3 3 9 9 9 9 1.4 1.4 Left Notch Length 1.550 1.550 1 1 . 3 3 6 6 1 1 . 3 3 9 9 9 9 1.39 1.4 Right Notch Height 0.113 0.113 0 0 . 0 0 8 8 4 4 0 0 . 0 0 8 8 4 4 0.085 0.086 Left Notch Height 0.111 0.113 0 0 . 0 0 8 8 4 4 0 0 . 0 0 8 8 4 4 0.086 0.086 a (For both Left and Right Notch) 0.055 0.055 0 0 . 0 0 5 5 6 6 0 0 . 0 0 5 5 6 6 0 0 . 0 0 5 5 5 5 0 0 . 0 0 5 5 5 5 b (For both Left and Right Notch) 0.226 0.226 0.168 0.168 0.172 0.172

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51 Height Width Thickness Notch Length Notch Height Notch Radius a b Figure 4-3. Dimension of the specimens. 4.3 Numerical Model Characteristics 4.3.1 Elements and Meshing The ANSYS elements chosen for the FE M were PLANE2 (2-D 6 node triangular element with quadratic displa cement functions) and SOLID95 (3 -D structural solid with 20 nodes) capable of incorporating anisotro pic properties (Figure 4-4 and Figure 4-5). After the 3-D solid model was created, th e front face was meshed with the PLANE2 elements. This front face has precise elemen t sizing along the defined radial lines around the notch tip at 5 intervals [Figure 4-6A]. Once the front face was meshed with the desired element sizing, 3-D elements were swept through the volume to complete the meshing of the model and the two-dimensional mesh was deleted. Working in conjunction with the 2-D elements on the front face, the 3-D elements retain their sizing definitions.

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52 Figure 4-4. PLANE2 2-D 6 NODE Triangular Structural Solid (ANSYS 8.1 Elements Reference, 2003). Figure 4-5. SOLID95 3-D 20Node Structural Solid (ANSYS 8.1 Elements Reference, 2003). 4.3.2 Solution Location To observe the stresses in the vicinity of the notch (radial and angular), sixteen concentric arcs were created between the radii r = 0.5 and r = 8 in equal increments; where is the notch height (Figure 46B). The six component stresses ( x, y, z, xy, zx, and yz) were then calculated at each arc in 5 increments. The element sizing of the FEM allows data to be collected on any of the seven separate xy planes including the front, middle, and back planes (Figure 4-7B).

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53 (A) (B) Figure 4-6. Close-up view of the notch. A) Close-up view of element sizing on the specimen front face near the left notch B) Radial and angular coordinates used for producing slip sector plots. (A) (B) Figure 4-7. Figure shows the 3-D FEA model. A) Front view of th e 3-D FEA model. B) Isometric view of the m odel showing the different x-y planes through the thickness. MidPlane SurfacePlane 90 deg 0 deg 8

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54 CHAPTER 5 RESULTS AND DISCUSSION In this chapter, the experimental and nu merical results of a SCNBS are presented. Predictions based upon the numerical analysis of the effects of various parameters such as the secondary orientation, thickness and temperature on the development of plasticity at the notch tip of FCC single crystals are discussed and validated against the experimental findings. This chap ter is divided into several to pics. First we describe the procedure for the analysis of the numerical re sults and our approach in describing stresses at the notch tips. We then compare the experimental and numerical results individually for samples loaded along the <100> orient ation with three sec ondary orientations (specimens A, B and C). Comparisons are only made for the specimen surface results. Emphasis is also placed on the discussion of the effects of elastic anisotropy on the plastic zone evolution. Subsequently, the e ffects of secondary orie ntation, thickness and temperature on the plasticity evolution at notch tips of FCC single cr ystals are discussed in detail. Finally we conclude this chapte r by comparing the experimental and numerical results in a copper single crys tal subjected to a four-point bending load which has been tested by Crone et al. (2003). 5.1 Definition of Dominant Slip Systems In chapter 4, it was shown that the mesh design was created in a manner so as to enable the calculation of the six compone nt stresses at a distance r and angle from the tip of the notch (Figure 4-6). These compone nt stresses were then transformed into twelve RSS values by using Equation 2-7. Each of these twelve shear stresses were then

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55 plotted as a function of (from 0 to the top of the notch) at radii ranging from 0.5 to 8 in a 0.5 interval, in the form of x-y plots (a total of 16 x-y plots were created for each surface plane, for the region above the notch growth axis). Figures 5-1, 5-2 and 5-3 show the x-y plots of RSS values of the tw elve slip systems as functions of at radius r = 2 on the surface of specimen A at 4982 N (KI = 50 MPam1/2), 2100 N (KI = 20 MPam1/2) and 1500 N (KI = 14 MPam1/2) loads respectively. Figure 5-1. Figure shows the RSS values at r=2 on the surface of specimen A at 1500 N load. CRSS = 324MPa NOACTIVATION

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56 Figure 5-2. Figure shows the RSS values at r=2 on the surface of specimen A at 2100 N load. Figure 5-3. Figure shows the RSS values at r=2 on the surface of specimen A at 4982 N load. CRSS = 324 MPa 3 1 9108 88 137 108 2 CRSS = 324MPa 2 388 12 11 3 6 4

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57 Slip activation in single cr ystals is usually governed by Schmids law according to which when the RSS on a slip system r eaches a critical value (CRSS), plastic deformation occurs. This implies that in figures 5-1, 5-2 and 5-3, a ny point that lies on or above the dark horizontal line representing the constant CR SS value, is expected to deform plastically. For example in figure 5-1 at load = 1500 N, none of the slip systems cross the CRSS of the material and hence at th is load the material will not slip (deform plastically) at r = 2 on the surface of the specimen. As the load increases from 1500 N to 2100 N, some of the slip systems cross the CRSS of the material and become activated (Figure 5-2). 2 becomes activated from = 88 108 and 3 becomes activated from = 108 120 These are the slip systems which are first expected to appear and become visible on the surface of the e xperimental specimen at r = 2 at 2100 N load. From figure 5-3 one observes that as the load increases from 2100 N to 4982 N, more number of slip systems become activated for the same and also over a wider range of For example, observe that in figure 5-2, th e range of activation is from = 88 120 which it increases to = 0 137 in figure 5-3. Moreover in figure 5-3, from = 88 108, 3, 4, 6, 11 and 12 become activated with 2 whereas 2wasthe only activated slip system at 2100 N load from = 88 108Therefore, in this example according to Schmids law along with 2, these other activated slip systems must al so appear on the surface of the experimental specimen from = 88 108 at 4982 N load. However, according to our analysis, we predict that those slip systems, which first become activated with increasing load, will become persistent and will inhibit the activation of new slip systems with any further increase in the load, and will be the ones expected to be seen on the surface of the experimental specimen. Due to the elasti c nature of the results, the slip systems

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58 represented by the highest RSS will be the one s which first become activated with load at various values of For example from figure 5-3 one notes that 1, 2, 3 and 9 are the slip systems with highest RSS, and ther efore these are the slip systems which will become activated first at r = 2 on the surface of the specimen These slip systems with highest RSS are referred by us as the domin ant slip systems. In the next section, we shall see whether the numerical predictions based upon the dominant slip system theory can predict the slip activation in the experimental specimens. 5.2 Development of the Polar Plots From the above analysis we have seen that each x-y plot provides information about the dominant slip systems at a given ra dius, yet fails to give a complete scenario around the notch. In order to get a complete pi cture of the dominant slip systems near the notch with the help of the x-y plots, all of the information from the16 x-y plots must be analyzed simultaneously, which may become a confusing and cumbersome process. To overcome this limitation of x-y plots and to be tter visualize the slip evolution near the notch, the dominant slip systems were represen ted in the form of polar plots. These polar plots also provided a comparison between the experimental and the numerical slip field results (a slip field is a co mbination of various slip sect ors, each of which contain a different orientations of activated slip lines) on the surface of the specimen. This procedure is illustrated with the help of the following examples. Figures 5-4, 5-5, and 5-6 show the x-y plots at r = 1 1.5 and 4 respectively and figure 5-7 shows the polar plot for the surface of specimen A at a 1600 N load (KI = 15MPam1/2). These radii are randomly selected for the purpose of illustration.

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59 Figure 5-4. Figure shows the RSS values at r = 1 on the surface of specimen A at 1600 N load. Figure 5-5. Figure shows the RSS values at r = 1.5 on the surface of specimen A at 1600 N load. 74 108 96 108 117 23 2 3 NOACTIVATION NOACTIVATION NO ACTIVATION 11 3

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60 Figure 5-6. Figure shows the RSS values at r = 4 on the surface of specimen A at 1600 N load. Figure 5-7. Polar plot shows the dominant s lip systems around the not ch on the surface of specimen A loaded to 1600 N. 2 3 NOACTIVATION

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61 Figure 5-4 indicates that at 1600 N load ther e is no activation of slip systems from = 0 -74 after which the slip systems start to activate. The dark horizontal line on the x-y plots represents the CRSS of the material and the dark vertical lines represent the boundary of a particular domain. Fi gure 5-4 also shows that from = 74 -108 2, 3 and 11 experience RSS above the CRSS of the material, however only 2 which has the highest RSS and hence termed as a dominant slip systems, will be observed to be activated on the polar plot from = 74 -108 (Figure 5-7). From = 108 -110 3 acts as a dominant slip system as seen in figures 5-4 and 5-7. Figure 5-5 show the x-y plot at r = 1.5 which indicates that from = 0 -96 no slip systems were activated, from = 96 -108 2 is the only one activated and hence the dominant slip system, from = 108 -117 3 is the activated/dominant slip system and, from = 117 -140 there is again no activation. At r = 4 none of the slip systems cross the CRSS line and therefore there are no activ ated/dominant slip systems (Figure 5-6). Comparing the x-y plots at r = 1 1.5 and 4 we note that as one moves away from the notch tip the absolute RSS values of the dominant slip systems go down. Also the slip system patterns and the boundaries of the domin ant slip systems domains change with the radius. The region of the dominant slip systems at various radii occurring in the 16 x-y plots, were plotted in the form of a polar plots as seen (Fig ure 5-7). Sectors with different hatched lines indicate regions of the do minant slip systems around the notch. In summary, figure 5-7 indicates that at a load of 1600 N, 2 and 3 are the only dominant slip systems on the surface of specimen A and there is no activation after r = 1.5 radius.

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62 Figures 5-8, 5-9, 5-10 and 5-11 shows the x-y plots at r = 1 1.5 and 4 and corresponding polar plot on the surfa ce of specimen A at a 3200 N load (KI = 30MPam1/2). Figure 5-8. Figure shows the RSS values at r = 1 on the surface of specimen A at 3200 N load. Figure 5-9. Figure shows the RSS values at r = 1.5 on the surface of specimen A at 3200 N load. CRSS = 324MPa 23 37108 1 23 1 9 58 108 124 130 CRSS = 324MPa

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63 Figure 5-10. Figure shows the RSS values at r = 4 on the surface of specimen A at 3200 N load. Figure 5-11. Polar plots shows the dominant slip systems around the notch on the surface of specimen A loaded to 3200 N. Comparing figures 5-4, 5-5, 5-6 and 5-7 with figures 58, 5-9, 5-10 and 5-11 we can observe that with the increase in load from 1600 N to 3200 N, more number of slip 2 3 1 6 9 21 617 5067 110 95 NO ACTIVATION (NA)

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64 systems get activated at a given radius and, more number of dominant slip systems are seen around the notch. For example, a comp arison between the x-y plot at r = 1 for 1600 N and 3200 N load (Figure 5-4 and Figure 5-8) reveals that at the 1600 N load, there was a region of no activation from = 0 74 but at 3200 N, we can see that slip systems are activated all the way from 0 to the top of the notch (that is till 110 ). Similarly at r =1.5 more number of slip systems are activated at 3200 N than at 1600 N load (Figure 5-5 and Figure 5-9). At r = 4 there was no activation at 1600 N load, but at 3200 N some of the slip systems became activated. This implies that with an increase in the load, slip systems get activated at greater distances from the not ch tip. It should be noted that due to the elastic nature of the results, a change in th e applied load only changes the absolute RSS values of the slip systems but does not cha nge their relative positions (that is slip patterns) (compare figures 5-4 and 55 with 5-8 and 5-9, respectively). Figure 5-11 shows the polar plot corres ponding to the load 3200 N. The active dominant slip systems according to figure 510 at r = 4 are 1 from = 17 50 6 from = 67 95 and 2 from = 95 110 which are also seen as the dominant slip systems in the polar (figure 5-11) These polar plots thus give a comprehensive picture of the dominant slip systems around the notch whic h shows that the size of the plastic zone around the crack tip increases with increasing load (compare Figure 5-7 and Figure 5-11). 5.3 Comparison of Experimental and Numerical Results During a tensile testing of the notched sa mples, slip lines were observed on the surface of the samples. These slip lines pr ovide the information about the active slip planes within a sector and delineate the s ector boundary angles in the experimental specimen. Initially slip lines occurred in one sector, and additional slip lines developed at

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65 further distances from the notch tip in the sa me sector with continued loading. As the load increased, the number of slip lines and their length continued to grow until the sector was filled. Sector boundaries were identified by the change in the orientation of the slip lines. Figure 5-12 compares of the experimental plastic zone and nume rical results on the surface of specimen A at a 4982 N load. Appare nt consistency between the observed slip bands and the trace of the dominant slip systems predicted by the FEA was found. A good correlation between the expe rimental and numerical slip fields is observed by comparing the results at r = 5 Figure 5-13 shows the x-y plot of the RSS values as a function of for the 12 slip systems at radius r = 5 The active dominant slip systems, according to figure 5-13, are found to be 1 from = 0 -56 2 from = 56 -60 6 from = 60 -95 2 again from = 95 -125 and 3 from = 125 -127 These are the dominant slip systems which will be obser ved as slip bands on the experimental specimen at r = 5 (see Figure 5-12). Table 5-1 compar es the experimental and numerical sector boundaries at r = 5 It indicates that th e slip traces related to the (111) slip plane are activated from 0 to 75 on the expe rimental specimen whereas according to FEA predictions, the dominant slip systems 1 and 2 (which are also rela ted to the (111) slip plane) are activated only till 60. It is also observed that th e numerical analysis predicts the 6 system to be dominant between 60 and 95 however the slip traces related with ( 111) slip plane extend from 75 to 108, wh ich is approximately 15 off from the expected domain boundaries. Furthermore, th e slip traces on the experimental specimen related to the (111) slip plane are observed from = 96 -117 whereas the numerical analysis predicts the activation of 2 and 3 from = 95 -127 The above evaluation

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66 confirms that within approximately 15, the numerical analysis predicts the development of the slip bands near the notch tip for speci men A. It may be noted that from figure 512B that the experimental results presented he re show a slight asymmetry about the notch growth axis. This asymmetry is attributed to the difference in the two-crack lengths cut in this sample (Table 4-2), as well as any misa lignment in the loading fixture. Other factors that contribute to the lack of symmetry are the fact that the specimen orientations were a few degrees off the assumed crystallographic orientations and also the presence of specific irregularities in the notch alig nments in the experimental specimen. The comparison between the experimental observations a nd the FEA results suggest that only the dominant slip systems (t hat is the systems with the highest RSS) and not all the slip systems with a RSS above the CRSS are activated at a given load level. For example, in Figure 5-13, there are other slip systems that experience stresses above the CRSS but slip bands corresponding to their traces were not observed. Table 5-2 summarizes the numerical prediction of th e dominant slip systems on the surface of specimen A, for varying radii r from the notch.

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67 (A) (B) Figure 5-12. Comparison between numerical and experimental results from r = 0.5 to 8 on surface of specimen A at load=4982 N (KI = 50MPam1/2). A) Numerically generated slip fields. B) Experimentally generated slip fields. Figure 5-13. Figure shows RSS values at r = 5 on the surface of specimen A at 4982 N load. 21 6 2 3 12 6 9 23 11 108 10 12 4 [ 001 ] [ 110] 125 56 95 127

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68 Table 5-1. Comparison of numerical a nd experimental results on the surface of the specimen A, at r = 5 Dominant Sli p S y stem Sectors On Surface (r = 5 Numerical Solution Experimental Results Sector (deg) max Slip System (deg) Slip Plane I 0-56 1 (111) [ 101] II 56-60 2 (111) [ 011] 0-75 (111) III 60-95 6 ( 111) [011] 75-108 ( 111) IV 95-125 2 (111) [ 011] V 125-127 3 (111) [ 110] 96-117 (111) Table 5-2. Numerical prediction of dominant slip systems on the surface of specimen A, for varying radii, r, from the notch. Dominant Slip System Sectors r = 1 r = 2 r = 3 r = 4 Sector (deg)max (deg)max (deg)max (deg)max I 0-40 10-60 10-60 10-58 1 II 40-110 260-108 260-68 258-63 2 III 108-130 368-90 663-94 6 IV 130-137 990-113 294-120 2 V 113-138 3 120-135 3 r = 5 r = 6 r = 7 r = 8 Sector max max max max I 0-56 10-55 10-52 10-49 1 II 56-60 255-59 252-58 249-56 2 III 60-95 659-95 658-98 656-100 6 IV 95-125 295-124 298-121 2100-117 2 V 125-127 3 As can be seen in Figure 5-12B there is a region were both 2 and 6 systems are activated (from = 96108), which at first glance seems to be contradictory with the concept of the activation of only the domi nant systems. This observation can be

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69 explained by the evolution of the slip as a f unction of load. Let us consider the evolution of the slip traces at distance r = 5 as the specimen A is load ed. Because of the elastic nature of the calculation, the shape of the st ress distribution curves given in Figure 5-13 is independent of the load level. As the load is increased, 6 and 1 systems are activated. At larger loads, 2 and 3 systems become operationa l in regions outside the 6 and 1 domains. However as the load is increased, the initial dominant slip systems that is 6 and 1, persist in regions where they are not dom inant but they experience a RSS above the CRSS. This persistence results in the observa tion of domains with more than one set of slip bands. Therefore, the overlap of the 2 and 6 domains in the = 96108 section at r = 5 (Table 5-1) can be attributed to the extension of the 6 system beyond its dominance. Specimen B was also evaluated using techniques similar to those used for specimen A. A comparison of the FEM results with the ob served slip traces at two load levels for specimen B is shown in Figure 5-14. The result s suggest that initiall y one set of planes were activated (Figure 5-14a) and with incr easing load, the second set became activated at larger distances from the tip (Figure 514b). The FEM results identify various domains, where in each a slip system is dominated. Because of the crystallographic symmetry of the notch only domains for = 0-180 are presented. It should be noted that two slip systems (e.g. 1 or 11) are mentioned in each numerical domain the first (e.g. 1) represents the dominant system for the bottom (e.g. 11) half which corresponds to the optical (experimental) picture, and the sec ond corresponds to the top half. Thus, there is an excellent agreement between the observed slip traces and thos e predicted by the FEM calculations at both load levels The experimentally observed s lip field results are seen to

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70 be very different from those of Specimen A, highlighting the importance of crystal orientation in activating specific slip sy stems (compare figure 5-12 and figure 5-14). Table 5-3 lists the numerical prediction of th e dominant slip systems on the surface of the specimen B, for varying radii r from the notch. Table 5-3. Numerical predictions of domina nt slip systems on the surface of specimen B for varying radii r, from the notch. Dominant Slip System Sectors r = 1 r = 2 r = 3 r = 4 Sector (deg)max (deg)max (deg)max (deg)max I 0-50 40-50 45-50 416-48 4 II 50-110 150-60 1150-68 1148-57 11 III 60-65 10 58-66 10 57-65 10 IV 65-132 1 66-132 1 65-127 1 r = 5 r = 6 r = 7 r = 8 Sector (deg)max (deg)max (deg)max (deg)max I 24-46 430-45 485-102 9II 46-55 1145-50 11III 55-58 50-68 IV 58-63 10 68-85 11 V 63-121 1 85-110 4 VI 110-113 11

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71 Figure 5-14. Comparison between numerical an d experimental results on the surface of specimen B loaded to A) 1780 N (KI = 20MPam1/2) and B) 3456 N (KI = 40MPam1/2).

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72 Recall that specimen C has the notch plane as (001), the notch growth direction as [ 310] and the notch tip direction as [130] (see figure 3-3). This specimen has crystallographic symmetry with resp ect to only notch growth axis ([ 310]). This is demonstrated by the numerical results in fi gure 5-15, which show s the front and back surface slip fields on the left and right notches of specimen C. This orientation also has diagonal symmetry between the left and right notches with respect to the midplane (one which is perpendicular to the [130] directio n). By diagonal symmetry we mean that the slip patterns on the left notch on the front surf ace are similar to the slip patterns of the right notch on the back surface and vice-ve rsa (Figure 5-15). Figur e 5-15 also suggests that the plastic zone sizes at the left and ri ght notches on the same surface (front or back) are not equal. These differences in the shape a nd size of the slip fields are due to changes in the stress distribution around the left and right notches, which again are caused by the asymmetry of specimen C with respect to the loading axis.

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73 Left Notch Right Notch Front Surface Left Notch Right Notch Back Surface Figure 5-15. Plot shows the front and back surf ace slip fields on the left and right notches of specimen C. Figures 5-16 and 5-17 show the stress distributions at r = 2 on the surface of specimen C at the left and right notches resp ectively. The figures demonstrate that not 2 96 10 3 2 10 4 10 12 28 10 8 2 2 10 6 10 8 2 2 10 6 2 9 6 10 3 2 10 4 10 12 2 8

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74 only are the slip patterns at the left an d right notches different, but the RSS corresponding to the vari ous values of r and are also different. Moreover, the left notch is experiencing more stress than the right not ch and hence the plastic zone size is bigger at the left than at the right (Figure 5-18). Figure 5-16. Stress distribution on the surface of specimen C, at the left notch at r = 2 CRSS

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75 Figure 5-17. Stress distribution on the surface of specimen C, at the right notch, at r = 2 Figure 5-18 shows a comparison of the experi mental and numerical slip fields at the left and right notches on the front surf ace of specimen C. Figure 5-19 shows the trace of the different {111} slip planes on the plan e of observation (that is the (130) plane). Similar to specimen B, two slip systems are mentioned in each sector of the numerical slip fields: the first represents the dominant system for the bottom half, which corresponds to the optical pi cture, and the second, corre sponds to the top half. The comparison reveals that the slip fields betw een the experimental and numerical results match extremely well at the right notch (Figure 5-18B), however we see a small discrepancy at the left notch (F igure 5-18A). For example, the numerical results at the left notch predict to be dominant between and and to be dominant between and These domains of and were not seen in the experimental results. CRSS

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76 Figure 5-18. Comparison between numerical an d experimental results on the surface of specimen C loaded to 3500 N (KI = 30MPam1/2). A) Left notch, B) Right notch. 38 58

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77 Figure 5-19. Traces of different {111} slip pl anes, on the plane of observation (that is (130) plane). Also, is seen to be activated (t he dotted sector in the experimental picture of figure 18A) in the domain of2 in the experimental specimen however, the activation of can be explained by the slip evolution as a func tion of load by the nume rical results. Besides these discrepancies, the domains of (near the notchandonthe left notch and the domains of andon the right notch were correctly predicted by the numerical results. Furthermore, the plastic z one sizes predicted by the numerical results match extremely well with the experimental results at both the notches. The match between the numerical and expe rimental results for specimen C, which has asymmetric slip fields at both the notches, again proves that the dominant slip systems theory is extremely useful in the prediction of the experimentally generated slip fields. 5.4 Evolution of Slip Sectors as a Function of Load Slip fields of specimen A have been insp ected for three different tensile loads. Figure 5-20 shows the slip field evoluti on on the surface of specimen A at 1600 N, 3200 N, and 4982 N loads. The size of the plastic zone (slip field) ar ound the notch and the

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78 number of dominant slip systems increases wi th load. The results suggest that the slip planes activated at lower load levels persist at higher load levels, which is consistent with the recent experimental findings (Ebrahimi et. al; 2005). Numerical and experimental results demonstrate that there is a significan t change in the sector boundary angles with increasing load and hence a single number can not be used for defining a sector boundary for various load levels, as done by Rice (1987) and Shield (1996). (A) (B) Figure 5-20. Polar plots showi ng the evolution of slip fi elds around the notch of the specimen A loaded to A) 1600 N (KI = 15MPam1/2), B) 3200 N (KI = 30MPam1/2) and C) 4982 N (KI = 50MPam1/2). 2 3 1 2 6 3 9

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79 (C) Figure 5-20. Continued 5.5 Comparison of Anisotropic and Isotropic Results Rice (1987) previously noted that the incl usion of anisotropy in the numerical or analytical models is important for predicti ng the experimental resu lts accurately. Here, the comparison of anisotropic a nd the isotropic results furthe r validate this importance. Fundamentally it should be possible to calcula te the elastic modulii for a polycrystal from a weighted average of the elastic behavior of all orientations of crystals present in the polycrystal. However, the appropriate way to cal culate this average is not apparent. There have been several methods proposed in the lit erature for a random polycrystal. The Voigt (Reuss, 1929) average was based on assumi ng uniform local strain and averaging the modulii over all the orientations. The Reuss (1 929) average assumes uniform local stress and is based on averaging the compliances ove r all the orientations. Finally Hill (1952) found that the Voigt and Reuss averages co rrespond to upper and lower bounds to the true behavior and suggested th at the arithmetic average of the Reuss and Voigt averages is a good approximation for the polycrystal. La ter Hashin and Shtrikman came up with a good approximation of the upper and the lowe r bounds for the cubic polycrystals (1962). 1 2 6 9 2 3

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80 In this study, the isotropic properties were calculated using Hashin and Shtrikmans method. We took the arithmetic average of the upper and lower bounds, for calculating the isotropic properties of our model. Equati on 5-1 and 5-2 were used to calculate the shear and elastic modulus. -1 lower11 21 -1 upper22 12 11125 G= G+ 34 GG 5 G= G+ 26 GG 1 B = C2C 3 lowerupperG+ G G = 2 (5-1) 9BG E = G + 3B (5-2) where 12 12 1122 11112244-3B + 2G-3B + 2G = ; = 5G3B + 4G5G3B + 4G 1 G= CC; G= C 2 C11, C12 and C44 are the stiffness constant of the si ngle crystal material used in this study. Table 5-4 shows the calculated isotropi c properties. These isotropic properties have been incorporated into a 3D-FE model. A direct comparison of results between the isotropic and anisotropic models for single crystals was made possible by calculating RSS on the 12 primary octahedral slip systems of an FCC single crystal.

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81 Table 5-4. Material properties used for the isotropic model Isotropic Model Elastic Modulus (E) 2.39 x 1011 Pa Shear Modulus (G) 8.354 x 1010 Pa Load applied 4982 N Figure 5-21 compares the isotropic and an isotropic results for specimen A. For triaxial stress state, since onl y the dominant slip systems become activated rather than all slip systems that experience a stress above the CRSS, there is significant difference between the isotropic and anisotropic results. However, for the uniaxial tension case the activated slip systems predicted by both models are the same. It is seen from figure 521 that the area covered by 1 is almost identical in both the specimens, but the slip fields of slip systems 2, 3, and 6 are dissimilar. Also, 9 was not dominant in the isotropic case, whereas it was dominant in the anisotropic case. We observe that the anisotropic model slip field of 6 is converging towards the notch and is very similar to the experimentally generated slip field. Conversely, in the isotropic model 6 initiated at a radius of 6. These results demonstrate that elastic anisotropy has a prominent effect on the dominance of the s lip systems and show the necessity of the inclusion of anisotropy in the FE model. Anisotropic Model Isotropic Model Figure 5-21. Comparison between the anisotr opic and isotropic slip fields of the specimen A. 9 3 2 1 6 6 1 3 2

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82 Figure 5-22 shows the comparison between an isotropic and isotropic slip fields on the surface of specimen B, at a 4982 N load. Similar to specimen A, specimen B also shows dissimilarity between its anisotropic and isotropic results. The comparison shows that the incorporation of anisotropy not only modifies the boundaries of the dominant slip systems, but also changes the number of do minant slip systems. For example, in anisotropic case, we have five (9, 1 2, 4 and 11) dominant slip systems while in the isotropic case we have only three (9, 4 and 11) dominant slip systems. Figures 5-23 and 5-24 show the stress distributions at r = 5 for the anisotropic and isotropic cases respectively. These indicate that the RSS corre sponding to various values of r and are larger in the isotropic case. C onsequently the plastic zone of 11 is greater in the isotropic case as compared to anisotropic case (see figure 5-22). Anisotropic Model Isotropic Model Figure 5-22. Comparison between the anisotropic and isotropic slip fields of specimen B. 211 9 1 4 11 9 4

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83 Figure 5-23. Stress distributions on the surface of specimen A (anisotropic case) at r = 5at load = 4982 N. Figure 5-24. Stress distributions on the surf ace of specimen A (isotropic case) at r = 5at load = 4982 N. 2 119 1 119 Max RSS =370MPa Max RSS =387MPa

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84 5.6 Summary A comparison of the numerical results with the experimental results for all the three specimens reveals reasonable correlation betwee n the slip activation in the experimental specimen and those predicted numerically by the dominant slip system sectors. Good agreement between the numerical model and the experiment suggest that 3-D linear elastic, orthotropic assumpti ons are appropriate for pred icting the experimentally observed persistent slip bands In the experimental specimen, we were only able to observe those planes on which slip took place, since the identification of activated slip systems would require knowledge of the direction of the Burger vectors. However, correlating the known slip planes to the numer ical slip systems is a good measure of the models accuracy. The numerical procedure wa s able to predict the experimentally observed slip planes extremely well, while giving additional insight into the actual direction of slip (that is Bu rgers vector). The slip sector boundaries are shown to have complex curved shapes rather than straight sector boundaries, as predicted in the past literature (Rice, 1987). Furthermore, activated slip planes and s ector boundaries are shown to be a strong function of the crysta llographic orientation. Comparisons between isotropic and anisotropic result s justify the importance of th e inclusion of anisotropy in the theoretical models. 5.7 Effect of Secondary Orientation on Slip Systems of Single Crystal Nickel Base Superalloy It was discussed in the first chapter that secondary orientation has a major impact on fatigue properties of nickel base superall oys. In order to unders tand the influence of secondary orientation on slip evolution under tr iaxial state of stress and fatigue resistance of single crystal material, we examine four di stinct secondary orient ations in this study.

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85 All the numerical specimen models considered here will utilize the same geometry and material properties as that of specimen A (arb itrarily selected), to observe the effects of orientation without othe r defects/size considerations. A range of = 045 was selected for the secondary orientation as it has been seen in the literature that the variation in the stiffness constant has a periodicity of 45 in (0 is <100> orientation). Keeping the loading or primary orientation same, sec ondary orientation will be varied from = 045 in 15 increments (Figure 5-25). Initially th e results for each secondary orientation will be discussed individually. Slip evoluti on and change in maximum RSS as a function of thickness will also be discussed for each s econdary orientation. The changes in the slip field patterns as the secondary orientation of the specimen is rotate d will be discussed later. We will also discuss the change in maximum RSS as a function of secondary orientation. A load of 4982 N (KI = 50 MPam1/2) has been used for all the orientations in this analysis. Figure 5-25. Figure showing the vari ation of secondary orientation from 0 to 45. Left Notch Right Notch Front Face Back Face

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86 5.7.1 Zero Degree Secondary Orientation Zero degree secondary orientation has a f our fold symmetry hence the slip fields pattern on the left and right notch on both surf aces are similar to each other (only the slip systems change). Due to symmetry of this or ientation we analyze only the left notch of the specimen. Here the evolution of slip se ctors was analyzed thr ough the thickness of the specimen and plotted in increments of 0.3mm fr om 0 (at the front face) to 1.8mm (at the back face) as shown in figure 5-26. Figure 5-27 shows the polar plots of the dominant slip systems acting at various thicknesses for a 0 orientation specimen. The slip systems are mirror images of each other about the midpl ane (the one which is normal to the plane of observation) that is 1, 3, and 5 are the mirror images about the midplane of 9, 7, 11, and 12, respectively. Since this particular orientation has a four-fold symmetry, the slip patterns above and below the notch are al so symmetric, with a change in the slip systems that is 1, 3, 4, and 5 are the mirror images about the notch of 11, 12, 9 and 7 respectively.

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87 Figure 5-26. Figure showing the elements through the thickness in a 3D finite element model.

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88 Figure 5-27. Polar plots showing the evolution of slip fields around the left notch of the 0 orientation specimen at various thicknesses at load = 4982N. 114 1 9 5 12 3 7 19 4 11 4 11 9 1 12 5 7 3 9 1114 9 11 1 12 1 411 3 5 4 91 9 11 7 4 12 3 9 11 11 411 5 71 49 9114 A) Front Surface D) 1.8mm (Back Surface) B) 0.3mm F) 1.2mm C) 0.6mm E) 1.5mm

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89 Figure 5-27. Continued Figure 5-27 indicates that the number of dominant slip systems varies as a function of thickness. The front and the back surfaces, have twelve dominant slip systems (six above and six below the notch) whereas the s lip systems on the planes in between these surfaces have eight dominant slip systems. Furthermore, slip systems occur in pairs at the midplane for this specimen. The paired slip systems are indicative of two dominant slip systems acting simultaneously in the same lo cation over the entire midplane. The paired dominant slip systems fo r this orientation are 1& 9, 4& 11, 5& 12 and 3& 7 (Figure 5-27g). Figure 5-27 also indicate s that the slip fields on th e surface and interior of the 4& 11 1& 9 4& 115& 12 1& 93& 7 G) 0.9mm (Middle Surface)

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90 specimen are very different. For example, when we compare the slip fields on the front surface with those at 0.3mm we find that 4& 5 vanish and the major area around the notch is covered by 9& 11 at 0.3mm (compare figure 5-27a and 5-27b). On the contrary, the slip fields at 0.3mm and 0.6mm are not ve ry different from each other and also the number of dominant slip systems at 0.3mm and 0.6mm are equal (see figure 5-27b and 527c). When the notched samples are loaded, yi elding begins first on those locations where the RSS is a maximum. For this orientation, the maximum RSS values are obtained at the positions on the notch surface. In order to evaluate which slip system is activated first, the values of RSS were ca lculated on the notch surface at locations shown in Figure 5-28 (since the results are symmetri c with respect to not ch growth axis, the maximum RSS results are only calculated on the notch surface above the notch growth axis). After finding the maximum RSS at each thickness it was discovered that the value of the maximum RSS changes, as one varies the thickness. Figure 5-28. Figure shows the nomenclature of the finite element nodal points along the specimen notch.

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91 Figures 5-29 and 5-30 show the variation of RSS with respect to the location on the surface of the notch, for the specimens fr ont surface and at mid-plane respectively. It is seen from the plot that the maximum R SS lies on point 10 at the surface and at the midplane. Similar calculations were conducte d through the thickness of the sample and it was found that for all seven planes, maximu m RSS was achieved at position 10. Table 55 presents the maximum RSS values, their location on the notch surface on a specific plane, slip systems on which maximum RSS occurs, and the stress normal to the maximum shear stress plane. It is seen fr om the table 5-5 that the highest of the maximum RSS occurs simultaneously at 0.3mm and 1.5mm thickness (on the left and right notch) of the specimen with 0 orientation which indicates th at the slip will initiate first on this thicknesses for this orientation; however for other notch orientations this may not be valid (the row of the highest of the ma ximum RSS is in bold font in the table 5-5). A simple calculation indicates that the lo ad at which the slip will initiate at 0.3mm/1.5mm thickness and at the midplane of the specimen is 612N and 630N, respectively. The important point to notice here is that there is not much difference in the loads that initiate the slip at 0.3mm/1.5mm thickness and at the midplane of the specimen. On the other hand, slip at the midpl ane takes place in pairs and the stress state at this location is more complex than at 0.3mm/1.5mm thickness. It has been observed that the normal stress on the crack plane tends to expedite the fatigue crack nucleation process (Suresh, 1998). In the light of this, from table 5-5 one obs erves that the stress normal to the maximum shear stress plane is also greater on the midplane as compared to that at 0.3mm/1.5mm thickness plane. The co mparisons between the stress state, maximum RSS and normal stress have interesti ng implications on fatigue crack initiation

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92 which imply that for the 0 orientation, the fatigue crack will likely initiate at 0.3/1.5mm thickness or at the midplane of the sp ecimen for released tension loading. Table 5-5. Summary of maximum RSS on the surf ace of the notch at several thicknesses, for 0 orientation. Planes Thickness at which plane is located (in mm). Location on the notch surface Slip System Maximum Values of RSS (MPa) Corresponding Values of normal stress (MPa) 1 0 10 11 2226 3369 2 0.3 10 11 2636 5770 3 0.6 10 11 2591 6324 4 0.9 10 4 11 2559 6528 5 1.2 10 4 2591 6324 6 1.5 10 4 2636 5770 7 1.8 10 4 2226 3369 Figure 5-29. Plot showing the maximum RSS on the surface of the notch, on the specimens front surface, for 4982 N load. CRSS = 324MPa Max RSS=2226

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93 Figure 5-30. Plot showing the maximum RSS on the surface of the notch, on the specimens midplane, for 4982 N load. 5.7.2 Fifteen Degree Secondary Orientation In the previous section (5.3), we discu ssed that the slip fields around the notch change with a change in the orientation of the specimen. Likewise from figure 5-27 and figure 5-32 we find that there is a major change in slip fields with a 15 change in the secondary orientation. In the 15 orientation, there is an abse nce of symmetry in the slip patterns with respect to the midplane (the one which is normal to the plane of observation) and the loading or primary axis of the specimen. In this orientation, there exists symmetry with respect to the notch gr owth axis and diagonal symmetry in the left and right notch. By diagonal symmetry we mean that the slip patterns on the left notch on the front surface is similar to the slip patte rns of the right notch on the back surface and vice-versa (Figure 5-31). Hence th e evolution of the slip fields at the left notch from the front to the back surface is similar to the e volution of the slip fiel ds at the right notch from the back to the front surface (this implies that the slip fields at the left and right notches will be similar on the midplane). Theref ore it is sufficient to examine either one CRSS = 3 24MPa Max RSS=2559

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94 of the two notches at various thicknesses. From figure 5-31, one observes that for this particular orientation since the slip field patterns and stress distributions on a particular thickness are different at both the left and righ t notches at the same load, the deformation behavior of this orientation is more comple x as compared to the one corresponding to the 0 orientation. Left Notch Right Notch Front Surface Figure 5-31. Plots show the front and back surf ace slip fields on the left and right notches of the15 orientation specimen. 4 11 112 9 1 11 3 1 1 4 9 11 12 11 3 4 1 9 1

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95 Left Notch Right Notch Back Surface Figure 5-31. Continued In addition to the dissimilar deformation beha vior of the left and right notches, we also see from figure 5-32a and 5-32d that for the same load, both the notches deform differently on their front and back surfaces. Not only is the pattern of slip fields different, but the number of activated dominant slip sy stems also differs (see figure 5-32a and 532d). Furthermore, at the left notch as one goe s inside the interior of the specimen from the front surface, the complex slip field patter n collapses to a simple slip field pattern. For instance, there are fewer dominant slip systems and simple slip fields at 0.3mm as compared to the front surface (see figure 5-32a and 5-32b). On the contrary, as one goes inside the interior of the specimen from the back surface at the left notch, there is only a modest change in the complex ity of the slip fields and the number of activated dominant slip systems (see figure 5-32a and 5-32b). Fr om figure 5-32g, it is evident that in this orientation, slip systems are not occurring in pairs a bout the midplane (which is related to the asymmetry of the specimen orientation). The maximum RSS in this specimen is 1 114 12 911 1 11 43 9 1 4 11 112 9 1 11 3

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96 occurring on the surface of the notch at the location 11 for 0mm, 0.3mm, 0.6mm, 0.9mm, 1.2mm and 1.5mm thicknesses and at the lo cation 9 for 1.8mm thickness (Table 5-6). Unlike the 0 orientation, the highest of th e maximum RSS in this orientation occurrs only at one thickness plane (1.5 mm ) at the left notch and hence the slip will initiate at this thickness at the left notch. In table 56, note that the highest maximum normal stress also occurs at the same location where the highest maximum RSS is occurring. This implies for this orientation, fatigue crack initiation will most likely occur at 1.5 mm thickness at the left notch for released te nsion loading. Subseque ntly because of the diagonal symmetry of this specimen, the st ress values in the specimen also have a diagonal symmetry. Thus we infer that at th e same load, slip and fatigue crack will initiate at the 1.5mm thickness plane at the left notch, and at the 0.3mm thickness at the right notch for released tension loading. Figure 5-32. Polar plots showing the evolution of slip fields around the left notch of the 15 orientation specimen at various thicknesses at load=4982N. 4 11 112 9 1 11 3 1 114 12 9 11 111 4 39 1 A) Front Surface D) 1.8mm (Back Surface)

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97 Figure 5-32. Continued 11 1 1 11 B) 0.3mm F) 1.2mm C) 0.6mm E) 1.5mm 11 1 12 1 11 3 1 4 12 11 3 9 1 11 1 4 12 1111 3 9 1 1 11

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98 G) 0.9mm (Middle Surface) Figure 5-32. Continued Comparison of the highest maximum RSS of the 0 and 15 orient ations shows that the 15 orientation experiences a greater hi ghest maximum RSS than the 0 orientation. Moreover the 15 orientation ha s asymmetric slip fields at the left and right notches, whereas the 0 orientation has symmetric ones. These asymmetric slip fields in the15 orientation will create a more complex stre ss distribution around the notch as compared to those in the 0 orientation. From the a bove comparisons we conclude that the 15 112 11 11 3 1

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99 orientation is more vulnerable to fatigue cr ack initiation than the 0 orientation when subjected to released tension loading. Table 5-6. Summary of maximum RSS on the su rface of the notch, at several thicknesses for the 15 orientation. Planes Thickness at which plane is located (in mm). Location on the notch surface Slip System Maximum Values of RSS (MPa) Correspondi ng Values of normal stress (Mpa) 1 0 11 11 2140 2254 2 0.3 11 12 2650 4390 3 0.6 11 12 2730 5060 4 0.9 11 12 2770 5394 5 1.2 11 12 2810 5622 6 1.5 11 12 2820 5771 7 1.8 9 4 2370 3999 5.7.3 Thirty Degree Secondary Orientation Similar to the15 orientation, the 30 orientation also lack s symmetry with respect to both, the midplane and the primary axis and is symmetric about the notch growth axis and has diagonal symmetry in the left and the right notch. Like the case of the 15 orientation, we do not see slip systems occurri ng in pairs at the midplane of the specimen, which is due to the asymmetry of this orie ntation (Figure 5-33g) The maximum RSS in this orientation occurs at position 11 on the su rface of the notch for all the seven planes (Table 5-7). It is seen fr om table 5-7 that the highes t of the maximum RSS and the highest maximum normal stress occurs at th e same location (1.5mm thickness) of the specimen, at the left notch. This implies that fa tigue crack for this orientation will initiate at 1.5mm thickness at the left notch for released tension loading.

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100 A) Front Surface D) 1.8mm (Back Surface) B) 0.3mm F) 1.2mm C) 0.6mm E) 1.5mm 4 11 19 12 2 11 11 1 9 11 13 4 1 10 2 111 11 1 10 111 11 112 111 3 111 124 11 111 39 1 11 1 12 111 3 11 1 12 111 3

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101 Figure 5-33. Polar plots showing the evolution of slip fields around the left notch of the 30 orientation specimen at various thicknesses at 4982N load. Table 5-7: Summary of maximum RSS on the surface of the notch at several thicknesses, for the 30 orientation. Planes Thickness at which plane is located (in mm). Location on the notch surface Slip System Maximum Values of RSS (MPa) Corresponding Values of normal stress (MPa) 1 0 11 11 21011524 2 0.3 11 12 24632930 3 0.6 11 12 26043434 4 0.9 11 12 26973700 5 1.2 11 12 27873893 6 1.5 11 12 28954067 7 1.8 11 12 26973572 G) 0.9mm (Middle Surface) 11 1 11 112 3

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102 5.7.4 Forty Five Degree Secondary Orientation Similar to the 0 orientation, this orient ation also has symmetry about the midplane (the one which is normal to the plane of observation), the loading axis and the notch growth axis of the specimen a nd hence results for only the le ft notch are presented here. As seen in figure 5-34 in this orientation, the slip fields and th e number of activated dominant slip systems change with the th ickness of the specimen. Since this is a symmetric orientation, slip systems occur in pairs on the midplane of the specimen (see figure 5-34g). Also the maximum RSS occurs on location 11 on the surface of the notch, on all seven thickness planes (T able 5-8). The highest of the maximum RSS occur on the front and back surfaces of the speci men. The highest maximum normal stress simultaneously occurs at the 0.3 and the 1.5mm thickness planes. Similar to the 0 orientation, the location of the highest ma ximum RSS and the highest maximum normal stresses occur at a different location in this orientation. Therefore for this orientation also we calculated the loads required to initiat e the slip at the 0/1.8 mm and 0.3/1.5mm thicknesses of the specimen. Furt hermore, the occurance of slip systems in pairs on the midplane makes the midplane critical for fati gue crack initiation. Consequently the load to initiate slip is also calculated for th e miplane. The loads are found to be 611N, 635N and 680N for the 0/1.8mm, 0.3/1.5mm and midplane, resp ectively. The differences between the loads to initiate slip are not significant. On the basis of the above discussions, we conclude th at the fatigue crack can e ither initiate at 0/1.8mm or 0.3/1.5mm or at the midplane of the speci men for released tension loading.

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103 Figure 5-34. Polar plots showing the evolution of slip fields around the left notch of the 45 orientation specimen at various thicknesses at 4982N load. 9 3 2 1 12 6 10 11 4 8 12 A) Front Surface D) 1.8mm (Back Surface) B) 0.3mm F) 1.2mm C) 0.6mm E) 1.5mm 2 1 10 11 11 10 1 2 10 11 1 22 1 10 11 2 110 11 1110 12 2 1 10 11 1110 12 12 1110 8 4 12 6 9 3

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104 Table 5-34. Continued Table 5-8. Summary of maximum RSS on the su rface of the notch at several thicknesses, for 45 orientation. Planes Thickness at which the plane is located (in mm). Location on the notch surface Slip System Maximum Values of RSS (MPa) Corresponding Values of normal stress (MPa) 1 0 11 3 2640 2391 2 0.3 11 3 2540 2575 3 0.6 11 3 2441 2555 4 0.9 11 3,12 2371 2500 5 1.2 11 12 2441 2555 6 1.5 11 12 2540 2575 7 1.8 11 12 2640 2391 2& 11 1& 10 2& 11 1& 10G) 0.9mm (Middle Surface)

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105 A) 0 Orientation B) 15 Orientation C) 30 Orientation D) 45 Orientation On the Front Surface Figure 5-35. Polar plots showi ng the evolution of slip fi elds as a function of the secondary orientation at the front surface of the specimen around the left notch at 4982N load. 114 1 9 5 12 3 7 1 9 4 11 4 11 1 12 9 1 11 3 4 11 19 12 2 11 11 1 9 11 13 4 1 10 9 3 21 12 6 1011 48 12

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106 A) 0 Orientation B) 15 Orientation C) 30 Orientation D) 45 Orientation At 0.3mm Figure 5-36. Polar plots showi ng the evolution of slip fi elds as a function of the secondary orientation at 0.3mm thic kness of the specimen around the left notch at 4982N load. 9 11 1 12 1 411 3 11 1 1 11 2 111 11 1 10 111 2 1 10 11 11 101 2

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107 A) 0 Orientation B) 15 Orientation C) 30 Orientation D) 45 Orientation At 0.6mm Figure 5-37. Polar plots showi ng the evolution of slip fi elds as a function of the secondary orientation at 0.6mm thic kness of the specimen around the left notch at 4982N load. 12 3 9 11 11 411 11 1 12 1 11 3 11 112 111 3 10 11 1 22 1 10 11

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108 A) 0 Orientation B) 15 Orientation C) 30 Orientation D) 45 Orientation At Midplane Figure 5-38. Polar plots showi ng the evolution of slip fi elds as a function of the secondary orientation at the midplane of the specimen around the left notch at 4982N load. 4& 11 1& 94& 11 5& 12 1& 9 3& 7 1 12 11 11 3 1 11 1 111 12 3 2& 111& 10 2& 11 1& 10

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109 A) 0 Orientation B) 15 Orientation C) 30 Orientation D) 45 Orientation At 1.2mm Figure 5-39. Polar plots showi ng the evolution of slip fi elds as a function of the secondary orientation at 1.2mm thic kness of the specimen around the left notch at 4982N load. 5 7 1 4 99 1141 4 12 11 3 9 1 11 11 112 111 3 2 1 10 11 11 10 1 2

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110 A) 0 Orientation B) 15 Orientation C) 30 Orientation D) 45 Orientation At 1.5mm Figure 5-40. Polar plots showi ng the evolution of slip fi elds as a function of the secondary orientation at 1.5mm thic kness of the specimen around the left notch at 4982N load. 5 4 9 1 9 11 7 4 1 4 12 1111 3 9 1 1 11 11 112 111 3 2 110 11 11 10 1 2

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111 A) 0 Orientation B) 15 Orientation C) 30 Orientation D) 45 Orientation At 1.8mm Figure 5-41. Polar plots showi ng the evolution of slip fi elds as a function of the secondary orientation at 1.8mm thic kness of the specimen around the left notch at 4982N load. 4 11 9 1 12 5 7 3 9 1 11 4 1 114 12 9 11 111 4 39 1 11 112 4 11 1 113 9 1 12 1110 84 12 6 9 3

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112 Figures 5-35 through 5-41 show a detailed numerical exploration of the relationship between the secondary orientation and a plas tic deformation (slip) that develops at a notch at various thicknesses. Fi gures indicate that at any gi ven thickness, none of the slip fields and sector boundaries are similar to each other when other factors such as boundary conditions, specimen geometry a nd material properties are ke pt constant. Figures also show that in each orientation, the activated slip systems and sector boundaries drastically change from the surface to th e interior of the specimen. Since experimental techniques generally rely on the surface observations fo r the determination of slip fields, an experimental technique should be developed to measure the deformation fields in the interior of the specimens. 5.7.5 Calculation of Failure Life for Different Orientations Fatigue induced failures in aircraft gas turbine engines are pervasive problems affecting a wide range of com ponents and materials. Single cr ystal turbine blades are the components most likely to fail by fatigue (Cowles, 1996). To address this problem, Arakere et al.(2001) came up with a simple fatigue failure parameter for SCNBS which was based on the uniaxial low cycle fatigue te st data. They conducted a strain controlled LCF tests for PWA 1480 uniaxial smooth specim ens for four different orientations. All the specimens were loaded to different values of stress ratio(R). They computed the shear stresses for each data point, for maximum and minimum test strain values and arrived at a simple fatigue parameter given by the following equation = 397758 x N-0.1598 (5-1) where, = max min and,

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113 max Maximum shear stress for maximum sp ecimen test strain value in psi min Maximum shear stress for minimum specimen test strain value in psi This fatigue parameter was used in this an alysis to calculate the failure life at the critical locations for the four orientations. Fatigue crack nuclei from which the cracks grow, are thought to be formed due to the repeated cyclic straining of the material which leads to slip on different glide planes. Irreversibility of shear displacem ent along the slip bands then results in the roughening of the surface of the material. This roughening is seen as microscopic hills and valleys at sites where slip bands emerge at the free surface (Figure 5-42). Figure 5-42. Extrusion and intr usion on the surface of a coppe r single crystal (Ma et al. 1989) These valleys act as micronotches and th e effect of stress co ncentration at the root of the valleys promotes additional slip and fatigue crack initia tion. In single crystals, a quasi steady state of deformation known as saturation is reached after thousands of cycles. One of the most visibl e features of cyclic saturation is the localization of slip bands. The localized slip bands are knows as persistent slip bands. Figure 5-43 shows a

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114 high cycle fatigue specimen (Ni-base superallo y), tested for thousands of cycles and loaded in the [001] direction. It has been established that due to cyclic loading, slip accumulates and get localized over a period, to form persistent slip bands (PSB) which lead to crack initiation and later to failure (Deluca,). Figure 5-44 also shows the fatigue crack initiation at PSB in a copper single crystal, fatigued for 60,000 cycles at 20C. Figure 5-43. Persistent Slip band observed in a SCNBS HCF specimen (Deluca, 1995) Persistent Slip Band <001>

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115 Figure 5-44. Fatigue crack initia tion at PSB in a copper single crystal at 20C (Ma et al. 1989). Recall that the location where the maximu m RSS occurs in the material is the location where the slip will init iate, during static loading. Now let us discuss a scenario where instead of applying a sta tic load, we apply a cyclic lo ad (which is below the yield stress of the material) of released tension, that is the load is cycled from zero to maximum. One expects that with increasing number of cycles, slip is likely to initiate at a point where maximum RSS occurs in the mate rial. Eventually with increasing number of cycles, PSBs are expected to form followed by material failure. On the basis of this theory the maximum RSS location found by the st atic loading is the expected location of crack initiation in the single crystal material during cyclic loading. However, this may not be true for 0 and 45 orientation as discu ssed previously. Yet for calculating the failure life for these orientations, we shall assume that they fail at the locations of maximum RSS. On the basis of this theory and with the help of fatigue parameter developed by Fatigue crack initiation

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116 Arakere et al. (2001) one can predict the li fe of the specimens of various crystal orientations, at their critical locations. In order to calculate the life of the specimen material, the max for cyclic loading is taken to be max of the static load, and the min for the cyclic load is se t to zero since we are assuming a released tension loading. Thus, for cyclic load is equal to max. Since fatigue analyses are generally performed below the yield limit of the material, the maximum RSS is recalculated for an arbitrarily selected load of 500N (KI = 5 MPam1/2) (which falls below the yield st ress of the material) for all th e four orientations. Table 5-9 summarizes the maximum RSS, and the predicted number of cycles for failure for the 500N cyclic load. Table 5-9. Summary of the maximum RSS, and the predicted number of cycles for failure for 500N cyclic load Secondary Orientation (deg) Thickness at which the plane of max RSS is located (in mm). Location on the notch surface Slip System Maximum Values of RSS for cyclic load, i.e. max (MPa/ksi) Minimum Values of RSS for cyclic load, i.e. min (MPa/ksi) Fatigue parameter (ksi) Predicted cycles to failure (x105) 0 0.3 and 1.5 10 4 and 11 264/38.3 0 38.3 23 15 1.5 11 12 283/41 0 41 15 30 1.5 11 12 291/42.2 0 42.2 12.9 45 0 and 1.8 11 3 and 12 265/38.5 0 38.5 22 Figure 5-45 shows the plot of the highest maximum RSS as a function of the four different secondary orientations. Figure 45 a nd Table 5-9 indicate th at the highest among these highest maximum RSS occurs for the 30 orientation. Table 5-9 also indicates that the 0 secondary orientation is the most adva ntageous and the 30 se condary orientation is the worst crystal orientation for number of cycl es to failure. Table 59 also indicates that

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117 the choice of secondary orientation has the pote ntial to increase the li fe of a single crystal turbine blade by up to 80% without adding additional weight or cost. Max RSS v.Secondary Orientation260 265 270 275 280 285 290 295 051015202530354045Beta(deg)l rssl (Mpa) Figure 5-45. Figure shows the highest maxi mum RSS as a function of secondary orientation for 500N load. 5.8 Effect of Temperature on Slip Systems of Single Crystal Nick el Base Superalloys The previous topic discussed about the e ffects of secondary orientation on the evolution of plasticity in SC NBS specimens at room temper ature. In this section, the influence of temperature on the slip evol ution in SCNBS, particularly in PWA 1480 SCNBS will be studied. PWA 1480 is selected for the study because its nominal composition is very close to the single crys tal used for the experimental study, and also due to the availability of its high temperatur e properties in the literature. Specimens with 0 and 45 secondary orient ations were studied at 38C and 927C. The geometry of the

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118 notched specimen used in this analysis is se lected to be the same as that of Specimen A (Table 4-2). Milligan and Antolovich (1987) reported that below 600C, deformations in SCNBS occur by slip in the primary oc tahedral slip systems (slip numbers 1-12 in Table 2-2). They also state that between 600C and 950C, deformations occur simultaneously by octahedral and cube slips. Table 5-10 a nd Figure 5-46 show six cube slip systems <110>{111} and three cube planes ({100}) in FCC crystals, respectively. Keeping this in mind, the results were calculated for the primary octahedral slip systems ({111} <110>) at the room temperature (38C) and for both primary octahe dral and cube slip systems ({100} <110>) at high temperature (927C) in this analysis. Table 5-10. Cube slip systems in FCC crystals (Stouffer,1996). Slip System Slip Plane <110>{111} Slip Direction 13 (100) [011] 14 (100) [ 011] 15 (010) [101] 16 (010) [ 101] 17 (001) [110] 18 (001) [ 110]

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119 Figure 5-46. Three cube slip planes in FCC crystal (Stouffer, 1996) For analysis at high temperature, one must know the elastic properties E, and G of the single crystal, the CRSS values on both octahedral {111} and cube {100} planes at high temperature, and the transformation matr ix required to resolve the six component stresses ( ij) on the cube planes. The CRSS and the elastic properties of PWA 1480 at 38C and 927C were taken from the re ference Milligan et al; 1987, and the transformation matrix required to resolve the six component stresses is given by equation 2-8. Table 5-11 shows the material properties and the loads used in the FEA model at 38C and 927C. The slip field results reported in th is study are in the form of polar plots

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120 generated in a manner discussed earlier (section 5.2). Results for each orientation, at 38C and 927C will be presented at first and will be compared and discussed later. Table 5-11. Material pr operties used at 38C and 927C in this analysis. Material Properties At 38C At 927C Elastic Modulus (Ex=Ey=Ez) 1.21 x 1011 Pa 8.76 x 1010 Pa Shear Modulus (Gx=Gy=Gz) 1.29 x 1011 Pa 9.70 x 1010 Pa Poissons Ratio (x=y=z) 0.395 0.413 Critical Resolved Shear Stress at {111} 410 x 106 Pa 151 x 106 Pa Critical Resolved Shear Stress at {100} 172 x 106 Pa Load Applied P =1600 N or KI = 15 MPam1/2 P =1600 N or KI = 15 MPam1/2 5.8.1 Zero Degree Secondary Orientation Figure 5-47 shows the surface and midpl ane results (polar plots) at 38C and 927C for the 0 secondary orientation sp ecimen. Due to the symmetry of this orientation, results are presented only for the region above the notch. Figures 547a and 5-47c show that at 1600N, the slip systems on {111} slip planes are dominant only at a small region near the notch both on the surface and the midplane of the specimen, at 38C. Furthermore, only one slip system is found to be dominant on the surface, and two on the midplane. However at the same load, the sl ip field results on {111} slip planes at 927C (both on the surface and midplane) indicate that the dominant slip systems are activated at a larger area near the notch (Figur es 5-47b and 5-47d). Additionally at 927C more number of dominant slip systems are activated than that at 38C.

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121 At 38C At 927C (a) (b) (c) (d) (e) (f) Figure 5-47. Slip field at the surface of the specimen with zero degree secondary orientation on a) {111} slip plane at 38C, b) {111} slip plane at 927C, Slip fields on the midplane of the specimen on c) {111} slip plane at 38C, d) {111} slip plane at 927C, e) Polar plot showing no activation on {100} slip plane on the surface of the specimen at 38C, f) Slip fields on the surface of the specimen on {100} slip plane at 927C, g) Polar plot showing no activation on {100} slip plane on th e midplane of the specimen at 38C, h) Slip field on the midplane of the specimen on {100} slip plane at 927C. 11 11 49 1 5 4 & 11 4 & 11 1 & 9 5 & 12 15 18 NO ACTIVATION

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122 (g) (h) Figure 5-47. Continued Figures 5-48 and 5-49 show x-y plots on {111} slip plane at r = 1on the surface of the specimen at 38C and 927C, respectively. The dark lines on the figures indicate the CRSS at 38C (410MPa) and 927C (151 MPa) of the si ngle crystal material. Comparisons between the graphs indicate no majo r change in the absolute RSS values of slip systems and their relative positions with each other with an increase in temperature. The only change is in the numbe r of activated slip systems and dominant slip systems boundaries, which can be explained by the change in the CRSS (the dark lines on figures) of the material. Table 5-11 shows that CRSS of the single crystal mate rial decreases with an increase in temperature. Furthermore tabl e 5-11 shows that with an increase in the temperature from 38C to 927C, the decrease in the values of elastic modulus (E) and shear modulus (G) are 28% and 25% respectivel y, and the increase in the value of the poison ratio () is 5%, yet there is a very minor change in the absolute RSS values of the corresponding slip systems in both the plot s (compare figure 5-48 and figure 5-49). For instance, the highest value of RSS at 38C is 647 MPa at = 110 (Figure 5-48) and the corresponding value of RSS at the same loca tion at 927 C is 640 MPa (a decrease of 1%). Although the absolute values of R SS are not affected by an increase in the temperature, the load required to init iate slip on {111} slip plane at r = 1 is greatly 15, 16, 17 & 18 NO ACTIVATION

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123 affected by the change in the CRSS. For example the load required to initiate slip at this radius(r = 1) on the surface of the specimen at 38C is calculated to be 1014N and the load required to initiate slip at 927C is calculated to be 378N. Figure 5-48. Plot shows the maximum RSS at r = 1 on {111} slip plane, on the surface of the specimen at 38C for 1600N load. CRSS = 410 MPa 647 MPa

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124 Figure 5-49. Plot shows the maximum RSS at r = 1 on {111} slip plane, on the surface of the specimen at 927C for 1600N load. Figures 5-47c, 5-47d and 547h show that at both 38C and 927C the slip systems occur in pairs at the midplane of th e specimen, on both {111} and {100} slip planes. However figure 5-47-h shows one uncommon ch aracteristic: on {100} slip plane, four dominant slip systems are simultaneously ac ting at the same location on the midplane of the specimen. It was seen in the previous section that slip initiates at lo cations where RSS is maximum. In this orientation at both 38C and 927C, the maximum RSS occurs on the surface of the notch. Furthermore, with the increase in temperature from 38C to 927C, there is no major change in the absolute maximum RSS values. However the load required for initiating slip at both the temper atures is different due to the difference between the CRSS at these temperatures Table 5-12 summarizes the maximum RSS values on the surface and midpl ane of the specimen at 38C and 927C. CRSS = 151 MPa 640 MPa

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125 Table 5-12. Summary of the maximum RSS and the corresponding load for slip initiation on the surface and midplane of the specimen. Temperature (C) Plane Location on the notch surface Slip System Maximum Values of RSS (MPa) Load required for slip initiation (N) 38 Surface 10 11 715 917 927 Surface 11 15, 18 853 323 38 Midplane 11 411 821 799 927 Midplane 12 15 16, & 18 1080 255 From the table 5-12 it is seen that at both temperatures, the maximum RSS occurs on the midplane of the specimen for this particul ar orientation. Also the load required to initiate slip at 38C is approximately three times gr eater than that needed at 927C. Moreover, at 927C the maximum value of RSS on one of the {100} slip planes is greater than the corresponding value on one of the {111} slip planes, both at the surface and midplane of the specimen. This i ndicates that the slip will in itiate at one of the {100} slip planes at 927C. This is demonstrated in figures 5-50 and 5-51 which show the x-y plots of RSS values versus the position on the surface of the notch at 927C on {111} and {100} slip planes, respectively. These figures indicate that the maximum RSS value on one of the {111}slip plane is 711 MPa and the load required to initiate slip is 340 N, whereas the maximum RSS value on one of th e {100} slip plane is 853 MPa and the load required to initiate slip is 323 N. Th is analysis indicates that at 38C, slip will initiate at the midplane of the specimen on {111} slip plane; and at 927C, slip will initiate at the midplane of the specimen on {100} slip plane fo r the 0 secondary orientation specimen.

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126 Figure 5-50. Plot shows the maximum RSS on the surface of the notch on {111} slip plane on the specimen surface, at 927C. Figure 5-51. Plot shows the maximum RSS on the surface of the notch on {100} slip plane at the specimen surface, at 927C. CRSS=151MPa CRSS=172MPa 711 MPa 853 MPa

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127 5.8.2 Forty Five Degree Secondary Orientation Figure 5-52 displays the surf ace and midplane results at 38C and 927C, for the 45 secondary orientation specimen. Due to the symmetry of this or ientation, results are presented only for the region above the notch. Si milar to the 0 seconda ry orientation, slip systems on {111} slip planes are dominant onl y at a small region near the notch, both on the surface and midplane of the specimen. Here also at 927C, slip lines are extended to a larger radius in comparison to that at 38C. Also we see more nu mber of dominant slip systems activated at 927C than at 38C, both on the surface and midplane. Figures 5-53 and 5-54 show the excel plots on {111} slip plane at 38C and 927C plotted at r = 1. At 38C At 927C (a) (b) Figure 5-52. Slip field at the surface of th e specimen with forty five degree secondary orientation on a) {111} slip plane at 38C, b) {111} slip plane at 927C; Slip field at the midplane of the spec imen on c) {111} slip plane at 38C, d) {111} slip plane at 927C, e) Polar plot showing no activation on {100} slip plane at surface of the specimen at 38C, f) Slip field at the surface of the specimen on {100} slip plane at 927C, g) Polar plot showing no activation on {100} slip plane at the midplane of the specimen at 38C, h) Slip field at the midplane of the specimen on {100} slip plane at 927C. 2 6 2 1 3

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128 (c) (d) (e) (f) (g) (h) Figure 5-52. Continued Similar to the 0orientation, in this orient ation also the absolute RSS values of the slip systems and their relative positions with each other do not change much with temperature. However in this orientation, the maximum RSS value of the slip system increases with an increase in temperature (F igures 5-53and 5-54), in contrast with the 0 secondary orientation where the maximum RSS value of the slip system decreases with an increase in temperature (Figures 5-50 and 5-51). 2 & 1 2 & 11 1&10 1& 10 18 15 NO ACTIVATION NO ACTIVATION 18 14& 15

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129 Figure 5-53. Plot shows the maximum RSS at r=1 on {111} slip plane, at the surface of the specimen at 38C, for 1600N load. Figure 5-54. Plot shows the maximum RSS at r=1 on {111} slip plane at the surface of the specimen at 927C, for1600N load. In this orientation, the slip systems on {111} slip plane occur in pairs at the midplane of the specimen, at 38C and 927C (Figures 5-52c and 5-52d). However on the 782 MPa 805 MPa CRSS=410MPa CRSS=151MPa

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130 {100} slip plane, two slip systems act as a pa ir and one slip system acts alone, that is 14 and 15 occur in pairs and 18 acts alone (Figure 5-52h). In this orientation also, the maximum RSS are found on the surface of the notch at 38C and 927C. Table 5-13 summarizes the maxi mum RSS values on the surface and midplane of the specimen at 38C and 927C, for the 45 orientation specimen. Table 5-13. Summary of maxi mum RSS and corresponding loadsrequired for slip initiation on the surface and midplane of the specimen. Temperature (C) Plane Location on the notch surface Slip System Maximum Values of RSS (MPa) Load required for slip initiation (N) 38 Surface 11 3 847 775 927 Surface 11 18 1210 227 38 Midplane 11 312 761 862 927 Midplane 12 18 1380 199 From table 5-13 we see that at 38C the maximum RSS occurs at the surface of the specimen, while at 927C the maximum RSS occurs at the midplane of the specimen. Also the load required to initiate slip at 38C is approximately three times greater than that at 927C at the surface of the specimen, and approximately four times greater than that at 927Con the midplane of the specimen. In this orientation also it is seen that at 927C, the maximum value of RSS on one of the{100} slip plane is greater than the corresponding value on one of the {111} slip pl ane, both at the surface and midplane of the specimen, which implies that at 927C slip will always initiate at one of the{100} slip plane. After analyzing the s lip systems on {111} and {100} slip plane for this particular orientation, we conclude that at 38C slip will initiate at the surface of the specimen on {111} slip plane, and at 927C slip initiation will take place at the midplane of the specimen on {100} slip planes.

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131 5.8.3 Summary Figures 5-47 and 5-52 show that slip fields on {111} and {100} slip planes are orientation dependent not only at a lower temperature (38C), but also at a high temperature (927C). In addition to this, the number of dominant slip systems and the size of the plastic zones (slip fields) are also f unctions of orientations. For example, figure 547b and 5-52b shows that at equal load (1600N) five dominant slip systems (1, 4, 5, 9, 11) are active on the surface of the specimen for 0 orientation whereas for 45 orientation four different dominant slip systems(1, 2, 3, 6) are activated with different region of activation. It should also be noted that in the 0 orie ntation slip fields related to {111} slip planes are activated to a larger radius as compared to those in the 45 orientation. Furthermore, slip fields related to {100} slip plane are extended to smaller radii in the 0 orientation than those in the 45 or ientation (see figures 5-47 and 5-52). Also the load required to initiate slip at the critical location is different in the 0 orientation than that in the 45 orientation, at both temperatures. For example, the load required to initiate slip at 38 C and 927C in the 0orientatio n is calculated to be 799 N and 255 N, respectively. At both the temperatur es, the slip is found to initiate at the midplane of the specimen for the 0orientati on. For the 45 orienta tion, the load required to initiate slip at 38C and 927C is found to be 775 N and 199 N respectively. At 38C, slip initiation starts at the surface of the sp ecimen and at 927C, the slip is found to start at the midplane of the specimen for the 45orie ntation. Also the locati on of slip initiation on the surface of the notch for both the orient ations, at both temperatures is found to be more or less at the same location, which indicat es that the location of slip initiation on the notch surface depends very sli ghtly on the notch orientation.

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132 5.9 Comparison of Experimental and Numeri cal Results in a Copper Single Crystal subjected to Four Point Bending In the previous topic, the effect of temperature on the slip fields of SCNBS was discussed. Here the dominant slip systems theory will be applied to predict slip activation near notches in copper single cr ystal specimen subjected to a four point bending test. Two symmetric or ientations are treated in this analysis. Finally the experimental results of the copper single cr ystal will be compared with the numerical results. 5.9.1 Experimental Procedure Crone et al. (2003) experimentally inves tigated plastic deformation around a notch tip within copper single crystals. Two symmetric crystallographic orientations tested by them were considered for this study. The fi rst orientation defined the notch plane as ( 101), the notch growth direction as [010], and the notch tip direction (normal to the plane of observation) as [ 101]. The second orientation define d the notch plane as (010), the notch growth direction as [101], and the notch tip direction as [ 101]. Orientations I and II are related by a 90 rotation about the [ 101] axis (Figure 5-55). Both orientations were also analytically invest igated by Rice (1987). (A) Orientation I (B) Orientation II Figure 5-55. Figure shows orientations I and II.

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133 Crone and his co-workers used electrical discharge machining (EDM) for all their cutting operations, such as for specimen and notch cutting. A four point bending test was conducted on the copper single crystal by usi ng Instron 4502 universal testing machine. A four point bending jig was used to apply load to th e specimen through 6.35 mm hardened steel dowels. The distance betw een the two sets of dowels was 25.4 and 38.1mm. The four point bend specimens used in the te sting have 6.35 mm square crosssection and a total length of 5.1cm with a central single crys tal section of approximately 2cm and and the rest made up of polycrystal line copper. Extensions of polycrystalline copper were attached with epoxy. The notch was cut to a depth of 2.465 mm. Both the specimens have the same notch and body geom etry. Figure 5-56 shows the geometry and dimensions of the specimen. Figure 5-56. Figure showing the dimensions of the specimen. When the load is applied on single cr ystal copper, specimens, they deform plastically and develop slip lines on the surf ace. These slip lines are observed optically and provide a significant amount of information about the active slip systems within a sector and sector boundary angles. Figures 5-57 through 5-60 show the optical micrograph of the slip line fields around the notch tip and the slip line trace for

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134 orientations I and II. The inne r circle on the micrographs has a radius of 0.35 mm and the outer circle has a radius of 0.7 mm. We compare our numerical results (secto r boundary angles) at these radii. From figures 5-57 and 5-59, note that though both orientations (I and II) are symmetric, their slip line patterns show slight asymmetry. It has been seen that orientation II show more asymmetry than orientation I. Also the number of activated slip sectors and slip fields in both the orientations are not si milar. In Orientation I six s lip sectors are found, whereas in orientation II there are five slip sectors. Figure 5-57. Optical micrograph of the slip lin e field around the notch tips in orientation I.

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135 Figure 5-58. Slip trace around a notch tip in orientation I. Figure 5-59. Optical micrograph of the slip lin e field around notch tips in orientation II.

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136 Figure 5-60. Slip traces around a notch tip in orientation II. 5.9.2 Numerical Analysis The finite element model created for the numerical analysis of the specimen is a linear-elastic, orthotropic model. The mode l was created in a manner analogous to the numerical model for the single crystal supe ralloy specimen. The material properties of copper single crystals used in this anal ysis are given in Ta ble 5-14. Since we are interested in comparing the experimental result s with our dominant slip field results, an arbitrary load of 445 N has been selected, in this analysis. Table 5-14. Material properti es used in the analysis of copper single crystal. Elastic Modulus (Ex=Ey=Ez) 6.7 x 1010 Pa Shear Modulus (Gx=Gy=Gz) 7.5 x 1010 Pa Poissons Ratio (x=y=z) 0.4161 Load Applied P = 445 N The model was first created around a global specimen coordinate system and then the direction cosines were introduced to de fine the material coordinate system. To calculate the stresses in the vicinity of the notch, four concentric ar cs were created around the notch with radii 0.2mm, 0.35mm, 0.7mm and 1.4mm. Six component stresses were

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137 then calculated in the material coordinate system on each arc at 5 intervals. By using equation 2-7, six component stresses (ij) were converted into twelve RSS. These twelve RSS were finally plotte d at 0.35 mm and 0.7 mm from = 0 170 (that is to the top of the notch), see figure 5-61. In this analysis, we compare numerical results with experimental results at 0.35mm and 0.7mm radius. Similar to our previous tensilenotched model, this numerical model also uses PLANE2 and SOLID95 elements. Figure 5-62 shows the FEA model of a four point bending specimen. The only simplification made in this numerical model is that the en tire specimen is treated as a single crystal. Since we are only interested in the vicinity of the notch, this simplification does not affect the numerical results. Figure 5-61. Radial and angul ar coordinates used for pr oducing slip sector plots.

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138 Figure 5-62. Figure shows the 3-D FEA model of the four point bending specimen with boundary conditions. Figures 5-63 and 5-64 show the plots of the RSS values as functions of for the twelve slip systems at r = 0.35 mm and r = 0.7 mm, respectively, for orientation I. Earlier we showed that the slip systems which we re represented by the highest RSS(dominant slip systems), are the slip systems that will be observed as slip bands on the surface of the specimen. For example, at r = 0.35mm, the active dominant slip systems, according to figure 5-63, are 10 from = 026, 5 from = 2655, 3 from = 55100, 9 from = 100120, 8 from = 120165 and 7 from = 165170. At r = 0.7 mm the active dominant slip systems according to figure 5-64, are 10 from = 032, 5 from = 3263, 3 from = 63110, 9 from = 110132 and 8 from = 132170.

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139 Figure 5-63. Plot shows the st ress distribution for orientat ion I at r = 0.35 mm distance from the notch tip. Figure 5-64. Plot shows the stress distri bution for orientation I at r = 0.7mm. 3 10 5 9 8 3 10 5 9 8

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140 Tables 5-15 and 5-16 summarize the comp arisons between experimental and numerical results at r = 0.35 mm and r = 0.7 mm for orienta tion I. From the tables, we find that for the first three sectors the experimental results match very well with the numerical predictions, at r = 0.35 mm and r = 0.7 mm. On the surface of the experimental specimen, there are two elastic sectors (sectors of no activation) wh ere there is no slip activity. Although we do not see any sectors wit hout slip in our numerical results, we do see that the slip systems 9 8 and 7 are activated from = 100 170, which belong to the group of slip systems activated on the surface of the experime ntal specimen from = 120-150. Furthermore, both the experimental and numerical results indicate that the same slip systems (10, 5, 3, 9 and 8) are activated at both r = 0.35 mm and r = 0.7 mm (Figures 5-57, 5-63 and 5-64, and Tables 5-15 and 5-16). Table 5-15. Comparison between experiment al, numerical and an alytical sector boundaries at r = 0.35mm for orientation I. Sector Experimental Numerical Analytical (Rice Solution) (deg) Slip systems (deg) Slip Systems (deg) 1 0-32 7 or 0-26 54.7 2 32-50 or 26-55 90 3 50-97 or 55-100 125.3 4 97-120 No Activation 100-120 5 120-150 7 or 120-165 6 150-170 No Activation 165-170

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141 Table 5-16. Comparison between experiment al, numerical and an alytical sector boundaries at r = 0.7 mm for orientation I Sector Experimental Numerical Analytical (Rice Solution) (deg) Slip systems (deg) Slip Systems (deg) 1 0-33 7 or 0-32 54.7 2 33-54 or 32-63 90 3 54-102 or 63-110 125.3 4 102-121 No Activation 110-132 5 121-152 7 or 132-170 6 152-170 No Activation Figures 5-65 and 5-66 are the plots of the RSS values as functions of for the 12 slip systems at r = 0.35 mm and r = 0.7mm re spectively, for orientation II. At r = 0.35 mm, the active dominant slip systems are 8 from = 053, 7 from = 5360, 3 from = 6079, 7 from = 7996, 9 from = 96142, 6 from = 142160 and 9 from = 160170. At r = 0.7 mm, the active dominant slip systems are 8 from = 044, 6 from = 4461, 3 from = 61104 and 9 from = 104170. Tables 5-17 and 5-18 show the comparison between numerical, experimental and analytical results for orientation II. Simila r to orientation I, in this or ientation also the experimental results match well with the numerical predicti ons. Furthermore, in th is orientation also the only point of discrepancy between the experi mental and numerical re sults is related to the two elastic sectors (sectors of no activa tion), which were observed on the surface of the experimental specimen, but were not pr edicted by the numerical results. The most noticeable characteristic captured by the numer ical results in this orientation is the prediction of 6 at r = 0.7 mm. For example, from figure 5-59 and table 5-17 one observes

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142 that at r = 0.35 mm, from = 0 64, one of the s lip systems from the group of 7, 8, 9, 10, 11 and 12 is activated, while from = 64 77 one of the slip systems from the group of 1, 2 and 3 is activated on the surface of the experimental specimen. As one goes further from r = 0.35mm to r = 0.7mm, the domain of one of the slip systems from the group 7, 8, 9, 10, 11 and 12 decreases from = 0 64 to = 0 48 and a new slip system becomes activated from = 48 66 from the group of 4, 5 and 6. The same evolution is captured and predicted by the numerical results. Figure 5-65 and table 5-17 show that at r = 0.35mm, 8 is activated from = 053, 7 from = 5360 and 3 from = 60 79 according to nu merical results. Similar to the experimental results, the numerical results at r = 0.7 mm indicate that the domain of 8 decreases from = 053 to = 044 as one moves from r = 0.35mm to r = 0.7mm. In addition to this, numerical results also predict that at r = 0.7 mm, a new slip system 6 becomes activated (dominant) from = 44 61. These are the same charac teristics which were also seen in the experimental results, which further validates the ability of the dominant slip systems theory based 3-D linear elastic anisotropic m odel to accurately predic t the slip evolution around notches of FCC single crystals. Fu rthermore, the good match between the experimental and numerical results also imp lies that the dominant slip system theory works very well not only for single crystal supe ralloys, but is also very helpful for other FCC single crystals.

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143 Figure 5-65. Plot shows the stress distri bution for orientati on II at r = 0.35mm. Figure 5-66. Plot shows the stress distri bution for orientation II at r = 0.7mm. 3 8 7 9 8 7 6 9 8 36 9

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144 Table 5-17. Comparison between experiment al, numerical and an alytical sector boundaries at r = 0.35mm for orientation II. Sector Experimental Numerical Analytical (Rice Solution) (deg) Slip systems (deg) Slip Systems (deg) 0-53 1 0-64 7 or 53-60 54.7 2 64-77 or 60-79 90 3 77-100 7 or 79-96 125.3 4 100-139 No Activation 96-142 142-160 5 139-170 or 160-170 Table 5-18. Comparison between experiment al, numerical and an alytical sector boundaries at r = 0.7mm for orientation II. Sector Experimental Numerical Analytical (Rice Solution) (deg) Slip systems (deg) Slip Systems (deg) 1 0-48 7 or 0-44 54.7 2 48-66 or 44-61 90 3 66-80 or 4 80-100 7 or 61-104 125.3 5 100-170 No Activation 104-170 Besides comparing the numerical and expe rimental results, comparisons of the numerical and experimental resu lts are also made with Rice s analytical results for both

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145 orientations I and II. Tables 5-15 to 5-18 show a poor agreement between the experimental sector boundary angles a nd Rices analytical predictions for both orientations. Rices analytical solution pred icts the sector boundary angles of two different orientations to be the same. Howeve r, experimental and num erical results show that the sector boundary angles of both orientations are di fferent. Moreover unlike Rices analytical predictions, both the numerical and experimental results indicate that sector boundaries have a complex curved shape and hence a single angle cannot be used to define sector boundary angles.

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146 CHAPTER 6 CONCLUSIONS A comprehensive numerical and experiment al investigation of the evolution of plasticity (slip) under a tri-ax ial stress state near notches was conducted in single crystal nickel base superalloy specimens. A 3-D linea r elastic finite elem ent model including the effect of material anisotropy, was developed to predict slip activa tion near notches in three double-notched tensile specimen orient ations with [001] loading direction and [010], [110] and [-310] not ch directions, designated as samples A, B and C, respectively. The validity of the 3D elastical ly anisotropic finite element analysis was confirmed through comparison with experime ntally tested samples. The comparison between isotropic and anisotropic results was made in order to highlight the importance of the inclusion of anisotropy in the numerical and analytic al models. Effects of load, crystallographic orientation, thickness and temp erature on the slip ev olution near notches were also studied. Following conclusions were drawn based on these analyses. The excellent correlation between the 3D anisotropi c FEA results on the surface and experimental slip evolu tion near notches in each of the three specimens for a given load level, confirms the ability of the model to predict slip activation under triaxial stress states. The numerical results suggest that slip pers ists on the slip systems with the highest resolved shear stresses (dominant slip systems) in Ni-based superalloy single crystal tested. Slip on the dominant slip sy stems persists as the load is increased, and new slip systems do not become activated. Based on the dominant slip system theory, the 3-D linear elastic anisotropic FEA model was also able to predict slip activa tion near notches in c opper single crystals subjected to four point bending load. G ood correlation between the numerical and experimental results in c opper single crystals indicat es that the dominant slip

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147 system theory not only works for single cr ystal superalloys, but can also be applied to other FCC single crystals as well. Both numerical and experimental findings i ndicate that the slip sectors boundaries have complex curved shapes rather than st raight lines predicte d by Rice analytical results. Comparison between the anisotropic a nd isotropic results shows that the incorporation of elastic anisotropy has a not iceable effect on the slip evolution at discontinuities. The incorpor ation of anisotropy not only modifies the boundaries of the dominant slip systems, but also change s the number of dominant slip systems. An analysis of the stresses as a function of thickness revealed that the activated slip systems and sector boundaries drastically ch ange from the surface to the interior of the specimens. These numerical results s uggest that experiment al observation of slip lines on the surface is not representa tive of plasticity within the samples Slip sectors predicted near notches are seen to be str ong functions of the secondary orientation (notch direction), not only on the surface of the specimen but also at various thickness planes of the specimen. Slip evolution thro ugh the thickness of the specimen is very different in all th e four secondary orientations. The maximum RSS in all the four secondary orientations occurs at the surfac e of the notches. In the15 and 30 orientations the highest of the maximum RSS and the highest of the normal stress occur at the same location on the surface of the notches, whereas in the 0 and 45 orientations, they occur at different locations on the surface of the notches. Furthermore in the 0 and 45 sec ondary orientations, slip systems occur in pairs on the midplane of the sp ecimen whereas in the 15 and 30 orientations, they do not occur in pairs. The maximum resolved shear stress (RSS) at a given load level is achieved on the surface of the notch for all specimen s investigated. Among the secondary orientations studied, the not ch at 30 from a <100> or ientation shows the highest level of maximum RSS and the minimu m occurs for a <100> orientation. The effect of temperature on elastic anis otropy and hence stress field was evaluated for two secondary orientations of SCNBS. Re sults indicated that th at slip fields are orientation dependent not only at low temperature (38C), but also at high temperature (927C). The load required to in itiate slip is found to be lower in the 45 secondary orientation than in the 0 secondary orient ation, at both temperatures. Also at 927C, the maximum RSS on the notch surface is found to be greater on {100} slip planes as compare to {111} slip planes in both secondary orientations.

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148 LIST OF REFERENCES ANSYS Elements Reference, ANSYS Re lease 5.6; ANSYS, Inc. November 1999. Arakere, N. K., Siddiqui, S., Magnan, S., Eb rahimi, F., Forero, L., Investigation of Three Dimensional Stress Fields and Slip Systems for fcc Single-Crystal Superalloy Notched Specimens, ASME J ournal of Gas Turbines and Power, 127 (3), 2005, pp. 629-637. Arakere, N. K., Swanson, G., Effect of Crys tal Orientation on Fati gue Failure of Single Crystal Nickel Base Turbine Blade Supera lloys, ASME Journal of Gas Turbines and Power, 124, 2002, pp. 161-176. Bilby, B. A., Cottrell, A. H., Swinden, K. H., The Spread of Plastic Yield from a Notch, Proc. R. Soc., A 272, 1962, pp. 304-314. Chandler, W. T., Materials for Advanced Ro cket Engine Turbo Pumps Turbine Blades, NASA CP-2372, 1985. Chang, S. J., Ohr, S. M., Dislocation Free Z one Model of Fracture, Journal appl. Phys. 52, 1981, pp. 7174. Cho, J.W., Yu, J., Near Crack Tip Defo rmation in Copper Single Crystals, Philosophical Magazine Letters, 64 (4), pp.175-182, 1991. Cowles, B.A., High Cycle Fatigue Failure in Aircraft Gas Turbines: An Industry Perspective, International J ournal of Fracture, 80, pp.147-163, 1996. Crone, W., Shield, T.W., Experimental Study of the Deformation near a Notch Tip in Copper and Copper-Beryllium Single Crysta ls, Journal of the Mechanics and Physics of Solids, 49, 2001, pp.2819-2838. Crone, W., Shield, T.W., An experimental st udy of the effect of hardening on plastic deformation at notch tips in metallic single crystals, Journal of the Mechanics and Physics of Solids, 51, 2003, pp.1623-1647. Crone, W., Shield, T.W., Creuziger, A., He nneman, B., Orientation Dependence of the Plastic Slip near Notches in Ductile FCC Single Crystals, Journal of the Mechanics and Physics of Solids, 52, 2003, pp.85-112.

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149 Cuitino, A., and Ortiz, M., Three-Dimensional Crack Tip Fields in Four-Point Bending Copper Single-Crystal Specimens, Journa l of the Mechanics and Physics of Solids, 44 (6), 1996, pp. 863-904. Davis, J.R., ed., Heat Resi stant Materials, ASM Special ty Handbook, ASM International, 1997, pp.256-263. Dalal, R. P., Thomas, C. R., Dardi, L. E., The Effect of Crysta llographic Orientaion on the Physical and Mechanical Properties of Investment Cast Single NickelBase Superalloy, Superalloy, 5th International Symposium on Superalloys, ASM, Metals Park, Ohio, 1984, pp. 185. Decker, R.F., Sims, C.T., The Metallurgy of Nickel-BaseAlloys, The Superalloys, C.T. Sims and W.C. Hagel,eds., Wiley, NY, 1972, pg 33. Decker, R.F., Strengthening Mechanisms in Nickel-Base Alloys, International Nickel Co., NY, 1970. Deluca, D., and Annis, C., Fatigue in Si ngle Crystal Nickel Superalloys, Office of Naval Research, Department of the Navy FR23800, August 1995. Dieter, G., Advanced Mechanical Metallurgy, 3rd Ed., McGraw-Hill, 1986, pp.106, 130. Dollar, M., Bernstein, I. M., The Effect of Temperature on the Deformation Structure of Single Crystal Nickel-Base Supera lloy Superalloys, AIME, 1988, pp.275. Dreshfield, R. L., Parr, R. A., Application of Single Crystal Supe ralloys for Earth-toOrbit Propulsion Systems, NASA TM-89877, 1987 Drugan, W. J., Asymptotic solutions for tensil e crack tip fields w ithout kink-type shear bands in elastic-ideally plastic single cr ysatls, Journal of the Mechanics and Physics of Solids, 49, 2001, pp.2155-2176. Ebrahimi, F., Forero, L., Symposium on Fati gue of High Temperature Materials, TMS annual Meeting, Seattle, Feb 17-22, 2002. Ebrahimi, F., Kalwani, K., Fracture Anis otropy in Silicon Single Crystal, Materials Science and Engineering, 268A, 1999, pp. 116-126. Ebrahimi, F., Siddiqui, S., Forero, L. E., Arak ere, N., Effect of No tch Orientation on the Evolution of Plasticity in Superalloy Single Crystal, Materials Science and Engineering A, 426(1-2), June 2006, pp.214-220. Ebrahimi F., Westbrooke E., Forero, L.E., S lip Analysis in Ni-Base Superalloy, Acta Materialia, 53(7), April 2005, 2137-2147. Ezz, S., Pope, D., Paidar, V., The Tens ion/Compression Flow Stress Asymmetry in Ni3 (Al,Nb) Single Crystal, Acta Metallurgica, 30, 1982, pp. 921.

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150 Flouriot, S., Forest, S., Cailletaud, G., Kost er, A., Remy, L., Burgardt, B., Gros, V., Mosset, S., Delautre, J., Strain Localizati on at the Crack Tip in Single Crystal CT Specimens under Monotonous Loading: 3D Finite Element Analyses and Application to Nickel Base Superalloys, International Journal of Fracture, 124, 2003, pp. 43-77. Forest, S., Pascal, B., Sievert, R., Strain loca lization patterns at a cr ack tip in generalized single crystal plastic ity Scripta Materia lia, 44(6), 2001, pp. 953-958. Fritzemeier, L. G., Advanced Single Cr ystal for SSME Turbopumps, NASA CR182244, 1989. Gell, M., Duhl, D. N., Giamei, A. F., The development of Single Crystal Superalloy Turbine Blades, Superalloys, Proceedings of Fourth International Symposium on Superalloys, ASM International, 1980, pp.205-214. Gerberich, W. W., Oriani, R. A., Lii, J. J ., Chen, X., Foecke, T., Necessity of Both Plasticity and Brittleness in the Fractur e Thresholds of Iron, Phil. Mag., 63A, 1991, pp. 363-376. Gumbsch, P., Riedle, J., Hartma ier, A., Fischmeister, H. F., Controlling Factor for the Brittle-to-Ductil Transition in Sing le Crystals, Science 282, 1998, pp. 1293-1295. Hashmin, Z., Shtrikman, S., On some variational principle in anisotropic and non homogenous elasticity, Jour nal of the Mechanics and Physics of Solids, 10, 1962(a), pp. 335-42. Hashmin, Z., Shtrikman, S., A variational appro ach to the theory of elastic behaviour of polycrystals, Journal of the Mechanic s and Physics of Solids, 10, 1962(b), pp. 343-52 Heredia, F., Pope, D., The Tension/ Compression Flow Asymmetry in a High Volume Fraction Nickel Base Alloy, Acta Metall., 34, 1986, pp.279-285. Hill, R., The elastic behavior of crysta lline aggregate Proc. Phys. Soc, A65, 1952, pp. 349-354. Honeycombe, R., The Plastic Deformation of Metals, Edward Arnold, Ltd.; 2nd Ed., 1984, pp.15. Jones, R., Mechanics of Composite Ma terials, McGraw Hill, New York, 1975. Kakehi, K., Influence of primary and s econdary crystallographic orientations on strengths of nickel-based superalloy singl e crystals, Mater. Trans. A, 45(6), 2004, pp. 1824-1828.

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151 Kalluri, S., Abdul-Aziz, A., McGaw, M.A., Elastic Response of [001]-Oriented PWA 1480 Single Crystal-The influence of Secondary Orientation, NASA TM-103782, 1991. Kear, B., Piearcey, B., Tensile and Creep Pr operties of Single Crystal of the NickelBase Superalloy Mar-M200, Trans,AIME, 239, 1967, pp.1209. Kelly, A., Tyson, W. R., Contrell, A. H., D uctile and Brittle Crystals, Phil. Mag. 15, 1967, pp. 567. Kim, Y., Chao, Y.J., Zhu, X. K., Effect of Specimen Size and Crack Depth on 3D Crack-Front Constraint for SENB Specimens , International Journal of Solid and Structures, 40, 2003, pp. 6267-6284. Kobayashi, S., Ohr, S. M., In-situ Fract ure Experiments in BCC Metals, Phil. Mag., 42A,1980, pp. 763-772. Kysar J.W., Briant, C.L., Crack Tip Deform ation Fields in Ductile Single Crystals, Acta Materialia, 50, 2001, pp.2367-2380. Lall, C., Chin, S., Pope, D., The Orientat ion and Temperaturee Dependence of the Yield Stres of Ni3 (Al, Nb) Single Crystals, Metall. Trans, 10A, 1979, pp.1323. Lekhnitskii, S.G., Theory of Elasticity of an Anisotropic Elastic Body, Holden-Day, Inc., 1963, pp. 1-40. Li, X. M., Chiang, F.P., Experimental Measur ement of the Crack Tip Strain Field in a Single Crystal Engineering Fr acture Mechanics, 43, 1991, pp. 171-184. Lieberman, D. S., Zirinsky, S., A Simplified Calculation for the Elastic Constants of Arbitrarily Oriented Single Cr ystals Acta Cryst, 9, 1956, pp. 431-436. Majumdar, B. S., Burns, S. J., Crack Tip Shielding Anelastic Theory of Dislocation and Dislocation Arrays Near a Shar p Crack, Acta Metall., 29, 1981, pp. 579-588. Meetham, G.W. and Voorde, M.H., Mater ials for High Temperature Engineering Applications. Springer-Verlag, 2000, pp.78-80. Miner, R., Voigt, R., Gayda, J., Gabb, T., Orientation and Temperature Dependance of Some Mechanical Properties of the Single Crystal Nickel base superalloy Rene N4: Part I Tensile Behavior, Metall. Trans, 17A, 986, pp.491. Miner, R., Voigt, R., Gayda, J., Gabb, T., Orientation and Temperature Dependance of Some Mechanical Properties of the Single Crystal Nickel base superalloy Rene N4: Part III Tensile/Compression Ansi otropy, Metall. Trans, 17A,1986, pp.507 Milligan, W. W., Antolovich, S. D., Yieldi ng and Deformation Behaviour of the Single Crystal Superalloy PWA 1480, Metall. Trans., 18A, 1987, pp.85.

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152 Mohan, R.; Ortiz, M., and Shih, C., An Analysis of Cracks in Ductile Single Crystals II. Mode I loading, Journa l of the Mechanics and Phys ics of Solids, 40 (2), 1992, pp.315-337. Mollenhauer, D.; Ifju, P., and Han, B., A Compact, Robust and Versatile Moir Interferometer, Optics and La sers in Engineering, 23, 1995, pp. 29-40. Moroso, J., Control of Secondary Crystallogr aphic Orientation in Single Crystal Nickel Superalloys for Increased Resistance to Fa tigue Crack Growth, Master of Science Thesis, Mechanical Engineering Depart ment, University of Florida, 1999. Ohr, S. M., Chang, S. J., Dislocation Free Zone Model of Fracture: Comparison with Experiment, Journal of Applied Physics, 53, 1982, pp. 5645. Pope, D., Ezz, S., Mechanical Properties of Ni3Al and Nickel-Base Alloys with High Volume Fraction of , International Meta ls Reviews, 29, 1984, pp. 136. Reuss, A., Berechnung der flie ssgrenze von mischkristallen au f der plastizitatsbedingen fur einkristalle, Z.Angew.Math, 9, 1929, 49-58. Rice, J.R., Tensile Crack Tip Fields in El astic-Ideally Plastic Crystals, Mechanics of Materials, 6, 1987, pp.317-335. Rice, J, R., Thomson, R., Ductile versus Brit tle Behavior of Crystals, Phil. Mag., 29, 1974, pp. 73-97. Rice, J. R., Saeedvafa, M., Crack Tip Singular Fields in Ductile Crystals with Taylor Power-Law Hardening, Journal of the Mechanics and Physics of Solids, 36, 1987, pp. 189-214. Robinson, P. M., Scott, H. G., Plastic Defo rmation of Anthracene Single Crystals, Acta Metallurgica, 15, 1967, pp.1581. Saeedvafa, M., and Rice, J.R., Crack Tip Singul ar Fields in Ductile Crystals with Taylor Power-Law Hardening, II: Plane Strain, J ournal of the Mechan ics and Physics of Solids, 37 (6), 1988, pp.673-691. Schmid, E., Boas, W., Kristallplas tizitat, Springer Verlag, Berlin, 1935 Schulson, E. M., Xu, Y., Notch Tip de formation of Ni3Al Single Crystals, Mat.Res.Soc.Symp.Proc, 460, 1997, pp. 555-560. Shah, D., Duhl, D., The Effect of Orie ntation, Temperature and Gamma Prime size on the Yield Strength of a Single Crystal Nick el Base Superalloy, Proceedings of the Fifth International Symposium on Supe ralloys, ASM, Metals Park,Ohio, 1984. Shield, T. W., Experimental Study of the Plas tic Strain Fields Near a Notch Tip in a Copper Single Crystal during Loadin g, Acta Materialia, 44, 1996, pp. 1547-1561.

PAGE 173

153 Shield, T.W., and Kim, K., Experimental Meas urement of the Near Tip Strain Field in an Iron-Silicon Single Crystal, Journal of the Mechanics and Physics of Solids, 42 (5), 1993, pp.845-873. Shrivastava, S., Ebrahimi, F., Effect of Crystallographic Orient ation on the Fracture Toughness of NiAl Single Crystals, MR S Proceedings 460, Materials Research Society, Pittsburgh, 1997, pp. 393-398. Siddiqui, S., Arakere, N. K., Ebrahimi, F., Evolution of slip th rough the thickness of a single-crystal nickel-base s uperalloy notched specimen, ASME Turbo Expo Conference, May 2006, Barcelona, Spain, 2005. Stouffer, D., and Dame, L., Inelastic Defo rmation of Metals: Models, Mechanical Properties, and Metallurgy, John Wiley & Sons, Inc., New York, NY 1996, pp.387417. Suresh, S., Fatigue of Materials, Cambridge Publication, second edition, New York, NY, 1998 Umakoshi, Y., Pope, D.,.Vitek, V., The Asymmetry of the Flow Stress in Ni3 (Al,TA) Single Crystals, Acta Metallurgica, 32,1984, pp.449. VerSnyder, F., Piearcey, B., Single Crystal Alloy Extends Turbine Blade Service Life Four Times, SAE Journal, 74, 1966, pp.36. Weertman, J., Lin, I. H., Thomson, R., Doubl e Slip Plane Crack Model, Acta Metall., 31, 1983, pp. 473-482. Westbrooke, E., Effect of Crystallographic Or ientation on Plastic Deformation of Single Crystal Nickel Base Superalloy, Ph.D. Dissertation, University of Florida, 2005. Yang, S., Elastic Constants of a Monocryst alline Nickel Base Superalloy, Metall Trans, 16A, 4, 1984, pp. 661.

PAGE 174

154 BIOGRAPHICAL SKETCH Shadab Siddiqui was born on the 10th of December, 1977, in Nagpur (known as the city of oranges), in India. He lived in Na gpur for nearly 23 years where he completed his early education and earned a B achelor of Engineering (BE) de gree from Y. C. College of Engineering. In the year 2000, he traveled to the United States of America to pursue a masters degree in mechanical engineering at the University of Florida. He successfully completed his master's degree in the Spring of 2003 and started his PhD in the Fall of 2003.


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Title: Numerical and Experimental Investigation of Plasticity (Slip) Evolution in Notched Single Crystal Superalloy Specimens
Physical Description: Mixed Material
Copyright Date: 2008

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NUMERICAL AND EXPERIMENTAL INVESTIGATION OF PLASTICITY (SLIP)
EVOLUTION IN NOTCHED SINGLE CRYSTAL SUPERALLOY SPECIMENS















By

SHADAB SIDDIQUI


A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA


2006

































Copyright 2006

by

Shadab A Siddiqui

































To my parents and to my wife Rukshana















ACKNOWLEDGMENTS

I owe thanks and gratitude to many people in my life, whose help and support have

led to the successful consolidation of this research. I will begin with thanking both my

mentors, Dr. Nagaraj and Dr. Ebrahimi, who have relentlessly provided me with more

than I could ask in terms of their time, advice, encouragement and support to facilitate

this work. I thank Dr. Nagaraj for encouraging me to pursue doctoral research after the

completion my master's degree under his own guidance, and for giving me the support

and freedom to undertake exploratory paths in my research. I thank Dr. Ebrahimi, first of

all for teaching a wonderful course on "Advanced Metallurgy" from which we started a

collaborative effort and journey towards this research. Words are just not enough to

express the many ways in which she has influenced my life; I consider her more than

being a mentor, and think of her as a close friend. I would also like to thank my

supervisory committee members for their contributions to the completion of this project.

Next I would like to thank my friends and colleagues Eric, George, Jeff, Matt,

Srikant, and TJ for not just the valuable discussions of research and course work, but also

for patiently bearing with my grunts and nods during those hard moments in my research

when I kept my attention on my screen monitor. I owe a lot to Srikant who played a very

important role in my success and was always there for me whenever I needed him.

Special thanks also go to Jeff for helping me in my research. I would also like to thank

my experimentalist colleagues Yanli, Luis, Mike, Shankara, Ian, Eboni and Krishna for

their warm welcome and for all their help and suggestions.









Last but not the least, the one person who has been constantly by me throughout all

the ups and downs of not just research but every aspect of my life, is my wife. I thank her

not just for her endless love and patience, but for being my biggest critic and source of

motivation, and also for her editorial help in this thesis.
















TABLE OF CONTENTS



A C K N O W L E D G M E N T S ................................................................................................. iv

LIST O F TA BLE S ............. .... .............................. ................. ... viii

LIST OF FIGURES ............................... ... ...... ... ................. .x

A B ST R A C T ................................................................................................................... xviii

CHAPTER

1 IN TR OD U CTION ............................................... .. ......................... ..

1.1 P project B background ................................................. ........................ ....... 1
1.2 O bjectiv es ................................................................. ............................... . 8

2 LITERATURE REVIEW ........................................................................... 10

2.1 E plastic A nisotropy ...................................................... ... ... .......... .... 10
2.2 Plastic (Slip) D eform ation ............................................................... ............... 13
2.2.1 Slip System s in FCC Single Crystals ............................... ................19
2.3 Evaluation of Plasticity at Crack Tips .............. ............................................. 22

3 MATERIAL AND EXPERIMENTAL PROCEDURE..............................37

3 .1 M a te ria l ........................................................................................................... 3 7
3.2 Experim ental Procedure................................................ ............................ 43

4 THREE-DIMENSIONAL ELASTIC ANISOTROPIC FEA OF A NOTCHED
SIN GLE CRY STAL SPECIM EN ....................................................... ................46

4.1 Development of a Three-Dimensional Linear Elastic FEA Model ....................47
4.2 Geom etries of the Specimen A, B and C ................................... .................49
4.3 N um erical M odel Characteristics .............................................. .............. 51
4.3.1 Elem ents and M eshing ........................................ .......... ............... 51
4.3.2 Solution Location ....................................... .......................52









5 RESULTS AND DISCU SSION .......................................... ........................... 54

5.1 Definition of "Dominant Slip Systems" .................... ..................... 54
5.2 Development of the Polar Plots ................. .....................58
5.3 Comparison of Experimental and Numerical Results .......................................64
5.4 Evolution of Slip Sectors as a Function of Load .............................................77
5.5 Comparison of Anisotropic and Isotropic Results ..........................................79
5.6 Sum m ary ....................... .......... ......... ...... ....... ................................ ....... 84
5.7 Effect of Secondary Orientation on Slip Systems of Single Crystal Nickel
B a se S u p erallo y ................... .................................................................. 8 4
5.7.1 Zero Degree Secondary Orientation....................................................86
5.7.2 Fifteen Degree Secondary Orientation ....................................... .......... 93
5.7.3 Thirty Degree Secondary Orientation ................................ ............... 99
5.7.4 Forty Five Degree Secondary Orientation....................................102
5.7.5 Calculation of Failure Life for Different Orientations .............................112
5.8 Effect of Temperature on Slip Systems of Single Crystal Nickel Base
Superalloys ................................ ......... ........... ........ ............. 117
5.8.1 Zero Degree Secondary Orientation........... .... ............ ............... 120
5.8.2 Forty Five Degree Secondary Orientation....................................127
5.8.3 Sum m ary ...................................................................... 13 1
5.9 Comparison of Experimental and Numerical Results in a Copper Single
Crystal subjected to Four Point Bending ................................................132
5.9.1 Experim ental Procedure ........................................ ........ ............... 132
5.9.2 N um erical A analysis ............................................................................ 136

6 CONCLUSIONS ................................. ............... .. .............146

L IST O F R E FE R E N C E S ......... ..................... .......................................... ......................148

BIOGRAPHICAL SKETCH ......... .................. ......... .... ....................... 154
















LIST OF TABLES


Table page

2-1 D direction cosines ......... .......................... ... ........ .. ... .. .. ........ .... 11

2-2 Slip planes and slip directions in an FCC crystal...............................................20

2-3 Comparison of sector boundary angles ........................................ ............... 28

2-4 Comparisons of experimental sector boundary angle with numerical and
analytical solutions of Orientation 2 ............................................. ............... 29

2-5 Comparison among the plane strain, plane stress and experimental data ..............33

4-1 Material properties used in the analysis of the notched single crystal specimens ...48

4-2 Actual and finite element specimen dimensions of specimens A, B and C
in m m ........................................................................... 5 0

5-1 Comparison of numerical and experimental results on the surface of the
specim en A at r = 5p. ......................................... ................. .. ......68

5-2 Numerical prediction of dominant slip systems on the surface of specimen A, for
varying radii, r, from the notch. ........................................ ......................... 68

5-3 Numerical predictions of dominant slip systems on the surface of specimen B
for varying radii r, from the notch ................................................ ....... ........ 70

5-4 Material properties used for the isotropic model ........................................... 81

5-5 Summary of maximum RSS on the surface of the notch at several thicknesses,
for 0 orientation .......... .. .... ......................... .. ... .......... .. ........ .... 92

5-6 Summary of maximum RSS' on the surface of the notch, at several thicknesses
for the 15 orientation .......... .... ........................ ............................... 99

5-7: Summary of maximum RSS' on the surface of the notch at several thicknesses,
for the 300 orientation. .............................. ................ ................ ............. 101

5-8 Summary of maximum RSS' on the surface of the notch at several thicknesses,
for 450 orientation. ....................... ...................... ..................... .. .... .. 104









5-9 Summary of the maximum RSS, Ac, and the predicted number of cycles for
failure for 500N cyclic load .......................................................................... .. 116

5-10 Cube slip systems in FCC crystals ............................... ..................118

5-11 Material properties used at 380C and 9270C in this analysis.............................120

5-12 Summary of the maximum RSS and the corresponding load for slip initiation on
the surface and midplane of the specimen...................... ................... 125

5-13 Summary of maximum RSS' and corresponding loadsrequired for slip initiation
on the surface and midplane of the specim en....................................................... 130

5-14 Material properties used in the analysis of copper single crystal........................136

5-15 Comparison between experimental, numerical and analytical sector boundaries
at r = 0.35m m for orientation I. ........................................ .......................... 140

5-16 Comparison between experimental, numerical and analytical sector boundaries
at r = 0.7 m m for orientation I.......................................... ........... ............... 141

5-17 Comparison between experimental, numerical and analytical sector boundaries
at r = 0.35mm for orientation II............................. .. ................................ 144

5-18 Comparison between experimental, numerical and analytical sector boundaries
at r = 0.7mm for orientation II............................. .. .................................. 144
















LIST OF FIGURES


Figure page

1-1 Figure showing the microstructure of SCNBS........................................................

1-2 Different crystals used for the manufacturing of turbine blades..............................2

1-3 Blade leading edge crack location and orientation for the SSME AHPFTP 1st
Stage Turbine B lade ........... .. ................................. ........ .. ...... ........ .. ..

1-4 Figure showing the {001} orientation of the turbine blade....................................

1-5 Convention for defining single crystal orientation in turbine blades......................

1-6 Slip systems predicted by Rice. The thick lines indicate the sector boundary
an g le s ........................................................... .............. .... 6

1-7 Sector boundaries at the crack tip. A) Slip sector boundary. B) Kink sector
boundary ................................. .. .................................. ......... 8

1-8 Figure showing dislocations A) shearing the y' precipitate, B) by-passing the y'
precipitate. .............................................................................. 9

2-1 Slip lines observed on the surface of copper crystal ............................................. 15

2-2 Slip elements in a specimen subjected to uniaxial tension............... ...............16

2-3 Yield behavior of anthracene single crystals................................ ..................... 17

2-4 Schematic diagram of the deformation characteristics of a single-crystal in a
tensile machine .................... ................... 18

2-5 Shear stress/strain curves of metal crystals.................................... .......... ........ 19

2-6 Four octahedral slip planes of FCC crystal showing primary slip directions ..........20

2-7 Effect of orientation on the number of active slip systems { 111 }<101> for the
FC C crystal lattice structure ......................................................... ............... 21

2-8 Effect of orientation on the shape of the flow curve for FCC single crystals..........22

2-9 Orientations of specimens used by Rice. ..................................... ............... 23









2-10 Orientation of specimen used by Shield et al. (1993). ...........................................25

2-11 The E22 strain components near a notch in an iron silicon single crystal.............26

2-12 Orientations of specimens used by Crone and Shield for their experiment............29

2-13 E xperim ental slip sectors .............................................................. ............... 31

2-14 Orientations of specimens used by Schulson et al. (1997) .....................................32

2-15 Optical micrographs of slip sectors around a notch tip of Ni3Al ...........................33

2-16 Figure shows the results for the specimen having (011) as the crack plane and
[100] as the notch grow th direction................................. ........................ .. ......... 35

2-17 Figure shows the results for the specimen having its crack plane as (001) and
the notch grow th direction as [100]................................... ..................... .. .......... 36

3-1 SEM pictures showing the tested superalloys' microstructure which consists of
the y' precipitate and y channels. ........................................ ......................... 38

3-2 Figure shows the dog-bone specimen, A) schematic, B) actual and C) including
double notch. ........................................................................ 39

3-3 Figures showing orientations of specimens. ................................. .................39

3-4 Engineering stress-strain curves for specimen A and B ........................................40

3-5 Engineering stress-strain curves for specimen C............................................ 40

3-6 The evolution of slip lines in the tensile specimen of nickel-base superalloy
immediately, a) after yielding and b) fracture ......................................................41

3-7 SEM pictures showing the shearing of y' precipitate in the SCNBS........................42

3-8 A 001 standard stereographic projection of cubic crystals showing different
poles or directions .....................................................................43

3-9 Fixture designed for notch cutting. ........................................ ....... ............... 44

3-10 Figure showing the specimen grips and the dog-bone specimen used in the
experim ents. ....................................................... ................. 44

4-1 Figure showing the material and specimen co-ordinate system ............. ...............47

4-2 Flow chart for the analysis of the slip fields. ....................................................49

4-3 D im pension of the specim ens. .......................... .............................................. 51









4-4 PLANE2 2-D 6 NODE Triangular Structural Solid................................ ......... 52

4-5 SOLID95 3-D 20Node Structural Solid........................ ............................. 52

4-6 Close-up view of the notch........... ............................ ................. .............. 53

4-7 Figure shows the 3-D FEA model. ............... ............................ .. 53

5-1 Figure shows the RSS values at r=2p on the surface of specimen A at 1500 N
load ................................................................................55

5-2 Figure shows the RSS values at r=2p on the surface of specimen A at 2100 N
load ................................................................................ 56

5-3 Figure shows the RSS values at r=2p on the surface of specimen A at 4982 N
load....... .............. ................................ ............ 56

5-4 Figure shows the RSS values at r = Ip on the surface of specimen A at 1600 N
load....... .............. ................................ ............ 59

5-5 Figure shows the RSS values at r = 1.5p on the surface of specimen A at 1600 N
load....... .............. ................................ ............ 59

5-6 Figure shows the RSS values at r = 4p on the surface of specimen A at 1600 N
load....... .............. ................................ ............ 60

5-7 Polar plot shows the dominant slip systems around the notch on the surface of
specimen A loaded to 1600 N. ............. ............ ........... .... 60

5-8 Figure shows the RSS values at r = Ip on the surface of specimen A at 3200 N
load ................................................................................62

5-9 Figure shows the RSS values at r = 1.5p on the surface of specimen A at 3200 N
load ................................................................................62

5-10 Figure shows the RSS values at r = 4p on the surface of specimen A at 3200 N
load....... .............. ................................ ............ 63

5-11 Polar plots shows the dominant slip systems around the notch on the surface of
specim en A loaded to 3200 N ............................................................................ 63

5-12 Comparison between numerical and experimental results from r = 0.5p to 8p, on
surface of specimen A at load=4982 N (KI = 50MPam/2) ....................................67

5-13 Figure shows RSS values at r = 5p on the surface of specimen A at 4982 N
lo ad .................................................................................6 7









5-14 Comparison between numerical and experimental results on the surface of
specimen B loaded to A) 1780 N (KI = 20MPam1/2) and B) 3456 N (KI =
4 0M P am /2) ...................................... .............................. ................ 7 1

5-15 Plot shows the front and back surface slip fields on the left and right notches of
specim en C .......................................... ............................ 73

5-16 Stress distribution on the surface of specimen C, at the left notch at r = 2p............74

5-17 Stress distribution on the surface of specimen C, at the right notch, at r = 2p. .......75

5-18 Comparison between numerical and experimental results on the surface of
specimen C loaded to 3500 N (K = 30MPaml/2)............... ........................76

5-19 Traces of different { 111 } slip planes, on the plane of observation (that is (130)
p la n e ) .................................................................................................................. 7 7

5-20 Polar plots showing the evolution of slip fields around the notch of the
specimen A loaded to A) 1600 N (K = 15MPaml/2), B) 3200 N (KI =
30MPaml/2) and C) 4982 N (KI = 50M Pam /2)........................................... ........... 78

5-21 Comparison between the anisotropic and isotropic slip fields of the
specim en A .................................................................... 81

5-22 Comparison between the anisotropic and isotropic slip fields of specimen B.........82

5-23 Stress distributions on the surface of specimen A (anisotropic case) at r = 5p at
load = 4982 N ..........................................................................83

5-24 Stress distributions on the surface of specimen A isotropicc case) at r = 5p, at
load = 4982 N ..........................................................................83

5-25 Figure showing the variation of secondary orientation 3 from 0 to 450. ................85

5-26 Figure showing the elements through the thickness in a 3D finite element
m odel. ................................................................................87

5-27 Polar plots showing the evolution of slip fields around the left notch of the 00
orientation specimen at various thicknesses at load = 4982N ..................................88

5-28 Figure shows the nomenclature of the finite element nodal points along the
specim en notch. ........................................................................90

5-29 Plot showing the maximum RSS on the surface of the notch, on the specimen's
front surface, for 4982 N load. ..... ...................................................................... 92

5-30 Plot showing the maximum RSS on the surface of the notch, on the specimen's
m idplane, for 4982 N load.............................................. .............................. 93









5-31 Plots show the front and back surface slip fields on the left and right notches of
thel5 orientation specim en. ...... ....................................................................... 94

5-32 Polar plots showing the evolution of slip fields around the left notch of the 15
orientation specimen at various thicknesses at load=4982N............................... 96

5-33 Polar plots showing the evolution of slip fields around the left notch of the 300
orientation specimen at various thicknesses at 4982N load..............................101

5-34 Polar plots showing the evolution of slip fields around the left notch of the 450
orientation specimen at various thicknesses at 4982N load..............................103

5-35 Polar plots showing the evolution of slip fields as a function of the secondary
orientation at the front surface of the specimen around the left notch at 4982N
lo ad ................................................................................ 10 5

5-36 Polar plots showing the evolution of slip fields as a function of the secondary
orientation at 0.3mm thickness of the specimen around the left notch at 4982N
lo ad ................................................................................ 10 6

5-37 Polar plots showing the evolution of slip fields as a function of the secondary
orientation at 0.6mm thickness of the specimen around the left notch at 4982N
lo ad ................................................................................ 10 7

5-38 Polar plots showing the evolution of slip fields as a function of the secondary
orientation at the midplane of the specimen around the left notch at 4982N
lo ad ................................................................................ 10 8

5-39 Polar plots showing the evolution of slip fields as a function of the secondary
orientation at 1.2mm thickness of the specimen around the left notch at 4982N
lo ad ................................................................................ 10 9

5-40. Polar plots showing the evolution of slip fields as a function of the secondary
orientation at 1.5mm thickness of the specimen around the left notch at 4982N
lo ad ............................................................................................. 1 10

5-41. Polar plots showing the evolution of slip fields as a function of the secondary
orientation at 1.8mm thickness of the specimen around the left notch at 4982N
load. .............................................................................................. 111

5-42. Extrusion and intrusion on the surface of a copper single crystal ..........................113

5-43. Persistent Slip band observed in a SCNBS HCF specimen..................................114

5-44. Fatigue crack initiation at PSB in a copper single crystal at 200C .........................115

5-46. Three cube slip planes in FCC crystal ............. ............................ ..... ............. 119









5-47. Slip fields on the surface of the specimen with zero degree secondary
orientation ............................................................................................... ........12 1

5-48. Plot shows the maximum RSS at r= Ip on {111 } slip plane, on the surface of
the specimen at 380C for 1600N load. ....................................... ............... 123

5-49. Plot shows the maximum RSS at r= Ip on {111 } slip plane, on the surface of
the specimen at 9270C for 1600N load. ...................................... ............... 124

5-50. Plot shows the maximum RSS on the surface of the notch on { 111} slip plane
on the specim en surface, at 927 C....................................................................... 126

5-51. Plot shows the maximum RSS on the surface of the notch on {100}
slip plane at the specimen surface, at 927C. ................................. .................126

5-52. Slip field at the surface of the specimen with forty five degree secondary
orientation .......................................................................................................127

5-53. Plot shows the maximum RSS at r=lp on {111} slip plane, at the surface of
the specimen at 380C, for 1600N load. ...................................... ............... 129

5-54. Plot shows the maximum RSS at r=lp on { 111 } slip plane at the surface of
the specimen at 9270C, forl600N load. ...................................... ............... 129

5-55. Figure shows orientations I and II. ........................................................................132

5-56. Figure showing the dimensions of the specimen.................................................133

5-57. Optical micrograph of the slip line field around the notch tips in orientation I. ....134

5-58. Slip trace around a notch tip in orientation I. ................................. ............... 135

5-59. Optical micrograph of the slip line field around notch tips in orientation II..........135

5-60. Slip traces around a notch tip in orientation II. .................. ................... .......... 136

5-61. Radial and angular coordinates used for producing slip sector plots ...................137

5-62. Figure shows the 3-D FEA model of the four point bending specimen with
boundary conditions. .......................... ...................... ... .. ...... .... ........ 138

5-63. Plot shows the stress distribution for orientation I at r = 0.35 mm distance
from the notch tip. ................................................ .................... 139

5-64. Plot shows the stress distribution for orientation I at r = 0.7mm. ........................139









5-65. Plot shows the stress distribution for orientation II at r = 0.35mm ......................143

5-66. Plot shows the stress distribution for orientation II at r = 0.7mm ........................143















KEY TO ABBREVIATIONS


SCNBS Single Crystal Nickel Base Superalloy

HCF High Cycle Fatigue

LCF Low Cycle Fatigue

FCC Face Centered Cubic

BCC Body Centered Cubic

PSB Persistent Slip band

SSME Space Shuttle Main Engine

EDM Electron Discharge Machine

CRSS Critical Resolved Shear Stress

RSS Resolved Shear Stress















Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy

NUMERICAL AND EXPERIMENTAL INVESTIGATION OF PLASTICITY (SLIP)
EVOLUTION IN NOTCHED SINGLE CRYSTAL SUPERALLOY SPECIMENS

By

Shadab Siddiqui

August 2006

Chair: Dr. Nagaraj Arakere
Cochair: Dr. Fereshteh Ebrahimi
Major Department: Mechanical and Aerospace Engineering

Single crystal nickel base superalloys (SCNBS) are being used increasingly for

high temperature turbine blade and vane applications in aircraft and rocket engines. As a

first step toward developing a mechanistically based fatigue life prediction system for

SCNBS components, an understanding of the evolution of plasticity in regions of stress

concentration, under the action of triaxial stresses, is necessary.

A detailed numerical and experimental investigation of the evolution of plasticity

and slip sector boundaries near notches in SCNBS double-notched tensile specimens was

conducted. The evolution of plasticity in the vicinity of notches in three specimens with a

<100> loading orientation, and having their notches parallel to one of the <010>, <110>

and <310> directions (secondary orientation), were studied. A three dimensional (3D)

linear elastic anisotropic finite element model of the specimens was developed using

ANSYS. Ni-base superalloys which deform by the shearing of the y' precipitate, were

selected for the experimental study to insure that slip bands followed the slip planes,


xviii









similar to single-phase materials. The tensile testing of the notched specimens was

carried out using a 1125 Instron system, and optical microscopy was utilized to observe

the slip bands on the surface of the specimens near notches. The experimental tests were

conducted at room temperature to limit the plastic deformation to { 111 } planes, similar to

FCC metals.

In this study, we demonstrate that a 3D linear elastic anisotropic finite element

model is able to predict the activated slip planes and sector boundaries accurately on the

surface of the specimens. The experimental and numerical results suggest that the

dominant slip planes activated at low load levels persist even at high load levels, and the

activation of other slip bands within a domain is initially inhibited. Results reveal that

slip sector boundaries have complex curved shapes, rather than straight sector boundaries

as predicted previously. Moreover, both the experimental and numerical results indicate

that sector boundaries change with increasing load. A comparison between the isotropic

and anisotropic results demonstrates that elastic anisotropy has a noticeable effect on the

slip evolution near notches.

The numerical model was further exploited to systematically evaluate the effects of

crystallographic orientation, thickness and test temperature, on the evolution of plasticity

in SCNBS. An analysis of the stresses as a function of thickness revealed that the

activated slip systems and sector boundaries drastically change from the surface to the

interior of the specimens. These numerical results suggest that experimental observation

of slip lines on the surface is not representative of plasticity within the samples. Slip

plane and sectors predicted near notches are found to be strong functions of the notch

orientation, not only on the surface of the specimen but also at various thickness planes.









Furthermore, results indicate that the slip fields are orientation dependent not only at low

temperature (38C), but also at high temperature (927C).

Based on the dominant slip system concept developed here, good correlation

between numerical and experimental results was also found in copper single crystals

subjected to four point bending load. This finding confirms that the dominant slip system

concept not only works for single crystal superalloys, but is also applicable to other FCC

single crystals, as well as for other loading modes.














CHAPTER 1
INTRODUCTION

1.1 Project Background

Fracture behavior of single crystalline materials is an important area of research. In

addition to the applicability of single crystals at micro scales, the fracture behavior in

strong and relatively brittle polycrystalline alloys is controlled by the response of the

single grain encompassing the crack tip. Therefore from a fundamental point of view,

knowledge of fracture behavior of single crystals is important for understanding and

predicting the toughness of polycrystalline alloys. Furthermore, the deformation

mechanisms and failure modes of face centered cubic (FCC) single crystal components

subjected to triaxial states of static and fatigue stress are very complicated to predict. This

is because plasticity precedes fracture in regions of stress concentration, and the

evolution of plasticity on the surface and through the thickness is influenced by elastic

and plastic anisotropy. Thus in order to design single crystal material against fatigue and

fracture failure, more experimental and numerical testing will be necessary.

This study deals with single crystal nickel base superalloy (SCNBS) in particular.

SCNBS' are precipitation strengthened cast monograin alloys based on the Ni-Cr-Al

system. The microstructure of this alloy consists of approximately 60% by volume

(depends on the superalloy generation) of ordered y'-precipitates (L12 structure)

coherently set in a FCC nickel-base solid solution y matrix (Figure 1-1).






















(A) (B)
Figure 1-1. Figure showing the microstructure of SCNBS. A) Schematic of y' precipitate
in a y matrix (Arakere and Swanson, 2002). B) SEM micrograph showing y'
precipitate in a y matrix (Ebrahimi, et al. 2005).

Due to their creep, thermal fatigue and corrosion resistance properties over

polycrystalline and columnar crystal alloy, these alloys are widely used in the production

of turbine blades and vanes, used in the aircraft and rocket engine (Figure 1-2)


I Columnar-crystal


O Polycrystal


D Single Crystal


10
9
8
7
S6
5
4
3
2
1
0


Creep Thermal
strength fatigue
resistance


Polycrystal Columnar crystal Single crystal
(A) (B)
Figure 1-2. Different crystals used for the manufacturing of turbine blades. A)
Polycrystal, columnar crystal and single crystal turbine blades. B)
Comparative mechanical properties and surface stability of polycrystalline,
columnar crystal and single crystal superalloys (Fermin, 1999).


Corrosion
resistance









The use of single crystal superalloys for the turbine blade materials has led to

complex relationships between load, temperature and deformation. This is exacerbated by

the complex geometry and anisotropy of blade components, and the triaxial stress state

they experience at the blade tip leading edge. Many of these blades have failed during

operation due to the nucleation and propagation of fatigue cracks from an area of high

concentrated stress at the blade tip leading edge (Arakere and Swanson, 2002). Figure 1-3

shows the blade leading edge prone to fatigue crack growth failure. In an attempt to study

the state of stress in a component with complex geometry (e.g. turbine blades), notched

specimens are often used to represent the areas of stress concentration or the theoretical

fracture condition. As a first step towards understanding the effect of triaxial states of

static and fatigue stress in complex structural alloys, a complete numerical and

experimental investigation of nickel-base single crystal notched specimen is essential.




Crack length




Typical Tip Crack

Crack depth


Figure 1-3. Blade leading edge crack location and orientation for the SSME AHPFTP 1st
Stage Turbine Blade (Arakere and Swanson, 2002).

The growth direction of the single crystal turbine blade is controlled in the

preferred low modulus [001] crystallographic direction to enhance the thermal fatigue

resistance and creep strength of the alloy (Dreshfield and Parr; 1987). The [001] axis is









parallel to the span of the blade, which is also in the direction of the centrifugal loading

(Figure 1-4).



[001]












Figure 1-4. Figure showing the {001} orientation of the turbine blade.

There is some variation between the [001] primary crystallographic direction and

the airfoil stacking line of the single crystal (commonly referred as a the primary

orientation angle) from one blade to another due to the manufacturing process (Figure 1-

5). However, current manufacturing capability permits control of a to within 50 of the

stacking line (Arakere and Swanson, 2002). Past studies indicate that the influence of the

primary orientation angle, when constrained between 00 and 100 on the elastic stresses

generated within the nickel-base single crystal superalloy, is substantially lower (Deluca

and Annis, 1995). Hence a variation of the primary orientation angle (if kept within the

limit of 0-10) will have an insignificant impact on the life of the turbine blade. Due to

this, the effect of primary orientation on the resolved shear stresses is not included in this

study.

In most turbine blade castings, the secondary crystallographic direction is neither

specified nor controlled and is randomly oriented with respect to the fixed geometric axes









in the turbine blade. The orientation of the secondary direction may be controlled by

using a seed crystal during its solidification (Kalluri et al., 1991). The control of the

secondary orientation was not considered necessary, until recent reviews of space shuttle

main engine (SSVME) turbine blade lifetime data, which indicate that the secondary

orientation has a significant impact on the high-cycle fatigue resistance (Arakere et al.,

2002). Furthermore, the creep properties of single crystals are greatly influenced by the

secondary crystallographic orientation (Kakehi, 2004). Therefore an optimization of the

secondary orientation angle (commonly referred as 3 which is the angle between

secondary crystallographic direction and airfoil mean chord line) has the potential to

increase the fatigue life of the single crystal material, without additional weight or cost.

Although extensive research has been done on the effect of primary orientation, little is

known about that of the secondary orientation.


Airfoil
Stacking
Line
Airfoil
Primary
Chord Angle a Crystallographic
Line Orientation
Angle p






Secondary
Cryst all graphic
Orientation

Figure 1-5. Convention for defining single crystal orientation in turbine blades (Arakere
and Swanson, 2002).









For single crystal materials, the analysis of the stress and deformation fields near a

crack tip goes back to the work of Rice (1987). Rice proposed the first asymptotic (i.e., as

the radius from the crack tip approaches zero) solution of the crack tip stress field in FCC

and BCC single crystals for two specific crack orientations under plane strain conditions

using a small strain plasticity framework. He predicted plastic deformation in the form of

patchy "fan shaped sectors" around the crack tip of single crystals (Figure 1-6).



90
125.30 5470

Sector 3 Sector 2



Sector 4 Sector I

crack X
Figure 1-6. Slip systems predicted by Rice. The thick lines indicate the sector boundary
angles (Crone et al. 2001).

Later his work was investigated analytically (Drugan, 2001), experimentally (Cho

et al. 1991; Li et al. 1991; Shield et al. 1993; Shield, 1996; Schulson et al. 1997; Crone

and Shield, 2001; Kysar et al. 2001; Crone and Shield, 2003; Crone et al. 2003) and

numerically (Mohan et al. 1992; Cuitino and Ortiz, 1996; Forest et al., 2000; Flouriot et

al. 2003). Some of the models (Rice, 1987; Mohan et al. 1992; Cuitino and Ortiz 1996)

were able to predict certain features seen in the experiments, however there existed

significant areas of discrepancies. For example, Rice's analytical solution does not

distinguish between the two orientation's sector boundaries and also the sector

boundaries between FCC and BCC crystals (Rice, 1987). However, Shield et al., (1993,

1996) later discovered from their experimental analysis that the sector boundaries of FCC









and BCC crystals, and of two different orientations, are dissimilar. The non-hardening

analytical solution provided by Rice (1987) and Drugan (2001) provides the closed form

solution for the stress field around the crack, but are asymptotic in nature with an

unspecified region of dominance. Moreover, Rice's analytical solution predicts either

kink bands (slip direction perpendicular to the band) or slip bands (slip direction along

the band) on the sector boundaries and requires the presence of both the bands in the slip

fields (Figure 1-7), whereas Drugan came up with an analytical solution that only

involves localized slip bands. This indicates that the non-hardening plasticity solution is

not unique. Besides, the majority of published results use plane stress or plane strain

assumptions in their analytical and numerical models (Rice, 1987; Mohan et al., 1992;

Drugan, 2001; Crone et al., 2003). In fact, the main drawback of the 2D models is their

inability to accommodate displacements along the burgers vectors that do not lie in the

plane of analysis. Moreover many experimentalists (Shield et al., 1993, Shield et al.,

1996, Crone et al., 2003) have measured deformation fields using moire' interferometry,

which allows precise determination of in-plane normal and shear strains. However, since

the technique is applied to a free surface, the resulting measurements are under conditions

of neither plane strain nor plane stress, which introduces ambiguity when comparing

experimental measurements with theoretical and numerical predictions under plane strain

conditions. Therefore in order to evaluate the usual plane strain condition at the specimen

center or to determine the stress-strain fields at the free surface, usage of a three-

dimensional numerical model will be required instead of using two dimensional plane

stress or plane strain models.










(a) (b)







Figure 1-7. Sector boundaries at the crack tip. A) Slip sector boundary. B) Kink sector
boundary (Rice, 1987).

1.2 Objectives

An accurate modeling of the anisotropic material behavior and the fatigue damage

process is fundamental to the development of a mechanistically based life-prediction

system for single crystals. An ideal numerical test would incorporate the parameters for

the specimen size, type of test, plasticity, hardening, lattice rotation and so on, and

eventually a fatigue crack rather than a notch. However before incorporating such

complexity, a basic model must be developed. Keeping this in mind, the objectives for

the study is motivated by the need to understand the deformation behavior of single

crystals under a triaxial state of stress. The first step towards realizing this goal is to

conduct a comprehensive experimental and numerical investigation of the evolution of

plasticity (slip) near notches in one of the SCNBS. A double edged notched rectangular

specimen is chosen for the study. There are several goals in this project. The first is to

develop a 3-D linear elastic anisotropic finite element model (using ANSYS) which can

predict slip activation near notches and the second is to validate the same by

experimentally generating the slip bands. The next goal is to understand how

crystallographic orientation, thickness, load level, and test temperature affect the

evolution of plasticity in this single crystal material.








The reason for choosing SCNBS for our analysis is due to their application in

turbine blades of commercial and military aircraft engines and also because of their high

strength, which makes the handling of the specimen little easier. Moreover their

availability and cost also play a major role in their selection.

As is already shown in the previous section, SCNBS is a two phase materials.

Depending on the composition and heat treatment of SCNBS dislocation in some of the

SCNBS shear (cut) through the y' precipitate (Figure 1-8A) or with the help of cross slip

by-pass the y' precipitate (Figure 1-8B) at room temperature. SCNBS in which a shearing

of the y' precipitate takes places during yielding, behaves like a single phase FCC crystal.

Since we model this SCNBS as a single phase crystal in ANSYS, we select the SCNBS

which will deform by shearing of the y' precipitate so that the results between the FEA

and experiments may be directly compared.


Dislocation I










(A) (B)

Figure 1-8. Figure showing dislocations A) shearing the y' precipitate, B) by-passing the
y' precipitate.














CHAPTER 2
LITERATURE REVIEW

This chapter will provide the background for understanding the deformation

mechanism in cubic single crystal materials. A review of recent literature regarding the

slip mechanism and plastic deformation near a notch tip in cubic single crystals will be

provided. Elastic anisotropy found in cubic single crystals will be explained in a greater

detail. In this chapter we will also discuss the different approaches (analytical,

experimental and numerical) taken by researchers in the past to understand the

phenomena within the region close to the tip of a crack in single crystals. Advantages and

limitation of their results and approaches will also be highlighted to justify the need of a

complete model which can explain the slip evolution near stress concentrations in single

crystals.

2.1 Elastic Anisotropy

The generalized Hooke's law relating stress and strain tensors in linear elastic

solids is shown in Equation (2-1). The Sij are compliance coefficients. For the general

case of an elastic solid, there are 21 independent material constants.

{} = [Si] {} (2-1)

Since the elastic properties of FCC crystals exhibit cubic symmetry also described

as cubic syngony, only three independent elastic compliance constants are required to

describe the elastic properties of the crystal. In material coordinate system, Si can be

expressed as shown in Equation (2-2).









S,, S12 S12 0 0 0

S12 Si, S12 0 0 0

S. S12 S12 S,, 0 0 0
0 0 0 S44 0 0 (2-2)
0 0 00 S44 0
0 0 0 0 0 S44

The compliance constants of the single crystal vary with the direction of the

coordinate axis (that is the Si matrix varies with the crystal orientation). The elastic

behavior in the specimen coordinate system can be described by a transformation of the

compliance tensor from the material coordinate system to the specimen coordinate

system according to the law of transformation of fourth rank tensors. Lieberman and

Zirinsky developed a method for transforming the compliance and stiffness constants of a

single crystal between two coordinate systems that are related by a matrix of direction

cosines as shown in Table 2-1(Lieberman and Zirinsky, 1956).

Table 2-1. Direction cosines
x y z
X' (1 a2 a3
y' PI P2 P3
z Y1 Y2 73

Consider a Cartesian coordinate system (x', y', z') that has rotated about the origin

O of(x, y, z). The elastic constant matrix [S'i] in the (x', y', z') coordinate system that

relates {E'} and {'}: [{'} = [' [S'ij] {' }] is given by the following transformation,.


[si]= [][sj][r]T (2-3)

The transformation matrix [ Y ] is a 6x6 matrix that is a function of the direction

cosines between the (x, y, z) and (x', y', z') coordinate axes (Equation 2-4).









2 2 2
a, a2 a3 aa3 aga, a~al
PI P2 P3 PAP3 PAPl P2PI
[Y] Yi Y2 Y3 Y2Y3 Y3Y1 Y2Y1 (2-4)
2p1y, 2p2,y 2P3Y3 (P323 +P32) (P13 + P31Y) (P2Y1 +P1y2)
2ay,1 2a2Y2 2a3Y3 (0a32 + 023) (0a3j1 +a013) (1'y2 + a2Y )
2aP1, 2a2p2 203P3 (a23 +G2) (1P3 +a3P1 ) (a2P1 +(P2)

Knowing the state of stress at a given location in the material coordinate system (x,

y, z), the resolved shear stresses on the 12 primary octahedral slip systems denoted by c1,

2 ,...,12 can be readily obtained using the following method. The stress vector S acting

on some plane with an outward normal vector N is given as:

SI C 312 13
S2 = 21 C 2 G 23 m (2-5)
S3_ 31 n32 ]3 n

where N= li+mj+nk is a unit vector normal to the octahedral plane and cy are the stress

components in the principal coordinate of the material at the point of interest. The

components of the stress r, in some direction p (which will be the slip directions), is then

calculated using,

T=[S 2 S S3][pI 2 p 3] (2-6)

where p is also a unit vector. Equation 2-7 will resolve the component stresses in the

primary directions as listed in Table 2-2 from 1 to 12.









1 0 -1 1 0 -1
T2 0 -1 1 -1 1 0
T3 1 -1 0 0 1 -1
T4 -1 0 1 1 0 -1 C1
5 -1 1 0 0 -1 -1 2
6 1 0 1 -1 -1 -1 0 (2-7)
'T7 -6 1 -1 0 0 -1 -1 GC12
T8 0 1 -1 -1 1 0 C31
T9 1 0 -1 -1 0 -1 (23
T10 0 -1 1 -1 -1 0
T 1 -1 0 1 -1 0 1
T12 -1 1 0 0 1 -1

To resolve the component stresses on cube planes, equation 2-8 will be used.

T13 0 0 0 1 0 1 C
114 0 0 0 1 0 -1 GC2
1 1 0 0 0 1 1 0 (2-8)
'C 1000 1 3 (2-8)
T16 0 0 0 1 1 0 G12
17 0 0 0 0 1 1 C31
18 0 0 0 0 1 -1 C23


2.2 Plastic (Slip) Deformation

Having discussed the elastic properties of cubic single crystals, we now proceed

towards the plastic deformation in single crystals. Slip, twinning and diffusion assisted

plastic deformation are the three main mechanisms responsible for inelastic deformation

in metals. When the temperature is less than approximately 0.5 Tm, where Tmis the

absolute melting temperature, crystalline metals deform primarily by the propagation of

dislocations through the lattice. The plastic deformation occurs in metals due to the glide

of dislocations on the slip plane (the plane of high atomic density) in the close packed

direction (which represents the shortest distance between two atoms equilibrium









positions). The type of slip systems varies with the material's crystal lattice. At higher

temperatures, deformation occurs by diffusion-controlled processes such as dislocation

climb. Twinning, a rotation of atoms in the lattice structure is not as important, as strains

are very small compared to slip and climb, however the process becomes important at

very low temperatures.

The resistance of a crystal to slip, as well as the level of resolved shear stress along

the slip direction in the slip plane controls the activation of slip in single crystals. The

resistance to slip depends on the atomic bonding and crystal structure, while the level of

resolved shear stress (RSS) depends on the level of loads and their direction relative to

the crystal axes. Slip begins when the resolved shear stress on the slip plane in the slip

direction reaches a threshold value called the critical resolved shear stress (CRSS)

(Dieter, 1986). The value of the critical resolved shear stress depends chiefly on the

material composition, strain rate, and temperature.

During plastic deformation, slip bands can be observed on the polished surface of

specimens. Each atom in the slipped part of the crystal moves forward the same integral

number of lattice spacings and causes changes in the surface elevation, which at high

magnification can be seen as slip lines. Figure 2-1 shows slip lines on the polished

surface of copper single crystal (Dieter, 1986).
























Figure 2-1. Slip lines observed on the surface of copper crystal (Dieter, 1986).

As dislocations can only glide under the effect of shear stresses, these shear stresses

have to be determined. Figure 2-2 shows the orientation of the slip plane and the slip

direction in the crystal relative to the loading axis in a simple tensile test. According to

Figure 2-2, we see that the cross sectional area of the slip plane is given by:


Ashp plane = (2-9)
cos)

where A cross-sectional area perpendicular to the loading axis,

4 angle between the rod axis and the normal to the slip plane.

Furthermore, the load on this plane resolved in the slip direction, is given by:

Fresolved = Fcosk (2-10)

where F applied axial force and k angle between load axis and slip direction.

Dividing equation 2-10 by 2-9, we get the resolved shear stress acting on the slip plane.

S_ resolved Fcoscos) (2-11)
RSS s op plane
A slip plane A









Norm


Slip Plane


al n F Applied force F=oA


Cross-sectional area A







I Diction
I .1


Figure 2-2. Slip elements in a specimen subjected to uniaxial tension.

The factor m = cos 4 cosk represents an orientation factor which is known as the

Schmid factor. From equation 2-11, we can see that slip will occur on the slip system

possessing the greatest Schmid factor. Equation 2-11 shows that in some circumstances

TRSS will be zero (viz. if the tension axis is normal to the slip plane ( =90) or if it is

parallel to the slip plane (4 =900)). Thus, slip deformation will not take place for these

two extreme orientations, as there is no shear stress on the slip plane. On the other hand,

TRSS will be maximum for k= =45.

Single crystals are elastically anisotropic, whereby each crystallographic direction

may respond differently to similar loading conditions. Schmid and his co-workers

(Schmid et al. 1935) experimentally confirmed that in hexagonal crystals (such as

cadminum, zinc and magnesium), the tensile yield stress varies greatly with the










crystallographic orientation. However, when the tensile yield stress is converted to

resolved shear stress by using equation 2-11, it is found that the calculated resolved shear

stress is constant for a particular material. This simple yield criterion for crystallographic

slip is called Schmid's Law. For example, it can be seen in Figure 2-3 that the axial stress

necessary for yielding anthracene crystals varies dramatically with crystal orientation,

while the critical resolved shear stress is unchanged (Robinson and Scott 1967). From

Figure 2-3A we can see that the elastic modulus of the crystal also depends on the

crystallographic orientation. It is also worth noting that the yield stress is minimum for

M=0.5. Consequently, it is very important to specify the orientation of the load.


1.4 .21
1,4 I- .21
1.2 -/ .18
= 83; X = 9.5* Slip system .18
1,2 -l (001) 10101 -
1.0 Sip system .15 (001) 110)
Slip system
(001) 1010] 1.0 .15
S.8 -- ..12 I
g .12
.6 .09




.2 .03 2 .3

0 0 0.2 0.4 0.4 0.2 0
0.01 0.02 0.03 0.04 0.05 o cos
Strain <4 5 0s > 45*
(A) (B)
Figure 2-3. Yield behavior of anthracene single crystals. A) Axial stress-strain curves for
crystals possessing different orientations relative to the loading axis. B) Axial
stress for many crystals versus respective Schmid factors (Robinson and Scott,
1967).

It has been seen in the literature that the deformation mechanism of single crystals

are mostly studied by loading them in simple uniaxial tension. During the experiment, the

tensile specimen is fully constrained at both the ends as the specimen is in grips, which

are attached to the crossheads. Therefore when the load is applied, the specimen is not









allowed to deform freely by uniform glide on every slip plane along the gage length of

the specimen (Figure 2-4). Due to this, the slip planes rotate towards the tensile axis of

the specimen. For example, consider the deformation of the single crystal tensile

specimen oriented in a single slip orientation. The grips hold the ends of the specimen in

a fixed alignment. During application of the load, slip will initiate on a number of parallel

slip planes located close to the maximum shear stress, and slip bands will be observed on

the surface of the specimen. As deformation progresses, continued slip will not be

possible without rotation of the slip planes to accommodate the change in length of the

specimen. Bending will occur near the grips to maintain continuity and alignment of the

specimen. During elongation, the slip planes rotate towards the tensile axis and the angle

between the slip plane and tensile axis decreases. As rotation continues a second slip

plane will become oriented in a position to permit slip. When the stress on this plane

reaches the CRSS, slip will occur in a second slip system. Eventually, the slip on the

primary and secondary planes will become equal and duplex slip (identical slip in both

systems) will occur.




N










Figure 2-4. Schematic diagram of the deformation characteristics of a single-crystal in a
tensile machine (Dieter, 1986).









Since slip in single crystals takes place in a specific plane in a specific direction,

the increase in length of the specimen for a given amount of slip will depend on the

orientations of the slip plane and its direction with the specimen loading axis.

In single crystals the fundamental measure of plastic strain is the shear strain or glide

strain which is defined as the relative displacement of two slip planes that are unit

distance apart. Typical resolved shear stress vs. resolved shear strain curve can be seen in

Figure 2-5 for different metal crystals. It can be seen that the curves for FCC and HCP

metals are different although in all the cases the resolved shear stress increases as

deformation proceeds. This phenomenon is known as strain hardening in metals. Also

from the graph we infer that the rate of work hardening in FCC single crystals is much

greater than in HCP crystals, which is due to the difference in the number of slip systems

that get activated during deformation in FCC and HCP crystals.

Cu


Al







9Zn

Glide shear strain %
Figure 2-5. Shear stress/strain curves of metal crystals (Dieter, 1986).

2.2.1 Slip Systems in FCC Single Crystals

For FCC crystals, the high-density planes are the octahedral {111} planes and the

close-packed directions are the primary <110> slip directions. There are eight {111}

planes in the FCC unit cell. Since the planes at opposite faces of the octahedron are









parallel to each other, there are only four sets of octahedral planes. Each { 111} plane

contains three <110> directions. The slip plane with slip direction constitutes a slip

system. Therefore the FCC single crystal consists of 12 slip systems. Table 2-2 and

Figure 2-6 show the 12 slip directions for an FCC single crystal.

Table 2-2. Slip planes and slip directions in an FCC crystal (Stouffer and Dame, 1996).
Slip Slip Plane Slip
System <110>{111} Direction
T (111) [101]
T2 (111) [oil]
3 (111) [110]
T4 (111) [101]
T5 (111) [110]
6 --
T6 (I ) [oll]
T7 (i) [110]
8 (il) [Oil]

10 (il) [011]
T11 (i ) [101]
12 )
T (111 [110]


Figure 2-6. Four octahedral slip planes of FCC crystal showing primary slip directions
(Stouffer and Dame, 1996).










The number of active slip systems in FCC single crystals depends on the

orientation and magnitude of the applied load. If the loading orientation of the FCC

single crystals lies within the stereographic triangle, the deformation will take place on

the slip system with highest Schmid factor of all the twelve {111 }<101> slip systems.

However if the loading axis of the crystal lies on the boundaries of the stereographic

triangle, then the CRSS values will be the same on more than one slip systems and plastic

deformation will occur simultaneously on the slip systems with equivalent Schmid

factors. It has been seen that for { 111 } <101> slip system, loading in the [001] direction

will activate eight slip systems simultaneously if the stress is equivalent to the CRSS.

Loading in the [110] will activate four slip systems simultaneously, whereas only one slip

system will be activated for loading in the [123] direction. Figure 2-7 summarizes the

number of slip systems similar to the {111 }<101> slip system with equal stresses as a

function of orientation.

6 Slip
systems [Till



S2 Slip
Systems 2 Slip.
Systems
1 Slip
System

001 ~[011]
8 Slip 2 Slip 4 Slip
Systems Systems Systems
Figure 2-7. Effect of orientation on the number of active slip systems (111 )<101> for the
FCC crystal lattice structure (Stouffer and Dame, 1996).

The effect of crystallographic orientation on the flow curve of FCC single crystals

can also be explained by the number of activated slips for different loading directions.

Figure 2-8 shows that the increase in shear stress with shear strain is different for









different loading orientations. When the specimen in loaded in the <011> direction, one

slip system is experiencing appreciably more shear stress than any other and the flow

curve shows a less strain hardening. When the specimen is loaded in the <111> or <100>

direction, the stress on several slip systems is not very different and the material

experiences more strain hardening.

8


V / 'z




o 001 Oil




Resolved shear strain
Figure 2-8. Effect of orientation on the shape of the flow curve for FCC single crystals
(Dieter, 1986).

2.3 Evaluation of Plasticity at Crack Tips

Rice in 1987, provided the foundation for much recent and current work in the area

of crack/notch tip stress and strain analysis by examining the mechanics of both FCC and

BCC notched specimens loaded in tension. Although the analysis techniques are

applicable to other orientations, he paid attention to two specific crack orientations in

FCC and BCC crystals. The first orientation defined the notch plane as (101), the notch

growth direction as [010], and the notch tip direction as [101]. The second orientation

defined the notch plane as (010), the notch growth direction as [101], and the notch tip

direction as [101]. Due to symmetry of the sample, only half of the notched sample is

shown in Figure 2-9.










[101] [010]







[010] [101]



[i01] [101]

Orientation I Orientation 2
Figure 2-9. Orientations of specimens used by Rice.

He presented an asymptotic analysis of the plane stress field at a crack tip in an

elastic-ideally plastic crystal, which predicts a stress field with strong strain localization

along certain radial directions around the crack tip. His work predicted that the cartesian

components of stress remain constant within each sector, and change discontinuously

from sector to sector, that is sector boundaries are necessarily stress and displacement

discontinuities. Further he claimed that the orientation of a sector boundary is constrained

to lie either parallel or perpendicular to plastic slip planes that intersect the crack tip.

Accordingly, two different types of displacement discontinuities at the sector boundaries

exist which Rice postulated, are the result of different types of dislocation structures, as

illustrated in Figure 1-7. Those sector boundaries which are parallel to a slip plane are

referred to as slip discontinuities and those sector boundaries perpendicular to a slip plane

are referred to as kink discontinuities. He also predicted the angles at which these sectors

meet, for two crack orientations, for both FCC and BCC single crystals. However, the

solution does not distinguish between the two orientations sector boundaries or between









FCC or BCC crystal structures. Both crystal orientations and structures predict

boundaries at 55, 90, and 125 (Figure 1-6). Rice notes the weakness of this attribute,

based on contradictory experimental studies, related to the rotation of the crystal lattice.

He acknowledges the simplification of the plane strain assumption and encourages

incorporating anisotropy, strain hardening, and 3D effects into future models. In 1988,

Saeedvafa and Rice extended this analysis and considering hardening effects, assumed

that the crystals obeyed Taylor power law hardening, and proposed HRR- type solutions

for crack tip singular fields (Rice, 1987; Saeedvafa and Rice, 1988).

On the footsteps of Rice, Drugan (Drugan, 2001) also derived the asymptotic

solutions for the near-tip stress fields for stationary plane strain tensile cracks in elastic-

ideally plastic single crystals. He also analyzed the same orientations (Figure 2-9), which

were studied by Rice (1987) for both FCC and BCC crystals. His analytical solution was

different than Rice's solution in regard to the type of sector boundaries around a crack tip

of a symmetric orientation. Rice predicted kink and slip type of sector boundary for a

symmetric orientation whereas Drugan predicted only slip type of sector boundary for the

symmetric orientation. Also Drugan's solution had two interesting differences with

Rice's solution: first one is that Rice's solution predicted identical sector boundaries for

cracks having the same orientations in FCC and BCC crystals whereas Drugan's solution

for the FCC and BCC cases differed substantially; second is that Rice's near-tip stress

field solutions were identical for both the orientations whereas Drugan's solutions

differed substantially from one orientation to another. Drugan also compared his

analytical solution with Crones' experimental observations and measurements and

showed that his asymptotic solutions agreed quite well with the experiments.









Following the theoretical work of Rice, Cho et al. in 1991 performed the

experimental analysis of a notched copper single crystal. They did a 3-point bending test

and examined the surfaces near the crack tips by an optical microscope and a stylus

profilometer. They verified the theoretical predictions of Rice's by confirming that the

plastic field around a crack tip (in ductile single crystals) consists of fan-shaped sectors

with each sector characterized by a family of dominant slip lines. From their

experimental findings they also concluded that the crack tip slip field was different for

the two orientation tested, contradicting Rice's theoretical prediction.

Shield and Kim (1993) followed the work of Rice to correlate their experimental

solution with Rice's analytical solution. Results were presented for determining the

plastic deformation fields near a crack tip (200[tm wide notch) in an iron 3% silicon

single crystal (FE-11). The notch in Shield's and Kim's specimen was in a (011) plane

with prospective crack growth in a [100] direction. Since this orientation is symmetric

about the [100] axis, only the upper half-plane was considered. The specimen was loaded

in four-point bending with measurements made at zero load after extensive plastic

deformation had occurred.

[100]









S[011] 011



Figure 2-10. Orientation of specimen used by Shield et al. (1993).










The specimen they considered had dimensions of 7.45 mm x 6.00mm x 26.05 mm.

The bar was extended to a length of 51.95 mm by welding 12.95 mm long polycrystalline

bars of the same cross-section to each end. They introduced a single-edge notch at the

center of the crystal with a depth of 2.05mm and a width of 200tlm. To verify that the

surface strains reflect the behavior of the material in the interior of the specimen, the

specimen was sectioned and etched. They presented strains as a function of angle, since

the strains do not vary much with radial distances from the notch tip. The angle was

measured from the crack propagation direction and was taken as positive in the

counterclockwise direction.

800.0


.0800
.0eso
S.0700

LO .0600
.0800
Z .0400

.0300
S.0200
U .0100

-750,0 E 0.0
-1250. 0.000
micrometers
Figure 2-11. The E22 strain components near a notch in an iron silicon single crystal
(Shield et al. 1993).

Shield et al. (1993) predicted slip sectors similar to Rice based on the plastic strain

field data. They assumed that the total strain was equal to the plastic strain neglecting the

elastic strain. From the experiment, a pattern of four (eight symmetric) sectors was found.

This pattern is shown in figure 2-11, which displays the strain components. Sectors 1 and









2 had constant strains, although sector 2 had some small variations in strain. The third

sector had the largest strain values which varied with radius in an approximately 1/r

manner. The fourth sector had roughly constant strain, though the strain levels were too

low to make an absolute statement. These sectors were separated by transition regions

where the strains are changing very rapidly. A good agreement between the interior

dislocation pattern and surface strains was found. Thus, it was suggested that the surface

measurements accurately reflect the deformations that occurred in the interior of the

specimen and a comparison with the plane strain result of Rice was justified. However on

the contrary with the help of some numerical and experimental analysis (Kysar et al.

2001; Cuitino and Ortiz, 1996; Flouriot et al. 2003; Siddiqui et al. 2005), it was later

discovered that the surface and interior slip fields are not similar.

Following his work on iron-silicon single crystals, Shield extended his work to

copper single crystals (C1-B1). He chose to study the (011) as the crack plane with the

prospective crack growth direction in the [100] direction. This is the same orientation as

the iron-silicon (FE-11) specimen. Because iron-silicon has a BCC structure and copper

has a FCC structure, the slip systems in the two materials are different. However since

the orientations are the same, it was possible to directly compare on the basis of

orientation. Also the effect of the different slip systems on the strain fields was assessed.

Shield compared this work with his previous work and concluded that the discrete

strain fields observed in FE-11 were also present in this specimen. The sector boundary

angles were similar to, but not exactly the same as, those observed in FE-11, which had

the same orientation but different slip systems (Table 2-3).









Table 2-3. Comparison of sector boundary angles (Shield, 1996).
FE-11 (BCC) Cl-B1 (FCC) Rice (1987)
Sector boundary angles (degrees)
1-2 boundary 35 43 55
2-3 boundary 65 62 90
3-4 boundary 110 100 125

The greatest difference in sector boundary angles occurred in the 1-2 sector

boundary. The angle of the maximum strains (in sector 3) was almost identical in both the

specimens, suggesting that this angle may be related more to the notch tip geometry than

the crystal structure. Shield also observed that load level had no effect on the sector

boundary angles. However as the load increased, the amount of plasticity near the notch

tips also increased. It was found that the results obtained for low loads showed

similarities to Rice's model but the experimental results did not correlate to Rice's model

at high plastic strain (Table 2-3). The boundary angles between the copper and iron-

silicon samples were similar but not constant. This disagreement was explained to be due

to the material structure alone or due to flaws that might be present in the material

structure, regardless of a constant specimen orientation and test condition. This

disagreement can also be due to the geometry of the notch, which is very difficult to

duplicate accurately. The contradictory results of Shield's experiment and Rice's results

provoke the need to replace the existing model, which can provide more accurate

solutions.

Crone and Shield continued experimental studies of notch tip deformation in two

different orientations of single crystal copper and copper-beryllium tensile specimens.

Two crystallographic orientations were considered in this research. Orientation 1 was

defined as the orientation containing a crack or notch on the (101) plane and its tip along

the [101] direction. This orientation was investigated experimentally by Shield (1996) and









Shield et al. (1993). Orientation 2 was defined as the orientation containing a crack or

notch on the (010) plane with a tip along the [101] direction. Slip sector boundaries were

determined experimentally, using Moire interferometry. The plane of observation was the

same for these two orientations with the crack or notch being rotated by 900. Both of

these orientations were also analytically investigated by Rice (Rice, 1987).

[010] [101]




rioll] 10J



[101] [1011
Figure 2-12. Orientations of specimens used by Crone and Shield for their experiment
(Crone et al. 2001)

The visible slip patterns determine slip activity, but as the authors note, a lack of

visible slip does not rule out any activity. Slip systems may be activated internally, rather

than at the surface, or may show varying patterns on the surface as deformation

continues. They compared their experimental results with Rice's analytical solution, as

well as numerical FEA solutions by Mohan et al. (1992) and Cuitino and Ortiz (1996).

Table 2-4. Comparisons of experimental sector boundary angle with numerical and
analytical solutions of Orientation 2 (Crone et al. 2001).
Sector boundary Experimental Analytical Numerical
In degrees Crone and Shield Rice Mohan, et al. Cuitino and Ortiz
(2001) (1987) (1992) (1996)
1-2 50-54 55 40 45
2-3 65-68 90 70 60
3-4 83-89 125 112 100
4-5 105-110 130 135
5-6 150









The numerical solutions were based on plane strain assumptions, however Cuitino

and Ortiz later concluded that the problem under consideration is not a plane strain

problem because of the differences between the interior and surface fields. Even with the

plane strain assumption, the results from Cuitino et al's and Mohan's numerical model

and those from Rice's analytical models do not match well with the experimental results

(Table 2-4).

The experimental results are somewhat ambiguous due to the annuluss of validity,"

where Crone and Shield (2001) take their measurements (Figure 2-13). This annulus

corresponds to an area spanning the region between the radial distances of 350 through

750tm from the notch tip. The notch width is between 100-200tm, making the notch

radius to be between 50-100tm. Therefore the annulus, and the region where the sectors

are measured by Crone et al. (2001), was anywhere between 3.5 7.0 and 7.5-15.0 times

the notch radius from the tip. They chose this annulus so that they could avoid the

material close to the notch which is dominated by the notch geometry and the material in

the farfield, which is affected by the surface boundary. They note that the observed slip

activity begins in a single sector and as the deformation proceeds, more slip lines become

visible in the same sector at larger radial distances from the notch.









4 x2-[oll]


Figure 2-13. Experimental slip sectors (Crone et al. 2001).

They also clearly observed horizontal slip traces directly ahead of the notch,

however they discounted their observations and labeled the slip as "elastic" in order to

compare their solution to other perfectly plastic sharp crack solutions (Figure 2-13) (It is

also interesting to note that perfectly plastic models do not give stress distributions, and

yielding is judged based on elastic stresses although the model is plastic in nature).

Contrary to the equivalent sectors predicted by Rice, Crone and Shield's observed sectors

show a marked difference with orientation, varying in both specific boundary angles and

in the number of sectors. The FEA plane strain results from Cuition et al. (1996)

appeared to correspond more closely to the experimental results of Crone et al. (2001).

Schulson and Xu (1997) examined the state of stress at a notch tip for single crystal

Ni3Al, the y'-component of single crystal superalloys using three-point bending


notch


350-750pml
350-750iim









specimens. The orientation they analyzed has the notch plane as (101), the notch growth

direction as [010], and the notch tip direction as [ 01 ].















[io[J
--101






Figure 2-14. Orientations of specimens used by Schulson et al. (1997).

An analytical model based on elastic isotropic assumptions was used to calculate

stress field around the notch, based on the equations for a sharp notch. Two solutions

based on plane stress and plane strain assumptions were evaluated. The solution based on

plane stress predicted four sectors to be activated around the notch, however the plane

strain solution predicted the activation of five sectors around the notch. They also found

that within each sector, one slip system dominates. Experimental results from Schulson et

al (1997) show that the notch tip deformation field of Ni3Al is characterized by fan

shaped sectors (Figure 2-15). Results also indicated that the primary octahedral slip

systems, { 111 }<110>, were activated in these sectors. Experimental results after

significant plastic deformation reveal results that deviate from those predicted by either

plane stress or plane strain assumption, but are closer to the plane stress assumption

(Table 2-5). However they note that since the notch causes a triaxial state of stress, both

plane stress and plane strain assumptions ignoring anisotropy are approximations.






















,.I-[ 1o il


IVli l








Figure 2-15. Optical micrographs of slip sectors around a notch tip ofNi3Al (Schulson et.
al. 1997).

Table 2-5. Comparison among the plane strain, plane stress and experimental data
(Schulson et al. 1997).
Sector Plane Strain Plane Stress Experimental
0 (deg) Slip systems 0 (deg) Slip systems 0 (deg) Slip systems
I 0-23 (111) [110] 0-43 (111) [110] 0-38 (111)

II 23-60 (111)[110] 43-60 (111)[110] 38-58 (111)

III 60-107 (111)[110] 60-103 (l1)[11 ] 58-100 (111)

IV 107-133 (111)[101] 103-180 (111)[011] 100-max (111)

V 133-180 (111)[Oil] --

Kysar (2002) published his experimental results regarding the crack tip

deformation fields in ductile aluminum single crystal. He applied Mode I loading to his

specimen and sectioned it to map in plane rotation field using Electron Backscatter









Diffraction. His observations provided evidence of the main features of the deformation

fields predicted by Rice (1987) using continuum single crystal plasticity, especially the

existence of kink shear sector boundaries which had not been unambiguously identified

in previous studies.

Numerical investigations to study the evolution of slip systems from the surface to

the midplane of notched FCC single crystals subjected to triaxial states of stress have

been limited. Experimental techniques generally depend on surface observations for the

determination of deformation field in the vicinity of notches and cracks. Experimentalists

have often compared the surface slip results with stress states computed on the basis of

plane strain assumptions (Shield et al. 1994; Shield, 1996; Crone et. al. 2001). There is

ample evidence that (Arakere et al. 2005; Siddiqui et al. 2005; Ebrahimi et. al; 2005;

Cuitino et al. 1996; Flouriot et al. 2003; Kim et al. 2003) the state of stress, changes very

rapidly from the surface to interior and neither plane stress nor plane strain isotropic

assumptions adequately describe the surface or midplane stress states respectively.

To address this problem Cuitino and Ortiz (Cuitino et al. 1996) developed a 3-D

finite element analysis (FEA) model of a copper single crystal specimen loaded in four

point bending. Their results suggested that there is a large discrepancy between surface

and interior fields. They also found that the plane strain condition is not fully attained at

the specimen midplane.

Forest et al. (2000) also investigated the effect of a generalized continuum theory

on the localized deformation patterns arising at the crack tip in elastoplastic FCC single

crystals by using a plane strain finite element model. He used the material properties of

single crystal nickel base superalloy SC 16 for his FEA model. He also included lattice









rotation and hardening factor in his model and varied the hardening parameter to see the

effect of lattice rotation on localization. He noticed that when he increased the hardening

parameter in his model, the slip localization bands started disappearing (Figure 2-16). He

justified this behavior due to an increase in the local stresses, which is causing limited

localization. Figure 2-16 shows the results for the specimen having (011) as the crack

plane and [100] as the notch growth direction.


i / / ,\ I. ,

S. .

*"* I "" ..... .....* '"" ........ -"



-Shlp locahzatrid
S. -. hd :-hnd

S ICk r crack
(A) (B)
Figure 2-16. Figure shows the results for the specimen having (011) as the crack plane
and [100] as the notch growth direction. A) Strain localization at the crack tip
of the FEA model with lower hardening parameter. B) Strain localization for
higher hardening parameter.

Flouriot et al. (2003) presented 3-D finite element simulations of mode I crack tip

fields for elastic ideally plastic FCC single crystal tensile specimens. The analysis was

specifically applied to the single crystal nickel-base superalloy AM1 at low temperatures.

They discussed their FEA result on the surface and the midplane of the specimen and

found that the plastic strain fields predicted by the 3-D FEA are highly three-dimensional

and strongly rely on crack orientations. They reported that there is strong disparity

between the plastic strain field obtained at the free surface and the one in the mid-section

of the specimen, which indicated that the slip activity is different in the bulk as compared









to the surface of the specimen (Figure 2-17). In addition to this, they also discovered that

the strain localization found in the plane strain model is different as compared to the

midplane results of the 3-D FEA model, which again shows the necessity of having 3-D

models. Figure 2-17 shows the results for the specimen having its crack plane as (001)

and the notch growth direction as [100].









crack








(A) (B)
Figure 2-17. Figure shows the results for the specimen having its crack plane as (001) and
the notch growth direction as [100]. A) Strain localization at the crack tip of
the plane strain model. B) Strain localization at the free surface and the mid
plane of the 3-D FEA model.

The existing analytical and numerical works provide some insight into the

deformation behavior of ductile single crystals in the presence of a crack or notch, but the

results available do not completely predict the behavior seen in the experiments.

Additionally, experimental and numerical work available in this area is limited and only

begins to elucidate the complex behavior that occurs at a notch in single crystals. A

single-crystal model that incorporates 3-D elastic anisotropy and near-notch plasticity

effects necessary to accurately predict the evolution of slip sectors in 3-D stress fields, is

far from complete.














CHAPTER 3
MATERIAL AND EXPERIMENTAL PROCEDURE

The literature review presented in chapter 2 indicates that a complete understanding

of the mechanics of single crystal materials, their failure mechanisms, and methods of life

prediction have yet to be achieved. Requirement of an accurate modeling of the

deformation of single crystals, occurring under a triaxial state of stress has been

highlighted for optimizing the efficiency in design of components with respect to both,

preventing failure and avoiding over design. Keeping this in mind, this research involves

a comprehensive experimental and numerical investigation of SCNBS material. This

chapter will serve to explain the experimental aspect of this research.

3.1 Material

Pratt and Whitney provided single crystals of two nickel-base superalloys for this

research. Figure 3-1 shows the microstructure of these alloys, which consists of cubical y'

precipitates (L12 structure) coherently set in a FCC nickel-base solid solution y matrix.

The crystallographic orientations of the crystals were known when they were first

received. Specimens with different orientations were cut using an electron discharge

machine (EDM) and were tested experimentally. Two of the specimens (specimen A and

specimen B) were machined from the same alloy and therefore have similar nominal

composition and same CRSS while the third one (specimen C) was machined from the

other alloy and therefore has a different nominal composition and CRSS than specimens

A and B.




















Figure 3-1. SEM pictures showing the tested superalloys' microstructure which consists
of the y' precipitate and y channels.

The specimens used in this study were designed as flat dog-boned samples. Figure

3-2 shows the schematic and actual pictures of the dog-bone samples used in this study

The experimental testing of two (specimens A and B) out of the three single crystal

nickel base notched tensile specimens had been carried out by Luis Forrero (a Ph.D.

student from Dr. Ebrahimi's Lab), in which he obtained the stress strain curves and the

optical pictures of the slip bands of both the specimens, while the third specimen

(specimen C) was analyzed in this thesis. The specimens defined as A/ B/ C have the

notch planes as (001) / (001)/ (001), the notch growth directions as [110] / [010]/ [310]

and the direction along the notch tip (i.e. direction normal to the plane of observation) as

[110] / [100]/ [130] respectively (Figure 3-3A/3-3B/3-3C). Only gauge length and half of

the specimens are shown in the Figure 3-3.





























(A) (B) (C)
Figure 3-2. Figure shows the dog-bone specimen, A) schematic, B) actual and C)
including double notch.



[001] [001] [001]






[ilo10] [010] [31
[ 0 [31(


[110] [100] [13

Specimen A Specimen B Specimen C
(a) (b) (c)
Figure 3-3. Figures showing orientations of specimens. A) Specimen A, B) specimen B
and C) specimen C.

The un-notched dog bone tensile specimens were loaded to evaluate the CRSS of

the material at room temperature. All the tests were carried out at a strain rate of 3.33 x










105s. The observation of the slip lines confirmed that this Ni-base superalloy deformed

on the octahedral slip planes. The CRSS was calculated based on the yield stress and

schmid factor for each orientation. The average yield strength was found to be 794 MPa

and the CRSS on the octahedral planes was calculated to be 324 MPa for specimen A and

specimen B (Figure 3-4) and the average yield strength and CRSS of specimen C was

found to be 868 MPa and 378 MPa respectively (Figure 3-5).

1200

1000

A: 800

600

S400

200

0 I I
0 0.05 0.1 0.15 0.2 0.25

Strain
Figure 3-4. Engineering stress-strain curves for specimen A and B (Ebrahimi et al. 2002).

2000-

1800

1600

1400

-1200

-1000

800

600

400

200

0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
Strain
Figure 3-5. Engineering stress-strain curves for specimen C (Westbrooke, 2005)









From the stress-strain curve of specimens A and B (Figure 3-4) we note that the

strain-hardening rate in this particular SCNBS was relatively small. Figure 3-6 shows the

evolution of slip lines in the tensile specimen of a SCNBS (Ebrahimi et al. 2005) one

immediately after the yield point, and the other after fracture. Initially when the load was

applied, deformation occurred by the formation of Luder band (Figure 3-6a) which

eventually traversed the length of the sample with increasing load. Moreover at higher

strain levels, traces of other slip systems were detected, which were consistent with the

deformation on the { 111 }<110> slip systems (Figure 3-6b). For the <001> loading

orientation, multiple slip activation (8 slip systems) is expected based on the Schmid's

law. The observation of the Lider band which is consistent with the lack of strain

hardening, suggests that localization occurs easily in the superalloy investigated. Detailed

analysis of tensile specimens also revealed that the slip bands propagated by shearing of

the y' precipitates (Figure 3-7).
















Figure 3-6. The evolution of slip lines in the tensile specimen of nickel-base superalloy
immediately, a) after yielding and b) fracture (Ebrahimi et al. 2005).



























Figure 3-7. SEM pictures showing the shearing of y' precipitate in the SCNBS (Ebrahimi
et al. 2002).

SCNBS's when grown in the [001] crystallographic direction have good thermal

fatigue resistance and creep strength; this is the reason why single crystal turbine blades

are directionally solidified in [001] direction. So from application point of view we chose

[001] as our loading or primary orientation for all three specimen. Figure 3-8 shows a 001

standard stereographic projection of a cubic crystal. In this figure, different numbers

represents different crystallographic directions. The secondary directions of the

specimens ([110] / [010]/ [310 ]) are shown in the red circles on the 001 stereographic

projection. These secondary crystallographic directions of the specimens A, B and C

were decided on the basis of their four fold, two fold and mirror symmetries respectively.













5Cy- -7i 2i sao



231. ? 201 .,1 3,

0.
"0 3j 2* 2-" \
120 2 1"o Ar e }3 -* "*31 O
/ i'12* i o 32





3 0 .21
lc"s 2.e *2 3 *2 ,i30
ISO 22* i *22 315o
/JJ -.sl .M3 i/ -* / .* V
1 1* l 123 133 *

^31 02* 2

0 2 1* 3 2'2 0*I 31S 12







poles or directions (Honeycombe, 1984).
S3.2 Experimenl Pe 2 22dur
special fixture (Figure 3-9) was created2 2for notch1 cutting,31 230which helped create straight
mechanically polished. initially the specimen was polished to remove at 20least 5032m
re 313* 0 *1

51 0e *I11
32 s .* SI' *2 13-
2io 210

50 -510

Figure 3-8. A 001 standard stereographic projection of cubic crystals showing different
poles or directions (Honeycombe, 1984).

3.2 Experimental Procedure

After calculating the yield properties of the specimens, two symmetric notches

were created in the specimens to analyze the results near stress concentrations. A slow

speed saw with a thin diamond blade was used to cut the notches in the samples. A

special fixture (Figure 3-9) was created for notch cutting, which helped create straight

and symmetric notches. Following the notch cutting, both the sides of the specimens were

mechanically polished. Initially the specimen was polished to remove at least 50[m

(thickness of the damage layer) from the specimen surface during the EDM cutting of the

specimen.























Figure 3-9. Fixture designed for notch cutting.


Figure 3-10. Figure showing the specimen grips and the dog-bone specimen used in the
experiments.

The tensile testing of the notched specimens was carried out using a 1125 Instron

system. Special specimen grips were created to accommodate the design of the dog-bone

sample during tensile testing (Figure 3-10). The design of the grips allows the head of the

specimen to sit flat inside the grips so that it remains straight during testing. Finally

optical microscopy was utilized to observe the deformation or slip bands on the surface of

the specimen near notches.

In this analysis, the specimens are loaded to different load levels of apparent

isotropic stress intensity factors calculated on the sharp crack assumption. Equations 3-1

and 3-2 are used to calculate the stress intensity factor, Ki.

K, = YCYVa 3-1






45


Y =1.99 + 0.76 -8.48 a +27.36 a 3-2
W W W

Where c is the far field stress, Y is dimensionless quantity (which depends on the

geometry and type of loading) and "a" is the crack length.














CHAPTER 4
THREE-DIMENSIONAL ELASTIC ANISOTROPIC FEA OF A NOTCHED SINGLE
CRYSTAL SPECIMEN

In the field of linear elastic fracture mechanics, various approaches such as

analytical, numerical and experimental approach have been taken in order to study the

elastic response of isotropic-notched specimens under the action of a tensile load. These

methods developed for isotropic specimens pose many difficulties when applied to three

dimensional anisotropic specimen models. For example, in the case of isotropic analytical

models, the current solution relies on many simplifications/approximations that lead to

inaccurate results when compared with the experimental results. However, these

limitations are overcome by the use of a three-dimensional FEA approach, which yields

solutions that correlate well with actual experimental results. Moreover, unlike the

analytical solutions, both the numerical and experimental model specimens have the

capability of introducing notches, which act as very simplified cracks to model the

fracture behavior.

This chapter serves to discuss the procedure to develop a 3-D linear anisotropic

elastic numerical model, which has the ability to predict slip initiation in single crystal

superalloys. We will also discuss the geometries of the specimen used in the numerical

and experimental analyses and the characteristics of the numerical model such as material

properties, meshing and coordinate systems.









4.1 Development of a Three-Dimensional Linear Elastic FEA Model

In a polycrystalline material each grain has its own primary and secondary

crystallographic orientation but because of the random orientation of grains, these

materials are isotropic in nature (except some textured materials). Moreover in a

polycrystalline material, two elastic constants govern the transformation from stress to

strain, unlike the case with single crystal materials. A single crystal has specific

crystallographic orientation in which each direction may respond differently to similar

loading conditions. These materials (with cubic symmetry) require three independent

elastic constants (elastic modulus, shear modulus and poisson ratio) for stress-strain

transformation. Therefore while developing a finite element model of a single crystal,

special care must be taken while defining the material properties of the 3D model.

V= [001]
I V-= loading direction
I 'A






sx= [010] .robn-i notch direction

Z [100] Z
z I



(A) (B)
Figure 4-1. Figure showing the material and specimen co-ordinate system. A) The
alignment of specimen (x', y', z') and material coordinate system (x, y, z), B)
Finite element model is created around specimen coordinate system and the
material coordinate system is specified later.

The commercial software ANSYS (Finite Element Software Version 8.1) was

employed to model the specific geometries and orientations of the double-notched tensile









test specimens. ANSYS has various 3D elements available to account for the anisotropic

material properties. These elements in conjunction with the three independent stress

tensors (S11, S12, S44), or the three independent directional properties (G, E and v), can be

used to model a single crystal material. The Si values are always reported along the

<100> direction, here defined as the material coordinate system.

In experiments, specimens with specific crystallographic orientations are machined

from a solid single crystal material. Therefore, while modeling a single crystal specimen

in finite element software, we defined two co-ordinate systems: the material and the

specimen coordinate systems (Figure 4-1). The model was created around a specimen

coordinate system in the finite element software. Then the material coordinate system

was aligned with the specimen coordinate system using direction cosines (Figure 4-1B).

Finite element software aligns the material properties with the element coordinate

system; therefore the element coordinate system must be aligned with the material

coordinate system in order that the directional material properties are suitably applied.

The stress can now be transformed in any required coordinate system as the properties

have been defined in the material coordinate system. The material properties used in the

analysis of the single crystal material are given in Table 4-1. These values are for the

PWA 1480, a typical Ni-base superalloy used in the manufacturing of blades for turbine

engines.

Table 4-1. Material properties used in the analysis of the notched single crystal specimens
(Milligan et al., 1987).
Elastic Modulus (Ex=Ey=Ez) 1.21 x 1011Pa
Shear Modulus (Gx=Gy=Gz) 1.29 x 1011 Pa
Poisson's Ratio (vx=vy=vz) 0.395









Before creating the 3D notched finite element model, the stress transformation

between the material and specimen coordinate system should be properly verified. A

solid anisotropic rectangular specimen was first created and the six component stresses

(yxx, ayyy, z, cxz, zx, and ayz) were calculated from the material coordinate system of the

specimen. Subsequently, the numerically obtained component stresses were compared

with the component stresses from an analytical solution. After ensuring the correct stress

transformation in the solid model, two notches were incorporated in the model. After

creating the notched FEA model, component stresses was calculated from the material

coordinate system of the finite element model at the desired location, and used in the

transformation equations (Equation 2-7) to calculate the individual resolved shear stresses

(RSS) on the octahedral planes. Data was analyzed over a wide range of radial and

angular distances around the notch tip to create a complete stress field, which eventually

was used to draw conclusions on sectors and slip activation. Figure 4-2 describes the

entire procedure for the analysis in terms of a flow chart.

FEM Componenti
Stiesses8
Figure 4-2. Flow chart for the analysis of the slip fields.

4.2 Geometries of the Specimen A, B and C

The geometries of the numerical model of specimen A, B and C are based on their

experimental counterparts. The specimens used in the experimental analysis possessed

end shoulders for gripping the samples (Figure 3-2). However, FEM will not take into

account the entire geometry of the experimental specimen but will be limited to the gauge

length of the specimen considering that the objective of the study, is to evaluate the

evolution of slip near the notch rather than the shoulders of the samples. An elliptical









notch tip is used in the finite element model of the specimen, which closely approximates

the experimental notch geometry. We also performed a systematic investigation on the

sensitivity of the computed notch stress fields for several notch tip geometries and found

minimal variation in stress fields. Moreover the length of the notches in each sample

were not exactly the same on both sides while for the FEA model both notch lengths and

heights were set equal to those of the largest actual dimensions. Table 4-2 shows the

actual geometry of the specimens, as well as the finite element specimen geometries.

Also, Figure 4-3 shows the dimension of the specimens.

Table 4-2. Actual and finite element specimen dimensions of specimens A, B and C in
mm.
Specimen A Specimen B Specimen C

Dimensions Actual FEM Actual FEM Actual FEM

Width 5.100 5.100 5.04 5.04 5.1 5.1

Height 19.000 19.000 17.594 17.594 19.0 19.0

Thickness 1.800 1.800 1.82 1.82 1.77 1.77

Right Notch Length 1.300 1.550 1.399 1.399 1.4 1.4

Left Notch Length 1.550 1.550 1.36 1.399 1.39 1.4

Right Notch Height 0.113 0.113 0.084 0.084 0.085 0.086

Left Notch Height 0.111 0.113 0.084 0.084 0.086 0.086
a (For both Left and
(ForbothLeftand 0.055 0.055 0.056 0.056 0.055 0.055
Right Notch)
b (For both Left and
(ForbothLeftand 0.226 0.226 0.168 0.168 0.172 0.172
Right Notch)

















Notch Radius



Notch Length Height a b





Notch Height





Figure 4-3. Dimension of the specimens.

4.3 Numerical Model Characteristics

4.3.1 Elements and Meshing

The ANSYS elements chosen for the FEM were PLANE2 (2-D 6 node triangular

element with quadratic displacement functions) and SOLID95 (3-D structural solid with

20 nodes) capable of incorporating anisotropic properties (Figure 4-4 and Figure 4-5).

After the 3-D solid model was created, the front face was meshed with the PLANE2

elements. This front face has precise element sizing along the defined radial lines around

the notch tip at 50 intervals [Figure 4-6A]. Once the front face was meshed with the

desired element sizing, 3-D elements were swept through the volume to complete the

meshing of the model and the two-dimensional mesh was deleted. Working in

conjunction with the 2-D elements on the front face, the 3-D elements retain their sizing

definitions.

















Y
(or Axial) L




Figure 4-4. PLANE2 2-D 6 NODE Triangular Structural Solid (ANSYS 8.1 Elements
Reference, 2003).

p xw D

7I -/ o
M V

IN A
L S

t 4<







Figure 4-5. SOLID95 3-D 20Node Structural Solid (ANSYS 8.1 Elements Reference,
2003).

4.3.2 Solution Location

To observe the stresses in the vicinity of the notch (radial and angular), sixteen

concentric arcs were created between the radii r = 0.5p and r = 8p in equal increments;

where p is the notch height (Figure 4-6B). The six component stresses (Cx, oy, Oz, Oy,

Czx, and cyz) were then calculated at each arc in 5 increments. The element sizing of the

FEM allows data to be collected on any of the seven separate x-y planes including the

front, middle, and back planes (Figure 4-7B).









90 deg
I


~8p



*\ It--O0 deg
F44








(A) (B)
Figure 4-6. Close-up view of the notch. A) Close-up view of element sizing on the
specimen front face near the left notch. B) Radial and angular coordinates
used for producing slip sector plots.


Surface Plane


Mid Plane


Figure 4-7. Figure shows the 3-D FEA model. A) Front view of the 3-D FEA model. B)
Isometric view of the model showing the different x-y planes through the
thickness.














CHAPTER 5
RESULTS AND DISCUSSION

In this chapter, the experimental and numerical results of a SCNBS are presented.

Predictions based upon the numerical analysis of the effects of various parameters such

as the secondary orientation, thickness and temperature on the development of plasticity

at the notch tip of FCC single crystals are discussed and validated against the

experimental findings. This chapter is divided into several topics. First we describe the

procedure for the analysis of the numerical results and our approach in describing stresses

at the notch tips. We then compare the experimental and numerical results individually

for samples loaded along the <100> orientation with three secondary orientations

(specimens A, B and C). Comparisons are only made for the specimen surface results.

Emphasis is also placed on the discussion of the effects of elastic anisotropy on the

plastic zone evolution. Subsequently, the effects of secondary orientation, thickness and

temperature on the plasticity evolution at notch tips of FCC single crystals are discussed

in detail. Finally we conclude this chapter by comparing the experimental and numerical

results in a copper single crystal subjected to a four-point bending load which has been

tested by Crone et al. (2003).

5.1 Definition of "Dominant Slip Systems"

In chapter 4, it was shown that the mesh design was created in a manner so as to

enable the calculation of the six component stresses at a distance r and angle 0 from the

tip of the notch (Figure 4-6). These component stresses were then transformed into

twelve RSS values by using Equation 2-7. Each of these twelve shear stresses were then









plotted as a function of 0 (from 0 to the top of the notch) at radii ranging from 0.5p to 8p

in a 0.5p interval, in the form of x-y plots (a total of 16 x-y plots were created for each

surface plane, for the region above the notch growth axis). Figures 5-1, 5-2 and 5-3 show

the x-y plots of RSS values of the twelve slip systems as functions of 0, at radius r = 2p,

on the surface of specimen A at 4982 N (KI = 50 MPam/2), 2100 N (KI = 20 MPam1/2)

and 1500 N (KI = 14 MPam1/2) loads respectively.

I Resolved Shear Stress v. Theta


-- 75
,324 CRSS= 324 MPa -*-76
S--77

'T79
I _.^ ^^ -- -- 710

^-.-- 711
712



0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150
Theta(deg)
Figure 5-1. Figure shows the RSS values at r=2p on the surface of specimen A at 1500 N
load.
































Figure 5-2. Figure shows the RSS values at r=2 on the surface of specimen A at 2100 N
load.


Figure 5-3. Figure shows the RSS values at r=2 on the surface of specimen A at 4982 N
load.









Slip activation in single crystals is usually governed by Schmid's law according to

which when the RSS on a slip system reaches a critical value (CRSS), plastic

deformation occurs. This implies that in figures 5-1, 5-2 and 5-3, any point that lies on or

above the dark horizontal line representing the constant CRSS value, is expected to

deform plastically. For example in figure 5-1 at load = 1500 N, none of the slip systems

cross the CRSS of the material and hence at this load the material will not slip (deform

plastically) at r = 2p on the surface of the specimen. As the load increases from 1500 N to

2100 N, some of the slip systems cross the CRSS of the material and become activated

(Figure 5-2). c2 becomes activated from 0 = 88- 1080 and T3 becomes activated from 0 =

108- 1200. These are the slip systems which are first expected to appear and become

visible on the surface of the experimental specimen at r = 2p at 2100 N load. From figure

5-3 one observes that as the load increases from 2100 N to 4982 N, more number of slip

systems become activated for the same 0, and also over a wider range of 0. For example,

observe that in figure 5-2, the range of activation is from 0= 880 1200 which it increases

to O= 00 1370 in figure 5-3. Moreover in figure 5-3, from 0= 880 1080, C3, T4, T6, T11

and T12 become activated with C2 whereas'2wasthe only activated slip system at 2100 N

load from 0= 880 108oTherefore, in this example according to Schmid's law along with

T2, these other activated slip systems must also appear on the surface of the experimental

specimen from 0 = 88- 1080 at 4982 N load. However, according to our analysis, we

predict that those slip systems, which first become activated with increasing load, will

become persistent and will inhibit the activation of new slip systems with any further

increase in the load, and will be the ones expected to be seen on the surface of the

experimental specimen. Due to the elastic nature of the results, the slip systems









represented by the highest RSS' will be the ones which first become activated with load

at various values of 0. For example from figure 5-3 one notes that c1, C2, 3 and T9 are

the slip systems with highest RSS', and therefore these are the slip systems which will

become activated first at r = 2p on the surface of the specimen. These slip systems with

highest RSS' are referred by us as the "dominant slip systems". In the next section, we

shall see whether the numerical predictions based upon the dominant slip system theory

can predict the slip activation in the experimental specimens.

5.2 Development of the Polar Plots

From the above analysis we have seen that each x-y plot provides information

about the dominant slip systems at a given radius, yet fails to give a complete scenario

around the notch. In order to get a complete picture of the dominant slip systems near the

notch with the help of the x-y plots, all of the information from the 16 x-y plots must be

analyzed simultaneously, which may become a confusing and cumbersome process. To

overcome this limitation of x-y plots and to better visualize the slip evolution near the

notch, the dominant slip systems were represented in the form of polar plots. These polar

plots also provided a comparison between the experimental and the numerical slip field

results (a slip field is a combination of various slip sectors, each of which contain a

different orientations of activated slip lines) on the surface of the specimen. This

procedure is illustrated with the help of the following examples. Figures 5-4, 5-5, and 5-6

show the x-y plots at r = Ip, 1.5p and 4p respectively and figure 5-7 shows the polar plot

for the surface of specimen A at a 1600 N load (KI = 15MPam1/2). These radii are

randomly selected for the purpose of illustration.






59


Resolved Shear Stress v. Theta


3 ----- ----1
972





648 A 3


2 3
SNO ACTIVATION --







74 10




N load.

Resolved Shear Stress v. Theta








NO ACTIVATION NO 3 o
324 A 78












972
410
o AC ATO 11-" A I12
0 10 20 30 4 0 5 0 60 70 80 90 100 110
Theta(deg)
Figure 5-4. Figure shows the RSS values at r = Ip on the surface of specimen A at 1600





N load.







"34------------------ j -. -- --- ^-- x81





Theta(deg)

Figure 5-5. Figure shows the RSS values at r = 1.5p on the surface of specimen A at 1600
N load.










Resolved Shear Stress v. Theta
972 --'ci
-U----c2

c3
6- NO ACTIVATION >. 4
,x,648
ct




324 -T
34--------------------------- T8- z


T9
-10



0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 17( A T12
Theta(deg)

Figure 5-6. Figure shows the RSS values at r = 4p on the surface of specimen A at 1600
N load.


Figure 5-7. Polar plot shows the dominant slip systems around the notch on the surface of
specimen A loaded to 1600 N.


150


160


170


[80


20




10
II0









Figure 5-4 indicates that at 1600 N load there is no activation of slip systems from

o = 00 -740, after which the slip systems start to activate. The dark horizontal line on the

x-y plots represents the CRSS of the material and the dark vertical lines represent the

boundary of a particular domain. Figure 5-4 also shows that from 0 = 740-108,'2, 23 and

11 experience RSS above the CRSS of the material, however only T2 which has the

highest RSS and hence termed as a dominant slip systems, will be observed to be

activated on the polar plot from 0 = 74-1080 (Figure 5-7). From 0 = 1080-1100 3 acts as

a dominant slip system as seen in figures 5-4 and 5-7.

Figure 5-5 show the x-y plot at r = 1.5p which indicates that from 0 = 00-96, no

slip systems were activated, from 0 = 96-1080 2 is the only one activated and hence the

dominant slip system, from 0 = 1080-1170, c3 is the activated/dominant slip system and,

from 0 = 117-1400, there is again no activation. At r = 4p, none of the slip systems cross

the CRSS line and therefore there are no activated/dominant slip systems (Figure 5-6).

Comparing the x-y plots at r = Ip, 1.5p and 4p, we note that as one moves away from the

notch tip the absolute RSS values of the dominant slip systems go down. Also the slip

system patterns and the boundaries of the dominant slip systems domains change with the

radius.

The region of the dominant slip systems at various radii occurring in the 16 x-y

plots, were plotted in the form of a polar plots as seen (Figure 5-7). Sectors with different

hatched lines indicate regions of the dominant slip systems around the notch. In

summary, figure 5-7 indicates that at a load of 1600 N, c2 and c3 are the only dominant

slip systems on the surface of specimen A and there is no activation after r = 1.5p radius.







62


Figures 5-8, 5-9, 5-10 and 5-11 shows the x-y plots at r = Ip, 1.5p and 4p and

corresponding polar plot on the surface of specimen A at a 3200 N load (K =

30MPam/2).


1620


1296


n. 972


S648


324


0


Resolved Shear Stress v. Theta


c3
c4
---IE--- t 5


---I--- t 7
-,8
t9
- --, 10
- -IlO
----, 11
A 12


0 10 20 30 40 50 60 70 80 90 100 110
Theta(deg)


Figure 5-8. Figure shows
N load.


972





,648





- 324





0


the RSS values at r = Ip on the surface of specimen A at 3200


Resolved Shear Stress v. Theta


Theta(deg)
Figure 5-9. Figure shows the RSS values at r = 1.5p on the surface of specimen A at 3200
N load.










648






B 324
p


0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170
Theta(deg)
Figure 5-10. Figure shows the RSS values at r = 4p on the surface of specimen A at 3200
N load.


170 10


180


Figure 5-11. Polar plots shows the dominant slip systems around the notch on the surface
of specimen A loaded to 3200 N.

Comparing figures 5-4, 5-5, 5-6 and 5-7 with figures 5-8, 5-9, 5-10 and 5-11 we

can observe that with the increase in load from 1600 N to 3200 N, more number of slip


100 90 80
110


160









systems get activated at a given radius and, more number of dominant slip systems are

seen around the notch. For example, a comparison between the x-y plot at r = Ip for 1600

N and 3200 N load (Figure 5-4 and Figure 5-8) reveals that at the 1600 N load, there was

a region of no activation from 0 = 0- 740 but at 3200 N, we can see that slip systems are

activated all the way from 0 to the top of the notch (that is till 1100). Similarly at r =1.5p

more number of slip systems are activated at 3200 N than at 1600 N load (Figure 5-5 and

Figure 5-9). At r = 4pthere was no activation at 1600 N load, but at 3200 N some of the

slip systems became activated. This implies that with an increase in the load, slip systems

get activated at greater distances from the notch tip. It should be noted that due to the

elastic nature of the results, a change in the applied load only changes the absolute RSS

values of the slip systems but does not change their relative positions (that is slip

patterns) (compare figures 5-4 and 5-5 with 5-8 and 5-9, respectively).

Figure 5-11 shows the polar plot corresponding to the load 3200 N. The active

dominant slip systems according to figure 5- 10 at r = 4p, are T1 from 0 = 17- 50, c6

from 0 = 67- 950 and c2 from 0 = 95- 1100, which are also seen as the dominant slip

systems in the polar (figure 5-11). These polar plots thus give a comprehensive picture of

the dominant slip systems around the notch which shows that the size of the plastic zone

around the crack tip increases with increasing load (compare Figure 5-7 and Figure 5-11).

5.3 Comparison of Experimental and Numerical Results

During a tensile testing of the notched samples, slip lines were observed on the

surface of the samples. These slip lines provide the information about the active slip

planes within a sector and delineate the sector boundary angles in the experimental

specimen. Initially slip lines occurred in one sector, and additional slip lines developed at









further distances from the notch tip in the same sector with continued loading. As the

load increased, the number of slip lines and their length continued to grow until the sector

was filled. Sector boundaries were identified by the change in the orientation of the slip

lines.

Figure 5-12 compares of the experimental plastic zone and numerical results on the

surface of specimen A at a 4982 N load. Apparent consistency between the observed slip

bands and the trace of the dominant slip systems predicted by the FEA was found. A

good correlation between the experimental and numerical slip fields is observed by

comparing the results at r = 5p. Figure 5-13 shows the x-y plot of the RSS values as a

function of 0 for the 12 slip systems at radius r = 5p. The active dominant slip systems,

according to figure 5-13, are found to be T1 from 0 = 00-56, c2 from 0 = 560-60, c6 from

0 = 600-95, c2 again from 0 = 95-125, and T3 from 0 = 125-127. These are the

dominant slip systems which will be observed as slip bands on the experimental

specimen at r = 5p (see Figure 5-12). Table 5-1 compares the experimental and numerical

sector boundaries at r = 5p. It indicates that the slip traces related to the (111) slip plane

are activated from 0 to 750 on the experimental specimen whereas according to FEA

predictions, the dominant slip systems T1 and c2 (which are also related to the (111) slip

plane) are activated only till 600. It is also observed that the numerical analysis predicts

the T6 system to be dominant between 600 and 950, however the slip traces related with

(111) slip plane extend from 750 to 1080, which is approximately 150 off from the

expected domain boundaries. Furthermore, the slip traces on the experimental specimen

related to the (111) slip plane are observed from 0 = 96-1170 whereas the numerical

analysis predicts the activation of 2 and T3 from 0 = 95-1270. The above evaluation









confirms that within approximately 150, the numerical analysis predicts the development

of the slip bands near the notch tip for specimen A. It may be noted that from figure 5-

12B that the experimental results presented here show a slight asymmetry about the notch

growth axis. This asymmetry is attributed to the difference in the two-crack lengths cut in

this sample (Table 4-2), as well as any misalignment in the loading fixture. Other factors

that contribute to the lack of symmetry are the fact that the specimen orientations were a

few degrees off the assumed crystallographic orientations and also the presence of

specific irregularities in the notch alignments in the experimental specimen.

The comparison between the experimental observations and the FEA results

suggest that only the dominant slip systems (that is the systems with the highest RSS) and

not all the slip systems with a RSS above the CRSS are activated at a given load level.

For example, in Figure 5-13, there are other slip systems that experience stresses above

the CRSS but slip bands corresponding to their traces were not observed. Table 5-2

summarizes the numerical prediction of the dominant slip systems on the surface of

specimen A, for varying radii r from the notch.













1'3

I .. ." '..

170, "- -


10011

5t!
0lio]




Io


350


340
'30

8." 320
.... 300
250 -.. 290
260 270 280


10011









-i_---g g i*"
S..S -3 4 5 6 7 8


y 3&10 o
\ no0


(A) (B)
Figure 5-12. Comparison between numerical and experimental results from r = 0.5p to
8p, on surface of specimen A at load=4982 N (KI = 50MPaml/2). A)
Numerically generated slip fields. B) Experimentally generated slip fields.


Resolved Shear Stress v. Theta


\ 1-
15 127 1 160 170
130 140 150 160 170


Theta(deg)
Figure 5-13. Figure shows RSS values at r = 5p on the surface of specimen A at 4982 N
load.


1907 ..
200
210
220
)30


648








324








0









Table 5-1. Comparison of numerical and experimental results on the surface
of the specimen A, at r = 5p.
Dominant Slip System Sectors
On Surface (r = 5p)
Numerical Solution Experimental Results
Sector 0 (deg) -max Slip System 0 (deg) Slip Plane
I 0-56 T1 (111)[101]
-- 0-75 (111)
II 56-60 T2 (111) [011]

III 60-95 T6 (111) [011] 75-108 (111)

IV 95-125 C2 (111) [0il]
96-117 (111)

V 125-127 T3 (111)[110]_

Table 5-2. Numerical prediction of dominant slip systems on the surface of specimen A,
for varying radii, r, from the notch.
Dominant Slip System Sectors

r= Ip r=2p r= 3p r =4p
Sector 0 (deg) cmax 0 (deg) cmax 0 (deg) cmax 0 (deg) cmax
I 0-40 T1 0-60 T1 0-60 T1 0-58 T1
II 40-110 T2 60-108 T2 60-68 T2 58-63 T2
III 108-130 T3 68-90 T6 63-94 T6
IV 130-137 T9 90-113 T2 94-120 T2
V 113-138 T3 120-135 T3


r=5p r=6p r= 7p r= 8
Sector 0 cmax 0 cmax 0 cmax 0 cmax
I 0-56 1 0-55 T1 0-52 T1 0-49 1
II 56-60 T2 55-59 T2 52-58 T2 49-56 2
Ill 60-95 T6 59-95 T6 58-98 T6 56-100 T6
IV 95-125 T2 95-124 T2 98-121 T2 100-117 T2
V 125-127 3 -

As can be seen in Figure 5-12B, there is a region were both C2 and T6 systems are

activated (from 0 = 96- 1080), which at first glance seems to be contradictory with the

concept of the activation of only the dominant systems. This observation can be









explained by the evolution of the slip as a function of load. Let us consider the evolution

of the slip traces at distance r = 5p as the specimen A is loaded. Because of the elastic

nature of the calculation, the shape of the stress distribution curves given in Figure 5-13

is independent of the load level. As the load is increased, T6 and T1 systems are activated.

At larger loads, c2 and T3 systems become operational in regions outside the T6 and T1

domains. However as the load is increased, the initial dominant slip systems that is c6 and

T1, persist in regions where they are not dominant but they experience a RSS above the

CRSS. This persistence results in the observation of domains with more than one set of

slip bands. Therefore, the overlap of the T2 and T6 domains in the 0 = 96- 1080 section at

r = 5p (Table 5-1) can be attributed to the extension of the T6 system beyond its

dominance.

Specimen B was also evaluated using techniques similar to those used for specimen

A. A comparison of the FEM results with the observed slip traces at two load levels for

specimen B is shown in Figure 5-14. The results suggest that initially one set of planes

were activated (Figure 5-14a) and with increasing load, the second set became activated

at larger distances from the tip (Figure 5-14b). The FEM results identify various domains,

where in each a slip system is dominated. Because of the crystallographic symmetry of

the notch only domains for 0 = 0-180 are presented. It should be noted that two slip

systems (e.g. T1 or T"1) are mentioned in each numerical domain the first (e.g. T1)

represents the dominant system for the bottom (e.g. "11) half which corresponds to the

optical (experimental) picture, and the second corresponds to the top half. Thus, there is

an excellent agreement between the observed slip traces and those predicted by the FEM

calculations at both load levels. The experimentally observed slip field results are seen to









be very different from those of Specimen A, highlighting the importance of crystal

orientation in activating specific slip systems (compare figure 5-12 and figure 5-14).

Table 5-3 lists the numerical prediction of the dominant slip systems on the surface of the

specimen B, for varying radii r from the notch.

Table 5-3. Numerical predictions of dominant slip systems on the surface of specimen B
for varying radii r, from the notch.
Dominant Slip System Sectors

r = lp r= 2p r= 3p r =4p
Sector 0 (deg) cmax 0 (deg) -max 0 (deg) cmax 0 (deg) -max
I 0-50 T4 0-50 T4 5-50 T4 16-48 T4
II 50-110 T1 50-60 T11 50-68 T11 48-57 T11
III 60-65 10 58-66 T10 57-65 10
IV 65-132 -c 66-132 cT 65-127 T1


r= 5p r 6p r= 7p r = 8p
Sector 0 (deg) cmax 0 (deg) cmax 0 (deg) cmax 0 (deg) cmax
I 24-46 T4 30-45 T4 85-102 T9
II 46-55 T11 45-50 T11
III 55-58 50-68 -
IV 58-63 T10 68-85 T11 -
V 63-121 T1 85-110 T4 -
VI 110-113 11 -









100 90 80
110 70

13 20 50
140 40











[011] = trace t and t (a)
,15(X)0 r 1

120..-- \ ,-

[011][0 1] t and



n B l t A 1 N ( 35



41 0 MP 2)0




S[011] trace Oo, rT1 andr 4
[01 1]= and -g 3
,,, ,, (b )

Figure 5-14. Comparison between numerical and experimental results on the surface of
specimen B loaded to A) 1780 N (KI = 20MPaml/2) and B) 3456 N (KI
40MPaml/2a.









Recall that specimen C has the notch plane as (001), the notch growth direction as

[310 ] and the notch tip direction as [130] (see figure 3-3). This specimen has

crystallographic symmetry with respect to only notch growth axis ([310 ]). This is

demonstrated by the numerical results in figure 5-15, which shows the front and back

surface slip fields on the left and right notches of specimen C. This orientation also has

diagonal symmetry between the left and right notches with respect to the midplane (one

which is perpendicular to the [130] direction). By diagonal symmetry we mean that the

slip patterns on the left notch on the front surface are similar to the slip patterns of the

right notch on the back surface and vice-versa (Figure 5-15). Figure 5-15 also suggests

that the plastic zone sizes at the left and right notches on the same surface (front or back)

are not equal. These differences in the shape and size of the slip fields are due to changes

in the stress distribution around the left and right notches, which again are caused by the

asymmetry of specimen C with respect to the loading axis.


























Left Notch Right Notch


Front Surface


Left Notch Right Notch


Back Surface

Figure 5-15. Plot shows the front and back surface slip fields on the left and right notches
of specimen C.

Figures 5-16 and 5-17 show the stress distributions at r = 2p on the surface of

specimen C at the left and right notches respectively. The figures demonstrate that not







74


only are the slip patterns at the left and right notches different, but the RSS'

corresponding to the various values of r and 0 are also different. Moreover, the left notch

is experiencing more stress than the right notch and hence the plastic zone size is bigger

at the left than at the right (Figure 5-18).


756








0378








0


1 1'


0 10 20 30 40 50 60 70 80 90
Theta(deg)


100 110 120 130 140 150


Figure 5-16. Stress distribution on the surface of specimen C, at the left notch at r = 2p.










Resolved Shear Stress v. Theta
756



\ 4
-- 4


S378. CRSS 7


m A- 9

A T12



0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150

Theta(deg)
Figure 5-17. Stress distribution on the surface of specimen C, at the right notch, at r = 2p.

Figure 5-18 shows a comparison of the experimental and numerical slip fields at

the left and right notches on the front surface of specimen C. Figure 5-19 shows the trace

of the different {111} slip planes on the plane of observation (that is the (130) plane).

Similar to specimen B, two slip systems are mentioned in each sector of the numerical

slip fields: the first represents the dominant system for the bottom half, which

corresponds to the optical picture, and the second, corresponds to the top half. The

comparison reveals that the slip fields between the experimental and numerical results

match extremely well at the right notch (Figure 5-18B), however we see a small

discrepancy at the left notch (Figure 5-18A). For example, the numerical results at the left

notch predict T4 to be dominant between T2 and Tz, and z10 to be dominant between zs and

T12. These domains of C4 and z10 were not seen in the experimental results.




























































Figure 5-18. Comparison between numerical and experimental results on the surface of
specimen C loaded to 3500 N (KI = 30MPam1/2). A) Left notch, B) Right
notch.


170

180









[001]


Figure 5-19. Traces of different { 111 } slip planes, on the plane of observation (that is
(130) plane).

Also, c10 is seen to be activated (the dotted sector in the experimental picture of figure -

18A) in the domain of 22 in the experimental specimen however, the activation of 10 can

be explained by the slip evolution as a function of load by the numerical results. Besides

these discrepancies, the domains of 2, 1-10 (near the notch), "c and T12 on the left notch

and the domains of c10~, c2 and T6 on the right notch were correctly predicted by the

numerical results. Furthermore, the plastic zone sizes predicted by the numerical results

match extremely well with the experimental results at both the notches. The match

between the numerical and experimental results for specimen C, which has asymmetric

slip fields at both the notches, again proves that the dominant slip systems' theory is

extremely useful in the prediction of the experimentally generated slip fields.

5.4 Evolution of Slip Sectors as a Function of Load

Slip fields of specimen A have been inspected for three different tensile loads.

Figure 5-20 shows the slip field evolution on the surface of specimen A at 1600 N, 3200

N, and 4982 N loads. The size of the plastic zone (slip field) around the notch and the









number of dominant slip systems increases with load. The results suggest that the slip

planes activated at lower load levels persist at higher load levels, which is consistent with

the recent experimental findings (Ebrahimi et. al; 2005). Numerical and experimental

results demonstrate that there is a significant change in the sector boundary angles with

increasing load and hence a single number cannot be used for defining a sector boundary

for various load levels, as done by Rice (1987) and Shield (1996).


170

180


(B)
Figure 5-20. Polar plots showing the evolution of slip fields around the notch of the
specimen A loaded to A) 1600 N (KI = 15MPam1/2), B) 3200 N (KI =
30MPam1/2) and C) 4982 N (KI = 50MPam/2).









100 90 80





0180 070
120 ... ,.. ,. 60












(C)
Figure 5-20. Continued

5.5 Comparison of Anisotropic and Isotropic Results

Rice (1987) previously noted that the inclusion of anisotropy in the numerical or

analytical models is important for predicting the experimental results accurately. Here,

the comparison of anisotropic and the isotropic results further validate this importance.

Fundamentally it should be possible to calculate the elastic modulii for a polycrystal from

a weighted average of the elastic behavior of all orientations of crystals present in the

polycrystal. However, the appropriate way to calculate this average is not apparent. There

have been several methods proposed in the literature for a random polycrystal. The Voigt

(Reuss, 1929) average was based on assuming uniform local strain and averaging the

modulii over all the orientations. The Reuss (1929) average assumes uniform local stress

and is based on averaging the compliances over all the orientations. Finally Hill (1952)

found that the Voigt and Reuss averages correspond to upper and lower bounds to the

true behavior and suggested that the arithmetic average of the Reuss and Voigt averages

is a good approximation for the polycrystal. Later Hashin and Shtrikman came up with a

good approximation of the upper and the lower bounds for the cubic polycrystals (1962).
.. ".... ........ ...


160 '.. "- ".,
t '--(. :....... '' ". "



180 1 10

(C)
Figure 5-20. Continued

5.5 Comparison of Anisotropic and Isotropic Results

Rice (1987) previously noted that the inclusion of anisotropy in the numerical or

analytical models is important for predicting the experimental results accurately. Here,

the comparison of anisotropic and the isotropic results further validate this importance.

Fundamentally it should be possible to calculate the elastic modulii for a polycrystal from

a weighted average of the elastic behavior of all orientations of crystals present in the

polycrystal. However, the appropriate way to calculate this average is not apparent. There

have been several methods proposed in the literature for a random polycrystal. The Voigt

(Reuss, 1929) average was based on assuming uniform local strain and averaging the

modulii over all the orientations. The Reuss (1929) average assumes uniform local stress

and is based on averaging the compliances over all the orientations. Finally Hill (1952)

found that the Voigt and Reuss averages correspond to upper and lower bounds to the

true behavior and suggested that the arithmetic average of the Reuss and Voigt averages

is a good approximation for the polycrystal. Later Hashin and Shtrikman came up with a

good approximation of the upper and the lower bounds for the cubic polycrystals (1962).









In this study, the isotropic properties were calculated using Hashin and Shtrikman's

method. We took the arithmetic average of the upper and lower bounds, for calculating

the isotropic properties of our model. Equation 5-1 and 5-2 were used to calculate the

shear and elastic modulus.


Gl= G + 3 5 40,
Glower G+3 G 4p


Gpper = + 2 632

1
B= -(C,,- 2C12)
3

G +G
G= lower upper (5-1)
2

9BG
E 9 (5-2)
G+3B

where

-3(B + 2G,) -3(B + 2G2)
11 5G (3B+4G,1) 5G2(3B+4G2)
1
Gl= (C,,- C12); G2= C44
2

C11, C12 and C44 are the stiffness constant of the single crystal material used in this

study. Table 5-4 shows the calculated isotropic properties. These isotropic properties

have been incorporated into a 3D-FE model. A direct comparison of results between the

isotropic and anisotropic models for single crystals was made possible by calculating

RSS' on the 12 primary octahedral slip systems of an FCC single crystal.