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PREDICTION OF SLIP SYSTEMS IN NOTCHED FCC SINGLE CRYSTALS USING 3D FEA By NIRAJ SUDHIR BIDKAR A THESIS PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE UNIVERSITY OF FLORIDA 2003 I would like to dedicate this work to my parents for their everextending support and encouragement. ACKNOWLEDGMENTS I would like to express deep gratitude to Dr. Nagaraj Arakere for his encouragement and enthusiastic assistance and instructions. I would also like to extend special thanks to Dr. Ashok Kumar and Dr. John Schueller for their time and support. Thanks are also due to Dr. Ebrahimi and Luis Forrero for providing the necessary information and data. I am thankful to my friends and lab mates Jeff and Shadab for helping me in understanding many concepts and ideas. Finally I am grateful to the Almighty God for giving me the strength for the completion of this work. TABLE OF CONTENTS page A C K N O W L E D G M E N T S ................................................................................................. iii LIST OF TABLES ............................................................ ... ............ .. vi L IST O F F IG U R E S .... ...... ................................................ .. .. ..... .............. vii ABSTRACT .............. .......................................... xi CHAPTER 1 IN TRODU CTION ............................................... ..... ..... .............. .. Slip Deform ation in Single Crystal Superalloys ....................... ................ ........... 2 M material C haracteristics.............................................................. ....................... 7 Temperature Effects ................. .............. .. .......... .... ............. 8 T est M ethods .................................................................. .................................. 9 A nalytical A approach ..................................................... .. .......... 9 N um erical A pproach.................................................... ......................... 10 Experim ental A approach ........................................................ ......... ..... 10 2 REVIEW OF LITERA TURE ........................................................ ............... 12 A nisotropy of E lasticity ............................................ ......................................... 12 E la stic ity ........................................................................ 12 Elasticity M atrix .............. ...................................... ...... ............ .. 13 N otch Tip D eform ation ......................................................................... 16 Rice (1987) ...... ......................................... .......... 16 C rone and Shield (200 1)...................................................... ... ................. 18 Schulson and X u (1997) ....................................................................... ... 23 Slip A ctivation and D eform ation ........................................ ......... ............... 27 3 ANALYTICAL APPROACH ......................................................................29 Coordinate Axes Transformation Using Miller Indices...................................31 E xam ple Transform ation ............................................................... ... 34 Transformation Matrices for Stress and Strain Fields........................................ 35 Shear Stresses and Strains in the Slip Systems ................................................38 4 NUMERICAL APPROACH: FINITE ELEMENT METHOD ..............................41 Verification of the Finite Element Model Results......................................... 42 Specimen Orientation and Dimensions ............................................... 43 Specim en G eom etry.................................................... .......................... 44 Specim en G eom etry.................................................... .......................... 45 Finite Element Model Characteristics ........................................ ...............45 M material P roperties................................................... ............. ........... 45 E lem ents and M eshing ......................................................... ... ................. 46 F inite E lem ent S solution ......................................... .............................................4 9 A ssum options .................................................................................................. ....... 50 5 RESULTS AND DISCU SSION ........................................ ......................... 52 S p ecim en I .............................................................5 2 Specim en II .................................................................. ..................70 Identifying the Slip Systems from the Experimental Results ................................99 Sp ecim en C om p arison ............................................. ......................................... 103 Specim en I N otch G eom etry ........................................................ ............... 104 6 CONCLUSIONS AND RECOMMENDATIONS...............................................107 L IST O F R E F E R E N C E S ......... .. ............... ................. .............................................. 108 BIOGRAPHICAL SKETCH ........... ..... ......... .. ..........................111 v LIST OF TABLES Table page 1.1 Slip systems in FCC crystal. Source: Stouffer and Dame, 1996 .................................3 2.1 Atomic density on FCC crystal planes Source: Dieter (1986)..................................13 2.2 Symmetry in various crystal structures Source: Dieter (1986)............................... 13 2.3 Sector boundary angle comparisons for Orientation II Source: Crone and Shield (2 0 0 1) ..........................................................................................1 9 2.4 Sector boundary angles from the experimental tests by Crone and Shield Source: C rone and Shield (2001) ............................................... ............................. 22 2.5 Slip sectors from plane stress and plane strain assumptions. Source: Schulson and Xu (1996) ..................................... .................. ............. ........... 26 4.1 Actual and specimen geometry for Specimen I .......................................................44 4.2 Actual and specimen geometry for Specimen II ............................................... 45 5.1 Specimen I dominant slip system sectors on the surface...........................................62 5.2 Specimen I dominant slip system sectors on the on the midplanes............................69 5.3 Specimen II dominant slip system sectors on the surface on the upper portion of the notch grow th direction ......... ................. ................... ................... ............... 77 5.4 Specimen II dominant slip system sectors on the surface on the lower portion of the notch grow th direction ......... ................. ................... ................... ............... 84 5.5 Specimen II dominant slip system sectors on the midplanes on the upper portion of the notch grow th direction ............................................ ............................. 91 5.6 Specimen II dominant slip system sectors on the midplanes on the lower portion of the notch grow th direction ............................................ ............................. 98 5.7 Load Levels applied to specimen I during the experiments .....................................99 5.8 Intersection of the slip planes with the notch plane.............................. ..............100 5.9 Specim en I experimental results ................................................. ..................... 102 vi LIST OF FIGURES Figure page 1.1 Convention for defining primary and secondary crystallographic orientations in turbine blades.Source: Swanson and Arakere, 2000....................................2 1.2 Slip sectors observed under plastic deformation. Source: Crone and Shield, 2001. .....7 1.3 Microstructure of material A. The y' precipitate forms in the y matrix and has a volume fraction of about 60%. Source: Superalloys II by Sims, Stoloff and H ag el. ................................................................................ 8 2.1 Schem atic of FCC crystal structure ........................................ ......................... 13 2.2 N otch direction term inology. ................................................................................ .....17 2.3 Yield surface based on plane strain state of stress ..................................................17 2.4 Orientations used by Crone and Shield in their tests. .............................................19 2.5 Slip sectors observed in the experimental tests by Crone and Shield. Source: Crone and Shield (200 1).................................................. ...................... .... 20 2.6 Slip sectors and slip lines from the experimental tests by Crone and Shield. Source: C rone and Shield (2001) ............................................... ............................. 2 1 2.7 Specimen Orientation used by Schulson and Xu............................... ............... 23 2.8 Slip sectors obtained by Schulson and Xu Source: Schulson and Xu, 1997. ............25 2.9 L oad and slip direction............................................. .................. ............... 27 3.1 M material and Specimen coordinate system ....................... ................................... 30 3.2 First rotation of the coordinate system about the zaxis ................ ............... 31 3.3 Projections from the original axes on the projected axes ................. ............ .....32 3.4 Second rotation of the coordinate system about the zaxis...................................32 3.5 Third rotation of the coordinate system about the zaxis ................................. 33 4.1 The notched specimens analyzed using finite element method ...............................42 4 .2 Specim en specifications...................................................................... ...................44 4.3 The specimen is first defined with respect to the global coordinate system and then the material coordinate system is specified later. .............................................46 4.4 M eshing around the notch in AN SY S .............................................. .....................47 4.5 Schematic of the SOLID95 element in ANSYS. Source ANSYS 6.1 Elements R reference, 2002. .....................................................................48 4.6 Radial arcs used for element location and sizing; centered at notch tip....................48 4.7 Radial arcs for stress field calculations...................... .... ......................... 49 5.1 RSS on the 12 primary slip systems for the upper portion of the notch growth direction for R=0.25*p .... ...... ...................................... 54 5.2 RSS on the 12 primary slip systems for the upper portion of the notch growth direction for R =0.5*p ......................... ............ ........ ... .. .. .. ........ .... 55 5.3 RSS on the 12 primary slip systems for the upper portion of the notch growth direction for R = l*p p.......... ................ ................................ .. ...... 56 5.4 RSS on the 12 primary slip systems for the upper portion of the notch growth direction for R=2*p .......... ..... ...... .... .... .................. .. ...... 57 5.5 RSS on the 12 primary slip systems for the upper portion of the notch growth direction for R=5*p ......... ...... .. ..... ........ ...... .. .. .. .... .. ...... .... 58 5.6 Slip system sectors on the surface for specimen I ................................................. 59 5.7 Upper portion of the radar plot showing the surface stresses for the [100] orientation ........................................................................... 60 5.8 Lower portion of the radar plot showing the surface stresses for the [100] orientation ........................................................................... 6 1 5.9 RSS on the 12 primary slip systems for the upper portion of the notch growth direction for R=0.25*p .......... .... .......... .. ......... .... ........................ 63 5.10 RSS on the 12 primary slip systems for the upper portion of the notch growth direction for R =0.5*p ......................... ............ ........ ... .. .. .. ........ .... 64 5.11 RSS on the 12 primary slip systems for the upper portion of the notch growth direction for R = l*p p.......... ................ ................................ .. ...... 65 5.12 RSS on the 12 primary slip systems for the upper portion of the notch growth direction for R=2*p .......... ...... ...... .... ................... .. ...... 66 5.13 RSS on the 12 primary slip systems for the upper portion of the notch growth direction for R=5*p.......... .... ......... ................. .. ...... 67 5.14 Slip system sectors on the midplanes for specimen I............... ....... ............68 5.15 RSS on the 12 primary slip systems for the upper portion of the notch growth direction for R=0.25*p .... ...... ... ................................... 71 5.16 RSS on the 12 primary slip systems for the upper portion of the notch growth direction for R =0.5*p ................. ..... ...... .. .... ...... ... .. ..................72 5.17 RSS on the 12 primary slip systems for the upper portion of the notch growth direction for R = p............. ... .................................... .......... ........ 73 5.18 RSS on the 12 primary slip systems for the upper portion of the notch growth direction for R=2*p .......... ...... ...... .... ................... .. ...... 74 5.19 RSS on the 12 primary slip systems for the upper portion of the notch growth direction for R=5*p.......... .... ......... ................. .. ...... 75 5.20 Slip system sectors on the surface on the upper portion of the notch growth direction for specim en II.............................................. .............................. 76 5.21 RSS on the 12 primary slip systems for the lower portion of the notch growth direction for R=0.25*p .......... .... ...... .. .... ......... .... ........................ 78 5.22 RSS on the 12 primary slip systems for the lower portion of the notch growth direction for R =0.5*p ................. ..... ...... .. .... ...... ... .. ..................79 5.23 RSS on the 12 primary slip systems for the lower portion of the notch growth direction for R = p............. ... .................................... .......... ........ 80 5.24 RSS on the 12 primary slip systems for the lower portion of the notch growth direction for R=2*p .......... ...... ...... .... ................... .. ...... 81 5.25 RSS on the 12 primary slip systems for the lower portion of the notch growth direction for R=5*p.......... .... ......... ................. .. ...... 82 5.26 Slip system sectors on the surface on the lower portion of the notch growth direction for specim en II.............................................. .............................. 83 5.27 RSS on the 12 primary slip systems on the midplanes for the upper portion of the notch growth direction for R=0.25*p ....................................... ............... 85 5.28 RSS on the 12 primary slip systems on the midplanes for the upper portion of the notch growth direction for R=0.5*p ........................................... ............... 86 5.29 RSS on the 12 primary slip systems on the midplanes for the upper portion of the notch grow th direction for R=p ........................................ ........................ 87 5.30 RSS on the 12 primary slip systems on the midplanes for the upper portion of the notch growth direction for R=2*p ............................. ................................. 88 5.31 RSS on the 12 primary slip systems on the midplanes for the upper portion of the notch growth direction for R=5*p ............... .... .... ...................................89 5.32 Slip system sectors on the midplanes on the upper portion of the notch growth direction for specim en II.............................................. .............................. 90 5.33 RSS on the 12 primary slip systems on the midplanes for the lower portion of the notch growth direction for R=0.25*p ....................................... ............... 92 5.34 RSS on the 12 primary slip systems on the midplanes for the lower portion of the notch growth direction for R=0.5*p ........................................... ............... 93 5.35 RSS on the 12 primary slip systems on the midplanes for the lower portion of the notch grow th direction for R=p ........................................ ........................ 94 5.36 RSS on the 12 primary slip systems on the midplanes for the lower portion of the notch growth direction for R=2*p ............................. ................................. 95 5.37 RSS on the 12 primary slip systems on the midplanes for the lower portion of the notch growth direction for R=5*p ................. .... .... ...................................96 5.38 Slip system sectors on the surface on the upper portion of the notch growth direction for specim en II.............................................. .............................. 97 5.39 Picture showing the slip lines around the right notch of specimen I for level 6........99 5.40 Picture show ing the slip traces.......................................... ........................... 101 5.41 R SS V s Theta plot scaled to load level # 6....................................... .................... 102 5.42 RSS Vs. Theta plots obtained from the semicircular notch geometry..................104 5.43 RSS Vs. Theta plots obtained from the semicircular notch geometry..................105 5.44 RSS Vs. Theta plots obtained from the elliptical notch geometry.........................105 Abstract of Thesis Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Master of Science PREDICTION OF SLIP SYSTEMS IN NOTCHED FCC SINGLE CRYSTALS USING 3D FEA By Niraj Sudhir Bidkar May 2003 Chair: Dr. Nagaraj K. Arakere Department: Mechanical and Aerospace Engineering Nickelbase single crystal superalloys are the current choice for hightemperature jet engine applications such as turbine blades and vanes. Since these materials are grown as single crystals, there are no grain boundaries present and as a result the properties of these materials are highly directiondependent. This makes it necessary to test these materials for different load orientations. Notched tensile test specimens are typically used to study the evolution of slip systems in 3D stress fields. Many researchers in the past have tried to come up with solutions for prediction of the slip systems in notched test specimens for specific load orientations. The two notchedspecimens considered in this work differ in geometry as well as in load orientation. An elastic finite element model has been created which predicts the behavior in the plastic regime quite accurately. The linear elastic model is used to predict only the onset of yield. The stress concentrated region near the notch, with the triaxial state of stress, is of particular interest. Slip systems in this region are predicted using 3D linear elastic FEA, including material anisotropy. This model does not account for effects such as strain hardening, crystal lattice rotation, creep, hightemperature conditions. These results were correlated with the experimental tests performed by the MSE department for similar specimens. The first of the two specimens had [001] load direction, [010] notch growth direction and [100] as the notch plane direction. The other specimen had a load direction of [111], notch growth direction [121] and [303] as the notch plane direction. The slip systems predicted using 3D FEA had excellent correlation with those observed experimentally. This demonstrates that 3D linear elastic FEA that includes the effect of material anisotropy can be used to effectively predict the onset of yield and hence slip systems in FCC single crystals. CHAPTER 1 INTRODUCTION As one of the most important classes of hightemperature structural materials, Nickel base superalloys exhibit a truly unique combination of properties at elevated temperatures. These desirable properties enable these materials to enhance the performance and efficiency of turbine engines designed for both aircraft and power generation applications. In high temperature polycrystalline alloys grain boundaries provide passages for diffusion and oxidation. Eliminating grain boundaries and grain boundary strengthening elements produces materials with superior high temperature fatigue and creep properties compared to conventional superalloys. However, the absence of grains makes these alloys orientation dependent or anisotropic with tension compression asymmetry. As components are cast in single crystal form the entire component inherits the anisotropic properties of the crystal lattice. The slip deformation mechanisms are strong functions of orientation and deformation occurs in specific crystallographic directions rather than in the direction of the applied load. During the manufacturing of the superalloys, the primary growth direction is oriented in the <001> orientation as this orientation has the best combination of mechanical properties. However in practice, due to the difficulties encountered in casting and due to cost considerations blades that are oriented away from the <001> orientation by up to 150 is used. Also in the operating environment the blades are subjected to loading in a variety of orientations due to the hot impinging gases causing bending and torsion. The formation of hotspots leads to thermal gradients, which also contribute to multiaxial loading. Hence the deformation mechanisms depend on the high temperature effects and loading in different orientations. Airfoil Stacking Line Relative Angle ac Airfoil Primary Mean Relative Crystailographic Chord Angle s Orientation Line Secondary Crystallographic Orientation Figure 1.1 Convention for defining primary and secondary crystallographic orientations in turbine blades. Source: Swanson and Arakere, 2000. Slip Deformation in Single Crystal Superalloys Slip deformation in nickelbase superalloys is governed by the sliding of separate layers of the crystal structure over one another along definite crystallographic planes or slip planes in specific directions, according to Dieter (1986). It is considered a stress controlled process. Table 1.1 Slip systems in FCC crystal. Source: Stouffer and Dame, 1996 Slip Number Slip Plane Slip Direction Octahedral Slip a/2{ 111 }<110> 1 (111) [101] 2 (111) [011] 3 (111) [110] 4 (111) [101] 5 (111) [110] 6 (111) [011] 7 (111) [110] 8 (111) [011] 9 (111) [101] 10 (111) [011] 11 (111) [101] 12 (111) [110] Octahedral Slip a/2{ 111}<112> 13 (111) [121] 14 (111) [211] 15 (111) [112] 16 (111) [121] 17 (111) [112] 18 (111) [211] 19 (111) [112] 20 (111) [211] 21 (111) [121] 22 (111) [211] 23 (111) [121] 24 (111) [112] Cubic cells a/2{ 100}<110> 25 (100) [011] 26 (100) [011] 27 (010) [101] 28 (010) [101] 29 (001) [110] 30 (001) [110] The CRSS is therefore the controlling value and it is the shear stress at which the slip was initiated. The CRSS is a function of the applied load and direction, specimen geometry, and crystal structure, though it is not directly related to the material's anisotropy. Depending on the direction of the applied load, certain slip systems get activated first. These are termed as the 'easy glide' systems or the primary octahedral slip systems. Usually, the planes with greatest atomic densities are the ones that are activated first. This whole process of slip occurs so as to relieve the energy of the high shear stress or the resolved shear stress (RSS) within the 12 primary slip systems. The slip thus helps to obtain a more energetically stable system. As this slip activity and the deformation process continues, the indications of slip at the surface can be observed. With the increase in the applied stress, some of the 12 secondary slip systems may also get activated. Extreme temperature or load conditions may also cause the activation of the six cubic slip planes. But the activation of secondary or cubic slip planes does not occur first because the slip activity along the primary slip planes requires less energy to relieve the high RSS. Certain microstructural behaviors like pinning or locking of dislocations too prevent the shift in slip from the primary to secondary or cubic slip systems. Usually slip occurs when the RSS exceeds the yield strength of the material. Nickelbase superalloys, however exhibit different yield strength in tension and compression. This kind of behavior is termed as tension/compression asymmetry. For instance, for the <001>orientation these alloys exhibit highest yield strength in tension, while the compressive yield strength is lower for this orientation. On the other hand, for the <110>orientation, the yield strength is higher in compression than in tension. For a sample loaded in the <111> direction there is practically no tension/compression asymmetry. In some cases, parts of the dislocations get separated to reach a lower energy state. These separated parts are termed as 'extended dislocations'. The extended dislocations, on octahedral slip planes must recombine into a single dislocation before cross slip can take place. Cross slip is the phenomenon where a dislocation moves from one plane to another. Cross slip also occurs to reduce the energy of the system. Again the material orientation affects the process of recombination of the dislocation parts as it does in case of the magnitude of the resolved shear stress. An applied compressive load helps to overcome the force separating two extended dislocations at an interface where cross slip can occur and can combine them into a single dislocation. The combined dislocation can then move as a cross slip. Conversely, an applied tensile load aids the force separating the dislocation and makes cross slip more difficult to occur. This asymmetrical load behavior should be addressed as an important issue in the design of a part that is subjected to both tensile and compressive loads. As far as the failure analysis research is concerned, priority is given to tensile stress analysis as compared to compressive stress analysis as compressive stresses usually prove to be beneficial. Instead of causing cracks to initiate or propagate, compressive stresses usually either have no effect or may even arrest the cracks that develop under tensile stresses. However, in the case of triaxial stress states, such as those occurring at the notch tip, the asymmetric behavior becomes even more complex. Depending on orientation, some slip directions may hinder partial dislocation recombination, while others may aid this process. Nickelbase superalloys exhibit another abnormal yield characteristic that is related to temperature called 'anomalous yield behavior'. A normal trend shows a decrease in the yield strength with increasing temperature. These superalloys however have been shown to exhibit an increase in their yield strength with rising temperature, up to a certain point. Here it is essential to take into account the process of superdislocations that is used to relieve the stress. Superdislocation can be defined as a dislocation composed of two dislocations separated by an antiphase boundary (APB) that glide along { 111} octahedral planes during slip. This superdislocation always attempts to reduce its energy, which can be brought about by lowering its APB energy if it crossslips to a {100} plane. Now when the first dislocation cross slips onto the { 100} plane, it gets locked due to the higher stress required for it to move on a {100} plane. It gets locked on two separate planes in such a way that one dislocation is left on a { 111 } plane and the other on a { 100} plane separated by the APB. Because of this lock, no further motion can take place. With increasing temperature, thermally activated crossslip to { 100} planes occur easily, forming locks. These locks keep on multiplying, thus preventing further motion and increasing the CRSS. In addition to this, the 'Peierl stress' required to move the locked dislocation on the {100} plane, is lowered with the increase in temperature. At a certain point of temperature, the thermal activation helps the dislocation to overcome the Pierel stress. At this critical point the entire dislocation is able to move to the { 100} plane resulting in a cubic slip. At this point, more of the previously locked dislocations are released on the {100} planes and consequently the CRSS begins to fall. Thus in high temperature applications, anomalous yield behavior is an important consideration for fracture and fatigue analysis. This consideration helps understand the mechanisms that increase strength and their limitations. Figure 1.2 Slip sectors observed under plastic deformation. Source: Crone and Shield, 2001. This work will focus on the variation of the 12 primary resolved shear stresses in a notched specimen. The activity of the specific slip systems with respect to the radial and angular distances surrounding the notch tip will be determined for specific orientations. The slip systems will be studied both near the notch and at far field to observe the changes associated with high stress gradients prevalent in close proximity of the notch tip. The maximum RSS values and slip systems are expected to shift along a line of constant radius. This, in a way indicates a shift in the state of stress. In the actual tensile testing, these different slip systems can be clearly visualized as "sectors" surrounding the notch tip. In short, the goal of this work is the prediction of slip system activity as a function of the radial and angular distances from the notch tip and the resulting slip sectors. Material Characteristics Nickelbase superalloys have a microstructure that consists of a ymatrix and a fine dispersion of hard y'precipitates. The matrix is considerably alloyed with other elements that may vary, including cobalt, chromium, tungsten and tantalum, though it is mainly composed of nickel. The precipitate however is the intermetallic compound Ni3Al. Figure 1.3 Microstructure of material A. The y' precipitate forms in the y matrix and has a volume fraction of about 60%. Source: Superalloys II by Sims, Stoloff and Hagel. So far these superalloys have evolved in three generations. Rene N4, CMSX4, and others are some of the most advanced or thirdgeneration superalloys. Usually, these alloys have a high volume fraction of y', around 60%. The majority of the deformation occurs in the softer matrix as suggested by Svoboda and Lukas (1998). However, it should also be noted that at such high volume fractions, the precipitate has a considerable effect on the overall performance of the superalloy's performance. Temperature Effects These materials are designated to provide unique strength and/or corrosion properties at elevated temperatures (i.e., greater than 6000C). The experimental notched tensile tests conducted to study slip systems activity have been performed at room temperature, which is well below the transition temperature for superalloys. As observed by Stouffer and Dame (1996), in the low temperature regime, the octahedral slip system is predominant and basically controls low temperature deformation. The secondary planes are activated only at high temperatures. At temperatures above 6000C the cubic slip systems get activated and act along with the octahedral planes. When the temperature reaches above 8500C, the deformation increases rapidly and the material strength is almost free of the orientation. Test Methods For the isotropic notch specimens in tension many analytical and numerical solutions have been developed for their elastic response, especially in the field of linear elastic fracture mechanics. However, it is very difficult to develop threedimensional analytical models for anisotropic notched specimens. The current solutions have been derived only after many simplifications and therefore give inaccurate results when compared to experimental data. However, the threedimensional finite element analysis (FEA) is capable of taking into account the limitations in the elastic models and it does provide a solution comparable to the actual experimental results. In numerical and experimental specimens, the notches can be considered as simplified cracks to develop a realistic model to study fracture behavior. In many applications, it may so happen that though the material is designed for primary strength in one direction, it must withstand multiaxial loading. This makes the study of effect of anisotropy all the more interesting. Analytical Approach Generally analytical solutions provide exact solution to any problem. But this is not true in case of very complex problems. For such problems, the analytical solutions are arrived at after close approximations or by the use of theoretical and empirical solutions. For the notched specimen in tension, an analytical solution is not available that correctly predicts slip system activity around the notch tip. The ones that are available make use of several assumptions like the plane strain. This issue will be discussed in detail later. Numerical Approach As far as finite element analysis is concerned, it does account for gross material properties such as modulus of elasticity and Poisson's ratio in isotropic materials. It even accounts for the directional properties while analyzing anisotropic materials. However, FEA is unable to account for microstructural properties that govern yield strength, such as dislocation mechanisms or other microstructural behavior. It may be possible to predict dislocation mechanisms by using smallscale atomistic simulations, but it would be too much expensive to conduct analysis for such small dimensions. Also, reducing the size to atomic level would distort the model and the results would no longer be realistic. Therefore smallscale simulations are not capable of providing accurate results for single crystal notched specimens. As such, FEA is the most feasible kind of computer simulation to analyze these specimens. As it neglects the microstructural behavior, FEA can also determine the effect of the specimen's geometry and anisotropy on material property behavior, without considering the atomic interactions. Experimental Approach The doubleedged notched tensile specimen loaded to study stress and strain fields and the slip line deformation, in particular, is the most commonly used test sample in many material testing laboratories. A triaxial state of stress is created around the notch, which provides the opportunity to study the slip system activity in threedimensional stress fields. Experimental tests have been carried out to determine the effect of the load orientation on the slip systems around the notch. These results are compared to those obtained by FEA for the same set of conditions. A constant load can be applied to the specimen in different crystallographic orientations and the different active slip planes can be observed. This approach is 11 altogether different than the elastic response measured by FEA, the magnitude of the applied elastic stress gives an idea of which planes will slip first. CHAPTER 2 REVIEW OF LITERATURE The stress calculations of a specimen without notches can be done by an analytical approach. However, as far as the modeling and stress calculations of the specimen with notches is concerned, it involves a lot of complexities and still an accurate single crystal analytical model has not yet been established. Quite a few researchers have investigated the stresses and sectors around the notch tip, but no method to get the precise results has been proven to be valid. To analyze the stressfield at a single crystal notch involves calculating the accurate stress components and then predicting and defining the slip zones around the notch. Anisotropy of Elasticity Elasticity As defined by Dieter (1986) elasticity can be described by specific elastic constants that relate to atomic strength and spacing of the material. The spacing between the atoms in an FCC is different in different directions. Comparing the atomic densities of the material in different directions can show this. Clearly, the atomic density of the material is not same in all the directions and so the spacing also varies with direction. From this, we can conclude that the elasticity is a function of orientation. The greatest atomic density corresponds to the least atomic spacing and most susceptible planes for slip. For example, planes such as the ones belonging to the { 111} family are termed as the 'closepacked planes' as they have the least distances between the atoms. Consequently, the atoms in these planes do not have to travel greater distances to reach another atomic position and therefore slip occurs most commonly on these planes. This is the case for closepacked directions; the atomic spacing being less in these directions makes them most likely for slip occurrence. Figure 2.1 Schematic of FCC crystal structure Table 2.1 Atomic density on FCC crystal planes Source: Dieter (1986) FCC Plane Atoms per unit area Atoms per unit cell {100} 2/ao2 2 {110} 2/(/2. ao2) 1.414 {111} 4/(/3. ao2) 2.309 Elasticity Matrix The elastic properties for any material can be completely defined by 36 elastic constants. However, in most of the materials, some kind of symmetry exists thereby reducing the number of independent constants. As seen in table 2.2, there are only two Table 2.2 Symmetry in various crystal structures Source: Dieter (1986) Crystal structure Number of elastic constants Tetragonal 6 Hexagonal 5 Cubic 3 Isotropic 2 independent constants for an isotropic material. Any two of the three constants E (modulus of elasticity), G (shear modulus) and v (Poisson's ratio) are enough to define the properties of the material in a particular direction. The symmetry of the cubic structures highly reduces the number of elastic constants, which in turn reduces the elasticity matrix for these structures. To reduce the number of independent constants from 36 to the final three for cubic structures, the original full elasticity matrix [aij] has to be considered: a11 a12 a13 a14 a15 a16 a21 a22 a23 a24 a25 a26 S] a31 a32 a33 a34 a35 36 (2.1) a41 a42 a43 a44 "45 a46 a51 a52 a53 a54 a55 a56 a61 a62 a63 a64 a65 a66 The strain matrix is then given by: E, = a cra (2.2) Assuming elastic potential exists (i.e isothermal deformation) the following relationship is achieved in equilibrium: [a ]= [a, ] (2.3) On account of this symmetry, the general matrix gets reduced to: a11 a12 "13 "14 "15 a16 12 a22 a23 a24 a25 a26 [a]= a13 a23 a33 a34 35 36 (2.4) a14 a24 a34 a44 a45 a46 a15 a25 a35 a45 a55 a56 a16 a26 a36 a46 a56 a66 Nickelbase superalloys exhibit orthotropic characteristics; so they have three orthogonal planes of elastic symmetry. Considering the effects of cubic elastic symmetry (Lekhnitskii, 1963), the number of elastic constants reduces to three for the final elasticity matrix: a11 a12 a12 0 0 0 a12 a1 a12 0 0 0 1 2 a a12 all 0 0 0 [a ]= 12 a12 (2.5) 0 0 0 a44 0 0 0 0 0 0 a44 0 0 0 0 0 0 a44 These elasticity constants can be defined using the modulus of elasticity, the shear modulus and the Poisson's ratio in specific directions as follows: al  (2.6) E, 1 a44 (2.7) Gyz V V a12 (2.8) E= E Exx Eyy For material that has been used to conduct these tests, the values of the elasticity constants are (Swanson and Arakere, 2000): all = 6.494 E 8 a44= 6.369 E 8 a12 =2.063 E 8 (psi) Making use of these elastic constants and the direction cosines, the modulus of elasticity for any material in any direction can be found as follows (Dieter, 1986): S=all 2. (a1 a12) a44 (a2 3 + i32 27 +a2 ) (2.9) Euvw 2 Where a3, 83 and 73 are the direction cosines from the load orientation to the x, y and z axes respectively (also sometimes referred to as 1, m and n). Materials like tungsten always maintain their isotropic properties, even in singlecrystal form and their elasticity is constant in all directions. Some materials, on the other hand, like nickelbase superalloys have varying elastic properties with varying directions. Notch Tip Deformation The stress distribution and analysis of specimens having uniform geometry and subjected to ideal load and temperature conditions is a straightforward task. However, in practical applications of superalloys, this is not the case. A common application of superalloys, for example, is a turbine blade, which has a complex geometry and is subjected to multiaxial, centrifugal and contact stresses at very high temperature gradients. In an attempt to study the state of the stress for more complex specimens, notched tensile specimens are often used to represent either areas of stress concentration or theoretical fracture conditions. Rice (1987) Rice discussed the crack/notch tip stress and strain fields both for FCC and BCC crystal structures. In his work, Rice assumed the plane strain condition and plotted the CRSS in specific angular zones or 'sectors', as he termed these zones. He also stated that the stress state is uniform (independent of the angle) within finite angular sectors near the tip and the stress state jumps discontinuously at boundaries between sectors. Basically, he considered two specific orientations and derived an analytical solution for these orientations and also specified angular boundaries between various sectors. Rice set the convention for referring to notched samples in terms of notch plane, notch growth direction and notch tip direction (as shown in figure 2.2). Load Direction Notch Growth Direction Normal to notch plane Figure 2.2 Notch direction terminology. Since the assumption of plane strain is made, the yielding process can be expressed in terms of a 'yield area'. This helps to define the yield stress in different areas that are separated by boundaries such that the yield stress is uniform throughout a particular area. The slip activity occurs only in some slip systems, which combine to produce a large strain. The state of stress near the crack tip in the plastic zone that Rice has created is based on a plane state of stress in an isotropic material. Definitely, this is not a characteristic of single crystal materials. Therefore, the results obtained using this assumption of plane strain will not correspond well with the experimental results. AP yield 2 T surface 4\(g3rzg "/L~a~2 W Figure 2.3 Yield surface based on plane strain state of stress Another simplification that rice made was the assumption of zero out of plane stress and strain near the notch tip. However, a state of triaxial stress does exist near the notch. This approach therefore, cannot be used to solve for detailed strain field near the notch. The orientations Rice chose for his study produced symmetric results about the notch growth axis. He inferred that there were two conditions under which the notch surface boundary conditions could be met for all angles. The first one was that the stresses in some sectors should be below the yield values. The second condition was that there are discontinuities at some angles. The first condition is difficult to compromise, as Rice had assumed a perfectly plastic state of stress, so all sectors must be at yield or past. The solution Rice obtained had no differences for the FCC and BCC crystal structures or between the two orientations. However, there are differences as regards to the changes in the slip systems with respect to the orientations and there is a switch between the slip systems in the shear and normal directions. Thus orientation does not seem to predict the effect on the yield surface or sector boundaries. This contradicts the experimental results and Rice noticed that this contradiction was because he had neglected the actual rotation of the crystal lattice. Rice has also neglected the effect of strain hardening. However, he recommends incorporating all simplifications in future study of single crystal models. Crone and Shield (2001) In light of Shield's previous work (1996), Crone and Shield conducted further experimental tests to study the notch tip deformation in two different orientations of single crystal copper and copperberyllium tensile specimens, as shown in figure 2.8. Out of these, many other researchers for study of notchtip deformation study have considered Orientation II. Crone and Shield also used Moire interferometry to obtain the strain fields to determine the sectors and the obvious visible changes from one area to another were defined as sector boundaries. Visible slip patterns do indicate slip activity, but the absence of a visible slip does not indicate the absence of any slip activity. The slip systems that do not show up on the surface may be activated internally, rather than at the surface. Even on the surface, the slip systems may show varying patterns as the load and consequently the deformation increases. [010] [101] [101] [010] Orientation I Orientation II Figure 2.4 Orientations used by Crone and Shield in their tests. Table 2.3 Sector boundary angle comparisons for Orientation II Source: Crone and Shield (2001) Sector boundary Experimental Analytical Numerical Crone and Rice Mohan Cuitino and Shield (1987) et al. Ortiz (2001) (1992) (1996) 12 5054 54.7 40 45 23 6568 90 70 60 34 8389 125.3 11 100 45 105110 130 135 56 150 Crone and Shield also compared their experimental results with the solutions obtained by other researchers like Rice' analytical solution as well as numerical Finite Element solutions by Mohan et al. (1992) and Cuitino and Ortiz (1996), as shown in Table 2.4. Both the numerical solutions were obtained by plane strain assumption. However, Cuitino and Ortiz stated that this problem couldn't be really considered a plane strain problem on account of the large variations in strains internally and on the surface. Even with the plane strain assumption, there is no similarity between the numerical and analytical results and moreover any of the three do not match with the experimental results (Figure 2.9 and Figure 2.10). The experimental results obtained by Crone and Shield are justifiable only in the annuluss of validity', where the actual measurements are taken. This annulus is a zone at radial distances from 350 tm to 750 tm from the notch tip. It should be noted here that the notch width is about 100200 [m and the notch radius 50100 am. This makes the annulus and the region where the sectors are measured to be 3.57.0 times the notch width and 7.515.0 times the notch radius from the tip. These sectors clearly allow the stresses and strains to be studied only in the plastic zone. x = 10110 notch i x 350750pm Figure 2.5 Slip sectors observed in the experimental tests by Crone and Shield. Source: Crone and Shield (2001). Crone and Shield chose this annulus also to avoid areas too close to the notch tip. This is because the areas in the vicinity of the notch tip are dominated by geometry. Crone and Shield initially start with a single sector of slip activity and later go on adding the slip lines as the deformation proceeds and so more than one slip lines become evident in the same sector at higher radial distances from the notch tip. The slip lines keep growing in numbers until the sector is 'filled'. It is difficult to state what 'filled' implies here. However, it can be stated for sure that either the initially visible slip system or another with the same visible trace remains active throughout the process. S[2 010] 83890 Sector 4 r Sector 3 105110 e / / 6568o B, B Se or2 cr S or A 50A540 Sector 5 o / F 1500 elastic Sector SSector D / .Dor / ..... eastic Dcor ....j.. elastic castic Sector 6 .... Sector 1 Sector A  or F F \ or B SSector 2 Sector 4 Slip at 550 Slip at 125  Slip at 1800 ***" Figure 2.6 Slip sectors and slip lines from the experimental tests by Crone and Shield. Source: Crone and Shield (2001) Crone and Shield have disregarded the horizontal slip lines observed near the notch in Sector 1 for Orientation II and they have assumed that these slip lines as elastic so that they can compare their solutions to other perfectly plastic crack solutions, as seen in Figure 2.10. Sector 5 is believed to be the only sector showing elastic behavior on the basis of lack of any visible slip activity as well as low strain values. Therefore the determination by Crone and Shield of an elastic region exactly ahead of the notch tip is not consistent with the visible experimental results that they have obtained. As assumed, the sectors observed are actually symmetric about the [101] axis except some variations in the sectors 3/3 and 4/4 (Figure 2.10). This asymmetry may be due to deviation from the ideal conditions such as irregularities in the notch geometry, material defects, inaccurate notch plane direction and so on. Orientation I is a rotation of the Orientation II by 90 about the notch tip direction in a way that the notch tip growth direction and the notch plane are interchanged. It was mentioned previously that Rice's solution did not vary with orientation. However, Crone and Shield's sectors did show prominent variation with orientation, exhibiting changes in specific boundary angles as well as in the number of sectors (Table 2.5). Table 2.4 Sector boundary angles from the experimental tests by Crone and Shield Source: Crone and Shield (2001) Sector Boundary Orientation I Orientation II 12 5054 3540 23 6568 5459 34 8389 111116 45 105110 138 56 150 Although neither the analytical nor the numerical solutions match with the experimental results, Crone and Shield uphold a plane strain assumption on the basis of the numerical solution by Cuitino and Ortiz (1996) because the finite element results on the central plane of the specimen seemed to correspond well with the experimental results. However, Crone and Shield make this clear that the plane strain assumption is valid only for specific locations. In fact, the results from central plane do not depict the actual specimen being modeled and the stresses at the surface are preferred to get more accurate results. Schulson and Xu (1997) Schulson and Xu studied the state of stress near the notchtip for single crystal Ni3Al, which constitutes the y'component of the material used in this study. They considered specimens with two different orientations for this study. They used the three point bending technique to deform the crystals and the deformation was carried out through different zones like elastic, plastic and then through initial stages of crack propagation. [101] [010] [101] Figure 2.7 Specimen Orientation used by Schulson and Xu For both of the samples, a growth direction of [010], a notch plane direction of [ 101] and a notch tip direction of [101] was used. Schulson and Xu calculated the state of stress (assuming isotropic material conditions) using the equations for a sharp notch by Anderson (1995). The purpose behind this was to establish an analytical model to compare with their experimental results. However, isotropic conditions do not hold good for single crystal materials. They first calculated the state of stress on the basis of plane stress conditions. This assumption is valid for large thin plates subjected to inplane loading. Consequently, the thickness of the specimen should be negligible as compared to its height and width and the only stress components present are Cx, Gy and Txy. However, Schulson and Xu observe here that a triaxial state of stress is created around the notch tip and as a result, out of plane material stresses are present. Then they made a second set of calculations based on the plane strain assumption, which again can be well applied for the isotropic conditions (where the out of plane strain is zero) and is not valid for the case of single crystal materials. In an anisotropic single crystal material, plane strain is a function of material orientation. The only condition when the simplified stress tensor matrix is valid is when the specimen is loaded parallel to the FCC lattice edges (Swanson and Arakere, 2000). Schulson and Xu may use this basic transformation if their specimens are oriented accurately. Even then the plane stress is not valid because the true component stresses have out of plane components. The strain components can be obtained by multiplying the stress tensor matrix by the stress as follows: x a a12 a12 0 0 0 ox y a12 all a12 0 0 0 ay s, a12 a12 a 0 0 0 0 O0 0 0 a44 0 0 y (2 .10) Y, 0 0 0 0 a44 0 rz 0Yxy 0 0 0 0 a44 Txy 11 'all +a +a12 + al2 y /a 12 Cx + all OCy + a12 *Oz Ez a= 1t2 x +a12 Oy +all (2.11)Z >=1 ; yz (a44 yz Y zx (144 *a zx Y, a 44Z Z It can be seen that out of plane strain components are present; so the plane strain assumption is not valid.  Figure 2.8 Slip sectors obtained by Schulson and Xu Source: Schulson and Xu, 1997. Schulson and Xu calculated the 12 primary resolved shear stresses on the basis of plane stress and plane strain assumptions. Then using the mode I stress intensity factor K and radial distance r from the tip of the notch, they normalized the resolved shear stresses. They obtained similar slip system results for every individual assumption, however exact systems do differ in some sectors like II, III, IV and V and sector angles show significant variations for certain areas like I/II and IV/V. The sector I slip systems are same for both the assumptions, but the stress values are different. Also there is a considerable difference in the specific slip systems for sectors IIIII, though the systems under the plane strain assumption are just the symmetrical systems from those of the plane stress. There is an exception though in case of sector lib and in sector IV; the maximum stresses are resolved on the same slip system, but the directions do not match. In addition, the plane strain stresses in sector V jump on to { 111) planes as indicted in Table 2.6, while in sector II the plane strain stresses are still on the { 111 planes for the same angles. This data is in conjunction with the elastic region conditions, that is for sectors II and III in the range 43100. However, it diverges significantly later in the sectors I, I/II and IV/V. The experimental results obtained after plastic deformation, vary from the results predicted by either plane stress or plane strain, but are considerably close to plane strain assumption. Usually the slip plane sectors can be observed visually. However some specific slip systems cannot be distinguished at the surface. In such cases, electron microscopes are used to determine the specific slip systems. The results obtained by Schulson and Xu correlate well to the plane stress model for the [011] and [011] dislocations. Table 2.5 Slip sectors from plane stress and plane strain assumptions. Source: Schulson and Xu (1996) Sector Plane Stress Plane strain 0 (deg) Slip systems 0 (deg) Slip systems I 043 (111)[110] 023 (111)[110] (111)0111 (111)0111 II 4360 (111)[110] 2360 (111)[110] (111 )0111 (111)0111 III 60103 (111)[110] 60107 (111)[110] (111)[0111 (111)[0111 IV 103180 (111)[011] 107133 (111)[101] (111)1101 (111)1011 V  133180 (111)[011] Slip Activation and Deformation In case of an isotropic material, all the twelve primary slip systems get activated simultaneously, as is seen from figures 2.12 and 2.13. The Schmid factor is defined using the load orientation, the slip plane orientation and the slip direction, as follows: m = cos Acos 6 Tr = mcr Here a is the load applied, Zrss is the RSS component along a given slip plane and direction, X is the angle between the slip direction and the applied load and finally P is the angle between the normal to the slip plane and the applied load. Slip Plane Normal Slip Direction Figure 2.9 Load and slip direction. According to the Schmid's equation, when the RSS is equal to the yield stress of the material, the CRSS is reached. Also the slip systems with the highest Schmid's factor will reach the CRSS first. This is true for isotropic materials. However, it cannot be 28 correlated to the single crystals and another method needs to be used for the prediction of slip plane activation of single crystal materials. CHAPTER 3 ANALYTICAL APPROACH The overall objective of this work is to find the state of stress in the material coordinate system of the specimen, which eventually is used to calculate the resolved shear stresses in the 12 primary slip systems. For an anisotropic specimen with cubic symmetry, the stressstrain relation is governed by three independent constants in the material coordinate system and a stress tensor matrix that is dependent on the orientation. Thus it is not just governed by a single elasticity constant as in the case of an isotropic material. As the specimen orientation may not be fixed, we need to establish a relationship between the specimen and material coordinate systems, which facilitates the conversion of stress and strain fields from one coordinate system to another. Practically, it is very difficult to cut a sample such that its x, y and z axes are perfectly aligned to the material axes: [100], [010] and [001] respectively. This creates a need for transformation matrix to convert the observed stresses, which are in the specimen coordinate system into the material coordinate system. First of all, to define the elasticity matrix, it is necessary to determine the exact orientation of the specimen under study in terms of the material Miller indices (direction indices). There are two methods to bring about the transformation of the stress components from specimen coordinate system to the material coordinate system. The first method is a straightforward method, in which the angles between the material and specimen coordinate axes are directly measured to obtain the direction cosines. This method can be employed if the angles between the coordinate systems can be easily found. The second method can be used for more complex orientations, where the angles between the two systems are difficult to determine, just by inspection. Here, the transformation equations can be obtained if the Miller indices of the specimen coordinate axes are known with respect to the material coordinate axes. In this technique, the original axes are transformed through a certain number (usually three) of 2D rotations to obtain the final transformed axes. In analytical approach, any of the two methods can be used. However, in experimental specimens, it is difficult to determine the exact Miller indices of the transformed axes and it is preferable to use the first method. As stated earlier, the original coordinate system will be termed as the material coordinate system: xo= [100], yo= [010] and zo= [001] (figure 3.1). The transformed coordinate system is termed as the specimen coordinate system. It will be denoted by x", y" and z". All the properties associated with this system will be denoted by the doubleprime symbol. yyo[010] ^)^ ~ ^ x" zo[001] x" 00/ o[100] z" Figure 3.1 Material and Specimen coordinate system. Coordinate Axes Transformation Using Miller Indices As discussed earlier, it is possible to obtain the final transformed axes if their Miller indices are known. A stepbystep transformation process can accomplish this process. The first transformation can be brought about by rotating the xo, yo and zo axes through an angle P about the zo axis. It is obvious that this is a 2D rotation with the original and final zaxis being the same. The equations of transformation for this first step are: x = x, cos(O) + y, sin(O) (3.1) x = xo, sin() + y, cos(O) (3.2) Sz (3.3) The above equations in matrix form can be written as: x cos(O) sin( ) 0 xO y = sin(O) cos()) 0 y(3 0 zo (3.4) I) yo[010] y  Load Direction zo[001] I xo[100] Figure 3.2 First rotation of the coordinate system about the zaxis 0 yo[010] y ' BB x A A xo100] Figure 3.3 Projections from the original axes on the projected axes The second transformation is again a 2D rotation, but this time the axes are rotated by an angle D about the yaxis. Therefore in this transformation, the yaxis is same before and after the rotation. Proceeding in the same way as for the first transformation, we get the second following transformation matrix for the second step: x' cos(q') 0 sin(o') y'= 0 1 0 y z' sin('>) 0 cos(S') z (3.5) y, y X' ^"""" Figure 3.4 Second rotation of the coordinate system about the zaxis Finally, the axes have to be rotated about the xaxis through an angle This third and final step gives us the complete transformation to the specimen coordinate system , Y and z". The transformation matrix for the third step is: x" 1 0 0 x, y = 0 cos(I") sin(") Yo z" 0 sin(O") cos( )_ (3.6) y ,y z Figure 3.5 Third rotation of the coordinate system about the zaxis The overall transformation from the material to specimen coordinate system can then be calculated multiplying the three transformation matrices obtained in the three transformations. o Y 1 1 io y K f 72 A y0 I" a23 3 3 zo (3.7) where a, w hr 1 0 0 cos(') 0 sin ') cos() sin@) 0 a, 7 2/ =0 cos(O") sin(O") 0 1 0 sin() cos() 0 a3 83 73 0 sin(O") cos(O") sin(O') 0 cos(O') 0 0 1 (3.8) The values obtained through the above equation are the direction cosines of the specimen coordinate system with respect to the material coordinate system. The table below summarizes the above equation: Table3.1 The direction cosines of the specimen coordinate system with respect to the material coordinate system xo Yo zo x" A A Y7 y a2 A 72 z a3 7A '3 The values of the direction cosines or the final transformation matrix will be: a2 A 72 (a3 A/3 73 (3.9) F cos(').cos) cos(') sino) sin() sino") sino') cos) coso") sino) sino"), sino'), sin) +coso"), coso) sino"). coso') cosY"). sino'). cos) +sinno"). sino) cosN"). sino'). sino)sino") .cos5) coso") cos5') In some cases, fewer than three steps are involved to arrive at the final direction cosines. The sequence of rotation is arbitrary. However, it is important to check the accuracy of the results obtained. There are several methods to ensure this. Example Transformation To demonstrate this method of arriving at the direction cosines, we will consider a case in which a load is applied in the [3 1 2] direction. This transformation also illustrates a case in which only two steps are involved to get the overall transformation. Here the first step is eliminated (i.e. 0 = 0) and step two begins by reflecting the load vector onto the xz plane. The reflection is simply a triangle whose sides comprise of the x and z Miller indices: x = 2 and z = 3. The first rotation 0' can be evaluated as: =' tan' 3 '= 56.309 (3.10) Again, the second angle of rotation can be obtained as follows: = tan1 1 Stan 1! 2 +(2)2 "= 15.501 (3.11) Therefore, the direction cosines for the overall transformation can be obtained by using the previously established equation: aL r, 1 0 0 cos(o') 0 ') a,2 /2 = 0 cos(0") sin((") 0 1 0 (3.12) a3 73 73 0 sin(") cos(O") sin(o') 0 cos( ') Substituting the values for 0' and q", we get: La, 1 /, y, 0.5547 0 0.832 a2 2 /2 = 0.2223 0.9636 0.1482 (3.13) a,3 /3 /3 0.8017 0.832 0.5345 Transformation Matrices for Stress and Strain Fields After arriving at the values of the direction cosines between the material and specimen coordinate system, we can establish the matrices to transform the stresses and strains. Using these transformed matrices, we can finally evaluate the resolved stresses and strains in any of the slip systems. 36 The stress transformation, as in Lethniskii is given as: {() = [Q'](. (3.14) {c}= [Q]' 1r "= [Q]. {C" (3.15) [Q] is the stress transformation matrix is a function of the direction cosines: a12 2 3 2a3 2 2a1 a3 2a2 "1 12 22 A32 2.A#32 2.A A#3 2.#2 .1 2 2 2 [Q] 2 3 2" 73 "72 271"73 2. 1 2 1 A /yi A2/y2 Ay/3 (A2/y3+Ay/2) (A1y/3+A83y/) (18y2 +82 y/) a, y2 a2 ,a3 (7 2 a + 73 a) (y, a + 73 a,) (y7 a2 + y2 a,) a 11 a2 182 a,33 (a23 +a3 ,22) (a,1 3+a,3 ) (a,.2 +a2. 1) (3.16) The state of stress is defined in terms of the specimen {(c} or material {ao"} as: 'ff ox ox oy oy {=}< z {= (3.17) Tyz yz Z zx zx T Ziff Same procedure is followed to get the strain transformation matrix: {"}= [Q { (3.18) t_= [10'1. t"}= [Q1 ]. 1"} (3.19) Like the stress transformation matrix, the stress transformation matrix [Q, is also a function of the direction cosines, but it does differ from the stress matrix: 2 2 2 2, a2 03 03 *2 d *"3 2 *"1 A12 A622 A2 AA62 AA A62"1 2 2 2 [ 71 72 73 Y3Y2 71 3 Y2 1 (3.20) 2A1y. 2A2.y2 2P33. ( 3 22 3+A 2) (fl 73 +A3 3.) (fl + 72 ,2 1) 27, q 272 "a2 27'3 "3 (y2 "3 +7'3 2) (1 3 a1 ) (71 R2 + 72 1ai) 2 .) A 2a2 A 2a3 3 ( .3+03 ) (c 3 1) ( 3A) (, +2 1 ) The stressstrain relationship for an isotropic material for a uniaxial state of stress, using Hooke's law can be given as: a=E. (3.21) However, the generalized Hooke's law for a homogeneous anisotropic material can be given as: a = [A,j. (3.22) where, [A] = [aj (3.23) Here [A, and [aj are the elastic constant matrices. It should be noted here that [a, is a symmetric matrix such that: a, = lal (3.24) So we can state: { )= [aJ ] o (3.25) and {"}= [a .{a (3.26) The elasticity matrix also has to be transformed. However, it does not lose its symmetry. The number of constants limit to 21 depending on the direction cosines or specimen orientation. [a ]=Q] [ a ]. [] (3.27) Therefore, we can conclude that if the component stresses are known in the specimen coordinate system, then it is possible to find the stress components in the material coordinate system by using the above equations. Shear Stresses and Strains in the Slip Systems The component stresses are sufficient to define the state of stress of the material. However, in anisotropic materials, these stresses do not give any idea about the slip system activity. It is therefore imperative to transform these component stresses into the resolved shear stresses along the 12 primary slip systems. Thus after arriving at the component stresses, we have to again perform a transformation to determine these resolved shear stresses. Resolved Shear Stress Components The resolution of stresses along the slip systems is calculated as: {r}= c. [S]. {c} (3.28) Here c and [s] are constants depending on slip plane and direction: c = (3.29) h 2 f2 ,2 2 f2 f2 Vh + k + u + v + w S, = h, u, k, .v 1, .w, w, v,: u: (3.30) Here, [u'' v' w'] is the slip direction and (h' k' 1') is the slip plane. For the primary octahedral planes, c is a constant and then the Resolved Shear Stress (RSS) matrix is given by: hi *u1 k1 vI [1 "w1 h, u, k, v, I, *w, h2 U2 k2 *v2 [2 *w2 h3 U3 k3 V3 13 W3 h4 u4 k4 v4 14 "W4 h, u, k, *v, I, *w, h5 u5 k5 v5 15 "w5 h6 u6 k6 V6 16 W6 h7 *u7 k7 V7 /7 6W7 h, u, k, v, I, *w, h8 u8 k8 v8 18 "w8 h, u, k, *v, I, *w, h9 u9 k9 v9 0 9 w9 h1o *u1 k1o *vo 11 *w10 hl u11 ki v,1 1, *w,1 h12 *u12 k12 v12 '12 "w12 Solving the above equation for the 12 primary systems, we get: 1 = . 1 0 1 1 0 1 0 1 1 1 1 0 1 1 0 0 1 1 1 0 1 1 0 1 1 1 0 0 1 1 0 1 1 1 1 0 1 1 0 0 1 1 0 1 1 1 1 0 1 0 1 1 0 1 0 1 1 1 1 0 1 0 1 1 0 1 1 1 0 0 1 1 Similar procedure has to be followed for calculating the shear strains: (3.33) Ox Cy TX Txy Ux y Cz Tyz Tax (3.32) (r)= c [S] 40 The resolved stress and strain fields along the 12 primary octahedral planes are determined following the above steps. This gives a lead for predicting the slip systems for a particular specimen orientation and applied load. CHAPTER 4 NUMERICAL APPROACH: FINITE ELEMENT METHOD The finite element technique can be used to model the specimens having a given geometry and loaded in given orientations. This helps to simulate the conditions in tensile tests. Specimens with different geometry were modeled in two different orientations using the ANSYS finite element software and they were analyzed to predict the stress fields and sectors around the notch of the specimen. One of the two specimens are based on previous work (Rice, 1987; Schulson and Xu, 1997; Crone and Shield, 2001) and both of these correlate to collaborative work between the Mechanical Engineering and Materials Science departments of the University of Florida (UF). In this numerical approach, slipdeformation is predicted by the highest resolved shear stress obtained in the numerical model. The slip systems having the highest resolved shear stress should be the slip lines in the experimental model. These results should match with the experimental results that are obtained in the tensile test. First of all, a solid specimen (without notches) was modeled for a particular load and orientation using ANSYS. These results were then compared to the analytical solution (that was set up in chapter) to verify the accuracy of the results obtained through the finite element analysis. Proceeding on these lines, two notches were introduced in the specimen to simulate the exact tensile test specimen. The component stresses obtained for this model from the finite element solution, which were in the material coordinate system, were incorporated into the analytical solution to get the RSS in the 12 slip systems. This was done for a wide range of radial and angular distance around the notch. This data was used to predict the slip activation for various sectors in the vicinity of the notch. [001] [111] S [010] [121] [100] [303] Specimen I Specimen II Figure 4.1 The notched specimens analyzed using finite element method. Verification of the Finite Element Model Results The specimen without notches can be analyzed using the procedure established in Chapter 3 and these results can be compared with the finite element model to verify the accuracy of the results. This was done using the same dimensions (except the notches) as the actual specimen for consistency. The applied load for all the cases was consistently 100 lbs. The reason for choosing this load was that this load causes stresses well below the yield stress of the material thereby giving the opportunity to study the slip plane activity purely in the elastic zone. For the specimen without notches, the component stresses evaluated from analytical methods seem to match pretty closely to those obtained by the FEM model. So this initial test of the comparing the results between the two methods does work. However, it is better to verify these results on the basis of strain components also. In the analytical technique, the strain components can be evaluated by making use of the stress tensor matrix. The strain components also seem to match consistently for the two techniques. As the results obtained through the analytical and numerical methods agree with each other, it is safe to proceed to obtain results for the specimens with notches. Specimen Orientation and Dimensions Here two specimens with different orientation and geometry will be studied. While modeling the specimen geometry, only the specimen body is considered and the end grips involved in the actual tensile test are neglected. There are certain reasons for neglecting these end grips. First of all, the mechanics at the grips involves tensile rig contact pressure, loading rate and this makes it significantly different than the mechanics from the center of the tensile specimen. In the experimental model, different deformation mechanisms are seen at the grips and these mechanisms can cause fracture even there. But the aim of these tests is to study the deformation mechanisms at the center of the specimen and the stresses generated in this region. Second of all, the grips may be changed in the pursuit of getting more accurate results. Accordingly, the effect and deformation mechanisms at the ends will change. Here, the numerical model will not be subjected to the effect caused by the grips. As far as the geometry of the specimens is concerned, both the specimens have a different geometry. However, in both the specimens, the notch is modeled as a combination of a rectangle and a semicircle. However, in the actual specimen, the notch has some offset with the horizontal and also an offset in the ydirection from the center of the specimen. In addition, the notch tip is not a perfect semicircle, but a smaller arc. In the numerical technique, all the notch dimensions for both right and left notches were modeled equal, although there are slight differences between the dimensions of the two notches. Also, the notch radius was set equal to the notchheight making it a perfect semi 44 circle. In this work, the specimen orientation is of primary importance. This makes the minor changes in the notch dimensions negligible. However, the model can be made perfect geometrically if there is a need for more specific results. Thickness Notch Radius Notch Length Width Notch Width Notch Height Height _W_ Figure 4.2 Specimen specifications Table 4.1 Actual and specimen geometry for Specimen I Specimen Geometry (mm) Actual FEM Width 5.100 5.100 Height 19.000 19.000 Thickness 1.800 1.800 Right Notch Length 1.300 1.550 Left Notch Length 1.550 1.550 Right Notch Height 0.113 0.113 Left Notch Height 0.111 0.113 Right Notch Radius 0.045 0.055 Left Notch Radius 0.055 0.055 Table 4.2 Actual and specimen geometry for Specimen II Specimen Geometry (mm) Actual FEM Width 5.04 5.04 Height 17.594 17.594 Thickness 1.82 1.82 Right Notch Length 1.399 1.399 Left Notch Length 1.36 1.399 Right Notch Height 0.084 0.084 Left Notch Height 0.0845 0.113 Right Notch Radius 0.056 0.056 Left Notch Radius 0.056 0.056 Finite Element Model Characteristics While modeling the specimen in the Finite Element software, all the details of the actual tensile test were incorporated so as to obtain accurate results. Accordingly, the material properties, orientation and boundary conditions were specified to simulate the test conditions. Since the stresses near vicinity of the notch are of prime importance, the areas near the notches were meshed more densely than the rest of the areas. Material Properties To define the stresses or strains in any particular direction, it is convenient to define the material properties with respect to the material coordinate first. However, in the FEM, the specimen is modeled such that its properties are defined with respect to the global coordinate system. Therefore, proper adjustments must be made while defining the direction cosines so as to create the material coordinate system. In case of ANSYS, the material properties are aligned in the same orientation as the element coordinate system. Therefore, the element coordinate system must be defined for the same orientation as the material coordinate system so that the material properties are applied in the right orientation. z x Zo Figure 4.3 The specimen is first defined with respect to the global coordinate system and then the material coordinate system is specified later. The model created in the finite element software is a linearelastic, orthotropic model. ANSYS has a wide range of threedimensional elements to take care of anisotropic and material properties. To model the single crystal material, one of these elements needs to be used. Also, either the three independent stress tensors (a11,a12 a44) or the three independent directional properties (E, G, v) need to be defined to accurately model the single crystal material. Elements and Meshing There are two different elements chosen to mesh the specimen model created in ANSYS: PLANE2 and SOLID95. The front face of the threedimensional solid model is meshed with the PLANE2 elements. The front face has an exact element sizing along the specified radial lines around the notch tip at intervals of 5.The PLANE2 element is a twodimensional, sixnode triangular solid. This element has a quadratic displacement behavior and can be used to good results for modeling irregular meshes. It has six nodes and two degrees of freedom, which helps to incorporate orthotropic material properties. Figure 4.4 Meshing around the notch in ANSYS Sweeping the threedimensional elements through the volume can bring about the meshing of the entire model. It should be noted that the threedimensional elements retain the sizing definitions specified for the twodimensional elements defined on the xy planes (front face) through the thickness of the specimen. The SOLID95 element used for the threedimensional meshing is a 20node element with each three degrees of freedom for each node. This element also works well for irregular shapes and curved boundaries without significant loss of accuracy. This is why SOLID95 is selected as the areas near the notches are of prime interest. This element also takes care of the orthotropic material properties. Here, the original element structure is left intact instead of using the pyramidal shape, which reduces run time. The pyramidalshaped element does reduce the analysis and solution time, but at the cost of accuracy. Using the original element structure also makes sense in the fact that the stress gradient near the notches is very large. In addition, SOLID95 also has features like plasticity, creep, stress stiffening, large deflection, and large strain capabilities. Figure 4.5 Schematic of the SOLID95 element in ANSYS. Source ANSYS 6.1 Elements Reference, 2002. Figure 4.6 Radial arcs used for element location and sizing; centered at notch tip. Through ANSYS the stress tensor can be transformed to any particular orientation to get the stress and strain values. To get a nearfield as well as farfield state of stress in terms of radial and angular distances, six concentric arcs were generated at the following radial distances form the notch tip: 0.25*p, 0.5* p, p, 2* p, 5* p; where p is the radius of the notch. The reason for creating these arcs is that in ANSYS, the stress calculations can be obtained only at the corner nodes of the elements and not at all the nodes. 90 5*p 2*p 0.5*"p 0.25*p Figure 4.7 Radial arcs for stress field calculations Finite Element Solution The nodeselection along the selected concentric arcs around the notch is done manually using the ANSYS Graphical User Interface (GUI). Initially all the nodes starting from 0 to 1800 are selected. However, on account of the symmetry, the calculations of the stresses in the 0 to 1800 need not be done. This assumption was validated by calculating the stresses all over the notch for the p arc, wherein symmetry was found in the stress calculations in the two semicircular regions. Though the resolved shear stresses for the negative angles change direction in conjunction with the changing direction of shear deformation between the top and lower halves of the test specimen, the magnitudes of the maximum resolved shear stresses are same for the positive and negative angles. Another point of interest in the FEM model is regarding the plane of data collection. The element sizing was done in such a way that data collection was possible on five different planes including the front, middle and back planes of the specimen. However, in the collaborative interests with the MSE department, the stress calculations on the front and middle planes were focused on. It should be noted that here we are not assuming a state of plane strain. Assumptions In the finite element analysis, we have considered the model to be a linear elastic model. This assumption is useful to predict the initial slip, but it fails to accurately give us the idea about the forthcoming plastic behavior around the notch. In FEM, the effects due to the temperature can be taken into account. However, in order to be consistent with the MSE department, the material properties at a constant room temperature were considered in the FE analysis. The effects of microstructural mechanisms are also been disregarded in this model. The last assumption is regarding the omission of the effect of crystal lattice orientation. This assumption suits well for these conditions because according to Stouffer and Dame (1996), crystal lattice rotation only takes place if the there is some strain in a single slip system and the load applied to the specimen is low enough to prevent this effect from occurring. This FE model only considers the deformation in the elastic zone. Here, only the elastic stresses and strains are emphasized rather than the values at the point of fracture. 51 This model can be the basis for further study that includes plasticity, creep and other behavior caused by very high stress and strain values. CHAPTER 5 RESULTS AND DISCUSSION As discussed in the previous chapter, the FEM is used to simulate the tensile test conditions so that the slip system activity and subsequently the sectors can be obtained. The results for both the specimens are plotted in terms of the RSS as a function of radial and angular distances. Here, the direction of the slip is not of any concern. As such, only the absolute values of the RSS are considered. The stresses are symmetric about the notch growth direction in case of specimen I. Therefore only the stresses on the upper portion of the notch growth direction are plotted. However, the orientation of specimen II is such that the stress distribution is not symmetric about the notch growth direction and so the stresses on the upper as well as lower portion of the notch growth direction have been plotted. The slip sectors will be plotted as the dominant slip system at any angle theta. The results for both the specimens (with different geometry and load orientation) will be analyzed separately. Specimen I As stated earlier, specimen I was loaded in the [001] direction with a [010] growth direction and a [100] notch plane. The results have been plotted for the RSS on the 12 primary slip systems. The radial distances vary from 0.25*p to 5* p and the angular distances vary from 0 to the top or bottom of the notch (1000 for 0.25* p to 1650 for 5* p). The maximum RSS at any location is 16= T8=31,140 psi at R=0.5* p and 1050 on the midplanes. The slip system with the maximum RSS varies with varying radial and angular distances. On this basis, sectors were determined for the stresses on the surface and the midplanes. The RSS varies as the state of stress changes away from the notch tip. However, the variation of RSS for each slip system is different for different slip systems. So the relative magnitudes of RSS on different slip systems may vary with respect to each other. Therefore, it should be noted for a given angle, the dominant slip system is not constant with changing radius. The plots between the RSS and theta highlight the effect of theta only on the RSS values. However, the effect of radius on the RSS can be understood by combining the effects of all the radii. Figure (5.6) and (5.12) give a picture of the combined effect of the radial and angular distances on the RSS. For example, consider an angle of 700. The radar plots (Figure 5.7 and 5.8) also give an idea of the combined effect of stresses at various radial and angular distances. However, the radar plots are difficult to comprehend. So they are plotted only for a single set of data. Moreover, the sector plots give a much simple picture of the combined effect of theta and varying radii on the RSS. RSS Vs. Theta for 0.25*rho 0 0 10 20 30 40 50 Theta (deg) 70 80 90 Figure 5.1 RSS on the 12 primary slip systems for the upper portion of the notch growth direction for R=0.25*p 25000 20000 15000 10000 5000  T1 W T2  T3 T4 WT5  T6 T7 T8 T9  T10  T11  T12 RSS Vs. Theta for 0.5*rho 30000 25000 20000 a. 15000 10000 5000 0 0 20 40 60 80 100 Theta (deg) Figure 5.2 RSS on the 12 primary slip systems for the upper portion of the notch growth direction for R=0.5*p  T1 W T2  T3 T4 ST5  T6 1T7 T8 T9  T10  T11 T12 RSS Vs. Theta for 1*rho 18000  16000 14000   T1 T2 12000 T3 T4 10000 . T5 a. 40 T6 8000 z  T8 T9 6000 T1 4000 T12 2000 0 0 20 40 60 80 100 120 Theta (deg) Figure 5.3 RSS on the 12 primary slip systems for the upper portion of the notch growth direction for R=l*p RSS Vs. Theta for 2*rho 12000 10000 ST1 8000 T3 T4 6000 ,,T7 T8 T9 2000 0 0 20 40 60 80 100 120 140 160 Theta (deg) Figure 5.4 RSS on the 12 primary slip systems for the upper portion of the notch growth direction for R=2*p RSS Vs. Theta for 5*rho 8000 7000 6000 5000 4000 3000 2000 1000 0 0 20 40 60 80 100 120 140 160 Theta (deg) Figure 5.5 RSS on the 12 primary slip systems for the upper portion of the notch growth direction for R=5*p  T1 W T2  T3 T4 ST5 T6 1T7 T8 T9  T10  T11 T12 *T8 * T6 ST10 * T2 * T3 * T4 * T9 100 110 160 170 7Q^ ' 180 Figure 5.6 Slip system sectors on the surface for specimen I 10 J0 5 Figure 5.7 Upper portion of the radar plot showing the surface stresses for the [100] orientation. Figure 5.8 Lower portion of the radar plot showing the surface stresses for the [100] orientation. Table 5.1 Specimen I dominant slip system sectors on the surface R=0.25* p R=0.5* p R=1.0* p Sector 0 Tmax Slip system 0 Tmax Slip system 0 Tmax Slip system I 041 s8 (111)[011] 056 s8 (111)[011] 052 Tio (111)[011] II 41100 T6 (111)[011] 56105 z6 (111)[011] 5256 z2 (111)[011] III 56116 T6 (111)[011] IV 116120 T4 (111)[101] V VI R=2.0* p R=5.0* p Sector 0 Tmax Slip system 0 Tmax Slip system I 050 zio (111)[011] 046 Tio (111)[011] II 5060 z2 (111)[011] 4656 r2 (111)[011] III 6068 z3 (111)[110] 5664 "3 (111)[110] IV 68150 T6 (111)[011] 64131 T6 (111)[011] V 131164 "9 (111)[011] VI 164165 T6 (111)[110] RSS Vs. Theta for 0.25*rho 20 30 40 50 Theta (deg) 90 100 Figure 5.9 RSS on the 12 primary slip systems for the upper portion of the notch growth direction for R=0.25*p 24000 20000 16000 12000 8000 4000  T1 U T2 T3 T4 T5 T6 T8 T9  T10  T11 T12 RSS V. Theta for 0.5*rho 35000 30000  tl 25000 W t2 t3 t4 S20000 t5 V 0 t6 t7 i 15000 t8 10000  t11 t12 5000 5000  "......_ 0 0 20 40 60 80 100 RSS (psi) Figure 5.10 RSS on the 12 primary slip systems for the upper portion of the notch growth direction for R=0.5*p RSS V Theta for 1*rho 20000 18000 16000 1 t2 14000 t t3 t4 12000 t5 0 r  t6 1000 0 t9 6000 t ti1 4000 I t12 2000 0 0 20 40 60 80 100 120 Theta (deg) Figure 5.11 RSS on the 12 primary slip systems for the upper portion of the notch growth direction for R=l *p RSS V. Theta for 2*rho 10000 9000  8000  t1 7000 _____t2 t3 6000 " i"d t4 M E t5 \ 0 t6 5000 *t6 3 000 t12 4 0 20 40 60 80 10 120 140 160 RSS (psi) 3000 f 7A 1000 0 20 40 60 80 100 120 140 160 RSS (psi) Figure 5.12 RSS on the 12 primary slip systems for the upper portion of the notch growth direction for R=2*p RSS V. Theta for 5*rho 0 20 40 60 80 100 120 140 160 RSS (psi) Figure 5.13 RSS on the 12 primary slip systems for the upper portion of the notch growth direction for R=5*p  t1 W t2  t3 t4 W t5 0 t6 + t7 t8 t9  t10  tll  t12 * T3,T5,T7,T12 * T6,T8 T2,T10 T4,T9 110 70 120 60 130 "',50 140 "'\ 40 150 \ ,30 160 \.200 180 0 .25 .5 1 2 5 Figure 5.14 Slip system sectors on the midplanes for specimen I Table 5.2 Specimen I dominant slip system sectors on the on the midplanes R=0.25* p R=0.5* p R=1.0* p Sector 0 Tmax Slip system 0 Tmax Slip system 0 Tmax Slip system (111)[110], (111)[110], 038 T2, 1io (111)[011], (111)[110], (111)[110], (111)[011] I 044 T3, z5, 17, z12 (111)[110], 0105 13, 15, 17, z12 (111)[110], (111)[110] (111)[110] (111)[011], 38112 T6, 18 (111)[011], II 4495 T6, 8 (111)[011] (111)[011] (111)[110], 112120 T4, z9 (111)[101] (111)[110], (111)[101] III 95100 T3, T5, T7, z12 (111)[110], (111)[110] R=2.0* p R=5.0* p Sector 0 Tmax Slip system 0 Tmax Slip system 053 12, 1io (111)[011], 058 12, Tio (111)[011], (111)[011] (111)[011] 53120 16, 18 (111)[011], 58130 16, 18 (111)[011], II (111)[011] (111)[011] 120150 14, 19 (111)[101] 130157 4, 19 (111)[101] (111)[101] (111)[101] 157163 16, 18 (111)[011], IV (111)[011] V 163165 12, T0o (111)[011], V (111)[011] Specimen II As stated earlier, specimen I was loaded in the [111] direction with a [1 21] growth direction and a [3 03] notch plane. The load axis in this case is such that the RSS values will not be symmetrical above and below the notch. Therefore results for both the upper and bottom parts of the notch have been plotted. The results have been plotted for the RSS on the 12 primary slip systems. The radial distances vary from 0.25*p to 5* p and the angular distances vary from 0 to the top or bottom of the notch (1000 for 0.25* p to 1650 for 5* p). The maximum RSS at any location is T6=36,090 psi at R=0.5* p and 1050 on the midplanes. The slip system with the maximum RSS varies with varying radial and angular distances. On this basis, sectors were determined for the surface stresses. The RSS varies as the sate of stress changes away from the notch tip. However, the variation of RSS for each slip system is different for different slip systems. So the relative magnitudes of RSS on different slip systems may vary with respect to each other. Therefore, it should be noted that for a given angle, the dominant slip system is not constant with changing radius. Again, the sector plots in Figure (5.20), (5.26), (5.32) and (5.38) give a picture of the combined effect of theta and radii on the RSS values. RSS V. Theta for 0.25*rho 20000  T1 16000 T2  T3 T4 * T5 12000 T6 T T8 8000 T9 T10 000 T12 4000 0 10 20 30 40 50 60 70 80 90 100 Theta (deg) Figure 5.15 RSS on the 12 primary slip systems for the upper portion of the notch growth direction for R=0.25*p Error! RSS V. Theta for 0.5*rho 30000 25000 ST1 BT2 U T2 / a T3 20000 ST4 T5 aL 0 T6 15000 ST8 T9 10000 T  T11 5000 0 0 10 20 30 40 50 60 70 80 90 100 Theta (deg) slip systems for the upper portion of the notch growth direction for R=0.5*p Figure 5.16 RSS on the 12 primary RSS V. Theta for 1*rho 18000 14000 T2 WT2 S/ I\IT3 10000 T4 1000 T5 a. 4* T6  OiT7 l I T8 6000 T9 T10 frT12 2000 2 20 40 60 80 100 10 2000 Theta (deg) Figure 5.17 RSS on the 12 primary slip systems for the upper portion of the notch growth direction for R= p RSS V. Theta for 2*rho 12000 10000 8000 6000 4000 2000 0 0 20 40 60 80 100 120 140 Theta (deg) Figure 5.18 RSS on the 12 primary slip systems for the upper portion of the notch growth direction for R=2*p ST1  T2  T3 T4 ET5 e T6  T7 T8 T9  T10  T11  T12 RSS V. theta for 5*rho 8000 7000 6000  T1 T2 5000 T3 T4 S/  T5 00, M T7 4000 2 4 T T8 3000 T9 T10 Ti 1  T12 2000 1000 / M OO 0 20 40 60 80 100 120 140 160 Theta (deg) Figure 5.19 RSS on the 12 primary slip systems for the upper portion of the notch growth direction for R=5*p OT10 * T6 T5 * T9 *T11 ST3 TT2 T12 1 1 100 90 80 60 170 ' 80 Figure 5.20 Slip system sectors on the surface on the upper portion of the notch growth direction for specimen II Table 5.3 Specimen II dominant slip system sectors on the surface on the upper portion of the notch growth direction R=0.25* p R=0.5* p R=1.0* p Sector 0 Tmax Slip system 0 Tmax Slip system 0 Tmax Slip system I 013 Tio (111)[011] 050 Tio (111)[011] 070 Tio (111)[011] II 1327 T6 (111)[011] 50105 T5 (111)[110] 70110 T5 (111)[110] III 27100 z5 (111)[110] 110118 T3 (111)[110] IV 118120 z9 (111)[101] V VI R=2.0* p R=5.0* p Sector 0 Tmax Slip system 0 Tmax Slip system I 063 Tio (111)[011] 018 T9 (111)[101] II 6367 Tir (111)[101] 1858 Tio (111)[011] III 67119 z3 (111)[110] 58102 z5 (111)[110] IV 119127 r6 (111)[011] 102107 r6 (111)[011] V 127137 T2 (111)[011] 107113 "5 (111)[110] Vi 137138 z5 (111)[110] 113117 z3 (111)[110] VII 138150 z3 (111)[110] 117167 z2 (111)[011] VIII 167170 z12 (111)[110] RSS V. Theta for 0.25*rho T1  T2 A T3 T4  T5 T6 T7 T8 T9  T10 T11  T12 90 100 Figure 5.21 RSS on the 12 primary slip systems for the lower portion of the notch growth direction for R=0.25*p 20000 16000 12000 8000 4000 0 0 10 20 30 40 50 60 70 80 Theta (deg) RSS V. Theta for 0.5*rho 24000 20000 16000 12000 8000 4000 0 20 40 60 80 100 Theta (deg) Figure 5.22 RSS on the 12 primary slip systems for the lower portion of the notch growth direction for R=0.5*p 4T1  T2 T3 T4  T5  T6 IT7 T8 T9  T10 T11  T12 RSS V. Theta for 1*rho 16000 T1 12000 T4 0 T6 8000  T8 ST9 T9 T1O 4000 , T1 1 o0 T12 0 20 40 60 80 100 120 Theta (deg) Figure 5.23 RSS on the 12 primary slip systems for the lower portion of the notch growth direction for R= p RSS V. Theta for 2*rho 12000 10000  T1 Sm T2 A T3 8000 S T5 C .T6 6000 W I ^ "I/ / T8 00 4000 T9 T10 A . T1 1 2000 T12 0 t 0 20 40 60 80 100 120 140 Theta (deg) Figure 5.24 RSS on the 12 primary slip systems for the lower portion of the notch growth direction for R=2*p RSS V. Theta for 5*rho 8000 * T1 6000  T3 T4 A T5  T6 4000  T7 TO ( T8 00 T9 T10 .T11 2000 T11 / T1 2 0 20 40 60 80 100 120 140 160 Theta (deg) Figure 5.25 RSS on the 12 primary slip systems for the lower portion of the notch growth direction for R=5*p *T7 *T10 T3 ST6 *T11 T2 * T5 * T9 *T12 1: 160 170 180 100 110  90 go Figure 5.26 Slip system sectors on the surface on the lower portion of the notch growth direction for specimen II Table 5.4 Specimen II dominant slip system sectors on the surface on the lower portion of the notch growth direction R=0.25* p R=0.5* p R=1.0* p Sector 0 Tmax Slip system 0 Tmax Slip system 0 Tmax Slip system I 0100 T7 (111)[110] 01 Ti (111)[011] 01 zlo (111)[011] II 1105 T7 (111)[110] 1111 z7 (111)[110] III 111117 z3 (111)[110] IV 117120 "6 (111)[011] V VI R=2.0* p R=5.0* p Sector 0 Tmax Slip system 0 Tmax Slip system I 063 zTo (111)[011] 018 z9 (111)[101] II 6367 zTi (111)[101] 1858 zio (111)[011] III 67119 T3 (111)[110] 58102 T5 (111)[110] IV 119127 T6 (111)[011] 102107 T6 (111)[011] V 127137 T2 (111)[011] 107113 T5 (111)[110] VI 137138 T5 (111)[110] 113117 T3 (111)[110] VII 138150 T3 (111)[110] 117167 T2 (111)[011] VIII 167170 T12 (111)[110] RSS V. Theta for 0.25*rho 28000 24000 20000 T2 , T3 T4 16000 W T5 8.' 0T6 CO AAIiT7 12000 T8 00 oo S* T9 0 20 40 60 80 100 Theta (deg) Figure 5.27 RSS on the 12 primary slip systems on the midplanes for the upper portion of the notch growth direction for R=0.25*p RSS V. Theta for 0.5*rho 40000 36000 32000 4 T1 +WT2 28000 , T3 T4 24000 W T5 S0T6 S20000 T7 T9  TI 16000 oo T11 12000 WT12 A1 A T1 2 8000 4000 . . WO * 0 20 40 60 80 100 RSS (psi) Figure 5.28 RSS on the 12 primary slip systems on the midplanes for the upper portion of the notch growth direction for R=0.5*p RSS V. Theta for 1* rho 20000 16000 S12000 a. 8000 4000 0  T1 * T3  T2 .a T3 T4 t T5  T6 IT7 T8 T9  T10  T11 A T12 20 40 60 80 100 120 Theta (deg) Figure 5.29 RSS on the 12 primary slip systems on the midplanes for the upper portion of the notch growth direction for R=p RSS V. Theta for 2*rho 12000 10000 ' ' T1 l T2 . T3 8000 T4 i T5 . T6 ._ \ \ ~T9 4000  T1 *.uT12 2000 T 0 ,' O O0 q q & . 0 20 40 60 80 100 120 140 Theta (deg) Figure 5.30 RSS on the 12 primary slip systems on the midplanes for the upper portion of the notch growth direction for R=2*p 