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FINITE ELEMENT ANALYSIS OF SLIP SYSTEMS IN SINGLE CRYSTAL SUPERALLOY NOTCHED SPECIMENS By SHADAB A. SIDDIQUI 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 Copyright 2003 by Shadab A. Siddiqui Dedicated to my parents, because of whom I am here, and to my fiancee Rukshana who guided my steps in the most difficult moments and gave me her support for completing this task. ACKNOWLEDGMENTS I would like to thank Dr. Nagaraj Arakere, my research advisor, for giving me the opportunity to work in his group and for his valuable assistance and advice during the different stages in the development and successful completion of this research. I would also like to thank my supervisory committee members for their contributions to the completion of this project. I would also like to thank Srikant, Rukshana, Jeff and my lab mates for their help and support. Finally, I thank God for helping me complete this task. TABLE OF CONTENTS page A C K N O W L E D G M E N T S ................................................................................................. iv TA B LE O F C O N TEN T S.................................................................. .......................... v LIST O F TA B LE S ......... ............... ......... ..................... .. .............. .. vii LIST OF FIGURES ......................... ........ .... ........ ............. viii ABSTRACT .............. ..................... .......... .............. xii CHAPTERS 1 SINGLE CRYSTAL SUPPERALLOY ................. ....................... S u p erallo y s ................... ................... ...................1....... M icrostructure of Superalloy s............................................................................ ...... 2 SingleCrystalNickel Base Superalloys................... ...... ... .................. 3 NickelBase Superalloys Evolution: A General Literature Review........................ 3 MicroStructural Properties of Nibased Single Crystals. .................................... 5 M manufacturing of Single Crystal................................ ......... ................ 6 D eform ation M echanism s ...................... .... ............ ..................... .............. 7 M otivation for the Thesis.................... .......................................... ......................... 10 Need for the Study .. ................. ............................ .. .......... .. 12 2 PLASTIC DEFORMATION AROUND A NOTCH TIP ...........................................13 Slip M mechanism ............... ..... .............................. 13 Work of Prominent Researchers in the Field..................... .... ............... 15 Rice (1987)........................ ..... ........ 15 Shield and K im (1993).................... ....................................... ......................... 17 Shield (1995)..................... ...... ........ 21 C rone and Shield (2001) ............................................... ............................ 22 3 3D STRESS ANALYSIS OF SINGLE CRYSTAL NOTCHED SPECIMENS U S IN G F E M ...................................... ................................. ................ 2 7 In tro d u ctio n ........................................ ...... ..... ....... ...... .......... ................. 2 7 Close Form Solution for a Uniaxially Loaded, Smooth, Single Crystal Specimen...... 27 Coordinate Transformation for Orthotropic Material ........................................ 28 Coordinate Axes Transformation.................... ...... .......................... 29 Exam ple Problem .......... .... ......... .... ...................... 34 Transformation of Stress and Strain Tensors................................... .... .......... 36 Calculation of Shear Stresses and Strains on Crystallographic Slip Systems. ......... 40 Finite Element Solution .......................................................... .............. 41 The Finite Elem ent M odel ....................................................................... 42 Criteria of Testing the Numerical Model .............. ..................................... 44 In itia l C h e ck ............................................................................. 4 4 F in al C h eck ............................................................................... 4 4 M odel Features and Characteristics .......... ............ .............. ..... ............ 45 Spatial Characteristics .......................... ... ............... .......... ...... .. 45 Sim plified Geom etry of the N otch....................................................... ........... ... 46 M material P properties ..................................... ................................ ................. 47 A ssum ption U sed in M odeling ........................................ .......... .............. 51 Experim ental M ethod.................... ................................................ ......................... 52 4 RESULTS AND DISCU SSION .............................. .......................... ............... 54 Specim en A ...................... .... ... ...... ............... .......... 54 Surface R results for Specim en A ........................................ ......... .............. 55 Mid Plane Results for Specimen A ................................................. 67 Specim en B ................................. .......................... ........... 76 Surface Results on Upper Half of Specimen B ............. .................................. 76 Surface Results on Lower Half of Specimen B .............................................. 84 Mid Planes Results on Upper Half of Specimen B............... .................... 92 Mid Planes Results on Lower Half of Specimen B .......................................... 100 C om prison of Specim en R results. .............................................................................. 108 E xperim mental R results ..................................................... ................................. 109 Conclusions ................. ................................. ......... 112 Recommendations For Future W ork....................................................... ....... 113 APPENDIX COORDINATE AXES TRANSFORMATION AND ACCURACY CHECKS.............14 R E F E R E N C E S ..................................................................................... .......... .. .. 1 16 B IO G R A PH IC A L SK E T C H ................................................................ .....................118 LIST OF TABLES Table p 11 Slip planes and slip directions in an FCC crystal. ................................................ 10 21 Orientation II sector boundary angle comparisons................................ ............... 21 22 Comparisons of experimental sector boundary angle with numerical and analytical solutions of orientation II. ........................................ ............... 24 23 Experimental sector boundary angles for copper samples. ......................................24 31 D direction cosines ...................... ...................... .. .. .......................34 32 Comparison of analytical and numerical component stresses for specimen A. (Material coordinate system) ............ ........ .......................... 44 33 Comparison of analytical and numerical component strains for specimen A. (Material coordinate system) ............ ........ .......................... 45 34 Actual (Specimen A) and finite element specimen geometry. ..................................47 41 Specimen A dominant slip systems (Surface plane). .................................................66 42 Specimen A dominant slip systems (M id plane)........................................... ........... 75 43 Specimen B dominant slip system. (Upper half on surface). ....................................83 44 Specimen B dominant slip system (Lower half on surface)...........................91 45 Specimen B dominant slip system sectors (Upper half on mid plane)....................99 46 Specimen B dominant slip system sectors (Lower half on mid plane) ..................107 47 Comparison of numerical and experimental results of specimen A........................111 LIST OF FIGURES Figure p 11 Temperature capability of superalloys with approximate year of introduction. .....2 12 Comparative mechanical properties and surface stability of polycryatalline, directionally solidified and single crystal superalloys...........................................5 13 Microstructure showing y matrix and y' precipitate................. ............... .............6 14 Single crystal blade production with: a) Crystal selector and b) Seed crystal. ............7 15 Straight up slip lines in copper ...................................................................... 9 16 Representative jet engine component distress mode statistics. ................................11 17 Representative distributions of HCF problems by components. ............................. 11 21 Calculation of resolved shear stress parallel to slip direction, from tensile force F...14 22 Orientations of specimens used by Rice. (1987)..................................................... 16 23 Rice's perfectly plastic analytical solution for orientation 2............................... 17 24 Orientation of specimens used by Shield (1993) ...................................................18 25 Specimen loaded in fourpoint bending by Crone and Shield.................................19 26 The E22 strain components near a notch in an iron silicon single crystal ...............20 27 Orientations of specimens used by Crone and Shield. (2001)...............................23 28 Experimental slip sectors from Crone and Shield .............................................. 25 31 Material (xi, yi, zi) and specimen (x"', y"', z"') coordinate system...........................30 32 First rotation of the material coordinate system by angle +Y1 about the z, axis.......31 33 Second rotation by angle +Y2 about the y' axis ............. ........................................ 32 34 Third rotation by angle +Y3 about the x" axis. ................................ ..................33 35 Figure showing the load direction. ........................................ ........................ 34 36 Figure showing total transformation.................... ...... ........................... 35 37 Orientations of specimen A and specimen B................................... .................43 39 Specimen and material coordinate system s .................................... .................48 310 PLANE2 2D 6NODE triangular structural solid .................................................48 311 Figure showing the meshing near the notch tip in specimen A .............. ...............50 312 SO LID 95 3D 20node structural solid ........................................ .....................50 313 Element through the thickness................................ ......... ................. 51 314 Radial arcs around the notch tip. ........................................ ......................... 52 41 Figure showing mid plane and surface plane on which RSS is calculated for specim en A and B ........................ .............................. .. ......... ................55 42 Twelve primary resolved shear stresses on the surface of specimen A; r = 0.25*p...57 43 Twelve primary resolved shear stresses on the surface of specimen A; r = 0.5*p.....58 44 Twelve primary resolved shear stresses on the surface of specimen A; r = 1*p........59 45 Twelve primary resolved shear stresses on the surface of specimen A; r = 2.0*p.....60 46 Twelve primary resolved shear stresses on the surface of specimen A; r = 5.0*p.....61 47 Maximum resolved shear stress on each radius occurring on the surface of specimen A. RSS scaled for each radius to plot as one ......................................62 48 Active slip sectors on the surface of specimen A from 0.25*p to 5*p.......................63 49 Complete RSS field on the surface of Specimen A ............... ..... ............... 65 410 Twelve primary resolved shear stresses on the mid plane of specimen A; r = 0 .2 5 p ........................................................................... 6 8 411 Twelve primary resolved shear stresses on the mid plane of specimen A; r = 0 .5 p ............................................................................. 6 9 412 Twelve primary resolved shear stresses on the mid plane of specimen A; r = 1 .0 p ............................................................................ 7 0 413 Twelve primary resolved shear stresses on the mid plane of specimen A; r = 2 .0 p ............................................................................ 7 1 414 Twelve primary resolved shear stresses on the mid plane of specimen A; r = 5 .0 p ....................................................................... 7 2 415 Maximum resolved shear stress on each radius occurring on the mid plane of sp e c im e n A .............................................................................................................7 3 416 Active slip sectors on the mid plane of specimen A from 0.25*p to 5*p. ...............74 418 Twelve primary resolved shear stresses on the upper half of the surface of specim en B ; r = 0.25* p ........................ ............ .. .................. ........ .... 77 419 Twelve primary resolved shear stresses on the upper half of the surface of specim en B ; r = 0.5* p ............................................ .................. ........ .... 78 420 Twelve primary resolved shear stresses on the upper half of the surface of specim en B; r = p ................................. .. ... ... .... ............ 79 421 Twelve primary resolved shear stresses on the upper half of the surface of specim en B; r = 2* p................................. .. ... ... .... ............ 80 422 Twelve primary resolved shear stresses on the upper half of the surface of specim en B; r = 5* p................................. .. ... ... .... ......... .. 81 423 Maximum resolved shear stress on each radius occurring on the upper half of the surface of specim en B. ...... ........................... ......................................82 424 Twelve primary resolved shear stresses on the lower half of the surface of specim en B ; r = 0.25* p ............................ ................ ................ ............ 85 425 Twelve primary resolved shear stresses on the lower half of the surface of specim en B ; r = 0.5* p ...................................... ................. ......... 86 426 Twelve primary resolved shear stresses on the lower half of the surface of specim en B; r = p ................................. .. ... ... .... ............ 87 427 Twelve primary resolved shear stresses on the lower half of the surface of specim en B; r = 2* p................................. .. ... ... .... ............ 88 428 Twelve primary resolved shear stresses on the lower half of the surface of specim en B; r = 5* p................................. .. ... ... .... ............ 89 429 Maximum resolved shear stress on each radius occurring on the lower half of the surface of specim en B. ...... ........................... ......................................90 430 Twelve primary resolved shear stresses on the upper half of the mid plane of specim en B ; r = 0.25* p ............................ ................ ................ ............ 93 431 Twelve primary resolved shear stresses on the upper half of the mid plane of specim en B ; r = 0.5* p .......................................... .................... ........ .... 94 432 Twelve primary resolved shear stresses on the upper half of the mid plane of specim en B; r = p ................................. .. ... ... .... ............ 95 433 Twelve primary resolved shear stresses on the upper half of the mid plane of specim en B; r = 2* p................................. .. ... ... .... ............ 96 434 Twelve primary resolved shear stresses on the upper half of the mid plane of specim en B; r = 5* p................................. .. ... ... .... ............ 97 435 Maximum resolved shear stress on each radius occurring on the upper half of the m id plane of specim en B .......................................... ............................ 98 436 Twelve primary resolved shear stresses on the lower half of the mid plane of specim en B ; r = 0.25* p ...................................... .................... ........ .... 101 437 Twelve primary resolved shear stresses on the lower half of the mid plane of specim en B ; r = 0.5* p .......................................... .................. ........ .... 102 438 Twelve primary resolved shear stresses on the lower half of the mid plane of specim en B ; r = p ................................................. .............. ... .......... 03 439 Twelve primary resolved shear stresses on the lower half of the mid plane of specim en B ; r = 2* p ................................................. .............. ............. 04 440 Twelve primary resolved shear stresses on the lower half of the mid plane of specim en B ; r = 5* p ................................................. .............. ... .......... 05 441 Maximum resolved shear stress on each radius occurring on the lower half of the m id plane of specim en B ....................................... ........................... 106 442 RSS for specimen A when experimental load is applied. The dashed line indicates the yield stress of the material. Any RSS curves above this line represent slip systems that are activated. ........... ..................... ....................110 443 Slip planes in experimental tensile test specimen tested by material science departm ent..................................................... ....... .. .. ................ 111 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 FINITE ELEMENT ANALYSIS OF SLIP SYSTEMS IN SINGLE CRYSTAL SUPERALLOY NOTCHED SPECIMENS By Shadab A. Siddiqui May 2003 Chairman: Nagaraj Arakere. Major Department: Mechanical And Aerospace Engineering Fatigue failure of turbine engine components is a pervasive problem. Among the most demanding structural applications for high temperature materials are those of aircraft engines and power generating industrial gas turbines. In particular, the turbine blades and vanes used in these applications are probably the most demanding, due to the combination of high operating temperature, corrosive environment, high monotonic and cyclic stresses, long expected component lifetimes and the enormous consequences of structural failure. However, the need to maximize efficiency results in the need to minimize component weight and forces design margins to be as small as possible. To develop a mechanistically based life prediction system, an understanding of the evolution of slip systems in regions of stress concentration, under the action of 3D fatigue stresses, is necessary. A study of slip systems in 3D anisotropic (single crystal) stress fields is presented as a function of crystal orientation. Threedimensional anisotropic stress fields are examined by analyzing the stress field in a single crystal doubleedged notched rectangular specimen with varying crystal orientation. Slip fields activated near the notch during tensile loading are observed experimentally (Material Science And Engineering Department, University of Florida). Detection of the specific slip systems is possible through the study of the visible traces left on the surface of the specimens. Threedimensional FEA of the specimen is used to predict the slip systems. Analysis results are verified by comparing them with experimentally generated slip fields. The load and the crystallographic orientations govern the activation of specific slip systems. Slip systems are examined for two different crystal orientations at the surface and on the mid planes. For both orientations, the resulting slip sectors are different. Maximum resolved shear stresses for both the orientations are found to be on their mid planes. The past research, which used 2D isotropic plane stress or plane strain model, predicts sectors with straight boundaries. The present 3D anisotropic FEA analysis with accurate representation of specimen geometry and load predicts curved slip sectors boundaries with complex shapes. Overall, the slip systems predicted by 3D FEM show very good agreement with experimentally measured slip systems on the surface. CHAPTER 1 SINGLE CRYSTAL SUPPERALLOY Superalloys Superalloys are a group of nickel, ironnickel, and cobalt base alloys that are used at temperature above about 5400C. Initially superalloys were developed for use in aircraft piston engine turbo superchargers. Their development over the last 60 years is because of their usage in advancing gas turbine engine technology. Superalloys exhibit a combination of high strength at high temperatures, excellent creep and stress rupture resistance, toughness and metallurgical stability; useful thermal expansion characteristics and strong resistance to thermal fatigue and corrosion. The high temperature strength of all superalloys depends on the principle of a stable, face centered cubic (FCC) matrix, combined with either precipitation strengthening and/or solid solution hardening. In general, superalloys have an austenitic (y phase) matrix and contain a wide variety of secondary phase. The most common secondary phases are y' and metal carbides. Nickel base superalloys are the most widely used alloy for the hottest parts. The high phase stability of the FCC nickel matrix and the capability to be strengthened by a variety of direct and indirect means are the principal characteristics of nickel superalloys. The introduction of directional solidification and single crystal casting technology are the additional aspect of nickel base superalloys (Davis, 1997). These alloys exhibit better high temperature properties than polycrystalline wrought or cast alloys (Figure 11). 1200 Mechanical Single Directional lloying crytals structure \ eMA CMSX10 CMSX4 110 0 +T a + Ht D S e ci DS euacrtics Cast alloys +/W and Nb IN591 9 CMSX.2 MA MARM22 J MARM002 DS I MARM200 R.M 200 0 / M21 M246 MARM200 + HIOS ,1 N  I6202 lo IN1 *TRWVIA ..+Re 1 t 0 1 90 B 90 R 0 I l IN S203 B190 aR80 IN 792 IN 6201 Smelting"^, 713CI N R77 ,r ,N939 + Udimet 700 N15 8 N 935 ) \ N,105 MARM509 Udimet720 \Niol +Co N100 105 . \ DWaslwy152 N1)1 X40 N90 ..  sB16 ... / X750 N8A ought N alloys cast o v Ibase  +AI /LC "se 800 HS21 *Na1 r fo Nibase / 8N1 Wugh 0 Cobase N80_ +Ti Cast A DS and SC Haslelloy B P/M A 00DS Nibase 700 I 1940 1950 1960 1970 1980 1990, Year Introduced Figure 11 Temperature capability of superalloys with approximate year of introduction. (Davis, 1997) Microstructure of Superalloys The major phases that are present in superalloys are, gamma matrix (y), gamma prime (y'), gamma double prime (y"), grain boundary, carbides and borides. y matrix, in which the continuous matrix is an FCC nickel base nonmagnetic phase, usually contains a highpercentage of solid solution elements. y is present as the matrix, in all nickel base alloys. y' is present when aluminum and titanium are added in adequate amounts required to precipitates within the austenitic gamma matrix. The nature of the y' precipitate is of primary importance in obtaining optimum high temperature properties (Davis, 1997). SingleCrystalNickel Base Superalloys Nickel base single crystal superalloys are precipitation strengthened cast monograin superalloys based on the NICrAl system. They have attracted considerable attention for use in rocket and gas turbine engines because of their high temperature properties. In high temperature application grain boundaries are typically the weak link, which provide passages for diffusion and oxidation, which results in failures at this location. Grain boundary strengtheners are added to the alloy chemistry to increase capability, which results in lowering the melting point of the alloy. Because of this entire single crystal components are produced from one large grain. Removal of grain boundaries and grain boundary strengthening elements raise the incipient melting temperature of the alloy by 1500F and results in improved high temperature fatigue and creep capabilities (Stouffer and Dame, 1996). This increase in melt temperature permits higher heat treatment temperature that in turn yields improved creep capability. These single crystal superalloys are orthotropic and have highly directional material properties. The <001> direction is the most common primary growth direction for nickel base superalloys (Davis, 1997). NickelBase Superalloys Evolution: A General Literature Review Nickel Base superalloys have evolved over a period of 82 years. The first landmark in their evolution was laid with the development of Nimonic 80A (NiCr alloy) a polycrystal / wrought superalloy (Dreshfield, 1986). To an extent, variations in chemical compositions through increasing element additions and introducing refractory metals improve the elevated temperature mechanical properties and surface stability of nickel base superalloys. During the 1940's an innovative process called the investment casting process was adopted from dental prosthesis technology. This technology enabled the production of high precision components with complicated shapes. However, due to impurities problems, there were not many improvements in the mechanical properties and creep resistance, in the initial components processed by investment casting. Falih N. Darmara solved this problem in the 1950's with the invention of a new melting process called vacuum induction melting(VIM). This process, considered by many to be one of the most important advances in the evolution of superalloys, allowed for the development of alloys with increased quantities of reactive elements. Reactive elements such as aluminum and titanium, participate in the precipitation of coherent intermetallic precipitate gammaprime phase Ni3 (Al,Ti) (Fermin, 1999). Between the 1960's and 1970's, a further development in superalloy processing was introduced in order to increase the efficiency of the turbine performance by increasing the operating temperature and rotational speeds and reducing clearance between static and rotational components. This development was through the introduction of the directional solidification process developed by Frank Versnyder and others at Pratt and Whitney. This process produced significant improvements in the rupture life and thermal fatigue resistance ofNibase superalloys. After about 10 years of research and investigation, the single crystal solidification process was developed, which was the result of minor variation in the directional solidification process. This minor variation in the directional solidification process yielded a significant increase in the thermal capability of nickelbase superalloys due to increased mechanical properties and thermal stability (Fermin, 1999). Figure 12 illustrates a comparison between polycrystalline, single crystal and columnarcrystal superalloys. 4 i Polycrystal U Columnarcrystal 30 13 single crystal 2 1 Creep Thermal Corrosion strength fatigue resistance resistance Figure 12 Comparative mechanical properties and surface stability of polycryatalline, directionally solidified and single crystal superalloys (Fermin, 1999) MicroStructural Properties of Nibased Single Crystals. The description of the macrostructure of Nibased single crystals primarily consists of primary and secondary dendrites. Primary dendrites are parallel, continuous and span the casting without interruption in the direction of solidification. The secondary dendrite arms on the other hand are perpendicular to the direction of solidification and define the interdendritic spacing .The <001>family of direction is the one along which the solidification for both primary and secondary dendrite arms proceed. The principal hardening mechanism in single crystal nickelbase superalloys is precipitation of gammaprime, y' (Deluca and Annis, 1995). The gamma prime precipitate is a face centered cubic (FCC) structure, composed of the intermetallic compound Ni3Al .The y' precipitate is suspended within the y matrix, which is also of FCC structure and comprised of nickel with cobalt, chromium, tungsten and tantalum in solution (Figure 13). y' Precipitate Figure 13 Microstructure showing y matrix and y' precipitate (Moroso, 1999). Manufacturing of Single Crystal Single crystals are manufactured by using techniques similar to those of directionally solidified castings with one important difference; that is by selecting a single grain with desired orientation. The first method for growing the single crystal uses a helical mold. In this method, a helical section of mold is placed between a chill plate and the part casting. A single grain is selected by the helix or spiral grain selector, which acts as a filter. This is because superalloys solidify by dendritic growth and each dendrite can grow only in three mutually orthogonal<00 >directions. Due to the combination of an orthogonal nature of dendritic growth and continually changing direction of the helix, a single crystal is emitted from the top of the helix. The second method is called seeding, which is capable of controlling both primary and secondary orientation. The seed crystal should be made of an alloy, which has equivalent or a higher melting temperature than molten alloy. The seed is placed on a chill plate where the temperature at the top of the seed is tightly controlled. This is done so that the seed crystal does not melt completely thereby allowing the molten alloy in the mold cavity to solidify with the same orientation as the seed (Davis, 1997). Both the methods are shown in Figure 14. Blade Blade cavity (a) 77T(b) / Shell mouli Crystal Selector Seed crystal Starter block Chill Chill Figure 14 Single crystal blade production with: a) Crystal selector and b) Seed crystal. (Meetham, Voorde, 2000). Deformation Mechanisms Slip, climb and twinning are three main factors responsible for inelastic deformation in metals. The main reason for deformation of crystalline metals is the propagation of dislocations through the metal's lattice when temperatures are less than 0.5 of the absolute melting temperature. At higher temperatures, deformation occurs by dislocation climb (which is a diffusion controlled process). Twinning, a rotation of atoms in the lattice structure, is not as important as strains are very small as compared to slip and climb. Plastic flow may result in an ideal crystal when one plane of atoms slides over another by the simultaneous breaking of all the metallic bonds between the atoms. However the actual yield stress is much lower than the theoretical shear stress, due to the presence of dislocation in the lattice structure. Dislocation is defined as a disruption in the crystal lattice structure of material (Stouffer And Dame, 1996). Slip deformation is a stresscontrolled process. The geometry of the crystal structure, magnitude of the shearing stress produced by external loads, and the orientation of the active slip planes with respect to the shearing stresses decides the extant of slip in single crystals. Slip begins when the shearing stress on the slip plane in the slip direction reaches a threshold value called the critical resolved shear stress (Dieter, 1986). The value of critical resolved shear stress depends chiefly on the material composition and temperature. It is also the function of applied load and direction, crystal structure and specimen geometry. During the application of a load to an FCC single crystal specimen, the first planes to get activated are the planes of high atomic densities and called the primary octahedral slip systems. A slip line or step is observed in experiments involving polished single crystals specimens (Figurel5). Table 11 shows the 30 possible slip systems in an FCC crystal. Figure 15 Straight up slip lines in copper (Dieter, 1986) Slip will occur when the resolved shear stress (RSS) exceeds the yield strength of the material. However, in single crystal alloys the yield strength depends on the material orientation relative to the applied uniaxial load. Single crystal materials also have tensioncompression asymmetry of the yield stress that is a function of orientation. A Sample loaded in the [001] direction, shows higher yield strength in tension than compression. In the [011] case, it is just the opposite.For the [111] orientation, the tension  compression asymmetry and orientation effects become negligible. When temperature increases above 7000C to 7500C, there is a sharp drop in the yield stress, and tension  compression asymmetry and orientation effects disappear. Test specimens near the [011] orientation generally show the lowest tensile strength and greatest ductility while specimens near the [111] orientation generally have the high tensile strength (Stouffer And Dame, 1996) Table 11 Slip planes and slip directions in an FCC crystal.(Stouffer and Dame, 1996) Slip Number Slip Plane Slip Direction Octahedral Slip a/2{111}<110> Primary Slip Directions 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{1 1 1}<112> Secondary Slip Directions 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 Slip a/2{100}<110> Cube Slip Directions 25 (100) [011] 26 (100) [011] 27 (010) [101] 28 (010) [101] 29 (001) [110] 30 (001) [110] Motivation for the Thesis In modern military gas turbine engines, various causes of component failure are low cycle fatigue, corrosion, overstress, manufacturing processes, mechanical damage and the types of materials used. However the single largest cause of component failure is attributed to high cycle fatigue (HCF) (Cowles, 1996) (Figure 16). 13% 24% HCF SLCF 8% El Overstress 1%o I ::":::] Mechanical Damage 7% Manufacturing Process Materials 12% D Thermal 11%" Corrosion 12% All Other Fatigue 12% Figure 16 Representative jet engine component distress mode statistics.(Cowles, 1996). The problem of high cycle fatigue is a pervasive one, affecting all turbine engine parts made from a wide range of materials (Figure 17). The components most likely to fail by HCF are turbine and compressor blades. 20% 33% 1 Blades E Vanes 3%a Disk and Spacers 6 Sheet Metal Airseals S. Cases and Housings S , .Others 13% Figure 17 Representative distributions of HCF problems by components.(Cowles, 1996). Need for the Study Potential cause of HCF in gas turbines is presence of vibratory stresses. Turbine blades are particularly susceptible to damage by vibratory high cycle fatigue due to the large range in frequency responses excited by rotation. Such vibratory stresses are further superimposed by cyclic and steady stresses induced by thermal and mechanical loads. Due to these loads, cracks fatigue can nucleate in regions of high stress concentrations, and eventually propagate and cause blade failure. To develop a mechanistically based life prediction systems, an understanding of evolution of slip systems in regions of stress concentration, under the action of 3D fatigue stresses, is necessary. The planes, on which crystallographic fatigue cracks initiate in 3D stress fields and their crack propagation rates, are essential for life prediction. At a first step towards realizing this goal, the understanding of evolution of slip systems in a single crystal specimen, under the action of 3D static stress is important. A doubleedged notched rectangular tension test single crystal is used to simulate 3D stress fields. Slip fields activated near the notch during tensile loading are observed experimentally (Material Science And Engineering Department, University of Florida). Threedimensional FEA of the specimen is used to predict the slip systems. Analysis results are verified by comparing with experimentally generated slip fields. CHAPTER 2 PLASTIC DEFORMATION AROUND A NOTCH TIP This chapter contains a general review of the literature published in the area of stresses and sectors around a notch tip. The information will provide the necessary a background to understand the slip mechanism and plastic deformation near a notch tip. Investigation of the plastic deformation at a crack tip in a single crystal is important to the development of the understanding of single crystal failure. Plastic fields, with sectors of deformation and crystallographically dependent radial boundaries are produced by the plastic deformation around a crack tip in a single crystal material. In 1987, Rice had predicted plastic deformation around a crack tip in metallic single crystals. These results were confirmed experimentally (Shield and Kim, 1994; Shield, 1996) and investigated numerically (Rice et al., 1990; Mohan et al., 1992). Currently, no analytical or numerical work exists which provides insight into the slip system behavior of ductile anisotropic single crystals in the presence of a crack or notch or which can completely predict the behavior observed in experiments. Slip Mechanism In a single crystal specimen, the extent of slip depends on the magnitude of the shearing stress produced by external loads, the crystal structure geometry and the orientation of the active slip planes with respect to the shearing stresses. Therefore the stress strain behavior of a material, which is a function of the number of activated slip systems, varies with orientation (Figure 21). SApplied force F=oA Normal n Slip Plane Figure 21 Calculation of resolved shear stress parallel to slip direction, from tensile force F.( Dieter,1986). When a load is applied slip systems get activated based on Schmid factor, m. Where m = cos A x cos 0 (21) Where 4 is the angle between the tensile axis and the normal to the slip plane and , is the angle between the slip direction and the tensile axis. From eq.21 its clear that the Schmid factor is also a function of the load orientation and slip plane orientation. It has been observed experimentally that a single crystal will slip when the resolved shear stress on the slip plane reaches the critical resolved shear stress for that material. TRss = m x c Crosssectional area A Sli p Direction (22) Where oy is the load applied and CRSS is the resolved shear stress. This behavior is known as Schmid's law. However, agreement to Schmid's law has been proven only with isotropic materials, and its correlation to single crystals is not yet known. Therefore, another method must be used to predict RSS values and slip activation for these anisotropic materials. Work of Prominent Researchers in the Field Rice (1987) Rice analyzed crack tip stress and deformation fields for ideally plastic tensile loaded crystals, by examining the mechanics of both FCC and BCC notched specimens. Rice presented the analysis for plane strain tensile cracks. He also used critical resolved shear stress criteria to predict sectors. He paid attention to two specific crack orientations in FCC and BCC crystals, although the analysis techniques are applicable to other orientations too. One orientation defined the notch plane as (010), the notch growth direction as (101) and the notch tip direction as (101). The second orientation defined the notch plane as (101); the notch growth direction as (010) and the notch tip direction as (101). [101] [010] [010] [101] [101] [101] Orientation 1 Orientation 2 Figure 22 Orientations of specimens used by rice. (1987) Rice analyzed both orientations and derived analytical solutions to predict the active slip systems and to determine sectors around the notch. Rice considered only the positive half of the plane for both of the cases since the solution is symmetric about the notch growth axis. He found a continuous solution with respect to the radius and angular displacement. The main drawbacks in rice solutions are 1) He did not include anisotropy in his model, so his solution cannot validate the experimental results, 2) His solutions cannot determine detailed strain field data based on the state of stress near the tip, 3) His solutions show no difference between either orientation's sector boundaries or between FCC and BCC crystal structure. Both orientations predict boundaries at 550, 990 and 1250. 125.3 sector 3 125.3 Sector 4 x [0101 Sector 2  s7o Slip at 550 Slip at 1250 Slip at 180" 0 . Figure 23 Rice's perfectly plastic analytical solution for orientation 2. (Crone and Shield, 2001). The families of slip plane traces in the FCC and BCC case are identically oriented relative to one another, except that the slip plane traces for the BCC families are rotated by 900 relative to the traces for the FCC families. For example ifN=X and S=Y describe a particular FCC family, then N=Y and S= X describe a corresponding BCC family, where N is the unit normal to a slip plane and S is a unit vector. Thus orientation had no effect on the sector boundaries. Rice also neglects strain hardening, because of which there is no effect of plastic deformation on yield locus. Shield and Kim (1993) Shield and Kim followed the work of Rice to correlate their experimental solution with Rice's analytical solution. Results are presented for determining the plastic deformation fields near a crack tip (200jtm wide notch) in an iron 3% silicon single crystal (Specimen FE11). The notch in Shield and Kim's specimen was in a (011) plane with prospective crack growth in a [100] direction. [011] [100] [011]/ Figure 24 Orientation of specimens used by shield (1993). Since the solution is symmetric about the [100] axis, only the upper half plane was considered. The specimen was loaded in fourpoint bending with measurements were made at zero loads after extensive plastic deformation has occurred (Figure 25). Rice presented an asymptotic analysis of the plane strain stress field at a crack tip in a perfectly plastic crystal. Subsequently, Saeedvafa and Rice (1998) extended this analysis to include Taylor power law hardening and presented an asymptotic solution of the plane strain crack tip stress field which we will refer to as an HRR type solution. Figure 25 Specimen loaded in fourpoint bending by crone and shield. (Crone and Shield, 2001). Shield et al. predicts slip sectors, similar to Rice, based on plastic strain field data. They assumed that the total strain was equal to plastic strain and neglected the elastic strain. The specimen they considered had dimension of 7.45 mm x 6.00mm x 26.05 mm. The bar was extended to a length of 51.95 mm by welding 12.95 mm long polycrystalline bars of the same crosssection to each end. They introduced a single edge notch at the center of the crystal to a depth of 2.05mm and a width of 200[tm. To verify that the surface strains reflect the behavior of the material in the interior of the specimen, the specimen was sectioned and etched. Shield and Kim present strains as a function of angle, since the strains do not vary much with radial distances from the notch tip. The angle was measured from the crack propagation direction and was taken as positive in the counterclockwise direction. From the experiment a pattern of four (eight symmetric) sectors were found. This pattern is shown in Figure (26). 800.0 Q E .2 E 750,0 1' 1250. U m m U m m U n m U * Im * * 0.000 micrometers Figure 26 The E22 strain components near a notch in an iron silicon single crystal. (Shield, 1995). The figure displays the E22 strain component. Sectors 1 and 2 have constant strains, although section 2 had some small variations. The third sector has the largest strain values and they vary with radius in an approximately 1/r manner. The fourth sector has roughly constant strain, though the strain levels are too low to make an absolute .1000 .0900 .0800 .0700 .0600 .0500 .0400 .0300 .0200 .0100 0.0 statement. An excellent agreement between the interior dislocation pattern and surface strains was found. Thus the surface measurements accurately reflect the deformations that are occurring in the interior of the specimen and a comparison with the plane strain result of Rice is justified. Shield (1995) Following his work on ironsilicon single crystal Shield extended his work to copper single crystals. He chose the study of { 110} as a crack plane with the prospective crack growth direction in the [100] direction. This is the same orientation as the iron silicon specimen (FE11). Because, iron silicon has a BCC structure and copper has a FCC structure, the slip systems in the two materials are different. However, since the orientations are the same, it will be possible for a direct comparison to be made on the basis of orientation. Also, the effect of the different slip systems on the strain fields can be assessed. Shield compared this work with his previous work and concluded that the discrete sectors observed in FE11 are also present in this specimen. The sector boundary angles are similar to, but not exactly the same as those observed in FE11, which has same orientation but different slip systems (Table21). Table 21 Orientation II sector boundary angle comparisons. __FE11 (bcc) C1B (fcc) Rice (1987) Sector boundary angles (degrees) 12 boundary 35 43 55 23 boundary 65 62 90 34 boundary 110 100 125 (Modified from Crone and Shield, 2001) The greatest difference in sector boundary angles occurs in the 12 sector boundary. The angle of maximum strains (in sector 3) is almost identical in both specimens, suggesting that this angle may be more related to the notch tip geometry than the crystal structure. Shield also observed that load had no effect on sector boundary angles. However, as the load increased, the amount of plasticity near the notch tips also increases. Shield also observed that results obtained for low loads show similarities to Rice's model but Shield's experimental results do not correlate to Rice's model at high plastic strain (Table 21). The boundary angles between the two samples were similar but not constant. This disagreement can be due to the material structure alone or due to flaws that may be present in the material structure, regardless of a constant specimen orientation and test condition. This can also be due to the geometry of the notch, which is very difficult to duplicate accurately. Shield observed slip lines which are caused by plastic deformation at large strains. He then compared results with strain sectors determined by a Moire interferometer. He found that the sector boundary angles determined by Moire's interferometer matched well with the strain field images. The contradictory results of Shield's experiment and Rice's results provoke the need to replace the existing model, which can provide more accurate solutions. Crone and Shield (2001) Crone and Shield extend the work of Shield (1995) by experimentally studying notch tip deformation in two different orientations. Moire microscopy was used to measure the strain field on the surface of the bending samples. Two crystallographic orientations were considered in this research. Orientation 1 is defined as the orientations containing a crack or notch on the (101) plane and its tip along the [101] direction. This orientation was investigated experimentally by Shield (1995) and Shield and Kim (1994). Orientation 2 is defined as the orientation containing a crack or notch on the (010) plane with a tip along the [101] direction. Experimental results for these orientations were found. The plane of observation was the same for these two orientations with the crack or notch being rotated by 900. Both orientations were analytically investigated by Rice (1987). [101] [010] [010] [101] [101] / \[101] Orientation 1 Orientation 2 Figure 27 Orientations of specimens used by crone and shield. (2001) Crone and Shield compared their experimental results with Rice's analytical solution as well as with a numerical solution by Mohan et al. (1992) and Cuitino and Ortiz (1996) (Table 22). Rice's analytical solution was applied to both orientations 1 and 2 where the plane of observation was the same with the notch being rotated by 900. The numerical solutions are based on plane strain assumptions even though Cuitino and Ortiz concluded that the problem under consideration is not a plane strain because of differences between the interior and surface fields. Even with the plane strain assumptions Rice's analytical solutions do not match with Crone and Shield's experimental results. Table 22 Comparisons of experimental sector boundary angle with numerical and analytical solutions of orientation II. Sector boundary Experimental Analytical Numerical In degrees Crone and Shield Rice Mohan, et al. Cuitino and Ortiz (2001) (1987) (1992) (1996) 12 5054 54.7 40 45 23 6568 90 70 60 34 8389 125.3 112 100 45 105110 130 135 56 150 _ (Modified from Crone and Shield, 2001) Orientations 1 and 2 are related by a 900 rotation about the x3 axis. This means that orientation 1 represents orientations 2 rotated by 900 about the notch tip direction, such that the notch growth direction and notch plane directions are switched. The experimentally determined sector boundary angles for both orientations are compared in Table 23. Table 23 Experimental sector boundary angles for copper samples. Sector boundary Orientation I Orientation II Boundary angles (101) Plane (010) Plane in degrees 12 3540 5054 23 5459 6568 34 111116 8389 45 138 105110 56 150 (Modified from Crone and Shield, 2001). Contrary to the equivalent sectors predicted by Rice, there are several clear differences between orientations 1 and 2 predicted by Crone and Shield. Here, the experimental results are fairly unclear due to the annuluss of validity" where Crone and Shield take their measurements. The annulus chosen following Shield and Kim (1994) corresponds to the area from 350750[tm from the notch tip. This annulus was chosen to avoid inclusion of material close to the notch. Behaviour of material close to the notch is dominated by the notch geometry while behaviour of material in the far field may be affected by specimen boundaries.The annulus was also chosen to place the sectors well out of the range of any plastic deformation and can be used where only elastic deformation is taking place. X2 [101] notch nuus o = Validity I 350750flm Figure 28 Experimental slip sectors from crone and shield. (Modified from Crone and Shield, 2001). Here, the research presented has further confirmed that the structure of the deformation field near a notch in a metallic single crystal is linked to crystallographic orientation. Although Rice (1987) captures the main features of the deformation 26 experimentally observed near a notch in FCC copper and copperberyllium single crystal, a more complex 3 Dimensional anisotropic analytical solution is required to account for the sector boundary angles and the elastic sectors noted in these experiments. Research is currently underway to develop an analytical solution that more closely correlates with the experimental findings. CHAPTER 3 3D STRESS ANALYSIS OF SINGLE CRYSTAL NOTCHED SPECIMENS USING FEM Introduction In the field of linear elastic fracture mechanics, various test methods have been developed in order to study the elastic response of isotropicnotched specimens under the action of tensile load. The following are the test methods developed for the above study: Analytical Approach, Numerical Approach and Experimental Approach. The methods developed for isotropic specimens pose many difficulties when applied to three dimensional anisotropic specimen models. For example, In case of the isotropic analytical models, the current solution relies on many simplifications/approximations that lead to inaccurate results when compared with the experimental results. However these limitations in the elastic models are overcome by the use of the threedimensional FEA approach, which enables solutions that correlate well with actual experimental results. Moreover, unlike the analytical solutions, both the numerical and experimental model specimen have the capability of using notched specimens which act as very simplified cracks to model fracture behavior. The study of the elastic response of anisotropic specimens is also useful to find the multiaxial loading strength of the specimen. Close Form Solution for a Uniaxially Loaded, Smooth, Single Crystal Specimen. Anisotropic materials play an important role in many phases of modem technology. They are used widely in areas like material sciences, solidstate physics, missile and aircraft manufacturing and many others. Thus this sophisticated technology requires the study of the properties of anisotropic materials, particularly the elastic properties of these materials in various directions. Unlike in the past, where the materials were consider to be homogenous and isotropic in order to simplify calculations, it is unfeasible today to make such oversimplified assumptions as these will lead to inadequate and incorrect results. The test specimen chosen for the study is a notched single crystal super alloy. The objective of the study is to find the state of stress in the material coordinate system of the specimen and consequently calculate the resolved shear stresses in the 12 primary slip systems of critical locations. In an isotropic material a single elastic constant governs the transformation from stress to strain and properties of materials do not vary with directions. But this is not the case with anisotropic material, in this the elastic constant of a crystal vary markedly with orientations. The stress strain relationship for an anisotropic solid with cubic symmetry has three independent constant (elastic modulus, shear modulus and Poisson ratio) in the material coordinate system, as well as stress tensor matrix instead of single elasticity constant that varies with orientation (Lekhnitskii, 1963). Coordinate Transformation for Orthotropic Material The definition of the elasticity matrix requires the determination of the precise orientation of the actual specimen. This may be done either in terms of the material miller indices or angular measurements. A coordinate transformation is essential in the case of physical material test specimen, due to the difficulty encountered while cutting the sample such that the x, y and z test axes be perfectly aligned along the material axes: [100], [010] and [001] respectively. Such a coordinate transformation will translate the known specimen stresses in terms of the material coordinate systems. Outlined below are the transformation procedure derived from Lekhnitskii and Stouffer and Dame. There are two approaches to the transformation from the specimen to the material coordinate system. First approach is the direct measurement of the angles between the original and the transformed coordinate systems to find the directions cosines. This approach is suitable if the angles are easily found. Second approach is to find the miller in dieses of the transformed axes, which are rotated through a series of steps to arrive at the final transformed destination. This method is based on rigid body rotations and is more suitable for complex orientations, where the angles between the two coordinate systems are difficult to find. Although neither method is preferred over the other, the first method (angle measurement) turns out to be more convenient in the case of experimental specimens. Coordinate Axes Transformation Knowing the orientation of the sample, one can perform the coordinate transformation and the transformation matrices can then be used to determine the stresses and strains resolved on any given plane and slip system. Here the material coordinate system is denoted by xl, yi, and zi and the specimen coordinate system is denoted by x'", y"', and z"'. The original coordinate system is the material coordinate system and the transformed coordinate system is defined as the specimen coordinate system, and is at some angular displacement from the original axes. yi [010] /" x"' zi [001] x [100] z"' Figure 31 Material (xi, yi, zi) and specimen (x"', y"', z"') coordinate system. By breaking the total transformation into several rigid rotations, transformations from material coordinate system to the specimen coordinate system is done. The first transformation, to the x', y' and z' axes, is performed by rotating by yi about the ziaxis (Transformation from xl toward yi is defined as positive). y, [010] X1 Load Direction Zi [001] x1 [100] Figure 32. First rotation of the material coordinate system by angle +x1 about the zl axis. Here the x', y' and z' represents the transformed coordinates, in terms of the original coordinates after first rotation by angle +1~. x'= xicos(vi) + y1.sin(vi) (31) y'= xlsin(vi) + YI.cos(i) (32) z'= z1 (33) Writing the transformation in matrix form is: (34) The second transformation, to the x", y" and z" axes, is done by rotating by W2 about the y' axis (Transformation from z' toward x' is defined as positive). y' y" x" Load Direction x' Figure 33. Second rotation by angle +Y2 about the y' axis. Now the second transformation in matrix form is: x"L 'cos(=2) 0 sin(V2)" 'x') y" = 0 1 0 I y'/ z" ) sin(v2) 0 cos(W2) z') (35) X') COS(TIV) Si4TI) 0) 'XI1 Y I osi4TI) COS(TI1) 0 1 \I ZI The final transformation to the specimen coordinate system (x"', y"' and z"' axes) is done by rotating the y" and z" axes by angle W3 about the x"axis (Transformation from y" toward z" is defined as positive). Load Direction Figure 34. Third rotation by angle +Y3 about the x" axis. The third transformation in matrix form can be written as: 1 0 0 x y'" = 0 cos ( 3) sin (v3) l y" (36) z'", 0 sinm(I3) cos (W3) z" By multiplying the three transformations (Individual step matrices) the total transformation can be calculated. (The first transformation becomes the last one multiplied): Y U 2 P2 Y2 I. Y1 i"'111 U3 P3 Y3) Y Z (37) Where il1 i 71 1 0 0 cos( 2) 0 sin(22)" rcos(li) sin( lI) 0" L22 P2 2= cos( V3) sin(v3) 0 1 0 sin(\1) cos(T1) 0 (38) u3 3 Y3 0 sin(3) cos(\3)) sin(T2) 0 cos( 2) ) 0 0 1) The following results represent the direction cosines between the material and the specimen coordinate system axes. Table 31. Direction cosines X1 yi zi x' ao pi Yi y'" 2 P2 Y2 '" a3 3 Y3 Several checks based on perpendicularity can be performed to make sure that a proper orthogonal coordinate transformation has been done (Appendix). Example Problem Consider a specimen loaded in the [314] direction (Figure 36) Y1 [010] z [0] section [314] [l] Zi [001] xl [100] Figure 35. Figure showing the load direction. The coordinate transformation is reduced to two rigid body rotations T2 and T3 since j1 = O.In second step load vector is reflected onto the xlzi plane. The reflection shows a triangle whose sides are the u and w Miller indices: u= 3, w = 4. The first angular translation, W2, is: P2 atan 1) w)2 2= 36.87' (39) yi [010] zi [001] xl [100] Load Direction [314] Figure 36. Figure showing total transformation. In the same way, the second angle, p3, forms a triangle with the hypotenuse, h, of the first reflection and the ytranslation: h =u + 2 v = 1. Therefore, the second angular translation is: 13 := atanl ) F3 + w11 3 = 11.31 (310) Therefore the directions cosines will be lcq P1 Y71 '1 0 0 cos(V2) 0 si(2) co's(Wv1) sin(W1) 0" u2 32 Y2 :=0 cos(w3) sin(W3) 0 1 0 sin(w1) cos(v1) 0 (311) P3 33 Y3) s in(v3) cos(Wv3)) sin(W2) 0 cos(W2) 0 0 1) 1 P1 i 71 0.8 0 0.6 a2 P2 Y2 1= 0.118 0.981 0.157 a3 P3 Y3) 0.588 0.196 0.784) A proper orthogonal transformation can be confirmed from Appendix. Transformation of Stress and Strain Tensors After finding the direction cosines between the material and specimen coordinate systems transformation of stress and strain tensors between the material and specimen co ordinate systems can be done by applying the proper load conditions. Later on resolved shear stresses and strains can be found from these transformed matrices on the crystallographic planes. Following Lekhnitskii (1963) the stress transformation is: {o'} = [Q'o] {to (312) {y = [Q',]lY') = [Q,] {o'} (313) Here [Q,] is the stress transformation matrix 2P32 2Y3Y2 (P2Y3 + P3Y2) (Y23 + 73"2) (U23 + a3P2) 2P1 3 271 Y3 (P11Y3 + 3Y1) (71.3 +Y3 1) (U1P3 +a3P1) 2P2P1 2Y2Y1 (P1Y2 + P2Y1) (71*2 + Y2*1) (U1.P2 + 2'P1) The state of stress is defined in terms of the specimen {a} or material {o'} stresses by: {C} = Gx 7 CTy CTZ Tyz Tzx Ixy {C'} = G'x 7 1C'y Gz Z'yz Z'zx t xy) (315) The strain transformation is carried out by following the same approach: {'} = [Q'J] {8} {} =[Q' {'}= IQ] {'1 2 O"3 P32 2 Y3 P3.Y3 Y3'a3 "3P3 2 2 al "2 P12 P22 2 2 71 Y2 P1Y71 P2Y2 71Ya1 Y2'a2 a1.P a2232 (316) (317) 2u2lq Here [Q,] is the strain transformation matrix 2 2 2 al ~2 "3 a3c2 UIPU3 i12 22 32 2 3P12 13133 [QE] = 2 2 2 71 Y2 Y3 Y3'Y2 Y1'Y3 213171 2P32Y2 2P3.Y3 (132Y3 + 33Y2) (11Y3 + P371) 271Y1 2Y2.2 273.3 (72."3 + 73."2) (r7"3 +Y3.1l) 2all31 2a2P2 2013.3 (2P13+"3P32) (U133+"331) 132"131 Y2'Y1 (11Y2 + 32Y1) (Y71*2 + Y2*1) (U1P2+ U2'P1) Isotropic material's stress and strain for a uniaxial state of stress According to Hooke's law is given by: c = Es (319) According to Hooke's law, stress and strain relationship for a homogeneous anisotropic body is given by: {} = [A1] {s} (320) [A ] = [a,]' (321) Where constant aj is the coefficient of deformation and constant Ai is the moduli of elasticity, which are the function of orientation and [aij] is a symmetric matrix such that: Therefore [ai] = [aji]. {} = [a] {o} And {'} = [a',] {o'} (318) (322) (323) (324) a 12 a22 a23 a24 a25 a26 a13 a23 a33 a34 a35 a36 where [a ij] = a14 a24 a34 a44 a45 a46 (325) (325) a15 a25 a35 a45 a55 a56 a16 a26 a36 a46 a55 a66 ) The elastic properties of FCC crystals exhibit cubic symmetry, also described as cubic syngony. The majority of pure metalsiron, copper, nickel, silver, gold, and others form crystals of cubic syngony. Materials with cubic syngony have only three independent elastic constants designated as the elastic modulus, shear modulus, and poisson ratio. Here the elastic constants are defined as: 1 1 Vyx Vxy all 14 a44 = al2 Exx Gyz Exx Eyy all a 12 a 12 0 0 0 a 12 a 11 a 12 0 0 0 a 12 a 12 a 1 0 0 0 Therefore [a ij ] = 0 0 0 a 44 0 0 (326) 0 0 0 0 a44 0 0 0 0 0 0 a44 ) [a'] = [Q]T[aJ][Q] (327) By using the component stresses in the specimen coordinate system, the above equations can be applied for working out the component stresses in the material coordinate system. Calculation of Shear Stresses and Strains on Crystallographic Slip Systems. Now we have all the component stresses, but since they don't give the clear picture about the individual slip systems, we have to calculate RSS on 12 primary slip systems. Here both the slip plane and slip direction define the primary slip systems. Below outlined are the calculation procedure inspired from Stouffer and Dame. {} = c[S]{a} (328) Where ci (329) S2 2 2 /2 2 2 (329) h'i + k'i + l'i u + vi + w'i Si= (h'i.u'i k'iv'i l'i.w'i w'i v'i u'i) (330) Here, [u' v' w'] is the slip direction and (h' k' 1') is the slip plane (Recall Figure 21). Constant c is constant for all the 12 primary systems. Combining all the result in one matrix we have: 11 r h'lu'1 k'1v'1 1,1.w'1 w'1 v'1 u'1 r2 h'2u'2 k'2.v'2 12 w'2 w'2 v'2 u'2 13 h'3u'3 k'3v'3 1'3. w'3 w'3 '3 u'3 14 h'4.u'4 k'4.v'4 1'4. w4 w'4 '4 u'4 Ox 15 h'5.u'5 k'5.v'5 1'5 w'5 w'5 V'5 u'5 CT 6 h'6u'6 k'6.v'6 1'6 w'6 w'6 '16 u'6 az = c. (331) 7 h'7.u'7 k'7v'7 117. w'7 w'7 '7 u'7 Txy 8 h'8.u'8 k'8v'8 1'8.w'8 w'8 v'8 u'8 zx 19 h'9u'9 k'9v'9 1'9 w'9 w'9 v'9 u'9 vyz Z10 h'10u'10 k'10v'10 l'10w10 w10 v'10 u'10 tIl h'llu'll k'll.Vl 1'111. w'1 v u'll 12) I h'12u'12 k'12v'12 1'12w'12 w'12 v'12 u'12) Solving Eq. 330 for the 12 primary slip systems we have: 14 12 13 t4 15 16 1 17 % " 18 19 110 Ill 112) 7x) Cy Tz Txy tzx Zy) (332) We can calculate shear strains in the same way as the shear stresses described above: {y} =c[S]{} (333) After knowing shear stresses and strains on 12 primary octahedral slip systems, we can use them to predict slip within a particular system. Finite Element Solution The finite element method is a numerical method for solving problems of engineering and mathematical physics. For problems involving complicated geometries, loadings and materials properties, it is generally not possible to obtain analytical mathematical solutions. Following the study and inspection of various approaches available for modeling, the method most appropriate for a notched single crystal specimen is the FEA (finite element method). It is also the only feasible type of computer simulation available for this purpose. The micro structural properties, such as dislocation mechanism for instance, play an important role in determining the yield strength of the material specimen. In order to account for the micro structural properties, small scale atomic simulations may be employed which can predict dislocation generation, interaction etc; however the cost factor for such an approach would be considerably high even for the analysis of very small specimen with actual dimension, on the atomic level. Another limiting factor of using the atomic simulation method is the distortion of the model to an extent that may lead to invalid prediction. Thus, FEA is the most appropriate tool for such an analysis as it can predict the influence of the geometry and anisotropy of the specimen on the behavior of its material properties without considering atomic interactions. FEA proves itself in its capability of accounting for gross material properties such as modulus of elasticity and Poisson's ratio; and in addition also the directional counterparts of these properties in the case of anisotropic materials. The Finite Element Model The commercial software ANSYS (Finite Element Software Version 5.7) is used to model the specific geometries and orientations of the tensile test specimen. The analysis consists of the modeling of two different samples for prediction of slip activity and sectors around the notch (Figure 37), which correlate to collaborative work between the Mechanical And Aerospace Engineering and Materials Science And Engineering Department of University of Florida. The prediction of the slip deformation is based on numerical model's highest individual resolved shear stresses. There is a definite corelation between the slip lines observed in the experimental test samples and the slip systems that are represented by the highest resolved shear stresses in the numerical model. [001] [111] [110] 101] [110] [01] Specimen A Specimen B Figure 37 Orientations of specimen A and specimen B. MathCAD 2000 professional was used to make a comparison between the numerically modeled specimen for the given load condition and the analytical solution of an unnotched specimen. Thereafter, in order to make a comparison of the numerical model with the experimental test specimen, a double notch was introduced in the model specimen. The FEA component stresses were taken from the material coordinate system, around the notch, and then used in the transformation equations to calculate the individual resolved shear stresses. The analysis of the data was done, for a complete stress field including a wide range of radial and angular distances. The results of this analysis were subsequently used to make predictions about sectors and slip activity. Criteria of Testing the Numerical Model The Finite element model of the entire tensile specimen without any notches was verified using the following two step procedure. Initial Check The chapter on analytical method describes the process required to verify the initial finite element model. The complete tensile specimen, without any notches has been analyzed in accordance with this process. Any dimension given for the model would result in a correct solution through the analytical method. For consistency, the same dimensions as those used in the experimental model were used for the numerical method. Provided that the material stays within the elastic range, the stresses will vary linearly with the load. The load of 100 lbs, which is used for all numerical models, results in stresses lower than the yield point. The stresses are scaled proportionally for other loads. The initial check gave the following results, which are tabulated below. (Table 32). Table 32 Comparison of analytical and numerical component stresses for specimen A. Material coordinate system) C (psi) CTX CTy Cz Cxy CTvz Cxz Analytical 0 0 7027.9 0 0 0 Numerical 3.6642 3.6643 7028.4 3.7176 4.86E03 2.49E02 % Error 0.007% The component stresses are in excellent agreement according to the preliminary check. The percentage error is within acceptable limits, confirming the accuracy of the coordinate and stress transformation of the model. Final Check After the initial check, the model was tested for the strain components because that would provide a better test of a correct anisotropic model. Such testing incorporates the use of the transformed stress tensor matrix. The component strain is tabulated below. The negligible amount or error shown below, confirms the correctness of the model. Table 33 Comparison of analytical and numerical component strains for specimen A. (Material coordinate system) SEx Sy ESz Sxy vz Exz Analytical 1.83E04 1.83E04 4.56E04 0 0 0 Numerical 1.83E04 1.83E04 4.56E04 2.37E07 3.09E10 1.59E09 % Error 0.016% 0.016% 0.044% The above twostep testing procedure was repeated for each orientation before the introduction of the notch in the specimen. Finally, the notch geometry was incorporated onto the existing model, which completed the actual specimen model. Model Features and Characteristics Spatial Characteristics. The FEM does not take into account the entire geometry of the specimen and is limited to the body of the specimen. The specimen end grips are excluded mainly because the mechanics at the grips differ from those at the specimen center and include other effects such as loading rate and tensile rig contact pressure. The grip is most susceptible to early deformations of different kinds and also to fracture, as seen in the experimental model. Also, most often it is seen that grips are changed/updated in order to gain better and accurate results. The numerical model will therefore not be subject to the type of grip used or any variations that it may cause, keeping in mind that our main focus is to model those specimens, which fail at the central area of the specimen and thereafter analyze those stresses. Both the numerical models utilize the same geometry (including the notch geometry), to observe the effects of orientations without other defect / size considerations. The geometry of both the models was simplified, based on the experimental counter part to Specimen, A which are shown below (Figure 38). Also Table 34 shows the actual specimen geometry of Specimen A. Thickness Width Notch Radius  Notch Length Height Notch Width Notch Height Figure 38 Dimensions of the Specimen. Simplified Geometry of the Notch The modeled notch consists of a combination of a rectangle and a semicircle. In the experimental model, the notch is likely to have an angular offset within the horizontal, and also some y displacement offset from the specimen center. In addition, the actual notch tip has an arc smaller than a semicircle. The geometrical; simplifications used for the notch model were 1) Setting both notch lengths and heights equal to those of the largest actual dimension.2) Setting the notch radius equal to the notch height (one half of the notch width) to form a half circle. This simplification provides accurate results in our limited scope of focus on orientation. To study specific test results, these geometrical simplifications may be removed. Table 34 Actual (Specimen A) and finite element specimen geometry. Specimen Geometry (mm) Actual FEM Height 19.000 19.000 Width 5.100 5.100 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 Material Properties An accurate model of a single crystal material can be created in ANSYS from our finite element model, which is linear, elastic and orthotropic. The three dimensional elements available in ANSYS can be used to justify orthotropic or anisotropic material properties. These elements in conjunction with the three independent stress tensors (all, a12, a44) or the three independent directional properties (G, E and v) can be used to model a single crystal material. The model is created around a global specimen coordinate system in ANSYS (Figure 39). The use of proper direction cosines will create the material coordinate system. The stress can now be calculated in any direction as the properties have been defined in the material coordinate system. The directional material properties are duly applicable as in ANSYS the element coordinate system is aligned with the material coordinate system. y A, yi z x Z1 Figure 39 Specimen and material coordinate systems. (The specimen is created around the global system (x"', y"', z"') and the material system (xi, yi, zi) is later specified) Meshing Technique Initially the three dimensional solid model is created and the front face is meshed with the PLANE2 elements (ANSYS 5.7 Element Reference, 1999). It is a two dimensional six node triangular structural solid. K N M Y (or Axial) L X (or Radia) Figure 310 PLANE2 2D 6NODE triangular structural solid. (ANSYS 5.7 elements reference, 1999). The element has a quadratic displacement behavior and is well suited to model irregular meshes. This element also has plasticity, creep, large deflection and large strain capabilities. Besides the nodes it also includes orthotropic material properties. However their material properties are not applicable for the modeling purpose it can be deleted later. Front face has precise element sizing along the defined radial lines around the notch tip at a 50 intervals. As soon as the front face is meshed, three dimensional element, SOLID 95 (Prism option) is swept through the volume of the threedimensional model to complete meshing of the model. SOLID 95 is a three dimensional structural solid with 20 nodes. Each node has three degree of freedom with translation in the x, y and z directions. It works in combination with the two dimensional elements (PLANE 2) on the front face, and retain their sizing definitions on each of the xy planes through the specimen thickness. It has same basic structure as of SOLID 45(Three dimensional element) but it is chosen since it can accurately model the area around the notch because of the presence of the mid size nodes. Also it includes the orthotropic material properties. Among others SOLID 95 also supports plasticity, creep, large deflections and large strains and is well suited for the future development of the model. Notch Tip Figure 311 Figure showing the meshing near the notch tip in specimen A. Figure 312 SOLID95 3D 20node structural solid.(ANSYS 5.7 elements reference, 1999). Figure 313 Element through the thickness. Six concentric arcs were created to examine stresses at angular and at radial distances from the notch (to examine near field and far field stresses). The six concentric circles are as follows: 0.25*p, 0.50*p, 1.00*p, 2.00*p, 3.00*p, and 5.00*p; where p is the notch radius (Figure 314). Assumption Used in Modeling Following are the assumptions used for the modeling purpose 1) Low temperature deformation 2) Microstructural behavior are not considered 3) There is no crystal lattice rotation in the model. Finite element analysis is capable of calculating the results for changing temperature. However to simplify the problem, and to collaborate more closely with the MSE department, material properties at a constant room temperature is applied in the numerical model. We also assume that the elastic deformation is taking place and .. ... ... . there is no plasticity near the notch tip. Plasticity can be included in the current model for future research. 90 2*p 3*p 5*p 1*p 0.5*p 0.25*p Figure 314 Radial arcs around the notch tip. Experimental Method The FEA results presented in my thesis will be compared with the results of experimental testing, which have been carried out (Forerro et al, 2002) for the single crystal superalloy. The specimen is tested to observe the effect of load orientation on the "sectors"(the active slip region), about the notch. This experimental approach, based on tensile testing is used to measure various material properties like stress strain behavior and yield strength etc .It employs a doubleedged notched tensile specimen as the test sample. The tensile specimen is loaded to observe and study the stress and stain fields, particularly slip line deformation and also the effect of overall displacement of the material. The reason of introducing a notch in the specimen is the production of triaxial state of stress in its proximity, which provides an environment to study slip systems formation in 3dimensional stress field. In the experimental approach, unlike the FEA; a compact load is applied to several test specimen with different crystallographic orientations and the response to failure is studied in order to observe the active slip planes. On the contrary, The FEA essentially studies an elastic response where in the magnitude of the applied stress is an indication of which planes will first allow plastic deformation. The correlation between the most highly stressed planes in the elastic analysis and the slip lines observed in the experimental approach determines the degree of influence of others dislocation mechanisms on fracture. CHAPTER 4 RESULTS AND DISCUSSION This chapter will present results of resolved shear stresses on principal octahedral planes [{111 }<110>] and evolution of slip systems in 3Dimensional anisotropic stress fields. A rectangular single crystal notched specimen in two crystallographic orientations is analyzed and the slip systems predictions are compared with the experimental results. The two specimens are, Specimen A with [001] load orientation and [110] notch growth direction and Specimen B with [111] load orientation and [101] notch growth direction (Figure 3.7). Results on the surface and the mid planes of the specimen are compared and contrasted to examine condition of plane stress and plane strain. Specimen A The states of stress and slip systems were analyzed both on the surface and on the mid plane of the specimen in the vicinity of the notch. Only the upper half of the specimen was analyzed (from 0 to 1800) since stress fields and hence the RSS in Specimen A are symmetric about the growth axis [110]. Figure 41 Figure showing mid plane and surface plane on which RSS is calculated for specimen A and B. Surface Results for Specimen A Twelve primary resolved shear stresses are calculated from 0.25*p to 5*p (in the radial direction) and from 0 to the top of the notch (in angular directions; 100 for 0.25*p up to 170 for 5*p) (Figures 42 to 46). The slip system with the maximum RSS varies with radial and angular position. The maximum RSS near the notch is T2 =25,000 psi and occurs at a radius = 0.5*p at + 105 angle. The dominant slip system and sectors were determined for each radius, by the overall maximum RSS at that radius (Table 41). The stress gradients are very steep in the vicinity of the notch and hence the RSS also vary strongly as a function of the position near the notch. The effects of variations of theta on RSS can be clearly seen in figures 42 to figure 46. To observe the effect of theta at a given radius, consider the variation of maximum RSS as a function of radial position. At r = 0.25*p, T11, T2 and T6 are indicating the dominant slip systems and have maximum RSS from 0 to 170, 170 to 820 and 820 to 1000 respectively.. However, at r = 0.5*p, Tl and T2 are the dominant slip systems from 00 to 350 and from 350 to 1050,respectively, demonstrating that slip system activation is not uniform for all radial distances from a notch. The effect of radius on resolved shear stresses can be seen by plotting the results with the 12 primary stresses for the entire range of radii (Figure 49) In order to show stresses on different radii (from 0.25*p to 5*p), all stresses at each radius have been scaled so they appear in ascending order (with respect to radial distance) from the origin. To observe the effect of radius at a given angle, look at how the RSS changes along 0 = 50. For example, at r = 0.25*p, T2 is the maximum RSS; T2 remains the maximum RSS at r = 0.5*p and then quickly shifts to Tr at r = 5*p. By analyzing the results we can see the variation in dominant slip systems, though some angles do maintain a single dominant slip systems for all radii (z2 at 590 680). The RSS field is dominated by TI, T2, and T3 on the (111) plane and by T6, T9 and zn on the (111), (111) and (111) planes, respectively. Specimen A (On surface) r = 0.25*p 20000 15000 Q. 10000 5000 Resolved Shear Stress v. Theta   ^ A A A A a 0 0 10 20 30 40 Th .......... ........... 50 60 70 80 90 1 eta (deg) Figure 42 Twelve primary resolved shear stresses on the surface of specimen A; r = 0.25*p T.i =.= Specimen A(On Surface) r= 0.5*p 25000 20000 15000 0. U) 10000 5000 Resolved Shear Stress v. Theta  till Uri I 0 10 20 30 40 50 60 70 80 Theta (deg) Figure 43 Twelve primary resolved shear stresses on the surface of specimen A; r = 0.5*p 90 100 Specimen A (On Surafce) r= l*p 16000 14000 12000 10000 0. S 8000 I 6000 4000 2000 0 Resolved Shear Stress v. Theta ' r 4 r4 m rI I A ,n2 0 10 20 30 40 50 60 70 80 90 100 110 120 Theta (deg) Figure 44 Twelve primary resolved shear stresses on the surface of specimen A; r = l*p Specimen A(On Surface) r =2*p 10000 8000 6000 4000 2000 0, Resolved Shear Stress v. Theta U r4P Figure 45 Twelve primary resolved shear stresses on the surface of specimen A; r = 2.0*p 20 40 60 80 100 120 140 Theta (deg) Specimen A(On surface) r = 5*p 6000 5000 4000 3000 2000 1000 0 Resolved Shear Stress v. Theta 1" U II Figure 46 Twelve primary resolved shear stresses on the surface of specimen A; r = 5.0*p 20 40 60 80 100 120 140 160 Theta (deg) 170 10 180 0 .25 .5 1 2 5 Magnified View Figure 47 Maximum resolved shear stress on each radius occurring on the surface of specimen A. RSS scaled for each radius to plot as one. TI .." 90 11W 0 3111 I~90 xo 30 \ 20 10 60 50 40 > 0 , Ij 1 911 1"11 '9 110 140' 150 1 M5 I 25.5 1 2 5 Figure 48 Active slip sectors on the surface of specimen A from 0.25*p to 5*p. If7 40 30 S20 60 SI) 20 A 10 SI 5 40 20 10 Resolved Shear Stress v. Theta 2'35 ~7 U4 J I . IS N Z N 4~ _b l~~ ic 60 ' SK S 7') S. ~ ~L4 t&  *r "e 1w  rl  T2 r3  r5  Tb  r7 r9 710  rll T12 B0 85 90 S90 95  100 105 110 _1 ... 1 71 180 175 Figure 49 Complete RSS field on the surface of Specimen A. Table 41 Specimen A dominant slip systems (Surface plane). Dominant Slip System Sectors Specimen A r = 0.25*p r = 0.5*p r= 1*p Sector 0 Tmax Slip System 0 Tmax Slip System 0 Tmax Slip System I 017 Til (111)[101] 035 TI (111)[101] 057 TI (111)[101] III 82100 c6 (111)[011] 113120 T3 (111)[110] r = 2.0*p r = 5.0*p Sector 0 rmax Slip System 0 Tmax Slip System I 059 Ti (111)[101] 054 Ti (111)[101] II 59116 T2 (111)[011] 5468 T2 (111)[011] III 116150 T3 (111)[110] 6886 T6 (111)[011] IV 86122 T2 (111)[011] V 122145 T3 (111)[110] VI 145165 'C9 (111)[011] Mid Plane Results for Specimen A Twelve primary resolved shear stresses are calculated from 0.25*p to 5*p and from 0 to the top of the notch in order to calculate stresses in angular directions (100 for 0.25*p up to 170 for 5*p) (Figures 410 to 414). The maximum RSS near the notch is T4 and T6=28,380 psi at a radius = 0.5*p at an angle of+ 105 angle. The maximum RSS on the mid plane is greater than the maximum RSS on the surface plane. The dominant slip system and sectors were determined for each radius by the overall maximum RSS (Table 42). Results obtained at the mid plane were obtained and analyzed similarly to the results obtained at the surface of specimen A. The number of activated slip systems is higher here as compared to the slip systems on the surface. It shows dominant systems on each of the four possible primary slip planes in the RSS fields. Overall the RSS field is dominated by T1, T2, and T3 on the (111) plane and by T4, T6, T8, T9, T10, T11 and T12 on the {111} family of planes. Specimen A ( On Mid Plane) Resolved Shear Stress v. Theta r= 0.25*r 32000 28000 24000 i mI.T  m1 S12000. == ' '= = 20000 S T74 16000 X n ' U U m .. Em  12000 Tlr . illI 8000 A A A A A A A A  I A T1V 8000 A A 4000 A A 0 10 20 30 40 50 60 70 80 90 100 Theta(deg) Figure 410 Twelve primary resolved shear stresses on the mid plane of specimen A; r = 0.25* p Specimen A( On Mid Plane) r = 0.5*r Resolved Shear Stress v. Theta A A A A A 10 20 30 40 50 60 Theta(deg) 70 80 90 100 Figure 411 Twelve primary resolved shear stresses on the mid plane of Specimen A; r = 0.5* p 28000 24000 20000 , 16000 12000 8000 4000 * r4 A ri? Specimen A (On Mid Plane) r= 1*r 16000 14000 12000 10000 8000 6000 4000 2000 Resolved Shear Stress v. Theta H ,1  nil 0 10 20 30 40 50 60 Theta(deg) 70 80 90 100 110 120 Figure 412 Twelve primary resolved shear stresses on the mid plane of specimen A; r = 1.0* p Resolved Shear Stress v. Theta Specimen A (On Mid Plane) r =2*r 9000 8000 7000 6000 S5000 S4000 3000 2000 1000 A A 0 NEN(* 0 20 40 60 80 100 120 140 Theta(deg) Figure 413 Twelve primary resolved shear stresses on the mid plane of specimen A; r = 2.0* p T. i il_ r4 r4 A ri2 Specimen A ( On Mid Pane) r = 5*r 7000 6000 5000 4000 3000 2000 1000 0 Resolved Shear Stress v. Theta >'** *\ U U .7  /. 20 40 60 80 100 120 140 160 Theta(deg) Figure 414 Twelve primary resolved shear stresses on the mid plane of specimen A; r = 5.0* p r7 7 1 4&7II 170 10 180 .25 .5 1 2 5 0 I Magnified View Figure 415 Maximum resolved shear stress on each radius occurring on the mid plane of specimen A. T 2. IC it 0 li, 80 Figure 416 Active slip sectors on the mid plane of specimen A from 0.25*p to 5*p. 4, 6 40 30 20 I0 2 5 r9, 8 Table 42 Specimen A dominant slip systems (Mid plane). Dominant Slip System Sectors Specimen A r= 0.25*p r = 0.5*p r= 1*p Secto 0 Tmax Slip System 0 Tmax Slip System 0 Tmax Slip System I 0100 C4 (111)[101] 035 T2 (111)[011] 038 Ti (111)[101] II 0100 C6 (111)[011] 035 Tli (111)[101] 038 Tio (111)[011] III 35105 C4 (111)[101] 38107 12 (111)[011] IV 35105 C6 (111)[011] 38107 Til (111)[101] V 107117 T3 (111)[110] VI 107117 12 (111)[110] VII 117120 T8 (111)[011] VlII 117120 q9 (111)[101] r = 2.0*p r =5.0*p Secto 0 Tmax Slip System 0 Tmax Slip System I 054 TJ (111)[101] 057 TJ (111)[101] II 054 To0 (111)[011] 057 To0 (111)[011] III 54120 T2 (111)[011] 57159 T2 (111)[011] IV 54120 Til (111)[101] 57159 Til (111)[101] V 120129 T3 (111)[110] 159170 Ti (111)[101] VI 120129 T12 (111)[110] 159170 o10 (111)[011] VII 129140 T8 (111)[011] VlII 129140 c9 (111)[101] IX 140150 T2 (111)[011] X 140150 __l (111)[101] Specimen B The states of stress and slip systems of Specimen B were also analyzed both on the surface and on the mid plane of the specimen, in the vicinity of the notch. Both the upper and the lower half of the specimen were analyzed (from 0 to 1800 and from 0 to  1800) since RSS is not symmetric about the growth axis [101]. The results of Specimen B on surface and mid planes are given below. Surface Results on Upper Half of Specimen B Twelve primary resolved shear stresses are calculated here from 0.25*p to 5*p and from 0 to the top of the notch. (Figures 418 to 422). The maximum RSS near the notch is T9=27,100 psi at radius = 0.5*p and at 105. This stress is a fair amount higher and in different slip systems than the maximum RSS for Specimen A on the surface (max RSS T2=25,000 psi), though it does occur at the same location. The dominant slip system and sectors were determined for each radius by the overall maximum RSS, (Table 43). Like Specimen A, The RSS change values and shifts positions relative to each other with respect to theta. The effects of theta on resolved shear stresses can be seen in figure 418 to figure 422. It also shows dominant systems on each of the four possible primary slip planes in the RSS fields .In conclusion the RSS field is dominated by TI on the (111) plane and by T4,T5, T6, T,9, T10 and Tll on the {111} family of planes. Specimen B (Upper part on Surface) Resolved Shear Stress v. Theta r = 0.25*r 20000 18000 16000 14000 12000 4 10000 8000 m <"  6000  ti h, 4000 2000 0 A A A A A 0 10 20 30 40 50 60 70 80 90 100 Theta(deg) Figure 418 Twelve primary resolved shear stresses on the upper half of the surface of specimen B; r = 0.25* p Specimen B (Upper part on Surface) r = 0.5*r 30000 25000 20000 S15000 10000 5000 A A A Resolved Shear Stress v. Theta U K S r4I 4l 70 80 90 100 Figure 419 Twelve primary resolved shear stresses on the upper half of the surface of specimen B; r 0 10 20 30 40 50 60 Theta(deg) 0.5* p Specimen B (Upper part on Surface) r = 1*r 20000 18000 16000 14000 12000 10000 8000 6000 4000 2000 0 Resolved Shear Stress v. Theta U r2 I T, 4 T4 _T' ilj.  Till A TI_" 0 10 20 30 40 50 60 Theta(deg) 70 80 90 100 110 120 Figure 420 Twelve primary resolved shear stresses on the upper half of the surface of specimen B; r = 1* p Specimen B (Upper part on Surface) r = 2*r 14000 12000 10000 8000 6000 4000 2000 0 Resolved Shear Stress v. Theta r4 4 UHiI 20 40 60 80 100 120 140 Theta(deg) Figure 421 Twelve primary resolved shear stresses on the upper half of the surface of specimen B; r = 2* p Specimen B (Upper part on Surface) r = 5*r 9000 8000 7000 6000 5000 4000 3000 2000 1000 0 Resolved Shear Stress v. Theta r4  ' r II U ril A ti' Figure 422 Twelve primary resolved shear stresses on the upper half of the surface of specimen B; r = 5* p 20 40 60 80 100 120 140 160 Theta(deg) Specimen B (Upper half on the surface) 't5,t11 0 0 0 1 3 10 10 160 170 180 I Figure 423 Maximum resolved shear stress on each radius occurring on the upper half of the surface of specimen B. Table 43 Specimen B dominant slip system. (Upper half on surface). Dominant Slip System Sectors Specimen B r= 0.25*p r = 0.5*p r= 1*p Sector 0 rmax Slip System 0 rmax Slip System 0 rmax Slip System I 07 Ts (111)[110] 042 T5 (111)[110] 019 T5 (111)[110] II 7100 c9 (111)[101] 4262 T6 (111)[011] 019 Til (111)[101] III 62105 T9 (111)[101] 1956 5 (111)[110] IV 5694 T6 (111)[011] V 94117 q9 (111)[101] VI 117120 Ts (111)[011] r = 2.0*p r = 5.0*p Sector 0 rmax Slip System 0 rmax Slip System I 043 'l1 (111)[101] 050 Tl (111)[101] II 4357 T5(111)[110] 5053 T5 (111)[110] III 5778 T6 (111)[011] 5361 6 (111)[011] IV 78138 'c9 (111)[101] 61103 'r9 (111)[101] V 138150 T4 (111)[101] 103160 T (111)[101] VI 160170 o10 (111)[011] Surface Results on Lower Half of Specimen B Twelve primary RSS are calculated from 0.25*p to 5*p and from 0 to the bottom of the notch (Figures 424 to 428). The maximum RSS near the notch is n11=26,290 psi at radius = 0.5*p and at 105 angle. This stress is lower than maximum RSS of upper half of Specimen B and quite higher than the maximum RSS for Specimen A on the surface, but it occurs at the same location in the RSS field. Here also, the dominant slip system and sectors were determined for each radius. (Table 44). Here also the RSS change values and shifts positions relative to each other with respect to theta. Overall the RSS field is dominated by TI and T3 on the (111) plane and by T4, T5, T6, T7, Z8, T9, Z10 T11, and T12 on the {111 } family of planes. Specimen B (Lower part on Surface) r = 0.25*r 24000 20000 16000 0  12000 U_ 8000 4000 Oa Resolved Shear Stress v. Theta S r4 ii, 90 100 Figure 424 Twelve primary resolved shear stresses on the lower half of the surface of specimen B; r = 0.25* p 10 20 30 40 50 60 70 80 Theta(deg) Specimen B (Lower part on Surface) r = 0.5*r 28000 24000 20000 S16000 S12000 8000 4000 0r Resolved Shear Stress v. Theta I'  I I' UilI 70 80 90 100 Figure 425 Twelve primary resolved shear stresses on the lower half of the surface of specimen B; r = 0.5* p 0 10 20 30 40 50 60 Theta(deg) Specimen B (Lower part on Surface) r=1*r 16000 12000 J 8000 4000 0 yd*1* o '^^L,,__ Resolved Shear Stress v. Theta Ui: T1" A rl: ilil ISll  Til 0 10 20 30 40 50 60 70 80 90 100 110 Theta(deg) Figure 426 Twelve primary resolved shear stresses on the lower half of the surface of specimen B; r = 1* p 