<%BANNER%>

Prediction of slip systems in notched FCC single crystals using 3D FEA

University of Florida Institutional Repository

PAGE 1

PREDICTION OF SLIP SYSTEMS IN NOTCHED FCC SINGLE CRYSTALS USING 3D FEA By NIRAJ SUDHIR BIDKAR A THESIS PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLOR IDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE UNIVERSITY OF FLORIDA 2003

PAGE 2

I would like to dedicate this work to my parents for their ever-extending support and encouragement.

PAGE 3

iii ACKNOWLEDGMENTS I would like to express deep gratitude to Dr. Nagaraj Arakere for his encouragement and enthusiastic assistance a nd instructions. I woul d also like to extend special thanks to Dr. Ashok Kumar and Dr. John Schueller for their time and support. Thanks are also due to Dr. Ebrahimi and Luis Forrero for providing the necessary information and data. I am thankful to my friends and lab mates Jeff and Shadab for helping me in understanding many concepts and ideas. Finally I am grateful to the Almighty God for giving me the strengt h for the completion of this work.

PAGE 4

iv TABLE OF CONTENTS page ACKNOWLEDGMENTS.................................................................................................iii LIST OF TABLES.............................................................................................................vi LIST OF FIGURES..........................................................................................................vii ABSTRACT....................................................................................................................... xi CHAPTER 1 INTRODUCTION......................................................................................................1 Slip Deformation in Single Crystal Superalloys........................................................2 Material Characteristics..............................................................................................7 Temperature Effects...................................................................................................8 Test Methods..............................................................................................................9 Analytical Approach........................................................................................9 Numerical Approach......................................................................................10 Experimental Approach.................................................................................10 2 REVIEW OF LITERATURE...................................................................................12 Anisotropy of Elasticity...........................................................................................12 Elasticity........................................................................................................12 Elasticity Matrix............................................................................................13 Notch Tip Deformation............................................................................................16 Rice (1987)....................................................................................................16 Crone and Shield (2001)................................................................................18 Schulson and Xu (1997)................................................................................23 Slip Activation and Deformation.............................................................................27 3 ANALYTICAL APPROACH..................................................................................29 Coordinate Axes Transforma tion Using Miller Indices...........................................31 Example Transformation...............................................................................34 Transformation Matrices for Stress and Strain Fields..............................................35 Shear Stresses and Strains in the Slip Systems........................................................38

PAGE 5

v 4 NUMERICAL APPROACH: FINITE ELEMENT METHOD...............................41 Verification of the Finite Element Model Results.........................................42 Specimen Orientation and Dimensions.........................................................43 Specimen Geometry.......................................................................................44 Specimen Geometry.......................................................................................45 Finite Element Model Characteristics......................................................................45 Material Properties.........................................................................................45 Elements and Meshing...................................................................................46 Finite Element Solution............................................................................................49 Assumptions.............................................................................................................50 5 RESULTS AND DISCUSSION..............................................................................52 Specimen I................................................................................................................52 Specimen II..............................................................................................................70 Identifying the Slip Systems from the Experimental Results..................................99 Specimen Comparison............................................................................................103 Specimen I Notch Geometry..................................................................................104 6 CONCLUSIONS AND RECOMMENDATIONS.................................................107 LIST OF REFERENCES.................................................................................................108 BIOGRAPHICAL SKETCH...........................................................................................111

PAGE 6

vi LIST OF TABLES Table page 1.1 Slip systems in FCC crystal. Source: Stouffer and Dame, 1996...................................3 2.1 Atomic density on FCC crystal planes Source: Dieter (1986).....................................13 2.2 Symmetry in various crystal st ructures Source: Dieter (1986)....................................13 2.3 Sector boundary angle comparisons for Orientation II Source: Crone and Shield (2001).....................................................................................................................19 2.4 Sector boundary angles from the experi mental tests by Crone and Shield Source: Crone and Shield (2001)........................................................................................22 2.5 Slip sectors from plane stress and plane strain assumptions. Source: Schulson and Xu (1996)...............................................................................................................26 4.1 Actual and specimen geometry for Specimen I...........................................................44 4.2 Actual and specimen geometry for Specimen II..........................................................45 5.1 Specimen I dominant slip system sectors on the surface.............................................62 5.2 Specimen I dominant slip system sectors on the on the mid-planes............................69 5.3 Specimen II dominant slip system sector s on the surface on the upper portion of the notch growth direction...........................................................................................77 5.4 Specimen II dominant slip system sectors on the surface on the lower portion of the notch growth direction...........................................................................................84 5.5 Specimen II dominant slip system sectors on the mid-planes on the upper portion of the notch growth direction.....................................................................................91 5.6 Specimen II dominant slip system sectors on the mid-planes on the lower portion of the notch growth direction.....................................................................................98 5.7 Load Levels applied to spec imen I during th e experiments........................................99 5.8 Intersection of the slip planes with the notch plane...................................................100 5.9 Specimen I experimental results................................................................................102

PAGE 7

vii LIST OF FIGURES Figure page 1.1 Convention for defining primary and sec ondary crystallographic orientations in turbine blades.Source: Swanson and Arakere, 2000................................................2 1.2 Slip sectors observed under plastic defo rmation. Source: Crone and Shield, 2001......7 1.3 Microstructure of material A. The ’ precipitate forms in the matrix and has a volume fraction of about 60%. Source: Superalloys II by Sims, Stoloff and Hagel........................................................................................................................8 2.1 Schematic of FCC crystal structure.............................................................................13 2.2 Notch direction terminology........................................................................................17 2.3 Yield surface based on plane strain state of stress.......................................................17 2.4 Orientations used by Crone and Shield in their tests...................................................19 2.5 Slip sectors observed in the experimental tests by Crone and Shield. Source: Crone and Shield (2001)...................................................................................................20 2.6 Slip sectors and slip lines from the expe rimental tests by Crone and Shield. Source: Crone and Shield (2001)........................................................................................21 2.7 Specimen Orientation used by Schulson and Xu.........................................................23 2.8 Slip sectors obtained by Schulson a nd Xu Source: Schulson and Xu, 1997..............25 2.9 Load and slip direction.................................................................................................27 3.1 Material and Specimen coordinate system...................................................................30 3.2 First rotation of the coordina te system about the z-axis..............................................31 3.3 Projections from the original axes on the projected axes............................................32 3.4 Second rotation of the coordinate system about the z-axis..........................................32 3.5 Third rotation of the coordina te system about the z-axis.............................................33

PAGE 8

viii 4.1 The notched specimens analyzed using finite element method...................................42 4.2 Specimen specifications...............................................................................................44 4.3 The specimen is first defined with respec t to the global coordina te system and then the material coordinate sy stem is specified later...................................................46 4.4 Meshing around the notch in ANSYS.........................................................................47 4.5 Schematic of the SOLID95 element in ANSYS. Source ANSYS 6.1 Elements Reference, 2002.....................................................................................................48 4.6 Radial arcs used for element locati on and sizing; centered at notch tip......................48 4.7 Radial arcs for stress field calculations........................................................................49 5.1 RSS on the 12 primary slip systems for the upper portion of the notch growth direction for R=0.25* ...........................................................................................54 5.2 RSS on the 12 primary slip systems for the upper portion of the notch growth direction for R=0.5* .............................................................................................55 5.3 RSS on the 12 primary slip systems for the upper portion of the notch growth direction for R=1* ................................................................................................56 5.4 RSS on the 12 primary slip systems for the upper portion of the notch growth direction for R=2* ................................................................................................57 5.5 RSS on the 12 primary slip systems for the upper portion of the notch growth direction for R=5* ................................................................................................58 5.6 Slip system sectors on the surface for specimen I.......................................................59 5.7 Upper portion of the radar plot showi ng the surface stresses for the [100] orientation..............................................................................................................60 5.8 Lower portion of the radar plot showi ng the surface stresses for the [100] orientation..............................................................................................................61 5.9 RSS on the 12 primary slip systems for the upper portion of the notch growth direction for R=0.25* ...........................................................................................63 5.10 RSS on the 12 primary slip systems fo r the upper portion of the notch growth direction for R=0.5* .............................................................................................64 5.11 RSS on the 12 primary slip systems fo r the upper portion of the notch growth direction for R=1* ................................................................................................65

PAGE 9

ix 5.12 RSS on the 12 primary slip systems fo r the upper portion of the notch growth direction for R=2* ................................................................................................66 5.13 RSS on the 12 primary slip systems fo r the upper portion of the notch growth direction for R=5* ................................................................................................67 5.14 Slip system sectors on the mid-planes for specimen I...............................................68 5.15 RSS on the 12 primary slip systems fo r the upper portion of the notch growth direction for R=0.25* ...........................................................................................71 5.16 RSS on the 12 primary slip systems fo r the upper portion of the notch growth direction for R=0.5* .............................................................................................72 5.17 RSS on the 12 primary slip systems fo r the upper portion of the notch growth direction for R= ...................................................................................................73 5.18 RSS on the 12 primary slip systems fo r the upper portion of the notch growth direction for R=2* ................................................................................................74 5.19 RSS on the 12 primary slip systems fo r the upper portion of the notch growth direction for R=5* ................................................................................................75 5.20 Slip system sectors on the surface on the upper portion of the notch growth direction for specimen II........................................................................................76 5.21 RSS on the 12 primary slip systems fo r the lower portion of the notch growth direction for R=0.25* ...........................................................................................78 5.22 RSS on the 12 primary slip systems fo r the lower portion of the notch growth direction for R=0.5* .............................................................................................79 5.23 RSS on the 12 primary slip systems fo r the lower portion of the notch growth direction for R= ...................................................................................................80 5.24 RSS on the 12 primary slip systems fo r the lower portion of the notch growth direction for R=2* ................................................................................................81 5.25 RSS on the 12 primary slip systems fo r the lower portion of the notch growth direction for R=5* ................................................................................................82 5.26 Slip system sectors on the surface on the lower portion of the notch growth direction for specimen II........................................................................................83 5.27 RSS on the 12 primary slip systems on th e mid-planes for the upper portion of the notch growth direction for R=0.25* ....................................................................85 5.28 RSS on the 12 primary slip systems on th e mid-planes for the upper portion of the notch growth direction for R=0.5* ......................................................................86

PAGE 10

x 5.29 RSS on the 12 primary slip systems on th e mid-planes for the upper portion of the notch growth direction for R= .............................................................................87 5.30 RSS on the 12 primary slip systems on th e mid-planes for the upper portion of the notch growth direction for R=2* .........................................................................88 5.31 RSS on the 12 primary slip systems on th e mid-planes for the upper portion of the notch growth direction for R=5* .........................................................................89 5.32 Slip system sectors on the mid-planes on the upper portion of the notch growth direction for specimen II........................................................................................90 5.33 RSS on the 12 primary slip systems on th e mid-planes for the lower portion of the notch growth direction for R=0.25* ....................................................................92 5.34 RSS on the 12 primary slip systems on th e mid-planes for the lower portion of the notch growth direction for R=0.5* ......................................................................93 5.35 RSS on the 12 primary slip systems on th e mid-planes for the lower portion of the notch growth direction for R= .............................................................................94 5.36 RSS on the 12 primary slip systems on th e mid-planes for the lower portion of the notch growth direction for R=2* .........................................................................95 5.37 RSS on the 12 primary slip systems on th e mid-planes for the lower portion of the notch growth direction for R=5* .........................................................................96 5.38 Slip system sectors on the surface on the upper portion of the notch growth direction for specimen II........................................................................................97 5.39 Picture showing the slip lines around the right notch of specimen I for level 6........99 5.40 Picture showing the slip traces.................................................................................101 5.41 RSS Vs Theta plot scaled to load level # 6..............................................................102 5.42 RSS Vs. Theta plots obtained from the semi-circular notch geometry....................104 5.43 RSS Vs. Theta plots obtained from the semi-circular notch geometry....................105 5.44 RSS Vs. Theta plots obtained from the elliptical notch geometry...........................105

PAGE 11

xi Abstract of Thesis Presen ted to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Master of Science PREDICTION OF SLIP SYSTEMS IN NOTCHED FCC SINGLE CRYSTALS USING 3D FEA By Niraj Sudhir Bidkar May 2003 Chair: Dr. Nagaraj K. Arakere Department: Mechanical and Aerospace Engineering Nickel-base single crystal superalloys ar e the current choice for high-temperature jet engine applications such as turbine blad es and vanes. Since these materials are grown as single crystals, there are no grain boundaries present and as a resu lt the properties of these materials are highly direction-depende nt. This makes it necessary to test these materials for different load orientations. Notc hed tensile test specimens are typically used to study the evolution of slip systems in 3D stress fields. Many re searchers in the past have tried to come up with solutions for pred iction of the slip systems in notched test specimens for specific load orientations. The two notched-specimens considered in this work differ in geometry as well as in load orientation. An elastic finite element model has been created which predicts the behavior in the plastic regime quite accuratel y. The linear elastic model is used to predict only the onset of yield. The stre ss concentrated region near the notch, with the triaxial

PAGE 12

xii state of stress, is of particul ar interest. Slip systems in this region are predicted using 3D linear elastic FEA, including material anisotr opy. This model does not account for effects such as strain hardening, crystal latti ce rotation, creep, high-te mperature conditions. These results were correlated with the experimental tests performed by the MSE department for similar specimens. The first of the two specimens had [001] load direction, [010] notch growth direction and [100] as the not ch plane direction. The other specimen had a load direction of [111], notch growth direction [-12-1] and [30-3] as the notch plane direction. The slip systems predicted using 3D FE A had excellent correlation with those observed experimentally. This demonstrates th at 3D linear elastic FEA that includes the effect of material anisotropy can be used to effectively predict the onset of yield and hence slip systems in FCC single crystals.

PAGE 13

1 CHAPTER 1 INTRODUCTION As one of the most important classes of high-temperature structural materials, Nickel base superalloys exhi bit a truly unique combination of properties at elevated temperatures. These desirable properties enable these materials to enhance the performance and efficiency of turbine e ngines designed for both aircraft and powergeneration applications. In high temperature polycrystalline all oys grain boundaries provide passages for diffusion and oxidation. Eliminating grain boundaries and grain bounda ry strengthening elements produces materials with superior hi gh temperature fatigue and creep properties compared to conventional superalloys. Howe ver, the absence of grains makes these alloys orientation dependent or anisotropi c with tensioncompression asymmetry. As components are cast in single cr ystal form the entire compone nt inherits the anisotropic properties of the crystal lattice. The slip de formation mechanisms are strong functions of orientation and deformation occurs in specific crystallographic directi ons rather than in the direction of the applied load. During the manufacturing of the superall oys, the primary growth direction is oriented in the <001> orientation as this orientation has the best combination of mechanical properties. However in practice, due to the difficulties encountered in casting and due to cost considerations blades that are oriented away from the <001> orientation by up to 15 is used. Also in the operating environment the blades are subjected to loading in a variety of orientations due to the hot impinging gases causing bending and

PAGE 14

2 torsion. The formation of hotspots leads to th ermal gradients, which also contribute to multiaxial loading. Hence the deformation m echanisms depend on the high temperature effects and loading in different orientations. Figure 1.1 Convention for defining primary and secondary crystallographic orientations in turbine blades.Source: Swanson and Arakere, 2000. Slip Deformation in Single Crystal Superalloys Slip deformation in nickel-b ase superalloys is governed by the sliding of separate layers of the crystal structur e over one another along definite crystallographic planes or slip planes in specific directions, according to Dieter (1986). It is considered a stresscontrolled process.

PAGE 15

3 Table 1.1 Slip systems in FCC crystal. Source: Stouffer and Dame, 1996 The CRSS is therefore the controlling valu e and it is the shear stress at which the slip was initiated. The CRSS is a function of the applied load a nd direction, specimen geometry, and crystal structure, though it is not directly related to the material’s anisotropy. Slip Number Slip Plane Slip Direction Octahedral Slip a/2{111}<110> 1 2 3 4 5 6 7 8 9 10 11 12 (111) (111) (111) (-11-1) (-11-1) (-11-1) (1-1-1) (1-1-1) (1-1-1) (-1-11) (-1-11) (-1-11) [10-1] [0-11] [1-10] [10-1] [110] [011] [110] [0-11] [101] [011] [101] [1-10] Octahedral Slip a/2{111}<112> 13 14 15 16 17 18 19 20 21 22 23 24 (111) (111) (111) (-11-1) (-11-1) (-11-1) (1-1-1) (1-1-1) (1-1-1) (-1-11) (-1-11) (-1-11) [-12-1] [2-1-1] [-1-12] [121] [1-1-2] [-2-11] [-11-2] [211] [-1-21] [-21-1] [1-2-1] [112] Cubic cells a/2{100}<110> 25 26 27 28 29 30 (100) (100) (010) (010) (001) (001) [011] [01-1] [101] [10-1] [110] [-110]

PAGE 16

4 Depending on the direction of the applied load, certain slip systems get activated first. These are termed as the ‘easy glide’ sy stems or the primary octahedral slip systems. Usually, the planes with greatest atomic densitie s are the ones that are activated first. This whole process of slip occurs so as to relieve the energy of the high shear stress or the resolved shear stress (RSS) with in the 12 primary slip systems. The slip thus helps to obtain a more energetically stable system. As this slip activity and the deformation pro cess continues, the indi cations of slip at the surface can be observed. With the increase in the applied stress, some of the 12 secondary slip systems may also get activat ed. Extreme temperature or load conditions may also cause the activation of the six cubic slip planes. But the activation of secondary or cubic slip planes does not occur first be cause the slip activity along the primary slip planes requires less energy to relieve the high RSS. Certain microstructural behaviors like pinning or locking of disloca tions too prevent the shift in slip from the primary to secondary or cubic slip systems. Usually slip occurs when the RSS exceed s the yield strength of the material. Nickel-base superalloys, however exhibit different yield strength in tension and compression. This kind of behavior is termed as tension/compression asymmetry. For instance, for the <001>orientation these alloys exhibit highest yield strength in tension, while the compressive yield st rength is lower for this orie ntation. On the other hand, for the <110>orientation, the yield strength is higher in compression than in tension. For a sample loaded in the <111> direction th ere is practically no tension/compression asymmetry.

PAGE 17

5 In some cases, parts of the dislocations ge t separated to reach a lower energy state. These separated parts are termed as ‘extende d dislocations’. The extended dislocations, on octahedral slip planes must recombine into a single dislocation be fore cross slip can take place. Cross slip is th e phenomenon where a dislocatio n moves from one plane to another. Cross slip also occurs to reduce the energy of the system. Again the material orientation affects the process of recombinati on of the dislocation pa rts as it does in case of the magnitude of the resolved shear stre ss. An applied compressive load helps to overcome the force separating two extended disl ocations at an interface where cross slip can occur and can combine them into a singl e dislocation. The combined dislocation can then move as a cross slip. Conversely, an appl ied tensile load aids the force separating the dislocation and makes cross slip more difficult to occur. This asymmetrical load behavior should be addressed as an important issue in th e design of a part that is subjected to both tensile and compressive loads. As far as th e failure analysis research is concerned, priority is given to tensile stress analysis as compared to compressive stress analysis as compressive stresses usually prove to be bene ficial. Instead of causing cracks to initiate or propagate, compressive stresse s usually either have no eff ect or may even arrest the cracks that develop under tensile stresses. However, in the case of tri-axial stress states, such as those occurring at the notch tip, the asymmetric behavior becomes even more complex. Depending on orientation, some slip directions may hinder partial dislocation recombination, while others may aid this process. Nickel-base superalloys exhibit another abnor mal yield characteristic that is related to temperature called ‘anomalous yield behavior’. A normal tr end shows a decrease in the yield strength with increasing temperature. These superalloys however have been shown

PAGE 18

6 to exhibit an increase in thei r yield strength with rising temper ature, up to a certain point. Here it is essential to take into account the process of superdislocati ons that is used to relieve the stress. Superdislo cation can be defined as a di slocation composed of two dislocations separated by an antiphase boundary (APB) that glide along {111} octahedral planes during slip. This superdis location always attempts to reduce its energy, which can be brought about by lowering its APB energy if it cross-slips to a {100} plane. Now when the first dislocation cross slips on to the {100} plane, it gets locked due to the higher stress required for it to move on a {100} plane. It gets locked on two separate planes in such a way that one dislocati on is left on a {111} plane and the other on a {100} plane separated by the APB. Because of this lock, no further motion can take place. With increasing temperature, thermally activated cross-slip to {100} planes occur easily, forming locks. These locks keep on multiplying, thus preventing further motion and increasing the CRSS. In addition to this the ‘Peierl stress’ required to move the locked dislocation on the {100} plane, is lowe red with the increase in temperature. At a certain point of temperature, the th ermal activation helps the dislocation to overcome the Pierel stress. At this critical point the entire dislocation is able to move to the {100} plane resulting in a c ubic slip. At this point, more of the previously locked dislocations are released on the {100} planes and consequently the CRSS begins to fall. Thus in high temperature applications, a nomalous yield behavior is an important consideration for fracture and fatigue analys is. This considerati on helps understand the mechanisms that increase strength and their limitations.

PAGE 19

7 Figure 1.2 Slip sectors observed under plas tic deformation. Source: Crone and Shield, 2001. This work will focus on the variation of th e 12 primary resolved shear stresses in a notched specimen. The activity of the specific sl ip systems with respect to the radial and angular distances surrounding the notch tip w ill be determined for specific orientations. The slip systems will be studied both near the notch and at far field to observe the changes associated with high stress gradients prevalent in close proximity of the notch tip. The maximum RSS values and slip systems are expected to shift along a line of constant radius. This, in a way indicates a shift in the state of stress. In the actual tensile testing, these different slip systems can be clearly visualized as “sectors” surrounding the notch tip. In short, the goal of this work is the prediction of slip system activity as a function of the radial and angular distances from the notch tip and the resulting slip sectors. Material Characteristics Nickel-base superalloys have a micr ostructure that consists of a -matrix and a fine dispersion of hard ’-precipitates. The matrix is considerably alloyed with other elements that may vary, including cobalt, chromium tungsten and tantalum, though it is mainly composed of nickel. The precipitate how ever is the intermetallic compound Ni3Al.

PAGE 20

8 Figure 1.3 Microstructure of material A. The ’ precipitate forms in the matrix and has a volume fraction of about 60%. Source: Superalloys II by Sims, Stoloff and Hagel. So far these superalloys have evolved in three generations. Rene N4, CMSX-4, and others are some of the most advanced or third-generation superalloys. Usually, these alloys have a high volume fraction of ’, around 60%. The majority of the deformation occurs in the softer matrix as suggest ed by Svoboda and Lukas (1998). However, it should also be noted that at such high volume fractions, the precipitate has a considerable effect on the overall performance of the superalloy’s performance. Temperature Effects These materials are designated to provi de unique strength and/or corrosion properties at elevated temperatures (i.e., gr eater than 600C). The experimental notched tensile tests conducted to study slip system s activity have been performed at room temperature, which is well below the transi tion temperature for superalloys. As observed by Stouffer and Dame (1996), in the low temper ature regime, the octahedral slip system is predominant and basically controls lo w temperature deformation. The secondary planes are activated onl y at high temperatures.

PAGE 21

9 At temperatures above 600C the cubic slip systems get activated and act along with the octahedral planes. When the temperature reaches above 850C, the deformation increases rapidly and the material stre ngth is almost free of the orientation. Test Methods For the isotropic notch specimens in tension many analytical and numerical solutions have been developed for their elastic response, especially in the field of linear elastic fracture mechanics. However, it is very difficult to develop three-dimensional analytical models for anisotropic notched sp ecimens. The current solutions have been derived only after many simplif ications and therefore give inaccurate results when compared to experimental data. However, th e three-dimensional finite element analysis (FEA) is capable of taking into account the limitations in the elastic models and it does provide a solution comparable to the actua l experimental results. In numerical and experimental specimens, the notches can be considered as simplifie d cracks to develop a realistic model to study fracture behavior. In many applications, it may so happen that though the material is designed for primary st rength in one directi on, it must withstand multi-axial loading. This makes the study of effe ct of anisotropy all the more interesting. Analytical Approach Generally analytical solutions provide exact solution to any probl em. But this is not true in case of very complex problems. For such problems, the analytical solutions are arrived at after close approximations or by th e use of theoretical a nd empirical solutions. For the notched specimen in tension, an analy tical solution is not available that correctly predicts slip system activity around the notch tip. The ones that are available make use of several assumptions like the plane strain. This issue will be discussed in detail later.

PAGE 22

10 Numerical Approach As far as finite element analysis is co ncerned, it does account for gross material properties such as modulus of elasticity and Pois son’s ratio in isotropic materials. It even accounts for the directional prope rties while analyzing anisotropic materials. However, FEA is unable to account for microstructural pr operties that govern yield strength, such as dislocation mechanisms or other microstructural behavi or. It may be possible to predict dislocation mechanisms by using small-scale atomistic simulations, but it would be too much expensive to conduct analysis fo r such small dimensions. Also, reducing the size to atomic level would dist ort the model and the results would no longer be realistic. Therefore small-scale simulations are not cap able of providing accurate results for single crystal notched specimens. As such, FEA is the most feasible kind of computer simulation to analyze these specimens. As it neglects the microstructural behavior, FEA can also determine the effect of the speci men’s geometry and anisotropy on material property behavior, without consid ering the atomic interactions. Experimental Approach The double-edged notched tensile specimen lo aded to study stress and strain fields and the slip line deformation, in particular, is the most commonly used test sample in many material testing laboratories. A triaxial state of stress is cr eated around the notch, which provides the opportunity to study the sl ip system activity in three-dimensional stress fields. Experimental tests have been car ried out to determine th e effect of the load orientation on the slip systems around the not ch. These results are compared to those obtained by FEA for the same set of conditions. A constant load can be applied to th e specimen in different crystallographic orientations and the different active slip planes can be observed. This approach is

PAGE 23

11 altogether different than the elastic res ponse measured by FEA, the magnitude of the applied elastic stress gives an idea of which planes will slip first.

PAGE 24

12 CHAPTER 2 REVIEW OF LITERATURE The stress calculations of a specimen wit hout notches can be done by an analytical approach. However, as far as the modeling a nd stress calculations of the specimen with notches is concerned, it involves a lot of complexities and s till an accurate single crystal analytical model has not yet been establishe d. Quite a few researchers have investigated the stresses and sectors around the notch tip, but no method to get the precise results has been proven to be valid. To analyze the st ress-field at a single crystal notch involves calculating the accurate stress components and then predicting and defining the slip zones around the notch. Anisotropy of Elasticity Elasticity As defined by Dieter (1986) elasticity can be described by specific elastic constants that relate to atomic strengt h and spacing of the material The spacing between the atoms in an FCC is different in different direc tions. Comparing the atomic densities of the material in different directions can show this. Clearly, the at omic density of the material is not same in all the directions and so the spacing also varies with direction. From this, we can conclude that the elasticity is a function of orientation. The greatest atomic density corresponds to the least atomic spacing and most susceptible planes for slip. For example, plan es such as the ones belonging to the {111} family are termed as the ‘close-packed planes ’ as they have the least distances between the atoms. Consequently, the atoms in these pl anes do not have to travel greater distances

PAGE 25

13 A B to reach another atomic position and theref ore slip occurs most commonly on these planes. This is the case for close-packed dire ctions; the atomic spacing being less in these directions makes them most likely for slip occurrence. Figure 2.1 Schematic of FCC crystal structure Table 2.1 Atomic density on FCC crys tal planes Source: Dieter (1986) FCC Plane Atoms per unit area Atoms per unit cell {100} 2/ao 2 2 {110} 2/( 2. ao 2) 1.414 {111} 4/( 3. ao 2) 2.309 Elasticity Matrix The elastic properties for any material can be completely defined by 36 elastic constants. However, in most of the materi als, some kind of symmetry exists thereby reducing the number of indepe ndent constants. As seen in table 2.2, there are only two Table 2.2 Symmetry in various crystal structures Source: Dieter (1986) Crystal structure Number of elastic constants Tetragonal 6 Hexagonal 5 Cubic 3 Isotropic 2

PAGE 26

14 independent constants for an isotropic ma terial. Any two of the three constants E (modulus of elasticity), G (shear modulus) and (Poisson’s ratio) are enough to define the properties of the material in a particular direction. The symmetry of the cubic structures highly reduces the number of elastic constants, which in turn reduces the elasticity matrix for these structures. To reduce the number of independent constants from 36 to the final three for cubic structures, the original full elasticity matrix [aij] has to be considered: 66 65 64 63 62 61 56 55 54 53 52 51 46 45 44 43 42 41 36 35 34 33 32 31 26 25 24 23 22 21 16 15 14 13 12 11a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a aij (2.1) The strain matrix is then given by: j ij ia (2.2) Assuming elastic potential exists (i.e isothermal deformation) the following relationship is achieved in equilibrium: ji ija a (2.3) On account of this symmetry, the general matrix gets reduced to: 66 56 46 36 26 16 56 55 45 35 25 15 46 45 44 34 24 14 36 35 34 33 23 13 26 25 24 23 22 12 16 15 14 13 12 11a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a aij (2.4)

PAGE 27

15 Nickel-base superalloys exhibit orthotropi c characteristics; so they have three orthogonal planes of elastic symmetry. Consider ing the effects of cubic elastic symmetry (Lekhnitskii, 1963), the number of elastic constants reduces to three for the final elasticity matrix: 44 44 44 11 12 12 12 11 12 12 12 110 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 a a a a a a a a a a a a aij (2.5) These elasticity constants can be defined using the modulus of elasticity, the shear modulus and the Poisson’s ratio in specific directions as follows: xxE a 111 (2.6) yzG a 144 (2.7) yy xy xx yxE E a 12 (2.8) For material that has been used to conduct these tests, the values of the elasticity constants are (Swanson and Arakere, 2000): 11a= 6.494 E –8 44a= 6.369 E –8 12a=-2.063 E –8 (psi) Making use of these elastic constants and the direction cosine s, the modulus of elasticity for any material in any direc tion can be found as follows (Dieter, 1986): ) ( 2 1 ) ( 2 12 3 2 3 2 3 2 3 2 3 2 3 44 12 11 11 a a a a Euvw (2.9)

PAGE 28

16 Where 3 ,3 and 3 are the direction cosines from the load orientation to the x, y and z axes respectively (also sometimes referred to as l, m and n). Materials like tungsten always maintain their isotropic properties, even in single-crystal form and their elasticity is constant in all directions. Some mate rials, on the other hand, like nickel-base superalloys have varying elastic pr operties with varying directions. Notch Tip Deformation The stress distribution and analysis of specimens having uniform geometry and subjected to ideal load and te mperature conditions is a strai ghtforward task. However, in practical applications of superalloys, this is not the case. A common application of superalloys, for example, is a turbine blad e, which has a complex geometry and is subjected to multiaxial, centrifugal and contact stresses at very high temperature gradients. In an attempt to study the stat e of the stress for more complex specimens, notched tensile specimens are often used to re present either areas of stress concentration or theoretical fracture conditions. Rice (1987) Rice discussed the crack/notch tip stress and strain fields both for FCC and BCC crystal structures. In his work, Rice assume d the plane strain c ondition and plotted the CRSS in specific angular zones or ‘sectors’, as he termed these zones. He also stated that the stress state is uniform (independent of the an gle) within finite angular sectors near the tip and the stress state jumps discontinuously at boundaries between se ctors. Basically, he considered two specific orientations and derived an analytical solution for these orientations and also specif ied angular boundaries between va rious sectors. Rice set the

PAGE 29

17 convention for referring to notched samples in terms of notch plane, notch growth direction and notch tip direc tion (as shown in figure 2.2). Figure 2.2 Notch direction terminology. Since the assumption of plane strain is made, the yieldi ng process can be expressed in terms of a ‘yield area’. This helps to defi ne the yield stress in different areas that are separated by boundaries such th at the yield stress is uniform throughout a particular area. The slip activity occurs only in some slip systems, which combine to produce a large strain. The state of stress near the crack tip in the plastic zone that Rice has created is based on a plane state of stress in an isot ropic material. Definitely, this is not a characteristic of single crystal materials. Therefore, the results obtained using this assumption of plane strain will not corre spond well with the experimental results. Figure 2.3 Yield surface based on plane strain state of stress Load Direction Notch Growth Direction Normal to notch plane

PAGE 30

18 Another simplification that rice made wa s the assumption of zero out of plane stress and strain near the notch tip. However, a state of triaxial stress does exist near the notch. This approach therefore, cannot be used to solve for detailed strain field near the notch. The orientations Rice chose for his study produced symmetric results about the notch growth axis. He inferred that there were two conditions under which the notch surface boundary conditions could be met for all angles. The first one was that the stresses in some sectors should be below th e yield values. The second condition was that there are discontinuities at some angles. The first condition is difficu lt to compromise, as Rice had assumed a perfectly plastic state of stre ss, so all sectors must be at yield or past. The solution Rice obtained had no diffe rences for the FCC and BCC crystal structures or between the two orientations. Ho wever, there are differences as regards to the changes in the slip systems with respect to the orientations and there is a switch between the slip systems in the shear and normal directions. Thus orientation does not seem to predict the effect on the yield surf ace or sector boundaries. This contradicts the experimental results and Rice noticed that this contradiction was because he had neglected the actual rotation of the crystal lattice. Rice has also neglected the effect of strain hardening. However, he recommends in corporating all simplifications in future study of single crystal models. Crone and Shield (2001) In light of Shield’s previous work (1996), Crone and Shield conducted further experimental tests to study the notch tip de formation in two differe nt orientations of single crystal copper and copper-beryllium tensil e specimens, as shown in figure 2.8. Out of these, many other researchers for study of notch-tip deformation study have considered Orientation II. Crone and Shield also used Moir interferometry to obtain the strain fields

PAGE 31

19 [101] [010] [010] [101]to determine the sectors and the obvious visi ble changes from one area to another were defined as sector boundaries. Visible slip pa tterns do indicate slip activity, but the absence of a visible slip does not indicate the absence of any slip activity. The slip systems that do not show up on the surface may be activated internally, rather than at the surface. Even on the surface, the slip systems may show varying patterns as the load and consequently the deformation increases. Figure 2.4 Orientations used by Cr one and Shield in their tests. Table 2.3 Sector boundary angle comparisons fo r Orientation II Source: Crone and Shield (2001) Sector boundary Experimental Analytical Numerical Crone and Shield (2001) Rice (1987) Mohan et al. (1992) Cuitino and Ortiz (1996) 1-2 2-3 3-4 4-5 5-6 50-54 65-68 83-89 105-110 150 54.7 90 125.3 40 70 11 130 45 60 100 135 Crone and Shield also compared their experimental results with the solutions obtained by other researchers li ke Rice’ analytical solution as well as numerical Finite Orientation I OrientationII

PAGE 32

20 Element solutions by Mohan et al. (1992) and Cuitino and Ortiz (1996), as shown in Table 2.4. Both the numerical solutions were obtained by plane strain assumption. However, Cuitino and Ortiz stated that this pr oblem couldn’t be really considered a plane strain problem on account of the large variations in strains in ternally and on the surface. Even with the plane strain assumption, there is no similarity between the numerical and analytical results and moreover any of the three do not match with the experimental results (Figure 2.9 and Figure 2.10). The e xperimental results obtained by Crone and Shield are justifiable only in the ‘annulus of validity’, where the actual measurements are taken. This annulus is a zone at radial distances from 350 m to 750 m from the notch tip. It should be noted here that the notch width is about 100-200 m and the notch radius 50-100 m. This makes the annulus and the regi on where the sectors are measured to be 3.5-7.0 times the notch width and 7.5-15.0 tim es the notch radius from the tip. These sectors clearly allow the stress es and strains to be studied only in the plastic zone. Figure 2.5 Slip sectors observed in the expe rimental tests by Crone and Shield. Source: Crone and Shield (2001).

PAGE 33

21 Crone and Shield chose this annulus also to avoid areas too clos e to the notch tip. This is because the areas in the vicinity of the notch tip are dominated by geometry. Crone and Shield initially start with a singl e sector of slip activity and later go on adding the slip lines as the deformation proceed s and so more than one slip lines become evident in the same sector at higher radial distances from the notch tip. The slip lines keep growing in numbers until the sector is ‘f illed’. It is difficult to state what ‘filled’ implies here. However, it can be stated for sure that either the initially visible slip system or another with the same visible trace remains active throughout the process. Figure 2.6 Slip sectors and slip lines from the experimental tests by Crone and Shield. Source: Crone and Shield (2001) Crone and Shield have disregarded the hor izontal slip lines observed near the notch in Sector 1 for Orientation II and they have assumed that these slip lines as elastic

PAGE 34

22 so that they can compare their solutions to other perfectly plastic crack solutions, as seen in Figure 2.10. Sector 5 is be lieved to be the only sector showing elastic behavior on the basis of lack of any visible slip activity as well as low strain values. Therefore the determination by Crone and Shield of an elas tic region exactly ahead of the notch tip is not consistent with the visible experimental results that they have obtained. As assumed, the sectors observed are actually symmetric a bout the [101] axis ex cept some variations in the sectors 3/-3 and 4/-4 (Figure 2.10). This asymmetry may be due to deviation from the ideal conditions such as irregularities in the notch geometry, material defects, inaccurate notch plane direction and so on. Orientation I is a rotation of the Orientation II by 90 about the notch tip direction in a way that the notch tip growth direction and the notch plane are interchanged. It was mentioned previously that Rice’s solution did not vary with orientation. However, Crone and Shield’s sectors did show prominent variation with orientation, exhibiting changes in specific boundary angles as well as in the number of sectors (Table 2.5). Table 2.4 Sector boundary angles from the experimental tests by Crone and Shield Source: Crone and Shield (2001) Sector Boundary Orientation I Orientation II 1-2 2-3 3-4 4-5 5-6 35-40 54-59 111-116 138 50-54 65-68 83-89 105-110 150 Although neither the analyti cal nor the numerical so lutions match with the experimental results, Crone and Shield uphold a plane strain assumption on the basis of the numerical solution by Cuitino and Ortiz ( 1996) because the finite element results on the central plane of the specimen seemed to correspond well with the experimental

PAGE 35

23 results. However, Crone and Shield make this clear that the plane strain assumption is valid only for specific locations. In fact, the results from central pl ane do not depict the actual specimen being modeled and the stresses at the surfa ce are preferred to get more accurate results. Schulson and Xu (1997) Schulson and Xu studied the state of stress near the notch-tip for single crystal Ni3Al, which constitutes the ’-component of the material used in this study. They considered specimens with two different orient ations for this study. They used the threepoint bending technique to de form the crystals and the deformation was carried out through different zones like el astic, plastic and then thr ough initial stages of crack propagation. Figure 2.7 Specimen Orientati on used by Schulson and Xu For both of the samples, a growth directi on of [010], a notch pl ane direction of [101] and a notch tip direction of [-10-1] wa s used. Schulson and Xu calculated the state of stress (assuming isotropic material conditions) using the equations for a sharp notch by Anderson (1995). The purpose behind this was to establish an analytical model to [010] [-10-1] [-101]

PAGE 36

24 compare with their experimental results. Ho wever, isotropic cond itions do not hold good for single crystal materials. They first calculat ed the state of stress on the basis of plane stress conditions. This assumption is valid fo r large thin plates subjected to in-plane loading. Consequently, the thickness of the sp ecimen should be negligible as compared to its height and width and the onl y stress components present are x, y and xy. However, Schulson and Xu observe here th at a triaxial state of stress is created around the notch tip and as a result, out of plane material stresses are present. Then they made a second set of calculations based on the plane strain assumpti on, which again can be well applied for the isotropic conditions (where the out of plane strain is zero) and is not valid for the case of single crystal materials. In an anisotropic sing le crystal material, plane strain is a function of material orientation. The only condition when the simplified stress tensor matrix is valid is when the specimen is loaded pa rallel to the FCC lattice edges (Swanson and Arakere, 2000). Schulson and Xu may use this basic transfor mation if their specimens are oriented accurately. Even then the plane st ress is not valid because the true component stresses have out of plane components. The strain components can be obtained by multiplying the stress tensor matr ix by the stress as follows: xy zx yz z y x xy zx yz z y xa a a a a a a a a a a a 44 44 44 11 12 12 12 11 12 12 12 110 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 (2 .10)

PAGE 37

25 xy zx yz z y x z y x z y x xy zx yz z y xa a a a a a a a a a a a 44 44 44 11 12 12 12 11 12 12 12 11 (2.11) It can be seen that out of plane strain components are present; so the plane strain assumption is not valid. Figure 2.8 Slip sectors obtained by Schuls on and Xu Source: Schulson and Xu, 1997. Schulson and Xu calculated the 12 primary resolved shear stresses on the basis of plane stress and plane strain assumptions. Then using the mode I stress intensity factor K and radial distance r from the tip of the notch, they normalized the resolved shear stresses. They obtained similar slip system results for every individual assumption, however exact systems do differ in some sector s like II, III, IV and V and sector angles show significant variations for certain areas like I/II and IV/V The sector I slip systems

PAGE 38

26 are same for both the assumptions, but the st ress values are different. Also there is a considerable difference in the specific slip systems for sectors II-I II, though the systems under the plane strain assumption are just th e symmetrical systems from those of the plane stress. There is an exception though in case of sector IIb and in sector IV; the maximum stresses are resolved on the same slip system, but the directions do not match. In addition, the plane strain st resses in sector V jump on to {111} planes as indiacted in Table 2.6, while in sector II the plane strain stresses are still on the {-111} planes for the same angles. This data is in conjunction with the elastic region c onditions, that is for sectors II and III in the range 43 -100 However, it diverges significantly later in the sectors I, I/II and IV/V. The experimental results obtai ned after plastic deformati on, vary from the results predicted by either plane stress or plane stra in, but are considerably close to plane strain assumption. Usually the slip plane sector s can be observed visually. However some specific slip systems cannot be distinguished at the surface. In such cases, electron microscopes are used to determine the speci fic slip systems. The results obtained by Schulson and Xu correlate we ll to the plane stress mode l for the [01-1] and [011] dislocations. Table 2.5 Slip sectors from plane stress and plane strain assumpti ons. Source: Schulson and Xu (1996) Sector Plane Stress Plane strain (deg) Slip systems (deg) Slip systems I 0-43 (11-1)[-110] ( -111 )[ 01-1 ] 0-23 (11-1)[-110] ( -111 )[ 01-1 ] II 43-60 (111)[1-10] ( 111 )[ 01-1 ] 23-60 (-1-1-1)[-110] ( -1-1-1 )[ 01-1 ] III 60-103 (1-11)[110] ( 1-11 )[ 011 ] 60-107 (-11-1)[-1-10] ( -11-1 )[ 0-1-1 ] IV 103-180 (11-1)[011] ( -111 )[ 110 ] 107-133 (11-1)[101] ( -111 )[ 101 ] V -133-180 (-1-1-1)[0-11]

PAGE 39

27 Slip Activation and Deformation In case of an isotropic material, all the twelve primary slip systems get activated simultaneously, as is seen from figures 2.12 a nd 2.13. The Schmid factor is defined using the load orientation, the slip plane orient ation and the slip direction, as follows: cos cos m mrss Here is the load applied, rss is the RSS component al ong a given slip plane and direction, is the angle between the slip direc tion and the applied load and finally is the angle between the normal to th e slip plane and the applied load. Figure 2.9 Load and slip direction. According to the Schmid’s equation, when the RSS is equal to the yield stress of the material, the CRSS is reached. Also the slip systems with the highest Schmid’s factor will reach the CRSS first. This is true for isotropic materials. However, it cannot be Slip Direction Slip Plane Normal

PAGE 40

28 correlated to the single crysta ls and another method needs to be used for the prediction of slip plane activation of single crystal materials.

PAGE 41

29 CHAPTER 3 ANALYTICAL APPROACH The overall objective of this work is to find the state of stress in the material coordinate system of the specimen, which even tually is used to calculate the resolved shear stresses in the 12 primary slip system s. For an anisotropic specimen with cubic symmetry, the stress-strain relation is govern ed by three independent constants in the material coordinate system and a stress tensor matrix that is depende nt on the orientation. Thus it is not just governed by a single elasticity constant as in the case of an isotropic material. As the specimen orientation may not be fixed, we need to establish a relationship between the specimen and material coordinate systems, which facilitates the conversion of stress and strain fields fr om one coordinate system to another. Practically, it is very difficult to cut a sample such that its x, y and z axes are perfectly aligned to the materi al axes: [100], [010] and [001] respectively. This creates a need for transformation matrix to convert the observed stresses, which are in the specimen coordinate system into the material coordinate system. First of all, to define the elasticity matrix, it is necessa ry to determine the exact or ientation of the specimen under study in terms of the material Mi ller indices (direction indices). There are two methods to bring about th e transformation of the stress components from specimen coordinate system to the materi al coordinate system. The first method is a straightforward method, in which the angl es between the material and specimen coordinate axes are directly measured to obt ain the direction cosines. This method can be employed if the angles between the coordina te systems can be easily found. The second

PAGE 42

30 method can be used for more complex orientations, where the angles between the two systems are difficult to determine, just by inspection. Here, the transformation equations can be obtained if the Miller indices of the specimen coor dinate axes are known with respect to the material coordinate axes. In this technique, the original axes are transformed through a certain number (usually th ree) of 2-D rotations to obtain the final transformed axes. In analytical approach, any of the two methods can be used. However, in experimental specimens, it is difficult to determine the exact Miller indices of the transformed axes and it is preferable to use th e first method. As stated earlier, the original coordinate system will be termed as the material coordinate system: xo= [100], yo= [010] and zo= [001] (figure 3.1). The transformed coordi nate system is termed as the specimen coordinate system. It will be denoted by x”, y” and z”. All the prope rties associated with this system will be denoted by the double-prime symbol. Figure 3.1 Material and Sp ecimen coordinate system. x y" z xo[100] yo[010] zo[001]

PAGE 43

31 Coordinate Axes Transforma tion Using Miller Indices As discussed earlier, it is possible to obtain the final transformed axes if their Miller indices are known. A step-by-step transformation process can accomplish this process. The first transformation can be brought about by rotating the xo, yo and zo axes through an angle about the zo axis. It is obvious th at this is a 2-D rotation with the original and final zaxis being the same. The equations of transformation for this first step are: ) sin( ) cos( o oy x x (3.1) ) cos( ) sin( o oy x x (3.2) oz z (3.3) The above equations in matrix form can be written as: o o oz y x z y x 1 0 0 0 ) cos( ) sin( 0 ) sin( ) cos( (3.4) Figure 3.2 First rotation of the coor dinate system about the z-axis yo[010] xo[100] zo[001] x y LoadDirection

PAGE 44

32 Load Direction Figure 3.3 Projections from the orig inal axes on the projected axes The second transformation is again a 2-D rota tion, but this time the axes are rotated by an angle ’ about the y-axis. Therefore in this transformation, the y-axis is same before and after the rotation. Proceeding in the same way as for the first transformation, we get the second following transformation matrix for the second step: z y x z y x ) cos( 0 ) sin( 0 1 0 ) sin( 0 ) cos( (3.5) Figure 3.4 Second rotation of the coor dinate system about the z-axis yo[010] xo[100] x y A A ’ B ’ B y, y’ z x x’ ’ ’ z’

PAGE 45

33 z” y’ y” ” ” Finally, the axes have to be rotate d about the x-axis through an angle ”. This third and final step gives us the complete transf ormation to the specimen coordinate systemx y andz The transformation matrix for the third step is: o o oz y x z y x ) cos( ) sin( 0 ) sin( ) cos( 0 0 0 1 (3.6) Figure 3.5 Third rotation of the c oordinate system about the z-axis The overall transformation from the materi al to specimen coordinate system can then be calculated multiplying the three transformation matrices obtained in the three transformations. o o oz y x z y x3 3 3 2 2 2 1 1 1 (3.7) where 1 0 0 0 ) cos( ) sin( 0 ) sin( ) cos( ) cos( 0 ) sin( 0 1 0 ) sin( 0 ) cos( ) cos( ) sin( 0 ) sin( ) cos( 0 0 0 13 3 3 2 2 2 1 1 1 (3.8) x’, x” z’

PAGE 46

34 The values obtained through the above e quation are the direction cosines of the specimen coordinate system with respect to the material coordinate system. The table below summarizes the above equation: Table3.1 The direction cosines of the specime n coordinate system with respect to the material coordinate system The values of the direction cosines or the final transformation matrix will be: ) cos( ) cos( ) cos( ) sin( ) sin( ) sin( ) cos( ) sin( ) sin( ) cos( ) sin( ) cos( ) cos( ) sin( ) cos( ) cos( ) sin( ) sin( ) sin( ) sin( ) cos( ) cos( ) sin( ) sin( ) sin( ) sin( ) cos( ) cos( ) cos(3 3 3 2 2 2 1 1 1 (3.9) In some cases, fewer than three steps are involved to arrive at the final direction cosines. The sequence of rotation is arbitrar y. However, it is important to check the accuracy of the results obtained. There are several methods to ensure this. Example Transformation To demonstrate this method of arriving at the direction cosines, we will consider a case in which a load is applied in the [3 1 -2] direction. This transformation also illustrates a case in which only two steps are involved to get the overall transformation. Here the first step is eliminated (i.e.0 ) and step two begins by reflecting the load ox oyoz x y z 1 2 3 1 2 3 1 2 3

PAGE 47

35 vector onto the x-z plane. The reflection is simply a triangle whose sides comprise of the x and z Miller indices: x = 2 and z = 3. The first rotation can be evaluated as: 2 3 tan1 309 56 (3.10) Again, the second angle of rota tion can be obtained as follows: 2 2 1) 2 ( 3 1 tan 501 15 (3.11) Therefore, the direction cosines for th e overall transformation can be obtained by using the previously established equation: ) cos( 0 ) sin( 0 1 0 ) sin( 0 ) cos( ) cos( ) sin( 0 ) sin( ) cos( 0 0 0 13 3 3 2 2 2 1 1 1 (3.12) Substituting the values for and we get: 5345 0 832 0 8017 0 1482 0 9636 0 2223 0 832 0 0 5547 03 3 3 2 2 2 1 1 1 (3.13) Transformation Matrices for Stress and Strain Fields After arriving at the values of the di rection cosines between the material and specimen coordinate system, we can establish the matrices to transform the stresses and strains. Using these transformed matrices, we can finally evaluate the resolved stresses and strains in any of the slip systems.

PAGE 48

36 The stress transformation, as in Lethniskii is given as: Q (3.14) Q Q1 (3.15) Q is the stress transformation matrix is a function of the direction cosines: ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( 2 2 2 2 2 2 2 2 21 2 2 1 1 3 3 1 2 3 3 2 3 3 2 2 1 1 1 2 2 1 1 3 3 1 2 3 3 2 3 3 2 2 1 1 1 2 2 1 1 3 3 1 2 3 3 2 3 3 2 2 1 1 1 2 3 1 2 3 2 3 2 2 2 1 1 2 3 1 2 3 2 3 2 2 2 1 1 2 3 1 2 3 2 3 2 2 2 1 Q (3.16) The state of stress is defi ned in terms of the specimen or material as: xy zx yz z y x xy zx yz z y x (3.17) Same procedure is followed to ge t the strain transformation matrix: Q (3.18) Q Q1 (3.19) Like the stress transformation matrix, the stress transformation matrix Q is also a function of the direction cosines, but it does differ from the stress matrix:

PAGE 49

37 ) ( ) ( ) ( 2 2 2 ) ( ) ( ) ( 2 2 2 ) ( ) ( ) ( 2 2 21 2 2 1 1 3 3 1 2 3 3 2 3 3 2 2 1 1 1 2 2 1 1 3 3 1 2 3 3 2 3 3 2 2 1 1 1 2 2 1 1 3 3 1 2 3 3 2 3 3 2 2 1 1 1 2 3 1 2 3 2 3 2 2 2 1 1 2 3 1 2 3 2 3 2 2 2 1 1 2 3 1 2 3 2 3 2 2 2 1 Q(3.20) The stress-strain relationship for an isotropi c material for a uniaxial state of stress, using Hooke’s law can be given as: E (3.21) However, the generalized Hooke’s law for a homogeneous anisotropic material can be given as: ijA (3.22) where, 1ij ija A (3.23) Here ijA and ija are the elastic constant matrices. It should be noted here that ija is a symmetric matrix such that: ji ija a (3.24) So we can state: ija (3.25) and ija (3.26) The elasticity matrix also has to be tr ansformed. However, it does not lose its symmetry. The number of constants limit to 21 depending on the direction cosines or specimen orientation.

PAGE 50

38 Q a Q aij T ij (3.27) Therefore, we can conclude that if the component stresses are known in the specimen coordinate system, then it is po ssible to find the stre ss components in the material coordinate system by using the above equations. Shear Stresses and Strains in the Slip Systems The component stresses are sufficient to defi ne the state of stress of the material. However, in anisotropic materials, these stresses do not give a ny idea about the slip system activity. It is therefor e imperative to transform thes e component stresses into the resolved shear stresses along the 12 primary s lip systems. Thus after arriving at the component stresses, we have to again pe rform a transformation to determine these resolved shear stresses. Resolved Shear Stress Components The resolution of stre sses along the slip syst ems is calculated as: S c (3.28) Here c and S are constants depending on s lip plane and direction: 2 2 2 2 2 21 i i i i i i iw v u l k h c (3.29) i i i i i i i i i iu v w w l v k u h S (3.30) Here, w v u is the slip direction and l k h is the slip plane. For the primary octahedral planes, c is a constant and then the Resolved Shear Stress (RSS) matrix is given by:

PAGE 51

39 xy zx yz z y xu v w w l v k u h u v w w l v k u h u v w w l v k u h u v w w l v k u h u v w w l v k u h u v w w l v k u h u v w w l v k u h u v w w l v k u h u v w w l v k u h u v w w l v k u h u v w w l v k u h u v w w l v k u h 12 12 12 12 12 12 12 12 12 11 11 11 11 11 11 11 11 11 10 10 10 10 10 10 10 10 10 9 9 9 9 9 9 9 9 9 8 8 8 8 8 8 8 8 8 7 7 7 7 7 7 7 7 7 6 6 6 6 6 6 6 6 6 5 5 5 5 5 5 5 5 5 4 4 4 4 4 4 4 4 4 3 3 3 3 3 3 3 3 3 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 1 12 11 10 9 8 7 6 5 4 3 2 1 (3.31) Solving the above equation for the 12 primary systems, we get: xy zx yz z y x 1 1 0 0 1 1 1 0 1 1 0 1 0 1 1 1 1 0 1 0 1 1 0 1 0 1 1 1 1 0 1 1 0 0 1 1 0 1 1 1 1 0 1 1 0 0 1 1 1 0 1 1 0 1 1 1 0 0 1 1 0 1 1 1 1 0 1 0 1 1 0 1 6 112 11 10 9 8 7 6 5 4 3 2 1 (3.32) Similar procedure has to be followe d for calculating the shear strains: S c (3.33)

PAGE 52

40 The resolved stress and strain fields al ong the 12 primary octahedral planes are determined following the above steps. This give s a lead for predicting the slip systems for a particular specimen orie ntation and applied load.

PAGE 53

41 CHAPTER 4 NUMERICAL APPROACH: FINITE ELEMENT METHOD The finite element technique can be used to model the specimens having a given geometry and loaded in given orientations. This helps to simulate the conditions in tensile tests. Specimens with different geometry we re modeled in two different orientations using the ANSYS finite element software and they were analyzed to predict the stress fields and sectors around the notch of the specimen. One of the two specimens are based on pr evious work (Rice, 1987; Schulson and Xu, 1997; Crone and Shield, 2001) and both of these correlate to collaborative work between the Mechanical Engineering and Materials Science departments of the University of Florida (UF). In this numerical approach, slip-deforma tion is predicted by the highest resolved shear stress obtained in the numerical m odel. The slip systems having the highest resolved shear stress should be the slip lines in the experimental model. These results should match with the experimental result s that are obtained in the tensile test. First of all, a solid specimen (without notches) was modeled for a particular load and orientation using ANSYS. These results were then compared to the analytical solution (that was set up in chapter3) to ve rify the accuracy of the results obtained through the finite element analysis. Pro ceeding on these lines, two notches were introduced in the specimen to simulate the exact tensile test specimen. The component stresses obtained for this model from the fi nite element solution, which were in the material coordinate system, were incorporated into the analytical solution to get the RSS

PAGE 54

42 in the 12 slip systems. This was done for a wide range of radial and angular distance around the notch. This data was used to predic t the slip activation fo r various sectors in the vicinity of the notch. Figure 4.1 The notched specimens analyzed using finite element method. Verification of the Finite Element Model Results The specimen without notches can be analy zed using the procedure established in Chapter 3 and these results can be compared w ith the finite element model to verify the accuracy of the results. This was done using th e same dimensions (except the notches) as the actual specimen for consistency. The app lied load for all the cases was consistently 100 lbs. The reason for choosing this load wa s that this load causes stresses well below the yield stress of the material thereby gi ving the opportunity to study the slip plane activity purely in th e elastic zone. For the specimen without notches, the component stresses evaluated from analytical methods s eem to match pretty cl osely to those obtained by the FEM model. So this initial test of the comparing the results between the two methods does work. However, it is better to ve rify these results on the basis of strain components also. In the analytical technique, the strain components can be evaluated by making use of the stress tensor matrix. Th e strain components also seem to match [010] [-12-1] [001] [100] [111] [30-3] Specimen I Specimen II

PAGE 55

43 consistently for the two techniques. As th e results obtained thr ough the analytical and numerical methods agree with each other, it is safe to proceed to obtain results for the specimens with notches. Specimen Orientation and Dimensions Here two specimens with different orient ation and geometry will be studied. While modeling the specimen geometry, only the speci men body is considered and the end grips involved in the actual tensile te st are neglected. There are certain reasons for neglecting these end grips. First of all, the mechanics at the grips involves tensile rig contact pressure, loading rate and this makes it significantly different than the mechanics from the center of the tensile specimen. In the experimental model, different deformation mechanisms are seen at the grips and these mechanisms can cause fracture even there. But the aim of these tests is to study the de formation mechanisms at the center of the specimen and the stresses generated in this region. Second of all, the grips may be changed in the pursuit of getting more accu rate results. Accordingly, the effect and deformation mechanisms at the ends will change. Here, the numerical model will not be subjected to the effect caused by the grips. As far as the geometry of the specimens is concerned, both the specimens have a different geometry. However, in both th e specimens, the notch is modeled as a combination of a rectangle and a semi-circle. However, in the actual specimen, the notch has some offset with the horizontal and also an offset in the y-direction from the center of the specimen. In addition, the notch tip is not a perfect semi-circle, but a smaller arc. In the numerical technique, all the notch dimens ions for both right and left notches were modeled equal, although there are slight differences between the dimensions of the two notches. Also, the notch radius was set equal to the notch-height ma king it a perfect semi-

PAGE 56

44 circle. In this work, the specimen orientati on is of primary importance. This makes the minor changes in the notch dimensions negligible. However, the model can be made perfect geometrically if there is a need for more specific results. Figure 4.2 Specimen specifications Table 4.1 Actual and specimen geometry for Specimen I Specimen Geometry (mm) Actual FEM Width 5.100 5.100 Height 19.000 19.000 Thickness 1.800 1.800 Right Notch Length 1.300 1.550 Left Notch Length 1.550 1.550 Right Notch Height 0.113 0.113 Left Notch Height 0.111 0.113 Right Notch Radius 0.045 0.055 Left Notch Radius 0.055 0.055 Thickness Height Notch Height NotchWidth Width Notch Length No tchRadius

PAGE 57

45 Table 4.2 Actual and specimen geometry for Specimen II Specimen Geometry (mm) Actual FEM Width 5.04 5.04 Height 17.594 17.594 Thickness 1.82 1.82 Right Notch Length 1.399 1.399 Left Notch Length 1.36 1.399 Right Notch Height 0.084 0.084 Left Notch Height 0.0845 0.113 Right Notch Radius 0.056 0.056 Left Notch Radius 0.056 0.056 Finite Element Model Characteristics While modeling the specimen in the Finite Element software, all the details of the actual tensile test were incorporated so as to obtain accurate results. Accordingly, the material properties, orientation and boundary conditions were specified to simulate the test conditions. Since th e stresses near vicinity of the notch are of prime importance, the areas near the notches were meshed more densely than the rest of the areas. Material Properties To define the stresses or stra ins in any particular directi on, it is convenient to define the material properties with respect to the mate rial coordinate first. However, in the FEM, the specimen is modeled such that its prope rties are defined with respect to the global coordinate system. Therefore, proper adju stments must be made while defining the direction cosines so as to create the materi al coordinate system. In case of ANSYS, the material properties are aligned in the same orientation as the element coordinate system. Therefore, the element coordinate system must be defined for the same orientation as the material coordinate system so that the material properties are applied in the right orientation.

PAGE 58

46 x” y” Figure 4.3 The specimen is first defined with respect to the global coordinate system and then the material coordinate system is specified later. The model created in the finite element software is a linear-elastic, orthotropic model. ANSYS has a wide range of threedimensional elements to take care of anisotropic and material properties. To mode l the single crystal material, one of these elements needs to be used. Also, either the three independe nt stress tensors (44 12 11, ,a a a) or the three independent directional properties ( ,G E) need to be defined to accurately model the single crystal material. Elements and Meshing There are two different elements chosen to mesh the specimen model created in ANSYS: PLANE2 and SOLID95. Th e front face of the three-dimensional solid model is meshed with the PLANE2 elements. The front face has an exact element sizing along the specified radial lines around the notch tip at intervals of 5.The PLANE2 element is a two-dimensional, six-node triangular solid. This element has a quadratic displacement xo yo zo z” Applied Load

PAGE 59

47 behavior and can be used to good results for modeling irregular meshes. It has six nodes and two degrees of freedom, which helps to in corporate orthotropic material properties. Figure 4.4 Meshing around the notch in ANSYS Sweeping the three-dimensional elements through the volume can bring about the meshing of the entire model. It should be noted that the three-dimensional elements retain the sizing definitions specified for the tw o-dimensional elements defined on the x-y planes (front face) through the thickness of the specimen. The SOLID95 element used for the three-dimensional meshing is a 20-node el ement with each three degrees of freedom for each node. This element also works well for irregular shapes and curved boundaries without significant loss of accur acy. This is why SOLID95 is selected as the areas near the notches are of prime interest. This element also takes care of the orthotropic material properties. Here, the original element struct ure is left intact instead of using the pyramidal shape, which reduces run time. The pyramidal-shaped element does reduce the analysis and solution time, but at the cost of accuracy. Using the original element structure also makes sense in the fact that the stress gradient near the notches is very

PAGE 60

48 large. In addition, SOLID95 al so has features like plasticity, creep, stress stiffening, large deflection, and large strain capabilities. Figure 4.5 Schematic of the SOLID95 element in ANSYS. Source ANSYS 6.1 Elements Reference, 2002. Figure 4.6 Radial arcs used for element lo cation and sizing; cent ered at notch tip.

PAGE 61

49 90 Through ANSYS the stress tensor can be tran sformed to any particular orientation to get the stress and strain values. To get a nea r-field as well as far-field state of stress in terms of radial and angular distances, six con centric arcs were generated at the following radial distances form the notch tip: 0.25* 0.5* , 2* 5* ; where is the radius of the notch. The reason for creating these arcs is that in ANSYS, the stress calculations can be obtained only at the corn er nodes of the elements and not at all the nodes. Figure 4.7 Radial arcs fo r stress field calculations Finite Element Solution The node-selection along the selected con centric arcs around the notch is done manually using the ANSYS Graphical User Interface (GUI). Initially all the nodes starting from 0 to 180 are selected. However, on account of the symmetry, the calculations of the stresses in the 0 to -180 need not be done. This assumption was validated by calculating the stre sses all over the notch for the arc, wherein symmetry was found in the stress calculati ons in the two semi-circular regions. Though the resolved shear stresses for the negative angles change direction in conjunction with the changing 5* 2* 0.5* 0.25* 0

PAGE 62

50 direction of shear deformation between the top and lower halves of the test specimen, the magnitudes of the maximum resolved shear stresses are same for the positive and negative angles. Another point of interest in the FEM model is regarding the plane of data collection. The element sizing was done in su ch a way that data collection was possible on five different planes incl uding the front, middle and back planes of the specimen. However, in the collaborative interests with the MSE department, the stress calculations on the front and middle planes we re focused on. It should be noted that here we are not assuming a state of plane strain. Assumptions In the finite element analysis, we have c onsidered the model to be a linear elastic model. This assumption is useful to predict th e initial slip, but it fail s to accurately give us the idea about the forthcoming plastic be havior around the notch. In FEM, the effects due to the temperature can be ta ken into account. However, in order to be consistent with the MSE department, the material propertie s at a constant room temperature were considered in the FE analysis. The effects of microstructural mechanisms are also been disregarded in this model. The last assumpti on is regarding the omi ssion of the effect of crystal lattice orientation. This assump tion suits well for these conditions because according to Stouffer and Dame (1996), crysta l lattice rotation onl y takes place if the there is some strain in a single slip system and the load applied to the specimen is low enough to prevent this effect from occurring. This FE model only considers the deformation in the elastic zone. Here, only the elastic stresses and strains are emphasized rather than the values at the point of fracture.

PAGE 63

51 This model can be the basis for further st udy that includes plastic ity, creep and other behavior caused by very hi gh stress and strain values.

PAGE 64

52 CHAPTER 5 RESULTS AND DISCUSSION As discussed in the previous chapter, the FEM is used to simulate the tensile test conditions so that the slip system activity a nd subsequently the sectors can be obtained. The results for both the specimens are plotted in terms of the RSS as a function of radial and angular distances. Here, the direction of the slip is not of any concern. As such, only the absolute values of the RSS are consider ed. The stresses are symmetric about the notch growth direction in case of specimen I. Th erefore only the stresse s on the upper portion of the notch growth direction are plotted. However, the orientation of specimen II is such that the stress distribution is not symmetric about the notch growth direction and so the stresses on the upper as well as lower portion of the notch growth direction have been plotted. The slip sectors will be plotted as th e dominant slip system at any angle theta. The results for both the specimens (with differe nt geometry and load orientation) will be analyzed separately. Specimen I As stated earlier, specimen I was loaded in the [001] direction with a [010] growth direction and a [100] notch pl ane. The results have been plotted for the RSS on the 12 primary slip systems. The ra dial distances vary from 0.25* to 5* and the angular distances vary from 0 to the top or bottom of the notch (100 for 0.25* to 165 for 5* ). The maximum RSS at any location is 6= 8=31,140 psi at R=0.5* and 105 on the mid-planes. The slip system with the maxi mum RSS varies with varying radial and angular distances. On this basis, sectors we re determined for the stresses on the surface

PAGE 65

53 and the mid-planes. The RSS varies as the st ate of stress changes away from the notch tip. However, the variation of RSS for each s lip system is different for different slip systems. So the relative magnitudes of R SS on different slip systems may vary with respect to each other. Therefore, it should be noted for a given angle, the dominant slip system is not constant with changing radius. The plots between the RSS and theta highlig ht the effect of theta only on the RSS values. However, the effect of radius on the RSS can be understood by combining the effects of all the radii. Figure (5.6) and (5.12) give a picture of the combined effect of the radial and angular distances on the RSS. For ex ample, consider an angle of 70. The radar plots (Figure 5.7 and 5.8) also give an idea of the combined effect of stresses at various radial and angular distances. However, the radar plots are difficult to comprehend. So they are plotted only for a si ngle set of data. Moreover, th e sector plots give a much simple picture of the combined effect of theta and vary ing radii on the RSS.

PAGE 66

54 RSS Vs. Theta for 0.25*rho0 5000 10000 15000 20000 25000 0102030405060708090100 Theta (deg)RSS (psi) T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 T12 Figure 5.1 RSS on the 12 primary slip systems for the uppe r portion of the notch gr owth direction for R=0.25*

PAGE 67

55 RSS Vs. Theta for 0.5*rho0 5000 10000 15000 20000 25000 30000 020406080100 Theta (deg)RSS (psi) T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 T12 Figure 5.2 RSS on the 12 primary slip systems for the uppe r portion of the notch gr owth direction for R=0.5*

PAGE 68

56 RSS Vs. Theta for 1*rho0 2000 4000 6000 8000 10000 12000 14000 16000 18000 020406080100120 Theta (deg)RSS (psi) T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 T12 Figure 5.3 RSS on the 12 primary slip systems for the upper portion of the notch growth direction for R=1*

PAGE 69

57 RSS Vs. Theta for 2*rho0 2000 4000 6000 8000 10000 12000 020406080100120140160 Theta (deg)RSS (psi) T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 T12 Figure 5.4 RSS on the 12 primary slip systems for the upper portion of the notch growth direction for R=2*

PAGE 70

58 RSS Vs. Theta for 5*rho0 1000 2000 3000 4000 5000 6000 7000 8000 020406080100120140160 Theta (deg)RSS (psi) T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 T12 Figure 5.5 RSS on the 12 primary slip systems for the upper portion of the notch growth direction for R=5*

PAGE 71

59 Figure 5.6 Slip system sectors on the surface for specimen I

PAGE 72

60 Figure 5.7 Upper portion of the radar plot showin g the surface stresses fo r the [100] orientation.

PAGE 73

61 Figure 5.8 Lower portion of the radar plot showi ng the surface stresses for the [100] orientation.

PAGE 74

62Table 5.1 Specimen I dominant slip system sectors on the surface R=0.25* R=0.5* R=1.0* Sector max Slip system max Slip system max Slip system I 0-41 8 (1-1-1)[0-11] 0-56 8 (1-1-1)[0-11] 0-52 10 (-1-11)[011] II 41-100 6 (-11-1)[011] 56-105 6 (-11-1)[011] 52-56 2 (111)[0-11] III 56-116 6 (-11-1)[011] IV 116-120 4 (-11-1)[10-1] V VI R=2.0* R=5.0* Sector max Slip system max Slip system I 0-50 10 (-1-11)[011] 0-46 10 (-1-11)[011] II 50-60 2 (111)[0-11] 46-56 2 (111)[0-11] III 60-68 3 (111)[1-10] 56-64 3 (111)[1-10] IV 68-150 6 (-11-1)[011] 64-131 6 (-11-1)[011] V 131-164 9 (-11-1)[011] VI 164-165 6 (111)[1-10]

PAGE 75

63 RSS Vs. Theta for 0.25*rho0 4000 8000 12000 16000 20000 24000 0102030405060708090100 Theta (deg)RSS (psi) T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 T12 Figure 5.9 RSS on the 12 primary slip systems for the uppe r portion of the notch gr owth direction for R=0.25*

PAGE 76

64 RSS V. Theta for 0.5*rho0 5000 10000 15000 20000 25000 30000 35000 020406080100 RSS (psi)Theta (deg) t1 t2 t3 t4 t5 t6 t7 t8 t9 t10 t11 t12 Figure 5.10 RSS on the 12 primary slip systems for the upper portion of the notch gr owth direction for R=0.5*

PAGE 77

65 RSS V Theta for 1*rho0 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000 020406080100120 Theta (deg)RSS (psi) t1 t2 t3 t4 t5 t6 t7 t8 t9 t10 t11 t12 Figure 5.11 RSS on the 12 primary slip systems for the upper portion of the notch gr owth direction for R=1*

PAGE 78

66 RSS V. Theta for 2*rho0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 020406080100120140160 RSS (psi)Theta (deg) t1 t2 t3 t4 t5 t6 t7 t8 t9 t10 t11 t12 Figure 5.12 RSS on the 12 primary slip systems for the upper portion of the notch gr owth direction for R=2*

PAGE 79

67 RSS V. Theta for 5*rho0 1000 2000 3000 4000 5000 6000 7000 020406080100120140160180 RSS (psi)Theta (deg) t1 t2 t3 t4 t5 t6 t7 t8 t9 t10 t11 t12 Figure 5.13 RSS on the 12 primary slip systems for the upper portion of the notch gr owth direction for R=5*

PAGE 80

68 Figure 5.14 Slip system sectors on the mid-planes for specimen I

PAGE 81

69 Table 5.2 Specimen I dominant slip system sectors on the on the mid-planes R=0.25* R=0.5* R=1.0* Sector max Slip system max Slip system max Slip system I 0-44 3, 5, 7, 12 (111)[1-10], (-11-1)[110], (1-1-1)[110], (-1-11)[1-10] 0-105 3, 5, 7, 12 (111)[1-10], (-11-1)[110], (1-1-1)[110], (-1-11)[1-10] 0-38 2, 10 (111)[0-11], (-1-11)[011] II 44-95 6, 8 (-11-1)[011], (1-1-1)[0-11] 38-112 6, 8 (-11-1)[011], (1-1-1)[0-11] III 95-100 3, 5, 7, 12 (111)[1-10], (-11-1)[110], (1-1-1)[110], (-1-11)[1-10] 112-120 4, 9 (-11-1)[10-1] (1-1-1)[101] R=2.0* R=5.0* Sector max Slip system max Slip system I 0-53 2, 10 (111)[0-11], (-1-11)[011] 0-58 2, 10 (111)[0-11], (-1-11)[011] II 53-120 6, 8 (-11-1)[011], (1-1-1)[0-11] 58-130 6, 8 (-11-1)[011], (1-1-1)[0-11] III 120-150 4, 9 (-11-1)[10-1] (1-1-1)[101] 130-157 4, 9 (-11-1)[10-1] (1-1-1)[101] IV 157-163 6, 8 (-11-1)[011], (1-1-1)[0-11] V 163-165 2, 10 (111)[0-11], (-1-11)[011]

PAGE 82

70 Specimen II As stated earlier, specimen I was loaded in the [111] direction with a [-1 2-1] growth direction and a [3 0-3] notch plane. The load axis in this case is such that the RSS values will not be symmetrical above and be low the notch. Therefore results for both the upper and bottom parts of the notch have been plotted. The results ha ve been plotted for the RSS on the 12 primary slip systems. The radial distances vary from 0.25* to 5* and the angular distances vary from 0 to the top or bottom of the notch (100 for 0.25* to 165 for 5* ). The maximum RSS at any location is 6=36,090 psi at R=0.5* and 105 on the midplanes. The slip system with the maximum RSS varies with varying radial and angular distances. On this basi s, sectors were determined for the surface stresses. The RSS varies as the sate of stre ss changes away from the notch tip. However, the variation of RSS for each sl ip system is different for different slip systems. So the relative magnitudes of RSS on different slip syst ems may vary with respect to each other. Therefore, it should be noted that for a give n angle, the dominant slip system is not constant with changing radius. Again, the sector plots in Figure (5.20), (5 .26), (5.32) and (5.38) give a picture of the combined effect of theta and radii on the RSS values.

PAGE 83

71 RSS V. Theta for 0.25*rho0 4000 8000 12000 16000 20000 0102030405060708090100 Theta (deg)RSS (psi) T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 T12 Figure 5.15 RSS on the 12 primary slip systems for the upper portion of the notch gr owth direction for R=0.25*

PAGE 84

72Error! RSS V. Theta for 0.5*rho0 5000 10000 15000 20000 25000 30000 0102030405060708090100 Theta (deg)RSS (psi) T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 T12 Figure 5.16 RSS on the 12 primary slip systems for the upper portion of the notch gr owth direction for R=0.5*

PAGE 85

73 RSS V. Theta for 1*rho-2000 2000 6000 10000 14000 18000 020406080100120 Theta (deg)RSS (psi) T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 T12 Figure 5.17 RSS on the 12 primary slip systems for the upper portion of the notch gr owth direction for R=

PAGE 86

74 RSS V. Theta for 2*rho0 2000 4000 6000 8000 10000 12000 020406080100120140 Theta (deg)RSS (psi) T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 T12 Figure 5.18 RSS on the 12 primary slip systems for the upper portion of the notch gr owth direction for R=2*

PAGE 87

75 RSS V. theta for 5*rho0 1000 2000 3000 4000 5000 6000 7000 8000 020406080100120140160 Theta (deg)RSS (psi) T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 T12 Figure 5.19 RSS on the 12 primary slip systems for the upper portion of the notch gr owth direction for R=5*

PAGE 88

76 Figure 5.20 Slip system sectors on th e surface on the upper portion of the not ch growth direction for specimen II

PAGE 89

77 Table 5.3 Specimen II dominant slip system sectors on the su rface on the upper portion of the notch growth direction R=0.25* R=0.5* R=1.0* Sector max Slip system max Slip system max Slip system I 0-13 10 (-1-11)[011] 0-50 10 (-1-11)[011] 0-70 10 (-1-11)[011] II 13-27 6 (-11-1)[011] 50-105 5 (-11-1)[110] 70-110 5 (-11-1)[110] III 27-100 5 (-11-1)[110] 110-118 3 (111)[1-10] IV 118-120 9 (1-1-1)[101] V VI R=2.0* R=5.0* Sector max Slip system max Slip system I 0-63 10 (-1-11)[011] 0-18 9 (1-1-1)[101] II 63-67 11 (-1-11)[101] 18-58 10 (-1-11)[011] III 67-119 3 (111)[1-10] 58-102 5 (-11-1)[110] IV 119-127 6 (-11-1)[011] 102-107 6 (-11-1)[011] V 127-137 2 (111)[0-11] 107-113 5 (-11-1)[110] VI 137-138 5 (-11-1)[110] 113-117 3 (111)[1-10] VII 138-150 3 (111)[1-10] 117-167 2 (111)[0-11] VIII 167-170 12 (-1-11)[1-10]

PAGE 90

78 RSS V. Theta for 0.25*rho0 4000 8000 12000 16000 20000 0102030405060708090100Theta (deg)RSS (psi) T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 T12 Figure 5.21 RSS on the 12 primary slip systems for the lo wer portion of the notch gr owth direction for R=0.25*

PAGE 91

79 RSS V. Theta for 0.5*rho0 4000 8000 12000 16000 20000 24000 020406080100 Theta (deg)RSS (psi) T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 T12 Figure 5.22 RSS on the 12 primary slip systems for the lo wer portion of the notch gr owth direction for R=0.5*

PAGE 92

80 RSS V. Theta for 1*rho0 4000 8000 12000 16000 020406080100120Theta (deg)RSS (psi) T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 T12 Figure 5.23 RSS on the 12 primary slip systems for the lower portion of the notch growth direction for R=

PAGE 93

81 RSS V. Theta for 2*rho0 2000 4000 6000 8000 10000 12000 020406080100120140Theta (deg)RSS (psi) T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 T12 Figure 5.24 RSS on the 12 primary slip systems for the lower portion of the notch growth direction for R=2*

PAGE 94

82 RSS V. Theta for 5*rho0 2000 4000 6000 8000 020406080100120140160 Theta (deg)RSS (psi) T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 T12 Figure 5.25 RSS on the 12 primary slip systems for the lower portion of the notch growth direction for R=5*

PAGE 95

83 Figure 5.26 Slip system sectors on the surface on the lower portion of the not ch growth direction for specimen II

PAGE 96

84 Table 5.4 Specimen II dominant slip system sectors on the surface on the lower portion of the notch growth direction R=0.25* R=0.5* R=1.0* Sector max Slip system max Slip system max Slip system I 0-100 7 (1-1-1)[110] 0-1 10 (-1-11)[011] 0-1 10 (-1-11)[011] II 1-105 7 (1-1-1)[110] 1-111 7 (1-1-1)[110] III 111-117 3 (111)[1-10] IV 117-120 6 (-11-1)[011] V VI R=2.0* R=5.0* Sector max Slip system max Slip system I 0-63 10 (-1-11)[011] 0-18 9 (1-1-1)[101] II 63-67 11 (-1-11)[101] 18-58 10 (-1-11)[011] III 67-119 3 (111)[1-10] 58-102 5 (-11-1)[110] IV 119-127 6 (-11-1)[011] 102-107 6 (-11-1)[011] V 127-137 2 (111)[0-11] 107-113 5 (-11-1)[110] VI 137-138 5 (-11-1)[110] 113-117 3 (111)[1-10] VII 138-150 3 (111)[1-10] 117-167 2 (111)[0-11] VIII 167-170 12 (-1-11)[1-10]

PAGE 97

85 RSS V. Theta for 0.25*rho0 4000 8000 12000 16000 20000 24000 28000 020406080100 Theta (deg)RSS (psi) T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 T12 Figure 5.27 RSS on the 12 primary slip systems on the mid-planes for the upper portion of the notch growth direction for R=0.25

PAGE 98

86 RSS V. Theta for 0.5*rho0 4000 8000 12000 16000 20000 24000 28000 32000 36000 40000 020406080100 RSS (psi)Theta (deg) T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 T12 Figure 5.28 RSS on the 12 primary slip systems on the mid-planes for the upper portion of the notch growth direction for R=0.5*

PAGE 99

87 RSS V. Theta for 1* rho0 4000 8000 12000 16000 20000 020406080100120 Theta (deg)RSS (psi) T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 T12 Figure 5.29 RSS on the 12 primary slip systems on the mid-planes for the upper portion of the not ch growth direction for R=

PAGE 100

88 RSS V. Theta for 2*rho0 2000 4000 6000 8000 10000 12000 020406080100120140 Theta (deg)RSS (psi) T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 T12 Figure 5.30 RSS on the 12 primary slip systems on the mid-planes for the upper portion of the not ch growth direction for R=2*

PAGE 101

89 RSS V. Theta for 5*rho0 1000 2000 3000 4000 5000 6000 7000 020406080100120140160180Theta (deg)RSS (psi) T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 T12 Figure 5.31 RSS on the 12 primary slip systems on the mid-planes for the upper portion of the not ch growth direction for R=5*

PAGE 102

90 Figure 5.32 Slip system sectors on the mid-planes on the upper portion of the notch growth direction for specimen II

PAGE 103

91 Table 5.5 Specimen II dominant slip system sectors on the mi d-planes on the upper portion of the notch growth direction R=0.25* R=0.5* R=1.0* Sector max Slip system max Slip system max Slip system I 0-100 5, 6 (-11-1)[110], (-11-1)[011] 0-4 7 10 (1-1-1)[110], (-1-11)[011] 0-44 7 10 (1-1-1)[110], (-1-11)[011] II 4-105 5, 6 (-11-1)[110], (-11-1)[011] 44-100 5, 6 (-11-1)[110], (-11-1)[011] III 100-120 2 3 (111)[0-11], (111)[1-10] IV R=2.0* R=5.0* Sector max Slip system max Slip system I 0-57 7 10 (1-1-1)[110], (-1-11)[011] 0-63 7 10 (1-1-1)[110], (-1-11)[011] II 57-108 5, 6 (-11-1)[110], (-11-1)[011] 63-118 5, 6 (-11-1)[110], (-11-1)[011] III 108-137 2 3 (111)[0-11], (111)[1-10] 118-143 2 3 (111)[0-11], (111)[1-10] IV 137-150 8 12 (1-1-1)[0-11] (-1-11)[1-10] 143-154 8 12 (1-1-1)[0-11] (-1-11)[1-10] V 154-170 7 10 (1-1-1)[110], (-1-11)[011]

PAGE 104

92 RSS V. Theta for 0.25*rho0 4000 8000 12000 16000 20000 0102030405060708090100 Theta (deg)RSS (psi) T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 T12 Figure 5.33 RSS on the 12 primary slip systems on the mid-planes for the lower portion of the notch growth direction for R=0.25

PAGE 105

93 RSS V. Theta for 0.5*rho0 4000 8000 12000 16000 20000 24000 28000 32000 020406080100Theta (psi)RSS (psi) T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 T12 Figure 5.34 RSS on the 12 primary slip systems on the mid-planes for the lower portion of the notch growth direction for R=0.5*

PAGE 106

94 RSS V. Theta for 1*rho0 4000 8000 12000 16000 20000 020406080100120 Theta (deg)RSS (psi) T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 T12 Figure 5.35 RSS on the 12 primary slip systems on the mid-planes for the lower portion of the notch growth direction for R=

PAGE 107

95 RSS V. Theta for 2*rho0 2000 4000 6000 8000 10000 12000 020406080100120140Theta (deg)RSS (psi) T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 T12 Figure 5.36 RSS on the 12 primary slip systems on the mid-planes for the lower portion of the not ch growth direction for R=2*

PAGE 108

96 RSS V. Theta for 5*rho0 1000 2000 3000 4000 5000 6000 020406080100120140160 Theta (deg)RSS (psi) T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 T12 Figure 5.37 RSS on the 12 primary slip systems on the mid-planes for the lower portion of the not ch growth direction for R=5*

PAGE 109

97 Figure 5.38 Slip system sectors on th e surface on the upper portion of the not ch growth direction for specimen II

PAGE 110

98 Table 5.6 Specimen II dominant slip system sectors on the midplanes on the lower portion of the notch growth direction R=0.25* R=0.5* R=1.0* Sector max Slip system max Slip system max Slip system I 0-100 9, 11 (1-1-1)[101], (-1-11)[101] 0-81 7 10 (1-1-1)[110], (-1-11)[011] 0-94 7 10 (1-1-1)[110], (-1-11)[011] II 81-105 2, 3 (111)[0-11], (111)[1-10] 94-118 2 3 (111)[0-11], (111)[1-10] III 118-120 5 6 (-11-1)[110], (-11-1)[011] IV V R=2.0* R=5.0* Sector max Slip system max Slip system I 0-20 7 10 (1-1-1)[110], (-1-11)[011] 0-29 7 10 (1-1-1)[110], (-1-11)[011] II 20-48 5, 6 (-11-1)[110], (-11-1)[011] 29-50 5, 6 (-11-1)[110], (-11-1)[011] III 48-129 2 3 (111)[0-11], (111)[1-10] 50-170 7 10 (1-1-1)[110], (-1-11)[011] IV 129-131 5, 6 (-11-1)[110], (-11-1)[011] V 131-150 8, 12 (1-1-1)[0-11] (-1-11)[1-10]

PAGE 111

99 Identifying the Slip Systems fr om the Experimental Results Forero et al. have performed the tensile-t est experiments for eight different load levels. The loads have been increased gradually for each succeeding level. Pictures, showing the slip activity o ccurring around the notch are ta ken during the experimental tests for the sixth, seventh and eighth load levels. The specimen is subjected to a maximum load of 949.8 lbs during the eighth load level (Table 5.7). Table 5.7 Load Levels applied to specimen I during the experiments Load Level Load Applied (lbs) Stress Applied (ksi) 1 345.4 18.198 2 431.7 22.75 3 518.1 27.3 4 604.4 31.8 5 690.8 36.4 6 777.1 40.9 7 863.5 45.5 8 949.8 50.0 Figure 5.39 Picture showing the slip lines around the right not ch of specimen I for level 6 As seen from figure 5.39, slip lines ar e visible on the notch surface around the notch. These slip lines represent different active slip systems. The visible slip systems

PAGE 112

100 can be determined by getting the intersections between the octahedral planes and the notch plane. However, for every visible slip line, two possible intersections exist. Planes 1 and 2 have identical intersections with the notch plane. For plane 1, 2 is intersecting with the notch plane. Similarly, for planes 2,3 and 4, 6 8 and 10 are intersecting with the notch plane. Any intersecting slip system gives the direction in which that slip plane is causing the slip trace. Though from table 5.8, plane 1 appe ars to be activated for only one slip direction (2 ), it may be activated in the any of the three slip directions (2 1, or 3 ). Table 5.8 Intersection of the slip planes with the notch plane Slip Plane number Slip Plane Slip systems represented Intersection with (001) plane Slip system represented by the intersection 1 (111) 2 1, ,3 [-110] 2 2 (1-11) 5 4, ,6 [110] 6 3 (-111) 8 7, ,9 [110] 8 4 (11-1) 11 10, ,12 [-110] 10 By drawing arcs on the pictures taken during the tensile test, the slip plane causing the slip trace at any location around the notch can be found out. At a particular radial and angular distance, the picture shows two possibl e slip planes causing the slip trace giving the possibility of six slip systems being activat ed at that location. Now going back to the RSS Vs. Theta plots obtained from the FEA solution, the active slip systems can be determined by scaling the plots depending on the load level under consideration. The active slip systems obtained by the FEA solu tion match fairly accurately with those obtained from the experimental results.

PAGE 113

101 Slip Trace for 2 10 Slip Trace for 6, 8 30 2* For example at 30 and R=2* we see that the slip trace representing 2 and 10 is observed in the experimental resu lts (Figure 5.40). If we look at the FEA results, then at 30, we see the slip systems that lie above the CRSS or which are activated are again 2 and 10 Figure 5.40 Picture showing the slip traces

PAGE 114

102 RSS Vs. Theta for 2*rho for load level #6 0.00 10.00 20.00 30.00 40.00 50.00 60.00 70.00 80.00 90.00 100.00 0102030405060708090100110120130140150 Theta (deg)RSS (ksi) T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 T12 47.00 Figure 5.41 RSS Vs Theta plot scaled to load level # 6 Table 5.9 Specimen I experimental results R=1.0* Numerical Solution Experimental Results Sector max Slip system Slip Systems I 0-52 10 (-1-11)[011] II 52-56 2 (111)[0-11] 0-64 2 or 10 III 56-116 6 (-11-1)[011] IV 116-120 4 (-11-1)[10-1] 64-120 6 or 8 R=2.0* Numerical Solution Experimental Results Sector max Slip system Slip System I 0-50 10 (-1-11)[011] II 50-60 2 (111)[0-11] 0-70 2 or 10 III 60-68 3 (111)[1-10] IV 68-150 6 (-11-1)[011] 70-140 6 or 8 From Table 5.9, it can be seen that the num erical solution matches pretty closely to the experimental results. For example, for R=1.0* the experimental results show that

PAGE 115

103 for the region of 0-64, the slip systems that are activated are either 2 or 10 and for the region between 64-120, either 6 or 8 is the slip system causing the slip trace. In the numerical solution also for the region, 0-52, 10 is the dominant slip system and then for the next four degrees, 2 is dominant. Then, in the region 56-116, 6 is the slip system that is dominant and then for the last four degrees, 4 is the dominant slip system. It can be seen that 4 does not appear in the experime ntal results, but we know that 6 is causing the slip trace for that region and it is on the same slip plane as 4. Thus the results match pretty closely with the experime ntal results. The slight diffe rences that are seen in the results is because of the fact that the specimen coordinate syst em is off by 4.5 with the material coordinate system. Specimen Comparison As discussed earlier, there is difference in both the orientation and dimensions of the two specimens. For both the specimens, a semi-circular profile has been used to incorporate the notch geometry. It is observ ed that the stresses on the mid-planes are higher than those on the surface for the respec tive locations in both the specimens. Also, from the tables showing the sectors, it is cl ear that unlike on the surface, at any location, the dominant slip systems on the mid-planes occur in pairs or gr oups of four. The RSS values are symmetrical about the notch growth direction for specimen I. However, in specimen II, the RSS values are higher on the upper portion of the notch than on the lower portion of the notch. Specimen I ha s much lower maximum RSS value than specimen II. Its value for specimen I is 6= 8=31,140 psi and for specimen II is 6=36,090 psi. These maximum values for the two specimens occur at the same location:

PAGE 116

104 R=0.5* and 105 from the notch tip on the midplanes. Therefore, on a purely stressbased criterion, specimen I is the obvious choice for the de sign of notched specimens. However, certain other factor s such as cross-slip are wo rth considering. This is because though they may cause more deformati on, but they also at times avoid fracture by releasing energy in the form of ductile deformation. Therefore, the ultimate choice can only be made after detailed considerations of dislocation and other atomic mechanisms. Specimen I Notch Geometry The semi-circular profile could not in corporate the notch geometry for the specimen exactly. So, results using the elliptical profile and the filleted profile have also been obtained. However, it is observed that there is not much difference between the RSS Vs. Theta for 2*rho 0 2000 4000 6000 8000 10000 12000 020406080100120140160 Theta (deg)RSS (psi) T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 T12 Figure 5.42 RSS Vs. Theta pl ots obtained from the semi-circular notch geometry

PAGE 117

105 2*rho Nodal Results for Surface Filleted Notch Corners 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 020406080100120140 Theta (degrees)Stress (psi) T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 T12 Figure 5.43 RSS Vs. Theta pl ots obtained from the semi-circular notch geometry 2*rho Nodal Results for Surface Elliptical Notch Tip Approximation -20000 0 20000 40000 60000 80000 100000 120000 140000 020406080100120140160 Theta (degrees)Stress (psi) T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 T12 Figure 5.44 RSS Vs. Theta pl ots obtained from the elli ptical notch geometry

PAGE 118

106 results obtained by the three profiles. But the fill eted profile seems to give results that are a bit closer to the experimental results than the other two profiles. Figures 5.42, 5.43 and 5.44 give the comparison between the three notch profiles for the RSS values for the same radius (2* ).

PAGE 119

107 CHAPTER 6 CONCLUSIONS AND RECOMMENDATIONS 1. The linear elastic finite element model that accounts for the material anisotropy is effective in accurately pred icting the active slip system s and slip sectors on the surface of the notched specimen of FCC si ngle crystals. The model also predicts slip systems in the mid-planes of the specimen. However, these results cannot be verified experimentally. 2. For specimen I, semi-circular profile has b een used to incorporate notchgeometry. To investigate the variation of notch geom etry, an elliptical and a semi-circular profile with end fillets were used for incorporating the notch-geometry. However for specimen II, only the semi-circular profil e was used to investigate the results. 3. The maximum stresses for both the specimens occur on the mid-planes. This indicates that the slip systems perhaps get activated first on the mid-planes. This lends further evidence to conclusions dr awn by other research ers (Schulson and Xu)) that yield is not always initiated at th e surface. Therefore, even though the slip lines may not be visible on the surface for low loads, the slip activity might have initiated because this phenomenon may have initiated in the inte rior regions of the specimen. Slip lines observed on the surface, therefore may not be considered as the only indication of slip activity. 4. The slip sectors are a functi on of the particular load a pplied and they are not fixed for a material under all conditions. The slip sectors obtained for the varying radial and angular distances are lobe d sectors rather th an straight sectors with constant boundaries, as predicted by 2D isotropic an alyses of many researchers (Crone and shield). 5. The specimens though modeled in the elastic regime, are capable of predicting the fatigue life. The RSS values arrived at are cogent here as the fatigue life is determined from the stresses in the elastic regime. 6. For studying crack initiation, crystal plasticity and fatigue properties near the notch, an accurate model is required, to eff ectively model crystal lattice rotation. 7. In fatigue loading, materials are subjected to tensile as well as compressive loads. This study, however, considers the effect of tensile loads only. Future work should consider effects due to compressive loads also for these specimens to make it more relevant to practical situa tions. Tests for cyclic load ing should also follow this.

PAGE 120

108 LIST OF REFERENCES Alden, D., “An Analysis of the Yield Phen omenon in Ren N4+ Single Crystals with respect to Orientation and Temperature, ” Ph.D. dissertation, University of Cincinnati, 1990. Anderson, T., Fracture Mechanics: Fundamentals and Applications, 2nd Ed., Boca Raton: CRC Press, Inc.; 1995, p.54. ANSYS (http://www.ansys.com), Elements reference, ANSYS Release 5.6; ANSYS, Inc. May 2002. Bickford, W., Advanced Mechanics of Materials, Reading, Massachusetts: AddisonWesley, 1998, pp. 10, 280-1. Crone, W., and Shield, T., “Experimental Study of the Deformation near the Notch Tip in Copper and Copper-Beryllium Single Crystals,” Journal of the Mechanics and Physics of Solids, 49, 2001, pp. 2819-2838. Cuitino, A., and Ortiz, M., “Three-Dimensiona l Crack Tip Fields in Four-Point Bending Copper Single-Crystal Specimens,” Journa l of the Mechanics and Physics of Solids, 44 (6), 1996, pp. 863-904. Davis, J.R., ed., Heat Resi stant Materials, ASM Special ty Handbook; Materials Park, Ohio: ASM International, 1997, pp. 256-263. Dieter, G., Advanced Mechanical Metallurgy, 3rd Ed., New York: McGraw Hill, 1986, pp. 106,114. Forero, L., Ebrahimi, F., Magnan, S., and Arak ere, N., “Effect of Crystal Orientation on Strain Localization in Nickel-Base Singl e Crystal Superalloys”, February 2002. Huang, X., Borrego, A., and Panyleon, W., “P olycrystal Deformation and Single Crystal Deformation: Dislocation Structur e and Flow Stress in Coppper”, Materials Science and Engineering, A319-321, 2001, pp. 237-241. Lall, C., Chin, S., and Pope, D., “The Orie ntation and Temperature Dependence of the Yield Stress of Ni3(Al, Nb) Single Crystals”, Metallurgical Transactions, 10A, 1979, pp. 1323-1332.

PAGE 121

109 Lethnitskii, S.G., Theory of Elasticity of an Anisotropic Elastic Body, San Francisco: Holden-Day, Inc., 1963, pp. 1-40. Magnan, S., “Three-Dimensional Stress Fiel ds and Slip Systems for Single Crystal Superalloy Notched Specimens”, Master of Science Thesis, University of Florida, 2002. Miner, R., Voigt, R., Gayda, J., and Gabb, T., “Orientation and Temperature Dependence of Some Mechanical Properties of the Ni ckel Base Single Crystal Alloy Ren N4, Part 1: Tensile Behavior”, Metallurgical Transactions A, 17A (3), 1986, pp.491. Mohan, R., Ortiz, M., and Shih, C., “An Analysis of Cracks in Ductile Single Crystals-II. Mode I loading”, Journal of the Mechanics and Physics of Solids, 40(2), 1992, pp. 315-337. Mollenhauer, D., Ifju, P., and Han, B., “A Compact, Robust and Versatile Moir Interferometer”, Optics and Lasers in Engineering, 23, 1995, pp. 29-40. Moroso, J., “Control of Secondary Crystallogr aphic Orientation in Single-Crystal Nickel Superalloys For Increased Resistance to Fa tigue Crack Growth”, Master of Science Thesis, University of Florida, 1999. Nitz, A and Nembach, E., “Anisotropy of th e Critical Resolved Shear Stress of a ’ (47 vol.)Hardened Nickel-Base Superalloy and its Constituent and ’ –Single Phases”, Materials Science and Engineering, A243-236, 1997, pp. 684-686. Rice, J.R., “Tensile Crack Tip Fields in Elastic-Ideally Plastic Crystals”, Mechanics of Materials, 6, 1987, pp. 317-335. Saeedvafa, M., and Rice, J.R., “Crack Tip Singul ar Fields in Ductile Crystals with Taylor Power-Law Hardening, II: Plane Strain”, Journal of The Mechanics and Physics of the Solids, 37 (6), 1989, pp. 673-691. Sass, V., and Feller-Kneipmeier, M., “Orienta tion Dependence of Dislocation Structures and Deformation Mechanics in Creep Deformed CMSX-4 Single Crystals”, Materials Science and Engineering, A245, 1988, pp. 19-28. Schulson, E., and Xu, Y., “Notch-Tip Deformation of Ni3Al Single Crytals”, Materials Research Society Symposium Proceedings, 460, 1997, pp.555-560. Shield, T., “An Experimental Study of the Plas tic Strain Fields Near a Notch Tip in a Copper Single Crystal During Loading”, Acta Materialia, 44 (4), 1996, pp. 15471561. Shield, T., “Microscopic Moir Interferometr y, Retrieved November 14, 2002 from the Internet (http://www.aem.umn.edu/ people/faculty/shield/ ).

PAGE 122

110 Shield, T., and Kim, K., “Experimental Measur ement of the Near Tip Strain Field in an Iron-Silicon Single Crystal”, Journal of the Mechanics and Physics of Solids, 42 (5), 1994, pp. 845-873. Sims, Stoloff., and Hagel “Superalloys II”, New York, Wiley-Interscience [1972]. Stouffer, D., and Dame, L., Inelastic Deformation of Metals: Models, Mechanical Properties, and Metallurgy, New York: J. Wiley, c1996. Svoboda, J., and Lukš, P., “Model of Creep in <001>Oriented Superalloy Single crystals.” Acta Materialia, 46 (10), 1998, pp.3421-3431. Swanson, G., and Arakere, N., “Effect of Crystal Orientation on Analysis of SingleCrystal, Nickel-Based Turbine Blade Superalloys,” National Aeronautics and Apace Administration-Marshall Space Fli ght Center Technical Publication, February 2000. Xu, Y., and Schulson, E., “On the Notch Se nsitivity of the Ductile Intermetallic Ni3Al Containing Boron”, Acta Materialia, 44, 1996, pp.1601. Zhu, W., Fort, D., Jones, I., and Smallman R., “Orientation Dependence of Creep of Ni3Al at Intermediate Temperature”, Acta Materialia. 46 (11), 1998, pp. 38733881.

PAGE 123

111 BIOGRAPHICAL SKETCH Niraj Sudhir Bidkar was born in Nagpur, India, in July 1978. He attended the Y.C.College of Engineering at Nagpur from 1996 to 2000. After getting the bachelor’s degree in mechanical en gineering in 2000, he enrolled for graduate studies in Mechanical Engineering at the University of Florida at Gainesville. He will be conferred the Master of Science degree by the Univer sity of Florida in May 2003.


Permanent Link: http://ufdc.ufl.edu/UFE0000645/00001

Material Information

Title: Prediction of slip systems in notched FCC single crystals using 3D FEA
Physical Description: Mixed Material
Creator: Bidkar, Niraj Sudhir ( Author, Primary )
Publication Date: 2003
Copyright Date: 2003

Record Information

Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
System ID: UFE0000645:00001

Permanent Link: http://ufdc.ufl.edu/UFE0000645/00001

Material Information

Title: Prediction of slip systems in notched FCC single crystals using 3D FEA
Physical Description: Mixed Material
Creator: Bidkar, Niraj Sudhir ( Author, Primary )
Publication Date: 2003
Copyright Date: 2003

Record Information

Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
System ID: UFE0000645:00001


This item has the following downloads:


Full Text











PREDICTION OF SLIP SYSTEMS IN NOTCHED FCC SINGLE CRYSTALS USING
3D FEA



















By

NIRAJ SUDHIR BIDKAR


A THESIS PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
MASTER OF SCIENCE

UNIVERSITY OF FLORIDA


2003






























I would like to dedicate this work to my parents for their ever-extending support and
encouragement.















ACKNOWLEDGMENTS

I would like to express deep gratitude to Dr. Nagaraj Arakere for his

encouragement and enthusiastic assistance and instructions. I would also like to extend

special thanks to Dr. Ashok Kumar and Dr. John Schueller for their time and support.

Thanks are also due to Dr. Ebrahimi and Luis Forrero for providing the necessary

information and data. I am thankful to my friends and lab mates Jeff and Shadab for

helping me in understanding many concepts and ideas. Finally I am grateful to the

Almighty God for giving me the strength for the completion of this work.
















TABLE OF CONTENTS
page

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

LIST OF TABLES ............................................................ ... ............ .. vi

L IST O F F IG U R E S .... ...... ................................................ .. .. ..... .............. vii

ABSTRACT .............. .......................................... xi

CHAPTER

1 IN TRODU CTION ............................................... ..... ..... .............. ..

Slip Deform ation in Single Crystal Superalloys ....................... ................ ........... 2
M material C haracteristics.............................................................. ....................... 7
Temperature Effects ................. .............. .. .......... .... ............. 8
T est M ethods .................................................................. .................................. 9
A nalytical A approach ..................................................... .. .......... 9
N um erical A pproach.................................................... ......................... 10
Experim ental A approach ........................................................ ......... ..... 10

2 REVIEW OF LITERA TURE ........................................................ ............... 12

A nisotropy of E lasticity ............................................ ......................................... 12
E la stic ity ........................................................................ 12
Elasticity M atrix .............. ...................................... ...... ............ .. 13
N otch Tip D eform ation ......................................................................... 16
Rice (1987) ...... ......................................... .......... 16
C rone and Shield (200 1)...................................................... ... ................. 18
Schulson and X u (1997) ....................................................................... ... 23
Slip A ctivation and D eform ation ........................................ ......... ............... 27

3 ANALYTICAL APPROACH ......................................................................29

Coordinate Axes Transformation Using Miller Indices...................................31
E xam ple Transform ation ............................................................... ... 34
Transformation Matrices for Stress and Strain Fields........................................ 35
Shear Stresses and Strains in the Slip Systems ................................................38









4 NUMERICAL APPROACH: FINITE ELEMENT METHOD ..............................41

Verification of the Finite Element Model Results......................................... 42
Specimen Orientation and Dimensions ............................................... 43
Specim en G eom etry.................................................... .......................... 44
Specim en G eom etry.................................................... .......................... 45
Finite Element Model Characteristics ........................................ ...............45
M material P roperties................................................... ............. ........... 45
E lem ents and M eshing ......................................................... ... ................. 46
F inite E lem ent S solution ......................................... .............................................4 9
A ssum options .................................................................................................. ....... 50

5 RESULTS AND DISCU SSION ........................................ ......................... 52

S p ecim en I .............................................................5 2
Specim en II .................................................................. ..................70
Identifying the Slip Systems from the Experimental Results ................................99
Sp ecim en C om p arison ............................................. ......................................... 103
Specim en I N otch G eom etry ........................................................ ............... 104

6 CONCLUSIONS AND RECOMMENDATIONS...............................................107

L IST O F R E F E R E N C E S ......... .. ............... ................. .............................................. 108

BIOGRAPHICAL SKETCH ........... ..... ......... .. ..........................111




























v
















LIST OF TABLES


Table page

1.1 Slip systems in FCC crystal. Source: Stouffer and Dame, 1996 .................................3

2.1 Atomic density on FCC crystal planes Source: Dieter (1986)..................................13

2.2 Symmetry in various crystal structures Source: Dieter (1986)............................... 13

2.3 Sector boundary angle comparisons for Orientation II Source: Crone and Shield
(2 0 0 1) ..........................................................................................1 9

2.4 Sector boundary angles from the experimental tests by Crone and Shield Source:
C rone and Shield (2001) ............................................... ............................. 22

2.5 Slip sectors from plane stress and plane strain assumptions. Source: Schulson and
Xu (1996) ..................................... .................. ............. ........... 26

4.1 Actual and specimen geometry for Specimen I .......................................................44

4.2 Actual and specimen geometry for Specimen II ............................................... 45

5.1 Specimen I dominant slip system sectors on the surface...........................................62

5.2 Specimen I dominant slip system sectors on the on the mid-planes............................69

5.3 Specimen II dominant slip system sectors on the surface on the upper portion of the
notch grow th direction ......... ................. ................... ................... ............... 77

5.4 Specimen II dominant slip system sectors on the surface on the lower portion of the
notch grow th direction ......... ................. ................... ................... ............... 84

5.5 Specimen II dominant slip system sectors on the mid-planes on the upper portion of
the notch grow th direction ............................................ ............................. 91

5.6 Specimen II dominant slip system sectors on the mid-planes on the lower portion of
the notch grow th direction ............................................ ............................. 98

5.7 Load Levels applied to specimen I during the experiments .....................................99

5.8 Intersection of the slip planes with the notch plane.............................. ..............100

5.9 Specim en I experimental results ................................................. ..................... 102

vi
















LIST OF FIGURES


Figure page

1.1 Convention for defining primary and secondary crystallographic orientations in
turbine blades.Source: Swanson and Arakere, 2000....................................2

1.2 Slip sectors observed under plastic deformation. Source: Crone and Shield, 2001. .....7

1.3 Microstructure of material A. The y' precipitate forms in the y matrix and has a
volume fraction of about 60%. Source: Superalloys II by Sims, Stoloff and
H ag el. ................................................................................ 8

2.1 Schem atic of FCC crystal structure ........................................ ......................... 13

2.2 N otch direction term inology. ................................................................................ .....17

2.3 Yield surface based on plane strain state of stress ..................................................17

2.4 Orientations used by Crone and Shield in their tests. .............................................19

2.5 Slip sectors observed in the experimental tests by Crone and Shield. Source: Crone
and Shield (200 1).................................................. ...................... .... 20

2.6 Slip sectors and slip lines from the experimental tests by Crone and Shield. Source:
C rone and Shield (2001) ............................................... ............................. 2 1

2.7 Specimen Orientation used by Schulson and Xu............................... ............... 23

2.8 Slip sectors obtained by Schulson and Xu Source: Schulson and Xu, 1997. ............25

2.9 L oad and slip direction............................................. .................. ............... 27

3.1 M material and Specimen coordinate system ....................... ................................... 30

3.2 First rotation of the coordinate system about the z-axis ................ ............... 31

3.3 Projections from the original axes on the projected axes ................. ............ .....32

3.4 Second rotation of the coordinate system about the z-axis...................................32

3.5 Third rotation of the coordinate system about the z-axis ................................. 33









4.1 The notched specimens analyzed using finite element method ...............................42

4 .2 Specim en specifications...................................................................... ...................44

4.3 The specimen is first defined with respect to the global coordinate system and then
the material coordinate system is specified later. .............................................46

4.4 M eshing around the notch in AN SY S .............................................. .....................47

4.5 Schematic of the SOLID95 element in ANSYS. Source ANSYS 6.1 Elements
R reference, 2002. .....................................................................48

4.6 Radial arcs used for element location and sizing; centered at notch tip....................48

4.7 Radial arcs for stress field calculations...................... .... ......................... 49

5.1 RSS on the 12 primary slip systems for the upper portion of the notch growth
direction for R=0.25*p .... ...... ...................................... 54

5.2 RSS on the 12 primary slip systems for the upper portion of the notch growth
direction for R =0.5*p ......................... ............ ........ ... .. .. .. ........ .... 55

5.3 RSS on the 12 primary slip systems for the upper portion of the notch growth
direction for R = l*p p.......... ................ ................................ .. ...... 56

5.4 RSS on the 12 primary slip systems for the upper portion of the notch growth
direction for R=2*p .......... ..... ...... .... .... .................. .. ...... 57

5.5 RSS on the 12 primary slip systems for the upper portion of the notch growth
direction for R=5*p ......... ...... .. ..... ........ ...... .. .. .. .... .. ...... .... 58

5.6 Slip system sectors on the surface for specimen I ................................................. 59

5.7 Upper portion of the radar plot showing the surface stresses for the [100]
orientation ........................................................................... 60

5.8 Lower portion of the radar plot showing the surface stresses for the [100]
orientation ........................................................................... 6 1

5.9 RSS on the 12 primary slip systems for the upper portion of the notch growth
direction for R=0.25*p .......... .... .......... .. ......... .... ........................ 63

5.10 RSS on the 12 primary slip systems for the upper portion of the notch growth
direction for R =0.5*p ......................... ............ ........ ... .. .. .. ........ .... 64

5.11 RSS on the 12 primary slip systems for the upper portion of the notch growth
direction for R = l*p p.......... ................ ................................ .. ...... 65









5.12 RSS on the 12 primary slip systems for the upper portion of the notch growth
direction for R=2*p .......... ...... ...... .... ................... .. ...... 66

5.13 RSS on the 12 primary slip systems for the upper portion of the notch growth
direction for R=5*p.......... .... ......... ................. .. ...... 67

5.14 Slip system sectors on the mid-planes for specimen I............... ....... ............68

5.15 RSS on the 12 primary slip systems for the upper portion of the notch growth
direction for R=0.25*p .... ...... ... ................................... 71

5.16 RSS on the 12 primary slip systems for the upper portion of the notch growth
direction for R =0.5*p ................. ..... ...... .. .... ...... ... .. ..................72

5.17 RSS on the 12 primary slip systems for the upper portion of the notch growth
direction for R = p............. ... .................................... .......... ........ 73

5.18 RSS on the 12 primary slip systems for the upper portion of the notch growth
direction for R=2*p .......... ...... ...... .... ................... .. ...... 74

5.19 RSS on the 12 primary slip systems for the upper portion of the notch growth
direction for R=5*p.......... .... ......... ................. .. ...... 75

5.20 Slip system sectors on the surface on the upper portion of the notch growth
direction for specim en II.............................................. .............................. 76

5.21 RSS on the 12 primary slip systems for the lower portion of the notch growth
direction for R=0.25*p .......... .... ...... .. .... ......... .... ........................ 78

5.22 RSS on the 12 primary slip systems for the lower portion of the notch growth
direction for R =0.5*p ................. ..... ...... .. .... ...... ... .. ..................79

5.23 RSS on the 12 primary slip systems for the lower portion of the notch growth
direction for R = p............. ... .................................... .......... ........ 80

5.24 RSS on the 12 primary slip systems for the lower portion of the notch growth
direction for R=2*p .......... ...... ...... .... ................... .. ...... 81

5.25 RSS on the 12 primary slip systems for the lower portion of the notch growth
direction for R=5*p.......... .... ......... ................. .. ...... 82

5.26 Slip system sectors on the surface on the lower portion of the notch growth
direction for specim en II.............................................. .............................. 83

5.27 RSS on the 12 primary slip systems on the mid-planes for the upper portion of the
notch growth direction for R=0.25*p ....................................... ............... 85

5.28 RSS on the 12 primary slip systems on the mid-planes for the upper portion of the
notch growth direction for R=0.5*p ........................................... ............... 86









5.29 RSS on the 12 primary slip systems on the mid-planes for the upper portion of the
notch grow th direction for R=p ........................................ ........................ 87

5.30 RSS on the 12 primary slip systems on the mid-planes for the upper portion of the
notch growth direction for R=2*p ............................. ................................. 88

5.31 RSS on the 12 primary slip systems on the mid-planes for the upper portion of the
notch growth direction for R=5*p ............... .... .... ...................................89

5.32 Slip system sectors on the mid-planes on the upper portion of the notch growth
direction for specim en II.............................................. .............................. 90

5.33 RSS on the 12 primary slip systems on the mid-planes for the lower portion of the
notch growth direction for R=0.25*p ....................................... ............... 92

5.34 RSS on the 12 primary slip systems on the mid-planes for the lower portion of the
notch growth direction for R=0.5*p ........................................... ............... 93

5.35 RSS on the 12 primary slip systems on the mid-planes for the lower portion of the
notch grow th direction for R=p ........................................ ........................ 94

5.36 RSS on the 12 primary slip systems on the mid-planes for the lower portion of the
notch growth direction for R=2*p ............................. ................................. 95

5.37 RSS on the 12 primary slip systems on the mid-planes for the lower portion of the
notch growth direction for R=5*p ................. .... .... ...................................96

5.38 Slip system sectors on the surface on the upper portion of the notch growth
direction for specim en II.............................................. .............................. 97

5.39 Picture showing the slip lines around the right notch of specimen I for level 6........99

5.40 Picture show ing the slip traces.......................................... ........................... 101

5.41 R SS V s Theta plot scaled to load level # 6....................................... .................... 102

5.42 RSS Vs. Theta plots obtained from the semi-circular notch geometry..................104

5.43 RSS Vs. Theta plots obtained from the semi-circular notch geometry..................105

5.44 RSS Vs. Theta plots obtained from the elliptical notch geometry.........................105















Abstract of Thesis Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Master of Science

PREDICTION OF SLIP SYSTEMS IN NOTCHED
FCC SINGLE CRYSTALS USING 3D FEA

By

Niraj Sudhir Bidkar

May 2003


Chair: Dr. Nagaraj K. Arakere
Department: Mechanical and Aerospace Engineering

Nickel-base single crystal superalloys are the current choice for high-temperature

jet engine applications such as turbine blades and vanes. Since these materials are grown

as single crystals, there are no grain boundaries present and as a result the properties of

these materials are highly direction-dependent. This makes it necessary to test these

materials for different load orientations. Notched tensile test specimens are typically used

to study the evolution of slip systems in 3D stress fields. Many researchers in the past

have tried to come up with solutions for prediction of the slip systems in notched test

specimens for specific load orientations.

The two notched-specimens considered in this work differ in geometry as well as in

load orientation. An elastic finite element model has been created which predicts the

behavior in the plastic regime quite accurately. The linear elastic model is used to predict

only the onset of yield. The stress concentrated region near the notch, with the triaxial









state of stress, is of particular interest. Slip systems in this region are predicted using 3D

linear elastic FEA, including material anisotropy. This model does not account for effects

such as strain hardening, crystal lattice rotation, creep, high-temperature conditions.

These results were correlated with the experimental tests performed by the MSE

department for similar specimens.

The first of the two specimens had [001] load direction, [010] notch growth

direction and [100] as the notch plane direction. The other specimen had a load direction

of [111], notch growth direction [-12-1] and [30-3] as the notch plane direction.

The slip systems predicted using 3D FEA had excellent correlation with those

observed experimentally. This demonstrates that 3D linear elastic FEA that includes the

effect of material anisotropy can be used to effectively predict the onset of yield and

hence slip systems in FCC single crystals.














CHAPTER 1
INTRODUCTION

As one of the most important classes of high-temperature structural materials,

Nickel base superalloys exhibit a truly unique combination of properties at elevated

temperatures. These desirable properties enable these materials to enhance the

performance and efficiency of turbine engines designed for both aircraft and power-

generation applications.

In high temperature polycrystalline alloys grain boundaries provide passages for

diffusion and oxidation. Eliminating grain boundaries and grain boundary strengthening

elements produces materials with superior high temperature fatigue and creep properties

compared to conventional superalloys. However, the absence of grains makes these

alloys orientation dependent or anisotropic with tension- compression asymmetry. As

components are cast in single crystal form the entire component inherits the anisotropic

properties of the crystal lattice. The slip deformation mechanisms are strong functions of

orientation and deformation occurs in specific crystallographic directions rather than in

the direction of the applied load.

During the manufacturing of the superalloys, the primary growth direction is

oriented in the <001> orientation as this orientation has the best combination of

mechanical properties. However in practice, due to the difficulties encountered in casting

and due to cost considerations blades that are oriented away from the <001> orientation

by up to 150 is used. Also in the operating environment the blades are subjected to

loading in a variety of orientations due to the hot impinging gases causing bending and









torsion. The formation of hotspots leads to thermal gradients, which also contribute to

multiaxial loading. Hence the deformation mechanisms depend on the high temperature

effects and loading in different orientations.

Airfoil
Stacking
Line Relative
Angle ac


Airfoil Primary
Mean Relative Crystailographic
Chord Angle s Orientation
Line







Secondary
Crystallographic
Orientation




Figure 1.1 Convention for defining primary and secondary crystallographic orientations
in turbine blades. Source: Swanson and Arakere, 2000.

Slip Deformation in Single Crystal Superalloys

Slip deformation in nickel-base superalloys is governed by the sliding of separate

layers of the crystal structure over one another along definite crystallographic planes or

slip planes in specific directions, according to Dieter (1986). It is considered a stress-

controlled process.









Table 1.1 Slip systems in FCC crystal. Source: Stouffer and Dame, 1996
Slip Number Slip Plane Slip Direction
Octahedral Slip a/2{ 111 }<110>
1 (111) [10-1]
2 (111) [0-11]
3 (111) [1-10]
4 (-11-1) [10-1]
5 (-11-1) [110]
6 (-11-1) [011]
7 (1-1-1) [110]
8 (1-1-1) [0-11]
9 (1-1-1) [101]
10 (-1-11) [011]
11 (-1-11) [101]
12 (-1-11) [1-10]
Octahedral Slip a/2{ 111}<112>
13 (111) [-12-1]
14 (111) [2-1-1]
15 (111) [-1-12]
16 (-11-1) [121]
17 (-11-1) [1-1-2]
18 (-11-1) [-2-11]
19 (1-1-1) [-11-2]
20 (1-1-1) [211]
21 (1-1-1) [-1-21]
22 (-1-11) [-21-1]
23 (-1-11) [1-2-1]
24 (-1-11) [112]
Cubic cells a/2{ 100}<110>
25 (100) [011]
26 (100) [01-1]
27 (010) [101]
28 (010) [10-1]
29 (001) [110]
30 (001) [-110]


The CRSS is therefore the controlling value and it is the shear stress at which the

slip was initiated. The CRSS is a function of the applied load and direction, specimen

geometry, and crystal structure, though it is not directly related to the material's

anisotropy.









Depending on the direction of the applied load, certain slip systems get activated

first. These are termed as the 'easy glide' systems or the primary octahedral slip systems.

Usually, the planes with greatest atomic densities are the ones that are activated first. This

whole process of slip occurs so as to relieve the energy of the high shear stress or the

resolved shear stress (RSS) within the 12 primary slip systems. The slip thus helps to

obtain a more energetically stable system.

As this slip activity and the deformation process continues, the indications of slip at

the surface can be observed. With the increase in the applied stress, some of the 12

secondary slip systems may also get activated. Extreme temperature or load conditions

may also cause the activation of the six cubic slip planes. But the activation of secondary

or cubic slip planes does not occur first because the slip activity along the primary slip

planes requires less energy to relieve the high RSS. Certain microstructural behaviors like

pinning or locking of dislocations too prevent the shift in slip from the primary to

secondary or cubic slip systems.

Usually slip occurs when the RSS exceeds the yield strength of the material.

Nickel-base superalloys, however exhibit different yield strength in tension and

compression. This kind of behavior is termed as tension/compression asymmetry. For

instance, for the <001>orientation these alloys exhibit highest yield strength in tension,

while the compressive yield strength is lower for this orientation. On the other hand, for

the <110>orientation, the yield strength is higher in compression than in tension. For a

sample loaded in the <111> direction there is practically no tension/compression

asymmetry.









In some cases, parts of the dislocations get separated to reach a lower energy state.

These separated parts are termed as 'extended dislocations'. The extended dislocations,

on octahedral slip planes must recombine into a single dislocation before cross slip can

take place. Cross slip is the phenomenon where a dislocation moves from one plane to

another. Cross slip also occurs to reduce the energy of the system. Again the material

orientation affects the process of recombination of the dislocation parts as it does in case

of the magnitude of the resolved shear stress. An applied compressive load helps to

overcome the force separating two extended dislocations at an interface where cross slip

can occur and can combine them into a single dislocation. The combined dislocation can

then move as a cross slip. Conversely, an applied tensile load aids the force separating the

dislocation and makes cross slip more difficult to occur. This asymmetrical load behavior

should be addressed as an important issue in the design of a part that is subjected to both

tensile and compressive loads. As far as the failure analysis research is concerned,

priority is given to tensile stress analysis as compared to compressive stress analysis as

compressive stresses usually prove to be beneficial. Instead of causing cracks to initiate

or propagate, compressive stresses usually either have no effect or may even arrest the

cracks that develop under tensile stresses. However, in the case of tri-axial stress states,

such as those occurring at the notch tip, the asymmetric behavior becomes even more

complex. Depending on orientation, some slip directions may hinder partial dislocation

recombination, while others may aid this process.

Nickel-base superalloys exhibit another abnormal yield characteristic that is related

to temperature called 'anomalous yield behavior'. A normal trend shows a decrease in the

yield strength with increasing temperature. These superalloys however have been shown









to exhibit an increase in their yield strength with rising temperature, up to a certain point.

Here it is essential to take into account the process of superdislocations that is used to

relieve the stress. Superdislocation can be defined as a dislocation composed of two

dislocations separated by an anti-phase boundary (APB) that glide along { 111}

octahedral planes during slip. This superdislocation always attempts to reduce its energy,

which can be brought about by lowering its APB energy if it cross-slips to a {100} plane.

Now when the first dislocation cross slips onto the { 100} plane, it gets locked due to the

higher stress required for it to move on a {100} plane. It gets locked on two separate

planes in such a way that one dislocation is left on a { 111 } plane and the other on a

{ 100} plane separated by the APB. Because of this lock, no further motion can take

place. With increasing temperature, thermally activated cross-slip to { 100} planes occur

easily, forming locks. These locks keep on multiplying, thus preventing further motion

and increasing the CRSS. In addition to this, the 'Peierl stress' required to move the

locked dislocation on the {100} plane, is lowered with the increase in temperature.

At a certain point of temperature, the thermal activation helps the dislocation to

overcome the Pierel stress. At this critical point the entire dislocation is able to move to

the { 100} plane resulting in a cubic slip. At this point, more of the previously locked

dislocations are released on the {100} planes and consequently the CRSS begins to fall.

Thus in high temperature applications, anomalous yield behavior is an important

consideration for fracture and fatigue analysis. This consideration helps understand the

mechanisms that increase strength and their limitations.






















Figure 1.2 Slip sectors observed under plastic deformation. Source: Crone and Shield,
2001.

This work will focus on the variation of the 12 primary resolved shear stresses in a

notched specimen. The activity of the specific slip systems with respect to the radial and

angular distances surrounding the notch tip will be determined for specific orientations.

The slip systems will be studied both near the notch and at far field to observe the

changes associated with high stress gradients prevalent in close proximity of the notch

tip. The maximum RSS values and slip systems are expected to shift along a line of

constant radius. This, in a way indicates a shift in the state of stress. In the actual tensile

testing, these different slip systems can be clearly visualized as "sectors" surrounding the

notch tip. In short, the goal of this work is the prediction of slip system activity as a

function of the radial and angular distances from the notch tip and the resulting slip

sectors.

Material Characteristics

Nickel-base superalloys have a microstructure that consists of a y-matrix and a fine

dispersion of hard y'-precipitates. The matrix is considerably alloyed with other elements

that may vary, including cobalt, chromium, tungsten and tantalum, though it is mainly

composed of nickel. The precipitate however is the intermetallic compound Ni3Al.























Figure 1.3 Microstructure of material A. The y' precipitate forms in the y matrix and has
a volume fraction of about 60%. Source: Superalloys II by Sims, Stoloff and
Hagel.

So far these superalloys have evolved in three generations. Rene N4, CMSX-4, and

others are some of the most advanced or third-generation superalloys. Usually, these

alloys have a high volume fraction of y', around 60%. The majority of the deformation

occurs in the softer matrix as suggested by Svoboda and Lukas (1998). However, it

should also be noted that at such high volume fractions, the precipitate has a considerable

effect on the overall performance of the superalloy's performance.

Temperature Effects

These materials are designated to provide unique strength and/or corrosion

properties at elevated temperatures (i.e., greater than 6000C). The experimental notched

tensile tests conducted to study slip systems activity have been performed at room

temperature, which is well below the transition temperature for superalloys. As observed

by Stouffer and Dame (1996), in the low temperature regime, the octahedral slip system

is predominant and basically controls low temperature deformation. The secondary

planes are activated only at high temperatures.









At temperatures above 6000C the cubic slip systems get activated and act along

with the octahedral planes. When the temperature reaches above 8500C, the deformation

increases rapidly and the material strength is almost free of the orientation.

Test Methods

For the isotropic notch specimens in tension many analytical and numerical

solutions have been developed for their elastic response, especially in the field of linear

elastic fracture mechanics. However, it is very difficult to develop three-dimensional

analytical models for anisotropic notched specimens. The current solutions have been

derived only after many simplifications and therefore give inaccurate results when

compared to experimental data. However, the three-dimensional finite element analysis

(FEA) is capable of taking into account the limitations in the elastic models and it does

provide a solution comparable to the actual experimental results. In numerical and

experimental specimens, the notches can be considered as simplified cracks to develop a

realistic model to study fracture behavior. In many applications, it may so happen that

though the material is designed for primary strength in one direction, it must withstand

multi-axial loading. This makes the study of effect of anisotropy all the more interesting.

Analytical Approach

Generally analytical solutions provide exact solution to any problem. But this is not

true in case of very complex problems. For such problems, the analytical solutions are

arrived at after close approximations or by the use of theoretical and empirical solutions.

For the notched specimen in tension, an analytical solution is not available that correctly

predicts slip system activity around the notch tip. The ones that are available make use of

several assumptions like the plane strain. This issue will be discussed in detail later.









Numerical Approach

As far as finite element analysis is concerned, it does account for gross material

properties such as modulus of elasticity and Poisson's ratio in isotropic materials. It even

accounts for the directional properties while analyzing anisotropic materials. However,

FEA is unable to account for microstructural properties that govern yield strength, such

as dislocation mechanisms or other microstructural behavior. It may be possible to

predict dislocation mechanisms by using small-scale atomistic simulations, but it would

be too much expensive to conduct analysis for such small dimensions. Also, reducing the

size to atomic level would distort the model and the results would no longer be realistic.

Therefore small-scale simulations are not capable of providing accurate results for single

crystal notched specimens. As such, FEA is the most feasible kind of computer

simulation to analyze these specimens. As it neglects the microstructural behavior, FEA

can also determine the effect of the specimen's geometry and anisotropy on material

property behavior, without considering the atomic interactions.

Experimental Approach

The double-edged notched tensile specimen loaded to study stress and strain fields

and the slip line deformation, in particular, is the most commonly used test sample in

many material testing laboratories. A triaxial state of stress is created around the notch,

which provides the opportunity to study the slip system activity in three-dimensional

stress fields. Experimental tests have been carried out to determine the effect of the load

orientation on the slip systems around the notch. These results are compared to those

obtained by FEA for the same set of conditions.

A constant load can be applied to the specimen in different crystallographic

orientations and the different active slip planes can be observed. This approach is






11


altogether different than the elastic response measured by FEA, the magnitude of the

applied elastic stress gives an idea of which planes will slip first.














CHAPTER 2
REVIEW OF LITERATURE

The stress calculations of a specimen without notches can be done by an analytical

approach. However, as far as the modeling and stress calculations of the specimen with

notches is concerned, it involves a lot of complexities and still an accurate single crystal

analytical model has not yet been established. Quite a few researchers have investigated

the stresses and sectors around the notch tip, but no method to get the precise results has

been proven to be valid. To analyze the stress-field at a single crystal notch involves

calculating the accurate stress components and then predicting and defining the slip zones

around the notch.

Anisotropy of Elasticity

Elasticity

As defined by Dieter (1986) elasticity can be described by specific elastic constants

that relate to atomic strength and spacing of the material. The spacing between the atoms

in an FCC is different in different directions. Comparing the atomic densities of the

material in different directions can show this. Clearly, the atomic density of the material

is not same in all the directions and so the spacing also varies with direction. From this,

we can conclude that the elasticity is a function of orientation.

The greatest atomic density corresponds to the least atomic spacing and most

susceptible planes for slip. For example, planes such as the ones belonging to the { 111}

family are termed as the 'close-packed planes' as they have the least distances between

the atoms. Consequently, the atoms in these planes do not have to travel greater distances









to reach another atomic position and therefore slip occurs most commonly on these

planes. This is the case for close-packed directions; the atomic spacing being less in these

directions makes them most likely for slip occurrence.


Figure 2.1 Schematic of FCC crystal structure

Table 2.1 Atomic density on FCC crystal planes Source: Dieter (1986)
FCC Plane Atoms per unit area Atoms per unit cell
{100} 2/ao2 2

{110} 2/(/2. ao2) 1.414

{111} 4/(/3. ao2) 2.309


Elasticity Matrix

The elastic properties for any material can be completely defined by 36 elastic

constants. However, in most of the materials, some kind of symmetry exists thereby

reducing the number of independent constants. As seen in table 2.2, there are only two

Table 2.2 Symmetry in various crystal structures Source: Dieter (1986)
Crystal structure Number of elastic constants
Tetragonal 6
Hexagonal 5
Cubic 3
Isotropic 2










independent constants for an isotropic material. Any two of the three constants E

(modulus of elasticity), G (shear modulus) and v (Poisson's ratio) are enough to define

the properties of the material in a particular direction.

The symmetry of the cubic structures highly reduces the number of elastic

constants, which in turn reduces the elasticity matrix for these structures. To reduce the

number of independent constants from 36 to the final three for cubic structures, the

original full elasticity matrix [aij] has to be considered:


a11 a12 a13 a14 a15 a16
a21 a22 a23 a24 a25 a26
S] a31 a32 a33 a34 a35 36 (2.1)
a41 a42 a43 a44 "45 a46
a51 a52 a53 a54 a55 a56

a61 a62 a63 a64 a65 a66

The strain matrix is then given by:

E, = a cra (2.2)


Assuming elastic potential exists (i.e isothermal deformation) the following

relationship is achieved in equilibrium:

[a ]= [a, ] (2.3)


On account of this symmetry, the general matrix gets reduced to:


a11 a12 "13 "14 "15 a16
12 a22 a23 a24 a25 a26

[a]= a13 a23 a33 a34 35 36 (2.4)
a14 a24 a34 a44 a45 a46
a15 a25 a35 a45 a55 a56
a16 a26 a36 a46 a56 a66









Nickel-base superalloys exhibit orthotropic characteristics; so they have three

orthogonal planes of elastic symmetry. Considering the effects of cubic elastic symmetry

(Lekhnitskii, 1963), the number of elastic constants reduces to three for the final

elasticity matrix:

a11 a12 a12 0 0 0
a12 a1 a12 0 0 0
1 2 a a12 all 0 0 0
[a ]= 12 a12 (2.5)
0 0 0 a44 0 0
0 0 0 0 a44 0
0 0 0 0 0 a44

These elasticity constants can be defined using the modulus of elasticity, the shear

modulus and the Poisson's ratio in specific directions as follows:


al -- (2.6)
E,

1
a44 (2.7)
Gyz

V V
a12- (2.8)
E= E
Exx Eyy

For material that has been used to conduct these tests, the values of the elasticity

constants are (Swanson and Arakere, 2000):

all = 6.494 E -8 a44= 6.369 E -8 a12 =-2.063 E -8 (psi)

Making use of these elastic constants and the direction cosines, the modulus of

elasticity for any material in any direction can be found as follows (Dieter, 1986):

S=all -2. (a1 -a12) a44 (a2 3 + i32 27 +a2 ) (2.9)
Euvw 2









Where a3, 83 and 73 are the direction cosines from the load orientation to the x, y

and z axes respectively (also sometimes referred to as 1, m and n). Materials like tungsten

always maintain their isotropic properties, even in single-crystal form and their elasticity

is constant in all directions. Some materials, on the other hand, like nickel-base

superalloys have varying elastic properties with varying directions.

Notch Tip Deformation

The stress distribution and analysis of specimens having uniform geometry and

subjected to ideal load and temperature conditions is a straightforward task. However, in

practical applications of superalloys, this is not the case. A common application of

superalloys, for example, is a turbine blade, which has a complex geometry and is

subjected to multiaxial, centrifugal and contact stresses at very high temperature

gradients. In an attempt to study the state of the stress for more complex specimens,

notched tensile specimens are often used to represent either areas of stress concentration

or theoretical fracture conditions.

Rice (1987)

Rice discussed the crack/notch tip stress and strain fields both for FCC and BCC

crystal structures. In his work, Rice assumed the plane strain condition and plotted the

CRSS in specific angular zones or 'sectors', as he termed these zones. He also stated that

the stress state is uniform (independent of the angle) within finite angular sectors near the

tip and the stress state jumps discontinuously at boundaries between sectors. Basically, he

considered two specific orientations and derived an analytical solution for these

orientations and also specified angular boundaries between various sectors. Rice set the









convention for referring to notched samples in terms of notch plane, notch growth

direction and notch tip direction (as shown in figure 2.2).



Load
Direction



Notch Growth Direction




Normal to notch plane


Figure 2.2 Notch direction terminology.

Since the assumption of plane strain is made, the yielding process can be expressed

in terms of a 'yield area'. This helps to define the yield stress in different areas that are

separated by boundaries such that the yield stress is uniform throughout a particular area.

The slip activity occurs only in some slip systems, which combine to produce a large

strain. The state of stress near the crack tip in the plastic zone that Rice has created is

based on a plane state of stress in an isotropic material. Definitely, this is not a

characteristic of single crystal materials. Therefore, the results obtained using this

assumption of plane strain will not correspond well with the experimental results.

AP yield
2- T---- surface
4\(g3rzg
"-/-L~a~2
W


Figure 2.3 Yield surface based on plane strain state of stress









Another simplification that rice made was the assumption of zero out of plane

stress and strain near the notch tip. However, a state of triaxial stress does exist near the

notch. This approach therefore, cannot be used to solve for detailed strain field near the

notch. The orientations Rice chose for his study produced symmetric results about the

notch growth axis. He inferred that there were two conditions under which the notch

surface boundary conditions could be met for all angles. The first one was that the

stresses in some sectors should be below the yield values. The second condition was that

there are discontinuities at some angles. The first condition is difficult to compromise, as

Rice had assumed a perfectly plastic state of stress, so all sectors must be at yield or past.

The solution Rice obtained had no differences for the FCC and BCC crystal

structures or between the two orientations. However, there are differences as regards to

the changes in the slip systems with respect to the orientations and there is a switch

between the slip systems in the shear and normal directions. Thus orientation does not

seem to predict the effect on the yield surface or sector boundaries. This contradicts the

experimental results and Rice noticed that this contradiction was because he had

neglected the actual rotation of the crystal lattice. Rice has also neglected the effect of

strain hardening. However, he recommends incorporating all simplifications in future

study of single crystal models.

Crone and Shield (2001)

In light of Shield's previous work (1996), Crone and Shield conducted further

experimental tests to study the notch tip deformation in two different orientations of

single crystal copper and copper-beryllium tensile specimens, as shown in figure 2.8. Out

of these, many other researchers for study of notch-tip deformation study have considered

Orientation II. Crone and Shield also used Moire interferometry to obtain the strain fields









to determine the sectors and the obvious visible changes from one area to another were

defined as sector boundaries. Visible slip patterns do indicate slip activity, but the

absence of a visible slip does not indicate the absence of any slip activity. The slip

systems that do not show up on the surface may be activated internally, rather than at the

surface. Even on the surface, the slip systems may show varying patterns as the load and

consequently the deformation increases.

[010] [101]





[101]
[010]




Orientation I Orientation II
Figure 2.4 Orientations used by Crone and Shield in their tests.


Table 2.3 Sector boundary angle comparisons for Orientation II Source: Crone and Shield
(2001)
Sector boundary Experimental Analytical Numerical
Crone and Rice Mohan Cuitino and
Shield (1987) et al. Ortiz
(2001) (1992) (1996)
1-2 50-54 54.7 40 45
2-3 65-68 90 70 60
3-4 83-89 125.3 11 100
4-5 105-110 130 135
5-6 150


Crone and Shield also compared their experimental results with the solutions

obtained by other researchers like Rice' analytical solution as well as numerical Finite









Element solutions by Mohan et al. (1992) and Cuitino and Ortiz (1996), as shown in

Table 2.4. Both the numerical solutions were obtained by plane strain assumption.

However, Cuitino and Ortiz stated that this problem couldn't be really considered a plane

strain problem on account of the large variations in strains internally and on the surface.

Even with the plane strain assumption, there is no similarity between the numerical and

analytical results and moreover any of the three do not match with the experimental

results (Figure 2.9 and Figure 2.10). The experimental results obtained by Crone and

Shield are justifiable only in the annuluss of validity', where the actual measurements are

taken. This annulus is a zone at radial distances from 350 tm to 750 tm from the notch

tip. It should be noted here that the notch width is about 100-200 [m and the notch radius

50-100 am. This makes the annulus and the region where the sectors are measured to be

3.5-7.0 times the notch width and 7.5-15.0 times the notch radius from the tip. These

sectors clearly allow the stresses and strains to be studied only in the plastic zone.


x = |10110















notch i x

350-750pm
Figure 2.5 Slip sectors observed in the experimental tests by Crone and Shield. Source:
Crone and Shield (2001).










Crone and Shield chose this annulus also to avoid areas too close to the notch tip.

This is because the areas in the vicinity of the notch tip are dominated by geometry.

Crone and Shield initially start with a single sector of slip activity and later go on

adding the slip lines as the deformation proceeds and so more than one slip lines become

evident in the same sector at higher radial distances from the notch tip. The slip lines

keep growing in numbers until the sector is 'filled'. It is difficult to state what 'filled'

implies here. However, it can be stated for sure that either the initially visible slip system

or another with the same visible trace remains active throughout the process.


S[2 010]
83-890
Sector 4
r Sector 3
105-110 e / / 65-68o
B, B Se or2
cr S or A 50-A540
Sector 5 o / F

1500 elastic Sector

SSector


D / .Dor /- ..... eastic
Dcor ....j.. elastic
castic
Sector 6 ....
Sector 1

Sector A -
or F F \ or B
SSector 2

Sector 4 Slip at 550
Slip at 125 -
Slip at 1800 ***"

Figure 2.6 Slip sectors and slip lines from the experimental tests by Crone and Shield.
Source: Crone and Shield (2001)

Crone and Shield have disregarded the horizontal slip lines observed near the

notch in Sector 1 for Orientation II and they have assumed that these slip lines as elastic









so that they can compare their solutions to other perfectly plastic crack solutions, as seen

in Figure 2.10. Sector 5 is believed to be the only sector showing elastic behavior on the

basis of lack of any visible slip activity as well as low strain values. Therefore the

determination by Crone and Shield of an elastic region exactly ahead of the notch tip is

not consistent with the visible experimental results that they have obtained. As assumed,

the sectors observed are actually symmetric about the [101] axis except some variations

in the sectors 3/-3 and 4/-4 (Figure 2.10). This asymmetry may be due to deviation from

the ideal conditions such as irregularities in the notch geometry, material defects,

inaccurate notch plane direction and so on.

Orientation I is a rotation of the Orientation II by 90 about the notch tip direction

in a way that the notch tip growth direction and the notch plane are interchanged. It was

mentioned previously that Rice's solution did not vary with orientation. However, Crone

and Shield's sectors did show prominent variation with orientation, exhibiting changes in

specific boundary angles as well as in the number of sectors (Table 2.5).

Table 2.4 Sector boundary angles from the experimental tests by Crone and Shield
Source: Crone and Shield (2001)
Sector Boundary Orientation I Orientation II
1-2 50-54
35-40
2-3 65-68
54-59
3-4 83-89
111-116
4-5 105-110
138
5-6 150


Although neither the analytical nor the numerical solutions match with the

experimental results, Crone and Shield uphold a plane strain assumption on the basis of

the numerical solution by Cuitino and Ortiz (1996) because the finite element results on

the central plane of the specimen seemed to correspond well with the experimental









results. However, Crone and Shield make this clear that the plane strain assumption is

valid only for specific locations. In fact, the results from central plane do not depict the

actual specimen being modeled and the stresses at the surface are preferred to get more

accurate results.

Schulson and Xu (1997)

Schulson and Xu studied the state of stress near the notch-tip for single crystal

Ni3Al, which constitutes the y'-component of the material used in this study. They

considered specimens with two different orientations for this study. They used the three-

point bending technique to deform the crystals and the deformation was carried out

through different zones like elastic, plastic and then through initial stages of crack

propagation.

[-10-1]







[010]


[-101]


Figure 2.7 Specimen Orientation used by Schulson and Xu

For both of the samples, a growth direction of [010], a notch plane direction of [-

101] and a notch tip direction of [-10-1] was used. Schulson and Xu calculated the state

of stress (assuming isotropic material conditions) using the equations for a sharp notch by

Anderson (1995). The purpose behind this was to establish an analytical model to









compare with their experimental results. However, isotropic conditions do not hold good

for single crystal materials. They first calculated the state of stress on the basis of plane

stress conditions. This assumption is valid for large thin plates subjected to in-plane

loading. Consequently, the thickness of the specimen should be negligible as compared to

its height and width and the only stress components present are Cx, Gy and Txy. However,

Schulson and Xu observe here that a triaxial state of stress is created around the notch tip

and as a result, out of plane material stresses are present. Then they made a second set of

calculations based on the plane strain assumption, which again can be well applied for the

isotropic conditions (where the out of plane strain is zero) and is not valid for the case of

single crystal materials. In an anisotropic single crystal material, plane strain is a function

of material orientation. The only condition when the simplified stress tensor matrix is

valid is when the specimen is loaded parallel to the FCC lattice edges (Swanson and

Arakere, 2000). Schulson and Xu may use this basic transformation if their specimens are

oriented accurately. Even then the plane stress is not valid because the true component

stresses have out of plane components. The strain components can be obtained by

multiplying the stress tensor matrix by the stress as follows:

x a a12 a12 0 0 0 o-x
y a12 all a12 0 0 0 ay
s, a12 a12 a 0 0 0 0
O0 0 0 a44 0 0 y (2 .10)
Y, 0 0 0 0 a44 0 rz
0Yxy 0 0 0 0 a44 Txy









11 'all +a +a12 + al2
y /a 12 Cx + all OCy + a12 *Oz
Ez a= 1t2 x +a12 Oy +all (2.11)Z
>=1 ; yz (a44 yz
Y zx (144 *a zx
Y, a 44Z- Z

It can be seen that out of plane strain components are present; so the plane strain

assumption is not valid.



---



















Figure 2.8 Slip sectors obtained by Schulson and Xu Source: Schulson and Xu, 1997.

Schulson and Xu calculated the 12 primary resolved shear stresses on the basis of

plane stress and plane strain assumptions. Then using the mode I stress intensity factor K

and radial distance r from the tip of the notch, they normalized the resolved shear

stresses. They obtained similar slip system results for every individual assumption,

however exact systems do differ in some sectors like II, III, IV and V and sector angles

show significant variations for certain areas like I/II and IV/V. The sector I slip systems









are same for both the assumptions, but the stress values are different. Also there is a

considerable difference in the specific slip systems for sectors II-III, though the systems

under the plane strain assumption are just the symmetrical systems from those of the

plane stress. There is an exception though in case of sector lib and in sector IV; the

maximum stresses are resolved on the same slip system, but the directions do not match.

In addition, the plane strain stresses in sector V jump on to { 111) planes as indicted in

Table 2.6, while in sector II the plane strain stresses are still on the {- 111 planes for the

same angles. This data is in conjunction with the elastic region conditions, that is for

sectors II and III in the range 43-100. However, it diverges significantly later in the

sectors I, I/II and IV/V.

The experimental results obtained after plastic deformation, vary from the results

predicted by either plane stress or plane strain, but are considerably close to plane strain

assumption. Usually the slip plane sectors can be observed visually. However some

specific slip systems cannot be distinguished at the surface. In such cases, electron

microscopes are used to determine the specific slip systems. The results obtained by

Schulson and Xu correlate well to the plane stress model for the [01-1] and [011]

dislocations.

Table 2.5 Slip sectors from plane stress and plane strain assumptions. Source: Schulson
and Xu (1996)
Sector Plane Stress Plane strain
0 (deg) Slip systems 0 (deg) Slip systems
I 0-43 (11-1)[-110] 0-23 (11-1)[-110]
(-111)01-11 (-111)01-11
II 43-60 (111)[1-10] 23-60 (-1-1-1)[-110]
(111 )01-11 (-1-1-1)01-11
III 60-103 (1-11)[110] 60-107 (-11-1)[-1-10]
(1-11)[0111 (-11-1)[0-1-11
IV 103-180 (11-1)[011] 107-133 (11-1)[101]
(-111)1101 (-111)1011
V -- 133-180 (-1-1-1)[0-11]









Slip Activation and Deformation

In case of an isotropic material, all the twelve primary slip systems get activated

simultaneously, as is seen from figures 2.12 and 2.13. The Schmid factor is defined using

the load orientation, the slip plane orientation and the slip direction, as follows:

m = cos A-cos 6

Tr = m-cr

Here a is the load applied, Zrss is the RSS component along a given slip plane and

direction, X is the angle between the slip direction and the applied load and finally P is

the angle between the normal to the slip plane and the applied load.





Slip Plane
Normal











Slip
Direction



Figure 2.9 Load and slip direction.

According to the Schmid's equation, when the RSS is equal to the yield stress of

the material, the CRSS is reached. Also the slip systems with the highest Schmid's factor

will reach the CRSS first. This is true for isotropic materials. However, it cannot be






28


correlated to the single crystals and another method needs to be used for the prediction of

slip plane activation of single crystal materials.














CHAPTER 3
ANALYTICAL APPROACH

The overall objective of this work is to find the state of stress in the material

coordinate system of the specimen, which eventually is used to calculate the resolved

shear stresses in the 12 primary slip systems. For an anisotropic specimen with cubic

symmetry, the stress-strain relation is governed by three independent constants in the

material coordinate system and a stress tensor matrix that is dependent on the orientation.

Thus it is not just governed by a single elasticity constant as in the case of an isotropic

material. As the specimen orientation may not be fixed, we need to establish a

relationship between the specimen and material coordinate systems, which facilitates the

conversion of stress and strain fields from one coordinate system to another.

Practically, it is very difficult to cut a sample such that its x, y and z axes are

perfectly aligned to the material axes: [100], [010] and [001] respectively. This creates a

need for transformation matrix to convert the observed stresses, which are in the

specimen coordinate system into the material coordinate system. First of all, to define the

elasticity matrix, it is necessary to determine the exact orientation of the specimen under

study in terms of the material Miller indices (direction indices).

There are two methods to bring about the transformation of the stress components

from specimen coordinate system to the material coordinate system. The first method is a

straightforward method, in which the angles between the material and specimen

coordinate axes are directly measured to obtain the direction cosines. This method can be

employed if the angles between the coordinate systems can be easily found. The second









method can be used for more complex orientations, where the angles between the two

systems are difficult to determine, just by inspection. Here, the transformation equations

can be obtained if the Miller indices of the specimen coordinate axes are known with

respect to the material coordinate axes. In this technique, the original axes are

transformed through a certain number (usually three) of 2-D rotations to obtain the final

transformed axes. In analytical approach, any of the two methods can be used. However,

in experimental specimens, it is difficult to determine the exact Miller indices of the

transformed axes and it is preferable to use the first method. As stated earlier, the original

coordinate system will be termed as the material coordinate system: xo= [100], yo= [010]

and zo= [001] (figure 3.1). The transformed coordinate system is termed as the specimen

coordinate system. It will be denoted by x", y" and z". All the properties associated with

this system will be denoted by the double-prime symbol.



yyo[010]







^)^ ~------ ^ x"


zo[001] x"
00/ o[100]


z"


Figure 3.1 Material and Specimen coordinate system.









Coordinate Axes Transformation Using Miller Indices

As discussed earlier, it is possible to obtain the final transformed axes if their

Miller indices are known. A step-by-step transformation process can accomplish this

process. The first transformation can be brought about by rotating the xo, yo and zo axes

through an angle P about the zo axis. It is obvious that this is a 2-D rotation with the

original and final z-axis being the same.

The equations of transformation for this first step are:

x = x, cos(O) + y, sin(O) (3.1)

x = -xo, sin() + y, cos(O) (3.2)

Sz (3.3)

The above equations in matrix form can be written as:

x cos(O) sin( ) 0 xO
y = sin(O) cos()) 0 y(3
0 zo (3.4)



I) yo[010]
y -



Load Direction



zo[001] I xo[100]


Figure 3.2 First rotation of the coordinate system about the z-axis










0 yo[010]
y '

BB
x



A
A xo100]
Figure 3.3 Projections from the original axes on the projected axes

The second transformation is again a 2-D rotation, but this time the axes are rotated

by an angle D about the y-axis. Therefore in this transformation, the y-axis is same

before and after the rotation.

Proceeding in the same way as for the first transformation, we get the second

following transformation matrix for the second step:

x' cos(q') 0 -sin(o')
y'= 0 1 0 y
z' sin('>) 0 cos(S') z (3.5)

y, y

X'


^""""


Figure 3.4 Second rotation of the coordinate system about the z-axis








Finally, the axes have to be rotated about the x-axis through an angle This third

and final step gives us the complete transformation to the specimen coordinate system ,

Y and z". The transformation matrix for the third step is:

x" 1 0 0 x,
y = 0 cos(I") sin(") Yo
z" 0 sin(O") cos( )_ (3.6)



y ,y






z





Figure 3.5 Third rotation of the coordinate system about the z-axis

The overall transformation from the material to specimen coordinate system can

then be calculated multiplying the three transformation matrices obtained in the three

transformations.
o Y



















1 1 io
y K f 72 A y0
I" a23 3 3 zo (3.7)

where

a, w hr 1 0 0 cos(') 0 sin ') cos() sin@) 0
a, 7 2/ =0 cos(O") sin(O") 0 1 0 -sin() cos() 0
a3 83 73 0 -sin(O") cos(O") sin(O') 0 cos(O') 0 0 1 (3.8)









The values obtained through the above equation are the direction cosines of the

specimen coordinate system with respect to the material coordinate system. The table

below summarizes the above equation:

Table3.1 The direction cosines of the specimen coordinate system with respect to the
material coordinate system
xo Yo zo


x" A A Y7

y a2 A 72

z a3 7A '3



The values of the direction cosines or the final transformation matrix will be:



a2 A 72
(a3 A/3 73 (3.9)
F cos(').cos) cos(') sino) -sin()
sino")- sino')- cos) -coso")- sino) sino"), sino'), sin) +coso"), coso) sino"). coso')
cosY"). sino'). cos) +sinno"). sino) cosN"). sino'). sino)-sino") .cos5) coso")- cos5')

In some cases, fewer than three steps are involved to arrive at the final direction

cosines. The sequence of rotation is arbitrary. However, it is important to check the

accuracy of the results obtained. There are several methods to ensure this.

Example Transformation

To demonstrate this method of arriving at the direction cosines, we will consider a

case in which a load is applied in the [3 1 -2] direction. This transformation also

illustrates a case in which only two steps are involved to get the overall transformation.

Here the first step is eliminated (i.e. 0 = 0) and step two begins by reflecting the load









vector onto the x-z plane. The reflection is simply a triangle whose sides comprise of the

x and z Miller indices: x = 2 and z = 3. The first rotation 0' can be evaluated as:


=' tan-' 3


'= -56.309 (3.10)

Again, the second angle of rotation can be obtained as follows:


= tan-1 1
Stan 1! 2 +(-2)2


"= -15.501 (3.11)

Therefore, the direction cosines for the overall transformation can be obtained by

using the previously established equation:



aL r, 1 0 0 cos(o') 0 ')
a,2 /2 = 0 cos(0") sin((") 0 1 0 (3.12)
a3 73 73 0 sin(") cos(O") sin(o') 0 cos( ')

Substituting the values for 0' and q", we get:

La, 1 /, y, 0.5547 0 -0.832
a2 2 /2 = -0.2223 0.9636 -0.1482 (3.13)
a,3 /3 /3 0.8017 -0.832 0.5345

Transformation Matrices for Stress and Strain Fields

After arriving at the values of the direction cosines between the material and

specimen coordinate system, we can establish the matrices to transform the stresses and

strains. Using these transformed matrices, we can finally evaluate the resolved stresses

and strains in any of the slip systems.






36


The stress transformation, as in Lethniskii is given as:

{() = [Q'](.- (3.14)

{c}= [Q]' 1r "= [Q]. {C" (3.15)

[Q] is the stress transformation matrix is a function of the direction cosines:

a12 2 3 2-a3 -2 2-a1 a3 2-a2 "1
12 22 A32 2.A-#32 2.A A#3 2.-#2 .1
2 2 2

[Q] 2 3 2" 73 "72 271"73 2. 1 2 1
A /yi A-2/y2 A-y/3 (A2/y3+A-y/2) (A1y/3+A83-y/) (18-y2 +82 -y/)
-a, y2 a2 ,a3 (7 2 a + 73 a) (y, a + 73 -a,) (y7 a2 + y2 -a,)
a 11 a2 182 a,3-3 (a2-3 +a3 ,22) (a,1 3+a,3 ) (a,.2 +a2. 1)
(3.16)

The state of stress is defined in terms of the specimen {(c} or material {ao"} as:

'ff
o-x o-x
oy oy

{=}< z {= (3.17)
Tyz yz
Z zx zx
T Ziff


Same procedure is followed to get the strain transformation matrix:

{"}= [Q { (3.18)

t_= [10'-1. t"}= [Q1- ]. 1"} (3.19)

Like the stress transformation matrix, the stress transformation matrix [Q, is also

a function of the direction cosines, but it does differ from the stress matrix:








2 2 2
2, a2 03 03 *2 d *"3 2 *"1
A12 A622 A2 A-A62 A-A A62"1
2 2 2
[ 71 72 73 Y3Y2 71 3 Y2 1 (3.20)
2A-1y. 2A2.y2 2P33. ( 3 22 3+A 2) (fl 73 +A3 3.) (fl + 72 ,2 1)
27, q 272 "a2 27'3 "3 (y2 "3 +7'3 2) (1 3 a1 ) (71 R2 + 72 1ai)
2 .) A 2a2 A 2a3 3 ( .3+03 -) (c 3 1) ( 3-A) (, +2 -1 )

The stress-strain relationship for an isotropic material for a uniaxial state of stress,

using Hooke's law can be given as:

a=E.- (3.21)

However, the generalized Hooke's law for a homogeneous anisotropic material can

be given as:

a = [A,j. (3.22)

where,

[A] = [aj (3.23)

Here [A, and [aj are the elastic constant matrices. It should be noted here that

[a, is a symmetric matrix such that:

a, = lal (3.24)

So we can state:

{ )= [aJ- ] o (3.25)

and

{"}= [a .{a- (3.26)

The elasticity matrix also has to be transformed. However, it does not lose its

symmetry. The number of constants limit to 21 depending on the direction cosines or

specimen orientation.









[a ]=Q] [ a- ]. [] (3.27)

Therefore, we can conclude that if the component stresses are known in the

specimen coordinate system, then it is possible to find the stress components in the

material coordinate system by using the above equations.

Shear Stresses and Strains in the Slip Systems

The component stresses are sufficient to define the state of stress of the material.

However, in anisotropic materials, these stresses do not give any idea about the slip

system activity. It is therefore imperative to transform these component stresses into the

resolved shear stresses along the 12 primary slip systems. Thus after arriving at the

component stresses, we have to again perform a transformation to determine these

resolved shear stresses.

Resolved Shear Stress Components

The resolution of stresses along the slip systems is calculated as:

{r}= c. [S]. {c} (3.28)

Here c and [s] are constants depending on slip plane and direction:

c = (3.29)
h 2 f2 ,2 2 f2 f2
Vh + k + u + v + w

S, = h, -u, k, -.v 1, .w, -w, -v,: -u: (3.30)

Here, [u'' v' w'] is the slip direction and (h' k' 1') is the slip plane. For the

primary octahedral planes, c is a constant and then the Resolved Shear Stress (RSS)

matrix is given by:










hi *u1 k1 vI [1 "w1
h, -u, k, -v, I, *w,
h2 U2 k2 *v2 [2 *w2
h3 U3 k3 V3 13 W3
h4 u4 k4 v4 14 "W4

h, -u, k, *v, I, *w,
h5 u5 k5 v5 15 "w5
h6 u6 k6 V6 16 W6
h7 *u7 k7 V7 /7 6W7
h, -u, k, -v, I, *w,
h8 u8 k8 v8 18 "w8
h, -u, k, *v, I, *w,
h9 u9 k9 v9 0 9 -w9
h1o *u1 k1o *vo 11 *w10

hl u11 ki v,1 1, *w,1
h12 *u12 k12 v12 '12 "w12


Solving the above equation for the 12 primary systems, we get:


1
= -.


1 0 -1 1 0 -1
0 -1 1 -1 1 0
1 -1 0 0 1 -1
-1 0 1 1 0 -1
-1 1 0 0 -1 -1
0 1 -1 -1 -1 0
1 -1 0 0 -1 -1
0 1 -1 -1 1 0
1 0 -1 -1 0 -1
0 -1 1 -1 -1 0
-1 0 1 -1 0 -1
-1 1 0 0 1 1


Similar procedure has to be followed for calculating the shear strains:


(3.33)


Ox
C-y


TX

Txy


Ux

y
Cz

Tyz

Tax


(3.32)


(r)= c [S]-






40


The resolved stress and strain fields along the 12 primary octahedral planes are

determined following the above steps. This gives a lead for predicting the slip systems for

a particular specimen orientation and applied load.














CHAPTER 4
NUMERICAL APPROACH: FINITE ELEMENT METHOD

The finite element technique can be used to model the specimens having a given

geometry and loaded in given orientations. This helps to simulate the conditions in tensile

tests. Specimens with different geometry were modeled in two different orientations

using the ANSYS finite element software and they were analyzed to predict the stress

fields and sectors around the notch of the specimen.

One of the two specimens are based on previous work (Rice, 1987; Schulson and

Xu, 1997; Crone and Shield, 2001) and both of these correlate to collaborative work

between the Mechanical Engineering and Materials Science departments of the

University of Florida (UF).

In this numerical approach, slip-deformation is predicted by the highest resolved

shear stress obtained in the numerical model. The slip systems having the highest

resolved shear stress should be the slip lines in the experimental model. These results

should match with the experimental results that are obtained in the tensile test.

First of all, a solid specimen (without notches) was modeled for a particular load

and orientation using ANSYS. These results were then compared to the analytical

solution (that was set up in chapter) to verify the accuracy of the results obtained

through the finite element analysis. Proceeding on these lines, two notches were

introduced in the specimen to simulate the exact tensile test specimen. The component

stresses obtained for this model from the finite element solution, which were in the

material coordinate system, were incorporated into the analytical solution to get the RSS









in the 12 slip systems. This was done for a wide range of radial and angular distance

around the notch. This data was used to predict the slip activation for various sectors in

the vicinity of the notch.

[001] [111]






S [010] [-12-1]


[100] [30-3]


Specimen I Specimen II

Figure 4.1 The notched specimens analyzed using finite element method.

Verification of the Finite Element Model Results

The specimen without notches can be analyzed using the procedure established in

Chapter 3 and these results can be compared with the finite element model to verify the

accuracy of the results. This was done using the same dimensions (except the notches) as

the actual specimen for consistency. The applied load for all the cases was consistently

100 lbs. The reason for choosing this load was that this load causes stresses well below

the yield stress of the material thereby giving the opportunity to study the slip plane

activity purely in the elastic zone. For the specimen without notches, the component

stresses evaluated from analytical methods seem to match pretty closely to those obtained

by the FEM model. So this initial test of the comparing the results between the two

methods does work. However, it is better to verify these results on the basis of strain

components also. In the analytical technique, the strain components can be evaluated by

making use of the stress tensor matrix. The strain components also seem to match









consistently for the two techniques. As the results obtained through the analytical and

numerical methods agree with each other, it is safe to proceed to obtain results for the

specimens with notches.

Specimen Orientation and Dimensions

Here two specimens with different orientation and geometry will be studied. While

modeling the specimen geometry, only the specimen body is considered and the end grips

involved in the actual tensile test are neglected. There are certain reasons for neglecting

these end grips. First of all, the mechanics at the grips involves tensile rig contact

pressure, loading rate and this makes it significantly different than the mechanics from

the center of the tensile specimen. In the experimental model, different deformation

mechanisms are seen at the grips and these mechanisms can cause fracture even there.

But the aim of these tests is to study the deformation mechanisms at the center of the

specimen and the stresses generated in this region. Second of all, the grips may be

changed in the pursuit of getting more accurate results. Accordingly, the effect and

deformation mechanisms at the ends will change. Here, the numerical model will not be

subjected to the effect caused by the grips.

As far as the geometry of the specimens is concerned, both the specimens have a

different geometry. However, in both the specimens, the notch is modeled as a

combination of a rectangle and a semi-circle. However, in the actual specimen, the notch

has some offset with the horizontal and also an offset in the y-direction from the center of

the specimen. In addition, the notch tip is not a perfect semi-circle, but a smaller arc. In

the numerical technique, all the notch dimensions for both right and left notches were

modeled equal, although there are slight differences between the dimensions of the two

notches. Also, the notch radius was set equal to the notch-height making it a perfect semi-






44


circle. In this work, the specimen orientation is of primary importance. This makes the

minor changes in the notch dimensions negligible. However, the model can be made

perfect geometrically if there is a need for more specific results.


Thickness


Notch Radius




Notch Length


Width


Notch Width






Notch Height


Height







_W_


Figure 4.2 Specimen specifications

Table 4.1 Actual and specimen geometry for Specimen I
Specimen Geometry
(mm) Actual FEM
Width 5.100 5.100
Height 19.000 19.000
Thickness 1.800 1.800
Right Notch Length 1.300 1.550
Left Notch Length 1.550 1.550
Right Notch Height 0.113 0.113
Left Notch Height 0.111 0.113
Right Notch Radius 0.045 0.055
Left Notch Radius 0.055 0.055









Table 4.2 Actual and specimen geometry for Specimen II
Specimen Geometry
(mm) Actual FEM
Width 5.04 5.04
Height 17.594 17.594
Thickness 1.82 1.82
Right Notch Length 1.399 1.399
Left Notch Length 1.36 1.399
Right Notch Height 0.084 0.084
Left Notch Height 0.0845 0.113
Right Notch Radius 0.056 0.056
Left Notch Radius 0.056 0.056

Finite Element Model Characteristics

While modeling the specimen in the Finite Element software, all the details of the

actual tensile test were incorporated so as to obtain accurate results. Accordingly, the

material properties, orientation and boundary conditions were specified to simulate the

test conditions. Since the stresses near vicinity of the notch are of prime importance, the

areas near the notches were meshed more densely than the rest of the areas.

Material Properties

To define the stresses or strains in any particular direction, it is convenient to define

the material properties with respect to the material coordinate first. However, in the FEM,

the specimen is modeled such that its properties are defined with respect to the global

coordinate system. Therefore, proper adjustments must be made while defining the

direction cosines so as to create the material coordinate system. In case of ANSYS, the

material properties are aligned in the same orientation as the element coordinate system.

Therefore, the element coordinate system must be defined for the same orientation as the

material coordinate system so that the material properties are applied in the right

orientation.























z- x

Zo



Figure 4.3 The specimen is first defined with respect to the global coordinate system and
then the material coordinate system is specified later.

The model created in the finite element software is a linear-elastic, orthotropic

model. ANSYS has a wide range of three-dimensional elements to take care of

anisotropic and material properties. To model the single crystal material, one of these

elements needs to be used. Also, either the three independent stress tensors (a11,a12 a44)

or the three independent directional properties (E, G, v) need to be defined to accurately

model the single crystal material.

Elements and Meshing

There are two different elements chosen to mesh the specimen model created in

ANSYS: PLANE2 and SOLID95. The front face of the three-dimensional solid model is

meshed with the PLANE2 elements. The front face has an exact element sizing along the

specified radial lines around the notch tip at intervals of 5.The PLANE2 element is a

two-dimensional, six-node triangular solid. This element has a quadratic displacement









behavior and can be used to good results for modeling irregular meshes. It has six nodes

and two degrees of freedom, which helps to incorporate orthotropic material properties.




















Figure 4.4 Meshing around the notch in ANSYS

Sweeping the three-dimensional elements through the volume can bring about the

meshing of the entire model. It should be noted that the three-dimensional elements retain

the sizing definitions specified for the two-dimensional elements defined on the x-y

planes (front face) through the thickness of the specimen. The SOLID95 element used for

the three-dimensional meshing is a 20-node element with each three degrees of freedom

for each node. This element also works well for irregular shapes and curved boundaries

without significant loss of accuracy. This is why SOLID95 is selected as the areas near

the notches are of prime interest. This element also takes care of the orthotropic material

properties. Here, the original element structure is left intact instead of using the

pyramidal shape, which reduces run time. The pyramidal-shaped element does reduce the

analysis and solution time, but at the cost of accuracy. Using the original element

structure also makes sense in the fact that the stress gradient near the notches is very









large. In addition, SOLID95 also has features like plasticity, creep, stress stiffening, large

deflection, and large strain capabilities.


Figure 4.5 Schematic of the SOLID95 element in ANSYS. Source ANSYS 6.1 Elements
Reference, 2002.


Figure 4.6 Radial arcs used for element location and sizing; centered at notch tip.









Through ANSYS the stress tensor can be transformed to any particular orientation

to get the stress and strain values. To get a near-field as well as far-field state of stress in

terms of radial and angular distances, six concentric arcs were generated at the following

radial distances form the notch tip: 0.25*p, 0.5* p, p, 2* p, 5* p; where p is the radius of

the notch. The reason for creating these arcs is that in ANSYS, the stress calculations can

be obtained only at the corner nodes of the elements and not at all the nodes.

90
5*p


2*p







0.5*"p


0.25*p

Figure 4.7 Radial arcs for stress field calculations

Finite Element Solution

The node-selection along the selected concentric arcs around the notch is done

manually using the ANSYS Graphical User Interface (GUI). Initially all the nodes

starting from 0 to 1800 are selected. However, on account of the symmetry, the

calculations of the stresses in the 0 to -1800 need not be done. This assumption was

validated by calculating the stresses all over the notch for the p arc, wherein symmetry

was found in the stress calculations in the two semi-circular regions. Though the resolved

shear stresses for the negative angles change direction in conjunction with the changing









direction of shear deformation between the top and lower halves of the test specimen, the

magnitudes of the maximum resolved shear stresses are same for the positive and

negative angles.

Another point of interest in the FEM model is regarding the plane of data

collection. The element sizing was done in such a way that data collection was possible

on five different planes including the front, middle and back planes of the specimen.

However, in the collaborative interests with the MSE department, the stress calculations

on the front and middle planes were focused on. It should be noted that here we are not

assuming a state of plane strain.

Assumptions

In the finite element analysis, we have considered the model to be a linear elastic

model. This assumption is useful to predict the initial slip, but it fails to accurately give

us the idea about the forthcoming plastic behavior around the notch. In FEM, the effects

due to the temperature can be taken into account. However, in order to be consistent with

the MSE department, the material properties at a constant room temperature were

considered in the FE analysis. The effects of microstructural mechanisms are also been

disregarded in this model. The last assumption is regarding the omission of the effect of

crystal lattice orientation. This assumption suits well for these conditions because

according to Stouffer and Dame (1996), crystal lattice rotation only takes place if the

there is some strain in a single slip system and the load applied to the specimen is low

enough to prevent this effect from occurring.

This FE model only considers the deformation in the elastic zone. Here, only the

elastic stresses and strains are emphasized rather than the values at the point of fracture.






51


This model can be the basis for further study that includes plasticity, creep and other

behavior caused by very high stress and strain values.














CHAPTER 5
RESULTS AND DISCUSSION

As discussed in the previous chapter, the FEM is used to simulate the tensile test

conditions so that the slip system activity and subsequently the sectors can be obtained.

The results for both the specimens are plotted in terms of the RSS as a function of radial

and angular distances. Here, the direction of the slip is not of any concern. As such, only

the absolute values of the RSS are considered. The stresses are symmetric about the notch

growth direction in case of specimen I. Therefore only the stresses on the upper portion

of the notch growth direction are plotted. However, the orientation of specimen II is such

that the stress distribution is not symmetric about the notch growth direction and so the

stresses on the upper as well as lower portion of the notch growth direction have been

plotted. The slip sectors will be plotted as the dominant slip system at any angle theta.

The results for both the specimens (with different geometry and load orientation) will be

analyzed separately.

Specimen I

As stated earlier, specimen I was loaded in the [001] direction with a [010] growth

direction and a [100] notch plane. The results have been plotted for the RSS on the 12

primary slip systems. The radial distances vary from 0.25*p to 5* p and the angular

distances vary from 0 to the top or bottom of the notch (1000 for 0.25* p to 1650 for 5*

p). The maximum RSS at any location is 16= T8=31,140 psi at R=0.5* p and 1050 on the

mid-planes. The slip system with the maximum RSS varies with varying radial and

angular distances. On this basis, sectors were determined for the stresses on the surface









and the mid-planes. The RSS varies as the state of stress changes away from the notch

tip. However, the variation of RSS for each slip system is different for different slip

systems. So the relative magnitudes of RSS on different slip systems may vary with

respect to each other. Therefore, it should be noted for a given angle, the dominant slip

system is not constant with changing radius.

The plots between the RSS and theta highlight the effect of theta only on the RSS

values. However, the effect of radius on the RSS can be understood by combining the

effects of all the radii. Figure (5.6) and (5.12) give a picture of the combined effect of the

radial and angular distances on the RSS. For example, consider an angle of 700. The radar

plots (Figure 5.7 and 5.8) also give an idea of the combined effect of stresses at various

radial and angular distances. However, the radar plots are difficult to comprehend. So

they are plotted only for a single set of data. Moreover, the sector plots give a much

simple picture of the combined effect of theta and varying radii on the RSS.

















RSS Vs. Theta for 0.25*rho


0
0 10 20 30 40 50
Theta (deg)


70 80 90


Figure 5.1 RSS on the 12 primary slip systems for the upper portion of the notch growth direction for R=0.25*p


25000





20000





15000


10000





5000


--- T1
-W- T2
- -T3
T4
W--T5
-- T6
---T7
-T8
T9
- -T10
- -T11
- -T12

















RSS Vs. Theta for 0.5*rho


30000




25000




20000



a.
15000




10000




5000




0


0 20 40 60 80 100
Theta (deg)



Figure 5.2 RSS on the 12 primary slip systems for the upper portion of the notch growth direction for R=0.5*p


- T1
-W- T2
- -T3
T4
S--T5
--- T6
-1T7
-T8
T9
- -T10
- -T11
--T12

















RSS Vs. Theta for 1*rho


18000 -


16000


14000 -
-- T1
---T2
12000 -T3
T4

10000 -. -T5
a. -40- T6

8000 -z -- T8

T9
6000- -T1


4000 -T12


2000


0
0 20 40 60 80 100 120
Theta (deg)


Figure 5.3 RSS on the 12 primary slip systems for the upper portion of the notch growth direction for R=l*p
















RSS Vs. Theta for 2*rho


12000




10000

ST1


8000 --T3
T4


6000
,,T7
T8
T9





2000




0
0 20 40 60 80 100 120 140 160
Theta (deg)



Figure 5.4 RSS on the 12 primary slip systems for the upper portion of the notch growth direction for R=2*p

















RSS Vs. Theta for 5*rho


8000



7000



6000



5000



4000



3000



2000



1000



0


0 20 40 60 80 100 120 140 160
Theta (deg)



Figure 5.5 RSS on the 12 primary slip systems for the upper portion of the notch growth direction for R=5*p


- T1
-W- T2
- -T3
T4
S--T5
---T6
-1T7
-T8
T9
- -T10
- -T11
--T12











*T8
* T6
ST10
* T2
* T3
* T4
* T9


100
110


160

170 7Q^ '

180


Figure 5.6 Slip system sectors on the surface for specimen I


10

-J0
5











































Figure 5.7 Upper portion of the radar plot showing the surface stresses for the [100] orientation.












































Figure 5.8 Lower portion of the radar plot showing the surface stresses for the [100] orientation.












Table 5.1 Specimen I dominant slip system sectors on the surface
R=0.25* p R=0.5* p R=1.0* p
Sector 0 Tmax Slip system 0 Tmax Slip system 0 Tmax Slip system
I 0-41 s8 (1-1-1)[0-11] 0-56 s8 (1-1-1)[0-11] 0-52 Tio (-1-11)[011]
II 41-100 T6 (-11-1)[011] 56-105 z6 (-11-1)[011] 52-56 z2 (111)[0-11]
III 56-116 T6 (-11-1)[011]

IV 116-120 T4 (-11-1)[10-1]

V
VI


R=2.0* p


R=5.0* p


Sector 0 Tmax Slip system 0 Tmax Slip system
I 0-50 zio (-1-11)[011] 0-46 Tio (-1-11)[011]
II 50-60 z2 (111)[0-11] 46-56 r2 (111)[0-11]
III 60-68 z3 (111)[1-10] 56-64 "3 (111)[1-10]

IV 68-150 T6 (-11-1)[011] 64-131 T6 (-11-1)[011]
V 131-164 "9 (-11-1)[011]
VI 164-165 T6 (111)[1-10]

















RSS Vs. Theta for 0.25*rho


20 30


40 50

Theta (deg)


90 100


Figure 5.9 RSS on the 12 primary slip systems for the upper portion of the notch growth direction for R=0.25*p


24000



20000



16000



12000



8000



4000


- T1
-U- T2
--T3
T4
T5
T6


T8
T9
- -T10
- -T11
--T12

















RSS V. Theta for 0.5*rho


35000



30000


-- tl
25000 -W-- t2
-t3
t4
S20000 --t5
V -0- t6
t7
i 15000 ---t8



10000 --- -t11
-t12


5000
5000-- ------ ---"------......_



0
0 20 40 60 80 100
RSS (psi)


Figure 5.10 RSS on the 12 primary slip systems for the upper portion of the notch growth direction for R=0.5*p

















RSS V Theta for 1*rho


20000


18000


16000-
1- t2

14000- t
-t3
t4
12000
t5
0 -r --- t6

1000 0

t9

6000 --t
-ti1

4000 I -t12


2000


0
0 20 40 60 80 100 120

Theta (deg)


Figure 5.11 RSS on the 12 primary slip systems for the upper portion of the notch growth direction for R=l *p
















RSS V. Theta for 2*rho


10000


9000 -


8000 -
t1
7000 _____t2
-t3

6000 -" i"d t4
M E t5
-\ -0- t6
5000 -*-t6




3 -000 -t12




4 0 20 40 60 80 10 120 140 160





RSS (psi)
3000 f 7A




1000



0 20 40 60 80 100 120 140 160
RSS (psi)


Figure 5.12 RSS on the 12 primary slip systems for the upper portion of the notch growth direction for R=2*p

















RSS V. Theta for 5*rho


0 20 40 60 80 100 120 140 160
RSS (psi)



Figure 5.13 RSS on the 12 primary slip systems for the upper portion of the notch growth direction for R=5*p


- t1
-W- t2
- -t3
t4
W-- t5
-0- t6
-+- t7
-t8
t9
- -t10
- -tll
- -t12











* T3,T5,T7,T12
* T6,T8
T2,T10
T4,T9
110 70
120 60
130 "',50
140 "'\ 40

150 \ ,30

160 \.200



180 0
.25 .5 1 2 5


Figure 5.14 Slip system sectors on the mid-planes for specimen I












Table 5.2 Specimen I dominant slip system sectors on the on the mid-planes
R=0.25* p R=0.5* p R=1.0* p
Sector 0 Tmax Slip system 0 Tmax Slip system 0 Tmax Slip system
(111)[1-10], (111)[1-10], 0-38 T2, 1io (111)[0-11],
(-11-1)[110], (-11-1)[110], (-1-11)[011]
I 0-44 T3, z5, 17, z12 (1-1-1)[110], 0-105 13, 15, 17, z12 (1-1-1)[110],
(-1-11)[1-10] (-1-11)[1-10]
(-11-1)[011], 38-112 T6, 18 (-11-1)[011],
II 44-95 T6, 8 (1-1-1)[0-11] (1-1-1)[0-11]
(111)[1-10], 112-120 T4, z9 (-11-1)[10-1]
(-11-1)[110], (1-1-1)[101]
III 95-100 T3, T5, T7, z12 (1-1-1)[110],
(-1-11)[1-10]


R=2.0* p


R=5.0* p


Sector 0 Tmax Slip system 0 Tmax Slip system
0-53 12, 1io (111)[0-11], 0-58 12, Tio (111)[0-11],
(-1-11)[011] (-1-11)[011]
53-120 16, 18 (-11-1)[011], 58-130 16, 18 (-11-1)[011],
II (1-1-1)[0-11] (1-1-1)[0-11]
120-150 14, 19 (-11-1)[10-1] 130-157 4, 19 (-11-1)[10-1]
(1-1-1)[101] (1-1-1)[101]
157-163 16, 18 (-11-1)[011],
IV (1-1-1)[0-11]
V 163-165 12, T0o (111)[0-11],
V (-1-11)[011]









Specimen II

As stated earlier, specimen I was loaded in the [111] direction with a [-1 2-1]

growth direction and a [3 0-3] notch plane. The load axis in this case is such that the RSS

values will not be symmetrical above and below the notch. Therefore results for both the

upper and bottom parts of the notch have been plotted. The results have been plotted for

the RSS on the 12 primary slip systems. The radial distances vary from 0.25*p to 5* p

and the angular distances vary from 0 to the top or bottom of the notch (1000 for 0.25* p

to 1650 for 5* p). The maximum RSS at any location is T6=36,090 psi at R=0.5* p and

1050 on the midplanes. The slip system with the maximum RSS varies with varying

radial and angular distances. On this basis, sectors were determined for the surface

stresses. The RSS varies as the sate of stress changes away from the notch tip. However,

the variation of RSS for each slip system is different for different slip systems. So the

relative magnitudes of RSS on different slip systems may vary with respect to each other.

Therefore, it should be noted that for a given angle, the dominant slip system is not

constant with changing radius.

Again, the sector plots in Figure (5.20), (5.26), (5.32) and (5.38) give a picture of

the combined effect of theta and radii on the RSS values.

















RSS V. Theta for 0.25*rho





20000



--- T1
16000 T2
--- T3
T4
-*- T5
12000
T6


T T8

8000 T9
-T10


000-- T12
4000






0 10 20 30 40 50 60 70 80 90 100
Theta (deg)



Figure 5.15 RSS on the 12 primary slip systems for the upper portion of the notch growth direction for R=0.25*p















Error!


RSS V. Theta for 0.5*rho


30000



25000
ST1
---B--T2
-U- T2
/ -a- T3
20000
ST4
T5
aL -0- T6
15000

S-T8

T9
10000 T

---- T11

5000



0
0 10 20 30 40 50 60 70 80 90 100
Theta (deg)


slip systems for the upper portion of the notch growth direction for R=0.5*p


Figure 5.16 RSS on the 12 primary

















RSS V. Theta for 1*rho


18000





14000


T2
--W---T2
S/ I\IT3


10000 T4
1000 T5
a. -4-* T6
- Oi-T7
l I T8
6000 T9
--T10


fr-T12

2000



2 20 40 60 80 100 10

-2000
Theta (deg)


Figure 5.17 RSS on the 12 primary slip systems for the upper portion of the notch growth direction for R= p

















RSS V. Theta for 2*rho


12000




10000




8000


6000


4000




2000




0


0 20 40 60 80 100 120 140

Theta (deg)


Figure 5.18 RSS on the 12 primary slip systems for the upper portion of the notch growth direction for R=2*p


ST1
--- T2
--- T3
T4
ET5
-e- T6
-- T7
-T8
T9
- -T10
-- T11
-- T12















RSS V. theta for 5*rho


8000


7000


6000 ---
T1
T2
5000 --T3
T4
S/ --- -T5

00, M T7
4000 2 4 T
-T8

3000 T9
-T10
---Ti 1
-- T12
2000


1000 /
M-- OO


0 20 40 60 80 100 120 140 160
Theta (deg)


Figure 5.19 RSS on the 12 primary slip systems for the upper portion of the notch growth direction for R=5*p












OT10
* T6
T5
* T9
*T11
ST3
TT2
T12



1


1


100 90 80

60


170 '

80


Figure 5.20 Slip system sectors on the surface on the upper portion of the notch growth direction for specimen II












Table 5.3 Specimen II dominant slip system sectors on the surface on the upper portion of the notch growth direction
R=0.25* p R=0.5* p R=1.0* p
Sector 0 Tmax Slip system 0 Tmax Slip system 0 Tmax Slip system
I 0-13 Tio (-1-11)[011] 0-50 Tio (-1-11)[011] 0-70 Tio (-1-11)[011]
II 13-27 T6 (-11-1)[011] 50-105 T5 (-11-1)[110] 70-110 T5 (-11-1)[110]
III 27-100 z5 (-11-1)[110] 110-118 T3 (111)[1-10]
IV 118-120 z9 (1-1-1)[101]
V
VI


R=2.0* p


R=5.0* p


Sector 0 Tmax Slip system 0 Tmax Slip system
I 0-63 Tio (-1-11)[011] 0-18 T9 (1-1-1)[101]
II 63-67 Tir (-1-11)[101] 18-58 Tio (-1-11)[011]
III 67-119 z3 (111)[1-10] 58-102 z5 (-11-1)[110]

IV 119-127 r6 (-11-1)[011] 102-107 r6 (-11-1)[011]

V 127-137 T2 (111)[0-11] 107-113 "5 (-11-1)[110]
Vi 137-138 z5 (-11-1)[110] 113-117 z3 (111)[1-10]

VII 138-150 z3 (111)[1-10] 117-167 z2 (111)[0-11]
VIII 167-170 z12 (-1-11)[1-10]















RSS V. Theta for 0.25*rho


--T1
-- T2
-A- T3
T4
-- T5
--T6
T7
T8
T9
- -T10
--T11
- T12


90 100


Figure 5.21 RSS on the 12 primary slip systems for the lower portion of the notch growth direction for R=0.25*p


20000




16000


12000




8000


4000




0


0 10 20 30 40 50 60 70 80
Theta (deg)















RSS V. Theta for 0.5*rho


24000


20000


16000


12000


8000


4000


0


20 40 60 80 100


Theta (deg)


Figure 5.22 RSS on the 12 primary slip systems for the lower portion of the notch growth direction for R=0.5*p


-4--T1
- T2
--T3
T4
- T5
-- T6
-I-T7
-T8
T9
- -T10
---T11
-- T12















RSS V. Theta for 1*rho




16000
--T1



12000 T4


0 -T6

8000
-- T8
ST9 T9
-T1O
4000 --,-- T1 1
o0 T12



0 20 40 60 80 100 120
Theta (deg)


Figure 5.23 RSS on the 12 primary slip systems for the lower portion of the notch growth direction for R= p















RSS V. Theta for 2*rho


12000



10000 -- T1
S--m- T2
-A- T3
8000

S--- T5
C -.T6
6000
W I ^ "I/ / T8 00

4000 T9
-T10
A- -.- T1 1
2000 T12



0 t
0 20 40 60 80 100 120 140
Theta (deg)


Figure 5.24 RSS on the 12 primary slip systems for the lower portion of the notch growth direction for R=2*p
















RSS V. Theta for 5*rho


8000



-*- T1

6000
-- -T3
T4
-A T5
--- T6
4000 -- T7
TO (- T8
00
T9
-T10
-.-T11
2000 --T11
-/ T1 2






0 20 40 60 80 100 120 140 160

Theta (deg)


Figure 5.25 RSS on the 12 primary slip systems for the lower portion of the notch growth direction for R=5*p












*T7
*T10
T3
ST6
*T11
T2
* T5
* T9
*T12


1:

160

170

180


100
110 -


90 go


Figure 5.26 Slip system sectors on the surface on the lower portion of the notch growth direction for specimen II












Table 5.4 Specimen II dominant slip system sectors on the surface on the lower portion of the notch growth direction
R=0.25* p R=0.5* p R=1.0* p
Sector 0 Tmax Slip system 0 Tmax Slip system 0 Tmax Slip system
I 0-100 T7 (1-1-1)[110] 0-1 Ti (-1-11)[011] 0-1 zlo (-1-11)[011]
II 1-105 T7 (1-1-1)[110] 1-111 z7 (1-1-1)[110]
III 111-117 z3 (111)[1-10]
IV 117-120 "6 (-11-1)[011]

V
VI


R=2.0* p


R=5.0* p


Sector 0 Tmax Slip system 0 Tmax Slip system
I 0-63 zTo (-1-11)[011] 0-18 z9 (1-1-1)[101]

II 63-67 zTi (-1-11)[101] 18-58 zio (-1-11)[011]
III 67-119 T3 (111)[1-10] 58-102 T5 (-11-1)[110]

IV 119-127 T6 (-11-1)[011] 102-107 T6 (-11-1)[011]
V 127-137 T2 (111)[0-11] 107-113 T5 (-11-1)[110]
VI 137-138 T5 (-11-1)[110] 113-117 T3 (111)[1-10]
VII 138-150 T3 (111)[1-10] 117-167 T2 (111)[0-11]

VIII 167-170 T12 (-1-11)[1-10]
















RSS V. Theta for 0.25*rho


28000



24000



20000 T2
--,-- T3

T4
16000 -W T5
8.' --0-T6
CO A-AI-iT7
12000 T8 00 oo
S* T9











0 20 40 60 80 100
Theta (deg)



Figure 5.27 RSS on the 12 primary slip systems on the mid-planes for the upper portion of the notch growth direction for R=0.25*p

















RSS V. Theta for 0.5*rho


40000


36000


32000 --4-- T1
+-W-T2
28000 -,- T3
T4
24000 -W T5
S-0--T6

S20000 T7
T9
-- --TI
16000- oo
----T11
12000 WT12
A1 A- T1 2

8000


4000 ----. .


WO *--
0 20 40 60 80 100
RSS (psi)


Figure 5.28 RSS on the 12 primary slip systems on the mid-planes for the upper portion of the notch growth direction for R=0.5*p


















RSS V. Theta for 1* rho


20000





16000





S12000
a.




8000





4000





0


- T1
-*-- T3
---- T2
-.a- T3
T4
--t- T5
- T6
-I-T7
---T8
T9
- -T10
-- T11
-A- T12


20 40 60 80 100 120

Theta (deg)


Figure 5.29 RSS on the 12 primary slip systems on the mid-planes for the upper portion of the notch growth direction for R=p
















RSS V. Theta for 2*rho


12000



10000 -'
'--- T1
-l- T2
--.- T3
8000
T4
i- T5
-.- T6



.--_ \ \ ~T9
4000
-- T1

*.-u-T12
2000 T


0 ,' O O0 q q & .-
0 20 40 60 80 100 120 140
Theta (deg)


Figure 5.30 RSS on the 12 primary slip systems on the mid-planes for the upper portion of the notch growth direction for R=2*p