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SUBSURFACE STRESSES IN ANISOTROPIC CYLINDRICAL CONTACTS By SANGEET SUBHASH SRIVASTAVA 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

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Copyright 2003 By Sangeet Subhash Srivastava

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ACKNOWLEDGMENTS I am grateful to Dr. Nagaraj K. Arakere, my committee chairman, who has given me full guidance throughout this project. Also, I am thankful to the members of my thesis committee, Dr. Ashok V. Kumar and Dr. John K. Schueller, for the time they spent in reviewing and commenting on this study. Special thanks go to my family members and friends for giving me support and help throughout the study. Finally, I would like to thank the Almighty God for giving me power and courage to complete this thesis. iii

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TABLE OF CONTENTS page ACKNOWLEDGMENTS.................................................................................................iii LIST OF TABLES.............................................................................................................vi LIST OF FIGURES..........................................................................................................vii ABSTRACT.........................................................................................................................x CHAPTER 1 INTRODUCTION............................................................................................................1 History.................................................................................................................................1 The Objective of This Study...............................................................................................3 Motivation for the Problem.................................................................................................6 Isotropy...............................................................................................................................7 Anisotropy...........................................................................................................................8 2 LITERATURE REVIEW AND THEORETICAL FRAMEWORK..............................10 Hertz Theory of Elastic Contact.......................................................................................10 A Review of Articles on Plane Contact Problems............................................................14 Contact Problems Neglecting Forces of Friction...................................................14 Contact Problems for the Case When the Punches are Rigidly Connected to the Elastic Body..............................................................................................16 Contact Problems in the Theory of Elasticity Considering Friction......................17 Problems of Contact Between Two Elastic Bodies................................................20 A Review of Articles on Three-Dimensional Contact Problems......................................22 Examples of Numerical Approaches to Contact Mechanics Problems............................25 An Overview of Lekhnitskiis Work................................................................................27 3 ANISOTROPIC ELASTICITY EQUATIONS FOR FCC SINGLE CRYSTALS........28 Slip Activation and Deformation in FCC Single Crystal..................................................29 Elasticity...........................................................................................................................31 Elasticity Matrix................................................................................................................32 Coordinate System Transformation..................................................................................35 Steps in Coordinate Transformation.................................................................................37 Stress Transformation.......................................................................................................40 Transformation of the Elastic Constants...........................................................................41 iv

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Slip System Shear Stresses...............................................................................................42 Resolved Shear Stresses....................................................................................................42 4 GENERALIZED PLANE STRAIN EQUATIONS FOR A HOMOGENEOUS ANISOTROPIC ELASTIC SOLID...................................................................................43 Plane Strain Conditions for Isotropic Contacts.................................................................43 Generalized Plane Strain Conditions for Anisotropic Contact Problems.........................45 Generalized Plane Strain Equations..................................................................................47 The Distribution of Stresses in an Elastic Half-Space Under the Influence of Stresses Applied to the Bounding Plane.............................................................................52 5 FINITE ELEMENT SOLUTION OF THE SUBSURFACE STRESSES......................55 Finite Element Model.......................................................................................................55 Material Properties and Model Characteristics.................................................................58 6 RESULTS AND DISCUSSION.....................................................................................62 Crystallographic Orientation 1..........................................................................................63 Crystallographic Orientation 2..........................................................................................68 Crystallographic Orientation 3..........................................................................................73 Crystallographic Orientation 4..........................................................................................79 Crystallographic Orientation 5..........................................................................................85 Comparison of FEM Solution with the Analytical Results for Case 1 Neglecting Tangential Traction...............................................................................................92 Comparison of FEM Solution with the Analytical Results..............................................96 Case1 ...................................................................................................................96 Case 4 .................................................................................................................102 7 CONCLUSIONS AND RECOMMENDATIONS FOR FUTURE WORK.................109 Conclusions.....................................................................................................................109 Recommendations for Future Work................................................................................110 APPENDIX SAMPLE COORDINATE TRANSFORMATION WITH ACCURACY CHECKS.....112 Sample Coordinate Transformation................................................................................112 Checks for Accuracy.......................................................................................................113 LIST OF REFERENCES.................................................................................................114 BIOGRAPHICAL SKETCH...........................................................................................118 v

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LIST OF TABLES Table page 3-1 Atomic densities on FCC crystal planes......................................................................32 3-2 Symmetry in various crystal structures.......................................................................33 3-3 Direction Cosines........................................................................................................40 6-1 Orientation 1 activated slip system sectors..................................................................67 6-2 Orientation 2 activated slip system sectors..................................................................72 6-3 Orientation 3 activation slip system sectors................................................................77 6-4 Orientation 4 activated slip system sectors..................................................................83 6-5 Orientation 5 activated slip system sectors..................................................................89 vi

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LIST OF FIGURES Figure page 1-1 Isotropic cylindrical indenter in contact with a flat isotropic body...............................4 1-2 Isotropic cylindrical indenter in contact with a flat anisotropic body...........................5 1-3 Pressure profile due to the indenter on the flat body (a stress problem).......................6 1-4 Blade orientation............................................................................................................7 2-1 Two bodies in Hertzian Contact..................................................................................11 3-1 Orientation-dependant stress-strain behavior..............................................................29 3-2 Load and slip directions and angles.............................................................................30 3-3 Primary resolved shear stress planes and directions....................................................30 3-4 FCC crystal structure...................................................................................................31 3-5 Material (xo, yo, zo) and specimen (x, y, z) coordinate systems...........................37 3-6 First rotation of the material coordinate axes about the zo-axis..................................37 3-7 Second rotation about the y-axis.................................................................................38 3-8 Third rotation about the x-axis...................................................................................39 4-1 Isotropic elastic half-space..........................................................................................44 4-2 Distribution of forces in an anisotropic half-space......................................................46 4-3 Homogeneous Anisotropic Body bounded by a cylindrical surface...........................48 4-4 Elastic equilibrium of an infinite homogeneous body with a parabolic profile..........53 5-1 Crystallographic Orientations of the anisotropic body................................................56 5-2 Global and material coordinate systems......................................................................57 5-3 Isotropic cylindrical indenter and the anisotropic substrate in contact.......................58 5-4 Dimensions of the anisotropic contact model..............................................................59 vii

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5-5 ANSYS SOLID 45 element.........................................................................................60 5-6 Refined meshing in the contact zone...........................................................................60 5-7 Meshed anisotropic FCC single crystal substrate........................................................61 6-1 Schematic of the polar coordinate system used in the subsurface contact region.......62 6-2 Orientation 1 primary resolved shear stresses; r = 0.3*a.............................................64 6-3 Orientation 1 primary resolved shear stresses; r = 0.8*a.............................................65 6-4 Orientation 1 primary resolved shear stresses; r = 3*a................................................66 6-5 Orientation 2 primary resolved shear stresses; r = 0.3*a.............................................69 6-6 Orientation 2 primary resolved shear stresses; r = 0.8*a.............................................70 6-7 Orientation 2 primary resolved shear stresses; r = 3*a................................................71 6-8 Orientation 3 primary resolved shear stresses; r = 0.3*a.............................................74 6-9 Orientation 3 primary resolved shear stresses; r = 0.8*a.............................................75 6-10 Orientation 3 primary resolved shear stresses; r = 3*a..............................................76 6-11 Orientation 4 primary resolved shear stresses; r = 0.3*a...........................................80 6-12 Orientation 4 primary resolved shear stresses; r = 0.8*a...........................................81 6-13 Orientation 4 primary resolved shear stresses; r = 3*a..............................................82 6-14 Orientation 5 primary resolved shear stresses; r = 0.3*a...........................................86 6-15 Orientation 5 primary resolved shear stresses; r = 0.8*a...........................................87 6-16 Orientation 5 primary resolved shear stresses; r = 3*a..............................................88 6-17 Contour plot for sigma (x)--Analytical solution........................................................92 6-18 Contour plot for sigma (x)--FEM solution................................................................92 6-19 Contour plot for sigma (y)--Analytical solution........................................................93 6-20 Contour plot for sigma (y)--FEM solution................................................................93 6-21 Contour plot for tau (xy)--Analytical solution..........................................................94 6-22 Contour plot for tau (xy)--FEM solution...................................................................94 viii

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6-23 Contour plot for sigma (z)--Analytical solution........................................................95 6-24 Contour plot for sigma (x)--FEM solution................................................................95 6-25 Comparison of stresses in the X-direction for orientation 1......................................96 6-26 Comparison of stresses in the Y-direction for orientation 1......................................97 6-27 Comparison of stresses in the Z-direction for orientation 1......................................98 6-28 Comparison of shear stresses in the XY-direction for orientation 1.........................99 6-29 Comparison of shear stresses in the YZ-direction for orientation 1........................100 6-30 Comparison of shear stresses in the XZ-direction for orientation 1........................101 6-31 Comparison of stresses in the X-direction for orientation 4....................................102 6-32 Comparison of stresses in the Y-direction for orientation 4....................................103 6-33 Comparison of stresses in the Z-direction for orientation 4....................................104 6-34 Comparison of shear stresses in the XY-direction for orientation 4.......................105 6-35 Comparison of shear stresses in the YZ-direction for orientation 4........................106 6-36 Comparison of shear stresses in the XZ-direction for orientation 4........................107 A-1 Two step coordinate transformation.........................................................................113 ix

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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 SUBSURFACE STRESSES IN ANISOTROPIC CYLINDRICAL CONTACTS By Sangeet Subhash Srivastava May 2003 Chair: Dr. Nagaraj K Arakere Major Department: Mechanical and Aerospace Engineering This study deals with an analytical procedure for calculating subsurface stresses in anisotropic cylindrical contacts. The procedure uses the method of complex potentials with the assumptions of generalized plane strain at the contact. This assumption allows for five independent stress components to be evaluated in the half space ( x y xy xz zx ). z is a function of the five independent stress components. Stresses are not the functions of z-coordinate. The generalized plane strain assumptions allow for modeling the shear coupling present in anisotropic materials. The analysis procedure developed is used to examine subsurface stresses in contact between a cylindrical isotropic indenter and a single crystal anisotropic substrate. Subsurface stresses are studied for various crystallographic orientations. The resolved shear stresses on the twelve primary octahedral planes [(111) <110>] are computed in the vicinity of the contact, to predict evaluation of slip systems in the contact region. x

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Subsurface stresses are also computed using FEA, for a few cases, for comparison purposes. The analytical solution procedure is shown to be an efficient and accurate procedure for computing subsurface stresses in anisotropic contacts. xi

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CHAPTER 1 INTRODUCTION Contact Mechanics is a branch of engineering that deals with the explanations about the nature of contact between solid bodies and its detailed study. It is concerned primarily with the stresses and deformation that arise when the surfaces of two solid bodies are brought into contact [1]. Contact problems can be classified into the following two categories: Problems where one body is elastic and the other is rigid Problems involving two elastic bodies In the first class of problems, termed punch problems, the contact region is known a-priori. In the second class, termed elastic contact problems, the contact region is initially unknown and needs to be determined. History It may surprise those who venture into the field of contact mechanics that the subject of contact mechanics was first started in 1882 with the publication by Heinrich Hertz of his classical paper On the Contact of Elastic Solids [2]. At first glance it appears as if the nature of the contact between the two elastic bodies has nothing to do with the electricity. But Hertz realized that mathematics was the same and so founded the field, which has retained a very small but loyal following over the past one hundred years. Hertz always wanted to be an engineer and so in 1877, at age 20, he traveled to Munich for further studies in engineering. But he was really more interested in 1

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2 mathematics, mechanics, and physics. Under his parents guidance he chose physics course and found himself in Berlin a year later to study under Hermann von Helmholtz and Gustav Kirchhoff. In October 1878, Hertz began attending Kirchhoffs lectures and came across an advertisement for a prize for solving a problem involving electricity. He asked for permission from Helmholtz to research the matter and got a room for conducting the experiments. He wrote his first paper, Experiments to determine an upper limit to the kinetic energy of an electric current, and won the prize [1]. Next Hertz worked on The distribution of electricity over the surface of moving conductors, which would become his doctoral thesis. This work impressed Helmholtz so much that Hertz was awarded Acuminis et doctrine specimen laudabile with an added magna cum laude. In 1880, he became an assistant to Helmholtz after which he became interested in the phenomenon of Newtons rings. It occurred to Hertz that, although much was known about the optical phenomenon when two lenses were placed in contact, not much was known about the deflection of the lenses at the point of contact. Hertz was particularly concerned with the nature of the localized deformation and the distribution of pressure between the contacting surfaces [1]. In January 1881, at the age of 24, Hertz presented his theory, which will be discussed in detail in the later chapters of this study to the Berlin Physical Society. The members of the audience were quick to perceive its technological importance and persuaded him to publish a second paper in a technical journal that gave Hertz notoriety in the technical circles. But the theory first appeared in the literature in the beginning of the last century, stimulated by engineering developments on the railways, in marine reduction gears and in the rolling contact bearing industry [1].

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3 The Hertz theory is restricted to frictionless surfaces and perfectly elastic solids. Progress in contact mechanics in the second half of the 20 th century has been largely associated with the removal of these restrictions. A proper analysis of friction at the interface of bodies in contact has enabled elastic theory to be extended to both slipping and rolling contact in a realistic way. At the same time development in theories of plasticity and linear viscoelasticity have enabled the stresses and deformations at the contact of inelastic bodies to be examined. Surprisingly, there are a very few books written purely on contact mechanics. In 1953 the book by L. A. Galin [3], Contact Problems in the Theory of Elasticity appeared in Russian summarizing the pioneering work of N. I. Muskhelishvili in elastic contact mechanics. In 1980, an updated treatment by G. M. L. Gladwell, Contact Problems in the Classical Theory of Elasticity [4] was published. These books excluded the rolling contacts and are restricted to perfectly elastic solids. In 1985 Contact Mechanics by K. L. Johnson [5], was published which aimed at providing an introduction to most aspects of mechanics of contact of non-conforming surfaces. This book excluded the restrictions of previous books. Recently, in the end of the 20 th century and the beginning of the 21 st century a few more books have been published. But on the whole the subject of Contact Mechanics after being started with the theory of Hertz was inactive for the first half of the 20 th century and has become active in the latter half of the century. The Objective of This Study The objective of this study is to develop an analytical procedure (described in detail in chapter 3 and 4 of this study) for evaluating the subsurface contact stresses between an anisotropic (single crystal superalloy) elastic body (half-space) and an isotropic elastic body (half-space) in contact. Using the stress results obtained from the

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4 analytical approach, the resolved shear stresses (RSS) acting on the twelve primary slip systems are calculated. The RSS values determine the dominant and the activated slip planes as a function of radial and angular coordinates. The analytical technique used in this study is adapted from Theory of Elasticity of an Anisotropic Elastic Body by S. G. Lekhnitskii [6]. Here, a Hertzian-type distribution is selected to simulate contact. The solution for Hertzian contact problems of the type described in Fig 1-1, where an isotropic cylindrical indenter is in contact with a flat isotropic body, is described in many books such as Contact Mechanics by K.L. Johnson [5]. For the assumption of plane strain to be justified, the thickness of the solid (in the Z-direction) should be large compared with the width (2a) of the contact region [5]. The contact width a is generally very small compared to the length of the body for practical problems; hence the plane strain conditions are valid. The loading profile and the shape of the contact region are calculated under the plane strain conditions [5]. The subsurface stresses can also be evaluated under these conditions, the formulas for which are given [5] and discussed in Chapter 4. Figure1-1 Isotropic cylindrical indenter in contact with a flat isotropic body

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5 This thesis deals with evaluating the subsurface stresses for an isotropic cylindrical indenter in contact with an anisotropic (single crystal superalloy) elastic substrate (Fig 1-2). The problem of evaluating subsurface stresses for a general anisotropic half-space with applied normal and tangential traction forces has been solved by Lekhnitskii [6] using an analytical stress function approach. This solution is valid under the assumption of plane strain. In this study the analytical procedure outlined in Lekhnitskii [6] for a general anisotropic elastic solid has been adapted for evaluating subsurface stresses for an FCC single crystal substrate. The normal and tangential traction forces are obtained under the assumptions of Hertzian cylindrical isotropic contact. It has been shown that normal traction forces for anisotropic contacts are also parabolic in nature as in Hertzian isotropic contacts. This simplification will allow the use of an analytical solution [6] of an otherwise very complicated problem. Figure1-2 Isotropic cylindrical indenter in contact with a flat anisotropic body

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6 Figure 1-3 Pressure profile due to the indenter on the flat body (a stress problem) The accuracy and results of the analytical approach are further verified in chapter 5 by numerical method using a commercial FEA program, ANSYS. The subsurface stresses computed by analytical solution are verified using FEA for a few crystallographic orientations of the substrate. Motivation for the Problem Contact between anisotropic (single crystal superalloy) and isotropic elastic bodies are of practical interest in turbine blade applications in aircraft and rocket engines. Single crystal turbine blades are used in high temperature applications. These blades are attached to turbine disks, made of isotropic nickel-based materials, using dovetail joints. In many instances the turbine blades also have friction damping devices that are tuned to rub against the blade of certain frequencies, to add Coulomb damping. The vibratory Hertzian subsurface contact stresses induced at the damped surfaces and blade attachments initiate crystallographic fatigue cracks. The goal of this thesis is to develop an analytical procedure for computing subsurface stresses at the contact of a cylindrical anisotropic body and a single crystal anisotropic substrate.

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7 Figure 1-4 Blade orientation Source: Modified from [7] S.G.Lekhnitskii [6] offers a good detail of complex potentials and integral transform methods. Large classes of problems in the plane theory of elasticity including those problems, which are reducible to this theory, are treated by the method of complex variables and other ingenious techniques. It is impossible to develop the close form solution to these problems. Thus, an analytical solution is developed for a problem involving a state of stress of a homogenous anisotropic body bounded by a cylindrical surface. In this case, the stresses do not vary along the width. Isotropy An isotropic material has identical properties in all the directions at a point, any plane being a plane of elastic symmetry (infinite planes of symmetry). Many polycrystalline materials exhibit isotropy. Isotropy results when the crystal size is small relative to the size of the sample provided nothing has acted to disturb the random

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8 distribution of crystal orientations within the aggregate. Mechanical processing operations such as cold rolling may contribute to minor anisotropy, which in practice is often disregarded. Such materials have only two independent variables or elastic constants in their stiffness and compliance matrices. These two elastic constants are generally expressed as the Youngs modulus, E and the Poissons ratio, However the alternative elastic constants bulk modulus, K and /or shear modulus, G can also be used and can be found from E and by a set of equations, and vice versa. These equations are as follows: GE21 (1.1) KE312 (1.2) Anisotropy E. I. Givargizov [8] describes anisotropy as the dependence of structure and properties on direction in space. A non-isotropic or anisotropic solid displays direction-dependant properties. For example, a solid exhibits greater strength in a direction parallel with the fiber (grain) than perpendicular to the fiber (grain). It is a result of the discrete nature of the crystal lattice, and it is the property that distinguishes the crystalline state from another solid state of matter, the amorphous. Single crystal also displays pronounced anisotropy, manifesting different properties along various crystallographic directions. Some polycrystalline materials may also exhibit anisotropy. Composite like wood is an example of anisotropy. Recent man-made materials that have joined wood are fiberglass, metal matrix fibrous composites sandwich constructions, and fortisan composites. S.G. Lekhnitskii in his book [6] gives a

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9 detailed study of anisotropic materials. He gives a few examples of anisotropic materials of non-crystalline nature like pinewood, delta wood and plywood along with the numerical values of their elastic constants. Anisotropic materials play an important role in modern technology. Missile and aircraft designers, specialists in mining problems, solid-state physicists, geophysicists, manufacturers of certain parts and materials, and in general, people engaged in all material sciences-all has to deal with a variety of anisotropic problems. Such materials have 21 elastic constants in their stiffness and compliance matrices. A material without any planes of symmetry is fully anisotropic whereas a material that has at least two orthogonal planes of symmetry and where material properties are independent of direction within each plane is orthotropic. Such materials require 9 elastic constants in their constitutive matrices. Special classes of orthotropic materials are those that have the same properties in one plane (e.g. the x-z plane) and different properties in the direction normal to this plane (e.g. the y-axis). Such materials are called transverse isotropic, and they are described by 5 independent elastic constants, instead of 9 for fully orthotropic.

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CHAPTER 2 LITERATURE REVIEW AND THEORETICAL FRAMEWORK A review of relevant literature in contact mechanics is covered in this chapter. It also covers the detailed explanation of the introduction of contact mechanics by Hertz in the late 19 th century and all the assumptions he made in his theory. After contact mechanics was introduced by Hertz in 1882, the field was surprisingly dormant in the early first half of the 20 th century and then research started again in the latter half. In the last 70 years the field has gained pace. Hertz Theory of Elastic Contact The first satisfactory analysis of the stresses at the contact of two elastic solids is due to Heinrich Hertz in his paper, On the contact of elastic solids in 1882 [2]. At that time Hertz was studying Newtons optical interference fringes in the gap between two glass lenses. Primarily, he was concerned at the possible influence of elastic deformation of the surfaces of the lenses due to the contact pressure between them. His theory, which was worked out during the Christmas vacation in 1880 aroused considerable interest when it was first published. Hertz made certain assumptions in his theory that are: The surfaces are continuous and non-conforming: a R; The strains are small: aR; The bodies are in frictionless contact: q x = q y = 0; Each solid can be considered as an elastic half-space: aR 1,2 1; The displacements and stresses must satisfy the differential equations of equilibrium for elastic bodies, and the stresses are localized. 10

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11 Figure 2-1 Two bodies in Hertzian Contact Source: Modified from K.L. Johnson [5] Here a is the contact area, R is the relative radius of curvature, R 1 and R 2 are significant radii of each body and l is significant dimensions of the bodies both laterally and in depth. In addition to static loading Hertz also investigated the quasi-static impacts of spheres that is mentioned in his paper, On the contact of rigid elastic solids and on hardness [1]. In this paper, Hertz also attempted to use his theory to give a precise definition of hardness of a solid in terms of the contact pressure to initiate plastic yield in the solid by pressing a harder body into contact with it. This definition has proved unsatisfactory, as it is very difficult to detect the point of first yield under the action of contact stress. Consider the deformation as a normal load P is applied as shown in Fig 2.1 [5]. Two solids are chosen that are of general shape and convex for convenience. Before

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12 deformation the separation between two corresponding surface points S 1 (x,y,z1) and S 2 (x,y,z 2 ) is given by equation: hAx2By2 12R' x21R'' y2 (2.1) where A and B are positive constants and R' and R'' are defined as principle relative radii of curvature. From the symmetry of this expression about the center, the contact region must extend an equal distance on either side of the center. During the compression distant points in the two bodies T 1 and T 2 move towards the center, parallel to the z-axis, by displacements 1 and 2 respectively. If the solids did not deform their profiles would overlap as shown by dotted lines in Fig 2.1. Due to contact pressure the surface of each body is displaced parallel to z by an amount u z1 and u z2 (measured positive into each body) relative to the distant points T 1 and T 2 If, after deformation, the points S 1 and S 2 are coincident within the contact surface then, uz1uz2h12 (2.2) Writing = 1 + 2 and making use of Eq. (2.1) we obtain an expression for elastic displacements: uz1uz2Ax2By2 (2.3) where x and y are common coordinates of S 1 and S 2 projected onto the x-y plane. If S 1 and S 2 lie outside the contact area so that they do not touch then, uz1uz2Ax2By2 (2.4) Hertz formulated the conditions expressed by equations (2.3) and (2.4) which must be satisfied by the normal displacements on the surface of the solids. He first made

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13 the hypothesis that the contact area is elliptical. Later he introduced that for calculating the local deformations, each body can be regarded as an elastic-half space loaded over a small elliptical region of its plane surface. By this simplification, generally followed in contact stress theory, the highly concentrated contact stresses are treated separately from the general distribution of stress in two bodies that arises from their shape and the way in which they are supported. It also gives the well-developed methods to solve boundary-value problems for the elastic half-space for the solution of contact problems. In order for this simplification to be justifiable two conditions must be satisfied: The significant dimensions of the contact area must be small compared with the dimensions of each body with the relative radii of curvature of the surfaces. A little caution should be taken while applying the results of the theory to low modulus materials like rubber where it is easy to produce deformations that exceed the restriction to small strains. Finally, the surfaces are assumed to be frictionless so that only a normal pressure is transmitted between them. Thus Hertz can be called as the father of contact mechanics whose theory has proved very important to the mathematicians, scientists, and physicists for future research. They used Hertz theory in the second half of the 20 th century and removed its restrictions. Hertz generalized his analysis by attributing a quadratic equation to represent the profile of two opposing surfaces and gave particular attention to the case of contacting spheres. For this case, the required distribution of normal pressure z is:

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14 zpm 32 1r2a2 ra (2.5) The distribution of pressure reaches a maximum (1.5 times the mean contact pressure p m ) at the center of the contact and falls to zero at the edge of the circle of contact (r = a). Hertz did not calculate the magnitudes of the stresses at points throughout the interior but offered a suggestion as to their character by interpolating between those he calculated on the surface and along the axis of symmetry. In 1953 the book by L.A. Galin [3] appeared in Russian summarizing the pioneering work of Muskhelishvili in elastic contact mechanics. An up-to-date and thorough treatment of the same field by Gladwell [4] was published in 1980. These books exclude rolling contacts and are restricted to perfectly elastic solids. A Review of Articles on Plane Contact Problems The plane contact problems are of the following types [3]: Contact problems taking no account of forces of friction, Contact problems for the case when the punches are rigidly connected to the elastic body, Contact problems of the theory of elasticity considering friction and Problems of contact between two elastic bodies. The detailed reference study of plane contact problems done here has been written by dividing them into the above four categories. Contact Problems Neglecting Forces of Friction The solution of contact problems is simple if we neglect the forces of friction between the contacts [3]. This neglect is justified in most of the cases of contact problems that arise in calculating details of machines. There is a layer of lubricant between

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15 machine parts in contact that lowers the force of friction drastically. If the velocity of the movement of one part is very small as compared to the other, the hydrodynamic phenomenon taking place in this layer can be neglected. The presence of lubricant actually means that the forces of friction between the bodies are very small. Therefore it is possible to equate them to zero with sufficient degree of accuracy and that was one of the assumptions made by Hertz. M.A. Sadovsky [9] has studied several cases of a rigid body, exerting pressure on an elastic semi-infinite plane in the frictionless case. He considered the pressure of the punch with a plane base on an elastic semi-infinite plane, and also the case when there are infinitely many punches and they repeat themselves periodically. Actually, S.A.Chaplygins [10] second edition of the collected works contains the solution of the pressure of the punch with a plane base but he never published his work. The manuscript was dated 1900, i.e. much earlier than the work by M.A. Sadovsky, which appeared in 1928. A.I. Begiashvili [11] in 1940 generalized the conclusions obtained in the course of solving the contact problems for one punch to include the case where the number of punches is any number desired. He cited examples of punches with a plane base. All problems, described above were applied to an isotropic semi-infinite plane. Contact problems for an anisotropic semi-infinite plane neglecting friction were discussed by G.N. Savin [12] in 1939. He gave the solution of problems about the pressure of a punch with a plane base, and also with a base bounded by the arc of a circle, on an anisotropic semi-infinite plane.

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16 If the punch moves with a constant speed along the boundary of an isotropic elastic semi-infinite plane, then the mixed problem, which has to be solved, turns out to be close to the one which has to be dealt with as in the case of an anisotropic semi-infinite plane. L.A. Galin considered this problem in 1943 and gave the solution for it in [3]. Here, if friction is absent between the punch and the elastic body, the distribution of pressure turns out to be the same as in the case of an immobile punch. In 1976, the report by US DOT [13] presented a general numerical method of solution to conformal frictionless contact problems. In particular, the method developed was to be used in future research on the analysis of interfacial contact stresses between a railway wheel and rail. Numerical influence functions needed for the solutions of problems with cylindrical and spherical surface geometries were generated and their accuracy was verified by comparison to exact analytical solutions when they existed. Also, it was shown that the method in which sphere indents a spherical seat and a cylinder indents a cylindrical seat produces accurate values of contact pressure approach, displacements, strains, and applied force. Moreover, a problem of a pitted sphere indenting a sphere was solved for the first time and the appropriate boundary iteration for multiply connected contact region was established. Contact Problems for the Case When the Punches are Rigidly Connected to the Elastic Body In this category of problems, it is assumed that if the coefficient of friction between the bodies in contact is very large, then the bodies turn out to be rigidly connected to each other. In some cases this type of boundary conditions can approximate those, which occur between foundations and ground. But in most of these cases, friction is neglected.

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17 A problem of the type for an elastic semi-infinite plane, when displacements on one part of the boundary and stresses on another part are given, was first considered by N.I. Muskhelishvili [14] in 1935. It was reduced to the solution of an infinite system of simultaneous linear equations. Effective solutions have been developed for the case when the region occupied by the elastic body is bounded by a circumference. Here, the work of I.N. Kartzivadze [15,16] in 1943, who gave the solution of the problem for the inside of the circle, and the work of B.L. Mintsberg [17] in 1948, who gave the solution for the outside of the circle should be mentioned. Solutions of mixed problems for an anisotropic semi-infinite plane in the case when the displacements are given on one part of the boundary and stresses on another part are given by L.A. Galin [3]. Contact Problems in the Theory of Elasticity Considering Friction Friction often occurs when the elastic bodies touch each other. Here two cases can arise. In the first case, one elastic body moves with respect to the other, and the movement is so slow that the dynamic effect can be neglected. In this case, it can be supposed that under the action of the displacing force, the punch is situated on the surface of the elastic body in a state of limited equilibrium. In the second case, no displacement of the punch as a whole with respect to the elastic body takes place. However at the points where the tangential effort is less than the normal pressure multiplied by the coefficient of friction, rigid linkage takes place, i.e. stick occurs and the points where the tangential effort is sufficient to achieve motion, a displacement of the elastic body with respect to the punch takes place i.e. slip occurs.

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18 The contact problem when forces of friction obeying Coulombs law act over the whole area of contact has been solved by N.I. Muskhelishvili [18] in 1942. This problem as well as the contact problems discussed earlier can be reduced to finding one function of complex variable. When the punch moves along the boundary of the elastic body, the force of friction acts in a constant direction over the whole area of contact. Consider the problem of a punch moving with constant speed along the boundary of the elastic semi-infinite plane. The solution of this problem allows us to establish the extent to which it is important to consider the dynamic phenomena taking place here. L.A. Galin solved this kind of problem and the problem with forces for friction for an anisotropic semi-infinite plane in 1943 and the results are discussed in his book [3]. If the punch is pressed into the elastic semi-infinite plane and there occurs the forces of friction obeying Coulombs law, then area of contact is divided into several sectors. At places where tangential forces are inadequate for shifting particles of the elastic body with respect to the punch, these bodies are joined by means of a rigid linkage and they stick. In this case, TN (2.6) where N() and T() are the normal and tangential components of the stress and is the coefficient of friction. TN (2.7)

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19 Then sliding movement of the punch with respect to the elastic body will occur and the direction of this sliding will be different at different sectors. One problem of this type with a plane base has been solved by L.A. Galin in 1945. The solution is given in [3]. Somewhat different assumptions are made in the work by S.V. Falkovich [19] in 1946. He mentioned that in some sectors friction is absent and so those are the regions of slip while in others it is present and so those are the regions of stick. However, in case of machine constructions, a layer of lubricant separates the bodies in contact. Thus in such a case it is necessary to solve a problem of a theory of elasticity as a problem in the hydrodynamic theory of lubricants. Seireg [20] treats friction, lubrication, and wear as empirical phenomena and relies heavily on the experimental studies done by him to develop practical tools for design. He summarizes the relevant relationships necessary for the analysis of contact mechanics in smooth and rough surfaces, as well as the evaluation of the distribution of the frictional resistance over the contacting surfaces due to the application of tangential loads and twisting moments. He also presents an overview of the mechanism of the transfer of frictional heat between rubbing surfaces and gives equations for estimating the heat partition and the maximum temperature in the contact zone. The problem of friction and lubrication in rolling/sliding contacts is also given by Seireg [20] where he gives empirical equations for calculating the coefficient of friction from the condition of pure rolling to high slide-to-roll ratios. The effect of surface layers is taken into consideration in the analysis. Seireg and Weiter [21] conducted experiments to investigate the load-displacement and displacement-time characteristics of friction contacts of a ball between

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20 two parallel flats under low rates of tangential load application. The tests showed that the frictional joint exhibited creep behavior at room temperatures under loads below the gross slip values, which could be described by a Boltzmann model of viscoelasticity. They also found that the static coefficient of friction in Hertzian contacts was independent of the area of contact, the magnitude of the normal force, the frequency of the oscillatory tangential load, or the ratio of the static and oscillatory components of the tangential force [22,23]. Walter Sextros book [24] considers the dynamical contact problems with friction. Here the dynamics of the elastic bodies in contact are described by a reduced order model through the so-called modal description. Dynamics of oscillators with elastic contact and friction and rolling contact is also considered. Abdallah A. Elsharkawy [25] presented a numerical scheme based on Fourier transformation approach to investigate the effect of friction on subsurface stresses arising from the two-dimensional sliding contact of two multilayered elastic solids. The analysis incorporated bonded and unbonded interface boundary conditions between the coating layers. Two line contact problems were presented. The first one was the contact problem between a rigid cylinder and a two-layer half space and the second one is the indentation of a multilayered elastic half space by a rigid flat punch. He presented the effect of surface coating on the contact pressure distribution and subsurface stress fields. Problems of Contact Between Two Elastic Bodies In the works that are mentioned in the previous sections, it was supposed that one of the bodies is absolutely rigid. It is called as a punch. This assumption imposes certain limitations, as in reality each of the bodies in contact is elastic. Thus the problems of contact when both the bodies are elastic have to be considered.

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21 The problem of the contact of elastic bodies was first considered by Hertz in 1882 and it has already been discussed earlier in this section in detail. Here, the solution of the plane problem where two parabolic cylinders whose axes are parallel touch each other can be found. Later the problem of the contact of elastic bodies was considered by A.N. Dinnik [26] in 1906. I.J. Sjtaerman [27] considered the problem of the contact of two cylinders whose radii are nearly equal in 1940. Here one of the bodies is the outside and the other is the inside of the cylinder. He considered the problem twice. A general method of solution of non-Hertzian, non-conformal elastic contact problems was developed by Singh and Paul [28] in 1974. They considered the classical contact criterion (which includes that of Hertz) for arbitrary surface geometries. In order to solve the governing integral equation of the first kind they introduced three different numerical schemes. The first simply-discretized method was found to be relatively unstable for the particular problems they investigated. In order to overcome this difficulty, Singh and Paul [29] in 1973-74 introduced two other techniques of solving ill posed integral equations, called Redundant Field Point method and the Functional Regularization method; the latter of which is based on Tychonovs regularization procedure. In the group of conformal problems an elastic sphere indenting an elastic seat has been solved by Goodman and Keer [30] in 1965. They presented the results for the half angles of contact up to 20 degrees and provided experimental results that generally agree with their solutions. It is noted that there are higher order terms in the exact formulation of the sphere problem that do not appear in the formulation if the half space assumption is

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22 used without truncating terms. These terms are particular to the spherical geometry. Goodman and Keer justify their extension of the Hertzian theory through analysis of these second order terms. With the advent of the digital computer several numerical techniques have been developed to analyze a more general class of contact problems. Conry and Seireg [31] in 1971 have examined elastic contact in terms of a linear programming model. Their method is general in scope; however, the only examples that were analyzed were Hertzian or one-dimensional beam problems. Kalker and Van Randen [32] in 1972 derived a variational principle for both linear and non-linear elastic contact problems. For the case of linear elasticity the principle takes the form of an infinite dimensional convex quadratic programming problem. They successfully solved both Hertzian and non-Hertzian problem. It was concluded that the solution yielded accurate values of approach, maximum pressure and applied force; however, the actual contact area was not determined with great accuracy. A Review of Articles on Three-Dimensional Contact Problems The solutions of the three-dimensional problem are comparatively difficult to solve as compared to the plane contact problems. They have characteristics of all spatial problems of the theory of elasticity. In such problems it is assumed that one of the elastic bodies occupies a semi-infinite space and the second body in contact, which maybe absolutely rigid or elastic, can also be replaced by a semi-infinite space. Usually friction is supposed to be absent between the elastic bodies. It was possible only for certain class of problems to find a solution of a space axis-symmetrical contact problem for the cases when friction is considered.

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23 If it is assumed that there is no force of friction between the contacting bodies then the problem indicated is reduced to a certain space problem in the theory of potential. Here it is necessary to find harmonic function, which tends to zero at infinity and which has a value on both sides of a plane region of the outline of the area of contact. Most of the results of the three-dimensional contact problems were obtained by Russian scientists. A number of problems were investigated by A.N. Dinnik [26] I.J. Sjtaerman [33] in 1939 gave the solution of the contact problem, in which one of the contacting bodies is a paraboloid whose degree is greater than the second body. Hertz in his theory gave the assumption that the radii of curvature of the surfaces bounding the touching bodies are large in comparison with the area of contact. Therefore only the first terms were preserved in the equations of these surfaces, and the problem was reduced to the contact of two bodies bounded by the surfaces of the second order like elliptical paraboloids or elliptical cylinders. But these considerations are useful for comparatively smooth bodies. In the case of less smooth bodies; the unevenness on the surfaces might be of the same order as the dimensions of the area of contact. Therefore the initial assumption about the approximation of the equation of the surface does not hold. Moreover, area of contact can be very large in some problems. Thus, a more exact expression for the shape of the surface of the touching bodies should be taken. The generalization of the problem considering the pressure of a punch having a circular cross-section on an elastic semi-infinite body is given in L.A. Galin in [3]. Here the case is considered where the base of the punch is bounded by a surface that is represented by a function of two coordinates. The case in which the punch exerting the pressure on an elastic body is a solid of rotation was investigated in detail by L. A. Galin.

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24 In addition, the axi-symmetrical problem with the forces of friction for such punches has been considered. In this case, the punch, which is pressed against the elastic body, rotates about its own axis, while the forces of friction arising possessing axial symmetry. Also, the influence of a load (acting outside of the punch) on the distribution of pressure arising under the punch, as applied to punches of circular cross-section, has been considered. Punches of elliptical cross-section have also been investigated. Here it was shown that if the equation of the base of a punch is a certain polynomial, then the distribution of pressure under the punch could also be expressed in the form of a polynomial of the same degree, multiplied by a simple algebraic function (1947). Also the solution to the problem of the pressure on an elastic semi-infinite body of a wedge-shaped punch has been given by L.A.Galin [3]. This solution permits the establishment of a law of distribution of pressure in the vicinity of a vertex point of a punch of a polygonal cross-section The case where the punch has a narrow cross-section has also been investigated by L.A. Galin [3]. An example of such a problem is when a narrow beam exerts pressure on a semi-infinite elastic body. In all the problems indicated earlier it was assumed that the elastic body on which the punch exerts pressure is sufficiently large to be represented by a semi-infinite body. Another case is when a rigid body presses on a thin lamina that was investigated by L.A Galin again in 1948. L.A. Galins work [3] considers a thorough discussion of the problems in the field of elastic contact mechanics that summarized the work of Muskhelishvili. Even the work by Gladwell [4] involves a detailed of discussion of it. But these books are restricted to perfectly elastic solids only.

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25 Yen-Yih Ni [34] in his dissertation utilized the mathematical programming approach for evaluating the Hertzian elliptical contact parameter for conditions with very large ratio between the major and minor axes. The problem of transition from elliptical to rectangular contact for the case of crowned cylinders was investigated. The pressure distribution between short cylinders on semi-infinite solids and between the layers of leaf springs was also considered. The study basically dealt with developing a computational procedure for the analysis of pressure distribution between elastic beams in contact as well as the surface modifications necessary to generate uniform pressure between them. The contact between elastic solids with finite dimensions is also considered, and a scaling relationship is developed to improve the computational accuracy when a limited number of nodes are used as well as when the elements in the grid have large aspect ratios. A methodology for surface and sub-surface stress calculation of nominally flat on flat rough surface contact was developed by Maria M. -H. Yu and Bharat Bhushan [35]. This methodology is applicable for both Hertzian contact (large area contact) and point load contact (small area of asperity contact) with and without surface friction. A three-dimensional numerical model was presented by Wei Peng and Bharat Bhushan [36] to investigate the contact behavior of layered elastic/plastic solids with rough surfaces. The surface and subsurface stresses in the layer and the substrate were determined and von Mises yield criterion was used to determine the onset of yield. The surface deformation and pressure distributions were obtained based on a variational principle with a fast Fourier transform (FFT)-based technique. Examples of Numerical Approaches to Contact Mechanics Problems Finite element techniques have also been developed to solve contact problems by Chan and Tuba [37] and by Chaud, Haug and Rim [38]. Both methods are general as they

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26 handle problems that fall into the domain of FE analysis such as analyzing non-isotropic, non-homogeneous media or problems with plasticity and creep, however, both works report only examples which are composed of isotropic materials stressed within the range of linear elasticity. Chan and Tuba compared their computed results to photo-elastic studies and concluded that trends were identical but the results lacked close agreement. Chaud et al, analyzed the non-Hertzian problem of a human knee joint and the contact between two half spaces where one half space has three bumps on the surface. The contact area in the latter case found in photo elastic studies had good general agreement with their computed results. In the case of conformal contacts a number of problems involving a disc in an infinite plate under tension have been solved by FEM. Chan and Tuba [37] again analyzed a plate under tension with a shrink fit disc located in the center. They presented results that showed good agreement between their computed values of circumferential stress and the exact solution, however, there is a larger discrepancy between the computed value of compressive stress and the exact solution. Chaud et al in 1974 have analyzed the problem of a plate under tension with either a loose or full inclusion. They show good agreement between their predicted contact stress and experimental results for a contact angle of 20 degrees. Sen, Aksakal and Ozel [39] solved an elastic-plastic problem in elastic-work hardening layered half-space indented by an elastic sphere using FEM. The case of a surface layer stiffer than the substrate was considered and general solutions for the subsurface stresses and deformation fields were presented for a relatively thin elastic

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27 layer. Differences between the elastic and elastic-plastic solutions for the contact pressure distribution have been investigated for various layer thickness. K.L. Johnsons book [5] covers the theoretical development on normal contact of elastic solids (Hertzian Contacts) as well as non-Hertzian Contacts. It also includes the details of dynamic effects and impact, thermo elastic contact, and study of rough surfaces. Thus, K.L. Johnsons [5] book covers a detailed and comprehensive study of the field of contact mechanics. An Overview of Lekhnitskiis Work S.G. Lekhnitskiis work [6] is discussed separately as this study adapts his analytical solution method to solve anisotropic contact problems. One of his basic contributions is his extension of N.I. Muskhelishvilis work in the plane theory of isotropic elasticity to the anisotropic case. Here, the author utilizes the theory of analytic functions of several complex variables in an elementary and systematic manner in order to solve some boundary value problems. It is of interest to point out that the theory of functions of several complex variables that ordinarily is considered to be in the domain of pure mathematics now finds application to problems in modern physics as well as to problems of technology including contacting mechanics. Lekhnitskii in his book [6] gives a detailed account of certain parts of the theory of anisotropic bodies, which have been studied, but not organized systematically. The problems of plane deformation and generalized plane stress are discussed briefly. In all cases it is assumed that the deformations are elastic and small, and the material satisfies a generalized Hookes law. The analytical solution described in Lekhnitskii [6] is adapted for this study and explained in detail in the next chapter.

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CHAPTER 3 ANISOTROPIC ELASTICITY EQUATIONS FOR FCC SINGLE CRYSTALS Analytical solutions have the advantage of providing a closed-form solution for some contact problems. However, very complex problems often do not have exact analytical solutions; many approaches represent a combination of theoretical and empirical solutions, or close approximations. For the problem considered in this study, a semi-analytical solution procedure with complex functions is outlined for which closed form solution is not possible. The complex parameters are the quantities, which depend on the elastic constants. These functions determine the directions and deformations in a homogeneous body bounded by a cylindrical surface. The problems, which consider the equilibrium of such a body, are reduced to determination of functions that satisfy differential equations of the second, fourth, or sixth orders of the type: ),(......a 2221221220yxfyuayxuxuammmmmmm (3.1) Here m=1, 2, or 3; a 0 a 1 ,.a 2m are constant coefficients which depend on the elastic constants; u is an unknown stress function; f (x, y) is a given function of the coordinates. The roots kk, that are always complex or purely imaginary are called the complex parameters of the equation: 0.....0121222aaammmm (3.2) 28

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29 Slip Activation and Deformation in FCC Single Crystal In a single crystal material, with a FCC crystal structure, the material exhibits cubic symmetry with a particular kind of anisotropy. The slip activation depends on the loading orientation and the stress-strain pattern in a material depends on the activated slip systems. Thus stress-strain behavior is a function of load orientation and varies with it. (Fig 3.1). In an isotropic material the twelve primary slip planes (or fewer depending on orientation), should be activated simultaneously based on equal Schmid factors (Fig 3.2 and Fig 3.3). The Schmid factor, is a function of the load orientation, the slip plane orientation and the slip direction: = cos.cos (3.3) rss = m. (3.4) where is the applied load, rss is the RSS (resolved shear stress) component in a given slip plane and direction, is the angle between the direction of the applied load and the shear direction, and is the angle between the applied load and the normal to the slip plane (Fig 3.2). Figure 3-1 Orientation-dependant stress-strain behavior Source: Modified from Shannon Magnans Masters thesis [40], 2002

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30 Figure 3-2 Load and slip directions and angles Source: Modified from Shannon Magnans Masters thesis [40], 2002 Figure 3-3 Primary resolved shear stress planes and directions Source: Modified from Stouffer and Dame [41], 1996

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31 The cosine of the angle between two directions [h 1 k 1 l 1 ] and [h 2 k 2 l 2 ] can be found using the direction indices [40]: cosh1h2k1k2l1l2h12k12l12 h22k22l22 (3.5) Since the CRSS is reached when the RSS is equal to the yield stress of the material, the slip systems with the highest Schmid factors will reach the CRSS first. This is called as Schmids Law (Eq 3.5). But this law is true only for isotropic materials, as its correlation to non-isotropic materials is not yet known. Elasticity Elasticity of anisotropic FCC crystal varies with orientation. In any given direction the spacing between atoms in an FCC crystal is different. For example, consider Fig 3.4. A is the center of the front face; B of the right face and C is the center of the FCC cube. If a o is the side of the cube or the unit atomic spacing then we have the distance between A to B as a o /2 and the distance from A to C as a o /2. Another way of expressing the relative spacing between atoms is given in the Table 3.1. Figure 3-4 FCC crystal structure Source: Modified from Shannon Magnans Master thesis [40], 2002

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32 Table 3-1 Atomic densities on FCC crystal planes FCC Plane Atom/Area Atom/Area {100} 2/a o 2 2 {110} 2/(2.a o 2 ) 1.414 {111} 4/(3.a o 2 ) 2.309 Source: Dieter [42], 1986 Planes like {111} are called closed-packed planes, because they minimize the spacing between atoms; they are the most common slip planes because the atoms do not have to travel great distances to reach another atomic position. Closed-packed directions, like closed-packed planes, minimize the distance between atoms. Therefore slip often occurs along the close-packed directions in close-packed planes to minimize the amount of energy needed for displacement. Elasticity Matrix The energy needed to deform in any direction is related to the elastic constants, and any material can be completely defined with 36 separate elastic constants. However, most material structures obey some type of symmetry, which reduces the number of independent constants. Dieter [42] has given the rotational symmetry of various crystal structures (Table 3.2).

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33 Table 3-2 Symmetry in various crystal structures. Crystal structure Rotational symmetry No. of Independent Elastic Constants Tetragonal 1 fourfold rotation 6 Hexagonal 1 sixfold rotation 5 Cubic 4 threefold rotation 3 Isotropic 2 Source: Dieter [42], 1986 As indicated in Table 3.2 isotropic materials have only two independent elastic constants (E and G, E and or and G for instance). Cubic structures require three independant elastic constants (E, and G). According to Lekhnitskii [6] when no elements of elastic symmetry are present in the general case of a homogeneous anisotropic body, the original elasticity matrix [a ij ] is given by, aija11a21a31a41a51a61a12a22a32a42a52a62a13a23a33a43a53a63a14a24a34a44a54a64a15a25a35a45a55a65a16a26a36a46a56a66 (3.6) The subscripts i and j correlate to the stress and strain components. The strain is related to stress by, iaijj (3.7)

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34 The constants a ij are called as the coefficients of deformation. If an elastic potential exists, the number of elastic constants in the most general case of anisotropy is reduced to 36. Such an elastic potential exists when the variation of the body under deformation occurs isothermally or adiabatically. Considering equilibrium, if we assume isothermal deformation, i.e. the temperature of each element remains constant, we have, aijaji (3.8) Therefore the matrix is reduced to 21 components: aija11a12a13a14a15a16a12a22a23a24a25a26a13a23a33a34a35a36a14a24a34a44a45a46a15a25a35a45a55a56a16a26a36a46a56a66 (3.9) This is the general elasticity matrix for an anisotropic material under isothermal deformation. For FCC crystals exhibiting cubic syngony, the number of independent constants reduces to three for the final elasticity matrix [6]: aija11a12a12000a12a11a12000a12a12a11000000a44000000a44000000a44 (3.10) The constants are defined by E, the modulus of elasticity, G, the shear modulus and Poissons ratio along the given directions:

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35 a111Exx (3.11) a441Gyz (3.12) a12yxExx xyEyy (3.13) The main aim of this study is to find the state of subsurface stresses in the material coordinate system of single crystal material, and then using those stresses to calculate the resolved shear stresses (RSS) on the twelve primary slip systems. For an isotropic material a single elastic constant governs the transformation from stress to strain as given in Eq. 3.7. However, for an anisotropic material, the analysis of stress and strain fields is not so straightforward. First of all the material properties in a desired orientation is determined. If the material and the specimen coordinate axes are in the same directions then the problem is easy, as the transformations are not needed. The stress-strain relation for an anisotropic solid with cubic symmetry has three independent constants in the material coordinate system, as well as a stress tensor matrix, instead of a single elasticity constant as in the case of an isotropic material. Thus, if the specimen orientation varies, then it is necessary to convert between the specimen and material coordinate systems. Coordinate System Transformation The first step in defining the elasticity matrix is to determine the precise orientation of the specimen or part, either in terms of the material Miller indices (direction indices) or angular measurements. Physically it is nearly impossible to cut a sample such that the x, y, and z test axes are perfectly aligned to the material axes: [100],

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36 [010] and [001] respectively. Thus it is necessary to transform the known specimen stresses to the material coordinate system. The stresses vary by slip plane and direction according to a particular planes cross-sectional area and its given orientation within a unit cube of the material. Therefore, these stresses are not affected by anisotropy. However, only two material properties are independent without anisotropic effects; the shear coupling induced in the three-dimensional model, and resulting component stresses, will not be properly accounted for without the third independent constant to define the single crystal material. Lekhnitskii [6] gives a detail explanation on this topic. The transformation equations presented here follow the procedures mentioned in Lekhnitskii [6] (1963) and Stouffer and Dame [41] (1996). The transformation from the specimen to the material coordinate system can be accomplished by two methods. In the first one, the angles between the original and transformed coordinate systems may be directly measured to find the direction cosines. This method is possible only if the angles can be easily found out. The second approach, based on the rigid body rotations, may be used for more complex orientations, where the angles between the two coordinate systems are not as obvious. But here one should know the Millers indices of the transformed axes. In this method, the axes are rotated through a series of steps to arrive at the final transformed matrix. Coordinate transformations can be performed as long as the orientation of the angle is known, and then the final transformed matrices can be used to determine the stresses and strains resolved on any given plane or slip system. The original coordinate system is referred as the material coordinate system and the transformed coordinate system is called as the specimen coordinate system. The specimen coordinate system is

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37 offset by some angular displacement from the material coordinate system. The material axes are denoted by: x o = [100], y o = [010], and z o = [001] while the specimen axes are denoted by x, y and z (Fig 3.5). x Figure 3-5 Material (xo, yo, zo) and specimen (x, y, z) coordinate systems. Steps in Coordinate Transformation Shannon Magnan [40] has clearly explained the steps involved in transformation. The easiest way to do the transformation is to break it into several rigid body rotations. The process we have used is a three-step process. The first transformation, to the x, y and z axes, is performed by rotating the material coordinate axes by o about the z o -axis (positive direction is defined as x o towards y o )(Fig 3.6). + Figure 3-6 First rotation of the material coordinate axes about the zo-axis.

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38 The transformed coordinates, in terms of the original coordinates, are: xxocosoyosino (3.14) yxosinoyocoso (3.15) zzo (3.16) Writing the first step in matrix form: xyz cososino0sinocoso0001 xoyozo (3.17) The second transformation is obtained by reflecting the load vector to the x-z plane and rotating the x, y, and z-axes by 1 to x, y and z-axes about the y-axis (positive direction is defined as z towards x) (Fig 3.7). Figure 3-7 Second rotation about the y-axis. Writing the second step in matrix form: x'y'z' cos10sin1010sin1 0cos1 xyz (3.18)

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39 The third and the final step occur by rotating the y and z-axes by 2 to the x, y and z axes about the x axis (positive direction is defined as the y towards z) (Fig 3.8). Figure 3-8 Third rotation about the x-axis. The matrix for the step is: x''y''z'' 1000cos2sin20sin2cos2 x'y'z' (3.19) The total transformation can then be calculated by multiplying the three individual step matrices together (Note: the first transformation becomes the last one multiplied): x''y''z'' 123123123 xoyozo (3.20) where, 123123123 1000cos2sin20sin2cos2 cos1 0sin1010sin1 0cos1 cos0sin00sin0 cos00001 (3.21)

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40 The resultant values represent the cosines of the angles between the material and specimen coordinate system axes (Table 3.3). Table 3-3 Direction Cosines x o y o z o x y z 1 2 3 1 2 3 1 2 3 Source: Modified from Lekhnitskii [6] When fewer than three steps are used, the selection of primary and secondary rotation axes is somewhat arbitrary and it is important to verify the results. Several checks based on perpendicularity can be calculated to ensure a proper orthogonal coordinate transformation has been performed (Appendix). Stress Transformation Once the direction cosines between the material and specimen coordinate axes are calculated, the load conditions can be applied and incorporated into separate matrices to correctly transform the individual stresses and strains. These transformed matrices are then used to solve for the resolved stresses and strains in each desired slip system. Lekhnitskii [1] gives the stress transformation as: {} = [Q]{} (3.22) {} = [Q] -1 {}= [Q] {} (3.23) Here [Q] is the stress transformation matrix, a function of the direction cosines calculated above:

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41 Q121212111111222222222222323232333333232232232233223322332213213213133113311331221221221122112211221 (3.24) The state of stress is defined in terms of the specimen {} or material {} stresses by: xyzyzzxxy ''''x''y''z''yz''zx''xy (3.25) The strain transformation is carried out in more or less the same manner but we are not concentrating on strains in this study, so we are not going to discuss the strain transformation here in detail. Lekhnitskii [6] and Shannon Magnan [40] have given the detailed explanation of the strain transformation. Transformation of the Elastic Constants The elasticity matrix also undergoes transformation but remains symmetric and is given by: [c ij ] = [Q] T [a ij ] [Q] (3.26) A maximum of 21 individual constants may be present, depending on the specimen orientation. Now this [c ij ] matrix gives us the component stresses in the

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42 specimen coordinate system. The above equations thus can be further utilized to solve for the component stresses in the material coordinate system. Slip System Shear Stresses The component stresses define the complete state of stress for the material, but these stresses alone do not reveal much about individual slip systems or RSS. The twelve primary slip planes are defined by both a slip plane and direction. Resolved Shear Stresses The resolved shear stresses on twelve primary slip planes {(111)[110]} can be calculated from the following: 123456789101112 16 101110101011011011110101110101011110110101011110011011110101101110101011 xyzxyzxyz (3.27) The stresses resolved onto the primary slip systems are known and can be used to predict slip behavior within a particular system. The analyses of some crystal orientations are discussed in detail in the chapter 6

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CHAPTER 4 GENERALIZED PLANE STRAIN EQUATIONS FOR A HOMOGENEOUS ANISOTROPIC ELASTIC SOLID Plane Strain Conditions for Isotropic Contacts The assumption of the plane strain condition is justified when the thickness of the solid is large compared with the width a of the contact region. Many books such as Contact Mechanics by K.L. Johnson [5] explains these categories of problems in detail. For practical problems contact width is very small as compared to the length of the isotropic body and hence the plane strain conditions are valid. The loading profile and the shape of the contact region are calculated under these plane strain conditions [5]. The subsurface stresses can also be evaluated under these conditions, the formulas for which are given [5]. Under the conditions of plane strain, the equations of equilibrium are given as: 0yxxyx (4.1a) 0xyxyy For compatibility, the corresponding strains x y and xy must satisfy the equation yxxyxyyx22222 (4.1b) 43

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44 Figure 4-1 Isotropic elastic half-space The boundary value problem for evaluating subsurface stresses under the action of traction forces N() and T(), will involve solving equations subjected to the conditions: ,0xyy a x a x (4.2) )( Ny )( Txy a x a The formulas for calculating subsurface stresses for any pressure profile is given as [2]: x2y sNxs()2xs()2y2 2 d2 sTxs()3xs()2y2 2 d (4.3a) y2y3 sNxs()2y2 2 d2y2 sTxs()xs()2y2 2 d (4.3b)

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45 xy2y2 sNxs()xs()2y2 2 d2y sTxs()2xs()2y2 2 d (4.3c) where N() is the normal loading and T() is the tangential loading (Fig. 4-1) and )()( NT (4.4a) where is any point on the X-axes In this study, 2201*)(apN (4.4b) (4.4c) aPp20 the maximum pressure (4.4d) *4EPRa where R is composite radius of curvature and E* is composite modulus of elasticity. Generalized Plane Strain Conditions for Anisotropic Contact Problems For the case of a cylindrical isotropic body contacting anisotropic elastic half-space, the conditions of plane strain given by eq.4.1a and eq.4.1b will not be met. The reason for this is that anisotropic materials can have out of plane stresses induced because of shear coupling. To account for this effect additional equilibrium equation has to be included to set of eq.4.1 as shown in eq.4.5 and this describes the condition of generalized plane strain. It should be noted that even though stresses are not functions of z there are five independent stresses that can be calculated and z is a function of these five stresses. Lekhnitskii has outlined a solution for solving these sets of equations.

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46 0zyxxzxyx 0 yxxyx 0zxyyzxyy 0 xyxyy 0zyxzyzxz 0 yxyzxz (4.5) All the stress values except z are independent in the solution [6] that will be discussed later in this chapter. The boundary conditions on the surface for this study are (Fig 4-2): yN xyT xz0 (4.6) where N() is the function for normal loading and T() is the function for tangential loading. The set of equations Eq. 4.4 are valid for the anisotropic case also [5]. Figure 4-2 Distribution of forces in an anisotropic half-space The stress-strain relations for a general anisotropic solid are given as follows [6]:

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47 xa11xa12ya13za16xy ya12xa22ya23za26xy za13xa23ya33za36xy yza44yza45xz xza45yza55xz xya16xa26ya36za66x y (4.7) The above equations include the elastic coefficients a ij due to which it is possible to calculate stresses in different orientations. These a ij values are calculated through the coordinate transformation discussed in the previous chapters and are functions of 3 variables E, G and depending on the orientations of the specimen (Eq. 3.9). Generalized Plane Strain Equations Consider the elastic equilibrium of a body bounded by a cylindrical surface. The body is under the influence of body forces and stresses that are distributed along the surface. The region of the cross section can be either finite or infinite. The body forces and the surface stresses are assumed to act in planes normal to the generators of the cylindrical surface and do not vary along the generators. (Fig 4.3)

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48 Figure 4-3 Homogeneous Anisotropic Body bounded by a cylindrical surface Source: Modified from Lekhnitskii [6] The body is referred to a system of Cartesian coordinates x, y, z which has z-axis parallel to the generators and the xand y-axes directed arbitrarily. The components of the stresses applied to the cylindrical surface are denoted by X n and Y n per unit area and the components of the body forces are denoted by X and Y per unit volume. P z is the axial force; M 1 and M 2 are the bending moments while M t is the twisting moment. Let us assume that the body forces are derivable from a potential U then, XUx YUy (4.8) The system of differential equations, which the stress functions must satisfy and that has the general solution [1] are given as: F = F + F 0 = + 0 (4.9) Here F, is the general solution of the system of homogeneous equations: L 4 F + L 3 = 0 L 3 F + L 2 = 0 (4.10)

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49 and F 0 and 0 are particular solutions of the non-homogeneous system for which general solution is given by Eq. (4.9). For detailed analysis please refer Lekhnitskii [6]. The general solution for (4.10) is obtained by solving for those equations simultaneously. If is eliminated we get: (L 4 L 2 L 3 2 ) F = 0 (4.11) The equation for is obtained in exactly the same way. The operator of the sixth order L 4 L 2 L 3 2 can be decomposed into six linear operators of the first order. Then Eq (4.11) can be represented by: D 6 D 5 D 4 D 3 D 2 D 1 F = 0 (4.12) Here, Dky kx (4.13) k are the roots of the algebraic equation which corresponds to the differential equation (4.11) and is: l4l2l320 (4.14) where we have: l255224544l315314562254624l4114216321266222622 (4.15) Here in Eq. (4.15)

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50 ijcijci3cj3c33 ij1 6() (4.16) where ij are the elastic constants or coefficients of deformation for the rotated coordinate system. If there are no transformations, i.e. the specimen coordinate axes are same as the material coordinate axes then we can use the following formula also: ijaijai3aj3a33 (4.17) The roots of the system of Eq (4.15) are of the form + i where > 0 and the root order is selected with the first root having the highest positive or negative value of and the second one having the next highest value if there is one. Finally the last root should have value of if there is one Lekhnitskii [6] has stated a theorem according to which the roots of the Eq. (4.14) are always complex and imaginary. Let us assume roots k as distinct: D 1 F = 2 D 2 2 = 3 D 3 3 = 4, D 4 4 = 5 D 5 5 = 6 (4.18) The function 6 satisfies the equation: D 6 6 = 0 (4.19) The general integral is equal to an arbitrary function of the argument x + 6 y and is denoted by f V 6 (x + 6 y). 6 = f V 6 (x + 6 y) (4.20) Thus, D 5 5 = f V 6 (x + 6 y) (4.21) By integrating this equation, we get:

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51 5f5IVx5yf6IVx6y 65 (4.22) In the same way we define 4 3 2 and finally F. Changing the notations of the arbitrary functions, we get the general expressions for F and analogous expression for : F'16kFkxky '16kkxky (4.23) The functions F and satisfy both Eq. (4.23) and Eq. (4.10). Thus: kxkyl3kl2k F'kxkyakxkybk (4.24) The general expressions for the stress functions are now given as: F2ReF1z1F2z2F3z3 F0 2Re1F'1z12F'2z213 F'3z3 0 (4.25) Here Re is the notation for the real part of the complex expression in the brackets, F k (z k ) are analytic functions of the complex variables z k = x + k y (k = 1,2,3) and 1 2 3 are the complex numbers equal to: 1l31l21 2l32 l22 3l33 l43 (4.26)

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52 Now let us assume that the body forces are absent. In this case Lekhnitskii [6] has given formulas that connect the components of stress and displacement with the functions k where, kzkF'kzk 3z313 F'3z3 where (k=1,2) (4.27) The component of stresses are given as: x2Re12'122'2323'3 y2Re'1'23'3 xy2Re1'12'233'3 xz2Re11'122'23'3 yz2Re1'12'2'3 z1c33 c13xc23yc34yzc35xzc36xy (4.28) The Distribution of Stresses in an Elastic Half-Space Under the Influence of Stresses Applied to the Bounding Plane Consider the elastic equilibrium of an infinitely homogeneous body bounded by a plane (an elastic half space). The body is in a state of generalized plane deformation under the influence of stresses applied to the bounding plane. Considering the x-z plane, and the y-axis is directed outward (Fig 4.3). Assuming: The material possesses rectilinear anisotropy of the most general form; The stresses act on the planes normal to the z-axis and do not vary along this axis;

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53 The resultant vector of the stresses distributed in any strip of infinite width parallel to the z-axis is finite and tends toward a definite limit as the ends of the segment tend toward infinity. Figure 4-4 Elastic equilibrium of an infinite homogeneous body with a parabolic profile Modified from Lekhnitskii [1] Let f(z) be a function of the complex variable z = x + iy, holomorphic in the lower half-plane y 0 and continuous up to the boundary where f() = 0. Then, if z is a point in the lower half-plane and is a point on the boundary (the abscissa), the following equalities hold according to Lekhnitskii [6]: 12i fz dfz() 12i f''z d0 (4.29) Denote the normal and tangential components of the external stresses (per unit area) by N() and T() respectively. The boundary conditions for y = 0 can be written as given by Eq. (4.6). Hence after solving we get the final conditions for the functions k and k :

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54 '1z112i N3232T231z1 d '2z212i N1313T113z2 d '3z312i N2112T12z3 d (4.30) Here 212313 1332 Npo1x2a2 ,Tpo1x2a2 zixiy (4.31) y = 0 and x = and z, z 1 z 2 z 3 take one and the same value equal to Thus the components of stresses can be calculated making use of these systems of equations (4.30). Further using these stresses and Eq. 3.27 primary RSS can be calculated.

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CHAPTER 5 FINITE ELEMENT SOLUTION OF THE SUBSURFACE STRESSES Finite Element Model The finite element method can be used to model specific geometries and orientations of the bodies in contact. Here, the contact has been simulated by using parabolic loading and no gap elements have been used in any of the models. Instead, loading is directly applied which reduces the time for pre-processing and also for solving the model. Using the ANSYS finite element software (Version 6.1), a few cases have been solved, for which the results are discussed in the next chapter. As a stress based process, slip deformation is well explained by the numerical models highest individual resolved shear stresses. The results of a few cases that are solved analytically using Lekhnitskiis [6] solution mentioned in the previous chapter is further used to compare with the ANSYS solution by applying the same loading conditions in the finite element model. The assumptions made here are as follows: The contact is sliding The contact width is calculated from the Hertzian solution, which is used in isotropic case as Lekhnitskii solves for stress solutions and not for contact problems. The actual contact using gap elements is replaced by the parabolic profile loading on the body in contact. 55

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56 Case 0: = 0 0 = 0 0 = 0 0 Casting Coordinate System Figure 5-1 Crystallographic Orientations of the anisotropic body

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57 Figure 5-2 Global and material coordinate systems The material properties of anisotropic (single crystal superalloys) vary significantly with direction relative to the crystal lattice. The anisotropic model for the crystallographic orientation 1, in which the material and global coordinate axes are the same and X is [100], Y is [010] and Z is [001] (details explained in chapter 6) is the first case to be analyzed using FEM to verify the analytical finite element model (Fig. 5-1). The solution of this anisotropic model is compared with the accuracy of the analytical solution. The second crystallographic orientation (crystallographic orientation 4 in chapter 6) is the one in which the body is oriented such that the material and global coordinate axes are not the same and thus transformations are needed (Fig.5.2). In this case = 40 0 and = = 10.61 0 (Fig 5.1). The subsurface stresses are evaluated after proper coordinate transformations. The results are then compared with the results of the analytical solution.

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58 Figure 5-3 Isotropic cylindrical indenter and the anisotropic substrate in contact The applied load P on the cylindrical indenter is 4080lbs for all the cases, contact width a is 0.02in, maximum pressure p o is calculated as 2P/a, and x is the distance of the point on the surface of the body (on the X-axis) from the origin. The contact model is shown in Fig 5.3. The single crystal turbine blades used in high temperature applications in aircraft and rocket engines are attached to turbine disks made of isotropic materials. The contact between the blade and the disk mostly has the width of 0.02in approximately. Hence the considered value for contact width is of practical interest. As the load, P = 4080lbs generates a contact width a of 0.02in, this value of load is selected. Material Properties and Model Characteristics The finite element model requires modulus of elasticity E, modulus of rigidity G and Poissons ratio as the only input material property parameters in ANSYS. By defining these parameters we can model a single crystal anisotropic substrate. The length, width and height of the model are taken as 0.25in. The dimensions of the model have

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59 been selected accordingly due to the limitation to the maximum number of nodes in ANSYS. Anisotropic (FCC single crystal) substrate cannot be modeled in 2-D as it can have out of plane stresses induced because of shear coupling. Thus, the plane strain conditions are not valid. Hence it is modeled in 3-D that increases the number of nodes in the model. The coordinate transformations are possible only in the 3-D model. Figure 5-4 Dimensions of the anisotropic contact model The element chosen in this study is SOLID 45. ANSYS reference manual [43] gives the description for the elements. SOLID 45 (Fig 5.5) is used for the three-dimensional modeling of solid structures. The element is defined by eight nodes having three degrees of freedom at each node: translations in the nodal x, y and z directions.

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60 Figure 5-5 ANSYS SOLID 45 element Source: ANSYS 6.1 Elements Reference, 2002 The area near the contact is meshed separately for accurate results (Fig.5.6) as compared to the whole body in order to restrict the maximum number of nodes in the body. Fig. 5-6 shows the front view of the highly meshed contact region and Fig. 5-7 shows the fully meshed anisotropic single crystal substrate in 3-D. Figure 5-6 Refined meshing in the contact zone

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61 Figure 5-7 Meshed anisotropic FCC single crystal substrate

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CHAPTER 6 RESULTS AND DISCUSSION The main purpose of this study is to analyze subsurface stresses in an anisotropic (FCC single crystal) substrate subjected to contact loading under various crystallographic orientations. The stresses are further used to evaluate the RSS values to determine the dominant and activated slip planes. The RSS values are evaluated on the primary octahedral slip systems in the subsurface region as a function of radial and angular position as shown in Fig 6.1. Each of the five crystallographic orientations examined show different slip activations at different locations, emphasizing the effect of the materials anisotropy. Each orientation is discussed individually. Further the finite element solution is compared for orientations 1 and 4 with the analytical results. 2a Figure 6-1 Schematic of the polar coordinate system used in the subsurface contact region 62

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63 Crystallographic Orientation 1 In this crystallographic orientation the specimen coordinate axes and the material coordinate axes are the same given by x [100], y [010], and z [001]. Figures 6.2-6.4 show that the maximum RSS at any location is 2 = 10 = 7.496*10 4 psi at r = 0.8*a and 40 0 (leading edge of the contact subsurface). Results are presented for the twelve primary RSS values for r = 0.3*a, r = 0.8*a and r = 3*a and from 0 0 to 180 0 The dominant slip system with the maximum RSS varies with radial and angular position. Notice that for a given angle the dominant slip system (and often the other activated systems) is not constant over the range of the radii. As the state of stress changes away from the contact region, the RSS also changes. In general the slip systems are highly variable throughout the RSS field though some angles do maintain a single dominant slip system for all radii; 2 and 10 at 40 0 -70 0 and 3 5, 7 and 12 at 100 0 -110 0 Overall the RSS field is dominated by 2 10, 3 5, 7 and 12 as well as 6 and 8 in some areas. Table 6.1 gives a detail explanation of the activated slip systems for the orientation 1. The activated slip systems are known from the Fig 6.2-6.4. The resolved shear stress values, which are higher than the materials critical resolved shear stress value of 47ksi, gets activated. 2 and 10 are activated for all the angles at r =0.3*a and from the angles 0 0 -90 0 for r = 0.8*a. At r = 3*a no slip systems are activated.

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64 00.511.522.531104210431044104510461047104 RSS Vs Theta Specimen 1 (r = 0.3*a)Theta (rad)RSS (psi)6.273104 213.523 ()00 ()10 ()20 ()30 ()40 ()50 ()60 ()70 ()80 ()90 ()100 ()110 2.967 0.175 CRSS v alue = 47ksi Figure 6-2 Orientation 1 primary resolved shear stresses; r = 0.3*a

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65 00.511.522.5311042104310441045104610471048104 RSS Vs Theta Specimen 1 (r = 0.8*a)Theta (rad)RSS (psi)7.496104 109.702 ()00 ()10 ()20 ()30 ()40 ()50 ()60 ()70 ()80 ()90 ()100 ()110 2.967 0.175 CRSS v alue = 47ksi Figure 6-3 Orientation 1 primary resolved shear stresses; r = 0.8*a

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00.511.522.53500011041.510421042.510431043.5104 RSS vs Theta Specimen 1 (r = 3*a)Theta (rad)RSS (psi)3.321104 48.116 ()00 ()10 ()20 ()30 ()40 ()50 ()60 ()70 ()80 ()90 ()100 ()110 2.967 0.175 66 Figure 6-4 Orientation 1 primary resolved shear stresses; r = 3*a

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Table 6-1 Orientation 1 activated slip system sectors Activated Slip System Sectors Orientation 1 x [100] y [010] z [001]r = 0.3*a r = 0.8*a r = 3*a Sector Activated Slip Systems Activated Slip Systems 1 0-55 2 and 10 (111)[0-11], (-1-11)[011] 0-40 2 and 10 (111)[0-11], (-1-11)[011] 2 55-145 2, 10, 3, 5, 7 12 (111)[0-11], (-1-11)[011], (111)[1-10], (-11-1)[110], (1-1-1)[110], (-1-11)[110] 40-46 2 10 4, 9 (111)[0-11], (-1-11)[011], (-11-1)[10-1], (1-1-1)[101] 3 145-180 2 and 10 (111)[0-11], (-1-11)[011] 46-55 2 10 4, 9, 3 5, 7 12 (111)[0-11], (-1-11)[011], (-11-1)[10-1], (1-1-1)[101], (111)[1-10], (-11-1)[110], (1-1-1)[110], (-1-11)[110] 4 55-90 2 10 3 5, 7 12 (111)[0-11], (-1-11)[011], (111)[1-10], (-11-1)[110], (1-1-1)[110], (-1-11)[110] 5 90-95 3 5 7 12 (111)[1-10], (-11-1)[110], (1-1-1)[110], (-1-11)[110] 6 95-142 3 5, 7 12, 6, 8 (111)[1-10], (-11-1)[110], (1-1-1)[110], (-1-11)[110], (-11-1)[011], (1-1-1)[0-11] 7 142-180 6 and 8 (-11-1)[011], (1-1-1)[0-11] No slip systems are activated 67

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68 Crystallographic Orientation 2 Orientation 2 is the off-axes case of anisotropy where the specimens coordinate axes and the material coordinate axes are not the same and where = 15 0 = = 0 (Fig 5.1) (Case 17). Figures 6.5-6.7 shows that the maximum RSS at any location is 3 = 5 = 7 = 12 = 7.304*10 4 psi at r = 0.8*a and 100 0 Notice that this value is slightly lower than the first case value but occurs at a different location. Results are again presented for the twelve primary RSS values for r = 0.3*a to r = 0.8*a and r = 3*a and from 0 0 to 180 0 The dominant slip system with the maximum RSS varies with radial and angular position. Notice that again for a given angle the dominant system (and often the other activated systems) is not constant over the range of the radii. In general the slip systems are highly variable throughout the RSS field though some angles do maintain a single dominant slip system for all radii; 2 and 10 at 45 0 -70 0 Overall the RSS field is dominated by 2 and 10 as well as 6 and 8 in some areas. Table 6.2 gives a detail explanation of the activated slip systems for the orientation 2. The activated slip systems are known from the Fig 6.5-6.7. The resolved shear stress values, which are higher than the materials critical resolved shear stress value of 47ksi, gets activated. 2 and 10 are activated for all the angles at r =0.3*a and from the angles 0 0 -95 0 for r = 0.8*a. At r = 3*a no slip systems are activated.

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00.511.522.531104210431044104510461047104 69 RSS vs Theta Specimen 2 (r = 0.3*a)Theta (rad)RSS (psi)6.374104 556.771 ()00 ()10 ()20 ()30 ()40 ()50 ()60 ()70 ()80 ()90 ()100 ()110 2.967 0.175 CRSS v alue = 47ksi Figure 6-5 Orientation 2 primary resolved shear stresses; r = 0.3*a

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00.511.522.5311042104310441045104610471048104 70 RSS vs Theta Specimen 2 (r = 0.8*a)Theta (rad)RSS (psi)7.304104 174.588 ()00 ()10 ()20 ()30 ()40 ()50 ()60 ()70 ()80 ()90 ()100 ()110 2.967 0.175 CRSS v alue = 47ksi Figure 6-6 Orientation 2 primary resolved shear stresses; r = 0.8*a

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00.511.522.53500011041.510421042.510431043.5104 RSS vs Theta Specimen 2 (r = 3*a)Theta (rad)RSS (psi)3.454104 407.417 ()00 ()10 ()20 ()30 ()40 ()50 ()60 ()70 ()80 ()90 ()100 ()110 2.967 0.175 71 Figure 6-7 Orientation 2 primary resolved shear stresses; r = 3*a

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Table 6-2 Orientation 2 activated slip system sectors Activated Slip System Sectors Orientation 2 = 15 0 = = 0 r = 0.3*a r = 0.8*a r = 3*a Sector Activated Slip Systems Activated Slip Systems 1 0-18 2 10 1, 11 (111)[0-11], (-1-11)[011], (111)[10-1], (-1-11)[101] 0-35 2 10 1, 11 (111)[0-11], (-1-11)[011], (111)[10-1], (-1-11)[101] 2 18-70 2 and 10 (111)[0-11], (-1-11)[011] 35-55 2 and 10 (111)[0-11], (-1-11)[011] 3 70-150 2 10 3, 5, 7, 12 (111)[0-11], (-1-11)[011], (111)[1-10], (-11-1)[110] (1-1-1)[110], (-1-11)[110] 55-90 2 10 3 5 7, 12 (111)[0-11], (-1-11)[011], (111)[1-10], (-11-1)[110] (1-1-1)[110], (-1-11)[110] 4 150-180 2 and 10 (111)[0-11], (-1-11)[011] 90-95 2 10 3, 5, 7, 12, 6, 8 (111)[0-11], (-1-11)[011], (111)[1-10], (-11-1)[110] (1-1-1)[110], (-1-11)[110], (-11-1)[011], (1-1-1)[0-11] 5 95-180 3, 5, 7, 12, 6, 8 (111)[1-10], (-11-1)[110] (1-1-1)[110], (-1-11)[110], (-11-1)[011], (1-1-1)[0-11] No slip systems are activated 72

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73 Crystallographic Orientation 3 Orientation 3 is the case of anisotropy where = = 10.61 0 = 0 0 (Fig 5.1) (Case 19). Figures 6.8-6.10 shows that the maximum RSS at any location is 5 = 7.635*10 4 psi at r = 0.8*a and 90 0 Notice that this value is slightly higher than the first case value but occurs at a different location. Results are again presented for the twelve primary RSS values for r = 0.3*a, r = 0.8*a and r = 3*a and from 0 0 to 180 0 The dominant slip system with the maximum RSS varies with radial and angular position. Notice that again for a given angle the dominant system (and often the other activated systems) is not constant over the range of the radii. The dominant slip system changes one or more times for a particular value of theta. In this case a single dominant slip system for all radii is only maintained at approximately 70 0 by 2 In general the slip systems are highly variable throughout the RSS field. Overall the RSS field is dominated by 2 5 6 and 10 Table 6.3 gives a detail explanation of the activated slip systems for the orientation 3. The activated slip systems are known from the Fig 6.8-6.10. The resolved shear stress values, which are higher than the materials critical resolved shear stress value of 47ksi, gets activated. 2 and 10 are activated for all the angles at r =0.3*a. 2 is activated from 10 0 to 100 0 10 from 0 0 to 96 0 5 from 52 0 -180 0 and 6 from 85 0 -180 0 for r = 0.8*a. At r = 3*a no slip systems are activated.

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74 00.511.522.531104210431044104510461047104 RSS Vs Theta Specimen 3 (r = 0.3*a)Theta (rad)RSS (psi)6.829104 513.17 ()00 ()10 ()20 ()30 ()40 ()50 ()60 ()70 ()80 ()90 ()100 ()110 2.967 0.175 CRSS v alue = 47ksi Figure 6-8 Orientation 3 primary resolved shear stresses; r = 0.3*a

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00.511.522.5311042104310441045104610471048104 CRSS v alue = 47ksi RSS Vs Theta Specimen 3 (r = 0.8*a)Theta (rad)RSS (psi)7.635104 1.431103 ()00 ()10 ()20 ()30 ()40 ()50 ()60 ()70 ()80 ()90 ()100 ()110 2.967 0.175 75 Figure 6-9 Orientation 3 primary resolved shear stresses; r = 0.8*a

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76 00.511.522.53500011041.510421042.510431043.5104 RSS Vs Theta Specimen 3 (r = 3*a)Theta (rad)RSS (psi)3.444104 8.03 ()00 ()10 ()20 ()30 ()40 ()50 ()60 ()70 ()80 ()90 ()100 ()110 2.967 0.175 Figure 6-10 Orientation 3 primary resolved shear stresses; r = 3*a

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Table 6-3 Orientation 3 activation slip system sectors Activated Slip System Sectors Orientation 3 = = 10.61 0 = 0 r = 0.3*a r = 0.8*a r = 3*a Sector Activated Slip Systems Activated Slip Systems 1 0-18 2 10 1 11 (111)[0-11], (-1-11)[011], (111)[10-1], (-1-11)[101] 0-10 10 1 11 (-1-11)[011], (111)[10-1], (-1-11)[101] 2 18-25 2 10 1 (111)[0-11], (-1-11)[011], (111)[10-1] 10-35 2 10 1 11 (111)[0-11], (-1-11)[011], (111)[10-1], (-1-11)[101] 3 25-70 2 10 (111)[0-11], (-1-11)[011 35-42 2 10 1 (111)[0-11], (-1-11)[011], (111)[10-1] 4 70-75 2 10 12 (111)[0-11], (-1-11)[011], (-1-11)[1-10] 42-50 2 10 (111)[0-11], (-1-11)[011] 5 75-85 2 10 12 5 (111)[0-11], (-1-11)[011], (-1-11)[1-10], (-11-1)[110] 50-52 2 10 12 (111)[0-11], (-1-11)[011], (-1-11)[1-10] 6 85-90 2 10 12 5 3 (111)[0-11], (-1-11)[011], (-1-11)[1-10], (-11-1)[110], (111)[1-10] 52-60 2 10 12 5 (111)[0-11], (-1-11)[011], (-1-11)[110], (-11-1)[110], 7 90-125 2 10 12 5 3 7 (111)[0-11], (-1-11)[011], (-1-11)[1-10], (-11-1)[110], (111)[1-10], (1-1-1)[110] 60-62 2 10 12 5 3 (111)[0-11], (-1-11)[011], (-1-11)[110], (-11-1)[110], (111)[1-10] 8 125-145 2 10 12 5 3 (111)[0-11], (-1-11)[011], (-1-11)[1-10], (-11-1)[110], (111)[1-10] 62-85 2 10 12 5 3 7 (111)[0-11], (-1-11)[011], (-1-11)[110], (-11-1)[110], (111)[1-10], (1-1-1)[110] No slip systems are activated 77

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Table 6-3. Continued 78 9 145-155 2 10 5 (111)[0-11], (-1-11)[011], (-11-1)[110 85-88 2 10 12 5 3 7 6 (111)[0-11], (-1-11)[011], (-1-11)[110], (-11-1)[110], (111)[1-10], (1-1-1)[110], (-11-1)[011] 10 155-180 2 10 (111)[0-11], (-1-11)[011] 88-96 2 10 12 5 3 7 6, 8 (111)[0-11], (-1-11)[011], (-1-11)[110], (-11-1)[110], (111)[1-10], (1-1-1)[110], (-11-1)[011], (1-1-1)[0-11] 11 96-100 2 12 5 3 7 6 8 (111)[0-11], (-1-11)[110], (-11-1)[110], (111)[1-10], (1-1-1)[110], (-11-1)[011], (1-1-1)[0-11] 12 100-148 12 5 3 7 6, 8 (-1-11)[110], (-11-1)[110], (111)[1-10], (1-1-1)[110], (-11-1)[011], (1-1-1)[0-11] 13 148-155 12 5 3 6 8 (-1-11)[110], (-11-1)[110], (111)[1-10], (-11-1)[011], (1-1-1)[0-11] 14 155-170 5 3 6, 8 (-11-1)[110], (111)[1-10], (-11-1)[011], (1-1-1)[0-11] 15 170-180 5 3 6 (-11-1)[110], (111)[1-10], (-11-1)[011]

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79 Crystallographic Orientation 4 Orientation 4 is again the off-axes case of anisotropy where = = 10.61 0 = 40 0 (Fig 5.1). Figures 6.11-6.13 shows that the maximum RSS at any location is 1 = 1.189*10 5 psi at r = 0.3*a and 0 0 which is at the surface. Notice that this value is comparatively higher than the previous case maximum values and occurs totally at a different radius and angle. Results are again presented for the twelve primary RSS values for r = 0.3*a, r = 0.8*a and r = 3*a and from 0 0 to 180 0 The dominant slip system with the maximum RSS varies with radial and angular position. Notice that again for a given angle the dominant system (and often the other activated systems) is not constant over the range of the radii. In this case a single dominant slip system for all radii is maintained from 75 0 to 110 0 only by 5 In general the slip systems are highly variable throughout the RSS field. Overall the RSS field is dominated by 1 and 5 and somewhat by 2 and 12 Table 6.4 gives a detail explanation of the activated slip systems for the orientation 3. The activated slip systems are known from the Fig 6.11-6.13. The resolved shear stress values, which are higher than the materials critical resolved shear stress value of 47ksi, gets activated. 2 is activated for all the angles at r = 0.3*a and 5 is activated from 60 0 -158 0 for the same radius. 2 is activated from 0 0 to 64 0 5 from 46 0 to 180 0 and 12 from 60 0 -180 0 for r = 0.8*a. At r = 3*a no slip systems are activated.

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00.511.522.53210441046104810411051.2105 CRSS v alue = 47ksi RSS Vs Theta Specimen 4 (r = 0.3*a)Theta (rad)RSS (psi)1.028105 168.964 ()00 ()10 ()20 ()30 ()40 ()50 ()60 ()70 ()80 ()90 ()100 ()110 2.967 0.175 80 Figure 6-11 Orientation 4 primary resolved shear stresses; r = 0.3*a

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00.511.522.5321044104610481041105 RSS Vs Theta Specimen 4 (r = 0.8*a)Theta (rad)RSS (psi)8.339104 1.413103 ()00 ()10 ()20 ()30 ()40 ()50 ()60 ()70 ()80 ()90 ()100 ()110 2.967 0.175 CRSS v alue = 47ksi 81 Figure 6-12 Orientation 4 primary resolved shear stresses; r = 0.8*a

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00.511.522.53500011041.510421042.510431043.51044104 RSS Vs Theta Specimen 4 (r = 3*a)Theta (rad)RSS (psi)3.518104 27.379 ()00 ()10 ()20 ()30 ()40 ()50 ()60 ()70 ()80 ()90 ()100 ()110 2.967 0.175 82 Figure 6-13 Orientation 4 primary resolved shear stresses; r = 3*a

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Table 6-4 Orientation 4 activated slip system sectors Activated Slip System Sectors Orientation 4= = 10.61 0 = 40 0 r = 0.3*a r = 0.8*a r = 3*a Sector Activated Slip Systems Activated Slip Systems 1 0-15 1 2 10 11 7 (111)[10-1], (111)[0-11], (-1-11)[011], (-1-11) [101], (1-1-1)[110] 0-10 1 2 10, 7 (111)[10-1], (111)[0-11], (-1-11)[011], (1-1-1)[110] 2 15-23 1 2 10 11 (111)[10-1], (111)[0-11], (-1-11)[011], (-1-11) [101] 10-22 1 2 10 (111)[10-1], (111)[0-11], (-1-11)[011] 3 23-35 1 2 10 (111)[10-1], (111)[0-11], (-1-11)[011] 22-35 1 2 (111)[10-1], (111)[0-11] 4 35-60 1 2 (111)[10-1], (111)[0-11] 35-40 2 (111)[0-11] 5 60-90 2 5 (111)[0-11], (-11-1)[110] 40-46 2 4 (111)[0-11], (-11-1)[10-1] 6 90-140 2 5 12 (111)[0-11], (-11-1)[110], (-1-11)[1-10] 46-60 2 4 5 (111)[0-11], (-11-1)[10-1], (-11-1)[110] 7 140-155 2 5 (111)[0-11], (-11-1)[110] 60-64 2 4, 5, 12 (111)[0-11], (-11-1)[10-1], (-11-1)[110], (-1-11)[1-10] 8 155-158 1 2 5 (111)[10-1], (111)[0-11], (-11-1)[110] 64-72 4, 5, 12 (-11-1)[10-1], (-11-1)[110], (-1-11)[1-10] No slip systems are activated 83

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Table 6-4. Continued 84 9 158-180 1 2 10 (111)[10-1], (111)[0-11], (-1-11)[011] 72-85 3 4, 5, 12 (111) [1-10], (-11-1)[10-1], (-11-1)[110], (-1-11)[1-10] 10 85-100 3 5 12 (111) [1-10], (-11-1)[110], (-1-11)[1-10] 11 100-115 5 12 11 (-11-1)[110], (-1-11)[1-10], (-1-11)[101] 12 115-120 5 12 11, 6 (-11-1)[110], (-1-11)[1-10], (-1-11)[101], (-11-1)[011] 13 120-150 5 12 6, 8 (-11-1)[110], (-1-11)[1-10], (-11-1)[011], (1-1-1)[0-11] 14 150-160 5 12 8 (-11-1)[110], (-1-11)[1-10], (1-1-1)[0-11] 15 160-180 5 12 (-11-1)[110], (-1-11)[1-10]

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85 Crystallographic Orientation 5 Orientation 5 is the last case considered and is again the off-axes case of anisotropy where = = 0 0 = 40 0 (Fig 5.1). Figures 6.14-6.16 shows that the maximum RSS at any location is 10 = 7.7*10 4 psi at r = 0.8*a and 17 0 Notice that this value is higher than all the previous case maximum values except the fourth case. Results are again presented for the twelve primary RSS values from 0.3*a to 3*a and from 0 0 to 180 0 for each radius as was presented till now. The dominant slip system with the maximum RSS varies with radial and angular position. Notice that again for a given angle the dominant system (and often the other activated systems) is not constant over the range of the radii. In this case there is no single dominant slip system for all radii and some particular angles. So the slip systems are highly variable throughout the RSS field. Overall the RSS field is dominated by 10 for smaller radius of 0.3*a and for higher radii the domination is by multiple Table 6.5 gives a detail explanation of the activated slip systems for the orientation 3. The activated slip systems are known from the Fig 6.14-6.16. The resolved shear stress values, which are higher than the materials critical resolved shear stress value of 47ksi, gets activated. 10 is activated for the angles 0 0 -120 0 and then 142 0 -180 0 .at r = 0.3*a and 2 is activated from 0 0 -120 0 and then 147 0 -180 0 for the same radius. 3 is activated from 40 0 to 149 0 7 from 40 0 to 148 0 5 from 42 0 -153 0 and 12 from 42 0 -148 0 for r = 0.8*a. At r = 3*a no slip systems are activated.

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00.511.522.5311042104310441045104610471048104 RSS Vs Theta Specimen 5 (r = 0.3*a)Theta (rad)RSS (psi)7.411104 694.093 ()00 ()10 ()20 ()30 ()40 ()50 ()60 ()70 ()80 ()90 ()100 ()110 2.967 0.175 CRSS v alue = 47ksi 86 Figure 6-14 Orientation 5 primary resolved shear stresses; r = 0.3*a

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00.511.522.5311042104310441045104610471048104 RSS Vs Theta Specimen 5 (r = 0.8*a)Theta (rad)RSS (psi)7.7104 252.609 ()00 ()10 ()20 ()30 ()40 ()50 ()60 ()70 ()80 ()90 ()100 ()110 2.967 0.175 CRSS v alue = 47ksi 87 Figure 6-15 Orientation 5 primary resolved shear stresses; r = 0.8*a

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00.511.522.53500011041.510421042.510431043.5104 RSS Vs Theta Specimen 5 (r = 3*a)Theta (rad)RSS (psi)3.418104 11.974 ()00 ()10 ()20 ()30 ()40 ()50 ()60 ()70 ()80 ()90 ()100 ()110 2.967 0.175 88 Figure 6-16 Orientation 5 primary resolved shear stresses; r = 3*a

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Table 6-5 Orientation 5 activated slip system sectors Activated Slip System Sectors Orientation 5 = = 0, = 40 0 r = 0.3*a r = 0.8*a r = 3*a Sector Activated Slip Systems Activated Slip Systems 1 0-18 1 2 10, 11 (111)[10-1], (111)[0-11], (-1-11)[011], (-1-11)[101] 0-27 1 2 10, 11 (111)[10-1], (111)[0-11], (-1-11)[011], (-1-11)[101] 2 18-20 2 10, 11 (111)[0-11], (-1-11)[011], (-1-11)[101] 27-30 1 2 10, 11, 9 (111)[10-1], (111)[0-11], (-1-11)[011], (-1-11)[101], (1-1-1)[101] 3 20-45 2 10 (111)[0-11], (-1-11)[011] 30-33 2 10 11 9 4 (111)[0-11], (-1-11)[011], (-1-11)[101], (1-1-1)[101] (-11-1)[10-1] 4 45-48 2 10 7 (111)[0-11], (-1-11)[011], (1-1-1)[110] 33-40 2 10 9, 4 (111)[0-11], (-1-11)[011], (1-1-1)[101](-11-1)[10-1] 5 48-52 2 10 7 3 12 (111)[0-11], (-1-11)[011], (1-1-1)[110], (111) [1-10], (-1-11)[1-10] 40-42 2 10 9 4 3, 7, (111)[0-11], (-1-11)[011], (1-1-1)[101](-11-1)[10-1], (111) [1-10], (1-1-1)[110] 6 52-120 2 10 7 3 12, 5 (111)[0-11], (-1-11)[011], (1-1-1)[110], (111) [1-10], (-1-11)[1-10], (-11-1)[110] 42-75 2 10 9 4 3, 7, 12, 5 (111)[0-11], (-1-11)[011], (1-1-1)[101](-11-1)[10-1], (111) [1-10], (1-1-1)[110], (-1-11)[1-10], (-11-1)[110] No slip systems are activated 89

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Table 6-5. Continued 90 7 120-142 7 3 12, 5 (1-1-1)[110], (111) [1-10], (-1-11)[1-10], (-11-1)[110] 75-78 2 9 4 3 7 12 5 (111)[0-11], (1-1-1)[101], (-11-1)[10-1], (111) [1-10], (1-1-1)[110], (-1-11)[1-10], (-11-1)[110] 8 142-147 10 7 3 12, 5 (-1-11)[011], (1-1-1)[110], (111) [1-10], (-1-11)[1-10], (-11-1)[110] 78-90 9 4 3 7 12, 5 (1-1-1)[101], (-11-1)[10-1], (111) [1-10], (1-1-1)[110], (-1-11)[1-10], (-11-1)[110] 9 147-150 2 10 7 3 12, 5 (111)[0-11], (-1-11)[011], (1-1-1)[110], (111) [1-10], (-1-11)[1-10], (-11-1)[110] 90-93 9 4 3 7 12, 5, 1,11 (1-1-1)[101], (-11-1)[10-1], (111) [1-10], (1-1-1)[110], (-1-11)[1-10], (-11-1)[110], (111)[10-1], (-1-11)[101] 10 150-152 2 10 7 12 5 (111)[0-11], (-1-11)[011], (1-1-1)[110], (-1-11)[1-10], (-11-1)[110] 93-120 3 7 12 5 1, 11 (111) [1-10], (1-1-1)[110], (-1-11)[1-10], (-11-1)[110], (111)[10-1], (-1-11)[101] 11 152-155 2 10 7, 12 (111)[0-11], (-1-11)[011], (1-1-1)[110], (-1-11)[1-10] 120-124 3 7 12 5 1 11 6 (111) [1-10], (1-1-1)[110], (-1-11)[1-10], (-11-1)[110], (111)[10-1], (-1-11)[101], (-11-1)[011] 12 155-180 2 10 (111)[0-11], (-1-11)[011 124-148 3 7 12 5 1 11 6, 8 (111) [1-10], (1-1-1)[110], (-1-11)[1-10], (-11-1)[110], (111)[10-1], (-1-11)[101], (-11-1)[011], (1-1-1)[0-11]

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Table 6-5. Continued 91 13 148-149 3 12 5 1 11 6 8 (111) [1-10], (-1-11)[1-10], (-11-1)[110], (111)[10-1], (-1-11)[101], (-11-1)[011], (1-1-1)[0-11] 14 149-153 12 5 1 11 6, 8 (-1-11)[1-10], (-11-1)[110], (111)[10-1], (-1-11)[101], (-11-1)[011], (1-1-1)[0-11] 15 153-163 1 11 6, 8 (111)[10-1], (-1-11)[101], (-11-1)[011], (1-1-1)[0-11] 16 163-168 11 6 8 (-1-11)[101], (-11-1)[011], (1-1-1)[0-11] 17 168-180 6 8 (-11-1)[011], (1-1-1)[0-11]

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92 Comparison of FEM Solution with the Analytical Results for Case 1 Neglecting Tangential Traction Figure 6-17 Contour plot for sigma (x) -Analytical solution Figure 6-18 Contour plot for sigma (x) -FEM solution

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93 Figure 6-19 Contour plot for sigma (y) -Analytical solution Figure 6-20 Contour plot for sigma (y)-FEM solution

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94 Figure 6-21 Contour plot for tau (xy)-Analytical solution Figure 6-22 Contour plot for tau (xy)-FEM solution

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95 Figure 6-23 Contour plot for sigma (z)-Analytical solution Figure 6-24 Contour plot for sigma (x)-FEM solution

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Comparison of FEM Solution with the Analytical Results Case1 -0.025-0.02-0.015-0.01-0.0050-2.50E+05-2.00E+05-1.50E+05-1.00E+05-5.00E+040.00E+005.00E+04x psiDistance along the subsurface depth (in) Sx-analytical Sx-numerical coarse mesh Sx-numerical fine mesh 96 Figure 6-25 Comparison of stresses in the X-direction for orientation 1

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-0.025-0.02-0.015-0.01-0.0050-3.50E+05-3.00E+05-2.50E+05-2.00E+05-1.50E+05-1.00E+05-5.00E+040.00E+00y psiDistance along the subsurface depth (in) Sy-analytical Sy-numerical-coarse mesh Sy numerical fine mesh 97 Figure 6-26 Comparison of stresses in the Y-direction for orientation 1

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-0.025-0.02-0.015-0.01-0.0050-2.50E+05-2.00E+05-1.50E+05-1.00E+05-5.00E+040.00E+00zDistance along the subsurface depth (in) Sz-analytical Sz-numerical-coarse mesh Sz numerical fine mesh 98 Figure 6-27 Comparison of stresses in the Z-direction for orientation 1

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-0.025-0.02-0.015-0.01-0.0050-7.00E+04-6.00E+04-5.00E+04-4.00E+04-3.00E+04-2.00E+04-1.00E+040.00E+00xy psiDistance along the subsurface depth (in) Txy-analytical Txy-numerical-coarse mesh Txy-numerical fine mesh 99 Figure 6-28 Comparison of shear stresses in the XY-direction for orientation 1

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-0.02-0.018-0.016-0.014-0.012-0.01-0.008-0.006-0.004-0.0020-2.50E+03-2.00E+03-1.50E+03-1.00E+03-5.00E+020.00E+005.00E+021.00E+03yz psiDistance along the subsurface depth (in) Tyz-analytical Syz-numerical-coarse mesh Syz-numerical fine mesh 100 Figure 6-29 Comparison of shear stresses in the YZ-direction for orientation 1

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-0.025-0.02-0.015-0.01-0.0050-1.20E+03-1.00E+03-8.00E+02-6.00E+02-4.00E+02-2.00E+020.00E+002.00E+024.00E+026.00E+028.00E+02xz psiDistance along the subsurface depth (in) Sxz-analytical Sxz-numerical-coarse mesh Sxz-numerical fine mesh 101 Figure 6-30 Comparison of shear stresses in the XZ-direction for orientation 1

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Case 4 -0.02-0.018-0.016-0.014-0.012-0.01-0.008-0.006-0.004-0.0020-3.00E+05-2.50E+05-2.00E+05-1.50E+05-1.00E+05-5.00E+040.00E+005.00E+04x psidistance along the subsurface depth (in) Sx analytical Sx numerical coarse mesh Sx numerical fine mesh 102 Figure 6-31 Comparison of stresses in the X-direction for orientation 4

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-0.02-0.018-0.016-0.014-0.012-0.01-0.008-0.006-0.004-0.0020-3.00E+05-2.50E+05-2.00E+05-1.50E+05-1.00E+05-5.00E+040.00E+00y psidistance along the subsurface depth (in) Sy analytical Sy numerical coarse mesh Sy numerical fine mesh 103 Figure 6-32 Comparison of stresses in the Y-direction for orientation 4

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-0.02-0.018-0.016-0.014-0.012-0.01-0.008-0.006-0.004-0.0020-2.50E+05-2.00E+05-1.50E+05-1.00E+05-5.00E+040.00E+00z psidistance along the subsurface depth (in) Sz analytical Sz numerical coarse mesh Sz numerical fine mesh 104 Figure 6-33 Comparison of stresses in the Z-direction for orientation 4

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-0.02-0.018-0.016-0.014-0.012-0.01-0.008-0.006-0.004-0.0020-7.50E+04-6.00E+04-4.50E+04-3.00E+04-1.50E+040.00E+001.50E+04xy psidistance along the subsurface depth (in) Sxy analytical Sxy numerical coarse mesh Sxy numerical fine mesh 105 Figure 6-34 Comparison of shear stresses in the XY-direction for orientation 4

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-0.02-0.018-0.016-0.014-0.012-0.01-0.008-0.006-0.004-0.0020-1.50E+040.00E+001.50E+043.00E+044.50E+046.00E+04yz psidistance along the subsurface depth (in) Syz analytical Syz numerical coarse mesh Syz numerical fine mesh 106 Figure 6-35 Comparison of shear stresses in the YZ-direction for orientation 4

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-0.02-0.018-0.016-0.014-0.012-0.01-0.008-0.006-0.004-0.00200.00E+005.00E+031.00E+041.50E+042.00E+042.50E+04xz psidistance along the subsurface depth (in) Sxz analytical Sxz numerical coarse mesh Sxz numerical fine mesh 107 Figure 6-36 Comparison of shear stresses in the XZ-direction for orientation 4

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108 The stress contour plots produced by analytical and FEM solutions in the case of normal traction force for the case 1 are found to be in good agreement. As the tangential traction force is zero the plots are symmetric about the vertical Y-axis. The FEM results are evaluated for two different meshes for all the stresses and compared with the Lekhnitskiis analytical solution. First the results are plotted for case 1 where = = = 0 0 The second comparison is for the case 4 which is = = 10.61 0 and = 40 0 The first mesh size is coarse and the second one is more refined for both the cases. As seen in all the plots of stresses as a function of subsurface depth the results for the refined mesh are closer to the analytical solution as compared to the coarse mesh which shows that FEM results are approximately close and can be more accurate if the mesh is further refined. The mesh is refined by dividing the region around the contact with more number of nodes. The stresses in the FEM solution are compared with the analytical stress values as a function of subsurface depth.

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CHAPTER 7 CONCLUSIONS AND RECOMMENDATIONS FOR FUTURE WORK The analysis procedure developed is used to examine subsurface stresses in contact between a cylindrical isotropic indenter and a single crystal anisotropic substrate. Subsurface stresses are studied for various crystallographic orientations. The resolved shear stresses on the twelve primary octahedral planes [(111) <110>] are computed in the vicinity of the contact, to predict evaluation of slip systems in the contact region. Subsurface stresses are also computed using FEA, for a few cases, for comparison purposes. The results are applicable to the load of 4080lbs only as the contact area varies non-linearly with load. This load value was used to get the contact width of 0.02in and the maximum contact pressure of 260ksi on the anisotropic body in contact that has some practical applications. Conclusions The analytical solution presented for the calculation of subsurface stresses in anisotropic solids is shown to be an effective and efficient procedure for evaluating subsurface stresses in anisotropic cylindrical contacts. The resolved shear stresses calculated from the subsurface stresses in FCC single crystal substrate provides a clear explanation of active slip planes and sectors for the load used. Slip sectors determined by the resolved shear stresses are not constant for the same orientation, but depend on the value of the applied load and the yield stress (CRSS) of the material. 109

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110 The plots for resolved shear stresses as a function of radial and angular components under the surface of the body, considering the normal and tangential traction force, are not symmetrical for anisotropic solids. The results of ANSYS finite element model used are comparable with the results of the analytical solution. Mesh refinement is needed in the FEM model for more accurate results. The analytical approach is a fast and efficient technique as compared to the FEM approach. Recommendations for Future Work The anisotropic elastic models used in this study are models that predict the slip accurately, and can be further used to analyze the fatigue behavior with the help of resolved shear stresses. The RSS values are calculated for the elastic region in this study. The dominant slip system in the elastic region is only one of the parameters to predict fatigue failure. Fatigue and fracture mechanics have to deal with the plastic models in order to predict failure properly. In the plastic region failure occurs due to many parameters such as creep, strain hardening, temperature effects, etc. Thus, analyzing the stresses in the plastic regime is an important area to be researched in the future to predict actual causes of failures in materials. The study takes into account the bodies in contact that are non-conforming and where the pressure profile due to contact is parabolic (like a Hertzian model) under a normal load of 4080lbs. The results can be studied under various loads as contact area changes non-linearly with the applied normal loads. The tangential traction force in this study is considered to be in one direction only. It is important to consider the reversal of loads as in fatigue loading. In the case of reversal loading the effective resolved shear stress values on the all the twelve primary

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111 octahedral slip planes at a point would be the difference between the values of the resolved shear stresses in both the directions at that point.

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APPENDIX SAMPLE COORDINATE TRANSFORMATION WITH ACCURACY CHECKS Sample Coordinate Transformation x'y'z' cos10sin1010sin1 0cos1 xyz x''y''z'' 1000cos2sin20sin2cos2 x'y'z' x''y''z'' 123123123 xoyozo 123123123 1000cos2sin20sin2cos2 cos1 0sin1010sin1 0cos1 123123123 cos1 sin2sin1cos2sin10cos2sin2sin1 sin2cos1cos2cos1 y [010] 4 1 x [100] 2 2 z [001] z, Load direction [214] 112

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113 Fig.App-1 Two step coordinate transformation 1atan12 10.4642atan125 20.22 123123123 cos1 sin2sin1cos2sin10cos2sin2sin1 sin2cos1cos2cos1 123123123 0.8940.0980.43600.9760.2180.4470.1950.873 Checks for Accuracy All of the below should be equal to zero: 1 2 + 1 2 + 1 2 = 0 1 3 + 1 3 + 1 3 = 0 3 2 + 3 2 + 3 2 = 0 1 1 + 2 2 + 3 3 = 0 1 1 + 2 2 + 3 3 = 0 1 1 + 2 2 + 3 3 = 0 All of the below should be equal to one: 1 2 + 1 2 + 1 2 = 1 2 2 + 2 2 + 2 2 = 1 3 2 + 3 2 + 3 2 = 1 1 2 + 2 2 + 3 2 = 1 1 2 + 2 2 + 3 2 = 1 1 2 + 2 2 + 3 2 = 1 All the checks are valid and confirm the accuracy of the transformation

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LIST OF REFERENCES 1. Fischer-Cripps, A.C. (2000). Introduction to Contact Mechanics New York: Springer-Verlag New York, Inc. 2. Hertz, Heinrich. (1882). On the Contact of Elastic Solids. Miscellaneous papers [microform], London: Macmillan, pp. 146-158. 3. Galin, L.A. (1961). Contact Problems in the Theory of Elasticity Translated by H. Moss. Edited by I.N. Sneddon. Raleigh: North Carolina State College, Mathematics Department. 4. Gladwell, M.L. (1980). Contact Problems in the Classical Theory of Elasticity Germantown, Maryland: Sijthoff and Noordhoff International Publishers. 5. Johnson, K.L. (1985). Contact Mechanics New York: Cambridge University Press. 6. Lekhnitskii, S.G. (1963). Theory of Elasticity of an Anisotropic Elastic Body San Francisco, California: Holden-Day Inc. 7. Swanson, G.R. and Arakere, N.K. (2000). Effect of Crystal Orientation on Analysis of Single Crystal, Nickel-Based Turbine Blade Superalloys. NASA/TP-2000-210074, FEB 2000. 8. Givargizov, E.I. (1987). Highly Anisotropic Crystals Norwell, Massachusetts: D. Reidel Publishing Co. 9. Sadovskii, M.A. (1928). Two-dimensional Problems of the Theory of Elasticity. ZAMM, Vol.8 N2, pp. 107-121. 10. Chaplygin, S.A. (1950). Pressure of a Rigid Stamp on an Elastic Foundation. Collected works, Vol.3, pp. 317-323. 11. Begiashvili, A.I. (1940). Solution of the Problem of the Pressure of a System of Rigid Profiles on the Rectangular Boundary of Elastic Half-Plane. SSSR, Vol.27 N9, pp. 914-916. 12. Savin, G.N. (1939). Pressure of a Perfectly Rigid Stamp on an Elastically Anisotropic Base. Dopovidi Akad. Nauk. SSSR, Viddil, Tekh. Nauk. No.6, pp. 22-34. 13. Woodward, W. and Paul, B. (1976). Contact Stresses For Closely Conforming Bodies-Applications to Cylinders and Spheres. Final Report Under Contract DOT-OS-40093. 114

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BIOGRAPHICAL SKETCH Sangeet Subhash Srivastava was born at Sewagram, Wardha, in India on April 10, 1978. He graduated from the Nagpur University with his B.E. in mechanical engineering in July 2000. Then he joined the University of Florida in the Mechanical Engineering Department in August 2000 to pursue his graduate studies. 118