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DEVELOPMENT OF A NUMERICAL PROCEDURE FOR MIXED MODE KSOLUTIONS AND FATIGUE CRACK GROWTH IN FCC SINGLE CRYSTAL SUPERALLOYS By SRIKANT RANJAN A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2005 To My parents Prof. Yamuna Prasad and Mrs. AM ,ti, Devi for aiv, loving me. ACKNOWLEDGMENTS This dissertation is the result of 5 years of work whereby I have been accom panied and supported by many people. It is a pleasant aspect that I have now the opportunity to express my gratitude to all of them. The first person I would like to thank is my supervisor, Dr. N1 ,1i ij Arakere. I have been under his supervision since 2000 when I moved to the University of Florida. His enthusiasm and integral view on research and his mission for providing "only highquality work and not less," has made a deep impression on me. I owe him lots of gratitude for having me shown this way of research. I would also like to thank the other members of my PhD committee who took the effort in reading and providing me with valuable comments on earlier versions of this thesis: Dr. Ashok Kumar, Dr. Bhavani Sankar, Dr. Fereshteh Ebrahimi and Dr. John Ziegert. I thank them all. My colleagues in the Fatigue and Tribology (FAT) LabErik Knudsen, Jeff Leismer, Shadab Siddiqui and TaeJoong YUall gave me the feeling of being at home at work. The friendly ambience of FAT lab alvb, kept me in a cheerful mood. The discussions and the interactions with Shadab and Jeff had a direct impact on the final form and quality of this thesis. I would like to thank Shadab for providing me tips that helped me a lot whenever I went off track. It was not possible to continue my work without his companionship and cheerful support. Erik, Jeff and Amitoj Likhari have read parts of my thesis and provided me valuable comments. Alexander Pacheco has helped me tremendously in getting the thesis in correct format. My friend and colleague since my undergrad d ,v Guruditta Golani, have been of great help in the last d4,v, of thesis writing to meet all the crucial deadlines. I feel a deep sense of gratitude for my loving father and mother who formed part of my vision and taught me to be a good human being first and then anything else. I thank my three great loving brothers, Nishant Kumar R li, i i Prashant Agrawal and Ravikant R ,lii in for ahv,x bestowing their love on me and alvi believing in me. I thank my brother Prashant Agrawal for constantly encouraging me to pursue excellence and providing me emotional and financial support in my hardest time. He has been a constant source of motivation and inspiration since my childhood d,4 I am fortunate to have a big brother like him. Last but not least, I would like to express my gratitude to my friend Prashant Kumar Singh from my undergrad d i, who took time out of his busy schedule to provide unwavering support and help when I needed most. TABLE OF CONTENTS page ACKNOW LEDGMENTS ............................. iii LIST OF TABLES ................................. vi LIST OF FIGURES ................................ vii ABSTRACT .... ............................ xi CHAPTER 1 INTRODUCTION .............................. 1 2 COMPUTATION OF STRESS INTENSITY FACTORS FOR SINGLE CRYSTAL: LITERATURE REVIEW .................. 17 3 SIF EQUATION FORMULATION FOR MIXED MODE LOADING 23 4 MODELING AND MESHING ........................ 30 5 MIXED MODE SIF: RESULTS AND DISCUSSION .......... 36 6 FCG GROWTH IN FCC SINGLE CRYSTAL MATERIALS ....... 58 6.1 Isotropic m materials .. .. .. .. .. ... .. .. .. .. .. .. 58 6.2 FCC single ivI .1 m materials ..................... 61 7 CONCLUSIONS ............................... 78 APPENDIX A DISPLACEMENT FIELD EVALUATION FOR AN ANISOTROPIC ELASTIC SOLID ............................. 81 B DETAILS OF SIF EQUATION FORMULATION ............ 95 C ANSYS PROGRAM ............................. 102 REFERENCES .. .. ... .. .. .. .. .. .. ... .. .. .. .. ... 121 BIOGRAPHICAL SKETCH ............................ 126 LIST OF TABLES Table page 11 Direction cosines of material coordinate axes (xyz) with universal co ordinate axes (x'y'z') . . . . . . 8 12 Direction cosine of (x', y', z') with (x, y, z) coordinate axes, when x' axis is aligned along [213] orientation. ....... . . 16 31 Geometrical and material properties of the two specimens analyzed 28 61 The geometry and loading condition of the three Brazilian Disk speci m en tested . . . . . . . . 68 62 Krs8 for 12 primary slip systems with increasing crack length for spec im en 95830 . . . . . . . . 75 63 K.rs for 12 primary slip systems with increasing crack length for spec im en 96842 . . . . . . . . 76 64 Krss for 12 primary slip systems with increasing crack length for spec im en 98C 21 . . . . . . . 77 LIST OF FIGURES Figure page 11 Schematic of the 7' precipitate in a 7 matrix .. . .... 2 12 Cuboidal 7' precipitates (0.350.6 pm) in PWA1480. . .... 3 13 Temperature capability of superalloys with approximate year of in troduction . . . . . . . .. 4 14 Aeroengine blades are nominally oriented in the (001) orientation. 4 15 Material coordinate system (xyz) relative to universal coordinate sys tem (x'y'z') . . . . . . . 11 16 Two separated portions of a single crystal showing a model for calcu lating the resolved shear stress in a singlecrystal specimen. . 11 17 Slip lines and fracture plane in experimental tensile test specimen tested by Materials Science and Engineering Department, UF. 12 18 Primary (Closepack) and Secondary (Nonclosepack) slip directions on the octahedral planes for an FCC iI 1 . ...... 14 19 Cube slip planes and slip directions for an FCC crystal. . ... 15 31 Crack tip nodal displacement of isotropic elastic material . ... 24 32 The (101) and (121) family of slip directions are superimposed on { 111} plane showing how the hexagonal crack front is delineated by the slip directions . . . . . . 27 33 Brazilian disk having center crack lying in {111} slip plane and aligned along [T21] direction . . . . . . 28 34 Rectangular specimen 'A' meshed with solid95 element (left) and the geometrical details (right) . . . . . 29 35 Brazilian disk (Specimen 'B') with center crack lying in (111) plane and oriented along [10l] direction . . . . 29 41 Brazilian disk meshed with triangular element having isotropic prop erties (left) and zoomed view of crack tip (right). . .... 31 42 (a) 20 node isoparametric element in natural coordinate system and (b) Quarter point singular element with the ( = 1 face collapsed in local Cartesian coordinate system. ....... . . 33 43 Arrangement of quarterpoint wedge elements along segment of crack front with nodal lettering convention. ....... . . 34 44 The symmetric meshing of BD specimen with solid95 element and enlarged picture of crack tip elements ..... . . 34 51 K, versus Crack Length/Width ratio for [10t] and [T21] orientation of Specimen 'A' at 0 00. . . . . . 37 52 KI1 versus Crack Length/Width ratio for [101] and [T21] orientation of Specimen 'A' at 0 00 . . . . . ... 38 53 KII, versus Crack Length/Width ratio for [101] and [121] orienta tion of Specimen 'A' at 0 00 . . . . .... 38 54 K, versus crack angle with force for [10t] and [12t] orientation of Specimen 'A' at 2a/W 0.4. . . . . . 39 55 KH versus crack angle with force for [10t] and [T21] orientation of Specimen 'A' at 2a/W 0.4. . . . . . 39 56 KmI, versus crack angle with force for [101] and [T21] orientation of Specimen 'A' at 2a/W 0.4. . . . . . 40 57 K, versus Crack Length/Diameter ratio for [101] and [121] orienta tion of Specimen 'B' at 00 . . . . . 41 58 KI1 versus Crack Length/Diameter ratio for [10t] and [12t] orienta tion of Specimen 'B' at 00 . . . . . 42 59 KmI versus Crack Length/Diameter ratio for [10t] and [121] orien tation of Specimen 'B' at 00. . . . . 42 510 K, versus crack angle with force for [101] and [T21] orientation of Specimen 'B' at 2a/W 0.55. . . . . ... 43 511 K11 versus crack angle with force for [101] and [121] orientation of Specimen 'B' at 2a/W 0.55. . . . . ... 43 512 Km, versus crack angle with force for [101] and [T21] orientation of Specimen 'B' at 2a/W 0.55 . . . . . 44 513 Half meshed model of Brazilian disk specimen and the crack coordi nate system . . . . . . . 45 514 Variation of SIF K, along isotropic BD specimen thickness at differ ent Crack angle . . . . . . .. 515 Variation of SIF K, along orthotropic BD specimen thickness at dif ferent Crack angle . . . . . . .. 516 Variation of SIF KI1 along isotropic BD specimen thickness at differ ent Crack angle . . . . . . .. 517 Variation of SIF KI1 along orthotropic BD specimen thickness at dif ferent Crack angle . . . . . . .. 518 Variation of SIF K1I, along isotropic BD specimen thickness at dif ferent Crack angle . . . . . . .. 519 Variation of SIF KII, along orthotropic BD specimen thickness at different Crack angle. . . . . . . . 520 a)Unsymmetry about mid plane for crack oriented along {111}(101); b) Symmetry for crack lying along {111}(121) . . . 521 Variation of SIF K, along BD specimen thickness at different Crack angle for [10T] orientation . . . . . .. 522 Variation of SIF K, along BD specimen thickness at different Crack angle for [T21] orientation . . . . . .. 523 Variation of SIF K11 along BD specimen thickness at different Crack angle for [101] orientation . . . . . .. 524 Variation of SIF Kn along BD specimen thickness at different Crack angle for [T21] orientation . . . . . .. 525 Variation of SIF KmI, along BD specimen thickness at different Crack angle for [101] orientation . . . . . .. 526 Variation of SIF KII, along BD specimen thickness at different Crack angle for [T21] orientation . . . . . .. 527 Variation of SIF K, along BD specimen thickness at different Crack Length/Diameter ratio for [lOT] orientation. . . . . 528 Variation of SIF K, along BD specimen thickness at different Crack Length/Diameter ratio for [121] orientation. . . . . 529 Variation of SIF K11 along BD specimen thickness at different Crack Length/Diameter ratio for [101] orientation. . . . . 530 Variation of SIF K1 along BD specimen thickness Length/Diameter ratio for [T21] orientation. .. at different Crack 531 Variation of SIF K11, along BD specimen thickness at different Crack Length/Diameter ratio for [10l] orientation.. . . 57 532 Variation of SIF K11, along BD specimen thickness at different Crack Length/Diameter ratio for [121] orientation.. . . 57 61 Schematic Fatigue Crack Growth Curve ..... . . 59 62 Fatigue Crack Length versus Applied Cycles. Fracture is Indicated by the x . . . . . . . . 61 63 Microscopic slip observed on two {111} slip planes inclined 520 and 380 to the starter notch . . . . . 62 64 Details of crack tip displacements and stresses at a distance r and 0 from the crack tip in the crack coordinate system . . 64 65 Burgers vector b is along slip direction (011) and slip plane direc tion is normal vector n along (111) . . . . 65 66 Fatigue crack growth rate as a function of AKrs8 and AK for 2D rectangular specimen . . . . . . 65 67 Crack growth of BD specimen 95830 with no. of cycles . ... 69 68 Crack growth of BD specimen 96842 with no. of cycles . ... 69 69 Crack growth of BD specimen 98C21 with no. of cycles . ... 69 610 Crack growth rate of BD specimen 95830 with increasing crack length 70 6 11 Crack growth rate of BD specimen 96842 with increasing crack length 70 612 Crack growth rate of BD specimen 98C21 with increasing crack length 70 613 Trace of primary slip planes on the plane normal to the crack plane 71 614 Crack growth on {111} slip plane can be observed for 96842 BD spec im en . . . . . . . . 73 615 Fatigue crack growth rate of 3 specimens 95830, 96842, 98C21, as a function of AK ms . . . . . . 74 616 Fatigue crack growth rate of 3 specimens 95830, 96842, 98C21, as a function of AKr . . . . . . . 74 A1 Forces acting on a rectangular element with dimension dx x dy . 81 Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy DEVELOPMENT OF A NUMERICAL PROCEDURE FOR MIXED MODE KSOLUTIONS AND FATIGUE CRACK GROWTH IN FCC SINGLE CRYSTAL SUPERALLOYS By Srikant R lii in August 2005 C!i ,i: N. K. Arakere M i"r Department: Mechanical and Aerospace Engineering Fatigueinduced failures in aircraft gas turbine and rocket engine turbopump blades and vanes are a pervasive problem. Turbine blades and vanes represent perhaps the most demanding structural applications due to the combination of high operating temperature, corrosive environment, high monotonic and cyclic stresses, long expected component lifetimes and the enormous consequence of structural failure. Single crystal nickelbase superalloy turbine blades are being utilized in rocket engine turbopumps and jet engines because of their superior creep, stress rupture, melt resistance, and thermomechanical fatigue capabilities over p., ,i ii~l 111iiw. alloys. These materials have orthotropic properties making the position of the crystal lattice relative to the part geometry a significant factor in the overall analysis. Computation of stress intensity factors (SIFs) and the ability to model fatigue crack growth rate at single crystal cracks subject to mixedmode loading conditions are important parts of developing a mechanistically based life prediction for these complex alloys. A general numerical procedure has been developed to calculate SIFs for a crack in a general anisotropic linear elastic material subject to mixedmode loading conditions, using threedimensional finite element analysis (FEA). The procedure does not require an apriori assumption of plane stress or plane strain conditions. The SIFs KI, KII, and KIII are shown to be a complex function of the coupled 3D crack tip displacement field. A comprehensive study of variation of SIFs as a function of i l11. graphic orientation, crack length, and modemixity ratios is presented, based on the 3D elastic orthotropic finite element modeling of tensile and Brazilian Disc (BD) specimens in specific crystal orientations. Variation of SIF through the thickness of the specimens is also analyzed. The resolved shear stress intensity coefficient or effective SIF, Krs, can be computed as a function of crack tip SIFs and the resolved shear stress on primary slip planes. The maximum value of Krs and AK,.s was found to determine the crack growth direction and the fatigue crack growth rate respectively. The fatigue crack driving force parameter, AKs8, forms an important multiaxial fatigue damage parameter that can be used to predict life in superalloy components. CHAPTER 1 INTRODUCTION A superalloy is a group of nickel, ironnickel, and cobaltbase metallic alloys which can be used at high temperatures, often in excess of 0.7 of the absolute melting temperature. Exceptional creep and high temperature oxidation resistance are two of the prime design criteria of superalloys, but apart from these properties they exhibit a combination of high strength at high temperature, stress rupture resistance, toughness and metallurgical stability, useful thermal expansion characteristics and strong resistance to thermal fatigue. They exhibit exceptional corrosion resistance at room temperature. The high temperature strength of Ni based superalloys depends on a stable, face centered cubic (FCC) matrix, combined with either precipitation strengthening and/or solid solution hardening. Iron, nickel, and cobalt are generally face centered cubic (FCCaustenitic) in crystal structure when they are the basis for superalloys. However, the normal room temperature structures of iron and cobalt elemental metals are not FCC. Both iron and cobalt undergo transformations and become FCC at high temperatures or in the presence of other elements allo, d with iron and cobalt. Nickel, on the other hand, is FCC at all temperatures. The microstructure of an FCC superalloy consists of an austenitic (7 phase) matrix and a wide variety of secondary phase. The most common secondary phases are 7' and metal carbides. The microstructure consists of ~60 percent by volume of 7' precipitates in a 7 matrix. These alloys obtain their hightemperature strength from the presence of the 7' precipitate phase within the primary 7 matrix. The 7' precipitate is a face centered cubic (FCC) structure and composed of the intermetallic compound Ni3Al, with Al atoms occupying the corners and Ni the faces of the unit cell. The 7' precipitate is Figure 11: Schematic of the 7' precipitate in a 7 matrix suspended within the 7 matrix, which is also of FCC structure and comprised of Ni with cobalt, chromium, tungsten, and tantalum in solution. Typically the 7 phase consists of the elements from groups V, VI and VII of the periodic table. The role of 7' phase within the solid solution strengthened alloy is to increase the resistance to dislocation motion. The magnitude of the strengthening effect of 7' is governed by the degree of 7/ 7' mismatch. The greater the mismatch, the higher is the strength increment (Fig. 11 shows a schematic diagram of the 7' phase within the 7 matrix [1] and Fig. 12 on page 3 shows a picture of cuboidal 7' precipitate in 7 matrix [2]). Hence shearing of the precipitate is required in order for a dislocation to move through the 7 matrix. Shearing of the 7' precipitate relative to the 7 matrix requires significantly higher energy, resulting in a stronger material. Nickel base single crystal superalloys are precipitation strengthened cast monograin superalloys based on the NiCrAl system. These alloys exhibit better high temperature properties than p, i, i il 1Iiiw. wrought or cast alloys (Fig. 13 on page 4). In high temperature application grain boundaries are typically the weak link, which provide passages for diffusion and oxidation, which results in Figure 12: Cuboidal 7' precipitates (0.350.6 pm) in PWA1480. failures at this location. Grain boundary strengtheners are added to the alloy chemistry to increase capability, which results in lowering the melting point of the alloy. Removal of grain boundaries and grain boundary strengthening elements raise the incipient melting temperature of the alloy by 150 F and result in improved high temperature fatigue and creep capabilities [4]. This increase in melt temperature permits higher heat treatment temperature that in turn yields improved creep capability. Unlike the more commonly used pf1l. i' iI 11 11, alloys, these single crystal superalloys are orthotropic and have highly directional material properties with exceptional thermo mechanical fatigue properties at high temperature. The most common primary growth direction for the nickelbase superalloys is the (001) direction. This is not only the most easily grown but also is the direction with the most desirable combined strength properties. This is advantageous because many parts are cast, or grown, rather than machined from a larger single crystal sample with a specific orientation. 4 Mechanical ng DIOollonal ioying crys strctus \ MAIMBO CMSX 10 Cast lloy 4,W and Nb / IN59 S 0 / MARM*22 MARM l2S CMSX2 MAR1200 M21 M246 M2 +HlDS 0 SVuM21 M248 MAM200 )HIM 97 1N100 T A C .0 I w N I1 ,at a i ng1 71, 0 1 N77 ,1 I NIP B  I W rIWrought X5stUNy 0i P. A Nit NI bNe 001940 19 i0 m 1960 19170 130 19 Year IntroducedM509 Ime72 auction. [3] X750 N80AWrought .A[ Co'bas. OD at rest* DO S aund SC 1940 195 19110 1070 1 98D09 Year Introduced Figure 13: Temperature capability of superalloys with approximate year of intro duction. [3] Figure 14: Aeroengine blades are nominally oriented in the (001) orientation.[5] [aij] is the matrix of 36 elastic coefficients, of which only 21 are independent, since [aij] = [agj]. Depending on the the material structure, the crystalline material di pl'i different forms of geometric symmetry. There are 32 forms of geometric symmetry of crystals, which can be further divided into seven crystal systems called syngony: 1) triclinic, 2) monoclinic, 3) rhombic, 4) tetragonal, 5) trigonal, 6) hexagonal, and 7) cubic [6]. For an isotropic material the mechanical properties (E, v, G) are the same at each point of the material because it can have only two independent elastic constants as all = a22 a= 33 a12 a= 13 a= 23 a44 a= 55 a= 66 = 2(all a12) (1.2) and the rest of the coefficients of deformations are zero. all a12 a12 0 0 0 a12 all a12 0 0 0 ] a12 a12 all 0 0 0 [a,1 (1.3) 0 0 0 a44 0 0 0 0 0 0 a44 0 0 0 0 0 0 a44 An isotropic material subjected to multiaxial lti:. under mechanical equilibrium, has three principal stresses. These principal stresses act on orthogonal planes, which are free of shear stresses. The Von Mises and Tresca criterions are two of the most widely used yield criteria for ductile isotropic materials. Anisotropic Elastic Deformation of FCC Single Crystal: The generalized Hooke's law for a homogeneous anisotropic elastic body in a cartesian coordinate system (x, y, z with origin at point 0, Fig. 15 on page 11) is given by equation (1.4) [6] {} = [ai]{} (. (1.4) The elastic properties of FCC i ,I J1 exhibit cubic syngony; i.e., it has three orthogonal planes of elastic symmetry at every point, which is called orthogonallyanisotropic or, for brevity, orthotropic. Therefore cubic sym metry can be described with three independent constants designated as the elastic modulus, shear modulus, and Poisson's ratio [4] and hence [a1j] can be expressed as shown in equation (1.3), in the material coordinate system (FCC crystal axes are parallel to x, y and z coordinate axes) and the coefficient of deformation is given as 1 1 Vyx 1 xy ( a11 = t, a44 C a12 F (1.5) Exx Gyx Exx Eyy The elastic constants in the generalized Hooke's law of an anisotropic body, [aij], vary with the direction of the coordinate axes. For orientations other than the (x, y, z) axes, the [a1j] matrix varies with the crystal orientation. In the case of an isotropic body the constants are invariant in any orthogonal coordinate system. Consider a Cartesian coordinate system (x', y', z) that has rotated about the origin 0 of (x, y, z). The elastic constant matrix [at] in the (x', y', z') coordinate system that relates {e'} and {oa'} is {c'} = '.]{'} (1.6) where [at] is given by the following transformation 6 6 ] [Q][a] [Q] = a mQm (ij = 1,2, ......,6) (1.7) rrn in 1 where [Q] is the transformation matrix, which is a function of the direction cosines between the(x, y, z) and (x', y', z') coordinate axes (Fig. 15 on page 11, Table 11 on page 8). Here a, /3 and 7 are the direction cosines of the material coordinate system relative to the universal coordinate system (specimen coordinate system). 2a2 a3 2020 3 2/32/33 272773 3273 + /3372 72a3 + 73a2 a2/3 + a332 2 ,1 2/33/31 27371 /3173 + /3371 7103 + 730a1 a1/33 + a3/31 2a1a2 2/31/32 27172 /3172 + /3271 71a2 + 72al a10/32 + a2/31 Table 1 1: Direction dinate axes (x'y'z') cosines of material (< : :: : axes (. ) with universal coor x y  X / 'i (a2 ( 2 S71 72 .3 Also the transformation of the stress and strain tensors between the material and specimen coordinate systems is necessary for implementing the failure theories. The components of stresses and strains in the (x', y', z') system in terms of the (x, y, z) system are given by {} fa / [Q'] {}7 ; {'} [ 1{ [Q]I 1 {1} [Q] {1'} IQ(] W1 { ' [Od fc' (1.9) 2 1 71 /171 710a1 al/3i 2 a2 f3J 7 2 2 7202 /3272 72 a2 a 2/32 2 a3 33 73 73a3 (1.8) where 01 a2 a3 0203 (' "'1 0102 /3 3 13 /32/33 /33/31 /31/32 712 71 2 2 73 7273 7371 7172 2/3171 2/3272 2/33<73 +0372 /Y173 + 371 /3172 +0271 271ai 272a2 273a3 72a3 + 73a2 7la3 + 73al 71a2 + 72al 2a1/31 2a2/32 2a3/33 a233 + a332 a13 + a331 a1/32 + a231 (1.10) The transformation matrix [Q] is orthogonal and hence [Q] [Q] [Q,]. Now the resolved shear stresses on the 12 primary octahedral slip systems (Table 12 on page 16, Fig. 18 on page 14), denoted by ri, T2,..., 712 can be readily obtained using the transformation given by equation (1.13) [4]. Only the primary octahedral slip systems get activated at room temperature whereas high temperature is required for the activation of the other slip systems. Inelastic Deformation of FCC Single Crystals: Slip and twinning are two main factors responsible for inelastic deformation in metals. The main reason for deformation of crystalline metals is the propagation of dislocations through the metal's lattice when temperatures are less than 0.5 of the absolute melting temperature. At higher temperatures, deformation occurs by dislocation climb (which is a diffusion controlled process). Twinning, a rotation of atoms in the lattice structure, is not as important as strains produced by this mechanism are very small as compared to slip and climb. On an atomic scale, plastic deformation involves the net movement of large numbers of atoms in response to an applied stress. Actual strength to deform a metal is much lower than that predicted for theoretically perfect ( i I J1 This difference is explained by y <010> yO z <001> z' x <100> Figure 15: Material coordinate system (xyz) relative to universal coordinate sys tem (x'y'z') Figure 16: Two separated portions of a single crystal showing a model for calcu lating the resolved shear stress in a singlecrystal specimen. [7] Figure 17: Slip Lines and Fracture Plane in experimental tensile test specimen tested by Materials Science and Engineering Department, UF. [8] slip when the resolved shear stress on the slip plane reaches the critical resolved shear stress for that material. To calculate the stress needed to exceed this level (TCRSS) is calculated by TCRSS cos Q. cos A a will be a minimum when = 45 Therefore when a load is applied slip systems get activated based on Schmid factor, m. where m = cos 4. cos A (1.12) This behavior is known as Schmid's law. The value of critical resolved shear stress depends chiefly on the material composition and temperature. It is also a function of applied load and direction, crystal structure and specimen geometry. During the application of a load to an FCC single crystal specimen, the first planes to get activated are the planes of high atomic densities and called the primary octahedral slip systems. Each octahedral plane has six slip directions associated with it. Three of these are termed easy slip or primary slip directions and the other three are secondary slip directions. Thus, there are 12 primary and 12 secondary slip directions associated with the four octahedral planes (Fig. 18 on page 14). In addition, there are six possible slip directions in the three cube planes, as shown in Fig. 19 on page 15. There are 30 possible slip systems in an FCC crystal (Table 12 on page 16). The resolved shear stress on the 12 primary slip systems, based on kinematic relations, can be shown to be 51 1 0 1 1 0 1 72 0 1 1 1 1 0 73 1 1 0 0 1 1 4 1 0 1 1 0 1 aJ 5 1 1 0 0 1 1 o7yy 76 1 0 1 1 1 1 0 azz < > => (1.13) 76 1 1 0 0 1 1 (7Y S0 1 1 1 1 0 az1 79 1 0 1 1 0 1 o7z 710 0 1 1 1 1 0 711 1 0 1 1 0 1 T12 1 1 0 0 1 1 Plane 1 F rim lry: "l, T2, 0 Secondary: 013, 014, 15 Ulu C16 T6 r Plane 2 Primary: :4, 5, 6 S radnrarv ,l16 ,17. rl8  100 Plane 3 Frirm i r. ,7., T , Secondary: t 20, 2 ,21 100 Plane 4 Primary: 0 1, r 21 :2 Secondary: C22, r'z, t4 Figure 18: Primary (Closepack) and Secondary (Nonclosepack) slip directions on the octahedral planes for an FCC ( i I 1 [)] Secondary:~ 0 7T6 T4 r17 Plaie2 Figure 19: Cube slip planes and slip directions for an FCC i ivl [9] Plane I Table 1 2: Direction cosine of (x, ', is aligned along [213 orientation. z') with (x, y, z) coordinate axes, when x' axis " Number ... FI ...., "1.. Direction Octahedral :: a/2 {111} (110) Primary :, Directions I (Iln [IOT 2(111) 101] 2 (111) (011 3 (111) [110] 4 (111) 101] 5 (111) [110] 6 (111) [OI11] 7 (111) [110o] 8 (111) [011] 9 (111) [101] 10 (111) [011] Octahedral :. a/2{ 111 }(112) Secondary .': Directions 13 (111) 121] 14 (111) 211] 15 (111) 1112 16 (111) [1211 17 (111) [1121 18 (111) [211] 19 (111T) [t112] (100) [211] 21 (111) [121] 22 (111) [211] (I ) 121 24 (111) 1121 Cubic i. a/2{100}(110) Cube 1 *.Directions 25 (100) 011 (100) 01T1 27 (010) 101 (010) [101]OT (001) 1101 ____________(001) [110] CHAPTER 2 COMPUTATION OF STRESS INTENSITY FACTORS FOR SINGLE CRYSTAL : LITERATURE REVIEW Stress intensity factor (SIF) about a crack tip p.1i' a significant role in the propagation of the crack. The SIF is a measure of intensity near the crack tip under linear elastic conditions. The knowledge of SIF is necessary to predict the growth of a fatigue crack or to determine the residual strength of a cracked struc ture. This factor characterizes the intensity of the stress field in the neighborhood of the crack tip and depends substantially on specimen geometry, material proper ties, external loads and crack size. In cases with idealized geometry and loads the SIF can be found in handbooks for isotropic elastic materials. In the case of more complex structures the SIF has to be computed by numerical methods. Isotropic materials have uniform material properties in all directions, whereas anisotropic materials are directional in nature (i.e., they have different material properties in different directions). Because of this property SIF is direction invariant for isotropic materials while SIF varies in anisotropic materials due to change in material coordinate system, while keeping all the other parameters same. A study of the effect of anisotropy and inhomogeneity on the SIF using the FEA was done by K rn,v and Kitamura [10]. It was found that in a pcli, iI 1 specimen, the SIF is influenced by the deformation constraint due to .,I.i Ient < I ,I 1 only for very small cracks; whereas for anisotropic single and the magnitude deviates from that in the isotropic body for any size crack. But as crack size increases in the pf,1 ivI ,1 materials, SIF tends to converge to values seen in homogeneous/isotropic bodies. Many methods have been proposed, to calculate SIFs for cracks subjected to mixedmode loading conditions in isotropic elastic solids. Some commonly used methods are J integral [11, 12], virtual crack extension [13, 14], modified crack closure integral and displacement extrapolation methods [15] etc. None of these proposed methods are able to provide the complete solution for all the three modes (Mode I, II and III) of SIF for anisotropic material. Atkinson et al. [16] presented the idea of calculating mixed mode SIF using Fredholm equation transformation. They used a center cracked Brazilian disc (BD) test specimen made of isotropic material. The mode mixity ratio for the BD specimen is a function of the crack angle with respect to the load vector. Results were presented for varied crack angles and hence mode mixity ratios. Small crack approximation was also taken and the results were found to be in accordance with Awaji and Sato [17], but it did not incorporate anisotropy in the model and was limited to Mode I and II SIFs. Su and Sun [18] studied various kinds of 2D anisotropic cracks to evaluate SIF under mixedmode loading condition. Fractal finite element method (FFEM) [19] was used to calculate Mode I and II SIFs for 2D anisotropic plate. The variation of the SIFs with material properties and orientations of a crack was presented. It was shown that SIFs were not sensitive to the variation of shear modulus. Hwu and Liang [20] used remote boundary data to calculate SIFs for 2D anisotropic material. It eliminated the error in SIF calculation, caused by abrupt change in the stresses near crack tips, by finding equivalent formulation for SIF by using only remote boundary responses (displacements, stresses and strains), cooperating with the necessary geometric data. A special boundary element was developed which removed the requirement of meshing around the crack boundary. Through this boundary element, all the internal stresses and strains could be expressed in terms of displacements and tractions on the boundaries excluding the hole, crack and inclusion boundaries. These results could be applied to any kind of linear anisotropic materials but was restricted to the 2D problems which included generalized plane stress, generalized plane strain and antiplane problems. Denda and Marante [21] developed a crack tip singular element (CTSE) for the general anisotropic solids in 2D with the built in V/ and 1/,r singular stress behavior at its tip, which provided the SIF as an integral part of the main solution, no post processing was required. It was an ideal fracture analysis tool for 2D multiple curvilinear cracks in general anisotropic solids, but as it was based on plane strain assumption, it could not be incorporated into a generalized 3D anisotropic model. The solution provided by Heppler and Hansen [22] for combined mode (I and II) SIFs in case of planar, rectilinear, anisotropic structures using a 12 node singular finite element was accurate, but was restricted to a 2D model. Likewise, Sosa and Eischen [12] calculated SIF for a plate containing a through crack subjected to bending loads using the J integral. TwoD eight node element was selected for this purpose where K = K111 = 0. Mews and Kuhn [23] used the Green's function approach to calculate mixed mode SIF without any crack discretization in an isotropic plate, which used an ., iiii 1l tic displacement field at the crack tip. They calculated SIFs for plate having multiple cracks, for various inclinations, and found SIFs very close to that of Shih et al. [24]. Ishikava [14] presented the idea of the strain energy release rate (virtual crack extension method) to calculate SIF for mode I and II, and described SIF calculation based on only one use of virtual crack extension. Huang and Kardomateas [25] described the continuous dislocation technique to calculate mixedmode stress intensity factors in an anisotropic infinite strip. This method was limited in its use to Mode I and II SIFs, and it was found to be most suitable for cracks of relatively small dimensions. The method was verified by calculating SIFs for isotropic material and comparing the results with readily available formulas given in Tada et al. [26], which resulted in excellent agreement. Sun et al. [27] used the boundary element method (BEM) to analyze cracked anisotropic bodies under antiplane shear. The new boundary formulation used dislocation density as an unknown on the crack surface, and K111 was determined near the crack tip. The equation and method could be directly used for antiplane problems with cracks of any geometric shapes. It did not give a complete solution under mixedmode loading conditions, but it did give an idea about the behavior of K111 under antiplane shear loading. Shih et al. [24] have calculated SIFs for 2D isotropic materials using quarter point element nodal displacements at the crack tip based on Finite Element Method (FEM). The Modes I, II and III have been decoupled because of the isotropic nature of the material. Sih et al. [28] defined SIFs as a function of stress at the crack tip. Following the work of Shih et al., Ingraffea and Manu [29] showed how to compute SIF from 3D quarter point nodal displacement, for cracked isotropic elastic bodies for all three modes. They used a quarter point isoparametric element, which has been accurate in computing SIF [30]. Saouma and Sikiotis [31] introduced anisotropy in the above model [29] and proposed a method to calculate SIF for 3D anisotropic elastic material based on the model of Shih et al. The computed SIFs, when compared with 2D anisotropic bodies with known exact solutions yielded an error of 61'. for K, and K11. However the orientation of the elastic constants was not incorporated. The expression of K111 was not correct as it resulted in almost zero value, irrespective of the geometry, orientation and the load applied to an anisotropic component. Dhondt [32] analyzed two methods, interaction integral method (IINT) and the quarter point element stress method (QPES), to calculate SIF for single edge notch specimen (SEN) and a slant crack in a 3 point bending (3PB) specimen for isotropic material. For the regular mesh in the SEN specimen both methods yielded similar results. For the irregular mesh in the 3PB specimen the QPES methods seemed to be more robust and accurate. Later Guido [33] presented a method to calculate SIF for anisotropic material using the finite difference method along an arbitrary crack. Although the model was more robust to irregular meshes in comparison to interaction integral method, this method was limited to single edge specimen and corner crack specimen. Tweed et al. [34] used the specific case of an edge crack of an anisotropic elastic half space under generalized plane strain conditions to determine the stress intensity factors using integral transform techniques. Pan and Yuan [35] used single domain BEM to calculated mixed mode SIFs for both bounded and unbounded 3D anisotropic cracked solids. Denda [36] used BEM to determine mixed mode SIFs (KI, K11 and K111) of 3D anisotropic material with multiple cracks. It addressed the issue of coupling effect of the three modes of fracture controlled by Mode I, II and III SIFs. Though very accurate, these formulations were based on the plane strain assumption. Although a substantial body of literature describes computation of SIF, a generalized numerical solution to calculate SIF for 3D anisotropic material is unavailable. The objective of the present work is to model a 3D orthotropic specimen having a through crack and to calculate the SIFs for all three modes assuming linear elastic properties at the crack tip. A mathematical model has been developed to calculate mixed mode SIF for FCC single crystal orthotropic material for different orientations, which is also applicable to generalized anisotropic material. Looking at the crack tip nodal displacements it is possible to calculate stress intensity factor for any crystallographic orientation of the material. It was observed that the material orthotropy results in coupled crack tip (x, y, z) displacements, leading to the interdependence of SIFs for Modes I, II, and III. Results are presented for a centercracked BD specimen, with two specific crystallographic orientations. The crack plane for the first specimen is (111) and the crack direction is [o101. For the second specimen the crack plane is again (111), while the crack direction is [121]. There are important reasons for choosing these two specific BD specimen configurations. These two crack directions typically represent the fastest and the slowest crystallographic crack growth rates, respectively, on the { 111} family of octahedral planes, and hence have important implications on estimating fatigue crack growth life for single crystal components [2, 37]. Even if cracks nucleate on other planes, because of local influence from intrinsic defects such as micropores, carbide particles and undissolved eutectics, they tend to migrate to the octahedral planes, in the primary slip directions ((101) family of directions), since they represent the paths of least resistance for crack propagation [2]. Results presented show that SIF values are consistently higher for the (101) crack direction, compared to the (121) direction, for same crack angles and loads. CHAPTER 3 SIF EQUATION FORMULATION FOR MIXED MODE LOADING The stress intensity solution, for all the three modes for isotropic materials, using crack tip nodal displacements method is given by [38]: K, = P 2 [4(vB _D) +vE V] Kil P /2 [4(UB_ UD) + E UC] (3.1) K +1 L Here u, v and w are the displacements of the nodes B, C, D and E at the crack tip along x, y and z directions respectively. L is the length of the element at the crack tip normal to the crack front (Fig. 31 on page 24). These equations show that all three stress intensity factors KI, K1 and K111 are decoupled. For anisotropic materials it is not possible to use the same equation, as their material properties are direction dependent. Because of this directional dependence the stress intensity factor for the same material changes according to the orientation of the crack plane with material orientation. As has been discussed earlier, an orthotropic material has three independent elastic constants, E, v and G. The elastic constants in material coordinate system get transformed to specimen coordinate system by equation (1.7). The interde pendence of displacements in anisotropic materials due to shear coupling results in coupled stress intensity factors. Anisotropy was incorporated in the 3D model to calculate SIF for mixed mode loading. It can be shown that mixed mode stress intensity factors can be z Figure 31: Crack tip nodal displacement of isotropic elastic material for calcula tion of Stress Intensity Factors of Mode I, II and III computed by equation (3.2) [28] [29]1. K, K i [B]1 [A] 2l Kill Where L1 is the element length along the crack face and {A} is given as 2uB uc + 2UE uF +UD + T(4UB + UC + 4UE UF)+ 2(UF + UC 2D) 2VB vc + 2VE F+ VD + Tl(VB + VC + 4VE VF)+ 2WB Wc + 2WE WpF + WD + T(4WB + WCE + 4WE WF)+ lr>2(wF + wc 2wD) (3.2) (3.3) 1 The details of the equation are given in Appendix B Here u, v and w are the nodal displacements of nodes B, C, D, E and F at the crack tip relative to B', C', D', E', and F' as shown in Fig. 43 on page 34. TI is the natural coordinate system value defined as 2z S ( + 1) L2 L2 is the length of the element along the crack front (Fig. 42 on page 33) and [B]1 is defined as Re [' (q2 Re[ '(piq2 qi)] Re [ P (p2 291)2]' Re[ / (1P2 12P)] I 0 0 /C44C55 C(5 (3.5) As it is apparent from the [B]1 matrix that K, and KI1 are only (function of u and v) and KII, is function of w only. D is the determinant of the equation coupled SIFs Re\[ (pip2 D R[ /12 (/1P2 Re[ 12 (pijq2 1^192 ^'li /2Pl)] Re[ P2 (P2 2qi)] Re[ 1 2 (q2 pi)] qil)] /i1 and P2 are the roots of the fourth order characteristic equation all/4 2aI6/p3 + (2a12 + a66)2 2a26p + a22 = 0 and given by pij aj + i30,and fj > 0 (as pij can be only a complex no.[6]2 ) 2 The fourth order equation formulation was followed from the work of Lekhnit skii [6], refer to Appendix A. (3.4) [B]1 (3.6) (3.7) pj = aipj + a12 a16/j qj a= 12 +a22 a26 (3.8) For Plane stress, 1 1 V12 1 all E ; a22 22 a21 a12 ; a66 1 (3.9) Ell E22 El1 G12 For Plane strain, 7 ,"I., aj = aj (3.10) a33 Equation (3.2) is used to calculate the SIF at the crack tip by displacement method; one of the most used methods to get the value accurately. Finite Element Method was used to calculate the displacements at the crack tip. The commercial software, ANSYS, was used for FEA modeling. The crack tip nodal displacements were then extracted from FEA model and fed to the analytical equations explained above to calculate all the three modes of SIFs. Single Crystal Specimen Geometries Used for MixedMode Loading. Two specimen geometries were used to investigate the effects of mode mixity at the crack tip. One was a rectangular tension specimen (Fig. 34 on page 29) with a center crack loaded such that the crack lied in the {111} plane. The crack directions used were (101) and (121) family of directions. The second specimen modeled is a round Brazilian disk (BD) specimen, loaded in compression. This specimen with center crack has a mode mixity at the crack tip, which varies as function of the crack angle '0', shown in Fig. 35 on page 29. The crack lies on the {111} plane and crack directions used are (101) and (121) family of directions. These specimen and crack orientations have been checked very carefully, based on experimentally observed fatigue crack growth rates (FCGR) [39]. At low temperature (< 4270C(), the dominant mode of FCG in FCC single crystal superalloys is crystallographic crack propagation on octahedral planes Figure 32: The (101) and (121) family of slip directions are superimposed on {111} plane showing how the hexagonal crack front is delineated by the slip directions. [37] [37]. Extensive FCGR studies have shown that the fastest crack growth rates are observed for cracks propagating on the {111} plane, in the (101) family of directions. A slower FCGR has been observed with the crack on the {111} plane, in the (121) family of directions. Hence the FCGR on the {111} plane, in these two directions typically provide the fastest and the slowest ( il I 11 graphic crack propagation rates. This variation in FCGR in the (101) and (121) directions, on the {111} is illustrated in dramatic fashion in Fig. 32 on page 27. A penny shaped crack on the {111} plane evolved into a hexagonal crack front, because of different crack growth in the (101) and (121) directions. These six fastest crack growth directions (hexagonal corners) were found to be (101) family of slip directions on {111} plane. Details of the two specimens analyzed are presented in Table 31 on page 28. Where, Crack Cordinate System Material Coordibue System aligned Material Coordinate System to Crack Cooninate System Figure 33: Brazilian disk having center crack lying in {111} slip plane and aligned along [12t] direction Table 31: Geometrical and material properties of the two specimens analyzed Rectangular Specimen 'A' (Fig. 34) Brazilian Disk B (Fig. 35) E 106.2GPa H 20.32cm E 106.2GPa W 2.794cm G 108.2GPa W 5.08cm G 108.2GPa t 0.254cm v 0.4009 t 1.016cm v 0.4009 2a/W 0.20.8 F 4.448KN 2a/W 0.10.9 F 4.448KN W  Width or Diameter of the specimen, H  Height of the Specimen, t  Thickness of the specimen, F  Uniaxial load applied,  Angle of inclination between crack and load applied. Fig. 33 on page 28 shows that the crack plane lies on (111) plane and is directed along (121) direction. This slip system has been explained with the help of octahedral slip planes. w [10T] \ 2a 121] A .  (9 e) H F Figure 34: Rectangular specimen 'A' meshed with solid95 element (left) and the geometrical details (right) Figure 35: Brazilian disk (Specimen 'B') with center crack lying in (111) plane and oriented along [101] direction F t/ t  Crack Plane CHAPTER 4 MODELING AND MESHING In order to study the effect of orthotropy on SIFs, specimens with geometries mentioned in Table 31 on page 28 were modeled, meshed and analyzed for different crack lengths, crack angles, material properties and i i vI orientations. A rectangular specimen with isotropic properties and without any crack was modeled, and its accuracy was ascertained. A far field stress was calculated analytically and was found to be matched with meshed FEA result. The same model was used with a through thickness center crack and SIF at the crack tip was calculated and verified with standard table. A Brazilian disk with center crack having isotropic properties was modeled and compared with the SIFs extracted from FEA model. The values were verified with the available tabulated values for the isotropic Brazilian disk. Equation (3.1) was used to calculate the SIFs to verify the equation used by FEM software. As expected, the FEA results were found to be in excellent agreement with published results [26]. The Brazilian disk was selected because of its ability to vary mode mixity at the crack tip by varying load angle '0'. The next modeling was done using the orthotropic rectangular specimen with single edge crack and through thickness center crack. The orthotropic stress intensity factors calculated through analytical method were also compared with isotropic results obtained by FEA software. The difference between these results are discussed. Likewise the orthotropic Brazilian disk, with center crack, was modeled and a comprehensive parametric ,a i1 ,~i i was performed. The analysis was done for the following cases: Figure 41: Brazilian disk meshed with triangular element having isotropic proper ties (left) and zoomed view of crack tip (right). The model was checked for errors using isotropic (G ) and or thotropic (G E m) material properties while keeping all other parameters same; The crack length was changed from minimum to maximum in 10 steps; The crack angle with the applied force direction was changed from 00 to 900; The material coordinate systems used were {111} (101) and {111} (121). {111} being the crack plane and (101), (121) being the crack growth direc tions on {111} plane (Figs. 35 on page 29, 33 on page 28). After defining the geometry of the material in FEA software, free meshing was done on one of the surfaces (front or back), and was swept through the thickness. The mesh density was optimized to save time and effort (Fig. 34 on page 29). The crack tip element size was also varied to check the accuracy of the result. Once standardized, the same model was used for various other analyses. The FEA model was meshed using solid95 element (Fig. 41), which is a quadratic isoparametric brick element with 8 corner nodes and 12 midside nodes (Fig. 42 on page 33). The details of this element have been discussed in the following section. Crack Tip Element. A number of special crack tip finite elements have been developed applicable to the displacement method, also the hybrid method has been used in developing special crack tip elements. These special crack tip elements contain a singularity off the strain field at the crack tip equal to the theoretical singularity. For twodimensional geometry the 8node quadratic isoparametric element can be made to simulate the 1//r singularity in stress. That is done by placing the midside nodal coordinates on any side (connected to the crack tip) at the quarter point .,.1i 'ent to the crack tip [40]. Lynn and Ingraffea [41] have shown computation of SIF for isotropic material with high accuracy can be achieved by placing midside nodes between quarter and midpoint of the elements at the crack tip. This information further led to the development of the transition element, which resulted in improved SIF calculations when the ratio of the quarter point element and crack length is decreased. Quadratic isoparametric elements with 20 nodes have proven to be excellent elements for the calculation of SIF for elastic crack problems [31], when the mid side nodes are put near the crack tip at the quarter point. Also this element has inverse square root ( ) singularity, which provides a stress field in agreement with the theoretical stress singularity of linear fracture mechanics. While meshing the model, the size of the element was of significant importance. Tarasovs [42] has shown the effect of quarter point singular and transition element's size on several SIF computation techniques. It was determined that computed SIF was not sensitive to the singular element internal angle and size except for the quarter point displacement technique and the displacement correlation technique. Therefore finer mesh around the crack tip gives better results for the quarter point displacement method. Twodimensional triangular elements [30], formed by collapsing one side of rectangular elements, give better results for stress intensity factor. Also, it has N JR Figure 42: (a) 20 node isoparametric element in natural coordinate system and (b) Quarter point singular element with the ( = 1 face collapsed in local Cartesian coordinate system. [31] been found that rectangular quadratic isoparametric elements (and brick elements) have singular stiffness (singular total strain energy) at the crack tip [30]. For the threedimensional case, 20node isoparametric elements were used. 20node brick elements collapse and form wedge shaped elements at the crack tip. One of the element faces collapses at the crack tip to form the wedge shape as shown in the Fig. 42. Fig. 42(a) shows a 20 node isoparametric element in universal coordinate system, and Fig. 42(b) depicts its quarter point singular version with the = 1 face collapsed in local Cartesian coordinate system. This element is the best crack tip element for FEA solution of mixed mode SIF analysis of orthotropic materials [24] [42]. Around the crack tip the size of the elements should be very small to reduce the error in calculating SIFs. A typical example of crack tip element length (Li) is 0.002in. The enlarged figure of the meshing around the crack tip can be seen in Fig. 44 on page 34. All the nodes had to be aligned along the material coordinate system to get the desired result. Two different orientations (101) and (121) (on {111} plane) were used to analyze the effect on SIF. The crack lies on the {111} plane (Fig. 31 on page 24), and the crack direction was aligned along family of either (101) or (121) directions. Figure 43: Arrangement of quarterpoint wedge elements along segment of crack front with nodal lettering convention. [31] I.LL".LII" ' Figure 44: The symmetric meshing of BD specimen with solid95 element and enlarged picture of crack tip elements The resolved shear stress in the direction of (101) is maximum and in the direction of (121) is minimum on the {111} plane as the Schmid's factor is maximum along (101) and minimum along (121) direction. Since the maximum and minimum resolved shear stress occur along the (101) and (121) directions respectively, these directions are a family of primary or 'easy slip' directions and secondary or 'hard slip' directions, on the {111} plane. 7000 6 *[ Isotropic Solution for <121> *6Anisotropic Solution for <121> " Isotropic Solution for <101> 5000 Anisotropic Solution for <101> 4000 3000 2000 * 1000 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Crack Length Width ratio Figure 51: K, versus Crack Length/Width ratio for [101] and [T21] orientation of Specimen 'A' at 00. gradual increase in K, was observed with an increase in the 2a/W ratio. Also the values of K, for the [101] orientation were ahbv~ found to be greater than the corresponding values for the [121] orientation (Fig. 51 on page 37), at the same far field stress. The same trend was also observed for K11, though the values were very small for both the orientations (Fig. 52 on page 38). This is because the loading is normal to the (111) plane, thus inducing zero shear in the plane of the crack. Therefore K11 is expected to be very small and within the numerical calculation error. The values of K111 for the [101] orientation are negative (Fig. 53 on page 38) because the relative displacement of the two crack faces along the zdirection (Fig. 43 on page 34) was negative. This is simply a consequence of sign convention definition. However, the magnitude of the SIF for [101] is found to be greater than for [121]. This validates that the (101) family of slip directions are primary or easy slip directions as compared to those of (121). The change in SIFs with the crack angle '0' was analyzed for both orienta tions. K, decreases for both orientations with increasing crack angle as is evident 140 *Anisotropic Solution for <121 120_ Isotropic Solution for <121> Anisotropic Solution for <101> 100 6. Isotropic Solution for <101> ~0 0.1 0.2 0.3 0.4 0.5 0.6 .7 0.8 0 Crack Length / Width ratio Figure 52: KI1 versus Crack Length/Width ratio for [10t] and [121] orientation of Specimen 'A' at 00. 200 0 T 300 r 400 500 600 700 800 .4 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9  Anisotropic Solution for <121"  4 Isotropic Solution for <121>  Anisotropic Solution for <101>  * Isotropic Solution for <101> Crack Length / Width Ratio Figure 53: K11, versus Crack Length/Width ratio for [101] and [121] orientation of Specimen 'A' at 00. Anisotropic Solution for <121:  ** Isotropic Solution for <121> Anisotropic Solution for <101>  * Isotropic Solution for <121> 0 10 20 30 40 50 60 70 80 Crack angle with Force (degree) Figure 54: K, versus crack angle Specimen 'A' at 2a/W 0.4. 700 600 500 400  300 200  100  with force for [10t] and [121] orientation of 0 10 20 30 40 50 Crack angle with Force (degree) 60 70 80 Figure 55: KI1 Specimen 'A' at versus crack angle with force for [iot] and [121] 2a/W 0.4. orientation of 3000 2500 S2000 5 1500 1000 500 0 500 50  10 20 30 40 50 60 70 80 500 4* Isotropic Solution for <121> S* Anisotropic Solution for <121> 1000 Isotropic Solution for <101> "10 J AAnisotropic Solution for <101> S1500 2000 2500 Crack angle with Force (degree) Figure 56: K111 versus crack angle with force for [101] and [121] orientation of Specimen 'A' at 2a/W 0.4. from the plots (Fig. 54 on page 39), but the value of K, is alvb, greater for [10t] than that for the [12t] orientation. For isotropic material, K, becomes zero upon crack closure at a certain angle. For an orthotropic material, it can be seen that K, is still nonzero due to the coupling of the nodal displacements. The value of K1 initially increases with an increase in Q due to an increase in shear stress, which after reaching a maximum value starts decreasing (Fig. 55 on page 39). The value of K111 for the [10t] orientation is negative (Fig. 56 on page 40) as the relative nodal displacements of the crack faces along the zdirection were negative. However, it was observed that the magnitude of K111 for [10t] was ah ,v greater than for the [121] orientation. From the Brazilian Disk specimen (Specimen 'B'), it can be seen that the magnitude of SIF for the [10t] orientation is ahv greater than the corresponding values of the [121] orientation, thus enabling a crack to move faster on the [101] plane than on the [T21] plane (Figs. 57, 58 on page 42, 59, 510 on page 43, 511, 512 on page 44), under fatigue loading for same AK values. Even though 25000 *Anisotropic Solutions for <101> 20000 . Isotropic Solution for <101> 20000  SAnisotropic Solutions for <121> * Isotropic Solutions for <121> '5 15000 10000 ................ 0 0  0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Crack Length / Diameter ratio Figure 57: K, versus Crack Length/Diameter ratio for [10] and [121] orientation of Specimen 'B' at Q 00. the magnitude of Kn1 for the [121] orientation is greater than that for the [101] orientation, it may not affect the effective SIF very much, as the magnitude of K111 for the [101] orientation is far greater than that for the [121] orientation. The values of K, for the [101] orientation rapidly decrease as compared to the [121] orientation and reach crack closure angle at around 180, whereas the crack closure angle is 300 for the [172] orientation. The magnitude of K/i for [121] is alv, greater than that for the [101] orientation, but the difference is far less in comparison to the magnitude of K111 for the [121] and [t10] orientations. Following the work of Sauma and Sikiotis [31], the calculated values for Knli were found to be negligible and after much analysis and rederiving of relevant equations, the equation given by Sauma and Sikiotis was found to be wrongly formulated. On the other hand, equation (3.5) correctly gave rise to all the three modes of SIFs. As a result of the coupling of displacements at the crack tip due to anisotropy, non negligible values of K111 were found, and they varied with respect to the applied force, crack length and crack angle. 5 1000 5 2000 *~0 0~~ S p U U 2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 3000 *Anisotropic Solutions for <101> Isotropic Solution for <101> 4000 Anisotropic Solutions for <121 * Isotropic Solutions for <121> Crack Length / Diameter ratio Figure 58: KI1 versus Crack Length/Diameter ratio for [10t1 and [121] orienta tion of Specimen 'B' at 0. 2000 0 0 2000  4000 6000 r 8000 10000 12000 2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 $Anisotropic Solutions for <101>  Isotropic Solution for <101>  Anisotropic Solutions for <121  * Isotropic Solutions for <121> 14000 Crack Length / Diameter ratio Figure 59: K11 versus Crack Length/Diameter ratio for [101] and [121] orienta tion of Specimen 'B' at Q 0. A,_ 43 15000 .. *. * 10000 *" ''  5000 ..  S 0 0 10 30 40 50 60 70 80 90 5000 10000 15000 20000 _ Isotropic Solution for <101> 25000 Anisotropic Solutions for <121> 25000  * Isotropic Solutions for <121> 30000 Crack angle with Force (deg) Figure 510: K, versus crack angle with force for [10T] and [T12] orientation of Specimen 'B' at 2a/W 0.55. 10000 5000 * q 0 5000  *Isotropic Solutions for 121 10 20 30 40 50 60 70 80 90 10000 Anisotropic Solutions for <101> Anisotropic Solutions for <121> @ Isotropic Solutions for <121> 15000 Crack angle with Force (deg) Figure 511: KI1 versus crack angle with force for [101] and [T2T] orientation of Specimen 'B' at 2a/W 0.55. 20000 Anisotropic Solutions for <101> .B Isotropic Solution for <101> 15000Anisotropic Solutions for <121> 15000 . Isotropic Solutions for <121> 10000 5000 0 10 20 30 40 50 60 70 80 90 5000 10000 Crack angle with Force (deg) Figure 512: K111 versus crack angle with force for [10lt and [F21] orientation of Specimen 'B' at 2a/W 0.55. The BD specimen SIFs (KI, K1i and K1il) were further analyzed along the thickness for the [10Lt and [T21] crack orientations with change in crack length and crack angle. They were calculated at 5 different points along the thickness at the crack front as shown in the Fig. 5 13 on page 45. A plane stress assumption was made at the surface of the BD specimen (Plane 1 & 5) and a plane strain assumption was made at all the interior points (Plane 2, 3 & 4). In order to check the difference in SIF of isotropic BD specimen and or thotropic BD specimen with change in load angle with crack, under similar loading condition and specimen geometry, two models were analyzed. The orthotropic BD specimen analyzed had the material coordinate system aligned with specimen coordinate system. Therefore a symmetry was expected in the SIF values across the thickness (Figs. 515 on page 47, 517 on page 48, 519 on page 49). For isotropic material, properties of pure nickel was used. The trends of KI, K11 and K111, because of change in force angle with crack, for orthotropic BD specimen and isotropic BD specimen were found to be similar, but the magnitude of SIFs calculated x Crack tip front Figure 513: Half meshed model of Brazilian disk specimen and the crack coordi nate system for both the specimen were significantly different, because of the material prop erties. The crack closure angle was the same (~ 200, Figs. 514, 515 on page 47). Negative K, was found because no contact elements were used at the crack interface. After reaching the crack closure angle, further increase in force angle lead to penetration of crack faces, resulting in negative K1. At the edges, K, was less than that towards midplane (Figs. 514, 515 on page 47). K1,max was found to be 12 Kpsivin for orthotropic BD specimen, whereas it was less than 10 Kpsivin for isotropic BD specimen. The values of K11 can be seen to be negative because of the sign convention followed at the crack tip coordinate system (Fig. 513 on page 45, 64 on page 64). The absolute value of Kt reached maximum at ~ 300 for both the specimens(Figs. 516, 517 on page 48), then starts slowly decreasing towards zero. The absolute Knlmax was 18 Kpsi for orthotropic BD specimen, whereas it was 14 Kpsi in for isotropic BD specimen. The variation of K1 across the thickness was more distinguished for the orthotropic case than for the isotropic specimen (Fig. 516, 517 on page 48). Higher K7). values were observed for the 10 * Angle 00 SU Angle 06  Angle 12  Angle 18 S 6  Angle 24 4 4 0 ( 0.2 0.4 0.6 0.8 1 1.2 2 4 Normalized Thickness Figure 514: Variation of SIF K, along isotropic BD specimen thickness at differ ent Crack angle. orthotropic specimen than for isotropic specimen (Figs. 518, 519 on page 49). For load aligned with crack plane, KIII was found to be zero across the thickness. Since K1II is a function of out of plane relative displacements (normal to the disk plane) of nodes at the crack tip and because of perfect symmetry in loading and geometry of the material, it ought to zero. But as the force angle was introduced, because of poisson's effect, there were small out of plane relative displacements on both sides of the mid plane of the disk (plane 3, Fig. 513 on page 45), which was equal and opposite to each other. The absolute value of K1I reaches maximum between 30360 and it is ~ 15Kpsiin for isotropic and ~ 22Kpsivn for or thotropic specimen. Among all the SIFs K1I is more dominant at the crack edges at higher force angles, whereas it is zero at the midplane for all the angles (Figs. 518, 519 on page 49). The average value of K1II is zero across the thickness for both specimen. From Figs. 521, 522 on page 52, it can be seen that K, decreases with an increase in crack angle and crack closure angle (where K, is almost zero) was *Angle 00  Angle 06 eAngle 12  Angle 18 + Angle 24 I N 0.2 0.4 0.6 0.8 1 Normalized Thickness Figure 515: ferent Crack 2 0 2 4 o 61 S8 I 10 Variation of SIF K, along orthotropic BD specimen thickness at dif angle. Normalized Thickness Figure 516: Variation of SIF KI1 along isotropic BD specimen thickness at differ ent Crack angle. 1.2 2 0 2 i *a  12 4' 16 18 0.2 0.4 0.6 0.8 1  SAngle uu UAngle 06 sAngle 12 xAngle 18 Angle 24  Angle 30 Angle 36 Angle 42 Angle 48 Angle 54 *Angle 60 Angle 66 4Angle 72 Angle 78 Angle 84 Normalized Thickness Figure 517: Variation of SIF KI1 along orthotropic BD specimen thickness at different Crack angle. * Angle 00  Angle 06 eAngle 12 Angle 18 + Angle 24 0.2 0.4 Normalized Thickness Figure 518: Variation of SIF KII, along isotropic BD specimen thickness at differ ent Crack angle. 25 * Angle 00 2 Angle 06 20 sAngle 12 Angle 18 + Angle 24 15 Angle 30  Angle 36 10 ~ Angle 42 x Angle 48 5 x Angle 54 0* Angle 60  Angle 66 0 Angle 72 S2 04 Angle 78 5s  Angle 84 10  15 20 25 Normalized Thickness Figure 519: Variation of SIF K111 along orthotropic BD specimen thickness at different Crack angle. reached at ~ 180 for the [10t] orientation, whereas it was ~ 300 for the [121] orientation, as was observed by Fig. 510 on page 43. K, can be seen symmetric for [t21 orientation across the thickness, whereas it is not for the case of [10T] orientation. In general K, inside the surface, (plane 2, 3, 4, Fig. 513 on page 45) is alvb, greater than those at the crack edges (plane 1, 5, Fig. 513). The absolute value of K11 increases and becomes maximum at an angle ~ 240 for the [10t] orientation, whereas for the [T21] orientation it is ~ 360 (Figs. 523, 524 on page 53), as also illustrated in Fig. 511 on page 43. K11 then starts slowly decreasing with further increases in crack angle, as was observed in Fig. 511. It can be seen that at one of the faces (Plane 1, thickness = 0; Fig. 521 on page 52), absolute values of K, are alvb greater than those at the other face (Plane 5, thickness = 1) of the BD specimen for the [10l] orientation. But absolute values of Kl (Fig. 523 on page 53) are alvb, greater at Plane 5, than those at Plane 1 for the [101] orientation. It is interesting to observe that SIFs (Figs. 522 on page 52, 524 on page 53, 526 on page 54)(KI, Kl and K1il) are symmetric 14 12 10 * Angle 00 8 Angle 06 eAngle 12 6 Angle 18 S  Angle 24 4 2 II 0.2 .OA S 0.8 1 1.2 4 6 Normalized Thickness Figure 521: Variation of SIF K, along BD specimen thickness at different Crack angle for [101] orientation. 12 0.8 1 F 0.2 * Angle 00  Angle 06 Angle 12 Angle 18  Angle 24  Angle 30 Normalized Thickness Figure 522: Variation of SIF K, along BD specimen thickness at angle for [121] orientation. different Crack Angle 30  Angle 36 a Angle 42 .94Angle 48 SAngle 60 Angle 60 6  Angle 66 Angle 72 M 8 Angle 78  uAngle 84 10 12 14 Normalized Thickness Figure 523: Variation of SIF KI along BD specimen thickness at different Crack angle for [101 orientation. 0.2 0.4 0.6 0.8 1 1.2 _ "^a ~^ *Angle 00 2 Angle 06 2 Angle 12 Angle 18 4Angle 24 5 ^ Angle 30 *a Angle 36 Angle 42 Angle 48 6 Angle 54 4Angle 60 ^ hAngle 66 SAngle 72 M . 8Angle78 oAngle 84 10 12 Normalized Thickness Figure 524: Variation of SIF KI along BD specimen thickness at different Crack angle for [12t] orientation. * Angle 00 Angle 06  Angle 12 60 Normalized Thickness Figure 525: Variation of SIF KII, along BD specimen thickness at different Crack angle for [o10] orientation. * Angle 00  Angle 06 eAngle 12 Angle 18  Angle 24 0.2 0.4 Normalized Thickness Figure 526: Variation of SIF KII, along BD specimen thickness at different Crack angle for [121] orientation. 40 35 .5 30 j 25 15 10 5 0 0.2 0.4 0.6 0.8 Normalized Thickness Figure 527: Variation of SIF K, along BD specimen thickness Length/Diameter ratio for [101] orientation. 30 25 .a S20 S15 10 5 A CU I 1 1.2 at different Crack *Ratio 0.3  Ratio 0.4 Ratio 0.5 e Ratio 0.6  Ratio 0.7 * Ratio 0.8   0.6 Normalized Thickness Figure 528: Variation of SIF K, along BD specimen thickness at Length/Diameter ratio for [121 orientation. different Crack 5 i 4 S3 S2 0 1 0.6 0.8 Normalized Thickness Figure 529: Variation of SIF KI1 along BD specimen thickness at different Crack Length/Diameter ratio for [101] orientation. 2 3 S4  6 7 *Ratio 0.3 1  Ratio 0.4 aRatio 0.5  Ratio 0.6 * Ratio 0.7 *Ratio 0.8 0 Ratio 0.8 Normalized Thickness Figure 530: Variation of SIF KI1 along BD specimen thickness at different Crack Length/Diameter ratio for [T21] orientation.  Ratio 0.3  Ratio 0.4  Ratio 0.5 e Ratio 0.6  Ratio 0.7 * Ratio 0.8 0.2 0.4 * Ratio 0.3  Ratio 0.4 10 S30 40 50 Normalized Thickness Figure 531: Variation of SIF Ksss along BD specimen thickness at different Crack Length/Diameter ratio for [101] orientation. 2 * Ratio 0.3  Ratio 0.4 1.5  Ratio 0.5 e Ratio 0.6 Ratio 0.7 1  Ratio 0.8 0 0' 1 0.2 0.4 1 1.2 0.5 1 1.5 Normalized Thickness Figure 532: Variation of SIF Ksss along BD specimen thickness at different Crack Length/Diameter ratio for [12T] orientation. Sl > S2 > S3 Applied cycles, N Figure 62: Fatigue Crack Length versus Applied Cycles. Fracture is Indicated by the x. [43] given initial crack size, the life to fracture depends on the magnitude of the applied stress and the fracture resistance of the material. 6.2 FCC single < I 1 materials Fatigue crack driving force parameter AK, as described in previous section is well suited for isotropic materials, but it may not be the best parameter to use for very large grain or single crystal alloys. Earlier studies have shown that for crack size smaller than the grain size or of the same order of magnitude, threshold stress intensities are lower and FCG rate is accelerated in comparison to long cracks when compared at similar values of AK [4448]. This shows that the parameter AK does not incorporate microstructure and the grain orientation and the associated deformation mechanisms in controlling the FCG behavior. An FCC single < i i1I J1 alloy offers the best opportunity to study in detail the effect of microstructure and the deformation mechanisms on the FCG behavior. Several Studies have been conducted on FCG of Nibase single crystals [4953] and all of these studies have shown that FCG is highly sensitive to the orientation of the crystal and that the crack plane is crystallographic and follows a single slip plane or a combination of slip planes. Since shear decohesion on a slip plane is caused by dislocation motion, many researchers have r . 1. 1 that the resolved shear stress acting on the active slip plane ahead of a crack tip must be responsible for the propagation of the fatigue crack [51, 54, 55]. The dislocation motion is controlled by the forces at the crack tip, which is directly related to the resolved shear stress on the slip plane. Therefore the rate of shear decohesion must be related to the resolved shear stress 'intensity' (RSSI) at the crack tip. The active shear decohesion plane or planes must be the slip planes) with a high RSSI. If the RSSI on a plane is much higher than all the other slip planes, then the plane must be the primary plane for shear decohesion and the slip plane becomes the crack plane. But if the resolved shear stresses on two or more of the slip systems are comparable, then the shear decohesion will take place on all of those slip planes and the macro crack plane will not follow a single slip plane. This was shown by Telesman and Ghosn [56] as shown in the Fig. 63 on page 62. Even though macroscopic crack was observed along (001) plane inclined 70 to the starter notch, the microscopic slip was observed on 111 slip planes, inclined 520 and 380 to the starter notch. ,' 50 pm Figure 63: Microscopic slip observed on two {111} slip planes inclined 520 and 380 to the starter notch. [56] If a crack surface is a slip plane, it is logical that the crack growth rate on that slip plane will correlate with its RSSI. Chen and Liu [57] proposed a crack driving force parameter for correlating FCG data, which is based on the resolved shear stresses on the active slip planess. This parameter may be better than AK for the correlation of FCG data since it takes into consideration of the deformation mechanisms, grain orientation and the actual crack path. The resolved shear stress field of a slip system is defined by its intensity coefficient, which can be calculated once the Mode I, II and III crack tip fields are obtained. The resolved shear stress is given by [58] Tss = bibjjj (6.7) where bi and b are the Burgers vector and its magnitude; nj is the unit normal vector of the slip plane; and ryj is the crack tip stress tensor field given by [57] [1yi] [Kf1 (0) + K1fi' (0) + Ki f i' (0)] (6.8) where r and 0 are the local polar coordinates at the crack tip as shown in the Fig. (64 on page 64); fij(0) are the angular component of the stress field. Substituting equation (6.8) into equation (6.7), the resolved shear stress is 1 s ] [KIf' (0) + Ki f i (0) + Ki if"(0)][n ] (6.9) where b' and nj are the unit Burgers vectors and unit normal vectors of the slip planes respectively. The above equation indicates that Trss preserves the 1//r singularity, and the intensity of T,ss, is dependent on the (i iI 1 orientation relative to the crack surface. For a given crystal orientation and crack geometry, the angle 0 is equal to the angle tx Crack Surface Figure 64: Details of crack tip displacements and stresses at a distance r and 0 from the crack tip in the crack coordinate system between the trace of a particular slip plane on the plane normal to the slip plane and the horizontal axis. The intensity of Trs, is linearly proportional to the quantity resolved shear stress intensity coefficient, K.s which for a given slip system can be defined as the limiting value of the resolved shear stress, T.rs, multiplied by 27r, as r approaches zero [56, 57] Krs, = limia 8.v2r (6.10) rO where r is the distance of the crack tip and Tr~, is defined as the projection of the stress tensor [oa] on a plane whose outward normal is S in the direction of slip b (Fig. 65 on page 65). The two distinct advantages in using K.rs are: (1) the dependency of Tr, on r is eliminated; (2) the angle 0 has a definite physical ri ii. which is directly related to the orientation of the slip system. Telesman and Ghosn [56] used the above model to calculate AKrs for 2D model and checked the validity of this parameter by plotting it against da/dN data (Fig. 66 on page 65). Also AK for mode I has been plotted against da/dN to n 4 {111} Figure 65: Burgers vector b is along slip direction (011) and slip plane direction is normal vector S along (111) /ExtrapoLation of / Paris region ,o 3 4 6 810 20 0 AK, AKr.. MPoVrm Figure 66: Fatigue crack growth rate as a function of AKrs8 and AK for 2D rectangular specimen. [56] show similarity between both the curves. They look virtually identical and it can be seen that a linear relationship exists between AKs88 and da/dN, which is in accordance with the Paris law in region II. But in this method mode III SIF was not considered as the specimen was not tested under complete multiaxial fatigue loading. State of stress on a slip plane, under mixed mode loading, whose trace on a plane normal to crack plane makes an angle 0 with the horizontal axis (Fig. 64 on page 64 and Fig. 613 on page 71), can be defined as1 cr dil d12 0 y d21 d22 0 K, tz 1 d31 d32 d33 (6. Ty 0 0 d43 Ki TZ 0 0 d53 Ty d61 d62 0 where, b b d1F2 22 1 ( p d11 = Re r12 t P1 d12 = e Pt (P2 j LI [2 b2 bl [1l p2 b2 bl 1 Re t 1 1 : 2] 1 (I 1) d21 Re ld22 Re  11 2 b2 b 1l [1 2 b2 bl 0 Plane Stress d3j dj1a13 + d2ja23 + d6j36 Plane Strain1, 2 a33 Plane Strain 833 1 The details of the equation derivation can be seen in Appendix B. d33 { 0 d43a34 + d53a35 a33 d43= Re b] d61 Re !12 / /1 cL(1 [12 bl b2 n 6i V/cos(0) + sin(0) Plane Stress Plane Strain d53 = Re [b] d62 Re 11 (I /12)1 62 = 3 P1 [2 bl b2 i 1,2,3 /il and /2 are the two roots with positive imaginary parts as defined by the equation all/4 2a16 /3 + (2a12 + a66)_2 2a26p + a22 = 0 (6.12) and /P3 is the root of the characteristic equation [26] a55/2 2a45/ + a44 0 (6.13) It should be noted that the compliance constants (ai,3(i 1, 2,... 6)) used for the expression (d3,j(j 1,2, 3)) are the ones as given by equation (3.9). The above equations, when used in conjunction with equations (6.7) & (6.10) gives Krs. on all the 12 slip systems for FCC single i ivI 1 superalloy. /' [nj] K, KIl Kil (6.14) K/rss 0 10 20 30 40 50 60 No. of Cycles, N in Millions Figure 67: Crack growth of BD specimen 95830 with no. of cycles 12 n 10 6 4 0.05 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 No. of Cycles, N in Millions Figure 68: Crack growth of BD specimen 96842 with no. of cycles 16 n 14 12 10 6 4 0 0.1 0.2 0.3 0.4 0.5 0.6 No. of Cycles, N in Millions Figure 69: Crack growth of BD specimen 98C21 with no. of cycles 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9 Crack Length, a in mm Figure 610: Crack growth rate of BD specimen 95830 with increasing crack length 4 5 6 7 8 9 Crack length, a in mm 10 11 12 Figure 611: Crack growth rate of BD specimen 96842 with increasing crack length 4.E04 I 3.E04 S3.E04 2.E04 2.E04 1.E04 2 4 6 8 10 Crack length, a in mm 12 14 Figure 612: Crack growth rate of BD specimen 98C21 with increasing crack length 9.E05 8.E05 7.E05 6.E05 5.E05 *9 4.E05 3.E05 The mixed mode SIFs were calculated at those points shown in the Figs. (610, 611, 612 on page 69) for all the three specimens on the mid plane of the Brazilian disk specimens, assuming plane strain assumption. AKrms was calculated using the following equation in order to compare with AKss. AKrms = AKj + AK}, + AK}JJ (6.15) K..s was calculated using the equation (6.10), where state of stress obtained from equation (6.11) was multiplied by Schmid's factor for FCC single i i,l I as defined in equation (1.13) to get the resolved shear stress intensity on all the 12 primary slip systems. The trace of the two slip planes (111) and (111) were found to be making equal angles with the (111) plane (67.780), while the trace of the plane (111) was making 00 with the crack plane (111) as shown in the Fig. 613. (111) {1To0} Plane along depth {1 1 2} Trace of Slip Planes 11) Trace SC Trace Plane y 0 (Plane normal to 0= 670 Crack Plane) 0 = 670 z Trace of Slip Planes Crack Surface ( 1 1) Figure 613: Trace of primary slip planes on the plane normal to the crack plane The load ratio R (min load/max load) was 0.1 (Table 61 on page 68), as the load on the BD specimen alv,x compressive. As SIFs are linearly proportional to the load applied, therefore Krs. was multiplied with R to get the AKrss. The max K,.s was found to be on the (111) slip plane for all the three specimens (First row of Tables 62, 63 on page 76 and Table 64 on page 77), as observed in the experiment test results (Fig. 614 on page 73). The calculated AKr.s was plotted against da/dN on a loglog scale to check the validity of the model (Fig. 616 on page 74). After AK,.r reaches 10 MPa /m a linear plot can be seen, which might correspond to region II (Paris region) where the crack growth rate is directly proportional to the the applied AK, on a loglog basis. But not enough data is available to support the theory.2 However below a AK,.rs of 8 MPa\/m an accelerated crack growth can be seen between 5 MPa\/m and 7 MPa /m, in the threshold region; region or stage I. The lines drawn through those points yield AKth on the AK,.rs axis, which is called fatigue threshold stress intensity factor. The average AKth of the three specimens has been taken ( 4.3MPa V/), at the given load ratio R(0.1). Below this value (AKr.s < AKt, crack growth either does not occur by cyclic loading i. e. da/dN = 0 and specimen can have infinite life or it grows at undetectable rates. The value of AKth can be affected by alterations in the material microstructure, load ratio, environment and crack size 3 However the cyclic stress intensity factor (AK, r.) is not the only load parameter to control the FCG rate, because da/dN, at a given AKrs, can increase with increase in load ratio 'R' as max (1 R) 2 Because of the proprietary nature of specimens, only three specimens' data were made available. But we are expecting some more data in June'05 to verify our work. 3 The effect of these parameters on cyclic stress intensity factor is beyond the scope of my research. However the detail can be found in 'Fatigue of Materials' by S. Suresh [59] 00 * o 0 *0 Figure 615: Fatigue crack growth function of AKrms rate of 3 specimens 95830, 96842, 98C21, as a 1.E03 A A 1.E04 1.E05 1.E06 1.E07 I 1.OE+00 o 95830 n 96842 A 98C21 1.OE+01 1.OE+02 AK.,. MPa m1/2 1.OE+03 Figure 616: Fatigue crack growth rate of 3 specimens 95830, 96842, 98C21, as a function of AKrss 1.E03 1.E04 A 95830 * 96842 o 98C21 1.E05 1.E06 1.E07  1.0OE+00 1.0OE+01 AKms, MPa m1/2 1.0OE+02 75 .g J, L b c 1 CM c. o, ,, o I  ~C/2 ' I a o C I 'cu u C  0  v ri L' icx 2mc cl i,.... i.... i.... ,I' c: CV 0 M cmom ," i 5 'm' ;0 10 'Nn CO C ~ T'. o 0 0 I^ (LI. cm 0^ rV rV <<,  C '?III ci ci S o C ;; ^ ,_ i rc i i v' 1' 1~ zz0T I  t> >0r '" 1 I'' XIT. ^' ~ c" '''  o XI oo ' ci^ t S & ^ o  . CJf a o 0 } a *M 0. O0 "0 i I cij C0C r" 0C ,' I DD "  __ >> III  ! II d ^.i a to J~ a ,. ~' a t L7, I " A 12@ O i0 >i .> M V c c B o coc a c5 y 0  cI Go 1 ^ ^7^ o. c I o0 ^ *' ci at  ' to A CI  G6 ' C1 0c, C CC    c c I K I * '. 0 + Qt ; W ; ID. D t_ 0 ~ D  C 00^^^ r ^ I as C.*, ci .00 0c 0^ wl F ~ "^ "^ 0 '~ l*i ~1 t t ^ ^^ I I I* '^^ ^ CHAPTER 7 CONCLUSIONS The goal was to estimate the fatigue life of single crystal nickelbase superal loys, based on fatigue crack driving force parameter, for which a numerical model was developed to compute mixed mode SIFs at the crack tip. The results obtained can be summarized as 1. An analytical method has been developed for the calculation of all the three modes of stress intensity factors as a function of crystallographic orientation for an orthotropic material, which can be applied to any anisotropic material, if all the material constants are known. No prior assumption of plane stress or plane strain was made in developing this theory. 2. Mode I, (KI), was ah. found to be greater for [10T] than for the [121] orientation. For an orthotropic material, K, was found to be nonzero at crack closure due to the coupling of the nodal displacements, whereas for an isotropic material, it was zero. 3. The magnitude of K1i for [121] was found to be alvb greater than that for [101] orientation, but the difference was not much. 4. Mode III SIF (Kill) existed because of the coupling of displacements at the crack tip due to anisotropy. K111 for [101] was found to be much bigger than that for [121] orientation for the Brazilian disk specimen. This p .1i  an important role in calculation of effective K to predict the living of an anisotropic material. 5. The K, and K111 values calculated were much higher for [101] orientation, than those for [121] orientation. Therefore a high value of K.rs can be expected for the [101] direction than that for the [121] direction, if we ignore required to get the accurate value of parameter C and n of the Paris law. The equation obtained can be used to calculate the life of the material. 11. The method developed is not related to any specific geometry. Therefore it can be used for any kind of geometry and for any general anisotropic material, to calculate mixed mode SIFs at the crack tip and therefore crack growth rate and hence life of any material can be predicted. 12. Additional experimental data are required to study FCG on {111} plane, (101) orientation. More data are needed in region II to get the accurate result to predict lifing. Also experimental data for crack plane orientations other than slip planes, e.g., (100), can be used to check the validity of the model. Possible Directions for Future Work are given below: Incorporation of crack tip plasticity. FCG testing under mixed mode loading for a wider range of specimen geometries and orientations. FCG testing at high temperatures. APPENDIX A DISPLACEMENT FIELD EVALUATION FOR AN ANISOTROPIC ELASTIC SOLID UNDER GENERALIZED PLANE STRAIN CONDITION   dx f)Cx STxy+ dx x +y dy aTy+ aoydy Figure A 1: Forces acting on a rectangular element with dimension dx x dy in equi librium condition. A generalized plane strain condition has been adopted for an anisotropic solid applicable to that of an FCC single crystal. [6] Let us consider a rectangular pipe like element with dimension dx, dy, 1. We assume that this body is under equilibrium under various forces. If body forces are X, Y per unit volume, then the equation of equilibrium of force along x and y direction can be written as followed, Along xaxis. Ix + aXdy dx axdy + TsY + Txydy\ dx Txydx + Xdxdy = 0 (A.1) 