PLACE IN RETURN Box to remove this checkout from your record. To AVOID FINES return on or before date due. MAY BE RECALLED with earlier due date if requested. DATE DUE DATE DUE DATE DUE 3203111 6/01 cJCIRC/Dataouopss-sz TENSILE CRACK INITIATION IN 7-TiAl By Benjamin Andrew Simkz'n A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Chemical Engineering and Materials Science 2003 ABSTRACT TENSILE CRACK INITIATION IN ’y-TiAl By Benjamin Andrew Sim/an Using electron channeling methods, fracture initiation in 7-TiAl based alloys has been studied with the overall objective of determining the principal fracture initia- tion mode. Electron channeling methods reveal sufficient information to determine true grain orientation as well as deformation modes for the tetragonal ’y-TiAl crystal structure. Fracture initiation is found to occur principally at 7-7 grain boundaries, with the second most common initiation site occurring at 7—012 interphase boundaries. For both initiation modes, boundary crack initiation appears to be related to impingement of deformation twins from the y-TiAl phase. For the 7-7 grain boundary fracture initiation, a factor that incorporates com- ponents relating to deformation twinning, grain boundary conformation to incident deformation twins, and the relationship between twin shear and macroscopic stress state is reported. Interpretation of this factor implies that the driving mechanism of 7-7 boundary fracture is the accumulation of grain boundary strain energy associated with deformation twin accommodation by the grain boundary. Further refinement of this grain boundary fracture factor, F , is discussed. The current study suggests an enhancement of the fracture correlation with a modified form of F that incorporates grain boundary orientation; in addition, further avenues of grain boundary fracture factor refinement are suggested, toward the ultimate goal of describing a threshold for which grain boundary fracture will occur. Ultimately, this study suggests that any y-TiAl based alloy that deforms principally through deformation twinning is unlikely to display extensive tensile ductility, due to the nature of accommodation of the grain boundary to deformation twins. Copyright by Benjamin Andrew Simkin 2003 ACKNOWLEDGEMENTS I would like to thank my entire committee, Drs. Thomas Bieler, Eldon Case, Martin Crimp, and Phillip Duxbury. I would also like to thank Reza Loloee for his assistance with the AFM, Drs. Jun Nogami and David Grummon for their helpful discussions and advice, and Boon-Chai (Steve) Ng for protracted discussions about TiAl. Most especially, I would like to thank Takako for giving me a reason to finish. This research has been conducted with funding from the Air Force Office of Sci- entific Research (grant #F49620—01- 1-0116), the Michigan State University Compos- ite Materials and Structures Center, and The National Science Foundation (grant #DMR9302040). TABLE OF CONTENTS LIST OF TABLES LIST OF FIGURES 1 Introduction 1.1 TiAl—Based Intermetallic Alloys ..................... 1.2 Deformation Behavior of y-TiAl ..................... 1.2.1 Alloying of the y-TiAl alloys ................... 1.3 Electron Channeling ........................... 1.4 Electron Scattering Effects Due to the Superlattice Structure ..... 1.5 7-7 Deformation Transfer ........................ 1.5.1 Deformation transfer across special 7-7 boundaries ...... 1.5.2 Twin deformation transfer across the general 7-7 boundary . . 1.6 Grain Boundary Effects on Fracture ................... 1.7 Overview .................................. 2 Experimental Procedure 2.1 Sample Preparation ............................ 2.1.1 Material .............................. 2.1.2 Bend test samples ......................... 2.2 Sample Loading .............................. 2.3 Electron Microscopy ........................... 2.4 Sample Loading into SEM ........................ 2.5 Grain Orientation Determination .................... 2.6 Plane Trace Identification ........................ 3 Results 3.1 Sample Microstructure .......................... 3.2 Deformed samples ............................. 3.2.1 Fractured samples ......................... 3.2.2 Unfractured sample ........................ vi ix CXJKICJ‘OOr—li-l 10 12 14 16 17 18 18 18 18 19 21 22 25 31 33 33 33 35 37 3.2.3 Crack initiation sites in the unfractured sample ........ 37 3.2.4 Deformation twinning and microcracking ............ 40 3.2.5 Topographic contrast and crack opening from deformation twinning .............................. 49 3.3 Sample Regions Descriptions ....................... 51 3.3.1 Region 1 .............................. 52 3.3.2 Region 2 .............................. 54 3.3.3 Region 3 .............................. 55 3.3.4 Region 4 .............................. 57 3.3.5 Region 5 .............................. 60 3.3.6 Region 6 .............................. 61 3.3.7 Region 7 .............................. 66 3.3.8 Region 8 .............................. 66 3.3.9 Region 9 .............................. 68 3.3.10 Region 10 ............................. 70 3.3.11 Region 11 ............................. 70 3.3.12 Region 12 ............................. 73 3.3.13 Region 13 ............................. 73 3.3.14 Region 14 ............................. 74 3.3.15 Region 15 ............................. 75 3.3.16 Region N ............................. 78 3.4 Grain Orientations and Grain Boundaries ................ 79 4 Discussion 85 4.1 Data Selection ............................... 85 4.2 Geometric Factors and Hacture ..................... 86 4.2.1 An existing compatibility factor ................. 86 4.2.2 A new fracture factor ....................... 88 4.2.3 Fmax and its constituents. .................... 90 4.2.4 The factor F ............................ 93 4.2.5 Other expected contributions to F ............... 102 4.2.6 The association between the Fmax plane and fracture ..... 104 5 Conclusions 107 6 References 109 A Mathematica code for calculation of Schmid factors 119 vii B Mathematica code for calculation of {111} plane traces 124 C Mathematica code for calculation of F 128 D Grain SACP information 136 viii 1.1 2.1 3.1 3.2 3.3 3.4 3.4 3.5 3.6 4.1 4.1 4.2 4.2 4.3 4.4 LIST OF TABLES Superlattice scattering for planes with (h2+k2+l2) S 6. ........ 10 Distinguishing characteristics of low-index zone axes in y-TiAl. A Intersections of 2 or more ’narrow’ {111} or {200} bands; B Other readily identifiable zone axes. ...................... 28 Location of the regions on the sample. Sample width is ~3.65 mm, and sample length is ~35 mm. The bend loading points were at ~8 mm and ~28 mm (see Figure 3.5). ........................ 41 Grain 38 orientation, crystal directions relative to the image coordinate system, and Schmid factors for the deformation systems on the (I11)33 plane. ................................... 45 Schmid factors for all deformation systems in grains 21 and 22 assuming uniazial tensile stress state. .................. 64 Grain Orientations ............................ 81 Grain Orientations (continued) ..................... 82 Traces of all the {111} planes for all grains. The angle of the trace is measured in the counterclockwise direction from the positive x-axis. ......................... 83 All Grain Boundaries Considered .................... 84 Fmax and its components as grain boundary fracture .......... 91 Fmax and its components as grain boundary fracture (continued) . . . 92 The maxima of the components of F .................. 94 The maxima of the components of F (continued) ............ 95 Incorporation of grain boundary orientation into F .......... 103 Association Between Fmax and Grain Boundary Microcracks ..... 106 ix 1.1 1.2 1.3 1.4 1.5 1.6 1.7 LIST OF FIGURES Phase diagram of the Ti-Al system. (Adapted from [14-16].) ..... Model of the L10 unit cell of 7-TiAl, including Burgers vectors of the dominant deformation modes. Note the polarity of the b=é <112 mode, and the structural similarity to the FCC structure ........ Deformation directions available to the upper (small dots) (111) plane in TiAl over the lower (large dots) (111) plane. The compositional anisotropy results in different ordinary and superdislocation lattice re- peat vectors, and imposes nearest-neighbor changes on any but one of the % <211>-type shears. (See text.) .................. View of twinning by the passage of 5112] partial (twinning) dislo- cations through the 7-TiAl structure on successive (111) planes. A: Untwinned crystal prior to the passage of partial dislocations shown in the right side of the figure. B: The twinned crystal after the passage of the partial dislocations through the crystal. The twin action shifts the line A-B-C in the untwinned crystal to the line A-B-C in the twinned crystal. The view is along the [1T0] direction normal to both the (111) twin habit plane and the [112] twinning direction. ........... Examples of SACPs a) showing superlattice bands; b) showing the use of superlattice bands for determining orientation. The black line on the inset, c), shows the approximate range of the SACP data shown in b). ..................................... Stereographic octant for the TiAl L10 structure showing the plane traces for which h2+k2+l2 S8. The planes with superlattice structure are shown in light gray (1T0), medium gray {201), and dark gray {112), while the non-superlattice {202) (thin lines), {111} (dashed lines), and the bounding {200} planes are shown without shading. The (001) superlattice plane is not indicated. ................... Projection of the atom positions across the ordered domain interface, viewed along the [II-Oh direction (parallel to the [10mg direction). 11 13 1.8 2.1 2.2 2.3 2.4 2.5 2.6 2.7 Atom positions across the twin boundary interface, viewed in projec- tion along the [1T0]1 direction (parallel to the [I10]2 direction) . . . . The apparatus used for loading the sample in 4-point bending. The hollow cylinder to the right stabilizes the top and bottom punches (to the left), and load is applied to the top punch through the ball bearing. A typical 35 mm 4—point bend sample is shown to the bottom right for scale. A cut-away sketch of the loading is shown in Figure 2.2 ..... Cross sectional line drawing of the 4-point bending apparatus shown in Figure 2.1, with a line drawing of the critical dimensions and loading points of the sample included below. The stabilizing collar is omitted for clarity. Copper foil was used as a buffer between the loading points and the sample ............................... The alignment of the as—deformed 4point bend sample with the mi- croscope axes after mounting to the sample holder (inset). ...... Sample mounted into the sample holder, showing curvature due to 4- point bending ................................ Schematic view of the sample mounted into the sample holder, showing some of the deviations possible between the local sample surface normal 14 19 20 23 23 and the microscope z-axis. The view is down the long axis of the sample. 24 A typical SACP used to determine 'y-TiAl grain orientations. The central ~60% of the SACP comes from the grain of interest, while the rest comes from the surrounding grains. Slight waviness of the band edges is due to localized variations in the grain orientation across the grain surface, while the features labeled ’A’ near the center of the SACP are due to grain surface topography. Note the superlattice band within the band (labeled ’8’) crossing the center of the SACP. . . . . An example of a SACP composite used to determine 'y-TiAl grain ori- entations. Only the portions of the SACPs corresponding to the grain of interest are included in the composite. (The SACP in Figure 2.6 is near the upper right of the composite.) ................. xi 25 26 2.8 2.9 2.10 3.1 3.2 3.3 3.4 3.5 Schematic representation of a SACP composite showing the location of the sample surface normal (n, centered on the SACP taken at 0° sample tilt), the line perpendicular to the tilt axis connecting the centers of all the component SACPS, and various zone axes, marked by + symbols. Various zone axes (Z1-Z4) are labeled, only one of which (Z4) is in the field of view of the composite. All others (Zl-Z3) are located at channeling band intersections that are located off the edge of the composite .................................. Stereographic octant for the TiAl L10 structure showing the plane traces for which h2+k2+l2 38, along with the traces of the planes with superlattice structure. (Gray lines are (1T0), {201), and {112) superlattice traces; thin lines are {202) traces, dotted lines are {111} traces. The bounding {200} planes are shown without shading, and the (001) superlattice plane is not indicated.) ............. Schematic drawing showing how to determine approximate indices for an arbitrary location within a SACP (see section 2.5 in text). . . . . Example of the undeformed microstructure of the Ti-48Al-2Cr-2Nb alloy. The majority of the microstructure is equiaxed 'y, with clusters of irregular a2. .............................. Example of the as-deformed microstructure of the Ti-48Al-2Cr-2Nb alloy. The majority of the microstructure is equiaxed 'y, with clusters of irregular (12. Note the prevalence of deformation twins in the as- deformed deformation structure (see Section 3.2.4.) .......... Examples of cracks in the fractured samples. A: Fracture initiated at the boundary between a lamellar colony (’L’) and a '7 grain (’7’). The crack has propagated into the lower '7 grain. B: Fracture initiated in an a2 grain. The crack has propagated slightly into the neighboring '7 grains. C: Fracture initiated between two 7 grains (circled). D: Fracture initiated at the 7-012 interphase boundary (circled). ..... Sample mounted into the sample holder, showing curvature due to 4- point bending ................................ View of the back of the sample after bending. A: Roughened surface resulting from deformation. The light ’speckled’ appearance is due to specular reflection from individual grains aligned to reflect the light. 8: Sample angled slightly so no specular reflection. C: Sample aligned for specular reflection off the region of the sample indented by the back loading point of the 4-point bend frame (arrowed) ............ xii 27 29 30 34 34 36 37 38 3.6 3.7 3.8 3.9 3.10 3.11 3.12 3.13 3.14 3.15 3.16 Example of a 7-7 grain boundary fracture. A: Backscattered electron image showing the grain boundary and crack. B: Secondary electron image showing better resolution of the crack opening. ......... Example of a 7-02 interphase boundary fracture (circled). A: Backscat- tered electron image showing atomic number and channeling contrast. The grain boundary a2 is bright due to higher atomic number, while the 7 grains above and below have different shades due to channeling contrast. B: Secondary electron image. ................. Backscattered electron image of a region including apparent deforma- tion twins. This is region 11, described further in Section 3.3.11. . . . Atomic force microscopy image of the same region as Figure 3.8. Image shade indicates local elevation; boxed area corresponds to the scan field of Figure 3.11 ................................ Three dimensional projection of the AFM data shown in Figure 3.9. Scale varies across the projection; the Z—axis scale is exaggerated for effect. ................................... Two dimensional projection of elevation from the boxed region of Fig- ure 3.9. The line in the center of the figure indicates the approximate location of the line profile of Figure 3.14. ................ Three dimensional projection of the AFM data shown in Figure 3.11. Scale varies across the projection; the Z-axis scale is exaggerated for effect. ................................... Three dimensional projection of the AFM data shown in Figure 3.11. Scale varies across the projection; the Z-axis is presented at approxi- mate true scale ............................... Line profile taken across the twin-like feature shown in Figure 3.11. Line drawing showing the relationship between the apparent twin ap- parent, the true width, the twin plane inclination, and surface relief due to twinning ............................... Discrepancy between expected twin step and observed twin step ex- plained. Twin shear for a twin of width w would produce a surface relief step shown by the dotted line, however ordinary dislocation shear on these same slip planes produces the opposite sense of surface step, so that the total surface step is that shown by the solid line. ..... xiii 39 39 41 42 43 43 44 44 46 48 49 3.17 3.18 3.19 3.20 3.21 3.22 3.23 3.24 3.25 3.26 3.27 3.28 3.29 The mechanism for bright-dark topographic BSE contrast arising from a surface step under the conditions used for ECCI. When the beam is at position A a portion of the BSE are blocked; when the beam is at position B, there is enhanced BSE emission. .............. Crack opening resulting from twin impingement upon grain boundary. An unconstrained deformation twin in grain 1 (shown schematically by the dotted grain boundary line) is constrained by grain 2 at the grain boundary by traction forces on the twinned grain face. In this example case, the grain boundary experiences a compressive force to the right of the twin and a tensile traction to the left of the twin. Crack opening occurs on the tensile (left) side of the twin, as shown in the inset. Overview of region 1 with the various grains labeled with their ID. The 1—3 grain boundary is fractured. ..................... Topographic image of the 1-3 grain boundary showing the cracks opened at the grain boundary. Note the surface relief caused by de- formation twinning in grain 1 (upper grain), particularly in the upper left of the micrograph. .......................... Backscattered electron image of the same area of the 1-3 grain bound— ary as shown in Figure 3.20, showing the cracks opened at the grain boundary. The deformation system associated with cracking is uncer- tain (see text). .............................. Overview of region 2 with the various grains labeled with their iden- tification numbers. The 7-8 grain boundary is fractured, as shown in greater detail in Figures 3.23 and 3.24. ................. Detail of the 6-8 and 7-8 grain boundaries from region 2. ....... The microcracked 7-8 grain boundary ................... Overview of region 3 with the various grains labeled with their identi- fication numbers. The 10—12 grain boundary is fractured, as shown in greater detail in Figures 3.26 and 3.27. ................. Multiple microcracking of the 10-12 grain boundary ........... Detail of one of the 10-12 grain boundary microcracks showing cracking at the impingement of a deformation twin on the (I11) plane from grain 12. ..................................... Overview of region 4 with the various grains labeled with their ID. The 14-15 grain boundary is fractured ..................... Detail of grain 14 and the fractured 14-15 grain boundary. ...... xiv 51 52 53 53 54 55 56 56 57 58 58 59 59 3.30 3.31 3.32 3.33 3.34 3.35 3.36 3.37 3.38 3.39 3.40 3.41 Overview of region 5 with the various grains labeled with their identi— fication numbers. The 18-19 grain boundary is fractured. ....... Overview of region 6 with the various grains labeled with their ID numbers. The 21-22 grain boundary is fractured, as shown in greater detail in Figure 3.32. ........................... Detail of the fractured 21-22 grain boundary showing the active defor- mation system traces. The channeling contrast conditions for grain 21 are shown .................................. BSE (+left-right) difference image which highlights topographic con- trast while suppressing ECG and atomic number contrast. Surfaces tilted downward towards the left appear bright, while surfaces tilted downward towards the right appear dark. The deformation twins in grain 22 show a local tilt to the left, while the slip driven by them in grain 21 gradually shows a lower surface tilt going into the grain, implying dislocation pile-up within grain 21. .............. Schematic showing how confirmation of the 4-point bend sample to the loading points of the bend frame can result in stress states and material flow significantly different from that which would be predicted by pure bending. In the end view (A), the uneven-surfaced sample (1) deforms when it is pressed against the loading point (2), resulting in shear flow of material to ’fill in the gaps’ between the sample and the loading point. The side view (8), showing one end of the sample with two of the four loading points, shows the location (grey markings) of the end view relative to the rest of the sample. ................. Overview of region 7, for which no orientation information was col- lected due to the small size of the grains above the crack (which are collectively labeled ’25’). ......................... Overview of region 8 which consists of grains 28 through 30. ..... Detail of the 28-29 grain boundary identifying the active deformation traces in grains 28 and 29. ........................ Overview of region 9, consisting of grains 31 through 33. The 32-33 grain boundary is fractured in the circled area .............. View of region 9 under differing channeling contrast conditions as F ig- ure 3.38, highlighting the 31-32 grain boundary. ............ Detail of the 32—33 grain boundary crack ................. Overview of region 10 with grains 34 and 35, and the fractured grain boundary between them. ......................... XV 60 61 62 63 65 67 67 68 69 69 70 71 3.42 3.43 3.44 3.45 3.46 3.47 3.48 3.49 3.50 3.51 3.52 3.53 4.1 4.2 4.3 4.4 The fractured 34-35 grain boundary and the traces of the active defor- mation systems ............................... Region 11, including grains 36 through 38, showing the fractures on the 37-38 grain boundary associated with twin deformation on the (I11)38 plane. ................................... ’Secondary electron’ image showing the opening of the microcracks in the 37-38 grain boundary (circled). ................... Overview of region 12. .......................... Overview of region 13, encompassing grains 40 through 43. ...... Detail of the fractured 40-41 and 40—43 grain boundary ......... Overview of region 14, including the fracture (circled) in the 45—47 grain boundary. .............................. Detail of the 45-47 grain boundary, with labeling of the active defor- mation systems in grains 45 and 47. Strong non-{111} deformation for accommodation of strain at the grain boundary is particularly apparent (see text). ................................. Detail of the fracture circled in Figure 3.49. .............. Overview of region 15 with grains 48—50 labeled. Grain boundary 48—49 is fractured (circled), and the active deformation planes are identified in grains 48 and 49 ............................. Detail of the fractured region of the 48-49 grain boundary. ...... Overview of region N, containing grains n1-n3. Note the (221) defor- mation bands in grain n2 due to deformation transfer from the (1T1) planes in grain n1. ............................ Geometric compatibility factor of Luster and Morris [70] plotted against Schmid factor for each of the two bounding grains. ...... The twin-twin geometric compatibility factor m' plotted against the maximum Schmid factor for deformation twinning in each of the two bounding grains. ............................. The cumulative fraction of the grain boundary population, and the numberoffract uredgrainboundaries totalnumbero f f racturedg‘rai nboundaries ’ running total of both plotted against Fmax- ................................... The displacement at a grain boundary due to an impinging twin (A), and the requisite deformation of grain ’b’ to avoid grain boundary decohesion (B). Note that this figure only addresses deformation by a single grain (’b’) , while the general case typically involves deformation by both grains (’a’ and ’b‘). ....................... xvi 71 72 73 74 75 76 76 77 77 78 79 87 88 93 96 4.5 4.6 D.1 D2 D3 D.4 D5 D6 D7 D8 D9 The components of F and their relationship to physical quantities. . Schematic of deformation transfer from an incident deformation twin (shown by the the array incident twinning dislocations Em, bottom) to ordinary dislocations (Bord, top). This schematic is simplified to only two ordinary dislocation systems in a single grain from 8 systems in both grains. In both cases, however, there is a local accumulation of strain in the grain boundary (shown as the total accumulated strain in the graph at bottom center). Because of the vector mismatch between the the incident and outgoing dislocations, there is an additional resid- ual deformation (R in the vector sum to the right) that resides in the grain boundary. .............................. SACP composite for grain 1. The crosses indicate the center of the SACP at a) 0° tilt and b) ~26° tilt .................... SACP composite for grain 3. The crosses indicate the center of the SACP at a) 0° tilt and b) ~21° tilt .................... SACP composite for grain 4. The crosses indicate the center of the SACP at a) 0° tilt and b) ~24° tilt .................... SACP composite for grain 5. The crosses indicate the center of the SACP at a) 0° tilt and b) ~28° tilt .................... SACP composite for grain 6. The crosses indicate the center of the SACP at a) 0° tilt and b) ~30° tilt .................... SACP composite for grain 7. The crosses indicate the center of the SACP at a) 0° tilt and b) ~30° tilt .................... SACP composite for grain 8. The crosses indicate the center of the SACP at a) 0° tilt and b) ~30° tilt .................... SACP composite for grain 9. The crosses indicate the center of the SACP at a) 0° tilt and b) ~25° tilt .................... SACP composite for grain 10. The crosses indicate the center of the SACP at a) 0° tilt and b) ~25° tilt .................... D.10 SACP composite for grain 11. The crosses indicate the center of the SACP at a) 0° tilt and b) ~30° tilt .................... D.11 SACP composite for grain 12. The crosses indicate the center of the SACP at a) 0° tilt and b) ~30° tilt .................... D.12 SACP composite for grain 13. The crosses indicate the center of the SACP at a) 0° tilt and b) ~35° tilt .................... D.13 SACP composite for grain 14. The crosses indicate the center of the SACP at a) 0° tilt and b) ~33° tilt .................... xvii 101 137 138 139 140 141 142 143 144 145 146 147 148 149 D.14 SACP composite for grain 15. The crosses indicate the center of the SACP at a) 0° tilt and b) ~35° tilt .................... D.15 SACP composite for grain 16. The crosses indicate the center of the SACP at a) 0° tilt and b) ~25° tilt .................... D.16 SACP composite for grain 17. The crosses indicate the center of the SACP at a) 0° tilt and b) ~30° tilt .................... D.17 SACP composite for grain 18. The crosses indicate the center of the SACP at a) 0° tilt and b) ~33° tilt .................... D.18 SACP composite for grain 19. The crosses indicate the center of the SACP at a) 0° tilt and b) ~25° tilt .................... D.19 SACP composite for grain 20. The crosses indicate the center of the SACP at a) 0° tilt and b) ~33° tilt .................... D20 SACP composite for grain 21. The crosses indicate the center of the SACP at a) 0° tilt and b) ~25° tilt .................... D.21 SACP composite for grain 22. The crosses indicate the center of the SACP at a) 0° tilt and b) ~25° tilt .................... D.22 SACP composite for grain 23. The crosses indicate the center of the SACP at a) 0° tilt and b) ~25° tilt .................... D.23 SACP composite for grain 28. The crosses indicate the center of the SACP at a) 0° tilt and b) ~33° tilt .................... D.24 SACP composite for grain 29. The crosses indicate the center of the SACP at a) 0° tilt and b) ~25° tilt .................... D.25 SACP composite for grain 30. The crosses indicate the center of the SACP at a) 0° tilt and b) ~25° tilt .................... D.26 SACP composite for grain 31. The crosses indicate the center of the SACP at a) 0° tilt and b) ~25° tilt .................... D.27 SACP composite for grain 32. The crosses indicate the center of the SACP at a) 0° tilt and b) ~30° tilt .................... D.28 SACP composite for grain 33. The crosses indicate the center of the SACP at a) 0° tilt and b) ~30° tilt .................... D.29 SACP composite for grain 34. The crosses indicate the center of the SACP at a) 0° tilt and b) ~28° tilt .................... D.30 SACP composite for grain 35. The crosses indicate the center of the SACP at a) 0° tilt and b) ~29° tilt .................... D.31 SACP composite for grain 36. The crosses indicate the center of the SACP at a) 0° tilt and b) ~25° tilt. . . . ............... xviii 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 D.32 SACP composite for grain 37. The crosses indicate the center of the SACP at a) 0° tilt and b) ~25° tilt .................... D.33 SACP composite for grain 38. The crosses indicate the center of the SACP at a) 0° tilt and b) ~26° tilt .................... D.34 SACP composite for grain 39. The crosses indicate the center of the SACP at a) 0° tilt and b) ~27° tilt .................... D.35 SACP composite for grain 39a. The crosses indicate the center of the SACP at a) 0° tilt and b) ~28.5° tilt ................... D.36 SACP composite for grain 40. The crosses indicate the center of the SACP at a) 0° tilt and b) ~28° tilt .................... D.37 SACP composite for grain 41. The crosses indicate the center of the SACP at a) 0° tilt and b) ~25° tilt .................... D.38 SACP composite for grain 42. The crosses indicate the center of the SACP at a) 0° tilt and b) ~30° tilt .................... D.39 SACP composite for grain 43. The crosses indicate the center of the SACP at a) 0° tilt and b) ~29° tilt .................... D.40 SACP composite for grain 44. The crosses indicate the center of the SACP at a) 0° tilt and b) ~35° tilt .................... D.41 SACP composite for grain 45. The crosses indicate the center of the SACP at a) 0° tilt and b) ~30° tilt .................... D.42 SACP composite for grain 46. The crosses indicate the center of the SACP at a) 0° tilt and b) ~25° tilt .................... D.43 SACP composite for grain 47. The crosses indicate the center of the SACP at a) 0° tilt and b) ~25° tilt .................... D.44 SACP composite for grain 48. The crosses indicate the center of the SACP at a) 0° tilt and b) ~25° tilt .................... D.45 SACP composite for grain 49. The crosses indicate the center of the SACP at a) 0° tilt and b) ~25° tilt .................... D.46 SACP composite for grain 50. The crosses indicate the center of the SACP at a) 0° tilt and b) ~25° tilt .................... D.47 SACP composite for grain ml. The crosses indicate the center of the SACP at a) 0° tilt and b) ~20° tilt .................... D.48 SACP composite for grain n2. The crosses indicate the center of the SACP at a) 0° tilt and b) ~25° tilt .................... D.49 SACP composite for grain n3. The crosses indicate the center of the SACP at a) 0° tilt and b) ~27° tilt .................... xix 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 D.50 SACP composite for grain n4. The crosses indicate the center of the SACP at a) 0° tilt and b) ~25° tilt .................... 186 D.51 SACP composite for grain n5. The crosses indicate the center of the SACP at a) 0° tilt and b) ~25° tilt .................... 187 SYMBOLS AND ABBREVIATIONS Em The Burgers’ vector for deformation twinning, one of four % <112 for each grain 51w The unit vector in the direction of the Burgers’ vector for deformation twinning bad The Burgers’ vector for ordinary dislocation slip Bord The unit vectors corresponding to the directions of each of the Burgers’ vectors for ordinary dislocation Slip 5 The slip plane normal direction .3 The slip plane unit normal vector 2 The unit vector describing the line of intersection of the slip plane with the surface plane mtw The Schmid factor for deformation twinning The grain surface normal expressed in terms of the grain direction The unit vector grain surface normal The grain tilt axis, corresponding to the positive x-axis of all microscopy images. The grain tilt axis unit vector The unit vector discribing the tensile direction The tensile direction in the crystal coordinate system The grain boundary unit normal vector A 2 -[/ a!) H" "I 3) J] The angle between two vectors, so that ALB = ’the angle between directions A and B’ F The grain boundary fracture factor; F: (m,.u,)|(b,w - t)| Ed |(b,u, - bo,.d)| xxi ip Electron probe current ,a Backscattered electron yield; the fraction of electrons backscattered from a sample relative to the scanning beam ECCI Electron Channelling Contrast Imaging or Electron Channeling Contrast Im- age ECC Electron Channeling Contrast SEM Scanning Electron Microscope SACP Selected Area Channeling Pattern BSE Backscattered Electron HIP Hot Isostatic Press or Hot Isostatic Pressing xxii CHAPTER 1 Introduction 1.1 TiAl—Based Intermetallic Alloys Engineering materials using alloys based on the 7-TiAl intermetallic phase composi- tion have been under consideration, development, and early application for a number of years (see for example [1-4]). The low density of the 7 structure, comparative abundance of deformation systems relative to many other intermetallic compounds, convenient phase transformations available [5—7] (see Figure 1.1), and environmental Temperature / ‘C Al 17 T I r I T 11102030405060? 8090A| at.%A| Figure 1.1. Phase diagram of the Ti-Al system. (Adapted from [14-16].) passivity all make this a potentially attractive structural phase [4,8—11]. A major limitation on the use of 7-TiAl-based materials is their low fracture toughness with consequent limited tensile ductility. (See for example [12,13].) TiAl has the L10 structure (Figure 1.2), a tetragonal structure reminiscent of the face centered cubic (FCC) structure, which consists of alternating (001) planes of Ti and Al. The resultant tetragonality (az4.00A, cz4.07A) and inhomogene— ity of the {111} planes (which consist of alternating rows of close-packed Al or Ti atoms along the <110]1directions) alters the {111} <110> dislocation defor- mation behavior found in the FCC system. Easy dislocation slip occurs on the {111} planes via the motion of -21- <110] dislocations, while the corresponding {111}<101] slip is more difficult due to the larger Burgers vector of the defect [8]. Slip of the E=<101] dislocations is usually achieved through the dissociation: <101]—+ ;— <101]+APB+§ <101] (where APB is an antiphase boundary), followed by glide of the dissociated Ezé <101] superpartial dislocations. One other slip system discribed in the literature is {111}% <112], which also represents a ’hard’ slip mechanism [8]. Twinning of the type {111}<112 is also present, and represents an easy deformation mode [8]. Alloys based on the 7-TiAl composition can be classified into two distinct cate- gories: the Al-rich above ~50 X, 23, which consist of single-phase 7, and the two-phase a2+7 alloys at lower Al compositions. The two—phase alloys tend to have greater 7 ductility due to the gettering effect of the (12 phase, which has a greater solubility for 7-embrittling impurities such as N, O, and C [17—19]. There is mounting evidence 1Mixed indices ( [ >, { ) ) are used throughout to indicate the non-equivalence of certain families of indices in the tetragonal TiAl crystal structure. 2 All compositions will be related in terms of atomic percent ( X, ) unless noted, in keeping with the literature. 3 Oxygen has a strong effect on the location of the 02 +7 - 7 phase boundary [18]. The addition of 5x10‘4 weight fraction oxygen shifts the phase boundary from ~48 % to ~52 X, at 800°C [19], so the exact location of the two phase region remains approximate depending on trace impurities. omec - 1 b=-2'<110] \. Figure 1.2. Model of the L10 unit cell of 7-TiAl, including Burgers vectors of the dominant deformation modes. Note the polarity of the b=~é <112 mode, and the structural similarity to the FCC structure. that the 7 structure undergoes ordering, forming Alng domains upon cooling for Al compositions greater than ~56 % Al [20-23]. This bounds the upper compositional range of the 7 structure at low temperatures. 1.2 Deformation Behavior of 7—TiAl The L10 structure of 7-TiAl causes an inhomogeneity on the {111} planes, and this alters the deformation systems available on these planes. Figure 1.3 shows two con- secutive (111) planes and the shear vectors of the available deformation systems. The shortest lattice repeat vectors in the [10D and [01:1] directions (the superdislocations) are approximately twice the length of the shortest lattice repeat vector in the [1T0] direction. The varying composition of the {111} planes also strongly limits the choice of twin shear; the twin (to A in Figure 1.3) results in the same nearest-neighbor en- vironment as the unsheared crystal, while either of the other two <211>-type shears (to B or C in Figure 1.3) results in a change to the nearest-neighbor environment. o 0. O . O ordTnary dislocation O O O O o in o o o o superdislocation u o o o o Figure 1.3. Deformation directions available to the upper (small dots) (111) plane in TiAl over the lower (large dots) (111) plane. The compositional anisotropy results in different ordinary and superdislocation lattice repeat vectors, and imposes nearest- neighbor changes on any but one of the % <211>-type shears. (See text.) Room temperature deformation of 7-TiAl in the Al-poor alloys occurs through the motion of $- <110] ordinary dislocations, % %[I10]+[101] [8]. Deformation twinning occurs through the motion of % <112] partial (twinning) dis- locations on successive {111} planes, as indicated in Figure 1.4. Because of the inher- ently polar nature of twinning, deformation twinning is restricted to the (111)% [112], I(11)% [I12,(1I1) g[1I2], and (H1); [I I2 deformation systems (or their (h k8) Fls'l“ vW] counterparts, (fiI)%[fi2], (1-1_1)%[1I2], (I1I)%[I12], and (11I)%[112]). These defor- mation systems will be referred to from now on by their (111)%[112, (I11)é—[I1Z, (1I1)[-1m, and (111)% [II2] variants. 1.2.1 Alloying of the 7-TiAl alloys Addition of Cr, Mn, and V have been found to improve ductility in the Al—poor 7 [8,27] phase. The addition of Cr appears to reduce 70335 (the critical resolved shear stress) for % <110] dislocation motion, while Mn and V appear to reduce 70353 for % <110] dislocations and the twin-boundary energy [27], both of which ease de- formation twinning [28]. Additionally, other elements have been added to improve characteristics such as oxidation resistance (Nb [2,8,11,29], Ta [8], Mo [10], Zr [29]), high temperature strength (Nb [30]), and creep resistance (Si [31-35], W [36], C[35]). These solutes often tend to favor residence on either the Al or the Ti sublattices. Cr appears to strongly partition to the Al sublattice in 7, making it essentially an Al substitute; Nb in turn strongly partitions to the Ti sublattice [37], making the addition of equal parts Cr and Nb to the 50 Ti-50 Al alloy an attractive choice for a more ductile, corrosion-resistant base alloy. In general, the Al-deficient alloys are heat-treatable by heating and cooling through various combinations of the 7, a, and (12 single— and multi-phase regions of the phase diagram. Different heating/ quenching/ annealing schedules can be used to produce fully (7+a2) lamellar, equiaxed 7 (containing small amounts of a2), and ’duplex’ equiaxed 7+lamellar 7+ag microstructures. Multiple heat treatments have 5 A .o.o.o.o.o.o.o.o.o.o.o.o.o A B 0.0.0.0...o.o.o.o.o.o.o.o. coco-00000.0. .__L 00000000000000 .__L coo-00000000000 *_L coco-00000000000 .__L 0.000000000000000 .__L oooooooooooooooooo «J- ooooooooooooooooooo .__L ooooooooooooooooooo .__L 00.......O.O.C..'0'..O.O.O.....O...O...O. A .0.0.0...0.0.0'0'..0.0.0.0...O.......O.C B B 0.0.0.0.....0.0.0.0.0.0.0.....0.0.0.0.0. io.o.o.o.o.o’0'..o.o.o.o.o.o.o.o.o.o.o.o.0. _L coo-00.000000000000- f Figure 1.4. View of twinning by the passage of 5112 partial (twinning) dislocations through the 7-TiAl structure on successive (111) planes. A: Untwinned crystal prior to the passage of partial dislocations shown in the right side of the figure. B: The twinned crystal after the passage of the partial dislocations through the crystal. The twin action shifts the line A-B-C in the untwinned crystal to the line A-B—C in the twinned crystal. The view is along the [1I0] direction normal to both the (111) twin habit plane and the [112 twinning direction. even been demonstrated as a practical means of refining grain sizes [5,7]. 1.3 Electron Channeling Electron channeling contrast (ECC) arises from the dependence of backscattered elec- tron (BSE) yield on the electron trajectory relative to the crystal orientation. First noted by Coates [38] and explained by Booker, et al. [39], electron channeling is characterized by a strong change in the backscatter yield (17) at electron trajectories close to the Bragg angles of the atomic planes of the crystal. It was immediately recognized that electron channeling contrast could be used to determine crystal orientation [38,39], and this idea was subsequently refined to include selected area channeling methods to obtain the same information for microscopic grain sizes. Booker, et al. quickly predicted that ECC could be used as a contrast mecha- nism in SEM imaging to see dislocations and other defects [39]. However it was not until a decade later that successful electron channeling contrast imaging (ECCI) was demonstrated [40], and even then only under arduous experimental conditions. The application of digital image manipulation [41] simplified the experimental conditions somewhat in the early 19905, followed by further simplification of the experimental procedure and near-routine application by the mid 19905 [42-52]. The current experi- mental constraints on ECCI require a high brightness electron gun, efficient collection of the backscattered electron signal, and good signal-to—noise in the collected signal. This last requirement usually requires some sort of time averaging of the BSE image and a substantial beam current [53]. Selected area channeling patterns (SACP) [54] (Figure 1.5) display crystal orien- tation information for individual grains using channeling contrast. SACPs are formed by rocking an electron beam across the surface of a grain and plotting the backscat- tered electron signal as a function of the rocking angle. The centers of the bands in a SACP correspond to the traces of planes within the crystal, and the band widths are twice the Bragg angle for those planes [54,55]. Because electron channeling is sen- sitive to ’diffraction’ information, SACP has the potential for detecting superlattice information (see section 1.4 below). Because the contrast from electron channeling is due to the crystal orientation relative to the electron beam, it can also provide information on local crystal distortion near the crystal surface. Use of this contrast mechanism forms the basis for ECCI, [40,42,43,56], which allows near-surface dislocations and crystal imperfections to be imaged in bulk crystals. 1.4 Electron Scattering Effects Due to the Super- lattice Structure The L10 structure of TiAl is a superlattice structure, and as such, the electron scatter- ing and diffraction behavior from this structure is more complex than would occur for a structurally identical monatomic crystal. Most importantly for grain orientation identification purposes, the alternating (001) layers of Ti and Al cause incomplete cancellation (extinction) of diffraction from (001) planes, because the magnitude of the scattered intensity from each (001)Ti plane family is not the same as the scattered intensity from the (001)A| plane family. Thus the observed intensity is non-zero, even though the waveforms scattered from the two plane families are exactly out of phase at the (001) Bragg angle [54,55,57]. This kind of incomplete destructive interference is called ’superlattice’ scattering, and in general occurs for any set of crystallographic planes for which the alternating nearest-neighbor planes do not have identical average chemical compositions [57,58]. For the Llo-TiAl structure, superlattice scattering occurs for the low-index planes Figure 1.5. Examples of SACPs a) showing superlattice bands; b) showing the use of superlattice bands for determining orientation. The black line on the inset, c), shows the approximate range of the SACP data shown in b). listed in Table 1. Because these superlattice bands (shown in in the stereographic projection of F ig- ure 1.6) have a wide distribution, they effectively act as ’markers’ for the orientation of a SACP within the possible orientation space for 7—TiAl. As a consequence, it is possible to determine the orientation of a crystal of 7-TiAl using a series of SACPs taken over a range of crystal tilts covering roughly ~30° [51]. Table 1.1. Superlattice scattering for planes with (h2+k2+12) S 6. Plane family superlattice planes {100} (001) {110} (110), (1I0) {111} - {210} (201),(§01),(021), 0 IT .1 {211} (112),( 1.5 7—7 Deformation Transfer The general case of deformation transfer across 7-7 interfaces in TiAl has been stud- ied for a few special cases that correspond to pure twist boundaries about the {111} plane normal [60,61]. The specific case of transfer of % <11? deformation twinning strain across general 7-7 grain boundaries has also been studied [59]. The {111} twist boundaries are reasonably common because of the three possible a[[7 phase ori- entation relationships that result from the a -—> a2+7 phase transformation. This transformation occurs upon cooling of alloys with ~38 X, ~52 X, Al. The three pos- sible domain boundaries between the three possible 7 phase orientations correspond to twist boundaries of multiples of 60° about the {111} surface normal. These high- 10 010 130 120 141 110 0310 "’, 021‘ I 121’ r 13 ’ / ,I’ I 210 011. / ‘x. '11 310 1 ‘2 / 311 012 113 . ' 411 013 “114 x 31 \ I \ [I 001 105 102 101 201 £01 100 Figure 1.6. Stereographic octant for the TiAl L10 structure showing the plane traces for which h2+k2+l2 S8. The planes with superlattice structure are shown in light gray (110), medium gray {201), and dark gray {112), while the non-superlattice {202) (thin lines), {111} (dashed lines), and the bounding {200} planes are shown without shading. The (001) superlattice plane is not indicated. 11 symmetry cases are designated ’pseudo—twin’ for a 60° rotation, ’ordered domain’ for a 120° rotation, and ’twin’ for a 180° rotation. These domain boundaries commonly occur either as the result of deformation twinning (the twin boundary), or as bound- aries between variants of the 7 phase nucleated from the same a grain. (This latter case describes essentially all the 7-7 orientation relationships in the lamellar 012-7 microstructure.) The twin-domain boundary seems to be the lowest energy interface of the three, and thus more common [8]. 1.5.1 Deformation transfer across special 7-7 boundaries Of the three {111} 60xn° rotational domain interface types mentioned above, defor- mation transfer across the ordered domain and twin interfaces has been studied in detail, and is described below. The ordered domain interface Deformation transfer across the ordered domain interface (Figure 1.7) has been ob- served to occur by three distinct deformation modes [60]. _1 In the first case, ordinary 31—5 <110]-type dislocations impinging on the interface tend to cause emission of 522% <112 Shockley partials across the domain boundary, with sessile 522% <121> or Egzé <212> dislocations left as debris at the interface. Emission of these partials occurs on the {111} plane that allows the E=é <112] Shockley partial to trail an intrinsic stacking fault. This {111} plane is not necessarily the {111} plane with the highest Schmid factor. If no {111} plane is available, these dislocations will be pinned at the interface [60]. In the second case, multiple 52% <112] Shockley partial dislocations impinging on the interface induce emission of 52% <110] and E=<101] dislocations on the {111} plane that is the projection of the initial slip plane across the interface. These 12 X I o. o. o o. x‘ 1* ++ x‘ . . 0° 0 .o‘ o. o'.o*++;xx" +:,,x"++: )l altomatzng T1, Al O . + o calm 1'1. O°OO°oo o +* ++ x O O 0 o 0 ++ a." ++ at" + . o o o o o o + n + x + - o' o o o' o + n + x + + alternaunq A1 T1 0 com-n A]. 3.0;:o3203.03203:x::+:;x::+::x:: ' o + ,. I + + + :g kth‘ 2 ‘1); 11" Figure 1.7. Projection of the atom positions across the ordered domain interface, viewed along the [1I0]1 direction (parallel to the [10fl2 direction). E=<101] dislocations tend to move rapidly after being emitted, followed by pinning through an unknown mechanism close to the interface [60]. In the third case, individual Shockley partials impinging the ordered domain boundary commonly cause emission of individual Shockley partials in the adjacent domain [60]. The twin interface Because the crystal lattices on opposite sides of a twin interface share a common < 110] crystal orientation across the twin interface, (see Figure 1.8, 180° rotational), some limited % <110] deformation transfer through this interface is possible without the pinning and re—nucleation that is common to deformation transfer across the ordered domain interface. 1 Ordinary (ID-=5 <110]-type) dislocations will easily cross slip from the parent to 13 ° 0 a .0 00 0.0. .000 a column '1'; 0° 0° 00000000000000.000000020 o o o o o . a calm A]. 3.00 o. o. o. o. .o 00.00.32.00. (xgjils Figure 1.8. Atom positions across the twin boundary interface, viewed in projection along the [1I0]1 direction (parallel to the [I10]2 direction) the twinned crystal if they are pure screw dislocations, where both 5 and the line direction lie in the interface plane. Dislocations with an edge component are pinned at the twin boundary [61]. Other shear transmission through the twin interface occurs by the production of twinning dislocations on the mirror plane across the twin interface, with the simulta- neous production of ordinary <110] dislocations within the interface [61]. In general, deformation transfer across twin (180° rotational) interfaces has been found to be easier than across ordered domain (120° rotational) interfaces [61]. 1.5.2 Twin deformation transfer across the general 7-7 boundary Gibson and Forwood [59] have addressed the early stages of twin strain transfer across more general 7-7 grain boundaries. In their study of two non-high-symmetry 14 7-7 boundaries, they observed successful strain accommodation at the interface by the emission of % <110]-type dislocations into a neighboring grain on {11x}4 planes, simultaneous with the emission of ’refiected’ i; <110] dislocations on {11y} planes in the parent crystal. In general, this process ends up leaving residual deformation shear at the grain boundary, which accommodates any residual c-axis component of the twinning shear. Gibson and Forwood did not observe any grain boundary accommodation via the emission of <101] superdislocations, even for the case where minimal strain transfer was observed via % <110] dislocations. This contrasts with the twin deformation transfer behavior reported by Zghal, et al. [60] for twin deformation transfer across the ordered domain ({111} 120° rotational) interface, where the emission of <101] superdislocations was reported in conjunction with ordinary % <110] dislocations. However, since superdislocations in that case were reported to slip only a short distance from the grain boundary, it is likely that the emission of superdislocations from the ordered domain interface is a special case. The persistency of these <101] dislocations near the interface may then be regarded as similar to the residual strain accumulation at the interface noted by Gibson and Forwood [59]. In general then, deformation twins incident on a 7-7 grain boundary are accom- modated via % <110] ordinary dislocations, leaving some residual deformation shear at [59] or near [60] the grain boundary in order to accommodate any residual c-axis component of the twinning shear [59]. 4 The use of {11x} is the author’s notation intended to indicate that all the slip planes were of some multiple of {11x}, where x varied between about 0.087 and 6 in Gibson and Forwood’s original work. 15 1.6 Grain Boundary Effects on Fracture The propensity of a grain boundary to fracture is related in part to the grain boundary energy, as noted by Watanabe and others [62-65]. In all these treatments, the grain boundary fracture stress is shown to be lower for grain boundaries with a greater sense of disorder, which results in a higher grain boundary energy. Conversely, grain bound- aries that are more strongly ordered (as low-angle dislocation array boundaries, or low-2 coincidence site lattice boundaries) also have a higher inherent fracture stress. This preference for high-energy grain boundary fracture is seen for both monolithic solids (Mo, Fe-6.5% Si, PbZrOngTiOg) [62,63] as well as multiphase (Fe-0.8% Sn, Zn-liquid) systems [62]. One effect of this grain boundary misorientation dependence on fracture strength is macroscopic component fracture. This fracture is influenced not only on the con- centration of lower-strength grain boundaries, but also upon their interconnectivity: not only do the low-strength boundaries tend to fracture at lower macroscopic load- ings, but also interconnected low-strength boundaries tend to form interconnected grain boundary cracks, which hasten component fracture [62,63]. This dependence of grain boundary fracture on grain boundary misorientation and interconnectedness has led to the development of ’grain boundary engineering’, which attempts to control the macroscopic fracture properties by thermal, mechanical, and thermomechanical treatments to control grain boundary misorientation distributions [63-65]. For solid-phase materials, this grain boundary energy approach to grain boundary fracture has thus far only been applied to materials that are relatively isotropically ductile, either because of a cubic crystal structure (Mo, Fe-6.5% Si, Fe-0.8 X, Sn), or because of a lack of slip deformation systems (PbZrO3PbTiOg). In other words, the deformation environment experienced by all grain boundaries is similar, so that it 16 is the grain boundary energy term that dominates grain boundary fracture. This is in contrast to 7-TiAl, which has multiple, strongly dissimilar deformation systems, so that local strain energies resulting from these deformations can dominate grain boundary fracture. 1.7 Overview This dissertation concerns the origin of fracture in 7 titanium aluminide based alloys. The principal fracture initiation sites are 7-7 grain boundaries, and are associated with deformation twinning in 7-TiAl grains. Scanning electron microscopic methods are used to determine grain orientations and to identify deformation processes, and relate these to fracture behavior. Atomic force microscopy is used to confirm deformation behavior. It is found that there is a correlation between grain boundary fracture and a factor describing deformation twinning, the conformability of the grain boundary to deformation twinning, and the twinning direction. This factor is interpreted and suggestions are made for improvements toward the ultimate goal of developing a grain boundary fracture prediction model. This work ultimately has implications for the design of 7-TiAl based alloys, and in addition, the methods employed here also open new ways of examining grain boundary fracture initiation in materials that deform through deformation twinning. The grain boundary fracture criterion presented (based on local deformation phenomina), is complementary to other grain boundary fracture initiation criteria that are based on grain boundary energy. 17 CHAPTER 2 Experimental Procedure 2. 1 Sample Preparation 2. 1. 1 Material The material used for these studies was provided by Howmet Corporation, and had a composition of Ti—47.9Al—2Cr-2Nb. The as-received material was an investment cast ingot which had been heat treated at 1364K for 5 hours, followed by hot isostatic pressing (HIPing) at 1376 K for 4 hours, with an additional 2 hours at 1376 K followed by rapid cooling [66]. 2.1.2 Bend test samples Samples of nominal dimensions 2mm x 4mm x 35mm were prepared for 4—point bending experiments by cutting with a high speed diamond wafering saw from the heat-treated ingot, followed by mechanical grinding using 240, 320, 400, 600, and 2000 grit SiC fixed abrasive papers, and then electropolished in a solution of 350ml methanol+175ml 1-butanol+30ml perchloric acid at -50i5°C and 30V for approxi- mately 10-15 minutes. Uniform electropolishing along the length of the samples was achieved by completely immersing the samples in the electrolyte, using tungsten rods 18 to grip the sample ends. 2.2 Sample Loading The electropolished samples were subjected to four point bending by loading in the apparatus shown in Figures 2.1 and 2.2. Multiple samples were loaded, although Figure 2.1. The apparatus used for loading the sample in 4-point bending. The hollow cylinder to the right stabilizes the top and bottom punches (to the left), and load is applied to the top punch through the ball bearing. A typical 351nm 4—point bend sample is shown to the bottom right for scale. A cut-away sketch of the loading is shown in Figure 2.2. only one sample was successfully halted subsequent to fracture initiation, but prior to catastrophic fracture. Because it is the unfractured sample that is of interest for fracture initiation studies, all subsequent treatment will deal principally with the unfractured sample. Samples were typically loaded at a crosshead deflection of 0.1mm/minute, al- though one initial sample was loaded at a crosshead deflection of 0.01 mm/minute. 19 H l\\\ top punch NWT L—l/sample base ~— 35mm T" F ”ml [L 30mm J>V 4mm Figure 2.2. Cross sectional line drawing of the 4-point bending apparatus shown in Figure 2.1, with a line drawing of the critical dimensions and loading points of the sample included below. The stabilizing collar is omitted for clarity. Copper foil was used as a buffer between the loading points and the sample. 20 All samples except one were loaded until catastrophic fracture. The sample that was not loaded to fracture was loaded until past yielding (shown by the curvature of the bar), but before fracture, and then unloaded. After unloading, the sample tensile surface was inspected via SEM for microcracking (of which none was observed), then loaded further in the same 4-point bending configuration, followed by unloading and re—inspected. Upon this second inspection microcracking was found. The approximate surface plastic strain at which microcracking was observed was ~1.4%, as determined by the radius of curvature of the unloaded sample (see Figure 3.4 in Section 3.2). 2.3 Electron Microscopy All sample observations were carried out using a CamScan 44F E scanning electron microscope with a Schottky field emission electron gun and the selected area channel- ing module for selected area channeling. Backscattered electron (BSE) imaging was accomplished using the signal from a polepiece—mounted, four quadrant, silicon diode type BSE detector. (Note that ECCI is a form of BSE imaging.) ’Secondary elec- tron’ imaging used was accomplished using the signal from a positionable Everheart— Thornley (ET) type detector which is sensitive to both secondary and backscattered electrons. All microscopy was carried out at a working distance of 12mm. Typical probe currents were on the order of 6nA-25 nA, with typical probe sizes on the order of 10100 11111 and beam convergence angles of ~42 2a Z~10 mrad. All SEM images (including both SACP and conventional scanning images) were captured and integrated over multiple frames by the framestore integral to the Cam- Scan 44FE SEM. These were then captured as images by an external personal com- puter with a frame grabber card reading the microscope television output signal, and stored as 640x480 pixel digital images. 21 Cracks were identified by scrutinizing the tensile surface of the sample in alter- nating BSE and ET image modes over an approximate 300 am scan field. Areas that appeared to be cracked in both image modes were later examined at much higher magnifications to verify any crack. This method of crack searching is expected to sig- nificantly undercount small cracks and cracks with minimal crack opening, because the crack contrast for both imaging methods is dependent on crack size and crack gape. 2.4 Sample Loading into SEM The previously bent (but not fully fractured) 4—point bend sample bar was loaded into the microscope and aligned as shown in Figure 2.3, with the sample surface normal h approximately parallel to the microscope beam axis (2), the sample width w parallel to the microscope stage x axis, and the sample long axis 1 parallel to the stage y direction. The alignment accuracy of the sample in the holder is inherently limited by a number of factors, which fall into two broad categories: sample clamping effects, and sample shape effects. In addition to these, grain surface normal measurements (determined by assuming (microscope-z)=(sample surface normal)) will tend to be somewhat inaccurate because the sample surface is rounded by the electropolishing process. The sample clamping errors arise from small tilts between the sample axes and the sample holder axes, due to small tilts between the two faces of the sample holder clamp as illustrated in Figure 2.5. This tilt error shows up primerally as a rotation of the sample about the sample l-axis. The sample shape errors are dominated by the bend in the sample along its length, shown in Figure 2.4, which produces a variance of both the the sample surface normal 22 £2 a. w E x hsample m z '0 . axes 5 microscope .o axes; - 4.. . .E O 9 v Figure 2.3. The alignment of the as-deformed 4-point bend sample with the micro- scope axes after mounting to the sample holder (inset). Figure 2.4. Sample mounted into the sample holder, showing curvature due to 4—point bending. 23 Electropolish rounded Sample axes corners [ w l AZ Y Holder axes Figure 2.5. Schematic view of the sample mounted into the sample holder, showing some of the deviations possible between the local sample surface normal and the microscope z-axis. The view is down the long axis of the sample. and the orientation of the local tensile direction along the length of the sample. In addition to rounding the sample corners, electropolishing also caused localized undulations in the sample surface, caused by differences in the electrolyte flow rate around different regions of the sample. Some of these can be seen in the inset of Figure 2.3, at the right most end of the sample. The sample was kept attached to the same sample holder during all observations. The holder with sample was mounted into and removed from the microscope between observations. Prior to each observation, the sample long axis was aligned parallel to the stage y axis using the stage rotation axis, and the microscope y—scan direction was in turn aligned parallel to the sample length using the scan coil rotation control. This ensured that variations in grain orientation determinations between microscopy runs were minimized, and any errors in true grain orientation determination were consistent for a given region of the sample. 24 2.5 Grain Orientation Determination Individual grain orientations were determined by collecting a series of SACPS (such as the example shown in Figure 2.6 from each grain at a variety of sample tilts about Figure 2.6. A typical SACP used to determine 7—TiAl grain orientations. The central ~60% of the SACP comes from the grain of interest, while the rest comes from the surrounding grains. Slight waviness of the band edges is due to localized variations in the grain orientation across the grain surface, while the features labeled ’A’ near the center of the SACP are due to grain surface topography. Note the superlattice band within the band (labeled ’B’) crossing the center of the SACP. the microscope x—axis (corresponding to the sample w-axis). These individual SACPS were typically collected at 5° increments, and combined overlapping into a single SACP composite map (examples shown in Figure 2.7 and 1.5). An SACP composite produced in this way has the sample normal direction at the center of the SACP collected at zero sample tilt; the normal to the tilt axis (and thus the sample w-axis) 25 Figure 2.7. An example of a SACP composite used to determine 7-TiAl grain orien- tations. Only the portions of the SACPs corresponding to the grain of interest are included in the composite. (The SACP in Figure 2.6 is near the upper right of the composite.) 26 of such a composite runs through the center of all the component SACPs, forming a line through the center of the composite (shown schematically in Figure 2.8). ‘ I ,flt . 7:1 CA . . ' 22 Normal to tilt axis superlattice band \ Figure 2.8. Schematic representation of a SACP composite showing the location of the sample surface normal (11, centered on the SACP taken at 0° sample tilt), the line perpendicular to the tilt axis connecting the centers of all the component SACPs, and various zone axes, marked by + symbols. Various zone axes (Zl-Z4) are labeled, only one of which (Z4) is in the field of view of the composite. All others (Z1—Z3) are located at channeling band intersections that are located off the edge of the composite. Bands within the SACP composite were identified through a combination of 1) relative band width, 2) inter-band angles, 3) the presence or absence of inscribed superlattice bands, and 4) the relative placement of bands and identified zone axes. 27 Typically this was done using the following procedure: 0 Note any ’narrow’ bands, which correspond to {111} or {200} bands. 0 Note any superlattice bands (which correspond to (1I0), (I12), (I12), (1I2), (201), or (021) bands). 0 Determine possible low index zone axes based on band types and inter-band angles. (See Figure 2.9 and Table 2.1). 0 Determine other (higher index) zone axes based on their proximity to already- identified bands and zone axes. 0 Check for consistency between the tentative identifications. Table 2.1. Distinguishing characteristics of low-index zone axes in 7-TiAl. A Inter- sections of 2 or more ’narrow’ {111} or {200} bands; B Other readily identifiable zone axes. Zone axis #’narrow’ ’narrow’ distinguishing superlattice bands bands interband angles A [100: 2 90° two {021) bands [001: 2 90° two {110) bands [101: 3 ~71°, ~55° none [110: 3 ~71°, ~55° two {112) bands, one {110) band Zone axis Distinguishing characteristics [111] Three bands at 60° to each other, one of which has a {110) super- lattice band and a {112) band .1. to the {110) band. <112> One narrow band 1 to {220} band, with two {113} bands ~31° to either side of the {220} band. The [112] zone axis has a { 110) B superlattice band within the {220} band, and two {021) superlattice bands. <512] Two broad superlattice bands ( {021) and {112) ), 90° to each other, and two {113} bands ~85° to each other. <312] Two broad superlattice bands ( {021) and {112) ), ~43° to each other, crossed by a ’narrow’ {111} band. 28 ——(110) 010 130 _____{201) 120 _' {112) ----- {111} “1 —{202).{200} 0311’ ,-.-....n.......- .. ’ 110 \. ‘ 111 / [ ‘. I211, 310 \x 12 5. / 311 012’ 113 . , 411 0130114 ‘. , \ I 001 105 I02 101 201 501 100 Figure 2.9. Stereographic octant for the TiAl L10 structure showing the plane traces for which h2+k2+l2 38, along with the traces of the planes with superlattice structure. (Gray lines are (1I0), {201), and {112) superlattice traces; thin lines are {202) traces, dotted lines are {111} traces. The bounding {200} planes are shown without shading, and the (001) superlattice plane is not indicated.) Once the bands and zone axes in an SACP composite were identified, the sample surface normal and sample tilt axis were determined and used to relate the grain orientation to the sample axes, and therefore to the loading condition and other grain orientations. The index of the sample surface normal (whose location in the SACP composite was identified using the procedure outlined in the previous section) was determined either by its fortuitous location on an already-identified zone axis, or (more commonly) by the following procedure: 1. Locate two identified zone axes to either side of the unknown (as in Figure 2.10). (A is the unknown, K3 and K4 are the known zone axes in Figure 2.10.) 2. Calculate the angle between these known zone axes. 29 Figure 2.10. Schematic drawing showing how to determine approximate indices for an arbitrary location within a SACP (see section 2.5 in text). 3. C) N] Calculate the angle between the unknown zone axis and the two known zone . . . z =|A K3|gK3ZK42 axes usmg the relationshlp A K3 [K3 K 4| . . Find a trial index for A along the K3-K4 tie line by: [h k ”trial: m1 x[h k l]K3+ m2 x[h k l]K4, where m1=m2=1. . Calculate the angle between [h k lltrial and [h k llK3' If the angle is less than ~0.15°, then accept the index. If the angle is greater, then find a new [h k lltrial by: [h k ”trial: m1 x[h k l]K3+ m2 x[h k llK41 where m1 and 1112 are integers. If the trial angle is larger than AZK3, then m1 should be increased more than m2; if the trial angle is smaller than AZK3, then m2 should be increased more than m1. . Repeat until AtrialZK3 <~0.15°. . If the unknown does not lie between two known zone axes (such as for B in Figure 2.10), then first find the index of C from two known zone axes following 30 steps 1-7, and then find the index of B from C and K1 following the same procedure. The sample tilt axis (and thus the sample w-axis) was found by taking the cross product between a convenient zone axis lying on the line connecting the centers of the individual component SACPs (P on line ’normal to tilt axis’ in Figure 2.8) and the sample normal (fi) determined in the previous paragraph. Care was taken to use the szxfi criterion as opposed to fixF, as the vector fixP is in the negative tilt direction (along the image -x axis). The sample principal tensile direction, T, (along the image y-axis) was found by T=fixf. The error introduced by using f=Fxfi (the cubic approximation) is on the order of 05° or less (the maximum difference between [h k lltetragional and [h k llcubic for §=1.0175, as is the case for L10 7- TiAl). Once the principal tensile direction was determined, calculation of Schmid factors was determined via Schmid factor, m=T - f) x T - 6, where T is the unit vector along the tensile axis, 6 is the unit vector along the slip direction, and f) is the slip plane surface normal unit vector. Schmid factor calculations were carried out using the Mathematica 4.1 code included as Appendix A. 2.6 Plane Trace Identification Once a grain orientation (as described by the nominal surface normal, fl, and the grain tilt axis, t) was determined following the procedures of Section 2.5, the intersection line of any plane in that crystal with the crystal surface plane was found using the method below: Let f be the unit vector describing the line of intersection of the plane of interest and the surface plane, where the plane of interest and surface planes are described by their unit normals, [3 and h. Then for a Cartesian coordinate system: pxfi=f 31 Provided that the sample has been loaded into the microscope stage with the tilt axis within the plane of the surface (which is satisfied by the prerequisite that the sample surface is normal to the microscope axis at zero stage tilt), the angle between the plane trace and the microscope tilt axis is: a = atan{L—WE '2} = atan{—F—E—(fixbffxrln ti t~ ("ytz-"ztyxpynz—Pzny)+(nztx—nxtz)(Pznx—Pxnz)+(nxty—"ytx)(Pxny-Pynx)} tx(Py"'z-PznyH-ty(l3znx—l3xnzl+tz(l3xny‘Pynxl If the microscope scan axes have been aligned to the stage axes prior to collection of = atan{ data (as in our current situation), then the angle of that trace with the micrograph x-axis is a. For a cubic crystal system, 6, f, and f) are defined by: h k | PX = ,P = , and pz = , etc... ;/h:+k,2,+l,2, y ]/ h:+k,2,+|: h:+k:+|: For lower symmetry crystal systems the definitions for the unit vectors become con- siderably more complex, although geometrically straightforward. Calculation of the plane traces was conducted via Mathamatica 4.1 using the code of Appendix B. Calculation of the grain boundary fracture factors, F , (see Section 4.2.2) was likewise carried out using the Mathematica code of Appendix C. For the present case of the 7-TiAl crystal structure, the <2% deviation of the g- from the cubic §=1.000 has been neglected. The errors introduced in this case (max- imum rotation between true plane normal and approximated plane normal ~0.5°) are on the order of the measurement errors in determination of the crystal orientation, as well the rounding errors associated with assignment of small integer indices for the surface plane and tilt axis. Thus, the L10 tetragonality and has been neglected for computational simplicity. 32 CHAPTER 3 Results 3.1 Sample Microstructure The sample microstructure resulting from the as-received cast and HIPed thermal and mechanical processing history resulted in a duplex sample microstructure of 7 equiaxed grains, second phase (12, and occasional colonies of 7—a2 lamellar regions, as shown in Figure 3.1. The deformed sample microstructures were essentially the same as the undeformed microstructures, with the exception of deformation twin features in many of the 7 grains, as shown in Figure 3.2. There are very few deformation twins in the undeformed sample. 3.2 Deformed samples Multiple samples were loaded in bending during the initial stages of investigation, all to the point of fracture. Because these samples have undergone two or more dissimilar loading conditions (quasi-static bending, shock loading during catastrophic fracture, and possibly the propagating crack-tip stress field during fracture), there is no way of unequivocally isolating the features characteristic of the fracture initiation conditions from the features characteristic of the fracture loading conditions. In addition, the 33 Figure 3.1. Example of the undeformed microstructure of the Ti—48Al-2Cr-2Nb alloy. The majority of the microstructure is equiaxed 7, with clusters of irregular a2. Figure 3.2. Example of the as-deformed microstructure of the Ti-48Al-2Cr-2Nb alloy. The majority of the microstructure is equiaxed 7, with clusters of irregular 012. Note the prevalence of deformation twins in the as-deformed deformation structure (see Section 3.2.4.) 34 large majority of cracking sites in the fractured samples involved multiple grains and grain boundaries, likely due to microcrack propagation subsequent to crack initiation. As a consequence, only the unfractured sample will be addressed in the Sections following Section 3.2. 1. 3.2. 1 Fractured samples The fractured samples typically fractured at ~2% plastic bending strain, as calculated from radius of curvature. A fracture initiation site was typically close to, or opposite, the inner loading point of the 4-point bending load frame. (In the one case where the sample was loaded at 0.01 mm/ minute, the sample fractured into 3 fragments, with only one of the two fractures occurring near a loading point.) With the exception of the fully-propagated fracture, the tensile—surface cracks in the fractured samples broadly fell into three categories: 7-7 cracks, 7-a2 cracks, and 02-02 cracks (see Figure 3.3). 7—7 cracks were the most common (Figure 3.30), followed by 7-a2 crack (Figures 3.3a,d)s, and then 012—012 cracks (Figure 3.3b) in terms of frequency. Crack propagation (Figures 3.3a,b) and branching (Figure 3.3a) was common. With the exception of the observed 02-02 cracking, all these microcracking types are the same as were seen in the unfractured sample, with the exception that cracks were typically larger than in the unfractured sample, and involved multiple grains due to apparent crack propagation. Crack branching and crack path deviations (such as shown in Figure 3.3a, and to a lesser extent in Figure 3.3b) were common in the fractured samples. Microcracking behavior seen in the unfractured sample (described the following sections) is entirely consistent with the cracking observed in the fractured samples, in that cracks in the fractured samples encompass a wide range of crack propagation conditions ranging from freshly initiated cracks (such as seen in the 7-7 grain boundary fractures of Figure 3.3c) up to the fully-propagated crack that led to fracture. 35 Figure 3.3. Examples of cracks in the fractured samples. A: Fracture initiated at the boundary between a lamellar colony (’L’) and a 7 grain (’7’). The crack has propagated into the lower 7 grain. B: Fracture initiated in an on grain. The crack has propagated slightly into the neighboring 7 grains. C: Fracture initiated between two 7 grains (circled). D: Fracture initiated at the 7-a2 interphase boundary (circled). 36 3.2.2 Unfractured sample As can be seen in Figure 3.4, the unfractured sample bar has been bent approxi- mately 21° over the length of the bar, indicating an average surface strain of ~1.4%. Examination of the back surface of the bar (the compressive surface, shown in F ig- ure 3.5) shows local deformation of the surface due to loading from the 4—point bend frame. Figure 3.4. Sample mounted into the sample holder, showing curvature due to 4-point bending. 3.2.3 Crack initiation sites in the unfractured sample Out of a total of 20 microcracks examined in the unfractured sample, all occurred at some form of grain or interphase boundary. Of these 20 microcracks, 15 occurred at 7—7 grain boundaries such as shown in Figure 3.6, and 5 occurred at 7—012 interphase boundaries, such as shown in Figure 3.7. No microcracks were found within any grains, only at boundaries. Many fractured grain boundaries and interphase bound— 37 4 0 NW v1 (:2 [[(Q,m5jlrli 1:1 Figure 3.5. View of the back of the sample after bending. A: Roughened surface resulting from deformation. The light ’speckled’ appearance is due to specular reflec- tion from individual grains aligned to reflect the light. B: Sample angled slightly so no specular reflection. C: Sample aligned for specular reflection off the region of the sample indented by the back loading point of the 4-point bend frame (arrowed). 38 Figure 3.6. Example of a 7-7 grain boundary fracture. A: Backscattered electron image showing the grain boundary and crack. B: Secondary electron image showing better resolution of the crack opening. 00.1171 Figure 3.7. Example of a 7—012 interphase boundary fracture (circled). A: Backscat- tered electron image showing atomic number and channeling contrast. The grain boundary ()2 is bright due to higher atomic number, while the 7 grains above and below have different shades due to channeling contrast. 8: Secondary electron image. 39 aries appear to be associated with deformation twins incident on the boundary, such as shown in Figures 3.6 and 3.7. Because of their apparent dominance in terms of fracture initiation sites, and the greater availability of data for the larger-grained 7, further crack initiation studies were restricted to the 7-7 grain boundaries. The area of the sample examined for microcracks extended over approximately 4mm~12 mm from the end of the sample, rather than at the center of the 4—point bend span. The result of this is that the local stress state was likely affected by the proximity to the bend loading points. (This will be discussed in Section 3.3.6.) The detected crack density in the region studied was on the order of ~1 crack/mm2 The locations of regions on the sample containing microcracked (and un-cracked) 7-7 grain boundaries are summarized in Table 3.1. Further information on grain orientations is summarized in Table 3.4, information concerning apparent grain boundary orientation and fracture information is shown in Table 3.6. Section 3.3 describes each of the regions of interest in greater detail. 3.2.4 Deformation twinning and microcracking As shown in Figures 3.2 and 3.8, long straight features resembling deformation twins are extremely common in the deformed microstructure. These deformation twins typically cross the entire grain. As will be shown in Sections 3.3.1 through 3.3.15, these features are frequently associated with grain boundary microcracking. Using atomic force microscopy, it is demonstrated in the next section that these features are deformation twins. Atomic force microscopy The atomic force microscopy (AFM) image of Figure 3.9 shows a strong topographic contrast on the linear features spanning the visible portion of the lower grain at about a 22° angle from the vertical. All these features are characterized by a left side pro- 40 Table 3.1. Location of the regions on the sample. Sample width is ~3.65 mm, and sample length is ~35 mm. The bend loading points were at ~8 mm and ~28 mm (see Figure 3.5). comments Figure 3.8. Backscattered electron image of a region including apparent deformation twins. This is region 11, described further in Section 3.3.11. 41 258.79nm Figure 3.9. Atomic force microscopy image of the same region as Figure 3.8. Image shade indicates local elevation; boxed area corresponds to the scan field of Figure 3.11. truding from the sample surface and a right side intruding into the surface. This same surface topography is seen in Figure 3.8, where the elevated regions produce bright BSE contrast and the lower regions produce dark BSE contrast. An examination of the same AFM data presented in a 3 dimensional projection is shown in Figure 3.10; a detail of the boxed region in Figure 3.9 is shown in Figures 3.11, 3.12. Figure 3.13 shows the true-scale projection of the Figure 3.12, as opposed to the exaggerated z-scale used for all other AFM images. Twin shear The grain orientation data for grain 38 determined via SACP analysis is presented in Table 3.2. The line direction of the presumed twin features lie along the trace of the intersection of the (I11)3.31 plane with the surface, indicating that the deformation 1 Throughout this and the following sections, subscripts on Miller indices are used to indicate the grain to which they apply. 42 Figure 3.10. Three dimensional projection of the AFM data shown in Figure 3.9. Scale varies across the projection; the Z-axis scale is exaggerated for effect. Figure 3.11. Two dimensional projection of elevation from the boxed region of Fig- ure 3.9. The line in the center of the figure indicates the approximate location of the line profile of Figure 3.14. 43 Figure 3.12. Three dimensional projection of the AFM data shown in Figure 3.11. Scale varies across the projection; the Z-axis scale is exaggerated for effect. Figure 3.13. Three dimensional projection of the AFM data shown in Figure 3.11. Scale varies across the projection; the Z—axis is presented at approximate true scale. 44 Table 3.2. Grain 38 orientation, crystal directions relative to the image coordinate system, and Schmid factors for the deformation systems on the (I11)33 plane. image X image Y (tensile direc- image Z tion) crystal [29 93 crystal [67 18587® crystal [41 I4 33] crystal direction vector in sample coordinates (x,y,z) (I11) normal (0.933,-0.354,0.064) (III) normal (~0.933,0.354,-0.064) g 112 (~0.212,-0.685,-0.697) I5:110: (-0.291,-0.637,0.714) slip direction Schmid factor é- I12 0.243 3110 0.223 process that created these features is occurring on the (I11)38 plane. The shear deformation vector for twinning on the (III) plane is HIIZ, which produces a sense of shear along the vector described by (-212,-685,-697) (in the figure coordinate system) on the right side of the (III) plane. Twinning shear on the (II I) plane (the opposite side of the (III) plane) is in the %[1I2] direction, or (0.212,0.685,0.697) in the figure coordinate system. This means that the left hand side of any twin would be expected to protrude from the sample surface, while the right hand side would be expected to intrude into the surface, consistent with what is seen in Figures 3.10, 3.8, and 3.12. Ordinary dislocation slip on the same (I11)38 plane occurs in the [110] direction2 (perpendicular to the twin shear vector), which produces a sense of shear in the (-0.291,-0.637,0.714) direction on the (III) face, producing a surface relief that is up on the right side, down on the left, opposite to what is observed. Deformation shear of the type that produces the surface relief seen in Figures 3.10, 3.8, and 3.12 2 In order for a deformation system to have a positive Schmid factor, the sum B-TXE-T must be positive. Thus the ordinary slip direction on the (III) plane must be [110] for a tensile direction T=W1858 705] as is the case for grain 38. 45 is also consistent also with the sense of shear produced by both the (T11)38[T0T] and (T11)38[01T] superdislocation slip systems, however superdislocation slip is not considered likely due to the high flow stress expected due to the larger Burgers’ vector. Figure 3.14 shows a line trace across the central feature of Figure 3.12. The feature 0.19pm 85.88 /_: 80.2nm Z[nm] 000 0.00 " ' 1.50 lem] Figure 3.14. Line profile taken across the twin-like feature shown in Figure 3.11. is characterized by a continuous slope across the feature width. This slope has a rise of AZ=80.2nm for a feature width of 0.19 pm? The continuous slope indicates that there is a continuous deformation throughout the volume at the resolution available to the AFM. This surface appearance is also consistent with deformation twinning, but not for deformation from dislocation slip occurring from dislocations multiplying through a Frank-Reid loop source [67,68], because dislocations produced in this way 3 This slope is taken from the central portion of the trace rather than the extremes because of the effects of the AFM tip radius, surface debris, and environn'iental degradation of the surface. 46 all tend to slip on the same plane, which produces a single step. This is counter to what is seen in Figures 3.10 and 3.12. Dislocation multiplication and slip from an active pole dislocation source pole dislocation source (such as described in [69]) will produce such a continuous deformation, however this mechanism is considered unlikely for superdislocations. For the case of deformation twinning, one twinning partial dislocation with btw=%[112 passes for each (111) plane, producing a shear of [btw|=0.1652 nm for every d{111}=0.2323 nm (T11) plane spacing. In the particular case of the feature shown in Figures 3.9-3.13, the linear portion of the slope shows AZ=82 nm for an apparent twin width, a, of 0.19 pm. Consider the line drawing of this in Figure 3.15. For the unit vectors [3 and 6, where f) and f! are the unit vectors in the 5 (habit plane normal) and fi (grain surface normal) directions, the angle between the habit plane normal and the surface normal is acos(fi - p), which equals 90°-6 (where 6:3.4° is the angle between the surface normal and the habit plane; see Figure 3.15), as well as the angle between the true thickness direction and the apparent thickness direction. As acos 62w, the true twin thickness w=cos(3.4°) x019 pm. This true width spans approximately Hit—1 (T11) planes, or about 816 {111} plane spacings. Because there is one shear of lbtw|=0.1652 nm for every {111} plane, the expected shear for the total width is 816x0.1652 nm=134.8nm along the [112 direction. To find the frac- tion of this shear in the z-direction, we must find the dot product between the shear direction and the z-direction, which is: %=0697. Therefore, the expected surface step height AZ=0.697X134.8 nmz94 nm, which corresponds reasonably well with the observed AZ of 80.2 nm. Based on the observations of Gibson and Forwood [59], it is likely that some of the impinging twin strain was relieved through the slip of ordinary dislocations in the twinning grain, and as already noted in Table 3.2, (T11)38%[110] ordinary dislocations shear results in a surface relief that is opposite that of twinning for 47 apparent width Figure 3.15. Line drawing showing the relationship between the apparent twin ap- parent, the true width, the twin plane inclination, and surface relief due to twinning. this grain. This is shown in Figure 3.16, where the twin shear would ordinarily produce a large step (dotted line), but the ordinary dislocation shear occurring on the same habit plane produces a shear that reduces the total observed step height, producing the step height shown by the solid line of Figure 3.16. Thus, the discrepancy between the theoretical twin step height and the observed step height is entirely consistent with some accommodation of the twin shear at the 37-38 grain boundary through the production of b=%[110] ordinary dislocations slipping on the (T11)33 plane, similar to what was noted by Gibson and Forwood in [59]. This accommodation via (T11)33 %[110] slip is also favored by the m=0.223 Schmid factor for (T11)38 %[110] deformation. 48 surface / j ‘ - —'_ ------ normal ,’ dislocation shear — / n twin slip and shear twin plane normal Figure 3.16. Discrepancy between expected twin step and observed twin step ex- plained. Twin shear for a twin of width w would produce a surface relief step shown by the dotted line, however ordinary dislocation shear on these same slip planes pro- duces the opposite sense of surface step, so that the total surface step is that shown by the solid line. 3.2.5 Topographic contrast and crack opening from deforma- tion twinning As already mentioned in Section 3.2.4, topographic contrast to the backscattered electron signal is produced by deformation twins (or any other shear mechanism) that produce significant surface relief. This contrast takes the form of the elevated portion of the sample surface being brighter than the background, and the recessed portion appearing darker than the background. This type of contrast is seen clearly in Figure 3.8, where the top of the twin traces appear bright, while the bottom of the twin traces appear dark (compare Figure 3.8 to the AFM image of Figure 3.10 which clearly shows the surface elevation of the area shown in Figure 3.8). This bright-dark topographic BSE contrast is a simple result of the wide angle of collection for the BSE detector when used under ECCI conditions, coupled with the 49 surface step that results from a deformation twin. Consider the twin step shown in Figure 3.17. In the figure, the collection angle of the BSE detector is shown in the dotted lines, and an electron beam in two different locations on the sample, near the top of the twin step, and near the base of the twin step. The approximate interaction volumes of the beam electrons within the sample for these two different beam loca- tions are shown with ovals under the beam-surface intersection points. Backscattered electrons emerge from the region of the sample surface where the interaction volume intersects the surface; it is the total number of BSE that reach the detector that de- termines BSE signal strength and that is eventually rendered as a BSE image. When the electron beam is at position A, near the bottom of the step, a fraction of the backscattered electrons are blocked from reaching the BSE detector by the twin step. This loss of signal gives rise to the dark contrast. Conversely, when the electron beam is in position B, there is an enhanced production of backscattered electrons because there is a larger area of intersection between the interaction volume and the surface, which leads to bright contrast. This sort of contrast is also seen for dust particles settled on a sample surface, for example in Figure 3.21 in Section 3.3.1. The shear produced by deformation twinning also results in a polarity to any cracks that open at a grain boundary under the influence of the twin shear. As shown in Figure 3.18, an unconstrained grain with a twin will result in a crystal surface with a step, as already noted for the grain free surfaces mentioned in Section 3.2.4. This unconstrained step is shown in Figure 3.18 for a twin in grain 2 by the dotted line. However, because the grain boundary is not free, but rather constrained by the neighboring grain (grain 1 of Figure 3.18), the grain boundary step will tend to be opposed by grain boundary stresses from the neighboring grain. In the case of Figure 3.18, these stresses will be compressive to the right of the twin, and tensile to the left of the twin. Because Type-I (opening) cracks occur under a tensile stress, any Type-I grain boundary crack that is opened under the influence of the deformation 50 A electron B Figure 3.17. The mechanism for bright-dark topographic BSE contrast arising from a surface step under the conditions used for ECCI. When the beam is at position A a portion of the BSE are blocked; when the beam is at position B, there is enhanced BSE emission. twin shear is likely to open on the tensile stress side of the grain boundary (the left hand side of the grain boundary, as shown by the inset figure of Figure 3.18). 3.3 Sample Regions Descriptions Specific descriptions of the all the grains and grain boundaries used in later analysis is presented in the following sections. In summary, of the sample area scrutinized, there are fifteen regions with a fractured grain boundary, and one region with no fractured grain boundaries. Of the fifteen regions with fractured grain boundaries, one of these has no data collected because of the difficulty of data collection, and three have been 51 Grain 1 constraint forces grain boundary 1'" “cofis'iraififfofces I l .1 3: twin I / tr:’:,1i0.99. In comparison, the correlation between fracture and the value of m is negligible (t=0.003), while both lb - 'l'l (t=2.43), and 0‘3 KBm - Bord)l(t = 0.98) show some correlation to fracture, although not as great as Fmax ; the details can be seen in Table 4.1. Because of the lack of correlation between the m of Fmax and grain boundary fracture, the usefulness of m in the definition of a predictive fracture factor might be raised, especially in the light that the product If) - 'l'l XE: |(f)tw - Bord)l Shows a stronger correlation to grain boundary fracture than does Fmax- However, both lb - "fl and 0%. [(52.0 - Bord)l have been pre-selected by their incorporation in Fmax in Table 4.1. If instead, the maximum values of lb - Tl x351. l(btw - bmdll are compared for the fractured and unfractured populations as shown in Table 4.2, then there is no correlation between lb - "fl x023 |(f)¢w - Badllmax and grain boundary fracture. 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Bum: ‘WOX _._. . m.— _~Eom. Bum: AWN _.._v . A: E meu_ kudUQSOfi 333303 2535 38552 58w 3 38:09:00 mfi was me... .3” mEfib 92 selection constituent. The Fmax data presented in Table 4.1 is also shown in Figure 4.3 to show the relative incidence of the fractured grain boundaries as a function of Fmax . The upper line of points shows the fractured and unfractured grain boundaries expressed by their relative ranking of Fmax in the population of grain boundaries, while the lower line of points shows the what fraction of the total fractured population of grain boundaries had fractured by that value of Fmax. Statistics vs. Fmax 1.0 I— X 3‘ x m E x A .0 x 2 ,3 A o unfractured e x A O 2' o A xfractured > 0 5 o 'g x ° ‘ A # fractured 3 o 5 tot. # fractured X E o A a a o O A 0 MA 0 9 v 0 0 5 1.0 1 5 Fmax Figure 4.3. The cumulative fraction of the grain boundary population, and the run- ning total of n“mbemffmauredgmmboundar'“ both plotted against Fmax- totalnumbero f f racturedgrainboundaries ’ 4.2.4 The factor F This factor F includes terms related to: 1) the magnitude of the twin shear in the tensile direction( “but, - 'l'l ); 2) the ease with which the ordinary dislocations in both 93 833$ 535202 58w 3 9530538 :2: one n. we manganese 2: mo £5wa BE 94 owo.m too owwo mmvo ~5on 8» mood wofim mono mvvo 91% mm» omo.m god omoo Emo Simv mm» owfim wood wood Hovo mem mom ago www.m onwo oovo mmdm max omwo omwd wood Sumo mm-mm 8% ~35 owo.m oowo bomo om-wm we» moon omo.m mmoo oovo oTwH max oood goo oowo oovo 3.3 8» wmmd Hmoo mmwo movo 2.3 mm» moo.m ooo.m «Ed owvo w.» mm» v28 wood wood ammo MA .83 58:3... . 3:: .mo F . m: V8533 . 2»: ‘mo me_.—. . A: meE @3552 n. .8 35:09:00 23 mo @858 25. Nd floral .8323 33358, Eme 3 Bamaosflou .205 one u. we 3:289:00 2: mo A3588 BEL II S.oA c Ange SdA c Ammo II 6 wood. maria ommé- Novoo- w mood woman wood mmvo :88 on owo.m mmw.m hood wflmo new: on oomo woho vmoo mono maéq on 30m Homo wood wood NF? 0: onwo vowo owwo nvvo ovév o: oumo ago mooo oomo mmém o: homo homo goo oovo omofl on omfio Sumo nooo oovo 5-3 0: ovoo ooo.m mmwo oovo onH 0: ago wvwo onoo vao o-o o: ommd mto ago vao mo os mmo.m hmoo oHoo ommo m; 85o me_Eom . 3:: ‘mo FA: me_Eom . 2.: .mo 56:. . n: meE Eggnog $35303 u mo manoaooEOQ 2: wo £5me £8 .m.v gear 95 A A parent and intersected grain can absorb the twin shear (o d szw - bmd)| ); and 3) the driving force on that particular twinning system (mtw). These are illustrated in Figures 4.4 and 4.5. A . _ .E _ graina btw E bm J I‘ II A P I C grainb B B .E 5 II graina grain b > O l C CD Figure 4.4. The displacement at a grain boundary due to an impinging twin (A), and the requisite deformation of grain ’b’ to avoid grain boundary decohesion (B). Note that this figure only addresses deformation by a single grain (’b’) , while the general case typically involves deformation by both grains (’a’ and ’b’). The range of F F can be considered in two independent segments, one dependent solely on the orien- tation of the twinning grain to the tensile direction, and the second dependent solely on the crystal orientation relationship at the grain boundary. The parent grain component incorporates the Schmid factor for twinning and the twinning Burgers’ vector relationship to the tensile direction. These can be expressed as (btw - T) (btw - p) and (Em - T), respectively, where [3 is the slip plane unit normal vector. 96 ’ 2allord.dislocationsl(btw ' ord)l :9 e d .5 e U Fv—vz 1' driving force opening displacement ability to conform Figure 4.5. The components of F and their relationship to physical quantities. 97 The maximum of (btw - T)(btw . [3) will occur when btw, T, and f) are coplanar. Considering only the btw, T, f) coplanar case, and utilizing the interdependence be- tween btw and f), the maximum of the parent grain factor can be expressed in terms of a single variable: the angle between btw and T, 6. The parent grain factor then becomes: cos(6 — 90°) cos2(6). Simplifying: cos(6 — 90°) cos2(6) = cos(90° -— 6) c052((5) = [cos(90) cos(6) + sin(90°) sin(6)] cosz(6) = [0 + sin(6)] cosg(6) = sin(6) cos2(6) The maxima and minima of sin(6) cosz(6) occur for the cases where: 53/86 = 0 = cos3(6) — 23in2(5) cos(6) Factoring this, 0 = cos((5) [cosQ(6) — 28in2(5); cos(6)=0 produces a minimum at 6:90°, while cos2(6) — 2 sin2(6)=0 produces a max- imum at tan2(6) = %, or 6 = arctan(71§) z35.2644°. This gives a potential maximum of the parent grain factor equal to "£03849. 2 The grain boundary orientation component (0 d Kb“, - bard” ) consists of 8 fac- tors. Four of these are constant, being [3th - bard, within the same grain. For the cubic approximation, these are equivalent to O, O, 752%, and 73275. The non-constant factors consist of 2(th 'Bllwlz) and 2-(btw, 'B[1‘1'0]2)- The sum of these has a maximum when btw is along the <100] directions of grain 2, when each of the four factors: 12. Thus, the maximum of of the grain boundary orientation component is 2.(722%)+ 4-(—\}=), or z3.983. IO 98 Combining the parent grain component and the grain boundary orientation com- ponents, therefore, yields a maximum possible value of F al.533. In the lower limit F becomes meaningless, as the % < 112(111} deformation twinning for which F is defined does not occur for mSO. Interpretation of F F may be considered in two different ways. In the first, F may simply be regarded as a measure of the potential magnitude of a local shear strain, and the larger the shear strain, the greater the local strain hardening, up until fracture. In the second interpretation of F , F may be regarded as how much shear a grain boundary will be called upon to transmit. This is because a large mm, will tend to produce more deformation twinning for any given constraint. In turn, the constraint on a defor- mation twin will tend to be inversely related to the ease with which the twinning deformation can be passed on to a neighboring grain, which is described by the sum 0%. |(btw - bard”. Thus, a large m and a large 0%. Kb“, - bord)| will tend to enable large deformation twins, with a strong driving shear (large m) and a weak constraint to further twinning (large 0%. Kb“, - bard)”. However, because shear transfer across the grain boundary almost always involves some mismatch in the vector sum of the incoming deformation and the outgoing deformation, the process of shear transfer tends to leave some residual strain or partial dislocation accumulation at the inter- face. This can be seen schematically in Figure 4.6, where the accumulated shear from three incoming twinning dislocations in the bottom grain is partially accommodated by the shear from two outgoing ordinary dislocations in the top grain. However, be- cause the vector sum of the two outgoing dislocations is not equal to the vector sum of the incoming dislocations, some residual shear strain (labeled ’R’ in the figure) remains at the grain boundary. What is worse, because of the nature of twinning, exactly one twinning dislocation is incident upon the boundary for each {111} plane 99 of the twin. This results in an accumulation of strain along the grain boundary until the shear from one or more ordinary dislocations accommodates part of that shear. However, because dislocation flow tends to cause a multiplication of dislocation lines within the same plane [63], the strain accommodation at the average grain boundary is likely much less uniform than is shown in Figure 4.6. It should be noted that even for the case where there is perfect alignment of both slip plane and shear direction across a grain boundary, perfect strain accommodation is impossible, simply because of the difference in the magnitude between % <110] ordinary dislocations and El; <112 twinning dislocations. Variants of F There is an implicit assumption to F, in that it examines all shear and accommodation processes in terms of a grain boundary that is normal to the tensile axis, and includes no accounting of the grain boundary inclination. Because of the apparent dependence of the propensity for grain boundary open- ing on the orientation of the tensile axis (expressed by the component (13m - T) in F), it might be expected that there would be a further refinement possible to F by incorporating a factor that accounts for the variation in normal tensile stress on the grain boundary. Indeed, if F is modified by the simple expedient of dividing through by the cosine of the apparent angle of the grain boundary relative to the tensile axis, )8 (measured from images of the grain boundaries), then the distinction between the fractured and unfractured populations is increased so that t increases from 2.64 for the comparison of Fmax, to 2.99 for the comparison of Fmax /cos(fl) as shown in Table 4.3. This is a crude incorporation of only a portion of the grain boundary tilt, and the distinction is expected to significantly improve if the entire grain boundary orientation is incor- porated. Although there is currently no convenient way to measure the true grain 100 outgoing dislocations incident twin dislocations local boundary fl: l—l—lfl strain ' II I ”U 0 displacement magnitude vector sums Figure 4.6. Schematic of deformation transfer from an incident deformation twin (shown by the the array incident twinning dislocations Em, bottom) to ordinary dislocations (Bord, top). This schematic is simplified to only two ordinary dislocation systems in a single grain from 8 systems in both grains. In both cases, however, there is a local accumulation of strain in the grain boundary (shown as the total accumulated strain in the graph at bottom center). Because of the vector mismatch between the the incident and outgoing dislocations, there is an additional residual deformation (R in the vector sum to the right) that resides in the grain boundary. 101 boundary inclination to the tensile axis, this finding would suggest a final form for F of: Ff“:'(B‘w'T)'(m“‘i)0§1-'(B‘w'B°’"“)' (T-N) ’ where N is the grain boundary unit normal. It is expected that this form of F ff" would show an even higher correlation to grain boundary fracture. An even further improvement in the correlation would be expected for a modification of F that not only incorporated the full grain boundary inclination to the tensile axis, but also the inclination of the twinning shear vector relative to the grain boundary. 4.2.5 Other expected contributions to F Besides the grain boundary orientation contribution to the final form of a grain bound- ary fracture criterion noted above in Section 4.2.4, there are a number of other po— tential orientation, and even chemical, contributions to a final version of F . One other geometric contribution to the fracture probability that would be ex- pected to affect both strain transfer and the residual stress state at the grain boundary is the predicted slip plane2 (and the resultant Schmid factor) for strain accommoda- tion by ordinary dislocation slip. A model that can successfully predict the slip planes of the accommodating dislocations would also provide greater insight into the ease with which the accommodation process can proceed. An understanding of the 7-7 grain boundary interface energy (which would be expected to be dependent on the lattice mismatch, boundary orientation, and the effect of composition and grain boundary solute concentration as noted in [62,63]) might provide a fracture criterion, such that any particular grain boundary might be able to reduce its total enthalpy through the creation of a free surface at the grain boundary. This would enable the prediction of the strain required to fracture a grain boundary as a function of the grain boundary parameters, all from first principles. 2 That is, the value of x in the {11x} slip plane for % <110] dislocations. 102 Table 4.3. Incorporation of grain boundary orientation into F F boundary Fmax 6515323) fracture 1-3 0.685 0.769 yes 7-8 1.317 1.327 yes 10—12 0.868 1.035 yes 14—15 1.414 1.464 yes 18-19 1.060 1.310 yes 28-29 1.155 1.161 yes 32—33 1.124 1.124 yes 34—35 1.389 1.533 yes 37—38 1.467 1.467 yes 45-47 0.907 0.908 yes 4849 1.108 1.406 yes mean 1.136 1.228 1-5 0.876 1.112 no 68 1.048 1.108 no 6-9 1.040 1.138 no 13-16 0.749 0.749 no 14-17 1.119 1.170 no 19-20 0.782 0.838 no 31-33 1.044 1.044 no 44—46 1.039 1.039 no n1-n2 0.715 0.718 no n1-n3 0.593 0.595 no n4-n5 0.827 0.827 no mean 0.894 0.940 t 2.6483 2.9901 (1 o.01> a >0.005 0.005> a >0.0005 103 Finally, the effects of grain orientation, neighboring grain elasticity, and grain plasticity would all be expected to play some role in grain boundary fracture. Stress concentrations, orientational-dependent stiffness, and orientational-dependent dislo- cation flow stress are some relevant phenomena that would be dependent on these components. thure research to refine F will likely require the determination of grain boundary orientation, or modeling of the grain elasticity and plasticity in the sur- rounding grains to careful design of experiments to carefully control the giving grain boundary However, despite the number refinements available, F by itself is expected to be useful for a simplified engineering approach to predict grain boundary fracture, as it combines a high significance probability for fracture (> 0.99) with a very small number of measurements (a single grain orientation per grain). This sort of ’sparse’ model easily lends itself to statistical fracture prediction modeling. 4.2.6 The association between the Fmax plane and fracture Table 4.4 shows the relationship between the maximum of F for fractured grain boundaries, and the location of the microcracking next to impinging plane traces. Of the 11 microcracked grain boundaries, six of these grain boundaries have a mi- crocrack that is proximal to only one twin deformation system, and therefore that microcrack can be attributed to that twin deformation system. For five of these six grain boundaries, the deformation system with the largest F (Fmax) is also the fracturing deformation system. Of the five microcracked grain boundaries that are not unequivocally attributable to only one twin deformation system, microcracking of two grain boundaries occurs where the Fmax deformation twinning system (that is, the deformation twinning system for which F is the largest for that grain boundary) impinges upon the grain boundary. (These are the 37—38 and 45-47 grain boundaries, shown in Figures 3.44 and 3.49, where microcracks touch both the Fmax deformation 104 twin and another deformation twin.) The remaining three cracked grain boundaries cannot be assigned a fracture twinning system because of the size or appearance of the crack. For the single case where the Fmax system is not also the fracture system, this may highlight the simplicities of the F model, namely a failure to account for constraints imposed by surrounding grains, and the relative geometry of the twinning plane and the grain boundary plane. For the case of the 14-15 grain boundary where the Fmax twinning system is not the twinning system associated with fracture, the Fmax system is (111)” éfllfl whose trace lies only ~22° from the grain boundary, a gross deviation from the simple model assumption of the twin being perpendicular to the grain boundary plane. In addition, this value of F was calculated in grain 14 (as well as in grain 15) without any determination of the effects of the surround- ing grains on the actual orientation of the principal tensile stress direction in either grain 14 or grain 15, potentially leading to an incorrect determination of the twinning deformation system for which F =Fmax- Therefore, even the simplified model for F seems to predict well the deforma- tion twinning plane associated with grain boundary fracture, where the deformation twinning system for which F=Fmax is also the deformation twinning system that is as- sociated with grain boundary fracture. It is the failure of this ’Fmax twinning system is the fracturing twinning system’ criterion (such as for the case of the fractured 14-15 grain boundary) that suggests a path for further refinement to the F grain boundary fracture factor. Future refinements to F might include 105 3.” .5 a» we: NEWVQE bflfizb are. EHWEHE 8.7%.?me encased Sad NEWEEV .Nfirfific tins NEWEEV $.25 20838“ $3 gamma: @2156 mg «3.5 80855 $2 fizweczv E. was.” at.” .wE as am: gamma: NEWECV ”mam Emma seizes £3 flawless: E. swam on.” .5 as 83 NEWEEV NE m 2le 2-2 a.” was 8 33 NEW. 2:6 We; 25.: 2.: R.” are. as $3 NEMNEE NFM 2:6 m2: am.” .5 ma :2 whims? NEW was 3. Emma mzozmfiss £3 NETS: E. 3 8:832 Sousa 888; 839mm EB» mem EBB Bosch hawwcsom $320822 zuccqsom 590 one meu 525mm :oEaBSmas Alt 28d. 106 CHAPTER 5 Conclusions Using the principal methods of scanning electron microscopy and selected area chan- neling, with additional information from atomic force microscopy, the fracture initi- ation mechanisms of 7-TiAl based alloys has been studied. Fracture initiation in a duplex 7-TiAl based alloy has shown that a substantial majority of fracture initiation sites occurs at 7—7 grain boundaries; that the second most common fracture initiation site is at ’7-02 interphase boundaries; that in both the 747 and 'y-ag fracture initiation sites, deformation twins were associated with the grain boundary fracture, where the fracture initiation was associated with any specific feature. For 747 grain boundaries, the propensity for fracture initiation was correlated with the maximum of the grain boundary fracture factor F : F = (mane... - in 0%, KB... - 60...». where mm, is the Schmid factor for deformation twinning, btw is the unit vector in the twin shear displacement direction, 'l' is the unit vector in the principal tensile direction, the bard are the unit vectors in the direction of the Burgers’ vectors for ordinary dislocation slip on each of the eight % <110] deformation systems in both the twinning and neighboring grain. There are eight values of F per grain boundary, one per deformation twinning system in each of the two 'y-TiAl grains. 107 The components of Fdescribe, respectively, the driving force for deformation twin- ning; the alignment of the twin deformation displacement with the principal tensile axis; and the ability of both grains to accommodate this twin displacement at the grain boundary. This formulation for F neglects a number of contributions which would be ex- pected to influence grain boundary fracture. One contribution that has been briefly addressed is the contribution of grain boundary orientation relative to the principal tensile axis, and it is shown that the factor F max/cosfi (where 6 is the appar- ent inclination of the grain boundary normal away from T) is more strongly cor- related with grain boundary fracture. The implication of this improvement is that Ffi'n.= “Btu!'T)l(mtw)o§i_l(5tw'fiord)l (T-N) (where N is the grain boundary unit normal) will have an even stronger correlation with grain boundary fracture. Other contributions that are expected to influence fracture are the orientation- dependent grain boundary cohesion energy and crystal elastic anisotropy, in addition to polycrystal plasticity and elasticity effects, and solute effects on grain boundary energy, deformation defect flow stresses, and crystal anisotropy. 108 CHAPTER 6 References [1] M. 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Data is input as a list in the code; the defor- mat ion systems are input as deformation plane at (A1) and direction at (A2), grain identification labels at (B), grain surface normal axes at (C), and grain tilt axes at (D)- (7‘ For calculating Schmid factors for all the twinning, ordinary, and superdislocations for a list of crystals given: 1) surface normal, and 2) tilt axis, following the methodology established by Ben Simkin *) 0" Approximates using a cubic unit cell. *) 0" (Remember, the initial tilt axis was found using the cubic approximation in the first place) *) K* Initial Definitions (define functions for vector mag. and finding a unit vector) *) mag[v_] :=Sqrt[(v[[1]])“2 +(v[[2]])"2 +(v[[3]])“2]; 119 lelitEV-]i={V[[1]],V[[2]],VI[3]]}/mag[v]; (* -- INPUT PLANES of deformation system (and count them); tilt and plans = integer plane indices, pln=unit vector surface normal, dimp=#planes -- *) THE PLANES GO WITH THE defor VECTOR DESCRIBED BELOW-- *) [ p1n[[i]]=unit[plansflilll, {1, 1, dimpI]; C at: -- INPUT DEFORMATION SYSTEMS ON THESE PLANES; defor=deformation shear vector *) (=k= the order is: twin, ordinary dislocation, two superdislocations for each of the 4 sets of {111} planes *) (As) defor:={{1,1,-2},{1,-1,0},{1,0,-1},{0,1,-1},{-1,1,-2},{1,1,0}, {1,0,1},{o,1,-1},{1,-1,-2},{1,1,0},{1,o,-1},{o,1,1}, {—1 , -1,—2},{1,-1,0},{1,o,1},{o,1,1}}; def=plans; (* dummy assignment so as to define array size *) D0 C def[[i]]=unit[defor[[i]]l, {1, 1, dimpl]; (* -- INPUT GRAIN ASSIGNMENT LABELS *) (B) id:={1,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,28, 29,30,31,32,33,34,35,36,37,38,39,39a,40,41,42,43,44,45,46,47, 48,49,50,n1,n2,n3,n4, n5}; 120 dimg=51; (* number of grains *) (* surface normals *) (* norm is integer, nor is unit *) (CD norm={{11,10,27},{0,4,1},{1,8,1},{16,23,6},{28,25,59},{15,16,5}, {10,6,23},{55,6,17},{17,12,28},{13,12,23},{3,3,2},{8,9,5}, {9,13,15},{9,12,1},{35,0,4}.{19,2,7},{13,15,8},{5,10,22}, {28,10,5},{71,86,75},{62,51,79},{9,10,9},{40,37,83}, {13,31,11},{10,7,25},{1,2,1},{30,12,31},{25,58,38}, {10,13,15},{13,27,58},{36,21,20},{70,58,20},{41,14,33}, {19,13,22},{147,175,271},{37,36,23},{19,27,7},{25,12,24}, {6,11,6},{41,15,22},{17,7,10},{6,41,48},{17,43,24}, {25,29,74},{10,31,21},{26,15,15},{4,13,22},{817,297,792}, {17,9,9},{17,12,13},{12,5,5}}; nor=norm; (* dummy assignment *) Do[ nor[[i]]=unit[norm[[i]]], {1, 1, dimg}]; (* tilt axes *) (* tilt is integer, tlt is unit *) (I)) tilt={{41,-91,17},{3,-1,4},{5,1,-13},{-353,430,-707}, {127,131,-116},{-7,-15,69},{6,16,-7},{‘3,-26,19}, {122,-66,-47},{61,-28,-20},{11,7,-27},{-21,-13,57}, {-896,-592,1072},{1,-1,3},{-28,177,245},{‘8,-141,62}, {-7,5,2},{836,847,-575},{5,-13,'2},{-370,745,-504}, {-115,383,-157},{-4,9,-6},{-85,56,16},{47,-96,215}, {-28,15,7},{-3,16,-29},{-23,-113,66},{-146,2,93}, {95,-20,-46},{-1387,975,-143},{1,-16,15},{1,-5,11}, {-24,9,26},{-27,31,5},{-57,-28,49},{2,-4,3},{-6,6,-7}, {-36,103,-14},{19,-18,14},{11,-11,-13},{‘1,-9, 8}, 121 {-7,-6,6},{-13,-1,11},{472,-168,-95},{-10,~11,21}, {-15, 27,-1},{-24,-106,67},{0,8,-3},{9,-10,-7}, {-33142-9}s{-30,17)55}}; tlt=tilt; (* dummy assignment *) Do[ t1t[[i]]=unit[tilt[[i]]]. {1, 1, dimg}]; (* -- CALCULATE TENSILE AXES - tens is integer, ten is unit *) tens=norm; ten=tens; (* dummy assignments *) Do[ tens[[i]]=Cross[norm[[i]], tilt[[i]]], (i, 1, dimg}]; Do[ ten[[i]]=unit[tens[[i]]], {1, 1, dimg}]; (* Now do for each grain... *) Do[ (* CALCULATE components of the SCHMID FACTORS *) 311=def; (* dummy assignment *) s21=def; (* dummy assignment *) Do[ sl1[[i]]=Dot[def[[i]],ten[[nn]]], {1,1,dimp}]; DoI 821[[i]]=Dot[pln[[i]],ten[[nn1]], {i.l.dimp}]; (* ---------- OUTPUT ------------ *) Print[" "J; Print[" Schmid Factors ------- "]; Print[ " Grain "<>ToString[id[[nn]]]<>" - "<>"normal= "<> ToStringEnorm[[nn]]]]; Print[ " "<>"tilt= "<>ToString[tilt[[nn]]]<>" tensile= "<> ToStringEtens[[nn]]]]; Do[Print[ 122 ToStringEdefor[[i]]]<>" on "<>ToString[p1ans[[i]]]<>", "<> ToString[N[(811[[i]]*s21[[i]])]]], {1, 1. dimp}] .{nn,1.dimg}]; 123 APPENDIX B Mathematica code for calculation of {111} plane traces Mathematica 4.1 code used to calculate the four {111} plane traces for all grains. The input consists of grain orientation (described by the grain surface normal, 5, and the grain tilt axis, f), and a list of pairs of grains for which to calculate F . Data is input as a list in the code; the planes for which plane traces are to be determined at (A), grain identification labels at (B), grain surface normal axes at (C), and grain tilt axes at (D). (* For calculating plane traces given 1) surface normal, and 2) tilt axis following the methodology established by Ben Simkin *) (* Approximates using a cubic unit cell. *) (* (Remember, the initial tilt axis was found using the cubic approximation in the first place) *) (* Initial Definitions (define functions for vector magnitude and finding a unit vector) *) magIV-]:=Sqrt[(v[[1]])“2 +(v[[2]])‘2 +(v[[3]])“2]; unit[v_]:={VI[1]1,v[[2]],v[[31]}/mag[v]; 124 ta=0; (* surface tilt angle in degrees *) (* -- INPUT PLANES OF INTEREST (and count them); tilt and plans = integer plane indices, pln=unit vector surface normal, dimp=#planes -- *) LA) p1ans:={{1,1,1}, {-1, 1,1}, {1, -1, 1}, {~1,-1,1}}; dimpp:=Dimensions[p1ans]; dimp=dimpp[[1]]; p1n=plans; (* dummy assignment so as to define array size *) Do[ plnIIiII=unit[planslfilll. {1, 1, dimp}]; (* -- INPUT GRAIN ASSIGNMENT LABELS. id=identification *) (I3) id:={1,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23, 28,29,30,31,32,33,34,35,36,37,38,39,39a,40,41,42,43,44,45, 46,47,48,49,50,n1,n2,n3,n4,n5}; (* surface normals *) (CU norm={{11,10,27},{O,4,1},{1,8,1},{16,23,6},{28,25,59},{15,16,5}, {10,6,23},{55,6,17},{17,12,28},{13,12,23},{3,3,2},{8,9,5}, {9,13,15},{9,12,1},{35,o,4},{19,2,7},{13,15,8},{5,1o,22}, {28,10,5},{71,86,75},{62,51,79},{9,10,9},{40,37,83}, {13,31,11},{10,7,25},{1,2,1},{30,12,31},{25,58,38}, {10,13,15},{13,27,58},{36,21,20},{7o,58,20},{41,14,33}, {19,13,22},{147,175,271},{37,36,23},{19,27,7},{25,12,24}, {6,11,6},{41,15,22},{17,7,10},{6,41,48},{17,43,24}, {25,29,74},{1o,31,21},{26,15,15},{4,13,22},{817,297,792}. {17,9,9},{17,12,13},{12,5,5}}; (* tilt axes *) (I)) tilt={{4l,-91,17},{3,-1,4},{5,1,-13},{-353,430,-707}, {127,131,-116},{-7,-15,69},{6,16,-7},{-3,-26,19}, {122,-66,-47},{61,-28,-20},{11,7,-27},{-21,‘13,57}, 125 {-896,-592,1072},{1,-1,3},{-28,177,245},{-8,-141,62}, {-7,5,2},{836,847,-575},{5,—13,-2},{-37o,745,-504}, {-115,383,-157}»{-4,9,-6},{-85,56,16},{47,-96,215}, {-28,15,7},{-3,16,-29},{-23,-113,66},{-146,2,93}, {95,-20,-46},{—1387,975,-143},{1,-16,15},{1,-5,11}, {-24,9,26},{-27,31,5},{-57,-28,49},{2,-4,3},{-6,6,-7}, {—36,103,-14},{19,-18,14},{11,—11,-13},{-1,-9, 8}, {-7,-6,6},{-13,-1,11},{472,-168,-95},{-10,-11,21}, {-15, 27,-1},{'24,-106,67},{0,8,-3},{9,-10,-7}, {—3,14,-9},{-3o,17,55}}; d1mgr=Dimensions[id]; dimg=dimgr[[1]]; (* norm is integer, nor is unit --*) nor=norm; (* dummy assignment *) Do[ nor[[i]]=unit[norm[[i]]], {1, 1, dimg}]; (* tilt is integer, tlt is unit *) tlt=tilt; (* dummy assignment *) Do[ t1t[[1]]=unit[tilt[[i]]], {1, 1, dimg}]; (* -- CALCULATE TENSILE AXES -- tens is integer, ten is unit *) tens=norm; ten=tens; (* dummy assignments *) Do[ tens[[1]]=Cross[norm[[i]], t11t[[i]]], ii, 1, dimg}]; Do[ ten[[i]]=un1t[tens[[i]]], {1, 1, dimg}]; (* -- do for each grain... -- *) Do[gn=gnct; (* ~- CALCULATE TRACE LINE INDICES -- *) 11=p1n; (* dummy assignment *) 126 Do[ 11[[i]]=Cross[pln[[i]], nor[[gn]]], {i, 1, dimp}]; (* ----CALCULATE TRACE LINE angles *) (* ngls is vector of angles relative to the tilt axes) of plane traces *) ngls=Range[N[dimp]]; Do[ {dot1=Dot[tlt[[gn]],ll[[i]]]; dot2=DotIten[[gn]],llffilll; tb=Cos[N[ta*Pi/180.00000000]]; num=tb*dot2; ngls[[1]]=ArcTan[dot1,num];} , {1, 1, dimp}]; (,1 ---------- OUTPUT ------------ 1.) Print["GRAIN "<>ToStr1nindIEgnllll; Do[Print[ ToString[p1ans[[i]]]<>" plane at "<> ToString[Round[(ngls[[1]]/Pi*180.0)]]<>", "<> ToString[Round[(ArcCos[Dot[p1n[[i]], norIfgnlllI/Pi*180.0)]]<>" deg from normal"], {i. 1. dimp}].{gnct.1.dimg}]; 127 APPENDIX C Mathematica code for calculation ofF Mathematica 4.1 code used to calculate the values of F , as well as a number of the components of F . The input consists of grain orientation (described by the grain surface normal, n, the grain tilt axis, f), and a list of pairs of grains for which to calculate F. Data is input as a list in the code; the deformation systems are input as deformation plane at (A1) and direction at (A2), grain identification labels at (B), grain pairs for calculation of F at (C), grain surface normal axes at (D), and grain tilt axes at (E). C * For calculating grain boundary fracture factor F and its components for a crystal given 1) grain surface normal, and 2) grain tilt axis. This assumes that grain tilt axis and grain normal are both perpendicular to the tensile axis. Follows the methodology established by Ben Simkin *) C "‘ Approximates using a cubic unit cell. at) < * (Remember, the initial tilt axis was found using the cubic approximation in the first place.) *) 128 (* Initial Definitions (define functions for vector magnitude and f inding a unit vector) *) mag Ev_] :=Sqrt[(v[[1]])“2 +(VE[2]])“2 +(v[[3]])“2]; unit [VJ :={VI[1]] ,VII211,VE[311}/mag[v]; (* —— INPUT PLANES of deformation system (and count them), tilt and plans are integer index, tlt and pln are unit vectors. d1mp=#planes ‘THE PLANES GO WITH THE defor VECTOR DESCRIBED BELOW-- *) (* —-do:(4)111, (4)-111, (4)1-11, (4)—1-11 in that order-- *) (A1) plans={{1,1,1},{1,1,1},{1,1,1},{1,1,1},{-1,1,1},{-1, 1,1},{-1,1,1}, {—1, 1,1},{1,—1,1},{1,-1,1},{1,—1,1},{1,-1,1},{-1,-1,1}, {—1,-1,1},{-1,-1,1},{-1,—1,1}}; dimpp=D1mensions [plans] ; dimp=dimpp[[1]] ; p1n=plans; (* dummy assignment so as to define array size *) DO I: p1n[[i]]=unit[plans[[i]]], {1, 1, dimp}]; (* —- INPUT DEFORMATION SYSTEMS ON THESE PLANES (same number 01f directions as there planes listed above), defor=def. system. Burgers’ vector-- *) (* —- in the order: twin, ord., Supers(2) in that order for each set of 4 {111} planes -- *) (A2) defor={{1,1,_2},{1,-1,o},{1,o,—1},{o,1,-1},{-1,1,-2},{1,1,0}, {1,0,1},{O,1,-1},{1,-1,-2},{1,1,0},{1,0,-1},{0,1,1}, {-1,-1,-2},{1,-1,0},{1,0,1},{0,1,1}}; def=p1ans; (* dummy assignment so as to define array size *) 129 (* ——-—- INPUT GRAIN ASSIGNMENT LABELS, and ASSIGN NUMBER OF GRAINS id=jdentif1cation -- *) (B) id:==-{:1,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,28,29, 130,31,32,33,34,35,36,37,38,39,39a,40,4l,42,43,44,45,46,47,48,49, 50,n1,n2,n3,n4,n5}; dimgr=Dimensions [id] ; dimg=dimgr I [1]] ; (* --INPUT PAIRS OF GRAINS FOR TESTING -- *) (* Ilote defined by position in id list above. dimpair=number of pairs . *) (C) pairs={{1,2},{1,4},{5,7},{5,8},{6,7}:{9, 11},{12, 15},{13, 14},{13, 16}, {17,18},{18,19},{23,24},{26,28},{27,28},{29,30},{32,33}, {40,42},{41,43},{44,45},{47,48},{47,49},{50,51}}; dimpair=22; (*‘ -- INPUT SURFACE NORMALS -- norm is integer, nor is unit vector -- *) (D) D££r1n={{11,10,27},{O,4,1},{1,8,1},{16,23,6},{28,25,59},{15,16,5}, {10,6,23},{55,6,17},{17,12,28},{13,12,23},{3,3,2},{8,9,5}, {9,13,15},{9,12,1},{35,o,4},{19,2,7},{13,15,8},{5,10,22}, {28,1o,5},{71,86,75},{62,51,79}.{9,10,9},{4o,37,83},{13,31,11}, {1o,7,25},{1,2,1},{30,12,31},{25,58,38},{1o,13,15},{13,27,58}, {36,21,20},{7o,58,2o},{41,14,33},{19,13,22},{147,175,271}, {37,36,23},{19,27,7},{25,12,24},{6,11,6},{41,15,22},{17,7,10}, {6,41,48},{17,43,24},{25,29,74},{1o,31,21},{26,15,15}, {4,13,22},{817,297,792},{17,9,9},{17,12,13},{12,5,5}}; noI‘=norm; (* dummy ass ignment *) DOE nor[[i]]=unit[norm[[i]]], {1, 1, dimg}]; 130 (* INPUT TILT AXES (tilt is integer, tlt is unit vector) *) (E) tilt;==r{{41,-91,17},{3,-1,4},{5,1,-13},{-353,430,-7o7},{127,131,-116}, {-7,-15,69},{6,16,-7},{-3,-26,19},{122,-66,-47},{61,-28,-20}, {11,7,-27},{-21,-13,57},{-896,-592,1072},{1,-1,3},{-28,177,245}, {-8,-141,62},{-7,5,2},{836,847,-575},{5,-13,-2},{-370,745,—504}, {-115,383,-157},{-4,9,-6},{-85,56,16},{47,-96,215},{-28,15,7}, {—3,16,-29},{-23,-113,66},{-146,2,93},{95,-2o,-46}, {-1387,975,-143},{1,—16,15},{1,-5,11},{-24,9,26},{-27,31,5}, {-57,-28,49},{2,-4,3},{-6,6,-7},{—36,103,-14},{19,-18,14}, {11,-11,-13},{-1,-9,8},{-7,-6,6},{-13,-1,11},{472,—168,—95}, {-1o,-11,21},{~15,27,-1},{—24,-106,67},{o,8,-3},{9,-1o,—7}, {—3,14,-9},{-30,17,55}}; tlt=tilt; (* dummy assignment *) Do I: t1t[[i]]=unit[t11t[[i]]], {1, 1, dimg}]; (* -— CALCULATE TENSILE AXES -- tens is integer, ten is unit vector -- *) tens=norm; ten=tens; (* dummy assignments *) DO I: tens[[i]]=Cross[norm[[i]], tilt[[i]]], {1, 1, dimg}]; (* reduce tensile to smallest form. *) D0 Cdivisor=GCD [tens[[1, 1]] ,tens[[i,2]] ,tens[[i,3]]] ; tens[[i]]=tens[[1]]/divisor,{i, 1,dimg}] ; (* produce unit vector *) 'DOI tenIIill=unitItens[[i]]], {1, 1, dimg}]; (* Now, iterate for all these pairs being the pair of interest... *) interest={0,0}; (* dummy assignment *) D0 [interest [ [1] ] =pairs [ [poi , 1] ] ; interest [ [2] ] =pairs [ [poi , 2] ] ; 131 (* the grains (NUMBERED BY POSITION IN LIST) of interest. *) (* -- CALCULATE SCHMID FACTORS for each of the 16 deformation systems in each grain. -- *) sch=Tab1e[0.0000,{i,1,dimp},{j,1,2}]; (* dummy assignment *) Do[ sch[[i, j]]=((Dot[def[[i]],ten[[interest[[j]]]]])(Dot[p1n[[i]], ten[[interest[[j]]]]])), {1,1,dimp},{j,1,2}]; (* -- Convert plane normals, Burgers’ vectors to sample space -- *) (* In is the transformed plane normals, bu is the transformed Burgers’ vectors *) bu=def; (* dummy assignment: sample axis Burgers’ vectors*) lu=pln; Dof dx=Dot[nor[[interest[[1]]]],pln[[i]]]; dy=Dot[tltIIinterestIIIIIII,plnIIilll; dz=Dot[ten[[interest[[1]]]],p1n[[i]]]; lu[[i,1]]={dx,dy,dz}; (* do parts, then finish the assignment *) dx=Dot[nor[[interest[[2]]]],pln[[i]]]; dy=Dot[tlt[[interest[[2]]]],plnIIilll; dz=Dot[ten[[interest[[2]]]],p1n[[i]]]; lu[[i,2]]={dx,dy,dz}; (* do parts, then finish the assignment *) dx=Dot[nor[[interest[[1]]]].def[[i]]]; dy=DotIt1tIIinterest[[111]],defIIiIII; dz=Dot[ten[[interest[[1]]]],def[[i]]]; bu[[i,1]]={dx,dy,dz}; (* do parts, then finish the assignment *) 132 dx=Dot[nor[[interest[[2]]]],def[[i]]]; dy=Dot[tlt[[1nterestII2llll,defIIiIJI; dz=Dot[ten[[interest[[2]]]],def[[i]]]; bu[[i,2]]={dx,dy,dz}, {1,1,dimp}]; (* OUTPUT *) Print[ "pair "<>ToStringIN[id[[interest[[1]]]]]]<>"-"<> ToString[N[1d[[interest[[2]]131]]; (* CALCULATE F and its components. *) (* Dummies. *) FF={0,0,0,0,0,0,0,0}; (* dummy assignment. This is the grain boundary fracture factor F *) FFbt={0,0,0,0,0,0,0,0}; (* dummy assignment. this is (twin b dotted with tens.) part of F *) FFsum={0,0,0,0,0,0,0,0}; (* dummy assignment. This is (sum of ord burg dotted with b-twin) part of F *) FFbts={0,0,0,0,0,0,0,0}; (* dummy assignment. this is (twin b dotted with tens. x sum) part of F *) (* outputs in the order of: lst grain 111, -111, 1-11, -1-11 twin systems, then for 2nd grain. *) (* First, calculate everything. *) Do[tmpcnt=(cr-1)*4+(sys+3)/4; FFbt[[tmpcnt]]=Abs[Dot[bu[[sys,cr]],{0,0,1}]],{cr,1,2}, {sys,1,13,4}]; Do[tmpcnt=(cr-1)*4+(sys+3)/4; 133 Do[PFsumEltmpcnt]]= FFsumIItmpcntJI+Abs[DothuIIsys,cr]],bu[[sys2,cr2]]]]. {sys2,2,14, 4},{cr2,1,2}],{cr,1,2},{sys,1,13,4}]; Do[tmpcnt=(cr-1)*4+(sys+3)/4; Do[FF[[tmpcnt]]= FF[[tmpcnt]]+(Abs[(Dot[bu[[sys,cr]],{0,0,1}])* (Dothqusys,cr]],bu[[sys2,cr2]]])]*sch[[sys,cr]]), {sys2,2,14,4},{cr2,1,2}],{cr,1,2},{sys,1,13,4}]; Do[FFbts[[tmpcnt]]=FFsum[[tmpcnt]]*FFbthtmpcntII.{tmpcnt,1,8}]; (* NOW PRINT IT *) Print[ "FF-b.t= "<>ToString[N[FFbt[[111]]<>", "<>ToStringlNEFFbt[[211]] <>", "<>ToString[N[FFbt[[3]]]]<>", "<>ToStringlNEFFbt[[411]]<> ", "<>ToString[N[FFbt[[511]]<>", "<>ToString[N[FFbt[[6]]]]<> H, "<>ToString[N[FFbt[[7111]<>", "<>ToStringINlFFbt[[81111]; Print[ "FF-b.t*sum= "<>ToString[N[FFbts[[1]]1]<>", "<> ToString[N[FFbts[[2]]]]<>", "<>ToStringENIFFbts[[3111]<>", " <>ToString[N[FFbts[[4]]]]<>", "<>ToString[N[FFbts[[511]]<> ", "<>ToString[N[FFbts[[611]]<>", "<>ToStringlNlFFbts[[711]]<> ", "<>ToString[N[FFbts[[8111]]; Print[ "FF= ”<>ToString[N[FF[[1]]]]<>", "<>ToStringlNEFF[[2]]]]<>", "<> ToStringlNlFF[[31]l]<>", "<>ToStr1ng[N[FF[[4]]]]<>", "<> ToStringENEFFll51111<>", "<>ToString[N[FF[[6]]]]<>", "<> ToStringINIFFII71111<>", "<>ToStringENIFFIE8lllll; Print[ 134 "FF-sum= "<>ToString[N[FFsum[[1]]]]<>", "<> ToString[N[FFsum[[2]]]]<>", "<>ToString[N[FFsum[[3]]]]<>", " <>ToString[N[FFsum[[4]]]]<>", "<>ToString[N[FFsum[[5]]]]<>", ' ()ToString[N[FFsum[[6]]J]<>", "<>ToStringENEFFsumE[711]]<>", ' <>ToString[N[FFSum[[8l1]]]; (* Finally, output the Schmid factors for each twinning system. *) Print[ "Schmids= "<>ToStr1ng[N[sch[[1,1]]]]<>", " <>ToString[N[sch[[5,1]]]]<>", "<>ToStringINIschII9,1]]]]<>", " <>ToString[N[sch[[13,1]]]]<>", "<>ToString[N[sch[[1,2]]]]<>", " <>ToString[N[sch[[5,2]]]]<>", "<>ToString[N[sch[[9,2]]]]<>". " <>ToString[N[sch[[13,2]]]]]; ,{poi,1,d1mpair}]; Print[]; (* Now everything is done for each pair. Output grain orientation *) (* summaries for the entire dataset available -- *) Do[Print["Gr."<>ToString[id[[j]]]<>" — "<>"normal= "<> ToString[norm[[j]]]<>", tilt= "<>ToString[tilt[[j]]]<> ", tensile= "<>ToString[tens[[j]]]<> ToString[N[D0t[{0,0,1},ten[[j]]]]]],{j,1,dimg}]; 135 APPENDIX D Grain SACP information 136 Grain 1 Figure D.1. SACP composite for grain 1. The crosses indicate the center of the SACP at a) 0° tilt and b) ~26° tilt. 137 Grain 3 Figure D.2. SACP composite for grain 3. The crosses indicate the center of the SACP at a) 0° tilt and b) ~21° tilt. 138 Grain 4 Figure D.3. SACP composite for grain 4. The crosses indicate the center of the SACP at a) 0° tilt and b) ~24° tilt. 139 Grain 5 Figure D.4. SACP composite for grain 5. The crosses indicate the center of the SACP at a) 0° tilt and b) ~28° tilt. 140 Figure D.5. SACP composite for grain 6. The crosses indicate the center of the SACP at a) 0° tilt and b) ~30° tilt. 141 Grain 7 Figure D.6. SACP composite for grain 7. The crosses indicate the center of the SACP at a) 0° tilt and b) ~30° tilt. 142 Grain 8 Figure D.7. SACP composite for grain 8. The crosses indicate the center of the SACP at 3.) 0° tilt and b) ~30° tilt. 143 Grain 9 Figure D.8. SACP composite for grain 9. The crosses indicate the center of the SACP at a) 0° tilt and b) ~25° tilt. 144 Grain 10 Figure D.9. SACP composite for grain 10. The crosses indicate the center of the SACP at a) 0° tilt and b) ~25° tilt. 145 Grain 11 Figure D.10. SACP composite for grain 11. The crosses indicate the center of the SACP at 3.) 0° tilt and b) ~30° tilt. 146 Grain 12 Figure D.11. SACP composite for grain 12. The crosses indicate the center of the SACP at a) 0° tilt and b) ~30° tilt. 147 Grain 13 Figure D.12. SACP composite for grain 13. The crosses indicate the center of the SACP at a) 0° tilt and b) ~35° tilt. 148 Grain 14 Figure D.13. SACP composite for grain 14. The crosses indicate the center of the SACP at a) 0° tilt and b) ~33° tilt. 149 Grain 15 Figure D.14. SACP composite for grain 15. The crosses indicate the center of the SACP at a) 0° tilt and b) ~35° tilt. 150 Grain 16 Figure D.15. SACP composite for grain 16. The crosses indicate the center of the SACP at a) 0° tilt and b) ~25° tilt. 151 Grain 17 Figure D.16. SACP composite for grain 17. The crosses indicate the center of the SACP at a) 0° tilt and b) ~30° tilt. 152 Grain 18 Figure D.17. SACP composite for grain 18. The crosses indicate the center of the SACP at a) 0° tilt and b) ~33° tilt. 153 Grain 19 Figure D.18. SACP composite for grain 19. The crosses indicate the center of the SACP at a) 0° tilt and b) ~25° tilt. 154 Grain 20 Figure D.19. SACP composite for grain 20. The crosses indicate the center of the SACP at 3.) 0° tilt and b) ~33° tilt. 155 Grain 21 Figure D.20. SACP composite for grain 21. The crosses indicate the center of the SACP at a) 0° tilt and b) ~25° tilt. 156 Grain 22 Figure D.21. SACP composite for grain 22. The crosses indicate the center of the SACP at a) 0° tilt and b) ~25° tilt. 157 Grain 23 Figure D.22. SACP composite for grain 23. The crosses indicate the center of the SACP at a) 0° tilt and b) ~25° tilt. 158 Grain 28 Figure D.23. SACP composite for grain 28. The crosses indicate the center of the SACP at a) 0° tilt and b) ~33° tilt. 159 Grain 29 Figure D.24. SACP composite for grain 29. The crosses indicate the center of the SACP at a) 0° tilt and b) ~25° tilt. 160 Grain 30 Figure D.25. SACP composite for grain 30. The crosses indicate the center of the SACP at a) 0° tilt and b) ~25° tilt. 161 Grain 31 Figure D.26. SACP composite for grain 31. The crosses indicate the center of the SACP at 3.) 0° tilt and b) ~25° tilt. 162 Grain 32 Figure D.27. SACP composite for grain 32. The crosses indicate the center of the SACP at a) 0° tilt and b) ~30° tilt. 163 Grain 33 Figure D.28. SACP composite for grain 33. The crosses indicate the center of the SACP at a.) 0° tilt and b) ~30° tilt. 164 Grain 34 Figure D.29. SACP composite for grain 34. The crosses indicate the center of the SACP at a) 0° tilt and b) ~28° tilt. 165 Grain 35 Figure D.30. SACP composite for grain 35. The crosses indicate the center of the SACP at a) 0° tilt and b) ~29° tilt. 166 Grain 36 Figure D.31. SACP composite for grain 36. The crosses indicate the center of the SACP at a) 0° tilt and b) ~25° tilt. 167 Grain 37 Figure D.32. SACP composite for grain 37. The crosses indicate the center of the SACP at a) 0° tilt and b) ~25° tilt. 168 Grain 38 Figure D.33. SACP composite for grain 38. The crosses indicate the center of the SACP at 3.) 0° tilt and b) ~26° tilt. 169 Grain 39 Figure D.34. SACP composite for grain 39. The crosses indicate the center of the SACP at a) 0° tilt and b) ~27° tilt. 170 Grain 39a Figure D.35. SACP composite for grain 39a. The crosses indicate the center of the SACP at a) 0° tilt and b) ~28.5° tilt. 171 Grain 40 Figure D.36. SACP composite for grain 40. The crosses indicate the center of the SACP at a) 0° tilt and b) ~28° tilt. 172 Grain 41 Figure D.37. SACP composite for grain 41. The crosses indicate the center of the SACP at 3.) 0° tilt and b) ~25° tilt. 173 Grain 42 Figure D.38. SACP composite for grain 42. The crosses indicate the center of the SACP at a) 0° tilt and b) ~30° tilt. 174 Grain 43 Figure D.39. SACP composite for grain 43. The crosses indicate the center of the SACP at a) 0° tilt and b) ~29° tilt. 175 Grain 44 Figure D.40. SACP composite for grain 44. The crosses indicate the center of the SACP at a) 0° tilt and b) ~35° tilt. 176 Grain 45 Figure D.41. SACP composite for grain 45. The crosses indicate the center of the SACP at a) 0° tilt and b) ~30° tilt. 177 Grain 46 Figure D.42. SACP composite for grain 46. The crosses indicate the center of the SACP at a) 0° tilt and b) ~25° tilt. 178 Grain 47 Figure D.43. SACP composite for grain 47. The crosses indicate the center of the SACP at a) 0° tilt and b) ~25° tilt. 179 Grain 48 Figure D.44. SACP composite for grain 48. The crosses indicate the center of the SACP at a) 0° tilt and b) ~25° tilt. 180 Grain 49 Figure D.45. SACP composite for grain 49. The crosses indicate the center of the SACP at a) 0° tilt and b) ~25° tilt. 181 Grain 50 Figure D.46. SACP composite for grain 50. The crosses indicate the center of the SACP at a) 0° tilt and b) ~25° tilt. 182 Grain n1 Figure D.47. SACP composite for grain D1. The crosses indicate the center of the SACP at a) 0° tilt and b) ~20° tilt. 183 Grain n2 Figure D.48. SACP composite for grain n2. The crosses indicate the center of the SACP at a) 0° tilt and b) ~25° tilt. 184 Grain n3 Figure D.49. SACP composite for grain n3. The crosses indicate the center of the SACP at a) 0° tilt and b) ~27° tilt. Grain n4 Figure D.50. SACP composite for grain n4. The crosses indicate the center of the SACP at a) 0° tilt and b) ~25° tilt. 186 Grain n5 Figure D.51. SACP composite for grain n5. The crosses indicate the center of the SACP at a) 0° tilt and b) ~25° tilt. 187 [Il][[l]]l][l[i][][[1]]]