ANALYSIS OF SLIP TRANSFER IN TI - 5AL - 2.5 (WT. %) AT TWO TEMPERATURES IN COMPARISON TO PURE ALUMINUM By Chelsea M. Edge A THESIS Submitted to Michigan State University i n partial fulfillment of the requirements f or the degree of M aterial Science and Engineering Master of Science 2020 ABSTRACT ANALYSIS OF SLIP TRANSFER IN TI - 5AL - 2.5 (WT. %) AT TWO TEMPERATURES IN COMPARISON TO PURE ALUMINUM By Chelsea M . Edge Understanding the deformation mechanisms present near grain bou ndaries in polycrystalline hexagonal alloys will aid in improving modeling methods. Ti - 5Al - 2.5Sn samples were tensile tested at 296K and 728K, and slip behavior was assessed near grain boundari es. From the EBSD measurements of grain orientations, vario us metrics related to the slip systems, traces, residual Burgers vectors, and grain boundary misorientation were computed for boundaries showing evidence of slip transfer and boundaries showing no evidence of slip transfer. This work is compared to a simi lar study of an Aluminum oligo - crystal to aid in understanding the differences in slip behavior near grain boundaries in HCP and FCC crystal structures. Slip transfer in Ti525 was generally obser ved in less geometrically compatible conditions than Al, and sl ip transfer occurs at high misorientation angles in Ti - 5Al - 2.5Sn much more frequently than in Al. iii AKNOWLEDGEMENTS First, I would like to thank my advisor, Dr. Thomas R. Bieler, for his continued support throughout my two years at Michigan State University. He taught me most of what I know about cryst all ography, and the processes used to further understand how material s behave. His previously developed code in Matlab, was the building block for my research, and his patience in helping me understand it was pivotal to my work. Dr. Bieler created a space for us to have open discussions about science and his approachability made for a great learning environment. I would also like to thank Dr. Phillip Eisenlohr for his invaluable insights , specifically regarding programming. He provided a critical role i n h elp ing me understand dislocations and other crystallographic topics. His detailed technical support and knowledge greatly improved the quality of the work presented. I would also like to express my gratitude toward my committee member Dr. Carl Boehlert, f or his participation in my thesis discussion, and constructive critic ism he has provided to complete the work. In addition, Dr. Per Askeland was a great help to me with the completion of EBSD analysis and use of the SEMs. Additionally, I would like to tha nk Hongmei Li, whom I have not met, but know through her work on Titanium alloys, which inspired this project. I would also like to thank the team of graduate students within the metals group that sacrificed their time to help me with my work and understan din g of concepts. Specifically, David Hernandez Escobar for his help with EBSD analysis and Harsha Phukan for h is help on understanding crystallography. iv Finally, I would like to express my deepest gratitude to my family , friends, and Alex, for their const ant support and encouragement throughout the course of this endeavor. v TABLE OF CONTENTS LIST OF TABLES ................................ ................................ ................................ ...................... vii LIST OF FIGURES ................................ ................................ ................................ ................... viii KEY TO ABBREVIATIONS ................................ ................................ ................................ .... xii Chapter 1: Introduction ................................ ................................ ................................ ............... 1 1.1 Rationale and research objective ................................ ................................ ........................... 1 Chapter 2: Literature Review ................................ ................................ ................................ ...... 3 2.1 Fundamentals of Titanium alloys ................................ ................................ .......................... 3 2.1.1 Titanium allotro pes ................................ ................................ ................................ ...................... 3 2.1.2 Dislocations ................................ ................................ ................................ ................................ .. 3 2.1.3 Crystal Structure ................................ ................................ ................................ .......................... 4 2.1.4 Deformation mechanisms ................................ ................................ ................................ ............ 5 2.2 Dislocations and slip tr ansfer across grain boundaries ................................ ......................... 6 2.2.1 Slip transmission ................................ ................................ ................................ .......................... 6 2.2.2 Metrics used to analyze slip transfer ................................ ................................ ............................ 8 2.3 Previous work ................................ ................................ ................................ ...................... 12 2.3.1 Titanium ................................ ................................ ................................ ................................ ..... 12 2.3.2 Aluminum ................................ ................................ ................................ ................................ .. 12 2.3.3 Aluminum and Titanium Crystal structure differences ................................ .............................. 13 Chapter 3: Experimental Procedure ................................ ................................ ......................... 15 3.1 Material ................................ ................................ ................................ ............................... 15 3.2 Mechanical testing ................................ ................................ ................................ ............... 15 3.2.1 Sample preparation ................................ ................................ ................................ .................... 15 3.2.2 Test set up ................................ ................................ ................................ ................................ .. 16 3.2.3 Tensile Test Results ................................ ................................ ................................ ................... 17 3.3 Microstructural Observations ................................ ................................ .............................. 18 3.3.1 Orientation evaluation ................................ ................................ ................................ ................ 19 3.4 Grain pair analysis ................................ ................................ ................................ ............... 21 3.4.1 Slip transfer ................................ ................................ ................................ ................................ 21 3.4.2 Slip transfer analysis ................................ ................................ ................................ .................. 30 Chapter 4: Results ................................ ................................ ................................ ....................... 33 4.1 Misorientation angle histogram ................................ ................................ ........................... 33 4.2 Metric combinations ................................ ................................ ................................ ............ 35 4.3 Misorientation angle vs. ................................ ................................ ................................ . 37 4.4 vs. residual Burger s vector ( ) ................................ ................................ ..................... 38 4.5 Misorientation angle vs. the sum of the Schmid factors ................................ ..................... 40 4.6 Misorientation angle vs. ................................ ................................ ............................ 43 4.7 Slip systems present ................................ ................................ ................................ ............ 45 vi Chapter 5: Discussion ................................ ................................ ................................ ................. 52 5.1 vs. ................................ ................................ ................................ .............................. 52 5.2 Misorientation angle vs. ................................ ................................ ................................ . 55 5.3 Misorientation angle vs. the sum of Schmid factors ................................ ........................... 57 5.4 Misorientation angle vs. ................................ ................................ ........................... 58 5.5 Slip system analysis ................................ ................................ ................................ ............ 60 5.6 Comparison between measured weakly textured grain orientations to random populations ................................ ................................ ................................ ................................ ................... 62 Chapter 6: Summary and conclusions ................................ ................................ ...................... 69 6.1 Summary ................................ ................................ ................................ ............................. 69 6.2 Conclusions ................................ ................................ ................................ ......................... 69 6.3 Future work suggestions ................................ ................................ ................................ ...... 71 APPENDICIES ................................ ................................ ................................ ........................... 72 APPENDIX A: Matlab hexagonal orientation code (written by Thomas R. Bieler) ................ 73 APPENDIX B: Matlab MPR hexagonal grain pair analysis code (written by Thomas R. Bieler) ................................ ................................ ................................ ................................ ................... 83 APPENDIX C: Sample reconstructed boundary and grain files ................................ ............... 99 BIBLIOGRAPHY ................................ ................................ ................................ ..................... 101 vii LIST OF TABLES Table 3.1: Measured composition of polycrystal Ti525 (Li, 2013) ................................ .............. 15 Table 3.2: Ti525 tensile properties obtained from room temperature and high temperature tensile tests (Li, 2013). ................................ ................................ ................................ ............................. 18 Table 3.3: m' values in the upper left - hand corner of the table indicates the slip system pair that is mo st likely transmitting slip. ................................ ................................ ................................ ........................... 25 Table 3.4: Comparison of random poll of data from each sample, and the number/percentages of the ful ly analyzed GBs used in the analysis. ................................ ................................ ................. 31 aries, 53 Table 5.2: Temperature comparison for Al and Ti525. RT Al and HT Ti525 have similar homologous temperatures. ................................ ................................ ................................ ............ 54 Table 5.3: Compensated Strength s and values used to obtain them. The pure aluminum compensated strength is much lower than the Titanium alloy (Aluminum 1100 - O and Li 2013). ................................ ................................ ................................ ................................ ....................... 55 viii LIST OF FIGURES Figure 1 .1: Schematic of a turbo pump showing where Ti525 is commonly used. The highlighted area points toward the part in the turbo pump that is typically Ti525 (Sutto n, 2017). ................... 2 Figure 2.1: (a) Disloc ation present within the Burgers circuit. (b) Burgers vector (purple arrow) from c to b in a perfe ct crystal. ................................ ................................ ................................ ....... 4 Figure 2.2 : Hexagonal crystal structure with shaded area indicating different type s of slip planes. The purple line represents the < > direction in which prism, basal and pyramidal planes can slip. The orange line represents the < > direction in which pyramida l planes can slip. ......... 5 Figure 2.3: Slip traces interacting with a grain boundary. Focusing on case (b), the dislocations are transmi tted from grai n 1 into grain 2, and a residual part of the dislocation in grain 1 is retained in the grain boundary (Bayerschen, 2016). ................................ ................................ ....... 7 Figure 2.4: Representation of slip planes (in blue and green) and the angles associated with them used to calculate various factors (Abuzaid, 2016). ................................ ................................ ......... 8 Figure 2.5: Representation of slip in a cylindrical crystal and the associated angles describing the slip plane a nd slip direction used to compute the Schmid factor. ................................ ................. 10 Fig ure 3.1: Geometry of tensile samples electro - discharge machined out of Ti525 (Li, 2013). .. 16 Figure 3.2: Tensile test set up within the Tescan Mira3 SEM (Li, 2013). ................................ .... 17 Figure 3.3: Stress vs. displacement curve for the room temperature (296K) and high temperature (728K ) samples. Load drops are stress relaxation points when the test was paused for imaging (Li, 2013). ................................ ................................ ................................ ................................ ..... 18 Figure 3.4: SEM image showing the microstructure of (a) the 296K and (b) the 728K tensile t ested Ti525 sample with fiduciary mark. ................................ ................................ ..................... 19 Figure 3.5: EBSD IPF map of the deformed Ti525 tensile samples. The intentional scratch is the fiduciary mark, which is visible in the IPF map for (b) the 728K sample, and is located out of the frame in (a) the 296K sample (black rectangular areas are eliminated data that have very small grains/questionable indexing). ................................ ................................ ................................ ...... 20 Figure 3.6: { } and { } p ole figures for (a) the 296K and (b) the 728K Ti525 samples. The black curved lines represent the 30° cones alon g the major axis. The material has orientations that are around 3 times random in both samples. (Li, 2013). ................................ ... 21 ix Figure 3.7: Computed slip system possibilities and corresponding slip traces for (a ) the left grain and (b) the right grain, drawn in actual orientation present in th e grain. ................................ ...... 23 Figure 3.8: Comparison of computed slip traces and observed slip traces in the SEM image. .... 24 Figure 3.9: Calculated prisms of each grain drawn relative to each other. ................................ ... 26 Figure 3.10: Topography at the grain boundary in (a) is small indicating that there is little to no heterogenous strain on either side. Topography at the grain boundary in (b) is large indicating that there is hetero genous strain on both sides. ................................ ................................ ............. 27 Figure 3.11: (a) The red arrows indicate an upward step sense in the grain (light slip traces in the lower part of grain 1 and dark traces in grain 2 and the upper corner of grain 1). (b) Sketch of cross section along the orange line showing opposite directional sense of shear (blue arrows). Though slip traces may appear to be correlated, their slip planes are far from parallel, indicating that slip transfer cannot account for the obser ved slip traces. ................................ ...................... 29 Figure 3.12: (a) Example of a grain boundary that is categorized as slip transfer, meaning that all five criter ia are met. (b) Exam ple of a grain boundary that is categorized as no slip transfer, meaning all five criteria for slip transfer were not met . ................................ ................................ 30 Figure 3.13: Probability density (the integral is 1) for the Ti525 7 28K and 296K data, and McKenzie probability distribution data for randomly generated hexagonal crystal orientations. 32 temperature (728K) sample and (b) the room temperature (296K) sample, binned by misor ientation angle. The slip transfer category for both samples have higher values at lower misorientation angles. ................................ ................................ ................................ ................... 34 Figure 4.2 ................................ ................... 36 r (a) the room temperature (296K) sample and (b) the high temperature (728K) s ample. There is a decreasing trend for misorientation angles below 20° in both samples. Above 20°, the trend is no l onger observed. ................................ ...................... 38 room temperature (296K) Ti525 sample and (b) the high t dotted lines, have high m' parameters and low residual Burgers vectors. The dashed black lines represent the boundary where are inside and outside of the box respectively. ................................ ................................ ............. 40 x Figure 4.5: The LG +SF RG ) parameter vs. misorientation angle for (a) th e room temperature (296K) Ti525 sample, and (b) the high temperature (728K) sample. High LG +SF RG ) is ransfe misorientation angle. ................................ ................................ ................................ ..................... 42 Figure 4.6: vs. misorientation angle for (a) the room temp erature (296K) sample and (b) the high temperature (728K) sample. In both samples, a strong decreasing ................................ ................................ ............. 44 Figure 4. vs. misorientation angle represented with computed s lip systems for each grain pair in (a) the room temperature (296K) sample and (a) the high temper ature (728K) sample. There is a cluster of prism to prism slip in grain pai rs with misorientation angle less than 30° in the high temperature sample and between 15° and 30° in the room temperature sample. ................................ ................................ ................................ ............. 46 vs. misorientation angle represented with observed active slip systems for each grain pair for (a) the room temperature (296K) tensile te sted sample and for (b) the high temperature (728K) tensile tested sample. Few clusters are observed in both data sets. ................................ ................................ ................................ ................................ ................ 48 Figure 4.9 : Representation of slip systems prevalent in the high temperature (728K) and room temperature sample. The most prevalent slip system pair in the room temperature (296K) data is basal to prism. The slip s ystem pair that shows up the least in all the data i s pyramidal to pyramidal. ................................ ................................ ................................ ........................... 50 Figure 5.1: vs. data for the (a) room temperature (296K) and (b) the high temperature (7 28K) tens ile tested Ti525 sample. Green boxes indicate comparable Al oligo - crystal tensile results at room temperature for both (a) and (b). ................................ ................................ .......... 53 Figure 5.2: Aluminum oligo - data for (a) the room temperature (2 96K) sample and (b) the high temperature (728K) sample. to the aluminum data. ................................ ................................ ................................ .................... 57 Figure 5.3: Mi LG +SF RG ) for the (a) room temperature Ti525 sample and t he (b) high temperature Ti525 sample. The aluminum boundaries represented by the green line show the difference between the aluminum data set which has a steep slope and th e titanium data set which slope is flatter. ................................ ................................ ................................ ....... 58 (b) the high temperature (728K) sample. The red sh aded area represents the aluminum oligo - xi e shaded area represents the aluminum oligo - crystal ........... 60 Figure 5.5: Misorientation angle cumulative p ercentages for Ti525 and randomly generated data sets separated by type of slip system observed in the grains. Pyramidal slip is behaving counter - intuitively with respect to more restrictive conditions. ................................ ................................ . 64 Figure 5.6: O verall Ti525 statistics in comparison to a random population . Random population data sets have a high frequency for pyramidal to pyramidal where experimental results are low for the same category. ................................ ................................ ........................... 65 Figure 5.7: Relative frequency of data represented as half - circles. Larger area half - circle represents a higher frequency. (a) Comparison of experimental Ti525 at room temperature and high temperature. (b) Compariso n of randomly generated d ata sets ................................ ............ 66 Figure 5.8: Randomly generated orientation pairs with the most l ikely slip systems present that facilitates slip across the grain boundaries. (a) Unmodified Schmid factors. (b) Schmi d factor for pyramidal is halved. A high frequency is present in slip system pairs with pyramidal slip systems in both cases. ................................ ................................ ................................ ................... 68 xii KEY TO ABBREVIATIONS Al Aluminum ASTM American Society for Testing and Materials BSE Back scattered electron CP Commercially Pure CPFE Crystal Pla sticity Finite Element Modeling CRSS Critically Resolved Shear Stress EBSD Electron Backscatter Diffraction EDM Electro discharge machined FCC Face centered cubic Fe Iron GBS Grain Bou ndary Sliding HCP Hexagonal close - packed HT High Tempera ture IPF Inverse Pole Figure m Geometric compatibility factor Ni Nickel RT Room Temperature SE Secondary electron SEM Scanning Electron Microscopy SF Schmid factor SFLG Schmid fac tor left grain SFRG Schmid factor right grain xiii SiC Silico n Carbon Sn Tin Ti Titanium Ti525 Ti - 5 - Al - 2.5Sn Wt.% Weight percent XRD X - ray Diffraction Zn Zinc Residual B urgers vector 1 Chapter 1: Introduction 1.1 R ationale and research objective The overarching goal of the research is to further understand th e deformation mechanisms in titanium alloys, specifically Ti - 5Al - 2.5Sn (Ti525). Knowledge of the deformations mechanisms can aid in improving the modeling meth ods of titanium alloys such as crysta l plasticity finite element (CPFE) modeling, to enable more predictive ability to model heterogeneous strain near grain boundar i es. Titanium is a good candidate for the aerospace industry for many factors. One of the mo st important factors are the possibil ities for weight reduction in an engine. Since titanium has a high strength - to - weight ratio, components traditionally made of steel or aluminum could be replaced with titanium, depending on the application. Titanium is also corrosion resistant meaning that coatings are not usually required unless it is in contact w ith an aluminum or a steel component in which galvanic corrosion may occur. Although titanium has many factors that make it appealing, it can cost from 3 to 10 times as much as aluminum or steel ( Boyer, 1996) . Ti525 is mainly used in cryogenic application s. It does not exhibit a ductile - to - brittle transition which aids in high ductility and fracture toughness at low temperatures. Ti525 has been used in the space shuttle, on the hydrogen side of th e high - pressure fuel turbo - pump (Boyer 1996) . Figure 1.1 sh ows a schematic of a turbo - pump, and the location of where Ti525 is commonly used (Sutton, 2017). At elevated temperatures Ti525 has excellent creep resist ance (Boyer, 1996) . 2 Figure 1 . 1 : Schematic of a tu rbo pump showing where Ti525 is commonly used. The highlighted area points toward the part in the turbo pump that is typically Ti525 (Sutton, 2017). 3 Chapter 2: Lit erature Review 2.1 Fundamentals of Titanium a lloys 2.1.1 Titanium allotropes allotrope s , - nominally co mmercially pure titanium, whereas near - - stabilizers such as Sn, or Al ( Li , 2013) . For the purposes of this paper, near - - alloy studied is Ti - 5Al - 2.5Sn (Ti525). 2.1.2 Dislocations Two im portant concepts that need to be u nderstood before moving forward in the paper are the dislocation and Burgers vectors. Dislocations are line defects within a crystal. The B urgers vector describes the relative displacement on the slipped plane . The Burgers circuit is a pathway along atoms, that form a closed loop within which dislocations are located. Figure 2.1 ( a ) shows the Burgers circuit (a - b - c - d - e) and a defect within the Burgers circuit. When you present the same circuit in a perfect crystal, there is a loop closure failure. In order to complete the circuit, a vector from c to b would be necessary, see Figure 2.1 ( b ) . This vector is the Burgers vector ( Hull, 2011) . 4 Figure 2 . 1 : (a) Dislocation present within the Burgers circuit. (b) Bu rgers vector (purple arrow) from c to b in a perfect crystal . 2.1.3 Crystal Structure The crystal structure of the titanium alpha phase is hexagonal close packed (HCP) . Figure 2.2 shows the hexagonal structure. The blue shaded a rea represents the basal planes. T he red shaded area represents prism planes. The yellow/green shaded area represents pyramidal planes. The a 1 , a 2 , a 3, and c directions are represented by black arrows. Slip occurs on close packed planes, in close packed di rections. For hexagonal crystal st ructures slip has been observed in the following directions: < > on { }, { }, and { } planes , and < 3 > on { } plane ( Li , 2013 and Bridier, 2005 ) . 5 Figure 2 . 2 : Hexagonal cryst al structure with shaded area indicating different types of slip planes. The purple line represents the < > direction in which prism, basal and pyramidal planes can slip. The orange line represents the < > direc tion in which pyramidal planes can slip. 2.1.4 Defo r mation m echanisms The critical resolved shear stress (CRSS) determines the ease of slip on a system. Generally, w ith a higher CRSS, the eas e of movement on that plane de creases. Hongemei et. a l. found that at both 296K and 728K for the Ti5 25 alloy, prismatic slip was more easily activated than all other slip systems, including basal, since prismatic slip was found at low Schmid factors. Basal slip was then considered to be the next slip system easily activa ted in both samples ( Li , 20 13) . With different alloys, different slip systems become more active. In near - slip is more active compared to commercially pure (CP) Ti. This is due to the addition of Al and c 6 Sn, in which both increase the c/a ratio from 1.587 in CP Ti to about 1.60, to wards the idea l c/a ratio (1.633) . With a higher c/a ratio, a more closely packed basal plane results, which increases the eas e of basal slip . In addition, with different temperatures, different slip systems become more active. With Al alloyed Ti, a decrease in the CRSS for both basal and prismatic slip systems is indicated with an increase of temperature. ( Li , 2013) . At high temperatures, climb is enabled. Climb is the process in which a dislocation moves out of its slip plane with the aid of diffusion (Hull, 20 13). This means that a dislocation can move to another, possibly more favorable slip plane during high temperature deformation. When dislocations move through the material, in one grain slip is often prevalent on a specific slip system with a spe cific Bur gers vector. If a dislocation moves across a grain boundary into another grain, it travels on another slip system with a different Burgers vector in the neighboring grain . Burgers vectors have a direction and length , depending on if the Burgers v ector is an vector, vector or a vector, the length will vary. For the purpose of this thesis , only the Burgers vector direction s were analyzed. That is, the Burgers vectors used are unit vectors. 2.2 Dislocations and slip transfer across grain boundarie s 2.2.1 Slip transmission One of the goals for this thesis is to identify and analyze slip transfer across grain boundaries. Slip traces interact with grain boundaries in a few different ways in polycrystalline metal s. Figure 2.3 depicts possible cases in which slip is transmitted. In case (a) dislocations are stopped by the grain boundary, do not transmit, but pile up at the grain boundary. Case (b) (our focus) is whe n dislocations are emitted from grain 1 in to grain 2 through the grain boundary , and 7 residual Burgers vector content is left in the boundary . In another example, dislocations can be dissociated into the grain boundary, leaving no dislocation s in grain 2 (c ase (c)). Case (d) shows full transmission o f slip. If a full transmission of slip has occurred (i.e. perfect alignment of the slip directions), theory suggests that no residual Burgers vector is left in the grain boundary. Case (e) shows where two disloca tions meet at a grain boundary and generate a new dislocation in the grain boundary. Case (f) depicts a reflection of the dislocation in grain 1 back into grain 1 and leaving a dislocation in the grain boundary (Bayerschen , 2016 ). Figure 2 . 3 : Slip traces interacting with a grain boundary. Focusing on c ase (b), the dislocations are trans mitted from grain 1 in to grain 2, and a residual part of the dislocation in grain 1 is retained in the grain boundary (Bayerschen, 2016) . The case that wil l be discussed is case (b) where perfect slip transfer does not occur across the grain boundary , implying that the Burgers vector changes direction from one grain to the next, resulting in some residual Burgers vector ( ) left in the grain boundary (see Figure 2.3 ( b ) ). The better the alignment of the slip systems in grain 1 and grain 2, the more likely the transmission of the dislocation in to grain 2 will occur , and a smaller residual component of slip would be retained in the grain boundary. This suggests that minimizing the size of the residual 8 Burgers vector would make a slip transmission event more likely , as observed and argued by Shen et al. (1989) and Lee et al. (1988) . 2.2.2 Metrics used to analyze slip transfer The direction and relative size of the residual Burgers vector can be estimated using data obtained from electron backscatter diffraction (EBSD) analysis. Knowing the grain orientation , the Burgers vec tor of the left grain ( ) , which we assume to be the initiating slip system, and the right grain Burgers vector ( ) is the transmitted slip , the residual Burgers vector, is estimated by the following dislocation reaction equation : (1) Figure 2.4 shows the Burgers vectors ( ) and ( ) on two different slip planes, and the angles associated with the transmission. Figure 2 . 4 : Representation of slip planes (in blue and green) and the angles associated with them used to calculate various factors (Abuzaid, 2016) . Grain Boundary 9 The geometric compatibility factor, , is used as a criterion to determine if slip transfer is likely to occur. The factor can be calculated from two angles and , the angle between the normal to the slip plane, and the angle between the two slip directions in grains 1 and 2, respectively (see Figure 2.4 ). Therefore, the factor resolves the strain from th e slip on grain 1 onto grain 2. The factor is calculated as follows (Luster, 1994) : (2) An value closer to 1 would imply that slip transfer is more likely to occur on the specified slip systems , as they would be nearly co l linear , as opposed to a slip system pair that has The misorientation angle is determined from the orientations of each grain through EBSD analysis. The misorientation between two grains is computed based upon the crystal orientations o f each grain, g A for the crystal orientation of grain 1 and g B for the crystal orientation of grain 2. The misorientation is defined as the rotation needed to bring the orientation of grain A into coincidence with the orientation of grain B. The calculatio n for misorientation ( ) is as follows: about which the rotation happens ([n 1 , n 2 , n 3 ]) is calculated, shown below ( Koc ks, 1998 ) . 10 For hexagonal crystal structures, there are 12 symmetric orientation matrices. The OIM software, from the above ([n 1 , n 2 , n 3 ]) for all 12 variants of symmetric rotations axes. The smallest misorientation angle is then chosen as the misorientation angle with the associated rotation axes. In pre vious work, the resolved shear stress based upon the global stress tensor was used as a criterion to determin e if slip is likely. The Schmid factor is used as a metric to determine the mos t likely slip system s that are activated in the two neighboring grains. A higher Schmid factor implies that slip is more likely to occur. The Schmid factor is determined by two angles, , when there is unidirectional stress as described in Figure 2.5 (Hull, 2011) . Figure 2 . 5 : Representation of slip in a cylindrical crystal and the associated angles describing the slip plane and slip direction used to compute the Schmid factor. 11 The Schmid factor is calculated by the following equation, using the angles described i n Figure 2.5 (Hull, 2011) : (3) Slip systems with higher Schmid factors will facilitate slip more easily. Since slip will be likely with a high and high Schmid factors based upon another metric worth considering ( Bieler, 2019, Aliz adeh 2020) , given by: (4) where SF LG is the Schmid factor in the left grain, and SF RG is the Schmid factor in the right grain. Another useful parameter is , as slip transfer is expected to be facilitated with high and low , where is the residual B urgers vector (Alizadeh 2020 , Bayerschen 2016, Shen 1988 , Lee 1989 ) . 12 2.3 Previous work 2.3.1 Titanium The work done in this thesis is built on the work published by Hongemei Li (2013) , r of the hexagonal close - packed alpha phase in titanium Tension and creep samples were tested at low a nd high temperatures for a variety of alloys . For the purpose of this paper, the interest lies in the Ti525 alloy. For each sample, d istributions of active slip systems in each grain were calculated. Basal and prism slip were found to be dominant in the Ti525 alloy. With the increase in tem perature , basal active slip systems increased in frequency . This increase in ease of slip in basal slip systems correlates to a lower CRSS . Prismatic slip systems for Ti - CRSS with high tem perature. Building upon Hongmei Li w ork, further investigation of the tensile tested Ti525 sample at 296K and 728K was considered. The focus was to look at grain pairs and determine if there was slip transfer across the grain boundary. Dividing the grain pairs into slip transfer and no slip transfer categories and using the various parameters that have been discussed , allowed for trends in slip transfer to be uncovered. 2.3.2 Aluminum A similar activity was done by Alizadeh (2020) deh found that slip transfer usually was associated with high values. Slip transfer is favored by low angle bou ndaries, which have high values due to the low misorientation geometry . Further , Alizadeh assessed the LG +SF RG ) , and the par ameter s vs. the misorientation angle to observed trends. Also, Linne and Daly examined a similar specimen of pure Al at 190 C using high resolution differential image correlation (DIC) 13 ( Linne 2020). It is hypothesized that similar trends observed in the Al data set s will be present in the Ti525 data set, with more complicated slip systems. The difference in cry stal structure may limit the similarities noticed between the Al face centered cubic ( FCC ) crystal structure and the Titanium hexagonal close pack ed ( HCP ) crystal structure. 2.3.3 Aluminum and Titanium Crystal structure differences Aluminum has face - centered cubic (FCC) crystal structure. Slip occurs on the close - packed planes along the close - packed directions, which in FCC is {111} and <110> respectivel y. Therefore, there are four close packed planes, each with 6 <110> directions, b ut positive and negative directional sense are not distinguish ed , meaning a reduction of a factor of t wo. Therefore , there are 12 slip systems in FCC crystal structure, all with equal ease of operating ( Jackson, 1991 ) . In contrast, titanium has a hexagonal crystal structure, where slip occurs on basal, pyramidal, pyramidal and prism slip systems . There are thre e slip systems each on basal and prism planes that are easily activated ( Li , 2013 ), but they do not enable changes in crystal dimension in the direction, so easy slip cannot enable needed shape changes in the direction . There are an additio nal 18 slip systems on pyramidal planes but slip on pyramidal planes is much mor e difficult to activate, especially in the directions, so higher resolved shear stresses are required ( Li , 2013) . Therefore, titanium alloys have fewer slip system s that facilitate eas y slip compared to FCC crys ta ls , even though there are many more slip systems available. To assess the difference between Al and Ti alloy slip transfer behavior, the methodology developed by Alizadeh et al. is used on a polycrystalli ne Ti alloy. This work will also examine if this methodology can yield simi lar statistical information in a standard polycrystal sample rather than an oligo - crystal foil . 14 The following chapters will explain the experiments done, the results obtained, and the observed trends in the 296 K and 728 K tensile tested Ti525 samples. Analysis was done on slip transfer data to further the investigation that Hongmei Li presented. The t rends observed in the lts of an aluminum alloy slip transfer data to compare differences in F CC and HCP materials. 15 Chapter 3: Experimental Procedure The description of the material and tensile tests on the Ti - 5 - Al - 2.5Sn alloy is summarized blications. 3.1 Material The Ti - 5 - Al - 2.5Sn (Ti525) alloy was provided by Pratt & Whitney, Rocketdyne. It was rial was then annealed at 1127K for 1 hour for recrystallization, followed directly by air cooling, followed by a vacuum annealing process at 1033K for 4 hours to reduce the hydrogen content. Table 3.1 denotes the measured bulk composition of the alloy in weight percent ( Li , 2013). Table 3 . 1 : Measured composition of polycrystal Ti525 ( Li, 2013) Element Al Sn Fe Zn Ti Weight % 4.7 2.7 0.2 0.1 balance 3.2 Mechanical testing 3.2.1 Sample preparation Two specimens were prepared by a mechanical polish, using 400, 600, 1200, 2400 and 4000 Silicon Carb ide (SiC) grinding papers sequentially for 5 to 10 minutes each. Between each grinding step, the sample was rinsed with water. After grinding, a final polishing step with polishing clo th from Buehler (catalog No. M500 - 12PS) and five parts colloidal silica wit h 0.06µm particle size and one part 30% hydrogen peroxide. This step took around an hour to obtain the desired finish. Water was again used to rinse the sample after polishing then 16 ultrasonically cleaned with acetone and methanol respectively to remove th e colloidal silica ( Li , 2013). Multiple dog - bone samples were electro - discharge machined (EDM) from the forging with a gage width of 3mm and 10mm length. Figure 3.1 shows the geometry of the EDM samples. Figure 3 . 1 : Geometry of tensile samples electro - discharge machined out of Ti525 ( Li , 2013 ) . 3.2.2 Test set up Th e samples examined were in - situ tensile tested within the Tescan Mira3 SEM, with a displacement rate of 0.004 mm/s (approximate strain rate of 10 - 3 s - 1 ). The tensile stage was built by Ernest F. Fullam, Incorporated. Displacement time and load data was rec orded using the MTESTW (version F 8.83) control software. To obtain the desired temperature of the sample, a 6mm diameter tungsten - based heating unit was used. One sample was tested at 296K and another a t 728K. The sample was held at this temperature for 3 0 minutes prior to loading. During testing, the temperature of the sample was monitored using a thermocouple that was welded to the side of the gage section. Throughout the test, a vacuum below 2 ×10 - 6 To rr was maintained. 17 Figure 3 . 2 : Tensile test set up within the Tescan Mira 3 SEM ( Li , 2013) . 3.2.3 Tensile Test Results Figure 3.2 shows the set up within the Tescan Mira3. The stress - displacement cur ves are presented in Figure 3 . 3 . The load drops shown are due to pausing the test for image acquisitions . Table 3.2 gives the yield strength and m aximum stress obtained from the stress displacement curves ( Li , 2013). 18 Figure 3 . 3 : Stress vs. displacement curve for the room temperature (296K) and high temperature (728K) samples. Load drops are st ress relaxation points when the test was paused for imaging ( Li , 2013). Table 3 . 2 : Ti525 tensile properties obtained from room temperature and high temperature tensile tests ( Li , 2013) . Tempera ture Yield Strength (MPa) Max Stress (MPa) 296K ~660 769 728K ~3 0 0 434 3.3 Microstructural Observations Figure 3.4 ( a ) and ( b ) show representative SE SEM images of the deformed Ti525 in - situ 296 K and 728 K tensile tested sample. The scratch in the figure is a fiduciary mark so the location of the EBSD analyzed area was easier to find. 19 Figure 3 . 4 : SEM image showing the microstructure of (a) the 296 K and (b) the 728 K tensile tested Ti525 sample with fiduciary mark. 3.3.1 Orientation evaluation After deformation, EBSD orientation maps were created from a portion of the gage section, marked with a fiduciary mark. The maps were obtained using OIM Analysis TM Version 7.3.1, by EDAX. Figure 3.5 ( a ) and ( b ) show the inverse pole figure (IPF) orientation maps with a variety of colors observed, meaning the samples are not st rongly textured. For the room temperature sample, ( Figure 3.5 ( a ) ) the fiduciary mark is located just out of the frame to the left of the image and in Figure 3.5 (b), the high temperature sample, the fiduciary mark is cropped out (shown as a black stripe) . The black rectangles in the room temperature sample are areas where the EBSD software determined many small grai ns that were probably artifacts of poor indexing , so they were cropped out of the analysis. Clean - up procedures including one or two rounds of near - neighbor correlation and one round of dilation was used to eliminate small grains from the analysis. This provided a convenient reconstructed boundary file to analyze. Fiduciary mark s (a) (b) 20 Figure 3 . 5 : EBSD IPF map of the deformed Ti525 tensile samples. The intentional scratch is the fiduciary mark , which is visible in t he IPF map for (b) the 728K sample, and is located out of the frame in (a) the 296K sample (black rectangular areas are eliminated data that have very small grains/questionable indexing) . Fiduciary mark (a) (b) 21 Figure 3.6 (a ) and (b) shows the pole figures for both the 296K and 728K data, respectively. The dark lines represent the 30° cones along the major axis. The maximum value in the legend indicates that the orientation of the material after deformation is ~3 times ran do m for both samples. Figure 3 . 6 : { } and { } p ole figures for (a) the 296K and (b) the 728K Ti525 samples. The black curved lines represent the 30° cones along the major axis . The material has or ientations that are around 3 times random in both samples. ( L i, 2013). 3.4 Grain pair analysis 3.4.1 Slip transfer To determine if slip transfer was present through grain boundaries, an in - depth look at slip traces near the grain boundary was necessary. If t here were correlated slip traces in one grain ( a ) Room Temperature ( b ) High Temperature 22 and a neighboring grain, a set of criteria were considered to deter mine if slip transfer accounted for the correlated slip traces. These criteria are: (1) D etermination of probable slip systems which would be consistent with observed slip traces . (2) A value associated with the observed correlated slip systems gen erally larger than 0.7 . (3) T he residual burgers vector ( ) associated with the observed correlated slip systems is generally smaller than 0.5b . ( 4 ) T he Schmid factor of each slip system is generally larger than 0.25, ( 5 ) the topography at the grain boundary is small indicating that the boundary does not lead to heterogenous strain on both sides . ( 6 ) T he observed slip traces on each sid e have a topographical directional sense that implies that the slip planes are approximately parallel. To determine if probable slip systems are consistent with observed slip traces (criterion (1)), the Euler angles for each grain in a grain pair were ent ered into a Matlab code (shown in Appendix A : Matlab hexagonal orientation code (written by Thomas R. Bieler) ) . This code draws the unit cell in the actual orientation that is present in the grain. The unit c ell is drawn multiple times with different slip systems in order of decreasing Schmid factor. It also draws the slip trace th at would be expected if this slip system was active. Figure 3.7 sh ows the output of this code for two grains . 23 Figure 3 . 7 : Computed slip system possibilities and corresponding slip traces for (a) the left grain and (b) the righ t grain , drawn in actual orientation present in the grain. Code generated s lip traces are then compared to the SEM image of the grain, to determine what computed slip trace best matches the observed slip trace in the SEM image. Figure 3.8 shows an example of the comparison, where the trace of slip system number 11 24 matches closely with the observed slip traces in the left grain and similarly, the trace of slip system number 5 matches the obser ved slip traces in the right grain. In cases where there are multiple slip systems on the same plane (on pyramidal and basal planes), the slip direction with the highest Schmid factor is chosen. Figure 3 . 8 : Comparison of computed slip traces and observed slip traces in the SEM image. criteria (4) (low residual B urgers vectors) are met, another code was used (also written by Dr. Thomas R. Bieler , shown in Appendix B ). The code reads the reconstructed boundary file (which includes the misorientation of each boundary) and grain file generated by the OIM Analysis software . Sample fil es are shown in Appendix C . The probable slip conditions present and the Schmid factors are determined for each grain. Next, the code computes grain boundary parameters, values and residual Burgers vectors ( ) for ea ch slip system pair possibility. Thresholds values are set up to filter out slip system possibilities that are unlikely to be 25 meaningful, such as Schmid factors lower than 0.2 and values below 0.6 . Tabl e 3.3 is then generated for each slip system pair in order of decreasing Schmid factor. The slip system number is shown on the far most left column for the left grain, and upper most row for the right grain. The corresponding number (right or below, respe ctively) is the corresponding Schmid factor. The values in the middle of the table are the metrics for each corresponding slip system pair, and high values of values are bolded. Values that are bolded, and in the upper left - hand corner of the t able, are the best candidates for slip transfer because they have high values, low values, and high Schmid factors. As with the other code, prisms of each grain are drawn in their relative positions to each other, and slip systems are illustrated for slip system pairs in order of decreasing values , as illustrated for the grain pair in Figure 3.9 . I f there are multiple slip systems that have slip traces that look like they could be present in th e SEM image of the grain, this table is used to determine which one is most likel y to account for the observation. Table 3 . 3 : v alues in the upper left - hand corner of the table indicates the slip system pair that is most likely transmitting slip . 26 Figure 3 . 9 : Calculated p risms of each grain drawn relative to each other. To det ermine if criterion (5) was met, further observations of the grain boundary were made to determine if a large ledge (indicated by topographic contrast) or small ledge was present. Figure 3.10 (a) shows an example of a small (or non - observable) ledge at the grain boundary, while Figure 3.10 (b) shows an example of a larger ledge at the grain boundary that indicates that heterogeneous strain occurred on bo th sides. 27 Figure 3 . 10 : Topography at the grain boundary in (a) is small indicating that there is little to no heterogenous strain on either side. Topography at the grain boundary in (b) is large indi cating that there is heterogenous strain on both sides. To determine if the observed slip traces on each side have a topographical directional sense that implies that the slip planes are nearly parallel (criteria ( 6 )), the surface topogra phy of the neighb oring grains w as assessed further. By considering surface topography, from the sense 1 , the topographic shape of the slip steps was identified in each grain. If the slip steps did not appear parallel, le ading to contrasting dark 1 or the image. Gra in boundary (large ledge) Grain bou ndary (small ledge) (a) (b) 28 and light steps on either side of the boundary, the likelihood of slip transfer was low because the sense of shea r is in different directions as illustrated in Figure 3.11 (a) and (b), and ther efore this criteria was not met . Figure 3.11 (a) shows the top view of two grains, as seen in the SEM. The red lines indicate the upward direction of steps. The orange line shows where a cross section is depicted in Figure 3.11 (b), where the same red arrows show upward steps. It also shows the sense of shear in each grain which are far from parallel. If the slip steps were generally parallel, the criteria would be met. 29 Figure 3 . 11 : (a) The red arrows indicate an upward step sense in the grain (light slip traces in the lower part of grain 1 and dark traces in grain 2 and the upper corner of grain 1). (b) Sketch of cross section along the orange line showing opposite directional sense of shear (blue arrows). Though slip traces may appear to be correlated, their slip planes are far from parallel, indicating that slip transfer cannot account for the observed slip trace s. Grain 1 Grain 2 Grain Boundary (a) ( b ) 30 If all five criteri a were met, the grain boundary was categorized as a case of slip transfer. If one or two criteria were not met, the boundary was categorized as possible slip transfer, or if more criteri a w ere not met, the grain boundary was categorized as a case of no slip transfer. An example of a case of slip transfer is shown in Figure 3.12 ( a ) . Figure 3. 12 ( b ) shows an example of a boundary where all five criteria are not met, therefor it is categorized as a case of no slip transfer. Figure 3 . 12 : (a) Example of a grain boundary that is categorized as slip transfer, meaning that all five criteria are met. (b) Example of a grain boundary that is categorized as no slip transfer, meaning all five criteria for slip tra nsfer were not met. 3.4.2 Slip transf er analysis To obtain a representative set of data from the samples, it was necessary to understand what types of grain boundaries were present in each sample. 50 random grain boundaries were chosen and identified as transfer slip transfer ip transfer as explained in section 3.4 : Grain pair analysis . Of these, 8% was determined to fall into the for both the room temperature and high temperature samples. The percentages of the random poll of grain b oundaries is shown in Table 3.4 . Grain boundaries were fully analyzed until the percentages matched the random sampling data , to ensure that the analyzed grain boundaries were representative of the w hole sample . Grain Boundary Grain Boundary (a) ( b ) 31 In the room temperature sample, 146 grains were analyzed and divided into the three groups , maybe slip transferred study and consisted of 10 grain boundaries. 73 grain boundaries were identified as slip transfer . The not slip transfer group consisted of around 6 8 grain boundaries. In the high temperature sample, around 173 grain boundaries were analyzed and d ivided into the three groups. The transferred grain boundary group consisted of 93 grain boundaries. transferred 78 gra in boundaries. The rest of the grains in the transferred was not used in the data analysis. Table 3 . 4 : Comparison of random poll of data f r o m each sample, and the number/percentages of the fully analyzed GBs used in the analysis. Slip transf er GBs Slip transfer GBs (%) Not slip transfer GBs Not slip transfer GBs (%) RT Random Poll from Ti525 24 52% 22 48% RT Ti525 fully analyzed 73 52% 68 48% HT Random Poll from Ti525 25 54% 21 46% HT Ti525 fully analyzed 93 54% 78 46% Figure 3.13 shows a probability of the misorientation angles collected from the Ti525 high temperature and room temperature data. In addition, random orientations likely for hexagonal crystal structure were generated and plotted against the Ti525 data. T here are some differences , specifically in misorientation angles less than 40°, with the Ti525 data having a higher probability. 32 Figure 3 . 13 : Probability density (the inte gral is 1) for the Ti525 728K and 296K data, and McKenzie probability distribution data for randomly generated hexagonal crystal orientations. 33 Chapter 4: Results This chapter presents the data from the grain boundaries that were fully analyzed, including metrics di scussed in the Chapter 2: Literature Review . 4.1 Misorientation angle histogram This section presents the data collected from the room temperature (296K) tensile tested sample and the high temperature (728K) tensile tested sample. From the orientation maps, the misorientation angles, values, Schmid factors, and residual Burgers vec tors ( ) for the grain pairs were extracted as explained in Chapter 3: Experimental Procedure . Each grain boundary was sorted to either slip transfer or no slip transfer categories. Figure 4.1 shows a histogram of the slip transfer and no slip transfer points plotted with respect to the misorientation angle between grains for (a) the room temperature Ti525 sample, and (b) the high temperature sample. Below 30° misorientation, slip transfer data are significantly higher than no slip transfer data in both sa mples. The cumulative percentage lines slip transfer no slip transfer tation angles. 34 Figure 4 . 1 : Histograms representing the slip transfer and no slip transfer data for the (a) high temperature (728K) sample and (b) the room temperature (296K) sample, binned by miso rientation angle. The s lip transfer category for both sample s have higher values at lower misorientation angles. 0.00% 20.00% 40.00% 60.00% 80.00% 100.00% 0 2 4 6 8 10 12 14 16 10 20 30 40 50 60 70 80 90 100 Misorientation angle (deg) (a) Room temperature sample 0.00% 20.00% 40.00% 60.00% 80.00% 100.00% 0 2 4 6 8 10 12 14 16 10 20 30 40 50 60 70 80 90 100 Number Misorientation angle (deg) (b) High temperature sample 35 4.2 M etric combinations T o determine which metric combinations were of interest, a matrix of all the combinations was created . Figure 4.2 shows a matrix with the possible correlations for (a) the room temperature Ti525 sample, and (b) the high temperature sample. In both matrices, metric pairs with interesting trends include vs. , vs. misorientation angle, and vs. misorientation angle. 36 Figure 4 . 2 : Possible combinations of factors, with (a) ( b ) 37 4.3 Misorien tation angle vs. Figure 4.3 shows the trend of the slip transfer and no slip transfer data for misorientation vs. the geometric compatibility factor ( ), for (a) the room temperature (296K) sampl e and (b) the high temperature (728K) sample. Below 20° misorientation angle in both the downward as the misorientat ion angle increases. Above 20° , the points begi n to spread out, and the downward sloping trend is no longer noticeable. The room temperature sample has 3 points with a sample does not have as strong of a trend comp no slip transfer no slip transfer data otherwise. 38 Figure 4 . 3 : vs. misorientation angle for (a) the room temperature (296K) sample and (b) the high temperature (728K) sample. There is a decreasing trend for misorientation angles below 20° in both samples. Above 20° , the trend is no longer observed. 4.4 vs. resi dual B urgers vector ( ) The Luster - Morris parameter ( ) vs. the residual Burgers vector ( ) is plotted in Figure 4.4 for (a) the room temperature (296K) sample and (b) the high temper ature (728K) sample. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 20 40 60 80 100 m' Misorientation angle (deg) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 20 40 60 80 100 m' Misorientation angle (deg) (a) Room t emperature (b) High t emperature 39 The black dotted lines are plotted such that the maximum number of slip transfer points are inside the box and the maximum number of no slip transfer point s are outside the box. For the room temperature sample , 64% of the slip tra nsfer points are located inside the black dashed box, and 94% of the no slip transfer points are outside the dashed box. For the high temperature sample , 91% of the slip transf er points are located inside the black dashed box, and 68% of the no slip transfer points are outside the dashed box. The high temperature data shows a much larger black dotted box compared to the room temperature data , and the relative percentag es of the two populations inside and outside the box are reversed . slip trans fer data are clustered in the lower right - hand corner of the graph, i.e. at no slip transfer out, and there are many points within the lower right box for the high temp erature data. This indicates that the slip transfer happens more frequently residual Burger vectors in the high temperature sample. 40 Figure 4 . 4 : vs. the residual Burgers vector for (a) the room temperature (296K) Ti525 sample and (b) the high temperature (728K) sample. The cluster slip transfer dotted lines, have high m' parameters and low residual Burgers vectors. The dashed black lines represen t the boundary where the maximum number of slip transfer and no slip transfer data are inside and outside of the box respectively. 4.5 Misorientation angle vs. the sum of the Schm id factors As described in Chapter 2: Literature Review , at high and high Schmid factors, slip transfer is likely to occur. The sum of the t ( LG +SF RG ) ) is plotte d against the misorientation angle in Figure 4.5 ( a) the room temperature (296K) data and (b) the high temperature (728K) data, where SF LG and SF RG is the Schmid factor of the left grain and right grain r espectively. In the room temperature sample above the purple lin e lies the maximum slip transfer data (~75%) and below lies the maximum no slip transfer data (~82%). In the high temperature sample above the purple solid line lies the maximum s lip transfer data (~94%) and below lies the maximum no slip transfer data 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 0 0.2 0.4 0.6 0.8 1 b m' (a) Room temperature 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 0 0.2 0.4 0.6 0.8 1 b m' (b) High temperature 41 (~56%). Both thresholds in the high temperature and room temperature sample are increasing lines, lyin g between 0.4 and 0.8 LG +SF RG ) no slip transfer a larger spread of LG +SF RG ) slip transfer LG +SF RG ) closer to 1 for slip transfer between 35° and 50° misorientation that have a LG +SF R G ) value below 0.6. This cluster is not present in the room temperature data. 42 Figure 4 . 5 : The LG +SF RG ) parameter vs. mi sorientation angle for (a) the room temperature ( 296K) Ti525 sample, and (b) the high temperature (728K) sample. High LG +SF RG ) is noted for slip transfer data in both cases. There is a decreasing trend in slip transfer data below 30° misorientatio no slip transfer above 30° misorienta tion angle. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 20 40 60 80 100 LG +SF RG ) Misorientation angle (deg) 75% 82% (a) Room temperature 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 20 40 60 80 100 LG +SF RG ) Misorientation angle (deg) 94% 56% 43 4.6 Misorientation angle vs. Chapter 3 identifies another factor worth looking at graphically, , where is the residual Burgers vector. Figure 4.6 shows the relationship between mis orientation angle and . In Figure 4.6 (a) the room temperature (296K) sample, above the purple line (which is constructed t o have maximum slip transfer points above, and maximum no slip transfer points below) is 90% of the slip transfer data, and below to which has 60% of the no slip tr ansfer data. Figure 4.6 (b), the high temperature ( 728 K) sample, above the purple solid line is 63% o f the slip transfer data, and below the purple solid line is 84% of the no slip transfer data. There is a decreasing trend for slip transfer data below the 30° misorientation angle for both the high temperature (728K) data and the room temperature (295K) data . The high temperature data set is observed to have a stronger trend in this region, as the data is more no slip transfer slip transfer data have a larger spread but there appe ars to be a threshold for the slip transfer data (above = 1 ), while the no slip transfer data are sca no slip transfer slip transfer to the high temperature data. 44 Figure 4 . 6 : vs. misorientation angle for (a) the room temperature (296K) sample and (b) the high temperature (728K) sample. In both samples, a strong decreasing trend for slip transfer data below 30° misorientation angle . 0.001 0.01 0.1 1 10 100 0 20 40 60 80 100 Misorientation angle (deg) (a) Room Temperature 74% 78% 0.001 0.01 0.1 1 10 100 0 20 40 60 80 100 m'/ Misorientation angle (deg) (b) High Temperature 76% 80% 45 4.7 Slip systems present Each datum point presented thus far has two grains associated with it, each with its own favored/active slip systems. Figure 4.7 and Figure 4.8 represents the same data presented in Figure 4.6 but with the slip systems computed to be present distinguished by color and shape. In Figure 4.7 (a ), the slip transfer data for the room temperature (296K) sample, there is a cluster of prism to prism slip transfer between 15° and 30° misorientation angle is noted. A cluster of prism to basal slip above 80° misorientat ion angle is also noted. Basal to basal favor ed slip is common at lower misorientation angles. In Figure 4.7 (b), the slip transfer for the high temperature (728K) sample, there is a cluster of prism to prism slip transfer in grain pairs with misorientation angle less than 45°. There is an observed cluster of prism to pyramidal slip transfer between 20° and 30° misorientation angle. Around 45° misorientation angle there is another cluster of prism to pyramidal. A cluster of basal to pyramidal between misorientation angles 45 ° to 60° is also observed . 46 Figure 4 . 7 Slip transfer vs. misorientation angle represented with computed slip systems for each grain pair in (a) the room temperature (296K) sample and ( a ) the high temperature (728K) sample. There is a cluster of prism to prism slip in grain pairs with mi sorientation angle less than 30° in the high temperature sample and between 15° and 30° in the room temperature sample. 0.1 1 10 100 0 10 20 30 40 50 60 70 80 90 100 m'/ b Misorientation angle (deg) basal basal + prism basal + pyr basal + pyr prism prism + pyr prism + pyr pyr (a) Room temperature 0.1 1 10 100 0 10 20 30 40 50 60 70 80 90 100 m'/ b Misorientation angle (deg) basal basal + prism basal + pyr basal + pyr prism prism + pyr prism + pyr pyr pyr + pyr pyr (b) High t emperature 47 Figure 4.8 no slip transfer with colors and shapes representin g observed slip systems in the two grains for (a) the room temperature ( 2 96K) sample, and (b) the high tempera ture (728K) sample . For the room temperature sample Figure 4.8 (a), there is a cluster of pyram idal slip systems between 40° and 50° misorientation angle. There are two clusters of prism + pyramidal between 50° and 60° miso rientation angle and 75° to 90° misorientation angle. In the high temperature sample Figure 4.8 (b), below 25° misorientation angle, both points are prism to prism slip. There is no other trend observed in the data, as the data are sporadic above 25° misorientation ang le. 48 Figure 4 . 8 : No slip transfer vs. misorientation angle represented with observed active slip systems for each grain pair for (a) the room temperature (296K) tensile tested sample and for (b) the high temperature (728K) ten sile tested sample. Few clusters are observed in both data sets. 0.001 0.01 0.1 1 10 100 0 20 40 60 80 100 Misorientation angle (deg) basal basal + prism basal + pyr basal + pyr prism prism + pyr prism + pyr pyr pyr + pyr pyr (a) Room temperature 0.001 0.01 0.1 1 10 100 0 20 40 60 80 100 m'/ b Misorientation angle (deg) basal basal + prism basal + pyr basal + pyr prism prism + pyr prism + pyr pyr pyr + pyr pyr (b) High t emperature 49 Figure 4.9 presents statistics comparing prevalent slip systems for slip transfer and no slip transfer data in the (a) room tem perature ( 296 K) sample and (b) high temperature (728K) samples. The relative area of the half circle represents the frequency of that slip system pair. For the room temperature sample, in both the slip transfer and no slip transfer data sets, basal to prism has the highest frequency. For the h igh temperature slip transfer data set, prism to prism and prism to pyramidal are both the most prevalent. In t he high temperature no slip transfer data, prism to pyramidal is most prevalent. As a group, the most prevalent slip system pair is basal to prism and the least prevalent slip system pair is pyramidal to pyramidal. The high temperature sample has a significant increase in pyramidal slip transfer data c ompared to the ro om temperature sample. 50 Figure 4 . 9 : R e presentation of slip systems prevalent in the high temperature (728K) and room temperature sample. The most prevalent slip system pair in the room temperature (296K) data is basal to prism. The slip system pair that shows up the least in all the data is pyramidal to pyramidal. 51 The chapter has provided an extensive assessment of slip transfer in the high temperature and room temperature tensile tested Ti525 alloy. Using parameters such as the misorientation angle ( ) , Schmid factor, and residual Burgers vector ( ) , the slip transfer and no slip transfer data were compared . Low , and high grain pairs are more li kely to have slip transfer present. The parameter vs. the misorientation angle shows a strong decreasing trend for slip transfer data below 30° misorientation angle. The trends found in this section have some differences from r oom temperature to high temperature tensile testing , which includes the high temperature sample having a general larger spread of data. 52 Chapter 5: Discussion This chapter provides an analysis of the results presented in Chapter 4: . This chapter will compare the results of the high temperature tensile tested sample, to the room temperature tensile tested sample. In addition, the results obtained in the Ti525 room temperature tensile (29 6K) sample and high temperature (728K) sample are espe cially useful when compared to the results investigation in an aluminum oligo - crystal tensile sample t ested at room temperature. Comparison between the two crystal structures will shed new understanding about what facilitates slip transfer. 5.1 vs . Figure 5.1 ( a ) and (b) compare the vs . residual Burg ers vector ( ) for the room temperature (296K) sample and high temperature (728K) sample, respectively. The green box identifies the cluster boundary determined by Alizadeh between slip transfer and no slip transfer for the Aluminum oligo - crystal. Ali zadeh maximized the percentage of slip transfer points within the box and percentage of no slip transfer points outside the box. Within the box lies 93% of slip transfer data for the Al sample. Outside the box lies 86% of no slip transfer data for the aluminum sample. A similar activity was completed for the Titanium sample, as explained in Chapter 4 : Results . A comparison of Al maximized boundaries and Titanium maximized boundaries in Table 5.1 . The aluminum sample has a boundary of lower and Figure 5.1 (b)). In comparing the room tem perature Ti525 sample and the room temperature aluminum sample ( Figure 5.1 (a)), the boundaries are very similar indicating their data sets are similar. Thi s shows the temperature dependence of the vs. factors. Higher values and lower values 53 enable slip transfer when the temperature in the material is hotter. In comparing the titanium room temperature sample to the titanium high temperature sample, more points with lower values are observed . Figure 5 . 1 : vs. data for the (a) room temperature (296K) and (b) the high temperature (728K) tensile tested Ti525 sample. Green box es indicate comparable Al oligo - crystal tensile results at room t emperature fo r both (a) and (b) . Table 5 . 1 : Percentages of slip transfer and no inside and outside of boundaries, respectively. RT and HT Ti5252 relationships are flipped with high percentage of no slip transfer outside of the box in RT and high percentage of slip transfer inside the box for HT. Slip transfer (Inside) No slip transfer (Outside) RT Ti525 64% 94% HT Ti525 91% 68% RT Al 93% 86% 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 0 0.2 0.4 0.6 0.8 1 b m' (b) High temperature 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 0 0.2 0.4 0.6 0.8 1 b m' (a) Room temperature 54 Dislocation climb in titanium alloys is facilitated by higher temperatures. If dislocations can climb near the boundary to align themselves with a lower geometry partner, then slip transfer is enabled under less favorable conditions. Higher diffusion r ates enable recovery processes to take place near and within grain boundaries, so that residual Burgers vector debris is more easily absorbed. The homologous temperatures of the two materials are different from each other. The melting temperature for Ti525 is 1863K (1590 C) and the m elting temperature for aluminum is 933K (660 C). Room temperature is 0.16 T m for Ti525 and 0.32T m for aluminum. In addition, the high temperature Ti525 experiment was performed at 0.39 T m (728K) which is equivalent to 364K (90 C) for Al. This fractio n is closer to that of the room temperature aluminum homologous temperature . Table 5.2 shows these relations. Table 5 . 2 : Temperature comparison for Al and Ti 525. RT Al and HT Ti525 have similar homologous temperatures. Homologous Temperature Melting Temperature (Kelvin) Test Temperature (Kelvin) RT Ti525 0.16 T m 1863 296 RT Al 0.32 T m 933 296 HT Ti525 0.39 T m 1863 728 Given that the slip transfer b ehavior in aluminum and Ti525 are more similar to each other at room temperature, the effect of alloying elements and/or the much lower CRSS in pure aluminum may lead to less stress assisted climb forces for a similarly high h omologous temperature, such th at the higher stresses in Ti525 facilitated climb more effectively than in Al. The stiffness normalized stre ngth in Ti525 was 0.006 for the room temperature sample and 0.003 55 for the high temperature sample while it wa s 0.00036 in pure Al. Table 5.3 shows the normalized strength and the values used to obtain the metric. The aluminum compensated strength is significantly lower than the Titanium alloy , due to the effects of alloying . Table 5 . 3 : Compensated Strengths and values used to obtain them. The pure aluminum compensated strength is much lower than the Titanium alloy ( Aluminum 1100 - O and Li 2013) . Modulus (GPa) Yield Strength (MPa) Compensated Strength ( Yield Strength / RT Ti525 110 660 0.006 HT Ti525 110 330 0.003 RT Al 70 25 0.0003 6 5.2 Misorientation angle vs. Figure 5.2 shows the misorientation angle vs. categories for both the titanium oligo - crystal and the aluminum alloy samples . T he room temperature (296K) Ti525 tensile sample (a) , and the high temperature (728K) Ti525 tensile sample (b) are overlaid with t he shaded areas represent ing the locus of most of the points in the aluminum polycrystal. Since aluminum has a cubic structure with a maximum d isorientations of 63° there is a much larger range of misorientation as well as value s for Ti525 slip transfer points . The black box shows the comparable area from the aluminum data . Alizadeh identified a threshold of about 20 that best separated the slip transfer and no slip transfer categories , which occurred in the mid dle of the slip transfer region (blue region). The titanium alloy shows similar behavior, but the threshold is not as distinct , as most observations are at misorientations larger than 20°. This disparity could arise from the fewer easy slip systems in hexagonal crystal structures that lead to more heterogeneous stress states in titanium. Another possibility is that the aluminum oligo - crystal grains have mostly free surfaces while the Ti525 56 sample is a polycryst al, where only one side of the grain has a free surface, which makes the stress state more complex. Also, unlike the Ti alloy, the aluminum oligo - crystal has a strong texture, so that most grains had a more similar stress state and strain response . W ith fewer easy slip systems available in the Ti525 , slip may be required on slip systems that do not facilitate slip easily, leading to a wider variation in the local stress state, which would lead to more spread in the data, as the Schmid factor is based upon the assumption of a uniform uniaxial st ress. Furthermore, the no slip transfer points are not present in the room temperature titanium sample below ~30° misorientation, while in the aluminum and high temperature Ti525 data sets, they are present at misorientation s as low as ~10° in Al and th e high temperature Ti525 data. These differences imply that differences in crystal structure and geometrical limitations need to be considered. 57 Figure 5 . 2 : Aluminum oligo - crystal slip transfer (blue shaded area) and no slip transfer (red shaded area) compared to data of the titanium polycrystal slip transfer and no slip transfer data for (a) the room temperature (296K) sample and (b) the high temperature (728K) sample. Titanium slip transfer and no slip transfer points do not follow as strict of a trend compared to the aluminum data. 5.3 Misorientation angle vs. the sum of Schmid factors Figure 5.3 shows comparison between the misorientation angle vs. LG +SF RG ) for the (a) room temperature Ti525 alloy, (b) high temperature Ti525 alloy. The green solid line represents the thresholds for aluminum, where slip transfer points was prevalent above the green line (for strongly textured Al with two different tensile a xis directions ). The same process done with the titanium data set as described in Chapter 4: Results . The trends between the boun daries of the aluminum and titanium slip tra nsfer data are significantly different, in that a shallow positive slope best separates prevalent slip transfer and no slip transfer populations , and the threshold is much lower threshold than that for Al. The boundaries for the high temperature and room temperature samples in the titanium alloy are very similar, but lower for 58 high temperature data. This also indicates that a threshold for LG +SF RG ) vs. misorientation an gle is heavily dependent on materi al and crystal structure. Clearly, the geometrical constraints for slip transfer are much smaller in the hexagonal crystal structure than in Al. Figure 5 . 3 : Misorientation angle vs. LG +SF RG ) for th e (a) room temperature Ti525 sample and the (b) high temperature Ti525 sample. The aluminum boundaries represented by the green line show the difference between the aluminum data set which has a steep slope and the titanium data set which slope is flatter . 5.4 Misorientation angle vs. Figure 5.4 presents the misorientation angle vs. for (a) the room temperature tensile tested sample and (b) the high temperature sample . The shaded areas are approximate representations for mi sorientation angle vs. for the aluminum oligo - crystal. The blue shaded region for the aluminum oligo - crystal slip transfer data and the blue cluster of data 59 f r om the titanium alloy below 30° mi sorientation angle line up well, indicating a strong co rrelation between the two data sets in this range . The red shaded area representing the aluminum no slip transfer data expands greatly below = 1 , as do the red x markers representing the titanium no slip transfer data. The black rectangle represe nts the data bounds for the FCC aluminum data set. Figure 5 . 4 also shows that there are less geometrical constraints for slip transfer in the hexagonal Ti alloy than the Aluminum FCC material. In comparing the 296K to 728K Ti525 data, the boundary lines separating maximum slip transfer and no slip transfer are ne arly the same, but the higher temperature has a slightly smaller slope indicat ing that temperature does not have a great effect on b vs. the misorientation angle. 60 Figure 5 . 4 : Misorientation vs. for (a) the room temperature (296K) tensile test sample and (b) t he high temperature (728K) sample. The red shaded area repres ents the aluminum oligo - crystal no slip transfer data set. The blue shaded area represents the aluminum oligo - crystal slip transfer data set and line up with the slip transfer points in the t itanium data sets. 5.5 Slip system analysis Figure 4.6 shows the favored slip systems for grain pairs that have slip transfer in (a) the room temperature (296K) Ti525 sample and in (b) the high temperature (728K) Ti525 sample. The clusters of slip systems described in the section c ould be attributed to geometric conditions that the hexagonal crystal system imposes on the slip tr ansfer process . At lower misorientation ranges, slip transfer between all slip systems are likely to be favored . The cluster of basal to pyramidal slip tr ansfer between misorientation angles 45° and 60° noted in the slip transfer high temperature (728 K) sample can be explained by the angle between the two 61 slip planes. At a misorientation of 45° to 60° the basal plane of one crystal is more likely to be al ign ed with the pyramidal plane of the second crystal. Figure 4.7 shows the slip systems in grain pairs that slip transfer for (a) the room temperature, and (b) the high temperature Ti525 sample. T here are no strong trends observed in either graph . This is consistent with that there should be many no slip transfer conditions between mismatched slip systems of any family , su ch that non - favored slip systems is ob served . Figure 4.8 presents the statistics around the slip systems observed in the slip transfer data and no slip transfer data. The main slip systems activated are basal and prism , consistent with the fact that these are the most easily activated systems . This is consis tent with literature that examines the relative activity of slip systems in Li , 2013 ). 62 5.6 Comparison between measured weakly textured grain orientations to random populations 5000 orientations pairs for a hexagonal crystal structure were randomly generated. In each pair the probable slip system s for uniaxial tension was determined for each grain. For each grain, the Schmid factors w ere calculated for each slip system and i n each grain pair, was calculated. Two different filters were applied to 5000 randomly generated orientation pairs to obtain two populations. The first (Ran 0.9) collects slip system pairs with an value above the threshold of 0.9 and grain pairs that have Schmid factors of a mini mum of 0.3 for each grain. The second data d a Schmid factor greater than 0.25, yielding a much larger set of nearly 25,000 data. Figure 5.5 s hows the corresponding cumulative misorientation fraction for the experimental and randomly generated data, separated by slip system pair type. In basal to basal slip, a ll t he Ran 0.9 data is below ~20° while all the Ran 0.8 data lies below ~35°. The green arrow indicates which direction the Ran 0.9 data set lies in comparison to the Ran 0.8 data set. It is known that Ran 0.9 is more restrictive than Ran 0.8. As i t is hyp othe sized that the room temperature Ti525 sample (Exp - RT) is more restrictive than the high temperature Ti525 sample (Exp - HT) , the hypothesis is supported by the room temperature data being mostly to the left (lower misorientations) of the high temperature obs ervations, but the two cross at the highest misorientations . The arrow is green because the experimental data does follow the same trend (the more restrictive data on the left ) compared to the randomly generated data until about 60° in which the Exp - RT an d Exp - HT switch . In prism to prism slip, the experimental data sets are much closer to each other, and to the simulation, but contrary to the hypothesis, the higher temperature data appear to be m ore restr icted than lower temperature data (for most of the misorientation range ) . In pyramidal to pyramidal slip, the high temperature Ti525, Ran 63 0.9, a nd Ran 0.8 data all follow a same general trend , and the small number of room temperature observations makes it im possible to compare with the high temperature data, but the simulations show a similar trend . In pyramidal to pyramidal slip, there are two observed peaks in the randomly generated data, one around 25° misorientation angle, and the other around 75° misorientation angle. Interestingly , the experimental data shows one peak between the randomly generated peaks , at around 45° misorientation angle. The purple arrow depicts this discrepancy. For the mixed slip system slip transfer, basal to prism slip shows the more restrictive of the random data sets (Ran 0.9) at highest misorientations ( on the right ) , and the hypothesized more restricted of the experimental data (Exp - RT) is also to the right of the higher temperature data . The green arrow depicts this , but the spread of slip tr ansfer misorientations of these two slip systems is much greater in the experimental data . In b asal to pyramidal slip, all data sets have a similar trend. In b asal to pyramidal slip, below 50° the more restrictive of the randomly generated data is on the right, while the hypothesized more restrictive experimental data set (Exp - RT) is on the left. Above 50°, the more restrictive randomly generated data set is now on the left, while the room temperature experimental data is on the right. This is depicted by the red arrows on the graph , indicating trends opposing the hypothesis . In prism to pyramidal slip, Ran 0.9 to the left of Ran 0.8, and Exp - RT to the right of Exp - HT, again contrary to the hypothesis . In p rism to pyramidal slip, the more restrictive Ran 0.9 to the right of Ran 0.8, and the more restrictive Exp - RT to the left of Exp - HT (except for between 50° and 70°). In pyramidal to pyramidal slip there is a relationship opposite of expected when concerning the restrictiveness of the data sets. 64 Figure 5 . 5 : Misorientation angle cumulative percentages for Ti525 and randomly generated data sets separated b y type of slip system observed in the grains . Pyramidal slip is behaving counter - intuitively with respect t o more restrictive conditions. It is important to note that the experim ental data does have a 3 times random texture, while the randomly generated d ata has a true to random texture. This difference could account for the behavior contrary to the hypothesis in Figure 5.5 . In general, pyramidal slip behav es counter - intuitively with respect to more restrictive conditions. Figure 5.6 p resents the overall Ti525 statistics for the type of slip system observed in the data compared to the two randomly generated data sets . It is hypothesized that the room temperature data would have more restrictive slip transfer conditions than the high temperature data set due to the higher ease of movement of dislocations by climb in the high temperature exp erimental data set. B asal - basal slip and basal - prism slip transfer have a higher number of observations at room temperature than elevated temperature, which could be related to the ease 65 of t hese slip systems. The variance in the relative activity of different combinations of slip systems may reflect different temperature dependence on the critical resolved shear stress for the four families of slip systems. The randomly generated data sets have a larger percentage shown in the pyramidal to pyramidal and pyramidal to pyramidal categories, but there are few experimental observations of these slip systems. The experimental data shows a large percentage with basal to prism slip while the random data sets have low percentages for this category. Figure 5 . 6 : Overa l l Ti525 statistics in comparison to a random population . Random population data sets have a high frequency for pyramidal to pyramidal where experimental results are low for the sam e category. Figure 5.7 show s the same data, but the frequencies are repre sented as half - circles. The larger the area of the half - circle, th e higher the frequency for that slip system pair. In comparing the two data sets, the randoml y generated data favors the lower right - hand corner, while the experimental data favors the upper left - hand corner. The randomly generated data hav e a large 0.00% 5.00% 10.00% 15.00% 20.00% 25.00% 30.00% Frequency (a) 66 frequency of slip system pairs including a pyramidal plane, and no distinction about the critical resolved shear stress, which could account for the large values for pyramidal slip systems . Figure 5 . 7 : Relative frequency of data represented as half - circles. Larger area half - circle represents a higher frequency. (a) Comparis on of experimental Ti525 at room temperature and high temperature. (b) Comparison of randomly generated data sets 67 Another analysis was done with the randomly generated data, to find the most likely slip system pair that would be present in the grain pairs. To determine the most likely slip system pair, a factor was calculated as follows: (5) The slip system pair with the highest factor for each orientation pair was deemed the slip system pair to most likely be present and facilitate slip. Figure 5.8 (a) shows the frequency of each slip system pair represented by relative area of a circle. There is a larger frequency associated with the right side of the chart, especially the pyramidal slip system. The multiplicity of the pyramidal plane give s a significant advantage to slip transfer possibilities . To compensate for this, the Schmid factor for the pyramidal slip system was halved , which would simulate a higher CRSS for this specific slip plane , and shown in Figu re 5.8 (b) . Reducing the likelihood for slip caused the generated data to have a similar distribution to the experimental data, with basal to prism , and prism to pyramidal sli p system pairs having a large frequenc y , and a smaller frequency in the lower right - hand corner of the chart in both sets . This shows that the factor in Equation 5 captures the overall phenomena that is observed experimentally. 68 Figure 5 . 8 : Randomly generated orientation pairs with the most likely slip systems present that facilitates slip across the grain boundaries. (a) Unmodified Schmid factors. (b) Schmid factor for pyramidal is halved. A h igh f requency is present in slip system pairs with pyramidal slip systems in both cases . 69 Chapter 6: Summary and conclusions 6.1 Summary This thesis describes the method and results of a study of Ti - 5Al - 2.5Sn defor mation behavior focusing on slip transfer across grain boundaries at two different temperatures (296K and 728K). The study investigates across grain boundaries and builds upon work done by Hongmei Li. The re sults are compared to a similar study that was done with pure aluminum by R. Alizadeh. In comparing the titanium results to an aluminum sample, some trends are similar, such as favorable sli p transfer of the same slip system family at low misorientations. D etails associated with geometrical constraints of more slip systems in the hexagonal crystal structure , but few facile ones, versus face centered cubic crystal structure in aluminum. Thes e results show that slip transfer is much more commonly accomplis hed in the Ti - 5Al - 2.5Sn deformation in and room and to a greater extent at high temperatures, which provides a basis for installing slip transfer criteria into CPFE modeling of the alloy. 6.2 Co nclusions 1. Slip transfer data is more prevalent than no slip tr ansfer data at misorientation angles < 30°. At these low misorientation angles, slip traces are categorized as slip transfer more than being categorized as no slip transfer at both 296K and 728K . 2. There is a decreasing trend of slip transfer trace s with increasing misorientation below 20° for both Al and Ti525. Many slip transfer cases occur at high misorientations in Ti tanium . 3. Literature suggests that h igh and low enable s slip transfer. This hypothesis was found true in Ti525. R oom temperature Ti525 and room temperature Al uminum show a vs. 70 4. Considering both the Schmid factor vs. , and vs. , geometrical constraints for slip transf er in hexagonal crystal structure are smaller than FCC. 5. Misorientation angles that favor a specific slip system pair (angles between slip systems are small) makes slip transfer easier and more prevalent. 6. In slip s ystem pairs where ther e is no slip transfer , slip is not facilitated easily as the alignment of slip systems is low (angles between slip systems are generally large). 7. The b asal + prism slip system pair is most common in Ti525. This r elates to the low CRSS of basal and prism slip systems in Ti525 . 8. Every type of slip system has a different kind of sensitivity with misorientation angle regarding the experimental data . This is partly due to the geometry of the slip systems, and partly unexplained. 9. Pyramidal slip behav es counter - intuitively with respect to high temperature and room temperature restrictiveness. 10. Using a random data set, t he factor and penalizing the likelihood for slip by a factor of two is able to capture the statistical trends similar ly to the observed distribution of slip transfer observations. 71 6.3 Future work suggestions 1. A random generation o f orientation pairs for the FCC crystal structure to observe if statistical trends are similar to that found in the Aluminum oligo - crystal experi me nts. 2. Investigate grain boundary sliding in correlation with the residual Burges vector between two grains an d the associate d slip transfer in the high temper ature Ti525 data . An atomic force microscope ( AFM ) or a profiliometer could be used to identify the ledge h eight at the grain boundary. 3. A similar investigation with a different deformation m ode , i.e. creep o r compression testing to further the accuracy of CPFE modeling by creating rules around slip transfer across grain boundaries . 4. The work of Line et al. could be analyzed with similar methods to determine the slip transfer relationships between high temper ature ( 464K) and the room temperature (296K) aluminum oligo - crystal . 72 APPENDICIES 73 APPENDIX A: Matlab hexagonal orientation code (written by Thomas R. Bieler) % HEXagonal orientation analysis by T.R. Bieler, November 22, 2010, updated 6 Nov 2018. % No guarantees that it is 100% correct, but it seems so. If you use this for your work, % it would be kind and ethical to acknowledge its use in published work. clear; clc; % dbstop if error; % Imported grains file requires Bunge Euler an gles in colu mns 3:5, (x,y can be put in 1:2, for example) % in as many rows as data set. Be wary about hidden assumptions; % plotting uses x -- > down, y right, so Euler angles are consistent with TSL maps, a1=86.764; b1=66.765; c1=259.31; % 3,4,5 are Eu ler angles Ang(1,:) = [1 0 a1+180 b1 c1]; ghkl = [ % for C3 - 1 - 1 2 8 0 - 1 1 7 1 - 2 1 8 0 - 1 1 4]; dum = size(ghkl); ghklrows = dum(1,1); dum = size(Ang); Angrows = dum(1,1); % Stress tensor is defined; put the one you want in last positino, or make a new one. ststens = [1 0 0 ; 0 0 0 ; 0 0 0]; LC = 'X' ; % tension in X % ststens = [0 0 0 ; 0 0 0 ; 0 0 1]; LC = 'Z' ; % tension in Z % ststens = [0 - 1 0 ; - 1 0 0 ; 0 0 0]; LC = ' - XY' ; % Shear in XY plane % ststens = [0 1 0 ; 1 0 0 ; 0 0 0]; LC = '+XY' ; % Shear in XY plane % ststens = [0 0 0 ; 0 0 - 1 ; 0 - 1 0]; LC = '+XZ' ; % Shear in XZ plane % ststens = [1 0 0 ; 0 1 0 ; 0 0 0]; LC = 'BiXY' ; % biaxial tension in X - Y % ststens = [ - 1 0 0 ; 0 1 0 ; 0 0 0]; LC = 'PSC - XY' ; % Plane Strain Compression in XY plane % ststens = [0 0 0 ; 0 1 0 ; 0 0 0]; LC = 'Y' ; % tension in Y % theta = 0; Rst = [cosd(theta) - sind(theta) 0; sind(theta) cosd(theta) 0; 0 0 1]; % rotates stress about Z axis %Rst = [cosd(theta) 0 - sind(theta); 0 1 0; sind(theta) 0 cosd(theta)]; % rotates stress about Y axis %Rst = [1 0 0; 0 cosd(theta) - sind(theta); 0 sind(th eta) cosd(theta)]; % rotates stress about X axis ststens_R = Rst*ststens*Rst'; % This rotates a desired stress tensor in the lab coord sys. str2 = ststens_R*ststens_R'; ststens_mag = (str2(1,1)+str2(2,2)+str2(3,3))^.5; ststens_n = ststens_R/ststens_mag; % normalized stress tensor to get generalized Schmid factor % % Inverse pole figures can be drawn by s etting values of ipfd; ipfd is % inverse pole figure direction, % iCTE is CTE direction, iEd is E direction, not checked recently, so bugs may exis t. stereo = 1; % plots stereographic projection, otherwise direct projection ipfd = 2; %inverse pole fig ure direction (x,y,z, = 1,2,3) will plot points from a group ihex = 1; % inverse pole figure for hexagonal crystal iEd = ipfd; % plot magnitude of E in this direction as red (high) - blue (low) % CTEd = ipfd; % plot magnitude of CTE in this direction as gray scale within symbol nslphex=69; % Slip sys: b p paa pya 1c+a 2c+a T1 T2 C2 C1 3c+a in first row, set 0 to skip, 1 to plot sschoice = [1 1 0 1 1 0 1 1 1 1 0 1 4 7 10 16 28 34 40 46 52 58 3 6 9 15 27 33 39 55 51 57 69 ]; sscol = 1; for isc = 1:1:nslphex pssf(isc) = sschoice(1,sscol); if isc == sschoice(3,sscol) sscol = sscol + 1; end end % % This section allows the point of view of the unit cell to be changed. Rotateview = [1 0 0 ; 0 1 0 ; 0 0 1]; iRotatev = 0; % Provide NUMBER of rotations 74 % rotation of observer from normal TSL point of view to another point of view (crystal stays put) % This right hand rotation matrix has a 4th row which has in columns 2,3, the angle a nd axis of the rotation. %Rotation(4,2:3,1) = [5,3]; % rotation to move axis to a tilted direction in the X - Y plane %Rotation(4,2:3,1) = [ - 45,1]; % about the X axis to look through detector for DAXM %Rotation(4,2 :3,1) = [180,2]; % rotate viewpoint about vertical (x) axis + to view from above/right, - to view from above/left Rotation(4,2:3,1) = [90,2]; % +90 to view from below, - 90 to view from above % Build the Euler angle rotation matrix starting with identit y if iRotatev ~= 0 LC = [ '! rot ! ' LC]; for i = 1:1:iRotatev rang = Rotation(4,2,i); if Rotation(4,3,i) == 3 Rotation(1:3,1:3,i)=[cosd(rang),sind(rang),0; - sind(rang),cosd(rang),0;0,0,1]; elseif Rotation(4,3,i) == 2 Rota tion(1:3,1:3,i)=[cosd(rang),0,sind(rang);0,1,0; - sind(rang),0,cosd(rang)]; elseif Rotation(4,3,i) == 1 Rotation(1:3 ,1:3,i)=[1,0,0;0,cosd(rang),sind(rang);0, - sind(rang),cosd(rang)]; end Rotateview = Rotateview*Rotation(1:3,1:3,i); end end % % Below, six points define plane, start point is start of Burgers vector b, 4th is end of b % 1st to 2nd or 2nd to 3r d or 3rd to 4th cross product identifines plane normal c_a=1.59; % 1.587 this is the last place for user input in this cell ... slpsys = cell(4,nslphex); O = [ 0 0 0 0]; % (I) (H) A = [ 2 - 1 - 1 0]/3; % C ------ - B B = [ 1 1 - 2 0]/3; % / \ / \ C = [ - 1 2 - 1 0]/3; % / a2 / \ D = [ - 2 1 1 0]/3; % / \ / \ E = [ - 1 - 1 2 0]/3; % (J)D ------ O(P) - a1 - > A(G) F = [ 1 - 2 1 0]/3; % \ / \ / P = [ 0 0 0 1]; % \ a3 \ / G = [ 2 - 1 - 1 3]/3; % \ / \ / H = [ 1 1 - 2 3]/3; % E -------- F I = [ - 1 2 - 1 3]/3; % (K) (L) J = [ - 2 1 1 3]/3; % K = [ - 1 - 1 2 3]/3; % where a1 = OA a2 = OC a3 = OE L = [ 1 - 2 1 3]/3; % DA || a1, FC || a2, BE || a3 % Slip system definitions set as of 1 Nov 2019 to be consistent with DAMASK % Hex plane direction 1st and 4th point is Burgers vector % basal - glide:&Be Mg Re Ti; Re slpsys{ 2,1} = [0 0 0 1; 2 - 1 - 1 0; D ; E ; F ; A ; B ; C ]; slpsys{2,2} = [0 0 0 1; - 1 2 - 1 0; F ; A ; B ; C ; D ; E ]; slpsys{2,3} = [0 0 0 1; - 1 - 1 2 0; B ; C ; D ; E ; F ; A ]; ibas = 1; fbas = 3; % prism - glide:Ti Zr RE; Be Re Mg slpsys{2,4} = [ 0 1 - 1 0; 2 - 1 - 1 0; E ; E ; E ; F ; L ; K ]; slpsys{2,5} = [ - 1 0 1 0; - 1 2 - 1 0; A ; A ; A ; B ; H ; G ]; slpsys{2,6} = [ 1 - 1 0 0; - 1 - 1 2 0; C ; C ; C ; D ; J ; I ]; iprs = 4; fprs = 6; % prism slpsys{2,7} = [ 2 - 1 - 1 0; 0 1 - 1 0; F; F; F; B ; H ; L ]; slpsys{2,8} = [ - 1 2 - 1 0; - 1 0 1 0; B; B; B; D ; J ; H ]; slpsys{2,9} = [ - 1 - 1 2 0; 1 - 1 0 0; D; D; D; F ; L ; J ]; i2prs = 7 ; f2prs = 9; % pyramidal - glide -- ** -- CORRECTED -- ** -- slpsys{2,10} = [ 1 0 - 1 1; - 1 2 - 1 0; E ; E ; E ; D ; I ; L ]; slpsys{2,11} = [ 0 1 - 1 1; - 2 1 1 0; F ; F ; F ; E ; J ; G ]; slpsys{2,12} = [ - 1 1 0 1; - 1 - 1 2 0; A ; A ; A ; F ; K ; H ]; s lpsys{2,13} = [ - 1 0 1 1; 1 - 2 1 0; B ; B ; B ; A ; L ; I ]; slpsys{2,14} = [ 0 - 1 1 1; 2 - 1 - 1 0; C ; C ; C ; B ; G ; J ]; slpsys{2,15} = [ 1 - 1 0 1; 1 1 - 2 0; D ; D ; D ; C ; H ; K ]; 75 ipyra = 10; fpyra = 15; % pyramidal - glide:; all? slpsys {2,16} = [ 1 0 - 1 1; - 2 1 1 3; A ; B ; I ; P ; L ; A ]; slpsys{2,17} = [ 1 0 - 1 1; - 1 - 1 2 3; B ; B ; I ; P ; L ; A ]; slpsys{2,18} = [ 0 1 - 1 1; - 1 - 1 2 3; B ; C ; J ; P ; G ; B ]; slpsys{2,19} = [ 0 1 - 1 1; 1 - 2 1 3; C ; C ; J ; P ; G ; B ]; sl psys{2,20} = [ - 1 1 0 1; 1 - 2 1 3; C ; D ; K ; P ; H ; C ]; slpsys{2,21} = [ - 1 1 0 1; 2 - 1 - 1 3; D ; D ; K ; P ; H ; C ]; slpsys{2,22} = [ - 1 0 1 1; 2 - 1 - 1 3; D ; E ; L ; P ; I ; D ]; slpsys{2,23} = [ - 1 0 1 1; 1 1 - 2 3; E ; E ; L ; P ; I ; D ]; slpsys{2,24} = [ 0 - 1 1 1; 1 1 - 2 3; E ; F ; G ; P ; J ; E ]; slpsys{2,25} = [ 0 - 1 1 1; - 1 2 - 1 3; F ; F ; G ; P ; J ; E ]; slpsys{2,26} = [ 1 - 1 0 1; - 1 2 - 1 3; F ; A ; H ; P ; K ; F ]; slpsys{2,27} = [ 1 - 1 0 1; - 2 1 1 3; A ; A ; H ; P ; K ; F ]; ipyrc = 16; fpyrc = 27; % pyramidal - 2nd order glide slpsys{2,28} = [ 1 1 - 2 2 ; - 1 - 1 2 3; (O+B)/2 ; C ; J ; (P+K)/2 ; L ; A]; slpsys{2,29} = [ - 1 2 - 1 2; 1 - 2 1 3; (O+C)/2 ; D ; K ; (P+L)/2 ; G ; B]; slpsys{2,30} = [ - 2 1 1 2; 2 - 1 - 1 3; (O+D)/2 ; E ; L ; (P+G)/2 ; H ; C]; slpsys{2,31} = [ - 1 - 1 2 2; 1 1 - 2 3; (O+E)/2 ; F ; G ; (P+H)/2 ; I ; D]; slpsys{2,32} = [ 1 - 2 1 2; - 1 2 - 1 3; (O+F)/2 ; A ; H ; (P+I)/2 ; J ; E]; slpsys{2,33} = [ 2 - 1 - 1 2; - 2 1 1 3; (O+A)/2 ; B ; I ; (P +J)/2 ; K ; F]; i2pyrc = 28; f2pyrc = 33; % *** Twin directions are opposite in Christian and Mahajan, and are not correcte to to be consistent with them % FROM Kocks SXHEX plane direction 1st and 4th point is Burgers vector, order of C1 differs fro m Kock's file % {1012}<1011> T1 twins 0.17; - 1.3 twins: all Twin Vector must g o in the % sense of shear, opposite C&M sense. slpsys{2,34} = [ 1 0 - 1 2; - 1 0 1 1; A ; B ; J ; K ; K ; K ]; slpsys{2,35} = [ 0 1 - 1 2; 0 - 1 1 1; B ; C ; K ; L ; L ; L ]; slpsys{2,36} = [ - 1 1 0 2; 1 - 1 0 1; C ; D ; L ; G ; G ; G ]; slpsys{2,3 7} = [ - 1 0 1 2; 1 0 - 1 1; D ; E ; G ; H ; H ; H ]; slpsys{2,38} = [ 0 - 1 1 2; 0 1 - 1 1; E ; F ; H ; I ; I ; I ]; slpsys{2,39} = [ 1 - 1 0 2; - 1 1 0 1; F ; A ; I ; J ; J ; J ]; iT1 = 34; fT1 = 39; % {2111}<2116> T2 twins: 0.63; - 0.4; Ti Zr Re RE]; Also does not follow C&M definition for shear direction slpsys{2,40} = [ 1 1 - 2 1; - 1 - 1 2 6; (O+B)/2 ; C ; (J+I)/2 ; P ; (L+G)/2 ; A]; slpsys{2,41} = [ - 1 2 - 1 1; 1 - 2 1 6; (O+C)/2 ; D ; (K+J)/2 ; P ; (G+H)/2 ; B]; slpsys{2,42} = [ - 2 1 1 1; 2 - 1 - 1 6; (O+D)/2 ; E ; (L+K)/2 ; P ; (H+I)/2 ; C]; slpsys{2,43} = [ - 1 - 1 2 1; 1 1 - 2 6; (O+E)/2 ; F ; (G+L)/2 ; P ; (I+J)/2 ; D]; slpsys{2,44} = [ 1 - 2 1 1; - 1 2 - 1 6 ; (O+F)/2 ; A ; (H+G)/2 ; P ; (J+K)/2 ; E]; slpsys{2,45} = [ 2 - 1 - 1 1; - 2 1 1 6; (O+A)/2 ; B ; (I+H)/2 ; P ; (K+L)/2 ; F]; iT2 = 40; fT2 = 45; % {1011}<101 - 2> C1 twins: 0.10; 1.1; Mg; Zr Ti]; agrees with C&M slp sys{2,46} = [ 1 0 - 1 1; 1 0 - 1 - 2; P ; L ; A ; (A+B)/2 ; B ; I ]; slpsys{2,47} = [ 0 1 - 1 1; 0 1 - 1 - 2; P ; G ; B ; (B+C)/2 ; C ; J ]; slpsys{2,48} = [ - 1 1 0 1; - 1 1 0 - 2; P ; H ; C ; (C+D)/2 ; D ; K ]; slpsys{2,49} = [ - 1 0 1 1; - 1 0 1 - 2; P ; I ; D ; (D+E)/2 ; E ; L ]; slpsy s{2,50} = [ 0 - 1 1 1; 0 - 1 1 - 2; P ; J ; E ; (E+F)/2 ; F ; G ]; slpsys{2,51} = [ 1 - 1 0 1; 1 - 1 0 - 2; P ; K ; F ; (F+A)/2 ; A ; H ]; iC1 = 46; fC1 = 51; % {2112}<211 - 3> C2 twins:; 0.22; 1.2 Ti Zr Re]; agrees with C&M slpsys{2,52} = [ 1 1 - 2 2; 1 1 - 2 - 3; (K+P)/2 ; L ; A ; (A+C)/2 ; C ; J]; slpsys{2,53} = [ - 1 2 - 1 2; - 1 2 - 1 - 3; (L+P)/2 ; G ; B ; (B+D)/2 ; D ; K]; slpsys{2,54} = [ - 2 1 1 2; - 2 1 1 - 3; (G+P)/2 ; H ; C ; (C+E)/2 ; E ; L]; slpsys {2,55} = [ - 1 - 1 2 2; - 1 - 1 2 - 3; (H+P)/2 ; I ; D ; (D+F)/2 ; F ; G]; slpsys{2,56} = [ 1 - 2 1 2; 1 - 2 1 - 3; (I+P)/2 ; J ; E ; (E+A)/2 ; A ; H]; slpsys{2,57} = [ 2 - 1 - 1 2; 2 - 1 - 1 - 3; (J+P)/2 ; K ; F ; (F+B)/2 ; B ; I]; iC2 = 52; fC2 = 57; % - 3rd order glide; slpsys{2,58} = [ 2 - 1 - 1 1; - 1 2 - 1 3; F ; (K+L)/2 ; P ; P ; (I+H)/2 ; B ]; slpsys{2,59} = [ 2 - 1 - 1 1; - 1 - 1 2 3; B ; (I+H)/2 ; P ; P ; (K+L)/2 ; F ]; slpsys{2,60} = [ 1 1 - 2 1; - 2 1 1 3; A ; (L+G)/2 ; P ; P ; (J+I)/2 ; C ]; 76 sl psys{2,61} = [ 1 1 - 2 1; 1 - 2 1 3; C ; (J+I )/2 ; P ; P ; (L+G)/2 ; A ]; slpsys{2,62} = [ - 1 2 - 1 1; - 1 - 1 2 3; B ; (G+H)/2 ; P ; P ; (K+J)/2 ; D ]; slpsys{2,63} = [ - 1 2 - 1 1; 2 - 1 - 1 3; D ; (K+J)/2 ; P ; P ; (G+H)/2 ; B ]; slpsys{2,64} = [ - 2 1 1 1; 1 - 2 1 3; C ; (H+I)/2 ; P ; P ; (L+K)/2 ; E ]; slpsys{2,65} = [ - 2 1 1 1; 1 1 - 2 3; E ; (L+K)/2 ; P ; P ; (H+I)/2 ; C ]; slpsys{2,66} = [ - 1 - 1 2 1; 2 - 1 - 1 3; D ; (I+J)/2 ; P ; P ; (G+L)/2 ; F ]; slpsys{2,67} = [ - 1 - 1 2 1; - 1 2 - 1 3; F ; (G+L)/2 ; P ; P ; (I+J)/2 ; D ]; slpsys{2,68} = [ 1 - 2 1 1; 1 1 - 2 3; E ; (J+K)/2 ; P ; P ; (H+G)/2 ; A ]; slpsys{2,69} = [ 1 - 2 1 1; - 2 1 1 3; A ; (H+G)/2 ; P ; P ; (J+K)/2 ; E ]; i3pyrc = 58; f3pyrc = 69; % cell 3 = Cartesian slip system unit vectors, cell 4 = Schmid matrix % for gg = 1:1:g hklrows ghklC(gg,:) = [ghkl(gg,1), (ghkl(gg,1)+2*ghkl(gg,2))/sqrt(3), ghkl(gg,4)/c_a]; %plane normal in cartesian unit_ghklC(gg,:) = ghklC(gg,:)/norm(ghklC(gg,:)); end for isc=1:1:nslphex; % isc is slip system counter n =[slpsys{2,isc}(1,1) , (slpsy s{2,isc}(1,1)+2*slpsys{2,isc}(1,2))/sqrt(3), slpsys{2,isc}(1,4)/c_a]; %plane normal in cartesian m =[3*slpsys{2,isc}(2,1)/2, (slpsys{2,isc}(2,1)+2*slpsys{2,isc}(2,2))*sqrt(3)/2, slpsys{2,isc}(2,4)*c_a]; % slip direction in cartesian p1=[3*sl psys{2,i sc}(3,1)/2, (slpsys{2,isc}(3,1)+2*slpsys{2,isc}(3,2))*sqrt(3)/2, slpsys{2,isc}(3,4)*c_a]; % point 1 p2=[3*slpsys{2,isc}(4,1)/2, (slpsys{2,isc}(4,1)+2*slpsys{2,isc}(4,2))*sqrt(3)/2, slpsys{2,isc}(4,4)*c_a]; % point 2 p3=[3*slpsys{2,isc}(5,1) /2, (slp sys{2,isc}(5,1)+2*slpsys{2,isc}(5,2))*sqrt(3)/2, slpsys{2,isc}(5,4)*c_a]; % point 3 p4=[3*slpsys{2,isc}(6,1)/2, (slpsys{2,isc}(6,1)+2*slpsys{2,isc}(6,2))*sqrt(3)/2, slpsys{2,isc}(6,4)*c_a]; % point 4 p5=[3*slpsys{2,isc}(7,1)/2, (slpsys{2,is c}(7,1)+ 2*slpsys{2,isc}(7,2))*sqrt(3)/2, slpsys{2,isc}(7,4)*c_a]; % point 5 p6=[3*slpsys{2,isc}(8,1)/2, (slpsys{2,isc}(8,1)+2*slpsys{2,isc}(8,2))*sqrt(3)/2, slpsys{2,isc}(8,4)*c_a]; % point 6 mag_m=(m(1,1)^2+m(1,2)^2+m(1,3)^2)^0.5; mag_n=(n(1, 1)^2+n(1 ,2)^2+n(1,3)^2)^0.5; unit_m = m/mag_m; unit_n = n/mag_n; dot0(isc,1) = unit_m*unit_n'; slpsys{1,isc} = isc; slpsys{3,isc} = [unit_n;unit_m;p1;p2;p3;p4;p5;p6]; % normal in first row, direction in next row, points in next 6 rows slpsys{4,isc} = 0.5*(unit_m'*unit_n + unit_n'*unit_m); % Schmid matrix unit_tau(isc,:) = [isc, cross(unit_m, unit_n)]; % edge dislocation line direction tau unit vectors are generated % if linedir(isc,2)<0 % linedir(isc,:) = - 1.*lin e dir(isc,:); % end end sortlinedir = sortrows(unit_tau,[ - 2, - 3, - 4]); Schm_f_ss = zeros(nslphex,11,Angrows); Schm_labvecA = zeros(nslphex+1,33); sortmv = zeros(nslphex+1,33,Angrows); xic = zeros(nslphex,3,ghklrows); xicg = zeros(nslphex,3,ghklrows); xi rg = zeros(nslphex,3,ghklrows); xig = zeros(nslphex,3,ghklrows); for gg = 1:1:ghklrows for isc=1:1:nslphex; xic(isc,1:3,gg) = cross(unit_tau(isc,2:4),unit_ghklC(gg,:)); % Streak direction vector in crystal frame, not normalized! % xi(i sc,1:3,isc) = xi(isc,1:3,gg) / norm(xi(isc,1:3,gg)); end end % code to set up inverse pole figure and labeling, probably has some inconsistencies sij = [0.9581 - 0.4623 - 0.1893 0.698 2.1413 2.8408]/100; % for Ti from Si mmons and Wang in units of 1 /GPa Elow = 83.2; Ehigh = 145.5; % Elastic Ccontants of Nb from Simmons and Wang for Ti CTE = [15.4 15.4 30.6]; % this is for Sn, not a cubic material ... such as this inconsistency f = figure( 'Position' , [0,0,500,500]); movegui(f, 'northwest' ); set(g cf, 'Color' , [1 1 1]); hold on ; hold on ; axis square ; xmax = 0; if c_a ~= 1 trideg = 45; if ihex == 1 trideg = 30; end angle = 0:1:trideg; 77 xang = cosd(angle); yang = sind(angle); borderx = [0 xang 0]; bordery = [0 yang 0]; xmax = 1.02; axis([0 xmax 0 xmax]), % TickDir, 'out' ??? plot(borderx,bordery, 'k - ' ); edgecolor = [0,0,1]; % perimeter of plotting symbol if ipfd == 2 edgecolor = [.9,.9,0]; end if ipfd == 1 edgecolor = [1,0,0]; end for p = 0:.05:1 % plot inverse pole figure gray scale symbols plot(.03+p*xmax*.7,.95*xmax, 'o' , 'LineWidth' ,4, 'MarkerEdgeColor' ,[1 1 - p p], ... 'MarkerFaceColor' ,[1 1 1], 'MarkerSize' ,8) if c_a ~= 1 plot(.03+p*xmax*.7,.85*xmax, 'o' , 'LineWidth ' ,1, 'MarkerEdgeColor' ,[p p p], ... 'MarkerFaceColor' ,[p p p], 'MarkerSize' ,6) end end text(0.02,.9*xmax, 'E direction, yellow (low) -- > magenta (high)' ); if c_a ~= 1 text(0.02,.8*xmax, 'CTE direction, black(low) -- > white(high)' ); end i f stereo == 0 text(0.02,.7*xmax, 'Z projection, not stereographic' ); else text(0.02,.7*xmax, 'Stereographic projection' ); end text(0.02,.5*xmax,[ 'IPF direction ' ,num2str(ipfd)]); text(0.02,.6*xmax,[ 'c/a ratio ' ,num2str(c_a)]); % Generate orientat ion matrices for each orientation in Ang R45 = [1,0,0;0,cosd(45), - sind(45);0,sind(45),cosd(45)]; % Rotation of 45 deg about X axis % % for iAng=1:1:Angrows; %1;%4; %2; % phidA = Ang(iAng,3:5); phidA(1) = phidA(1); %; +180% **** Rotating euler a ngles (e.g. to correct for 180 rotation) if phidA(1)>360 phidA(1) = phidA(1) - 360; end if phidA(1)<0 phidA(1) = phidA(1)+360; end phisA = phidA*pi/180; %Compute Bunge orientation matrix g gphi1=[cos( phisA(1,1)),sin(phisA(1,1)),0; - sin(phisA(1,1)),cos(phisA(1,1)),0;0,0,1]; gPhi=[1,0,0;0,cos(phisA(1,2)),sin(phisA(1,2));0, - sin(phisA(1,2)),cos(phisA(1,2))]; gphi2=[cos(phisA(1,3)),sin(phisA(1,3)),0; - sin(phisA(1,3)),cos(phisA(1,3)),0;0,0, 1]; gA= gphi2*gPhi*gphi1; gAR = gA*Rotateview; % to rotate point of view in plot g(:,:,iAng)=gA; R(:,:,iAng)=gA'; gsgTA = gA*ststens_n*gA'; %rotated stress tensor basalY(iAng,1) = abs(gA(3,2)); for i = 1:1:3 CTEm(:,i) = gA(:,i).*CTE'; end for i = 1:1:3 % This gives the CTE in the x (1), y (2), z (3) directions e1 = sij(1)*(gA(1,i)^4 + gA(2,i)^4) + sij(4)*gA(3,i)^4; e2 = (2*sij(2) + sij(6))*(gA(1,i)^2 * gA(2,i)^2); e 3 = (2*sij(3) + sij(5))*gA(3,i)^2 * (gA(1,i)^2 + gA(2,i)^2); EA(iAng,i)=1./(e1+e2+e3); end Eb = (EA(iAng,iEd) - Elow)/(Ehigh - Elow); pv = gA(:,ipfd)'; pv = abs(pv); if c_a == 1 x(1) = median(pv); x(2) = min(pv); 78 x(3) = max(pv); else x(1) = max(pv(1),pv(2)); x(2) = min(pv(1),pv(2)); x(3) = pv(3); end if ihex == 1 ang = atand(x(2)/x(1)); if ang > 30 % tangent of 30 deg na ng = 30 - (ang - 30); radius = (x(1)^2+x(2)^2)^.5 ; x(1) = cosd(nang)*radius; x(2) = sind(nang)*radius; end end if stereo == 1 plot(x(1)/(1+x(3)), x(2)/(1+x(3)), 'o' , 'LineWidth' ,2, 'MarkerEdgeColor' , ... edgecolor, 'MarkerFaceColor' ,[ 1 1 - Eb Eb], 'MarkerSize' ,14) text(x(1)/(1+x(3)), x(2)/(1+x(3))+.037,num2str(iAng)) else plot(x(1),x(2), 'o' , 'LineWidth' ,1, 'MarkerEdgeColor' ,edgecolor, ... 'MarkerFaceColor' ,[1 1 - Eb Eb], 'MarkerSize' ,8) text(x(1),x(2 )+.037,num2str(iAng)) en d for isc=1:1:nslphex Sf=0.; for i=1:1:3 for j=1:1:3 % Compute Schmid Factor, slpsys{4 = Schmid matrix} Sf=Sf+gsgTA(i,j)*slpsys{4,isc}(i,j); end en d if i sc >= 46 && Sf <0 Sf = 0.001 * Sf ; % this is to prevent anti - twin shears from being seriously considered later end rot_nA = slpsys{3,isc}(1,:)*gAR; rot_bA = slpsys{3,isc}(2,:)*gAR; rot_p1 = slpsys{3,isc}(3,:)*gAR; rot_p2 = slpsys{3,isc}(4,:)*gAR; rot_p3 = slpsys{3,isc}(5,:)*gAR; rot_p4 = slpsys{3,isc}(6,:)*gAR; rot_p5 = slpsys{3,isc}(7,:)*gAR; rot_p6 = slpsys{3,isc}(8,:)*gAR; rot_tA = cross(rot_bA',rot_nA); % This variable has slip system number, Schmid Factor, plane and Burgers in sample coord syst, and hkl,uvw Schm_f_ss(isc,1:1 1,iAng)=[isc, Sf, abs(Sf), slpsys{2,isc}(1,:),slpsys{2,isc}(2,:)]; % Schmid factors (1 - 3), rotated plane normal (4 - 6), Computed rotated Burgers vector (7 - 9), % , plane trace on Z surface (10 - 12) Computed rotated position vectors to points p1 - p4 (13 - 24) % 1 2 3 4 - 6 7 - 9 10 - 12 Schm_lab vecA(isc,:) = [isc, Sf, abs(Sf), rot_nA, rot_bA, cross(rot_nA',[0,0,1]), ... rot_p1, rot_p2, rot_p3, rot_p4, rot_p5, rot_p6, rot_tA]; end % 13 - 15 16 - 18 19 - 21 22 - 24 25 - 27 28 - 30 31 - 33 % % sortmv contains list from high to l ow Schmid factor, with associated infomation to draw slip system in unit cell % useful plotting unit cell vectors will sort to bottom row Schm_labvecA(nsl phex+1,:) = [0 1 - 1 [1 0 0]*gAR [0 1 0]*gAR [0 0 1]*gAR 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] ; % 33 columns sortmv(:,:,iAng) = sortrows(Schm_labvecA, - 3); if iAng == 1 % Rotated xi (streak) vectors for gg = 1:1:ghklrows ghklg(gg,1:3) = unit_ghklC(gg,:)*gAR; for isc=1:1:nslphex; xicg(isc,1:3,gg ) = xic(isc,1:3,gg) * gA; % streak direction vector, rotated to sample coordinates xirg(isc,1:3,gg) = cross(Sc hm_labvecA(isc,31:33),ghklg(gg,1:3)); % streak direction vector, from rotated sample coordinates xig(isc,1:3,gg) = xirg(isc,1:3,gg) * R45; % rotated streak vector rotated from sample to detector coordinate system defined by R45 end end end end 79 % plot the image of unit cell, slip vectors, planes, plane normals, and plane traces % Then choose what row (orientation) to analyze, % and set iAng to this orientation (row), and run this second cell. X down and Y to right!!! iAng = 1; ptpl = 1; % 0 = don't plot plane traces, ptss = 2; % 0 just the unit cell, and 1 slip planes, 2 and d irections, 3 and plane normals, 4 directions only, pta123o = 1; % 1 = a1 a2 a3 coordinates and origin (open circle), nplots = nslphex; %48; %1; % number of slip sorted systems to plot % no further use r input below here % Dashed lines give plane tr aces, (colors in groups based on slip % families) Shorter plane traces imply that the slip plane is nearly % parallel to page, longer traces imply that the plane is highly inclined. % dotted red, green, blue lines give the x,y,z edges of unit cell. % Turquiose/Teal line shows Burgers vector direction, with direction % away from the ball end ( -- ! -- > adjusted for the sign of the Schmid factor < -- ! -- ). % Slip plane is shaded light when the plane normal has out - of - page % component, darker when into the page. % Strategy: First extract useful vectors from slip system information to draw the hexagonal prisms % positions in sortmv p1:13 - 15 p2:16 - 18 p3:19 - 21 p4:22 - 24 p5:25 - 27 p6:28 - 30 % positio ns in pln p1:4 - 6 p2:7 - 9 p3:10 - 12 p4:13 - 15 p5:16 - 18 p6:18 - 21 for isc = 1:1:nplots if sortmv(isc,1,iAng) == 1; % locate the two basal planes pln(1,4:21) = sortmv(isc,13:30,iAng); % bottom basal plane pln(2,4:21) = sortmv(isc,13:30,iAng); % top basal plane rotc = sortmv(isc,4:6,iAng)*c_a; % basal plane normal * c/a for j = 4:3:19 pln(2,j:j+2) = pln(1,j:j+2) + rotc; % move top plane up by a unit of c end a1 = sortmv(isc,7:9,iAng); % locate a1 using SS1 elseif sortmv(isc,1,iAng) == 2; a2 = sortmv(isc,7:9,iAng); % locate a2 using SS2 elseif sortmv(isc,1,iAng) == 3; a3 = sortmv(isc,7:9,iAng); % locate a3 using SS3 end end for isc = 1:1:nplots; if sortmv(isc,1,iAng) == 4; % locate two prism planes on opposite sides using SS4 pln(3,4:21) = sortmv(isc,13:30,iAng); for j = 13:3:2 8 pln(4,j - 9:j - 7) = sortmv(isc,j:j+2,iAng) + a2 - a3; end elseif sortmv(isc,1,iAng) == 5; % locate t wo prism planes on opposite sides using SS5 pln(5,4:21) = sortmv(isc,13:30,iAng); for j = 13:3:28 pln (6,j - 9:j - 7) = sortmv(isc,j:j+2,iAng) + a3 - a1; end elseif sortmv(isc,1,iAng) == 6; % locate two prism planes o n opposite sides using SS6 pln(7,4:21) = sortmv(isc,13:30,iAng); for j = 13:3:28 pln(8,j - 9:j - 7) = sor tmv(isc,j:j+2,iAng) + a1 - a2; end end end for j = 1:1:2 % Find z elevation of basal planes for k = 1:1:3 pln(j,k) = (pln(j,3+k)+pln(j,6+k)+pln(j,9+k)+pln(j,12+k)+pln(j,15+k)+pln(j,18+k))/6; end end center = (pln(1,1:3)+pln(2,1:3))/2; for j = 3:1:8 % Find z elevation of prism planes pln(j,3) = (pln(j,12)+pln(j,15)+pln(j,18)+pln(j,21))/4; end so rtpln = sortrows(pln, - 3); minx = 0; miny = 0; minz = 0; maxx = 0; maxy = 0; maxz = 0; for j = 1:1:8 % assemble vectors for plotting faces of hex prism prsxplt(j,1:7) = [sortpln(j,4) sortpln(j,7) sortpln(j,10) sortpln(j,13) sortpln(j,16) sortpln(j,19) s ortpln(j,4)]; minx = min(minx,min(prsxplt(j,:))); maxx = max(maxx,max(prsxplt(j,:))); 80 prsyplt(j,1:7) = [sortpln(j,5) sortpln(j,8) sortpln(j,11) sortpln(j,14) sortpln(j,17) sortpln(j,20) sortpln(j,5)]; miny = min(miny,min(prsyplt(j,:))); maxy = max(maxy,max(prsyplt(j,:))); prszplt(j,1:7) = [sortpln(j,6) sortpln(j,9) sortpln(j,12) sortpln(j,15) sortpln(j,18) sortpln(j,21) sortpln(j,6)]; minz = min(minz,min(prszplt(j,:))); maxz = max(maxz,max(prszplt(j,:))); end sscol = 1; %decide if w an t to plot unit cells for this slip system ipl = - 8; % Strategy: Next, start isc loop for plotting slip systems ipc = 0; % plot counter for isc = 1:1:nplots ssn = sortmv(isc,1,iAng); % slip system number if pssf(ssn) == 1 ipc = ipc + 1; if ipl - ipc == - 9 % eight plots on a page f = figure( 'Position' , [0,0,1050,525]); movegui(f, 'northwest' ); set(gcf, 'Color' , [1 1 1]); hold on ; ipl=ipl+8; end subplot(2,4,ipc - ipl) hold on sp1 = sortmv(isc,13:15,iAng); % beginning of Burgers vector sp2 = sortmv(isc,16:18,iAng); % extract plotted points on perimeter of the slip plane sp3 = sortmv(isc,19:21,iAng); sp4 = sortmv(isc,22:24,iAng); sp 5 = sortmv(isc,25:27,iAng); sp6 = sortmv(isc,28:30,iAng); spx = [sp1(1) sp2(1) sp3(1) sp4(1) sp5(1) sp6(1) sp1(1)]; spy = [sp1(2) sp2(2) sp3(2) sp4(2) sp5(2) sp6(2) sp1(2)]; S f = sortmv(isc,2,iAng); % Schmid factor Sfs = 1; if Sf < 0 Sfs = - 1; end ; n = [0 0 0 sortmv(isc,4:6,iAng)]; % plane normal b = [sp1 sp4]; % p1+sortmv(isc,7:9,iAng)]; % Burgers vector nvec = sortmv(isc,4:6,iAng); bvec = sortmv (isc,7:9,iAng); pt = sortmv(isc,10:12,iAng); % plane trace minx = min(minx, sp1(1)+n(4)); maxx = max(maxx, sp1(1)+n(4)); % find appropriate range of x and y for plot miny = min(miny, sp1(2)+n(5)); maxy = max( maxy, sp1(2)+n(5)); midx = (minx+maxx)/2; midy = (miny+maxy)/2; del = 2; % These plots will match TSL with X down !!!! Plotting starts... axis square set(gca , 'ycolor' , 'w' ); set(gca , 'xcolor' , 'w' ); % make axes white for ease in later arranging. axis([midy - del midy+del - midx - del - midx+del]) if pta123o > 0 if Ang(iAng,4) < 90 % make the 3 coordinate axes visible below slip planes plot([0 a1(2)], - [0 a1(1)], ':' , 'Lin ewidth' ,3, 'Color' ,[1 0 .2]); % plot x = red plot([0 a2(2)], - [0 a2(1)], ':' , 'Linewidth' ,3, 'Color' ,[.6 .8 0]); % plot y = green - gold plot([0 a3(2)], - [0 a3(1)], ':' , 'Linewidth' ,3, 'Color' ,[0 0 1]); % plot z = blue end end if ipl - ipc== - 1 end if ptss > 0 && ptss < 4 % plot slip planes if sortmv(isc,6,iAng) > 0 % is k component of slip plane normal positive or negative? fill(spy, - spx, [.8 .8 .65]) % slip pl ane filled warm gray else % slip plane filled cool gray if normal has neg z component fill(spy, - spx, [.65 .65 .7]) % plot([n(2) n(5)], - [n(1) n(4)],'Linewidth',3,'Color',[.65 .65 .7]); end if n(6) > 0 pncolor = [0 0 0]; % positive plane normal color else pncolor = [.5 .5 .5]; % negative plane normal color 81 end end % plane is plotted if ptss == 2 || pt ss == 4 % plot Burgers vectors if sortmv(isc,6,iAng) > 0 Bvcolor = [0 .7 .7]; if ssn >= 55 Bvcolor = [.1 .6 0]; end if ssn >= 43 && ssn < 55 Bvcolor = [1 .6 0]; end else Bvcolor = [0 1 1]; if ssn >= 55 Bvcolor = [.3 .9 0]; end if ssn >= 43 && ssn < 55 Bvcolor = [1 .8 0]; end end Sf s = 1; if ssn < 43 % will reverse the sign of Schmid factor for dislocations end if Sf > 0 % plot Burgers vector direction if ssn >= 43 % this is for twins - the Burgers vector length is shown to be 1/2 of the usual length in the unit cell plot(b(2), - b(1), '.' , 'MarkerSize' , 24, 'Color' , Bvcolor) plot([b(2) (b(2)+b(5))/2], - ([b(1) (b(1)+b(4))/2]), 'Line width' ,4, 'Color' ,Bvcolor) else plot(b(2), - b(1), '.' , 'MarkerSize' , 24, 'Color' , Bvcolor) plot([b(2) b(5)], - [b(1) b(4)], 'Linewid th' ,4, 'Color' ,Bvcolor) % quiver(p1(1),p1(2),dp(1),dp(2) ,0,'Linewidth',2,'Color',Bvcolor) end else % plot Burgers vector in opposite direction if ssn >= 43 % this is for twins - the Burgers vector length is shown to be 1/2 of the usual length in the unit ce ll plot(b(2), - b(1), '.' , 'MarkerSize' , 24, 'Color' , Bvcolor) plot([b(2) (2*b(2)+b(5))/3], - ([b(1) (2*b(1)+b(4))/3]), 'Linewidth' ,4, 'Color' ,Bvcol or) else plot(b(5), - b(4), '.' , 'MarkerSiz e' , 24, 'Color' , Bvcolor) plot([b(5) b(2)], - [b(4) b(1)], 'Linewidth' ,4, 'Color' ,Bvcolor) % quiver(p2(1),p2(2), - dp(1), - dp(2),0,'Linewidth',4,'Color',Bvcolor) end end if ptss ==3 plot([b(2) (b(2)+n(5))], - ([b(1) (b(1)+n(4))]), 'Linewidth' ,4, 'Color' ,pncolor) end end % Burger s vector is plotted for j = 1:1:4 % plot the 4 top most surface prisms of the hex cell that have the highest z elevation plot(prsyplt(j,:), - prsxplt(j,:), 'Linewidth' ,2, 'Color' ,[.0 .0 .0]); end if pta123o > 0 if Ang(iAng,4) >= 90 % make the 3 coordinate axes visible above slip planes plot([0 a1(2)], - [0 a1(1)], ':' , 'Lin ewidth' ,3, 'Color' ,[1 0 .3]); % plot x = red plot([0 a2(2)], - [0 a2(1)], ':' , 'Linewidth' ,3, 'Color' ,[.5 .6 0]); % plot y = green - gold plot([0 a3(2)], - [0 a3(1)], ':' , 'Linewidth' ,3, 'Color' ,[0 0 1]); % plot z = blue end end if ptpl == 1 % plot plane traces if ssn>=55 % compression twin plane traces green plot([ - pt(2) pt(2)], - [ - pt(1) pt(1)], ' -- ' , 'Linewidth' ,3, 'Color' ,[.2 .8 0]) elseif ssn>42 && ssn<55 % extens ion twin plane traces orange plot([ - pt(2) pt(2)], - [ - pt(1) pt(1)], ' -- ' , 'Linewidth' ,3, 'Color' ,[1 .6 0]) elseif ssn>12 && ssn<43 % plane traces green - gold plot([ - pt(2) pt(2)], - [ - pt(1) pt(1)], ' -- ' , 'Linew idth' ,3, 'Color' ,[.95 .85 0]) elseif ssn<13 && ssn>6 % pyr green plot([ - pt(2) pt(2)], - [ - pt(1) pt(1)], ' -- ' , 'Linewidth' ,3, 'Color' ,[0 .9 .5]) elseif ssn<7 && ssn>3 % prism red plot([ - pt(2) pt(2)], - [ - pt(1) pt(1)], ' -- ' , 'Linewidth' ,3, 'Color' ,[1 .2 0]) else % {medium slip systems} blue plot([ - pt(2) pt(2)], - [ - pt(1) pt(1)], ' -- ' , 'Linewidth' ,3, 'C olor' ,[0 0 1]) end % ---- > NOTE tha t Schmid factor vector is plotted in correct direction, 82 end % plot plane traces if pta123o == 1 % plot coordinate axes plot(0,0, 'ko' ); end % ---- > Burgers vector is labeled and plotted with consistently signed b vector direction. if ptss >= 1 title({[ 'ssn' num2str(ssn) ' n' mat2str(slpsys{2,ssn}(1,:)) mat2str(Sfs*slpsys{2,ssn}(2,:)) 'b' ], ... [mat2str(sortmv(i sc,4:6,iAng),3), mat2str(sortmv(isc,7:9,iAng),3)], ... [ 'Eule rs = ' , mat2str(Ang(iAng,3:5),3)], ... [ 'c - axis = ' , mat2str(sortmv(nslphex+1,10:12,iAng),3)], ... [LC ' Or - ' num2str(iAng) ' m' num2str(isc) ' = ' num2str(Sfs* Sf, 3) ] }) end end end 83 A PPENDIX B: Matlab MP R hexagonal grain pair analysis code (written by Thomas R. Bieler) % T.R. Bieler - m' Schmid and fip calculator, 4 June - 14 1 July 2013 with input from Adam L Pilchak % contains pieces from prior codes that are probably right, but no guarantees, use at yo ur own risk. % Written and used in Matlab release R2009b, and R2016a, % substantially revised in 2018 during sabbatical at IMDEA Materiales % % Sources for these ideas are discussed in % Bieler et al. Int. J. Plasticity 25(9), 1655 1683, 2009, and % Kumar et al. J. Engineering and Materials Technology 130, 021012, 2008 % Bieler et al. Current Opinion in Solid State and Materials Science 18(4) 2014 212 - 226 % an d related prior work. % Important note - Euler angle computation coordinate system i s not % necessarily consistent with any other coordinate system. To obtain % consistent results to make slip planes and Burgers vectors come out % right, we pre - rotate a cquired data by 180deg in first Euler angle so that % the Euler angle coordinate syste m has X down and Y right. Plotting is % done with this perspective in mind. If your raw data is not pre - rotated % then setting hkl = 1 and adjusting first Euler angle when computing g % below is necessary. % Input data are default type 2 grains file a nd reconstructed grain boundary files from TSL % The reconstructed boundary file does not contain phase ID, so the grain % file is also needed, and it is the primary source. % This code expects phase ID to be 1 = hex, 2 = BCC, 3 = FCC, 4 = BCT (Sn). % If your Phase ID is different, then some adjustments need to be made - % probably easiest if you copy information in column 10 of IDgr and put it in % column 11, and then transform it to the phase number needed in this code in column 10. % To run o n particular boundary with two orientations, put values into IDGR % with estimates of relative grain center posi tions, and skip the file input in the next block % [ _ 25 13] is maximum Sf in single slip in Z clc; clear; IDgr = [1 8.0421 8 9.3031 354.1300 10 10 60.2 0.364 1.3 0 1 % 40.6+180 8.5 296 159.4, 8.8, 203.7, 36 IDgr = [1 0 0 0 354.27 88.49 352.07 60.2 0.364 0 0 1 % 40.6+180 8.5 296 159.4, 8.8, 203.7, 37 2 2.6823 101.5918 354.2043 15 4 60.2 0.364 1.3 0 1] % 270.6, 14.3, 96 delx = IDgr(2,5) - IDgr(1,5); dely = IDgr(2,6) - IDgr(1,6); mid = [(IDgr(2,5)+IDgr(1,5)) (IDgr(2,6)+IDgr(1,6))]*.5; gbvec = [0 1; - 1 0]*[delx;dely]*.5; x1 = mid(1,1)+gbvec(1,1); y1 = mid(1,2)+gbv ec(2,1); x2 = mid(1,1) - gbvec(1,1); y2 = mid(1,2) - gbvec(2,1); trang = atand( - (y2 - y1) / (x2 - x1) ); if trang < 0 trang = trang + 180; end RBdyn = [ 1 2 3 4 5 6 (delx^2+dely^2)^.5 trang x1 y1 x2 y2 2 1] % Y goes down; length, angle, x1 y1 x2 y 2 right grain # , left grain # fnameRCB = 'just_one_pair_of_points_RB.txt' ; % The code reads #(1), orientation(2:4), grain center(5,6), phase ID(10) from IDgr, and % grain boundary inclination from horizontal(8) and segment position(9:12) and right, left grain #s (13,14) % are read from the rec onstructed grain boundary file % If you want to enter your own data to explore using this code, you will want to populate the above % two variables in the same way as Grain ID and Recrystructed Boundary files ar e, as follows: % In IDgr, Column 1 is grai n number, columns 2 - 4 are Bunge Euler angles with X down and Y to the right % columns 5 and 6 contain x and y positions of the data points (normally grain center in Grains file) % In RBdy, in in row 1 (which is bou ndary #1) euler angles for the two grains identified in colums 13 and 14 % are provided in colums 1 - 6. colum 7 is grain boundary length, column 8 (H in excel), is the angle from the horizontal % to the apparent grain boundary direction e.g. 90deg if you want a vertical boundary. columns 9 - 12 have start and % end positions of the grain boundary. In columns 13 and 14 (J&K) put the % right,left grain numbers on either side of boundary 1 in that order (yes, the right one first). %% 84 clc; clear; G F2HeaderLines = 16; % 11 for single phase, 12 for two phase RCBHeaderLines = 11; % total number of header lines for the Reconstructed boundary file fnameGF2 = 'C: \ Users \ chels \ Desktop \ Titanium \ Ten sion \ Ti525_#10_EBSD Scans \ Ti525_10_GF.txt' ; GF2HeaderLines = 17; fnameRCB = 'C: \ Users \ chels \ Desktop \ Titanium \ Tension \ Ti525_#10_EBSD Scans \ Ti525_10_RB.txt' ; RCBHeaderLines = 11; GF2HeaderLines = 12; RCBHeaderLines = 8; fileID = fopen(fnameGF2) dataGF = importdata(fnameGF2, ' ' , GF2HeaderLines) IDgr = dataGF.data; f ileID = fopen(fnameRCB) dataRB = importdata(fnameRCB, ' ' , RCBHeaderLines) RBdyn = dataRB.data; fclose( 'all' ) fnlengthRCB = length(fnameRCB); chr=char(fnameRCB); %% Sfthr = 0.2; % higher schmid tolerance value used to limit serach for high m' values SflimL = 0.0; % lower schmid tolerance value used to limit filling out table mpthr = 0.75; % threshold v alue for considering m' values to be meaningful hkl = 1; % flag used to decide whether to adjust first euler angle for various reasons.. . see below nslphex = 39; c_a_hex = 1.59; % 1.587 for pure Ti ... not turning on compression twinning in Hexagonal nslpbcc = 24; c_a_bcc = 1.0; % could differ for metastable phases... not turning on 123 slip nslpfcc = 12; c_a_fcc = 1.0; % ~1.02 for TiAl% For FCC, it is 18 if cube slip is included. nslpbct = 32; c_a_bct = 0.5456; % for Sn nslp = [nslphex, nslpbcc, nslpfcc, nslpbct]; % number of slip systems used for phases 1, 2, 3, 4 c_a = [c_a_hex, c_a_bcc, c_a_fcc, c_a_bct]; % If your data is SINGLE PHASE, then you must put the correct phase number into the variable one_ss, HERE one_ss = 1; % e.g. , the if statement below will set phase = 1 for Hex, 2 for BCC(3 for FCC) in IDgr file column 10: numPhases = max(IDgr(:,10)); % check the number of phases in the dataset if numPhases >0 && (chr(fnlengthRCB - 4) == 'B' || chr(fnlengthR CB - 4) == 'b' ) % do nothing else % for a grain boundary trace, need to set IDGR to one_ss IDgr(:,10) = one_ss; % single phase, --- !!! set to 1 for HEX or 2 for BCC above !!! --- for i = 1:1:4 % else if it's two phase, do nothing, if phase 1 is hex, 2 is bcc. if i ~= one_ss nslp(i) = 0; end end end % Stress tensor is define d using TSL convensions with x down !!! put the one you want last sigma = [0,0,0; 0,1,0; 0,0,0]; nste n = 1; for i = 1:1:nsten str2 = sigma(:,:,i)*sigma(:,:,i)'; ststens_mag = (str2(1,1)+str2(2,2)+str2(3,3))^.5; sigma_n(:,:,i) = sigma(:,:,i)/s tstens_mag; % normalized stress tensor to get generalized Schmid factor sigma_v(:,i) = [sigma(1,1) sigma(2,2) sigma(3,3)]'; % vectorized version of trace end dIDgr = size(IDgr); % This is a default type 2 grain file, both are needed. dRBdyn = siz e(RBdyn); % This is a reconstructed grain boundary file good = 0; if dRB dyn(1,2) == 21; for ii = 1:1:dRBdyn(1,1) end else RBdy = RBdyn; end dRBdy = size(RBdy); % E(X,Y,Z) will be calculated from [S11 S12 S13 S33 S44 S66] in 1/GPa; % hexago nal stiffness chosen: sij(1,:) = [0.9581 - 0.4623 - 0.1893 0.698 2.1413 2.408]/100; % for Ti from Simmons and Wang in units of 1/GPa 85 % cubic stiffness chosen: sij(2,:) = [0.6862 - 0.2581 - 0.2581 0.6862 1.2123 1.2123]/100; % for Ta from Simmons and Wang in units of 1/GPa % dis p('E(X,Y,Z) in GPa; S11 S12 S13 S33 S44 S66 210 Rayne, J.A. and B.S. Chandrasekhar, % Elastic Ccontants of beta tin from 4.2K to 300K, Phys Rev. 118, 1545 - 49, 1960 sij(4,:) = [4.3627 - 3.3893 - 0.394 1.4501 4.5393 4.1667]/100; % for Sn in units of 1/GPa % Ti - 6Al, Ti - 15Cr, alpha/beta in Ti6242 from J. Kim and S.I. Rokhlin, J. Acoust. Soc. Am. 126 - 6 dec 2009 % % Set up vectors useful for plotting unit cells with slip systems O = [ 0 0 0 0]; % (I) (H) A = [ 2 - 1 - 1 0]/3; % C ------- B B = [ 1 1 - 2 0]/3; % / \ / \ C = [ - 1 2 - 1 0]/3; % / a2 / \ D = [ - 2 1 1 0]/3; % / \ / \ E = [ - 1 - 1 2 0]/3; % (J)D ------ O(P) - a1 - > A(G) -- > x F = [ 1 - 2 1 0] /3; % \ / \ / P = [ 0 0 0 1]; % \ a3 \ / G = [ 2 - 1 - 1 3]/3; % \ / \ / H = [ 1 1 - 2 3]/3; % E -------- F I = [ - 1 2 - 1 3]/3; % (K) (L) J = [ - 2 1 1 3]/3; % K = [ - 1 - 1 2 3]/ 3; % where a1 = OA a2 = OC a3 = OE L = [ 1 - 2 1 3]/3; % DA || a1, FC || a2, BE || a3 % Slip system definitions set as of 1 Nov 2019 to be consistent with DAMASK % Hex plane direction 1st and 4th point is Burgers vect or % basal - g lide:&Be Mg Re Ti; Re sshex(:,:,1) = [0 0 0 1; 2 - 1 - 1 0; D ; E ; F ; A ; B ; C ]; sshex(:,:,2) = [0 0 0 1; - 1 2 - 1 0; F ; A ; B ; C ; D ; E ]; sshex(:,:,3) = [0 0 0 1; - 1 - 1 2 0; B ; C ; D ; E ; F ; A ]; ibas = 1; fbas = 3; % prism - glide:Ti Zr RE; Be Re Mg sshex(:,:,4) = [ 0 1 - 1 0; 2 - 1 - 1 0; E ; E ; E ; F ; L ; K ]; sshex(:,:,5) = [ - 1 0 1 0; - 1 2 - 1 0; A ; A ; A ; B ; H ; G ]; sshex(:,:,6) = [ 1 - 1 0 0; - 1 - 1 2 0; C ; C ; C ; D ; J ; I ]; iprs = 4; fprs = 6; % pris m sshex(:,:,7) = [ 2 - 1 - 1 0; 0 1 - 1 0; F; F; F; B ; H ; L ]; sshex(:,:,8) = [ - 1 2 - 1 0; - 1 0 1 0; B; B; B; D ; J ; H ]; sshex(:,:,9) = [ - 1 - 1 2 0; 1 - 1 0 0; D; D; D; F ; L ; J ]; i2prs = 7; f2prs = 9; % pyramidal - glide -- ** -- CORRECTED - - ** -- sshex(:,:,10) = [ 1 0 - 1 1; - 1 2 - 1 0; E ; E ; E ; D ; I ; L ]; sshex(:,:,11) = [ 0 1 - 1 1; - 2 1 1 0; F ; F ; F ; E ; J ; G ]; sshex(:,:,12) = [ - 1 1 0 1; - 1 - 1 2 0; A ; A ; A ; F ; K ; H ]; sshex(:,:,13) = [ - 1 0 1 1; 1 - 2 1 0; B ; B ; B ; A ; L ; I ]; sshex(:, :,14) = [ 0 - 1 1 1; 2 - 1 - 1 0; C ; C ; C ; B ; G ; J ]; sshex(:,:,15) = [ 1 - 1 0 1; 1 1 - 2 0; D ; D ; D ; C ; H ; K ]; ipyra = 10; fpyra = 15; % pyramidal - glide:; all? sshex(:,:,16) = [ 1 0 - 1 1; - 2 1 1 3; A ; B ; I ; P ; L ; A ]; sshex(:,:, 17) = [ 1 0 - 1 1; - 1 - 1 2 3; B ; B ; I ; P ; L ; A ]; sshex(:,:,18) = [ 0 1 - 1 1; - 1 - 1 2 3; B ; C ; J ; P ; G ; B ]; sshex(:,:,19) = [ 0 1 - 1 1; 1 - 2 1 3; C ; C ; J ; P ; G ; B ]; sshex(:,:,20) = [ - 1 1 0 1; 1 - 2 1 3; C ; D ; K ; P ; H ; C ]; ss hex(:,:,21) = [ - 1 1 0 1; 2 - 1 - 1 3; D ; D ; K ; P ; H ; C ]; sshex(:,:,22) = [ - 1 0 1 1; 2 - 1 - 1 3; D ; E ; L ; P ; I ; D ]; sshex(:,:,23) = [ - 1 0 1 1; 1 1 - 2 3; E ; E ; L ; P ; I ; D ]; sshex(:,:,24) = [ 0 - 1 1 1; 1 1 - 2 3; E ; F ; G ; P ; J ; E ]; sshex(:,:,25) = [ 0 - 1 1 1; - 1 2 - 1 3; F ; F ; G ; P ; J ; E ]; sshex(:,:,26) = [ 1 - 1 0 1; - 1 2 - 1 3; F ; A ; H ; P ; K ; F ]; sshex(:,:,27) = [ 1 - 1 0 1; - 2 1 1 3; A ; A ; H ; P ; K ; F ]; ipyrc = 16; fpyrc = 27; 86 % pyramidal - 2nd ord er glide sshex(:,:,28) = [ 1 1 - 2 2; - 1 - 1 2 3; (O+B)/2 ; C ; J ; (P+K)/2 ; L ; A]; sshex(:,:,29) = [ - 1 2 - 1 2; 1 - 2 1 3; (O+C)/2 ; D ; K ; (P+L)/2 ; G ; B]; sshex(:,:,30) = [ - 2 1 1 2; 2 - 1 - 1 3; (O+D)/2 ; E ; L ; (P+G)/2 ; H ; C]; sshex(:,:,31) = [ - 1 - 1 2 2; 1 1 - 2 3; (O+E)/2 ; F ; G ; (P+H)/2 ; I ; D]; sshex(:,:,32) = [ 1 - 2 1 2; - 1 2 - 1 3; (O+F)/2 ; A ; H ; (P+I)/2 ; J ; E]; sshex(:,:,33) = [ 2 - 1 - 1 2; - 2 1 1 3; (O+A)/2 ; B ; I ; (P+J)/2 ; K ; F]; i2pyrc = 28 ; f2pyrc = 33; % *** Twi n directions are opposite in Christian and Mahajan, and are not correcte to to be consistent with them % FROM Kocks SXHEX plane direction 1st and 4th point is Burgers vector, order of C1 differs from Kock's file % {1012}<1011> T1 twins 0.17; - 1.3 t wins: all Twin Vector must go in the % sense of shear, opposite C&M sense. sshex(:,:,34) = [ 1 0 - 1 2; - 1 0 1 1; A ; B ; J ; K ; K ; K ]; sshex(:,:,35) = [ 0 1 - 1 2; 0 - 1 1 1; B ; C ; K ; L ; L ; L ]; sshex(:,:, 36) = [ - 1 1 0 2; 1 - 1 0 1; C ; D ; L ; G ; G ; G ]; sshex(:,:,37) = [ - 1 0 1 2; 1 0 - 1 1; D ; E ; G ; H ; H ; H ]; sshex(:,:,38) = [ 0 - 1 1 2; 0 1 - 1 1; E ; F ; H ; I ; I ; I ]; sshex(:,:,39) = [ 1 - 1 0 2; - 1 1 0 1; F ; A ; I ; J ; J ; J ]; iT1 = 34; fT1 = 39; % {2111}<211 6> T2 twins: 0.63; - 0.4; Ti Zr Re RE]; Also does not follow C&M definition for shear direction sshex(:,:,40) = [ 1 1 - 2 1; - 1 - 1 2 6; (O+B)/2 ; C ; (J+I)/2 ; P ; (L+G)/2 ; A]; sshex(:,:,41) = [ - 1 2 - 1 1; 1 - 2 1 6; (O+C)/2 ; D ; (K+J)/2 ; P ; (G+H)/2 ; B]; sshex(:,:,42) = [ - 2 1 1 1; 2 - 1 - 1 6; (O+D)/2 ; E ; (L+K)/2 ; P ; (H+I)/2 ; C]; sshex(:,:,43) = [ - 1 - 1 2 1; 1 1 - 2 6; (O+E)/2 ; F ; (G+L)/2 ; P ; (I+J)/2 ; D]; sshex(:,:,44) = [ 1 - 2 1 1; - 1 2 - 1 6; (O+F)/2 ; A ; (H+G)/2 ; P ; (J+K)/2 ; E ]; sshex(:,:,45) = [ 2 - 1 - 1 1; - 2 1 1 6; (O+A)/2 ; B ; (I+H)/2 ; P ; (K+L)/2 ; F]; iT2 = 40; fT2 = 45; % {1011}<101 - 2> C1 twins: 0.10; 1.1; Mg; Zr Ti]; agrees with C&M sshex(:,:,46) = [ 1 0 - 1 1; 1 0 - 1 - 2; P ; L ; A ; (A+B)/2 ; B ; I ]; sshex(: ,:,47) = [ 0 1 - 1 1; 0 1 - 1 - 2; P ; G ; B ; (B+C)/2 ; C ; J ]; sshex(:,:,48) = [ - 1 1 0 1; - 1 1 0 - 2; P ; H ; C ; (C+D)/2 ; D ; K ]; sshex(:,:,49) = [ - 1 0 1 1; - 1 0 1 - 2; P ; I ; D ; (D+E)/2 ; E ; L ]; sshex(:,:,50) = [ 0 - 1 1 1; 0 - 1 1 - 2; P ; J ; E ; (E+F)/2 ; F ; G ]; sshex(:,:,51) = [ 1 - 1 0 1; 1 - 1 0 - 2; P ; K ; F ; (F+A) /2 ; A ; H ]; iC1 = 46; fC1 = 51; % {2112}<211 - 3> C2 twins:; 0.22; 1.2 Ti Zr Re]; agrees with C&M sshex(:,:,52) = [ 1 1 - 2 2; 1 1 - 2 - 3; (K+P)/2 ; L ; A ; (A+C )/2 ; C ; J]; sshex(:,:,53) = [ - 1 2 - 1 2; - 1 2 - 1 - 3; (L+P)/2 ; G ; B ; (B+D)/2 ; D ; K]; sshex(:,:,54) = [ - 2 1 1 2; - 2 1 1 - 3; (G+P)/2 ; H ; C ; (C+E)/2 ; E ; L]; sshex(:,:,55) = [ - 1 - 1 2 2; - 1 - 1 2 - 3; (H+P)/2 ; I ; D ; (D+F)/2 ; F ; G]; sshe x(:,:,56) = [ 1 - 2 1 2; 1 - 2 1 - 3; (I+P)/2 ; J ; E ; (E+A)/2 ; A ; H]; sshex(:,:,57) = [ 2 - 1 - 1 2; 2 - 1 - 1 - 3; (J+P)/2 ; K ; F ; (F+B)/2 ; B ; I]; iC2 = 52; fC2 = 57; mnslp = max(nslp); ss = zeros(8,3,mnslp,4); for i=1:1:mnslp % Change n & m t o unit vector, if i <= nslphex n=[sshex(1,1,i) (sshex(1,2,i)*2+sshex(1,1,i))/3^.5 sshex(1,4,i)/c_a_hex]; % Plane normal /c_a_hex m=[sshex(2,1,i)*1.5 3^.5/2*(sshex(2,2,i)*2+sshex(2,1,i)) sshex(2,4,i)*c_a_hex]; % Slip direction *c_a_h ex ss(1,:,i,1) = n/norm(n); % alpha plane ss(2,:,i,1) = m/norm(m); % alpha direction PHASE 1 is HEX ss(3,:,i,1) = [3*sshex(3,1,i)/2, (sshex(3,1,i)+2*sshex(3,2,i))*sqrt(3)/2, sshex(3,4,i)*c_a_hex]; % hpoint 1 ss(4,:,i,1) = [3*sshex(4,1,i)/2, (sshex(4,1,i)+2*sshex(4,2,i))*sqrt(3)/2, sshex(4,4,i)*c_a_hex]; % hpoint 2 ss(5,:,i,1) = [3*sshex(5,1,i)/2, (sshex(5,1,i)+2*sshex(5,2,i))*sqrt(3)/2, sshex(5,4,i)*c_a_hex]; % hpoint 3 ss(6,:, i,1) = [3*sshex(6,1,i)/2, (sshex(6,1,i)+2*sshex(6,2,i))*sqrt(3)/2, sshex(6,4,i)*c_a_hex]; % hpoint 4 ss(7,:,i,1) = [3*sshex(7,1,i)/2, (sshex(7,1,i)+2*sshex(7,2,i))*sqrt(3)/2, sshex(7,4,i)*c_a_hex]; % hpoint 5 ss(8,:,i,1) = [3*sshex(8,1,i)/2 , (sshe x(8,1,i)+2*sshex(8,2,i))*sqrt(3)/2, sshex(8,4,i)*c_a_hex]; % hpoint 6 end if i <= nslpbcc n = [ssbcc(1,1,i),ssbcc(1,2,i),ssbcc(1,3,i)/c_a_bcc]; % slightly tetragonal has c/a <> 1.0 m = [ssbcc(2,1,i),ssbcc(2,2,i),ssbcc(2,3 ,i)*c_a_bcc]; ss(1,:,i,2) = n/norm(n); % bcc plane PHASE 2 is BCC 87 ss(2,:,i,2) = m/norm(m); % bcc direction ss(3,:,i,2) = [ssbcc(3,1,i),ssbcc(3,2, i),ssbcc(3,3,i)*c_a_bcc]; % point 1 ss(4 ,:,i,2) = [ssbcc(4,1,i),ssbcc(4,2,i),ssbcc(4,3,i)*c_a_bcc]; % point 2 ss(5,:,i,2) = [ssbcc(5,1,i),ssbcc(5,2,i),ssbcc(5,3,i)*c_a_bcc]; % point 3 ss(6,:,i,2) = [ssbcc(6,1,i),ssbcc(6,2,i),ssbcc(6 ,3,i)*c_a_bcc]; % point 4 ss(7,:,i,2) = [ssbcc(7,1,i),ssbcc(7,2,i),ssbcc(7,3,i)*c_a_bcc]; % point 5 ss(8,:,i,2) = [ssbcc(8,1,i),ssbcc(8,2,i),ssbcc(8,3,i)*c_a_bcc]; % point 6 end if i <= nslpfcc n = [ssfcc(1,1,i),ssfcc(1,2,i),ssfcc(1,3,i)/c_a_fcc]; % slightly tetrag onal has c/a <> 1.0 m = [ssfcc(2,1,i),ssfcc(2,2,i),ssfcc(2,3,i)*c_a_fcc]; ss(1,:,i,3) = n/norm(n); % fcc plane PHASE 3 is FCC ss(2,:,i,3) = m/norm(m); % fcc direction ss(3,:,i,3) = [ssf cc(3,1,i),ssfcc(3,2,i),ssfcc(3,3,i)*c_a_fcc]; % point 1 ss(4,:,i,3) = [ssfcc(4,1,i),ssfcc(4,2,i),ssfcc(4,3,i)*c_a_fcc]; % point 2 ss(5,:,i,3) = [ssfcc(5,1,i),s sfcc(5,2,i),ssfcc(5,3,i)*c_a_fcc]; % point 3 ss(6,:,i,3) = [ssfcc(6,1,i), ssfcc(6,2,i),ssfcc(6,3,i)*c_a_fcc]; % point 4 ss(7,:,i,3) = [ssfcc(7,1,i),ssfcc(7,2,i),ssfcc(7,3,i)*c_a_fcc]; % point 5 ss(8,:,i,3) = [ssfcc(8,1,i),ssfcc(8,2,i ),ssfcc(8,3,i)*c_a_fcc]; % point 6 end if i <= nslpbct n = [ ssbct(1,1,i),ssbct(1,2,i),ssbct(1,3,i)/c_a_bct]; m = [ssbct(2,1,i),ssbct(2,2,i),ssbct(2,3,i)*c_a_bct]; ss(1,:,i,4) = n/norm(n); % bct plane PHASE 4 is BCT ss(2,:,i,4) = m/norm(m); % bct direct ion ss(3,:,i,4) = [ssbct(3,1,i),ssbct(3,2,i),ssbct(3,3,i)*c_a_bct]; % point 1 ss(4,:,i,4) = [ssbc t(4,1,i),ssbct(4,2,i),ssbct(4,3,i)*c_a_bct]; % point 2 ss(5,:,i,4) = [ssbct(5,1,i),ssbct(5,2,i),ssbct(5,3,i)*c_a_bct]; % point 3 ss(6,:,i,4) = [ssbct(6,1,i),ssbct(6,2,i),ssbct(6,3,i)*c_a_bct]; % point 4 ss(7,:,i,4) = [ssbct(7,1,i),ssbct(7,2,i),ssbct(7,3,i)*c_a_bct]; % point 5 ss(8,:,i,4) = [ssbct(8,1,i),ssbct(8,2,i),ssbct(8,3,i)*c_a_bct]; % point 6 end end %% Loop for grains to establish slip conditions for each grain EY = zeros(int16(dIDgr(1,1)*1.1),3); Sfplbv = zeros(mnslp+1,30); sortmv = zeros(mnslp+1,30,int16(dIDgr(1,1)*1.1)); grcen = zeros(int16(dIDgr(1,1)*1.1)); listSf = zeros(8,dIDgr(1,1)); lists s = zeros(8,dIDgr(1,1)); grcen(:,1) = - 1; % that is a little bigger that needed because some grain numbers are skipped, % and are thus marked with - 1. Grains are processed by grain number, not array location grmax = 0; ngc ount = 0; fprintf( 'Numbers and vectors computed for Grain # ' ); fnlengthRCB = length(fnameRCB); chr=char(fnameRCB); Xr = [0;1;2;3]; for ng=1:1:dIDgr(1,1); % generalized Schmid factor calculation loop for each grain ng if ng>ngcount+dIDgr/10; n gcount=ngcount+dIDgr/10; fprintf( ' %d ' ,ng); end ig = IDgr(ng,1); if ig > grmax grmax = ig; end if ig > 0 % phase(1) grain center(2,3) eulers(4:6) grain ID grcen(ig, 1:7) = [IDgr(ng,10) IDgr(ng,5:6 ) IDgr(ng,2:4) IDgr(ng,1)]; phid = grcen(ig,4:6); % phid is Euler phi angles in degrees ph = grcen(ig,1); % phase ID set if hkl == 1 phid(1) = phid(1)+ 180 ; % + 180 or +90 t o convert hkl to TSL software d efault if phid(1)>360 % or +180 to modify TSL Euler angle coordinate system to have X down and Y right; phid(1) = phid(1) - 360; 88 elseif phid(1) < 0 phid(1) = phid(1) + 360; end end g1=[cosd(phid(1)),sind(phid(1)),0; - sind(phid(1)),cosd(phid(1)),0; 0,0,1]; g2=[1,0,0; 0,cosd(phid(2)),sind(phid(2)); 0, - sind(phid(2)),cosd(phid(2))]; g3=[cosd(ph id(3)),sind(phid(3)),0; - sind(phid(3)),cosd(phid(3)),0; 0,0,1]; g=g3*g2*g1; if nsten == 1 sigma_n(:,:,ig) = sigma_n(:,:,1); sigma_v(:,ig) = sigma_v(:,1); end gsgT = g*sigma_n(:,:,ig)*g'; %rotated stress tensor grcen(ig,7:9) = [0 0 1]*g; %c - axis dir ection grcen(ig,10:18) = [g(1,:) g(2,:) g(3,:)]; % Orientation matrix is stored % calculate elastic modulus to find compliance mismatch in three principal directions (from Nye textbook on Anisotropy) if ph == 5 % needs a differe nt structure for the sij matrix - needs more terms, not working in this version. elseif ph == 4 for i = 1:1:3 % This gives the modulus in the x (1), y (2), z (3) directions (as looped by i) e1 = sij(ph,1)*(g(1,i)^4 + g(2 ,i)^4) + sij(ph,4)*g(3,i)^4; e2 = (2*sij(ph,2) + sij(ph,6))*(g(1,i)^2 * g(2,i)^2); e3 = (2*sij(ph,3) + sij(ph,5))*g(3 ,i)^2 * (g(1,i)^2 + g(2,i)^2); EY(ig,i)=1./(e1+e2+e3); % NOTE: slip system and plane information not installed in this version for Sn or TiAl end elseif ph == 2 || ph == 3 for i = 1:1:3 EY(ig,i) = 1 ./(sij(ph,1) - 2*(sij(ph,1) - sij(ph,2) - sij(ph,5)/2) * ... (g(1,i)^2*g(2,i)^2 + g(2,i)^2*g(3,i)^2 + g(3,i)^2*g(1,i)^2) ); end elseif ph == 1 for i = 1:1:3 EY(ig,i) = 1./(sij(ph,1) * (1 - g(3,i)^2)^ 2 + sij(ph,4) * g(3,i)^4 + ... (sij(ph,5) + 2*sij(ph,3))*(1 - g(3,i)^2)*g(3,i)^2 ); end end for j=1:1:nslp(ph) % direction plane Sfplbv means Schmid factor, plane and Burgers vector (and points on plane) Sfplbv(j,1) = j; % m * sigma * n Sfplbv(j,2) = ss(2,:,j,ph)*gsgT*ss(1,:,j,ph)'; % generalized Schmid factor if ph == 1 && j>27 && Sfplbv(j,2)<0 Sfplbv(j,2) = 0.001*Sfplbv(j,2) ; % this is to prevent anti - twin shears from being seriously considered later end Sfplbv(j,3) = abs(Sfplb v(j,2)); % abs(generalized schmid factor) Sfplbv(j,4:6) = g'*s s(1,:,j,ph)'; % plane normal in lab coords Sfplbv(j,7:9) = g'*ss(2,:,j,ph)'; % bv direction in lab coords Sfplbv(j,10:12) = cross(Sfplbv(j,4:6),[0,0,1]); % plane trace for k = 1:1:6 is = 3*k+10; ie = is+2; Sfplbv(j,is:ie) = g'*ss(k+2,:,j,ph)'; % plane plotting vectors from origin to points in cell, in lab coords end end %useful plotting for hexahedral tetragonal unit cell vectors that sort to bottom row Sfplbv(mnslp+1,:) = [ph 1 - 1 [1 0 0]*g [0 1 0]*g [0 0 1*c_a(ph)]*g 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]; % don't change g to g' here! otherwise i t may make incorrect cubic prisms sortmv(:,:,ig) = sortro ws(Sfplbv, - 3); % Sort slip systems by Schimd factor % find slope of first four slip systems; y - intercept is close to max Schmid factor % XRank = [1 0; 1 1; 1 2; 1 3 ]; BL = XRank \ YLSf; BR = XRank \ YRSf; YSf(1:4,1) = sort mv(1:4,3,ig); mdl = fitlm(Xr,YSf); grcen(ig,19:21) = [mdl.Coefficients{1:2,{ 'Estimate' }}' mdl.Rsquared.Ordinary ]; listSf(1:8,ig) = sortmv(1:8,3,ig); listss(1:8,ig) = sor tmv(1:8,1,ig); % end end % ig > 0 check end % ng loop fprintf( ' %d \ n ' , ng); for ii = 1:1:dRBdy(1,1); gblist(ii,1) = ii; 89 end % to modify gblist with a short list, paste desired list into this variable as column and delete remaining r ows %% Now start processing by grain boundary... gbcount = 0; BL = [0 0 0]; BR = [0 0 0]; gbnorm = zeros(dRBdy(1,1),3); gbtrac = zeros(dRBdy(1,1),3); damp = zeros(24,dRBdy(1,1)); mpr = zeros(mnslp+2,mnslp+2,dRBdy(1,1)); pln = zeros(mnslp+2,mns lp+2,dRBdy(1,1)); Bvd = zeros(mnslp+2,mnslp+2,dRBdy(1,1)); rbvm = zeros(mnslp+2,mnslp+2,dRBdy(1,1)); rbvec = zeros(mnslp+2,3,mnslp+2,dRBdy(1,1)); nbcount = 0; fprintf( 'Computing grain boundary parameters for GB # ' ); jk4max = 1; top3d3 = 0; top3d2 = 0; top3d1 = 0; top3d0 = 0; gbcheck = sortrows(gblist,1); while gbcount < length(gbcheck) gbcount = gbcount+1; gbnum = gbcheck(gbcount,1); % for gbnum = 1:1:dRBdy(1,1); %gbnum is grain boundary number, will calculate m' and other damage parameter s if gbnum>nbcount+dRBd y/10; nbcount=nbcount+dRBdy/10; fprintf( ' %d ' ,gbnum); %, jk4max end grL = RBdy(gbnum,13); grR = RBdy(gbnum,14); % check and reset RBdy to 13 (left grain) and 14 (right grain) if grL > 0 && grR > 0 && grL <= grmax && grR <= grmax % dIDgr(1,1) && grR <= dIDgr(1,1) if grcen(grL,1) > 0 && grcen(grR,1) > 0 % this is to correct if left and right grains are wrong if grcen(grR,2) low tolerance jk4 = 1; % counter for the number of m' calculations made where Schmid factors are > high tolerance mpmax = 0; mploc = 0; dpsum = 0; dpsum4 = 0; mpsum = 0; mpsum4 = 0; damage = 0; damage4 = 0; mpr(2,1,gbnu m) = grL; mpr(1,2,gbnum) = grR; mpr(2,2,gbnum) = gbnum; % m - prime table label for grain numbers in mpr() if strcmp(num2str(sigma_v(:,ig)),num2str([1 0 0]')) % stress axis || [100] (X) EgrL = EY(grL,1); EgrR = EY(grR,1); elseif strcmp(num2str(sigma_v(:,ig)),num2str([0 1 0]')) % stress axis || [010] (Y) EgrL = EY(grL,2); EgrR = EY(grR,2); elseif strcmp(num2 str(sigma_v(:,ig)),num2str([0 0 1]')) % stress axis || [001] ( Z) EgrL = EY(grL,3); EgrR = EY(grR,3); elseif trace(sigma_n(:,:,ig)) == 0 % Crude estimate of shear effects follows, may not be meaningful EgrL = EY(grL,3)*abs(sigma_n(1,2,ig)) + EY(grL,2 )*abs(sigma_n(1,3,ig)) + EY(grL,1)*abs(sigma_n(2,3,ig)); EgrR = EY(grR,3)*abs(sigma_n(1,2,ig)) + EY(grR,2)*abs(sigma_n(1,3,ig)) + EY(grR,1)*abs(sigma_n(2,3,ig)); else fprintf( 'Can''t calculate modulus for th is stress state \ r' ); EgrL = 1; EgrR = 1 %pause end Eratio = min(EgrL,EgrR)/max(EgrL,EgrR); % always use Emin/Emax! F1A = zeros(1,mnslp); % F1 FIP, Simkin et al. 2003 for grain A % F14A = 0; % F1 FIP w/ restriction on Schmid factor value for grain A F1B = zeros(1,mnslp); % F1 FIP for grain B % F14B = 0; % F1 FIP w/ restriction on Schmid factor value for grain B F1 = 0; % F1 for grainA/grainB F14A = zeros(1,mnslp); % F14 for grainA (with restriction on Schmid fa ctor value) F14B = zeros(1,mnslp); avgdp4 = .75; % These values indicate instances where values avgmp4 = .55; % of dm' or m' are too l ow to take seriously % RBdy(gbnum,1:6) = (180/pi).*RBdy(gbnum,1:6); 90 gbnorm(gbnum,:) = [cosd(RBdy(gbnum,8)) sind(RBdy(gbnum,8)) 0]; gbtrac(gbnum,:) = sigma_n(:,:,grL)*gbnorm(gbnum,:)'; for k = 1:1:mnslp % Build table of m' values for each grain pair SchmL = sortmv(k,2,grL); % Use the correctly signed version of Schmid factor in position 2 mpr(k+2,1,gbnum) = sortmv(k,1,grL) ; mpr(k+2,2,gbnum) = abs(Schm L) ; % Schmid factor header grL is down (k goes with L) if abs(SchmL) > Sfthr kmax = k; end for j = 1:1:mn slp % j goes across (with R grain in columns) SchmR = sortm v(j,2,grR); % Use the correctly signed version of Schmid factor in position 2 if k == 1 && abs(SchmR) > Sfthr % find size of upper left corner of m' mat rix for which Schmid factors are higher than threshold j max = j; end mpr(1,j+2,gbnum) = sortmv(j,1,grR) ; mpr(2,j+2,gbnum) = abs(SchmR) ; % grL goes down, grR across, (j goes with R) if abs(SchmL) > SflimL && abs(SchmR) > SflimL mpl = sortmv(j,4:6,grR)*sortmv( k,4:6,grL)'; % plane This is a dot product mbv = sortmv(j,7:9,grR)*sortmv(k,7:9,grL)'*sign(SchmL*SchmR); % direction This is a dot product mprime = mpl*mbv; % if the b directions are pointing si milarly mb is positive mpr(k+2,j+2,gbnum) = mprime; pln(k+2,j+2,gbnum) = mpl; Bvd(k+2,j+2,gbnum) = mbv; % m', plane and Burgers vector table is filled in, rbva = sortmv(j ,7:9,grR) - sortmv(k,7:9,grL); % store residual Burgers vector rbvb = sortmv(j,7:9,grR) + sortmv(k,7:9,grL); % store residual Burgers vector if norm(rbva) > norm(rbvb) rbv = rbvb ; else rbv = rbva; end rbvm(k+2,j+2,gbnum) = norm(rbv); rbvec(k+2,1:3,j+2,gbnum) = rbv; % schmA F1A(k) = F1A(k)+abs(sortmv(k,3,grL)*dot(sortmv(k,7:9,grL)',sigma_v(:,ig))*dot(sortmv(k,7:9,grL)',sortmv(j,7:9,grR)')); % F1 for grain A F1B(j) = F1B(j)+a bs(sortmv(j,3,grR)*dot(sortmv(j,7:9,grR)',sigma_v(:,ig))*dot(sortmv(j,7:9,grR)',sortmv(k,7:9,grL)')); % F1 for grain B if mprime > mpthr % assumes no slip transfer (or damage) occurs if m' < m' threshold or when m' = 1 dampar = mprime + 0.1; % pushes m' up by 0.1, so that 1 is worst case (i.e. m'=0.9 is most damaging condition). if dampar > 1 dampar = 2 - dampa r; % ass ume dampar < 1 means less likely to generate damage due to less slip transmission activity end dpsum = dpsum + dampar; % damage parameter only, for all slip systems where m' > mpthr mpsum = mpsum + abs(mprime); % m' only, for all slip systems where m' > mpthr damage = damage + dampar*max(abs(SchmL),abs(SchmR)); % damage parameter modified by schmid factor if (abs(SchmL) > Sfthr && abs(SchmR) > Sfthr) || abs(SchmL) < 0.001 || abs(SchmR) < 0.001 % latt er condition is for twins, to enable seeing them later % assumes that slip transfer happens only if Sf > 0.Sfthr and both have high schmid factor and capturing effect of anti - twin himp4(jk4,: ,gbnum) = [k j abs(mprime), mprime SchmL SchmR]; % location and values of high m' values for grain pair; k(rows) is from left grain; j(columns) is from rig ht grain dpsum4 = dpsum4 + dampar; % running sum of slip system in teractions that have Sf > 0.4 mpsum4 = mpsum4 + abs(mprime); % running sum of m' for slip system interactions that have Sf > Sfthr damage4 = damage4 + dampar*max(abs(SchmL),abs(SchmR)); % damage parameter modified by schmid factor if jk4 > jk4max jk4max = jk4; end jk4 = jk4 + 1; F14A(k) = F14A(k)+abs(sortmv(k,3,grL)*dot(sortmv(k,7:9,grL)',sigma_v(:,ig))*dot(sortmv(k,7:9,grL)',sortmv(j,7:9,grR)')); % F14 for grain A F14B(j ) = F14B(j)+abs(sortmv(j,3,grR)*dot(sortmv(j,7:9,grR)',sigma_v(:,ig))*dot(sortmv(j,7:9,grR)',sortmv(k,7:9,grL)')); % F14 for grain B end if mpmax < abs(mpri me) % record maximum m' value % and its location in matrix ? mpmax = abs(mprime); end jk = jk + 1; end % of m' > mpthr if statement end % of if statement for values with schmid factors > SflimL end % slip system j loop for each B grain in pair end % slip system k loop for each A grain in pair 91 % Variables evaluated above in the l oops: dpsum mp sum jk4 dpsum4 mpsum4 F1 F14 gbodam = damage*abs(norm(gbtrac(gbnum,:))) ; % damage parameter modified by apparent GB inclination gbodam4 = damage4*abs(norm(gbtrac(gbnum,:))) ; % damage para meter for high schmid modified by GB inclination caxmis = (1 - (grcen(grL,7:9)*grcen(grR,7:9)')^2)^.5; % misorientation of c - axes (not meaningful for cubic) avgdp = dpsum/(jk - 1); % average value of dm' avgmp = mpsum/(jk - 1); % average value of m' F1Asort = sort(F1A, 'descend' ); F1Bsort = sort(F1B, 'descend' ); F14Asort = sort(F14A, 'descend' ); F14Bsort = sort(F14B, 'descend' ); maxF1 = max(F1Asort(1),F1Bsort(1)); maxF14 = max( F14Asort(1),F14Bsort(1)); if jk4> 1 avgdp4 = dpsum4/(jk4 - 1); %average value of damage for slip systems with Sf > 0.4 avgmp4 = mpsum4/(jk4 - 1); %average value of m' for slip systems with Sf > 0.4 end jk6 = 0; jktop3or6 = 0; % counters for number of m' values to average later. top3mpn = 0; top6mpn = 0; t3mpthr = 0; maxmpkj = 0; maxSfk = 0; maxSfj = 0; ma xkjSfs = 0; Sfsum = 2; pairsum = zeros(mnslp:mnslp); pairpro d = zeros(mnslp:mnslp); mpSflist = zeros(kmax*jmax:6); mpSfshlist = zeros(1,36); ikj = 1; for k = 1:1:kmax % Rather than finding all m' values for Sf > tolH, look only for top 3 or top 6 Sf value pairs for j = 1:1:jmax % find sum of schmid factors for each element of mpr array and put in pairsum(), similarly for pairprod(uct) pairsum(k,j) = mpr(k+2,2,gbnum) + mpr(2,j +2,gbnum); pairprod(k,j) = mpr(k+2,2,gbnum) * mpr(2,j+2,gbnum); mpSflist(ikj,1:6) = [k j abs(mpr(k+2,2,gbnum)) abs(mpr(2,j+2,gbnum)) abs(mpr(k+2,j+2,gbnum)) rbvm(k+2,j+2,gbnum)]; ikj = ikj + 1; end end sortmplist = sortrows(mpSflist, - 5); for ii = 1:1:6 mpSfshlist(1,6*(ii - 1)+1:6*ii) = sortmplist(ii,1:6); end % parameters for max m' are in first row; k j Sf(kL) Sf(jR) m' rbvm maxSfk = mpSfshlist(1,1); maxSfj = mpSfshlist(1,2); maxmpkj = mpSfshl ist(1,5); maxkjSfs = mpSfshlist(1,3)+mpSfshlist(1,4); maxkjSfp = mpSfshlist(1,3)*mpSfshlist( 1,4); maxkjsmp = maxkjSfs*maxmpkj; maxkjpmp = maxkjSfp*maxmpkj; maxmprbv = mpSfshlist(1,6); while (jk6 < 6 || jk top3or6 < 3) && Sfsum > 0 kjmax = [0 0 0 0]; for k = 1:1:kmax % find location of highest pairsum (and product) in current pairsum array for j = 1:1:jmax if pairsum(k,j)>kjmax(1) %finds largest Spair value in current kmax x jmax Pairsum() array kjmax = [pairsum(k,j) k j pairprod(k,j)]; %identifies location k j of sum of Schmid factors end end end Sfsum = kjmax(1); mpchk = abs(mpr(kjmax(2)+2 ,kjmax(3)+2,gbnum)); % puts (next) m' into mpchk if jk6 < 6 && Sfsum > 0 top6mpn = top6mpn + mpchk; pairsum(kjmax(2),kj max(3)) = - 1; % Now that the highest pairsum value is found and added to sum, m ake it unfindable jk6 = jk6 + 1; if jk6 == 3; top3mpn = top6mpn; % capture top 3 values in this variable end end if jktop3or6 < 3 && Sfsum > 0 pairsum(kjmax(2),kjmax(3)) = pairsum(kjmax(2),kjmax(3)) - 1; % mark position with - 2 if inside this query if mpchk > mpthr t3mpthr = t3mpthr + mpchk; % more stringe nt criterion for m' only > mpthr used. jktop3or6 = jktop3or6 + 1; % [gbnum t3mpthr mpchk jktop3or6 maxkj] end end end top6mpn = top6mpn/jk6; % These average values of m' are without regard to magnitude of m' if top3mpn > 0 top3mpn = top3mpn/3; else top3mpn = top6mpn; end if jktop 3or6 == 3 92 t3mpthr = t3mpthr/jktop3or6; top3d3 = top3d3 + 1; elseif jktop3or6 == 2 t3mpthr = t3mpthr/jktop3or6; top3d2 = top3d2 + 1; elseif jktop3or6 == 1 t3mpthr = t3mpthr/jktop3or6; top3d1 = top3d1 + 1; elseif jktop3or6 == 0 t3mpthr = 0; top3d0 = top3d0 + 1; end mpr(1,1,gbnum) = mpmax; % GB# , left gr, right gr, angle/axis of misorientation, average of top 3 m' values in upper kj box, maximum m' in upper kjbox, co rresponding Schmid factor sum, % product between Schmid factor sum and maximum m' value, row for k (left grain), col umn for j (rig ht grain), % left grain y - intercept, slope, and correlation coefficient, right grain y - intercept, slope, and correlation coefficient. slpint = [ - min(grcen(grL,20),grcen(grR,20)) - max(grcen(grL,20),grcen(grR,20)) ... % slope, inte rcept, and R^2 values for left and right grains max(grcen(grL,19),grcen(grR,19)), min(grcen(grL,19),grcen(grR,19)) grcen(grL,19:21) grcen(grR,19:21)]; mpmaxgrp = [t3mpthr maxmpkj maxkjSfs maxkjsmp maxkjSfp maxkjpmp maxmprbv maxSfk maxSfj]; tablepub(gbnum,:) = [gbnum grL grR RBdyn(gbnum, 7:10) mpmaxgrp slpint mpSfshlist] ; damp(:,gbnum) = [maxSfk maxmpkj maxSfj dpsum da mage gbodam caxmis dpsum4 damage4 gbodam4 avgdp avgdp4 avgmp avgmp4 ... maxF1 maxF14 ma xF1*Eratio maxF14*Eratio Eratio top3mpn top6mpn t3mpthr kmax jmax]; % this is a summary matrix used for plotting end % of if for grains with positive ID end % of if for valid grain pair end % of gbnum loop gbcount = 1; for ii = 1:1:length (tablepub) if tablepub(ii,1) > 0 tablepubshort(gbcount,:) = tablepub(ii,:); gbcount = gbcount+1; end end fprin tf( ' %d \ n ' , gbnum); plotname = { ' mp5k ' , ' maxmpkj ' , ' mp5j ' , ' dm''sum ' , ' m*dm''sum ' , ' gbo*m*dm''sum ' , ' cax - mis ' , ' dm''sum4 ' , ... ' m*dm''4 ' , ' gbo*dam4 ' , ' norm dm'' ' , ' norm dm''4 ' , ' norm m'' ' , ' norm m''4 ' , ... ' max(F1A,F1B) ' , ' max(F14A,F14B) ' , ' Emax(F1A,F1B) ' , ' Emax(F14A,F14B) ' , ' Eratio ' , 'top3mpn' , 'top6mpn' , 't3mpt hr' }; %% Choose what to plot along the grain boundary map from variables 4 - 19 noted above mpthr = mpthr; chsize = 16; plist = [22 2]; %13 14 11 12 15 16 17 18 19 20 21 for k = 1:1:length(plist); % 1:1:length(plist); % 4:4:4 % plnx = plist(k); if plnx == 13 || plnx == 14 bins = [.60 .64 .68 .72 .76 .80 .84 .88 .92 .96]; % for mp or mp4 or Eratio elseif plnx == 11 || plnx == 12 bins = [.80 .82 .84 .86 .88 .90 .92 .94 .96 .98]; % for normp or normp4 elseif plnx == 7 bins = [0 0.156 0.309 0.454 0.588 0.707 0.809 0.891 0.951 0.988 1.000]; % for c - axis plot % Corresponding degrees for bins above = [0 9 18 27 36 45 54 63 72 81 90]; % for c - axis plot elseif plnx == 22 || plnx == 2 bins = mpt hr :(1 - mpthr)/10:1; else bins = linspace(min(nonzeros(damp(plnx,:))), max(damp(plnx,:)), 11); % auto bin size for anything end binsdat = bins; % temporary storage while setting up plot scale max1 = max(RBdy(:,9)); % max x value for grR max2 = max(RBdy(:,11)); % max x value for grL maxx = max([max1; max2]); may1 = max(RBdy(:,10)); % max y value for grR may2 = max(RBdy(:,12)); % max y value for grL maxy = max([may1; may2]); f = fig ure( 'Position' , [0,0,750,500]); movegui(f, 'northwest' ); set(gcf, 'Color' , [1 1 1]); hold on ; % plot is based upon TV rastering, as given by TSL axis([0 maxx*2 - maxy .15*maxy ]); axis image ; %equal; set(gcf, 'Color' ,[1,1 ,1]) % surrounding field is this color text(maxx* - 0.12, 0.11*maxy, 'm'' ' , 'Fontsize' , chsize); 93 text(maxx* - 0.04, 0.11*maxy, num2str(binsdat(1,1), '%4.3f' ), 'Fontsize' , chsize); bins = [10 20 30 40 50 60 70 80 90 100]; % To set color key fo r boundarie s bcnt = 2; for gbnum = 1:1:100; wid = 10; widk = 2; % thickness for width bar if fix(gbnum/10) 90; wid = 4; widk=4; elseif gbnum > 80; wid = 3; widk=3; end else text(maxx/100*(gbnum - 1) - maxx*0.03, 0.11*maxy, num2str(binsdat(1,bcnt), '%4.3f' ), 'Fontsize' , chsize); bcnt = bcnt + 1; end vec = gbnum; if vec purple elseif vec>=bins(1) && vec blue elseif vec>=bins(2) && vec turquoise elseif vec>=bins(3) && vec dk grn elseif vec>=bins(4) && vec green elseif vec>=bins(5) && vec yellow elseif vec>=bins(6) && vec orange elseif vec>=bins(7) && vec red elseif vec>=bins(8) && vec pink elseif vec>=bins(9) && ve c pink elseif vec > bins(10); vRGB = [.6 0. .2]; %[.6 0. .3] m auv e bins(9) end plot(maxx/100*[gbnum - 1; gbnum],[0.06*maxy; 0.06*maxy], 'Linewidth' ,wid, 'color' ,vRGB); % place color key x coordinate at suitable place... end bins = binsdat; % replacing the necessary bins parameters for plotting along grain boundaries for gbnum = 1:1:dRBdy(1,1); % Plot parameter values on grain boundaries if RBdy(gbnum,13) > 0 && RBdy(gbnum,14) > 0 && RBdy(gbnum,13) <= grmax && RBdy(gbnum,14) <= grmax % wid = (RBdy(iGB,16) - .95*minF1m)/( maxF1 m - minF1m)*4; wid = 2; if damp(plnx,gbnum) > bins(9) wid = 4; elseif damp(plnx,gbnum) > bins(8) wid = 3; end kind = damp(1,gbnum); jind = damp(1,gbnum); kss = mpr(kind,1,gbnum); jss = mpr(1,jind,gbnum); if one_ss == 1 && kss < 7 && jss < 7 linetype = ':' ; % dotted lines for hexagonal slip systems of basal and/or prism elseif one_ss == 1 && kss < 7 || one_ss = = 1 & & jss < 7 linetype = ' - .' ; % dot dash lines for hexagonal slip systems if one ss is basal or prism else linetype = ' - ' ; end vec = damp(plnx,gbnum); if vec purple elseif vec>=bins(1) && vec blue elseif vec>=bins(2) && vec turquoise elseif vec>=bins(3) && vec dk grn elseif vec>=bins(4) && vec green elseif vec>=bins(5) && ve c yellow elseif vec>=bins(6) && ve c orange elseif vec>=bins(7) && vec red elseif vec>=bins(8) && vec pink elseif vec>=bins(9) && vec pink elseif vec > bins(10); vRGB = [.6 0. .2]; %[.6 0. .4] mauve bin s(9) en d plot([RBdy(gbnum,9);RBdy(gbnum,11)],[ - RBdy(gbnum,10); - RBdy(gbnum,12)],linetype, 'Linewidth' ,wid, 'color' ,vRGB); end end end % k for plist set(gca, 'FontSize' ,chsize); xlabel( 'Position, microns' ); ylabel( 'Position, microns' ); %% pl ot grain numbers chsize = 14; for ng = 1:1:grmax if grcen(ng,1) == 1 || grcen(ng,1) == 4 text(grcen(ng,2), - grcen(ng,3),int2str(ng), 'color' ,[0 .5 1], 'FontWeight' , 'bold' , 'FontSize' ,chsize); %[ng grcen(ng,4:6)] elseif grcen(ng,1) == 2 text(grcen(ng,2), - grcen(ng,3),int2str(ng), 'color' ,[0 .5 0], 'FontWeight' , 'bold' , 'FontSize' ,chsize); elseif grcen(ng,1) == 3 text(grcen(ng,2), - grcen(ng,3),int2str(ng), 'color' ,[.8 0 0], 'FontWeight' , 'bold' , 'F ontSize' ,chsize); end end % % plot gb numbers for gbnum = 1:1:dRBdy % plot grain boundary numbers if RBdy(gbnum,13) > 0 && RBdy(gbnum,14) > 0 && RBdy(gbnum,13) <= grmax && RBdy(gbnum,14) <= grmax text((2*RBdy(gbnum,9)+RBdy(gbnum,11))/3, - (2*RBdy(gbnum,10)+RBdy(gb num,12))/3, ... int2str(gbnum), 'color' ,[0 0 .5], 'FontWeight' , 'light' , 'F ontSize' ,chsize); end end %% Plot of rotated grain around grain 1... (for single intial grain pair with file of rotated orientations) clear m1 ; clear m2 ; clear m3 ; symbsize = 15; colorvec = [0 .2 .7 ; 1 0 0 ; 1 .8 0 ; .7 0 .6 ; 0 .8 0 ; 0 1 1 ; .6 0 .2 ; 0 0 1 ; 1 .4 0 ]; for ii = 1:1:360 m1(ii,:) = mpr(3,3,ii) ; m2(ii,:) = [mpr(3,4,ii) mpr(4,4,ii) mpr(4,3,ii) ]; m3(ii,:) = [mpr(3,5,ii) mpr(4,5,ii) mpr (5,5,ii) mpr(5,4,ii) mpr(5,3,ii) ]; m1s(ii,:) = round( symbsize * mpr(3,2,ii) * mpr(2,3,ii) + 1 ) ; m2s(ii,:) = [round( symbsize*1.3 * mpr(3,2,ii) * mpr(2,4,ii) + 1 ) ... r ound( symbsize*1.0 * mpr(4,2,ii) * mpr(2,4, ii) + 1 ) ... round( symbsize*0.7 * mpr(4,2,ii) * mpr(2,3,ii) + 1 ) ]; m3s(ii,:) = [round( symbsize*1.3 * mpr(3,2,ii) * mpr(2,5,ii) + 1 ) ... round( symbsize *1.3 * mpr(4,2,ii) * mpr(2,5,ii) + 1 ) ... round( symbsize*1.0 * mpr(5,2,ii) * mpr(2,5,ii) + 1 ) ... round( symbsize*0.7 * mpr(5,2,ii) * mpr(2,4,ii) + 1 ) ... round( symbsize*0.7 * mpr(5,2,ii) * mpr(2,3 ,ii) + 1 ) ]; 95 end f = figure( 'Position' , [ 0,0,800,600]); movegui(f, 'northwest' ); set(gcf, 'Color' , [1 1 1]); hold on ; plot(m1, 'LineWidth' ,3) plot(m2, ' -- ' , 'LineWidth' , 2) plot(m3, ':' , 'LineWidth' , 2) legend( 'mp11' , 'mp12' , 'mp22' , 'mp21' , 'mp13' , 'mp23' , 'mp33' , 'mp32' , 'mp31' , 'Location' , 'southw est' ); ylabel( 'm prime value' ); xlabel( [mat2str(grcen(1,4:6)) ' rotation incremented about Z axis of ' mat2str(grcen(2,4:6))] ); hold off ; f = figure( 'Position' , [0,0,800,600]); movegui(f, 'northwest' ); set(gcf, 'Color' , [1 1 1]); hold on ; for ii = 1 :1:360 plot(ii, m1(ii,1), '+' , 'MarkerSize' , m1s(ii,1), 'color' ,colorvec(1,:)); for jj = 1:1:3 plot(ii, m2(ii,jj), 'o' , 'MarkerSize' , m2s(ii,jj), 'color' ,colorvec(jj+1,:)); end for jj = 1:1:5 plot(ii, m3(ii,jj), '^' , 'MarkerSize' , m3s(ii,jj), 'color' ,colorvec(jj+4,:)); end end legend( 'mp11' , 'mp12' , 'mp22' , 'mp21' , 'mp13' , 'mp23' , 'mp33' , 'mp32' , 'mp31' , 'Location' , 'southwest' ); ylabel( 'm prime value (size = sum of SF)' ); xlabel( [mat2str(grcen(1,4:6)) ' rotation incremented a bout Z axis of ' mat2str(grcen(2,4:6))] ); hold off ; %% Choose your favorite grain boundary --------------------------------------- gbnum = 276 mpr_cur = mpr(:,:,gbnum); rbvm_cur = rbvm(:,:,gbnum); rbvec_cur = rbvec(:,:,:,gbnum); mplimit = .6 whiteanno tation = 1; % make gb trace and axes white ptpl = 1; % plots plane traces if = 1 Red dashed line in plot is perpendicular to the line connecting the centers of the two grains prsxpl t = zeros(8,7); % Black solid line in plot is the RC b oundary segment orientation prsyplt = zeros(8,7); % 13 and 14 lead to misplaced plane traces. prszplt = zeros(8,7); clear Sflist ; clear sortSflist ; mp4gr = himp4(:,:,gbnum); sorth imp4 = sortrows(mp4gr, - 4); % This sorts on basis of actual m' value mp limit = min(mplimit, sorthimp4(1,4)); gbcen = [RBdy(gbnum,11)+RBdy(gbnum,9) RBdy(gbnum,12)+RBdy(gbnum,10)]/2; v13cpos = [grcen(RBdy(gbnum,13),2) grcen(RBdy(gbnum,13),3)] - gbcen; % find vector from center of GB to grain center in raster coordinates v14cp os = [grcen(RBdy(gbnum,14),2) grcen(RBdy(gbnum,14),3)] - gbcen; v1314b = [(RBdy(gbnum,12) - RBdy(gbnum,10)) - (RBd y(gbnum,11) - RBdy(gbnum,9)) ]; % vector [dy, - dx] pointing perpendicular to GB if v13cpos * v1314b' < 0 v1314b = - v1314b; end v1314bn = v1314b/ norm(v1314b) % find unit vector pointing perpendicular to GB in raster coords plabel = [ 'Sfthr k j: ' num2str(Sfthr,3) ' ' num2str(damp(23,gbnum),2) ' ' num2str(damp(24,gbnum),2) ' t3mpthr ' num2str(damp(22,gbnum),3)]; g_gb_g = [num2str(RBdy(gbnum,13)) ' ' mat2str(grcen(RBdy(gbnum,13),7:9),4) ' ' num2str(gbnum) ' ' mat2str(grcen(RBdy(gbnum,14),7:9),4) ' ' num2str(RBdy(gbnum,14)) ]; ipl = - 6; imp = 0; % Strategy: Next, start isc loop for plotting slip systems while sorthimp4(imp+1,4) >= mplimit im p = imp + 1; if sorthimp4(imp,1) ~= 0 % evaluate only for recorded values (m'>.6) if ipl - imp== - 7 % six plots on a page f = figure( 'Position' , [0,0,1200,750]); movegui(f, 'northwest' ); set(gcf, 'Color' , [1 1 1]); hold on ; ipl=ipl+6; end subplot(2,3,imp - ipl); hold on ; set(gcf, 'Color' , [1 1 1]); if whiteannotation == 1 set(gca , 'ycolor' , 'w' ); set(gca , 'xcolor' , 'w' ); % make axes white for ease in later arranging. else plot([0 1.5*cosd(RBdy( gbnum,8))], [0 1.5*sind(RBdy(gbnum,8))], ' - k' ); % plots gb from map from angle given in normal x - y space end Sfsum = 0; kind = sorthimp4(imp,1)+2; jind = sorthimp4(imp,2)+2; kss = mpr(kind, 1,gbnum); jss = mpr(1,jind,gbnum); 96 for igr = 13:1:14 % 1 i.e. first for the left grain in column 13, then the right grain in column 14 in checked Reconstructed Boundary file. grnum = RBdy(gbnum,igr); issr = sorthimp4(imp,i gr - 12); % ss rank # in gr13 issr is slip system Schmid factor order # if igr == 13 del = v1314bn; cellcolor = [0 0 0]; %[.3 0 .5]; plot ([0 del(1)], - [0 del(2)]) else % del i s position vector from center of gb to 13 in raster coord system (ydown) del = - v1314bn; cellcolor = [0 0 0]; %[0.5 0 0]; plot ([0 del(1)], - [0 del(2)]) end % Plot the image of hexagonal unit cell, slip vectors, planes, plane normals, and plane traces % Strategy: First extract useful vectors to draw the hexagonal prisms from slip system information % positions in mvs p1:13 - 15 p2:16 - 18 p3:19 - 21 p4:22 - 24 p5:25 - 27 p6:28 - 30 % posi tions in hpln p1:4 - 6 p2:7 - 9 p3:10 - 12 p4:13 - 15 p5:16 - 18 p6:18 - 21 for isc = 1:1:nslphex if sortmv(isc,1,grnum) == 1; % locate basal planes using SS1 hpln(1,4:21) = sortmv(i sc,13:30,grnum); % bottom basal plane hpln(2,4:21) = sortmv(isc,13:30,grnum); % top basal plane rotc = sortmv(isc,4:6,grnum)*c_a_hex; % basal plane norma l * c/a for j = 4:3:19 hpln(2,j:j+2) = hpln(1,j:j+2) + rotc; % move top plane up by a unit of c end a1 = sortmv(isc,7:9,grnum); % locate a1 using SS1 elseif sortmv(isc,1,grnum) == 2; a2 = sortmv(isc,7:9,grnum); % locate a2 using SS2 elseif sortmv(isc,1,grnum) == 3; a3 = sortmv(isc,7:9,grnum); % locate a3 using SS3 end end for isc = 1:1:nslphex if sortmv(isc,1,grnum) == 4; % locate two prism planes on opposite sides using SS4 hpln(3,4:21) = sortmv(isc,13:30,grnum); for j = 13:3:28 hpln(4,j - 9:j - 7) = sortmv(isc,j:j+2,grnum) + a2 - a3; end elseif sortmv(isc,1,grn um) == 5; % locate two prism planes on opposite sides using SS5 hpln(5,4:21 ) = sortmv(isc,13:30,grnum); for j = 13:3:28 hpln(6,j - 9:j - 7) = sortmv(isc,j:j+2,grnum) + a3 - a1; end elseif sortmv(isc,1,grnum) == 6; % locate two prism pl anes on opposite sides using SS6 hpln(7,4:21) = sortmv(isc,13:30,grnum); for j = 13:3:28 hpln(8,j - 9:j - 7) = sortmv(isc,j:j+2,grnum) + a1 - a2; end end end for j = 1:1:2 % Find z elevation of basal planes for k = 1:1:3 hpln(j,k) = (hpln(j,3+k)+hpln(j,6+k)+hpln(j,9+k)+hpln(j,12+k)+hpln(j,15+k)+hpln(j,18+k))/6; end end center = (h pln(1,1:3)+pln(2,1:3))/2; for j = 3:1:8 % Find z elevation of prism planes hpln(j,3) = (hpln(j,12)+hpln(j,15)+hpln(j,18)+hpln(j,21))/4; end sortpln = sortrows(hpln, - 3); minx = 0; miny = 0; minz = 0; maxx = 0; maxy = 0; maxz = 0; for j = 1:1:8 % assemble vectors for plotting faces of hex prism prsxplt(j,1:7) = [sortpln (j,4) sortpln(j,7) sortpln(j,10) sortpln(j,13) sortpln(j,16) sortpln(j,19) sortpln(j,4)]; minx = min(minx,min(prsxplt(j,:))); maxx = max(maxx,max(prsxplt(j,:))); prsyplt(j ,1:7) = [sortpln(j,5) sortpln(j,8) sortpln(j, 11) sortpln(j,14) sortpln(j,17) sortpln(j,20) sortpln(j,5)]; miny = min(miny,min(prsyplt(j,:))); maxy = max(maxy,max(prsyplt(j,:))); prszplt(j,1:7) = [sortpln(j,6) sortpln (j,9) sortpln(j,12) sortpln(j,15) sortpln(j,1 8) sortpln(j,21) sortpln(j,6)]; minz = min(minz,min(prszplt(j,:))); maxz = max(maxz,max(prszplt(j,:))); end 97 sp1 = sortmv(issr,13:15,grnum); % identify plotted points on the slip plane sp2 = sortmv(issr,16:18,grnum); sp3 = sortmv(issr,19:21,grnum); sp4 = sortmv(issr,22:24,grnum); sp5 = sortmv(issr,25:27,grnum); sp6 = sortmv(is sr,28:30,grnum); spx = [sp1(1 ) sp2(1) sp3(1) sp4(1) sp5(1) sp6(1) sp1(1)]; spy = [sp1(2) sp2(2) sp3(2) sp4(2) sp5(2) sp6(2) sp1(2)]; ssn = sortmv(issr,1,grnum); % slip system number Sf = sortmv(issr,2,grnum); % Schmid factor Sfsum = Sfsum + abs(Sf); n = [0 0 0 sortmv(issr,4:6,grnum)]; % plane normal b = [sp1 sp4]; % p1+sortmv(issr,7:9,grnum)]; % Burgers vector pt = sortmv(issr,10:12,grnum); % plane trace midx = (minx+maxx)/2; midy = (miny+maxy)/2; cellcenter = [midx midy]; dx =(1.6*del(2) - cellcenter(1)); % del is raster, cellcenter is TSL coords dy =(1.6*del(1) - cel lcenter(2)); % so not dy = del(2) - cellcenter(2); [dx dy]; % diagnostic % These plots will match TSL with X down !!!! Plotting starts if grcen(grnum,5) < 90 % if PHI < 90, then make the 3 coo rdinate axes visible below slip planes plot([0 a1(2)]+dy, - ([0 a1(1)]+dx), ':' , 'Linewidth' ,3, 'Color' ,[1 0 .2]); % plot x = red plot([0 a2(2)]+dy, - ([0 a2(1)]+dx), ':' , 'Linewidth' ,3, 'Color' ,[.6 .8 0]); % plot y = green - gold plot([0 a3(2)]+dy, - ([0 a3(1)]+dx), ':' , 'Linewidth' , 3, 'Color' ,[0 0 1]); % plot z = blue end if sortmv(issr,6,grnum)>0 % is k component of slip plane normal positive or negative? fill(spy+dy, - (spx+dx), [.8 .8 .65]) % slip plane filled warm gray % plot([n(2) n(5)], - [n(1) n(4)],'Linewidth',3,'Color',[.8 .8 .65]); else % slip plane filled cool gray if normal has neg z component fill(spy+dy, - (spx+dx), [.65 .65 .7]) % plot([n(2) n(5)], - [n(1) n(4)],'Linewidth',3,'Color',[.65 .65 .7]); end if sortmv(issr,6,grnum)>0 Bvcolor = [0 .7 .7]; if ssn >= iC1 Bvcolor = [.1 .6 0]; end if ssn >= iT1 && ssn <= fT2 Bvcolor = [1 .6 0]; end else Bvco lor = [0 1 1]; if ssn >= iC1 Bvcolor = [.3 .9 0]; end if ssn >= iT1 && ssn <= fT2 Bvcolor = [1 .8 0]; en d end Sfs = 1; if ssn < iT1 Sfs = sign(Sf); end if Sf > 0 % plot Burgers vector direction if ssn >= iT1 % this is for twins - the Burgers vector length is shown to be 1/2 of the usual length in the unit cell plot(b(2)+dy, - (b(1)+dx), '.' , 'MarkerSize' , 24, 'Color' , Bvcolor) plot([b(2) (b(2)+b(5))/2]+dy, - ([b(1) (b (1)+b(4))/2]+dx), 'Linewidth' ,4, 'Color' ,Bvcolor) else plot(b(2)+dy, - (b(1)+dx), '.' , 'MarkerSize' , 24, 'Color' , Bvcolor) plot([b(2) b(5)]+dy, - ([b(1) b(4)]+dx), 'Linewidth' ,4, 'Color' ,Bvcolor) end else % plot Burgers vector in opposite direction if ssn >= iT1 % this is for twins - the Burgers vector length is shown to be 1/2 of the usual length in the unit cell plot(b(2)+dy, - (b(1)+dx), '.' , 'MarkerSize' , 24, 'Color' , Bvcolor) plot([b(2) (2*b(2)+b(5))/3]+dy, - ([b(1) (2*b(1)+b(4))/3]+dx), 'Linewidth' ,4, 'Color' ,Bvcolor) else plot(b(5)+dy, - (b(4)+dx), '.' , 'MarkerSize' , 24, 'Color' , Bvcolor) 98 plot([b(5) b(2)]+dy, - ([b(4) b(1)]+dx), 'Linewidth' ,4, 'Color' ,Bvcolor) end end for j = 1:1:4 % plot the 4 top most surface prisms of the hex cell that have the highest z elevation plot(prsyplt(j,:)+dy, - (prsxplt(j,:)+dx), 'Linewidth' , 2, 'Color' , cellcolor); end if gr cen(grnum,5) > 90 % if PHI < 90, make the 3 coordinate axes visible above slip planes plot([0 a1(2)]+dy, - ( [0 a1(1)]+dx), ':' , 'Linewidth' ,3, 'Color' ,[1 0 .2]); % plot x = red plot([0 a2(2)]+dy, - ([0 a2(1)]+dx), ':' , 'Linewidth' ,3, 'Color' ,[.6 .8 0]); % plot y = green - gold plot([0 a3(2)]+dy, - ([0 a3(1)]+dx), ':' , 'Linewidth' ,3, 'Color' ,[0 0 1]); % plot z = blue end if ptpl == 1 if ssn >= iC1 ptrcolor = [.2 .8 0]; % compression twin plane traces green elseif ssn >= iT1 && ssn <= fT2 ptrcolor = [1 .6 0]; % extension twin pla ne traces orange elseif ssn > ipyrc && ssn <= f2pyrc ptrcolor = [.95 .85 0]; % plane traces green - gold elseif ss n <= fpyra && ssn >= ipyra ptrcolor = [0 .9 .5]; % pyr green - blue elseif ssn <= f2prs && ssn >= iprs ptrcolor = [1 .2 0]; % prism red else ptrcolor = [0 0 1]; % basal blue end % ---- > NOTE that Schmid factor vector is plotted in correct direction, plot([ - pt(2) pt(2)]+dy, - ([ - pt(1) pt(1)]+ dx), ' -- ' , 'Linewidth' ,3, 'Color' ,ptrcolor) end if igr == 13 line1 = [ 'L g' num2str(grnum) ' m' num2str(issr) ' = ' num2str(Sfs*Sf, 2) ' ss' num2str(ssn) ... ' n' mat 2str(sshex(1,:,ssn)) mat2str(Sfs*sshex(2,:,ssn)) 'b' ]; Sflist(imp,1:3) = [grnum ssn Sf]; end if igr == 14 title({[line1] [ 'R g' num2str(grnum) ' m' num2str(issr) ' = ' num2str(Sfs*Sf , 2) ' ss' num2str(ssn) ... ' n' mat2str(sshex(1,:,ssn)) mat2str(Sfs*sshex(2,:,ssn)) 'b' ] [g_gb_g] [plabel ' m'' = ' , num2str(sorthimp4(imp,4),3)] ... [ 'rBvec = ' mat2str(rbvec(kind,:,jind,gbnum),3) 'mag = ' num2s tr(rbvm_cur(kind,jind),3)]} ); Sflist(imp,4:6) = [grnum ssn Sf]; end end % graincen for cubic or hex if (has middle else) end % igr - grains 13 and 14 rbvgry = [max (0,1 - rbvm_cur(k ind,jind)) 0 0] %; / max(max((rbvm_cur))); plot(0, 0, 'k+' ); % gray scale of residual Burgers vector plot([0 rbvec(kind,2,jind,gbnum)]*2, - [0 rbvec(kind,1,jind,gbnum)]*2, ' - ' , 'Color' , rbvgry, 'LineWidt h' , 2); axis square ; axis([ - 3 3 - 3 3]*1.5*(1.4 - Sfsum)); % plot representation of grain boundary 1.2 for cubic, 1.4 for hex Sflist(imp,7:9) = [imp Sfsum sorthimp4(imp,4)]; else fprintf( 'No slip system has Schmid factors > %4.2f fo r one of the two grains at GB %d %d \ n' , Sfthr, gbnum, imp) end % if statement for continuing the loop for non - zero himp4 values end % imp m' loop sortSflist = sortrows(Sflist, - 8); sortSfsize = size(sortSflist); 99 APPENDIX C: Sample reconstructe d boundary and grain files Sample reconstructed boundary file # Header: Project1::post scan rotated cleaned cropped::All data::Grain Size 2/12/2020 # # Column 1 - 3: right hand average orientation (phi1, PHI, phi2 in radians) # Column 4 - 6: left hand average orientation (phi1, PHI, phi2 in radians) # Column 7: Misorientation Angle # Column 8 - 10: Misorientation Axis in Right Hand grain # Column 11 - 13: Misorientation Axis in Left Hand grain # Column 14: lengt h (in microns) # Column 15: trace angle (in degrees) # Column 16 - 19: x,y coordinates of endpoints (in microns) # Column 20 - 21: IDs of right hand and left hand grains 1.367 1.287 4.784 0.863 1.274 5. 234 32.52 - 18 14 - 11 - 18 14 - 11 18.009 23.6 0.00 49.07 16.50 41.86 27 1 1.367 1.287 4.784 0.863 1.274 5.234 32.52 - 18 14 - 11 - 18 14 - 11 14.503 178.9 16.50 41.86 31.00 42.15 27 1 1.367 1.2 87 4.784 0.284 0.699 0.128 59.88 - 9 14 1 - 9 14 1 23.116 5.7 0.00 73.90 23.00 71.59 27 43 4.068 0.194 2.325 0.284 0.699 0.128 51.35 - 11 4 - 3 - 11 4 - 3 19.218 8.6 0.00 142.61 19.00 139.72 76 43 0.339 1.343 5.643 4.068 0.194 2.325 90.01 28 - 1 - 8 28 - 1 - 8 11.846 32.4 0.00 182.44 10.00 176.09 96 76 0.339 1.343 5.643 0.394 1.268 5.600 5.47 24 - 11 7 24 - 11 7 2.887 90.0 0.00 196.88 0.00 199.76 96 112 100 Sample of grain file # Header: Project1::post scan rotated cleaned cropped::All data::Grain Size 2/12/2020 # # Partition Formula: # Grain Tolerance Angle: 5.00 # Minimum Grain S ize: 2 # Minimum Confidence Index: 0.00 # Multiple Rows Requirement: Off # Column 1: Integer identifying grain # Column 2 - 4: Average orientation (phi1, PHI, phi2) in degrees # Column 5 - 6: Average Pos ition (x, y) in microns # Column 7: Average Image Quality (IQ) # Column 8: Average Confidence Index (CI) # Column 9: Average Fit (degrees) # Column 10: An integer identifying the phase # 1 - Titanium (Alpha) # 2 - Titanium (Beta) # Co lumn 11: Edge grain (1) or interior grain (0) 1 49 .464 72.992 299.89 18.911 20.765 234.9 0.470 0.80 1 1 2 105.059 4.763 247.52 54.452 3.723 216.5 0.623 1.06 1 1 3 199.783 57.925 157.96 81.238 12.2 49 176.8 0.539 1.18 1 1 4 190.524 86.037 1 66.98 93.086 3.490 222.9 0.515 1.05 1 1 5 59.181 67.583 287.44 108.472 27.021 183.5 0.497 0.93 1 1 6 138.098 80.576 209.67 129.297 6.448 303.9 0 .565 0.94 1 1 101 BIBLIOGRAPHY 102 BIBLIOGRAPHY Abuzaid W.Z, Sehitoglu H., Lambros J., 2016, Localisation of plastic strain at the microstructurlal level in Hastelloy X subjected to monotonic, fatigue, and creep loading: the role of grain boun daries and slip transmission, Materials at High Temperatures, Vol. 33 No. 4 - 5, pp. 384 - 400. Alizadeh R., Peña - Ortega M., Bieler T.R., LLorca J., 2020, A criterion for slip transfer at grain boundaries in Al, Scripta Materialia, Vol. 178, pp. 408 - 412. Alu minum 1100 - O. Matweb.com, N.p, 2020, Web. 2 July 2020. criteria in experiments and crystal plasticity models, Material Science, Vol. 52, pp. 2243 - 2258 Boyer R., 1996, A n overview on the use of titanium in the aerospace industry, Materials Science Bridier F., Villechaise P., and Mendez J., 2005, Analysis of the different slip systems entation, Acta Materialia, Vol. 53, No. 3, pp. 555 - 567. Britton T.J., Wilkinson A.J., 2012, Stress fields and geometrically necessary dislocation density distributions near the head of a blocked slip band, Acta Materialia, Vol. 60, pp. 5773 - 5782. Buchhei t T.E., Carrol l J.D., Clark B.G., Boyce B.L., 2015, Evaluating Deformation Induced Grain Orientation Change in a Polycrystal During In Situ Tensile Deformation using EBSD, Microscopy and Microanalysis, Vol. 21, pp. 969 - 984. Ding R., Gong J., Wilkinson A.J ., Jones I.P., 2015, A study of dislocation transmission through a grain boundary in hcp Ti 6Al using micro - cantilevers, Acta Materialia, Vol. 103, pp. 416 - 423. Hémery S., Tromas C., Villechaise P., 2019, Slip - stimulated grain boundary sliding in Ti - 6Al - 4 V at room tem perature, Materialia, Vol. 000, pp.100 - 189. L i H. , 2013, Analysis of the deformation behavior of the hexagonal close - packed alpha phase in titanium and titanium alloys [Doctor dissertation, Michigan State University]. Hull D., Bacon D.J., 2011, Introduction to dislocations, Fifth edition, Oxford, Butterworth - Heinemann. 103 Jackson A.G., 1991, Chapter 7: Slip Systems, Handbook of crystallography: for electron microscopists and others, pp.83 - 88. Kacher J., Robertson I.M., 2014, In s itu and tomographi c analysis of dislocation/grain - titanium, Philosophical Magazine, Vol. 94, No. 8, 814 - 829. Kocks U.F., Tomé C.N., Wenk H. - R., 1998, Texture and Anisotropy, Cambridge, United Kingdom, Cambridge University Press and Engineering A, Vol. 213, No. 1 - 2, pp. 103 - 114. Lee T.C., Robertson I.M., Birnbaum H.K., 1989, Prediction of slip transfer mechanisms across grain boundaries, Metallurgica, Vol. 23, pp. 799 - 803. Li H., 2013, Analysis of the deformation behavior of the hexagonal close - p acked alpha phase in titanium and titanium alloys, Michigan State University, PhD dissertation. Linne M., Bieler T.R., Daly S., 2020, The Impact of Microstructure on the Relationship between Grain Boundary Sliding and Slip Transmission i n High Purity Alum inum. Luster J., Morris M.A., 1994, Compatibility of deformation in Two - Phase Ti - AI Alloys: Dependence on microstructure and orientation relationships, Metallurgical and Materials Transactions A, Vol. 26A, pp. 1745 - 1756. Matsuki K., M orita H., Yamada M ., Murakami Y., 2013, Relative motion of grains during superplastic flow in an Al - 9Zn - 1 wt.% Mg alloy, Metal Science, Vol. 11, No. 5, pp. 156 - 163. Palomares - García A.J., Pérez - Prado M.T., Molina - Aldareguia J.M., 2018, Slip transfer acros s c - TiAl lamellae in tension, Materials and Design, Vol. 146, pp. 81 - 95. Peters M., Kumpfert J., Ward C.H., and Leyens, C., 2003, Titanium alloys for aerospace applications, Advanced Engineering Materials, Vol. 5, No. 6, pp. 419 - 427. Sutto n G., Biblarz O ., 2017. Rocket Pr opulsion Elements, Ninth Edition, Ch. 10. T.C. Lee, I.M. Robertson and H.K. Birnbaum, Prediction of Slip Transfer Mechanisms Across Grain Boundaries, Scripta Metall. 23, 799 - 803, 1989 . Ti - 6Al - 4V (Grade 5), Annealed. Matwe b.com, N.p, 202 0, Web. 2 July 202 0. Z. Shen, R. H. Wagoner and W. A. T. Clark, Dislocation nd Grain Boundary Interactions In Metals, Acta metall. 36(12) 3231 - 3242, 1988.