PREDICTING SHEAR TRANSMISSION ACROSS GRAIN BOUNDARY IN ALPHA TITANIUM By Yang Su A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of Material s Science and Engineering Doctor of Philosophy 2021 ABSTRACT PREDICTING SHEAR TRANSMISSION ACROSS GRAIN BOUNDARY IN ALPHA TITANIUM B y Yang Su The c apability to evaluat e and model how metal deforms has benefited the manufacture and processing of the metal s industry for decades. A predictive model that can be used to assess the evolution of stress/strain in a polycrystalline metals is desired as it will enable more accura te predictions of stress concentration s , damage nucleation, and material life span s . One of the major challenges in the development of such a model is understanding how plastic deformation flows through grain boundar ies (such phenomenon is called slip tra nsfer) and how the stress and strain field s are altered by slip transfer in the vicinity of the boundary . We attempted to overcome this challenge . I n the current study carefully designed experiments and calibrated simulations are used to address this challenge . A novel approach is used to s tudy the interaction between slip and grain boundaries using bi - crystal nanoindentation. By placing nanoindents at varying distances from grain boundaries and measuring the resultin g indent surface topographies using atomic force microscopy (AFM), the influence of grain boundaries on the development of indent surface topographies is assessed and related to a variety of slip transfer metrics. To further analyze the stress, strain, an d shear of individual slip system in the bi - crystal nanoindentation, a crystal plasticity finite element (CPFE) models w as built to simulate indentation near a grain boundary . The model was calibrated using experimental ly measure d parameters and successfu lly captured most of the features in the experiment s . Nonetheless, strict point - to - point comparison s between experimental ly measured and simulated indent topographies revealed some discrepancies , in that the model is less accurate in the vicinity of grain boundar ies than in the grain interior s . To evaluate slip transfer and the local stress evolution in a fully quantitative manner, a predictive model was developed that is capable of resolving slip accommodation of multiple systems i nvolved in the process . Slip trace analysis was combined with AFM and electron backscattered diffraction (EBSD) to analyz e the slip accommodation observed at multiple grain boundaries. Based on all the experimentally observed slip transfer cases, a new iterative stres s relief (ISR) model was developed. The ISR model , validated by experimental observations, features the ability to predicting multiple accommodating slip systems in a slip transfer and assessing the evolution of local stress state. In addition, a set of critical resolved shear stress (CRSS) ratios were obtained by minimizing the discrepancies between observations and model predictions. Th is work has furthered the understanding of slip transfer/accommodations and the influence of local stress evolution on the slip transfer in the community. The ISR model has been proved to be very successful in the studied material system , but has yet to be tested under different conditions. Copyright by YANG SU 2021 v ACKNOWLEDGEMENTS This work was carried out at the department of chemical engineering and material science at Michigan State University in East Lansing. I would like to thank my advisor Dr . Martin Crimp for his continuous support both in academi a and personal life during m y whole P h. D . career. All those invaluable discussions with him and a high amount of scientific freedom have made this dissertation possible. I would like to thank Dr. Philip Eisenlohr for those countless inspiring discussions and his consistent mentoring of how to be a consistent and organized researcher. Dr. Thomas Bieler is acknowledged for sharing his many original ideas both in the experimental design and simulation methods . I would also li ke to thank the two other professors in my committees. Dr. Richard Lunt for his tutoring in the X - ray diffraction class and some suggestions on my nanoindentation experiments during that time. Dr. Ronald Averill is thanked for bringing the commercial software marc Mentat , which is frequently used in the dissertation, to the engineering school . Dr. Askeland is acknowledged for his assistance with the use of SEM and FIB. Dr. D rown is thanked for fixing the nanoindenter. Dr. Baokang Bi is acknowledged for his assistance with the AFM work. T o my present and former graduate colleagues in the , many thanks for creating a friendly and stimulating environment. I am indebted to many colleagues in Max - Plank - Institut für Eisenforschung GmbH. Dr. Zambaldi is thanked for his help on the modeling of nanoindentation and many insightful ideas . Dr. Mercier is thanked for teaching me how to use s tabix to build a finite element indentation model . Dr. Raabe is thanked for hosting us at the MPIE and providing all the resources to complete our plans. I am also thankful to technicians at MPIE that provided useful suggestions on polishing titanium and helped me with many instruments. vi Last but not least, I owe my deepest gratitude to my parents for always believing in me. T heir support help ed me get through many frustrations and setbacks through the P h. D . and made this dissertation possible. vii TABLE OF CONTENTS LIST OF TABLES ................................ ................................ ................................ ........................ x LIST OF FIGURES ................................ ................................ ................................ ..................... xi KEY TO SYMBOLS AND ABBREVIATIONS ................................ ................................ .... xvii CHAPTER 1 INTRODUCTION AND LITERATURE REVIEW ................................ .......... 1 1.1 Understanding heterogeneous plastic deformation near grain boundaries .................. 1 1.2 Study of heterogeneous deformation at grain boundaries using slip transfer criteria . 2 1.2.1 Slip transfer criterion involving only one outgoing slip system ................................ .... 2 1.2.2 Multiple outgoing slip system accommodation predicted using tangential continuity theory ................................ ................................ ................................ ................................ ...... 7 1.3 State of the art experimental techniques and simulation methodology used in slip transfer study ................................ ................................ ................................ ............................ 8 1.3.1 Bi - crystal/quasi bi - crystal nanoindentation ................................ ................................ ... 9 1.3.2 Quantitative slip trace analysis ................................ ................................ .................... 20 1.3.3 Other experimental technique ................................ ................................ ...................... 35 1.3.4 Coupling experimental results with CPFE simulations ................................ ............... 39 1.4 Motivations of this study ................................ ................................ ................................ .. 44 1.5 Overview of this thesis ................................ ................................ ................................ ...... 46 CHAPTER 2 MATERIALS, EXPERIMENTAL DETAILS, AND THE CONSTITUTIVE LAW OF CPFEM ................................ ................................ ................................ ....................... 49 2.1 Description of material ................................ ................................ ................................ ..... 49 2.1.1 Material A ................................ ................................ ................................ .................... 49 2.1.2 Material B ................................ ................................ ................................ .................... 49 2.2 Sample prepa ration ................................ ................................ ................................ .......... 52 2.2.1 Sample A ................................ ................................ ................................ ...................... 52 2.2.2 Sample B ................................ ................................ ................................ ...................... 52 2.3 Experimental details ................................ ................................ ................................ ......... 52 2.3.1 Scanning electron microscopy ................................ ................................ ..................... 52 2.3.2 Focused Ion Beam (FIB) cross sectioning ................................ ................................ ... 55 2.3.3 Atomic Force microscopy (AFM) ................................ ................................ ............... 57 2.3.4 Nanoindentation ................................ ................................ ................................ ........... 57 2.3.5 Four - point bending ................................ ................................ ................................ ....... 60 2.4 Slip trace analysis ................................ ................................ ................................ .............. 62 2.5 Constitutive law in Crystal Plasticity Finite Element Method (CPFEM) ................... 64 CHAPTER 3 STUDY OF SLIP TRANSFER USING NANOINDENTATION AND CRYSTAL PLASTICITY MODELING ................................ ................................ .................. 66 3.1 Overview of the locations of nanoindentations ................................ .............................. 67 viii 3.2 Nanoindentation load - displacement curves ................................ ................................ .... 69 3.3 Differences of the topographies formed in single and bi - crystal nanoindentations .... 71 3.3.1 Quantifying the volume difference of two indent topographies ................................ .. 71 3.3.2 Reproducibility of nanoindentation topographies ................................ ........................ 72 3.3.3 Comparing single crystal indent topographies with the corresponding bi - crystal indent topographies ................................ ................................ ................................ .......................... 75 3.4 Simulations of single and bi - crystal nanoindentations using CPFEM ......................... 83 3.4.1 Computer - assisted processing of experimental data and automatic conversion into simulation files ................................ ................................ ................................ ...................... 83 3.4.2. - Ti ................................ ............................. 86 3.4.3. Three dimensional finite element simulation of indentation process ......................... 87 3.5. Comparing of experimental and simulated indent topographies ................................ 89 3.5.1 Single crystal indent topography comparison between experiment and simulation. ... 89 3.5.2 Bi - crystal indent topography comparison between experiment and simulation. ......... 91 3.6. Discussion ................................ ................................ ................................ .......................... 94 3.6.1 Quantifying the quality of single crystal and bi - crystal simulation using ................................ ................................ ................................ ................................ ............... 94 3.6.2 Correlating experiment and simulation results with slip transfer criterion and M (LRB criterion) ................................ ................................ ................................ ...................... 96 3.6.3 Grain boundary sensitive CPFE model for bi - crystal nanoindentation ..................... 101 3.7 Conclusions ................................ ................................ ................................ ...................... 108 CHAPTER 4 PREDICTING SLIP TRANSFER ACROSS GRAIN BOUNDARIES WITH AN ITERA TIVE STRESS RELIEF MODEL ................................ ................................ ....... 110 4.1 Observations of slip accommodation in - Ti ................................ ................................ 111 4.2 Determining the active slip systems, relative shears, and slip geometries of experimentally observed slip accommodations ................................ ................................ .. 113 4.3 Assessment of the tangential continuity theory based on slip transfer observations 117 4.3.1 Tangential continuity theory ................................ ................................ ...................... 117 4.3.2 Predicting accommodating slip systems using tangential continuity theory ............. 118 4.4. A new iterative stress relief (ISR) model based on slip accommodation observation s ................................ ................................ ................................ ................................ ................. 128 4.4.1 Algorithm of the iterative stress relief model ................................ ............................ 128 4.4.2 Predicting accommodating deformation systems using the iterative stress relief model ................................ ................................ ................................ ................................ ............. 131 4.5 Discussion ................................ ................................ ................................ ......................... 134 4.5.1 Optimizing the variables in the iterative stress relief model ................................ ...... 134 4.5.2 Limitations of the iterative stress relief model ................................ .......................... 139 4.5.3 Application of the ISR model to other materials and loading conditions .................. 142 4.6 Conclusions ................................ ................................ ................................ ...................... 143 CHAPTER 5 CONCLUSIONS ................................ ................................ ................................ 145 CHAPTER 6 OUTLOOK ................................ ................................ ................................ ........ 148 ix APPENDICES ................................ ................................ ................................ ........................... 150 APPENDIX A: AFM DATA OF INDENT TOPOGRAPHIES IN 3D ............................. 151 APPENDIX B: SEM AND AFM MEASUREMENTS OF ALL SLIP ACCOMMODATION CASES ................................ ................................ ............................ 152 BIBLIOGRAPHY ................................ ................................ ................................ ..................... 159 x LIST OF TABLES Table 1.1: Column 2 - 6: Slip geometry of all grain boundaries including disorientations (column 5) measured using the patterns in fig. 3 and ( column 4 ) which is the multiplication of column 2 and 3. Column 7 is the deformation behavior at grain boundaries where the type B, I, and T suggests slip band blocking, intermediate slip transmission, and slip transmission, respectively . [23] ........................................................................... 1 3 Table 3.1: Material parameters used for simulating the - Ti.[133] ............................................ . 8 6 Table 4.1: Slip systems types and geometric s , and number of dislocations in the slip bands for all 22 (16 single plus 6 double) slip accommodation obse rvations. .............................. 1 16 Table 4.2: Indices, shear direction d , and plane normal n of deformation systems considered for shear accommodation in the study . .......................................................................... . 1 19 Table 4.3: The center point of the search range used in the parameter optimization of the iterative stress relief model. The six parameters correspond to the six dimensions o f the optimization . ............................................................................................................. 1 35 xi LIST OF FIGURES Figure 1.1: Slip transfer geometry at a grain boundary . ................................................................ 6 Figure 1.2: Nanohardness as a function of the distance between each indentation and the grain boundary. Each data point represents one indentation measurement. [22] .............. 1 0 Figure 1.3 : SEM images of nanoindentations at three types of grain boundaries and the corresponding EBSD patterns in the indented and neighboring grain. (a): Slip band caused by nanoindentations were blocked by the grain boundary (note d as type B). (b): Intermediate slip band development in the neighboring grain (noted as type I). (c): Full slip transmission across the grain boundary (noted as type T). [23] ............1 2 Figure 1.4: Load - displacement curves of nanoindentations near two grain boundaries showing displacement jumps at different indentation depths and loads. [24] ..........................1 5 Figure 1.5: Fe - 14%Si boundary yielding events shown in a Hall - Petch type plot with the horizontal axis being the inverse square root of the distance between indent and boundary and vertical axis being the applied shear stress. [25] ..................................1 7 Fi gure 1.6: (a) Crystallographic details of the two grain boundaries including location of the indents and the orientation of the four indented grains. (b) The yielding stress calculated for each nanoindentations. [27] ............................................................... . 19 Figure 1.7: An optical image of the primary {011} slip bands and secondary {101} slip bands in the grain boundary vicinity in a compressed bi - crystal sample. [34] ...................... . . 2 2 blockage events as a data point, where dots and crosses means slip transfer and slip plot showing the same data set. [37] ..................................................................... .... 2 4 Figure 1.9. Crystal orientation map of the two aluminum samples with dramatic different microstructures, where the left being near - cube and right being rotated cube microstructure. [38] ............................................................. ...................................... 2 6 Figure 1.10: A vs grain boundary misorientation plot showing every of slip transfer and slip blockage events as a data point, where blue and brown means slip transfer and slip blockage, respectively. [38] ................................................................. ................... . 2 7 Figure 1.11: Plot of new slip transfer metric Mb as a function of Schmid factor. The three zones (1,2,3) indicate the direct slip transfer, slip band blockage with stress concentration, and blockage without stress concentration, r espectively. [39] ................................ 3 0 xii Figure 1.12: Schematic showing how the quantities in equation Eq. 1.9 are correlated with an AFM measurement. [44] ......................................................................................... . 3 2 Figure 1.13: A Backscattered image showing the impinging twin initiated micro - cracks at a grain boundary. [45] .................................................. ....................................................... 3 4 Figure 1.14: a) The four TEM images acquired at different tilt angles (marked at the top left corner) that were used to construct the tomograph shown in (b). b) The 3 - D tomograph revealing the r elative locations of each dislocations observed in the TEM image series in (a). [51] ........................................................................................... 3 7 Figure 1.15: a - c): CPFEM modeling of the Von Mises stress (a) and shear dis tribution (b,c) in the area of interest. d): Evolution of the simulated shear on two slip systems as a function of time. [49] .............................................................................................. 4 1 Figure 1.16: Left image: Quas i - 3D CPFE model shows the columnar microstructure of all the grains. Right image: 3D model with grain structure information obtained from DAXM. [120] ..................................................................................................... . .... 4 3 Figure 2.1: An EBSD inverse pole figure (IPF) map of the area of interest in material A is presented. The texture of the material A shows a moderately strong texture with the c - axes lying predominantly in the plane of the image [40]. The array o f black dots in the IPF map is a grid of nanoindents to mark the area of interest. ............................ 5 0 Figure 2.2: An EBSD inverse pole figure map of a random area in material B is presented. The texture of the material B is almost random. .. ............... ............................................. 5 1 Figure 2.3: A schematic showing the setup of EBSD inside of an SEM (left image) and a typical diffraction pattern (Kikuchi pattern in right image) acquired in the EBSD scan that can be used for determining the crystalline orientation of the diffracted regi on. (Revised based on the image on https://www.mpie.de/3077954/EBSD ) ................. . 5 4 Figure 2.4: Backscattered electron image showing a FIB cross - section cut used to determine the angle measured from image and is used for the calculation of real grain boundary inclination using Eq. 2.1 . ...................................................................................................................... 5 6 Figure 2.5: An SEM image of the region that contains the grid of nanoindentations (shown as black dots) in material A . .......................................................................................... 5 8 Figure 2.6: An SEM image of bi - crystal nanoindentations near a selected grain boundary with varying distances from the boundary in material A. The investigated grain boundary is colored in red dash line . ............................................................................... .......... 59 Figure 2.7: The picture of the four point stage with the sample . ................................................. 6 1 xiii Figure 2.8: The AFM measured surface profile of a slip trace (blue suggests lower height) overlaid with a unit cell of the corresponding grain orientation shown in top view (a) and side view (b, c). The assumed active slip plane in the slip trace analysis is color ed in grey in the unit cell . .................................................................................. 6 3 Figure 3.1: SEM images of bi - crystal nanoindents near the six selected grain boundaries and the corresponding single crystal indents in the two adjacent grains . ........ ...................... 6 8 Figure 3.2: Load - displacement curve of a bi - crystal indent (shown as the grey dotted line) and the corresponding single crystal indent in the originating grain (shown as the black solid line). A magnified inset is used to show t he displacement jump occurred in the bi - crystal indentation . ................................................................................................ 7 0 Figure 3.3: a,b): Example of two single crystal nanoindents from the same grain used for estab lishing the reproducibility and surface roughness of nominally identical nanoindentations, where and are the inner and outer radius of a ring centered on the indent and isolating the major indent topography features. c): Correlation V (short for or , calculated based on Eq. 3.1) and ring area A . d): Cumulative probabi lity of the surface roughness levels (dotted), established from the slope of the dotted line in (c), and reproducibility (solid) of single and bi - crystal indentation, as determined by the difference between slopes of the solid and dotted lines in (c). A r eproducibility better than 3 nm corresponds to negligibly small perceived differences, as demonstrated by shifting one part of the color bar by that amount (e). [126] ......................................................................................... .............. 7 4 Figure 3.4: Nanoindent topographies developed when the indents impression falls across grain boundaries (center column), in comparison to corresponding single crystal indent topographies from both sides of the grain boundaries (second an d fourth columns). The differences between the single and bi - crystal indents are mapped for the left grain and the right grain (first and last column) ........................................................ 7 6 F igure 3.5: Nine bi - crystal indents (cente r column) collected near six different grain boundaries (blue lines) compared with corresponding single crystal indents in both the originating grains (second column) and receiving grains (fourth column). Both single and bi - crystal indentation impression depths (h) are labeled near the indents, - crystal indent in decreasing order. Indent topography differences caused by the grain boundary are mapped in the outermost columns as the differe nce between grain boundary indent and single crystal indent. The central indent valleys have been removed in order to enhance visualization of the indent pile - ups around the valleys. [126] .................... 8 1 Figure 3.6: The GUI of stabix used for generating simulation input files by using grain orientation data from EBSD. Left image: EBSD crystallographic orientation data was first processed using the left GUI to determine the maximum parameter and to colo r the associated grain boundary. The hexagonal cells represent the orientations of each of the grains. Although the simulations include basal < a >, prismatic < a >, xiv and pyramidal < c + a >, the GUI only includes basal a and prismatic a calculation. Middle image: The second GUI was used to display the crystallographic and slip system relationships. In the upper part of the figure, grain boundary information, such as Euler angles, the orientation of the grain boundary line, the inclination of the gr ain boundary plane, and the active slip systems, can be entered to study the slip transfer of a desired grain boundary. Specific (or M) associated with each pair of slip systems are tabulated in lower part of the figure. Right image: Using the grain b oundary information entered in the second GUI, the third GUI builds finite element meshes for bi - crystal indentation simulations. Sample dimensions, mesh resolution, indentation depth, and indenter tip geometry are required as input. [126] ............... .............................................................................. 8 4 Figure 3.7: The finite element model used for a bi - crystal indentation tests where the black dotted line indicates the location of the grain boundary. The red color region indicates that the formation of surface topographies after the nanoindentation. [12 6] .......................................................................................................................... 88 Figure 3.8: Experimentally measured (first column) and simulated (second column) topographies of two single crystal indent and the corresponding point - wise differences between experiment and simulation shown in the third column. [126] .................................. 9 0 Figure 3.9: Comparisons between experimentally measured and simulated indent topographies near six grain boundaries. a) Indents that are located near the grain boundary. b) Indents that are right on the grain boundary. [126] ................................................... 9 3 Figure 3.10: Cumulative probability distributions of the differences bet ween simulated and experimentally measured indent topographies for single crystal and bi - crystal case are shown as dotted blue line and solid blue line, respectively. As a comparison, noise level in experimental measurements from Fig. 3.3 are plotted for s ingle crystal (dotted red line) and bi - crystal case (solid red line). [126] ........................ . 9 5 Figure 3.11: a): Schematic single and bi - crystal indents with gray areas representing (top) and (bottom). The grain boundary in the bi - crystal case is marked si - crystal and originating single crystal indents evaluated normalized volume di bi - crystal and originating single crystal indents evaluated within the originating grain (red) and the receiving grain (black). d): Volume ratio between simulated and measured bi - crystal indents evaluated within the originating grain (red circles) and the receiving grain (black circles). e): M vs. normalized volume difference (both sim and exp) between bi - crystal and originating single crystal indents evaluated within the originating grain (red ) and the receiving grain (black). [126] ............. ... 98 xv Figure 3.12: The grain boundary sensitive model proposed as a revision to the original model by inserting two layers of elements (Grain 1 boundary layer and grain 2 boundary layer) between the t wo adjacent grain 1 and 2 . ..................................................... 10 3 Figure 3.13: The physical meaning of the four slip parameters in the model is shown in a stress - strain curve in the left figure. The values of the and used in the original model are presented in the right graph . ................................. ................................ 10 4 Figure 3.14: Figure (a - c) show how the accuracy of the new model ( ) alters as a function of the change in each slip parameters ( , a, and . The red dotted line indicates the optimal value of (in the ratio form of ) that generates the most accurate simulation compared to experiment. Figure d presents all the optimal values of acquired at fou r grain boundaries as a function of the four . ............................................................... 1 07 Figure 4.1: Secondary electron images of the three categories of slip accommodation: non - correlated (left), one - to - one correlated slip (ce nter), one - to - two correlated slip (right), and corresponding topography maps measured by AFM (second row) . ................ 11 2 Figure 4.2: An example/key of a polar bar plot that presents all of the details of a shear transfer event and serves as key for Fig. 4.3, Fig. 4.4 and Fig. 4.5. A total of 30 (=3+3+6+12+6) potential deformation systems, color coded by type, are represented on the outside of the figure, with the numbers indicating the specific deformation system, as listed in Table 4.2. Each bar represents an incoming or outgoing slip system, with white indicating the incoming system, black indicating outgoing systems, and red denoting accommodating systems in the incoming grain. The direction each bar points to reflects its slip system type and the length of each bar represents its relative shear . .................................................................................... 12 2 Figure 4.3: Radar bar plots showing tangential continuity predictions compared to observations of single shear accommodation cases. The left column indicates that the predictions are identical (and never match the observations) when at least two self - accommodating systems are allowed. The center two columns show the response changing for less than two self - ac accurate, with about half agreeing with observations . ............................................ 1 25 Figure 4.4: Similar plot as Fig. 4.3 but for double shear accommodation cases. Tangential continuity mo del predictions are shown in columns 2 - 4 and observations are presented in column 5. Across all six cases, none of the three distinct self - accommodation conditions result in predictions that agree with the observed accommodating shear systems . ........... .................................................................... 1 27 Figure 4.5: The new (ISR) model predictions compared to observations for 16 single and 6 double accommodation cases . ................................................................................. 13 3 xvi Figure 4.6: Figure 4.6: Heat map showing the variation in the accuracy of the iterative stress model, shown in grey scale, is quantified using the fraction of correct predictions compared to observations. Lighter gray suggests higher accuracy of the model. .............................................. . 1 35 Figure 4.7: Heat map showing the accurac y variation of the iterative stress relief model as a function of CRSS ratios prism and pyramidal . The accuracy of the model is quantified using the fraction of correct predictions compared to observations. Lighter gray suggests higher accuracy o f the model at that point . .......................... 1 37 prism , pyramidal , and pyramidal in the model. The model accuracy is quantified using th e fraction of correct predictions . ............................................ 1 38 Figure A1: The AFM data of indent topographies a and b in 3D. a) Topography of indent a. b) Topography of indent b. c) The differences in topographies of a and b. ................ 15 1 Figure B 1: SEM images of single slip accommodation cases 1 - 4 . ........................................... . 1 5 2 Figure B 2 : SEM images of single slip accommodation cases 5 - 8 . ............................................ 1 53 Figure B 3 : SEM images of single slip accommodation cases 9 - 12 . .......................................... 1 54 Figure B 4 : SEM images of single slip accommodation cases 13 - 16 . ............................... ......... 1 55 Figure B 5: SEM images of double slip accommodation cases 1 - 4 . ......................................... . . 1 56 Figure B 6: SEM images of double slip accommodation cases 5 and 6 (Note that there is a twinning involved shown as yellow dotted line in the EBSD map inset, in the accommodating deformation system in case 6) . . ..................................................... . 1 57 Figure B 7: AFM measurements of double slip accommodation cases 1 - 6 . . .............................. . 1 58 xvii KEY TO SYMBOLS AND ABBREVIATIONS AFM atomic force microscopy A area of an nanoindenation a hardening exponent BFGS Broyden - Fletcher - Goldfarb - Shanno minimization routine BSE backscattered electron BX bi - crystal b Burgers vector residual Burgers vector in the grain boundary CPFEM crystal plasticity finite element model CRSS critical resolved shear stress fourth - order stiffness tensor relative activity of primary, secondary, and tertiary slip system DAXM differential aperture X - ray microscopy d unit vector of slip direction in current work EBSD electron backscattered diffraction EXP experiment e slip plane normal in Livingston and Chalmers FIB focused ion beam F d eformation gradient elastic deformation gradient xviii plastic deformation gradient a grain boundary fracture metric GND geometric necessary dislocation GUI graphical user interfaces g s lip direction in Livingston and Chalmers g vector in a TEM image self - hardening matrix hardening matrix initial hardening slope of slip system initial hardening slope for all slip systems height of an indent topography at a point ( r , ) in a polar coordinate system height of an indent topography at a point ( x , y ) in Cartesian coordinate system HR - EBSD high resolution electron backscattered diffraction ISR iterative stress relief I s econd order Identity tensor IPF inverse pole fig ure Hall - Petch slope l direction of the slip plane trace on the grain boundary. plastic velocity gradient M MPIE Max - Plank - Institut für Eisenforschung Schmid factor for a specific deformation twinning system xix Guo et al. geometric slip transfer parameter M s Shen et al. s geometric slip transfer parameter n stress exponent in the CPFE model n unit vector of slip plane normal in current work unit vector of specimen surface normal unit vector n ormal to a grain boundary number of dislocations in a slip band Livingston and Chalmers geometric slip transfer parameter orig originating (incoming grain in slip transfer) cross - hardening matrix recv receiving (outgoing grain in slip transfer) RSS resolved shear stress the approximate indentation impression radius SE secondary electron SEM scanning electron microscopy SiC silicon carbide SIM simulation SX single crystal S second Piola - Kirchhoff stress TEM transmission electron microscopy t unit normal of the habit plane of the twins tangential part of the plastic distortion tensor / plastic distortion rate in grain A/B xx volume of single crystal indent topography in the receiving grain volume of simulated indent topography in the receiving grain volume difference between two indentation topographies a and b over the area A estimated width of each slip/twin band weighting factor in the iterative stress relief model threshold of relative activity in the iterative stress relief model the angle between direction of the FIB cut and the grain boundary normal angle between the slip plane traces on the grain boundary angles between Burgers vectors in incoming and outgoing grains angle between slip plane normals of incoming and outgoing systems grain boundary inclination normalized global stress tensor global stress tensor local stress tensor resolved shear stress tensor initial slip resistance saturation slip resistance critical resolved shear stress of slip system saturation stress of slip system shear rate / shear rate of slip system i in grain A shear rate of slip system i in the incoming grain xxi the ratio of the circumference of a circle to the diameter 1 CHAPTER 1 INTRODUCTION AND LITERATURE REVIEW M aterial s are believed to define the limit of what modern engineering can achieve . B reakthrough s in material science or discovery of new materials is the foundation of many new emerging technolog ies . This is especially true in the development of new metals and alloys. D espite the over 2000 years of study of metal s /alloy s in history , there is still a strong need for metals with higher strength, ductility, and resistance to crack ing , as well as other superior physical properties . T he tradition al approach to developing new alloys or improving existing alloys has always been rather straightforward and constant : t hrough a careful examination of the microstructure and a n involved study of the deformation mechanism, an informed decision was made on how to adjust the material composition or develop new material processing conditions . Therefore, it is very beneficial to learn the material microstructure and understand the governing rule s of deformation in that material . Furthermore, with the rapid grow th of computing power , the ability to model material deformation in any industrial processing has become more crucial than ever . This can only be realized efficiently if deformation of metals and alloys in both macro - and micro - scale can be precisely characteriz e d . 1.1 U nderstanding heterogeneous plastic deformation near grain boundar ies The study of deformation mechanisms in metals is usu ally based on the dislocation theory proposed by G.I. Taylor [ 1 ] . The theory revealed that plastic deformation of metals is always realized by the movement of a fundamental line structure called a dislocation. Thus, the study of deformation mechanisms in metals can be achieved by learning the behavior of dislocations. T he disloc ation theory has been validated and proven to be sufficient to explain deformation in 2 metal s in most circumstances [1 - 3] . While current dislocation theory is sufficient to describe movement of dislocations under homogenous stress and strain field , there are still gaps in understanding how dislocations interact with grain boundaries where heterogenous stress/strain field s dominate [4] . A lack of knowledge of how dislocations behave in the vicinity of grain boundar ies will lead to inaccurate estimati on of how and where the stress concentrates and the crack s initiate on the grain boundary. As a result, i t is crucial that more effort is devoted into the study of how m etal deform s near a grain boundary in a quantitative manner . 1.2 Study of heterogeneo us deformation at grain boundaries using slip transfer criteria As the dislocation theory explains , when dislocations move towards a grain boundary, the stress field of the dislocation will be repelled by the stress field of the grain boundary (in common cases), creating a pile - up of dislocations towards the grain boundary. When the stress reaches a ce rtain threshold as the number of dislocations in the pile - up increases, the piling dislocation will then pass through the grain boundary. This explanation pictures a very sensible scenario of dislocation passing through grain boundary in a rather qualitat ive way. However, to quantitatively describe this dislocation slip transfer process, it is desirable to find the answer to these two questions: Do dislocations experience the same magnitude of repell ing stress at different types of grain boundaries, and if a dislocation slipped across a grain boundary , how will adjacent grain respond. 1.2.1 Slip transfer criterion involving only one outgoing slip system Effort s to resolv e these two questions have led to the establishment of various slip transfer criterion. The majority of the se slip transfer criterion are based on the assumption that one 3 incoming slip system triggers a unique outgoing slip system, despite the fact that a secondary outgoing slip system or even a tertiary outgoing slip system are often required to fully accommodate the shear of the incoming slip system. Such simplification of the slip transfer problem is mainly due to the fact that in most scenario s the primary outgoing slip system can already relieve the majority of the incoming shear, hence in those scenarios it is reasonable to approximate the slip transfer by assuming only one outgoing slip system is activated. Another reason for considering only one outgoing slip system is the incr easingly complexity of the problem as secondary and tertiary outgoing slip systems are considered. As a result, theories of slip transfer that concerns only one outgoing slip systems are firstly reviewed. T he first major breakthrough was made by Livingston and Chalmers in 1957 [ 5 ] . In the ir experiment, they conducted aluminum bi - crystal compression test and analyzed the dislocation activities near the grain boundary using the slip traces on the sample surface. Livingston and Chalmers found that the primary slip system, which is prominent both inside the grain and at the grain boundary, can activate multiple (up to three) slip systems in the vicinity of grain boundary , due to requirement of strain continuity at the grain boundary. In addition, Livingston and Chalmers proposed a geometric parameter that uses the information of primary slip system to predict the type of outgoing slip system(s). Eq . 1.1 where e and g In their theory, the slip transfer usually occurs between the incoming and outgoing slip systems that maximize the m agnitude of . A more general consideration of slip transfer criterion was proposed by Shen et al . [ 6 , 7 ] . Based on the observation of dislocation transfer across grain boundar ies in 304 stainless steel via 4 TEM , Shen et al . used four different criteri a to predict the outgoing slip system . ( 1) The first criteri on is identical to the geometric parameter used by Livingston and Chalmers. ( 2) Shen et al developed a new metric M s based on the assumption that the rotation of the dislocation line from the incoming to the outgoing slip plane is the rate limiting step in dislocation transmission. M s can be expressed as follows: Eq . 1.2 where the and are the lines of intersection between grain boundary and the slip planes of incoming and outgoing slip systems, respectively, and and are the slip directions of the incoming and outgoing slip systems. ( 3) The force acting on an emitted disloca tion across from the pile - up should be maximized for the slip propagation to take place. ( 4) This criterion combines the geometric and stress field contributions in that it used criterion 2 to predict the outgoing slip plane and criterion 3 to predict the outgoing slip directions. By testing all four models at five different grain boundaries, criterion 4 proves to be the most accurate. Shen et al work revealed that a precise slip transfer model cannot solely rely on the geometric relationship between i ncoming and outgoing slip systems, because the revolved shear stress on outgoing slip system is another component that strongly influence which outgoing slip system will be activated. In the light of the previous effort, Clark et al . [8] proposed a set of three conditions for determining the outgoing slip system: 1) the slip plane on which the dislocation will be emitted is chosen as that for which the angle between slip traces is a minimum. 5 2) the slip direction within the plane selected in 1) is chosen as that on which the resolved shear stress, on the emission side of the interface, is a maximum near the intersection of the slip plane and the boundary. 3) if the resolved shear stresses calculated in 2) are close f or two or more slip directions, the activated system is that which minimizes the residual grain boundary dislocation energy left in the interface. In Clark et al . the influence of residual Burgers vector was accounted for in the slip transfer for the first time . Lee, Robertson, and Birnbaum also independently proposed a similar slip transfer criterion [9 , 10] similar to , and this theory was proved to be the most comprehensive and accurate slip transfer criterion to date . T he theory was named LRB criterion and can be expressed as follow: 1) The angle between the traces of the slip planes on the grain boundary plane should be a minimum. This is generally expressed using the LRB parameter: Eq. 1.3 where is the angle between traces of slip planes in the grain boundary and is the angle between Burgers vectors of the incoming and outgoing slip system. An illustration of the two angles was shown in F ig . 1 .1 . 2) The resolved shear stress on the outgoing slip system: Eq .1.4 should be maximized , where the stands for the resolved shear stress on a slip system, is the local stress state in the vicinity of the slip transfer location, and and are the slip directions and slip plane normal of outgoing slip systems, respectively. 3) The magnitude of the Burgers vector in the grain boundary should be minimized. 6 In addition to the criterion proposed above , a s imple geometric criteri on was introduced by Luster and Morris [11] as , where and are the angles between slip plane normal and Burgers vectors, respectively. is advantageous due to its reasonable accuracy for describing most slip transfer events given its easy accessibility without the laborious effort of finding out the grain boundary inclination below surface. Figure 1 .1: Slip transfer geometry at a grain boundary. 7 1.2.2 Multiple outgoing slip system accommodation predicted using tangential continuity theory A common theme amongst the simp le slip accommodation criteria outlined above is the explicit assumption that slip from a single system in one grain is accommodated by a single system in its neighboring grain, despite the fact that it is quite possible, or even likely, that strain accommoda tion can occur on multiple slip systems. Livingston and Chalmers have attempted to determined what slip systems are involved in the accommodation by requiring continuity of the resulting grain boundary deformation [5] . This has been studied and improved by several studies [12 - 19] . Nonetheless, the overall underlying physics remains unchanged and can be compactly expressed by stating that the plastic distortion rates on either side of the grain boundary ( and ) must affect the grain boundary plane in the same way, i.e. or , Eq. 1.5 where is an arbitrary vector in the grain boundary plane (i.e. normal to the grain boundary normal n ) and I is the second rank identity tensor. Considering that the plastic distortion rates are superpositions of individual slip system activity in grain A and grain B ( Indexed by i and j ) , the overall activity in the vicinity of a grain boundary needs to be fulfilled: Eq . 1.6 While perfect tangential continuity may not be realistic because slip transfer generally leaves residual Burgers vector content in the grain boundary, one can nevertheless use Eq. 1.6 to identify such slip activity tha t minimizes discon tinuity for given incoming slip , i.e . Eq. 1.7 8 While the tangential continuity model has the potential to predict which deformation systems associated with shear transfer will be activated, it s pure kinematic nature suggests that it disregards the effect of an imposed state of stress as well as the influence of the residual Burgers vector left in the grain boundary 1 . Ther efore, it is still desirable to further study the physics of slip transfer across grain boundar ies of varying nature and develop a new model that can account for multiple outgoing slip systems with the consideration of local stress state and residual Burge rs vector. 1.3 State of the art experimental techniques and simulation methodology used in slip transfer study To determine which slip transfer /accommodation criterion most accurately describes the interaction between dislocation and grain boundary, experimental methods that are dedicated to the study of the heterogenous deformation in the vicinity of the grain boundary needs to be established. Past research has applied several experimental procedures to achieve this goal, which include s bi - crystal nanoindentations, micro - pillar compression /indentation , and orientation informed surface slip trace analysis. Each technique has its own limitations and advantages . N onetheless all of them have gained us insights into the mech a nisms of slip - grain boundary interplay from different length scales and aspects . 1 C. Fressengeas et al . has attempted in their latest work [12] to imple ment tangential continuity into a crystal plasticity finite element framework. In their methodology, the tangential continuity serves as a fitness function to modify the local stress tensor. 9 1.3.1 B i - crystal/quasi bi - crystal nanoindentation Bi - crystal nanoindentation, compared to the in - situ TEM, micro - pillar compression, and slip trace analysis, is faster and less laborious in studying slip transfer . It only requires accurate measurement of the indentation location in the grain boundary vicinity and the real - time load - displacemen t curves during the indenting (This can be achieved by the instrumented nanoindentation [ 20 , 21 ] ) . Nevertheless, due to the complex stress state caused by the indentation, the data analy sis of bi - crystal nanoindentation is always complicated and convoluted. Much effort has been dedicated to deconvolut ing the influence of grain boundar ies on slip transfer from bi - crystal nanoindentation. Y a . M . Soifer et al . [ 2 2 ] studied the nano - hardness of copper in the vicinity of the grain boundaries using instrumented nanoindentations and found that the nano - hardness is influenced by the distance between the indent location and the grain boundary . As shown in figure. 1. 2 the influence of the grain boundary on the nano - hardness in the right grain can be felt when the distance between the indent and grain boundary falls within 1.4 m. In addition, the influence of the grain boundaries on the nano - hardness in the left (negative part of the horizontal axis) and right (positive part of the horizontal axis) grains are significantly different. While the nano - hardness remains steady as the distance between indent and grain boundary reduces in the left grain, a dramatic rise of the nano - hardness in the right grain is observed as the indent approaches the grain boundary. This indicates that there is no significant resistance to plastic deformation during the indentation in the left grain , but a high resistance to plastic deformation in the right grain causes the increase of the nano - hardness. Th e different behaviors of the grain boundary in response to the plastic deformation caused by nanoindentation provides a unique way to study the role of grain boundaries using nano - hardness . 10 Figure 1. 2 : Nanohardness as a function of the distance between each indentation and the grain boundary. Each data point represents one indentation measurement. [ 22 ] 11 In the light of Soifer et al study on indentations near grain boundaries , P . C . W o et al. [ 23 ] studied the influence of various grain boundaries on plastic deformation in Ni 3 Al using nanoindentations carried out in the vicinity of those grain boundar ies . Instead of characterizing eac h indentation with nano - hardness, they categorized all grain boundary indents into three groups and examples of each group are shown in F ig . 1. 3 in three separate rows. The first row ( F ig . 1. 3a) gives an example of grain boundary blocking the plastic deformation generated by the indentation, which are manifested by absence of slip traces on the neighboring side of the grain boundary in the SEM backscattered imag e. I n contrast in the F ig . 1. 3b , limited slip lines developed across the grain boundary, suggesting a smaller grain boundary resistance to plastic strain compared to the first scenario. In the F ig . 1. 3c, abundant slip traces are observed in the neighboring grain and the presence of the grain boundary does appear to influence the development of those slip traces, indicating minim al grain boundary resistance to slip transfer in this case. Furthermore, they found a correlation between the three types of grain boundary indents and a slip t ransfer parameter , which are summarized in the table 1 .1 . They concluded from table 1 .1 that the ease of slip transmission in Ni 3 Al seems to correlate with the value of . At grain boundaries with low values (row 1 - 4), the slip trace of the indentation was blocked by the grain boundary (shown as type B). With higher values of , the slip trace of the indentation is able to travel across the grain boundary (shown as type I) even without much change of the direction of the slip trace (shown as type T). This study suggests that more insights can be gained by cor relat ing the slip traces developed in bi - crystal indentation to slip transfer geometry. 12 Figure 1. 3 : SEM images of nanoindentations at th ree types of grain boundaries and the corresponding EBSD patterns in the indented and neighboring grain . (a): Slip band caused by nanoindentations were blocked by the grain boundary (noted as type B) . (b): Intermediate slip band development in the neighboring grain (noted as type I) . (c): Full slip transmission across the grain boundary (noted as type T) . [ 23 ] 13 Table 1.1: Column 2 - 6: Slip geometry of all grain boundaries including disorientations (column 5) measured using the patterns in fig. 3 and ( column 4 ) which is the multiplication of column 2 and 3. Column 7 is the deformation behavior at grain boundaries where the type B, I, and T suggests slip band blocking, intermediate slip transmission, and slip transmission, respectively. [23] 14 In addition to the study of the influence of grain boundar ies on nano - hardness and slip trace development in bi - crystal nanoindentation, Wang et al . [ 2 4 ] studied the load - displacement curve s in a bi - crystal nanoindentation experiment and used displacement jumps on the load - displacement curves to characterize the plastic deformation near the grain boundary. They discovered that the yielding of a grain boundary can be observed in a load - displacement curve during a load controlled bi - crystal nanoindentation ( Load controlled mode means the indentation load was maintained constant throughout the process). When the nano - indent er presses the dislocations through a grain boundary , the yielding of the grai n boundary releases the excess energy that is stored in the dislocation pile - ups in the origin grain . This process leads to a displacement jump on a load - displacement curve. In theory, the size of the displacement jump is correlated with the energy of th e dislocations pile - ups. However, this correlation was not revealed in Wang et al . work and the grain boundary yielding was not observed near every grain boundary. Two examples of the displacement jumps are shown in F ig. 1. 4 , where the two nano indents were placed at different distances to the grain boundary and as a result the displacement jumps occurred at different load. With more detailed analysis, it was found out that displacement jumps tend to occur at grain boundaries with higher values, especially above 0.9. T hey argue d that the displacement jumps were caused by slip transmission across those grain boundar ies , and t he absence of the displacement jump s at grain boundaries with low values were attributed to the bl ockage of slip band by grain boundaries. 15 Figure 1. 4 : Load - displacement curves of nanoindentations near two grain boundaries showing displacement jumps at different indentation depths and loads. [ 2 4 ] 16 More generalized analysis of displacement jumps observed on the load - displacement curves of grain boundary nanoindentations were conducted by W. A. Soer et al . [ 25 ] . In their study , two type s of material , including commercially pure Mo bi - crystal with symmetric tile boundary and Fe - 14%Si alloy with general grain boundary , were used for bi - crystal nanoindentations. The displacement jumps, which is an indication of grain boundary yielding, were observed onl y in Fe - 14%Si alloy and none was observed in commercially pure Mo. The absence of grain boundary yielding in Mo bi - crystal was attributed to yield stress of a symmetric tile boundary being too low, in which case dislocation pile - ups cannot maintained agai nst the boundary. Furthermore, the grain boundary resistance to slip transmission was quantitatively related to the displacement jumps using a Hall - Petch type calculation using the equation: Eq. 1.8 where the and indicate the yielding stress of a single crystal nanoindentation and the yielding stress of a bi - crystal nanoindentation, respectively. The distance between the indent and the grain boundary is represented by d . The Hall - Petch slope is indicated by . The Fe - 14%Si grain boundary yielding events are summarized in F ig . 1. 5 . The Hall - Petch slope was determined to be 0.58, which agrees well with the macroscopically measured value, by measuring the slope of dotted line in F ig. 1. 5. Using this method ology, it was concluded that the grain boundary slip resistance can be obtained for various materials. T. B. Britton et al . [ 26 ] used similar approach to r esearch the slip resistance of Fe - 0.01 wt% C polycrystal and pure copper. They concluded that the grain boundary yielding events in bi - crystal nanoindentations are likely related to interstitials atoms pinning dislocations in the vicinity of the grain boundary. Furthermore, the measured values agrees well with macroscopic Hall - Petch effect observations. 17 Figure 1. 5 : Fe - 14%Si boundary yielding events shown in a Hall - Petch type plot with the horizontal axis being the inverse square root of the distance betwee n indent and boundary and vertical axis being the applied shear stress. [ 25 ] 18 Most recent research on bi - crsystal nanoindentations by Kalidindi et al . [ 27 ] and Vachhani et al . [ 28 , 29] w ere dedic a ted to accurately determin ing the grain boundary yielding stress by converting nanoindentation load - displacement curve s into a standard stress - strain curve s . In both of the se stud ies , a matrix of nanoindentations located at different distances to the grain boundaries were carried out near eight grain boundaries in high purity polycrystalline aluminum (99.999%). The acquired load - displacement curve of each grain boundary indent w as converted into stress - strain curv e using the procedures outlined by Kalidindi et al . [ 30 - 33] . The yield stress of the grain boundary was, then, quantified for each indentation. Figure 1. 6 shows the yielding stresses of two grain boundaries, where in the left case the misorientation acro ss the grain boundary is 12.1 degrees and in the right case the misorientation reaches 42.7. It can be observed that the grain boundary on the left side does not affect the yielding stress of all nanoindentations , as the yielding stress remains the same i n both grain interior and near grain boundary. This is due to the low misorientation angle across the left grain boundary, which leads to a negligible grain boundary resistance to slip transfer. Nonetheless for the case on the right side, the yielding st ress increase s significantly when the indent location approach es the grain boundary. The large difference in yielding stress between the grain interior and grain boundary vicinity was caused by the poor misalignment of slip systems on both sides , which is a result of the high misorientation angle between two grains. 19 Fi gure 1. 6 : (a) Crystallographic details of the two grain boundaries including location of the indents and the orientation of the four indented grains . (b) The yielding stress calculated for each nanoindentations. [ 27 ] 20 Although previous research has proved that the study of grain boundary resistance to slip transfer can be achieved using bi - crystal nanoindentations, there are a few disadvantages to this approach. The complicated stress and strain field in the deformed m aterial in a nanoindentation process is usually very difficult to quantify without powerful simulation tools. As a result, the determination of the active slip systems and their relative shear is impossible using bi - crystal nanoindentation method. 1.3.2 Q uantitative slip trace analysis Many studies ha ve taken a different route to investigate the role of grain boundary in plastic deformation using uniaxial tensile or four point bending test with polycrystalline samples . In th ese studies , a relativ ely small amount of uniaxial strain was applied to the sample , then the slip bands/traces that developed at grain boundaries on the surface of those polycrystalline samples can be captured using SEM in a post - mortem examination . With the crystalline orientation of each grain informed , the slip systems and their related Burgers vectors of each observed slip trace can be identified. The most obvious advantage of this methodology , compared to the bi - crystal nanoindentation, is the stress sta te near a grain boundary can usually be approximated with the global uniaxial tension stress state. This simple stress state facilitates the calculation of the resolved shear stress on each slip system, determining the relative magnitude of driving force on each slip system. Th is quantified knowledge of slip system s and the resolved shear stress is critical in understanding how heterogenous stress/strain evolves in the boundary vicinity , and can eventually enable one to build models that predict slip acti vities near all types of grain boundaries. 21 Highlights of the early studies of slip bands development and evolution near a grain boundary includes in the early work by R. E. Hook and J. P. Hirth [ 34 ] , who use d a bi - crystal of high purity Fe - 3%Si alloy to study the influence of grain boundar ies on small plastic deformation. T he sample was compressed along the grain boundary plane until significant amount of slip band were visible. T he slip system that is corre lated with each slip band formed near grain boundaries were determined using stereographic projection s . Figure 1.7 shows an example of the several types of slip bands developed in the grain boundary vicinity. Those slip bands were examined in terms of th e compatibility requirements imposed at the grain boundary plane. It was concluded that while the primary slip band is operated by the global uniaxial compression stress, the secondary slip systems are driven by the local stress state that is the result o f the elastic incompatibility at the grain boundary plane. Such activation of secondary slip systems is not normally expected in single crystal deformation. Similar studies regarding the slip bands development near a bi - crystal grain boundary were conduc ted by R. E. Hook and J. P. Hirth [ 35 ] and C. Rey and A. Zaoui [ 36 ]. All of these early researches offer insightful arguments on the slip transfer and strain compatibility near a grain boundary under plastic deformation. 22 Figure 1. 7 : An optical image of the primary {011} slip bands and secondary {101} slip bands in the grain boundary vicinity in a compressed bi - crystal sample. [ 34 ] 23 Recent research of slip transfer studied by slip trace analysis in Bieler et al . [ 37 ] and Alizadeh et al . [ 38 ] are based on a higher volume of slip transfer data compared to the early work from last century. This is mainly a result of the faster crystal orientation identification using EBSD and the higher resolution and magnification imaging of SEM. Biel er et al . [ 37 ] conduct ed slip trace analysis in an annealed polycrystalline Al with a near - cube microstructure deformed in uniaxial tension. One feature of the near - cube microstructure is that there are usually many active slip systems present at the grai n boundary due to the high Schmid factors. With 128 observations of either slip transfer or blockage by grain boundary, obvious slip transfer with correlated slip lines on the grain boundary was rarely observed, instead independent activation of slip was observed in each grain near the boundary. As a result, they concluded that within a near - cube microstructure the slip transfer is more difficult to activate than self - accommodation when there are plenty slip systems with high Schmid factors available in e ach grain. Both and the product of and global Schmid factor were used to evaluate the likelihood of slip transfer. The results in F ig. 1. 8a suggests that the slip transfer only occurs when the slip systems in the incoming and outgoing grains are highly aligned, e.g. > 0.97, which almost certain ly corresponds to a low angle grain boundary with misorientation less than 15º. It was also co ncluded in the study from F ig. 1. 8b that a product of Schmid factor and can be used as a threshold for slip transfer below which slip will be blocked by the grain boundary. This study revealed the importance of combining multiple slip transfer metric t o more accurately assess the likelihood of slip transfer. However, a near - cube texture is to some extent unique that multiple slip systems in each grain will be active due to the high Schmid factor. As a result, Alizadeh et al . [ 38 ] carried out similar e xperiments in both a near - cube and rotate - cube textures. 24 Figure 1. 8 : (a) A vs grain boundary misorientation plot showing every of slip transfer and slip blockage events as a data point, where dots and crosses means slip transfer and slip blockage, respectively. (b) times Schmid factor vs grain boundary misorientation plot sh owing the same data set . [ 37 ] 25 In R. Alizadeh et al . [38] , the slip transfer in high purity polycrystalline a l uminum with near - cube and rotated - cube microstructure (examples of the two microstructures are shown in F ig. 1. 9 ) w ere studied near ~ 250 grain boundaries , including both successful slip transfer and blockage of slip by grain boundary. The slip systems involved in each slip transfer were identified using the slip trace analysis. Due to the high symmetry of Al, the observ ed slip trace can always be correlated to multiple slip systems. Under such condition, the one that has the highest Schmid factor was selected to be the active slip system. Three metrics were used to assess the likelihood of slip transfer at the grain bo undary, the first two being the and the residual Burgers vector left in the grain boundary ( where the and is the Burgers vector of the active slip system in the incoming and outgoing grain, respectively ) and the third metric was defined by . By applying the three metrics to all 250 observations, they found out that the slip transfer tends to occur when is greater than 0.9 and is smaller than 0.35 b ( b is the unit B urgers vector in aluminum) . This study suggests that the residual Burgers vector plays a significant role in the slip transfer in aluminum. Furthermore, the third metric seems to be the most useful, where slip transfer usually occurs when the values of exceed a th reshold. This conclusion on can be generalized from F ig. 1. 10 , where the two groups of points indicated by blue and brown (slip transfer and slip blockage) can be separated by a threshold of , where above the threshold the data points are m ostly consisted of blue. 26 Figure 1. 9 : Crystal orientation map of the two aluminum samples with dramatic different microstructures, where the left being near - cube and right being rotated cube microstructure. [ 38 ] 27 Figure 1. 10 : A vs grain boundary misorientation plot showing every of slip transfer and slip blockage events as a data point, where blue and brown means slip transfer and slip blockage, respectively. [ 38 ] 28 Another significant contribution to the study of slip transfer by characterizing surface slip trace is made by Guo et al . [ 39 ] , where the slip band - grain boundary interaction was studied in commercially pure titanium using trace analysis and high resolutio n EBSD (HR - EBSD) . The focus of the study was not limited to the slip transfer event, but the stress field development near a slip band blockage by grain boundary was also characterized using HR - EBSD. To predict the possibility of a full slip transfer, th ey proposed a metric M b that incorporates the geometric alignment and the residual Burgers vector and is defined by: Eq. 1.8 where the n and b are the slip plane normal s and Burgers vector s , respectively. Using the metric M b , they concluded that direct slip transfer usually requires good alignment of slip systems, e.g. M b > 0.7, with high resolved shear stress on both the incoming and outgoing slip systems. Between the geometric alignment and resolved shear stress, the study claims that the former play s a more critical role. Poor geometric alignment, e.g. M b < 0.7, leads to blocked slip bands. Stress concentration characterized by HR - EBSD was observed only in some of the blocked slip bands vicinity . At th e grain boundaries that were blocked , the slip band did not cause obvious stress concentration and GND densities were found to increase in a diffuse distribution along the grain boundary. As a summary of the study, they attempted to corelate both slip transfer and slip ban d blockage with Schmid factor and slip transfer metric M b in F ig. 1. 11. All observations were classified into three groups (zone1, 2, 3) , which corresponds to slip transfer, blockage with stress concentration, and blockage without stress concentration. S uch plot s can be used for predicting the type of slip transfer with the knowledge of Schmid factor and metric M b . Nonetheless, the groups in F ig. 1. 11 w ere not clearly divided , with almost 20% outliers and a significant amount of data points located in th e vicinity of the division line s. A s a result, more 29 studies are necessary to better quantify the differences of the conditions, under which the slip transfer or the slip band blockage would occur. In addition to the studies outlined above , there are also plenty of studies that concern the slip transfer and heterogenous deformation near grain boundaries. L. Wan g et al. [ 40 ] examined the slip stimulated twinning in Ti using crystal orientation informed slip trace analysis and concluded that this type of slip transfer was possible at greater than 0.9. J. R. Seal et al . [ 4 1 ] studied the slip transfer across the / interface in a titanium alloy Ti - 5Al - 2.5Sn (wt.%) using slip trace analysis. It was discovered that the high Schmid factor in both incoming and outgoing grains usually leads to slip transfer alignment, while the alignment of Burgers vector in the and phase was not well correlated with slip transfer. T. R. Bieler et al . [ 4 2 ] studied the heterogenous deformation near grain boundaries in tantalum polycrystal. was shown to be a reasonable metric to assess the likelihood of slip transfer in this case. 30 Figure 1. 11 : Plot of new slip transfer metric M b as a function of Schmid factor. The three zones (1,2,3) indicate the direct slip transfer, slip band blockage with stress concentration, and blockage without stress concentration, respectively. [ 39 ] 31 Some slip trace analysis reported in the literature was applied to obtain the slip plane and Burgers vector of the active slip system, Y. Yang et al . [ 43 , 44 ] developed a new methodology that incorporat s AFM measurement into the slip trace analysis to quantify the shear involved in each deformation (slip or twin) band. In their study, slip/twin bands in a deformed commercially pure titanium w ere investigated using AFM, SEM, and EBSD. The slip plane s and Burgers vector s w ere identified using slip trace analysis, while the average shear of each slip band is calculated based on the equation: Eq. 1.9 where b and n stand for the Burgers vector and slip plane normal of the slip/twin band, h is the height of the slip/twin band measured by AFM (shown in F ig. 1. 12), is the sample surface normal, and is the estimated width of each slip/twin band. A detailed illustration of how the se quantities in the equation is related to AFM measurement is presented in F ig. 1. 12. With the shear quantified for each slip/twin band, the shear distribution both in the grain interior and near grain boundary were determined and used for comparison with the results from a CPFE simulation s . In their conclusions, it was stated that the magnitude of the measured shear agrees reasonab ly well with the simulation, while the spatial distribution of the shear is sometimes different. Although the study did not involve any analysis of the correlation between slip transfer and measured shear in each slip band, it provides a useful and conven ient way to quantify the shear of slip band. This methodology proved to be very useful in the present study. 32 Figure 1. 12 : Schematic showing how the quantities in equation Eq. 1.9 are correlated with an AFM measurement. [ 4 4 ] 33 In addition to the application of s lip trace analysis to slip band - grain boundary interaction, twin - grain boundary interaction can also be studied in similar approach, some research [ 45 - 49 ] focused on the study of the interaction between impinging deformation twins and - grain boundaries in a - TiAl alloy. In their studies, observations suggest that micro - cracks formed at a grain boundary are related to the impinging twins, an example is shown in F ig. 1. 13. With the slip plane and Burgers vector identified by slip trace analysis, a grain boundary fracture metric, that was used for predicting micro - crack formation at grain boundaries was proposed: where the is the Schmid factor for a specific deformation twinning system, and are the unit Burgers vector of the impinging twin and ordinary dislocation systems in each grain, respectively. is the unit normal of the habit plane of the twins. Further statistic al analysis of the magnitudes of at each grain boundaries sugges ted that mico - cracks tend to form at grain boundaries with larger magnitude of . L. Wang et al . [ 40 , 50 ] conducted related studies regarding the interaction s between twi ns and grain boundar ies in - Ti . T hey observed coincident twins at multiple grain boundaries that developed from the same boundary into each grain. With the analysis of slip transfer geometry metric , Schmid factor, and c - axis misorientation between two adjacent grains, they listed thre e conditions that are necessary for the formation of coincident twinning at a grain boundary: (1) the two grains have a pair of well - aligned twinning systems ( > 0.8) and at least one the system has a high Schmid factor . (2) The c - axis misorientation be tween the two grains should be less than ~25º . (3) The two adjacent grains should be relatively large in size. 34 Figure 1. 13 : A Backscattered image showing the impinging twin initiated a micro - crack at a grain boundary. [ 45 ] 35 1.3. 3 Other experimental technique Besides the bi - crystal nanoindentation and slip trace analys es reviewed above , there are also other alternative methodolog ies that have been used in the research of the interplay between dislocation s and grain boundary . A f ew examples are the study of individual dislocations in a in/ex - situ tensile test using TEM , analyzing the slip traces on a compressed mico - pillar that contains a grain boundary (a bi - crystalline micro - pillar), and the estimation of the heterogenous strain/stress field s near grain boundary in a deformed sample using high resolution EBSD . All of these techniques have provided insightful knowledge into the details of individual dislocation interaction with grain boundary on the nano - scale. J. Kacher et al . intensively studied slip transfer in 304 stainless steel [ 51 - 55] and titanium [ 5 6 , 5 7 ] using both in - situ and ex - situ straining in the TEM. By identifying the Burgers vector and the slip plane of individual dislocation that has participated in the slip transfer using TEM, the residual Burgers vector in the grain boundary and the resolved shear stress (RSS) on the potential outgoing slip systems was estimated using analysis and the equation respectively , for each slip transfer event ( t he b and n is the Burgers vector and the slip plane normal respectively, and the is the approximate d local stress state at the slip transfer location). It was concluded that in both materials the slip transmission across a grain boundary was governed mainly by the reduction of the residual Burgers vector , while the magnitude of the resolved shear stress on the outgoing slip system plays a minor role. In addition to the studies of slip transfer, a novel approach for visualization of grain boundary/dislocation interactions was proposed using tomographic rec onstruction. More specifically, a tomograph was constructed using a series of TEM images captured by continuously adjusting the tilt angles. An example of the construction of a tomograph from TEM images is shown in Fig. 1.14, where TEM images 36 acquired at tilt angles ranging from - 40 ° to 40 ° at 2 ° intervals. In general, t h is reconstructed tomograph might be helpful in correlat ing the spatial correlations between the incoming and outgoing bands of dislocations during the slip transfer process . 37 Figur e. 1.14: a) The four TEM images acquired at different tilt angles (marked at the top left corner) that were used to construct the tomograph shown in (b). b) The 3 - D tomograph revealing the relative locations of each dislocations observed in the TEM image series in (a). [5 1 ] 38 Using in - situ TEM, grain boundary/dislocation interaction s ha ve also been investigated by several other groups in various materials [ 58 - 65] . In general, the unique advantage of in - situ TEM is its capability to identify the Burgers vector and the slip plane of individual dislocation, which allows analysis of interaction between a single(few) dislocation(s) and a grain boundary. Nonetheless, such analysis of slip transfer that is based o n a rather small number of dislocations and may not be universally applicable, as the governing rule s of slip transfer may be different when a large number of dislocations transmitted across the grain boundary. C ompression test of micro - pillar s /micro - cantilever s is another approach to deform nano - scale samples. There are quite a few elegant experimental works using this approach to probe the interaction s between grain boundar ies and dislocation s [66 - 81] . Nonetheless, due to the difficulty of determining the stress state near the grain boundary in a deformed micro - pillar and the strong influence of the size effect of the sample on the material mechanical behavior, the conclusions of those stud ies on the slip transfer are generally quantitative and sometimes not appl icable to larger/bulk specimens and polycrystals . High resolution EBSD (HR - EBSD) is a relatively new methodology for estimating the strain/stress fields at a sample surface using the Kikuchi patterns from a n EBSD scan. Several studies have exploited th e advantage of knowing the full strain/stress fields and used them to analyze the heterogenous strain fields in the vicinity of the grain boundary [ 82 - 87] . It was , in general, found that grain boundaries with high (> 0.9) did not develop any stress concentration s , while those boundaries with (< 0.9) developed much higher magnitude s of strain/stress in the boundary vicinity relative to the grain interior s . 39 1.3. 4 Coupling experimental results with CPFE simulations A general trend in the study of metal s and alloy s is the increasing used of modeling techniques , such as crystal plasticity finite element modeling (CPFEM), dislocation dynamics simulation s (DD) , and atomistic simulation s . T hese simulation methods are advantageous to most experimental approach es, due to their capability of acquiring physical quantities that are almost impossible to measure directly in the experiment. In the study of disloca tion/grain boundary interactions , simulation tools can be very useful as they allow the assessment of the reaction s between dislocation s and a boundary at the atomistic level (atomistic simulations) [ 88 - 97] , the force on any individual dislocation using DD simulations [ 98 - 105] , and the stress/shear of any slip system using CPFEM [ 106 - 117] . With the help of simulation tools, one can coupl e experimental observations with properly built models to further understand the fundamental physics that governs the experiment al observations . Such knowledge , in return, can help impro ve and develop new experimental and simulation techniques. Latest developments of coupling the exper iment and simulation have been summarized in a number of reviews [ 117 - 119 ]. As only CPFEM is used in the present study of slip transfer, the review in this chapter will focus on the research that utilize d CPFEM to study the heterogenous deformation near g rain boundary. D. Kumar et al . [ 49 ] modeled the heterogenous deformation in a duplex near - TiAl alloy near a grain boundary where micro - crack nucleation was observed experimentally. The cause of the micro - crack nucleation was experimentally determined to be the result of interactions of deformation twinning with grain boundary. In order to un derstand how heterogenous stress and strain developed near the site of crack nucleation, CPFEM was used to model the deformation of the two adjacent grains that shared the cracked grain boundary . The microstructure of the 40 simulated region is shown in F ig . 1. 1 5 a where the white circle highlights where the micro - crack was observed in the experiment . Further analysis of the simulated shear is presented in F ig . 1. 1 5 (b - d), where the distribution of the shear on the two active twin systems in grain 14 is visual ized and plotted as a function of time. The evolution of the shear in F ig . 1. 1 5 d suggests that the major twin system near the micro - crack site switched from the ( - 111)[ - 11 - 2] system to the (1 - 11)[1 - 1 - 2] system. W hile the shear on the former system (primary system) was dominant in the early stage s of the deformation due to the high global Schmid factor (~0.5) , the shear on the latter system (secondary system) with a lower global Schmid factor accelerated , presumabl y caused by the need to maintain grain boundary continuity. As a result, the secondary system, though activated later, contributed more strain to the formation the micro - cracks. This study suggests that the CPFEM can significantly increase the ability to understand the evolution of shear on individual slip system and can potentially be a tool for predicting potential sites for micro - crack nucleation. 41 Figure 1. 1 5 : a - c): CPFEM modeling of the Von Mises stress (a) and shear distribution (b,c) in the area of interest. d): Evolution of the simulated shear on two slip systems as a function of time . [ 49 ] 42 A more elaborate CPFE model was used by C. Zhang et al . [ 120 ] to study the heterogenous deformation of commercially pure titanium in grain interior s and near grain boundaries. A 3 - D CPFEM model was built by using the surface microstructure characterized using EBSD and the sub - surface grain boundary inclination identified by the non - destructive Differentia l Aperture X - ray Microscopy (DAXM). For comparison, another simpler quasi 3 - D model was built using only surface grain orientation. Both models are shown in Fig. 1.1 6 , where the left model contains only columnar grains without any grain boundary morpholo gy under the surface, while the right model contains knowledge of the inclination of some of the sub - surface grain boundaries. By assessing the magnitude of the simulated stress/strain tensor in both models and compar ing them with the experimentally estim ated local strain, they found out that the 3 - D model more accurately captured the values of the local stress tensor than the quasi - 3D model. In addition, the comparison revealed that in order to accurately simulate the heterogenous deformation in the grai n interior s and near grain boundar ies , it is more important to have a n accurate geometrically realistic grain morphology than it is to fine - tun e the crystal plasticity constitutive parameters. This study indicates that the necessity of including grain boundary inclination as part of the study of heterogenous deformation near grain boundary. 43 Figure. 1.1 6 : Left image: Quasi - 3D CPFE model shows the columnar microstructure of all the grains. Right image: 3D model with grain structure information obtained from DAXM. [ 120 ] 44 Characterization of the heterogenous deformation near a grain bo undary can also be achieved through assessment of the general stress (such as Von Mises stress) and/or strain ( second Piola - Kirchhoff stress ). For example, CPFEM has been applied to the study of heterogenous strain field development and localization near grain boundaries at the macro - scale in pure Ni by Guan et al . [ 121 ]. Such CPFEM assisted in the identification of locations of stress concentrations near grain boundaries. Those predicted locations prove d to be in the vicinity of the cracks formed in the experiment. Similar study of the investigation of strain localizatio n using CPFEM can also be found in C. C. Tasan et al . [ 122 ] and W. Z. Abuzaid et al. [ 85 ]. In general, despite of numerous studies using CPFEM to characterize heterogenous deformation in general , t he application of CPFEM to the study of heterogenous deformation near individual grain boundar ies is much less common. Nevertheless, s uch site - specific stud ies can be very beneficial , as they can focus on specific type s of slip transfer (slip transfer at a g rain boundary with high can be significant ly different than with low ), resulting in more detailed knowledge . Therefore, it is sensible to conduct studies of the slip transfer behavior near individual grain boundaries, and to simulate the deformation process using CPFEM with the goal of learning the specific mechanics of slip transfer at various grain boundaries. 1.4 Motivations of this study In the light of previous studies, two major concerns regarding how to quanti fy the various interactions between slip systems and different types of grain boundaries still remain : A lack of reliable and comprehensive methodology for studying slip transfer with both experimental and CPFEM simulation components, and a model that quantit atively and accurately describe the interaction of multiple slip systems with various grain boundaries. This body of work seeks to 45 solve these two problems by analyzing slip transfer observations and comparing them with CPFEM simulations to obtain how ind ividual slip systems evolves , which will eventually lead to the development of a new model that can be applied to predicting slip transfer at all types of grain boundaries. P revious work that tried to combine the experimental observations with CPFEM simulations to study the effect of the grain boundary on the development of the heterogenous stress/strain is quite rare [ 120 - 122] . All of them investigated the deformed grain boundaries of a polycrystal in a uniaxial tensile post - mortem test and conducted corresponding simulat ion in the area of interest with CPFEM. The nature of the post - mortem experiment s and the complicated stress/strain development of polycrystalline material means that we are blind to the actual history of the evolution of slip in grain boundary vicinity. In other words, knowledge of the initial formation/nucleation of the slip band at the grain boundary and the direction of the slip transfer across the grain boundary are impossible to acquire. To overcome th ese disadvantages, precisely located nanoindentation near grain boundaries was conducted . Under such simple bi - crystal nanoindentation environment, it is obvious when and where the slip transfer will occur. In addition, the direction of the slip transfer can also be controlled by placing the nanoindent ation on either side of the grain boundaries. Attempts to theorize the nature of slip transfer across grain boundaries are also not common. Most of the work s are limited to the assumption of only one outgoing slip system is triggered by one incoming slip system. Such an assumption can significantly reduce the complexity of the problem, as the activation of a secondary and/or tertiary slip system increase s the potential combination of outgoing slip systems dramatically. Traditional t angential continuity theory [ 5 ] [12] , being the only model that can be used for quantifying multiple slip 46 system activations near grain boundary . But is also not accurate enough to predict the slip transfer that involves more than one slip system (testing of the traditional tangential continuity model will be conducted in Chapter 4). Improvement of the tangential continuity model and its implementatio n into crystal plasticity finite element framework is still under progress and not ready for experimental validation [ 12 ]. As a result, it is desirable to invent a new model that can be used in cases when multiple slip systems are active near grain bounda ry. Validation of the model using experimental observations (in our case slip trace analysis) is also critical. By solving the two problems outlined above, our knowledge of how strain/slip interacts with grain boundar ies will be expanded both expe rimentally and theoretically, meaning that we will be one step closer to build ing a predictive model that can capture both heterogenous deformation evolution and damage nucleation at grain boundary vicinities. 1.5 Overview of this thesis An outlin e is presented here to help reader navigate this dissertation. The m ain objective of the current work is to quantitatively study the slip transfer near grain boundaries using bi - crystal nanoindentations and slip trace analysis. Simulations using CPFEM ar e included to obtain additional insights of the deformation mechanisms of individual slip system. An iterative stress relief model (ISR M ) is proposed to predict multiple slip system accommodation near grain boundary and it is validated by the comparison w ith experiments. Chapter 2 covers the material s that were used in the study , as well as the procedure of sample preparation, the experimental details of the study, and the basic concept of the CPFEM used in the study. 47 In C hapter 3 , t he inf luence of grain boundaries on plastic deformation was studied by conducting nanoindentation near grain boundaries. Surface topographies of indentations near grain boundaries were characterized using atomic force microscopy (AFM) and compared to corresponding single crystal indent topographies collected from indentations in grain interiors. Comparison of the single crys tal indents to indents adjacent to low - angle boundaries shows that these boundaries have limited effect on the size and shape of the indent topography. Higher angle boundaries result in a decrease in the pile - up topography observed in the receiving grain, and in some cases increases in the topographic height in the indented grain, indicating deformation transfer across these boundaries is more difficult. A crystal plasticity finite element (CPFE) model of the indentation geometry was built to simulate both the single crystal and the near grain boundary indentation (bi - crystal indentation) deformation process. The accuracy of the model is evaluated by comparing the point - wise volumetric differences between simulated and experimentally measured topographies. T he discrepancies between experimental and simulation results will be discussed in terms of reverse plasticity, dislocation nucleation versus glide in the model , and in the physics of the slip transfer process. To remedy the discrepancies between experimen tal and simulation results, a revision of the original finite element model, which inserted a layer of grain boundary elements between two adjacent grains, was proposed. The new model and its associated modeling parameters are then briefly validated usi ng experimental observations. In Chapter 4, a new model (ISR) is proposed to quantitatively describe the accommodation/outgoing of slip/shear by multiple deformation systems at a grain boundary. The model uses an iterative approach to sequentially determine the accommodating/outgoing slip systems and their relative shear. The outcome of this iterative stress relief model is mainly 48 controlled by the continuity of Burgers vector in the grain boundary and the evolution of the impinging stress tensor at the grain boundary. The model was tested by comparing predictions with observations of shear accommodation in - Ti quantified using orientation informed slip trace analysis and quantitative atomic force microscopy. Similar comparisons were conducted b etween tangential continuity model predictions and the experimental observations. Critical resolved shear stress ratios used in this iterative stress relief model were optimized by maximizing the accuracy of the model predictions. Chapter 5 draws the conclusion of this dissertation and project s the possible ways of continuing the current study. 49 CHAPTER 2 MATERIALS, EXPERIMENTAL DETAILS, AND THE CONSTITUTIVE LAW OF CPFEM 2.1 Description of material The grade 2 commercially pure - i used in this study was acquired from two different sources, and will be designated as Material A and Material B. T he stud ies presented in Chapter 3 and 4 were conducted with Material A and Material B, respectively. 2.1.1 Material A Material A was provided by the Max - Plank - Institut für Eisenforschung, Düsseldorf, Germany (MPIE). This grade 2 commercially pure - i was determined to contain 0.17 wt% oxygen and 0.015 w t% carbon using infrared spectroscopy of combustion products. The material ha d an average grain size of ~130 m, and was also textured in such a way that the majority of the c - axis of the grains were parallel to the material surface , as shown in the EBSD map in Fig 2.1 . S ample A was cut from Material A using electron discharge machining into a size of approximately 5x5x5 mm 3 and used later for nanoindentation test s . 2.1.2 Material B Material B was provided by Chris Cowen . The material was textured and ha d an average grain size of 80 m as shown in Fi g. 2.2 . The sample B was cut from Material B with electron discharge machining into a size of approximately 25 x3x 2.5 mm 3 for four point bending test. 50 Figure 2.1 : An EBSD inverse pole figure (IPF) map of the area of interest in material A is presented . The texture of the material A shows a moderately strong texture with the c - axes lying predominantly in the plane of the image [4 0 ] . The array of black dots in the IPF map is a grid of nanoindents to mark the area of interest. 51 Figure 2.2 : An EBSD inverse pole figure map of a random area in material B is presented . The texture of the material B is almost random . 52 2.2 Sample preparation 2.2.1 Sample A To mitigate the difficulty in handling the small size of the sample A, it was metallographically mounted in Konductomet ® (Buehler, Lake Bluff, IL, USA). The sample was then mechanically gr ound with 400, 600, 1200, 2500 , 4000 grit silicon carbide (SiC) paper to achieve a mirror finish with minimal amount of scratch es , and then chemical - mechanical polished with a solution consist ing of 80 vol% colloidal silica and 20vol% hydrogen peroxide. The sample was then etched with HF. To minimize the residual dislocations caused by the grinding, the final polishing and etching were repeated three times. 2.2.2 Sample B The sample was mechanically ground from 400 to 4000 grit and polished in a solution consisted of 80 vol% colloidal silica and 20 vol% hydrogen peroxide. To further minimize the influence of the residual dislocations caused by grinding, the sample was electro - polished in a 59% methanol, 35% isopropanol, and 6% perchloric acid (in volume) solution at 38.5 vol ts and - 30° C for about 2 minutes until the channeling pattern s of the sample in the SEM was sharp and crisp . 2.3 Experimental details 2.3.1 Scanning electron microscopy 2.3.1.1 Electron imaging The two samples, A and B, were imaged using two different SEMs. All electron imaging of sample A was conducted on a high resolution JEOL 6500F (Tokyo, Japan) SEM located in the 53 D epartment of M icrostructure P hysics and M etal F orming, MPIE. All secondary elec tron (SE) images and backscattered electron (BSE) images were taken at an accelerating voltage of 15 kv and with 1024 x 768 pixel digital resolution. Sample B was imaged using a Tescan Mira III SEM located in D epartment of C hemical E ngineering and M aterial S cience, Michigan State University. All SE images were acquired using a conventional Everhart - Thornley detector at an accelerating voltage of 20 kv, and BSE images were acquired using a retractable 4 - quadrant Si diode detector at 20 kv. All imag es are captured with 1024x1024 pixel digital resolution. 2.3.1.2 EBSD /OIM scan Electron BackScattered Diffraction (EBSD) is a SEM - based technique used to determine crystallographic orientation, phase, and defect densities. Figure 2.3 a shows the s ample configuration in an SEM to capture the EBSD pattern ( F ig. 2.3 b ). In this study, t he crystallographic orientation of the grains was obtained using the either the high resolution JEOL 6500F SEM with EDAX - TSL orientation imaging system or the Te scan Mira III SEM also with an EDAX - TSL orientation imaging system. All EBSD scans was conducted at an accelerating voltage of 20 kv and a working distance between 15 - 25 mm. The EBSD scan was post - processed using OIM Analysis (TSL, Draper, UT, USA). 54 Figure 2.3 : A schematic showing the setup of EBSD inside of an SEM ( left image) and a typical diffraction pattern (Kikuchi pattern in right image ) acquired in the EBSD scan that can be used for determining the crystalline orientation of the diffracted region. ( Revised based on the image on https://www.mpie.de/3077954/EBSD ) 55 2.3.2 Focused Ion Beam (FIB) cross sectioning F ocused I o n B eam (FIB) is a nano - scale technique that allows one to carve/modify the topographies on a sample surface, which is achieved by shooting a focused beam consist ing of gallium ions to sputter away the unwanted material. In this work, FIB was used t o expose cross sections of material at grain boundary with the purpose of acquiring grain boundary inclination below sample surface . All FIB cross - sectioning was carried out using a Carl Zeiss (Oberkochen, Germany) Auriga Dual Column focused ion beam scan ning electron microscope (FIB - SEM). In addition, the FIB cross - sectioning was conducted in several steps with varying beam current s , from 4nA to 1nA , and end ing with 600 pA, to generate a smooth cross - section surface that reveal the location of the grain boundary. Finishing the milling with a low beam current also minimizes the beam damage to the cross - section surface, which may obscure the imaging of the grain boundary on the cross - section. An example of the FIB cross - sectioned grain boundary is shown in Fig. 2.4, where the grain boundary inclination in the image was measured and labeled as . The accurate grain boundary correct was acquired by correcting the using equation: Eq 2.1 where the is the angle between direction of the FIB cut and the grain boundary normal and the 54° repre sents the tilt angle of the sample in the SEM . 56 Fig ure . 2.4 : Backscattered electron image showing a FIB cross - section cut used to determine the grain boundary inclination below the surface, where is inclination angle measured from image and is used for the calculation of real grain boundary inclination using Eq. 2.1 . 57 2.3.3 Atomic Force microscopy (AFM) AFM is commonly used for measuring the surface topographies of the object at nano - scale. The AFM measurement s in this work w ere carried out using a VEECO Dimension 3100 (VEECO Instruments Inc. Plainview, NY, USA). The precision of this equipment on the Z axis (height) is around 0.1 nm, while being ~1nm on the two lateral axis. The AFM probes used were TESP etched silicon probes (Bruker, Billerica, Massachusetts, USA). All AFM scans were conducted in tapping mode. Under this mode, the surface damage from the AFM tip was minimized while the surface profiling still maintains highly accurate. The resolution of the AFM scan s was chosen between 256x256 or 512x512 , depending on the size of the scan area. Post - processing of the AFM scan was finished with the Gwyddi on software package 2 . 2.3. 4 Nanoindentation Nanoindentations were conducted only on sample A using a Hysitron TriboScope 900 (Minneapolis, MN, USA) nanoindenter equipped with a 1.4 m sphero - conical diamond tip indenter. All indentations were pe rformed in load - controlled mode. A maximum load of 6 mN was used with a load - hold - unload profile of 5 seconds for each segment. An array of indents (31 by 31) was placed at the area of interest on sample A. The indents in the array were spaced by 20 m v ertically and 30 m horizontally. An SEM image of the indentation array was shown in F ig . 2. 5 . Another round of nano indentations were placed within the array of the indents while at varying distances from 9 selected grain boundaries of interest. An exam ple of the indentations in the grain boundary vicinity is shown in F i g . 2. 6 . The surface topographies of selected nanoindents were measured using AFM. 2 Gwyddion is a free software for visualization and analysis of scanning probe microscopy such as AFM. 58 Figure 2. 5 : An SEM image of the region that contains the grid of nanoindentations (shown as black dot s) in material A. 59 Figure 2. 6 : An SEM image of bi - crystal nanoindentations near a selected grain boundary with varying distances from the boundary in material A. The investigated grain boundary is colored in red dash line. 60 2.3.5 Four - point bending Four - point bending was selected to be the method to deform Sample B, because it generate s maximum tensile stress states on the sample surface between the two inner pins while eliminating the triaxial stress states present in the uniaxial tensile tests. In addition, the plastic strain of on the sample surface between the two inner pins are roughly uniform according to the finite element simulation of four - p oint bending J. Seal [ 123 ]. The bending stage used in this work is shown in F ig. 2. 7 . The two outer and inner pins of the stage are 2 0 mm and 5 mm apart, respectively. Sample B was carefully strained to ~1.5%. The low magnitude of strain maint ains a flat sample surface. In addition, the low density of slip traces under low strain limited the occurrence of cross - slip which is not the focus of this study. Characterization of the sample surface was concentrated only in the regions between two in ner pins. SEM imaging and EBSD scans of the area of interest was conducted both before and after deformation. Surface profile s of the slip traces at selected grain boundaries was charactered using AFM. 61 Figure 2. 7 : The picture of the four point s tage with the sample. 62 2. 4 Slip trace analysis Slip trace analysis used an algorithm that allows the determination of slip plane, and in certain cases even Burgers vector associated with the slip band observed in experiment. An example slip trace analysis is shown in F ig. 2. 8 a, where the major slip traces can be observed sharing a common direction that goes from the lower left to upper right. To identify the slip system associated with these slip lines, the orientation of the unit cell is determined (the the known orientation of the unit cell, di rections of the slip traces formed by all 24 potential slip systems are determined. By matching the observed slip trace with each of the 24 calculated slip trace s , one can decide which slip system is the active one. In the case shown in F ig 2. 8 a, the sli p lines of the prismatic < a > slip system (the grey slip plane with the white Burgers vector on the unit cell) agrees perfectly with AFM measured slip lines. In some special cases where more than one slip systems can produce slip traces that agree well with the observations, the inclination of the slip plane was used as a secondary criterion to determine which slip system is active. For example, in F ig. 2. 8 b and c, two slip lines from both slip planes (shaded in grey) agree well with the observati on. Nonetheless, only the upper slip plane shares the same inclination as the AFM measured slip band. In addition to resolving the active slip plane, the active Burgers vector can be determined if the slip occurs on prismatic and pyramidal plane w here there is only one available Burgers vector. In the case of an active basal slip plane where there are three potential Burgers vector, the global Schmid factor needs to be calculated for each of the three slip systems. The Burgers vector with the hig hest global Schmid factor is chosen as the active slip system. 63 Figure 2. 8 : The AFM measured surface profile of a slip trace ( blue suggests lower heights ) overlaid with a unit cell of the corresponding grain orientation shown in top view (a) and side vi ew (b , c ). The assumed active slip plane in the slip trace analysis is color ed in grey in the unit cell. 64 2. 5 Constitutive law in Crystal Plasticity Finite Element Method (CPFEM) The CPFEM formulation used in the present study is based on previous work of Kalidindi et al . [ 124 ]. M odel s were built to simulate both the loading and unloading segment of the nanoindentations. All simulations in this work adopted a finite strain framework of elastoplasticity with a multiplicative decomposition of deformation gradient F into an elastic (including rigid - body rotation ) and plastic part: Eq. 2.2 The second Piola - Kirchhoff stress S , acting in the intermediate configuration, is the prod uct of and fourth - order stiffness tensor as: Eq . 2.3 The evolution of the plastic defor mation gradient is connected to the plastic velocity gradient through: Eq. 2.4 Plastic deformation is restricted to shear on defined slip systems, which means the plastic velocity gradient can be assembled from shear rates on every slip system by: Eq . 2.5 The shear rate on slip system is expressed in terms of resolved shear stress, , and the critical resolved shear stress, : Eq . 2.6 where and m are material parameters. The evolve with shear on other slip systems according to: Eq . 2.7 65 The hardening matrix, , is composed of the self - hardening, , and the cross - hardening matrix, . The self - harde ning, , is calculated by: Eq. 2.8 where and a are, respectively, the initial hardening slope and hardening ex ponent, which affects the shape of the self - hardening curve. The diagonal elements of cross - hardening matrix, , are all assigned to unity, while off - diagonal elements are taken as 1.4. 66 CHAPTER 3 STUDY OF SLIP TRANSFER USING NANOINDEN TATION AND CRYSTAL PLASTICITY MODELING In this chapter, a novel approach to studying the interaction between slip and grain boundar ies using bi - crystal nanoindentation is outlined . Previous research has tried to approach this problem by quantifying the differences in the nano - hardness, load - displacement curves between nanoindents in grain interior s and near the grain boundary. In this study, the influence of grain boundary on the formation of surface top ographies of nanoindentations 3 in the boundary vicinity compared to those in grain interior was observed . Such influence is found to be strongly dependent on the character of the grain boundary. To understand how a grain boundary alters the development of nanoindent surface topographies, AFM measurements of surface topographies of indents both near and away from grain boundary were compared, the differences were quantified using a new metric : the volume of each indent surface topograph y . In addition, th e differences between single and bi - crystal indention were related to slip transfer criterion such as and M ( LRB criterion) . To further obtain the stress, strain, and shear of individual slip system under both single and bi - crystal indentations, CPFE m odels were built for the two conditions using the M atlab toolbox Stabix developed by D. Mercier et al. [ 125 ]. The model successfully captured the surface topographies of all single crystal and most of the bi - crystal nanoindentations. For some high angle grain boundaries, the accuracy of the model decreases notably. Discussion of the discrepancies between observations and simulations suggest that the cause might be several underlying assumptions of CPFEM. (This work has been published . [126] ) 3 Previous work on the formation of surface topographies in single crystal nanoindentations were carried out in pure copper, - TiAl, and commercially pure - 67 3.1 Overview of the locations of nanoindentations A sample with large grain size (around 130 m) was used for this study . A n overview map of the microstructure of the indented region is shown in Figure 2.4. Nanoindentations away from the grain bound ary 4 in this region were deemed as single crystal nanoindentations as the indent is not influenced by any grain boundary. In addition, indents in the vicinity of grain boundary 5 were treated as bi - crystal nanoindentations. Both bi - crystal and correspon ding single crystal nanoindentations were placed near 6 different grain boundaries and inside 12 grains that shared those 6 boundaries. Figure 3.1 gives an overview map of all the nanoindents studied in this chapter. As a result of the difficulty of cont rolling the distance between the location of the indent and the grain boundary, multiple nanoindents were present at each grain boundary with only a fraction of them ending up close to the boundary. Only those nanoindents that are influenced by the grain boundary were treated as bi - crystal nanoindents. 4 An indent loc ated beyond 10 m from the grain boundary is generally considered a single crystal indent. 5 An indent within 2 m distance from the grain boundary is considered a bi - crystal indent. 68 Figure 3.1 : SEM images of bi - crystal nanoindents near the six selected grain boundaries and the corresponding single crystal indents in the two adjacent grains. 69 3.2 Nanoindentation load - displaceme nt curves For single crystal indentation, the maximum indentation depth s ranged from 250 nm to 300 nm, reflecting material property, such as nano - hardness, variations in different grain orientations. Nevertheless, load - displacement curves for nano indents within the same grain are highly reproducible, with variations in both load and maximum displacements of less than 1%. The load - displacement curves for single cry s tal indents and the corresponding bi - crystal indents are sometimes notably different . A n example of th is comparison is shown in F ig. 3.2 , where the indentation load - displacement curves show slightly enhanced hardening for indents placed near grain boundaries. This is evident in the slightly larger slope of the bi - crystal indent under th e condition that the same load profile was used. The curves also indicate that forward creep occurs during holding at peak load. This is consistent with observations of stress relaxation and creep at ambient temperature in - Ti ( [127 - 131] ). In only four of the nine grain boundaries studied, displacement jumps of 5nm to 15 nm were observed, an example of which is shown in F ig. 3.2. For those bi - crystal nanoindents that did not cause displacement jumps on the load displacement curves, no verified answer can be provided. One possible explanation is the uncertainty of the distribution of defects on the grain boundary , which can lead to the random formation of displacement jumps in any bi - crystal nanoindentation. 70 Figure 3.2 : Load - displacement curve of a bi - crystal indent (shown as the grey dotted line) and the corresponding single crystal indent in the originating grain (shown as the black solid line). A magnified inset is used to show the displacement jump occurred in the bi - crystal indentation. 71 3.3 D ifferences of the topographies formed in single and bi - crystal nanoindentations 3.3.1 Quantifying the volume difference of two indent topographies As both the AFM measurement of indent topographies and the simula tion results are pixelated data , i t is essential to establish a point - to - point comparison to q uantif y the differences between indent topographies , so that the comparison s are carried out in a systematic manner. Thus, the volume difference, , between two indent ation topographies a and b within a grain interior , as shown in F ig . 3.3, is defined as the integra l of the absolute value of the height difference using equation: E q. 3.1 where and is the height at corresponding point s with the coordinate of either of a radial coordinate system or of a Cartesian coordinate system, over the relevant area of interest A . In addition, the (absolute) volume of an indent topography a is, similarly, defined as: E q. 3.2 There are two major reasons for u sing volume differences as a measure to compare indentation topographies. The volume of an indentation topographies is an indication of the amount of plastic strain the indentation has generated. As a result, it facilitate s our study of the interaction b etween plastic deformation and grain boundary. The other reason of using volume representation is due to its simplicity and intuitive for visualization. 72 3.3. 2 Reproducibility of nanoindentation topographies The subsequent interpretation of bi - crystal indentation topographies on or close to grain boundaries relies on comparisons to corresponding single crystal indent topographies. To ensure that such comparisons are meaningful, the reproducibility of the inde nt topography needs to be established. To assess this, the absolute height difference between corresponding points of two indent topographies a and b is summed using eq.3.1, as illustrated in fig. 3.3a and b ( T he AFM data of indent topogr aphies a and b in 3D are shown in appendix A. ) . The resulting determined by radial integration over the ring region with varying is plotted as a function of A shown in Fig. 3.3c, where is the approximate indentatio n impression radius. In Fig. 3.3c, two distinct slopes are found. Within the area influenced by the indentation plastic deformation zone that is about three times the indent diameter [81] reas beyond the influence of the indent, a low slope is observed that represents the random mismatch between the two topographies due to surface roughness (and AFM noise). The high slope in smaller areas can be attributed to actual differences in the two indentation topographies (a and b). Thus, a measure of reproducibility of indent topography can be defined by the difference between these two slopes that corresponds to an average height difference between two topographies in excess of unavoidable surfac e roughness and measurement noise. This methodology was applied to nominally identical indentation pairs in eight single crystal orientations and two bi - crystal situations to assess their individual reproducibility. Fig . 3.3 d shows a plot of the cumulativ e probability of surface roughness (dotted line ) and topography reproducibility (solid line ) for the bi - crystal and single crystal indents. For both single and bi - crystal indents the surface roughness is very similar and falls between 1 nm to 3 nm, with t wo 73 outliers of exceptionally low roughness around 0.2 nm. The reproducibility of single crystal indent topography (black solid line) ranges between about 1 nm to 2 nm and is slightly better than the reproducibility observed for the two bi - crystals (grey solid line) . Comparisons between the two pairs of bi - crystal indents are somewhat more problematic due to the inability to accurately control the distance of the indent from the grain boundaries, resulting in differences in the distance d the two cases. Nevertheless, the reproducibility of bi - crystal indent topography is only slightly worse than that of the single crystal topographies, at approximately 3 nm. Overall, the reproduci bility of 2 nm to 3 nm compares very favorably to the overall indent pile - up heights that range from approximately 130 nm to 180 nm, and thus establishes a base noise level for comparison of topographies of indentations carried out under different conditio ns. 74 Fig ure 3.3 : a,b): Example of two single crystal nanoindents from the same grain used for establishing the reproducibility and surface roughness of nominally identical nanoindentations, where an d are the inner and outer radius of a ring centered on the indent and isolating the major indent topography features. V (short for or , calculated based on Eq. 3.1 ) and ring area A . d): Cumulative probability of the surface roughness levels (dotted), established from the slope of the dotted line in (c), and reproducibility (solid) of single and bi - crystal indentation, as determined by the difference between slopes of the solid and dot ted lines in (c). A reproducibility better than 3 nm corresponds to negligibly small perceived differences, as demonstrated by shifting one part of the color bar by that amount (e). [126] 75 3.3. 3 Comparing single crystal indent topographies with the corre sponding bi - crystal indent topographies To effectively evaluate the influence of grain boundaries on the formation of indent topographies, indentations affected by grain boundaries were categorized into two groups: indents with part of the residual impression in both grains i.e., crossing the boundary, as shown in Fig. 3.4, and indents that were placed sufficiently close to a grain boundary such that only their topographies were reached and/or exceeded the grain boundary, as exemplified in Fig. 3.5. In both cases, indentations are expected to involve a number of processes , including slip transfer across the boundary, grain boundary deflection of dislocations due to differences in the elastic behavior of the two grains, and shifting of the boundary f rom its original location. Indents located on the grain boundaries. The surface topographies of the three indents that fall across grain boundaries, along with the corresponding single crystal indents from both sides of the grain boundary, are shown in Fi g. 3.4 . In these cases, indentation topographies have develop ed on both sides of the grain boundaries. The location of those indent topographies are for the most part consistent with the corresponding single crystal topography of the grain on the respect ive sides of the boundaries. Nevertheless, the sizes of these indent topographies are typically smaller than those of the corresponding single crystal indents. In general, slip transfer across the grain boundary is not a dominant feature for this categor y of indents. 76 Fig ure 3.4 : N anoindent topographies developed when the indents impression fall s across grain boundaries (center column), in comparison to corresponding single crystal indent topographies from both sides of the grain boundaries (second and fourth columns). The differences between the single and bi - crystal indents are mapped for the l eft grain and the right grain (first and last column). 77 Indents near grain boundaries . When indent s w ere located entirely in one grain with only the indent topography reaching the grain boundary, the indent topography was generally reproducible, bu t significant differences in indent topogr a phy were observed between different grain pairs or for different boundary orientations between the same pair (Fig. 3.5 ). This suggests that the nature of the boundaries directly influences the indent topographies , and allows the indent topography to be used to study how strain is accommodated at and transferred across different boundaries. Nine indent topographies close to six grain boundaries with different values 6 are compared in Fig. 3.5 . Two of t hese, with = 0.62, show indents on either side of the same grain boundary. Two other pairs, for grain boundaries with = 0.98 and 0.56, each show two indents taken from the same side of the boundary. Circular areas, which are consistent with the obse rved indent impressions, have been used to mask the subsurface portion of the indents in Fig. 3.5 , such that only the relevant indent topographies are assessed. Dark vertical lines mark the grain boundaries for bi - crystal indents while white lines indicat e the corresponding position for single crystal indents (Note that the white lines are only fiducial lines used to facilitate the comparison between the corresponding regions between bi - crystal and single crystal indents) . Bi - crystal indent topographies a re shown in the central column, with the corresponding values shown in the upper right corner of each of these figures. The corresponding single crystal indent topographies collected from the grains to the left and right of the grain boundaries are shown in the two columns immediately to the left and right (second and fourth). The differences between 6 In the present study, a more involved method of determining the was carried out through 3 steps: 1. Slip syst e ms of both the originating and the receiving grain were ranked according to the accumulated shear calculated in simulation; 2. The most active slip systems on each side and in the vicinity of the grain boundary were picked for the calculation of the ; 3. If two systems had similar accumulated slip near the grain boundary, the one that rendered a higher was chosen. 78 the topographies of the bi - crystal indents and corresponding single crystal indents are mapped in columns 1 and 5. Positive and negative height differences are colored red and blue, respectively, to indicate a bi - crystal topography is higher or lower than the corresponding single crystal topography. Under c onstant indentation load (6 mN), the resulting bi - crystal indentation impression depths ( h in Fig. 3.5 ) were consistently less than the corresponding single crystal indents, reflecting the deformation resistance associated with the grain boundaries. In re sponse to the presence of a grain boundary, the height of the resulting pile - ups in the originating grain may either increase or decrease. This is illustrated by column 1 in Fig. 3.5 , which shows that difference plots may be either red or blue in the orig inating grain (left of grain boundary). In many cases in column 1, the bi - crystal indentation leads to higher topography (red) very close to the indent, perhaps indicating more resistance to plasticity as a result of the grain boundaries. This is particu larly evident in grain boundaries with higher slip system misorientation, as indicated by lower (arranged from top to bottom). The indent topography heights in the receiving grain of a bi - crystal indent are typically different from the heights in either of the two corresponding single crystal indents, but the locations of the topographies in the bi - crystal indents can always be associated with topography locations in at least one of the single crystal indents (Fig. 3.5 column 3 compared to colum ns 2 and 4). Upon initial inspection, it appears that these heights may simply be either higher or lower than the corresponding single crystal indent topographies (Fig. 3.5 columns 1 and 5) , h owever, more details were found in more specific observation s . That is, the bi - crystal indent topography in the receiving grain is always lower (indicated by blue in column 5) than the corresponding single crystal topography for that receiving grain ( i.e., comparing the blue 79 location in Fig. 3.5 column 5 to the topog raphy in column 4). Nevertheless, these same receiving grains often show areas of higher topography (indicated by red in column 5) that correspond to the indent topography of the respective originating single crystal grain in that area (topography locatio ns in column 2). Similar observation can be made regarding the receiving grain in column 1 of Fig. 3.5 . These topography differences are likely caused by the change in the stress field and/or by the change of the slip kinematics due to the different latti ce orientation. In the following, how the differences in indent topography can be cor related to grain boundary nature, as indicated by m , are rationalized . The top two rows of Fig. 3.5 show an example of a boundary that does not have a strong influence on the indent topography development. The topographies of the indentations in both the originating and receiving grains do not appear to be restricted by the low - angle grain boundary , an d a significant amount of indent topography has developed in both originating and receiving grain. This is consistent with the excellent alignment of all of the slip systems across the grain boundary, as indicated by the high value of 0.98. At the oth er extreme, as shown in the bottom row of Fig. 3.5 , the formation of a indent topography in the originating grain next to the grain boundary is reduced due to its high - angle character, indicated by an of 0.51. The seventh and eighth rows in Fig. 3.5 sh ow another high - angle grain boundary, with an value of 0.56, where two indents are located at the same distance from the grain boundary. In this case, both indents show virtually identical indent topography , both of which are strongly inhibited by the g rain boundary, with only limited pile - Indents near another high - angle grain boundary, with being 0.62, are displayed in rows four and five. Here, the slip systems are somewhat more aligned, and the indent topogr aphy transfer red into the receiving grain is moderately larger. It is worth noting that the two indents near this high - angle grain boundary are 80 on opposite sides of the same boundary. The limited indent topography transfer in both directions suggests tha t strain transfer is difficult at a high - angle boundary regardless of the directionality. Overall, it is clear that the indent topography development in the receiving grain is strongly affected by the slip system alignment on both sides of a grain boundar y . This analysis also indicates that the relative magnitude of the grain boundary resistance to slip can be assessed using the quantitative comparison between single and bi - crystal indent topographies illustrated in Fig. 3.5 . 81 Fig ure 3.5 : Nine bi - crystal indents (center column) collected near six different grain boundaries (blue lines) compared with corresponding single crystal indents in both the originating grains 82 (second column) and receiving grains (fourth column). Both single and bi - crystal indentation impression depths ( h ) are labeled near the indents, and the of the six grain boundaries are listed in the image of bi - crystal indent in decreasing order. Indent topography differences caused by the grain boundary are mapped in the outermost columns as the difference between grain boundary indent and single crystal indent. The central indent valleys have been removed in order to enhance visualization of the indent pile - ups around the valleys. [126] 83 3. 4 S imulat ion s of single and bi - crystal nano indent ation s using CPFEM 3 . 4 .1 Computer - assisted processing of experimental data and automatic conversion into simulation files A number of computational crystallographic codes were integrated into seve ral graphical user interfaces (GUIs) to support processing and analysis of the experimental data [125] . These data processing tools were crucial for minimizing possible sources of error in the numerous crystallographic calculations. The first of the modules was developed to analyze the lattice alignment of grain pairs based on orientations collected through EBSD and associated grain boundary networks. Evaluation based on different slip transfer parameters such as the maximum value of was carried out with no stress state imposed, and the resulting parameters were displayed on grain boundary maps, as shown in Fig. 3.6 (left). A second GUI was used to visualize slip system relationships across a grain boundary (see Fig. 3.6 (middle)). A third GUI (see Fig. 3.6 (right)) converted the bi - crystal geometry , including grain boundary inclination, and grain orientations into a finite element mesh and produced the associated input files for direct simulation of the grain boundary indentation pr ocess. 84 Figure 3.6 : The GUI of S tabix used for generating simulation input files by using grain orientation data from EBSD. Left image : EBSD crystallographic orientation data was first processed using the left GUI to determine the maximum parameter and to color the associated grain boundary. The hexagonal cells represent the orientations of each of the grains. Although the simulatio ns include basal < a > , prismatic < a > , and pyramidal < c + a > , the GUI only includes basal a and prismatic a for calculation. Middle image : The second GUI was used to display the crystallographic and slip system relationships. In the upper part 85 of the figure, grain boundary information, such as Euler angles, the orientation of the grain boundary line, the inclination of the grain boundary plane, and the active slip systems, can be entered to study the slip transfe r of a desired grain boundary. M) associated with each pair of slip systems are tabulated in lower part of the figure. Right image : Using the grain boundary information entered in the second GUI, the third GUI builds finite element meshes for bi - crystal indentation simulations. Sample dimensions, mesh resolution, indentation depth, and indenter tip geometry are required as input. [126] 86 3. 4 - Ti The crystal plasticity formulation used he re has been detailed in section 2.5 based on the work of Kalidindi et al . [124] - Ti with hexagonal lattice structure having c/a = 1.57. - Ti was simulated with 18 active slip systems. Three prismatic < a > slip directions are defined to be the most active slip systems, followed by 3 basal < a > and 12 pyramidal < c + a > slip systems. Twinning is not included in this model as it was not observed in the nanoindentation experiments in this study. The elastic constants used in the model were = 162.4 GPa, = 92.0 GPa, = 69.0 GPa, = 180.7 GPa, = 49.7GPa, = 76.5GPa ( [132] ). The initial slip resistance, , and saturation slip resistance, , were identified via a single crystal optimization method described in the work of Zambaldi et al. [133] , and are listed in Table 3.1 . The material hardening parameters, and a , were fixed at 0.2 GPa and 2.0, respectively. The stress exponent and refe rence shear rate, n and , were chosen to be 20 and 10 - 3 s - 1 , respectively Table. 3.1: Material parameters used for simulating the - Ti. [133] /MPa /MPa Prismatic 150 1502 Basal 349 568 Pyramidal 1107 3420 87 3. 4 .3. Three dimensional finite element simulation of indentation process The crystal plasticity formulation was implemented into the commercial finite element solver MSC.Marc 3 using the HYPELA2 user subroutine. For single crystal indentation, the simulation p rocess followed previous work of Zambaldi et al . [133] . To model the 3D nanoindentation deformation process near grain boundaries, bi - crystal specimen meshes ( an example of the mesh was shown in Fig. 3.7 ) with about 40000 elements, depending on th e indentation sample size, were generated using MSC.Menta t 3 . These meshes were developed using procedure files produced by the third GUI (Fig. 3.6 right), which uses the EBSD data and the FIB cross sectional measurements of grain boundary inclination. Me shing of the bi - crystal nanoindentation volumes was done so that the volume under the indenter and in the vicinity of the grain boundary was discretized by the smallest elements. The location of the nanoindentation was determined using the relative distan ce from the indent center to the grain boundary measured from the overlays of secondary electron images on AFM scans. The mesh size was adapted for simulations of different indents depending on the topographies produced by indentations, so that the influe nce of the mesh surface on indent topographies was minimized. Multiple simulations were carried out for the same experimental conditions of crystal orientation and indentation depth, but with a range of mesh sizes (from about 10 , 000 to 40 , 000 elements) . The resulting simulated topographies were found to be insensitive to mesh size, with V defined in section 3.3.1 and 3.3.2 ). In the following sections, meshes with element numbers in the range of 30 , 000 - 40 , 000 elements were used as they result in smoother representation of indent topographies. Following body, and the friction coefficient between the indenter and metal was set to 0.3 [133] . The 88 indentation process was modeled using displacement control. The residual indentation impression depth of simulations was set to match that of the corresponding experiments in order to facilitate comparison bet ween experimental and simulated indentation topography. Figure 3.7 : The finite element model used for a bi - crystal indentation tests where the black dotted line indicates the location of the grain boundary. The red color region indicates that the form ation of surface topographies after the nanoindentation. [ 126 ] 89 3. 5 . Comparing of experimental and simulated indent topographies With the capability of simulating both single and bi - crystal nanoindentations, it is important to determine whether the nature of slip transfer can be reliably modeled using the current phenomenological CPFE models. Thus, CPFE simulations of the five inde ntations that fell on grain boundaries and nine indentations near grain boundaries, as well as several corresponding single crystal simulations of the grains on both sides of the boundaries, were carried out. Differences between simulated and measured topo graphies resulting from single and bi - crystal indentations are examined. 3.5.1 Single crystal indent topography comparison between experiment and simulation . Single crystal indentation simulations of Ti, TiAl, and W ( [133 - 135] ) done in the past capture the general location and extent of the pile - up topography. In the present study, two examples of the single crystal indent topographies are shown for both experimentally measured and simulated cases in the first and second colum ns of Fig. 3.8, respectively. Close examination of the point - wise differences between experiment and simulation in the third column shows very good agreement between the shapes of the indent topographies and indent impression. In addition, the maximum in dent depth and topography height are also found to be very close in Fig. 3.8. Despite the generally good agreement, small systematic deviations between experimentally measured and si m ulated topographies are observed, illustrated by generally blue shades within the central area of the indent valley surrounded by a red - shaded ring in the central column of Fig. 3.8 . These systematic deviations may be a result of a number of factors: (1) reverse plasticity may occur as the experimental load is release d, resulting from the release of the back stress 90 associated with dislocation pile - ups; Such reverse plasticity is unlikely in the phenomenological model employed here, as it does not explicitly account for kinematic hardening resulting from dislocation pil e - up formation; (2) while the indenter in the simulation is modeled as a perfect sphero - conical tip, it is quite likely that the actual indenter deviates somewhat from this geometry; (3) there may be some friction/cohesion forces between the indenter tip a nd sample, resulting from basic atomic interaction or indenter surface roughness that is approximated for in the simulation; (4) the simulation treats the indenter tip as a rigid body, while in fact one would expect some elastic behavior of the tip. The ex act reasons for these deviations are presently not well established. Figure 3.8: Experimentally measured (first column) and simulated (second column) topographies of two single crystal indent and the corresponding point - wise differences between experim ent and simulation shown in the third column. [126] 91 3.5. 2 Bi - crystal indent topography comparison between experiment and simulation . Figures 3.9 a and 3. 9b show experimentally measured and simulated indent topographies of bi - crystal indent ations performed on grain boundaries and near grain boundaries respectively. The third column illustrates the difference between simulated and experimental topographies, where red (blue) shades indicate the simulated topography is higher (lower) than the experiment. A careful examination of the indent topography difference maps between simulations and experiments (third column in Figs. 3. 9a and 3. 9b) shows that in all but one case (row three of Fig. 3. 9b), the systematic red ring observed for the single crystal indents is again observed in these bi - crystal indentation comparisons . I n this case the red ring is more pronounced than in the single crystal cases in Fig. 3.8, indicating potential stronger reverse plasticity caused by the presence of the grain boundary in the experiments of bi - crystal indentations than single crystal indentations. Ignoring these systematic deviations, the locations and shapes of the simulated indent topographies are consistent with the experimental measurements. By careful exa mination of the impression (area inside the red ring), the maximum indentation depth between experiment and simulation are found to be very close. In addition, the comparison between the indent topography (regions with heights above the sample su rface) shows that the maximum heights of both experiment and simulated topographies do not always agree in the originating grain (two examples of large discrepancies shown in Fig. 3. 9 a row 3 and 4). In other words, shades of blue were frequently observed in the originating grain in the third columns of Fig. 3. 9 a, meaning higher surface topographies has been developed in the experiment s than in the simulation s in the originating grain. One possible explanation is that the grain boundary resistance to strai n transfer is higher in the experiment than in simulation , in other words the phenomenological model does not fully capture the actual grain boundary 92 resistance to plastic strain transfer. As a result of this larger resistance in experiment, the indent su rface topographies are prone to form in the originating grain rather than the receiving grain in the experiment, causing the height of indent topography to be larger in experiment. 93 Figure 3.9: Comparisons between experimentally measured and simulated indent topographies near six grain boundaries. a) Indents that are located near the grain boundary. b) Indents that are right on the grain boundary. [126] 94 3. 6 . Discussion 3.6.1 Quantifying the qu ality of s ingle crystal and bi - crystal simulation u sing In addition to the point - wise comparison of indent topographies between experiment s and simulation s shown in section 3.5 , i t is also desirable to generalize the overall quali ty of the simulations of both single and bi - crystal indentations. To quantify the overall accuracy of single crystal indentation simulations, the , calculated using Eq. 3.1 and 3.2, for single crystal indents were divided by the area over which the AFM measurements were taken and compared to the corresponding indentation noise level (same as noise level in Fig. 3.3 d) in Fig. 3.10 (dotted blue and dotted red respectively). It is clear that the differenc es between the simulated and experimentally measured topographies are, as expected, larger than the noise level , but are still only in the range of 1.7 nm, which is very small compared to the overall indentation heights of about 150 nm . This small discrep ancies of show s that the simulation matches the experimental indentations quite well in single crystal case . In the case of bi - crystal, the same routine was followed to calculate the using Eq. 3.1 and 3.2. The resulting is compared to the noise level of bi - crystal experiment in Fig. 3.10. The differences between the simulated and experimentally measured indent topography (solid blue in Fig. 3.10 ) are only slightly larger than the bi - crystal indentation noise level (solid red in Fig. 3.10 ), and are also consistently greater than the differences between the single crystal simulations and experiments (dotted blue in Fig. 7). Nonetheless, the average per area is still in the range of 2 - 4 nm, which is very small compared to the average heights of indent topographies of about 150 nm. This again shows that the general quality of the 95 simulations of bi - crystal indentations, though slightly worse than single crystal simulations are still quite high. Figure 3.10 : Cumulative probability distributions of the differences between simulated and experimentally measured indent topographies for single crystal and bi - crystal case are shown as dotted b lue line and solid blue line, respectively. As a comparison, noise level in experimental measurements from Fig. 3.3 are plotted for single crystal (dotted red line) and bi - crystal case (solid red line). [126] 96 3.6.2 Correlating experiment and simulation results with slip transfer criterion and M ( LRB criterion) It has been shown that in both experiment s and simulations (in Fig. 3.9 a ), the indent topographies development in the receiving grain is almost unhindered in cases where the slip systems of both adjacent grains are well aligned (first two rows in Fig. 3.9a), while in cases of poor alignment the indent topographies in the receiving grain are strongly limited (last two rows in Fig. 3.9a). To understand more on how different grain boundaries influence the development of bi - crystal indent topographies, a comprehensive analysis of V and V was carried out on the overall data set , including experiment s and simulations of both single and bi - crystal indent topographies. To facilitate the comparison, the V and V are evaluated separately for the indent topography in both the originating and receiving gr ains (i.e., left and right of the grain boundary shown in Fig. 3.11a, with indent topographies colored in red and impressions in blue) to quantify the influence of a grain boundary on the indent topography. All the results are shown in Fig. 3.11(b - d) as a function of slip transfer criterion or LRB. Figures 3.11 b and c plot the V V = 2( V a V b ) / ( V a + V b ) between bi - crystal and single crystal indents as a function of , with the experimentally measured values represented by crosses and the simulated values represented by circles, and the red and black representing the originating and receiving grain, res pectively. The grain boundaries do not have a significant effect on the indent topography in the originating grains, as indicated by the red symbols being clustered around V ratios of one in Fig. 3.11 b and normalized V of zero in Fig. 3.11 c. In contrast , the differences in indent topography in the receiving grains (black symbols) between bi - crystal and single crystal indents increase with decreasing m . 97 V versus M (LRB parameter) in Fig. 3.11 e. The displays a roughly linear correlation with M , (black crosses in Fig. 3.11 e), while the shows a more exponential relationship, (black crosses in Fig. 3.11 c). Furthermore, the ratios of the simulated (black circles), which are consistently smaller than the experimentally measured ratios (black crosses), do not show strong functional relationships with or M . V and V , are assessed across the entire positive and negative inde nt topography in Fig. 3.10 , the two measures have been evaluated for indent pile - up (positive) and impression (negative) separately. Similar trends and conclusions are observed in all of these analyses, indicating that the overall indent topography is suf ficient for quantifying the influence of grain boundaries on plastic strain transfer. 98 Figure 3.11: a): Schematic single and bi - crystal indents with gray areas representing (top) and (bottom). The grain boundary in the bi - crystal case. b) : vs. volume ratio ( markers - crystal and originating single crystal 99 indents evaluated within the originating grain (red) and the receiving grain (black). c) : vs. normalized volume difference ( markers between bi - crystal and originating single crystal indents evaluated within the originating grain (red) and the receiving grain (black). d) : Volume ratio between simulated and measured bi - crystal indents evaluated within the originating grain (red circles) and the receiving grain (black circles). e) : M vs. normalized volume difference (both sim and exp) between bi - crystal and originating single crystal indents evaluated within the originating grain (red) and the receiving grain (black). [126] 100 To interpret the difference between simulated and experimentally measured bi - crystal topographies, a number of factors that influence the kinematic response of a bi - crystal to an indentation can be considered. First, the difference in crystal orientation between the originating and receiving grain results in corresponding differences in the resolved shear stresses on the slip systems on either side of the boundary. This purely crystallographic effect is accurately captured by the crystal plasticity framework, so the differences between experiments and simulations cannot be rationalized simply on crystallographic characteristics. Second, a boundary can act as an intrinsic obstacle to plastic flow by hindering the passage of disloca tions through (or dislocation absorption and nucleation at) the boundary into the receiving grain, leading to pile - ups of same - sign dislocations and local stress build - up. Barriers to plasticity activati on the receiving side of a boundary are not accounte d for in the phenomenological constitutive description, and it is not entirely clear how to best incorporate them into the plasticity framework. A number of approaches have been taken by other researchers that can be computationally expensive and that hav e varying levels of effectiveness ( [110 , 111 , 114] ) . It is interesting to note that this intrinsic resistance to strain transfer at the boundaries increases with decreasing , as indicated by the growing discrepancy between the experiments (black crosses ) and simulations (black circles) at lower boundaries (Figs. 3.11 b and 3.11 c), so that improvements to such models need to consider the effects of disorientation on the boundary resistance to plastic flow. Third, the ease of dislocation nucleation in t he receiving grain (probably dislocation - starved) will also influence the kinematic response of the bi - crystal to an indentation. The present model effectively assumes an abundance of mobile dislocations at every material point to carry plasticity wherever the acting stress is sufficiently high, but this may not be the actual physical situation, where dislocation sources may be limited both at the boundary 101 and within the two grains. Such source limitations could be a reason for the consistently larger valu es of and in the experiments than in the simulations. Finally, the model also fails to account for the cooperative backward motion of blocked dislocations within the originating grain that could result in a co ncurrent back flow of plasticity on the receiving side of the grain boundary upon release of the indentation load, which would result in the lower experimentally measured topographies in the receiving grain than predicted by the simulations. Given that the current model is purely phenomenological and lacking any detailed dislocation mechanics, the generally good agreement suggests that the kinematic incompatibility, which is naturally taken into account by crystal plasticity, plays a key role in the deforma tion process near grain boundaries in commercially pure - Ti. Since the present comparison between experimental and simulated topographies is done at a convenient but arbitrary indentation depth, it remains open whether such good agreement holds also for the prior evolution of topography with indentation depth. 3.6.3 Grain boundary sensitive CPFE model for bi - crystal nanoindentation The comparison of simulated and experimentally measured bi - crystal indent topographies in the last section revea led an underestimation of grain boundary resistance to slip in the current CPFE model compared to the experiment al observations . In an attempt to remedy this discrepancy between model and observation, the original model for bi - crystal nanoindentation was modified by including a hardened layer of elements to act a s a grain boundary. The material properties of the grain boundary elements were assigned independently from grain interior, meaning that all the material parameters used for grain boundary element were set according to the grain boundary character. For those grain boundaries with high resistance to slip transfer 102 (low ), the boundary layer elements are modeled using a harder material, and for boundaries with low resistance (high ) vice versa. 3.6.3.1 modified bi - crystal nanoindentation model and slip parameters in the model The modified bi - crystal nanoindentation model, shown in Fig. 3.12, has two additional layer s of grain boundary elements compared to the original model. The crystal orientation of each grain boundary layer is consistent with the orientation of the corresponding grain interior ( for example, i n Fig. 3.12, the crystal orientation of the right layer is the same as that of the right grain). The inclination of the two grai n boundary layers was acquired from the FIB - cross section measurement detailed in section 2.3.2. The thickness of each of these two layer s was set to around 200 nm, which is a relatively small number compared to the size of the indentation (~2 - 3 m) while large enough to influence the strain transfer across the grain boundary in the simulation . The commonly used slip parameters that control individual slip system in the current phenomenological model (for both grain interior and grain boundar y) are the initial slip resistance , saturation slip resistance , initial hardening slope , and hardening exponent a . The se four slip parameters define the shape of strain hardening curve of the modeled material illustrated in Fig. 3.1 3 (left) . By changing the magnitude of each parameter, the hardening of the material undergoes a different path. For example, a larger value of causes the material to harden faster in the beginning of the plastic deformation , while a larger indicates that t he resolved shear stress required to initiate each slip system is higher. The values of slip parameters used in the original bi - crystal nanoindentation model is shown in Fig. 3.13 (right) ( a is set to 2 as constant). Each slip system ha d a different rang e of initial and saturation resistance, while the 103 and a were set to 1 and 2, respectively. Regarding to the slip parameters of the grain boundary layer, it is still unclear how to choose the optimal values. As a result, multiple groups of slip parameters were examined with the modified model, and the group that g enerate d the most accurate simulation results was chosen as the optimal slip parameters for th at grain boundary layer. Figure 3.12: The grain boundary sensitive model proposed as a revision to the original model by inserting two layers of elements (Gra in 1 boundary layer and grain 2 boundary layer) between the two adjacent grain 1 and 2. 104 Figure 3.13: The physical meaning of the four slip parameters in the model is shown in a stress - strain curve in the left figure. The values of the and use d in the original model are presented in the right graph. 105 3.6.3. 2 Optimizing the slip parameters of the grain boundary layer To determine the optimal slip parameters for the grain boundary layer elements, slip parameter s of all grain boundary elements , including , , , and a , were altered independently 7 (one dimensional optimization) to assess the influence of each slip parameter on the accuracy of the model . To accurately evaluate the influence of the grain boundary layer on the development of indent topography across the boundary in the bi - crystal indentation model, an objective function, used to optimize , , , and a , was determined to be , which is the ratio between the volume of the simulated and the measured indent topography in the receiving grain. The outcome of the optimization for each set of slip parameters ( , , , and a ) of a the studied grain boundary is shown in Fig. 3.14(a - c). The values of individual slip parameters were plotted against the objective function . As the and are modified simultaneously by the same ratio their results are shown in one plot (Fig. 3.14c). It can be concluded from Fig. 3.14(a,b) that both the initial hardening slope , , and hardening exponent , a , play no significant role s in providing addition al grain boundary resistance to slip transfer as the objective function remains almost constant at all values of , and a . As is shown in Fig. 3.14c, t he model is quite sensitive to the combined initial and saturated slip resistance, and . By i ncreasing the values of both and , the resistance to slip transfer rises significantly in the grain boundary region, leading to considerable drop of the objective function. In Fig. 3.14c a plateau is also observed at the end of the curve when and are increased 7 Due to the correlation b etween and and the requirement of , these two slip parameters cannot be adjusted independently. As a result, these two parameters are either increased or decreased simultaneously by the same ratio in this study. 106 more than ~ 2 .0 times of the original value. This suggest that there is a n upper limit of the resistance to strain transfer that the grain boundary layer is able to provide. In other words, the grain boundary resistance to strain transfer remains constant, regardless how much the and is raised beyond the limit. One possible expl anation is that when the grain boundary layer elements are hardened beyond this limit, those elements become so difficult to deform that the grain boundary layer is pushed away from the indenter . In such a scenario, and w ill not play a significant role in the strain transfer as the grain boundary is not heavily deformed. In general, the underestimation of grain boundary resistance to strain transfer can be best compensated at and above this limit, which is at . The same optimization procedure was conducted at another three grain boundaries. The limits that provide the most accurate simulations in the receiving grain were acquired for each grain boundary studied and are plotted as a function of in Fig. 3.14d. A positive correlation between the limits and the is observed . This suggest s that two adjoining grains with poor slip system alignment (low ) need higher magnitude of compensation of grain boundary resistance in the model , which is fulfilled by the larger limit of shown in the Fig. 3.14d. In summary, by incorporating a hardened grain boundary layer into the bi - crystal indentation model, some success has been achieved in provid i ng more accurate simulation results of the indent topographies. This study is a preliminary attempt to improve the CPFE model accuracy in the vicinity of grain boundary, and by no means the most consistent and physically meaningful method to simulate hete rogenous deformation near grain boundary. 107 Figure 3.14: Figure (a - c) show how the accuracy of the new model ( ) alters as a function of the change in each slip parameters ( , a , and . The red dotted line indica tes the optimal value of (in the ratio form of ) that generates the most accurate simulation compared to experiment. Figure d presents all the optimal values of acquired at four grain boundaries as a function of the four . 108 3. 7 Conclusions The deformation process of nanoindentations in both grain boundary interiors and near a number of low and high angle grain boundaries are characterized using instrume nted nanoindentation, EBSD, FIB, AFM and CPFE modeling in commercially pure - Ti. The deformation has been assessed using indent topographies. The reproducibility of this method was evaluated by determining the variation in indent topographies car ried out under nominally identical orientation conditions. The average topography deviation is only about 1 nm for single crystal indents and 2 - 4 nm for bi - crystal indents, compared to maximum topography elevations of approximately 150 nm, indicating that the indentation deformation is reproducible and hence it is reasonable to quantify the deformation processes using indent topography. Comparisons of bi - crystal and corresponding single crystal indentation topographies reveal that grain boundaries with poorly aligned slip systems, as indicated by low , tend to develop limited amount of topography across the boundary into the receiving g rain, suggesting that it is difficult to transfer slip across these boundaries, while high boundaries show only minimal reduction in the topography in the receiving grain. This proves that the study of bi - crystal indent topography can lead to the estim ation of resistance of different grain boundaries to slip tran s fer. Furthermore, q uantitative comparisons show good agreement between experimentally measured and CPFE simulated indent topographies, indicating that the crystal plasticity kinematics is a d ominant factor in both single and bi - crystal indentation deformation. Nevertheless, these comparisons also reveal two types of systematic differences between experiment measurements and simulation results. The consistent deviations near the indent impres sion observed in both single and bi - crystal indentation comparisons are potentially caused by insufficient accuracy of 109 modeling the tip and its interaction with the sample, or may be caused by disregarding reverse plasticity in the model. The simulations often overestimate the deformation transfer to a small degree, especially for high - angle grain boundaries. Such overprediction of topography heights in the simulations is likely due to a neglect of explicit dislocation - grain boundary interactions and disl ocation nucleation processes, particularly when considering the receiving side of the grain boundary being probably deprived of mobile dislocations before the indentation process. An attempt to resolve the underestimation of the grain boundary res istance to strain transfer in the model has been made by incorporating a hardened layer of elements as grain boundary. Optimized slip parameters such as , , , and a were obtained by minimizing the differences between modeled and simulated indent topography. The revised model with tuned slip parameters was proved to provide more accurate simulation of the indent topography compared to the original bi - crystal indentat ion model. 110 CHAPTER 4 PREDICTING SLIP TRANSFER ACROSS GRAIN BOUNDARIES WITH AN ITERATIVE STRESS RELIEF MODEL As discussed in section 1.2, predictive metrics/models for assessing slip transfer at grain boundary are a paramount compone nt to a complete description of the heterogenous plastic deformation in polycrystal. Most of the proposed slip transfer criterion/models, such as N LC [ 5 ] , , M ( LRB criterion [9] ) , and M s [ 7 ] , are based on the assumption that the shear (slip or twin) of the incoming slip system is accommodated by only one outgoing slip system in its neighboring grain. Nevertheless, it is quite possible or even common that the accommodation of the incoming shear requires the activation of multiple outgoing slip s ystems. Early research of Livingston and Chalmers [ 5 ] suggested that the active outgoing slip systems should be decided based on the requirement of continuity of grain boundary strain. This approach, more commonly nly model capable of resolving the accommodation of shear through multiple slip systems at a grain boundary. E xperimental validation of the tangential continuity model is rarely reported. Thus, it is desirable to further explore new theor ies /model s of sh ear accommodation at grain boundar ies and use tangential continuity model as a benchmark for comparison. In this Chapter, slip trace analysis combined with AFM and EBSD is used for analyzing the slip accommodation observed at multiple grain boundar ies in ~1.5% strained - Ti specimen . Two types of slip accommodation w ere observed most frequently: 1) the incoming slip system is accommodated by only one dominant outgoing slip system. 2) the incoming shear is accommodated by two outgoing slip systems with comparable amount of shears. To rationalize the two types of slip accommodations, a new iterative stress relief model was developed . This 111 new model , in addition to the tangential continuity model , was tested based on the slip accommodation observati ons. Optimizations of the model parameters, such as critical resolved shear stress (CRSS) ratios, were conducted to improve the accuracy of the new model. 4.1 Observations of slip accommodation in - Ti Slip band accommodation at grain boundar ies was not commonly observed due to the low level of strain approximately 1.5%. 8 After investigating hundreds of grain boundary pairs using SEM, three types of slip transfer were found , as shown in Fig. 4.1 . (SEM images of all slip transfer cases, as well as corresponding AFM measurements, are presented in Appendix A) Figure 4.1(top left) presents an example of a grain boundary where the slip line on both sides do not meet at grain boundary, indicating no apparent relationship between the slip lines on opposite side of the boundary. In most of these cases , including Fig 4.1(top left ), the spacing pattern between slip lines are different and the differing sense of contrast of slip band in the SEM image suggests an opposite sense of shear, further supporting the argument that there is no correlation between two sets of slip bands on op posite side of the boundary. Figure 4.1(top center) shows an example of well - correlated slip transfer across a grain boundary, where the incoming slip system is accommodated by a unique outgoing slip system in the neighboring grain . Due to the one slip b and - to - one slip band relation, the spacings between slip bands in both grains are correlated but not necessarily distributed evenly. events occurred at about 30% of those boundaries with slip lines in the vicinity . In contrast , the top right image of Fig 4.1 8 There are several advantages of using low strain to study slip transfer: 1) The surface remains relatively flat even at grain boundaries, which enables accurate AFM measurement and slip trace analysis in the vicinity of the grain boundary. 2) At low strain levels, the slip transfer plays a critical role in the activities of slip systems near grain boundaries (dislocations tend to nucleate at grain boundaries), while at high level of strain the nucleation of dislocations can also occur in the grain interior, complicating the whole analysis. 3) Cross - slip, which is rarely observed at low strain and more frequently found at high strain, is not desired in this study. 112 presents a scenario where a single set of slip lines in an incoming grain correlates with two sets of slip lines emanating from the boundary intersection in the outgoing grain. In these cases, the slip system in the incomi ng grain is well correlated at the grain boundary with the two outgoing slip bands. It is worth noting that at the grain boundaries where this double slip accommodation occurred, no signs of single slip accommodation were ever observed. Th ese suggests th e double were much less common than single slip accommodation, presenting only at a very small fraction of all grain boundaries. There were no obse rvations of triple accommodation. Figure 4.1 : S econdary electron images of the three categories of s lip accommodation: non - correlated (left), one - to - one correlated slip (center), one - to - two correlated slip (right), and corresponding topography maps measured by AFM (second row). 113 4. 2 Determining the active slip system s , relative shear s , and slip geometr ies o f experimentally observed slip accommodations To carry out quantitative analysis on the slip transfer/accommodation at various grain boundaries, it is necessary to identify the active deformation systems, their relative shear, and slip system align ment across a grain boundary. Active deformation systems were identified using slip trace analysis (detailed in section 2.4). To eliminate ambiguity in cases where the expected traces of more than one potential slip plane had traces close to that of the observed trace, the slip plane with an inclination closest to the slope of the surface slip step was chosen (as shown in Fig. 2.8 c ). A variety of slip plane types were observed to be associated with both single and double accommodation, includ ing prism, basal, and pyramidal planes. In cases where prism planes were identified as the active slip plane, the Burgers vector is fixed as there is only one slip direction on these planes. In cases where the basal and pyramidal planes were identified a s the active slip plane, the active Burgers vector was assumed to be the one with the largest resolved shear stress (under the global uniaxial tensile stress). In the present study, due to the material texture and the low level of strain (~1.5%), the deve lopment of deformation twinning was very limited. In no cases was twinning observed to be involved in single accommodation and in only one case was twinning observed in a double accommodation slip transfer. In that unique case the incoming slip band was accommodated by T1 deformation twinning in the outgoing grain in conjunction with dislocation slip on a secondary accommodating slip system. With the active slip systems determined, the alignment of the deformation systems across a grain boundary c an be assessed by quantifying the angles that are related to the geometry of the slip transfer: (angle between incoming and outgoing Burgers vectors), (angle between slip 114 plane normals of incoming and outgoing systems) , and (the angle between the slip plane traces on the grain boundary ). The relative shear on each identified deformation system was quantified using a quantitative AFM analysis proposed by Y. Yang et al . [ 43 ] [ 44 ] . This analysis was performed on some of the single accommo dation and all of the double accommodation cases, where surface elevation maps of the shear accommodation events were collected near the grain boundaries to quantify the step heights associated with the two or three deformation systems involved in the acco mmodation (shown in Fig. 4.1 bottom row) . The step heights along the specimen surface normal were transformed into the number of dislocations associated with the shear of the identified active deformation system (with known Burgers vector b ) using: Eq . 4.1 The identified slip system s , their alignment across a grain boundary, and for some of them the number of dislocations in each slip system are compiled in table 1. In table 1, c olumns 2 and 3 show the active Burgers vectors and planes of the incoming and outgoing deformation system s , columns 4 to 6 specify their geometric relationships across the grain boundary, while columns 7 and 8 list the number of dislocations associated with the shear that fo rmed the incoming and outgoing slip/twinning bands and resulted in the surface topography (Eq. 4.1 ). (It should be noted that was only determined for seven cases, as it requires FIB sections to be cut (see below), while AFM slip band measurement were o nly preformed on eight cases.) . In the vast majority of cases, the incoming deformation system was of prism < a > type. Only three out of the 16 single accommodation events observed exhibited activity by systems other than prism < a > . Four out of six of th e incoming deformation systems, as well as three out of six primary outgoing systems in double accommodation events, were prism < a > , while the secondary 115 outgoing deformation systems involved a variety of deformation systems. The relative shears on the inc oming and outgoing deformation systems were determined from the AFM quantified number of dislocations for two of the single and all of the double accommodation cases. From the two evaluated single accommodation cases, it was determined that the shear in t he incoming and outgoing slip band were always very close. 9 9 While the slip systems on both sides of the grain boundary are not perfectly aligned in case s 14 and 15, t h e number of dislocations in the incoming and the only outgoing slip band are very close . This suggests that in single slip accommodation it is possible, or even likely, to have a one - to - one ratio between the numbers of incoming and outgoing dislocation s . Nevertheless, this type of slip transfer will still leave grain boundary residual dislocations. In the case s 14 and 15 , the total residual Burgers vector s is 4% and 12% of the total incoming Burgers vectors , respectively . 116 Table 4.1: Slip systems, geometric parameters of slip transfer, and number of dislocations in the slip bands for all 22 (16 single plus 6 double) slip accommodation observations. 10 10 Columns 2 - 3: Burgers vector and slip plane normal of incoming and outgoing deformation systems for all shear accommodation observations (cases 1 16 are single accommodation and cases 17 22 are double accommodation). Columns 4 6: (angle between incoming and outgoing Burgers vectors), (angle between slip plane normals of incoming and outgoing systems), and (the angle between the slip plane traces on the grain boundary). was calculated based on Eq. 2.1 and requires a FIB cross section, which was done for seven cases. Columns 7 and 8: Number of Burgers vectors in each slip band was calculated based on Eq. 4.1 and requires surface topography measurements, which was done for eight cases. 117 4.3 A ssessment of the t angential continuity theory based on slip transfer observations 4.3.1 Tangential continuity theory As discussed in section 1.2.2, the concept of the tangential continuity was first brought forward by Livingston and Chalmers [ 5 requires that the plastic distortion tens or to satisfy: Eq. 4.2 m eaning that the integration of the distortion tensor along a closed circuit C across the grain boundary nets to zero sum. By collapsing the closed circuit C into a point, Eq 4.2 can be rewritten as a more compactly form : , or Eq. 4.3 where is the tangential part of the plastic distortion tensor . By differentiating Eq. 4.2, one can obtain formula similar to Eq. 4.3: Eq. 4.4 where the is the plastic distortion rate tensor. Assuming that the plastic deformation is result of the shear on each individual slip system, the can be related to the shear and geometry of each slip system: Eq. 4.5 where the is the shear rate on slip system i, and and are slip direction and slip plane normal of slip system i . 118 If an incoming slip system triggered the activation of slip systems in both grain A and grain B, then the tangential continuity requires that net effect of all slip systems on the grain boundary to be zero. This requirement can be quantified by combing Eq. 4.4 and 4.5: Eq. 4.6 Nonetheless , Eq. 4.6 will not hold in experiments , due to the disruption at the interface, such as residual Burgers vector, grain boundary sliding, formation of micro - cracks, etc. As a result, in this study the tangential continuity criterion used for predicting the accommodating slip systems at grain bo undary is adapted by minimiz ing the left part of the Eq. 4.6 . M ore specifically: Eq . 4.7 4.3. 2 Predicting accommodating slip systems using tangential continuity theory Based on Eq. 4. 7 , the measured crystal orientations, the identified incoming deformation systems, and the grain boundary plane normal were used as input for a Broyden - Fletche r - Goldfarb - Shanno (BFGS) minimization routine to determine the optimum combination of accommodating deformation systems (chosen among basal < a >, prism < a >, pyramidal < a >, pyramidal < c + a >, and T1 twin as listed in Table 4.2). 119 Table 4.2: Indices, shear direction d , and plane normal n of deformation systems considered for shear accommodation in the study. 120 As the tangential continuity theory indicates that the total number of accommodating systems can be a maximum of five , there are six different accommodation scenarios that can be considered: 1) All five accommodating slip systems are in the outgoing grain. 2) Four accommodating slip systems are in the outgoing grain, whil e the other one is in the incoming grain (self - accommodation). 3) Three accommodating slip system s are in the outgoing grain, and the other two are in the incoming grain. 4) Two accommodating slip systems are in the outgoing grain, and the other three ar e in the incoming grain. 5) Only one accommodating slip system is in the outgoing grain, and the other four are in the incoming grain. 6 ) All five accommodating slip system s are in the incoming grain, meaning the incoming shear is relieved completely thr ough self - accommodation. For later reference of each accommodating scenario, a short notation was used , where and are the number of potential 4 represents four accommodating systems in the incoming grain and one in the outgoing grain. To quantify and highlight the differ ences between each slip transfer event observed or predicted, polar bar plots are used to illustrate the deformation system activity, with an example/key shown in Fig. 4.2. Each deformation system family is categorized by the specific color of the underly ing sector. The six active deformation systems are represented by six polar bars, which are colored to distinguish their roles played in the shear transfer. The incoming slip system is represented in white, any self - accommodating deformation systems in t he incoming grain are represented in red, and deformation systems active in the outgoing grain are represented in black. The amount of shear on each system is proportional to the length of the bar. In the example in Fig. 3, the white radar bar that repre sents the impinging [2 - 1 - 10](01 - 10) slip system (ID number 4 in table. 4.2) lies in the red prism < a > region, the red bar represents the 121 self - accommodating slip in the incoming grain by the pyramidal < a > system (ID number 8), and the four black bars repres ent accommodating slip activity in the outgoing grain, with one basal < a > system (blue background, ID 2), one pyramidal < a > system (green background, ID 11) and the other two pyramidal < c + a > system s (tan background, ID number 18 and 21 ). 122 Figure 4.2 : An example/key of a polar bar plot that presents all of the details of a shear transfer event and serves as key for Fig. 4.3, Fig. 4.4 and Fig. 4.5 . A total of 30 (=3+3+6+12+6) potential deformation systems, color coded by type, are represented on the outside of the figure, with the numbers indicating the specific deformation system, as listed in Table 4.2 . Each bar represents an incoming or outgoing slip system, with white indicating the incoming system, black indicating outgoing systems, and red denoting accommodating systems in the incoming grain. The direction each bar points to reflects its slip system type . T he length of each bar represents its relative shear 11 . 11 The relative shear of incoming slip system is assigned to b e unity . A s a result the length of the white bar is constant in the radar bar plot . T he relative shear of any accommodating slip system (length of the black or red bar) is then a ratio based on the relative shear on the incoming slip system. 123 Using Fig. 4.2 as the standard way of plotting a slip transfer event (either observation or simulation), Figures 4.3 and 4.4 illustrate the various deformation system activities predicted by the tangential continuity model for the 16 experimentally characterized sing le shear and 6 double shear transfer cases. In cases where at least two self - accommodating deformation systems are allowed ( i.e., 5 , 4 ), tangential continuity can be fulfilled by those two systems alone. This trend can be observ ed in the second column of Fig. 4.3 and 4.4, as each inset has two red bars with the same length, indicating complete self - accommodation maximizes tangential continuity at the grain boundary. In fact, the tangential continuity optimization predicts identi cal accommodation for all those categories ( i.e., hence, they all are collected in the second column in Fig. 4.3 and 4.4. In the present study no instances of self - accommodation were observed. (We note that limited self - a ccommodation has been observed in a recent study in the same - Ti material used in this study, but not without major outgoing grain accommodation [ 141 ]. A closer analysis reveals that exclusive self - accommodation can happen under two scenarios in t he tangential continuity model. If the incoming slip is on the basal plane, then slip by the remaining two basal Burgers vectors is able to compensate for the incoming shape change (not observed in Figs. 4 .3 and 4.4 ). Similarly, any other (than basal) incoming shear can be accommodated by two additional slip systems that share the same Burgers vector with the incoming slip system but operate on different (here pyramidal) slip planes. This geometric peculiarity that sli p on three (specific) systems can (locally) result in zero net shape change will exist for the hexagonal lattices examined here, as well as for fcc and bcc, but might not be generalizable to other (lower symmetry) crystal structures. Nonetheless, the stre ss driving the incoming slip system will resolve to an opposing stress on at least one of the two required self - 124 accommodating systems. Therefore, it is conclude d that the tangential continuity solution, which does not account for externally imposed stress , is correct from a purely kinematic stand - point, but unlikely to be physically realized. 4 5 agree 12 much more often with the observed accommodating deformation system(s). Specifically, f or the 16 single shear accommodation cases , the 1 + 4 condition only predicted the correct outgoing system in 3 cases , while using the 0+ 5 condition led to a correct prediction in 7 out of the 16 cases. For the double shear accom modation cases, while the 0+ 5 condition had slightly more success than the 1+2 condition, which was unable to correctly predict any accommodating systems, the model still incorrectly predicted both accommodating slip systems in five of the six double accom modation cases, and in the two case s 17 and 18, only correctly predicted one of the two accommodating deformation systems. This lack of agreement results from the tangential continuity theory lacking a unique solution in its pure kinematic form. To overc ome the non - uniqueness, other biases such as imposed stress and total number of dislocations are necessary to be added as constraints to the solution. 12 Here a prediction is considered correct if the most active predicted deformation systems agree with the observation and any other systems fall below half of the primary shear magnitude. 125 Figure 4.3: Radar bar plots showing tangential continuity predictions compared to observations of 16 single slip accommodation cases. 126 Figure 4.3 : The left column indicates that the predictions are identical (and never match the observations) when at least two self - accommodating systems are allowed. The center two columns show the response changing for less than two self - accommodating systems. 0+5 predictions are most accurate, with about half agreeing with observations. 127 Figure 4.4: Similar plot as Fig. 4.3 but for double shear accommodation cases. Tangential continuity model predictions are shown in columns 2 - 4 and observations are presented in column 5. Across all six cases, none of the three distinct self - accommodation conditions result in predictions that agree with the observed accommodating shear systems. 128 4.4. A new iterative stress relie f (I SR ) model based on slip accommodation observati ons 4.4.1 Algorithm of the iterative stress relief model Due to the lack of consideration of an applied external stress in the tangential continuity model, a new approach is proposed that takes into account the combination of external stress and the stress that results from the local deformation band development. This new model, which is termed the used to predict (at most three) accommodating deformation systems based on a known impinging system and the global stress state. Contrary to the tangential continuity model, the reduction of the residual Burgers vector in the grain boundary (in other words the continuity of Burgers vector) serves as a decisive criterion in the proposed enhanced model. It is hypothesized that the accommodation of the impinging shear on a deformation system with normal and direction is afforded by multiple accommodating defor mation systems in both the incoming and the outgoing grain s . These three outgoing deformation systems are indexed by i activity is determined by the respective local stress s tate, which is approximated as a weighted superposition of the normalized externally applied stress and the sum of impinging, , and accommodating kinematics, , multiplied by a normalized stiffness tensor with an appropriate 13 choice for the stiffness to keep the local and global kinematic influence at the same magnitude. The and are acquired by rotating using the incoming and outgoing crystal orientation, re s pectively. 13 To reflect hexagonal titanium, values of are , , , , , , all in the units of GPa, were chosen based on [132] . 129 The stress state assumed to drive the activity of the primary outgoing deformation system , , is treated as a binary mixture : Eq. 4.8 of the global stress and only the impinging activity, where the weighting factor varies between zero and one and specifies the relative importance of the global applied stress and the local pile - up stresses. The primary outgoing deformation system is selected as that with the maximum resolved shear stress normalized by the respective critical resolved shear stress (CRSS): Eq. 4.9 where and cha racterize possible outgoing deformation systems. The secondary deforma tion system is identified using the same approach, but includes the modification to the stress resulting from the primary accommodating system: , or Eq. 4.10 Eq . 4.11 The activity of the primary accommodating system is found by expressing the incoming Burgers vector in a basis spanned by the (known) primary accommodating Burgers vector direction , the (to be determined) secondary Burgers vector direction , and their cros s product: Eq. 4.1 2 which is motivated by the assumption that the accommodating shear minimizes the residual Burgers vector in grain boundary. The same procedure is repeated for the tertiary accommodating system, i.e., the driving stress is given by: 130 Eq. 4.13 the activities and depend on the choice of the tertiary system: Eq. 4.1 4 and the tertiary outgoing deformation system again maximizes the normalized resolved shear stress: Eq . 4.15 Any impinging Burgers vector can be fully accommodated w ith the three identified outgoing deformation systems ( i.e. , any vector can be expressed by the combination of any three non - coplanar vectors , provided that the number of dislocations involved in the transmission event is large enough to properly approxima te arbitrary fractions ). Nevertheless, the activity on some of these systems may be very small, and in order to facilitate comparison with experimental results, it is reasonable to include a threshold of relative activity such that only secondary and te rtiary systems for which Eq. 4.16 are considered to be observable, with ranging from 0 to 1. 131 4. 4.2 Predicting accommodating deformation systems using the iterative stress relief model To a ssess its robustness, the iterative stress relief model was used to predict the accommodating deformation systems and their relative activity for the investig ated 22 grain fully agrees with the predicted ones irrespective of potential deviations of relative activity. In the new model proposed in section 4.4 .1 , there are multiple parameters (the weighting factor , the activity threshold , and the ratios of prism < a >, pyramidal < a >, pyramidal < c + a >, and T1 twin relative to basal slip) that could influence the predicted slip systems and/or the relative shear on each s lip systems. The determination of the optimized parameters will be outlined in the discussion (section 4.6). I n this section, the optimized parameters that generate the largest number of correct predictions in all the 22 cases, w ere used in the new model . The iterative stress relief model was evaluated on its ability to predict the type of outgoing deformation systems and the amount of accommodating shear on the predicted deformation systems. In the same manner as the tangential continuity resu lts were presented in Figs. 4.3 and 4.4 using polar bar plots, the predicted deformation systems and relative shear of all 22 cases are presented in Fig. 4.5 for single and double accommodation. On these figures, the white discs represent the threshold for secondary or tertiary activity. Thus, any bar extending beyond the white disc represents what will be considered an active deformation system. Fig. 4.5 shows that in all 16 single accommodation shear transfer cases, only the most active predicted def ormation system exceeds the activity threshold ; these most active systems are always identical to those experimentally observed and their activity generally falls within 10% of the observed amount of shear, with a worst case deviation of 25%. 132 Th e iterative stress relief model is somewhat less accurate in predicting the double shear accommodation events, as shown in Fig. 4.5 , where both outgoing deformation systems were correctly predicted in only four out of the six cases (with the secondary system in case 4 falling below the activity threshold in both the prediction and the experimental observation). The predicted magnitud e of shear in the four correct cases deviates from the experimental observations by between about 10% to 25%. In general, the predictions projected by the iterative stress relief model agree very well with the observations in both single and double accomm odation scenarios . 133 Figure 4. 5 : The new (I S R) model predictions compared to observations for 16 single and 6 double accommodation cases. 134 4.5 Discussion 4.5.1 Optimizing the variables in the iterative stress relie f model In this stress - driven shear accommodation model, the weighting factor , the activity threshold , and the CRSS ratios of prism < a >, pyramidal < a >, pyramidal < c + a >, and T1 twin relative to basal slip are essential to the model. Varying each of the five parameter s may change the predicted slip systems and/or the relative shear on each slip systems. The resulting six - dimensional parameter space was evaluated on a grid that is centered at the points given in Table 4.3 and varies between 0 and 1, while the five other values each span an order of magnitude. Each grid dimension is discretized into ten equidistant intervals, resulting in 116 gr id points being evaluated. Figure 4.6 presents the fraction of correctly identified cases (shades of gray) on the plane of the parameter space at one set of specific CRSS ratios based on the literature [ 136 - 140] and constitutes the center poin t within the four - dimensional subspace spanned by the CRSS ratios (Table 4.3 ). The highest fraction of correct predictions (20 out of 22) occurs at The fact that = 0.5 produces the most accurate predictions of the model implies tha t the importance of both global stress state and the local kinematics and kinetics. Moreover, in the cases studied in this chapter, the influences of both global and local stress are well balanced as each is accounted for half of the overall stress tensor . Nonetheless, the optimal may vary (deviate from 0.5) if the imposed global stress state changed or the crystal structure/texture of the material is different. 135 Table 4.3: The center point of the search range used in the parameter optimization of t he iterative stress relief model. The six parameters correspond to the six dimensions of the optimization. Figure 4.6: Heat map showing the variation in the accuracy of the iterative stress relief model as a function of both and . The accuracy of the model, shown in grey scale, is quantified using the fraction of correct predictions compared to observations. Lighter gray indicates higher accuracy of the model. 136 With the two coordinates of the six parameter space fixed at their optima ( = 0.5 and = 0.5 ) , the influence of CRSS ratios was evaluated within the range specified in Table 4.3 . This evaluation demonstrated that within the CRSS ratio ranges of basal < a > : prism < a > : pyramidal < a > : pyramidal < c + a > : T1 twin = 1 9.5) the fraction of correctly predicted outcomes remains at 20 out of 22. While the ranges of optimal CRSS ratios for prism < a > and pyramidal < a > are fairly narrow, the CRSS ratios of pyramidal < c + a > and T1 twin only need to exceed 1.6 and 3.0, respectively, to maintain the highest prediction accuracy. Figure 4. 7 illustrates the sensitivity of the correctly predicted fraction with respect to the CRSS ratios of prism < a > and pyramidal < a > as a heat map, which is a two - dimensional cut at = 0.5, = 0.5, and CRSS ratios of pyramidal < c + a > = 1.6 and T1 twin = 3.0. After determining the optimal values of all the six parameters, the sensitivity of and the CRSS ratios of prism < a > , p yramidal< a >, and pyramidal < c + a > can be visualized and understood separately , in Fig. 4.8, by varying one parameter at a time while keeping other parameters at their optimal values. The upper left plot of Fig. 4.8 show s that influences the accuracy of t he model. A t left and right end of the plot , where = 0 and 1, the model predict s with the worst accuracy . T he active slip systems predicted by considering only global or local stress state is incorrect in about 40% of the cases. In the upper right plot of Fig. 4.8, the optimal value of CRSS ratio of prismatic < a > is found at 1.0 , as expected . T he accuracy of the model drops significantly as the CRSS ratio of prismatic < a > increases beyond 1.0 , but only drops slightly as CRSS ratio of prismat ic < a > decreases below 1.0. A reverse trend can be observed for the CRSS pyramidal < a > value in lower left plot of Fig. 4.8. The influence of CRSS ratio of 137 pyramidal < c + a > on the model is shown in lower right plot of Fig. 4.8, where the model remains acc urate when CRSS ratio of pyramidal < c + a > is greater than 1.3. In general, the optimized CRSS ratios of all of the deformation systems agrees well with the literature. By visualizing how the CRSS ratio of individual slip system s influence the model , the sensitivity of the model to each CRSS ratio was quantified . Figure 4.7: Heat map showing the accuracy variation of the iterative stress relief model as a function of CRSS ratios prism < a > and pyramidal < a >. The accuracy of the model is quantified using the fraction of correct predictions compared to observations. Lighter gr ay suggests higher accuracy of the model at that point. 138 Figure 4.8: Plots of the model accuracy as a function of four parameters, and CRSS ratios of prism < a >, pyramidal < a >, and pyramidal < c + a > in the model. The model accuracy is quantified using th e fraction of correct predictions. 139 4.5.2 Limitations of the iterative stress relief model 4.5.2.1 Slip band blocking and threshold stress In addition to the correlated single and double shear accommodation events, three other types of interactions were observed: ( 1 ) apparent slip band blocking, ( 2 ) non - correlated slip activity, and ( 3) activation of localized plume - like/non - planar shear accommodation in the neighboring grain. In this latter case, electron channeling contrast imaging ( ECCI) has shown very limited shear accommodation, often in the form of a non - descript cloud of dislocations S. Han and M. A. Crimp [ 142 ] , which was not a focus of the present study . 14 Since the iterative stress relief model does not consider a stress threshold for excluding (any) accommodating deformation activity, none of these three additionally observed scenarios are presently predictable. A critical stress must be developed by the incoming dislocation slip band impinging on the boundary in order to nucleate and propagate accommodating dislocation or twinning shear in the neighboring grain. Clearly, reaching such a critical stress will be a function of the relative orientations of t he two grains in the context of the applied global stress. Furthermore, the stress developed by an incoming slip band will also be a function of the number of dislocations in that particular slip band ( i.e., the size of the pile - up). It is also likely th at the stress necessary to nucleate a grain boundary dislocation source will be controlled in some manner by the atomic level structure of the particular grain boundary. At this point, however, such factors are not incorporated into the model. 14 In such scenario, those cloud of dislocations in the vicinity of grain boundary are usually scattered and do not form a clear slip trace on the sample surf ace, as a result the current AFM based slip trace analysis has very limited capability of studying this type of slip accommodation at grain boundary. 140 4.5.2.2 Shear accommodation by lattice rotation Recent observations by F. Di Gioacchino et al. [ 142 ] on L 1 0 TiAl indicate that slip accommodation across grain boundaries depends on the continuity of the overall resulting kinematics. Specifically, they sh owed that, in cases where well - aligned deformation systems (in the sense of Livingston and Chalmers [ 5 ] or Luster and Morris [ 11 ]) are not easily available in the outgoing grain, the imposed kinematics can still be realized through spatially coordinated sl ip - grain) accommodates the imposed shear. Due to the fact that the final kinematics are a combination of shearing and lattice rotation, any slip prediction based solely on shear kinematics, such as those strictly based on the Luster and Morris approach, will disfavor such accommodation. In contrast, the approach by Livingston and Chalmers, as well as classical crystal plasticity, will likely correctly predi ct such an accommodation, provided that the affected volume contains a sufficient density of suitable dislocations and the resolved stress favors their activity over that of others. This latter provision might be one reason why this particular accommodati on mechanism appears to be rarely reported compared to activity of well - aligned deformation systems. The iterative stress relief model presented here selects accommodating deformation systems based on a combination of resolved stress (similar to the appro ach of Livingston and Chalmers) and a balancing of the incoming Burgers vector with multiple outgoing Burgers vector contributions. What effect the latter aspect of the iterative stress relief model will have on correctly predicting lattice rotation - domin ated accommodation events is presently unclear and could not be rigorously evaluated to date since, in the present study of hexagonal - Ti, no such events were observed. 141 4.5.2. 3 Influence of grain boundary trace angle on shear transfer An other factor that the iterative stress relief model does not consider is the angle between slip plane traces on the grain boundary. It is quite natural to rationalize that a smaller would facilitate the shear transfer process, since it wou ld be easier for a dislocation to climb or cross - slip from the impinging slip plane to the accommodating plane at the grain boundary. The role of has often been ignored in slip transfer studies, due to the difficulty in measuring this angle from surfac e information alone. In the present study, FIB cross - sections were used in a number of cases to determine the grain boundary inclination and estimate (Fig. 2 .4 ). Based on the measured grain boundary inclination, can be determined from : Eq. 4.17 where and are the slip plane normal of the impinging and responding slip systems, and is the normal of the grain boundary plane which can be obtained from the grain boundary inclination . The angle was determined for two of the single accommodatio n cases and for five of the six double accommodation cases (excluding the twinning case). The results in Table 4.1 range from 8° to 54°, indicating that shear transfer will occur between slip planes with large angles. This is in agreement with the ran ge measured by Lee et al. [ 10 ], which spans from 7° to 60°, and slightly larger than that observed for the primary slip systems Han and Crimp [ 141 ], who observed ranging from 5° to 42°. Given this wide range of in all slip transfer cases, it is diff icult or even impossible to determine a threshold above which the slip transfer will not occur and use such a threshold in a model. Consequently, at present the angle has not been incorporated in the model, as only including and is sufficient to accurately describe and quantify the slip transfer geometry and activity. 142 4.5.3 Application of the ISR model to other materials and loading conditions The iterative stress relief model has been applied successfully in predicting shear accommodatio n in commercially pure hexagonal titanium, with the adjustable model parameters The optimized values of the CRSS ratios are expected to be different if the model were appli ed to a different material, but should be insensitive to variations in the material microstructure, e.g., changes in grain size or texture. Nevertheless, as the material work hardens, the CRSS ratios have the potential to evolve, as the slip system - specif ic work hardening can be variable (and dependent on grain orientation). In the present study, we have only considered small strains, but one could envision extending the approach to evolving CRSS ratios. Likewise, one would expect changes in alloy chemis try, in particular in non - cubic systems such as the hexagonal Ti considered here, to change the CRSS ratios (as is well known in Ti [ 140 , 143 - 145] ). Regardless, we would anticipate the model to still be effective following optimization of the CRSS ratios. It is worth noting that the CRSS ratio optimization carried out in the present study resulted in ratios that fell well within the variations report ed in the literature for commercial purity titanium. In some non - cubic materials, such as titanium, it is difficult to accurately measure the CRSS ratios. The approach outlined here has the potential to be used as an indirect approach to determining thes e ratios. Overall, it is reasonable to expect strong predictions from the model if accurate CRSS ratios are available. The model predictions were found to be optimal for intermediate values of and and were found to be minimally sensitive arou nd these optima. It is difficult to imagine that material changes, such as lattice structure or alloy chemistry, as well as loading conditions, will substantially alter this optimal range, as it is reasonable to expect that both global and local stress 143 wi ll play a role in dictating the shear accommodation process. Nonetheless, this should be explored experimentally in future study . At present, the iterative stress relief model was applied to a single phase materials, i.e., only to grain boundarie s. It can be envisioned that the same underlying mechanisms control shear across phase boundaries. While beyond the scope of this present study, it would be relatively simple to apply the iterative stress relief model to inter - phase shear transfer, with the definitions of slip systems, their orientations, and their CRSS ratios simply defined for the phases on either side of a boundary. With such a generalization, the model could be applied to a wide range of microstructures. While the present st udy has examined a simple tensile external loading condition, the superposition principle would apply under any global stress condition. Nevertheless, it is not clear at the present time how different loading conditions will affect the and parameters, if at all. Overall, it would be reasonable to expect that the iterative stress relief model should be flexible over a broad range of materials, boundary types, and loading conditions. 4.6 Conclusions In light of previous studies of grain boundary slip transfer and in order to develop a model that is capable of resolving shear accommodation from multiple deformation systems , a unified model that incorporates the continuity of Burgers vector in the grain boundary and the modification of the impinging stress tensor iteratively is proposed to solve the shear accommodation via multiple deformation systems. This iterative stress relief model is able to predict outgoing deformation systems and their relative shear, based on grain orientations, the incoming deformation system, and the critical resolved shear stress ratios as input. The accuracy 144 of this model w as tested by comparing predictions with observations of slip transfers in - titanium quantified using orientation informed slip trace analysis and quantitative AFM. The comparison shows that the iterative stress relief model is accurate when used in the p rediction of correlated shear transfer, e.g., in single and double accommodation cases, and even in one slip - to - twinning event. Optimization of this model was accomplished by blending the local and global stress state using a factor and balancing the cr itical resolved shear stress ratios to best match the experimental observations. The optimized value of the blending factor was found to be around 0.5, suggesting comparable influences from the global stress tensor and the local shear stress developed f rom the incoming slip band. The optimized critical resolved shear stress ratios were determined as basal < a > : prism < a > : pyramidal < a > : pyramidal < c + a > : T1twin = The sensitivity of the model to each CRSS ratio was studied. Nonetheless, this model is limited in that it does not effectively deal with cases where direct shear transfer is not observed, as it assumes there is always sufficient stress in the incoming slip band to initiate outgoing ac commodation. Consequently, it is not possible to judge the likelihood of whether the outgoing grain will nucleate an outgoing slip band or not. 145 CHAPTER 5 CONCLUSIONS In t he present work, improvements in the understanding of heterogenous deformation in hexagonal - Ti were achieved in three ways . Experimentally, the crystal orientation informed bi - crystal nanoindentation and the quantitative slip trace analysis have been im proved and adapted to quantitatively study the influence of a grain boundary on the development of heterogenous deformation. Coupled to the experimental work, CPFE simulations of both single and bi - crystal nanoindentations were carried out, the result of which w ere analyzed for insights into the individual slip system s . Based on the knowledge obtained experimentally and from simulation s , a new model (iterative stress relief model) was proposed to resolve the shear accommodation of multiple deformation sys tems. By studying how nanoindentation surface topography was affected by the presence of a grain boundary, a novel methodology for quantifying the resistance of the grain boundary to slip transfer was developed . Both single and bi - crystal models w ere built using a carefully crafted procedure based on experimentally gathered data. As a result, the comparisons between CPFE simulations of nanoindentations and experiments are meaningful, which makes the evaluation of the current CPFE model possible. The slip trace analysis approach for identifying active slip systems was improved by adding the surface profile of each slip band quantified using AFM. Th is improvement allows a more accurate determination of slip system s with the additional slip p lane inclination information acquired in AFM. Furthermore, with quantitative AFM analysis, the shear on each slip band can be obtained. This information is critical in the analysis of shear accommodation/transfer at grain boundary. 146 The new iterat ive stress relief model was developed by integrating the influence of global and local stress state change, continuity of Burgers vector at grain boundary, and the CRSS ratios. This model is highly accurate when appl ied to predict experimentally observed slip transfer events. The significance of the new model resides in its capability of predicting shear accommodation involving multiple deformation systems , as well as the relative shear on each of these slip system s . Critical conclusions of the cu rrent study are summarized in the following, more detailed conclusions are given at the end of each chapter: Single crystal indents in the grain with same orientation and bi - crystal indents near the same grain boundary were highly reproducible. Therefore, quantitative study of individual grain and/or grain boundary can be achieved by comparing indentations under different conditions. A procedure for assessing grain boundary resistance to slip transfer was established. The procedure includes measuring surface topographies of single and bi - crystal indentations and comparing the two topographies to quantify the difference. A CPFE model was built to simulate both single and bi - crystal indentation s. The parameters in the model were based on experimental measurements , such as crystal orientation, maximum depth of the indent, the grain boundary inclination, and the distance between indent and grain bou ndary. Such a carefully calibrated model allows direct comparison between simulated indent topographies and experimental measurements. The grain boundary resistance to slip transfer in the CPFE simulation s is lower than that found experimentally, as indic ated by the fact that the simulations display larger amount 147 of indent topographies across the grain boundary, indicating a weaker grain boundary in the model. The slip trace analysis identification of slip system was improved by adding the inclination of the slip band and quantifying the shear in each slip band using AFM measurement. This enhancement is critical as it ensures the identification of slip system unambiguously and enables more detailed analysis of she ar accommodation/transfer at grain boundary. An iterative stress relief model was proposed to predict the shear accommodation of multiple slip systems at grain boundary. The new model was tested along with the tangential continuity theory using same exper imental data set. The iterative stress relief model predicts the accommodating slip systems at grain boundary with high accuracy, making it much more robust than the tangential continuity model. Optimization of the model parameters showed that in order to correctly quantify the heterogenous deformation near a grain boundary it is necessary to account for both global stress and local kinematics. The optimized CRSS ratios of the model was determined to be basal < a > : prism < a > : pyramidal < a > : pyramidal < c + a > : T1 twin = 1.0 : . 148 CHAPTER 6 OUTLOOK The ultimate goal of characterizing the evolution of the full stress and strain field in the grain boundary vicinity remain s to be fulfilled . Nonetheless, the work in this dissertation provides a significant step forward in the understanding and quantification of plastic deformation in the grain boundary vicinity via both novel experimental methods and simulations. The establishment of the reproducibility of both single and bi - crystal indentation topographies allows further assessment of individual grain and grain boundary mechanical/microstructural properties. It will hopefully help to decrease the time for characterizing mechanical properties of material as indentation is a relatively fast testing methods compared to others. The comparison of slip system activities between experiment and CPFE simulations in bi - crystal nanoindentation were not completed in this work. More ac curate comparison requires characterization of individual dislocations using ECCI or TEM. Future effort s can be devoted to study the activity of individual dislocations and compare the results with CPFE or dislocations dynamics (DD) simulations. This com parison will help develop a n understand ing of the link between the development of macro heterogenous strain field s and the movement of individual dislocation. The newly developed iterative stress relief model is very successful in predicting active slip systems in the grain boundary vicinity. By implementing this grain boundary sensitive model into the CPFE method, the prediction of activities of slip systems in the grain boundary vicinity may be improv ed. 149 It will also be interesting to test the newly developed iterative stress relief model with material of different texture and/or crystal structure. It is not unlikely that the optimized parameters used in - Ti do not make accurate predictions i n a new material system. Nonetheless, the success of the iterative stress relief model suggests that the consideration of only kinematic and partial kinetic in an iterative algorithm is sufficient to resolve the shear accommodation at grain boundary. The present study of slip band development in polycrystal s excluded those slip band that were blocked by the grain boundary . With the improved slip trace analysis, AFM measurement and high resolution EBSD, it may be possible to evaluate the small latt ice rotation in the outgoing grain where no obvious slip band is present. C ross slip which is more common at larger strain is another potential topic for future investigation. The quantitative slip trace analysis with high resolution map for AFM might be the solution to identify the role of cross slip in grain boundary shear accommodation . 150 A PPENDICES 151 A PPENDIX A: AFM DATA OF INDENT TOPOGRAPHIES IN 3D The AFM data of indent topographies a and b in 3D are shown below. Figure A1: The AFM data of indent topographies a and b in 3D. a) Topography of indent a . b) Topography of indent b . c) The differences in topographies of a and b . 152 A PPENDIX B: SEM AND AFM MEASUREMENTS OF ALL SLIP ACCOMMODATION CASES Sec ondary electron images of the 16 single slip accommodation cases and 6 double slip accommodation cases studied in the Chapter 4 followed by the AFM measurements of the 6 double slip accommodation cases. Figure B 1: SEM images of single slip accommodation cases 1 - 4 . 153 Figure B 2: SEM images of single slip accommodation cases 5 - 8 . 154 Figure B 3: SEM images of single slip accommodation cases 9 - 12 . 155 Figure B 4: SEM images of single slip accommodation cases 13 - 16 156 Figure B 5: SEM images of d ouble slip accommodation cases 1 - 4 157 Figure B 6: SEM images of d ouble slip accommodation cases 5 and 6 (Note that there is a twinning involved shown as yellow dotted line in the EBSD map inset, in the accommodating deformation system in case 6) . 158 Figure B 7: AFM measurements of double slip accommodation cases 1 - 6 . 159 B IBLIOGRAPHY 160 BIBLIOGRAPHY [1] G.I. Taylor, Plastic strain in metals, Journal of the Institute of Metals 62 (1938) 307 - 324. [2] U.F. Kocks, Polyslip in polycrystals, Acta Metallurgica 6 (1958) 85 - 94. doi:10.1016/0001 - 6160(58)90117 - 2. [3] J.P. Hirth and J. Lothe, Theory of Dislocations. NY, USA: John Wiley & Sons, 1981. [4] T. R. B ieler, P. 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