UPADHYAY-DISSERTATION-2014

422 Pages·2014·16.84 MB·English

ON THE ROLE OF DEFECT INCOMPATIBILITIES ON
MECHANICAL PROPERTIES OF POLYCRYSTALLINE
AGGREGATES: A MULTI-SCALE STUDY
A Thesis
Presented to
The Academic Faculty
by
Manas Vijay Upadhyay
In Partial Fulfillment
of the Requirements for the Degree
Doctor of Philosophy in the
School of Mechanical Engineering
Georgia Institute of Technology
December 2014
Copyright (cid:13)c 2014 by Manas Vijay Upadhyay
ON THE ROLE OF DEFECT INCOMPATIBILITIES ON
MECHANICAL PROPERTIES OF POLYCRYSTALLINE
AGGREGATES: A MULTI-SCALE STUDY
Approved by:
Professor Laurent Capolungo, Advisor Professor Claude Fressengeas
School of Mechanical Engineering Universit´e de Lorraine
Georgia Institute of Technology LEM3/CNRS France
Professor David McDowell Professor Hamid Garmestani
Department of Mechanical Department of Materials Science and
Engineering & Materials Science and Engineering
Engineering Georgia Institute of Technology
Georgia Institute of Technology
Professor Mohammed Cherkaoui Date Approved: 1 October 2014
Department of Mechanical
Engineering
Georgia Institute of Technology
To my parents,
Vijay Ishwarlal Upadhyay and Shweta Vijay Upadhyay,
and my sister,
Trusha Vijay Upadhyay,
for their unwavering support throughout this journey.
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ACKNOWLEDGEMENTS
I would like to thank: my dissertation advisor and mentor Dr. Laurent Capolungo
for his enthusiastic support for my work during the entire duration of this thesis. My
colleagues Pierre-Alexandre Juan, Nicolas Bertin, Aaron Dunn and Cameron Sobie
who have been the best lab mates and friends that one can ask for. My collaborators
Dr. VincentTaupinandDr. ClaudeFressengeasfortheirpatienceandvaluableinputs
during the various discussions that we had over the past five years. My dissertation
reading committee for their valuable inputs on the thesis objectives. The staff at
both Georgia Tech Lorraine and Georgia Tech Atlanta for their help and for being my
family away from home. The Georgia Tech library department for always providing
me with the literature that I asked for.
This PhD has resulted in several international collaborations for which I am very
grateful to Dr. Laurent Capolungo who has always encouraged me to pursue them.
I would like to thank them for their confidence in my work, especially Dr. Ricardo
Lebensohn.
And last but not the least, I would like to thank my parents Vijay Ishwarlal
Upadhyay and Shweta Vijay Upadhyay, and my sister Trusha Vijay Upadhyay for
their belief in my capabilities and their unwavering support during the pursuit of
PhD.
iv
TABLE OF CONTENTS
DEDICATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii
ACKNOWLEDGEMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . iv
LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix
LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x
SUMMARY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxiii
I INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.1.1 Challenges . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.2 Scope of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.3 Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.3.1 Superscripts and subscripts . . . . . . . . . . . . . . . . . . . 12
1.3.2 Mathetical notations and formulae . . . . . . . . . . . . . . . 12
1.3.3 Field variables . . . . . . . . . . . . . . . . . . . . . . . . . . 13
II INCOMPATIBLE THEORY OF LINE CRYSTAL DEFECTS . . 15
2.1 Discrete line defects . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.1.1 Statics of discrete dislocations . . . . . . . . . . . . . . . . . 21
2.1.2 Statics of discrete disclinations . . . . . . . . . . . . . . . . . 31
2.2 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
2.2.1 Geometric equivalence between disclinations and dislocations 40
2.2.2 Connecting disclinations to crystallography: State of the art 48
2.2.3 Experimental observation of disclinations . . . . . . . . . . . 61
2.2.4 Need for a fully continuous approach . . . . . . . . . . . . . . 64
2.3 Continuously distributed line defects . . . . . . . . . . . . . . . . . . 68
2.3.1 Geometric fields of a compatible body . . . . . . . . . . . . . 68
2.3.2 Geometric fields of continuously distributed dislocations . . . 72
v
2.3.3 Geometric fields of continuously distributed dislocations and
disclinations . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
2.4 Understanding incompatibility . . . . . . . . . . . . . . . . . . . . . 85
2.5 Summary and discussion . . . . . . . . . . . . . . . . . . . . . . . . 92
2.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
III MULTI-SCALE STATIC AND DYNAMIC FIELD THEORY OF
CONTINUOUSLY DISTRIBUTED DEFECTS . . . . . . . . . . . 96
3.1 Higher order/grade multi-scale elastic constitutive laws . . . . . . . 97
3.1.1 Elastic constitutive laws for a simply connected body sub-
jected to tractions and moments . . . . . . . . . . . . . . . . 98
3.1.2 Scale dependence of higher order elastic constants: non-locality108
3.1.3 Classification of higher order/grade elastic laws . . . . . . . . 109
3.1.4 Isotropic case - elasticity tensors and constitutive relationships 118
3.2 Fine scale static field theory of disclinations and dislocations . . . . 130
3.2.1 Governing equations . . . . . . . . . . . . . . . . . . . . . . . 130
3.3 Fine scale dynamic field theory of disclinations and dislocations . . . 132
3.3.1 Governing equations . . . . . . . . . . . . . . . . . . . . . . . 132
3.4 Incompatibilities at meso-scale . . . . . . . . . . . . . . . . . . . . . 141
3.5 Meso-scale dynamic field disclination and dislocation mechanics . . . 142
3.5.1 Phenomenological meso-scale fields . . . . . . . . . . . . . . . 143
3.5.2 Plasticity beyond dislocations . . . . . . . . . . . . . . . . . 149
3.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
3.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
IV TOWARDS A CONTIUOUS STRUCTURE SENSITIVE MODEL
OF GRAIN BOUNDARIES: APPLICATIONS TO <001> SYM-
METRIC TILT GBS AND TJS . . . . . . . . . . . . . . . . . . . . . 155
4.1 Incompatibility contribution to energy of <001> STGBs . . . . . . . 156
4.1.1 Disclination structural unit model . . . . . . . . . . . . . . . 156
4.1.2 GB energy in discrete static case: contribution of compatible
elastic strains and curvatures . . . . . . . . . . . . . . . . . . 163
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4.1.3 GB energy in continuous dynamic case: contribution of in-
compatible elastic strains and curvatures . . . . . . . . . . . 166
4.1.4 Continuous modelling at interatomic scale: rationalization . . 168
4.2 TJs from <001> STGBs . . . . . . . . . . . . . . . . . . . . . . . . 171
4.2.1 Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
4.2.2 Compatibility conditions . . . . . . . . . . . . . . . . . . . . 174
4.2.3 Relationship between TJ geometry and excess energy . . . . 180
4.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
4.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196
V TOWARDS A CRYSTAL PLASTICITY MODEL BEYOND DIS-
LOCATIONSLIP:MESO-SCALEAPPLICATIONSTOFCCPOLY-
CRYSTALS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198
5.1 Characterizing the microstructure: interfacial polar defect densities
and curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
5.2 PMFDDM using a continuous fast Fourier transform technique . . . 201
5.2.1 Modelling framework . . . . . . . . . . . . . . . . . . . . . . 202
5.2.2 Numerical scheme . . . . . . . . . . . . . . . . . . . . . . . . 206
5.3 Capturing curvature contribution to polycrystalline response: strategy 213
5.3.1 Microstructures: elastic/plastic properties and geometry . . . 213
5.3.2 Characterizing microstructures with initial curvatures . . . . 216
5.3.3 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . 220
5.3.4 Case studies . . . . . . . . . . . . . . . . . . . . . . . . . . . 221
5.4 Benchmarking and validation of PMFDDM FFT with initial curvatures223
5.4.1 Comparison with EVP FFT . . . . . . . . . . . . . . . . . . 223
5.4.2 Gibbs phenomenon: correction using DFT . . . . . . . . . . 225
5.5 Pure elastic tensile loading: impact of initial curvatures on local elas-
tic response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237
5.6 Relaxationofinitialstresses: understandingtheroleofcurvatureplas-
ticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244
5.6.1 Strain plasticity . . . . . . . . . . . . . . . . . . . . . . . . . 244
5.6.2 Combined strain and curvature plasticity . . . . . . . . . . . 247
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5.7 Uniaxial tension: impact of residual curvature on stress-strain response253
5.8 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256
5.9 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258
VI VIRTUAL DIFFRACTION: MULTI-SCALE CHARACTERIZA-
TION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260
6.1 Virtual diffraction peaks from a defected crystal: theoretical review . 261
6.2 A new average-strain based Fourier method . . . . . . . . . . . . . . 268
6.3 Establishing domains of applicability . . . . . . . . . . . . . . . . . . 275
6.3.1 Computational methodology . . . . . . . . . . . . . . . . . . 275
6.3.2 Applications to Dislocations in Single Crystal Microstructures 278
6.3.3 Dislocation discontinuity surface cut . . . . . . . . . . . . . . 292
6.4 Diffraction as a characterization technique . . . . . . . . . . . . . . . 295
6.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301
VIICONCLUSIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303
APPENDIX A — G-DISCLINATIONS: ELASTIC LAWS . . . . . 306
APPENDIX B — HIGHER ORDER ELASTICITY TENSORS . 313
APPENDIX C — MULTI-SCALE DESCRIPTIONS OF INTER-
FACES IN CONTINUOUS MEDIA . . . . . . . . . . . . . . . . . . 320
APPENDIX D — KELVIN DECOMPOSITION OF AN ELASTIC-
ITY TENSOR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332
APPENDIX E — STATIC FDDM FFT IN HETEROGENEOUS
ELASTICITY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 340
APPENDIX F — DISCRETE FOURIER TRANSFORMS . . . . 348
APPENDIX G — PMFDDM ALGORITHM . . . . . . . . . . . . . 357
REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361
viii
LIST OF TABLES
1 Free energy density and elastic laws of linear compatible models . . . 112
2 Free energy density and elastic laws of linear incompatible models . . 115
3 Isotropic elastic laws for compatible and incompatible media . . . . . 126
4 Structural unit decomposition of symmetric tilt GBs about the [001] axis158
5 CharacteristiclengthofthemajorityandminorityunitsandtheFrank’s
vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
6 Legend for flowchart 1 . . . . . . . . . . . . . . . . . . . . . . . . . . 211
7 Simulations to highlight the contribution of residual curvatures on the
local and bulk mechanical response of nc materials . . . . . . . . . . . 222
8 Green’s function and fictive body force . . . . . . . . . . . . . . . . . 345
9 Legend for algorithm 1 . . . . . . . . . . . . . . . . . . . . . . . . . . 357
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LIST OF FIGURES
1.1 Microstructure of nickel discs subjected to high pressure torsion for (a)
the central part of the disk after 5 rotations at 1 GPa pressure, (b) the
edge of the disk after 5 rotations at 1 GPa and (c) the edge of the
disk after 5 rotations at 6 GPa: Inset are the selective area electron
diffraction patterns that were taken with an aperture size of 1.8 µm.
(Adapted from Zhilyaev and Langdon et al. [483]) . . . . . . . . . . . 4
1.2 (a) View of the region along the Σ9 GB including the TJ and the
quadruple point. Different boundaries are indicated, (b) Rigid body
rotation measured in the form of a line profile along the Σ9 boundary
and averaged over the width of the box. Two gradients of opposite
sign emerge from the TJ and the quadruple point, respectively. The
dotted line indicates the average grain misorientation. Due to the
wavy character of the Σ9 boundary the rigid body rotation occurs as a
periodically modulated signal along its length. Note the jump in rigid
body rotation at ≈ 0.6 nm (adapted from R¨osner et al. [341]). . . . . 6
1.3 Schematic of different length scales and their associated plasticity.
(Adapted from Cherkaoui and Capolungo [72]) . . . . . . . . . . . . . 6
2.1 Volterra’s dislocations and disclinations (adapted from Romanov and
Kolesnikova [333]). Note here that |Ω(cid:126)| = ω . . . . . . . . . . . . . . . 17
2.2 Negative wedge disclination of strength |Ω(cid:126)| = −ω inserted into the
region between the cut faces ABA’B’ of figure 2.1 . . . . . . . . . . . 19
2.3 A curved dislocation line l with a Burgers vector b in an infinite con-
tinuous medium, bounded by an arbitrarily shaped defect surface S
and encircled by an arbitrarily shaped Burgers circuit λ, traverses the
Burgers surface σ at an arbitrary angle. The grey and black dotted
arrows indicate the possible extensions of S and l, respectively. The
line direction and the sense of λ make the crossings between l and σ
negative according to the right-hand rule. . . . . . . . . . . . . . . . . 23
2.4 Semi-infinite defect surface S along the −x axis and bounded by the
infinitelylongstraightdefectalongthez-axisin(a)3-dimensions. Blue
arrows indicate the directions of infinite extension of S and the red
dots indicate the critical points of intersection of the surface with the
coordinateaxes. (b)z = 0planeshowingthedefectsurfacerepresented
as a semi-infinite line bounded at the origin and extending along the
−x axis. φ defines the angular position of the point ρ in the xy plane 25
x