First published 2017 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc.

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ISBN 978-1-84821-375-3

Table of Contents

Cover

Title

Acknowledgements

Introduction

I.1. Background and motivation
I.2. Objectives
I.3. Organization
I.4. Notations

1 Continuous Dislocation Modeling

1.1. Introduction
1.2. Lattice incompatibility
1.3. Burgers vector
1.4. Compatibility conditions
1.5. Dislocation fields
1.6. Tangential continuity at interfaces
1.7. Curvatures and rotational incompatibiliy
1.8. Incompatibility tensor
1.9. Conclusion
1.10. Problems
1.11. Solutions

2 Elasto-static Field Equations

2.1. Introduction
2.2. Elasto-static solution to field equations
2.3. Straight screw dislocation in a linear isotropic elastic medium
2.4. Straight edge dislocation in a linear isotropic elastic medium
2.5. Conclusion
2.6. Problems
2.7. Solutions

3 Dislocation Transport

3.1. Introduction
3.2. Dislocation flux and plastic distortion rate
3.3. Coarse graining
3.4. Compatibility versus incompatibility of plasticity
3.5. Tangential continuity of plastic distortion rate
3.6. Transport equations
3.7. Transport waves
3.8. Numerical algorithms for dislocation transport
3.9. Conclusion
3.10. Problems
3.11. Solutions

4 Constitutive Relations

4.1. Introduction
4.2. Dissipation
4.3. Pressure independence
4.4. Dislocation climb versus dislocation glide
4.5. Viscoplastic relationships
4.6. Coarse graining
4.7. Contact with conventional crystal plasticity

5 Elasto-plastic Field Equations

5.1. Introduction
5.2. Fundamental field equations
5.3. Boundary conditions
5.4. Coarse graining
5.5. Resolution algorithm
5.6. Reduced field equations
5.7. Augmented crystal plasticity
5.8. Dynamics of a twist boundary
5.9. Conclusion
5.10. Problems
5.11. Solutions

6 Case Studies

6.1. Introduction
6.2. Dislocation core structure
6.3. Piezoelectricity and dislocations
6.4. Intermittent plasticity
6.5. Effects of size on mechanical response
6.6. Complex loading paths
6.7. Strain localization

7 Review and Conclusions

7.1. Comparisons with conventional crystal plasticity
7.2. Alternative approaches
7.3. Shortcomings and extensions
7.4. Final remarks

Appendix: Complements

A.1. Stokes’ theorem
A.2. Characterization of the compatibility of a tensor field
A.3. Stokes-Helmholtz decomposition
A.4. Second-order Riemann-Graves operator

Bibliography

Index

End User License Agreement

List of Illustrations

Introduction

Figure I.1. Dislocation walls in Si single crystal cyclically loaded in tension - compression at high temperature [LEG 04]
Figure I.2. Optical micrography of giant dislocation cells after GaAs crystal growth. Note that the average cell size varies in inverse proportion to stress. Inset: dislocation cells through X-ray imaging, dark areas are the images of lattice distortion around dislocations [NEU 01]
Figure I.3. Slip lines on the surface of Cu30at%Zn single crystal strained in tension at 19.4% and 77K [ZAI 06]

1 Continuous Dislocation Modeling

Figure 1.1. Cross-section view of a dislocation line terminating the elastic displacement discontinuity across a bounded surface S. S and F are the starting and finishing points of an oriented closed Burgers circuit. For a color version of this figure, see www.iste.co.uk/fressengeas/dislocation.zip
Figure 1.2. Tangential continuity versus normal discontinuity of the transformation gradient F along the interface I₀ of unit normal e₃. Bottom left panel: tangential continuity requires that the components F_ij, (i = 1, 2, 3; j = 1, 2) be continuous across I. All other panels represent the admissible normal discontinuities of the transformation gradient. For a color version of this figure, see www.iste.co.uk/fressengeas/dislocation.zip
Figure 1.3. Sketch of the initial, relaxed and current configurations of a crystal containing a screw dislocation. The Burgers circuit being described clockwise, the oriented dislocation line (dashed line) enters the crystal. The Burgers vector F′S′ is colinear to the dislocation line and points out of the crystal. For a color version of this figure, see www.iste.co.uk/fressengeas/dislocation.zip
Figure 1.4. Burgers circuit across interface I separating the crystalline domain D into D⁻ and D⁺. n: unit normal vector to the interface and l, n × l: unit vectors in the interface
Figure 1.5. Ice polycrystal subjected to 7% compression strain in the vertical direction, at −5°C. Left: planar EBSD orientation map. The colors indicate the orientation of the compression axis in the hexagonal crystal axes, according to the code shown in the inverse pole figure. For example, the compression axis lies along direction < 0001 > in red crystals, < > in blue crystals and < > in green crystals. Boundaries with more than 5 deg misorientation, i.e. grain boundaries, are depicted as black lines, and 1.55 deg sub-grain boundaries as grey lines. Right: map of the norm of the captured dislocation density components (rad.m⁻¹), as obtained from 50μm spatial resolution EBSD data. Note the presence of significant dislocation densities on sub-grain boundaries. The indicated length scale is 2 mm [MON 15]. For a color version of this figure, see www.iste.co.uk/fressengeas/dislocation.zip
Figure 1.6. Recrystallized titanium, 1% strain in tension. The map shows the density of wedge disclinations θ₃₃ (in rad.m⁻²) obtained from a planar EBSD orientation map. The arrows represent the Burgers vector resulting from the dislocation densities (α₁₃, α₂₃) (α₁₃ and α₂₃ are the horizontal and vertical components, respectively, in m⁻¹). The maximum Burgers vector length corresponds to a 5.19×10⁶m⁻¹ dislocation density. The step size of the grid for orientation data is 0.2μm [BEA 13]. For a color version of this figure, see www.iste.co.uk/fressengeas/dislocation.zip
Figure 1.7. Terminating discontinuity (a) and its regularization (b). For a color version of this figure, see www.iste.co.uk/fressengeas/dislocation.zip
Figure 1.8. Frank’s relation: Burgers circuit

2 Elasto-static Field Equations

Figure 2.1. Building a twist boundary with rotation axis normal to the figure: solid and opened circles represent atoms just above and just below the boundary plane, respectively. (a) Atom positions resulting from a rotation of angle θ; (b) accommodation of the rotation by two orthogonal screw dislocation sets labeled $-$ [REA 53]
Figure 2.2. Dislocation structure consisting of loops superimposed with the loop (α₁₁, α₁₂) in the plane (e₁, e₂) (for clarity, a single loop is shown in the figure), such that the associated stress field uniformly vanishes. For a color version of this figure, see www.iste.co.uk/fressengeas/dislocation.zip

3 Dislocation Transport

Figure 3.1. Tangential continuity versus normal discontinuity of the plastic distortion rate along the interface of unit normal e₃. Bottom left panel: tangential continuity requires that the components , (i = 1, 2, 3; j = 1, 2) be continuous across . All other panels represent the admissible normal discontinuities of the distortion rate. Note that pressure independence of the plastic distortion rate may additionally require = 0. For a color version of this figure, see www.iste.co.uk/fressengeas/dislocation.zip
Figure 3.2. Space–time diagram for the longitudinal fluctuations of the imposed strain rate on the surface of a Cu single crystal in tension, during the early stages of plastic deformation. Localized plastic spots propagate at the average speed 1 cm/s along straight characteristic lines, reflecting both intermittency and transport of plastic activity. The imposed strain rate is ˙_a = 5 × 10⁻⁴s⁻¹. Fluctuations in strain rate can be as high as 2.5 × 10⁻³s⁻¹ [FRE 09]. For a color version of this figure, see www.iste.co.uk/fressengeas/dislocation.zip
Figure 3.3. Characteristic curves and stationary shock during the annihilation of two sets of edge dislocation densities of opposite signs and moving toward each other at a constant speed
Figure 3.4. Numerical simulation of the annihilation of two opposite half-sinusoidal edge dislocation ensembles of opposite velocities, using a Galerkin least-squares finite element method [TAU 07a]. The two sets move toward each other in Panel 2, and start to meet in Panel 3. A stationary shock forms in Panel 4 and recedes in Panel 5. Annihilation is completed in Panel 6. For a color version of this figure, see www.iste.co.uk/fressengeas/dislocation.zip
Figure 3.5. Numerical simulation of a Frank–Read source using a Galerkin least-squares method (β = 1) [VAR 06]. For a color version of this figure, see www.iste.co.uk/fressengeas/dislocation.zip
Figure 3.6. Numerical simulation of the annihilation of two half-sinusoidal edge dislocation ensembles using an unfiltered spectral method [DJA 15]. Spurious Gibbs oscillations show up at corners of the dislocation profiles, well before annihilation starts
Figure 3.7. Numerical simulation of the annihilation of two half-sinusoidal edge dislocation ensembles (Panel 1) using a spectral method with an exponential low-pass filter [DJA 15]. The Gibbs oscillations in evidence in Figure 3.6 are removed. The developing/receding shock is accurately rendered (Panels 2,3) and the final result is obtained without any debris (Panel 4). The exact solution is derived by translating the two edge dislocation profiles along the characteristic lines. The filter parameters are p = 1, γ = 0.7. The number of pixels is N = 1024. For a color version of this figure, see www.iste.co.uk/fressengeas/dislocation.zip
Figure 3.8. Regularizing the dislocation velocity (Problem 3.10.1, question 4)
Figure 3.9. Expansion fan of characteristic lines in regularizing the propagation of a dislocation density discontinuity. The fan shows the occurrence of shocks at points such as A

4 Constitutive Relations

Figure 4.1. Vectorial sketch of the Peach-Koehler-type driving force F, glide force F, dilatant direction d and dislocation velocity V

5 Elasto-plastic Field Equations

Figure 5.1. Space–time diagram and characteristic curves of α₁₁ (solid lines) and α₃₃ (broken lines) screw dislocations [HEA 93]
Figure 5.2. Helical dislocation in a cuprite (copper oxyde) sample [VEB 83]

6 Case Studies

Figure 6.1. Relaxation of an edge dislocation core in its own internal stress field. The blue, black and red consecutive snapshots show progressive spreading of the core, until it uniformly vanishes. (a) Edge dislocation density field . (b) Internal shear stress field T₁₂ [GBE 16]. For a color version of this figure, see www.iste.co.uk/fressengeas/dislocation.zip
Figure 6.2. Initial and relaxed edge dislocation cores as obtained from the original Peierls sinusoidal potential. a) surface dislocation density . The blue curve defines an initially narrow dislocation core (amplitude divided by 30), the red curve is an initially wide dislocation core (amplitude multiplied by 5) and the black curve is the relaxed core profile obtained from both initial distributions. b) Time-evolution of the energy density Ψ associated with the self-relaxation of the narrow (blue) and wide (red) dislocation cores shown in panel (a) [GBE 16]. For a color version of this figure, see www.iste.co.uk/fressengeas/dislocation.zip
Figure 6.3. Relaxed configuration of an edge dislocation core as obtained from the original Peierls potential. (a) Perfect match between the edge dislocation density (blue curve) and the Peierls–Nabarro solution for (purple dotted line), and between the misfit η₁₂ (in units of 2b) (red curve) and the Peierls–Nabarro solution for the misfit (green dotted line). (b) Perfect match between the shear stress T₁₂ (GPa) (blue curve) and the Peierls–Nabarro solution (purple dotted line), and between the restoring stress (GPa) (red curve) and the Peierls–Nabarro solution (green dotted line) [GBE 16]. For a color version of this figure, see www.iste.co.uk/fressengeas/dislocation.zip
Figure 6.4. Log–Log plot of the average dislocation velocity as a function of applied shear stress during loading of an edge dislocation core obtained using the Peierls sinusoidal potential and linear viscous drag of dislocations. The dotted line shows the linear slope [GBE 16]
Figure 6.5. Piezoelectric polarization field (magnitude) of a screw dislocation α₃₃. The dislocation line and Burgers vector are normal to the figure. The arrow reflects the divergence-free rotational pattern of the polarization field associated with the dislocation density field [TAU 14a]. For a color version of this figure, see www.iste.co.uk/fressengeas/dislocation.zip
Figure 6.6. Piezoelectric polarization field component P₃ of the edge dislocation density α₁₃ in the (x₁, x₂) plane. The Burgers vector is horizontal and the line vector normal to the figure [TAU 14a]. For a color version of this figure, see www.iste.co.uk/fressengeas/dislocation.zip
Figure 6.7. Simulation of a threading edge dislocation α₁₃ in a three-dimensional GaN volume. The straight dislocation lies along the (x₁ = x₂ = 0) line in the x₃ direction (black line). Its Burgers vector is in the x₁ direction. The color-code in the (x₁, x₂) and (x₂, x₃) sections grades the electric potential induced by the edge dislocation at the external surfaces. The white arrows reflect the electric displacement field in the central (x₂, x₃) section [TAU 14a]. For a color version of this figure, see www.iste.co.uk/fressengeas/dislocation.zip
Figure 6.8. Square dislocation loop embedded in a Wurtzite GaN domain. The loop’s Burgers vector is in the x₃ direction (vertical in the figure). The loop is composed of two α₃₁ edge dislocation segments in the x₁ direction and two α₃₃ screw segments in the x₃ direction. Only the upper half of the loop is visible above the (x₁, x₂) cross-section. (A) Initial configuration. (B) Expanded loop after application of electro-mechanical loading. The color-code grades the charge density in the cross-section. Arrows indicate the orientation and magnitude of the electric displacement vector field in the cross-section [TAU 14a]. For a color version of this figure, see www.iste.co.uk/fressengeas/dislocation.zip
Figure 6.9. (a) Cu single crystal oriented for multislip under uniaxial tension (Gauge length: 30 mm, width: 5.5 mm, thickness: 5.5 mm, Schmid’s factor: 0.3, temperature: 20^◦C, driving strain rate: = 5 × 10⁻⁴s⁻¹), macroscopic force versus time (main graph) and displacement versus time in six locations distant by 1 mm (inset). (b) Variations in the axial strain rate about the driving strain rate = 5 × 10⁻⁴s⁻¹, as obtained from the lowest displacement curves in the stack in panel A. Note that the maximum size of the fluctuations 2.5 × 10⁻³s⁻¹ is much larger than . (c) Probability density (normalized to bin size) for event size (amplitude of strain rate jerks) in the time series shown in panel B. The dashed line indicates the powerlaw trend with slope τ = 2. [FRE 09]. For a color version of this figure, see www.iste.co.uk/fressengeas/dislocation.zip
Figure 6.10. Longitudinal fluctuations about the imposed strain rate in a space–time diagram during the elasto-plastic transition. Dotted characteristic lines run from the left and right of the gauge length, reflecting intermittency and transport. The imposed strain rate is = 5 × 10⁻⁴s⁻¹. Fluctuations can be as high as 2.5 × 10⁻³s⁻¹. For a color version of this figure, see www.iste.co.uk/fressengeas/dislocation.zip
Figure 6.11. Simulation of the tensile test of a flat Cu whisker of dimensions 200 μm×30 μm×2400 μm; stress versus time response during the elasto-plastic transition and stage II linear hardening. The highlighted portion corresponds to the strain rate plots shown in Figure 6.12
Figure 6.12. Successive frames of the strain-rate spatio-temporal field along the sample showing intermittent plasticity through dislocation transport, and general progression of plastic slip from left to right. The sequence highlights a single plastic burst shown in Figure 6.11. For a color version of this figure, see www.iste.co.uk/fressengeas/dislocation.zip
Figure 6.13. Model predictions for axial (x₁) strain-rate fluctuations in a space–time diagram. The sample is a 13 mm × 13 mm square in a glide plane subjected to equal shear rates 5 × 10⁻⁴s⁻¹ on both sides. The figure shows the evolution in time of the strain rate profile seen along the x₂ direction. For a color version of this figure, see www.iste.co.uk/fressengeas/dislocation.zip
Figure 6.14. (a) Excess dislocation density field showing self-organization in the form of loops in a square 100 μm×100 μm sample in the glide plane, at 0.5% plastic strain. The imposed strain rate is 5 × 10⁻⁴s⁻¹. (b) Time evolution along 50000 time steps of fluctuations about the applied strain rate at a given point in panel A. (c) Probability density of event size. The event size is defined as the maximum strain rate value during the event. The dotted line shows a τ = 2 slope [FRE 09]. For a color version of this figure, see www.iste.co.uk/fressengeas/dislocation.zip
Figure 6.15. Creep strain on outer surface versus time, for various diameter values. The average shear stress is = 0.12 MPa. In each sample, height and diameter are equal (their common value is given in mm), in order to avoid any bias due to end effects [CHE 08]. For a color version of this figure, see www.iste.co.uk/fressengeas/dislocation.zip
Figure 6.16. Distribution of screw dislocation density (m⁻²) in basal planes along the torsion axis/c-axis, as obtained from Laue hard X-ray diffraction [CHE 10]. The strain is γ = 56%. Note the strong heterogeneity of the distribution, despite smoothness of the macroscopic strain field. The mean spatial resolution is 1.7 μm. For a color version of this figure, see www.iste.co.uk/fressengeas/dislocation.zip
Figure 6.17. Excess screw dislocations α₁₁ piercing the plane with normal in the x₁ direction and α₂₂ piercing the plane with x₂ normal: (a) just before reversal in Figure 6.19 and (b) the structure developed after reversing the direction of creep, at the end of the curve [TAU 07b, FRE 11b]. For a color version of this figure, see www.iste.co.uk/fressengeas/dislocation.zip
Figure 6.18. Shear stress component T₁₃ shown at the beginning (a) and end (b) of the blue dashed curve in Figure 6.19. The figure highlights the development of stress due to the multiplication of excess dislocations, from the (effectively) elastic solution with low mobile density. Panel A also shows end effects in the distribution of stress [TAU 07b, FRE 11b]. For a color version of this figure, see www.iste.co.uk/fressengeas/dislocation.zip
Figure 6.19. Creep curves in forward/reverse torsion from experiments, one-dimensional and three-dimensional models. The blue dashed line shows the forward creep curve for a sample with halved radius and height. It is seen that the acceleration of creep increases when the sample size is reduced. The green continuous line, obtained for conventional elasto-viscoplastic (EVP) crystal plasticity, shows that the latter is unable to retrieve the acceleration of creep [FRE 11b]. For a color version of this figure, see www.iste.co.uk/fressengeas/dislocation.zip
Figure 6.20. Experimental data from the creep tests shown in Figure 6.15 for various sample diameters (solid lines) and simulated creep curves from the one-dimensional model (dashed lines). Consistency of the initial conditions on the screw dislocation density and the measured initial strain rate is ensured [CHE 08]. For a color version of this figure, see www.iste.co.uk/fressengeas/dislocation.zip
Figure 6.21. Sample diameter effects on acceleration of creep rate at torsion reversal and on creep rate deceleration during reverse torsion, from reversal to inflexion of the creep curve. Blue squares and diamonds: experimental data from creep tests. Red squares and diamonds: predictions from one-dimensional model. Experimental data and predictions coincide for the smallest diameter [FRE 11b]. For a color version of this figure, see www.iste.co.uk/fressengeas/dislocation.zip
Figure 6.22. Stress–strain curves during tension–aging–tension (solid line) and tension–aging–compression (dashed line) experiments on a 1020 mild steel. The aging period is marked with an orange dot. For convenience, the sign of the compression stress is reversed. Both tension and compression stresses are plotted against the same cumulative strain [TAU 08]. Note the presence of a Bauschinger effect in compression. For a color version of this figure, see www.iste.co.uk/fressengeas/dislocation.zip
Figure 6.23. Tensile stress (normalized by the initial slip system strength) versus strain curves in three-dimensional simulations of tension–aging–forward tension and tension–aging–compression. The applied strain rate is 6 × 10⁻⁵s⁻¹. For convenience, the stress sign is reversed in compression. For a color version of this figure, see www.iste.co.uk/fressengeas/dislocation.zip
Figure 6.24. Equivalent strain rate contours for the Lüders band corresponding to the stress plateau in Figure 6.23 at two instants A and B. After nucleation to the left, the band propagates to the right end of the sample. Note the inclination of the band, at an angle with both the axial and transverse directions [TAU 08]. For a color version of this figure, see www.iste.co.uk/fressengeas/dislocation.zip
Figure 6.25. Torque evolution during forward torsion–aging–forward torsion and forward torsion–aging–reverse torsion simulations. The aging period is marked with an orange dot. For convenience, the torque sign is reversed in reverse torsion. Both forward and reverse torques are plotted against the same cumulative strain. For a color version of this figure, see www.iste.co.uk/fressengeas/dislocation.zip
Figure 6.26. Solute stress evolution during forward torsion–aging–forward torsion and forward torsion–aging–reverse torsion. The aging period is marked with an orange dot [TAU 08]. For a color version of this figure, see www.iste.co.uk/fressengeas/dislocation.zip
Figure 6.27. Evolution of excess screw dislocation density during forward torsion–aging–forward torsion and forward torsion–aging–reverse torsion. The aging period is marked with an orange dot [TAU 08]. For a color version of this figure, see www.iste.co.uk/fressengeas/dislocation.zip
Figure 6.28. Back-stress evolution during forward torsion–aging–forward torsion and forward torsion–aging–reverse torsion simulations. The aging period is marked with an orange dot. For a color version of this figure, see www.iste.co.uk/fressengeas/dislocation.zip
Figure 6.29. Morphological texture in a rolled Al–Cu–Li sheet in the L–S plane. The rolling L and through-thickness S directions are horizontal and vertical, respectively [TAU 16]. For a color version of this figure, see www.iste.co.uk/fressengeas/dislocation.zip
Figure 6.30. Tensile tests on Al–Cu–Li rolled samples along the rolling and transverse directions L, T for different thicknesses. Engineering stress (MP a) versus engineering strain (%). The 1.2 mm-thick samples were obtained by further machining the 2.5 mm and 4.2 mm-thick rolled sheets [TAU 16]. For a color version of this figure, see www.iste.co.uk/fressengeas/dislocation.zip
Figure 6.31. Through-sample shear band in the necking region in tensile loading along the transverse direction T. Grains are distinctly elongated along the rolling direction L, normal to the tensile axis. The 1.2 mm-thick sample is obtained by machining a 2.5 mm-thick rolled sheet (black dashed line inFigure 6.30) [TAU 16]
Figure 6.32. Morphology of a slab comprising 19 grains of thickness 6 μm used in the simulations. (a) tension along L direction. (b) tension along T direction. The rolling frame is shown at the bottom. The color-code is relative to the Euler angle φ(rad) of the crystal orientation [TAU 16]. For a color version of this figure, see www.iste.co.uk/fressengeas/dislocation.zip
Figure 6.33. Left: in-plane plastic shear strain field, (a) in standard crystal plasticity, (b) with tangential continuity conditions implemented. Right: Excess dislocation density field (c) in standard crystal plasticity (norm maximum values in red are 0.3 μ⁻¹_m), and (d) with tangential continuity conditions implemented (component α_LS (μ⁻¹_m)). Simulation results in the L–S cross-section at a strain of 10% in simple tension in the direction L of the configuration shown in Figure 6.32(a). The L direction is horizontal and S is vertical. For a color version of this figure, see www.iste.co.uk/fressengeas/dislocation.zip
Figure 6.34. Effective strain field in the L–S cross-section, (a) in the absence of tangential continuity conditions, (b) with tangential continuity conditions implemented, at 10% applied strain in simple tension in the direction L of the configuration shown in Figure 6.32(a) with s = 3 μm. The L direction is horizontal and S is vertical. For a color version of this figure, see www.iste.co.uk/fressengeas/dislocation.zip
Figure 6.35. Strain field ₂₂, (a) in standard crystal plasticity, (b) with tangential continuity conditions implemented. Simulation results in the L–S cross-section at 10% applied strain in simple tension in the direction L of the configuration shown in Figure 6.32. The L direction is horizontal and S is vertical. For a color version of this figure, see www.iste.co.uk/fressengeas/dislocation.zip
Figure 6.36. Effective strain field at 10% applied strain. Left: tension along L direction of a slab composed of (a) 6 μm-thick grains, and (b) 3 μm-thick grains. Right: tension along T direction of a slab with (c) 6 μm-thick grains and (d) 3 μm-thick grains. The rolling frame is shown at the bottom. For a color version of this figure, see www.iste.co.uk/fressengeas/dislocation.zip

7 Review and Conclusions

Figure 7.1. Generalized stacking fault energy over the basal plane in hexagonal close packed titanium (a) and zirconium (b), as obtained from ab initio calculations and approximated by the Fourier series. The dashed white arrows show the minimum generalized stacking fault energy paths for dissociation of the Burgers vector into partials. The blue/green spots show paths corresponding to dissociated edge/screw dislocation cores. Panel (c) shows the energy landscape along < 1 − 100 > type dissociation paths in Ti (dotted line) and Zr (solid line). a is the lattice parameter [GBE 16]. For a color version of this figure, see www.iste.co.uk/fressengeas/dislocation.zip
Figure 7.2. Simulation of plastic activity in a spherically [111] nano-indented Cu single crystal (nanoindenter radius 40 nm) using a Discrete Dislocation Dynamics code. Prismatic dislocation loops develop along three directions under the indenter. Steps corresponding to the emergence of dislocations are seen at the surface of the crystal [CHA 10] (courtesy of M. Fivel). For a color version of this figure, see www.iste.co.uk/fressengeas/dislocation.zip
Figure 7.3. Wedge disclination density field θ₃₃ (rad.m⁻²) shown on top of the atomic structure of a Σ5(310) symmetrical < 001 > tilt boundary of misorientation 36.9° in a copper bicrystal [FRE 14a]. For a color version of this figure, see www.iste.co.uk/fressengeas/dislocation.zip
Figure 7.4. Disclination line: terminating curve of rotation discontinuity (left). G-disclination line: terminating curve of distortion (strain and rotation) discontinuity (right) [ACH 12]. For a color version of this figure, see www.iste.co.uk/fressengeas/dislocation.zip
Figure 7.5. (a): Shear strain field ϵ₂₃ and (b): generalized disclination density field π₃₂₁ (rad.m⁻²) shown on top of the atomic structure of a Σ37(610) symmetrical < 100 > tilt boundary of misorientation 18.9° in a copper bicrystal. Both fields are obtained from a continuous rendition of molecular dynamics descriptions of the tilt boundary [SUN 16]. The g-disclination field arises from a rotation discontinuity (not shown here) and the shear strain discontinuities seen along the boundary. For a color version of this figure, see www.iste.co.uk/fressengeas/dislocation.zip

Appendix: Complements

Figure A.1. Stokes’ theorem: orientation of bounding curves C in (a) simply and (b) multiply connected domains. In panel (b), the bounded surface S has a hole and C consists of the outer boundary C₁ and inner boundary C₂ with clockwise and counter-clockwise orientations, respectively. If S were cut by the oriented dashed lines extending from C₁ to C₂ such that the resulting surface would be simply connected, then its contour would be continuously oriented

Acknowledgements

I am indebted to my colleagues: Amit Acharya, Garani Ananthakrishna, Armand J. Beaudoin and Ladislas P. Kubin for the numerous discussions on the plasticity of crystalline materials that inspired the work which finally led to writing this book. I also warmly thank my numerous collaborators: first of all, Vincent Taupin who shared most of this work, and Benoît Beausir, Stéphane Berbenni, Laurent Capolungo, Juliette Chevy, Patrick Cordier, Sylvie Demouchy, Komlan S. Djaka, Paul Duval, Denis Entemeyer, Tatiana Lebedkina, Mikhail Lebyodkin, Russell J. McDonald, Maurine Montagnat, Thiebaud Richeton, Xiaoyu Sun, Satya Varadhan, Pascal Ventura, Guofeng Wang and Jérôme Weiss, who participated in joint publications, and whose work significantly contributed to the contents of the present book. Be they all ensured of my gratitude.