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The Matlab toolbox STABiX provides a unique and simple way to analyse slip transmission in a bicrystal. Graphical User Interfaces (GUIs) are implemented in order to import EBSD results, and to represent and quantify grain boundary slip resistance. Key parameters, such as the number of phases, crystal structure (fcc, bcc, or hcp), and slip families for calculations, are set by the user. With this information, grain boundaries are plotted and color coded according to the m′ factor [1] that quantifies the geometrical compatibility of the slip planes normals and Burgers vectors of incoming and outgoing slip systems. Other potential functions that could assess the potential to develop damage are implemented (e.g. residual Burgers vector [2] and [3], N factor [4], resolved shear stress [5], misorientation...). Furthermore, the toolbox provides the possibility to plot and analyze the case of a bicrystal, and to model sphero-conical indentation performed in a single crystal or close to grain boundaries (i.e. quasi bicrystal deformation). All of the data linked to the bicrystal indentation (indenter properties, indentation settings, grain boundary inclination, etc.) are collected through the GUI. A pythonTM file can be then exported in order to carry out a fully automatic 3D crystal plasticity finite element simulations of the indentation process using one of the constitutive models available in DAMASK [6] and [7]. The plasticity of single crystals is quantified by a combination of crystal lattice orientation mapping, instrumented sphero-conical indentation, and measurement of the resulting surface topography [8] and [9]. In this way the stress and strain fields close to the grain boundary can be rapidly assessed. Activation and transmission of slip are interpreted based on these simulations and the mechanical resistance of grain boundaries can be quantified. [1] J. Luster and M.A. Morris, “Compatibility of deformation in two-phase Ti-Al alloys: Dependence on microstructure and orientation relationships.”, Metal. and Mat. Trans. A (1995), 26(7), pp. 1745-1756. [2] M.J. Marcinkowski and W.F. Tseng, “Dislocation behavior at tilt boundaries of infinite extent.”, Metal. Trans. (1970), 1(12), pp. 3397-3401. [3] W. Bollmann, “Crystal Defects and Crystalline Interfaces”, Springer-Verlag (1970). [4] J.D. Livingston and B. Chalmers, “Multiple slip in bicrystal deformation.”, Acta Metallurgica (1957), 5(6), pp. 322-327. [5] T.R. Bieler et al., “The role of heterogeneous deformation on damage nucleation at grain boundaries in single phase metals.”, Int. J. of Plast. (2009), 25(9), pp. 1655–1683. [6] F. Roters et al., “Overview of constitutive laws, kinematics, homogenization and multiscale methods in crystal plasticity finite-element modeling: Theory, experiments, applications.”, Acta Materialia (2010), 58(4), pp. 1152-1211. [7] DAMASK — the Düsseldorf Advanced Material Simulation Kit. [8] C. Zambaldi et al., “Orientation informed nanoindentation of α-titanium: Indentation pileup in hexagonal metals deforming by prismatic slip”, J. Mater. Res. (2012), 27(01), pp. 356-367. [9] C. Zambaldi, “Anisotropic indentation pile-up in single crystals”.
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STABiX Documentation
Release 2.0.0
Mercier D., Zambaldi C. and Bieler T.R.
Nov 07, 2017
Contents
1 How to get STABiX code ? 3
2 How to cite STABiX in your papers ? 5
3 Reference paper 7
4 Contents 9
4.1 Motivation of this Work ......................................... 9
4.1.1 Strategy ............................................. 10
4.2 Getting started .............................................. 10
4.2.1 Source Code ........................................... 10
4.2.2 READ ME ........................................... 10
4.2.3 Path management ........................................ 11
4.2.4 The GUIs ............................................ 11
4.2.5 The YAML configuration files ................................. 12
4.2.6 MTEX toolbox ......................................... 12
4.2.7 OpenGL ............................................. 12
4.3 Bicrystal Definition ........................................... 12
4.3.1 Crystallographic properties of a bicrystal ............................ 12
4.4 Strain Transfer Across Grain Boundaries ................................ 15
4.4.1 Geometrical Criteria ...................................... 15
4.4.2 Stress Criteria .......................................... 21
4.4.3 Combination of Criteria ..................................... 22
4.4.4 Relationships between slip transmission criteria ........................ 23
4.4.5 Slip transmission parameters implemented in the STABiX toolbox .............. 23
4.4.6 Slip and twin systems implemented in the STABiX toolbox .................. 24
4.4.7 References ........................................... 24
4.5 Experimental data ............................................ 24
4.5.1 EBSD map GUI ......................................... 24
4.5.2 Bicrystal GUI .......................................... 26
4.5.3 Convention for bicrystal EBSD/indentation experiments ................... 27
4.6 EBSD map GUI ............................................. 27
4.6.1 Loading EBSD data ....................................... 28
4.6.2 Smoothing GBs segments .................................... 28
4.6.3 Misorientation angle ...................................... 29
4.6.4 m’ parameter .......................................... 29
4.6.5 Residual Burgers vector ..................................... 29
i
4.6.6 Schmid factor and slip trace analysis .............................. 29
4.7 Bicrystal GUI ............................................... 29
4.7.1 Loading Bicrystal data ..................................... 32
4.7.2 Plotting and analyzing a bicrystal ................................ 32
4.7.3 Distribution of all slip transmission parameters ........................ 32
4.8 CPFE simulation preprocessing GUIs .................................. 32
4.8.1 How to load crystallographic properties of the SX or of the BX ? ............... 34
4.8.2 Single crystal (SX) indentation ................................. 34
4.8.3 Bicrystal (BX) indentation ................................... 36
4.8.4 Scratch test on SX and BX ................................... 37
4.8.5 Indenter’s geometry ....................................... 37
4.8.6 Contact definition ........................................ 39
4.8.7 Mesh definition ......................................... 39
4.8.8 Python setup .......................................... 40
4.8.9 Adjusting the configuration settings .............................. 40
4.8.10 Installing DAMASK ...................................... 40
4.8.11 Writing the CPFE input files .................................. 40
4.8.12 Input files ............................................ 40
4.8.13 Using the CPFE input files ................................... 41
4.8.14 Running a job with DAMASK ................................. 41
4.8.15 See also ............................................. 42
4.9 Analysis of literature data ........................................ 43
4.9.1 Residual Burgers vector ..................................... 44
4.9.2 m’ factor ............................................ 47
4.10 A Matlab toolbox to analyze grain boundary inclination from SEM images .............. 47
4.10.1 How to use the toolbox ? .................................... 48
4.10.2 See also ............................................. 50
4.10.3 Links .............................................. 50
4.10.4 Authors ............................................. 50
4.10.5 Acknowledgements ....................................... 50
4.10.6 Keywords ............................................ 50
4.11 References ................................................ 51
4.11.1 Related Projects ......................................... 51
4.11.2 Institutions ........................................... 51
5 References 55
6 Contact 57
7 Contributors 59
8 Acknowledgements 61
9 Keywords 63
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STABiX Documentation, Release 2.0.0
Figure 1: Slip transmission analysis for an EBSD map of near alpha phase Ti alloy.
The
Mat-
lab
tool-
box
STABiX
pro-
vides
a
unique
and
sim-
ple
way
to
anal-
yse
slip
trans-
mis-
sion
in
a
bicrys-
tal.
Graph-
i-
cal
User
In-
ter-
faces
(GUIs)
are
implemented in order to import EBSD results, and to represent and quantify grain boundary slip resistance. Key
parameters, such as the number of phases, crystal structure (fcc, bcc, or hcp), and slip families for calculations, are
set by the user. With this information, grain boundaries are plotted and color coded according to the 𝑚factor1that
quantifies the geometrical compatibility of the slip planes normals and Burgers vectors of incoming and outgoing slip
systems. Other potential functions that could assess the potential to develop damage are implemented (e.g. residual
Burgers vector2and3,𝑁factor4, resolved shear stress5, misorientation...).
Furthermore, the toolbox provides the possibility to plot and analyze the case of a bicrystal, and to model sphero-
conical indentation performed in a single crystal or close to grain boundaries (i.e. quasi bicrystal deformation). All
of the data linked to the bicrystal indentation (indenter properties, indentation settings, grain boundary inclination,
etc.) are collected through the GUI. A PythonTM file can be then exported in order to carry out a fully automatic 3D
crystal plasticity finite element simulations of the indentation process using one of the constitutive models available in
1J. Luster and M.A. Morris, “Compatibility of deformation in two-phase Ti-Al alloys: Dependence on microstructure and orientation relation-
ships.”, Metal. and Mat. Trans. A (1995), 26(7), pp. 1745-1756.
2M.J. Marcinkowski and W.F. Tseng, “Dislocation behavior at tilt boundaries of infinite extent.”, Metal. Trans. (1970), 1(12), pp. 3397-3401.
3W. Bollmann, “Crystal Defects and Crystalline Interfaces”, Springer-Verlag (1970).
4J.D. Livingston and B. Chalmers, “Multiple slip in bicrystal deformation.”, Acta Metallurgica (1957), 5(6), pp. 322-327.
5T.R. Bieler et al., “The role of heterogeneous deformation on damage nucleation at grain boundaries in single phase metals.”, Int. J. of Plast.
(2009), 25(9), pp. 1655–1683.
Contents 1
STABiX Documentation, Release 2.0.0
DAMASK6and7. The plasticity of single crystals is quantified by a combination of crystal lattice orientation mapping,
instrumented sphero-conical indentation, and measurement of the resulting surface topography8and9. In this way the
stress and strain fields close to the grain boundary can be rapidly assessed. Activation and transmission of slip are
interpreted based on these simulations and the mechanical resistance of grain boundaries can be quantified.
6F. Roters et al., “Overview of constitutive laws, kinematics, homogenization and multiscale methods in crystal plasticity finite-element model-
ing: Theory, experiments, applications.”, Acta Materialia (2010), 58(4), pp. 1152-1211.
7DAMASK — the Düsseldorf Advanced Material Simulation Kit.
8C. Zambaldi et al., “Orientation informed nanoindentation of 𝛼-titanium: Indentation pileup in hexagonal metals deforming by prismatic slip”,
J. Mater. Res. (2012), 27(01), pp. 356-367.
9C. Zambaldi, “Anisotropic indentation pile-up in single crystals”.
2 Contents
CHAPTER 1
How to get STABiX code ?
First of all, download the source code of the Matlab toolbox.
Source code is hosted at Github.
Download source code as a .zip file.
3
STABiX Documentation, Release 2.0.0
4 Chapter 1. How to get STABiX code ?
CHAPTER 2
How to cite STABiX in your papers ?
5
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6 Chapter 2. How to cite STABiX in your papers ?
CHAPTER 3
Reference paper
“A Matlab toolbox to analyze slip transfer through grain boundaries” D. Mercier, C. Zambaldi, T. R. Bieler, 17th In-
ternational Conference on Textures of Materials (ICOTOM17), at Dresden, Germany (2014). IOP Conference Series:
Materials Science and Engineering Volume 82 conference 1. http://dx.doi.org/10.1088/1757-899X/82/1/012090
“Spherical indentation and crystal plasticity modeling near grain boundaries in alpha-Ti.D. Mercier, C. Zambaldi, P.
Eisenlohr, Y. Su, M. A. Crimp, T. R. Bieler, Poster presented at “Indentation 2014” Conference in Strasbourg (France)
(December 2014). http://dx.doi.org/10.13140/RG.2.1.3044.8486
“Grain Boundaries and Plasticity. D. Mercier, C. Zambaldi, P. Eisenlohr, M. A. Crimp, T. R. Bieler, R. Sánchez
Martín. Invited talk at “MTEX Workshop 2016” Conference in Chemnitz (Germany) (February 2016). http://dx.doi.
org/10.13140/RG.2.2.27427.66084
“Quantifying deformation processes near grain boundaries in 𝛼titanium using nanoindentation and crystal plasticity
modeling.” Y. Su, C. Zambaldi, D. Mercier, P. Eisenlohr, T.R. Bieler, M.A. Crimp, International Journal of Plasticity,
(2016). http://dx.doi.org/10.1016/j.ijplas.2016.08.007
“Evaluation of an inverse methodology for estimating constitutive parameters in face-centered cubic materials from
single crystal indentations.” A. Chakraborty, P. Eisenlohr, European Journal of Mechanics - A/Solids, (2017). http:
//dx.doi.org/10.1016/j.euromechsol.2017.06.012
“STABiX documentation.
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8 Chapter 3. Reference paper
CHAPTER 4
Contents
4.1 Motivation of this Work
The micromechanical behavior of grain boundaries is one of the key components in the understanding of hetero-
geneous deformation of metals1. To investigate the nature of the strengthening effect of grain boundaries, slip
transmission across interfaces has been investigated through bicrystal deformation experiments during the sixty past
decades2,3,4,5,6,7,8,9,10,11 ,12,13 ,14 and15. Originally, interactions between dislocations and grain boundaries have been
observed in the transmission electron microscope (TEM) after strain test or in situ4,5and15. Some authors observed as
1T.R. Bieler et al., “Grain boundaries and interfaces in slip transfer.”, Current Opinion in Solid State and Materials Science (2014), 18(4), pp.
212-226.
2K.T. Aust et al., “Solute induced hardening near grain boundaries in zone refined metals.”, Acta Metallurgica (1968), 16(3), pp. 291-302.
3J.D. Livingston and B. Chalmers, “Multiple slip in bicrystal deformation.”, Acta Metallurgica (1957), 5(6), pp. 322-327.
4Z. Shen et al., “Dislocation pile-up and grain boundary interactions in 304 stainless steel.”, Scripta Metallurgica (1986), 20(6), pp. 921–926.
5Z. Shen et al., “Dislocation and grain boundary interactions in metals.”, Acta Metallurgica (1988), 36(12), pp. 3231–3242.
6J. Luster and M.A. Morris, “Compatibility of deformation in two-phase Ti-Al alloys: Dependence on microstructure and orientation relation-
ships.”, Metal. and Mat. Trans. A (1995), 26(7), pp. 1745-1756.
7M.J. Marcinkowski and W.F. Tseng, “Dislocation behavior at tilt boundaries of infinite extent.”, Metal. Trans. (1970), 1(12), pp. 3397-3401.
8W. Bollmann, “Crystal Defects and Crystalline Interfaces”, Springer-Verlag (1970)
9L.C. Lim and R. Raj, “Continuity of slip screw and mixed crystal dislocations across bicrystals of nickel at 573K.”, Acta Metallurgica (1985),
33, pp. 1577.
10 T.C. Lee et al., “Prediction of slip transfer mechanisms across grain boundaries.”, Scripta Metallurgica, (1989), 23(5), pp. 799–803.
11 T.C. Lee et al., “An In Situ transmission electron microscope deformation study of the slip transfer mechanisms in metals”, Metallurgical
Transactions A (1990), 21(9), pp. 2437-2447.
12 W.A.T. Clark et al., “On the criteria for slip transmission across interfaces in polycrystals.”, Scripta Metallurgica et Materialia (1992), 26(2),
pp. 203–206.
13 W.Z. Abuzaid et al., “Slip transfer and plastic strain accumulation across grain boundaries in Hastelloy X.”, J. of the Mech. and Phys. of Sol.
(2012), 60(6) ,pp. 1201–1220.
14 J.R. Seal et al., “Analysis of slip transfer and deformation behavior across the 𝛼/𝛽interface in Ti–5Al–2.5Sn (wt.%) with an equiaxed
microstructure.”, Mater. Sc. and Eng.: A (2012), 552, pp. 61-68.
15 J. Kacher et al., “Dislocation interactions with grain boundaries.”, Current Opinion in Solid State and Materials Science (2014), in press.
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well slip transmission during indentation tests performed close to grain boundaries16,17 ,18,19 and20 .
To better understand the role played by the grain boundaries, we developed a Matlab toolbox with Graphical User
Interfaces (GUI), to analyze and to quantify the micromechanics of grain boundaries. This toolbox aims to link
experimental results to crystal plasticity finite element (CPFE) simulations23.
4.1.1 Strategy
Comparison of topographies of indentations at grain boundaries to simulated indentations as predicted by 3D CPFE
modelling.
The goals of this research are:
1 - Carry out indentation within the interiors of large grains of alpha-titanium to effectively collect single crystal
data coupled with extensive (three-dimensional) characterization of the resulting plastic defect fields surrounding the
indents21. By correlating with models of the indentation, a precise constitutive description of the anisotropic plasticity
of single-crystalline titanium shall be developed22 and23.
2 - Extension of this methodology to indentations close to grain boundaries, i.e. quasi bi-crystal deformation.
3 - Comparison of the measured characteristics of indentations at grain boundaries to simulated indentations as pre-
dicted by a constitutive model calibrated using the single crystal indentations.
4 - Based on this qualitative understanding, a grain boundary transmissivity description will be developed validated
against the collected indent characteristics.
4.2 Getting started
4.2.1 Source Code
First of all, download the source code of the Matlab toolbox.
Source code is hosted at Github.
Download source code as a .zip file.
4.2.2 READ ME
To have more details about the use of the toolbox, please have a look to :
16 P.C. Wo and A.H.W. Ngan, “Investigation of slip transmission behavior across grain boundaries in polycrystalline Ni3Al using nanoindenta-
tion.”, J. Mater. Res. (2004), 19(1), pp. 189-201.
17 W.A. Soer et al. ,”Incipient plasticity during nanoindentation at grain boundaries in body-centered cubic metals.”, Acta Materialia (2005), 53,
pp. 4665–4676.
18 T.B. Britton et al., “Nanoindentation study of slip transfer phenomenon at grain boundaries.”, J. Mater. Res., 2009, 24(3), pp. 607-615.
19 S. Patthak et al., “Studying grain boundary regions in polycrystalline materials using spherical nano-indentation and orientation imaging
microscopy.”, J. Mater. Sci. (2012), 47, pp. 815–823.
20 S.K. Lawrence et al., “Grain Boundary Contributions to Hydrogen-Affected Plasticity in Ni-201.”, The Journal of The Minerals, Metals &
Materials Society (2014), 66(8), pp. 1383-1389.
23 DAMASK — the Düsseldorf Advanced Material Simulation Kit
21 C. Zambaldi et al., “Orientation informed nanoindentation of 𝛼-titanium: Indentation pileup in hexagonal metals deforming by prismatic slip”,
J. Mater. Res. (2012), 27(01), pp. 356-367.
22 F. Roters et al., “Overview of constitutive laws, kinematics, homogenization and multiscale methods in crystal plasticity finite-element model-
ing: Theory, experiments, applications.”, Acta Materialia (2010), 58(4), pp. 1152-1211.
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README.txt
4.2.3 Path management
Run the following Matlab script and answer ‘y’ or ‘yes’ to add path to the Matlab search paths :
path_management.m
The Matlab function used to set the Matlab search paths is : path_management.m
4.2.4 The GUIs
Run one of these Graphical User Interfaces (GUIs) to play with the toolbox.
Matlab
function
Features YAML config. file
demo Start and run other GUIs.
EBSD map
GUI
Analysis of slip transmission across GBs for an EBSD
map.
con-
fig_gui_EBSDmap_defaults.yaml
Bicrystal GUI Analysis of slip transfer in a bicrystal.
preCPFE_SX Preprocess of CPFE models for indentation or scratch in
a SX.
config_CPFEM_defaults.yaml
preCPFE_BX Preprocess of CPFE models for indentation or scratch in
a BX.
config_CPFEM_defaults.yaml
GBinc Calculation of grain boundaries inclination.
Figure 4.1: The different GUIs of the STABiX toolbox.
Note: ‘SX’ is used for single crystal and ‘BX’ for bicrystal.
4.2. Getting started 11
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4.2.5 The YAML configuration files
“YAML is a human friendly data serialization standard for all programming languages.
Default YAML configuration files, stored in the folder yaml_config_files, are loaded automatically to set the
GUis :
config.yaml
config_CPFEM_defaults.yaml
config_CPFEM_material_defaults.yaml
config_CPFEM_materialA_defaults.yaml
config_CPFEM_materialB_defaults.yaml
config_gui_EBSDmap_defaults.yaml
config_gui_BX_defaults.yaml
config_gui_SX_defaults.yaml
config_mesh_BX_defaults.yaml
config_mesh_SX_defaults.yaml
You have to set your own YAML configuration files, by following instructions given in this README.
Warning: If you create your own YAML configuration files after running STABiX, you have to run again the
path_management.m Matlab function.
Visit the YAML website for more informations.
Visit the YAML code for Matlab.
4.2.6 MTEX toolbox
For some options and functions implemented in the STABiX toolbox, you have to download and install the
MTEX Toolbox.
4.2.7 OpenGL
If the OpenGL rendering is not satisfying, you can modify the corresponding option in the config.yaml file.
Visit the Matlab page about OpenGL rendering.
4.3 Bicrystal Definition
4.3.1 Crystallographic properties of a bicrystal
A bicrystal is formed by two adjacent crystals separated by a grain boundary.
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Five macroscopic degrees of freedom are required to characterize a grain boundary3,5,6and7:
3 for the rotation between the two crystals;
2 for the orientation of the grain boundary plane defined by its normal 𝑛.
The rotation between the two crystals is defined by the rotation angle 𝜔and the rotation axis common to both
crystals [𝑢𝑣𝑤].
Using orientation matrix of both crystals obtained by EBSD measurements, the misorientation or disorientation
matrix (∆𝑔)or (∆𝑔d)is calculated4and2:
𝑔=𝑔B𝑔1
A=𝑔A𝑔1
B(4.1)
𝑔d= (𝑔B*𝐶𝑆 )(𝐶𝑆1*𝑔1
A)=(𝑔A*𝐶𝑆)(𝐶 𝑆1*𝑔1
B)(4.2)
Disorientation describes the misorientation with the smallest possible rotation angle and 𝐶𝑆 denotes one of
the symmetry operators for the material1.
The Matlab function used to set the symmetry operators is : sym_operators.m
The orientation matrix 𝑔of a crystal is calculated from the Euler angles (𝜑1,Φ,𝜑2) using the following
equation :
𝑔=
cos(𝜑1) cos(𝜑2)sin(𝜑1) sin(𝜑2) cos(Φ) sin(𝜑1) cos(𝜑2) + cos(𝜑1) sin(𝜑2) cos(Φ) sin(𝜑2) sin(Φ)
cos(𝜑1) sin(𝜑2)sin(𝜑1) cos(𝜑2) cos(Φ) sin(𝜑1) sin(𝜑2) + cos(𝜑1) cos(𝜑2) cos(Φ) cos(𝜑2) sin(Φ)
sin(𝜑1) sin(Φ) cos(𝜑1) sin(Φ) cos(Φ)
(4.3)
The orientation of a crystal (Euler angles) can be determined via electron backscatter diffraction (EBSD)
measurement or via transmission electron microscopy (TEM).
The Matlab function used to generate random Euler angles is : randBunges.m
The Matlab function used to calculate the orientation matrix from Euler angles is : eulers2g.m
The Matlab function used to calculate Euler angles from the orientation matrix is : g2eulers.m
Then, from this misorientation matrix (𝑔), the rotation angle (𝜔) and the rotation axis [𝑢, 𝑣, 𝑤]can be obtained
by the following equations :
𝜔= cos1((𝑡𝑟(∆𝑔)1)/2) (4.4)
𝑢= ∆𝑔23 𝑔32
𝑣= ∆𝑔31 𝑔13
𝑤= ∆𝑔12 𝑔21
(4.5)
3L. Priester, “Grain Boundaries: From Theory to Engineering.”, Springer Series in Materials Science (2013).
5V. Randle, “A methodology for grain boundary plane assessment by single-section trace analysis.”, Scripta Mater., 2001, 44, pp. 2789-2794.
6V. Randle, “Five-parameter’ analysis of grain boundary networks by electron backscatter diffraction.”, J. Microscopy, 2005, 222, pp. 69-75.
7A.P. Sutton and R.W. Balluffi, “Interfaces in Crystalline Materials.”, OUP Oxford (1995).
4V. Randle and O. Engler, “Introduction to Texture Analysis : Macrotexture, Microtexture and Orientation Mapping.”, CRC Press (2000).
2A. Morawiec, “Orientations and Rotations: Computations in Crystallographic Textures.”, Springer, 2004.
1U.F. Kocks et al., “Texture and Anisotropy: Preferred Orientations in Polycrystals and Their Effect on Materials Properties.” Cambridge
University Press (2000).
4.3. Bicrystal Definition 13
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The Matlab function used to calculate the misorientation angle is : misorientation.m
The grain boundary plane normal 𝑛can be determined knowing the grain boundary trace angle 𝛼and the grain
boundary inclination 𝛽.
The grain boundary trace angle is obtained through the EBSD measurements (grain boundary endpoints coordi-
nates) and the grain boundary inclination can be assessed by a serial polishing (chemical-mechanical polishing
or FIB sectioning), either parallel or perpendicular to the surface of the sample (see Figure 4.3).
Figure 4.2: Schematic of a bicrystal.
Figure 4.3: Screenshot of the Matlab GUI used to calculate grain boundary inclination.
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4.4 Strain Transfer Across Grain Boundaries
The strain transfer across grain boundaries can be defined by the four following mechanisms (see Figure
4.4)50,79 ,89 and64 :
1. direct transmission with slip systems having the same Burgers vector, and the grain boundary is transparent to
dislocations (no strengthening effect) (Figure 4.4-a);
2. direct transmission, but slip systems have different Burgers vector (leaving a residual boundary dislocations)
(Figure 4.4-b);
3. indirect transmission, and slip systems have different Burgers vector (leaving a residual boundary dislocations)
(Figure 4.4-c);
4. no transmission and the grain boundary acts as an impenetrable boundary, which implies stress accumulations,
localized rotations, pile-up of dislocations... (Figure 4.4-d).
Figure 4.4: Possible strain transfer across grain boundaries (GB) from Sutton and Balluffi.
Several authors proposed slip transfer parameters from modellings or experiments for the last 60 years. A
non-exhaustive list of those criteria is given in the next part of this work, including geometrical parameter,
stress and energetic functions, and recent combinations of the previous parameters.
4.4.1 Geometrical Criteria
Based on numerous investigations of dislocation-grain boundary interactions, quantitative geometrical expres-
sions describing the slip transmission mechanisms have been developed. A non-exhaustive list of geometrical
criteria is detailed subsequently. The geometry of the slip transfer event is most of the time described by the
scheme given Figure 4.5.𝜅is the angle between slip directions, 𝜃is the angle between the two slip plane
intersections with the grain boundary, 𝜓is the angle between slip plane normal directions, 𝛾is the angle between the
direction of incoming slip and the plane normal of outgoing slip, and 𝛿is between the direction of outgoing slip and
the plane normal of incoming slip. 𝑛,𝑑and 𝑙are respectively the slip plane normals, slip directions and the lines of
intersection of the slip plane and the grain boundary.
𝑏is the Burgers vector of the slip plane and
𝑏ris the residual
Burgers vector of the residual dislocation at the grain boundary. The subscripts in and out refer to the incoming and
outgoing slip systems, respectively.
50 L.C. Lim and R. Raj, “Continuity of slip screw and mixed crystal dislocations across bicrystals of nickel at 573K.”, Acta Metallurgica (1985),
33, pp. 1577.
79 A.P. Sutton and R.W. Balluffi, “Interfaces in Crystalline Materials.”, OUP Oxford (1995).
89 S. Zaefferer et al., “On the influence of the grain boundary misorientation on the plastic deformation of aluminum bicrystals.”, Acta Materialia
(2003), 51(16), pp. 4719-4735.
64 L. Priester, “Grain Boundaries: From Theory to Engineering.”, Springer Series in Materials Science (2013).
4.4. Strain Transfer Across Grain Boundaries 15
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Figure 4.5: Geometrical description of the slip transfer.
𝑁factor from Livingston and Chalmers in 195752
𝑁= (𝑛in ·𝑛out)(
𝑑in ·
𝑑out)+(𝑛in ·
𝑑out)(𝑛out ·
𝑑in)(4.6)
𝑁= cos(𝜓)·cos(𝜅) + cos(𝛾)·cos(𝛿)(4.7)
Many authors referred to this criterion to analyze slip transmission32,19 ,34,35 ,71,72 ,47,48 ,17 and82. Pond et al.
proposed to compute this geometric criteria for hexagonal metals using Frank’s method63.
The Matlab function used to calculate the N factor is : N_factor.m
𝐿𝑅𝐵 factor from Shen et al. in 198671 and72
𝐿𝑅𝐵 = (
𝑙in ·
𝑙out)(
𝑑in ·
𝑑out)(4.8)
𝐿𝑅𝐵 = cos(𝜃)·cos(𝜅)(4.9)
52 J.D. Livingston and B. Chalmers, “Multiple slip in bicrystal deformation.”, Acta Metallurgica (1957), 5(6), pp. 322-327.
32 J.J. Hauser and B. Chamlers, “The plastic deformation of bicrystals of f.c.c. metals.”, Acta Metallurgica (1961), 9(9), pp. 802-818.
19 K.G. Davis et al., “Slip band continuity across grain boundaries in aluminum.”, Acta Metallurgica (1966), 14, pp. 1677-1684.
34 R.E. Hook and J.P. Hirth, “The deformation behavior of isoaxial bicrystals of Fe-3%Si.”, Acta Metallurgica (1967), 15(3), pp. 535-551.
35 R.E. Hook and J.P. Hirth, “The deformation behavior of non-isoaxial bicrystals of Fe-3% Si.”, Acta Metallurgica(1967), 15(7), pp. 1099-1110.
71 Z. Shen et al., “Dislocation pile-up and grain boundary interactions in 304 stainless steel.”, Scripta Metallurgica (1986), 20(6), pp. 921–926.
72 Z. Shen et al., “Dislocation and grain boundary interactions in metals.”, Acta Metallurgica (1988), 36(12), pp. 3231–3242.
47 T.C. Lee et al., “TEM in situ deformation study of the interaction of lattice dislocations with grain boundaries in metals.”, Philosophical
Magazine A (1990), 62(1), pp. 131-153.
48 T.C. Lee et al., “An In Situ transmission electron microscope deformation study of the slip transfer mechanisms in metals”, Metallurgical
Transactions A (1990), 21(9), pp. 2437-2447.
17 W.A.T. Clark et al., “On the criteria for slip transmission across interfaces in polycrystals.”, Scripta Metallurgica et Materialia (1992), 26(2),
pp. 203–206.
82 M. Ueda et al., “Effect of grain boundary on martensite transformation behaviour in Fe–32 at.%Ni bicrystals.”, Science and Technology of
Advanced Materials (2002), 3(2), pp. 171.
63 R.C. Pond et al., “On the crystallography of slip transmission in hexagonal metals.”, Scripta Metallurgica (1986), 20, pp. 1291-1295.
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The original notation of this 𝐿𝑅𝐵 factor is 𝑀, but unfortunately this notation is often used for the Taylor
factor11. Pond et al. proposed to compute this geometric criteria for hexagonal metals using Frank’s method63.
Recently, Spearot and Sangid have plotted this parameter as a function of the misorientation of the bicrystal
using atomistic simulations78.
46,47 ,48,17 ,41,42 ,2,28,29 and73 mentioned in their respective studies this geometrical parameter as a condition for
slip transmission.
The inclination of the grain boundary (𝛽) is required to evaluate this factor and the 𝐿𝑅𝐵 or 𝑀factor should
be maximized.
The Matlab function used to calculate the LRB factor is : LRB_parameter.m
𝑚parameter from Luster and Morris in 199553
𝑚= (𝑛in ·𝑛out)(
𝑑in ·
𝑑out)(4.10)
𝑚= cos(𝜓)·cos(𝜅)(4.11)
Many authors found that this 𝑚parameter, which takes into account the degree of coplanarity of slip systems,
is promising to predict slip transmission85,88 ,13,10 ,11 ,30,83 ,31 and [#Nervo_2016]. Both 𝑚and 𝐿𝑅𝐵 can be
easily assessed in computational experiments11. This 𝑚factor should be maximized (1 means grain boundary
is transparent and 0 means grain boundary is an impenetrable boundary).
A resistance factor of the grain boundary can be described by the following equation :
𝐺𝐵resfac = 1 𝑚(4.12)
11 T.R. Bieler et al., “Grain boundaries and interfaces in slip transfer.”, Current Opinion in Solid State and Materials Science, (2014), 18(4), pp.
212-226.
78 D.E. Spearot and M.D. Sangid, “Insights on slip transmission at grain boundaries from atomistic simulations.”, Current Opinion in Solid State
and Materials Science (2014), in press.
46 T.C. Lee et al., “Prediction of slip transfer mechanisms across grain boundaries.”, Scripta Metallurgica, (1989), 23(5), pp. 799–803.
41 T. Kehagias et al., “Slip transfer across low-angle grain boundaries of deformed titanium.”, Interface Science (1995), 3(3), pp. 195-201.
42 T. Kehagias et al., “Pyramidal Slip in Electron Beam Heated Deformed Titanium.”, Scripta Metallurgica et Materialia (1996), 33(12), pp.
1883-1888.
2W.M. Ashmawi and M.A. Zikry, “Prediction of Grain-Boundary Interfacial Mechanisms in Polycrystalline Materials.”, Journal of Engineering
Materials and Technology (2001), 124(1), pp. 88-96.
28 A. Gemperle et al., “Interaction of slip dislocations with grain boundaries in body-centered cubic bicrystals.”, Materials Science and Engineer-
ing A (2004), 378-389, pp. 46-50.
29 J. Gemperlova et al., “Slip transfer across grain boundaries in Fe–Si bicrystals.”, Journal of Alloys and Compounds (2004), 378(1-2), pp.
97-101.
73 J. Shi and M.A. Zikry, “Modeling of grain boundary transmission, emission, absorption and overall crystalline behavior in Σ1, Σ3, and Σ17b
bicrystals.”, J. Mater. Res., (2011), 26(14), pp. 1676-1687.
53 J. Luster and M.A. Morris, “Compatibility of deformation in two-phase Ti-Al alloys: Dependence on microstructure and orientation relation-
ships.”, Metal. and Mat. Trans. A (1995), 26(7), pp. 1745-1756.
85 M.G. Wang and A.H.W. Ngan, “Indentation strain burst phenomenon induced by grain boundaries in niobium.”, Journal of Materials Research
(2004), 19(08), pp. 2478-2486.
88 P.C. Wo and A.H.W. Ngan, “Investigation of slip transmission behavior across grain boundaries in polycrystalline Ni3Al using nanoindenta-
tion.”, J. Mater. Res. (2004), 19(1), pp. 189-201.
13 T.B. Britton et al., “Nanoindentation study of slip transfer phenomenon at grain boundaries.”, J. Mater. Res., (2009), 24(3), pp. 607-615.
10 T.R. Bieler et al., “The role of heterogeneous deformation on damage nucleation at grain boundaries in single phase metals.”, Int. J. of Plast.
(2009), 25(9), pp. 1655–1683.
30 Y. Guo et al., “Slip band–grain boundary interactions in commercial-purity titanium.”, Acta Materialia (2014), 76, pp. 1-12.
83 Wang F. et al., “In situ observation of collective grain-scale mechanics in Mg and Mg–rare earth alloys.”, Acta Materialia, (2014), 80, pp.
77–93.
31 Y. Guo et al., “Measurements of stress fields near a grain boundary: Exploring blocked arrays of dislocations in 3D.”, Acta Materialia (2015),
96, pp. 229-236.
4.4. Strain Transfer Across Grain Boundaries 17
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Figure 4.6: Distribution of m’ parameter in function of angles values.
Figure 4.7: Distributions of m’ parameter calculated for a) basal vs basal slip systems, b) basal vs prismatic <a> slip
systems and c) prismatic <a> vs prismatic <a> slip systems in function of misorientation angle.
This factor is equal to 0, when grain boundary is transparent to dislocations. This implies 𝑚parameter equal
to 1 (slip perfectly aligned).
The Matlab function used to calculate the m’ parameter is : mprime.m
𝑏rthe residual Burgers vector55,12,50 ,51,16 ,48 and17.
𝑏r=𝑔in ·
𝑏in 𝑔out ·
𝑏out (4.13)
The magnitude of this residual Burgers vector should be minimized.
Shirokoff et al., Kehagias et al., and Kacher et al. used the residual Burgers vector as a criterion to analyse slip
transmission in cp-Ti (HCP)74,41,42 and38 , Lagow et al. in Mo (BCC)44 , Gemperle et al. and Gemperlova et al.
in FeSi (BCC)28 and29, Kacher et al. in 304 stainless steel (FCC)37 , and Jacques et al. for semiconductors36 .
55 M.J. Marcinkowski and W.F. Tseng, “Dislocation behavior at tilt boundaries of infinite extent.”, Metal. Trans. (1970), 1(12), pp. 3397-3401.
12 W. Bollmann, “Crystal Defects and Crystalline Interfaces”, Springer-Verlag (1970)
51 L.C. Lim and R. Raj, “The role of residual dislocation arrays in slip induced cavitation, migration and dynamic recrystallization at grain
boundaries.”, Acta Metallurgica (1985), 33(12), pp. 2205-2214.
16 W.A.T. Clark et al., “The use of the transmission electron microscope in analyzing slip propagation across interfaces.”, Ultramicroscopy (1989),
30(1-2), pp. 76-89.
74 J. Shirokoff et al., “The Slip Transfer Process Through Grain Boundaries in HCP Ti.”, MRS Online Proceedings Library (1993), 319, pp.
263-272.
38 J. Kacher and I.M. Robertson, “In situ and tomographic analysis of dislocation/grain boundary interactions in 𝛼-titanium.”, Philosophical
Magazine (2014), 94(8), pp. 814-829.
44 B.W. Lagow, “Observation of dislocation dynamics in the electron microscope.”, Materials Science and Engineering: A, 2001, 309–310, pp.
445-450.
37 J. Kacher and I.M. Robertson, “Quasi-four-dimensional analysis of dislocation interactions with grain boundaries in 304 stainless steel.”, Acta
Materialia (2012), 60(19), pp. 6657–6672.
36 A. Jacques et al., “New results on dislocation transmission by grain boundaries in elemental semiconductors.”, Le Journal de Physique Collo-
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Patriarca et al. demonstrated for BCC material the role of the residual Burgers vector in predicting slip trans-
mission, by analysing strain field across GBs determined by digital image correlation62.
Misra and Gibala used the residual Burgers vector to analyze slip across a FCC/BCC interphase boundary56.
The Matlab function used to calculate the residual Burgers vector is : residual_Burgers_vector.m
The misorientation or disorientation (𝑔or 𝑔d)3,15 and88
It has been observed during first experiments of bicrystals deformation in 1954, that the yield stress and the
rate of work hardening increased with the orientation difference between the crystals3and15.
Some authors demonstrated a strong correlation between misorientation between grains in a bicrystal and the
grain boundary energy through crystal plasticity finite elements modelling and molecular dynamics simula-
tions79,54 ,49,5,67 and68 . Some authors studied the stability of grain boundaries by the calculations of energy
difference vs. misorientation angle through the hexagonal c-axis/a-axis26.
The misorientation and disorientation equations are given in the crystallographic properties of a bicrystal.
The Matlab function used to calculate the misorientation angle is : misorientation.m
𝜆function from Werner and Prantl in 199086
With this function, slip transmission is expected to occur only when the angle 𝜓between slip plane normal
directions is lower than a given critical value (𝜓𝑐= 15) and the angle 𝜅between slip directions is lower than
a given critical value (𝜅𝑐= 45).
𝜆= cos 90
𝜓𝑐
arccos(𝑛in ·𝑛out)cos 90
𝜅𝑐
arccos(
𝑑in ·
𝑑out)(4.14)
𝜆= cos 90𝜓
𝜓𝑐cos 90𝜅
𝜅𝑐(4.15)
The Matlab function used to calculate the 𝜆function is : lambda.m
The authors proposed to plot pseudo-3D view of the 𝜆map (see Figures 5 and 6) using the following equation86
:
𝜆=
𝑁
𝛼=1
𝑁
𝛽=1
cos 90
𝜓𝑐
arccos(𝑛in,𝛼 ·𝑛out,𝛽 )cos 90
𝜅𝑐
arccos(
𝑑in,𝛼 ·
𝑑out,𝛽 )(4.16)
With 𝑁the number of slip systems for each adjacent grains.
ques (1990), 51(C1), pp. 531-536.
62 L. Patriarca et al., “Slip transmission in bcc FeCr polycrystal.”, Materials Science&Engineering (2013), A588, pp. 308–317.
56 A. Misra and R. Gibala, “Slip Transfer and Dislocation Nucleation Processes in Multiphase Ordered Ni-Fe-Al Alloys”, Metallurgical and
Materials Trans. A (1999), 30A, pp. 991-1001.
3K.T. Aust and N.K. Chen, “Effect of orientation difference on the plastic deformation of aluminum bicrystals.”, Acta Metallurgica (1954), 2,
pp. 632-638.
15 W.A.T. Clark and B. Chalmers, “Mechanical deformation of aluminium bicrystals.”, Acta Metallurgica (1954), 2(1), pp. 80-86.
54 A. Ma et al., “On the consideration of interactions between dislocations and grain boundaries in crystal plasticity finite element modeling –
Theory, experiments, and simulations.”, Acta Materialia (2006), 54(8), pp.2181-2194.
49 Z. Li et al., “Strengthening mechanism in micro-polycrystals with penetrable grain boundaries by discrete dislocation dynamics simulation
and Hall–Petch effect.”, Computational Materials Science (2009), 46(4), pp. 1124-1134.
5D.V. Bachurin et al., “Dislocation–grain boundary interaction in <111> textured thin metal films.”, Acta Materialia (2010), 58, pp. 5232–5241.
67 M.D. Sangid et al., “Energy of slip transmission and nucleation at grain boundaries.”, Acta Materialia (2011), 59(1), pp. 283–296.
68 M.D. Sangid et al., “Energetics of residual dislocations associated with slip–twin and slip–GBs interactions.”, Materials Science and Engineer-
ing A (2012), 542, pp. 21–30.
26 H. Faraoun et al., “Study of stability of twist grain boundaries in hcp zinc.”, Scripta Materialia (2006), 54, pp. 865–868.
86 E. Werner and W. Prantl, “Slip transfer across grain and phase boundaries.”, Acta Metallurgica et Materialia (1990), 38(3), pp. 533-537.
4.4. Strain Transfer Across Grain Boundaries 19
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Figure 4.8: Pseudo-3D view of the lambda map for the FCC-FCC case.
Figure 4.9: Pseudo-3D view of the lambda map for the BCC-BCC case.
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The Matlab function used to plot pseudo-3D view of the the 𝜆function is : lambda_plot_values.m
This function is modified by Beyerlein et al., using the angle 𝜃between the two slip plane intersections with
the grain boundary, instead of using the angle 𝜓between the two slip plane normal directions9.
𝜆= cos 90
𝜃𝑐
arccos(
𝑙in ·
𝑙out)cos 90
𝜅𝑐
arccos(
𝑑in ·
𝑑out)(4.17)
𝜆= cos 90𝜃
𝜃𝑐cos 90𝜅
𝜅𝑐(4.18)
The Matlab function used to calculate the modified 𝜆function is : lambda_modified.m
4.4.2 Stress Criteria
Schmid Factor (𝑚)66,69 and1
The Schmid’s law can be expressed by the following equation:
𝜏𝑖=𝜎:𝑆0𝑖(4.19)
𝑆0𝑖=
𝑑𝑖𝑛𝑖(4.20)
𝜎is an arbitrary stress state and 𝜏𝑖the resolved shear stress on slip system 𝑖.𝑆0𝑖is the Schmid matrix defined
by the dyadic product of the slip plane normals 𝑛 and the slip directions
𝑑of the slip system 𝑖. The Schmid
factor, 𝑚, is defined as the ratio of the resolved shear stress 𝜏𝑖to a given uniaxial stress.
Knowing the value of the highest Schmid factor of a given slip system for both grains in a bicrystal, Abuzaid
et al.1proposed the following criterion :
𝑚GB =𝑚in +𝑚out (4.21)
The subscripts GB, in, and out refer to the grain boundary, and the incoming and outgoing slip systems,
respectively. This GB Schmid factor (𝑚GB ) factor should be maximized.
The Matlab function used to calculate the Schmid factor is : resolved_shear_stress.m
Generalized Schmid Factor (𝐺𝑆𝐹 )66 and11
The generalized Schmid factor, which describes the shear stress on a given slip system, can be computed from
any stress tensor 𝜎based on the Frobenius norm of the tensor.
9I. Beyerlein al., “Structure–Property–Functionality of Bimetal Interfaces.”, The Journal of The Minerals, Metals & Materials Society (TMS)
(2012), pp. 1192-1207.
66 C.N. Reid, “Deformation Geometry for Materials Scientists.”, Pergamon Press, Oxford, United Kingdom, 1973.
69 J.R. Seal et al., “Analysis of slip transfer and deformation behavior across the 𝛼/𝛽interface in Ti–5Al–2.5Sn (wt.%) with an equiaxed
microstructure.”, Mater. Sc. and Eng.: A (2012), 552, pp. 61-68.
1W.Z. Abuzaid et al., “Slip transfer and plastic strain accumulation across grain boundaries in Hastelloy X.”, J. of the Mech. and Phys. of Sol.
(2012), 60(6) ,pp. 1201–1220.
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𝐺𝑆𝐹 =
𝑑·𝑔𝜎𝑔 ·𝑛 (4.22)
𝑛 and
𝑑are respectively the slip plane normals and the slip directions of the slip system. The 𝑔is the orientation
matrix for a given crystal.
The Matlab function used to calculate the generalized Schmid factor is : generalized_schmid_factor.m
Resolved Shear Stress (𝜏)46,47,48 ,17,44 ,10,22 ,23 and24
The resolved shear stress 𝜏acting on the outgoing slip system from the piled-up dislocations should be maxi-
mized. This criterion considers the local stress state.
The resolved shear stress on the grain boundary should be minimized.
For Shi and Zikry, the ratio of the resolved shear stress to the reference shear stress of the outgoing slip system
(stress ratio) should be greater than a critical value (which is approximately 1)73.
For Li et al. and Gao et al. the resolved shear stress acting on the incoming dislocation on the slip plane
must be larger than the critical penetration stress. From the energy point of view, only when the work by the
external force on the incoming dislocation is greater than the summation of the GB energy and strain energy
of GB dislocation debris, it is possible that the incoming dislocation can penetrate through the GB49 and27.
It is possible to assess the shear stress from the geometrical factor 𝑁(Livingston and Chamlers) :
𝜏in =𝜏out *𝑁(4.23)
Where 𝜏out is the shear stress at the head of the accumulated dislocations in their slip plane and 𝜏in is the shear
acting on the incoming slip system52,34 and35 .
The Matlab function used to calculate the resolved shear stress is : resolved_shear_stress.m
4.4.3 Combination of Criteria
Geometrical function weighted by the accumulated shear stress or the Schmid factor11 :
Bieler et al. proposed to weight slip transfer parameters by the sum of accumulated shear 𝛾on each slip
system, knowing the local stress tensor. From a crystal plasticity simulation, the accumulated shear is the total
accumulated shear on each slip system for a given integration point. This leads to the following shear-informed
version of a slip transfer parameter:
𝑚
𝛾=𝛼𝛽𝑚
𝛼𝛽 𝛾𝛼𝛾𝛽
𝛼𝛽(𝛾𝛼𝛾𝛽)(4.24)
𝐿𝑅𝐵𝛾=𝛼𝛽𝐿𝑅𝐵
𝛼𝛽 𝛾𝛼𝛾𝛽
𝛼𝛽(𝛾𝛼𝛾𝛽)(4.25)
22 M.P. Dewald et al., “Multiscale modelling of dislocation/grain-boundary interactions: I. Edge dislocations impinging on Σ11 (1 1 3) tilt
boundary in Al.”, Modelling Simul. Mater. Sci. Eng. (2007), 15(1).
23 M.P. Dewald et al., “Multiscale modelling of dislocation/grain boundary interactions. II. Screw dislocations impinging on tilt boundaries in
Al.”, Phil. Mag. (2007), 87(30), pp. 1655–1683.
24 M.P. Dewald et al., “Multiscale modeling of dislocation/grain-boundary interactions: III. 60° dislocations impinging on Σ3, Σ9 and Σ11 tilt
boundaries in Al.”, Modelling Simul. Mater. Sci. Eng. (2011), 19(5).
27 Y. Gao et al., “A hierarchical dislocation-grain boundary interaction model based on 3D discrete dislocation dynamics and molecular dynam-
ics.” Science China Physics, Mechanics and Astronomy (2011), 54(4), pp. 625-632.
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𝑠𝛾=𝛼𝛽𝑠
𝛼𝛽 𝛾𝛼𝛾𝛽
𝛼𝛽(𝛾𝛼𝛾𝛽)(4.26)
𝑠= cos(𝜓)·cos(𝜅)·cos(𝜃)(4.27)
The Matlab function used to calculate the 𝑠function is : s_factor.m
Similarly, the 𝑚parameter can be weighted using the Schmid factor 𝑚on each slip system as a metric for the
magnitude of slip transfer:
𝑚
𝐺𝑆𝐹 =𝛼𝛽𝑚
𝛼𝛽 𝑚𝛼𝑚𝛽
𝛼𝛽(𝑚𝛼𝑚𝛽)(4.28)
In 2016, Tsuru et al. proposed a new criterion, based on the 𝑁factor, for the transferability of dislocations
through a GB that considers both the intergranular crystallographic orientation of slip systems and the applied
stress condition81 .
4.4.4 Relationships between slip transmission criteria
Some authors proposed to study relationships between slip transmission criteria30 and84. Thus, it is possible
to find in the litterature the 𝑚parameter plotted in function of the Schmid factor or the misorientation angle.
Such plots based on experimental values allow to map slip transmissivity at grain boundaries for a given
material.
4.4.5 Slip transmission parameters implemented in the STABiX toolbox
Slip transmission parameter Function Matlab function Refer-
ence
Misorientation angle (FCC and BCC
materials) (𝜔)
𝜔=𝑐𝑜𝑠1((𝑡𝑟(∆𝑔)1)/2) misorientation.m 79
C-axis misorientation angle (HCP
material) (𝜔)
c-axis
misorientation.m
79
𝑁factor from Livingston and Chamlers 𝑁=
cos(𝜓)·cos(𝜅)+ cos(𝛾)·cos(𝛿)
N_factor.m 52
𝐿𝑅𝐵 factor from Shen et al. 𝐿𝑅𝐵 = cos(𝜃)·cos(𝜅)LRB_parameter.m 71 /72
𝑚parameter from Luster and Morris 𝑚= cos(𝜓)·cos(𝜅)mprime.m 53
residual Burgers vector (
𝑏r)
𝑏r=𝑔in ·
𝑏in 𝑔out ·
𝑏out resid-
ual_Burgers_vector.m
55
𝜆function from Werner and Prantl 𝜆= cos(90𝜓
𝜓𝑐) cos( 90𝜅
𝜅𝑐)lambda.m 86
Resolved Shear Stress (𝜏𝑖) / Schmid
Factor
𝜏𝑖=𝜎:𝑆0𝑖with 𝑆0𝑖=𝑑𝑛re-
solved_shear_stress.m
66
Grain boundary Schmid factor 𝑚GB =𝑚in +𝑚out re-
solved_shear_stress.m
1
Generalized Schmid Factor (𝐺𝑆𝐹 )𝐺𝑆𝐹 =𝑑·𝑔𝜎𝑔 ·𝑛general-
ized_schmid_factor.m
66
81 T. Tsuru et al., “A predictive model for transferability of plastic deformation through grain boundaries.”, AIP ADVANCES, (2016), 6(015004).
84 Wang H. et al., “In situ analysis of the slip activity during tensile deformation of cast and extruded Mg-10Gd-3Y-0.Zr(wt.%) at 250°C”,
Materials characterization, (2016), 116, pp. 8-17.
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4.4.6 Slip and twin systems implemented in the STABiX toolbox
List of slip and twin systems for FCC phase material used in STABiX and DAMASK - FCC.
List of slip and twin systems for BCC phase material used in STABiX and DAMASK - BCC.
List of slip and twin systems for HCP phase material used in STABiX and DAMASK - HCP.
4.4.7 References
4.5 Experimental data
To use the STABiX toolbox, some experimental data are required :
average grain orientations (Euler angles (𝜑1,Φ,𝜑2) in degrees) or intragranular misorientation (misorientation
axis [𝑢𝑣𝑤]/ angle 𝜔);
grains boundaries positions (optional for the bicrystal analysis);
grains positions (optional for the bicrystal analysis);
geometry of grain boundaries (trace angle and grain boundary inclination) (optional).
TEM experiments can provide intragranular misorientation and EBSD measurements can provide average
grain orientations, grains boundaries and grains positions, and grain boundary trace angle.
Inclination of the grain boundary can be evaluated by serial polishing or focused ion beam (FIB) sectioning,
either parallel or perpendicular to the surface of the sample.
4.5.1 EBSD map GUI
To plot EBSD map in the EBSD map GUI, two types of TSL-OIM files are required :
Reconstructed Boundaries File ;
Grain File Type 2.
TSL-OIM data preparation
Open you .osc (or your .ctf) file in the TSL-OIM Analysis Software.
Warning: Set the TSL coordinates system !
Change data properties for the detection of grain boundaries (All data –> Properties).
Clean up your dataset (Filename –> Cleanup).
Reference : OIM ANALYSIS 6.0 (user manual) and OIM ANALYSIS 7.0 (user manual) / EDAX website
Reconstructed Boundaries File
Export “Reconstructed Boundaries File” of the cleaned dataset (All data –> Export –> Reconstructed
Boundaries), with the following options defined by default :
Right hand average orientation (𝜑1,Φ,𝜑2) in degrees ;
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Left hand average orientation (𝜑1,Φ,𝜑2) in degrees ;
Trace angle (in degrees) ;
• (𝑥, 𝑦) coordinates of endpoints (in microns) ;
IDs of right hand and left hand grains.
Note: Reconstructed boundary methodology is only applied to data collected on a hexagonal grid. Nevertheless, it is
possible to convert a square grid into an hexagonal grid in TSL-OIM software.
Warning: It is not possible to export a “Reconstructed Boundaries File”, containing “opened” grain boundaries.
Example of “Reconstructed Boundary File”: MPIE_cpTi_reconstructed_boundaries_2013.txt
The Matlab function used to read “Reconstructed Boundary File” is :
read_oim_reconstructed_boundaries_file.m
If some GBs segments are missing or some wrong segments are exported, play with partition properties in the
TSL-OIM software in order to export a more realistic Reconstructed Boundaries file:
decrease/increase “Grain Tolerance Angle” ;
decrease/increase “Minimum Grain Size” ;
decrease/increase the maximum deviation between reconstructed boundary and corresponding boundary seg-
ments.
Grain File Type 2
Export “Grain File Type 2” of the cleaned dataset (All data –> Export –> Grain File), with the following
options :
Integer identifying grain ;
Average orientation (𝜑1,Φ,𝜑2) in degrees ;
Average position (𝑥, 𝑦) in microns ;
An integer identifying the phase ;
Edge or interior grain (optional) ;
Diameter of the grain in microns (optional).
Note: Export the “Grain File Type 2” in the same location as the corresponding “Reconstructed Boundary File”.
Example of “Grain Gile Type 2”: MPIE_cpTi_grain_file_type2_2013.txt
The Matlab function used to read “Grain File Type 2” is : read_oim_grain_file_type2.m
Loading other type of EBSD data files...
It is possible to load other type of EBSD data files (e.g. : .ctf files), using the ‘import_wizard’ of the MTEX
toolbox.
First, download and install the MTEX Toolbox.
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Then, import your EBSD data (e.g.: .ang file) and set the coordinate systme, using the ‘import_wizard’ and
save the EBSD dataset in the Matlab workspace as a variable named ‘ebsd’, and press ‘Finish’.
The EBSD map is automatically plotted from the imported data. The coordinate system and the scan unit are
set from the properties of the imported data.
Note: For a single phase material, the phase number is 0 or 1. For a two phases material, the phase numbers are
respectively 1 and 2. For non-indexed pixels, the phase is numbered as -1.
How to generate a .ang file with TSL-OIM software ?
Export “Scan Data (.ang file)” of the cleaned dataset (Filename –> Export –> Scan Data) (optional).
Example of an .ang file.
The Matlab functions used to generate .ang file v6 and v7 are respectively:
write_oim_ang_file_v6.m ;
write_oim_ang_file_v7.m.
Possible errors introduced during files exportation from TSL-OIM
“Grain File Type 2” –> Missing integer identifying grain
Solved when file is imported via the GUI.
“Reconstructed Boundary File” –> Inversion of left and right grains for a given grain boundary
Cross product performed between GB vector and center of grains to check (if cross product < 0 : no inversion,
and if cross product > 0 : inversion).
“Reconstructed Boundary File” –> 𝑥-axis and 𝑦-axis not corrects. . .
𝑦coordinates is multiplied by -1 when file is imported via the GUI.
Note: All of these issues are taken into account and corrected automatically when user is loading his data via the
EBSD map GUI.
Issues with plot of EBSD maps
Sometimes, grain boundaries coordinates are too big compared to the grain size, because of the Voronoi tesse-
lation for example. Thus, the following plot can be obtained :
In this case, it is advised to use the ‘zoom’ function of Matlab to zoom in and zoom out in the center of the
EBSD map, to vizualize the grains. It is also possible to set directly the limits of axis (e.g.: xlim([0 1500]);
ylim([-1000 0]);) in the command window of Matlab.
4.5.2 Bicrystal GUI
The YAML configuration file provides a simple way to define a bicrystal.
An example of bicrystal configuration file is given here : config_gui_BX_defaults.yaml
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Figure 4.10: Screenshot of the EBSD map GUI with a problem of axis limits.
Copy this example file and modify it with your data. Be careful to put a space after the comma in a list (e.g.
[𝑥,𝑦,𝑧]).
Warning: Don’t change fieldnames and don’t round Euler angles. Euler angles are given in degrees.
Load your YAML bicrystal configuration file via the menu in the bicrystal GUI. You may have to run again the
path_management.m Matlab function, if your YAML bicrystal configuration file is not found by Matlab.
Visit the YAML website for more informations.
Visit the YAML code for Matlab.
4.5.3 Convention for bicrystal EBSD/indentation experiments
4.6 EBSD map GUI
This GUI allows to analyze quantitatively slip transmission across grain boundaries for an EBSD map.
The Matlab function used to run the EBSD map GUI is : A_gui_plotmap.m
This includes:
Loading EBSD data
Smoothing GBs segments
Misorientation angle
m’ parameter
Residual Burgers vector
Schmid factor and slip trace analysis
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Figure 4.11: Geometrical convention of a bicrystal.
4.6.1 Loading EBSD data
For more details about the format of the EBSD data, see also the page Experimental data.
Figure 4.12: The different steps to load data into tge EBSD map GUI.
4.6.2 Smoothing GBs segments
The smoothing algorithm allows to decrease the total number of grains boundaries in order to speed up calcu-
lations and plots.
The Matlab function used to smooth GBs is : interface_map_GB_segments_opti.m
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Figure 4.13: Screenshot of the EBSD map GUI with an EBSD map of near alpha phase Ti alloy a) before smoothing
and b) after smoothing.
4.6.3 Misorientation angle
Figure 4.14: Screenshot of the EBSD map GUI with an EBSD map of near alpha phase Ti alloy (GBs color-coded in
function of the maximum misorientation angle value).
4.6.4 m’ parameter
4.6.5 Residual Burgers vector
4.6.6 Schmid factor and slip trace analysis
4.7 Bicrystal GUI
This GUI allows to analyze quantitatively slip transmission across grain boundaries for a single bicrystal.
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Figure 4.15: Screenshot of the EBSD map GUI with an EBSD map of near alpha phase Ti alloy (GBs color-coded in
function of the maximum m’ value).
Figure 4.16: Screenshot of the EBSD map GUI with an EBSD map of near alpha phase Ti alloy (GBs color-coded in
function of the maximum m’ value obtained for slips with the highest generalized Schmid factor).
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Figure 4.17: Screenshot of the EBSD map GUI with an EBSD map of near alpha phase Ti alloy (GBs color-coded in
function of the maximum residual Burgers vector value).
Figure 4.18: Screenshot of the EBSD map GUI with an EBSD map of near alpha phase Ti alloy (slip plane plotted
inside grain and slip traces plotted around unit cells, both in function of the maximum Schmid factor calculated with
a given stress tensor).
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The Matlab function used to run the Bicrystal GUI is : A_gui_plotGB_Bicrystal.m
This includes:
Plotting and analyzing a bicrystal
Distribution of all slip transmission parameters
4.7.1 Loading Bicrystal data
It is possible to load bicrystal properties (material, phase, Euler angles of both grains, trace angle...) :
from the EBSD map GUI (by giving GB number and pressing the button ‘PLOT BICRYSTAL’) ;
from a YAML config. bicrystal (from the menu, by clicking on ‘Bicrystal, and ‘Load Bicrystal config. file’).
Figure 4.19: The different steps to set the Bicrystal GUI.
4.7.2 Plotting and analyzing a bicrystal
4.7.3 Distribution of all slip transmission parameters
It is possible to generate a new window, in which all values of the selected slip transmission parameter are
plotted in function of selected slip families.
4.8 CPFE simulation preprocessing GUIs
The preCPFE GUIs can rapidly transfer the experimental data into crystal plasticity finite element (CPFE)
simulation input files. The types of input files are :
scripts to generate the finite element models in MSC.Mentat (2008 to 2014) (procedure file format) or Abaqus
(6.12 to 6.14) (Python script) based on the experimental data and test geometry ;
the crystallographic orientations from the experimental data sets ;
material parameter files for the subroutines that implement the constitutive model.
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Figure 4.20: Screenshot of the Bicrystal GUI.
Figure 4.21: Screenshot of the distribution of all slip transmission parameters (e.g.: m’ parameter for a single phase
(HCP) bicrystal).
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A parametrized visualization of the bicrystal indentation model through the GUI allows tuning the geometry
and finite element discretization and the size of the sample and the indenter.
Currently the following models can be written:
Single crystal (SX) indentation (MSC.Mentat and Abaqus)
Bicrystal (BX) indentation (MSC.Mentat and Abaqus)
Scratch test on SX and BX (MSC.Mentat and Abaqus)
Please find here the Python package used to generate the SX and BX indentation models.
4.8.1 How to load crystallographic properties of the SX or of the BX ?
It is possible to set SX or BX properties (material, phase, Euler angles, trace angle...) :
from the Bicrystal GUI (by giving GB’s number and pressing the button ‘PLOT BICRYSTAL’);
from a YAML configuration file (from the menu, by clicking on ‘preCPFE-SX’ or ‘preCPFE-BX’, and ‘Load
Single Crystal config. file’ or ‘Load Bicrystal config. file’).
Figure 4.22: The different steps to set the preCPFE GUIs.
4.8.2 Single crystal (SX) indentation
Analysis of the orientation dependent pile-up topographies that are formed during single crystal indentation
provides insight into the operating deformation mechanisms. CPFE simulation of single crystal indentation
has an important role in clarifying the influence of the single-slip behaviour of different slip systems on the
resulting surface profiles.
The function used to run the preCPFE GUI for SX indentation is :
A_preCPFE_windows_indentation_setting_SX.m
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Figure 4.23: Screenshot of the preCPFE GUI for the single crystal indentation
Figure 4.24: Screenshot of the single crystal indentation model in Abaqus.
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Figure 4.25: Convention used to define the single crystal mesh.
Convention for the single crystal mesh
4.8.3 Bicrystal (BX) indentation
CPFE simulation of indentation close to grain boundaries can provide a good approximation of the local
micromechanics in this experiment. While models that take into account the micromechanical effect of the
boundary are the subject of ongoing research, most geometrical and kinematic factors are taken into account
by employing a local phenomenological crystal plasticity formulation in the simulations.
The function used to run the preCPFE GUI for BX indentation is :
A_preCPFE_windows_indentation_setting_BX.m
Figure 4.26: Screenshot of the preCPFE GUI for the bicrystal indentation.
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Figure 4.27: Convention used to define the bicrystal mesh.
Convention for the bicrystal mesh
4.8.4 Scratch test on SX and BX
CPFE simulation of scratch test in a single crystal or close to a grain boundary is implemented into this GUI.
Scratch length and scratch direction have to be set by the user.
Figure 4.28: Screenshot of the preCPFE GUI for the scratch test.
4.8.5 Indenter’s geometry
Currently the following geometries can be used for CPFE simulations :
cono-spherical indenter ;
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Berkovich indenter ;
Vickers indenter ;
cube corner indenter ;
flat punch ;
free topography (from an AFM measurement for instance).
For the Berkovich, Vickers, cube corner indenters and the free topography, the faces and vertices are saved
in a structure variable from a patch object. For the cono-spherical and the flat punch, geometries are already
implemented in the Python package for MSC.Mentat and Abaqus. It is possible as well to call the Matlab
function surf2patch, to return the faces and vertices from a surface object.
Then the function patch2inp is used to generate an Abaqus .inp file, which is used when the CPFE model is
created in MSC.Mentat or Abaqus.
It is possible to rotate directly into the GUIs, the Berkovich, Vickers, cube corner indenters and the free
topography before the generation of the Abaqus .inp file.
Figure 4.29: Screenshot of the preCPFE GUI for the bicrystal indentation with Berkovich indenter.
AFM topography
The topography from an Atomic Force Microscopy (AFM) measurement has to be saved into a .txt file in the
Gwyddion ASCII format.
The Matlab function used to load and read Gwyddion file is : read_gwyddion_ascii.m
Visit the Gwyddion website for more information.
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Figure 4.30: Screenshot of the preCPFE GUI for the bicrystal indentation with loaded AFM topography of the indenter.
4.8.6 Contact definition
MSC.Mentat
The indenter is modeled by a rigid body and the sample by a deformable body.
Contact is defined by a bilinear Coulomb friction model.
Abaqus
The indenter is modeled by a rigid body and the sample by a deformable body.
The external surface of the indenter is defined as the “master” region.
The top surface of the (multilayer) sample is defined as the “slave” region.
If the coefficient friction is different from 0, the classical isotropic Coulomb friction model is used to define
the contact between the indenter and the sample.
If the coefficient friction is set to 0, the contact is defined by a frictionless tangential behavior and a hard
normal behavior.
A friction coefficient of 0.3 is set by default for every CPFE simulation. It is possible to modify this parameter,
by changing its value in the preCPFE GUIs.
4.8.7 Mesh definition
MSC.Mentat
The mesh is defined by default by hexahedral eightnode elements (hex8).
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Abaqus
The mesh is defined by default by linear hexahedral eightnode elements (C3D8).
It is possible to set quadratic elements (e.g.: C3D20), by changing in the python code the value of the “lin-
ear_elements” variable from 1 to 0.
Note: Note that DAMASK incorporates a limited number of different types of element geometries. For a detailed
information about the characteristics of each element refer to MSC.Marc and Abaqus user’s manuals.
4.8.8 Python setup
For the generation of the CPFE preprocessing scripts an installation of Python is required together with the
Numpy 1.10.4 ans Scipy 0.16.0 packages. Often one of the scientific Python distributions is the easiest way to
get up and running (use a Python 2.x distribution). To make sure that STABiX can find the installed Python you
will have to either put it on the system’s PATH or put it’s exact location in the user configuration as detailed
below.
4.8.9 Adjusting the configuration settings
To write out the necessary files for finite element simulations it is likely that the user wants to adjust some
settings such as the used python installation or the path where the files are written to. This can be achieved
in the custom menu of the preCPFE GUis : Edit CPFEM config file. A user specific copy of the
default configuration YAML file is created and opened in the Matlab editor. To benefit from later changes in
the default settings, all configuration parameters that are not specific to the user’s setup should be deleted from the
user’s CPFE configuration file.
4.8.10 Installing DAMASK
For instructions on how to set up the DAMASK constitutive simulation code please visit http://DAMASK.
mpie.de.
4.8.11 Writing the CPFE input files
After everything is configured and the model geometry and discretization is optimized, all necessary files to
run a CPFE simulation can be generated by pressing the green button. All information will be written to a
newly created folder which also includes a timestamp for later reference.
4.8.12 Input files
MSC.Mentat
a procedure file containing the FEM model (*.proc)
a Python file containing parameters for FEM model (*.mat_FEM_model_parameters.py)
a Python file containing material configuration (*.mat_DAMASK_materialconfig.py)
a MAT-file (binary Matlab format file) storing Matlab workspace variables(*.mat)
a material configuration file (material.config)
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an input file for specific indenter’s geometry (*.inp) (optional)
Abaqus
a Python file containing the FEM model (*.py)
a Python file containing parameters for FEM model (*.mat_FEM_model_parameters.py)
a Python file containing material configuration (*.mat_DAMASK_materialconfig.py)
a MAT-file (binary Matlab format file) storing Matlab workspace variables(*.mat)
a material configuration file (material.config)
an input file for specific indenter’s geometry (*.inp) (optional)
4.8.13 Using the CPFE input files
MSC.Mentat ‘classic interface’
‘Files’ ==> ‘Current Directory’ ==> Select the folder containing input files
‘Utils’ ==> ‘Procedures’ ==> Select procedure file containing the FEM model (*.proc)
MSC.Mentat ‘new interface (> 2012)’
‘Files’ ==> ‘Current Directory’ ==> Select the folder containing input files
‘Tools’ ==> ‘Procedures’ ==> Select procedure file containing the FEM model (*.proc)
Abaqus
‘File’ ==> ‘Set Work Directory...’ ==> Select the folder containing input files
‘File’ ==> ‘Run Script’ ==> Select the Python file containing the FEM model (*.py)
4.8.14 Running a job with DAMASK
MSC.Mentat
In the JOB RUN menu choose USER SUBROUTINE FILE and select the interface routine
DAMASK_marc.f90.
Find the full documentation for the use of DAMASK with Marc here : http://damask.mpie.de/Usage/Marc.
Abaqus
In the Job Manager > Create... specify the User subroutine file (either DAMASK_abaqus_std.f or
DAMASK_abaqus_exp.f).
Find the full documentation for the use of DAMASK with Abaqus here : http://damask.mpie.de/Usage/Abaqus.
Note:
For Abaqus, you may have to modify the extension of the subroutine :
.f if the operating environment is Linux ;
.for if the operating environment is Windows.
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4.8.15 See also
Wang Y. et al., “Orientation dependence of nanoindentation pile-up patterns and of nanoindentation microtextures in
copper single crystals”, Acta Materialia (2004).
Liu Y. et al., “Combined numerical simulation and nanoindentation for determining mechanical properties of single
crystal copper at mesoscale”, Journal of the Mechanics and Physics of Solids (2005).
Zaafarani N. et al., “Three-dimensional investigation of the texture and microstructure below a nanoindent in a Cu
single crystal using 3D EBSD and crystal plasticity finite element simulations”, Acta Materialia (2006).
Zambaldi C. et al., “Modeling and experiments on the indentation deformation and recrystallization of a single-crystal
nickel-base superalloy”, Materials Science and Engineering A (2007).
Liu Y. et al., “Orientation effects in nanoindentation of single crystal copper”, International Journal of Plasticity (2008).
Zaafarani N. et al., “On the origin of deformation-induced rotation patterns below nanoindents”, Acta Materialia
(2008).
Casals O. and Forest S., “Finite element crystal plasticity analysis of spherical indentation in bulk single crystals and
coatings”, Computational Materials Science (2009).
Gerday A. F. et al., “Interests and limitations of nanoindentation for bulk multiphase material identification: Applica-
tion to the 𝛽phase of Ti-5553”, Acta Materialia (2009).
Britton T.B. et al., “The effect of crystal orientation on the indentation response of commercially pure titanium: ex-
periments and simulations”, Proc. R. Soc. A (2010).
Chang H.-J. et al., “Multiscale modelling of indentation in FCC metals: From atomic to continuum”, C. R. Physique
(2010).
Zambaldi C. and Raabe D., “Crystal plasticity modelling and experiments for deriving microstructure-property rela-
tionships in 𝛾-TiAl based alloys”, Journal of Physics (2010).
Zambaldi C., “Micromechanical modeling of 𝛾-TiAl based alloys”, PhD Thesis (2010).
Zambaldi C. and Raabe D., “Plastic anisotropy of c-TiAl revealed by axisymmetric indentation”, Acta Materialia
(2011).
Vu-Hoang S. et al., “Crystal Plasticity of Single Crystal and Film on Substrate Probed by Nano-lndentation: Simula-
tions and Experiments”, Mater. Res. Soc. Symp. Proc. (2011).
Eidel B., “Crystal plasticity finite-element analysis versus experimental results of pyramidal indentation into (001) fcc
single crystal”, Acta Materialia (2011).
Zambaldi C. et al., “Orientation informed nanoindentation of a-titanium: Indentation pileup in hexagonal metals
deforming by prismatic slip”, J. Mater. Res. (2012).
Zahedi A. et al., “Indentation in f.c.c. single crystals”, Solid State Phenomena (2012).
Liu M. et al., “Crystal Plasticity Study of the Effect of the Initial Orientation on the Indentation Surface Profile Patterns
and Micro-Textures of Aluminum Single Crystal”, Advances in Materials and Processing Technologies (2013).
Han F. et al., “Experiments and crystal plasticity finite element simulations of nanoindentation on Ti-6Al-4 V alloy”,
Materials Science & Engineering A (2014).
Choudhury S.F. et al., “Single Crystal Plasticity Finite Element Analysis of Cu6Sn5 Intermetallic”, Metall. and Mat.
Trans. A (2014).
González D. et al., “Numerical analysis of the indentation size effect using a strain gradient crystal plasticity model”,
Computational Materials Science (2014).
42 Chapter 4. Contents
STABiX Documentation, Release 2.0.0
Esqué-de los Ojos D. et al., “Sharp indentation crystal plasticity finite element simulations: Assessment of crystallo-
graphic anisotropy effects on the mechanical response of thin fcc single crystalline films”, Computational Materials
Science (2014).
Kitahara H. et al., “Anisotropic deformation induced by spherical indentation of pure Mg single crystals”, Acta Mate-
rialia (2014).
Kucharski S. et al., “Surface Pile-Up Patterns in Indentation Testing of Cu Single Crystals”, Experimental Mechanics
(2014).
Liu M. et al., “A crystal plasticity study of the effect of friction on the evolution of texture and mechanical behaviour
in the nano-indentation of an aluminium single crystal”, Computational Materials Science (2014).
Yao W.Z. et al., “Plastic material parameters and plastic anisotropy of tungsten single crystal: a spherical micro-
indentation study”, J. Mater. Sci. (2014).
Charleux L., “Abapy Documentation”.
CPFEM simulations of nanoindentation - ongoing research.
Zambaldi C. et al., “Orientation dependent deformation by slip and twinning in magnesium during single crystal
indentation”, Acta Materialia (2015).
Sánchez-Martín R. et al., “High temperature deformation mechanisms in pure magnesium studied by nanoindentation”,
Scripta Materialia (2015).
Mao L. et al., “Explore the anisotropic indentation pile-up patterns of single-crystal coppers by crystal plasticity finite
element modelling”, Materials Letters (2015).
Mao L. et al., “Crystal plasticity FEM study of nanoindentation behaviors of Cu bicrystals and Cu–Al bicrystals”, J.
Mater. Res., (2015).
Renner E. et al., “Sensitivity of the residual topography to single crystal plasticity parameters in Berkovich nanoin-
dentation on FCC nickel”, International Journal of Plasticity, (2015).
Juran P. et al., “Investigation of indentation-, impact- and scratch-induced mechanically affected zones in a copper
single crystal”, Comptes Rendus Mécanique, (2015).
Materna A. et al., “A Numerical Investigation of the Effect of Cubic Crystals Orientation on the Indentation Modulus”,
Acta Physica Polonica A, (2015).
Csanádi T. et al., “Nanoindentation induced deformation anisotropy in 𝛽-Si3N4 ceramic crystals”, Journal of the
European Ceramic Society, (2015).
Liu M. et al., “A combined experimental-numerical approach for determining mechanical properties of aluminum
subjects to nanoindentation”, Scientific Reports, (2015).
Su Y. et al., “Quantifying deformation processes near grain boundaries in 𝛼titanium using nanoindentation and crystal
plasticity modeling”, International Journal of Plasticity, (2016).
Weaver J.S. et al., “On capturing the grain-scale elastic and plastic anisotropy of alpha-Ti with spherical nanoindenta-
tion and electron back-scattered diffraction”, Acta Materialia, (2016).
Han F. et al., “Indentation Pileup Behavior of Ti-6Al-4V Alloy: Experiments and Nonlocal Crystal Plasticity Finite
Element Simulations”, Metallurgical and Materials Transactions A, (2017).
Chakraborty A. and Eisenlohr P., “Evaluation of an inverse methodology for estimating constitutive parameters in
face-centered cubic materials from single crystal indentations”, European Journal of Mechanics - A/Solids, (2017).
4.9 Analysis of literature data
Please, find here the Matlab functions to analyze results and to plot data from the following papers.
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4.9.1 Residual Burgers vector
Kacher and Robertson (2012)
Kacher and Robertson analyzed slip transfer in 304 stainless steel (FCC structure), using in situ TEM defor-
mation3. In this work, a bicrystal with a misorientation angle of 36° and misorientation axis of [-11, -22, -2]
is characterized. Dislocation/grain boundary interactions are analysed and knowing the incoming system, the
magnitude of residual dislocation Burgers vector is plotted in function of possible outgoing systems. Calcula-
tions are reproduced using the Matlab toolbox and obtained values are compared to Kacher’s results (see Figure 4.31).
Figure 4.31: a) Plot of a bicrystal with a misorientation angle of 36°and misorientation axis of [-11, -22, -2] from
Kacher’s paper. b) Magnitude of residual Burgers vector given in Kacher’s paper compared to values calculated with
the Matlab toolbox.
Patriarca et al. (2013)
Patriarca et al. analysed the deformation response of a FeCr polycrystal (BCC structure) by a combination
of EBSD and digital image correlation (DIC) characterizations6. The magnitude of residual dislocation Burg-
ers vector is plotted for numerous grain boundaries, knowing incoming and outgoing slips. Calculations are
reproduced using the Matlab toolbox and results are compared to Patriarca’s results (see Figure 4.32).
Kacher and Robertson (2014)
Kacher and Robertson analyzed slip transfer in alpha cp-Ti (HCP structure), using in situ TEM deformation4.
In this work, a bicrystal with a misorientation angle of 32° and misorientation axis of [1, 5, -6, 16] is character-
ized. Dislocation/grain boundary interactions are analysed and knowing the incoming system, the magnitude
of residual dislocation Burgers vector is plotted in function of possible outgoing systems. Calculations are
reproduced using the Matlab toolbox and obtained values are compared to Kacher’s results (see Figure 4.33).
3J. Kacher and I.M. Robertson, “Quasi-four-dimensional analysis of dislocation interactions with grain boundaries in 304 stainless steel.”, Acta
Materialia (2012), 60(19), pp. 6657–6672.
6L. Patriarca et al., “Slip transmission in bcc FeCr polycrystal.”, Materials Science&Engineering (2013), A588, pp. 308–317.
4J. Kacher and I.M. Robertson, “In situ and tomographic analysis of dislocation/grain boundary interactions in 𝛼-titanium.”, Philosophical
Magazine (2014), 94(8), pp. 814-829.
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Figure 4.32: Magnitude of residual Burgers vector given in Patriarca’s paper compared to values calculated with the
Matlab toolbox for numerous grain boundaries.
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Figure 4.33: a) Plot of a bicrystal with a misorientation angle of 32° and misorientation axis of [1, 5, -6, 16] from
Kacher’s paper. b) Magnitude of residual Burgers vector given in Kacher’s paper compared to values calculated with
the Matlab toolbox for the bicrystal #2.
Cui et al. (2014)
Cui et al. analyzed slip transfer in proton-irradiated 13Cr15Ni stainless steel (fcc structure), using in situ TEM
deformation1. In this work, two bicrystals with respectively a misorientation angles of 60° and 40° and a
misorientation axis of [1, 1, -1] and [1, 0, 1] are characterized. Dislocation/grain boundary interactions are
analysed and knowing the incoming system, the magnitude of residual dislocation Burgers vector is plotted in
function of possible outgoing systems. Calculations are reproduced using the Matlab toolbox and obtained values are
compared to Cui’s results (see Figure 4.34).
Figure 4.34: a) Plot of a bicrystal with a misorientation angle of 60° and misorientation axis of [1, 1, -1] from Cui’s
paper. b) Plot of a bicrystal with a misorientation angle of 40° and misorientation axis of [1, 0, 1] from Cui’s paper.
c) Magnitude of residual Burgers vector given in Cui’s paper compared to values calculated with the Matlab toolbox
for the two bicrystals.
1B. Cui et al., “Influence of irradiation damage on slip transfer across grain boundaries.”, Acta Materialia (2014), 65, pp. 150-160.
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4.9.2 m’ factor
Guo et al. (2014)
Guo et al. analyzed slip transfer in cp-Ti (HCP structure), by tensile test combined to in situ digital image
correlation (DIC)2. In this work, many bicrystals are characterized and slip band–grain boundary interactions
are analyzed in term of stress concentration along the slip plane direction. The 𝑚factor is used to quantify
the transmissivity across the GBs and calculations are reproduced using the Matlab toolbox (see Figure 4.35).
Figure 4.35: m’ factor values given in Guo’s paper compared to values calculated with the Matlab toolbox for 7
different bicrystals.
4.10 A Matlab toolbox to analyze grain boundary inclination from
SEM images
First of all, download the source code of the Matlab toolbox.
Source code is hosted at Github.
Download source code as a .zip file.
2Y. Guo et al., “Slip band–grain boundary interactions in commercial-purity titanium.”, Acta Materialia (2014), 76, pp. 1-12.
4.10. A Matlab toolbox to analyze grain boundary inclination from SEM images 47
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This toolbox helps to find the grain boundary inclination from two micrographs from serial polishing. At least
three marks such as microindents are needed for registration of the images.
Examples of micrographs from serial polishing.
To get started with this toolbox, clone the repository, then run Matlab, and cd into the folder containing this README
file. Then add the package path to the Matlab search path by typing “path_management”. Finally you can start the
launcher by typing demo or A_gui_gbinc at the Matlab command prompt.
4.10.1 How to use the toolbox ?
1. Run the function A_gui_gbinc.m.
2. Select your first image before serial polishing.
3. Do the calibration to get the factor scale.
4. Do the edge detection.
5. Repeat the same operation for the second image obtained after serial polishing.
6. Do the overlay :
If control points don’t exist (it’s the case for the 1st time), a window appears and it is possible to define control
points.
Define 3 control points per images.
Select a point on the figure on the left, then on the figure on the right, and repeat this operation 2 times.
Close the window for the selection of control points (Ctrl+W).
Control points are saved in .mat file (in the same folder than the 1st picture loaded).
7. If the control points are not satisfying, delete them and redo the step 6 to set new control points and to get a new
overlay.
8. Save the overlay in the same folder than the 1st picture loaded (as a screenshot.png) (optional).
9. Do the measurement of the distance between edges (Vickers faces) or ridges of a unique Vickers indent (see
Figure 4.37).
10. Do the measurement of the distance between edges of a unique grain boundary.
11. The value of the grain boundary inclination is finally given in degrees.
Calculation of the thickness of removed material after polishing
=𝑑
tan(90 𝛼)(4.29)
With 𝑑the distance between edges (Vickers faces) or ridges of a unique Vickers indent (obtained before and after
polishing), and 𝛼the angle between the Vickers indent and the surface of the sample (see Figure 4.37).
Calculation of grain boundary inclination
𝐺𝐵𝑖𝑛𝑐 = tan 𝑑𝐺𝐵
(4.30)
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Figure 4.36: Screenshot of the Matlab GUI used to calculate grain boundary inclination.
Figure 4.37: Schemes of a) the top view of a Vickers indent (before and after polishing) and of b) the cross-section
view.
4.10. A Matlab toolbox to analyze grain boundary inclination from SEM images 49
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With 𝑑𝐺𝐵 the distance between grain boundary traces (obtained before and after polishing), and the thickness of
removed material after polishing calculated
Note: Images should have the same scale factor.
Note: Distances and grain boundary inclination values are obtained with the mean scale factor of the two images.
4.10.2 See also
V. Randle and Dingley D., “Measurement of boundary plane inclination in a scanning electron microscope.”, Scripta
Metall., 1989, 23, pp. 1565–1569.
V. Randle, “A methodology for grain boundary plane assessment by single-section trace analysis.”, Scripta Mater.,
2001, 44, pp. 2789-2794.
4.10.3 Links
Matlab - Interactive Exploration with the Image Viewer App
Matlab - Distance tool
Matlab - Image conversions
Matlab - Image filtering
Matlab - Control Point Selection Tool
Matlab - Spatial transformation from control point pairs
Matlab - Edge detection
4.10.4 Authors
Written by D. Mercier [1] and C. Zambaldi [1].
[1] Max-Planck-Institut für Eisenforschung, 40237 Düsseldorf, Germany
4.10.5 Acknowledgements
Parts of this work were supported under the NSF/DFG Materials World Network program (DFG ZA 523/3-1 and
NSF-DMR-1108211).
4.10.6 Keywords
Matlab ; Graphical User Interface (GUI) ; Grain Boundaries ; Polycrystalline Metals ; Grain Boundary Inclination ;
Serial Polishing ; Scanning Electron Microscope (SEM).
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4.11 References
4.11.1 Related Projects
Figure 4.38: MTEX
Figure 4.39: DAMASK
4.11.2 Institutions
Figure 4.40: Max-Planck-Institut fuer Eisenforschung GmbH (Duesseldorf, Germany)
4.11. References 51
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Figure 4.41: Department of Chemical Engineering and Materials Science / Michigan State University (East Lansing,
MI, USA)
Figure 4.42: IMDEA Materials Institute (Madrid, Spain)
Figure 4.43: Structural Integrity, Institute of Materials Engineering Australian Nuclear Science and Technology Or-
ganisation (Australia)
Figure 4.44: Physikalische Metallkunde TU Darmstadt (Darmstadt, Germany)
Figure 4.45: RWTH Aachen University (Aachen, Germany)
Figure 4.46: Arizona State University, Solanki Research Group (Tempe, USA)
Figure 4.47: Queen’s University, NSERC Research Chair in Nuclear Materials (Kingston, Canada)
52 Chapter 4. Contents
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Figure 4.48: National University of Ireland (Galway, Ireland)
4.11. References 53
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54 Chapter 4. Contents
CHAPTER 5
References
55
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56 Chapter 5. References
CHAPTER 6
Contact
Authors David Mercier [1], Claudio Zambaldi [1] and Thomas R. Bieler [2].
[1] Max-Planck-Institut für Eisenforschung, 40237 Düsseldorf, Germany
[2] Chemical Engineering and Materials Science, Michigan State University, East Lansing 48824 MI, USA
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58 Chapter 6. Contact
CHAPTER 7
Contributors
Raúl Sánchez Martín (IMDEA, Madrid) contributed Python code to generate Abaqus indentation models.
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60 Chapter 7. Contributors
CHAPTER 8
Acknowledgements
This work was supported by the DFG/NSF Materials World Network grant references (DFG ZA 523/3-1 and NSF-
DMR-1108211).
The authors are grateful to Philip Eisenlohr, Martin Crimp and Yang Su of Michigan State University, and the Max-
Planck-Institut für Eisenforschung for support.
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62 Chapter 8. Acknowledgements
CHAPTER 9
Keywords
Matlab toolbox ; Graphical User Interface (GUI) ; Grain Boundary (GB) ; Polycrystalline Metals ; Slip Transmission
; Bi-Crystal (BX) ; Electron backscatter diffraction (EBSD) ; Instrumented indentation ; Crystal Plasticity Finite
Element Method (CPFEM) ; pythonTM toolbox ; DAMASK.
63
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