# Slip Transmission Matlab toolbox¶

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 Python^{TM} 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.

# How to get STABiX code ?¶

First of all, download the source code of the Matlab toolbox.

# How to cite STABiX in your papers ?¶

# Reference paper¶

“A Matlab toolbox to analyze slip transfer through grain boundaries” D. Mercier, C. Zambaldi, T. R. Bieler, 17th International Conference on Textures of Materials (ICOTOM17), at Dresden, Germany (2014). IOP Conference Series: Materials Science and Engineering Volume 82 conference 1. https://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). https://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). https://doi.org//10.13140/RG.2.2.27427.66084

“Quantifying deformation processes near grain boundaries in α titanium 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). https://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). https://doi.org//10.1016/j.euromechsol.2017.06.012

# Contents¶

## Motivation of this Work¶

The micromechanical behavior of grain boundaries is one of the key components in the understanding of heterogeneous deformation of metals [1]. 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 decades [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14] and [15]. Originally, interactions between dislocations and grain boundaries have been observed in the transmission electron microscope (TEM) after strain test or in situ [4], [5] and [15]. Some authors observed as well slip transmission during indentation tests performed close to grain boundaries [16], [17], [18], [19] and [20].

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) simulations [23].

### 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 indents [21]. By correlating with models of the indentation, a precise constitutive description of the anisotropic plasticity of single-crystalline titanium shall be developed [22] and [23].

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 predicted 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.

[1] | 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. |

[2] | K.T. Aust et al., “Solute induced hardening near grain boundaries in zone refined metals.”, Acta Metallurgica (1968), 16(3), pp. 291-302. |

[3] | J.D. Livingston and B. Chalmers, “Multiple slip in bicrystal deformation.”, Acta Metallurgica (1957), 5(6), pp. 322-327. |

[4] | (1, 2) Z. Shen et al., “Dislocation pile-up and grain boundary interactions in 304 stainless steel.”, Scripta Metallurgica (1986), 20(6), pp. 921–926. |

[5] | (1, 2) Z. Shen et al., “Dislocation and grain boundary interactions in metals.”, Acta Metallurgica (1988), 36(12), pp. 3231–3242. |

[7] | M.J. Marcinkowski and W.F. Tseng, “Dislocation behavior at tilt boundaries of infinite extent.”, Metal. Trans. (1970), 1(12), pp. 3397-3401. |

[8] | W. Bollmann, “Crystal Defects and Crystalline Interfaces”, Springer-Verlag (1970) |

[9] | 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. |

[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. |

[15] | (1, 2) J. Kacher et al., “Dislocation interactions with grain boundaries.”, Current Opinion in Solid State and Materials Science (2014), in press. |

[16] | P.C. Wo and A.H.W. Ngan, “Investigation of slip transmission behavior across grain boundaries in polycrystalline Ni3Al using nanoindentation.”, 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. |

[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. |

[23] | (1, 2) DAMASK — the Düsseldorf Advanced Material Simulation Kit |

## Getting started¶

### Source Code¶

First of all, download the source code of the Matlab toolbox.

### 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

### 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. | config_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. |

Note

‘SX’ is used for single crystal and ‘BX’ for bicrystal.

### 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.

### MTEX toolbox¶

For some options and functions implemented in the STABiX toolbox, you have to download and to install the MTEX Toolbox.

### OpenGL¶

If the OpenGL rendering is not satisfying, you can modify the corresponding option in the config.yaml file.

## Bicrystal Definition¶

### Crystallographic properties of a bicrystal¶

A bicrystal is formed by two adjacent crystals separated by a grain boundary.

**Five macroscopic degrees of freedom are required to characterize a grain boundary**[3], [5], [6] and [7] :- 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 \(\omega\) and the rotation axis common to both crystals \([uvw]\).

Using orientation matrix of both crystals obtained by EBSD measurements, the misorientation or disorientation matrix \((\Delta g)\) or \((\Delta g_\text{d})\) is calculated [4] and [2] :

Disorientation describes the misorientation with the smallest possible rotation angle and \(CS\) denotes one of the symmetry operators for the material [1].

The Matlab function used to set the symmetry operators is : sym_operators.m

The orientation matrix \(g\) of a crystal is calculated from the Euler angles (\(\phi_{1}\), \(\Phi\), \(\phi_{2}\)) using the following equation :

(3)¶\[\begin{split}g = \begin{pmatrix} \cos(\phi_{1})\cos(\phi_{2})-\sin(\phi_{1})\sin(\phi_{2})\cos(\Phi) & \sin(\phi_{1})\cos(\phi_{2})+\cos(\phi_{1})\sin(\phi_{2})\cos(\Phi) & \sin(\phi_{2})\sin(\Phi) \\ -\cos(\phi_{1})\sin(\phi_{2})-\sin(\phi_{1})\cos(\phi_{2})\cos(\Phi) & -\sin(\phi_{1})\sin(\phi_{2})+\cos(\phi_{1})\cos(\phi_{2})\cos(\Phi) & \cos(\phi_{2})\sin(\Phi) \\ \sin(\phi_{1})\sin(\Phi) & -\cos(\phi_{1})\sin(\Phi) & \cos(\Phi) \\ \end{pmatrix}\end{split}\]

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 (\(\Delta g\)), the rotation angle (\(\omega\)) and the rotation axis \([u, v, w]\) can be obtained by the following equations :

The Matlab function used to calculate the misorientation angle is : misorientation.m

The grain boundary plane normal \(n\) can be determined knowing the grain boundary trace angle \(\alpha\) and the grain boundary inclination \(\beta\).

The grain boundary trace angle is obtained through the EBSD measurements (grain boundary endpoints coordinates) 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 5).

[1] | U.F. Kocks et al., “Texture and Anisotropy: Preferred Orientations in Polycrystals and Their Effect on Materials Properties.” Cambridge University Press (2000). |

[2] | A. Morawiec, “Orientations and Rotations: Computations in Crystallographic Textures.”, Springer, 2004. |

[3] | L. Priester, “Grain Boundaries: From Theory to Engineering.”, Springer Series in Materials Science (2013). |

[4] | V. Randle and O. Engler, “Introduction to Texture Analysis : Macrotexture, Microtexture and Orientation Mapping.”, CRC Press (2000). |

[5] | V. Randle, “A methodology for grain boundary plane assessment by single-section trace analysis.”, Scripta Mater., 2001, 44, pp. 2789-2794. |

[6] | V. Randle, “Five-parameter’ analysis of grain boundary networks by electron backscatter diffraction.”, J. Microscopy, 2005, 222, pp. 69-75. |

[7] | A.P. Sutton and R.W. Balluffi, “Interfaces in Crystalline Materials.”, OUP Oxford (1995). |

## Strain Transfer Across Grain Boundaries¶

The strain transfer across grain boundaries can be defined by the four following mechanisms (see Figure 6) [51], [80], [90] and [65]:

- direct transmission with slip systems having the same Burgers vector, and the grain boundary is transparent to dislocations (no strengthening effect) (Figure 6-a);
- direct transmission, but slip systems have different Burgers vector (leaving a residual boundary dislocations) (Figure 6-b);
- indirect transmission, and slip systems have different Burgers vector (leaving a residual boundary dislocations) (Figure 6-c);
- no transmission and the grain boundary acts as an impenetrable boundary, which implies stress accumulations, localized rotations, pile-up of dislocations… (Figure 6-d).

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.

Note

Most of the time, following criteria are used to quantify slip transmission across grain boundaries in monophasic bicrystals. But in case of bimetal interfaces, it sounds that stress-based criteria are more relevant than geometrical criteria, given much higher stresses required for slip transmission [36].

### Geometrical Criteria¶

Based on numerous investigations of dislocation-grain boundary interactions, quantitative geometrical expressions 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 7. \(\kappa\) is the angle between slip directions, \(\theta\) is the angle between the two slip plane intersections with the grain boundary, \(\psi\) is the angle between slip plane normal directions, \(\gamma\) is the angle between the direction of incoming slip and the plane normal of outgoing slip, and \(\delta\) is between the direction of outgoing slip and the plane normal of incoming slip. \(n\), \(d\) and \(l\) are respectively the slip plane normals, slip directions and the lines of intersection of the slip plane and the grain boundary. \(\vec b\) is the Burgers vector of the slip plane and \(\vec b_\text r\) is the residual Burgers vector of the residual dislocation at the grain boundary. The subscripts \(\text{in}\) and \(\text{out}\) refer to the incoming and outgoing slip systems, respectively.

\(N\)

**factor from Livingston and Chalmers in 1957**[53]Many authors referred to this criterion to analyze slip transmission [32], [19], [34], [35], [72], [73], [48], [49], [17] and [83]. Pond et al. proposed to compute this geometric criteria for hexagonal metals using Frank’s method [64].

The Matlab function used to calculate the N factor is: N_factor.m

\(LRB\)

**factor from Shen et al. in 1986**[72] and [73]The original notation of this \(LRB\) factor is \(M\), but unfortunately this notation is often used for the Taylor factor [11]. Pond et al. proposed to compute this geometric criteria for hexagonal metals using Frank’s method [64]. Recently, Spearot and Sangid have plotted this parameter as a function of the misorientation of the bicrystal using atomistic simulations [79].

[47], [48], [49], [17], [42], [43], [2], [28], [29] and [74] mentioned in their respective studies this geometrical parameter as a condition for slip transmission.

The inclination of the grain boundary (\(\beta\)) is required to evaluate this factor and the \(LRB\) or \(M\) factor should be maximized.

The Matlab function used to calculate the LRB factor is: LRB_parameter.m

\(m'\)

**parameter from Luster and Morris in 1995**[54]Many authors found that this \(m'\) parameter, which takes into account the degree of coplanarity of slip systems, is promising to predict slip transmission [86], [89], [13], [10], [11] , [30], [84], [31] and [#Nervo_2016]. Both \(m'\) and \(LRB\) can be easily assessed in computational experiments [11]. This \(m'\) factor 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:

(7)¶\[GB_\text{resfac} = 1 - m'\]This factor is equal to 0, when grain boundary is transparent to dislocations. This implies \(m'\) parameter equal to 1 (slip perfectly aligned).

The Matlab function used to calculate the m’ parameter is; mprime.m

- \(\vec b_\text r\)
**the residual Burgers vector**[56], [12], [51], [52], [16], [49] and [17]. - (8)¶\[\vec b_\text{r} = \vec g_\text{in}\cdot\vec b_\text{in} - \vec g_\text{out}\cdot\vec b_\text{out}\]

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) [75], [42], [43] and [39], Lagow et al. in Mo (BCC) [45], Gemperle et al. and Gemperlova et al. in FeSi (BCC) [28] and [29], Kacher et al. in 304 stainless steel (FCC) [38], and Jacques et al. for semiconductors [37].

Patriarca et al. demonstrated for BCC material the role of the residual Burgers vector in predicting slip transmission, by analysing strain field across GBs determined by digital image correlation [63].

Misra and Gibala used the residual Burgers vector to analyze slip across a FCC/BCC interphase boundary [57].

The Matlab function used to calculate the residual Burgers vector is : residual_Burgers_vector.m

- \(\vec b_\text r\)
**The misorientation or disorientation**(\(\Delta g\) or \(\Delta g_\text d\)) [3], [15] and [89]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 crystals [3] and [15].

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 simulations [80], [55], [50], [5], [68] and [69]. Some authors studied the stability of grain boundaries by the calculations of energy difference vs. misorientation angle through the hexagonal c-axis/a-axis [26].

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

\(\lambda\)

**function from Werner and Prantl in 1990**[87]With this function, slip transmission is expected to occur only when the angle \(\psi\) between slip plane normal directions is lower than a given critical value (\(\psi_c = 15°\)) and the angle \(\kappa\) between slip directions is lower than a given critical value (\(\kappa_c = 45°\)).

The Matlab function used to calculate the \(\lambda\) function is: lambda.m

The authors proposed to plot pseudo-3D view of the \(\lambda\) map (see Figures 5 and 6) using the following equation [87] :

(11)¶\[\lambda = \sum\limits_{\alpha=1}^N \sum\limits_{\beta=1}^N \cos\left(\frac{90°}{\psi_c}\arccos(\vec n_{\text{in},\alpha} \cdot \vec n_{\text{out},\beta})\right)\cos\left(\frac{90°}{\kappa_c}\arccos(\vec d_{\text{in},\alpha} \cdot \vec d_{\text{out},\beta})\right)\]With \(N\) the number of slip systems for each adjacent grains.

The Matlab function used to plot pseudo-3D view of the the \(\lambda\) function is: lambda_plot_values.m

This function is modified by Beyerlein et al., using the angle \(\theta\) between the two slip plane intersections with the grain boundary, instead of using the angle \(\psi\) between the two slip plane normal directions [9].

The Matlab function used to calculate the modified \(\lambda\) function is : lambda_modified.m

### Stress Criteria¶

**Schmid Factor**(\(m\)) [67], [70] and [1]The Schmid’s law can be expressed by the following equation:

\(\sigma\) is an arbitrary stress state and \(\tau^i\) the resolved shear stress on slip system \(i\). \({S_0}^i\) is the Schmid matrix defined by the dyadic product of the slip plane normals \(\vec n\) and the slip directions \(\vec d\) of the slip system \(i\). The Schmid factor, \(m\), is defined as the ratio of the resolved shear stress \(\tau^{i}\) to 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. [1] proposed the following criterion:

(16)¶\[m_\text{GB} = m_\text{in} + m_\text{out}\]The subscripts \(\text{GB}\), \(\text{in}\), and \(\text{out}\) refer to the grain boundary, and the incoming and outgoing slip systems, respectively. This GB Schmid factor (\(m_\text{GB}\)) factor should be maximized.

The Matlab function used to calculate the Schmid factor is: resolved_shear_stress.m

**Generalized Schmid Factor**(\(GSF\)) [67] and [11]The generalized Schmid factor, which describes the shear stress on a given slip system, can be computed from any stress tensor \(\sigma\) based on the Frobenius norm of the tensor.

(17)¶\[GSF = \vec d \cdot g \sigma g \cdot \vec n\]\(\vec n\) and \(\vec d\) are respectively the slip plane normals and the slip directions of the slip system. The \(g\) is 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**(\(\tau\)) [47], [48], [49], [17], [45], [10], [22], [23] and [24]The resolved shear stress \(\tau\) acting on the outgoing slip system from the piled-up dislocations should be maximized. 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) [74].

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 GB [50] and [27].

It is possible to assess the shear stress from the geometrical factor \(N\) (Livingston and Chamlers):

(18)¶\[\tau_{\text{in}} = \tau_{\text{out}} * N\]Where \(\tau_{\text{out}}\) is the shear stress at the head of the accumulated dislocations in their slip plane and \(\tau_{\text{in}}\) is the shear acting on the incoming slip system [53], [34] and [35].

The Matlab function used to calculate the resolved shear stress is : resolved_shear_stress.m

### Combination of Criteria¶

**Geometrical function weighted by the accumulated shear stress or the Schmid factor**[11]:Bieler et al. proposed to weight slip transfer parameters by the sum of accumulated shear \(\gamma\) on 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:

(19)¶\[m_{\gamma}^{'} = \frac{\sum_{\alpha} \sum_{\beta} m_{\alpha\beta}^{'} \left(\gamma^{\alpha} \gamma^{\beta} \right)}{\sum_{\alpha} \sum_{\beta} \left(\gamma^{\alpha} \gamma^{\beta} \right)}\](20)¶\[LRB_{\gamma} = \frac{\sum_{\alpha} \sum_{\beta} LRB_{\alpha\beta}^{'} \left(\gamma^{\alpha} \gamma^{\beta} \right)}{\sum_{\alpha} \sum_{\beta} \left(\gamma^{\alpha} \gamma^{\beta} \right)}\](21)¶\[s_{\gamma} = \frac{\sum_{\alpha} \sum_{\beta} s_{\alpha\beta}^{'} \left(\gamma^{\alpha} \gamma^{\beta} \right)}{\sum_{\alpha} \sum_{\beta} \left(\gamma^{\alpha} \gamma^{\beta} \right)}\](22)¶\[s = \cos(\psi) \cdot \cos(\kappa) \cdot \cos(\theta)\]The Matlab function used to calculate the \(s\) function is: s_factor.m

Similarly, the \(m^{'}\) parameter can be weighted using the Schmid factor \(m\) on each slip system as a metric for the magnitude of slip transfer:

(23)¶\[m_{GSF}^{'} = \frac{\sum_{\alpha} \sum_{\beta} m_{\alpha\beta}^{'} \left(m^{\alpha} m^{\beta} \right)}{\sum_{\alpha} \sum_{\beta} \left(m^{\alpha} m^{\beta} \right)}\]In 2016, Tsuru et al. proposed a new criterion, based on the \(N\) factor, for the transferability of dislocations through a GB that considers both the intergranular crystallographic orientation of slip systems and the applied stress condition [82] .

### Relationships between slip transmission criteria¶

Some authors proposed to study relationships between slip transmission criteria [30] and [85]. Thus, it is possible to find in the litterature the \(m'\) parameter 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.

### Slip transmission parameters implemented in the STABiX toolbox¶

Slip transmission parameter | Function | Matlab function | Reference |
---|---|---|---|

Misorientation angle (FCC and BCC materials) (\(\omega\)) | \(\omega = cos^{-1}((tr(\Delta g)-1)/2)\) | misorientation.m | [80] |

C-axis misorientation angle (HCP material) (\(\omega\)) | c-axis misorientation.m | [80] | |

\(N\) factor from Livingston and Chamlers | \(N = \cos(\psi)\cdot\cos(\kappa) + \cos(\gamma)\cdot\cos(\delta)\) | N_factor.m | [53] |

\(LRB\) factor from Shen et al. | \(LRB = \cos(\theta)\cdot\cos(\kappa)\) | LRB_parameter.m | [72] / [73] |

\(m'\) parameter from Luster and Morris | \(m' = \cos(\psi)\cdot\cos(\kappa)\) | mprime.m | [54] |

residual Burgers vector (\(\vec b_\text{r}\)) | \(\vec b_\text{r} = g_\text{in}\cdot\vec b_\text{in} - g_\text{out}\cdot\vec b_\text{out}\) | residual_Burgers_vector.m | [56] |

\(\lambda\) function from Werner and Prantl | \(\lambda = \cos(\frac{90° \psi}{\psi_c})\cos(\frac{90° \kappa}{\kappa_c})\) | lambda.m | [87] |

Resolved Shear Stress (\(\tau^{i}\)) / Schmid Factor | \(\tau^{i} = \sigma : {S_0}^{i}\) with \({S_0}^{i} = d \otimes n\) | resolved_shear_stress.m | [67] |

Grain boundary Schmid factor | \(m_\text{GB} = m_\text{in} + m_\text{out}\) | resolved_shear_stress.m | [1] |

Generalized Schmid Factor (\(GSF\)) | \(GSF = d \cdot g \sigma g \cdot n\) | generalized_schmid_factor.m | [67] |

### 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.

### References¶

[1] | (1, 2, 3) 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. |

[2] | W.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. |

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[42] | (1, 2) T. Kehagias et al., “Slip transfer across low-angle grain boundaries of deformed titanium.”, Interface Science (1995), 3(3), pp. 195-201. |

[43] | (1, 2) T. Kehagias et al., “Pyramidal Slip in Electron Beam Heated Deformed Titanium.”, Scripta Metallurgica et Materialia (1996), 33(12), pp. 1883-1888. |

[44] | S. Kobayashi et al., “Grain boundary hardening and triple junction hardening in polycrystalline molybdenum.”, Acta Materialia (2005), 53, pp. 1051–1057. |

[45] | (1, 2) B.W. Lagow, “Observation of dislocation dynamics in the electron microscope.”, Materials Science and Engineering: A, 2001, 309–310, pp. 445-450. |

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[47] | (1, 2) T.C. Lee et al., “Prediction of slip transfer mechanisms across grain boundaries.”, Scripta Metallurgica, (1989), 23(5), pp. 799–803. |

[48] | (1, 2, 3) 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. |

[49] | (1, 2, 3, 4) 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. |

[51] | (1, 2) 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. |

[53] | (1, 2, 3) J.D. Livingston and B. Chalmers, “Multiple slip in bicrystal deformation.”, Acta Metallurgica (1957), 5(6), pp. 322-327. |

[56] | (1, 2) M.J. Marcinkowski and W.F. Tseng, “Dislocation behavior at tilt boundaries of infinite extent.”, Metal. Trans. (1970), 1(12), pp. 3397-3401. |

[57] | 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. |

[58] | L. Nervo et al., “A study of deformation twinning in a titanium alloy by X-ray diffraction contrast tomography”, Acta Materialia (2016), 105, pp. 417-428. |

[59] | K.A. Nibur and D.F. Bahr, “Indentation Techniques for the Study of Deformation Across Grain Boundaries.”, Mat. Res. Soc. Symp. Proc. (2003), 778, pp. 129-134. |

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[62] | S. Pathak 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. |

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## Experimental data¶

To use the STABiX toolbox, some experimental data are required :

- average grain orientations (Euler angles (\(\phi_{1}\), \(\Phi\), \(\phi_{2}\)) in degrees) or intragranular misorientation (misorientation axis \([uvw]\) / angle \(\omega\));
- 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.

### 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 (\(\phi_{1}\), \(\Phi\), \(\phi_{2}\)) in degrees ;
- Left hand average orientation (\(\phi_{1}\), \(\Phi\), \(\phi_{2}\)) 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 segments.

#### 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 (\(\phi_{1}\), \(\Phi\), \(\phi_{2}\)) 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.

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:

#### 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…
- \(y\) coordinates 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 tesselation 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.

### 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

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.

### Convention for bicrystal EBSD/indentation experiments¶

## 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

### Loading EBSD data¶

For more details about the format of the EBSD data, see also the page Experimental data.

### Smoothing GBs segments¶

The smoothing algorithm allows to decrease the total number of grains boundaries in order to speed up calculations and plots.

The Matlab function used to smooth GBs is : interface_map_GB_segments_opti.m

### Misorientation angle¶

### Residual Burgers vector¶

### Schmid factor and slip trace analysis¶

## Bicrystal GUI¶

This GUI allows to analyze quantitatively slip transmission across grain boundaries for a single bicrystal.

The Matlab function used to run the Bicrystal GUI is : A_gui_plotGB_Bicrystal.m

This includes:

### 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’).

### Plotting and analyzing a bicrystal¶

### 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.

## 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) (procedurefile format) orAbaqus(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.

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.

### 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’).

### 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

#### Convention for the single crystal mesh¶

### 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

#### Convention for the bicrystal mesh¶

### 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.

### Indenter’s geometry¶

- Currently the following geometries can be used for CPFE simulations :
- cono-spherical indenter ;
- 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.

#### 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

### 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.

### Mesh definition¶

*MSC.Mentat*- The mesh is defined by default by hexahedral eightnode elements (hex8).

*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 “linear_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.

### 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.

### 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.

### Installing DAMASK¶

For instructions on how to set up the DAMASK constitutive simulation code please visit http://DAMASK.mpie.de.

### 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.

### 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)
- 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)

### 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)

### 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.

## Analysis of literature data¶

Please, find here the Matlab functions to analyze results and to plot data from the following papers.

The small differences between values given in the litterature and values found with STABiX toolbox may come from the rounding of Euler angles and the rounding of slip transmission criteria.

### Residual Burgers vector¶

#### Kacher and Robertson (2012)¶

Kacher and Robertson analyzed slip transfer in 304 stainless steel (FCC structure), using in situ TEM deformation [3]. 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. Calculations are reproduced using the Matlab toolbox and obtained values are compared to Kacher’s results (see Figure 34).

#### 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) characterizations [6]. The magnitude of residual dislocation Burgers 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 35).

#### Kacher and Robertson (2014)¶

Kacher and Robertson analyzed slip transfer in alpha cp-Ti (HCP structure), using in situ TEM deformation [4]. In this work, a bicrystal with a misorientation angle of 32° and misorientation axis of [1, 5, -6, 16] 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. Calculations are reproduced using the Matlab toolbox and obtained values are compared to Kacher’s results (see Figure 36).

#### Cui et al. (2014)¶

Cui et al. analyzed slip transfer in proton-irradiated 13Cr15Ni stainless steel (fcc structure), using in situ TEM deformation [1]. 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 37).

### 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 \(m'\) factor is used to quantify the transmissivity across the GBs and calculations are reproduced using the Matlab toolbox (see Figure 38).

[1] | B. Cui et al., “Influence of irradiation damage on slip transfer across grain boundaries.”, Acta Materialia (2014), 65, pp. 150-160. |

[2] | Y. Guo et al., “Slip band–grain boundary interactions in commercial-purity titanium.”, Acta Materialia (2014), 76, pp. 1-12. |

[3] | 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. |

[4] | 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. |

[5] | T. Kehagias et al., “Slip transfer across low-angle grain boundaries of deformed titanium.”, Interface Science (1995), 3(3), pp. 195-201. |

[6] | L. Patriarca et al., “Slip transmission in bcc FeCr polycrystal.”, Materials Science&Engineering (2013), A588, pp. 308–317. |

## 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.

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.

### How to use the toolbox ?¶

- Run the function A_gui_gbinc.m.
- Select your first image before serial polishing.
- Do the calibration to get the factor scale.
- Do the edge detection.
- Repeat the same operation for the second image obtained after serial polishing.
- 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).

- 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.
- Save the overlay in the same folder than the 1st picture loaded (as a screenshot.png) (optional).
- Do the measurement of the distance between edges (Vickers faces) or ridges of a unique Vickers indent (see Figure 41).
- Do the measurement of the distance between edges of a unique grain boundary.
- The value of the grain boundary inclination is finally given in degrees.

Calculation of the thickness of removed material after polishing

(1)¶\[h = \frac{d}{\tan(90 - \alpha)}\]

With \(d\) the distance between edges (Vickers faces) or ridges of a unique Vickers indent (obtained before and after polishing), and \(\alpha\) the angle between the Vickers indent and the surface of the sample (see Figure 41).

Calculation of grain boundary inclination

(2)¶\[GB_{inc} = \tan \left(\frac{d_{GB}}{h}\right)\]

With \(d_{GB}\) the distance between grain boundary traces (obtained before and after polishing), and \(h\) 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.

### See also¶

### Links¶

### Authors¶

Written by D. Mercier [1] and C. Zambaldi [1].

[1] Max-Planck-Institut für Eisenforschung, 40237 Düsseldorf, Germany

### Acknowledgements¶

Parts of this work were supported under the NSF/DFG Materials World Network program (DFG ZA 523/3-1 and NSF-DMR-1108211).

### Keywords¶

Matlab ; Graphical User Interface (GUI) ; Grain Boundaries ; Polycrystalline Metals ; Grain Boundary Inclination ; Serial Polishing ; Scanning Electron Microscope (SEM).

# References¶

[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. |

[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”. |

# 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

# Contributors¶

- Raúl Sánchez Martín (IMDEA, Madrid) contributed Python code to generate
*Abaqus*indentation models.

# 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.

# 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) ; python^{TM} toolbox ; DAMASK.