# Investigating twin boundary-dislocation interaction in MgY using quasi in-situ nanoindentation and electron channeling contrast imaging

Fatim-Zahra Mouhib, Heixin Wang, Julien Guénolé, Stéphane Berbenni, Talal Al-Samman, Thiebaud Richeton, Zhuocheng Xie, Antoine Guitton

► To cite this version:

Fatim-Zahra Mouhib, Heixin Wang, Julien Guénolé, Stéphane Berbenni, Talal Al-Samman, et al. Investigating twin boundary-dislocation interaction in MgY using quasi in-situ nanoindentation and electron channeling contrast imaging. 2025. hal-05304214v3

HAL Id: hal-05304214

https://hal.univ-lorraine.fr/hal-05304214v3

Preprint submitted on 4 Nov 2025

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.
Investigating twin boundary-dislocation interactions in Mg-Y using quasi in-situ
nanoindentation and electron channeling contrast imaging

Fatim Zahra Mouhib<sup>a</sup>, Hexin Wang<sup>b</sup>, Julien Guénolé<sup>a</sup>, Stephane Berbenni<sup>a</sup>, Talal Al Samman<sup>b</sup>, Thiebaud Richeton<sup>a</sup>,
Zhuocheng Xie<sup>b</sup>, Antoine Guitton<sup>a</sup>,

<sup>a</sup>Université de Lorraine, CNRS, Arts et Métiers Institute of Technology, LEM3, F-57000, Metz, France

<sup>b</sup>RWTH Aachen University, Institute for Physical Metallurgy and Materials Physics (IMM), 52074, Aachen, Germany

Abstract

Defect–grain boundary interactions have been proven essential for the development of tailored microstructures for high-
performance materials. We aim to unravel the mechanical response of grain boundaries by investigating dislocation
activity in the vicinity of a tensile twin boundary (TB) in Mg–Y. A combined approach of quasi in-situ nanoindentation,
electron channeling contrast imaging (ECCI), and high-resolution electron backscattered diffraction (HR-EBSD) was
employed to induce localized deformation and analyze the resulting TB response. Indentation in the parent grain
near the TB produces distinct pop-ins, indicating deformation events close to the boundary. Observed slip includes
non-basal activity in the twin and prismatic slip in the parent grain. Dislocation pile-ups cause TB bulging rather than
transmission, consistent with low slip transmission factors. However, limited slip transmission is observed. Peach–Koehler
force calculations based on HR-EBSD measurements confirm twin boundary motion induced by dislocation-pile-ups.

**Keywords:** Twin boundary-dislocation interactions, Mg-Y alloy, Slip transmission, GB bulging, ECCI, nanoindentation, HR-EBSD

To meet the demands of a rapidly advancing industry,
modern materials require tailored properties, which only
a fundamental understanding of the interplay between mi-
crostructure, composition, and processing can provide.

In particular, grain boundaries (GBs) are crucial sites
for both material fracture and strength [1–4]. However,
reliable predictions of their response during deformation
remain challenging, are not yet fully understood [5, 6].
These range from pile-ups to multiple variations of slip
transfer and can depend on boundary properties and chem-
ical composition [7–10]. Typically, a twin boundary (TB)
that resists dislocation transmission will lead to dislocation
pile-ups, increasing the strength of the materials according
to Hall [11] and Petch [12], while it can ultimately render
the material brittle by impeding plastic deformation [13–
15]. In magnesium alloys, it was seen that piled-up basal
dislocations at {1012}-type twin boundary lead to strong
strain concentrations, that are at the origin of fracture
[16].

Magnesium alloys are a prominent choice for automo-
tive and aerospace applications, as they can significantly
decrease weight, thereby reducing energy consumption and
environmental pollution [17]. However, their limited forma-
bility at room temperature comes with high production
costs. The addition of Y (yttrium) was found to enhance
the activation of $\langle c + a \rangle$ slip at room temperature, in-

creasing the number of independent slip systems available
and thus improving low-temperature formability [18–21].
This makes Mg-Y a suitable material system to study GB-
controlled plasticity, enabling fine-tuning of the chemical
composition and GB-structure for low-temperature forma-
bility.

Investigating GB-controlled plasticity requires targeted
nano-scale testing and observation methods, such as nanoin-
dentation and electron channeling imaging (ECCI). Nanoin-
dentation is frequently used for nano-hardness measure-
ments or to induce localized deformation for investigations
of dislocation activation and behavior [7, 10, 15, 22–24].
From the resulting mechanical response curves, indica-
tions of dislocation related events can be obtained. During
depth-controlled indentation, a force drop at a constant
depth is called a pop-in and is caused by the penetration
of an oxide layer, twinning, phase transformation, dislo-
cation nucleation, or the glide of pre-existing dislocations
[10, 25–27].

Direct and nondestructive observation of dislocations
is possible by electron channeling imaging (ECCI) [28–
33], while high resolution electron backscattered diffrac-
tion (HR-EBSD) provides an accurate estimate of the local
densities of geometrically necessary dislocations (GNDs)
and informs about the variation of elastic strains [34–
36]. The combination of quasi-in situ nanoindentation,
ECCI, and HR-EBSD is a powerful tool for the investiga-
tion of GB-dislocation interactions, as it allows for both
targeted deformation and direct observation of dislocations

<footnote>Email address: antoine.guitton@univ-lorraine.fr (Antoine Guitton)</footnote>
[10, 15, 23]. ECCI can observe dislocations because, under specific orientations, electrons are channeled through the material rather than backscattered. In these conditions, only deviations from the perfect lattice produce a backscattered signal that increases with atomic number Z [28, 32, 33, 37]. Backscattered electron (BSE) micrograph contrast depends on topography, orientation, and composition, whereas SE images show only topography [37].

However, because Mg is prone to oxidization and has a low atomic mass, it is challenging to obtain a significant backscattered electron signal. Therefore, observations of dislocations in Mg are traditionally performed by TEM [38, 39]. To the best of our knowledge, this study provides for the first time, ECCI images of dislocations near a GB in a Mg alloy.

The GND density was obtained from HR-EBSD data using the Nye tensor, and the slip transmission across the TB was evaluated using geometrical criteria. Details and equations are provided in Appendices A and C, respectively.

To investigate dislocation-GB interactions, single-crystals with the orientation of $\langle 0001 \rangle$ were grown by the vertical Bridgman technique and subsequently compressed along $\langle 10\bar{1}0 \rangle$ by channel die up to 2% of deformation to induce $\langle \bar{1}011 \rangle \{10\bar{1}2\}$ tensile twinning. The resulting sample exhibited a parent and twin grain orientation of $\langle 0001 \rangle$ and $\langle 10\bar{1}0 \rangle$, respectively, with a TB inclined by 47° toward the twin. A coherent TB was chosen for this investigation due to its flat, straight structure and well-defined characteristics.

Metallographic sample preparation was performed by mechanical grinding and polishing up to 1 µm with a diamond-ethanol based suspension. Subsequently, a Struers AC-2 solution was used for electro-polishing for 90 s at 20 °C and - 20 V in a Lectro-Pol 5.

To induce deformation, nanoindentation was performed near and at least 30 µm away from the TB using a Berkovich tip and a FT-NMT04 depth controlled in-situ indenter (Femtotools AG / Oxford Instruments, Buchs, Switzerland) with an indentation depth of 0.4 µm.

ECCI and HR-EBSD were employed to investigate dislocation activity using a Zeiss Auriga microscope (Carl Zeiss AG, Oberkochen, Germany) operating at 20 kV. For ECCI, a working distance of 7 mm and a diffraction vector $g = \langle 0002 \rangle$ were used. For the BSE signal, a Crytur REBEKA detector (Crytur, Turnov, Czech Republic) suitable for materials with a low atomic mass was employed. HR-EBSD was performed with a sample tilt of 70° with a step size of 40 nm. Data were analyzed using the ATEX software package [34, 35].

To obtain the mechanical response from a reference region unaffected by the TB, nanoindentation was performed 30 µm away from the TB in the twin and 55 µm in the parent grain, respectively. Figures 1 (b) and (c) show representative force-displacement curves in the twin and parent grain regions, respectively.

In the twin region, a higher hardness of 1.19 +/- 0.04 GPa and multiple pop-in events are observed, while the parent grain exhibits a force-displacement response without pop-in and a lower hardness of 0.92 +/- 0.06 GPa. Multiple pop-ins in the twin may result from the successive nucleations of plastic events. The average hardness and the standard deviation were determined from nine force-displacement curves in each region.

The observed difference in hardness is likely due to the basal orientation of the twin, which restricts slip along the $\langle a \rangle$ direction under load in $\langle c \rangle$-direction. Hence, slip systems with a low critical resolved shear stress (CRSS) are geometrically unfavorable and high stress is required to activate $\langle c + a \rangle$ leading to a higher hardness in the twin. Table 1 shows the values of indentation Schmid factors (ISF), which were earlier defined in [27, 40], for the different slip and twinning systems. It is found that they are near zero for all $\langle a \rangle$ slip systems in the twinned grain, where tensile twinning is equally unfavorable. In contrast, the ISF for pyramidal $\langle c + a \rangle$ slip is near 0.3, hence making it the only viable deformation mechanism. In the parent grain, slip systems in $\langle a \rangle$ direction, with the exception of basal slip, exhibit high ISFs making prismatic slip the most favorable deformation mechanism.

To characterize dislocation-TB interactions, nanoindentation testing was performed near the GB in the twin (ROI A) and in the parent grain (ROI B) at a distance of approximately 1-1.5 µm from the TB. The corresponding force-displacement curves and secondary electron (SE) images are shown in Figure 2 (c) and in Figures 2 (a) and (b), respectively. The indent center was determined as the center of mass of a triangle, and the distance was measured from the center to the TB. Indentation in the twin did not show any pop-in event, in contrast with the observations made while indenting further away from the TB. Therefore, this may be due to the movement of preexisting dislocations near the TB, rather than the nucleation of new dislocations. In the parent grain, a distinct pop-in was observed. Compared to the absence of pop-in in the parent grain far from the TB, this indicates a deformation event near the TB, which could be due to dislocation-TB interactions.

Figure 2 (d) shows a schematic illustration of the deformation zone following Equations (1) to (3) [41], where $a_c$ and $h_c$ are the contact area and contact depth, c the radius of the plastic zone and d the depth of the highest stress concentration [42].

$$a_c = \sqrt{\frac{24.5}{\pi}} h_c \approx 2.79 h_c \quad (\text{Berkovich tip}) \quad (1)$$

$$c = 2.12 a_c \quad (2)$$

$$d = 0.48 a_c \quad (3)$$

At the final indentation depth of 0.4 µm the deformation zone radius was 2.37 µm and the depth of the maximum stress concentration 0.54 µm. As the TB is inclined by 47°
<figure><img src="image_1.png" /><figcaption>Figure 1: (a): Schematic illustration of channel die compression of a Mg-1%Y single crystal with a compression direction (CD) along 〈 10 ¯ 10 〉 and LD ‖ 〈 0001 〉 up to 2% and the resulting twinned microstructure. (b) and (c): Exemplary, force displacement curve for indentation in twin and parent grain regions, respectively.</figcaption></figure>

<table><caption>Table 1: Values of indentation Schmid Factor for the different slip and twinning systems</caption>
  <thead>
    <tr>
      <th rowspan="2">Grain</th>
      <th colspan="4">Slip Systems</th>
      <th colspan="2">Twinning Systems</th>
    </tr>
    <tr>
      <th>Basal &lt;a&gt;<br>{0001} &lt;1210&gt;</th>
      <th>Prismatic &lt;a&gt;<br>{1010} &lt;1210&gt;</th>
      <th>Pyramidal 1 &lt;a&gt;<br>{0111} &lt;1210&gt;</th>
      <th>Pyramidal 2 &lt;c+a&gt;<br>{2112} &lt;2113&gt;</th>
      <th>Tension Twin<br>{1102} &lt;1101&gt;</th>
      <th>Compression Twin<br>{1101} &lt;102&gt;</th>
    </tr>
  </thead>
  <tbody>
    <tr>
      <td>Twin</td>
      <td>0.047</td>
      <td>0.002</td>
      <td>0.024</td>
      <td>0.295</td>
      <td>-0.300</td>
      <td>0.282</td>
    </tr>
    <tr>
      <td>Parent</td>
      <td>0.013</td>
      <td>0.290</td>
      <td>0.260</td>
      <td>0.228</td>
      <td>0.309</td>
      <td>-0.046</td>
    </tr>
  </tbody>
</table>

towards the twin, the deformation zone in ROI A reaches into the parent grain even when indentation is performed in the twin. This suggests that dislocation nucleation may occur in both regions, i.e. twin and parent grain. Since deformation in the parent grain is likely accommodated by prismatic slip, which has a lower CRSS than the 〈 c + a 〉 slip required in the twin, nucleation of dislocations in the parent grain may be favored.

Figures 3 (a) and (b) present HR-EBSD results for ROI A and B as ND-IPF maps. In both cases, the TB exhibits bulging toward the twin grain near the indent. This is notable in ROI A as the boundary is curved towards the indent contrary to outward, as would be expected. The dislocation density (see Equation (A.1)) of ROI A and B, shown in Figure 3 (c) and (d), reveals a high dislocation

density in the bulged region on the parent grain side for ROI A, and zones of high localized dislocation density near the inflection points of the curved TB for ROI B. These dislocation pile-ups likely induce the observed boundary curvature.

To understand the dislocation activity underlying the observed boundary bulging, controlled ECCI was performed under the diffraction vector **g** = (0 0 0 2) with only the twin being in channeling conditions. Dislocations of type 〈 *a* 〉 are invisible under this diffraction vector (g ⋅ **b** = 0), while 〈 *c* + *a* 〉 remain visible. The results presented in Figure 4 (a) and (b), reveal a change of contrast, thus indicating dislocation activity. Figure 4 (b), (c), (e) and (f) show the projection of the net Burgers vector with a ND-IPF color code determined from the Nye tensor of ROI A and B and
<figure><img src="image_2.png" /><figcaption>Figure 2: (a) and (b): Secondary electron (SE) images of indents placed close to the TB in the twin (ROI A) and the parent grain (ROI B), respectively. The shape and the center of the indent, as well as the resulting distance from the TB are schematically marked. (c): Corresponding force-displacement curves for indentation in the twin and the parent grain near the TB. (d) Schematic deformation zones after indentation of ROI A and B.</figcaption></figure>

the norm of the Nye tensor, respectively.

It is noteworthy that HR-EBSD reveals a significant amount of net Burgers vectors with a c-component in the twin, making pyramidal $\langle c + a \rangle$ slip probable. In the parent grain, HR-EBSD suggests Burgers vectors on the basal plane, which is consistent with the ISF calculations (see Table 1) indicating the occurrence of prismatic slip.

Figures 5 (a) and (d) show ECCI of ROI A and ROI B, and (b) and (e) present possible mechanisms that may account for the observed GB motion. The symbol for edge dislocations is used as a representative for all types of dislocations. Due to the channeling conditions, only dislocation activity in the twin is visible. ROI A shows inward bulging of the boundary, which can be explained by the nucleation of dislocations in the parent grain emitted towards the boundary, moving it towards the twin (see Figure 5 (b)). In the case of ROI B, ECCI reveals same dislocation activity

in the twin, highlighted by a white circle, despite indentation being performed in the parent grain. This observation indicates the occurrence of slip transmission across the boundary. However, the Burgers vector could not be determined from the ECCI analysis. The probable mechanism behind the TB bulging in ROI B over time and the breakthrough of dislocations is illustrated in Figure 5 (e). A pile-up initiating from the indent corners creates a step and thereby moves the TB forward. As initially the indent is small, the corners are centered and close to each other. The resulting pile-ups will be directed at the center of the bulged region. With progressing indentation depth, the distance between the indentation corners increases, thereby extending the bulging region until it reaches its final extent, as illustrated in stages t1, t2, and t3 (see Figure 5 (e)). Ultimately, the increasing stress inflicted on the boundary by the pile-ups enables the transmission of
<figure><img src="image_3.png" /><figcaption>Figure 3: (a) and (b): HR-EBSD data presented as ND-IPF map of ROI A and B, respectively. (c) and (d): Resulting dislocation densities according to Equation (A.1). Zones of high dislocation density piling-up at TB are indicated by white circles.</figcaption></figure>
<figure><img src="image_4.png" /><figcaption>Figure 4: (a): Electron channeling imaging of ROI A with channeling pattern in the top right corner and diffraction vector $g = \langle 0002 \rangle$ indicated by a yellow arrow. The TB position is shown by a white dotted line and its displacement due to indentation by a white arrow. (b) and (c): Projection of the net Burgers vector with a ND-IPF color code and the norm of the Nye tensor determined from HR-EBSD data. (d)-(f): Respective analysis for ROI B.</figcaption></figure>

dislocations.

Based on this hypothesis, dislocation slip is expected to be directed toward the TB. For a continuous distribution of dislocations described by the Nye tensor $\alpha$ and a given stress field $\sigma$ (see Appendix B), the Peach-Koehler driving force, denoted $F$ [43–45] is given by:

$$F_i = \epsilon_{ijl} \alpha_{jk} \sigma_{lk} \quad (4)$$

where $\epsilon_{ijl}$ is the third-order Levi-Civita tensor.

Figures 5 (c) and (f) show the results for the Peach-Koehler forces represented by force vectors scaled according to the force magnitude. Data in the proximity of the indent should be regarded with caution as the underlying experimental data in this area are unreliable due to extreme deformation. This does play a minor role for ROI B, as the indent is sufficiently far away from the TB. The Peach-Koehler force represents the force acting on dislocations within a given stress field. Its direction indicates the path along which a dislocation will move. The direction of the Peach-Koehler forces, thus the direction of motion of the dislocations, points notably in the bulging direction. The indent in ROI A is close to the TB making the interpretation more challenging. However, most of the force arrows point in bulging direction as well. This strengthens the theory of TB motion induced by dislocation pile-ups

and excludes a rigid body motion as the reason for the observed TB bulging.

Although the TB shows a high resistance against slip transmission, a few transmitted dislocations can be observed from the ECC micrograph of ROI A (see Figure 5 (d) white circle). The bulging direction and Peach-Koehler force analysis indicate that dislocations are initially mainly active in the parent grain for both, ROI A and B. HR-EBSD and the indentation Schmid factor allow the assumption that deformation takes place via prismatic slip in the parent grain and pyramidal $\langle c + a \rangle$ slip in the twin (see Figure 4 and Table 1). Table 2 shows the transmission factor after Equation C.1 for slip transmission of prismatic to pyramidal slip across a coherent tensile TB. The observed transmission factors are < 0.52 and although the transmission factor after Shen et al. [46] is experimentally less studied and therefore there is no empirical threshold, these values can be considered very low with respect to the literature [46–49], indicating that slip transmission is geometrically unfavorable for the given slip systems and boundary. This corresponds well with the observed dislocation pile-ups causing TB bulging rather than slip transmission. However, the limited transmission is likely a result of stress accumulation induced by successive pile-ups.

In conclusion, combined use of nanoindentation, HR-EBSD, and ECCI revealed dislocation pile-ups, non-basal
<figure><img src="image_5.png" /><figcaption>Figure 5: (a) and (d): ECCI of ROI A and B marked region highlights dislocation contrast observed in the twin grain in ROI B. (b) an (e): Schematic illustration of TB bulging over time. For ROI B a growing indent and pile-ups originating from the indent corners are shown in 3 different stages $t_1$, $t_2$ and $t_3$. The symbol for edge dislocations is used as a representative for all types of dislocations. (c) and (f): Peach-Koehler forces calculated from the Nye and stress tensor obtained from HR-EBSD for ROI A and B, respectively.</figcaption></figure>

<table><caption>Table 2: Transmission factor M (Equation C.1) for slip transmission of prismatic ($(\overline{1}100)\{\overline{1}1\overline{2}0\}$) to pyramidal ($(1\overline{2}12)\{\overline{1}2\overline{1}3\}$) and basal slip ($(0002)\{\overline{1}1\overline{2}0\}$) across a TB.</caption><thead><tr><th>Twin/ Parent</th><th>(1100)[1120]</th><th>(1010)[1210]</th><th>(0110)[2110]</th></tr></thead><tbody><tr><td>(2112)[2113]</td><td>0.20</td><td>0.20</td><td>0.40</td></tr><tr><td>(1212)[1213]</td><td>0.15</td><td>0.31</td><td>0.15</td></tr><tr><td>(1122)[1123]</td><td>0.31</td><td>0.15</td><td>0.15</td></tr><tr><td>(2112)[2113]</td><td>0.26</td><td>0.26</td><td>0.52</td></tr><tr><td>(1212)[1213]</td><td>0.26</td><td>0.522</td><td>0.26</td></tr><tr><td>(1122)[1123]</td><td>0.05</td><td>0.03</td><td>0.03</td></tr></tbody></table>

slip, TB bulging, and limited slip transmission. Dislocation pile-ups were found to induce TB bulging rather than transmission, consistent with the low slip transmission factors. Two mechanisms of dislocation-TB interaction were identified: (i) TB bulging away from the indent due to successive dislocation pile-ups emitted from the indent corners, and (ii) towards the indent due to dislocations formed in the extremity of the plastic zone extending into the parent grain. Peach-Koehler force calculations based on the Nye tensor obtained from HR-EBSD data confirm the occurrence of twin boundary motion induced by dislocation-pile-ups.

## Acknowledgment

The authors acknowledge financial support from the Deutsche Forschungsgemeinschaft (DFG) - Projektnummer 505716422 and the French National Research Agency (ANR) grants MAMIE-NOVA (ANR-22-CE08-0018), HEIR-MAX (ANR-23-CE08-0005) and SILA (ANR-22-CE92-0058). The authors are thankful to the experimental facilities of MicroMat from LEM3 (Université de Lorraine-CNRS-Arts et Métiers ParisTech).
## Author contributions

Conceptualization: A.G.; Resources: H.W., T.A.S., A.G; Methodology: H.W., T.A.S., F.Z.M., A.G; Data curation: H.W., F.Z.M.; Formal analysis: F.Z.M.; Investigation: F.Z.M.; Validation: F.Z.M., J.G., T.R., S.B., Z.X., T.A.S., A.G.; Writing – original draft: F.Z.M.; Writing – review & editing: J.G., T.R., S.B., Z.X., T.A.S., A.G.; Project administration: A.G.; Funding acquisition: Z.X., T.R.

## Appendix A. Calculation of the GND density

The GND density can be estimated (approximation) according to Equation (A.1) from the Burgers vector magnitude b and HR-EBSD by calculating the Nye tensor α from the 2D measurements of elastic strain tensor ε, where five components are measured directly (ε₁₁, ε₂₂, ε₁₂, ε₁₃, and ε₂₃) and the out-of-plane component ε₃₃ is obtained assuming a traction-free surface (σ₃₃ = 0), and the lattice curvature tensor κ as presented in Equations (A.2) and (A.3). The curvatures can be calculated from a finite difference in rotation ωᵢ of two adjacent pixels x [34–36].

$$ \rho_{GND} = \frac{1}{b} \sqrt{\alpha_{12}^2 + \alpha_{13}^2 + \alpha_{21}^2 + \alpha_{23}^2 + \alpha_{33}^2} \quad (A.1) $$

where the Nye tensor α is given by:

$$ \alpha = curl\epsilon + tr(\kappa)I - \kappa^T \quad (A.2) $$

where I is the identity tensor and κ is the curvature tensor estimated by

$$ \kappa_{ij} \approx \frac{\Delta\omega_i}{\Delta x_j} \quad (A.3) $$

## Appendix B. Calculation of the stress tensor

The stress tensor used to compute the Peach-Koehler force was calculated using strain measurements obtained by HR-EBSD (see Appendix A) and an anisotropic stiffness matrix of single crystalline pure Mg (see Table B.3). The surface is assumed to be traction free, i.e. σ₃₃ = 0.

Equation B.1 denotes the relation of stress and strain in a fully elastic regime (Hooke’s law [50]), where Cᵢⱼₖ is the fourth-order stiffness tensor.

$$ \sigma_{ij} = C_{ijkl} \epsilon_{kl}, \quad (B.1) $$

## Appendix C. Calculation of slip transmission factors

Slip transmission is traditionally evaluated by geometrical transmission criteria, commonly the Luster and Morris transmission factor, which considers the alignment of incoming and outgoing slip systems [6, 51]. Shen et. al

proposes a transmission criteria incorporating the GB plane orientation [6, 46]. Equations (C.1) and (C.2) describe the transmission factor, M̃αβ, from a slip system α in grain A to a slip system β in grain B across a TB with the plane normal n<sub>Γ</sub>. The slip system is characterized by slip direction d<sub>α</sub><sup>A</sup> and slip plane normal n<sub>α</sub><sup>A</sup>.

$$ M_{\alpha\beta} = (l_{\alpha}^{A} \cdot l_{\beta}^{B}) (d_{\alpha}^{A} \cdot d_{\beta}^{B}) \quad (C.1) $$

$$ l_{\alpha}^{A} = \frac{n_{\alpha}^{A} \times n_{\Gamma}}{|n_{\alpha}^{A} \times n_{\Gamma}|} \quad (C.2) $$

## References

[1] X. Liu, J. Wang, Low-energy, mobile grain boundaries in magnesium, Scientific Reports 6 (1) (2016) 21393. doi:10.1038/srep21393.
URL https://doi.org/10.1038/srep21393

[2] D. McLean, A. Maradudin, Grain boundaries in metals, Physics Today 11 (7) (1958) 35–36. arXiv:https://pubs.aip.org/physicstoday/article-pdf/11/7/35/8252496/35\_1\_online.pdf, doi:10.1063/1.3062658.
URL https://doi.org/10.1063/1.3062658

[3] A. P. Sutton, R. W. Balluffi, Interfaces in Crystalline Solids, Oxford University Press, Oxford (UK), 1995.

[4] L. Priester (Ed.), Grain Boundaries-From Theory to Engineering, Springer Dordrecht, 2013.

[5] H. Su, X. Zhou, S. Zheng, H. Ye, Z. Yang, Atomic-resolution studies on reactions between basal dislocations and 101⁻² coherent twin boundaries in a mg alloy, Journal of Materials Science and Technology 66 (2021) 28–35. doi:10.1016/j.jmst.2020.06.017.

[6] E. Bayerschen, A. T. McBride, B. D. Reddy, T. Böhlke, Review on slip transmission criteria in experiments and crystal plasticity models (3 2016). doi:10.1007/s10853-015-9553-4.

[7] F. Javaid, H. Pourayevali, K. Durst, Dislocation–grain boundary interactions: recent advances on the underlying mechanisms studied via nanoindentation testing, Journal of Materials Research 36 (2021) 2545–2557. doi:10.1557/s43578-020-00096-z.

[8] D. An, X. Liu, Y. Liu, X. Li, J. Chen, Experimental investigation of dislocation-grain boundary interaction in coarse-grained high-manganese steels using quasi in situ electron channelling contrast imaging, Materials Characterization 195 (1 2023). doi:10.1016/j.matchar.2022.112545.

[9] S. T. Oyinbo, S. Singhania, R. Matsumoto, Exploring the basal/prismatic slip transfer at grain boundaries in magnesium: A molecular dynamic simulation, Vacuum 212 (2023) 111995. doi:10.1016/j.VACUUM.2023.111995.

[10] F. Habiyaremye, A. Guitton, F. Schäfer, F. Scholz, M. Schneider, J. Frenzel, G. Laplanche, N. Maloufi, Plasticity induced by nanoindentation in a crconi medium-entropy alloy studied by accurate electron channeling contrast imaging revealing dislocation-low angle grain boundary interactions, Materials Science and Engineering: A 817 (6 2021). doi:10.1016/j.msea.2021.141364.

[11] E. O. Hall, The deformation and ageing of mild steel: Iii discussion of results, Proceedings of the Physical Society. Section B 64 (9) (1951) 747. doi:10.1088/0370-1301/64/9/303.
URL https://doi.org/10.1088/0370-1301/64/9/303

[12] N. J. Petch, The cleavage strength of polycrystals, Journal of the Iron and Steel Institute (1953) 174.

[13] R. B. Figueiredo, M. Kawasaki, T. G. Langdon, Seventy years of hall-petch, ninety years of superplasticity and a generalized approach to the effect of grain size on flow stress, Progress in Materials Science 137 (2023) 101131. doi:https://doi.org/10.1016/j.pmatsci.2023.101131.
<table><caption>Table B.3: Anisotropic hexagonal pure Mg stiffness matrix $C_{ij}$ in Voigt notation (GPa)</caption><thead><tr><th>C<sub>ij</sub></th><th>1</th><th>2</th><th>3</th><th>4</th><th>5</th><th>6</th></tr></thead><tbody><tr><th>1</th><td>56.49</td><td>23.16</td><td>18.10</td><td>0</td><td>0</td><td>0</td></tr><tr><th>2</th><td>23.16</td><td>56.49</td><td>18.10</td><td>0</td><td>0</td><td>0</td></tr><tr><th>3</th><td>18.10</td><td>18.10</td><td>58.73</td><td>0</td><td>0</td><td>0</td></tr><tr><th>4</th><td>0</td><td>0</td><td>0</td><td>16.81</td><td>0</td><td>0</td></tr><tr><th>5</th><td>0</td><td>0</td><td>0</td><td>0</td><td>16.81</td><td>0</td></tr><tr><th>6</th><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>16.665</td></tr></tbody></table>

URL https://www.sciencedirect.com/science/article/pii/S0079642523000634

[14] C. A. Schuh, T. G. Nieh, Hardness and abrasion resistance of nanocrystalline nickel alloys near the hall-petch breakdown regime, MRS Online Proceedings Library (2003).

[15] A. Guitton, H. Kriaa, E. Bouzy, J. Guyon, N. Maloufi, A dislocation-scale characterization of the evolution of deformation microstructures around nanoindentation imprints in a tial alloy, Materials 11 (2) (2018). doi:10.3390/ma11020305. URL https://www.mdpi.com/1996-1944/11/2/305

[16] H. Somekawa, A. Singh, T. Mukai, Fracture mechanism of a coarse-grained magnesium alloy during fracture toughness testing, Philosophical Magazine Letters 89 (1) (2009) 2-10. doi:10.1080/09500830802537718.

[17] H. E. Friedrich, B. L. Mordike, Technology of magnesium and magnesium alloys, Magnesium Technology: Metallurgy, Design Data, Applications (2006) 219-430.

[18] A. Maldar, L. Wang, B. Liu, W. Liu, Z. Jin, B. Zhou, X. Zeng, Activation of (c) dislocations in mg with solute y, Journal of Magnesium and Alloys 12 (3) (2024) 1045-1053. doi:https://doi.org/10.1016/j.jma.2021.11.004. URL https://www.sciencedirect.com/science/article/pii/S2213956721002711

[19] J. Wu, S. Si, K. Takagi, T. Li, Y. Mine, K. Takashima, Y. L. Chiu, Study of basal (a) and pyramidal (c + a) slips in mg-y alloys using micro-pillar compression, Philosophical Magazine 100 (11) (2020) 1454-1475. doi:10.1080/14786435.2020.1725250. URL https://doi.org/10.1080/14786435.2020.1725250

[20] S. Sandlöbes, Z. Pei, M. Friák, L.-F. Zhu, F. Wang, S. Zaefferer, D. Raabe, J. Neugebauer, Ductility improvement of mg alloys by solid solution: Ab initio modeling, synthesis and mechanical properties, Acta Materialia 70 (2014) 92-104. doi:https://doi.org/10.1016/j.actamat.2014.02.011.

[21] S. Sandlöbes, M. Friák, J. Neugebauer, D. Raabe, Basal and non-basal dislocation slip in mg-y, Materials Science and Engineering: A 576 (2013) 61-68. doi:https://doi.org/10.1016/j.msea.2013.03.006. URL https://www.sciencedirect.com/science/article/pii/S1359645414000949

[22] M. S. Talaei, N. Nouri, S. Ziaei-Rad, Grain boundary effects on nanoindentation of fe bicrystal using molecular dynamic, Mechanics of Materials 102 (2016) 97-107. doi:10.1016/J.MECHMAT.2016.08.016.

[23] F. Habiyaremye, A. Guitton, X. Chen, T. Richeton, S. Berbenni, F. Schäfer, G. Laplanche, N. Maloufi, Influence of pre-existing configurations of dislocations on the initial pop-in load during nanoindentation in a croni medium-entropy alloy, Philosophical Magazine 104 (3) (2024) 137-160.

[24] L. Chamma, J.-M. Pipard, A. Arlazarov, T. Richeton, S. Berbenni, A plasticity-induced internal length mean field model based on statistical analyses of ebsd and nanoindentation data, International Journal of Plasticity 189 (2025) 104327. doi:https://doi.org/10.1016/j.ijplas.2025.104327.

[25] D. F. Bahr, D. J. Morris, Nanoindentation: Localized probes of mechanical behavior of materials, in: W. N. Sharpe (Ed.), Springer Handbook of Experimental Solid Mechanics, Springer, Boston, MA, 2008, pp. 389-408. doi:10.1007/978-0-387-30877-7_16. URL https://doi.org/10.1007/978-0-387-30877-7_16

[26] S. Baker, J. Liu, Nanoindentation techniques, in: Reference Module in Materials Science and Materials Engineering, Elsevier, 2016. doi:https://doi.org/10.1016/B978-0-12-803581-8_02586-8. URL https://www.sciencedirect.com/science/article/pii/B9780128035818025868

[27] C. Caër, E. Patoor, S. Berbenni, J.-S. Lecomte, Stress induced pop-in and pop-out nanoindentation events in cualbe shape memory alloys, Materials Science and Engineering: A 587 (2013) 304-312. doi:https://doi.org/10.1016/j.msea.2013.08.052. URL https://www.sciencedirect.com/science/article/pii/S0921509313009398

[28] H. Kriaa, A. Guitton, N. Maloufi, Fundamental and experimental aspects of diffraction for characterizing dislocations by electron channeling contrast imaging in scanning electron microscope, Scientific Reports 7 (1) (2017) 9742. doi:10.1038/s41598-017-09756-3. URL https://doi.org/10.1038/s41598-017-09756-3

[29] K. Venkatraman, M. Ben Haj Slama, V. Taupin, N. Maloufi, A. Guitton, Tuning critical resolved shear stress ratios for bcc-titanium ti21s via an automated data analysis approach, Modelling and Simulation in Materials Science and Engineering 29 (5) (2021) 055014. doi:10.1088/1361-651X/abfeb0. URL https://dx.doi.org/10.1088/1361-651X/abfeb0

[30] M. Ben Haj Slama, V. Taupin, N. Maloufi, K. Venkatraman, A. D. Rollett, R. A. Lebensohn, S. Berbenni, B. Beausir, A. Guitton, Electron channeling contrast imaging characterization and crystal plasticity modelling of dislocation activity in ti21s bcc material, Materialia 15 (2021) 100996. doi:https://doi.org/10.1016/j.mtla.2020.100996. URL https://www.sciencedirect.com/science/article/pii/S2589152920304129

[31] M. Ben Haj Slama, N. Maloufi, J. Guyon, S. Bahi, L. Weiss, A. Guitton, In situ macroscopic tensile testing in sem and electron channeling contrast imaging: Pencil glide evidenced in a bulk β-ti21s polycrystal, Materials 12 (15) (2019). doi:10.3390/ma12152479. URL https://www.mdpi.com/1996-1944/12/15/2479

[32] H. Kriaa, A. Guitton, N. Maloufi, Modelling electron channeling contrast intensity of stacking fault and twin boundary using crystal thickness effect, Materials 14 (7) (2021). doi:10.3390/ma14071696. URL https://www.mdpi.com/1996-1944/14/7/1696

[33] H. KRIAÄ, A. GUITTON, N. MALOUFI, Modeling dislocation contrasts obtained by accurate-electron channeling contrast imaging for characterizing deformation mechanisms in bulk materials, Materials 12 (10) (2019). doi:10.3390/ma12101587. URL https://www.mdpi.com/1996-1944/12/10/1587

[34] A. Mandal, B. Beausir, J. Guyon, V. Taupin, A. Guitton, Estimation of dislocation densities with nondestruc-tive scanning electron microscope techniques: Application to gallium nitride, Microscopy and Microanalysis 31 (1) (2025) ozac124. arXiv:https://academic.oup.com/ (Ed.), Springer Handbook of Experimental Solid Mechanics, Springer, Boston, MA, 2008, pp. 389-408. doi:10.1007/978-0-387-30877-7_16. URL https://doi.org/10.1007/978-0-387-30877-7_16
mam/article-pdf/31/1/ozae124/61429412/ozae124.pdf, doi: 10.1093/mam/ozae124.

URL https://doi.org/10.1093/mam/ozae124

[35] C. Ernould, B. Beausir, J.-J. Fundenberger, V. Taupin, E. Bouzy, Chapter one - measuring elastic strains and orientation gradients by scanning electron microscopy: Conventional and emerging methods, Vol. 223 of Advances in Imaging and Electron Physics, Elsevier, 2022, pp. 1-47. doi:https://doi.org/10.1016/bs.aiep.2022.07.001.

URL https://www.sciencedirect.com/science/article/pii/S1076567022000544

[36] J. Jiang, T. Britton, A. Wilkinson, Measurement of geometrically necessary dislocation density with high resolution electron backscatter diffraction: Effects of detector binning and step size, Ultramicroscopy 125 (2013) 1-9. doi:https://doi.org/10.1016/j.ultramic.2012.11.003.

URL https://www.sciencedirect.com/science/article/pii/S0304399112002823

[37] L. Reimer, Scanning electron microscopy: physics of image formation and microanalysis, Measurement Science and Technology 11 (12) (2000) 1826-1826.

[38] M. Xu, N. Li, X. Sha, Y. Liu, B. Gao, L. Xiao, X. Chen, H. Zhou, In-situ tom observations on interaction of basal (a) dislocations and β' phases in a mg-gd binary alloy, Materials Science and Engineering: A 841 (2022) 143017. doi:https://doi.org/10.1016/j.msea.2022.143017.

URL https://www.sciencedirect.com/science/article/pii/S0921509322004245

[39] Z. Yang, M. F. Chisholm, G. Duscher, X. Ma, S. J. Pennycook, Direct observation of dislocation dissociation and suzuki segregation in a mg-zn-y alloy by aberration-corrected scanning transmission electron microscopy, Acta Materialia 61 (1) (2013) 350-359. doi:https://doi.org/10.1016/j.actamat.2012.09.067.

URL https://www.sciencedirect.com/science/article/pii/S1359645412007008

[40] T. Li, Y. Gao, H. Bei, E. George, Indentation schmid factor and orientation dependence of nanoindentation pop-in behavior of nial single crystals, Journal of the Mechanics and Physics of Solids 59 (6) (2011) 1147-1162. doi:https://doi.org/10.1016/j.jmps.2011.04.003.

URL https://www.sciencedirect.com/science/article/pii/S0022509611000652

[41] M. Delincé, P. Jacques, T. Pardoen, Separation of size-dependent strengthening contributions in fine-grained dual phase steels by nanoindentation, Acta Materialia 54 (12) (2006) 3395-3404. doi:https://doi.org/10.1016/j.actamat.2006.03.031.

URL https://www.sciencedirect.com/science/article/pii/S1359645406002412

[42] L. Zhang, T. Ohmura, Plasticity initiation and evolution during nanoindentation of an iron – 3 % silicon crystal, Phys. Rev. Lett. 112 (2014) 145504. doi:10.1103/PhysRevLett.112.145504.

URL https://link.aps.org/doi/10.1103/PhysRevLett.112.145504

[43] M. Peach, J. S. Koehler, The forces exerted on dislocations and the stress fields produced by them, Phys. Rev. 80 (1950) 436-439. doi:10.1103/PhysRev.80.436.

URL https://link.aps.org/doi/10.1103/PhysRev.80.436

[44] F. R. N. Nabarro (Ed.), Dislocations in Solids, Vol. 1, Elsevier Science & Technology Books, 1979.

[45] A. El-Azab, Statistical mechanics treatment of the evolution of dislocation distributions in single crystals, Phys. Rev. B 61 (2000) 11956-11966. doi:10.1103/PhysRevB.61.11956.

URL https://link.aps.org/doi/10.1103/PhysRevB.61.11956

[46] Z. Shen, R. Wagoner, W. Clark, Dislocation pile-up and grain boundary interactions in 304 stainless steel, Scripta Metallurgica 20 (6) (1986) 921-926. doi:https://doi.org/10.1016/0036-9748(86)90467-9.

URL https://www.sciencedirect.com/science/article/pii/0036974886904679

[47] F. Schäfer, L. Weiter, M. Marx, C. Motz, Quantifying the grain boundary resistance against slip transfer by experimental combination of geometric and stress approach using stage-i-fatigue cracks, Philosophical Magazine 96 (32-34) (2016) 3524-3551. doi:10.1080/14786435.2016.1235289.

[48] S. Haouala, R. Alizadeh, T. Bieler, J. Segurado, J. LLoorca, Effect of slip transmission at grain boundaries in al bicrystals, International Journal of Plasticity 126 (2020) 102600. doi:https://doi.org/10.1016/j.ijplas.2019.09.006.

URL https://www.sciencedirect.com/science/article/pii/S0749641919303456

[49] R. Alizadeh, M. Peña-Ortega, T. Bieler, J. LLoorca, A criterion for slip transfer at grain boundaries in al, Scripta Materialia 178 (2020) 408-412. doi:https://doi.org/10.1016/j scriptamat.2019.12.010.

URL https://www.sciencedirect.com/science/article/pii/S1359646219307298

[50] R. Hooke, Lectures de Potentia Restitutiva, or of Spring, Printed for John Martyn, Printer to the Royal Society, London, 1678, available online: https://archive.org/details/bim_early-english-books-1641-1700_lectures-de-potentia-res_hooke-robert_1678.

[51] J. Luster, M. Morris, Compatibility of deformation in two-phase ti-al alloys: Dependence on microstructure and orientation relationships, Metallurgical and Materials Transactions A 26 (7) (1995) 1745-1756.