# A plasticity-induced internal length mean field model based on statistical analyses of EBSD and nanoindentation data

Layal Chamma, Jean-Marc Pipard, Artem Arlazarov, Thiebaud Richeton, Stéphane Berbenni

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## To cite this version:

Layal Chamma, Jean-Marc Pipard, Artem Arlazarov, Thiebaud Richeton, Stéphane Berbenni. A plasticity-induced internal length mean field model based on statistical analyses of EBSD and nanoindentation data. International Journal of Plasticity, 2025, 189, pp.104327. 10.1016/j.ijplas.2025.104327. hal-05048904

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ELSEVIER

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International Journal of Plasticity

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# A plasticity-induced internal length mean field model based on statistical analyses of EBSD and nanoindentation data

Layal Chamma<sup>a,b</sup>, Jean-Marc Pipard<sup>b</sup>, Artem Arlazarov<sup>b</sup>, Thiebaud Richeton<sup>a</sup>, Stéphane Berbenni<sup>a,*</sup>

## ARTICLE INFO

**Keywords:**

Plastic deformation

Dislocations

EBSD

Nanoindentation

Homogenization model

## ABSTRACT

A new plasticity-induced internal length mean field model (ILMF) is developed, based on statistical analyses of geometrically necessary dislocation (GND) densities and total dislocation densities estimated from EBSD and nanoindentation data, respectively. It is applied to a single phase ferritic Al-killed steel, which plastically deforms with the occurrence of heterogeneous intra-granular fields. During tensile tests up to 18 % of overall plastic strain, the deformation maps of GND densities due to intra-granular plastic strain gradients are obtained together with nano-hardness maps. The Nye tensor (or dislocation density tensor) is calculated from the 2D EBSD orientations to estimate the intragranular GND density, while a mechanistic model is used to estimate the intragranular total dislocation density from nano-hardness measurements. These data are quantified as a function of the distance to grain boundaries (GBs) to study the development of such plastic strain gradients in the vicinity of GBs. The novel methodology lies in extracting the evolution law of a single plasticity-induced internal length, denoted $\lambda$, from the statistical analysis of GND and total dislocation densities spatial distribution. Hence, it is introduced as an evolving variable in an elastoviscoplastic self-consistent model (EVPSC) for a two-phase composite as a new internal mean field (ILMF) approach. Both experimentally quantified microstructural internal lengths defined by the mean grain size and the evolving layer $\lambda$, are considered to more realistically describe the macroscopic and phase response in terms of stress, GND density evolution and total dislocation density in each phase. An experiment/model comparison is also discussed regarding GND density evolution with plastic deformation.

## 1. Introduction

Industrial materials like polycrystalline steels exhibit a heterogeneous microstructure, which contains interfaces like grain boundaries (GBs) and phase boundaries (PBs). Therefore, the design of new steels needs the understanding of internal length effects in addition to other more classic microstructural features such as phase volume fractions, alloying elements (i.e. chemistry), grain morphologies, crystallographic textures. These interfaces are the sources of internal lengths, which can be categorized into two distinct groups. The first one is inherent to the initial microstructure and is called *initial internal length*. The most important in this category for single-phase steels is the average grain size. This length scale often calibrates, at least phenomenologically, to the initial yield point due

E-mail address: stephane.berbenni@univ-lorraine.fr (S. Berbenni).

https://doi.org/10.1016/j.ijplas.2025.104327

Received 20 December 2024; Received in revised form 15 March 2025;

Available online 10 April 2025

0749-6419/© 2025 The Author(s). Published by Elsevier Ltd.

This is an open access article under the CC BY license

(http://creativecommons.org/licenses/by/4.0/).

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

<footnote><sup>b</sup> ArcelorMittal Global Research and Development, Voie Romaine, 57283 Maizières-lès-Metz, France</footnote>

<footnote>* Corresponding author.</footnote>
to the well-known Hall-Petch's law (Hall, 1951; Petch, 1953), for which the initial yield stress (or the flow stress) scales with the inverse of the square root of the mean grain size. An excellent review celebrating the 60th birthday of the Hall-Petch law can be found in Armstrong (2014). It is noteworthy that grain size distribution effects may be considered through this scaling law considering not only the mean grain size but also the grain size dispersion (Berbenni et al., 2007, 2014). Masumura et al. (1998) accounted for a grain size distribution with the simultaneous presence of Hall-Petch strengthening mechanism for "large" grains and Coble creep mechanism for "very fine" grains. From a mechanistic point of view, the Hall-Petch effect can be explained by dislocation-based models, especially pile-up models (Armstrong, 2014). First pioneering dislocation pile-up models (Eshelby et al., 1951) were able to find the same length scale dependence. Schouwenaars et al. (2010) extended this approach by calculating the stress field of array of dislocation pile-ups to quantify grain boundary hardening. Based on the analyses of extensive data available in the literature, Dunstan and Bushby (2013), Dunstan and Bushby (2014) and Cordero et al. (2016) concluded that the Hall-Petch exponent can be more general than ½. Recently, Schouwenaars and Kestens (2023) studied equilibrium configurations of different straight and circular dislocation pile-ups with various numbers of dislocations and explained how different geometries with multiple parallel dislocation pile-ups can provide different values for the Hall-Petch exponent. The second category of internal length scale is inherent to the deformed microstructure and is called plasticity-induced internal length due to Geometrically Necessary Dislocations (GND) intrinsically linked with plastic strain gradients (Ashby, 1970; Aifantis, 1987; Fleck et al., 1994).

Experimental evidence of this second category of internal length can be first investigated by the Electron Back Scattering Diffraction (EBSD) technique from the determination of the Nye tensor (Nye, 1953) as performed from 2D EBSD maps (El-Dasher et al., 2003; Pantleon, 2008; Gardner et al., 2010; Bachmann et al., 2011; Field et al., 2005, 2012; Kundu and Field, 2016, 2018, 2020) or from 3D EBSD maps (Calcagnotto et al. 2010; Bachmann et al., 2011; Konijnenberg et al., 2015; Witzen et al., 2020). For example, in the case of IF steels (Allain-Bonasso et al., 2012), or more recently for steel matrix composites like Fe-TiB₂ (Genée et al., 2021), large 2D EBSD maps with improved angular resolution were acquired to evaluate statistically the evolution of the GND density in the early stages of plastic deformation. In these studies, GNDs were found to accumulate preferentially at grain or phase boundaries. Using 2D EBSD on plastically deformed steel sheets, Signorelli et al. (2017) characterized two parameters: the "boundary effective thickness" and "gradient average severity" over a length in the range 1–2 µm, where a local linear increase of lattice orientation was observed as the distance to GB decreases. Later, De Vincentis et al. (2019) extended this study by considering the effects of neighboring grains on these parameters and considered dislocation pile-up at GBs as the cause of "gradient average severity". Neighborhood effects in a quenching and partitioning steel were also studied using in situ SEM/EBSD and in situ synchrotron X-ray diffraction tensile tests, phase-specific nanoindentation, and atom probe tomography by Kang et al. (2023). From 2D and 3D HR (High Resolution)-EBSD, Calcagnotto et al. (2010) quantified the lattice orientation gradients, i.e. lattice curvatures, and associated GND densities at both ferrite-ferrite and ferrite-martensite interfaces for dual phase steels. Due to stronger plastic incompatibilities between both phases, the GND densities at ferrite-martensite interfaces were higher than those at ferrite-ferrite interfaces. In addition to steels, internal scale effects were investigated for other polycrystalline metals and alloys. Jiang et al. (2013) also used the HR-EBSD technique to study GND storage in annealed copper during a tensile deformation. Using a statistical analysis, they reported a spatial correlation between the different regions in the polycrystal and GND density levels. It was observed that a GND layer with higher values near GBs develop and spread with plastic deformation from 0–2 µm at the onset of plasticity to 7 µm at large strains. In addition, the finest grains exhibited larger values of GND density, and, at low strains, GND density scaled with the inverse of grain size, which appears to be consistent with Ashby's model (Ashby, 1970). The GND evolution during strain hardening of Mg-4Al using in situ SEM tensile testing and HR-EBSD was recently studied by Song et al. (2024a). Higher GND densities were observed within a layer surrounding grain boundary, which were correlated with geometric and plastic incompatibility across grain boundary, to predict polycrystal's strain hardening following Ashby's type model. For the same alloy, Song et al. (2024b) used nanoindentation to propose a modified Ashby's model that considers the hardening effect of SSDs and GNDs separately.

Hall-Petch parameters can be extracted by nanoindentation experiments (Soer et al., 2005; Soer and De Hosson, 2005) and plasticity-induced internal length can also be investigated using the nanoindentation technique (Fischer-Cripps, 2011). First, it should be noted that intrinsic plastic strain gradients and GNDs inherent to non-homogeneous boundary conditions are at the origin of indentation size effect (ISE) obtained from nanohardness vs. penetration depth curves (Nix and Gao, 1998; Feng and Nix, 2004; Huang et al., 2006; Durst et al., 2006; Kysar et al. 2007; Kysar et al., 2010; Kucharski et al., 2024). The interactions between dislocations and GBs were studied using nanoindentation responses (Soifer et al., 2002; Soer and De Hosson, 2005; Soer et al., 2005; Aifantis and Ngan, 2007; Eliash et al., 2008; Voyiadjis and Zhang, 2015; Zhang and Voyiadjis, 2016; Lu et al., 2019). For ferritic interstitial-free (IF) steels, the effect of a single GB on nanoindentation-induced deformation was studied by Ohmura et al. (2005). The classic ISE, as derived by the model of Nix and Gao (1998), was modified by Voyiadjis and Zhang (2015) and Zhang and Voyiadjis (2016), who identified two length scales which are related to the interaction between dislocations beneath the indenter and the GB. In the presence of second phases (martensite), Delincé et al. (2006) extended the model of Nix and Gao, considering the grain/phase boundaries in dual phase steels with different grain sizes for ferritic grains. Previous studies concerned only prior undeformed specimen. For deformed tantalum after shock loading, Weaver et al. (2018) studied the heterogeneous deformation in the regions near GBs using a combination of 2D EBSD and nanoindentation. They characterized Berkovich nanohardness gradients as a function of the distance to GB and its inter-granular misorientation.

These experimental methods can be complemented for interpretation and validation by full-field crystal plasticity (CP) finite element (CP-FEM) (Roters et al., 2010) or fast Fourier transform (CP-FFT) (Lebensohn et al., 2012). These models can simulate micromechanical intragranular fields in addition to macroscopic stress-strain responses. In order to model plasticity-induced internal length effects on the mechanical behavior of polycrystalline materials, full-field non-local models based on strain gradient plasticity (SGP) have been developed (Gurtin, 2002; Kuroda and Tvergaard, 2008; Forest, 2008; Cordero et al., 2012; Mayeur and McDowell,
2014; Ryś and Petryk, 2018; Haouala et al., 2020; Jebahi et al. 2020; Hua et al., 2021; Zhao et al., 2021; Agius et al., 2022; Zhang et al., 2023). Size dependent strength and ductility in single and dual phase steels was simulated by the phenomenological SGP model of Pardoen and Massart (2012). It is noteworthy that GND density pile-ups at phase boundaries and GBs were predicted following the CP-FEM (Richeton et al., 2017) and CP-FFT predictions of mesoscale field dislocation mechanics simulations (Djaka et al., 2020; Berbenni et al., 2020; Genée et al., 2021; Berbenni and Lebensohn, 2021). More recently, Pai et al. (2022) have developed both an experimental framework based on the analysis of 2D EBSD maps, and a CP-FEM based SGP model, to study the development of lattice curvatures and misorientations near GBs during plastic deformation. The Kernel Average Misorientation (KAM) was used to quantify the regions with plastic strain gradients. Such regions were also called "Near Grain Boundary Zones" and their characteristic length was studied in comparison with the SGP model. It was demonstrated that the SGP model was able to reproduce these "Near Grain Boundary Zones", in contrast with classic CP-FEM models.

In addition to computationally intensive full-field models, physically-based mean-field homogenization models were also developed to study the grain size dependent mechanical behavior of polycrystalline materials. Mean-field homogenization self-consistent models were developed in the past decades (Hill, 1965; Berveiller and Zaoui, 1978; Lebensohn and Tomé, 1993; Jeong and Tomé, 2020). A new class of micromechanical approaches called Internal Length Mean Field (ILMF) model was developed starting from Pipard et al. (2009) to predict grain size effects on the overall hardening of IF steels. In this contribution, a simplified description of strong plastic strain gradients present close to GBs was described in a mean field approach. A new mechanical phase containing GND densities was defined with a single plasticity-induced internal length. The latter was only identified from macroscopic tensile curves and was supposed to be constant in course of plastic deformation. A two-phase composite homogenization model using the elasto-viscoplastic self-consistent (EVPSC) model based on the Translated Field (TF) method (Sabar et al., 2002; Mareau and Berbenni, 2015; Tsekpuia et al., 2023) was considered assuming this new mechanical phase containing both GNDs and SSDs (Statistically Stored Dislocations) and grain interior, as the other mechanical phase, containing only SSDs. The evolution equation for SSD density followed the Kocks-Mecking formulation (Kocks and Mecking, 2003). The ILMF model was later extended to a three-phase modeling with a generalized self-consistent scheme for applications to particle size effects in Al/SiC composites (Taupin et al., 2010) and ferrite/cementite steel (Taupin et al., 2013). In these first ILMF models, the grain size effects stemmed from the introduced internal length which is the GND layer thickness, the non-local character of the evolution equation for GND densities and the coupling with the evolution of SSDs.

In a previous experimental study by the present authors (Chamma et al., 2022), 2D EBSD and nanoindentation were combined to study spatial gradients and heterogeneous deformation in a single-phase ferritic Al-killed steel deformed under pure tension. Local configurations in particular grains were examined at 18% plastic strain, and it was observed that spatial gradients spread over a length scale of approximately 2 µm in these grains. However, this value was not derived in a statistical approach and therefore cannot be used as an average value in ILMF model. Furthermore, no information about the evolution of the plasticity-induced internal length, which will be denoted $\lambda$, was reported in this study. Going beyond the classic Ashby (1970)'s model mainly used in the literature as reviewed before, where only grain size and total dislocation densities were considered to predict polycrystal's hardening, the objective of the present study is to extract the value of the plasticity-induced internal length $\lambda$ from a statistical analysis, and to incorporate it in a homogenization-based (i.e. composite-based) micromechanical model. The strategy is to improve the physics of previous models based on total SSD and GND densities and the predictions of dislocation densities and stress/strain heterogeneities within grains together with overall material's hardening.

The paper is organized as follows. In Section 2, the experimental methods are presented. It includes the description of the material (single phase ferritic Al-killed steel), the tensile tests, characterization by Electron backscatter diffraction (EBSD) and nanoindentation testing. Section 3 first describes the methodology of the statistical analyses using both 2D EBSD and nanoindentation data to find the GND and total dislocation density profiles as a function of distance to GB and plastic strain. Then, these data are used in the new ILMF model which is described in Section 4. In Section 5, the results of the model are applied to the Al-killed steel. They are discussed in terms of overall and phase behaviors and compared to experimental data regarding GND density evolutions.

## 2. Experimental methods

### 2.1. Material and specimen preparation

An aluminum-killed steel (Al-k) was provided by the ArcelorMittal Research Center (Maizières-les-Metz, France). It is a single-phase, cold-rolled and batch annealed single phase polycrystalline ferritic steel used for deep drawing applications. The chemical composition of the Al-k steel studied is given in Table 1, and its mechanical properties are listed in Table 2. Fig. 1 (a) shows the conventional tensile curve of Al-k steel to failure. This steel was subjected to different uniaxial tensile tests along the rolling direction (RD) interrupted at different increasing true strain levels: 0.035–0.051–0.098 and 0.185 (Fig. 1 (b)). Tensile specimens with gauge length 50 mm and section $12.5 \times 1 \text{ mm}^2$ were prepared and tensile tests were performed at room temperature with a constant strain

<table><caption>**Table 1**
Chemical composition of Al-k steel (in wt. %).</caption><thead><tr><th>C</th><th>Mn</th><th>Al</th><th>Ni</th><th>Cr</th><th>Mo</th><th>Cu</th><th>P</th><th>S</th><th>N</th></tr></thead><tbody><tr><td>0.0353</td><td>0.1920</td><td>0.0400</td><td>0.0380</td><td>0.0230</td><td>0.0050</td><td>0.0210</td><td>0.0080</td><td>0.0070</td><td>0.0039</td></tr></tbody></table>
<table><caption>**Table 2**

Mechanical properties of Al-k steel.</caption><thead><tr><th>Mechanical properties</th><th>YS (MPa)</th><th>UTS (MPa)</th><th>UEL (%)</th><th>TEL (%)</th></tr></thead><tbody><tr><td>Values</td><td>170</td><td>310</td><td>22.6</td><td>41.4</td></tr></tbody></table>

<figure><img src="image_6.png" /><figcaption>**Fig. 1.** (a) Conventional tensile curve of Al-k steel, (b) Rational curves (true stress-true strain) of interrupted uniaxial tensile tests of Al-k steel. Four tensile specimens were stretched up to plastic strains: ε<sub>p</sub>=0.03, ε<sub>p</sub> =0.05, ε<sub>p</sub> =0.09, and ε<sub>p</sub> =0.18, respectively.</figcaption></figure>

rate of 0.0067 s<sup>-1</sup>.

The average grain size is determined from the EBSD measurements performed on about 140 grains as a weighted average of the area fractions of the equivalent grain diameters:

$$ \bar{D} = \frac{\sum_{i=1}^{N^B} D_i A_i}{\sum_{i=1}^{N^B} A_i} \qquad (1) $$

which gives $\bar{D} \approx 26 \mu m$, with $D_i$ the equivalent diameter of grain "i" corresponding to the diameter of a disk of the same area, $A_i$ the area of grain "i" and "N" the total number of grains detected on the EBSD map.

Surface preparation is a crucial preliminary step for the experimental characterization methods such as SEM/EBSD and nano-indentation. This guarantees the reliability of the data collected with both methods. Poorly controlled mechanical polishing of the specimens can enhance extra hardening of the analysed surface, which is detrimental for rigorous analyses. Therefore, in this study, the initial preparation (grinding) was performed with a polishing force of 10 N, followed by polishing with 3 µm then 1µm diamond disks still with a force of 10 N. Then, additional polishing with a 0.04 µm suspension of colloidal silica was performed for 20 min with a minimum force of 5 N to remove any surface deformation introduced during the previous polishing steps.

## 2.2. Electron backscatter diffraction (EBSD)

EBSD maps were acquired using the Symmetry CMOS (Complementary Metal-Oxide-Semiconductor) detector with a full camera resolution of 1244×1024 pixels and using the "Oxford Instruments Aztec" software on a Schottky JSM-F100 Field Emission Scanning Electron Microscope at 15 kV with 70° tilt. High resolution maps were acquired and processed in refined precision mode. The overall speed was approximately 48 Hz. Each map was acquired with a step size Δx of 100 nm and an angular resolution Δθ of 0.1° was achieved. The indexing rate was greater than 98.5 %.

### 2.2.1. Determination of GND densities from 2D EBSD data

Electron backscatter diffraction (EBSD) data were used by many researchers to obtain measurements of local lattice curvature to estimate GND densities (El-Dasher et al., 2003; Pantleon, 2008; Calcagnotto et al., 2010; Allain-Bonasso et al., 2012; Kundu and Field, 2020; Genée et al., 2021). These estimates are made possible by using the dislocation density tensor α also called Nye tensor (Nye, 1953). The latter can be determined from the lattice curvature tensor deduced from the spatial orientation gradients.

The local orientations are determined at each point on a surface EBSD map. We denote θ<sub>i</sub> the lattice rotation vector for switching from the global sample reference frame to the local crystal reference frame. The disorientation between two measurement points is denoted as Δθ<sub>i</sub> and corresponds to the minimum rotation required to change one orientation to the other. The lattice curvature tensor measured by EBSD is given by the following relation (Nye, 1953; Pantleon, 2008):
$$ \kappa_{ij} = \frac{\partial\theta_j}{\partial x_i} \approx \frac{\Delta\theta_j}{\Delta x_i} \qquad (2) $$

with $\Delta x_i$ the position difference between two measurement points. Neglecting the gradients of elastic strains, $\alpha$ is related to the lattice curvature tensor by (Nye, 1953):

$$ \alpha_{ij} = \kappa_{ij} - \delta_{ij}\kappa_{mm} \qquad (3) $$

The scalar density of GND can then be calculated from the Frobenius norm of the Nye tensor as follows:

$$ \rho_{\text{GND}}^{\text{Nye}} = \frac{1}{b} \sqrt{\alpha_{ij} \alpha_{ij}} \qquad (4) $$

The norm is divided by the Burgers vector length $b$ (=0.251 nm for $\alpha$-Fe) to obtain values in $m^{-2}$ comparable to the dislocation density values $\rho$ used in metallurgy (line per unit volume). Since the local lattice orientations determined by conventional EBSD are only in the 2D observation plane, only the six components $\kappa_{i1}$ and $\kappa_{i2}$ (with $i = 1, 2, 3$, denoting RD, ND, and TD, respectively) of the curvature tensor can be identified. In this case, five of the nine components of the dislocation density tensor can be calculated from the six accessible curvature components (Pantleon, 2008). Therefore, the scalar density of GND is approximated as follows from the accessible components using Eq. (4):

$$ \rho_{\text{GND}}^{\text{Nye}} = \frac{1}{b} \sqrt{\alpha_{12}^2 + \alpha_{13}^2 + \alpha_{21}^2 + \alpha_{23}^2 + \alpha_{33}^2} \qquad (5) $$

For the calculation of GND densities from Eqs. (2) and (5), a threshold of 5° of disorientation between two adjacent pixels was considered in our calculations. Therefore, neighbouring pixels across grain boundaries whose disorientation exceeds this value are not considered in the calculations.

The determination of the GND density from the differences in local orientation variations is highly sensitive to the angular resolution of the EBSD measurements (denoted $\theta_{threshold}$). These differences can only be considered for misorientation angles above the threshold value of the angular resolution (noise floor). Therefore, each misorientation value below this threshold value cannot be included in the GND measurement. Hence, it is necessary to estimate the noise floor below which the GND measurements are not accounted for. The GND density defining the noise floor $\rho_{\text{GND}}^{\text{noisethsd}}$ is extracted from the measurements on the undeformed ferritic specimen with an orientation uncertainty $\Delta\theta \approx 0.1°$. The GND density defining the noise floor $\rho_{\text{GND}}^{\text{noisethsd}}$ can be estimated as $\frac{1}{b} \frac{\theta_{threshold}(\text{rad})}{\Delta x}$ (Wilkinson and Randman, 2010) leading to a value of $7.0 \times 10^{13} \text{ m}^{-2}$ (Chamma et al., 2022). It is also important to estimate the proportion of measurement points above the noise floor. The fractions of measured GND densities that are above the noise floor, i.e., such that $\rho_{\text{GND}}^{\text{Nye}} > \rho_{\text{GND}}^{\text{noisethsd}}$, are 44 % for $\epsilon_p = 3 \%$, 47 % for $\epsilon_p = 5 \%$, 69 % for $\epsilon_p = 9 \%$, and reach about 79 % for $\epsilon_p = 18 \%$. In this study, only values above the noise floor are considered for the four deformation levels.

### 2.2.2. GND density maps and distributions

The orientation maps represented with inverse pole figures (IPF) obtained from EBSD maps in the rolling direction, which is identical to the tensile direction, are presented in Fig. 2 (a-e). During plastic deformation, lattice curvature development is observed with an increasingly pronounced colour gradient in grains with plastic deformation (Fig. 2 (c), (d) and (e)).

The GND density distributions were characterized by EBSD for both undeformed and deformed specimens to evaluate the evolution of GND density in Al-k steel. Fig. 2 (f-j) shows the GND density maps obtained by the Nye tensor method for undeformed specimen and plastically deformed specimens at 3 %, 5 %, 9 % and 18 %, respectively.

Fig. 3 (a) shows the GND density normalized probability distributions for all specimens. It can be seen that both GND density magnitude in average and standard deviation increase with plastic deformation. In addition, the evolution of the average GND density with plastic strain is quasi linear as illustrated in Fig. 3 (b) with a continuous increase from $7 \times 10^{13} \text{ m}^{-2}$ for the undeformed specimen to around $2.6 \times 10^{14} \text{ m}^{-2}$ at 18 % of plastic strain.

## 2.3. Nanindentation

### 2.3.1. Determination of nanohardness

Instrumented nanindentation using Berkovich indenter and Oliver and Phar's method (Oliver and Pharr 1992; Oliver and Pharr, 2004) is used to measure nanohardness and to estimate total dislocation densities in the material with a mechanistic model. The spatial gradients of total dislocation densities in the grain boundary affected zone (GBAZ) are analysed to estimate the internal length of the deformed microstructure.

In this study, a simple mechanistic model following the one proposed by Durst et al. (2006) is applied to estimate the total dislocation densities from nanohardness values. The locally measured nanohardness $H$ can be roughly estimated from the normal flow stress $\sigma_f$ of the material according to the Tabor (1970)'s law:

$$ H = k\sigma_f \qquad (6) $$

where $k \approx 3$ in a first-order approximation according to Dao et al. (2001), Durst et al. (2006) and Barnoush (2012). Then, the resolved shear stress for plastic flow $\tau_f$ can be related to the initial resolved shear stress $\tau_0$ and the total dislocation density $\rho_T$ due to forest-type
<figure><img src="image_7.png" /><figcaption>**Fig. 2.** Inverse pole figures along the RD direction and GND density maps, respectively, of the undeformed specimen (a, f) and the deformed specimen at 3 % (b, g), 5 % (c, h), 9 % (d, i) and 18 % (e, j) of plastic strain.</figcaption></figure>
hardening using a Taylor (1934)'s type law as follows:

$$ \tau_f = \tau_0 + \alpha_d \mu b \sqrt{\rho_T} \qquad (7) $$

where μ is the isotropic elastic shear modulus. Here, μ=80 GPa is commonly used for a ferritic steel following a previous study (Pipard et al., 2009). The constant αd is an empirical factor dependent on dislocation arrangements. This factor is generally between 0.2 and 0.4 (Mughrabi, 2016). Here, the value αd=0.3 is chosen. The total dislocation density is additively decomposed as:

$$ \rho_T = \rho_{GND} + \rho_{SSD} \qquad (8) $$

where ρSSD is the density of statistically stored dislocations. Using the Taylor factor M=3 (Durst et al., 2006) to relate the crystal's plastic flow shear stress τf to the mean normal plastic flow stress σf yields:

$$ \sigma_f = \sigma_0 + M \alpha_d \mu b \sqrt{\rho_T} \qquad (9) $$

with σ0 ≈ Mτ0 the initial tensile yield strength of the material calculated at εp=0.2 %. This value is identified as 175 MPa from Fig. 1 (b). It is noteworthy that M represents an average Taylor factor (Taylor, 1938), which is assumed to be constant for each grain in a polycrystal. Then, combining Eq. (6) together with Eq. (9) yields:

$$ H \approx H_0 + 3M\alpha_d\mu b\sqrt{\rho_T} \qquad (10) $$

where H0 represents the initial macro-hardness, i.e. at 0 % plastic deformation (H0 = kσ0 with k≈ 3). Therefore, according to Eq. (10), the total dislocation density can be roughly estimated from the measured nanohardness value H as follows:

$$ \rho_T \approx \left( \frac{H - H_0}{3M\alpha_d\mu b} \right)^2 \qquad (11) $$

### 2.3.2. Nanoindentation tests

Nanoindentation tests were carried out using a UltraNanoIndent (UNHT) from Anton Paar and a Berkovich diamond tip. Since the main objective is to determine an internal length related to the hardness gradient close to GBs, a constant and relatively low load was chosen to minimize the spacing between the indents. A load of 1mN was chosen for the nanoindentation tests. With a maximum load of 1 mN, maximum depths vary from 100 to 125 nm. Indent spacing is directly related to indentation depth (Zhao and Ovaert, 2010). Its value must be chosen in such a way as to avoid the possibility of mutual interaction between the plastic zones of subsequent indentations (Sudharshan Phani and Oliver, 2019). We considered a spacing of at least 17 times the indentation depth equivalent to indentation spacing set between 2 and 2.2 µm, in agreement with typical values of the literature (Zhao and Ovaert, 2010; Sudharshan Phani and Oliver, 2019; Jiang et al., 2020). Loading and unloading speeds were set at 0.2mN/s, with a holding time of 1 s at maximum load before unloading to minimize the effect of creep on the unloading curve. Indentations were made along several parallel lines, slightly offset from each other and roughly perpendicular to the grain boundary. The indentations in each of the lines are deliberately not aligned with each other in order to obtain several nanohardness values at a distance less than 2 µm from the grain boundary.

## 3. Statistical analysis

### 3.1. Method description

The procedure used to determine the internal length of the GBAZ from EBSD and nanoindentation data is described below and illustrated in Fig. 4 (a). It comprises 4 steps performed with a MATLAB code within the framework of the MTEX toolbox (Bachmann et al., 2011):

**Step 1** Determine the set of pixel coordinates of all grains and GBs detected from the EBSD maps at each strain level. Then position the indent marks on the EBSD maps from their corresponding coordinates. Table 3 summarizes the data set used for EBSD-2D and nanoindentation analysis in terms of number of pixels, number of grains, number of indents and number of indented grains.

**Step 2.** Determine the minimum distance, noted dᵢ, between each measurement point “j” and the closest GB using Eq. (12) where the pixels of the grains (xⱼ, yⱼ) and the GBs (xGB, yGB) are obtained from MTEX.

$$ d_{j(pixel,indent)} = \min_{(x_{GB}, y_{GB})} \sqrt{(x_j - x_{GB})^2 + (y_j - y_{GB})^2} \qquad (12) $$

**Step 3.** Define critical distances d<sub>cj</sub> within the interval [0, D/2], where D is the average diameter of all grains detected.

**Step 4.** Calculate the mean of the GND above the noise level of 7.0×10¹³ m⁻² and hardness densities for measurement points presenting a distance dⱼ less than or equal to each increment of d<sub>cj</sub>, such that dⱼ ≤ d<sub>cj</sub>. For a given d<sub>cj</sub>, these measurement points form the red zone in Fig. 4 (a).
<figure><img src="image_8.png" /><figcaption>Fig. 3. (a) Normalized probability distribution of GND density at 0 %, 3 %, 5 %, 9 % and 18 % of plastic strain, (b) Evolution of the average GND density over the EBSD map as a function of overall plastic strain during tensile test.</figcaption></figure>

The mean GND densities are calculated for the four strain states by using the cumulative measurement method described in Fig. 4. The objective is to statistically characterize the internal microstructural length $\lambda_s$ at each plastic strain and to study its evolution during deformation. Error bars, corresponding to the uncertainty in the estimation of the mean values $\mu_e$, were calculated considering the standard deviation $\sigma$, the number of measurement points $n_s$ and a parameter $t$ given by the table of normal distribution:

$$ \left[ \mu_e - t \frac{\sigma}{\sqrt{n_s}}; \mu_e + t \frac{\sigma}{\sqrt{n_s}} \right] \qquad (13) $$

For a 99 % confidence interval, the value of $t$ is equal to 2.58. Due to the fact that the number $n_s$ of EBSD points is very high at each strain level (see Table 3), the term $\left( t \frac{\sigma_{\text{GND}}}{\sqrt{n_s}} \right)$ is most often negligible in front of $\bar{\rho}_{\text{GND}}$ (example Fig. 3 (b)) where the error bars are not visible). Error bars are much more visible on nanoindentation figures where the number of indents is much smaller (see Table 3).

For comparison with the “cumulative analysis method” (Fig. 4), we have also considered another method based on the average of the GND density on thin intervals exploiting the same EBSD data, which is named “interval method”. In this method, the mean GND density of the set of pixels located in the blue colored layer is calculated, i.e. between two successive distances $d^{\text{int}} < d_j \le d^{\text{int}}+0.2 \, \mu\text{m}$, corresponding to only 0.2 $\, \mu\text{m}$ thick intervals (see Fig. 5).

## 3.2. EBSD and nanoindentation data analyses

We apply both the "cumulative measurement method" (Fig. 4) and the “interval method” (Fig. 5) on all grains and GB detected in the EBSD maps reported in Fig. 2 for the specimens plastically deformed at 3 %, 5 %, 9 %, and 18 %. Fig. 6 shows the variations in mean GND densities as a function of $d_{c_p}$ for the four studied deformation stages using the cumulative method. A clear evolution of the mean GND densities from 3 % to 18 % plastic deformation can be observed, which can also be noted on the maps in Fig. 2 (g-j). These profiles show that, as we approach the GB, the mean GND densities increase in all four specimens. Consequently, these profiles confirm that GNDs tend to be concentrated close to the GBs for all four strains. This increase of GND density near the GB is quantitatively confirmed by the second method based on the intervals as shown in Fig. 7 for the four plastic strains with approximately the same involved length scales and evolution. Therefore, the determination of the internal microstructural length $\lambda_s$ will be more thoroughly studied in the sequel only with the cumulative measurement method. The latter provides smoother transitions at all plastic strains and allows to better distinguish the GB affected zone from the grain interior since the latter is always characterized by the global mean GND density of the map.

With respect to the nanoindentation data, we have attempted to use the cumulative measurement method to calculate the means of the nanohardness and total dislocation density values. For the sample deformed at 5 % plastic strain, the number of indentations was not sufficiently representative (see Table 3). For this reason, we present only the results of the nanoindentation tests performed on the specimens deformed at 3 %, 9 % and 18 %. Fig. 8 (a) and (b) show the profiles of mean nanohardness and total dislocation densities $\rho_T$ for the selected plastic strains, derived from the mechanistic model, as a function of distance $d_{c_i}$ from the GB to the grain interior.

## 3.3. Determination of $\lambda$: methodology

The gradients of GND densities associated with areas near GBs show approximately linear behavior according to Fig. 6. These variations in GND densities are much less pronounced in the areas near the center of the grain.

The internal length $\lambda_s$ should be considered as the effective thickness over which a GND density gradient exists. The parameter $\lambda_s$
was determined from a least-squares fit of the linear profile. Based on the profiles shown in Figs. 6 and 8, the fit is considered acceptable if the square of the correlation coefficient R², is greater than a given value. Thus, the fit is made by considering the maximum number of points from the GB, such that R² remains above a threshold value. It should be noted that the threshold value of R² was chosen based on our profiles, but at the condition that the value of R² remains the same for all four specimens. To quantify the value of λs, the procedure presented in Fig. 9 was used. This figure shows an example illustrating the method used to determine λsEBSD based on the profile of the average GND densities of the sample deformed at 18 % plastic strain. A R² value of 0.98 was chosen for the four strains. The value of the microstructural internal length corresponds to the intersection of two dashed lines, the first describing the GND density gradient near the GB, and the second the constant value inside the grain. This graphical determination of λsEBSD is applied to the four profiles of average GND densities in Fig. 6 with the same criteria of R²=0.98. The λs values resulting from this procedure are 1.8 µm, 2.5 µm, 3 µm and 3.7 µm for the samples deformed at 3 %, 5 %, 9 % and 18 %, respectively. These statistical values confirm the existence of a GND layer near the GB, i.e. a GBAZ, and describe its evolution during plastic deformation.

For the nanoindentation measurement, the same fitting procedure was used to characterize the distance over which the gradient of total dislocation densities extends. Here, a threshold value of R² equal to 0.9 was considered for the three profiles in Fig. 8 (b) to estimate λs from the gradients of total dislocation density. The resulting λsNano values are 2.8 µm, 3 µm, and 3.4 µm for the samples deformed at 3 %, 9 %, and 18 % plastic strains, respectively. It is observed in Fig. 10 that the estimates of λs from both experimental methods (EBSD, nanoindentation) provide similar evolutions. These findings are now used to improve a homogenization-based Internal Length Mean Field model (ILMF) with physical and evolutive GND layer.

# 4. Plasticity-induced internal length mean field model (ILMF)

## 4.1. Two-phase composite representation

The experimental results and the previous statistical analysis showed that ferritic grains in this Al-k steel can be divided into two mechanically different regions. The first region is located close to the GB, where a dislocation density gradient has been clearly observed and extends over a distance defined by λs. The second region consists of the interior of the grain where no marked variation in dislocation densities has been reported. This microstructure can be considered as a "composite" material with two "phases" having different mechanical behaviors.

In order to predict the mechanical behavior of Al-k steel as a function of internal length (or based on experimental results), the starting point is the ILMF model (Internal Length Mean Field model) introduced by Pipard et al. (2009). This micromechanical model was initially developed to model the effect of grain size in single-phase, i.e. purely ferritic, polycrystalline steels. This model considers the material's microstructure as a two-phase composite by introducing a layer of finite thickness, which was restricted to be constant in Pipard et al. (2009). This finite layer represents the accumulation of geometrically necessary dislocations (GND) near the grain

<figure><img src="image_9.png" /></figure>

<figure><img src="image_10.png" /><figcaption>**Fig. 4.** (a) Description of the statistical analysis method to calculate the mean of dislocations densities and nanohardness in red zones, (b) Description of an example of the cumulative analysis method where the red area represents all pixels with a distance d<sub>j</sub> ≤ d<sub>cj</sub> = 1.9 µm, the white areas describe the interior of the grains and the black lines represent the GB.</figcaption></figure>
<caption>**Table 3**</caption>

Data set used for statistical analysis using 2D EBSD and nanoindentation measurements (GND densities at pixels only greater than $\rho_{GND}^{noisethsd}$).

<table><thead><tr><th rowspan="2">ε<sub>p</sub> (%)</th><th colspan="2">EBSD</th><th colspan="2">Nanoindentation</th></tr><tr><th>Number of grains</th><th>Number of pixels</th><th>Number of indented grains</th><th>Number of indents</th></tr></thead><tbody><tr><td>3</td><td>150</td><td>1 383 755</td><td>13</td><td>471</td></tr><tr><td>5</td><td>130</td><td>1 536 497</td><td>-</td><td>-</td></tr><tr><td>9</td><td>115</td><td>1 670 432</td><td>11</td><td>248</td></tr><tr><td>18</td><td>140</td><td>2 423 280</td><td>18</td><td>516</td></tr></tbody></table>

boundaries. Fig. 11 (a) and (b) shows the decomposition of the grain structure into "Phase 1", i.e. the GND layer surrounding "Phase 2", the grain core, where for the latter strain hardening only results from the evolution of the density of statistically stored dislocations (SSDs). SSDs are distributed in both phases, while GNDs are only present in "Phase 1".

In this ILMF model, plastic deformation in the grain core is assumed to be controlled solely by the evolution of SSD density, which is denoted $\rho_{SSD}^{(2)}$. The finite thickness layer defined by phase 1 represents the accumulation and storage of geometrically necessary dislocations (GND) near the GBs. The thickness of this layer defined by "λ" is an internal length scale of the model in addition to the grain diameter. This region is affected by the presence of both SSDs and GNDs. A one-site self-consistent scheme using the "Translated Field" (TF) method for heterogeneous elastoviscoplastic materials (Sabar et al., 2002; Berbenni et al., 2004) was used by Pipard et al. (2009) to determine both the macroscopic and the phase behaviors. Both phases completely fill the entire grain defined in Fig. 11 (b). Thus, $f^{(1)} + f^{(2)} = 1$, where $f^{(2)}$ and $f^{(1)}$ are the volume fractions of the grain core (region 2) and the layer (region 1), respectively. Assuming spherical grains, they are related to the grain size D and the layer thickness λ:

$$f^{(2)} = \left( \frac{D - 2\lambda}{D} \right)^3 \qquad (14)$$

$$f^{(1)} = 1 - \left( \frac{D - 2\lambda}{D} \right)^3 \qquad (15)$$

It should be noted that both internal lengths D and λ are included in both phase volume fractions. In Pipard et al. (2009), due to the lack of experimental data, λ was assumed to be constant and was identified from macroscopic material's response. In the present investigation, thanks to the novel experimental results and analyses provided in Section 3, the evolution of λ with plastic strain will be introduced in the model.

## 4.2. Phase constitutive equations

In this model, both phases are assumed to exhibit elastoviscoplastic behavior described by a total strain rate $\dot{\epsilon}^{(I)}$. Under the small strain hypothesis, this strain rate is additively decomposed into an elastic part $\dot{\epsilon}^{e(I)}$ and a viscoplastic part $\dot{\epsilon}^{vp(I)}$ for each phase "I" (I=1, 2):

$$\dot{\epsilon}^{(I)} = \dot{\epsilon}^{e(I)} + \dot{\epsilon}^{vp(I)} \qquad (16)$$

In the context of this model, the assumption of small strains is sufficient, since the theoretical study primarily focuses on plastic flow at low deformations in the elastic/plastic transition regime (i.e. the micro-plastic stage). The elastoviscoplastic behavior of both phases is therefore described by Maxwell-type constitutive equations. For each phase, the elastic part relates the elastic strain rate $\dot{\epsilon}^{e(I)}$ to the Cauchy stress rate tensor $\dot{\sigma}^{(I)}$ via the linear, isotropic and homogeneous elastic constitutive equation:

$$\dot{\sigma}^{(I)} = C : \dot{\epsilon}^{e(I)} \qquad (17)$$

<figure><img src="image_11.png" /><figcaption>**Fig. 5.** Description of the statistical analysis method based on thin intervals (interval method). The blue area depicts a thin layer of thickness 0.2 µm at a distance $d^{int}$ from GB, in which the GND density is averaged.</figcaption></figure>
<figure><img src="image_12.png" /><figcaption>Fig. 6. Cumulative measurement method: profile of mean GND densities as a function of cumulative distance to GB for the four strains. The increment of d<sub>c</sub> is 0.1 µm (corresponding to the spatial resolution of the EBSD measurements).</figcaption></figure>

<figure><img src="image_13.png" /><figcaption>Fig. 7. Profile of mean GND densities as a function of d<sup>int</sup> for the four strains. For each d<sup>int</sup>, the thickness of the layer to compute the average GND density corresponds to a thin layer of 0.2 µm.</figcaption></figure>

The viscoplastic strain rate tensor ˙ε<sup>vp(I)</sup> (with I=1, 2) follows the Norton's viscoplastic flow rule:

$$ \dot{\epsilon}_{ij}^{\text{vp(I)}} = \frac{3}{2} \dot{\epsilon}_0 \left( \frac{\sigma_{\text{eq}}^{(I)}}{\sigma_{\text{ref}}^{(I)}} \right)^n \frac{\sigma_{ij}^{(I)}}{\sigma_{\text{eq}}^{(I)}} \quad (18) $$

where ˙ε<sub>0</sub> is a reference plastic strain rate (here assumed equal to 1 s<sup>-1</sup>), σ<sub>ref</sub><sup>(I)</sup> is the reference flow stress, the exponent "n" characterizes the inverse of the material strain rate sensitivity "m" defined by m=1/n, σ<sub>eq</sub><sup>(I)</sup> is the equivalent Von Mises stress given by:

$$ \sigma_{\text{eq}}^{(I)} = \left( \frac{3}{2} \sigma_{ij}^{(I)} \sigma_{ij}^{(I)} \right)^{1/2} \quad (19) $$

where σ<sub>ij</sub><sup>(I)</sup> is the stress deviator tensor. Using a secant linearization, the rank-4 viscoplastic compliance tensor m is defined as:

$$ \dot{\epsilon}^{\text{vp}} = \mathbf{m}(\sigma) : \sigma \quad (20) $$

with:

$$ \mathbf{m}_{ijkl} = \frac{3}{2} \dot{\epsilon}_0 \left( \frac{\sigma_{\text{eq}}}{\sigma_{\text{ref}}} \right)^n \frac{\eta_{ijkl}}{\sigma_{\text{eq}}} \quad (21) $$

where η is a rank-4 tensor allowing the deviatoric part to be projected (here the viscoplastic strain rate tensor is deviatoric due to plastic incompressibility):
<figure><img src="image_14.png" /><figcaption>Fig. 8. Profiles of the mean of (a) nanohardness and (b) total dislocation density, with the cumulative distance-to-GB measurement method for the 3 strain levels. The increment of $d_{c_i}$ is 0.2 µm.</figcaption></figure>

<figure><img src="image_15.png" /><figcaption>Fig. 9. Profile of the average GND densities across the GB for the 18% deformed sample showing the method for determining the internal length $\lambda_S$. Here, $\lambda_S$ is estimated to 3.7 µm using the linear fit.</figcaption></figure>

$$ \eta_{ijkl} = I_{ijkl} - \frac{1}{3}\delta_{ij}\delta_{kl} \qquad (22) $$

In phase 1 (layer), we assume that both SSD and GND densities contribute to the total density of forest-like obstacles for mobile dislocations, whereas in phase 2 (grain core), only SSD densities contribute to the total number of forest-like obstacles. These differences lead to distinct flow laws for both phases. Consequently, based on the distribution of both populations of dislocation densities in both phases, the so-called reference flow stress $\sigma_{ref}^{(I)}$ can be written as follows:

$$ \sigma_{ref}^{(I)} = \sigma_{0}^{(I)} + M\alpha_{d}\mu b \sqrt{\rho_{GND}^{(I)} + \rho_{SSD}^{(I)}} \qquad (23) $$

where $\sigma_0^{(I)}$ is the part of $\sigma_{ref}^{(I)}$ controlled by the hardening due to solid solution elements and lattice friction (Peierls force), M is the
<figure><img src="image_16.png" /><figcaption>**Fig. 10.** Evolutions of $\lambda_s$ determined with both experimental methods (EBSD, nanoindentation) integrating error bars as function of plastic strain.</figcaption></figure>

<figure><img src="image_17.png" /><figcaption>**Fig. 11.** (a) Illustration of the two-phase composite model with two internal lengths: grain size D and GND layer defined by microstructural internal length $\lambda$. (b) Schematic figure illustrating the decomposition of the grain of diameter D into two regions. Region 2 is the grain core with only SSDs. It is surrounded by the layer (region 1) of thickness $\lambda$ with SSDs and GNDs. Only the GNDs are shown as polarized dislocations stored in the GBAZ, (c) Description of the material as a two-phase composite (RVE). Red: layer (phase 1). White: grain core (phase 2), (d) Principle of the 1-site self-consistent scheme for two-phase composite.</figcaption></figure>
Taylor factor (Taylor, 1938) generally taken equal to 3, $\alpha_d$ is a constant approximately describing the interactions between dislocations. Its value is generally taken between 0.2 and 0.5 according to Mughrabi (2016), $\mu$ is the isotropic shear modulus, $b$ is the magnitude of the Burgers vector.

### 4.3. Evolution laws for dislocation densities

The flow stress described in Eq. (23) contains both dislocation density populations $\rho_{GND}^{(I)}$ and $\rho_{SSD}^{(I)}$, as internal variables. The SSD densities intrinsically distributed in the whole grain of diameter D, are due to generation and annihilation processes between dislocations whose net Burgers vector is equal to zero. These are referred to as "non-polarized" dislocations.

In the framework of this model, the evolution law for the SSD density follows a Kocks-Mecking's law (Mecking and Kocks, 1981; Kocks and Mecking, 2003) modified by introducing the GND density only in the creation term (in forest hardening):

$$ \dot{\rho}_{SSD}^{(I)} = M \left( \frac{k}{b} \sqrt{\rho_{SSD}^{(I)} + \rho_{GND}^{(I)}} - f \rho_{SSD}^{(I)} \right) \dot{\epsilon}_{eq}^{vp(I)} \quad (24) $$

where $k$ and $f$ are material parameters to be identified that control dislocation creation and annihilation respectively. The equivalent viscoplastic strain rate $\dot{\epsilon}_{eq}^{vp}$ reads:

$$ \dot{\epsilon}_{eq}^{vp} = \left( \frac{2\epsilon_{ij}^{vp}\epsilon_{ij}^{vp}}{3} \right)^{1/2} \quad (25) $$

In phase 2, where GNDs are not present, Eq. (24) leads to a classical Kocks-Mecking's law, by setting $\rho_{GND}^{(2)} = 0$ (Kocks and Mecking, 2003). In phase 1, the GND density does not affect the annihilation term described by the "f" parameter, which controls the saturation of SSD densities. Annihilation is a recovery process that only involves SSD densities. The annihilation parameter "f" acts only as a multiplication coefficient on the SSD density ($\rho_{SSD}^{(I)}$). GND density is only related to the "k" parameter, which controls the creation of SSD densities.

The two-phase micromechanical model of Pipard et al. (2009) was inspired by the composite model of Mughrabi (1983 and 2004). This one described the crystal as a composite with a hard region of high dislocation density (dislocation walls) and a second soft region of low local dislocation density (channel-type zones between walls). Mughrabi (1983 and 2004) developed a two-phase composite model described by a laminate model. Here, the material is considered as a two-phase composite consisting of a hard phase, the layer (phase 1, where the GNDs are located), encompassing a soft phase (phase 2, the grain core) where the density of GNDs is zero. Here, the morphology of both phases is assumed to be spherical (Pipard et al., 2009), which appears more realistic to describe grains. GND densities are only localized in the layer, since slip bands are constrained at grain boundaries in the form of GND pile-ups. This constrained plasticity at grain boundary interfaces in Al-k steel allows us to write the following inequalities on equivalent plastic deformations in both phases:

$$ 0 < \epsilon_{eq}^{vp(1)} < \epsilon_{eq}^{vp(2)} \quad (26) $$

Deformation mechanisms linked to dislocation densities differ between the layer (phase 1) and the grain core (phase 2). These plastic deformation gradients are due to local incompatibilities of plastic deformation in the material and are the sources of associated internal stresses (Kröner, 1958; Kröner, 1981).

Using a static approach based on the incompatibility equation due to GND densities present at material point x, Ashby (1970) defined the GND density due to plastic slip gradients $\gamma_p^x$ by the equation $\rho_{GND}^{(1)} = \frac{1}{b} \frac{\partial \gamma_p^x}{\partial x}$. Assuming a Taylor factor M to relate the plastic slip gradient noted $\frac{\partial \gamma_p^x}{\partial x}$ to the equivalent plastic strain gradient $\frac{\partial \dot{\epsilon}_{eq}^{vp}}{\partial x}$ yields:

$$ \rho_{GND}^{(1)} = \frac{M}{b} \frac{\partial \epsilon_{eq}^{vp}}{\partial x} \quad (27) $$

To reasonably estimate the GND density in the layer of thickness $\lambda$, a linear variation of the equivalent plastic strain is assumed between both phases over the characteristic length $\lambda$. Therefore, assuming a radial spherical symmetry for the layer and the grain interior (Taupin et al., 2010) yields:

$$ \rho_{GND}^{(1)} \approx \frac{M}{b} \frac{\epsilon_{eq}^{vp(2)} - \epsilon_{eq}^{vp(1)}}{\lambda} \quad (28) $$

The time evolution of the GND density in the layer noted $\dot{\rho}_{GND}^{(1)}$ is determined from classic material time derivative formula (Malvern, 1969) and from Eq. (28):

$$ \dot{\rho}_{GND}^{(1)} \approx \frac{M}{b} \left[ \frac{\dot{\epsilon}_{eq}^{vp(2)} - \dot{\epsilon}_{eq}^{vp(1)}}{\lambda} - \frac{\dot{\lambda}}{\lambda^2} \left( \epsilon_{eq}^{vp(2)} - \epsilon_{eq}^{vp(1)} \right) \right] \quad (29) $$

where $\dot{\lambda}$ can be derived from the statistical analysis reported in Section 3.3. The original model of Pipard et al. (2009) is recovered for $\dot{\lambda} = 0$ and $\lambda$ constant in Eq. (29). Therefore, the novelty of the present ILMF model is that the evolution of the GND density in the layer
now depends on two terms: $\dot{\rho}_{GND}^{(1)} \approx G_{loc} + G_{conv}$, where $G_{loc} = \frac{M}{b} \frac{\dot{\epsilon}_{eq}^{(2)} - \dot{\epsilon}_{eq}^{(1)}}{\lambda}$ is a local rate of change of $\rho_{GND}^{(1)}$ related to the jump of equivalent plastic strain rate between both phases controlled by an evolving layer thickness $\lambda$, and, $G_{conv} = -\frac{M}{b} \frac{\dot{\lambda}}{\lambda^2} (\epsilon_{eq}^{vp(2)} - \epsilon_{eq}^{vp(1)})$ is a term depending on "$\dot{\lambda}$", which is a convective rate of $\rho_{GND}^{(1)}$ change related to layer growth during plastic deformation.

### 4.4. Internal length $\lambda$ inferred from experimental data and model adaptation

In Section 3, both methods based on EBSD data and nanoindentation measurements described an evolution of internal length $\lambda$ between 3 % and 18 % plastic deformation. In this section, only the results obtained from EBSD data are used, as they are more statistically representative (more data points were obtained with this method). Experimentally, it was observed that GNDs are distributed within grains with a higher concentration of GND density near GBs (see Fig. 2 (f-j)). Furthermore, the spatial profiles of GND density as a function of distance from the grain boundary (Fig. 6), obtained by the method of statistical analysis, showed a quasi-linear average variation in GND density near the grain boundaries for the four strain levels studied over an evolutionary distance equal to $\lambda_S$. At the same time, no variation in GND density was observed on average inside the core grains for any of the samples studied. Here, the GND density in the grain interior is essentially due to the formation of dislocation cells, which were not considered in the model. These observations are shown in Fig. 12 (a). This figure illustrates the hatched area of the triangle, which describes the linear variation of the GND density along the experimentally obtained internal length ($\lambda_S$) at a given strain. On the other hand, following Pipard et al. (2009), a zero GND density was considered outside the layer thickness $\lambda_M$, as shown in Fig. 12 (b). In order to better transfer the experimental data (spatial gradients along the layer of thickness $\lambda_S$) to the ILMF model (piecewise uniform fields with uniform GND density in the layer of thickness $\lambda_M$), $\lambda_S$ is divided by 2 to obtain $\lambda_M$, which is used in the model. This transformation preserves both the maximum GND density and the total number of GND dislocations, i.e. the area under the curve in the layer (see Fig. 5).

Following the procedure described above, the evolution of $\lambda_M$ is assessed at each strain level. The values are plotted in Fig. 13. Each evolution of $\lambda$ can be interpolated by a saturating exponential law type function which depends on the plastic deformation and is given by the following relationship:

$$ \lambda_M = (\lambda^\infty - \lambda^0)(1 - e^{-Cr_p}) + \lambda^0 \quad (30) $$

where $\lambda^\infty$ is the value of the saturating internal length (maximum expected value), $\lambda^0$ is the initial value and C is a constant related to the decay rate of the exponential term. These three parameters were determined by interpolation via the least-squares method of the 4 $\lambda_M$ values (red circles, as a function of $\epsilon_p$) obtained by the cumulative measurement method. Exponential interpolation of these values gave a better fit for the $\lambda_M$ values obtained by $\bar{\rho}_{d_p}^{GND}$.

Fig. 13 shows the curve of the obtained evolution law of $\lambda_M$. The evolution of $\lambda_M$ with plastic deformation clearly shows a rapid increase of $\lambda$ at low levels of plastic strain and a saturation of $\lambda$ at higher levels of plastic strain. However, according to Fig. 13, the interpolation remains uncertain below 3 % plastic strain due to the limitations of the EBSD method at low strains. It is therefore not certain that the trend follows exactly this steep exponential slope at low plastic strains between 0 and 3 %. The value of $\lambda_M$ extrapolated at zero plastic strain denoted $\lambda^0$ can be related to the width of the GND layer at the onset of plasticity (due to incipient plasticity).

Once the equation for the evolution law of "$\lambda_M$" (Eq. (30)) is determined, the three parameters ($\lambda^\infty$, $\lambda^0$ and C) are obtained from Fig. 13 which gives 0.5 µm for $\lambda^0$, 1.9 µm for $\lambda^\infty$ and C=12.70, with an error of 0.01. In addition, the evolution of the layer volume fraction (phase 1) with plastic deformation is given from Eq. (15) together with Eq. (30). The effect of the value of $\lambda^0$ will be studied in the numerical applications. In the sequel, we will use $\lambda=\lambda_M$ to simplify the notations. $\dot{\lambda}$ is determined by deriving Eq. (30) with respect to time, which reflects the growth rate of the layer:

$$ \dot{\lambda} = C(\lambda^\infty - \lambda^0)(e^{-Cr_p})\dot{\epsilon}_p \quad (31) $$

$\epsilon_p$ is the macroscopic equivalent plastic strain rate. In addition, the layer volume fraction is updated at each step through Eq. (15).

### 4.5. Homogenization-based model: elasto-visco-plastic self-consistent scheme (EVPSC)

The model of Pipard et al. (2009) uses and extends to non-local plasticity the elastoviscoplastic self-consistent scheme (EVPSC) based on the Translated Fields (TF) method developed by Sabar et al. (2002), Berbenni et al. (2004) and Berbenni et al. (2007) (see also Mareau and Berbenni, 2015 and Tsekpuia et al., 2023). Assuming homogeneous elasticity and a secant formulation for viscoplastic strain rate, the following stress interaction law is used:

$$ \dot{\sigma} = \dot{\Sigma} - C : (S^E - I) : (\dot{\epsilon}^{vp} - A^B : \dot{E}^{vp}) \quad (32) $$

where ‘:’ denotes the doubly contracted product (or scalar product) between two rank-2 tensors, $\dot{\sigma}$ is the rank-2 Cauchy stress rate tensor (in phase 1 or phase 2), $\dot{\Sigma}$ is the rank-2 macroscopic stress rate tensor, C is the rank-4 elastic stiffness tensor, $S^E$ is the rank-4 Eshelby tensor (Eshelby, 1957). I is the rank-4 unit tensor defined as $I_{ijkl} = \frac{1}{2}(\delta_{ik}\delta_{jl} + \delta_{il}\delta_{jk})$ with $\delta_{ij}$ is the Kronecker delta, $\dot{\epsilon}^{vp}$ is the rank-2 viscoplastic strain rate tensor (in phase 1 or phase 2), $\dot{E}^{vp}$ is the rank-2 macroscopic viscoplastic strain rate tensor defined as the
volume average over the composite: $\dot{E}^{vp} = \bar{\dot{\epsilon}}^{vp}$ (where (-) denotes the volume average over the composite), $A^e$ is the rank-4 visco-plastic strain concentration tensor associated with the rank-4 effective viscoplastic stiffness tensor $B^e$ of the composite, which is given by:

$$A^e = (I + \Gamma_1^e : \delta b^e)^{-1} \quad \text{with} \quad \delta b^e = b - B^e \qquad (33)$$

where $b$ is the rank-4 viscoplastic stiffness tensor, $\Gamma_1^e$ is the local part of the modified Green's tensor (Kröner, 1989) associated with $B^e$. The effective viscoplastic stiffness tensor $B^e$ is solved with a 1-site self-consistent loop using the fixed-point algorithm: $B^e = \overline{b : A^e : A^e^{-1}}$.

Fig. 11 (c) schematically represents a definition of RVE (Representative Volume Element) for two-phase composites, and Fig. 11 (d) represents the principle of the 1-site self-consistent scheme. The interaction law (Equation(32)) describes the mechanical interaction between each component (grain core or layer) and the Homogeneous Equivalent Medium (HEM), which represents the effective behavior of the composite. This is the scale transition between the phases and the RVE. The macroscopic behavior of the two-phase composite is determined using mechanical field averaging theorems and definitions on the RVE, using the principle of homogenization based on the macro-homogeneity condition (Qu and Cherkaoui, 2006):

$$\dot{E} = \bar{\dot{\epsilon}} \quad \text{and} \quad \dot{\Sigma} = \bar{\dot{\sigma}} \qquad (34)$$

# 5. Results and discussion

## 5.1. Model parameters

In this model, the parameter values of D=26 µm, $\lambda^0=0.5$ µm, $\lambda^\infty=1.94$ µm and C=12.7 are deduced from EBSD measurements and are used directly in the evolution laws of $\lambda$ (Eq. (30)), $\dot{\lambda}$ (Eq. (31)) and $f^{(1)}$ (Eq. (15)). In addition, the parameters k, f, $\sigma_0$ and n are identified from the experimental tensile curve (black curve in Fig. 14). Table 4 summarizes the values of these parameters as well as the values of the initial dislocation densities $\rho_{SSD}^0$ and $\rho_{GND}^0$. For simple tensile simulations, the macroscopic strain rate applied is $|\dot{E}|=0.0067$ s⁻¹ and the tensile is performed up to 20 % strain.

## 5.2. Overall responses

Fig. 14 shows the macroscopic equivalent stress-strain curves obtained by the model (blue and red) and the experimental stress-strain curve (black). The blue curve (case 1) represents the macroscopic response that considers only the $G_{loc}$ term in the evolution of the GND density, while the red curve (case 2) represents the macroscopic response that considers both the $G_{loc}$ and $G_{conv}$ terms in Eq. (29). The only difference between both cases is the value of the densities of GND and SSD in the phases, since we have the same volume fraction, keeping the same grain size and the same evolution law of $\lambda$. Fig. 14 shows that the macroscopic response is not the same in both cases. It is observed that a higher hardening rate is obtained by considering only the first term $G_{loc}$ in comparison with the experimental response. This represents the application of the non-local law developed by Pipard et al. (2009) but with an evolution law of $\lambda$. The introduction of the second $G_{conv}$ term into the GND density evolution law leads to a more pronounced saturation of the GND

<figure><img src="image_18.png" /><figcaption>Fig. 12. (a) Simplified geometric description of the experimentally obtained GND density profile showing linear variation of GND density over a distance $\lambda_S$, (b) GND density profile in the model showing constant GND density over a distance $\lambda_M = \lambda_S/2$ so that the hatched area in the layer is the same as in the experimental case.</figcaption></figure>
<figure><img src="image_19.png" /><figcaption>**Fig. 13.** Evolution of $\lambda_M$ during plastic deformation obtained from the cumulative measurement method, and the best fit obtained using an exponential law (Eq. (30)) of the evolution law of $\lambda_M$ using the profiles of the mean of the GND densities with the cumulative measurement method. The error bars were determined from the errors given by the linear fit on the gradients of the GND density at each strain level.</figcaption></figure>

density and hence of the equivalent macroscopic stress (red curve), which corresponds much better to the observed experimental behavior (dashed curve).

### 5.3. Phase responses

The aim of this section is to explain the hardening effect that occurs in the macroscopic behavior of case 1 (blue curve in Fig. 14) in terms of the evolution of local stresses. First, as shown in Fig. 15 (a) and (b), the local stresses of case 1 in the GB (layer) region are larger than those of case 2, while the stresses in the grain core are the same in both cases. For case 1, local stresses of up to 650 MPa are predicted in the GB region. The local hardening in case 1 is the only factor responsible for the hardening at the macroscopic stress level, since the volume fraction of the layer is the same in both cases.

### 5.4. Dislocation densities

The evolution of dislocation densities is also studied in order to understand local hardening in the layer and global hardening in case 1. Fig. 16 shows that the GND and SSD densities obtained in case 1 are larger than those obtained in case 2. We note that at the beginning of plastic deformation, when $\lambda$ is small, the GND density increases abruptly. As $\lambda$ evolves and becomes larger during plastic deformation, the second term $G_{conv}$ in the GND density evolution law slows the increase in GND density. Thus, this term counteracts the evolution of GND densities in the layer. Initially, a high density of GND is stored rapidly near the GB, but as the layer thickens, the $G_{conv}$ term will slow down this evolution according to Eq. (29).

### 5.5. Discussion: experiment/ILMF model dialogue

Fig. 17 shows the evolution of GND densities in the layer obtained experimentally and by the model. Experimental value of the GND density in the layer from the EBSD measurements is taken at $d_{cp} = 0.6$ µm (Fig. 17) which corresponds to the maximum value close to GB shown in Fig. 6. First, it should be emphasized that there is a good agreement, in terms of magnitude and trend, between the GND densities in the layer predicted by the ILMF model (red circles) and those measured experimentally (black disks). However, Fig. 17 shows also that the GND density in the layer computed by the model is higher than that determined experimentally. This difference can be explained by the fact that the EBSD measurements yield a 2D GND density, which is a lower bound for $\rho_{GND}$, since only 5 out of 9 components of the Nye tensor (Pantleon, 2008) are considered. Moreover, the observed evolution of the GND density between 3% and 5% deformation is very weak. At this deformation level, only 44% and 47% of the pixels, respectively, are considered in the statistical analysis by EBSD according to the criteria $\rho_{GND}^{Nye} > \rho_{GND}^{noise}$, suggesting some uncertainty in the obtained mean values. The third reason for these differences is that, in our model, we considered that all GND densities are localized in the layer thickness $\lambda$. This is not the case
<figure><img src="image_20.png" /><figcaption>Fig. 14. Simulated stress/strain curves versus experimental curve (dotted line): for D=26 µm with λ⁰=0.5 µm for ρ̇(1)GND ≈ Gᵢₐₗ (blue, case 1) and ρ̇(1)GND ≈ Gᵢₐₗ + Gᵣₐₗ (red, case 2) with D=26 µm and λ⁰=0.5 µm.</figcaption></figure>

<table><caption>**Table 4**

Set of parameters identified in the ILMF model for the studied Al-k steel.</caption><thead><tr><th>μ(MPa)</th><th>n</th><th>M</th><th>b(m)</th><th>α<sub>d</sub></th><th>ρ<sup>0</sup><sub>SSD</sub>(m<sup>-2</sup>)</th><th>ρ<sup>0</sup><sub>GND</sub>(m<sup>-2</sup>)</th><th>k</th><th>f</th><th>λ(μm)</th></tr></thead><tbody><tr><td>80000</td><td>10</td><td>3</td><td>2.5×10<sup>-10</sup></td><td>0.4</td><td>10<sup>10</sup></td><td>10<sup>10</sup></td><td>0.022</td><td>8</td><td>f(ε<sub>p</sub>)</td></tr></tbody></table>

<figure><img src="image_21.png" /></figure>

<figure><img src="image_22.png" /><figcaption>Fig. 15. (a) Local equivalent stress/macroscopic equivalent strain curves, (b) Local equivalent stresses (squares and circles) as a function of viscoplastic equivalent strain in each phase and for ρ̇(1)GND ≈ Gᵢₐₗ (case 1) and ρ̇(1)GND ≈ Gᵢₐₗ + Gᵣₐₗ (case 2).</figcaption></figure>
<figure><img src="image_23.png" /><figcaption>**Fig. 16.** Evolution of GND (squares) and SSD (triangles) densities in the layer (GB region) and SSD density (circles) in the grain core as a function of the equivalent viscoplastic strain of each phase for $\dot{\rho}_{GND}^{(1)} \approx G_{loc}$ (case 1) and $\dot{\rho}_{GND}^{(1)} \approx G_{loc} + G_{conv}$ (case 2).</figcaption></figure>

<figure><img src="image_24.png" /><figcaption>**Fig. 17.** Evolution of the GND densities in the layer obtained experimentally (black disks) and by the model (red circles).</figcaption></figure>

experimentally, as the GND densities are distributed throughout the grain, but still with higher average values in the GB region than in the grain core.

## 6. Conclusions

In this work, an experimental approach was first proposed based on a combined EBSD/nanoindentation study of the effects of microstructural internal lengths on plastic heterogeneities in a single-phase Al-k steel with a mean grain size of 26 µm. In the experimental study, we investigated the effect of grain boundary (GB) distance on geometrically necessary dislocation densities (GND) and total dislocation densities. GND densities were estimated by the Nye tensor from 2D EBSD following the method of Pantleon (2008). The total dislocation densities were estimated by a mechanistic model from nano-hardness measurements. To determine the existence of an evolutive plasticity-induced internal length $\lambda$ corresponding to the width of the zone affected by the GB (spatial plastic strain gradients), a statistical analysis of the experimental data was proposed as a new method. The results obtained by this method confirmed the statistical existence of a gradient of dislocation densities at GBs. Moreover, experimental variation of this internal length $\lambda$ with the plastic strain was obtained, and an evolution law for $\lambda$ was established. Consequently, the ferritic grains in Al-k steel could be divided into two mechanically different regions. The first region is located near the GBs, where a dislocation density gradient was clearly observed to spread over $\lambda$. The second region is the interior of the grain, where no spatial variation in average dislocation density was found. Such two-phase composite microstructure was then used to propose a new Internal Length Mean Field (ILMF) model, which accounts for the evolution of the GND density depending on a convective term related to the experimentally determined evolution law for $\lambda$. This new approach was able to more realistically describe the macroscopic and phase responses in terms of stress, GND and total dislocation density evolutions. The obtained prediction of the macroscopic behavior is very satisfactory. Furthermore, a good estimate of the evolution of dislocation densities compared to the experimental measurements and a realistic evolution of the local fields in each phase were obtained.

As a possible perspective to this work, tension-compression responses will be studied with the present ILMF model to study the relationship between the plasticity-induced internal length $\lambda$ and the Bauschinger effect. Furthermore, the present statistical analysis could be extended to deformed quenching and partitioning steels (Arlazarov et al., 2015) to tackle plasticity-induced internal length
scale effects in these multiphase microstructures.

## CRediT authorship contribution statement

**Layal Chamma:** Writing – original draft, Visualization, Validation, Software, Methodology, Investigation, Formal analysis, Data curation, Conceptualization.
**Jean-Marc Pipard:** Writing – review & editing, Writing – original draft, Visualization, Validation, Supervision, Resources, Methodology, Formal analysis, Conceptualization.
**Artem Arlazarov:** Writing – review & editing, Writing – original draft, Visualization, Validation, Supervision, Resources, Methodology, Formal analysis, Conceptualization.
**Thiebaud Richeton:** Writing – review & editing, Writing – original draft, Visualization, Validation, Supervision, Software, Resources, Methodology, Formal analysis, Data curation, Conceptualization.
**Stéphane Berbenni:** Writing – review & editing, Validation, Supervision, Software, Methodology, Resources, Conceptualization, Funding acquisition, Project administration.

## Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

## Acknowledgments

The authors are thankful to ArcelorMittal Global Research and Development for financial support. This work was supported by the ANRT (Association Nationale de la Recherche et de la Technologie) with a CIFRE fellowship granted to Layal Chamma. The authors also acknowledge the experimental facilities MicroMat from LEM3 (Université de Lorraine - CNRS UMR 7239), as well as the use of the Cassiopee Arts et Métiers Institute of Technology HPC Center made available for conducting the present research.

## Data availability

Data will be made available on request.

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