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# Tensile properties and work hardening in Al0.3CoCrFeNi: the role of L12 precipitates and grain size

Stéphane Gorsse, Florian Peyrouzet, Thierry Baffie, Christelle Navone, Julie Maisonneuve, François Saint-Antonin, Marion Descoins, Khalid Hoummada, Rajarshi Barnerjee, An-Chou Yeh, et al.

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## To cite this version:

Stéphane Gorsse, Florian Peyrouzet, Thierry Baffie, Christelle Navone, Julie Maisonneuve, et al. "Tensile properties and work hardening in Al0.3CoCrFeNi: the role of L12 precipitates and grain size." *Materialia*, 2024, 38, pp.102250. 10.1016/j.mtla.2024.102250. hal-04730451

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https://hal.science/hal-04730451v1

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# Tensile properties and work hardening in Al0.3CoCrFeNi: The role of L12 precipitates and grain size

Stéphane Gorsse<sup>a,*</sup>, Florian Peyrouzet<sup>a,b</sup>, Thierry Baffie<sup>b</sup>, Christelle Navone<sup>b</sup>, Julie Maisonneuve<sup>b</sup>, François Saint-Antonin<sup>b</sup>, Marion Descoins<sup>c</sup>, Khalid Hoummada<sup>c</sup>, Rajarshi Barnerjee<sup>d</sup>, An-Chou Yeh<sup>e,f</sup>, Mohamed Gouné<sup>a</sup>

## ARTICLE INFO

### Keywords:

- High entropy alloys

- Spark plasma sintering

- Tensile properties

- Work hardening

- Microstructure

- Modeling

## ABSTRACT

In this study, we investigate the FCC-Al0.3CoCrFeNi high entropy alloy fabricated via spark plasma sintering of atomized powders, focusing on its mechanical and work-hardening properties across three distinct microstructures: coarse-grained, fine-grained, and fine-grained with L12 nano-precipitates. Using a dislocation density-based model, we analyze the effects of grain size and L12 precipitates on these properties, achieving quantitative agreement between model predictions and experimental tensile and work-hardening behaviors. This exploration highlights the underlying deformation mechanisms at room temperature and their contributions to the strength/ductility trade-off. Significantly, our analysis reveals that twinning in HEAs manifests differently from that observed in steels. Furthermore, the incorporation of L12 precipitates emerges as a critical factor enhancing the alloy's mechanical attributes. Our findings underscore the essential roles of microstructural parameters in tailoring the mechanical properties of HEAs, offering insights that could guide the design of advanced alloys with optimized performance.

## 1. Introduction

The classic strategy for designing new alloys is to consider one base element and modulate its properties by adding a small amount of minor alloying elements. In 2004, the works of Cantor et al. [1] and Yeh et al. [2] defined a breakthrough strategy for alloy design by introducing the concept of high entropy alloys (HEAs). Their unconventional compositions characterize them since they contain multiple principal alloying elements with equal or near equiatomic ratios. The founding idea behind the development of such alloys is that they have high configurational entropies of mixing, which give the random solid solution phase high stability against the formation of potential embrittlement intermetallic phases. This led to a booming interest in HEAs [3-5]. Since then, most studies on HEAs have been mainly extended from two iconic alloys having an equimolar composition to favor the formation of a face-centered cubic (FCC) or body-centered cubic (BCC) single-phase

structure, i.e., FCC-CoCrFeMnNi known as the Cantor alloy [1], and BCC-HfNbTaTiZr from Senkov [6]. Single-phase FCC-HEAs can exhibit remarkable mechanical behavior, e.g., their yield stress, ultimate-tensile strength, ductility, and toughness are significantly improved with decreasing temperature and can reach exceptionally high values at cryogenic temperatures [3,4,7,8]. For example, in the case of equiatomic FCC-CoCrFeMnNi high-entropy alloy, strength levels above 1 GPa with ductility higher than 70 % have been reported [9]. Therefore, this HEA can be considered for cryogenic applications [10]. According to the literature, this mechanical behavior results from the high work-hardening rate induced by the so-called TWinning Induced Plasticity (TWIP) effect which increases work hardening and prevent the early onset of necking instability [9,11]. This phenomenon, prominent in TWIP steels, is rationalized as a dynamic Hall-Petch effect [12,13] that is the decrease of the mean free path of dislocations by the formation of nano-twins during deformation.

<footnote><sup>a</sup> Univ. Bordeaux, CNRS, Bordeaux INP, ICMCB, UMR 5026, F-33600 Pessac, France</footnote>

<footnote><sup>b</sup> Univ. Grenoble Alpes, CEA, LITEN, 38000 Grenoble, France</footnote>

<footnote><sup>c</sup> Aix-Marseille University, Toulon University, CNRS, IM2NP UMR 7334, 13397 Marseille, France</footnote>

<footnote><sup>d</sup> Department of Materials Science and Engineering, University of North Texas, Denton, TX 76207, USA</footnote>

<footnote><sup>e</sup> High Entropy Materials Center, National Tsing Hua University, Hsinchu 30013, Taiwan ROC</footnote>

<footnote><sup>f</sup> Department of Materials Science and Engineering, National Tsing Hua University, Hsinchu, 30013, Taiwan ROC</footnote>

<footnote>* Corresponding author.</footnote>

<footnote>E-mail address: stephane.gorsse@icmb.cnrs.fr (S. Gorsse).</footnote>
TWIP effect, prominent in austenitic steels with high content of Mn (so-called TWIP steels), has limited contribution at room temperature in most of FCC HEAs since plastic deformation is dominated by dislocation glide [9]. Furthermore, it is shown that when the twinning deformation is activated at room temperature, it occurs at higher deformations and therefore contributes less to the work hardening [14]. Therefore, the improvement of the mechanical properties of single-phase HEAs at room temperature appears limited because the primary activated deformation mechanism results from the interaction between dislocations and the solid solution. This is a strong incentive for a shift in focus from single-phase to multi-phase HEAs, also referred as complex concentrated alloys (CCAs) or multi-principal elements alloys (MPEAs) [15-17]. CCAs or MPEAs often contain an unequal proportion of principal elements and several phases, allowing the development of complex microstructures to benefit, for example, independently or simultaneously from the composite effect, the twinning/transformation-induced plasticity, and the precipitation hardening.

In this context, we have selected the FCC-Al<sub>0.3</sub>CoCrFeNi HEA, which offers a remarkably wide variety of microstructures and mechanical properties at room temperature depending on its thermo-mechanical history [18-20]. Furthermore, it possesses an FCC structure and is subject to a TWIP effect that can lead to a very good strength/ductility balance. Appropriate heat treatment can also control the formation of L1<sub>2</sub> precipitates in the FCC matrix. Under certain conditions, these precipitates can have a beneficial effect on both strength and ductility (refer to this excellent review [21], which shows that kinematic hardening is a powerful lever for achieving unusual mechanical properties in HEAs).

For example, the single-phase FCC microstructure HEA exhibits an ultimate tensile strength (UTS) of 400 MPa and a ductility of 65 %, while the formation of L1<sub>2</sub> precipitates leads to a UTS of 850 MPa and a ductility of 45 %. When the microstructure consists of FCC solid solution with B2 and σ precipitates, the strength and the ductility reach 1050 MPa and 35 %, respectively [19]. Published works evidence that the presence of secondary-phases (L1<sub>2</sub>, B2 or σ) in Al<sub>0.3</sub>CoCrFeNi also affects its work hardening. These results raise several questions: (i) what are the microstructure parameters controlling the plastic deformation mechanisms, (ii) what contributions to work hardening explain such balance between strength and ductility, (iii) are there still levers to improve the tensile behavior of FCC-Al<sub>0.3</sub>CoCrFeNi at room temperature?

The questions raised, which relate to the deformation mechanisms, are particularly relevant given recent findings that HEA alloys, like steels, can exhibit kinematic hardening which significantly affects strain hardening and the resulting mechanical properties. The demonstration of kinematic hardening in single HEA alloys is a very recent development [22] and could explain the extraordinary strength/ductility synergy observed [23]. Consequently, we believe it is important to gain a better understanding of both the contributions to plastic hardening and kinematic hardening, i.e., the back stress induced by grain boundaries, twin boundaries, and L1<sub>2</sub> precipitates in HEAs.

To provide some answers, we report an effort to decoupling the grain size effects of the FCC matrix from those linked to the presence of L1<sub>2</sub> precipitates and analyzing the mechanical and work-hardening properties using a dislocation density-based model. To this end, Al<sub>0.3</sub>CoCrFeNi HEA was fabricated by spark plasma sintering densification of atomized powders. Specific heat treatments were used to generate three different microstructures: a fully coarse-grained FCC structure, a fully fine-grained FCC structure, and a fine-grained FCC structure that contains L1<sub>2</sub> precipitates.

## 2. Materials and experimental methods

Al<sub>0.3</sub>CoCrFeNi prealloyed HEA powder supplied by DUCAL INTERNATIONAL was fabricated by gas atomization under nitrogen. The composition of the powder measured by inductively coupled plasma (ICP) was 6.0Al-23.9Co-23.4Cr-21.2Fe-23.6Ni (at.%). The oxygen and

<table><caption>**Table 1**
Experimentally measured microstructure parameters.</caption><thead><tr><th></th><th></th><th>asSPS1100</th><th>asSPS1200</th><th>Aged620</th></tr></thead><tbody><tr><td>D (μm)</td><td>Mean FCC grain diameter</td><td>9</td><td>18</td><td>10</td></tr><tr><td>f<sub>AlN</sub> (%)</td><td>AlN volume fraction</td><td>1</td><td></td><td></td></tr><tr><td>d<sub>AlN</sub> (nm)</td><td>Mean AlN particle diameter</td><td>120</td><td>220</td><td>120</td></tr><tr><td>ρ<sub>0</sub> (m<sup>-2</sup>)</td><td>Initials dislocation density</td><td>10<sup>13</sup></td><td></td><td></td></tr><tr><td>f<sub>L12</sub> (%)</td><td>L12 precipitate volume fraction</td><td></td><td></td><td>5</td></tr><tr><td>d<sub>L12</sub> (nm)</td><td>Mean L12 precipitate diameter</td><td></td><td></td><td>16</td></tr></tbody></table>

nitrogen levels are 0.10 and 0.43 at. %, respectively. The particle size distribution is characterized by D10, D50, and D90 values of 20, 29, and 42 μm, respectively.

The specimens were prepared by sintering the prealloyed powder in a graphite die (Ø 60 mm) between two graphite punches using spark plasma sintering (SPS) machine (FCT HPD25). Optimized densification parameters were determined by recording the densification rate as a function of temperature at a constant heating rate of 50 °C/min. The temperature was controlled using a pyrometer. Sintering was carried out under argon atmosphere, at a pressure of 30 MPa and temperatures of 1100 °C and 1200 °C for 10 mins to produce the fine-grain and coarse-grain structure, respectively. The densification of the specimens thus obtained was checked by Archimedes method and x-ray tomography to be higher than 99 %. Based on previously published investigations [18, 19], the alloy microstructure with L1<sub>2</sub> precipitates was produced by aging the fine-grain structure at 620 °C for 50 h under an argon atmosphere.

The phase structure of samples was characterized by X-ray diffraction (XRD). Specimens were mirror-polished using 0.03-μm colloidal silica. Their microstructures were analyzed by scanning electron microscope (SEM) using a MERLIN-ZEISS equipped with an EDAX X-ray Energy dispersive microscopy. Electron backscattered diffraction (EBSD) maps were carried out using Brüker instrument and the Esprit 2.2 software. The grain size was determined using equivalent circle diameter with area ponderation after image post-treatment and thresholding by M-TEX & ImageJ software. The average grain size measurements were performed with a grain misorientation tolerance angle of 15° and a step size of 0.8 μm.

The precipitates were characterized by transmission electron microscopy (TEM) and atom probe tomography (APT). TEM analysis was carried out using a TECNAI OSIRIS (Thermo Fisher Scientific, Eindhoven, Netherlands) microscope operated at 200 kV, equipped with a 4 K GATAN camera. The specimens for TEM were ground to a thickness of 400 mm from a 3 mm diameter disk. The thin foils were subsequently electropolished and then thinned with the ion beam technique.

APT specimens were prepared by focused ion beam (FIB) on a FEI Helios dual-beam via the lift-out technique. The micro-tips were prepared by means of the annular milling method to obtain an end-radius of ~100 nm. APT analyses were performed with an Imago Scientific Instruments LEAP 3000X HR in voltage mode. Specimen temperature during analysis was 80 K, the pulse repetition rate 200 kHz, the rate of evaporation 0.5 %, the pulse fraction 20 %, and a pressure of <3 × 10<sup>-9</sup> Pa.

Room-temperature tensile tests were performed at a strain rate of 5 × 10<sup>-4</sup> s<sup>-1</sup>, on dog-bone specimens with a gauge wide of 1.5 mm, 10 mm in length, and 1.5 mm in thickness. Three dog-bone specimens were extracted by electro-erosion cutting from the sintered compacts for each processing condition.

## 3. Experimental results and discussion

### 3.1. Microstructures

Key microstructural parameters of the Al<sub>0.3</sub>CoCrFeNi HEA prepared under different conditions are summarized in Table 1. After
<figure><img src="image_5.png" /><figcaption>**Fig. 1.** X-ray diffraction patterns of the Al₀.₃CoCrFeNi HEA for three different conditions: asSPS1100, asSPS1200, and Aged620.</figcaption></figure>

sintering by SPS, the samples are fully dense and their composition is 5.8Al-23.4Co-23.4Cr-21.4Fe-24.0Ni (at.%) with O, N and C levels of 0.14, 0.43 and 0.14 at.%. The XRD analysis (Fig. 1) evidences that the two Al₀.₃CoCrFeNi as-sintered alloys (asSPS1100 and asSPS1200) and the aged alloy (Aged620) exhibit an FCC lattice structure. EBSD inverse pole figure maps (Fig. 2) reveal the equiaxed grain structure of the three samples. The grain size distribution was measured and an average grain size of 9 ± 1, 18 ± 2, and 10 ± 1 µm for asSPS1100, asSPS1200, and Aged620 were obtained, respectively.

Fig. 3 shows the microstructure of the asSPS1200 sample at different scales. The EBSD phase map (Fig. 3a) at low magnification confirms the FCC microstructure. Backscatter SEM micrograph (Fig. 3b) and SEM-EDX elemental maps (Fig. 3d) show a low volume fraction (about 1 %) of coarse AlN particles, with an average size ranging from 120 to 220 nm, distributed in the FCC matrix. The formation of AlN particles, also observed in the prealloyed powder, results from the atomization process which was carried out under nitrogen. The 3D APT reconstruction

(Fig. 3c) does not reveal the presence of secondary phase precipitates.

The dark field TEM micrograph shown in Fig. 4a reveals a distribution of fine precipitates with a size of about 16 nm in the interior of the grain in the Aged620 sample. They show a “coffee-bean” contrast characteristic of misfit strain indicating that some degree of coherency with the surrounded FCC matrix is retained. The phase fraction of L1₂ precipitates was estimated to be ~5 % using image analysis after conversion of the area fraction. The insert of Fig. 4a shows the presence of ordered L1₂ superlattice reflections on the diffraction pattern recorded from a [001] FCC zone axis from one of the coarse grains. The formation of the (Ni,Al)-rich precipitates in the Al₀.₃CoCrFeNi alloy after aging at 620 °C (Aged620 sample) is confirmed by APT analysis (Fig. 4b). The Ni 45 at.% isoconcentration surface shows the partitioning of Al and Ni to the precipitate phase (Table 2), which is generally associated with L1₂ phase, as evidenced by multiple reports on FCC HEAs containing Al [24-33].

## 3.2. Tensile properties

Fig. 5(a and b) represent the nominal (engineering) stress as a function of the nominal strain and the true stress as a function of the true strain, respectively. The main mechanical properties in tension are summarized in Table 3. The Ultimate Tensile Strength (UTS) and the total elongation at fracture ($ε_t$) are parameters measured from engineering stress-strain curves. The maximum true stress ($σ_m$), indicative of the material's true mechanical strength, and the uniform elongation ($ε_u$), representing the maximum true strain before the onset of necking, were determined based on Considère's criterion [34]. The identification of the true yield stress is made difficult by the gradual transition from the elastic to plastic regime. Consequently, the yield stress at a strain of 0.2 %, denoted as $σ_{0.2}$ was considered. It is worth noting that $σ_{0.2}$ does not precisely represent the yield stress, as it includes additional stress contributions from the material's plastic deformation.

The mechanical properties obtained are of significant interest. Indeed, in a yield stress (YS) versus uniform deformation ($ε_u$) diagram, it

<figure><img src="image_6.png" /><figcaption>**Fig. 2.** EBSD inverse pole figure map and average gain size of the Al₀.₃CoCrFeNi HEA for three different conditions: (a) asSPS1100, (b) asSPS1200, and (c) Aged620. The measured grain size distributions are given below each EBSD image.</figcaption></figure>
<figure><img src="image_7.png" /><figcaption>**Fig. 3.** Microstructure of the Al<sub>0.3</sub>CoCrFeNi in the as-SPS condition. (a) EBSD phase map (using green color for FCC phase), (b) SEM-BSE micrograph showing the presence of dark AlN particles in the FCC matrix, (c) APT 3D reconstruction ion map for Al, Co, Cr, Fe, and Ni in the FCC solid solution, (d) SEM-EDX elemental maps of the FCC matrix and the AlN particles.</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>**Fig. 4.** Microstructure of the Al<sub>0.3</sub>CoCrFeNi HEA after aging at 620 °C for 50 h (Aged620 sample). (a) Dark field TEM image and diffraction pattern recorded from a [001] FCC zone axis showing FCC matrix and coherent L1<sub>2</sub> ordered precipitates surrounded by strain fields. (b) APT reconstruction with Ni ions from FCC matrix and Ni 45 at.% isoconcentration surface showing L1<sub>2</sub> precipitates.</figcaption></figure>

is demonstrated that the YS/ε<sub>u</sub> trade-offs achieved are positioned above the typical "banana curve," thus marking a breakthrough compared to most conventional metals, and even surpassing most Heterogeneous Nanostructured Metals (HNMs). Focusing specifically on the properties of HEAs, it is observed that the YS/ε<sub>u</sub> trade-off for the AsSPS1200 alloy is particularly noteworthy as it is located at the edge of the band that defines high-performance materials. These comparisons were made using data from a recent study (refer to Figs. 1 and 3 from ref. [35]).

The product of σ<sub>m</sub> and ε<sub>u</sub> has also been considered in our analysis, as

it reflects the total energy that the material can absorb before necking occurs. This product is a significant indicator of the material's strain hardening capability and its potential for fatigue resistance. The data in Fig. 5b were translated into the common Kocks-Mecking plot (Fig. 5c), which gives the evolution of the strain hardening rate θ = dσ/dε as a function of the true strain.

In Fig. 5b, it's clear that the tensile curve for the asSPS1100 condition does not simply translate from the asSPS1200 curve, indicating that grain size affects not only the yield strength but also the work hardening
<table><caption>**Table 2**
Compositions of the FCC matrix and the L1₂ precipitates from APT.</caption><thead><tr><th rowspan="2">at.%</th><th rowspan="2">asSPS<br>FCC</th><th colspan="2">Aged620</th></tr><tr><th>FCC matrix</th><th>L1<sub>2</sub> precipitates</th></tr></thead><tbody><tr><td>Al</td><td>9</td><td>7</td><td>37</td></tr><tr><td>Co</td><td>23</td><td>27</td><td>5</td></tr><tr><td>Cr</td><td>21</td><td>25</td><td>&lt; 1</td></tr><tr><td>Fe</td><td>19</td><td>24</td><td>1</td></tr><tr><td>Ni</td><td>26</td><td>17</td><td>56</td></tr><tr><td>Impurities</td><td>2</td><td>&lt; 1</td><td>1</td></tr></tbody></table>

behavior. This is further evidenced in Fig. 5c, where the strain hardening rate versus true strain is plotted. Both asSPS1200 and asSPS1100 conditions exhibit similar patterns, yet display distinct characteristics. Commonly, three regimes are observed in the strain hardening rates: (1) an initial quasi-linear decrease at low strains, (2) a leveling off and slight increase post a critical stress and strain, and (3) a gradual decline at higher strains. Such behavior, previously noted in TWIP steels and HEAs across temperatures ranging from 77 K to 298 K, is typical of FCC alloys with low SFE, highlighting twinning activity [13]. Initially, the strain hardening rate drops, but beyond a critical deformation and stress level, it either stabilizes or increases due to the activation of primary twins, a phenomenon explained by the dynamic Hall-Petch effect. The subsequent decrease in strain hardening rate at higher strains is attributed to the increasing difficulty in forming deformation twins.

Differences in the strain-hardening rates of asSPS1100 and asSPS1200 become particularly evident in the second regime. Here, larger grain sizes are associated with a more pronounced leveling off of the strain hardening rate at medium strains, indicative of increased twinning activity. This observation suggests that larger grains facilitate more extensive twin formation, impacting the material's work hardening behavior. In the case of the coarse-grain alloy, twin activation is initiated at a critical strain of approximately 10 % (evidenced by the onset of the leveling off observed in Fig. 5c) and a corresponding stress of around 650 MPa (the stress level at this critical strain, as depicted in Fig. 5b). Conversely, for the fine-grain HEA, the twin activation occurs at a higher critical strain of about 20 % and a stress level of approximately 950 MPa. This suggests that twinning is less active in smaller grain sizes, corroborating several studies that observed a shift in twinning behavior in HEAs with grain refinement [36,37]. Post-deformation analysis of the samples, as shown in Fig. 6, confirms twin formation during tensile testing in even the finest grain size alloy.

The analysis of the data from Fig. 5 provides insights into the interaction between the L1₂ precipitates and twinning. Indeed, the effect of twinning on the work-hardening rate appears to be relatively slight (i. e., the leveling off and the increase in work-hardening rate). Consequently, the twinning deformation mode seems to be inhibited in the presence of L1₂ precipitates. There are at least two clear reasons for the impact of precipitates on twinning deformation. First, the range of deformation in which twins can form is significantly reduced in the presence of L1₂ precipitates ($\Delta\epsilon = \epsilon_u - \epsilon_c = 0.25 - 0.2 = 5\%$, versus 12 % and 28 % for the asSPS1100 and asSPS1200, respectively). Second, the onset strain value at which twinning occurs corresponds to a level of deformation at which dynamic recovery is highly effective, since L1₂ precipitates influence plastic strain hardening. However, it is interesting to note that the L1₂ precipitates do not affect the onset strain value at which twinning occurs. This is clearly observable when comparing the work-hardening rate evolution for samples asSPS1100 and Aged 620 (Fig. 5c). This would seem to indicate that L1₂ precipitates do not directly affect the nucleation process of twinning.

The data in Table 3 shows that the increase in $\sigma_{0.2}$ with the decrease of the grain size from 18 µm (asSPS1200) to 9 µm (asSPS1100) is $\Delta\sigma_{0.2} = 85$ MPa. The relation between $\sigma_{0.2}$ and the grain size D can be approximated by the Hall-Petch equation:

$$\sigma_{0.2} \approx \sigma_0 + \frac{k_{slip}^{HP}}{\sqrt{D}}$$

where $k_{slip}^{HP}$ is the Hall-Petch coefficient and $\sigma_0$ includes different contributions such as the lattice friction stress, solid solution strengthening, AlN particle hardening, and the strain hardening contribution of the initial dislocation density. By subtraction, we can overcome the difficulty of determining $\sigma_0$ and we obtain the following relation for $k_{hp}$:

<table><caption>**Table 3**
Experimental data on tensile mechanical properties of the three considered microstructures: coarse-grain FCC (asSPS1200), fine-grain FCC (asSPS1100), and fine-grain FCC with fine L1₂ precipitates (Aged620).</caption><thead><tr><th>Sample</th><th>σ<sub>0.2</sub><br>(MPa)</th><th>UTS<br>(MPa)</th><th>ε<sub>t</sub><br>(%)</th><th>σ<sub>m</sub><br>(MPa)</th><th>ε<sub>u</sub><br>(%)</th><th>σ<sub>m</sub> × ε<sub>u</sub><br>(MPa.%)</th></tr></thead><tbody><tr><td>asSPS1200</td><td>337</td><td>714</td><td>54</td><td>1045</td><td>38</td><td>39,606</td></tr><tr><td>asSPS1100</td><td>422</td><td>782</td><td>44</td><td>1072</td><td>32</td><td>33,951</td></tr><tr><td>Aged620</td><td>565</td><td>928</td><td>32</td><td>1197</td><td>25</td><td>29,925</td></tr></tbody></table>

<figure><img src="image_9.png" /><figcaption>**Fig. 5.** Room-temperature tensile behavior for three microstructure states of Al<sub>0.3</sub>CoCrFeNi HEA. (a) engineering stress - strain curves, (b) true stress - true strain curves, (c) strain-hardening rates. The green, blue and orange curves correspond to a coarse-grain FCC structure (asSPS1200), a fine-grain FCC structure (asSPS1100), and fine-grain solid solution with fine L1₂ precipitates (Aged620).</figcaption></figure>
<figure><img src="image_10.png" /><figcaption>Fig. 6. Post-mortem EBSD-IPF map of the asSPS1100 sample and misorientation profile along the white line.</figcaption></figure>

$$k_{\text{slip}}^{\text{HP}} = \frac{\Delta\sigma_{0.2}}{\left(\frac{1}{\sqrt{D_1}} - \frac{1}{\sqrt{D_2}}\right)}$$

Considering $\Delta\sigma_{0.2} = 85$ MPa, $D_1 = 9$ µm and $D_2 = 18$ µm, $k_{\text{slip}}^{\text{HP}}$ is estimated to 871 MPa·µm$^{1/2}$, which lies between the extremum values (674 and 1014 MPa·µm$^{1/2}$) reported in the literature for Al$_{0.3}$CoCrFeNi [38] and close to that by Gwalani (824 MPa·µm$^{1/2}$) [39]. Interestingly, this HEA exhibits a higher Hall-Petch coefficient than many FCC alloys, including medium/high entropy alloys and TWIP steels [40,41], for which the upper-bound value of $k_{\text{slip}}^{\text{HP}}$ is about 600 MPa·µm$^{1/2}$ [42,43]. As a result, the grain boundary hardening efficiency in the Al$_{0.3}$CoCrFeNi HEA is greater than most FCC alloys. The physical origin of such a high value of $k_{\text{slip}}^{\text{HP}}$ cannot be established with certainty. However, we can infer from the dislocation theory of pile-up that a higher value of $k_{\text{slip}}^{\text{HP}}$ corresponds to larger critical shear stress for dislocation motion and propagating plastic flow across the grain boundary [44].

Furthermore, various studies [41,45] have demonstrated that the Hall-Petch coefficient is directly and inversely correlated with the shear modulus and stacking fault energy ($\gamma_{SFE}$), respectively. This implies that the Al$_{0.3}$CoCrFeNi High Entropy Alloy (HEA) exhibits a high shear modulus alongside a low to medium $\gamma_{SFE}$ value. To calculate the stacking fault energy ($\gamma_{SFE}$) of Al$_{0.3}$CoCrFeNi, we utilized the established relationship between the shear modulus ($\mu$), $\gamma_{SFE}$, and the separation distance ($w$) of the $1/6 < 112>$ Shockley partials. This relationship is expressed by the formula $\gamma_{SFE} = \frac{\mu b^2}{24\pi w} \left(1 - \frac{2\nu}{2-\nu} \cos 2\phi\right)$, where $\phi$ is the angle between the dislocation line and $\langle 101 \rangle$. Our calculation of the shear modulus of Al$_{0.3}$CoCrFeNi resulted in an approximate value of 83 GPa, derived using the expression $\mu = E/(2(1+\nu))$. Here, $E$ represents the Young's modulus, which was determined to be 220 GPa from tensile testing, similar to the value obtained from nano-indentation test by Ma et al. [46], and $\nu$, the Poisson's ratio, was assumed to be 0.3. Furthermore, we adopted a separation distance ($w$) value of 3.8 nm, in accordance with the findings of Yasuda et al. [47] for the same alloy composition. Consequently, the estimated $\gamma_{SFE}$ for Al$_{0.3}$CoCrFeNi is approximately 54 mJ/m$^2$. This estimation aligns closely with Yasuda's findings for the same HEA (51 mJ/m$^2$) and is notably higher than that of Al$_{0.1}$CoCrFeNi (26–35 mJ/m$^2$), CoCrFeNi (17–32 mJ/m$^2$), and CoCr-FeMnNi (18–27 mJ/m$^2$), yet lower than Al$_{0.57}$CoCrFeNi (73 mJ/m$^2$). These comparisons suggest that Al content in Al$_{x}$CoCrFeNi HEAs may hinder twinning. However, it is important to note that $\gamma_{SFE}$ is not the sole parameter influencing twinning.

### 3.3. Microstructure-dependent variations in tensile properties

The analysis of the Al$_{0.3}$CoCrFeNi HEA's microstructure, mechanical properties, and strain hardening behavior highlights the significant role of grain size and precipitates in determining the material's mechanical behavior, particularly in terms of twinning activity and work hardening. The observed variance in microstructures, particularly the differences in grain size and the presence or absence of precipitates, enables a detailed analysis of how these factors individually and collectively influence the tensile properties and work hardening behavior of the Al$_{0.3}$CoCrFeNi HEA. This distinction is crucial for accurately modeling the plastic flow stress and work hardening, as it allows us to isolate and understand the specific contributions of grain size and the presence of precipitates to the overall mechanical behavior. The following sections delve into the modeling of these phenomena, providing a comprehensive framework that correlates microstructural characteristics with the material's mechanical responses.

## 4. Modelling of plastic flow stress and work hardening: a general framework

Work-hardening corresponds to the increase in flow stress required to deform a material plastically. It results from the ability of mobile dislocations to move in the material. It depends on the microstructure because it is linked to the density and strength of the obstacles constituted, for example, by atoms in solid solution, forest dislocations, grains, twins, and precipitates.

### 4.1. Work hardening and plastic flow

The non-coplanar dislocations piercing the slip plan, so-called forest dislocations, represent an obstacle to moving dislocations. The critical stress for mobile dislocations to cross the forest dislocations, referred to as forest hardening, is given by the Taylor relation [48]:

$$\sigma_f = \alpha M \mu b \sqrt{\rho} \quad (1)$$

where $\alpha$ is the Taylor constant, $M$ the Taylor factor, $\mu$ the shear modulus of the FCC matrix, $b$ is the magnitude of the Burger's vector and $\rho$ is the density of statistically stored dislocations (SSD).

In the framework of the Mecking-Kocks-Estrin theory, the evolution of the dislocation density per increment of plastic deformation is given by the following general relation [49,50]:
$$ \frac{d\rho}{d\epsilon} = M \left[ \frac{1}{b} \left( \sum_{i} \frac{1}{L_i} + k_1 \sqrt{\bar{\rho}} \right) - k_2 \rho \right] \quad (2) $$

where $\epsilon$ is the plastic strain, $L_i$ is the mean free path that a mobile dislocation can cover before encountering an obstacle $i$, $k_1$ and $k_2$ are efficiency factors for dislocation storage and dynamic recovery.

The first term in the brakets represents the storage of dislocations that causes hardening. $k_1 \sqrt{\bar{\rho}}$ is the inverse of the mean free path associated with the forest dislocations (assuming that they are homogeneously and randomly distributed), while $L_i$ is associated to other obstacles such as grain and twin boundaries. In that case, $L_1 = D$, where $D$ is the mean grain size, and $L_2 = \lambda_t$, where $\lambda_t$ is the inter-twins spacing. It is worth noting that incorporating the Hall-Petch effect directly into Eq.1 is unnecessary, as it is already indirectly included through the term $L_1 = D$ in Eq.2, according to O. Bouaziz et al. [54]. The second term, proportional to the dislocation density, corresponds to the dynamic recovery related to the dislocation annihilation process. It leads, by nature, to a softening. The efficiency factor $k_2$ reflects that only dislocations of opposite Burgers vectors can annihilate by a thermally activated cross slip process.

In integrating Eqs. (1) and Eq. (2), the plastic flow stress derivatived is path-independent, leading to what is termed the isotropic component of strain hardening. The genesis of plastic hardening, however, can be attributed to two distinct sources of dislocation storage. Firstly, as previously discussed, dislocations accumulated via the forest mechanism are categorized as statistically stored dislocations (SSDs). Secondly, dislocations that are necessary to accommodate deformation inconsistencies among different microstructural elements are termed geometrically necessary dislocations (GNDs) [51]. These two dislocation populations are further distinguished by the stress fields they induce. In contrast to SSDs, GNDs generate a long-range, polarized (directional) stress field, which contributes to kinematic strain hardening. For a comprehensive understanding and description of the plastic behavior of most metallic materials, both isotropic and kinematic components of work hardening are often essential to consider [40,52,53]. When incorporating all components, the plastic flow stress can be expressed in a simplified form as:

$$ \sigma = \sigma_0 + \sigma^{iso} + \sigma^{kin} $$

where:

* $\sigma_0$ corresponds in the theoretical approach to the stress at zero plastic strain and not strictly to the yield stress,

* $\sigma^{iso}$ is the isotropic component of strain hardening. It can be written in a general way [54]:

$$ \sigma^{iso} = \left( (\sigma_f)^u + \left( \sum_i (\sigma_i^{iso})^u \right)^{\frac{1}{u}} \right) \quad (3) $$

where $\sigma_i^{iso}$ is the isotropic hardening arising from the obstacle $i$, $u$ is the stress additivity law between $\sigma_f$ and $\sigma_i^{iso}$, whose value depends on the density and strength of obstacles to dislocation motion,

* $\sigma^{kin}$ is the kinematic component of the work hardening which takes the following form:

$$ \sigma^{kin} = \sum_{i} \sigma_{i}^{back} $$

where $\sigma_i^{back}$ is the backstress generated by the obstacles $i$, inducing the kinematic hardening.

The plastic flow stress is then described from the following system of equations:

$$ \left\{ \begin{aligned} \sigma &= \sigma_0 + \left( (\alpha M \mu b \sqrt{\bar{\rho}})^u + \left( \sum_i (\sigma_i^{iso})^u \right)^{\frac{1}{u}} \right) + \sum_i \sigma_i^{back} \\ \frac{d\rho}{d\epsilon} &= M \left( \frac{1}{b} \sum_i \frac{1}{L_i} + \frac{k_1}{b} \sqrt{\bar{\rho}} - k_2 \rho \right) \end{aligned} \right. \quad (4) $$

## 4.2. Identification of the different contributions to work hardening

In addition to the forest dislocations already discussed, we have assembled in our model the description of other obstacles to dislocations motion that also captures the physically important contributions to work hardening, i.e. grain and twin boundaries, and precipitates.

### Stress additivity law and $\sigma_0$

The stress additivity law for $\sigma_f$ and $\sigma_i^{iso}$ (Eqs.3 & 4) depends on the density and strength of obstacles to dislocation motion. The value of $u$ in Eq.4 equals 1 if the scale of the obstacles is of the same order of magnitude [54]. This is the case, for example, for solid solution versus precipitation, or precipitation versus grain boundaries. In contrast, this is generally not the case between forest dislocations and precipitates because, even though these obstacles can be of the same strength, they have different densities. Then, $u$ takes a value of 2, and the additivity law of stresses for $\sigma_f$ and $\sigma_i^{iso}$ corresponds to the so-called Pythagorean addition rule [54,55].

The $\sigma_0$ stress accounts for several contributions (Peirls-Nabarro stress, solid solution, and AlN particle strengthening). Since it only shifts the tensile curve upward, $\sigma_0$ does not affect the work hardening.

### L12 precipitates

The strength enhancement due to L12 nanoprecipitates coherent with FCC matrix corresponds to $\sigma_p^{iso}$. In this case, the shear mechanism predominates, and the order strengthening related contribution is to be considered using the following expression [56] for coherent precipitates:

$$ \sigma_p^{iso} = 0.81 M \frac{\gamma_{apb}}{2b} \sqrt{\frac{3\pi f_p}{8}} \quad (5) $$

where $\gamma_{apb}$ is the antiphase boundary free energy of the L12 precipitate in FCC matrix, and $f_p$ is the volume fraction of L12 precipitates.

The kinematic component resulting from the formation of dislocation loops around the particles is written as [53]:

$$ \sigma_p^{back} = \frac{f_p E_p b}{M d_p} \varphi n_p \quad (6) $$

where $n_p$ represents the number of Orowan loops around the L12 precipitate, $\varphi$ is an efficiency factor related to the stability of the dislocation loops, $E_p$ and $d_p$ are the elastic modulus and the diameter of L12 precipitates. The stress relaxation phenomena around the precipitates is taken into account using the following equation:

$$ n_p = n_p^* \left( 1 - \exp \left( - \frac{M d_p}{b n_p^*} \epsilon \right) \right) \quad (7) $$

where $n_p^*$ depends on the stress relaxation mechanism and corresponds to the saturation value of the number of Orowan loops around the precipitates.

In this case, the mean free path related to the precipitates involves an effective inter-particle distance which can be written [57]:
$$L_p = \frac{d_p}{6f_p} \left(1 - \frac{n_p^*}{n_p}\right) \qquad (8)$$

It appears that $L_p$ is strain dependant since Orowan storage is expected to diminish with increasing deformation. It is also necessary to take into account the $\varphi$ efficiency factor in the evolution of the dislocation density per increment of plastic deformation ($1/bL_p$ becoming $\varphi/bL_p$ in Eq. 4). The parameter $\varphi$ expresses the character of the precipitates. For example, $\varphi = 1$ in the case of purely incoherent precipitates for which Orowan's mechanism operates, and $\varphi = 0$ in the case of purely coherent and shearable precipitates. The latter process can indeed be seen as a self-annihilation of Orowan loops, and in this case, the precipitates do not induce back stress in return.

## Grain and twin boundaries

The grain boundary strengthening (the so-called Hall-Petch effect) is implicitly contained in the isotropic component as the mean dislocation free path term equals the grain size $D$ [58]. As for twins, it is considered that they appear in grains in the form of fine platelets structure in a complex manner, and their volume fraction is expected to increase with deformation. The geometrical length scale $L_t$ is expected to decrease during deformation as new twins form [13,59]. Of course, in the absence of twins, $L_t$ is constant and equal to the mean grain size $D$. To account for this effect, $L_t$ is expressed as a function of strain using the following expression [60]:

$$\frac{1}{L_t} = \frac{1}{D} + \frac{1}{T} \qquad (9)$$

where $F_0$ is the maximum fraction of twins, $\varepsilon_{ini}$ is the onset strain value at which twinning occurs, $\beta$ and $m$ are calibration parameters to be evaluated.

The kinematic component, which results from the dislocations stacking at boundaries, is written in its simplest form [50,59]:

$$\sigma_t^{\text{back}} = \frac{M\mu b}{L_t} n_t \qquad (12)$$

where $n_t$ is the number of dislocations stopped at boundaries along a given glide plane.

It increases with deformation until saturation at a critical value for which cross-slip is favored and long-range internal stresses disappear. This phenomenon occurs, for example, when dislocations of opposite signs are stored at grain boundaries. To capture the strain dependence of $n_t$, we used the following equation [52]:

$$n_t = n_t^* \left( 1 - \exp\left(\frac{-\lambda}{bn_t^*} \varepsilon\right) \right) \qquad (13)$$

where $\lambda$ is the mean spacing between slip lines at the boundaries. The ratio $\lambda/b$ is the number of dislocations per slip band geometrically necessary to provide the deformation.

The process described above is not completely efficient because only a portion of the dislocation line is likely to reach the boundary. A corrective term ($1 - n_t/n_t^*$), related to the number of sites available for dislocations at the boundaries, must therefore be included in the storage term $1/L_t$ [52].

In summary, the summation of all the contributions to the plastic flow stress can be written with the following system of equations:

<figure><img src="image_11.png" /></figure>

where $D$ is the mean grain size and $T$ is the mean twin spacing.

According to the stereological analysis of Fullman [61], $T$ is given by the following relation:

$$\frac{1}{T} = \frac{1}{2t} \frac{F}{1 - F} \qquad (10)$$

where $t$ is the twins' width, supposed to be constant, and $F$ is the twin volume fraction. In the absence of a physical based model, $F$ is generally described by an empirical law of the form [59]:

$$F = F_0 [1 - \exp(-\beta(\varepsilon - \varepsilon_{\text{ini}}))]^m \qquad (11)$$

where $\rho$, $L_t$, $n_t$ and $n_p$ change during deformation as depicted by equations.

In Eq. 14, the additivity law was applied to both the isotropic and kinematic components. This approach is relatively classic and provides a good description of the Bauschinger tests [61]. However, within the isotropic component, we did not apply an additive law between the contributions of the forest dislocations and those of the $L_{12}$ precipitates. The method by which these different contributions are combined depends on their scale and amplitude (refer to the work of Kocks et al. [62]). In our case, it was assumed that the obstacles, i.e., $L_{12}$ precipitates and forest dislocations, possess identical strength but differ in densities.
The densities of obstacles on the gliding plane are cumulative, resulting in a quadratic additivity of stresses.

## 5. Computational results and discussion

In this section, we present the computational results and the optimization strategy employed for the strain hardening model parameters, as outlined in Equation 14. Given the non-uniqueness of solutions - where multiple parameter values can feasibly fit the same curve due to a large number of parameters relative to the dataset - we prioritized simplicity in our model selection for each of the three alloys. This approach was informed by the challenge of identifying a singular solution set that accurately captures the empirical data. However, a kinematic contribution was assumed based on evidence from multiple prior studies, despite the absence of Bauschinger effect tests to directly measure this kinematic contribution. To enhance the reliability of parameter estimation in light of these considerations, we conducted a simultaneous

optimization of the stress-strain curve and its derivative. This comprehensive methodology combined an initial global exploration via the Monte Carlo method with a subsequent, more focused optimization using the deterministic capabilities of the Limfit library [63]. This two-pronged approach not only facilitated a thorough exploration of the parameter space but also ensured a refined calibration of the model, closely mirroring empirical observations. Details of all constants derived from literature, along with the optimized model parameters, are systematically outlined in Table 4.

Using fitted values from Table 4, we computed each contribution to flow stress in the present HEAs, considering various microstructural mechanisms such as dislocation forests, grain boundaries, twins, and nanoprecipitates. Initially, we calculated the isotropic contribution of the dislocation forest ($\Delta\sigma_{df}^{iso}$) to the flow stress without considering the effects of grains and twins by assigning large values to the mean grain diameter and twin thickness, and by setting the fraction of twins and

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

Constantes and values of fitted model parameters used in the calculations.</caption><thead><tr><th colspan="4">Constants and parameters taken from the literature or estimated in this work</th></tr></thead><tbody><tr><td>M</td><td>Taylor factor</td><td>3.06</td></tr><tr><td>α</td><td>Taylor constant</td><td>0.3</td></tr><tr><td>μ (GPa)</td><td>Shear modulus of the FCC matrix</td><td>83</td></tr><tr><td>b (nm)</td><td>Magnitude of the Burger's vector</td><td>0.254</td></tr><tr><td>α<sub>apb</sub> (J/m<sup>2</sup>)</td><td>Antiphase boundary free energy of the L1<sub>2</sub> precipitate in FCC matrix</td><td>0.198 [64]</td></tr><tr><td>E<sub>p</sub> (GPa)</td><td>L1<sub>2</sub> Precipitate's Young modulus</td><td>216 (from rule of mixtures)</td></tr><tr><th colspan="4">Model fitting parameters evaluated in this work</th></tr><tr><td></td><td></td><th>asSPS1100</th><th>asSPS1200</th><th>Aged620</th></tr><tr><td>k<sub>1</sub></td><td>Efficiency factors for dislocation storage</td><td>0.020</td><td>0.030</td><td>0.024</td></tr><tr><td>k<sub>2</sub></td><td>Efficiency factors for dynamic recovery</td><td>1.36</td><td>3.88</td><td>1.22</td></tr><tr><td>t (nm)</td><td>Twin width</td><td>40</td><td>40</td><td></td></tr><tr><td>F<sub>0</sub> (%)</td><td>Maximum fraction of twins</td><td>1</td><td>10</td><td></td></tr><tr><td>ε<sub>ini</sub></td><td>Onset true strain value at which twinning occurs</td><td>0.2</td><td>0.1</td><td></td></tr><tr><td>β</td><td>Calibration parameters</td><td>5</td><td>5</td><td></td></tr><tr><td>m</td><td>Calibration parameters</td><td>2</td><td>2</td><td></td></tr><tr><td>n<sub>t</sub><sup>*</sup></td><td>Critical number of sites per slip band at the boundary</td><td>10</td><td>3</td><td>12</td></tr><tr><td>λ/b</td><td>Number of dislocations per slip band geometrically necessary to provide the deformation</td><td>58</td><td>10</td><td>69</td></tr><tr><th colspan="5">Model parameters to describe the L1<sub>2</sub> precipitate strengthening contribution</th></tr><tr><td>n<sub>p</sub><sup>*</sup></td><td>Maximum number of loops around a precipitate</td><td></td><td></td><td>10</td></tr><tr><td>φ (%)</td><td>Efficiency factor related to the stability of the dislocation loops</td><td></td><td></td><td>2</td></tr><tr><td>u</td><td>Stress additivity law</td><td></td><td></td><td>2</td></tr></tbody></table>

<figure><img src="image_12.png" /><figcaption>**Fig. 7.** Comparative visualization of plastic flow stress contributions and strain hardening rates for the fine-grain FCC asSPS1100 Al<sub>0.3</sub>CoCrFeNi HEA. The left panel presents experimental data points for true stress (blue) and strain hardening rate (gray) against true strain, overlaid with model predictions (black lines). The right panel details the incremental contributions to flow stress derived from the model, including base flow stress (silver fill), isotropic contribution from grain boundaries (light blue fill), kinematic contribution from grain boundaries (dark blue fill), isotropic contribution from twins (light green fill), and kinematic contribution from twins (dark green fill), illustrating the combined impact of microstructural mechanisms on material strength and ductility. Experimental data points (circles) highlight the fit of the model to observed values.</figcaption></figure>
<figure><img src="image_13.png" /><figcaption>Fig. 8. Comparative visualization of plastic flow stress contributions and strain hardening rates for the coarse-grain FCC asSPS1200 Al0.3CoCrFeNi HEA. The left panel presents experimental data points for true stress (green) and strain hardening rate (gray) against true strain, overlaid with model predictions (black lines). The right panel details the incremental contributions to flow stress derived from the model, including base flow stress (silver fill), isotropic contribution from grain boundaries (light blue fill), kinematic contribution from grain boundaries (dark blue fill), isotropic contribution from twins (light green fill), and kinematic contribution from twins (dark green fill), illustrating the combined impact of microstructural mechanisms on material strength and ductility. Experimental data points (circles) highlight the fit of the model to observed values.</figcaption></figure>

nanoprecipitates to 0. Subsequently, we included these effects stepwise: first by incorporating the isotropic effect of grains alone, then by adding both isotropic and kinematic effects of grains ($\Delta\sigma_{gb}^{iso}$ and $\Delta\sigma_{gb}^{back}$), and finally, by considering the combined effects of twins ($\Delta\sigma_{tw}^{iso}$ and $\Delta\sigma_{tw}^{back}$) for the as sintered alloys (asSPSP1100 and asSPS1200), or nanoprecipitates ($\Delta\sigma_{np}^{iso}$ and $\Delta\sigma_{np}^{back}$) for Aged620. This detailed breakdown allows for a deeper understanding of the material's mechanical behavior, particularly how different microstructural features influence its yield strength and hardening behavior. This can be summarized by the following equations of the flow stress for each alloy:

$$\sigma_{asSPS1100} = \Delta\sigma_0 + \Delta\sigma_{df}^{iso} + \Delta\sigma_{gb}^{iso} + \Delta\sigma_{gb}^{back} + \Delta\sigma_{tw}^{iso} + \Delta\sigma_{tw}^{back}$$

$$\sigma_{asSPS1200} = \Delta\sigma_0 + \Delta\sigma_{df}^{iso} + \Delta\sigma_{gb}^{iso} + \Delta\sigma_{gb}^{back} + \Delta\sigma_{tw}^{iso} + \Delta\sigma_{tw}^{back}$$

$$\sigma_{Aged620} = \Delta\sigma_0 + \Delta\sigma_{df}^{iso} + \Delta\sigma_{gb}^{iso} + \Delta\sigma_{gb}^{back} + \Delta\sigma_{np}^{iso} + \Delta\sigma_{np}^{back}$$

Finally, we layered the visual representation of flow stress contributions, starting with the base flow stress and sequentially adding isotropic contributions from forest dislocations, as well as isotropic and kinematic contributions from grains, twins, and nanoprecipitates. For each contribution, we first plotted a line representing the cumulative stress up to that point and then filled the area between the previous cumulative stress and the current one, using distinct colors for clarity. This approach not only highlights the individual contributions but also illustrates how they cumulatively build up the total flow stress experienced by the material (Figs. 7-9).

## 5.1. Effects of grain size – comparison between asSPS1100 and asSPS1200 HEAs

The model accurately simulates the tensile and work hardening behaviors of the alloys, separating the different contributions to work hardening (Figs. 7 and 8). The critical strain for twinning, $\epsilon_{ini}$, differentiates the coarse-grained alloy (asSPS1200) from the fine-grained HEA (asSPS1100), with twinning observed at 10% strain in asSPS1200 and 20% in asSPS1100. This difference highlights the role of grain size in influencing mechanical properties: twinning at lower strains in asSPS1200 contributes to work hardening, a mechanism that is negligible in asSPS1100 because it occurs at higher deformations. This result is consistent with a recent paper showing that twinning activity is strongly inhibited by grain refinement in Al0.1CoCrFeNi HEA [36].

Beyond $\epsilon_{ini}$ in asSPS1200, the strain hardening rate stabilizes, indicative of the twinning activity characteristic of TWIP alloys. This behavior results from the dynamic Hall-Petch effect (isotropic component) and the back-stress from storing GNDs at grain and twin boundaries (kinematic component). Although this phenomenon is well-documented in TWIP steels, its exploration in HEAs has been less extensive. The accumulation of GNDs during plastic deformation has been shown to introduce back-stress, as evidenced by recent studies on high/medium entropy alloys [65]. The model estimates a maximum back-stress of 112 MPa for a grain size of 18 µm (and only 12 MPa for a grain size of 9 µm in asSPS1100), slightly lower than the 190 MPa measured through a Bauschinger test on the CoCrFeMnNi Cantor alloy with a grain size of 18.2 µm [22].

The twins' size and volume fraction are the microstructural parameters that control the dynamic Hall-Petch effect (isotropic component). The size of twins in the fitted model is estimated to 40 nm. While for low deformation, thinner twins were measured in CoCrFeMnNi (12 nm) [14] and Al0.1CoCrFeNi HEA (13 nm) [36,54], the estimated value for the present alloy is close to that found for deformations above 16% (32 nm for CoCrFeMnNi HEA) [14]. The model shows that the volume fraction of twins at saturation (i.e., the maximum fraction of twins that could potentially form) is 10% in asSPS1200 and only 1% in asSPS1100. However, the volume fraction of twins formed during plastic deformation does not reach saturation and remains relatively low (6% in asSPS1200). These are significant differences with TWIP steels [40] which could explain why the Hall-Petch dynamic effect is not optimal in the Al0.3CoCrFeNi HEA.

Another significant phenomenon we examined was the influence of grain size on the critical stress for the onset of deformation twinning (i. e., the stress at $\epsilon_{ini}$). The fitted model evaluates the critical stress at 622 MPa in asSPS1200 and 917 MPa in asSPS1200, markedly higher than the yield strength of this alloy (337 and 422 MPa, respectively). The critical stress in asSPS1200 is comparable to values measured in several HEAs and significantly exceeds those noted in most TWIP steels [10,14,40,66]. This result suggests that twinning in HEAs does not occur until multiple slips are activated, leading to a delayed twinning activation, especially in alloys with smaller grains. Consequently, the coarse-grained Al0.3CoCrFeNi HEA (asSPS1200) demonstrates a superior combination of strength and ductility compared to the fine-grained version (asSPS1100). For asSPS1200, the onset of twinning occurs at a strain of 10% providing additional work hardening compared to asSPS1100, where twinning is absent at the same deformation. Notably, this 10%
critical twinning strain for the coarse-grain Al0.3CoCrFeNi HEA is lower than for the CrMnFeCoNi HEA at room temperature [14] and marches that of medium entropy CrCoNi alloys at room temperature [11].

While the influence of grain size on the critical stress for twinning in HEAs like FCC-Al0.3CoCrFeNi is complex and debated [16,40], the application of a general Hall-Petch relation [67], expressing the correlation between slip and twinning, offers insights into this relationship:

$$ \sigma_{tw} = M \frac{\gamma_{SFE}}{b_s} + \frac{k_{tw}^{HP}}{\sqrt{D}} \qquad (15) $$

where γSFE is the stacking fault energy, b_s is the Burgers vector for a Shockley partial, D is the grain size and ktw is the Hall-Petch constant for twinning.

The first term in Eq.15 represents the stress required to achieve partial dislocation separation to grow a twin, while the second term is the stress required to nucleate a twin at a grain boundary. With critical twinning stresses for grain sizes of 18 µm (622 MPa) and 9 µm (917 MPa), we estimate a twinning Hall-Petch coefficient of ktw = 3022 MPa.µm1/2. This coefficient is 3.5 higher than for the slip (kslip = 871 MPa.µm1/2 estimated in Section 3.2), highlighting that grain refinement might inhibit twinning more in Al0.3CoCrFeNi HEAs compared to traditional Fe-Mn-C TWIP steels [67,68]. This results suggests that grain refinement in Al0.3CoCrFeNi HEA is more likely to inhibit twinning in comparison to the latter TWIP steels.

## 5.2. Effects of precipitates – comparison between asSPS1100 and Aged620 HEAs

After undergoing aging treatment at 620 °C for 50 h, the Al0.3CoCrFeNi high-entropy alloy (HEA), referred to as Aged620, exhibits increased strength and reduced ductility (Fig. 5a) compared to its as-sintered state (asSPS1100). Despite these changes, Aged620 maintains an impressive balance between strength and ductility, as indicated by the product of ultimate tensile strength (σm) and uniform elongation (εu) shown in Table 3. Given the similar grain sizes of these two samples, the distinct behaviors are attributed to the presence of L12 precipitates, a phenomenon observed in various HEAs yet lacking a comprehensive explanation [69,70]. This section demonstrates that these differences arise from specific deformation mechanisms at the microscopic level.

Contrary to a simple displacement of the tensile curve from the FCC HEA (asSPS1100) to the FCC/L12 HEA (Aged620), the inclusion of L12

precipitates markedly alters the work hardening behavior of the Al0.3CoCrFeNi HEA. This results in a noticeably higher strain hardening rate over a wide range of deformation (Fig. 5c), indicating that the tensile curve of Aged620 cannot be derived from merely shifting the curve of asSPS1100. Both curves, however, follow a similar pattern with a decrease in strain hardening rate at lower strains.

Our model accurately replicates the tensile and strain hardening curves (Fig. 9 left), identifying various contributions to work hardening (Fig. 9 right). It reveals that L12 precipitates induce pronounced work hardening, contributing a substantial isotropic effect of up to 154 MPa. This enhancement is due to the distinct but equally potent obstacles posed by dislocation forests and L12 precipitates. The model adopts a value of u = 2 in the stress addition law (Pythagorean addition rule), indicating that precipitation hardening is not merely additive but significantly alters the flow stress and work hardening [54]. Furthermore, another contribution to the isotropic component results from the storage of dislocation loops around L12 precipitates. This is influenced by the parameter φ, reflecting the efficiency of dislocation loop storage. It varies from 0 in the case of purely shearing precipitates, typically observed when the particles are coherent with the surrounding matrix, to a maximum value of 1 for the by-pass mechanism, which is prevalent in incoherent precipitates. With φ valued at 0.01, our findings align with the coherent nature of L12 precipitates within an FCC matrix, suggesting a low efficiency of dislocation storage despite their significant role as obstacles to dislocation movement due to high number density.

However, the back stress contribution from dislocation loop formation around precipitates is minimal (< 10 MPa), minor compared to kinematic hardening from GNDs at twin boundaries in asSPS1200. This observation is consistent with the shearable character of coherent L12 precipitates, indicating that kinematic work hardening is not primarily due to L12 precipitates.

Our analysis suggests that enhancing the strength-ductility trade-off could be achieved by increasing the stress impact from precipitates, potentially through alloy composition adjustments to raise the volume fraction of L12 precipitates. Supporting this strategy, researchs by Yang et al. [70] and Dasari et al. [71] show that adding Ti and Al to FeCoNi-base alloys introduces high-density L12 precipitates, resulting in significant back-stress and remarkable ductility for a high strength level (1.5 GPa for 50 % ductility).

<figure><img src="image_14.png" /><figcaption>Fig. 9. Comparative visualization of plastic flow stress contributions and strain hardening rates for the fine-grain FCC/L12 Aged620 Al0.3CoCrFeNi HEA. The left panel presents experimental data points for true stress (orange) and strain hardening rate (gray) against true strain, overlaid with model predictions (black lines). The right panel details the incremental contributions to flow stress derived from the model, including base flow stress (silver fill), isotropic contribution from grain boundaries (light blue fill), kinematic contribution from grain boundaries (dark blue fill), isotropic contribution from L12 nanoprecipitates (light orange fill), and kinematic contribution from L12 nanoprecipitates (dark orange fill), illustrating the combined impact of microstructural mechanisms on material strength and ductility. Experimental data points (circles) highlight the fit of the model to observed values.</figcaption></figure>
# 6. Conclusion

The effects of grain size and $L_{12}$ precipitates on room temperature tensile properties and work hardening have been investigated in an FCC-Al$_{0.3}$CoCrFeNi high entropy alloy fabricated by SPS process. Sintering conditions and aging treatment were controlled to produce three microstructures: fine-grain and coarse-grain FCC structure, and fine-grain FCC with 5% $L_{12}$ precipitates. A model that combines the isotropic and kinematic contributions to the room temperature plastic flow stress, including forest dislocations, grains, twins and precipitates, has been used to successfully rationalize the tensile properties and work hardening behavior of Al$_{0.3}$CoCrFeNi HEA as a function of the microstructural state.

The following outcomes can be highlighted:

(1) The fully coarse-grained FCC structure promotes a TWinning Induced Plasticity (TWIP) effect, which results from an isotropic component linked to a dynamic Hall-Petch effect and a kinematic component linked to the storage of geometrically necessary dislocations (GNDs). The critical stress and strain at which twinning occurs are 625 MPa and 10%, respectively.

(2) As the grain size decreases in the fully FCC structure, the TWIP effect strongly fades since the critical stress and strain for twinning increase significantly. We then show that a Hall-Petch relation provides a good estimation of the influence of the grain size on the twinning stress. The strain hardening rate is mainly isotropic in nature and results from the dislocations stored at grain boundaries.

(3) Although $L_{12}$ precipitates have a minimal impact on kinematic hardening, they significantly contribute to work hardening through strong isotropic effects without influencing twinning activity.

(4) Given their coherent nature and the limited dislocation storage capacity, enhancing work hardening and the strength-ductility trade-off in FCC HEAs is best achieved by increasing the volume fraction of $L_{12}$ precipitates through alloy composition adjustments.

(5) The proposed analysis is mainly based on tensile and strain-hardening curves. A more exhaustive study in ESBD and TEM based on interrupted tests at different strain would clarify the twins formation during deformation and their interaction with $L_{12}$ precipitates.

## CRediT authorship contribution statement

**Stéphane Gorsse:** Writing – original draft, Visualization, Validation, Supervision, Software, Project administration, Methodology, Funding acquisition, Formal analysis, Data curation, Conceptualization. **Florian Peyrouzet:** Writing – review & editing, Validation, Investigation, Formal analysis, Data curation. **Thierry Baffie:** Writing – review & editing, Validation, Supervision, Resources, Project administration, Funding acquisition, Formal analysis. **Christelle Navone:** Supervision, Resources, Project administration, Funding acquisition. **Julie Maisonneuve:** Supervision, Resources, Project administration. **François Saint-Antonin:** Writing – review & editing, Investigation. **Marion Descoins:** Investigation. **Khalid Hoummada:** Investigation. **Rajarshi Barnerjee:** Investigation. **An-Chou Yeh:** Writing – review & editing, Validation. **Mohamed Gouné:** Writing – original draft, Validation, Methodology, Formal analysis, Conceptualization.

## 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.

# Acknowledgements

This work was financed by the AID (Agence de l'innovation de défense).

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