# Some advances on precipitation hardening in Waspaloy: Critical review on the contribution of shear, Orowan and coherency mechanisms

S.P. Murugan, C. Trainel, A. Oudriss, Guillaume Hachet, Frederic Danoix, Grégory Odemer, Benoît Malard, François Godet, Christine Blanc, Xavier Feaugas

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S.P. Murugan, C. Trainel, A. Oudriss, Guillaume Hachet, Frederic Danoix, et al. Some advances on precipitation hardening in Waspaloy: Critical review on the contribution of shear, Orowan and coherency mechanisms. Materials Characterization, 2024, 216, pp.114301. 10.1016/j.matchar.2024.114301. hal-04769485

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# Some advances on precipitation hardening in Waspaloy: Critical review on the contribution of shear, Orowan and coherency mechanisms

S.P. Murugan<sup>a,*</sup>, C. Trainel<sup>a,b</sup>, A. Oudriss<sup>a</sup>, G. Hachet<sup>c</sup>, F. Danoix<sup>c</sup>, G. Odemer<sup>b</sup>, B. Malard<sup>b</sup>, F. Godet<sup>a</sup>, C. Blanc<sup>b</sup>, X. Feaugas<sup>a,*</sup>

## ARTICLE INFO

**Keywords:**
Nickel base alloy
γ' precipitate
Lattice misfit
Hardening

## ABSTRACT

The precipitation hardening mechanisms and their dependence on precipitate size ($r_p$) and precipitate/matrix lattice misfit ($\delta$) in $\gamma'/\gamma$ systems were studied in nickel-base superalloy using micro-indentations experiments and theoretical models. Metallurgical states were characterized by TEM, SEM-EBSD, XRD and APT. For the precipitate size range studied (radius $r_p$ smaller than 110 nm), $\gamma'$ precipitates are ordered spherical Ni₃(Al,Ti) phase, the volume fraction remains relatively constant and equal to 20 % and lattice misfit $\delta$ is lower than 0.45 %. Prevalence of order strengthening by anti-phase boundary formation and Orowan bypassing was observed in various $r_p$ range, whereas chemistry and coherency strengthening were found insignificant in the nickel-base superalloy studied (Waspaloy®). Shearing by weakly coupled dislocations was dominant for fine precipitates, i.e. $r_p$ lower than 18 nm, shearing by strongly-coupled dislocations for an intermediate $r_p$ range, i.e. from 18 nm to 98 nm and Orowan bypassing for coarse precipitates with a $r_p$ larger than 98 nm.

## 1. Introduction

Precipitation hardening nickel-based superalloys are of great interest owing to their superior mechanical and corrosion properties at various temperatures. Waspaloy® is one of them with microstructure consisting of $\gamma'$-Ni₃(Al,Ti) precipitates in a $\gamma$-matrix, mainly used for turbine blade, vane and disk [1]. The search for high mechanical properties has led to the development of alloys through composition, time and temperature of aging [2–4]. Apart from the solid solution strengthening of $\gamma$ matrix, finely dispersed $\gamma'$ precipitate provides considerable increase in yield strength. According to this context, different hardening mechanisms have been proposed based on the nature of precipitate/dislocation interactions: Order-, chemical-, coherency- and modulus-strengthening, and Orowan bypassing [5–8]. Mainly studies have concluded that, order strengthening is the only significant mechanism in nickel-based alloys, and that the others are negligible [9,10]. Though, a few studies reported the contributions of coherency strengthening in superalloys hardened by the presence of $\gamma'$ and/or $\gamma''$ precipitates [11,12]. Additionally, some works question the possible contribution of chemical composition of $\gamma'/\gamma'$ phases in relation with steric and modulus effects

and conclude to a negligible impact on shear stress for $\gamma'/\gamma'$ alloys [3,4]. To summarize, the predominance of one mechanism over the others in relation with precipitate size is a fundamental aspect of the precipitation hardening mechanism as it was previously experimentally [5–13] questioned and investigated by modeling [14,15] for different alloys.

To the authors' best knowledge, a detailed experimental study on the individual contribution of the various strengthening mechanisms for Waspaloy® having with a wide range of precipitate size ($r_p$) and lattice misfit ($\delta$) has not yet been conducted. This work aims to investigate the effect of $r_p$ and $\delta$ on hardening mechanisms in Waspaloy® in relation with a possible evolution of the composition of $\gamma'$. Firstly, the material was subjected to different heat treatment conditions to generate various $r_p$ and $\delta$. Then the shear stress associated with the precipitation hardening of each sample with specific precipitation state was quantified by Vickers hardness measurement. Subsequently, the $r_p$ range for the predominance of different hardening mechanisms was quantified from the mechanistic models of order strengthening, Orowan bypassing and coherency strengthening, and compared to the behavior experimentally characterized with different $r_p$ and $\delta$.

<footnote><sup>a</sup> La Rochelle Université, LaSIE CNRS UMR, 7356, Avenue Michel Crépeau, La Rochelle F-17042, France</footnote>

<footnote><sup>b</sup> CIRIMAT, Université de Toulouse, CNRS, INP – ENSIACET, 4 Allée Emile Monso, 31030 Toulouse, France</footnote>

<footnote><sup>c</sup> Normandie Université, INSA Rouen, CNRS, Groupe de Physique des Matériaux, 76000 Rouen, France</footnote>

<footnote>* Corresponding authors.</footnote>

<footnote>E-mail addresses: smurugan@univ-lr.fr (S.P. Murugan), xfeaugas@univ-lr.fr (X. Feaugas).</footnote>
<caption>**Table 1**</caption>

Precipitation state (Wi), aging treatment conditions (temperature T, time t), precipitate size ($r_p$), volume fraction of precipitate ($f_v$) by TEM/XRD), grain size ($\phi$), Vickers Hardness (Hv, kg/mm²), chemical composition of $\gamma/\gamma'$ phases in at. %, and the size (g) around the precipitate where a chemical gradient was observed for various samples after heat treatments.

<table><thead><tr><th>State</th><th>T (°C)/ t (h)</th><th>r<sub>p</sub> (nm)</th><th>f<sub>v</sub> (%)</th><th>φ (μm)</th><th>CSL (%)</th><th>Hv<sub>1</sub></th><th>Cr</th><th>Co</th><th>Al</th><th>Ti</th><th>g (nm)</th></tr></thead><tbody><tr><td>W0</td><td>No aging</td><td>0</td><td>0</td><td>70</td><td>41</td><td>247</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>W1</td><td>550/4</td><td>5</td><td>13</td><td>170</td><td>41</td><td>325</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>W2</td><td>600/4</td><td>7.5</td><td>21</td><td>100</td><td>33</td><td>342</td><td>29/0</td><td>18/3</td><td>0/11</td><td>0/14</td><td>1.2</td></tr><tr><td>W3</td><td>650/4</td><td>10</td><td>27</td><td>125</td><td>28</td><td>356</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>W4</td><td>750/4</td><td>15</td><td>21</td><td>92</td><td>28</td><td>372</td><td>28/2</td><td>18/2</td><td>0/10</td><td>0/14</td><td>1.5</td></tr><tr><td>W5</td><td>850/4</td><td>20</td><td>29</td><td>100</td><td>19</td><td>373</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>W6</td><td>875/3</td><td>27</td><td>22</td><td>135</td><td>38</td><td>356</td><td>20/2</td><td>14/2</td><td>2/10</td><td>2/15</td><td>1.8</td></tr><tr><td>W7</td><td>875/4</td><td>32</td><td>14</td><td>197</td><td>51</td><td>362</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>W8</td><td>875/7</td><td>35</td><td>20/16</td><td>106</td><td>38</td><td>350</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>W9</td><td>875/10</td><td>43</td><td>16</td><td>84</td><td>46</td><td>335</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>W10</td><td>875/40</td><td>58</td><td>25</td><td>91</td><td>42</td><td>330</td><td>23/0</td><td>18/4</td><td>0/10</td><td>0/14</td><td>2</td></tr><tr><td>W11</td><td>875/120</td><td>85</td><td>21/28</td><td>165</td><td>45</td><td>320</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>W12</td><td>875/230</td><td>110</td><td>19/24</td><td>109</td><td>35</td><td>318</td><td>26/0</td><td>20/3</td><td>0/12</td><td>0/15</td><td>2.5</td></tr></tbody></table>

<figure><img src="image_3.png" /><figcaption>**Fig. 1.** Multi-scale microstructural representation of $W_4$: (a) EBSD-IPF; (b) EBSD-GBCD (blue line Random GB, red line $\Sigma3$); (c) TEM dark field image $\gamma$'-precipitates on $(001)\gamma$' spot and diffraction pattern in inset; (d) $\gamma$' crystallographic structure; (e) APT-chemical composition analysis of $\gamma$' matrix and $\gamma$' precipitates in a box size 40 $\times$ 40 $\times$ 160 nm³ ($W_4$). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)</figcaption></figure>

## 2. Experimental

Precipitation hardened Waspaloy® (Ni-19.05Cr-14Co-4.13Mo-1.35Al-3.13Ti) was first undergone a homogenization treatment in an air muffle furnace at 1080 °C for 4 h, followed by oil quenching. A subsequent aging treatment was then performed in a tubular furnace filled with argon at different temperatures and durations to develop different precipitation states while reducing the grain size variation. A heating rate of 50 °C/min was used for all the aging treatments and the cooling was in outside air. Grain size and grain boundaries character of all these samples were characterized and measured by using Zeiss 340 Scanning Electron Microscope (SEM) and electron backscattered diffraction analysis (EBSD). For statistical precision, EBSD scans were performed on an area larger than 4 mm² and data collection were conducted using TEAM™ EBSD Software Suite from EDAX. The EBSD Data were analyzed using the TSL OIM Analysis™ 8 software. The size, volume fraction and lattice parameter of the precipitate were evaluated by transmission electron microscopy (TEM) using a JEOL JEM 2011 and X-ray diffraction (XRD) using a Bruker D8 with a copper source. The composition of both matrix and precipitates, and the distribution of solute atoms were studied by atom probe tomography analysis (APT) on

<figure><img src="image_4.png" /><figcaption>**Fig. 2.** Mean grain size and the fraction of CSL as a function with precipitate size (radius $r_p$).</figcaption></figure>
<figure><img src="image_5.png" /><figcaption>Fig. 3. Volume fraction $f_v$ (a), inter-precipitate distance $d$ (b), lattice parameter $a$ (c) and lattice misfit $\delta$ (d) presented as a function with precipitate radius $r_p$.</figcaption></figure>

a CAMECA LEAP® 4000HR, while the reconstruction was conducted using IVAS 3.6 software package. The Vickers hardness measurements were performed under a load of 1 kg with a Q10M micro-indenter from Qness. The different samples studied are reported in Table 1 where homogeneous treatment $W_0$ is considered as the reference sample, while all the others are in precipitation hardened state.

## 3. Results and discussion

Fig. 1 shows the multi-scale microstructural features of $W_4$ including grain orientation with inverse pole figure (IPF) representation and grain boundaries character distribution (GBCD) maps (EBSD), dark field image and associated diffraction patterns (TEM), atom maps and element composition profiles (APT). These analyses were performed for each state reported in Table 1, except for APT where only selected conditions were investigated. The grain size and the fraction of special grain boundaries with coincidence site lattice (CSL) for $\Sigma < 29$ (mainly composed of twins) stay quite constant and equal respectively to 123 $\pm 37$ µm and $37 \pm 9$ % (Fig. 2). These results show that aging treatments have a low impact on grain size and twinning. The selected area electron diffraction patterns (SADP) obtained from [001] zone-axis on an area containing $\gamma'$ precipitates within the $\gamma$ matrix indicates, near the incident beam, large intense diffraction spots corresponding to {200} set of planes of the $\gamma$ matrix and weaker spots related to {100} and {110} atomic planes of the $\gamma'$ phase. $\gamma'$ precipitates are randomly distributed with the L1$_2$ structure (dark field on L1$_2$ substructure, Fig. 1d) and a typical orientations relationship: (100)$\gamma$ // (1 0 0)$\gamma'$ and [001]$\gamma$ // [001]$\gamma'$. The average values of precipitate radius $r_p$, volume fraction $f_v$ and inter-precipitate distance $d$ were evaluated from the dark field images on a population of 100 $\gamma'$ precipitates using ImageJ software (Fig. 3). The precipitate size increased from 5 to 110 nm (Table 1) while the volume fraction appears to be stable for the different heat treatment

conditions, and around 20 % (Fig. 3a). This volume fraction was confirmed by XRD measurements for W8, W11 and W12 (Table 1). Consequently, the inter-precipitate distance evolved linearly with precipitate size (Fig. 3b). The lattice parameter (a) of the two phases obtained from TEM are plotted as a function of $r_p$ in Fig. 3c. In all the heat treatment conditions, $a_{\gamma'}$ is larger than $a_{\gamma}$, which indicates that the precipitate is under compressive stress and matrix is at tensile stress. Lattice parameters of both the phases decreases for $r_p$ below 20 nm, then increase for a $r_p$ in the range 20 nm to 60 nm, and at last remain constant for $r_p$ above 60 nm. The lattice misfit ($\delta$) is deduced from the following relationship:

$$ \delta = \frac{2(a_{\gamma'} - a_{\gamma})}{(a_{\gamma'} + a_{\gamma})} \qquad (1) $$

The variation of $\delta$ with $r_p$ is plotted in Fig. 3d, where three domains can be observed: a first increase in $\delta$ for $r_p$ below 20 nm; followed by a decrease/drop in $\delta$ for 20 nm < $r_p$ < 40 nm; and a last domain of constant $\delta$ at $r_p$ > 40 nm. APT were performed in the three ranges of precipitate size to measure chemical composition of both phases. Only the main solutes are considered in the present work (Co, Cr, Al, Ti, Mo and Fe). No significant difference is observed between the different studied states (Table 1). Co and Cr are mainly localized in the $\gamma$ matrix and Al and Ti in the $\gamma'$ phase (Fig. 4a). Additionally, Mo and Fe are mainly distributed in the matrix (respectively 3 at.% and 0.4 at.%) as illustrated in Fig. 4b. In contrast, we observed an increase in gradient of composition near precipitate, g with the radius increase to reach a value 2.5 nm (Table 1). This value is quite similar to the one related for polycrystalline nickel-base superalloy RR1000 (2.5 nm) [16] and 626 Plus (2 nm) [2] for $\gamma'$ precipitate with a radius between 4 and 6 nm.

The Vickers Hardness ($H_v$) is used to determine the critical resolved shear stress (CRSS), i.e. $\tau_0$, based on the Tabor and Taylor equations ($\sigma_0$
<figure><img src="image_6.png" /><figcaption>Fig. 4. Element composition profiles based on atom probe tomography analysis (APT) for WP4.</figcaption></figure>

= k.H$_{V}$ and τ$_{0}$ = σ$_{0}$/M, k = 3.27 and M = 3.07 respectively the Tabor and the Taylor factor, with σ$_{0}$ (MPa) and H$_{V}$ (kg/mm$^{2}$)) [17–22,27].

The CRSS evolution as a function of precipitate size is presented in Fig. 5a; in addition to the experimental data previously published by Donachie et al. [23] and Whelchel et al. [24] to have a large set of data. A linearity of τ$_{0}$ versus √r$_{p}$ is obtained in a first regime (r$_{p}$ < 20 nm) with an intercept and a slope respectively equal to 240 MPa and 32 (Fig. 5b). The intercept value is close from to τ$_{0}$ obtained for W0 state (precipitate free: 247 MPa). The increase in shear stress due to solid solution addition in matrix and friction stress is expressed as follows:

$$ \tau_0(W_0) = \tau_s + A \sum_i (\varepsilon_{m,i})^{4/3} C_i^{2/3} \quad (2) $$

with i ∈ {Co, Cr, Mo}, τ$_{s}$ =12 MPa is the frictions stress of pure nickel, A is a fitting parameter equal to 288 GPa [22], ε$_{m,i}$ and C$_{i}$ (wt%) are respectively the misfit strain and concentration of solute composition (ε$_{m,Co}$ = − 0.21%, ε$_{m,Cr}$ = − 0.67%, ε$_{m,Mo}$ = − 0.76% [25]).

According to the Eq. (2), τ$_{0}$ (W$_{0}$) = 230 ± 14 MPa which is again close from the experimental value (247 MPa) and intercept one of Fig. 5a

(240 MPa). The progressively decrease of this stress near the precipitate in gradient zone associated with the decrease of Co, Cr and Mo content (g, Table 1) is partially compensated by the increase of solid solution strengthening associated to the increase of Al and Ti content τ$_{0}$ (Ti, Al) = 191 MPa (ε$_{m,Al}$ = 0.41% and ε$_{m,Ti}$ = 0.84%) [26]. Consequently chemical gradient has no significant contribution on the precipitation strengthening, but can modified.

The contribution of precipitates in the hardening process can be defined as Δτ$_{0}$ = τ$_{0}$ (W$_{i}$) − τ$_{0}$ (W$_{0}$), that is, the difference in CRSS of samples with and without precipitates. Fig. 5c shows the evolution of Δ τ$_{0}$ as a function of precipitate size assuming that the grain size is not much affected by the different heat treatments as in Table 1. Moreover, it should be noted that all the results are compared with the data from the corresponding literature [23,24]. Three regimes can be identified in Fig. 5c corresponding to the predominance of three classical plasticity processes (Fig. 5d) [5–8,15]. These three hardening mechanisms are further confirmed theoretically in the subsequent sections.

The γ' phase is based on ordered Ni$_{3}$(Al,Ti) lattice where Ni, Al and Ti atoms are arranged in {111} planes as in Fig. 1d. This requires the
<figure><img src="image_7.png" /><figcaption>Fig. 5. (a) Critical resolved shear stress $\tau_0$ vs the precipitate radius $r_p$, (b) linearity of $\tau_0$ versus $\sqrt{r_p}$ in a first regime ($r_p < 20$ nm), (c) contribution of precipitation on stress vs $r_p$, (d) three mechanisms of dislocation/precipitate interaction.</figcaption></figure>

glide of 1/2(110) dislocations in couples to retain ordered structure of the $\gamma'$ during cutting. The first dislocation disrupts the ordered arrangement of atoms, and the second dislocation restores the ordered structure. Thus, the shearing mechanism that governs hardening can further be divided into two categories: (i) shearing by weakly-coupled dislocations (WCD); (ii) shearing by strongly-coupled dislocations (SCD) [7]. Orowan bypassing is the third mechanism for coarse precipitates. The mechanism that offers least resistance to the movement of dislocations governs the hardening behavior.

Normally, shearing mechanism of $\gamma'$ is governed by the spacing between dislocation pairs (d) with respect to precipitate size, and this spacing is controlled by ($r_p, f_v$) and anti-phase boundary (APB) energy of the $\gamma'$ ($\gamma_{APB}$). When $r_p < d/2$, the two dislocations remain weakly coupled as they do not lie within the same precipitate. This scenario is called shearing by WCD, and the associated CRSS to shear precipitate by WCD is given by:

$$ \Delta\tau_{WCD} = \frac{\gamma_{APB}\sqrt{f_v}}{2b} \left( \left[ \frac{6\gamma_{APB}}{\pi T} \right]^{1/2} \sqrt{r_p} - \sqrt{f_v} \right) \quad (3) $$

where, b is Burgers vector and T is dislocation line tension, a function of shear modulus ($\mu$) and b ($T = \mu b^2/2$). In Fig. 5c, shear stress for shearing of precipitate by WCD mechanism is plotted as the red color curve (Eq. (3)). It increases sharply with $r_p$, as it is directly proportional to $r_p^{1/2}$ for a given volume fraction $f_v$. Here, the values of $\gamma_{APB}$, b, $f_v$ and $\mu$ were taken as 130 mJ/m$^2$, 0.254 nm, 20 % and 81 GPa, respectively.

When $r_p \ge d/2$, the second dislocation can enter the precipitate before the first dislocation leaves it and hence the pair is strongly coupled. This mechanism is termed as shearing by SCD in which the behavior of the second dislocation strongly depends upon the first dislocation due to the elastic repulsive force between them. The CRSS required for this mechanism is given by [7]:
<figure><img src="image_8.png" /><figcaption>Fig. 6. (a) Different contributions on critical resolved shear stress in comparison of the specific contribution of coherency strengthening in precipitation hardening, (b) distribution of interaction force around the precipitate for coherency strengthening and APB strengthening.</figcaption></figure>

$$ \Delta\tau_{\text{scd}} = \sqrt{\frac{3f_v}{2}} \left[ \frac{\mu b}{r_p} \right] \frac{w}{\pi^{3/2}} \left( \left[ \frac{\pi Y_{\text{APB}}}{wT} \right] r_p - 1 \right)^{1/2} \quad (4) $$

where, w is a parameter with value around 1, corresponding to the elastic repulsion between the strongly paired dislocations.

The shear stress associated with this mechanism is plotted in blue color in Fig. 5c. Different sets of values of w and f_v were considered to fit the curve with the experimental results of present and previous studies. For our data, the value of f_v in the currents samples is around 20 % and the best fit was obtained for w = 1.4, i.e. solid blue curve. In this case, the shear stress decreases with increasing r_p. Consequently, the transition between hardening and softening is observed at r_p = 18 nm is obtained as a transition between WCD and SCD mechanisms. The effect of shearing by SCD weakens as r_p increases, and the Orowan bypassing mechanism becomes predominant wherein the CRSS decreases with increasing r_p, i.e., τ ~ (1/r_p).

The expression for shear stress in this Orowan mechanism is given by:

$$ \Delta\tau_{\text{Orowan}} = \sqrt{f_v} \left[ \frac{\mu b}{2r_p} \right] \left( \sqrt{\frac{2\pi}{3}} - 2\sqrt{f_v} \right) \quad (5) $$

$\Delta\tau_{\text{Orowan}}$ is represented as green solid curve in Fig. 5c.

These relationships can be used to understand the hardening mechanisms and the precipitate size range where this mechanism is predominant. The experimental results (present and previous studies) in Fig. 5c are in good agreement with the theoretical shear stress calculated by considering only the shearing and Orowan processes. This means that the dominance of hardening mechanism and hence the CRSS are not significatively affected by δ though the value of δ changes between 0.15 % and 0.45 %. However, the strain field created by the lattice misfit between the precipitate and the matrix interacts with the dislocations and may contribute to the coherency strengthening [7]. The shear stress ($\Delta\tau_{\text{coh}}$) from the coherency strengthening is given by:

$$ \Delta\tau_{\text{coh}} = \alpha_{f}^{*} \mu \epsilon^{3/2} \left( \frac{f_v r_p}{b} \right)^{1/2} \quad (6) $$

$$ \alpha_{f}^{*} = 3.7 \left[ \beta_{1} \left( \frac{r_{p}}{b} \right)^{\beta_{2}} \right]^{3/2} \left[ 0.94(1 + 2.5\eta_{\text{SL}})^{1/3} \right] \quad (7) $$

$$ \eta_{\text{SL}} = 0.54 \left( \frac{f_v b}{\epsilon r_p \beta_1 \left( \frac{r_p}{b} \right)^{\beta_2}} \right)^{1/2} \quad (8) $$

where, $\alpha_{f}^{*}$ is constant, $\epsilon$ is lattice misfit parameter (0.67 δ), $\beta_1$ and $\beta_2$ are coefficients (0.328 and 0.239) and $\eta_{\text{SL}}$ is the normalized precipitate depth in the theory of Schwarz and Labusch [7].

Fig. 6a shows the shear strength of coherency strengthening with different δ values. Normally, coherency force reaches maximum value when the dislocation is at the precipitate interface, whereas, the APB energy force is maximum when the dislocation is at the middle of the precipitate as described in Fig. 6b [7]. Therefore, the effective shear stress for a given r_p is taken as the highest of the two contributions, i.e., $\Delta\tau_{\text{WCD/SCD}}$ and $\Delta\tau_{\text{coh}}$. In domain-I, $\Delta\tau_{\text{WCD}} > \Delta\tau_{\text{coh}}$ and, in domain-II, $\Delta\tau_{\text{SCD}} > \Delta\tau_{\text{coh}}$ for any δ values. Hence the coherency strengthening does not contribute to the overall shear strength of Waspaloy® in the given conditions. However, the coherency strengthening mechanism will be dominant in domain-I if δ > 0.6 %, and in domain-I and II if δ > 1 % which does not correspond to the experimental conditions.

## 4. Conclusions

The effect of r_p, δ and chemical gradient on hardening mechanisms in Waspaloy® was discussed using experimental results and theoretical models. Predominance of three strengthening mechanisms were revealed with different r_p ranges: (1) shearing by WCD when r_p < 18 nm; (2) shearing SCD for 18 nm < r_p < 98 nm; (3) Orowan bypassing when r_p > 98 nm. The chemical gradient of solutes observed near precipitate on zone of approximatively 2 nm does not significantly affects the strengthening. Additionally, variation in δ was negligible which suggests that no coherency strengthening was involved in the precipitation hardening effect in the given Waspaloy® even though maximum value of δ was 0.45 %.
CRediT authorship contribution statement

**S.P. Murugan:** Writing – original draft, Validation, Methodology, Investigation, Formal analysis, Data curation, Conceptualization. **C. Trainel:** Data curation, Formal analysis, Investigation. **A. Oudris:** Writing – original draft, Supervision, Project administration, Methodology, Funding acquisition, Formal analysis, Data curation, Conceptualization. **G. Hachet:** Investigation, Formal analysis, Data curation. **F. Danoix:** Investigation, Formal analysis, Data curation. **G. Odemer:** Supervision, Project administration, Investigation, Formal analysis, Data curation. **B. Malard:** Investigation, Formal analysis, Data curation. **F. Godet:** Investigation, Formal analysis, Data curation. **C. Blanc:** Supervision, Project administration, Investigation, Formal analysis, Data curation. **X. Feugas:** Writing – review & editing, Writing – original draft, Visualization, Validation, Supervision, Resources, Project administration, Methodology, Investigation, Funding acquisition, Formal analysis, Data curation, 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.

Data availability

No data was used for the research described in the article.

Acknowledgments

The authors would like to acknowledge our colleagues G. Lotte E. Conforto and S. Cohendoz for their contributions to the TEM and mechanical tests facilities in the LaSIE Laboratory. Funding from the French Research National Agency (ANR) through the SLHyCC project (Contract ANR-19-0CE08-0027) is gratefully acknowledged. APT experiments were carried out owing the experimental platform GENESIS. GENESIS is supported by the Région Normandie, the Métropole Rouen-Normandie, the CNRS via LEBEX EMC, and the ANR as part of the program 'Investissement d'Avenir' under reference ANR-11-EQPX-0020.

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