# In Situ Investigation of Plasticity Mechanisms of the β Phase in (Ni, Pt)Al Bond Coats During Thermal Cycling by High-Energy X-Ray Diffraction

Mathias Lamari, Lara Mahfouz, Damien Texier, Loïc Courtois, Sylvain Gailliegue, Andrew King, Henry Proudhon, Vincent Maurel

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## To cite this version:

Mathias Lamari, Lara Mahfouz, Damien Texier, Loïc Courtois, Sylvain Gailliegue, et al. In Situ Investigation of Plasticity Mechanisms of the β Phase in (Ni, Pt)Al Bond Coats During Thermal Cycling by High-Energy X-Ray Diffraction. Metallurgical and Materials Transactions A, In press, 10.1007/s11661-025-08103-w. hal-05479468

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# *In Situ* Investigation of Plasticity Mechanisms of the *β* Phase in (Ni, Pt)Al Bond Coats During Thermal Cycling by High-Energy X-Ray Diffraction

MATHIAS LAMARI, LARA MAHFOUZ, DAMIEN TEXIER, LOÏC COURTOIS, SYLVAIN GAILLIEGUE, ANDREW KING, HENRY PROUDHON, and VINCENT MAUREL

The durability of thermal barrier coating (TBC) systems is strongly influenced by the interaction between oxidation of the metallic bond coat and its mechanical behavior. While the response of bond coats under isothermal monotonic loading has been widely studied, the effect of thermal cycling remains poorly understood, even though cyclic loading naturally arises from the mismatch in coefficients of thermal expansion between the ceramic top coat, thermally grown oxide, metallic bond coat, and the superalloy substrate. In this work, high-energy X-ray diffraction was used to investigate the strain and stress evolution in the *β*- (Ni, Pt)Al bond coat of a standard TBC deposited on a nickel-based single-crystal superalloy during thermal cycling. Before *in situ* cycling, some of the studied specimens were aged through long furnace cycles. Strains and stresses in the *β* phase were quantified *in situ* using the $sin^2ψ$ method combined with micromechanical modeling. The results reveal that plastic deformation in *β* is strongly controlled by evolving interfacial effects and by the cyclic *β* ⇌ *γ'* phase transformation during thermal cycling. These mechanisms govern the accumulation of plastic strain in *β* and may promote rumpling, spallation, and ultimately TBC degradation. This study provides new mechanistic insight into bond-coat plasticity under thermal cycling.

https://doi.org/10.1007/s11661-025-08103-w
© The Author(s) 2026

## I. INTRODUCTION

THE continuous increase in target temperature in aeroengine applications mandates the use of thermal barrier coatings (TBC) to protect most high-temperature components, including high-pressure turbine blades, vanes and combustion chambers.¹,² TBCs are multilayer systems comprising the substrate material to be protected, typically a single-crystal or polycrystalline

superalloy, a bond coat (BC) layer designed as an Al reservoir to form a dense and protective thermal grown oxide (TGO) layer. This TGO layer provides both passivation and adhesion for the ceramic top coat (TC) layer. The latter is, in most applications, partially yttria-stabilized zirconia (YSZ). The longest lifetimes are achieved using the electron-beam physical vapor deposition (EB-PVD) process, in which the columnar microstructure of the TC limits the detrimental effects of the coefficient of thermal expansion (CTE) mismatch between layers by reducing the effective stiffness of the ceramic TC layer,³ as seen in Figure 1(a) extracted from our previous work.⁴,⁵

It has been shown that the lifetime of TBC against spallation is largely governed by a strong coupling between the oxidation rate and the mechanical behavior of the BC layer.⁶,⁷ Some authors have therefore developed a highly complex strategy of sample extraction from the TBC to characterize mainly in isothermal tensile test the BC behavior.⁸⁻¹⁰ However, target component, and TBC spallation are typically subjected to thermo-mechanical fatigue (TMF). The most common lifetime testing relies on standard furnace cycling test (FCT), in which CTE mismatch induces out-of-phase cyclic loading that can be regarded as a form of TMF

MATHIAS LAMARI, SYLVAIN GAILLIEGUE, HENRY PROUDHON, and VINCENT MAUREL are with the Mines Paris, PSL University, Centre des matériaux, CNRS UMR 7633, 78000, Versailles, France. Contact e-mail: mathias.lamari@minesparis.psl.eu
LARA MAHFOUZ is with the Safran Tech, Etablissement Paris Saclay, rue des Jeunes Bois-Chateaufort, 78772 Magny-les-Hameaux, France. DAMIEN TEXIER is with the Univ Toulouse, IMT Mines Albi, INSA Toulouse, ISAE-SUPAERO, CNRS, ICA, Albi, France. LOÏC COURTOIS is with the 3Dmagination, Atlas Building, Fermi Avenue, Harwell Oxford, Oxford, UK. ANDREW KING is with the Synchrotron SOLEIL, L'Orme des Merisiers Départementale 128, 91190 Saint-Aubin, France.
Manuscript submitted October 2, 2025; accepted December 30, 2025.
<figure><img src="image_3.png" /><figcaption>Fig. 1—Microstructure evolution of the studied TBC system, TGO layer appears as a dark line between TC and BC in (a) and (d), and is hardly visible in other figures; (a) SEM observation of the as-received condition with top coat (adapted from Ref. [13]), (b) OM observation of 1200 S-FCT without top coat (adapted from Ref. [14]) (c) OM observation of 1000 L-FCT without top coat (adapted from Ref. 14) (d) SEM observation of the 1200 S-FCT with top coat (adapted from Ref. 13), (e) OM observation of 1200 S-FCT with top coat (adapted from Ref. 13) and (f) γ' phase fraction in the bond coat as function of time cumulated at maximum temperature for different conditions (results from Refs. 4 and 5) (Color figure online).</figcaption></figure>

loading.¹¹ Moreover, varying the dwell time at maximum temperature has a drastic impact on the TBC damage rate. For a given cumulative time at maximum temperature, short furnace cyclic tests (S-FCT) involve a larger number of thermal cycles compared with long furnace cyclic tests (L-FCT). In this case, a higher damage rate has been observed for S-FCT than for L-FCT, further demonstrating the fatigue nature of damage in TBC.¹² Consequently, there is a critical need to investigate the cyclic behavior of the BC layer.

For aluminide coating, it has been observed that high temperature exposure induces a significant evolution of the microstructure.¹⁵,¹⁶ Initially, the outer layer of the diffusion zone consists of β-NiAl phase, formed as a result of Al vapor deposition and Ni diffusion from the substrate.¹⁷ In parallel, both an outward flux of Al, leading to the formation of a protective alumina layer, and an inward flux of Al, driven by concentration equilibration with the substrate, are observed, which is characteristic of an interdiffusion process (Figure 1(a)). As a direct consequence, prolonged high temperature exposure promotes the transformation of β-NiAl into γ'-Ni₃Al.¹⁸

In addition, the interdiffusion zone (IDZ) situated between the initial β layer and the substrate initially consists of a β matrix containing precipitates known as topologically close-packed (TCP) phases (mainly

Cr-rich phases). With increasing high-temperature exposure, the β matrix in the IDZ progressively transforms into a γ' matrix.⁵ Under FCT, γ' preferentially forms at the initial β grain boundaries, whereas isothermal aging promotes the growth of a γ' matrix within the IDZ.⁵,¹⁹

These differences are clearly evidenced by comparing L-FCT and S-FCT for a similar number of cycles. For L-FCT, the IDZ is fully transformed into a γ' matrix and has partially consumed the initial outer β layer (Figure 1(c)), whereas for S-FCT, the IDZ remains largely unchanged compared with the as-received state, despite significant γ' precipitation in the outer layer (Figures 1(b) and (e)). These evolutions of microstructure are a priori not directly modified by the presence or absence of YSZ top coat layer, as shown by comparing Figures 1(b) and (e) for a similar number of S-FCT. This result is confirmed by the measurement of surface fraction $f_s(γ')$ of γ' in the β outer layer in Figure 1(f): the evolution of $f_s(γ')$ are similar with or without top coat (red and yellow curves), while a clear difference is observed between S-FCT and L-FCT (red and blue curves).

Finally, it has been demonstrated that during FCT, the TGO growth associated with strain accumulation is a first order parameter driving a significant increase in TGO surface roughness in the absence of ceramic top-coat, commonly referred to as the rumpling effect.²⁰
This effect is enhanced in presence of Pt in the bond coat composition and mitigated by the presence of the YSZ top coat layer, as the latter mechanically constrains the oxide surface.

Regarding damage in TBC systems, rumpling is considered to be one of the main driving force for decohesion between the top coat and the TGO during thermal cycling,¹⁶ even though undulations are limited by the presence of the top coat. Moreover, it has been demonstrated that phase transformations could drastically modify the behavior of TBC systems. The initial β phase being brittle up to 700 °C to 800 °C, the β to γ' transformation increases significantly the ductility of the material.¹⁰,²¹

It is also observed that prolonged exposure at high temperature may induce, to some extent, the formation of TCP phases through diffusion processes. These TCP phases may act as crack initiation sites under high-temperature creep loading, but are not expected to directly drive the strain of the outer β layer.

Aging of the material also results in modifications of the mechanical behavior of TBC systems. Thus, beyond the need of a robust methodology to assess TBC behavior under FCT, the impact of aging on these properties must also be addressed.

Aging of the material also results in modifications of the mechanical behavior of the TBC system. Therefore, beyond the need for a robust methodology to assess TBC behavior under furnace cyclic testing (FCT), the impact of aging on these properties must also be addressed. It is also observed that prolonged exposure at high temperature may, to some extent, induce the formation of topologically close-packed (TCP) phases through diffusion processes. These TCP phases may act as crack initiation sites under high-temperature creep loading, but are not expected to directly drive the strain of the outer β layer.

In the field of non-destructive techniques, X-ray diffraction is a valuable tool to assess *in situ* phase evolution during thermal cycling in advanced coatings.²² A major limitation of laboratory X-ray analysis, however, lies in its limited penetration depth, which restricts measurements to the near-surface region²²⁻²⁴ and prevents access to stress states representative of the bulk material (free-surface effect). To overcome this limitation, high-energy X-ray diffraction (HEXRD) has become a key method to study the evolution of phase, strain, and stress, either in grazing²⁵,²⁶ or transmission mode.²⁷,²⁸ *In situ* HEXRD has been successfully applied, for instance, to track strain evolution in TBCs subjected to thermo-mechanical fatigue (TMF) loading.²⁹⁻³¹ In these studies, HEXRD enabled monitoring the evolution of the TBC during combined thermal

cycling and mechanical loading with dwell times, mainly focusing on the YSZ top coat. However, the analysis was largely limited to global strain responses, without establishing a link to stress states in the individual layers or to the underlying plastic deformation mechanisms.

Based on HEXRD in transmission mode, this study investigates the evolution of straining in a standard TBC system, focusing on the β phase of the bond coat under pure thermal cycling (S-FCT). In addition, pre-aging by furnace cyclic testing was performed to assess the impact of microstructural evolution on TBC behavior. Three TBCs were studied, differing in the number of prior cycles (L-FCT), with the most extensively cycled specimen exhibiting top-coat spallation. These conditions provide complementary scenarios to analyze β deformation mechanisms. Using *in situ* HEXRD, the microstructural evolution of the bond coat and IDZ was quantified by the ratio intensity reference (RIR) method. The $\sin^2\psi$ method was applied to obtain lattice strains in β, while thermal strains were simulated using thermal expansion coefficients. This approach enabled the determination of the full elastic strain tensor and the associated stress tensor in β. The total strain in β was then modeled through micromechanical calculations, allowing its plastic strain to be extracted and compared to the evolution of peak full-width at half maximum (FWHM). The discussion highlights the plastic deformation mechanisms of β as cycling progresses, and emphasizes the key role of the interactions between plasticity and the β ⇌ γ' phase transformation.

## II. MATERIALS AND METHODS

### A. Material: A Thermal Barrier Coating System

The samples were obtained according to the SAFRAN coating standard and consisted of a 150 µm-thick YSZ top coat deposited by EB-PVD, a (Ni, Pt)Al diffusion bond-coat layer and a first-generation single-crystal nickel-based superalloy, the so-called AM1 (see composition in Table I). The AM1 substrate was solution heat-treated prior to coating deposition. The (Ni, Pt)Al bond coat was produced by successive Pt electrodeposition followed by Al vapor-phase aluminizing using the standard SAFRAN route, namely the Aluminizing Phase Vapor Snecma/Safran (APVS) process.³² The bond-coat composition is reported in Table II for the as-received condition and after isothermal aging, with measurements performed within the external layer.

The specimens were button shape, with a diameter of 25 mm, and a substrate thickness of 1 mm. For the HEXRD experiments, small rectangular samples were

<table><caption>Table I. Chemical Composition of Substrate Superalloy, AM1 in Wt Pct and At. Pct Except for Hf and S in ppm (from Ref. 35)</caption><thead><tr><th></th><th>Ni</th><th>Ta</th><th>Cr</th><th>Co</th><th>W</th><th>Al</th><th>Mo</th><th>Ti</th><th>Hf (ppm)</th><th>S (ppm)</th></tr></thead><tbody><tr><td>Wt Pct</td><td>bal.</td><td>7.97</td><td>7.54</td><td>6.61</td><td>5.49</td><td>5.2</td><td>2.01</td><td>1.2</td><td>460 to 490</td><td>0.104 to 0.123</td></tr><tr><td>At. Pct</td><td>bal.</td><td>2.65</td><td>8.74</td><td>6.76</td><td>1.80</td><td>11.61</td><td>1.26</td><td>1.51</td><td>155 to 165</td><td>0.195 to 0.231</td></tr></tbody></table>
<table><caption>Table II. Chemical Composition of the (Ni, Pt)Al External Layer in At. Pct (from Ref. 36)</caption><thead><tr><th></th><th>Ni</th><th>Ta</th><th>Cr</th><th>Co</th><th>W</th><th>Al</th><th>Mo</th><th>Ti</th><th>Pt</th></tr></thead><tbody><tr><td>As-received</td><td>bal.</td><td>1.0</td><td>1.0</td><td>4.6</td><td>0.0</td><td>35.2</td><td>0.0</td><td>0.0</td><td>7.3</td></tr><tr><td>After 192 h at 1100 °C (Isothermal)</td><td>bal.</td><td>0.8</td><td>6.0</td><td>4.8</td><td>0.4</td><td>24.5</td><td>0.0</td><td>1.0</td><td>5.0</td></tr></tbody></table>

<table><caption>Table III. Samples Nomenclature and State Before HEXRD (Associated to L-FCT), and In Situ Cycling (Associated to S-FCT)</caption><thead><tr><th>Name</th><th>Pre-aging (Number of L-FCT)</th><th>Low Temperature S-FCT</th><th>High Temperature S-FCT</th><th>YSZ</th></tr></thead><tbody><tr><td>As-received (AR)</td><td>0</td><td>4</td><td>3</td><td>no spallation</td></tr><tr><td>Aged (N = 600)</td><td>603</td><td>2</td><td>3</td><td>no spallation</td></tr><tr><td>Spalled (N = 1000)</td><td>1029</td><td>2</td><td>3</td><td>full spallation</td></tr></tbody></table>

cut from these button-shaped specimens. The same TBC system has been extensively investigated in several previous studies, which detailed the microstructure evolution, the interfacial damage and top-coat spallation under various loading conditions.¹¹,¹³,³³,³⁴ The same substrate and bond coat have also been carefully analyzed in previous work, allowing a clear assessment of phase transformations under FCT with different dwell times at the maximum temperature (see, for example, Refs. 5 and 18).

Prior to the HEXRD experiments, long furnace cyclic tests (L-FCT) were performed to investigate the effect of phase transformations on the mechanical behavior of the bond coat (BC). Each L-FCT consisted of a 50 minutes dwell at 1100 °C followed by natural air cooling. Three samples were investigated in the present work. The first sample corresponds to the as-received state and did not undergo any thermal cycling after fabrication. The second sample was subjected to 603 L-FCT prior to the synchrotron experiments at 1100 °C. Finally, the third sample experienced 1029 L-FCT at 1100 °C leading to spallation of the top-coat layer. These samples are hereafter referred to as “as-received” (AR), “aged” (N = 600, where N denotes the approximate number of pre-cycles), and “spalled” (N = 1000), respectively, and are summarized in Table III.

## B. High Energy X-Ray Diffraction Setup (HEXRD)

The HEXRD experiments were performed at the SOLEIL synchrotron source, using the beamline Psiché. A schematic of the experimental setup is shown in Figure 2, along with a description of the sample and the attached reference frame. The incident X-ray beam was high-energy and monochromatic, with an energy of 45 keV. Its section was 0.5 × 0.5 mm². The experiments were conducted in transmission geometry, with the beam traversing all layers of the sample. The sample was placed inside a furnace. The direction normal to the sample is denoted as 2, while the in-plane directions are denoted as 1 and 3 and are assumed to be equivalent.

A home-made special furnace was designed for this experiment so as to carry out short furnace cyclic test (S-FCT) during *in situ* measurement by HEXRD.

Photographs of the furnace, the samples and the whole setup are given in Figure 3. The samples were glued with a ceramics glue on alumina rods. A S-type thermocouple was attached to the sample holder to ensure precise temperature control. The furnace core consists of stacked rigid cylindrical insulating elements, surrounded by flexible refractory insulation rated for 1600 °C, and enclosed in a rigid double-wall water-cooled casing. This casing was manufactured at the Centre des Matériaux by Laser Powder Bed Fusion (LPBF) on a Concept Laser M2 system using 316L stainless steel. Heating is provided by $MoSi_2$ resistances (350 W) with a maximum operating temperature of 1500 °C. At 1400 °C, the furnace achieves a temperature homogeneity of ±1 °C over 6 mm along the sample's Z axis. The design also allows vertical translation of the furnace (120 mm stroke) for sample release, as well as a slight tilt of the base to align the furnace and sample axes.

The thermal treatment consisted of two temperature ranges: 2 to 4 cycles between 400 °C and 900 °C and 3 cycles between 400 °C and 1100 °C. Each cycle lasted about 20 minutes, with dwell-time at maximum temperature of about 5 minutes. For each cycle, a heating rate of 2 1 °C/s and an average cooling rate of 1.35 1 °C/s were applied. The three experimental cycles are shown in Figure 4. For each cycle, a specific color code was chosen to enhance the readability of subsequent graphs. At the end of each cooling phase, the color shade gradually darkens as the thermal treatment progresses. Between the low-temperature and high-temperature cycles, the color scheme is reset, as indicated in Figure 4, to improve the clarity of temperature-dependent data presented later.

It is worth noting that S-FCT are assumed to induce similar amount of plastic straining during one cycle as compared to L-FCT, assuming that CTE mismatch between layers is the main driving force for straining. But the phase transformation for S-FCT promotes $\gamma'$ precipitation at the initial $\beta$ grain boundaries, whereas for L-FCT transformation is more homogeneous,⁴,⁵ see Figure 1.
<figure><img src="image_4.png" /><figcaption>Fig. 2—Schematics of the sample and the experimental setup. The possible layers within the sample are indicated. The elements in parentheses in the bond coat are not necessarily present. In the setup, the sample was placed in a furnace, orthogonally to the incident beam produced by the synchrotron.</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>Fig. 3—Photographs of the furnace, the samples mounted on their alumina rods, and the experimental setup.</figcaption></figure>

### C. *Integration of 2D Patterns*

The 2D diffraction patterns, consisting of Debye-Scherrer rings (Figure 2), were integrated using the pyFAI library to obtain 1D diffractograms.³⁷ The 2D plate detector operated at an acquisition rate of about 0.07 Hz, yielding roughly 500 images for each thermal treatment. These patterns capture the evolution of both the microstructure and the internal stress state throughout the entire TBC.

Full-radial integrations were performed to determine the phase fractions of the different samples. To evaluate internal stresses, partial integrations of the Debye-Scherrer rings were carried out over azimuthal

sectors (one such sector is sketched in Figure 2). Two sector apertures were selected: 4 deg (yielding 90 sectors) and 11.25 deg (32 sectors), which also allowed testing the robustness of the subsequent analyses. Each sector in an analysis is identified by an azimuthal angle $\chi$, defined as the angle between the vertical of the diffraction image and the center of the sector, as indicated in Figure 2. Such partial integrations enable the construction of $\sin^2 \psi$ curves by integrating the 2D pattern over a small azimuthal angle, and they also allow the determination of peak full width at half maximum (FWHM) while minimizing the influence of deviatoric internal stresses, which can artificially broaden the peaks in the case of full radial integration. Opposite sectors of the
<figure><img src="image_6.png" /><figcaption>Fig. 4—Cyclic thermal treatment for (a) as-received, (b) aged (N = 600), and (c) spalled (N = 1000) specimens. For each curve, every color change marks the end of the cooling step of a cycle (Color figure online).</figcaption></figure>

diffraction patterns were summed to reduce experimental noise in the $\sin^2 \psi$ analysis.

### D. Ratio Intensity Reference Method

Phase fractions of $\beta$ and $\gamma'$ were quantified using the Reference Intensity Ratio (RIR) method, which compares the ratios of integral breadths of selected diffraction peaks to a reference ratio for quantitative analysis of diffractograms.³⁸ For each phase, a single representative peak was chosen. For $\beta$ and $\gamma'$ phases, $[110]_\beta$ and $[111]_\gamma'$ were respectively selected, as they allow to avoid convolution issue with neighboring peak (all other peaks were convoluted with YSZ peaks). The integral breadth of each selected peak was determined by fitting a Pseudo-Voigt profile. Let $A_i$ denote the integral breadth of the peak associated with phase $i$ ($i = \beta$ or $\gamma'$) and $f_i$ its volume fraction. In the RIR formalism, the ratios between phase fractions and their corresponding integral breadths are assumed to be proportional. We define $P_{\beta\gamma'}$ as the proportionality factor linking the ratio of integral breadths $A_{\gamma'}/A_\beta$ to the ratio of phase fractions $f_{\gamma'}/f_\beta$³⁸:

$$ \frac{A_{\gamma'}}{A_{\beta}} = P_{\beta\gamma'} \frac{f_{\gamma'}}{f_{\beta}}. \quad [1] $$

The parameter $P_{\beta\gamma'}$ was determined from reference diffractograms for which the phase fractions were obtained by Rietveld refinement, following a similar methodology as in our previous work.³⁹,⁴⁰ Once calibrated, this factor allows direct calculation of $f_\beta$ and $f_{\gamma'}$ during the thermal treatments from the measured integral breadths of the selected peaks. For the aged sample (N = 600), a value of $P_{\beta\gamma'} = 1.05$ was obtained by refining a diffractogram midway through the treatment (2760 seconds, in the middle of the first high-temperature cycle at 1100 °C). For the spalled sample (N = 1000), refining a diffractogram at the same time point yields $P_{\beta\gamma'} = 1.23$. This change in $P_{\beta\gamma'}$ reflects the

evolution of texture of $\beta$ and $\gamma'$ phases after a high number of cycles.

In the as-received sample, however, very little $\gamma'$ phase is present, and the Rietveld refinement diverges because the $\gamma'$ peaks have intensities comparable to YSZ, $Al_2O_3$ and TCP phases). Therefore, we also chose $P_{\beta\gamma'} = 1.05$ to analyse this condition.

For both the aged and spalled samples, Rietveld refinements were performed at the start (550 seconds) and the end (5250 seconds) of the thermal treatments to evaluate the error associated with using the RIR method. Compared with the Rietveld refinement, the RIR method gives a maximum absolute error of 0.02 on $f_\beta$.

### E. $\sin^2 \psi$ Method

The stress state in the $\beta$ phase was determined using the $\sin^2 \psi$ method.⁴¹ The detailed procedure follows the approach described in previous work.³⁹,⁴⁰ This method requires measuring the $2\theta$ position of a family of lattice planes whose interplanar distance is representative of the elastic strain within the bulk. The $[110]_\beta$ peak was selected, as it is one of the few $\beta$ peaks that remains fully unconvoluted from other reflections. Peak positions $2\theta$ were obtained by fitting the experimental profiles with a pseudo-Voigt function.

For a given crystal plane with normal $n_{\phi\psi}$, we denote $\psi$ as the angle between direction 3 and $n_{\phi\psi}$, and $\phi$ as the angle between direction 1 and the projection of $n_{\phi\psi}$ onto the $O_{12}$ plane, Figure 2. The two angles are indicated in Figure 2. In transmission geometry, the strain component along $n_{\phi\psi}$ is then determined as:

$$ \epsilon_{\phi\psi} = \ln\left(\frac{\sin\theta_0}{\sin\theta}\right) \quad [2] $$

with $\theta_0$ the stress-free position of the considered peak. The associated value of $\sin^2 \psi$ is then⁴¹,⁴²:
$$ \sin^2 \psi = 1 - \cos^2 \theta(1 - \sin^2 \chi). \quad [3] $$

A typical $\sin^2 \psi$ ellipse obtained at the beginning of the treatment on the aged sample for the [110]$_{\beta}$ peak is shown in Figure 5(a) as black dots. Experimentally, since $\theta_0$ remains unknown, what is actually measured is $\varepsilon_{\phi\psi} + \varepsilon_0$, where $\varepsilon_0$ is a constant defined as $-\ln(\sin \theta_0)$. Each point of the ellipse characterizes the strain state of the [110]$_{\beta}$ plane, averaged over two sectors defined by the angles $\chi$ and $\chi + 180$ deg. The opening of the ellipse is related to the presence of shear strain. The mean deformation of the ellipse corresponds to the hydrostatic strain of the planes, while the slope of the extremal points mainly reflects the differences between the principal strain components. Accurate determination of $\varepsilon_0$ is thus critical for a reliable evaluation of the hydrostatic stress.

As no rotation was applied to the specimen during the HEXRD experiments, the $\sin^2 \psi$ method enables reconstruction of only three components of the strain tensor. However, since directions 1 and 3 are equivalent and thermal gradients are expected to mainly generate hydrostatic stresses, an equibiaxial state can be assumed in the $O_{13}$ plane (plane parallel to the different layers):

$$ \varepsilon_{11} = \varepsilon_{33}, \quad [4] $$

$$ \varepsilon_{12} = \varepsilon_{23}, \quad [5] $$

$$ \varepsilon_{13} = 0. \quad [6] $$

The full $\sin^2 \psi$ equation can be written as$^{41}$:

$$ \varepsilon_{\phi\psi} = [(\varepsilon_{11} \cos^2 \phi + \varepsilon_{12} \sin 2\phi + \varepsilon_{22} \sin^2 \phi) - \varepsilon_{33}] \sin^2 \psi + (\varepsilon_{13} \cos \phi + \varepsilon_{23} \sin \phi) \sin 2\psi + \varepsilon_{33}. \quad [7] $$

Considering the simplifications given by Eqs. [4] through [6], this reduces to:

$$ \varepsilon_{\phi\psi} = [((\varepsilon_{22} - \varepsilon_{11}) \sin^2 \phi + \varepsilon_{12} \sin 2\phi)] \sin^2 \psi + \varepsilon_{12} \sin \phi \sin 2\psi + \varepsilon_{11}. \quad [8] $$

In transmission geometry, the evolution of $\phi$ is linked with the angles $\theta$ and $\psi$ by$^{41}$:

$$ \phi = \arccos\left(\frac{\sin \theta}{\sin \psi}\right). \quad [9] $$

In Figure 5(b), the evolution of $\phi$ as a function of $\sin^2(\psi)$ is shown. In the range $0 \le \sin^2(\psi) \le 0.65$, the angle $\phi$ remains nearly constant, and we have:

$$ \sin(\phi) \approx 1 \quad \text{since} \quad \phi \approx 90^\circ \quad [10] $$

Therefore, only analysing the ellipse in the portion $0 \le \sin^2(\psi) \le 0.65$ is easier, since the $\sin^2(\psi)$ equation simply becomes:

$$ \varepsilon_{\phi\psi} = (\varepsilon_{22} - \varepsilon_{11}) \sin^2 \psi + \varepsilon_{12} \sin 2\psi + \varepsilon_{11}. \quad [11] $$

A $\sin^2(\psi)$ ellipse is characterized by its intercept $\varepsilon_{\phi\psi}^0$, its slope $\Delta\varepsilon_{\phi\psi}$, and its aperture. To determine the first two parameters, a linear regression of the ellipse is performed, as shown in red in Figure 5(a). It follows that:

<figure><img src="image_7.png" /><figcaption>Fig. 5—(a) A typical ellipse (black dots) obtained at the beginning of the treatment of the 600°C sample, for the [110]$_{\beta}$ peak. The red line corresponds to the linear regression of the ellipse, giving the intercept and the slope of the ellipse. The blue ellipse corresponds to the theoretical curve obtained after least squares regression. The slope $\Delta\varepsilon_{\phi\psi}$ (purple) corresponds to the difference between the value of the linear regression at $\sin^2(\psi) = 0$ and 1. (b) Evolution of $\phi$ as a function of $\sin^2(\psi)$. The area between $\sin^2(\psi) = 0$ and 0.65 corresponds to the region where $\sin(\phi)$ is approximately 1 (Color figure online).</figcaption></figure>
$$ \epsilon_{11} = \epsilon_{\phi\psi}^0 \quad \text{and} \quad \epsilon_{22} = \epsilon_{\phi\psi}^0 + \Delta\epsilon_{\phi\psi}. \qquad [12] $$

The aperture and $\epsilon_{12}$ is then obtained by applying a least-squares regression using Eq. [11]. The result is shown in blue in Figure 5(a). To obtain hydrostatic values, it is necessary to evaluate the constant $\epsilon_0$. As no reference are available to experimentally determine it, it was chosen to make the logical assumption that all the hydrostatic stresses are relaxed at the end of the first high temperature isothermal holding. At this time, we thus deduce that:

$$ \epsilon_0 = \epsilon_{\phi\psi}^0 + \Delta\epsilon_{\phi\psi}/3. \qquad [13] $$

Elastic strains were converted into stresses using Hooke's law. The directional Young's modulus $E_{[110]}$ of $\beta$ at room temperature was determined using the Reuss model$^{43,44}$ and from the components of the elastic tensor $C_{ij}$ calculated by density functional theory (DFT) on stoichiometric $\beta$-NiAl.$^{45}$ A value of 225 GPa was obtained. Since most of the thermal treatments occur between 400 °C and 1100 °C, an average value of $E_{[110]}$ representative of the experimental temperature range was adopted, namely the value at 800 °C. Direct calculation of $E_{[110]}$ at this temperature is not possible because the elastic constants $C_{ij}$ are not available. However, for the bulk modulus $E$ of the $\beta$ phase, a decrease from $E(RT) = 200$ GPa to $E(800 °C) = 145$ GPa is reported,$^{46-48}$ corresponding to a ratio of 1.37. Applying this ratio to $E_{[110]}$ gives an estimated value of $E_{[110]} \approx 163$ GPa, which is used in the following analysis. For the Poisson ratio $\nu$, a constant value of 0.26 is assumed$^{45}$ (based on Reuss model$^{43,44}$).

Numerous other values of $C_{ij}$ are reported in the literature, obtained from experiments or simulations,$^{49-52}$ depending on the stoichiometry and on the method used to determine the coefficients. Accounting for these alternative sets of $C_{ij}$ coefficients modifies the calculated hydrostatic stress by only $\pm 10$ pct, and may either leave the von Mises stress values essentially unchanged or reduce them by a factor of about 2, and up to a factor of 5 in the most extreme cases. As shown below, these stress variations do not alter the conclusions drawn in this article and, if anything, even reinforce them.

### III. RESULTS

#### A. Microstructural Evolution of the Thermal Barrier

The diffractograms obtained for the three experiments are shown in Figures 6(a) through (c). Peak indexing was performed on Diffrac.EVA software from Brucker, using the Crystallography open database and PDF-5+2026 database, and by comparing from diffrac-tograms found in the literature.$^{23,27,53-55}$ Several peaks could not be indexed in the diffractograms and are likely associated with minor phases such as TCP phases,$^{56}$ and their analysis is therefore neglected for simplicity in the following.

For the as-received sample, the thermal barrier consists primarily of $\beta$ in the external bond coat and in the interdiffusion zone (IDZ), and yttria-stabilized zirconia in the top coat. Small peaks related to alumina are visible, indicating a slight presence of the TGO even in the initial state. This point is fully consistent with the standard pre-oxidation treatment used before YSZ deposition.$^{57,58}$ Only a very small amount of $\gamma'$ phase is detected, and no martensitic NiAl is observed.

For the aged sample (exposed to 600 L-FCT before HEXRD), a significant amount of $\gamma'$ has nucleated in the bond coat and in the IDZ. Alumina-related peaks are now clearly visible. By zooming on the $[111]_{\gamma'}$ and $[110]_{\beta}$ peaks at the end of the last thermal cycle (1100 °C), a third peak appears between the two others, as shown in Figure 6(d), which can be attributed to $L_{10}$ martensitic NiAl, with $L_{10}$ crystallography.$^{53,59}$ This does not mean that there is no martensitic NiAl at lower temperature, only that it is too convoluted with to $[111]_{\gamma'}$ and $[110]_{\beta}$ peaks to be clearly distinguished. This observation is consistent with the literature for this temperature range.$^{53,59}$

Finally, for the spalled sample (exposed to 1000 L-FCT before HEXRD), even more $\gamma'$ is present relative to $\beta$ in the whole thermal barrier. Again, some martensitic NiAl is also found at high temperature (see Figure 6(e)). Peaks associated with the TGO are more pronounced. However, all peaks corresponding to tetragonal yttria-stabilized zirconia have disappeared, indicating a total spallation of the top coat.

To evaluate more quantitatively the evolution of the bond coat, the RIR method was applied using the $[111]_{\gamma'}$ and $[110]_{\beta}$ peaks. The evolution of the relative phase fractions of both phases inside the bond coat is shown in Figure 7. Although martensite is also present in the bond coat, it was not considered in this calculation.

Overall, the as-received specimen contains more $\beta$ phase in the bond-coat than the aged specimen, which itself contains more $\beta$ than the spalled specimen. This trend is consistent with previous studies reporting that the greater the time spent at high temperature, and consequently the number of thermal cycles, the more $\gamma'$ is formed, Figure 1.$^{5,18,60}$ At the initial stage of the aged and the spalled specimens, the bond coat and the IDZ are predominantly $\gamma'$, with fractions of 55 and 59 pct, respectively. Then, upon heating in all three experiments, the $\beta$ phase becomes predominant throughout the thermal barrier. For a given specimen, the higher the temperature during an applied short furnace cycle, the higher the fraction of $\beta$, in agreement with the binary Ni-Al equilibrium phase diagram.$^{53,61}$

Finally, during low-temperature cycles, the transformation kinetics are nearly symmetric between the cooling and heating stages (Figure 7(b)), whereas a clear hysteresis is observed for high-temperature cycles, particularly for the aged specimen (Figure 7(c)). In these cycles, during the heating stage the $\beta$ fraction increases abruptly only at temperatures above 1000 °C, whereas during cooling it decreases continuously from 1100 °C down to 700 °C.
<figure><img src="image_8.png" /><figcaption>Fig. 6—All diffractograms obtained for the (a) as-received, (b) aged (N = 600), and (c) spalled (N = 1000) samples during the thermal cycles. The temperature corresponding to each diffractogram is indicated by its color. A zoom on the [111]*<sub>v</sub> and [110]*<sub>β</sub> peaks at the end of the final high-temperature cycle is shown for (d) aged and (e) spalled samples, highlighting the presence of a [111]*<sub>mart</sub> peak (Color figure online).</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>Fig. 7—Evolution of β phase fractions f<sub>β</sub> as a function of (a) time and (b) temperature during the low temperature cycles and (c) high temperature cycles for the three specimens. For each curve, every color change marks the end of the cooling step of a cycle (see Fig. 4) (Color figure online).</figcaption></figure>

## B. *Strain Evaluation*

From synchrotron data, using the sin<sup>2</sup> ψ method, it is possible to extract the lattice strains, i.e., the sum of elastic and thermal strains. Plastic strains cannot be directly measured by synchrotron diffraction. For clarity, we denote ε<sub>11</sub><sup>XRD</sup> as the 11 component of the strain measured by XRD with the sin<sup>2</sup> ψ method, ε<sub>11</sub><sup>th</sup> as the thermal strain (assumed to be hydrostatic), and ε<sub>11</sub><sup>el</sup> as the 11 component of the elastic strain. Elastic and thermal strains cannot be separated directly from the

experimental measurements. Therefore, the thermal strain was modeled using the coefficients of thermal expansion (CTE) reported in Table IV and extracted from Refs. 46 through 48. For intermediate temperatures, a cubic spline interpolation was applied to obtain values of CTE. Since the thermal barrier coatings of the three specimens were initially formed at very high temperature, the thermal strain was set to 0 at 1100 °C in this work. Based on this modeling, the three components of the elastic strain tensor are then given by:
$$ \epsilon_{11}^{el} = \epsilon_{11}^{XRD} - \epsilon^{th}, \quad [14] $$

$$ \epsilon_{22}^{el} = \epsilon_{22}^{XRD} - \epsilon^{th}, \quad [15] $$

$$ \epsilon_{12}^{el} = \epsilon_{12}^{XRD}. \quad [16] $$

The evolutions of the 11 components of the three strain contributions are shown in Figure 8 as a function of time for the three specimens. For all samples, similar trends are observed. The measured strain closely follows the thermal strain, with the effects of the thermal cycles clearly visible: heating induces expansion, whereas cooling induces contraction. The systems remain globally in compression throughout the test, except during the dwell-time at high-temperature. The elastic strain exhibits more complex behavior, showing a less linear dependence on temperature. The mechanical strain, which originates from the difference in CTE between the β phase and the other phases of the system, tends to be positive in the β phase at high temperature and compressive at low temperature. This aspect will be analyzed in more detail in the discussion of hydrostatic and von Mises stresses below.

For conciseness, it should be noted that the strain components determined along the 33 direction are very close to those along the 11 direction, consistent with a nearly hydrostatic strain state within the probed volume, mainly induced by thermal expansion. Shear strains $ε_{13}$

<table><caption>Table IV. Evolution of the CTE Used in this Work as a Function of Temperature for Each Relevant Phase$^{46-48}$</caption><thead><tr><th rowspan="2">Phase</th><th colspan="5">CTE (10<sup>-6</sup> K<sup>-1</sup>)</th></tr><tr><th>RT</th><th>600 °C</th><th>800 °C</th><th>1000 °C</th><th>1100 °C</th></tr></thead><tbody><tr><td>β</td><td>13.6</td><td>15.2</td><td>16.1</td><td>17.2</td><td>17.6</td></tr><tr><td>AM1</td><td>8.0</td><td>13.0</td><td>14.0</td><td>15.4</td><td>15.8</td></tr></tbody></table>

are also detected (on the order of 10<sup>-4</sup>). Although small, these shear strains have a significant influence on the von Mises stresses, as discussed afterwards.

### C. Hydrostatic Stress

The hydrostatic stress deduced from the $\sin^2 \psi$ analysis is shown as a function of temperature for the three experiments in Figure 9, during the low-temperature short cycles (400 °C to 900 °C, Figure 9(a)) and the high-temperature short cycles (400 °C to 1100 °C, Figure 9(b)). At room temperature before the thermal treatment, the β phase of the as-received specimen is nearly stress-free, whereas the aged and spalled specimens exhibit a highly compressive hydrostatic stress. After the first heating, the initial compressive stresses inherited from the pre-cycles disappear, and the three specimens subsequently behave similarly during the low-temperature short cycles, oscillating between an almost zero stress at 400 °C and a tensile stress at 900 °C. A pronounced hysteresis is observed in all cycles: for a given temperature, the stress during heating is higher than during cooling. During the isothermal dwell-time at maximum temperature (for either 900 °C or 1100 °C), stress is observed to relax. The stress relaxation is partial at 900 °C, and is almost total at 1100 °C for the tested 5 minutes dwell-time. The hysteresis loops are more pronounced for the specimens that experienced a larger number of pre-cycles, which is consistent with a higher $\gamma'$ fraction in the bond coat, which increases the complexity of the loading state. When the top coat is delaminated (spalled specimen), the hydrostatic stress at 900 °C is higher, in agreement with a compressive effect of the top coat on the bond coat, as YSZ has a lower coefficient of thermal expansion than $\beta$.47,48,62,63

The evolution of hydrostatic stress during the high-temperature cycles is quite different from that observed in the low-temperature cycles, as shown in Figure 9(b). For the as-received specimen, the hydrostatic stress is initially tensile, before fully relaxing to zero at the end of the first isothermal dwell at 1100 °C. This behavior is expected, as it reflects the reference

<figure><img src="image_10.png" /><figcaption>Fig. 8—Evolution of the planar measured strain $\epsilon_{11}^{XRD}$, the thermal strain $\epsilon^{th}$ and the planar elastic strain $\epsilon_{11}^{el}$ of the β phase as a function of time for (a) as-received, (b) aged and (c) spalled samples.</figcaption></figure>
<figure><img src="image_11.png" /><figcaption>Fig. 9—Evolution of the hydrostatic stress in $\beta$ as a function of temperature during (a) the low temperature cycles and (b) the high temperature cycles for the three specimens. For each curve, every color change marks the end of the cooling step of a cycle (see Fig. 4) (Color figure online).</figcaption></figure>

value used in the $\sin^2 \psi$ method. After this step, the hydrostatic stress oscillates between –250 and 50 MPa, once again highlighting a hysteresis similar to that observed in the low-temperature cycles.

For the aged specimen, the hysteresis is more pronounced. The $\beta$ phase is under hydrostatic compression of approximately –50 MPa at 400°C, then transitions to hydrostatic tension at intermediate temperatures, reaching up to 400 MPa. At high temperatures, the stress approaches nearly complete relaxation by the end of the isothermal dwell.

For the spalled specimen, the hysteresis loops are even more pronounced, with compressive stresses at low temperature (400 °C) reaching –500 MPa and tensile peaks up to 500 MPa. Interestingly, the relaxation during heating begins around 1000 °C for both the aged and spalled specimens, corresponding to the dissolution of $\gamma'$ into $\beta$. This phase transformation logically reduces stresses associated with the difference in thermal expansion coefficients between the two phases, as well as transformation strains linked to the molar volume differences between $\beta$ and $\gamma'$.⁶⁴

### D. *von Mises Stress*

The von Mises stress for the three specimens is shown in Figure 10 as a function of temperature during the low-temperature cycles (Figure 10(a)) and high-temperature cycles (Figure 10(b)). For all specimens and all cycles, the von Mises stresses exhibit significantly smaller variations compared to the hydrostatic stresses previously presented.

For the as-received specimen, during low-temperature cycles, the von Mises stress is around 60 MPa. For a given temperature, it is slightly higher during heating than during cooling. During the high-temperature cycles, the hysteresis loop becomes more pronounced, with stresses oscillating between 60 and 70 MPa.

For the aged and spalled specimens, the von Mises stresses are very low after the first low-temperature annealing, on the order of 10 to 20 MPa. A pronounced hysteresis loop is observed for the aged specimen during both low- and high-temperature cycling, while for the spalled specimen, the hysteresis is present but more subtle and mainly noticeable during the high-temperature cycles.

For all specimens, it is interesting to note that during isothermal dwell-time at 900°C, no relaxation of deviatoric stresses occurs, whereas during dwell-time at 1100°C, the von Mises stresses clearly decrease by about 10 MPa. This behavior could be explained by recovery/recrystallization phenomena activated at higher temperatures.

### E. *Full Width at Half Maximum (FWHM)*

The evolution of the full width at half maximum (FWHM) for the three samples is shown as a function of temperature in Figure 11. It was determined on the $[1\bar{1}0]_{\beta}$ peak using full radial integration. Interestingly, we also verified that measuring the FWHM on all sectors during partial integrations, and then averaging, yields identical values. This demonstrates the absence of significant deviatoric stresses and confirms that the $\beta$ phase is primarily subjected to hydrostatic stress.

For the as-received sample, whether during low- or high-temperature cycles, the FWHM evolves very linearly, ranging from 0.054 deg at 400°C to 0.0501 deg at 1100°C. For the aged and spalled samples, hysteresis loops appear in the FWHM evolution, both at low and high temperatures. These loops are more pronounced for the aged specimen than for the spalled one. For a given phase, the FWHM of its peaks can increase due to two phenomena: first, the increase in the density of crystal defects, notably dislocations, and second, the reduction of crystallite sizes, for instance due to recrystallization. These aspects are further discussed below.
<figure><img src="image_12.png" /><figcaption>Fig. 10—Evolution of the von Mises stress in β as a function of temperature during (a) the low temperature cycles, and (b) high temperature cycles for the three specimens. For each curve, every color change marks the end of the cooling step of a cycle (see Fig. 4) (Color figure online).</figcaption></figure>

<figure><img src="image_13.png" /><figcaption>Fig. 11—Evolution of the full width at half maximum (in 1 deg) of [110]$_{β}$ peak as a function of temperature during (a) the low temperature cycles and (b) high temperature cycles for the three specimens. For each curve, every color change marks the end of the cooling step of a cycle (see Fig. 4) (Color figure online).</figcaption></figure>

IV. DISCUSSION

A. Plastic Deformation Associated with β Phase

Plastic strain is not directly accessible from X-ray diffraction measurements. However, in the case of thermal barrier coating systems, it is well established that the main source of mechanical loading in the thin layers, including the β layer, arises from the thermal strain of the Ni-base superalloy single crystal substrate, whose CTE is reported in Table IV for AM1. In other words, the total strain could be assumed to be prescribed by the substrate to the other layers. Thus, if we denote $\epsilon_{sub.}^{th}$ as the thermal strain tensor of the substrate, and $\epsilon^{pl}$, $\epsilon^{el}$ and $\epsilon^{th}$ as the plastic, elastic and thermal strain tensors of β, one has:

$$ \epsilon_{sub.}^{th} = \epsilon^{pl} + \epsilon^{el} + \epsilon^{th}. \quad [17] $$

In this work, $\epsilon_{sub.}^{th}$ is computed following the same approach as for $\epsilon^{th}$ described earlier. By inverting Eq. [17], the plastic strain tensor of β can thus be obtained. From these, the accumulated plastic strain $p_{cum}$ in β is determined following the classical relation$^{65}$:

$$ p_{cum} = \int_{0}^{t} \sqrt{\frac{2}{3} \dot{\epsilon}^{pl} : \dot{\epsilon}^{pl}} dt. \quad [18] $$

The evolution of $p_{cum}$ is shown in Figure 12(a) for the three specimens, for the whole in situ thermal cycling (low-temperature cycling, and high-temperature cycling).
<figure><img src="image_14.png" /><figcaption>Fig. 12—Evolution of the accumulated plastic strain in β as a function of time: (a) for the three specimens. For each curve, every color change marks the end of the cooling step of a cycle (see Fig. 4). (b) Zoom on the evolution of $p_{cum}$ during the first of the high temperature cycle of the as-received sample. Change in the stage of the cycle (holding at 400°C/heating/holding at 1100°C/cooling) are indicated by vertical bars (Color figure online).</figcaption></figure>

cycling). Plastic strain in the β phase increases in all samples but not in a linear manner. Three main regimes can be distinguished: (i) abrupt increases in cumulative plastic strain at the beginning of each heating stage, (ii) significant but slightly smaller increases during cooling, and (iii) stagnation during the plateaus at 400°C, 900°C, or 1100°C, see details for the as-received sample in Figure 12(b). For all specimens, the increase of $p_{cum}$ per cycle is lower during low-temperature cycling (about 0.006 strain increment per cycle) than during high-temperature cycling (between 0.007 and 0.009 strain increment per cycle). These plastic strain levels per cycle are consistent with simulation results reported in the literature.⁶⁶ Pre-cycling appears to reduce the plastic strain per cycle in the β phase when comparing as-received and aged samples, whereas YSZ delamination strongly increases it when comparing aged and spalled samples. These two aspects are discussed in more detail in the next sections.

When comparing the measured plastic strain with the von Mises stress, an apparent discrepancy emerges. As shown in Figure 12(b), plasticity mainly occurs during the transient stages of the thermal cycles (heating and cooling), at temperatures below 1000°C. However, for all samples, the von Mises stresses remain below 100 MPa, which is insufficient to induce plasticity below 1000°C.¹⁰,⁶⁷ The resolution of this apparent paradox lies in the fact that, at the synchrotron, the probed volume spans the entire BC over an area of about 0.5 × 0.5 mm², which averages out local variations. As a result, while the average von Mises stress may remain below the yield strength, local stresses, particularly near the AM1 and TGO interfaces, can be much higher. This strongly suggests that interfaces play a critical role in the plasticity of the β phase.

## B. Impact of Metallurgical Transformation on Plasticity

For a given sample, the FWHM remains roughly cyclic even though the cumulative plastic strain of the β phase increases steadily with each cycle. This indicates that the dislocation density does not continuously rise, which could be a priori attributed to metallurgical transformations such as recovery,⁶⁸ recrystallization,⁵ or $\beta \rightleftharpoons \gamma'$ transformation, all of which can absorb or annihilate dislocations. The β grain size is expected to remain the same in all tested samples, since even 1000 L-FCT at 1100°C are insufficient to alter the β morphology. Moreover, recrystallization should logically lead to a reduction of crystallite size. This is in contradiction with the evolution of FWHM between specimens observed at 1100°C, which increases between the as-received and the aged/spalled samples. Therefore, if recrystallization occurred and reduced the crystallite size after hundreds of cycles, the FWHM of the [110] peak of the β phase in the as-received specimen at 1100°C would be expected to be smaller than that of the β phase in the aged and spalled specimens, which is not the case.

Recovery is expected to occur at such high temperatures. However, the FWHM evolution is steeper for the aged and spalled samples than for the as-received sample, whereas recovery alone should not be more active. To explain this, it is necessary to closely examine the $\beta \rightleftharpoons \gamma'$ transformation through the evolution of the β fraction. During heating in each cycle, the β fraction increases, introducing fresh β into the system, free of crystalline defects. This is indeed associated with a reduction of the β FWHM, and thus with a lower defect density in its lattice. Conversely, upon cooling, the FWHM increases while the $\gamma'$ fraction grows. The latter can locally generate deviatoric stresses and trigger plasticity. Very interestingly, the fraction of β transformed in each cycle is proportional to the variation of
FWHM between the low and high ends of the thermal cycle. For instance, in the low-temperature cycles, the $β$ FWHM in the as-received sample varies only slightly (≈ 0.004 deg), while the $β$ fraction also changes only marginally (just above 1 pct per cycle). In contrast, for the aged and spalled samples, the FWHM variation is larger (≈ 0.018 deg) and corresponds to a proportionally higher increase of $β$ fraction during heating (≈ 6 pct). The cyclic $β ⇌ γ'$ transformation therefore appears as a key factor in limiting plasticity in $β$.

The fact that the $β ⇌ γ'$ transformation introduces dislocation-free $β$ into the system could explain why the FWHM is lower at 1100 °C for the aged and spalled samples compared to the as-received sample, the latter containing only $β$ that has undergone deformation during cycling. Another explanation to this reduction of broadening found in the literature is a reduction of the chemical heterogeneity in $β$ after multiple cycles, which has already been reported<sup>22,69</sup> and should indeed reduce the peak FWHM. Overall, a more detailed analysis of the synchrotron data would be required to draw firm conclusions on this aspect, considering not only the [110]$_β$ peak, but also other $β$ peaks. This has not yet been attempted due to the difficulty of deconvoluting the $β$ reflections from the overlapping YSZ peaks.

### C. Impact of Pre-aging by L-FCT on the TBC Behavior

The effect of pre-aging by long furnace cyclic test (L-FCT) prior to in-situ observation of short furnace cyclic test (S-FCT) on the thermal barrier coatings is most clearly observed by comparing the as-received sample and the aged sample (600 L-FCT pre-aging), since for the spalled sample, spallation occurred for the top coat layer, introducing additional effects. The main difference between the as-received and aged specimens lies in the presence of $γ'$ within the bond coat and the growing TGO, while the YSZ top coat is still adherent and not modified by thermal cycling.

Because $γ'$ has a slightly larger molar volume than $β$, its dissolution at high temperature to form $β$ eliminates the compressive stress it induces. This explains the appearance of a hydrostatic tensile stress peak at 800 °C during the low-temperature cycles and at 900 °C during the high-temperature cycles in the aged specimen, while no such peak is observed for the as-received specimen (Figure 9). This supplementary tensile hydrostatic stress could be an additional component leading to easier fracture of the TGO interface with the top coat during the latter stages of the thermal barrier coating life cycle.<sup>34,67</sup> Regarding the FWHM, the aged sample exhibits a steeper slope than the as-received sample (Figure 11). This likely reflects a higher density of dislocation nucleation sites in the aged bond coat, resulting from the presence of $β/γ'$ interfaces that are absent in the as-received bond coat (in addition to the absence of rumpling). Consequently, more dislocations can nucleate during each cycle, consistent with the increased slope of the FWHM curves.

To explain the slower increase of cumulative plastic deformation in the $β$ phase for the aged sample compared to the as-received one, despite the higher number and intensity of stress concentrators (rumpling and $γ'$), it is important to consider that the $β ⇌ γ'$ transformation is more intense in the aged sample than the as-received one, increasing the quantity of newborn $β$ grains with no plastic deformation (the fraction $f_β$ increases by 9 pct during the heating stage of high temperature cycling in the aged sample).

### D. Impact of the Top Coat on the Bond Coat Mechanical Behavior

The effect of top-coat delamination is clearly visible when comparing the aged and spalled samples. Since YSZ has a lower CTE than the $β$ phase,<sup>46</sup> it induces compressive hydrostatic stresses in the bond coat during thermal cycling. Consequently, after top-coat spallation, the $β$ phase experiences higher tensile hydrostatic stresses throughout the low-temperature cycles and during high-temperature cycles above 900 °C. The accumulated plastic strain $p_{cum}$ increases more rapidly in the spalled sample, which can be attributed to enhanced rumpling, when YSZ has spalled off, and a greater number of $β/γ'$ interfaces associated with the higher $γ'$ phase fraction. In contrast to the as-received vs aged samples comparison, the $β$ phase fraction increase per cycle in the aged and spalled samples are very similar. Additionally, when the top coat remains adherent it tends to inhibit rumpling and limit plastic deformation of the bond coat to regions locally affected by oxide growth. The spallation of the YSZ top coat may enhance rumpling and thereby promote plasticity in the bond coat and the $β$ phase.<sup>11,70</sup>

### E. Mechanisms of Plastic Deformation of $β$

Our results suggest that dislocations nucleate preferentially at the interfaces between the substrate, the bond coat, and the TGO, and then propagate into the $β$ phase, as schematized in Figure 13. Such a mechanism explains the observed increase in average plastic strain and its evolution as the bond coat undergoes thermal cycling. In the initial stage (as-received sample), the main contributors to plastic deformation are the substrate/BC and TGO/BC interfaces, which act as the primary deformation sites of the $β$ phase. The stress concentrations are relatively moderate at this stage.

As thermal cycling progresses, rumpling develops at the TGO/BC interface, amplifying the local stress concentration.<sup>34,67,71</sup> At the same time, the nucleation and growth of $γ'$ precipitates introduce new $β/γ'$ interfaces, activating additional, secondary plasticity mechanisms in the $β$ phase. However, the cyclic $β ⇌ γ'$ transformation amplifies, the increasing fraction of $β$ formed at each cycle, resulting in the repeated generation of dislocation-free $β$ grains, which plays a critical role in limiting the plasticity increase in $β$ (aged state). With further cycling and aging of the bond coat, the phase fraction of $β$ formed at each cycle remains the same, while $γ'$ precipitation continues and rumpling
<figure><img src="image_15.png" /><figcaption>Fig. 13—Schematics of the origin of plasticity in the β phase. Dislocations nucleate at stress concentrators located at interfaces associated with AM1, with the TGO (through rumpling) and with γ', and glide into the grain interior.</figcaption></figure>

intensifies. These combined effects ultimately result in an accelerated increase in cumulative plastic strain.

Further experiments are required to clarify the specific plasticity mechanisms active at each interface, which would necessitate detailed microstructural investigations such as transmission electron microscopy (TEM). The manner in which plastic deformation transfers from the interfaces into the bulk of the β phase also remains unclear, since dislocation glide alone cannot cross grain boundaries and additional mechanisms must therefore be involved. Another possibility is that plastic strain is largely confined to β grains adjacent to the interfaces, so that the measured bulk plastic strain predominantly reflects the response of these interfacial grains. An interplay with rumpling, which increases the surface of β grains in contact with the TGO interface, also seems logical.⁷⁰

V. CONCLUSION

Thermal cycling S-FCT experiments combined with high-energy X-ray diffraction were carried out on three thermal barrier systems differing by their pre-cycling history using L-FCT and by the presence or absence of the YSZ top coat. This approach enabled a detailed investigation of bond coat deformation, with particular emphasis on the β phase. By applying the $sin^2ψ$ method to XRD data, the measured strain was compared with

theoretical thermal strains, allowing us to separate the elastic contribution and to evaluate the associated hydrostatic and von Mises stresses in β. A micromechanical calculation further provided access to the evolution of total and plastic strains. From these analyses, the following conclusions can be drawn:

(1) On average, the von Mises stress levels remain too low to fully account for the observed plastic deformation of β, highlighting the essential role of interfaces in its plasticity.

(2) Plastic deformation of β first occurs at the substrate/β and TGO/β interfaces. With thermal cycling, secondary mechanisms emerge at the $β/γ'$ interfaces, while rumpling also contributes by enhancing stress concentrations at the TGO/β interface.

(3) The cyclic $β \rightleftharpoons γ'$ transformation occurring during thermal cycling appears as a key factor in limiting plasticity in β. In the early cycles, the increase in $f_{γ'}$ makes this transformation more pronounced, as a larger fraction of β is produced per cycle. The repeated formation of dislocation-free β grains initially slows down the rate of plastic strain accumulation despite the growth of interfacial area and rumpling. This effect gradually fades after extended cycling (at least 600 cycles at 1100 °C for 1 hour), once the increase in $f_{β}$ per cycle stabilizes, after which plastic strain accumulation accelerates again with aging.
(4) As the number of thermal cycles increases, β progressively experiences higher tensile hydro- static stresses, which may contribute to spallation and delamination processes.

This study paves the way of precise identification of the hardening behavior of high temperature coating, through thermal cycling, that is a key to address the TBC lifetime. Last but not least, the present work could also beneficial to clarify to which extent, the single crystal superalloy used as the substrate, evolves in the inter-diffusion zone, both to assess the volume affected by diffusion and the associated mechanical properties.

## ACKNOWLEDGMENTS

The authors are grateful to Psiche Beamline and SOLEIL Synchrotron for their support in this work. Charlotte Monteiro (now with Safran Aircraft Engi- nes) and Sylvain Depinoy (Mines Paris) are deeply acknowledged for their fruitful discussion. The authors also gratefully acknowledge Safran Group for provid- ing the samples.

## FUNDING

Open access funding provided by Mines Paris - PSL. This work was funded by the Centre des Matériaux of Mines Paris, by the CNRS (Centre National de la Recherche Scientifique). Synchrotron SOLEIL is acknowledged for providing beam time under proposal.

## CONFLICTS OF INTEREST

On behalf of all the authors, the corresponding author states that there is no conflict of interest.

## DECLARATION OF GENERATIVE AI AND AI-ASSISTED TECHNOLOGIES IN THE WRITING PROCESS

During the preparation of this work, the authors used ChatGPT (OpenAI) to improve the linguistic clarity and fluency of the manuscript. After using this tool, the authors reviewed and edited the content as needed and take full responsibility for the content of the publication.

## OPEN ACCESS

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## REFERENCES

1. N.P. Padture, M. Gell, and E.H. Jordan: *Science*, 2002, vol. 296(5566), pp. 280–84. https://doi.org/10.1126/science.1068609.

2. J.H. Perepezko: *Science*, 2009, vol. 326(5956), pp. 1068–69.

3. S. Bose: *High Temperature Coatings*, vol. 81, 1st edn, Butter- worth-Heinemann, 2007, pp. 1–312. https://doi.org/10.1007/s110 85-013-9462-3.

4. V. Maurel, R. Soulignac, L. Helfen, T.F. Morgeneyer, A. Koster, and L. Remy: *Oxid. Met.*, 2013, vol. 79(3–4, SI), pp. 313–23.

5. P. Sallot, V. Maurel, L. Remy, F. N'Guyen, and A. Longuet: *Metall. Mater. Trans. A*, 2015, vol. 46A(10), pp. 4589–4600. https://doi.org/10.1007/s11661-015-2857-9.

6. D.S. Balint and J.W. Hutchinson: *J. Mech. Phys. Solids*, 2005, vol. 53(4), pp. 949–73. https://doi.org/10.1016/j.jmps.2004.11.002.

7. M. Bogdan and I. Peter: *Metals*, 2024, vol. 14(5), p. 575. https://doi.org/10.3390/met14050575.

8. D. Pan, M.W. Chen, P.K. Wright, and K.J. Hemker: *Acta Mater.*, 2003, vol. 51(8), pp. 2205–17. https://doi.org/10.1016/S1359-6454 (03)00014-4.

9. M.Z. Alam, S. Kamat, V. Jayaram, and D.K. Das: *Acta Mater.*, 2013, vol. 61(4), pp. 1093–1105.

10. D. Texier, D. Monceau, S. Selezneff, A. Longuet, and E. Andrieu: *Metall. Mater. Trans. A*, 2020, vol. 51A, pp. 1475–80.

11. L. Mahfouz, B. Marchand, V. Guipont, F. Coudon, and V. Maurel: *Materialia*, 2023, vol. 28, p. 101728.

12. V. Maurel, V. Guipont, M. Theveneau, B. Marchand, and F. Coudon: *Surf. Coat. Technol.*, 2019, vol. 380, p. 125048.

13. R. Soulignac: PhD Thesis, Ecole Nationale Supérieure des Mines de Paris, 2014 (in French).

14. P. Sallot: PhD Thesis, Mines ParisTech, 2012 (in French).

15. V. Tolpygo and D. Clarke: *Acta Mater.*, 2000, vol. 48, pp. 3283–93.

16. N. Yanar, M. Helminiak, G. Meier, and F. Pettit: *Metall. Mater. Trans. A*, 2011, vol. 42A(4), pp. 905–21.

17. R. Mevrel and R. Pichoir: *Mater. Sci. Eng.*, 1987, vol. 88, pp. 1–9. https://doi.org/10.1016/0025-5416(87)90060-7.

18. V.A. Esin and F. Gaslain: *Surf. Coat. Technol.*, 2016, vol. 307, pp. 915–25. https://doi.org/10.1016/j.surfcoat.2016.10.014.

19. G. Cailletaud, J. Cormier, G. Eggeler, V. Maurel, and L. Nazé: *Nickel Base Single Crystals Across Length Scales*, Elsevier, Amsterdam, 2022.

20. V. Tolpygo and D. Clarke: *Acta Mater.*, 1998, vol. 46(14), pp. 5167–74.

21. C. Billard, D. Texier, M. Rambaudon, J.-C. Teissedre, N. Bour- hila, D. Marquie, L. Marcin, H. Singer, and V. Maurel: *International Symposium on Superalloys*, Springer, 2024, pp. 626–35.

22. N. Radi, S. Depinoy, C. Monteiro, F. Gaslain, V. Esin, A. Saboundji, V. Jaquet, and V. Maurel: *J. Alloys Compd.*, 2025, vol. 1010, p. 177871.
23. H. Ibegazene, S. Alperine, and C. Diot: *J. Mater. Sci.*, 1995, vol. 30(4), pp. 938–51.

24. Y. Chen, X. Zhao, Y. Dang, P. Xiao, N. Curry, N. Markocsan, and P. Nylen: *Acta Mater.*, 2015, vol. 94, pp. 1–14.

25. T. Reynolds, D. Collins, N. Soor, S. Street, N. Warnken, P. Mignanelli, M. Hardy, H. Evans, and M. Taylor: *Acta Mater.*, 2019, vol. 181, pp. 570–83.

26. S.F. Siddiqui, K. Knipe, A. Manero, C. Meid, J. Wischek, J. Okasinski, J. Almer, A.M. Karlsson, M. Bartsch, and S. Ragha-van: *Rev. Sci. Instrum.*, 2013, vol. 84(8), p. 083904.

27. R. Zhou, H. Zhang, X. Gao, B. Sun, C. Li, P. Wu, H. Cai, Y. Pei, S. Li, R. Tai, et al.: *J. Mater. Res. Technol.*, 2024, vol. 28, pp. 2831–44.

28. D. Collins, D. Crudden, E. Alabort, T. Connolley, and R. Reed: *Acta Mater.*, 2015, vol. 94, pp. 244–56.

29. K. Knipe, A. Manero, S.F. Siddiqui, C. Meid, J. Wischek, J. Okasinski, J. Almer, A.M. Karlsson, M. Bartsch, and S. Ragha-van: *Nat. Commun.*, 2014, vol. 5(1), pp. 1–7.

30. A. Manero, K. Knipe, J. Wischek, C. Meid, J. Okasinski, J. Almer, A.M. Karlsson, M. Bartsch, and S. Raghavan: *Coatings*, 2018, vol. 8(9), p. 320.

31. A. Manero, S. Sofronsky, K. Knipe, C. Meid, J. Wischek, J. Okasinski, J. Almer, A.M. Karlsson, S. Raghavan, and M. Bartsch: *JOM*, 2015, vol. 67(7), pp. 1528–39.

32. J. Benoist, K. Badawi, A. Malié, and C. Ramade: *Surf. Coat. Technol.*, 2004, vol. 182(1), pp. 14–23.

33. V. Guipont, G. Begue, G. Fabre, and V. Maurel: *Surf. Coat. Technol.*, 2019, https://doi.org/10.1016/j.surfcoat.2019.124938.

34. L. Mahfouz, V. Maurel, V. Guipont, B. Marchand, R. El Hourany, F. Coudon, D.E. Mack, and R. Vaßen: *J. Eur. Ceram. Soc.*, 2024, vol. 44(13), pp. 7867–82.

35. T. Perez, D. Monceau, and C. Desgranges: *Corros. Sci.*, 2022, vol. 206, p. 110485.

36. R. Molins, C. Guerre, and L. Remy: *Rev. métall.*, 2003, vol. 100(5), pp. 507–12.

37. J. Kieffer and J. Wright: *Powder Diffr.*, 2013, vol. 28(S2), pp. 339–50.

38. P. Gravereau: PhD Thesis, Sciences Chimiques, 2011.

39. M. Lamari, S.Y. Allain, G. Geandier, J.-C. Hell, A. Perlade, and K. Zhu: *Metals*, 2020, vol. 10(10), p. 1335.

40. M. Lamari, S.Y. Allain, G. Geandier, M. Ponçot, A. Perlade, and K. Zhu: *Int. J. Plast.*, 2024, vol. 173, p. 103866.

41. G. Geandier, L. Vautrot, B. Denand, and S. Denis: *Materials*, 2018, vol. 11(8), p. 1415.

42. F. Heidelbach, C. Riekel, and H.-R. Wenk: *Appl. Crystallogr.*, 1999, vol. 32(5), pp. 841–49.

43. A. Reuß: *J. Appl. Math. Mech.*, 1929, vol. 9(1), pp. 49–58.

44. V. Hauk: *Structural and Residual Stress Analysis by Nondestructive Methods: Evaluation-Application-Assessment*, Elsevier, Amsterdam, 1997.

45. C. Wang, J. Xu, X. Hu, D. Chen, H. Sun, and B. Yu: *Int. J. Mod. Phys. B*, 2011, vol. 25(27), pp. 3623–31.

46. J. Cormier, G. Cailletaud, V. Maurel, L. Nazé, and G. Eggeler: *Nickel Base Single Crystals Across Length Scales*, Elsevier, Amsterdam, 2022.

47. G. Simmons: *J. Grad. Res. Cent.*, 1965, vol. 34(1), p. 1.

48. C. Smithells: in *Smithells Metals Reference Book*. E.A. Brandes, ed., Butterworth-Heinemann, London, 1983.

49. N. Rusović and H. Warlimont: *Phys. Status Solidi (a)*, 1977, vol. 44(2), pp. 609–19.

50. I. Bleskov, E. Smirnova, Y.K. Vekilov, P.A. Korzhavyi, B. Johansson, M. Katsnelson, L. Vitos, I. Abrikosov, and E. Isaev: *Appl. Phys. Lett.*, 2009, vol. 94(16), p. 161901.

51. A. Kamara, A. Ardell, and C. Wagner: *Metall. Mater. Trans. A*, 1996, vol. 27A(10), pp. 2888–96.

52. H. Faraoun, H. Aourag, C. Esling, J. Seichepine, and C. Coddett: *Comput. Mater. Sci.*, 2005, vol. 33(1–3), pp. 184–91.

53. M.W. Chen, M.L. Glynn, R.T. Ott, T.C. Hufnagel, and K.J. Hemker: *Acta Mater.*, 2003, vol. 51(14), pp. 4279–94. https://doi.org/10.1016/S1359-6454(03)00255-6.

54. R. Gilles, D. Mukherji, M. Hoelzel, P. Strunz, D.M. Toebbens, and B. Barbier: *Acta Mater.*, 2006, vol. 54(5), pp. 1307–16.

55. K. Knipe, A. Manero, S.F. Siddiqui, C. Meid, J. Wischek, J. Okasinski, J. Almer, A.M. Karlsson, M. Bartsch, and S. Ragha-van: *Nat. Commun.*, 2014, vol. 5(1), p. 4559.

56. W. Xia, X. Zhao, Q. Yue, W. Xuan, Q. Pan, J. Wang, Q. Ding, H. Bei, and Z. Zhang: *Mater. Charact.*, 2022, vol. 187, p. 111855.

57. V. Tolpygo and D. Clarke: *Surf. Coat. Technol.*, 2005, vol. 200(5–6), pp. 1276–81.

58. M. Matsumoto, T. Kato, K. Hayakawa, N. Yamaguchi, S. Kitaoka, and H. Matsubara: *Surf. Coat. Technol.*, 2008, vol. 202(12), pp. 2743–48.

59. Y. Zhang, J. Haynes, B. Pint, I. Wright, and W. Lee: *Surf. Coat. Technol.*, 2003, vol. 163, pp. 19–24.

60. Z. Zhang, B. Gleeson, K. Jung, L. Li, and J.C. Yang: *Acta Mater.*, 2012, vol. 60(13–14), pp. 5273–83.

61. I. Ansara, N. Dupin, H.L. Lukas, and B. Sundman: *J. Alloys Compd.*, 1997, vol. 247(1–2), pp. 20–30.

62. J.B. Wachtman, W.R. Cannon, and M.J. Matthewson: *Mechanical Properties of Ceramics*, Wiley, Hoboken, 2009.

63. R. Hillery, B. Pilsner, R. McKnight, T. Cook, and M. Hartle: Technical Report, 1988.

64. D.B. Miracle: *Acta Metall. Mater.*, 1993, vol. 41(3), pp. 649–84. https://doi.org/10.1016/0956-7151(93)90001-9.

65. M. Lamari, P. Kerfriden, O.U. Salman, V. Yastrebov, K. Ammar, and S. Forest: *Int. J. Solid Struct.*, 2025, vol. 309, p. 113171.

66. M. He, J. Hutchinson, and A. Evans: *Acta Mater.*, 2002, vol. 50(5), pp. 1063–73.

67. L. Wang, Q. Fan, Y. Liu, G. Li, H. Zhang, Q. Wang, and F. Wang: *Mater. Des.*, 2015, vol. 86, pp. 89–97.

68. M.Z. Alam, S. Kamat, V. Jayaram, P.S. Karamched, P. Ghosal, and D. Das: *Mater. Sci. Eng. A*, 2014, vol. 604, pp. 18–22.

69. B. Denand, V. Esin, M. Dehmas, G. Geandier, S. Denis, T. Sourmail, and E. Aeby-Gautier: *Materialia*, 2020, vol. 10, p. 100664.

70. A.G. Evans, D.R. Mumm, J.W. Hutchinson, G.H. Meier, and F.S. Pettit: *Prog. Mater. Sci.*, 2001, vol. 46(5), pp. 505–53. https://doi.org/10.1016/S0079-6425(00)00020-7.

71. R. Panat, S. Zhang, and K.J. Hsia: *Acta Mater.*, 2003, vol. 51(1), pp. 239–49.

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