# Experimental full-field investigation of stress development in austenite during the thermally-activated transformation of a Cu-Al-Ni single crystal

Adrien Vinel, Xavier Balandraud, Benoît Blaysat, Michel Grédiac, Thomas Jailin

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Adrien Vinel, Xavier Balandraud, Benoît Blaysat, Michel Grédiac, Thomas Jailin. Experimental full-field investigation of stress development in austenite during the thermally-activated transformation of a Cu-Al-Ni single crystal. 2026. hal-05448205

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Preprint

Experimental full-field investigation of stress development in
austenite during the thermally-activated transformation of a
Cu-Al-Ni single crystal

Adrien Vinel¹, Xavier Balandraud¹*, Benoît Blaysat¹,², Michel Grédiac¹, Thomas Jailin¹

1: Université Clermont Auvergne, Clermont Auvergne INP, CNRS, Institut Pascal, F-63000 Clermont-Ferrand, France

2: Institut Universitaire de France, Paris, France

*: Corresponding author

Abstract

The paper presents an experimental analysis of the stress field developing within austenite in a Cu-Al-Ni single crystal during its thermally-activated transformation to martensite. Starting from a virgin sample in austenitic state, fifty homogeneous cooling-heating cycles allowed to reach about 10% of martensite at each cycle upon cooling. The in-plane strain fields were measured using Localized Spectrum Analysis (LSA) with a strain resolution of 2 × 10⁻⁴ and a spatial resolution of 0.26 mm. The evolution of the stresses out of the martensite zones was then deduced from these strains using known elastic properties of the austenite crystal. Compression normal stresses of about −80 MPa on average were evidenced in the middle of the sample, remaining fully austenitic, while positive values were revealed in the periphery around martensite needles. In addition, shear stresses of opposite signs were observed on either side of the martensitic needles. Zero-shear paths were identified between parallel and closely spaced needles, as well as in the extension of their tips. The presence of stresses in the austenite, even far from the martensitic zones, can be explained by the elastic strains required to achieve the crystallographic compatibility at the interfaces, and the necessity to achieve global static equilibrium of the entire sample. Among other results, e.g. residual stresses at the end of each thermal cycle and a “strongly symmetrical” kinetics of transformation between cooling and heating, this study quantifies the heterogeneity of the stress fields which contribute to the sample’s transformation although the macroscopic loading is purely thermal.

**keywords:** Shape memory alloy ; Austenite-to-martensite phase transformation ; Cu-Al-Ni ; Stress and strain ; Thermal cycling ; Localized Spectrum Analysis

# 1 Introduction

Shape memory alloys (SMAs) have emerged as a
highly promising class of smart materials due to
several unique abilities, namely, superelasticity, shape memory effect and strong damping properties depending on temperature conditions. Their distinctive thermomechanical behavior originates from a reversible solid-solid phase transformation triggered by mechanical and/or thermal loading, and less commonly by magnetic loading [1; 2]. The functionalities of SMAs, compared to conventional metals, make them ideal candidates for a wide range of high-tech applications [3]. Nickel-titanium alloys have enabled advanced applications in biomechanics, aerospace and robotics. Copper-based alloys offer more cost-effective alternatives, for other types of application, e.g. damping systems or elastocaloric cooling devices. Yet, despite the wide range of possible applications, the long-term reliability of SMAs, in particular copper-based alloys, remains a critical challenge. Indeed, in addition to classical structural fatigue (leading potentially to material failure), *functional fatigue* occurs in SMAs. This latter mechanism refers to the change in properties during the first cycles of transformation.

In all SMAs, the “high-temperature” phase, known as *austenite*, has an ordered cubic structure. In contrast, the “low-temperature” phase, referred to as *martensite*, adopts a lower-symmetry crystal structure, primarily depending on the alloy’s chemical composition. In addition, different *variants of martensite* can be created from a crystal of austenite due to its cubic symmetries. These martensite variants correspond to the same unit cell, but oriented differently in space with respect to the parent cubic cell. For instance, for the Cu-Al-Ni alloy considered in the present study, there are 6 variants of martensite with orthorhombic symmetry [4]. Each martensite variant is associated to a characteristic stress-free transformation strain tensor, which can be calculated from the atom displacements during the (stress-free) austenite-to-martensite transformation [5; 6]. The focus of the present study is on the existence of small additional strains, *i.e.* elastic strains, appearing even in the case of a temperature-driven transformation with no external forces applied to the SMA sample. Indeed, crystallographic compatibility at all the microstructure interfaces, *i.e.* continuity of

displacements in the sense of continuum mechanics, may require elastic strains. In practice, spatial distribution of the martensite variants within an SMA sample must ensure crystallographic compatibility at all the interfaces, while minimizing the total energy of the sample. References [6; 5] gather the theoretical results showing that perfect compatibility is generally not possible without additional strains. This induces internal stresses. The present study thus focuses on the measurement of these additional strains and the analysis of the deduced stresses.

Various studies have discussed low-energy microstructures (involving low elastic strains, *i.e.* low stresses) enabling reversibility of the phase transformation that gives SMAs their unique thermomechanical properties: see for instance [7; 8; 9; 10; 11; 12; 13; 14; 15; 16; 17; 18; 19; 20]. These studies rely on theoretical developments and simulations (including finite-element calculations) of different experimentally observed microstructures: primarily *habit planes* (interface between the austenite and a *twinning* of two martensite variants), as well as various other types of spatial organization of the martensite in austenite (branching of twins, martensite needle, wedge-microstructure, X-microstructure, five-domain microstructure...). It is worth noting that the above references are focused on “individual” microstructures. Besides, there is a lack of experimental full-field strain data for comparison purposes. The present study aims to fill this gap by investigating the experimental case of multiple microstructures existing within a large sample.

Strain measurements by Digital Image Correlation (DIC) are now classically carried out in the SMA community [21]. The method relies on a random pattern deposited onto the surface of the sample under study. Its ease of use and the large amount of data it provides made it possible advance our understanding of the mechanical responses of SMAs, especially in polycrystals. The 2021 review [21] mentions eight articles dealing with DIC on Cu-Al-Ni single crystals, which is the alloy considered in the present study. While DIC is a highly effective
technique, it does present certain limitations. One
key issue arises when balancing *spatial* resolution
 viewe d as the minimum distance between two
independent measurements) with *measurement*
resolution (defined as the standard deviation of
the noise), which is a crucial consideration for
capturing fine details in strain distributions. Indeed,
random speckle patterns are not optimal in terms
of sensor noise propagation [22]. Additionally, such
random patterns can introduce a pattern induced
bias, directly related to the randomness of the
pattern [23]. This is an issue in cases involving
high strain gradients [24; 25]. The present study
utilizes an alternative full-field measurement tech-
nique for obtaining surface strain maps: Localized
Spectrum Analysis (LSA) [26]. This technique
relies on a periodic pattern, e.g. a checkerboard
pattern, that is pre-applied to the surface of the
test sample using a laser engraving system. LSA
returns the phase modulations of carrier waves of
the reference (undeformed) and deformed images,
which correspond in fact to the checkerboard patter
periodicities. Images are processed using OpenLSA,
which is freely available [27]. It can be noted that
the application of LSA to SMAs remains uncommon,
having been reported only four times in the literature
(see [28; 29; 30; 31]).

The present paper analyses the stresses over the
surface of a disk-shaped Cu-Al-Ni single crystal
from strain measurements by LSA. This disk is
33 mm in diameter. The sample was initially virgin
at the beginning of the test, *i.e.* with no preced-
ing thermomechanical history. 50 homogeneous
cooling-heating cycles were applied to the sample to
evaluate the functional fatigue in terms of changes in
stress levels during the first cycles of transformation.
More specifically, the study focuses on the stress
evaluation within the austenite only. The study to
the martensite zones will be extended in a future
work. This choice was also made for metrological
reasons. Indeed, the spatial resolution was 0.26 mm,
and the strain resolution was 2 × 10⁻⁴. These
values enable small elastic deformations in austenite
to be reliably measured, except in the vicinity of
habit planes. Indeed, “jumps” in actual strain

distributions at the interfaces render the measured
strain data blurred over a 0.26 mm layer due to
processing. It can also be noted that this spatial
resolution does not allow for the identification of
austenite-martensite transition with the “zigzag with
flat facets” as modelled in [11]. In addition, strain
values measured inside “thin” martensite needles
surrounded by austenite are potentially lower than
actual values. Yet, this study still provides the
first high-definition, full-field estimation of stresses
in austenite using LSA, thereby opening up a new
avenue of analysis.

The paper is structured as follows. Section 2
describes the material and the surface preparation,
the experimental setup and the image processing
(Additional information about LSA is given in Ap-
pendix A). Section 3 shows the results, highlighting
the issue of the repeatability along the thermal
cycles and the description of the stress heterogeneity.
Results are then discussed in Section 4, before
concluding.

The following notations are used throughout the
paper: second-order tensors (matrices) are indicated
by a double underline, *e.g.* the stress tensor *g*; first-
order tensors (vectors) are indicated by a single un-
derline, *e.g.* the stress vector *T*; zero-order tensors
(scalars) are not underlined, *e.g.* the shear stress
*σ*<sub>XY</sub>.

2 Material and methods

## 2.1 Alloy and sample

The SMA specimen used in this work was extracted
from the same ingot as in [31]: see Fig. 1-a. This was
done to ensure that the sample was virgin, meaning
that neither mechanical nor thermal loading had
previously been applied to the sample. Its properties
are here briefly recalled. This is a Cu-Al<sub>13.9</sub>-Ni<sub>4.6</sub>
(nominal wt.%) single crystal elaborated in the form
of a cylindrical ingot by the Bridgman method by
Nimesis Technology, Metz, France. Its diameter
is 33 mm. The ingot is in the austenite state at
ambient temperature (around 25 °C). The DSC curve provided by the supplier (not shown here) gives the following transformation temperatures obtained from the tangent method: Mf = -20 °C, Ms = -12 °C, As = 35 °C and Af = 41 °C. However, observations of the microstructure during thermal loadings (in [31] as well as in the present study) show discrepancies with these DSC results, which may be explained by the presence of impurities, residual stresses or precipitated phases within the ingot. When cooling, the austenite of this alloy transforms into martensite γ'. This martensite is orthorhombic (≈ 2H). This can be deduced by comparing the number of electrons per atom of the alloy with the one at the eutectic point separating β' and γ' martensites [31; 32; 33]. This was also confirmed experimentally in [31] from the different strain states measured in the martensitic zones, which match the strains of the Habit-Plane Variants (HPV) associated to a cubic-to-orthorhombic transformation. Finally, the stretch parameters associated to this transformation [6] were identified in this previous study: α = 1.0507, β = 0.9272 and γ = 1.0239, which is consistent with the values found in the literature (α = 1.0619, β = 0.9178 and γ = 1.0231) for Cu-Al-Ni alloys with close compositions [34; 35].

The sample was cut perpendicularly to the ingot's axis and polished. The thickness is 3 mm. The normal to the disk's plane is a cubic axis of the austenite crystal, namely [001]. It is worth recalling that the sample was in a virgin state, i.e. with no preceding thermomechanical history (except associated to the cutting and polishing operations) at the beginning of the experiment. This is crucial because one of the objectives of the study is to evaluate the "training" of the alloy during the first thermal cycles of transformation (SMA functional fatigue).

Finally, the reference state for the strain measurement was defined at 60 °C from an initial homogeneous heating: see the initial temperature in Fig. 1-d.

<table><caption>Table 1: Imaging hardware parameters.</caption><tr><td>Camera</td><td>Prosilica GT 6600</td></tr><tr><td>Image resolution</td><td>6576 x 4384 pixels<sup>2</sup></td></tr><tr><td>Dynamic range, detector</td><td>8 bits</td></tr><tr><td>Acquisition rate</td><td>1 fps</td></tr><tr><td>Integration time</td><td>IT = 8.8 ms</td></tr><tr><td>Lens</td><td>TC16M048</td></tr><tr><td>Magnification</td><td>0.75</td></tr><tr><td>Aperture</td><td>f/16</td></tr><tr><td>Field of view</td><td>48.2 mm x 32.1 mm</td></tr><tr><td>Image scale</td><td>1 pixel = 7.3 µm</td></tr><tr><td>Stand-off distance</td><td>50 cm</td></tr><tr><td>Patterning technique</td><td>Laser-marked checkerboard</td></tr><tr><td>Pattern square size</td><td>4.09 pixel</td></tr></table>

## 2.2 Surface preparation, imaging system and strain measurement

Surface preparation consisted first in spray painting it in white. Next, a checkerboard pattern with a square size of 30 µm was engraved using a laser marker: see Fig. 1-c. The procedure followed here is described in detail in [36]. It was successfully employed in [31] on the same alloy.

Images of the checkerboard pattern were recorded during the thermal cycles of the sample using a Prosilica GT 6600 camera with a CCD sensor of 6576 × 4384 pixel² (28 Mpixel). The gray depth of the images was equal to 8 bits. A TC16M048 telecentric lens from Opto-engineering was used. Finally, the sample was illuminated using two LED light sources. The field of view was chosen to allow the observation of almost the entire sample: 48.2 mm × 32.1 mm. Only a very small zone at the bottom of the images is not visible, see Fig. 1-c. The image scale was 1 pixel = 7.3 µm. The width of the checkerboard squares corresponds thus to 4.09 pixels. The integration time was 8.8 ms and the acquisition rate was fixed to 1 frame per second (fps). The photo in Fig. 1-b shows the experimental setup. Following the recommendations of [37], Table 1 summarizes the imaging hardware parameters.
<figure><img src="image_2.png" /><figcaption>Figure 1: Material and method: a) disk-shaped sample, here partially transformed in martensite, cut from an ingot; b) experimental setup; c) photo of the sample covered with a checkerboard pattern created with a laser marker after pre-painting in white, with a close-up view showing 30 µm squares. ($x, y$) is the basis of the pixel grid of the camera, while ($\bar{X}, \bar{Y}$) is the basis of the austenite cubic cell, namely $\bar{X} = [100]$ and $\bar{Y} = [010]$; d) cyclic thermal loading.</figcaption></figure>

The LSA technique was used to extract the strain fields from the checkerboard images. It involves a Windowed Fourier Transform (WFT) of the reference image (austenite at 60 °C at the beginning of the test) and of the current images during the thermal cycles. The principle is described in depth in [26]. Appendix A briefly recalls the main points of the image processing enabling the measurement of the fields of the three in-plane strain components, namely $\varepsilon_{xx}$, $\varepsilon_{yy}$, and $\varepsilon_{xy}$, where ($x, y$) is the basis of the pixel grid of the camera. The strain components in the austenite cubic cell $\varepsilon_{XX}$, $\varepsilon_{YY}$ and $\varepsilon_{XY}$, with $\bar{X} = [100]$ and $\bar{Y} = [010]$, were deduced by rotation considering an angle of 22° between the two bases. This angle was identified by optimization, as described in [31], based on the strain values in the HPVs.

It is worth remembering that the strain fields were measured over the entire sample, including the martensite zones. However, the investigation performed in this study solely focuses on the austenitic zones, as mentioned in the introduction. Metrological points must be provided here to facilitate a meaningful analysis of the obtained maps. Indeed, a compromise must be defined between strain resolution and spatial resolution. As in [31], a spatial resolution of 0.26 mm was chosen by adjusting the distance between the camera and the sample. It nearly corresponds to the minimum value possible with the imaging hardware parameters in Table 1. In this configuration, the analysis of the noise in the strain maps was performed in the austenitic state, leading to a strain resolution of about $2 \times 10^{-4}$; see [31] for the procedure.

## 2.3 Thermal loading

In order to study the repeatability along the cycles, 50 thermal cycles were applied to the sample to create the appearance and disappearance of martensitic microstructures upon cooling and heating, respectively. The tests were performed using the same apparatus as in [31]. It mainly consists of a Peltier Cell (Adaptive 8.8 V, 13.1 A, 72 W, $40 \times 40 \text{ mm}^2$) and of a thin copper plate in contact with one face of
the sample. In addition, a cooling device designed for computer Central Processing Units (CPU), namely an Aquafusion 360 series, was used to ensure that the other face of the Peltier cell remained at room temperature. The voltage and current supplied to the Peltier cell were controlled using a setup that involved an Arduino card. Furthermore, this setup allows for triggering the camera for each image as well as recording the temperature.

Figure 1-d shows the variation in time of the temperature measured by a thermistor placed at the interface between the sample and the copper plate. The minimum temperature for each cycle was -1.7 °C, enabling us to avoid condensation at the sample's surface. More importantly, as the objective is to assess stresses by post-processing the strain data within the austenite only, this minimum temperature led to a small quantity of martensite (about 10%), thus leaving a large austenitic zone for investigation.

# 3 Experimental results and analysis

## 3.1 Repeatability along the cycles

The spatial distribution of the three in-plane components of the strain tensor $\varepsilon$ was retrieved all along the test by using LSA from the images captured by the camera. For an easy visualisation of the repeatability of the martensitic transformation over the cycles, the fields of equivalent strain $\varepsilon_{eq}$ were first computed: see Fig. 2. The quantity can be viewed as the von Mises norm of the in-plane strain tensor as follows:

$$ \varepsilon_{eq} = \sqrt{\frac{2}{3}((\varepsilon_{XX}^d)^2 + (\varepsilon_{YY}^d)^2 + 2\varepsilon_{XY}^2)}, \quad (1) $$

where $\varepsilon^d$ is the deviatoric part of the in-plane strain tensor. Its non-null components are defined by:

$$ \varepsilon_{ii}^d = \varepsilon_{ii} - \frac{1}{2}(\varepsilon_{XX} + \varepsilon_{YY}), \text{ with } ii = XX \text{ or } YY. \quad (2) $$

In Fig. 2, the field of equivalent strain $\varepsilon_{eq}$ is depicted at the lowest temperature (-1.7 °C) for

cycles #1, #2, #3, #4 and #50. Note that the color scale is thresholded at 0.01 to help distinguish the microstructure formed by martensite. This also allows us to begin to see heterogeneity in the strain distribution in austenite, which will be analyzed in the following sections. It appears that the obtained microstructures are nearly the same across all cycles, including the first two. The small differences will be discussed below. The kinetics of transformation within the cycles is also strongly “symmetrical” between cooling and heating: indeed, Video 1 in the supplementary material shows the order of appearance of the martensite needles during cooling is globally the same as that of their disappearance during heating. This repeatability and symmetry are quite surprising. It could be explained by a heterogeneous, strong and “symmetric” stress development along each cycle. Indeed, phase transformations in SMAs are both temperature- and stress-driven. In the present case of a thermally activated transformation with strong homogeneity of the temperature field, stresses are probably involved in the spatio-temporal kinetics of the transformation; see discussion in Section 4.

Nonetheless, slight changes in the microstructure can be observed in Fig. 2: they are circled and numbered in the maps. For example, an additional martensite needle is present in Zone 1 between the first and second cycles. The number of needles in Zone 2 varies from one cycle to another. The reader can continue to identify the (small) changes in the other areas circled. But overall, the observation here shows that the transformation kinetics is strongly repeatable from the first cycle onward.

Figure 3 shows the three in-plane components of the strain tensor ($\varepsilon_{XX}$, $\varepsilon_{YY}$ and $\varepsilon_{XY}$), again at the minimum temperature (-1.7 °C), for cycles #1, #2, #3 and #50. The color scale of the maps displayed here is centered around zero: blue for negative values and red for positive ones. Values are also thresholded to better evidence the gradients in the austenitic zones (bounds of ±1% for $\varepsilon_{XX}$ and $\varepsilon_{YY}$ and of ±0.5% for $\varepsilon_{XY}$). The same comments as for the analysis of Fig. 2 can be drawn in terms
<figure><img src="image_3.png" /><figcaption>Figure 2: Slight evolution of the microstructure along the thermal cycles, evidenced here through slight changes in the field of the equivalent strain $\varepsilon_{eq}$ at the lowest temperature (–1.7 °C) of the thermal cycles. The color scale is thresholded at 0.01 to help distinguish the martensite.</figcaption></figure>

of microstructure evolution. However, additional remarks can be made about the sign of the strain components in the austenitic zones. Clearly, both $\varepsilon_{XX}$ and $\varepsilon_{YY}$ exhibit negative values at the middle of the sample (blue color), whereas positive values (red color) are present at the periphery. The center of the sample thus seems to be subjected to compression along both the XX and YY directions. The value of $\varepsilon_{XY}$ is much smaller than that of $\varepsilon_{XX}$ and $\varepsilon_{YY}$. The heterogeneity of the $\varepsilon_{XY}$ field is more difficult to evaluate. A more precise characterization will be presented below from the stress fields, whose calculation is described and discussed in the next section.

## 3.2 Deriving stresses in the austenite zones

The stress fields during the test can be computed from the in-plane strain fields ($\varepsilon_{XX}$, $\varepsilon_{YY}$ and $\varepsilon_{XY}$) and the temperature change $\Delta T$ with respect to the reference temperature, i.e. 60°C. As the crystal structure of austenite is cubic, its stiffness tensor $\tilde{C}$ has a cubic symmetry and its thermal expansion is isotropic. A single coefficient of thermal expansion $\alpha$ is then necessary to model the thermal strain of this phase. Data from the literature are presented below:

* Concerning the austenite's stiffness components,

reference [38] provides values for a Cu-Al-Ni whose chemical composition (Cu-Al$_{14.3}$-Ni$_{4.1}$ wt.%) is close to that of our alloy: $C_{11}$ = 143 GPa, $C_{12}$ = 127 GPa and $C_{44}$ = 96 GPa in Voigt's notation, in the base of austenite crystal's cubic cell. Close values are reported in [39; 40; 41; 42; 43; 44; 45; 46]. This indicates a low sensitivity to chemical composition and heat treatment. Yet, note that stiffness changes with temperature: reference [39] shows an increase of a few GPa for $C_{44}$ during cooling down of about 50°C to Ms; more importantly, reference [47] shows that the shear elastic constant $C' = (C_{11} - C_{12})/2$ drops by about 5% just when approaching Ms (over some degrees). In any case, the above values for $C_{11}$, $C_{12}$, and $C_{44}$ are utilized for stress calculations in austenite in the present study. Potential errors due to chemical composition differences and the softening effect are below the uncertainty in the present stress assessment.

* A measurement of the austenite's coefficient of thermal expansion $\alpha$ is described in [46], giving a value of $17 \times 10^{-6} \text{ K}^{-1}$ for a Cu-Al$_{13.7}$-Ni$_{4}$ wt.% alloy. An identification of $\alpha$ was performed in the present study, considering that no stress develops when the sample remains purely austenitic in the early stage of cooling. The methodology is detailed in Appendix B. For
<figure><img src="image_4.png" /><figcaption>Figure 3: Heterogeneity in the austenite evidenced through the fields of the in-plane components of the strain tensor $\varepsilon_{\parallel}$ at the lowest temperature (–1.7 °C) of the thermal cycles. To observe the small thermoelastic deformations in the austenite, the color scale is set to the interval [-0.005, 0.005] for the *XY* component and [-0.01, 0.01] for the *XX* and *YY* components. The martensite needles appear in dark red or dark blue. See Video 1 in supplementary material.</figcaption></figure>
the first cycle, the following value was found: $\alpha = 4.2 \times 10^{-6} \text{ K}^{-1}$. A value of $4.5 \times 10^{-6} \text{ K}^{-1}$ was obtained for the last cycle. This difference is small, corresponding to a stress difference of approximately 0.9 MPa. The difference with the literature value ($17 \times 10^{-6} \text{ K}^{-1}$) is significant. However, the identification of the thermal expansion coefficient of our alloy using the present strain measurement technique is required to have no thermal stresses developing when the sample remains purely austenitic during the early stage of cooling.

The thermoelastic Hooke's law is used to calculate the stress tensor in the austenite, as shown in Eq. 3, where $\underline{I}$ represents the identity matrix. As the strain measurements were performed on a free surface, the hypothesis of a plane stress state is considered here: see Eq. 4, expressed with Voigt's notation.

$$ \underline{\underline{\sigma}} = \tilde{C} (\underline{\underline{\epsilon}} - \alpha \Delta T \underline{\underline{I}}). \quad (3) $$

$$ \begin{bmatrix} \sigma_{XX} \\ \sigma_{YY} \\ \sigma_{XY} \end{bmatrix} = \begin{bmatrix} C_{11} & C_{12} & 0 \\ C_{12} & C_{11} & 0 \\ 0 & 0 & C_{44} \end{bmatrix} \begin{bmatrix} \varepsilon_{XX} - \alpha \Delta T \\ \varepsilon_{YY} - \alpha \Delta T \\ 2\varepsilon_{XY} \end{bmatrix}. \quad (4) $$

An evaluation of the stress field assessment accuracy was carried out by checking the “global” static equilibrium of a portion of the sample. For that, a straight line crossing the austenite, denoted L and whose unit normal is $\underline{n}$ in Fig. 4-a, was chosen when the sample is at approximately 2.6 °C upon cooling for all the thermal cycles. The stress values along this line can be deduced from the measured strain components using Eq. 4, thus enabling us to calculate the distribution of the corresponding stress vectors $\underline{T} = \underline{\underline{\sigma}}\underline{n}$. Global equilibrium states that the integral of $\underline{T}$ over L should be equal to zero. The norm of this integral divided by the length of L, noted $|\underline{T}_L|$, is plotted in Fig. 4-b at 2.6 °C upon cooling at each cycle. For relative comparison, the norm of the stress vector at a given point P belonging to L (noted $|\underline{T}_P|$) is also plotted. The graph shows that $|\underline{T}_L|$, which should be equal to zero, is equal to about 10 MPa for all the cycles. This value can be considered as relatively small compared to the local value at P, as well as compared to the maximum values assessed during

<figure><img src="image_5.png" /><figcaption>Figure 4: Evaluation of the stress field assessment accuracy through the checking of the static equilibrium of a portion of sample: a) choice of a straight boundary L crossing only the austenite, where stress values are known. This is achieved at 2.6 °C during all the cooling stages of the thermal cycles; b) norm of the mean stress vector around point P and over the line L over the cycles.</figcaption></figure>

the test (about ±200 MPa locally), allowing us to validate the stress calculation approach. Finally, it can be noted that $|\underline{T}_P|$ globally increases along the fifty cycles, from about 20 MPa to 50 MPa. The next section is dedicated to the analysis of the stress fields, in particular along the cycles.

## 3.3 Analysis of the stress fields in the austenite

### 3.3.1 Stress heterogeneity

Figure 5 shows the maps of the three components $\sigma_{XX}$, $\sigma_{YY}$ and $\sigma_{XY}$ of the stress tensor in the austenite at the minimum temperature (−1.7 °C) of the first, second, third and last cycle. The martensitic zones are removed from the maps and correspond to the grey zones. The other cycles are not displayed because the maps are almost identical. These maps illustrate the heterogeneity of stresses within austenite: a compression state appears in the center of the sample, indicated by the blue areas in both the $\sigma_{XX}$ and $\sigma_{YY}$ maps; tension states are seen at the edges of the sample, shown by the red zones, although small peripheral areas also exhibit positive values. Locally, near the martensitic zones, stress levels rise to about ±200 MPa. In particular, high levels of normal stress are observed around the tips of the martensite nee-
dles: see the slightly darker blue indicated by the white arrows in the figure. The shear fields $\sigma_{XY}$ appear to fluctuate more on average than the other two components (this will be analysed in detail in Section 3.3.4) but are noisier. In terms of repeatability over the cycles, the $\sigma_{XX}$ and $\sigma_{YY}$ fields appear to remain virtually constant over the cycles. For the shear component $\sigma_{XY}$, the first cycles give broadly the same maps, but differences can be noted when comparing with the last cycle #50. Indeed, even though the signal is noisy, it can be seen that the intensity of the values is greater, whether positive or negative, for the last cycle. The following sections discuss the stress distributions: within the first cycle (Section 3.3.2), at the beginning and end of all the cycles (Section 3.3.3), and finally in terms of shear stress pattern around the martensite needles (Section 3.3.4).

### 3.3.2 Evolution during the first cycle

To track the phase transformation through the stress development, a square in the middle of the sample is considered: see Fig. 6. This square was chosen so that no martensite forms in it over all the cycles. The variation of the average in-plane stress components $\langle\sigma_{XX}\rangle$, $\langle\sigma_{YY}\rangle$ and $\langle\sigma_{XY}\rangle$ over this square is plotted with respect to the temperature all along the first cycle in Fig. 6-a. During the first stage of cooling, all the components are approximately equal to zero. This is consistent with purely thermoelastic behavior, without phase transformation. This actually has supported the identification of the austenite's thermal expansion coefficient. The onset of transformation in martensite can be identified at about 28 °C, when stresses start to develop. Since no martensite appeared in the square of interest, the stress in this area is thus a consequence of the elastic strains required to achieve the crystallographic compatibility between austenite and martensite, with a “long-distance” effect.

Other comments can be drawn from Fig. 6-a. The mean shear stress $\langle\sigma_{XY}\rangle$ remains nearly equal to zero during the thermal cycle. Also, very close values of $\langle\sigma_{XX}\rangle$ and $\langle\sigma_{YY}\rangle$ are observed, reaching about

-75 MPa at the minimum temperature. When the temperature increases, $\langle\sigma_{XX}\rangle$ and $\langle\sigma_{YY}\rangle$ first remain nearly constant, again in line with a purely thermoelastic response. The onset of reverse transformation in austenite, which occurs outside the square of interest, can be identified at about 40 °C, when the values start to decrease (in absolute value). Classically, this reverse transformation does not start immediately when the temperature increases due to hysteresis. It can be also noted that residual stresses are obtained when returning to 60 °C (-12 MPa for $\langle\sigma_{XX}\rangle$ and $\langle\sigma_{YY}\rangle$): they can be attributed to the presence of residual martensite (outside the square of interest), as can be seen in Video 2 in supplementary material.

### 3.3.3 Repeatability of the stress development

Figures 6-b and -c show the evolution of the mean stress components: at the end of each cycle (high temperature, 60 °C) and at the minimum temperature (-1.7 °C), respectively. The curves further support the “global” stability of the stress development over the 50 cycles, but perfect repeatability appears to be achieved from the 20th cycle. This can be seen in particular for the shear component $\langle\sigma_{XY}\rangle$. Its value slightly increases in magnitude: from about -2 MPa to about -15 MPa, both at the end of the thermal cycles (see vertical dashed line in Fig. 6-b) and at the minimum temperature (see horizontal dashed line in Fig. 6-c).

### 3.3.4 Shear stress patterns

This section finally focuses on the shear stress distribution $\sigma_{XY}$. Indeed, whereas the other components have smooth fluctuations in space (see Fig. 5), the XY one shows “rapid” fluctuations around the martensite needles: see Fig. 7-a showing an example at 2.7 °C during the cooling stage of the first cycle. Zooms are made in Figs 7-b and -c around martensite needles to better visualize the distribution. Note that the data presented in this figure are slightly spatially filtered using a Gaussian window of 21 x 21 pixel², corresponding to 0.153 mm². This filtering is done for readability purposes since the shear stresses are
<figure><img src="image_6.png" /><figcaption>Figure 5: In-plane stress fields in the austenite along the thermal cycles at the lowest temperature (−1.7°C) of the thermal cycles. The martensitic zones are in grey. See Video 2 in supplementary material.</figcaption></figure>
<figure><img src="image_7.png" /><figcaption>Figure 6: Evolution of the average stress over the square zone at the middle of the sample where the material remains fully austenite: a) time evolution of the mean in-plane stress components, evidencing stress development and disappearance during the austenite-to-martensite (A→M) and martensite-to-austenite (M→A) transformations occurring outside the square of interest, during the first thermal cycle; b) mean stress levels at the end of each cycle, i.e. at the return to high temperature; c) mean stress levels at the minimum temperature of each cycle. Recall that no martensite appears in the square of interest during the test: the stresses “far from the martensite” are the consequence of elastic strains required to achieve the crystalline compatibility between austenite and martensite.</figcaption></figure>
noisier than the other components (see Fig. 5). In both zoomed regions, alternating signs on either side of the needles can be observed. Three cases can be distinguished:

(a) the sign of the shear stress remains the same “far from the needle” if no other martensite needles are present: see Fig. 7-c showing a large red zone on the right part; see also the red zone on the left part of Fig. 7-b;

(b) when two parallel needles are close to each other, the alternation in the sign of the shear stress creates a “zero-shear line” in between: see the white lines in Fig. 7-b;

(c) interestingly, a zero-shear line is created in the extension of the needles' axis from their tips. This is a consequence of the above point. Several of these zero-shear paths are visible in Fig. 7-b: their length can be estimated to about 2 mm. In Fig. 7-c, it is also interesting to see that when two martensite needles face each other, their zero-shear lines from their tips “merge”, creating a longer zero-shear path.

We believe that the shear stress gradient between parallel and closely spaced martensite needles, as well as the extension of their tips, are involved in the kinetics of the transformation. This point, among others, is discussed in the next section.

# 4 Discussion

In Section 3.1, it is suggested that the stresses are strongly involved in the spatio-temporal kinetics of the transformation in the present case of a thermally-activated transformation with strong homogeneity of the temperature field. This statement is based on the following observations: (i) the martensitic microstructures are nearly the same between all the cycles; (ii) the kinetics of the transformation within the cycles is strongly symmetrical between cooling and heating in the sense that the order of appearance of the martensite needles during cooling is globally the same as the one of their disappearance during heating. These repeatability and symmetry

<figure><img src="image_8.png" /><figcaption>Figure 7: Experimental evidence of shear stress patterns in the austenite around martensite needles, here at 2.7 °C during the cooling stage of the first thermal cycle: a) location of two zones of interest; b) alternating signs of $\sigma_{XY}$ on either side of the martensite needles; c) zero-shear path between two facing martensite needles. The data are filtered using an averaging window of 21×21 pixels², corresponding to 0.153 mm².</figcaption></figure>
could be explained by the heterogeneous and high
stress development within the sample. Indeed, phase
transformations in SMAs are both temperature-
and stress-driven. In other words, the change in
the stress field that occurs when some martensite
appears at a given time during cooling drives the
location of the next martensite zone, which in turn
modifies the stress field and drives the location of
the following martensite zone, modifying the stress
field again, and so on. This reasoning is clearly
supported by the symmetry of the transformation
kinetics between cooling and heating. Given the
high levels of local stress reached, one can even
speak of strong coupling between the stress field
and the martensitic microstructure distribution in
the transformation kinetics, which is nevertheless
driven by temperature. Stresses certainly exist in
the virgin sample due to the alloy's processing route,
which may define the location of the first marten-
site zones upon cooling, among other reasons (e.g.
heterogeneity of composition, presence of impurities).

The concept of stress development in the austenite "far from the martensitic zones" is proposed in Section 3.3.2. It is worth remembering that small and elastic strains are required to achieve the crystallographic compatibility at the interfaces between the phases, i.e. continuity of the displacements in the continuum mechanics sense. The presence of stresses in the austenite is a consequence of these additional elastic strains, with a "long-distance" effect resulting from the need to achieve global static equilibrium throughout the entire sample.

A question may arise at this stage: why is the
space between parallel and closely spaced marten-
site needles not quickly filled by new martensite dur-
ing cooling? More specifically, why does martensite
wait for colder temperatures to fill the spaces be-
tween closely spaced martensite needles that have al-
ready appeared? One might initially think that ho-
mogeneous temperatures and homogeneous stresses
promote "homogeneous" phase transformation in the
sample. In the present case, where temperatures are
homogeneous but stresses are heterogeneous (due to
the mechanism mentioned above), the spatial stress

<figure><img src="image_9.png" /><figcaption>Figure 8: Norm of the gradient of the shear stress, $|∇σ_{XY}|$, here at 2.7 °C during the cooling stage of the first thermal cycle. Data are filtered using an averaging window of 21×21 pixels², corresponding to 0.153 mm².</figcaption></figure>

gradient is likely to influence the spatio-temporal ki-
netics of the transformation. Section 3.3.4 showed
specific patterns in the shear stress fields, particularly
on both sides of the martensite needles. Figure 8 dis-
plays a field of the norm of the shear stress gradient,
|∇σ<sub>XY</sub>|, around a group of parallel martensite nee-
dles. As expected, high values are observed between
the needles (see zones 'A' in the map), as well as in
the extension of their tips over about 2 mm (zones 'B'
in the map), whereas low values (on average) are ob-
served elsewhere. In particular, the requirement for a
shear stress variation from about –100 MPa to about
100 MPa in the distance between two parallel marten-
site needles causes high gradients when these needles
are close to each other. These gradients can hinder
the progression of phase transformation between par-
allel needles, among other mechanisms (e.g. stress
level itself, self-accommodation, energy dissipation).

5 Conclusion

In this study, the stress evolution within the austen-
ite phase of a Cu-Al-Ni single crystal undergo-
ing thermally-induced cubic-to-orthorhombic trans-
formation was experimentally investigated. Through
repeated cooling-heating cycles, strain fields were
first measured using Localized Spectrum Analysis (LSA), thus enabling the determination of the stress fields outside the martensitic regions, assuming that the thermoelastic properties of the austenite were known. These stresses in the austenite were explained by the elastic straining required to achieve perfect crystallographic compatibility at the interfaces, and the necessity to maintain global static equilibrium of the entire sample. The following results were obtained:

- Compression normal stresses of up to about -80 MPa on average were observed in the middle of the sample (zone remaining fully austenitic), while positive values were revealed in the periphery (around martensite needles);

- Stress levels of some hundred MPa were observed near the martensitic zones;

- Residual stresses were evidenced at the end of each thermal cycle, due to the presence of residual martensite (average normal stresses at the middle of the sample of -12 MPa);

- A strong reproducibility of the spatio-temporal kinetics of transformation was noted, especially after about 20 cycles in terms of stresses created. Besides, a strongly symmetrical kinetics of the transformation was noted between cooling and heating at each cycle;

- Shear stresses with alternating signs on either side of the martensite needles were revealed, with a high gradient between parallel and closely spaced needles as well as in extension of their tips (over about 2 mm).

In the end, this study reveals a high level of internal stress with strong gradients, which undoubtedly contribute to the kinetics of transformation though the macroscopic loading is purely thermal in nature. The prospects for further work concern the assessment of the stress in martensitic zones, which is, for instance, possible in large martensite wedges such as those observed in the same alloy in other experiments. In the longer term, the objective is to characterize/compare the durability of SMAs through the

level of transformation-induced stresses. Indeed, this stress level must be as low as possible to prevent plasticity from occurring.

## A Appendix A: Measuring strain fields using LSA

Localized Spectrum Analysis (LSA) is a full-field strain measurement technique that enables the determination of in-plane displacement fields at the surface of a material. Contrary to Digital Image Correlation, which relies on randomly speckled surfaces, LSA processes images of a periodic pattern. The checkerboard pattern has been shown to be optimal [22; 48] in terms of metrological performance. A variety of methods enables marking periodic patterns with spacing ranging from several millimeters to a few micrometers [49]. In 2004, Piro and Grédiac introduced a technique for producing grids with a period of approximately 50 µm using preprinted sheets combined with adhesive [50]. More recently, in 2022, Bouyra et al. explored the use of a laser marking system to engrave a checkerboard pattern onto a material sample [36], achieving checkerboards with squares about 50 µm wide. Further development enabled the creation of squares featuring a side as small as 30 µm. The primary benefit of employing a laser marking machine lies in its simplicity once the printing parameters have been properly optimized. It should be noted, however, that the process requires a thin layer of white paint to be sprayed beforehand.

The principle of LSA, as described in depth in [26], is briefly recalled here. A checkerboard consists of a repetition of squares. The spatial frequency of this periodic pattern is denoted here by *f*. LSA relies on the computation of the Windowed Fourier Transform (WFT) of the image of the pattern only for this frequency, and successively along the directions formed by each of the two diagonals of the unit squares.
Thus,

$$ \hat{s}_g(x, y, \theta) = \int_{-\infty}^{+\infty} \int_{-\infty}^{+\infty} s(\eta, \xi)g(x - \eta, y - \xi)e^{-2i\pi f(x\cos(\theta) + y\sin(\theta))}d\eta d\xi, \quad (5) $$

where $s$ is the gray level distribution of the image and $\theta$ the angle of the lines of diamonds with respect to the horizontal axis $\underline{x}$ (horizontal direction of the camera pixel grid). The angle $\theta$ defines the direction along which the displacements will be measured. $g$ is a Gaussian window centered at the pixel of coordinates $(x, y)$ where the quantity $\hat{s}_g(x, y, \theta)$ is computed. Thus,

$$ g(x, y) = \frac{1}{2\pi\sigma^2}\exp\left(-\frac{x^2 + y^2}{2\sigma^2}\right), \quad (6) $$

where $\sigma$ is the standard deviation of the Gaussian.

The WFT is estimated for the two values of $\theta$ defined by the two diagonals of the squares forming the checkerboard, leading to two distributions of complex numbers. The arguments of the latter correspond to the phases along these two directions. The sought displacements are deduced from the phase change between the reference state (here, the austenitic state at 60°C) and the current state. Thus:

$$ \tilde{u}(x, y) = -\frac{\tilde{\Phi}^{\text{curr}}((x, y) + \tilde{u}(x, y)) - \tilde{\Phi}^{\text{ref}}(x, y)}{2\pi f}, \quad (7) $$

where $\tilde{u}$ is the sought displacement, and $\tilde{\Phi}^{\text{curr}}$ and $\tilde{\Phi}^{\text{ref}}$ the current and reference phases, respectively. The tildes above the quantities denote the fact that the phases and displacements obtained are not exactly the true ones, but are the true ones convolved with the Gaussian window $g$ [51]. Since the displacement vector $\tilde{u}$ is involved in both sides of Eq. 7, a fixed-point algorithm is used to solve this equation. Then, the fields of the in-plane components of the strain tensor $\underline{\epsilon}$ are derived pixelwise from the displacements.

Finally, it is worth noting that small rigid-body motions of the sample occurred during the thermal cycles. This is because the sample was placed on a Peltier cell through a thin copper plate, which in turn rested on a cooling device, using thermal paste at both contact interfaces. The small rigid-body movements can be explained by changes in the viscosity of the paste, which allow the system to rotate slightly to release some of the tension in the various cables of the device. These small rigid-body motions do not alter the obtained strain fields.

## B Appendix B: Identification of the austenite's coefficient of thermal expansion

The coefficient of thermal expansion of the austenite, $\alpha$, was determined from the strains measured at the beginning of the test, from 60°C to 40°C, i.e., when the sample is still fully austenitic. In this temperature range, the strains are thus solely due to thermal contraction: $\underline{\epsilon} = \alpha \Delta T_I$, where $\Delta T$ is the temperature difference between the reference state (60°C) and the current one, and $I$ is the identity matrix. Let us recall that the thermal expansion of austenite is isotropic due to its cubic symmetry.

Let us note $\langle \epsilon_{XX} \rangle$ and $\langle \epsilon_{YY} \rangle$ respectively the XX and YY components of the strain tensor in the base of austenite crystal's cubic cell, averaged over the square zone at the middle of the sample shown in Fig. 6-a. For each time step during the considered cooling range, the following relations have to be verified:

$$ \langle \epsilon_{XX} \rangle_i = \alpha \Delta T_i, \quad (8) $$

$$ \langle \epsilon_{YY} \rangle_i = \alpha \Delta T_i, \quad (9) $$

where $i$ denotes the time steps verifying that $T_i \in [40°C, 60°C]$. This interval corresponds to 30 images, leading to 60 data points. The thermal expansion is then determined by a linear fit applied to these points.

The same identification was performed for other cycles. It should be remembered that residual
martensite was observed at the end of each thermal cycle when the sample returned to 60 °C. Consequently, the following points were taken into account for the identification of $\alpha$ from cycle #2:

* Only the stress fields located in the square area in the center of the sample in Figure 6-a should be taken into account. Indeed, this area remained fully austenitic during the thermal cycles;

* The reference state (at 60 °C) for the calculation of the strains to be used in Eq. 9 must be updated after each heating stage (for the identification of $\alpha$ only).

The identification gave $4.2 \times 10^{-6} \text{ K}^{-1}$ for the first cooling (cycle #1) and $4.5 \times 10^{-6} \text{ K}^{-1}$ for the last cooling (cycle #50). With these values, no stress is created during cooling to approximately 30 °C (see Figure 6 and Video 2 in the supplementary material), validating the choice of the [60 °C→40 °C] range for the identification of $\alpha$.

## Declaration of Competing interests

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.

## Funding

This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.

## Data availability

Data will be made available on request.

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