# Two-laser experiments to reproduce hypervelocity impacts on graphite at high temperature

B. Aubert, D. Hébert, J.L. Rullier, E. Lescoute, T. Doualle, L. Gallais

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## To cite this version:

B. Aubert, D. Hébert, J.L. Rullier, E. Lescoute, T. Doualle, et al., Two-laser experiments to reproduce hypervelocity impacts on graphite at high temperature. International Journal of Impact Engineering, 2025, 204, pp.105375. 10.1016/j.ijimpeng.2025.105375. hal-05519653

HAL Id: hal-05519653

https://cnrs.hal.science/hal-05519653v1

Submitted on 19 Feb 2026

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# Two-laser experiments to reproduce hypervelocity impacts on graphite at high temperature

B. Aubert<sup>a, *</sup>, D. Hébert<sup>a, ●</sup>, J.L. Rullier<sup>a</sup>, E. Lescoute<sup>b</sup>, T. Doualle<sup>c</sup>, L. Gallais<sup>d</sup>

<figure><img src="image_3.png" /></figure>

## ARTICLE INFO

### Keywords:

Hypervelocity impact

Laser heating

High temperature

Porous graphite

Laser/impact analogy

Crater

## ABSTRACT

Hypervelocity impacts (HVI) with millimeter or submillimeter projectiles can strike aerospace structures with relative velocities of several km/s. In the present paper, the effect of temperature, in the range of 300 to 3600 K, on the size of the craters produced by HVI on graphite targets is studied. This study was made possible by the development of an original experimental setup based on the use of two high-power lasers: a nanosecond shock laser delivering up to 40 J to simulate impacts of 260 µm diameter aluminum projectiles at velocities ranging from 2 to 4 km/s, and a continuous-wave heating laser in order to reach very high temperatures with a brief heating time and well-controlled heating ramp. This study shows that maximum depth and diameter of craters do not vary with the temperature up to 3400 K, but a slight increase of the crater volume is observed. It is explained by a more complete formation of the crater at high temperature. This behavior was not clear up to now in the literature, probably because of the effect of sublimation at high temperature which can be taken into account by the present methodology. Beyond 3400 K, a rapid decrease of crater dimensions is observed.

## 1. Introduction

Hypervelocity impacts (HVI) have to be taken into account in the design of aerospace structures such as spacecraft, satellites or jet engines. Millimeter or submillimeter projectiles can impact these structures with relative velocities of several kilometers per second [1]. In this particular regime, the size of the resulting crater depends essentially on the density of projectile and target materials but the mechanical properties of target materials also play a significant role. For specific applications, the temperature of the impacted target can reach several thousand Kelvins so it is crucial to take into account temperature in modeling of cratering processes because mechanical properties can be very different than those at ambient temperature. As an illustration, target material can be brittle at ambient temperature and ductile at very high temperature and it is well known that crater formation is very different for these two types of materials [2]. Two applications that raise this type of issue can be mentioned. The first one is design of heat shield for re-entry vehicles going to Mars. They are exposed to impacts of dust particles of a few tens of microns at velocities ranging from 1000 to 6000 m/s while their temperature exceeds 3000 K [3]. The second one, is the

problematic of ice crystals ingestion in jet-engine. The blades have to be able to withstand impacts of ice crystals whose diameter varies between 10 and 1000 µm at relative velocity that can reach 600 m/s at the end of the blade while their temperature exceeds 2000 K [4,5].

Crater formation in a brittle material such as graphite is the result of several mechanisms: the propagation and attenuation of stress waves (shock and release) from the impact point leads to combined compressive and shear loading in the depth of the specimen [6], resulting in crack nucleation and growth [7]. The propagation and attenuation of the stress waves is mainly governed by the equation of state (EOS) of the material, which may include porosity effects [8]. Crack nucleation and growth are governed by the strength of the material. Under confining pressure, crack growth is coupled to internal friction, resulting in quasi-plastic behavior [9,7]: due to this complex and coupled process, the compressive strength of porous brittle materials can be very sensitive to strain rate and confining pressure [10]. In the cratering process discussed in this paper, the temperature is likely to affect all the mechanisms discussed above.

Among the materials commonly used in the aerospace industry, graphite is particularly interesting because of its very good strength-to-

<footnote><sup>a</sup> CEA CESTA, 15 Avenue des Sablières, Le Barp, France</footnote>

<footnote><sup>b</sup> CEA DIF, Bruyères-le-Châtel, Arpajon, France</footnote>

<footnote><sup>c</sup> CEA, DES, IRESNE, DEC, Cadarache F-13108 Saint-Paul-Lez-Durance, France</footnote>

<footnote><sup>d</sup> Aix Marseille Univ, CNRS, Centrale Marseille, Institut Fresnel, Marseille, France</footnote>

<footnote>* Corresponding author.
E-mail address: Bertrand.aubert@cea.fr (B. Aubert).</footnote>
weight ratio, especially at high temperature [11]. For this reason, several aspects of the high-temperature mechanical behavior of graphite have been investigated in the quasi-static regime, such as tensile strength [11]. Although the material exhibits a brittle behavior [12], creep can be observed for stresses lower than the ultimate tensile stress [13]. Moreover, important mechanical properties (tensile strength, fracture toughness) of graphite increase from ambient temperature to 2800 K [12] and then decrease abruptly. According to [11], this abrupt decrease in strength observed at higher temperatures may be attributed to the onset of significant sublimation and of rapid creep. However, it remains unclear whether the mechanical behavior of high temperature graphite follows similar trends under dynamic loading, and more specifically under HVI.

The cratering process at ambient temperature on graphite is widely studied in the literature, both experimentally and numerically [14-17]. However, studies at high temperature are not very widespread because of the complexity of coupling an impact simulator and a heating device. Diedrich et al. used a two-stage gun to launch 2.4 mm glass spheres at ~7 km/s on graphite targets heated by induction to ~1000 K [18]. This work did not reveal any effect of the temperature but the studied temperature range was limited. Rubin and McClelland used a single-stage gun to launch 0.52 mm hollow glass spheres at up to ~3.6 km/s on graphite targets heated by induction to ~3350 K [19]. It is observed that mass loss decreases when target temperature increases. This work is very interesting but the dispersion and uncertainty associated with these data obtained in the 1970s prevent clear conclusions about the effect of temperature. Therefore, the use of induction heating to study temperature above 3000 K is questionable. Sublimation of carbon is significant in this temperature range so it is important to reduce the heating time as much as possible. This is why another innovative experimental system is proposed in this paper to study HVI in the range of temperature from 300 K to 3600 K.

Usually, gas or powder launchers are used to reproduce HVI in laboratory. However, this type of facility is not able to launch very small projectiles (typically particles of a few hundred microns) with sufficient accuracy. To simulate this type of impact, laser-driven shock is a good alternative [20-23]. Indeed, by focusing an intense laser beam on an absorbing surface, the temperature of the material located in the first few micrometers below this surface is rapidly increased until the material reaches a plasma state. This plasma expands violently and exerts a strong pressure on the surface of the target. The analogy between hypervelocity impact and laser shock is based on a comparison of the pressure fields generated by the two processes on the target surface. By tuning pulse duration, focal spot size and beam energy, it is possible to reproduce the same loading and damage as various projectiles in the range of interest [24-26]. The impact simulator chosen for the present study is the Générateur de Choc Laser Transportable (GCLT) high-power laser facility [27,28].

There are numerous options for the choice of the heating device: induction [19], oven [29], lamp [30], electron beam [31], laser [32-34] etc. To study temperature close to the sublimation temperature of target material, the heating time has to be as short as possible to limit the amount of material lost by evaporation but sufficiently long to limit the temperature gradient inside the target. Electron or laser heating are particularly suitable to this need. Laser heating is easier to combine with the GCLT facility so a high-power continuous Ytterbium fiber laser was chosen.

The laser shock experiments simulating hypervelocity impacts are described in Section 2. After presenting the target material, the experimental setup is detailed. The analogy between laser shock and HVI is described and used to determine the projectiles equivalent to these laser shots. Section 3 is dedicated to the coupling between the shock laser and the heating laser. A special effort is made to determine the temperature at the surface of the target but also inside the target by means of in situ diagnostics coupled with numerical simulation. In addition, the matter ejected during crater formation is observed with two 2-frame gated

cameras, using a laser shadowgraph system. Such a diagnostic allows us to compare formation and acceleration of ejecta as a complement to the crater investigation, which is done with an optical profilometer. The experimental results are presented and discussed in Section 4. Questions about the repeatability of the cratering process and its sensitivity to the temperature gradient are raised before analyzing the effect of the temperature on diameters, depths and volumes of the craters. Finally, the data obtained are compared with impact tests available in the literature.

## 2. Hypervelocity impact simulator

In this section, laser shock experiments on graphite are presented. The aim is to simulate hypervelocity impacts. The projectiles equivalent to these laser shots are determined using analogy relationships, which are described in detail.

### 2.1. Target materials

Heat shields protecting re-entry vehicles are commonly made of carbon-carbon composite materials. Thermal properties of such materials are very complex so a more generic but still representative material has been considered for the present work: the EDM-3 from POCO Graphite Inc., which is a polycrystalline graphite produced by sintering. This material is mainly used for its electrical properties, as an electrode for Electrical Discharge Machining (EDM) process. There are different grades of EDM graphite with significantly different properties, mainly related to the microstructure. EDM-3 is one of the so-called Ultrafine versions, characterized by an average grain size, which is <5 µm. It has a porosity of about 20 % and it is macroscopically homogeneous with a high degree of isotropy. All samples used in this work were disks of 30 mm diameter with a thickness of 5 mm. No specified surface finishing was done on the sample surface but its roughness measured thanks to an optical profilometer has been found < 10 µm. Mechanical properties of EDM-3 are strongly temperature dependent according to the literature [11,19,12]. Thermal properties also depend on the temperature. The values used in this work can be found in ref [32].

### 2.2. Laser shock experiments

The GCLT facility is a Nd-Glass Phosphate high power laser delivering energies up to 40 J with an adjustable pulse duration from 5 to 100 ns at 1057 nm wavelength. Several phase plates are available to shape the focal spot at the desired size. For the entire laser shots considered in this paper, the 100 ns impulsion was used along with a focal spot diameter of about 250 µm (Full Width at Half Maximum). Most of the tests were performed at 40 J, but three other energies (10, 20 and 30 J) were also used to investigate different simulated HVI. The laser beam impacts the target with an angle of 90° with respect to the sample surface. The temporal profile is chosen trapezoidal to generate on the sample a pressure field which is representative of a spherical projectile [24,25,35]. Indeed, in the case of a spherical projectile impact:

- pressure is at its maximum at the moment of impact,

- pressure then decreases slowly and progressively as the lateral expansion waves reach the impact axis,

- pressure finally decreases abruptly when the release wave, resulting from the shock reflected at the rear of the projectile, reaches the surface of the target again.

This profile is recorded for each laser shot together with the amount of energy E delivered on the target. Fig. 1.a shows a profile for a regular shot at 40 J. A Random Phase Plate (RPP) is used to obtain a well-reproducible and homogeneous spatial distribution of the energy, as shown in Fig. 1.b. The laser beam passes through the RPP, located 550 mm in front of the target, before being focused by a lens located 300 mm in front of the target.
<figure><img src="image_4.png" /></figure>

<figure><img src="image_5.png" /><figcaption>Fig. 1. Example of temporal profile (a) and spatial distribution of intensity (b) for a typical laser shot.</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>Fig. 2. 3D-map of the optical profilometer measurement (top) and horizontal extraction (along the blue line) of depth profile (bottom).</figcaption></figure>

The GCLT facility includes an experimental chamber, which has a great flexibility of use. The experiments have to be conducted under vacuum in order to avoid breakdown of laser beam in ambient air and reduce possible oxidation of carbon by oxygen, which is highly temperature dependent [36]. The pressure in the chamber was close to 5 × 10⁻² mbar, and the sample was placed at the center of the chamber on a holder made of calcarb, which is a high temperature insulating material.

After tests were completed, each crater was measured on a Contour GT profilometer (Brucker). Surfaces of 4 × 4 mm² were scanned and profiles crossing the craters' maximum depths were extracted. An example of such analysis is shown in Fig. 2. This crater results from a laser shock at 40 J on an EDM-3 sample heated at a maximum temperature of 3200 K. As mentioned in the literature on HVI [14,15], the crater generated by laser driven shock is also divided in two parts: a central volume surrounded by a subsurface spall. The diameter related to the central volume $\Theta_{int}$ is distinguished from the diameter of the total volume $\Theta_{ext}$ corresponding to the mean diameter of subsurface spall. Both diameters (interior and exterior) are presented on the profile together with the volume $V$ and the maximum depth $D_{max}$, which are defined with respect to the initial surface of the target. The spatial accuracy of measurement is < 20 µm and its vertical accuracy is < 2 µm.

### 2.3. Analogy with hypervelocity impacts

The analogy between laser shock and hypervelocity impact is based on a comparison of the pressure fields generated at the target surface by each of the two processes. In the case of HVI, it is the projectile itself that exerts this pressure. In the case of laser shock, it is the plasma that exerts this pressure as it expands against the target. In both cases, the increase

<caption>Table 1</caption>

Equivalence between energy of the laser shot and velocity of a projectile made of a spherical aluminum particle of about 260 µm diameter.

<table><tr><td>Laser energy [J]</td><td>10</td><td>20</td><td>30</td><td>40</td></tr><tr><td>Velocity of equivalent projectile [m/s]</td><td>2000</td><td>2600</td><td>3300</td><td>4100</td></tr></table>

<figure><img src="image_7.png" /><figcaption>Fig. 3. Comparison of pressure fields (top) and 2D-section of craters (bottom) in the case of a laser shock {$E = 40$ J, $\tau = 100$ ns, $\Theta = 250$ µm} (left) and an impact of a spherical aluminum projectile {$v = 4100$ m/s; $\Theta = 260$ µm} (right) on a semi-infinite graphite target.</figcaption></figure>
in pressure is so rapid that a shock wave propagates through the target, damaging it and forming a crater.

Knowing the laser parameters of the shots performed at GCLT facility, it is possible to determine the parameters of the equivalent projectile impacts using the analogy relationships detailed in Ref. [24]. These empirical relationships have been calibrated using gas gun experiments performed with aluminum spheres on EDM-3 targets and numerical simulations of these impacts. For this reason, the equivalent projectile is also assumed to be spherical and made of aluminum. The relationships are briefly summarized here in SI units.

First, gas gun experiments on EDM-3 have shown that the crater volume $V_{crater}$ grows proportionally to the projectile kinetic energy $E_{projectile}$:

$$V_{crater} = 2.25 \times 10^{-9} E_{projectile} \quad (1)$$

Knowing the volume of the crater generated by a laser shot, this equation provides the kinetic energy of the equivalent projectile.

In addition, numerical simulations of gas gun experiments were performed [14] and the pressure fields $P_{projectile}(r,t)$ exerted by the projectiles on the target surface were extracted [24,25]. This pressure depends on the radius $r$ and the time $t$. The average pressure in $r=0$ noted $\hat{P}_{projectile}$ associated to each pressure field is calculated excluding low pressures:

$$\begin{aligned} \hat{P}_{projectile} &= \int_{0}^{\infty} P_{projectile}(0, t) \times I(t) \times dt \text{ where } I(t) \\ &= \begin{cases} 0 & \text{si } P(0, t) < 0.1 \max(P(0, t)) \\ 1 & \text{si } P(0, t) \ge 0.1 \max(P(0, t)) \end{cases} \end{aligned} \quad (2)$$

This average pressure depends of the impact velocity as follow:

$$\hat{P}_{projectile} = 5.9 \times 10^3 v_{projectile}^{1.69} \quad (3)$$

In the same way, the laser shots carried out on the GCLT facility with the 100 ns impulsion and the 250 µm diameter focal spot were simulated numerically and the ablation pressure fields $P_{laser}(r,t)$ were determined using the numerical tools and the method described in Ref. [20]. In this simulation, the spatial distribution of pressure is assumed to be stable during laser deposition: lateral expansion of the plasma plume is neglected. The average pressure in $r=0$ noted $\hat{P}_{laser}$ depends on the laser energy $E_{laser}$:

$$\hat{P}_{laser} = 3.5 \times 10^8 E_{laser}^{0.80} \quad (4)$$

Eqs. (3) and (4) gives:

$$v_{projectile} = 6.7 \times 10^2 E_{laser}^{0.46} \quad (5)$$

Knowing the laser energy of each shot, this equation provides the velocity of the equivalent projectile. Since $E_{projectile} = 0.5 m_{projectile} v_{projectile}^2$, the mass $m_{projectile}$ of the equivalent projectile can be determined using equations (1). The diameter of the sphere $\Phi_{projectile}$

is then obtained by calculating $\Phi_{projectile} = (\frac{6 m_{projectile}}{\pi \rho_{projectile}})^{1/3}$ with $\rho_{projectile} =$

2700 kg.m$^3$. In practice, the shots performed at 10, 20, 30 and 40 J on the GCLT facility leads to $\Phi_{projectile} = 260 \pm 30 \mu m$. For the sake of simplicity, all the equivalent projectiles are supposed to be 260 µm diameter spheres whatever the laser energy. Consequently, the equivalent impact velocity is corrected as follow:

$$v_{260\mu m} = \sqrt{\frac{4 E_{projectile}}{m_{projectile} + m_{260\mu m}}} \quad (6)$$

Finally, the laser shot performed on the GCLT facility are equivalent to impacts of a spherical aluminum particle of about 260 µm diameter at a velocity depending on the laser energy, as shown in Table 1. These equivalent projectiles are representative of study cases mentioned in

introduction. This study highlights two advantages of the laser/IHV analogy: such small projectiles could not have been launched by a gas gun, and nearly fifty richly instrumented laser shots were carried out in one week.

The method used here to determine the parameters of the equivalent HVI of aluminum projectile on EDM-3 can be transposed to other materials. But, Eqs. (1), (3) and (4) should be recalibrated to deal with a different target material, and Eq. (3) should be recalibrated to deal with a different impactor material. Furthermore, a strong simplifying assumption has been considered here by assuming the equivalent projectile to be spherical. A more general method which requires more numerical simulation is available in [25].

In order to validate this equivalence, numerical simulations were performed with the Hesione hydrocode developed at CEA. All the details concerning these simulations can be found in Ref. [14,24,25]. On the one hand, four laser shots were simulated using the temporal profile and the focal spot represented in Fig. 1 and a beam energy of 10, 20, 30 or 40 J. On the other hand, impacts of 260 µm diameter spherical projectiles were simulated with impacts velocities of 2000, 2600, 3300 and 4100 m/s. Pressure fields at the surface of the samples and 2D sections of craters were extracted from each of these simulations. Fig. 3 shows the comparison in the case of the laser shot at 40 J and the impact at 4100 m/s. Overall, the two pressure fields are in accordance, both spatially and temporally. As a result, the two resulting craters are very similar. This good agreement validates the data presented in Table 1. It is also interesting to compare the shape of the simulated crater with the shape of the experimental laser crater shown in Fig. 2. The agreement on the central volume is very satisfactory, but the subsurface spall is not well reproduced. A crack initiation is simulated, but the crack does not reach the surface to form the spall. It is interesting to note that neither the laser shot simulation nor the projectile impact simulation reproduces this subsurface spall.

Later in this paper, the debris cloud ejected during the formation of the crater will be looked at with interest. It is interesting to note that debris clouds generated by laser shocks are very similar to debris clouds generated by real projectile impacts [35]. These similarities are visible in Fig. 4. After spatial and temporal resizing, three typical groups of debris are well visible in the two processes: very small debris ejected at high velocity along a cone, a large quantity of medium-sized debris ejected along the impact axis, and very large debris ejected at low velocity with no preferential angle. This is another similarity between laser shock and projectile impact.

The deposition of laser energy leads to a local elevation of temperature within the target. It is important to check if this elevation of temperature affect the formation of the crater. To do this, numerical simulations were carried out in a previous study [24]. These simulations showed that the area thermally affected by the laser is very limited: the thickness of the ablated material is of the order of a few micrometers, and the thickness of material subjected to a significant temperature elevation (typically a few hundred Kelvins) is less than a hundred micrometers. As these thicknesses are well below the depth of a crater (about 500 µm in this study), laser-induced thermal effects can legitimately be neglected.

### 3. Coupling to a heating device

The aim is to study impact on graphite at very high temperatures, so a heating device has been coupled to the GCLT facility to allow laser shock experiments on heated targets. The coupling of these two lasers is described in this section. In addition, numerical simulations were carried out to access the thermal gradient inside the target. These are also presented in this section. Finally, in-situ and post-mortem diagnostics adapted to high temperature experiments are described.
<figure><img src="image_8.png" /><figcaption>Fig. 4. Comparison of ejecta clouds resulting of a laser shock {E = 40 J, τ = 100 ns, Θ = 250 µm} (top) and an impact of a spherical aluminum projectile {v = 3350 m/s; Θ = 1 mm} (bottom) on a semi-infinite graphite target.</figcaption></figure>

<table><caption>**Table 2**
Imprecision in measured temperature for different maximum temperatures.</caption><thead><tr><th>Maximum temperature [K]</th><th>2300</th><th>2800</th><th>3000</th><th>3200</th><th>3400</th><th>3600</th></tr></thead><tbody><tr><th>Temperature uncertainty [± K]</th><td>30</td><td>40</td><td>45</td><td>50</td><td>55</td><td>60</td></tr></tbody></table>

<figure><img src="image_9.png" /><figcaption>Fig. 5. Principle diagram of the control loop.</figcaption></figure>

### 3.1. Full experimental set-up

A high-power continuous-wave (CW) laser was chosen to heat the target. The porous graphite under study is highly absorbing from the visible to infrared range so the choice of laser wavelength is free. A high-power CW Ytterbium fiber laser (SPI laser Qube 2000) was selected. It can deliver 2000 W of maximum power with a multimode laser beam at 1080 nm wavelength. To heat the sample surface, the laser beam diameter was set to 4.5 mm by means of a collimator and a lens at fixed position (diameter defined at $1/e^2$). The focal spot had a super Gaussian intensity profile of power 6. Some tests have been carried out by removing the lens changing the beam diameter to 10.5 mm. The laser beam was sent through laser window in the vacuum chamber with an angle of 40° with respect to the sample surface.

The temperature of the samples was measured during each shot with at least one of the two following pyrometers, which have a high optical resolution:

- a monochromatic pyrometer (SensorTherm, model Metis M313) working at 1270 nm for measurements in the 900–3800 K temperature range,

- a bichromatic pyrometer (SensorTherm, model Metis H322) operating at 1400 and 1640 nm for measurements in the 1600–3300 K temperature range.

The measurement spot size on the sample has a diameter of 0.8 mm, and a response time below 1 ms. An integrated laser pointer ensures that the measurement spot is positioned where the high-power laser spot is centered (alignment made with a CCD camera). The error budget on the temperature measurement can be made by summing the contributions of uncertainties in the emissivity $\epsilon$ of the material, the transmission $\tau$ of the vacuum window, and the intrinsic uncertainty of the pyrometer (given as 0.5 % by the manufacturer). The dependence on emissivity and transmission can be written as follow:

$$ \frac{\Delta T}{T} = \left| \frac{\lambda T}{C_2} \right| \left( \frac{\Delta \epsilon}{\epsilon} + \frac{\Delta \tau}{\tau} \right) \quad (7) $$

where $\lambda$ is the operational wavelength of the pyrometer and $C_2$ is Planck's second radiation constant. The emissivity was estimated to be in the range 0.91 ± 0.02 [32] and the antireflective-coated window had a transmission that can be estimated to 0.98 ± 0.01 based on measurements made with black body calibration source. In such conditions, the temperature uncertainty can be evaluated. The Table 2 provides these uncertainties for maximum temperatures of interest.

To measure the spatial temperature gradient at the sample surface, an infrared camera (FLIR, model A655sc) was also used in addition to pyrometers. This camera operates in the 7–14 µm band, with 300 to 2300 K temperature range. The pyrometers and the infrared camera

<figure><img src="image_10.png" /><figcaption>Fig. 6. Incident laser power and corresponding temperature obtained on a graphite sample in case of a regulated linear ramp.</figcaption></figure>
<figure><img src="image_11.png" /><figcaption>Fig. 7. Photo of a typical experimental set-up around the vacuum chamber.</figcaption></figure>

measure the temperature at the sample surface with an angle of 45°. Graphite is a very efficient absorber. Approximately 90 % of the heating laser radiation is absorbed and the remaining 10 % is scattered with Lambertian behavior, i.e. the reflected intensity decreases with the square of the distance from the sample. All instruments were protected with spectral filters to cut this reflection.

A PID (Proportional, Integral, Derivative) feedback loop has been implemented to ensure well-controlled heating ramps and stable temperature set-points. The maximum of temperature $T_{max}(t)$ at the center of the heated area is measured with a pyrometer and used as the input signal of a PID program controller (Sensotherm Regulus) that controls the laser power output $P(t)$. It is important to make sure that the pyrometer is pointed at the center of the heated area before each shock. Fig. 5 shows the configuration used to monitor the heating laser power with the PID. The shock laser controller provides the starting time of the shot sequence ($t_0$) to the PID.

In order to limit sublimation of the material on the target surface, the maximum heating rate (with some margin) was used to reach the maximum set-point temperature (3600 K). This resulted in a heating time of 30 s at maximum power. A linear heating ramp of 30 s was also used for all other lower set-point temperatures in order to limit the thermal gradient at the end of the heating sequence in the area of interest. This gradient was estimated thanks to the numerical simulations described in subSection 3.2.

This method makes it possible to precisely control the rise in temperature and therefore to trigger the shot at the desired temperature, then to stop the heating just after the impact. The case of a linear heating at 50 K/s up to 3000 K obtained on a graphite sample and the corresponding applied laser power are shown in Fig. 6. In the lower limit range of the pyrometer, the measurement is noisy and consequently the laser power fluctuates.

To complete these optical instruments deployed to monitor the temperature of the sample, gated cameras were used to observe the target surface and the ejected debris at a grazing angle. A typical arrangement around the experiment chamber is shown in Fig. 7. This flexible arrangement can change between two shots without any restrictions. Especially, the infrared camera can replace one of the pyrometers.

## 3.2. Temperature calculations

To supplement the measurements of temperature, simulations of the graphite heating have been conducted with commercial software, COMSOL Multiphysics, which uses a Finite Element Method. A 3D model is necessary to take into account the angle of 40° between the heating laser and the sample surface, the exact location of the shot on the sample (several test sites per sample), and the sample holder.

The nonlinear heat flow Eq. (8) is solved to obtain the temperature distribution and evolution $T(x,y,z,t)$, considering conduction as the only heat transfer mechanism in the material:

$$ \rho(T)C_p(T)\frac{\partial T}{\partial t} - \nabla \cdot (K(T)\nabla T) = 0 \qquad (8) $$

where $\rho(T)$, $C_p(T)$, and $K(T)$ represent the density, thermal conductivity, and specific heat for EDM-3, respectively. The laser coupling in the material is approximated by surface absorption and is treated as an incident heat flux $S(x,y,z,t)$ expressed as:

$$ S = A(T) I_N(x, y, z) \frac{P(t)}{\pi \omega^2} \qquad (9) $$

where $A(T)$ is the temperature dependent absorption ($A = 1-R$ where R is the total reflectivity), $I_N$ is the normalized distribution of the laser intensity on the sample surface taking into account the incidence, $P(t)$ is the laser power as a function of time, and $\omega$ is the waist of the incident laser beam.

Radiation losses are included as a boundary condition between the surface of the sample and the environment, taking into account the total emissivity and its temperature dependence. Convective cooling is neglected in the simulation. As the EDM-3 exceeds its sublimation point for the highest temperature reached in this study, evaporative cooling is included as a surface boundary condition. Numerical calculations were conducted taking into account the recorded laser power as a function of time for each test. All the details necessary to reproduce these simulations have been previously published by the authors in [32]. In particular, the thermal and optical properties of EDM-3 used are all listed in this reference.

Fig. 8 reports on two experiments where temperatures up to 2300 K and 3400 K are reached after 30 s of heating with linear rate of 50 K/s and 80 K/s, respectively. At the end of the heating sequence, the laser was shutdown resulting in very fast cooling of the sample: 1000 K lost in 10 ms. The thermal time scale in graphite is very long compared to the formation time of the craters generated during this campaign (~ 1 ms >> ~ 2 µs) so the temperature can be considered as constant during the whole cratering process. The very good agreement between experimental data and numerical simulation on surface temperature gives confidence in the calculated temperature under the surface that will be discussed in subSection 4.2.

## 3.3. In-situ and post-mortem analysis

If sublimation occurs during heating, the crater's dimensions cannot be found with a single measurement. A preliminary heating test under the same condition is required to determine the amount of matter lost by sublimation. An example is shown in Fig. 9 for a maximum temperature of 3600 K. On the left is the preliminary heating test and on the right is

<figure><img src="image_12.png" /><figcaption>Fig. 8. Laser heating experiments up to very high temperatures with a linear heating rate and their comparison to simulations.</figcaption></figure>
<figure><img src="image_13.png" /><figcaption>Fig. 9. 3D-map of the optical profilometer measurement and horizontal extraction of depth profile. Comparison between a crater formed only by sublimation (left) with a crater generated by both heating and shock laser (right).</figcaption></figure>

gated camera without filter

<figure><img src="image_14.png" /></figure>

gated camera with filter

<figure><img src="image_15.png" /><figcaption>Fig. 10. Influence of the filter for comparable lasers conditions. Images of the sample front 20 µs after the impact and 3D-map of the corresponding crater.</figcaption></figure>

the full test combining heating and shock. It is important to note the strong difference in color scale (x10) between the two 3D maps. The maximum depth and volume of the crater generated by the laser shock are calculated by subtracting the two measurements (blue – red). As the heating laser beam was sent with an angle of 40° with respect to the sample surface, heating induced both an elliptical crater formed by sublimation (left) and the incomplete formation of the spall along the longer axis (right).

As mentioned above, the matter ejected during crater formation was observed with two 2-frame gated cameras, using a 527 nm laser shadowgraph system. Such a diagnostic allows us to compare formation and acceleration of ejecta in addition to the crater analysis. In order to suppress the thermal radiation signal, an interference filter (532 ± 10 nm) can be located in front of the gated camera. Fig. 10 compares images obtained with/without filter for identical shock and heating laser settings. More fragments are identifiable using a filter, especially the warmer fragments. Subsequently, all tests done in this study were carried out with a filter.

## 4. Results and discussions

All the experimental results are presented in this section. First, the reproducibility of the shots is checked and the sensitivity of the result to the thermal gradient within the target is estimated. The effect of temperature on crater size is then detailed over the 300–2800 K temperature range for four different equivalent impact velocities. A complementary study in the 300–3600 K range is also presented. Finally, all these data are compared with glass projectile impacts on graphite in the 300–3350 K range available in the literature.

$$T_{max} = 3400 \text{ K}$$

<figure><img src="image_16.png" /><figcaption>Fig. 11. 3D-map of the optical profilometer measurement and horizontal extraction of depth profile. Example of reproducibility for crater formation at temperature near graphite sublimation.</figcaption></figure>
<figure><img src="image_17.png" /><figcaption>**Fig. 12.** Observation of ejecta at a maximum temperature of 2800 K for two shots with identical laser parameters.</figcaption></figure>

### 4.1. Tests reproducibility

To demonstrate the reproducibility, several tests were performed with exactly the same setting for heating and shock lasers. A first shot is performed with heating laser only (30 s up to $T_{max}$ = 3400 K with a heating diameter of 4.5 mm) as shown in Fig. 11 for the case $E = 0$ J. At such temperature, sublimation of the material is observed but only a few microns are sublimated. Using exactly the same heating procedure, three

shots were carried out with $E = 40 \pm 0.8$ J for the shock laser energy. Although some differences are visible in the surrounding subsurface spall (3D maps) and in the crater bottom (profiles), a very good reproducibility is observed.

In Fig. 12, two shots made with a laser energy of $E = 40 \pm 0.8$ J and similar heating conditions (30 s up to $T_{max}$ = 2800 K with a heating diameter of 4.5 mm) are compared. According to gated camera measurements, the distribution of ejecta although approximate is well reproducible. Images at $t = 20$ µs after the impact show that the ejecta clouds have approximately the same shape and velocity in the two tests. The 3D-maps, with same color code as Fig. 2, confirm the similarity of crater shapes after all ejecta is removed except for a few pieces of matter not detached (the most noticeable are surrounded by dashed yellow lines).

### 4.2. Effect of spatial thermal gradient

By means of COMSOL simulations, the temperature gradient inside the target is accessible. This gradient can be modified by changing the diameter of the heating laser beam while maintaining the same $T_{max}$. In order to quantify the influence of this gradient on the formation of the crater, several impacts were performed with two different heating beam diameters (4.5 mm and 10.5 mm) but with identical shock laser settings (energy 40 J, duration 100 ns and diameter 250 µm). The results obtained for two impacts on samples heated to $T_{max}$ = 2800 K in 30 s is shown in Fig. 13. From pyrometer and profilometer measurements combined with COMSOL calculations, the following data are shown for each beam diameter: the comparison of measured surface temperature with calculation, the 3D map of the crater (same color map as Fig. 2), and the extracted depth profile overlaid on the temperature gradient. Except for a small piece of matter visible at the bottom of the crater obtained with the heating beam diameter of 10.5 mm, the two craters are identical.

The ejecta generated during the two previous shots are compared in Fig. 14. Strong similarities are observed. The spreading of ejecta is very

<figure><img src="image_18.png" /><figcaption>Heating beam diameter: 10.5 mm</figcaption></figure>

<figure><img src="image_19.png" /><figcaption>**Fig. 13.** Comparisons between two heating beam diameters for experimental measurements and simulations; (a) pyrometer vs simulation at the beam center, (b) 3D-map from optical profilometer, (c) horizontal depth profile and calculated temperature gradient.</figcaption></figure>
reproducible and not sensitive to the temperature gradient. In both cases, images at $t = 20$ µs after the impact show that faster ejecta have similar speed. This result reinforces the good reproducibility of these experiments.

Other things being equal, it was concluded that the temperature gradient does not affect crater formation in the two specific cases studied. This point was not investigated further because it was strongly limited by the experimental setup. It was decided to use the 4.5 mm beam diameter for all the following studies because it is the largest beam diameter that allows a temperature of 3600 K to be reached at maximum power. To complete this sensitivity study without using a more powerful heating laser, it might be interesting to vary the heating rate for a given laser power to test a wider range of thermal gradients.

### 4.3. Heating effect for temperature up to 2800 K

In order to study the influence of temperature on crater formation, all craters were measured with the profilometer and the four dimensions defined in Fig. 2 were extracted. Data obtained with a maximum temperature of 2800 K are compared with those at ambient temperature in Fig. 15. Results are plotted as a function of the measured energy delivered by the shock laser. The exterior diameter is not mentioned at shock laser energy near 10 J because it is not discernable from the interior diameter on the 3D-map, as validated by the trend lines. In addition, the extrapolation of fit at $E < 10$ J is not suitable, suggesting that the trend in crater parameters has to be much faster at such low energies.

This figure shows growth of crater dimensions with energy of the shock laser when $E > 10$ J, excluding the interior diameter which requires $E > 30$ J. From the trend lines, a very small differences in depth and diameter is visible between heated and unheated samples. However, a difference is visible for volume when $E > 20$ J. The observation of ejecta at 40 J does not highlight this difference as shown in Fig. 16. This figure compares two series of images obtained at ambient temperature and at $T_{max} = 2800$ K, each of them is a combination of sequences from two shots with the same shock laser parameters. This combination is

<figure><img src="image_20.png" /><figcaption>Heating beam diameter: 10.5 mm</figcaption></figure>

<figure><img src="image_21.png" /><figcaption>**Fig. 14.** Observation of ejecta at a maximum temperature of 2800 K for two shots with different laser heating-beam diameters.</figcaption></figure>

<figure><img src="image_22.png" /></figure>

<figure><img src="image_23.png" /></figure>

<figure><img src="image_24.png" /><figcaption>**Fig. 15.** Measured crater dimensions shown as a function of the shock-laser energy; (a) diameters, (b) maximum depths, and (c) volumes.</figcaption></figure>

allowed by the excellent reproducibility which is illustrated by the images at $t = 20$ µs. Whatever the temperature, at $t = 5$ µs after the impact, the fastest pieces of material are ejected with a similar angle of about 60° compared to the normal to the sample and also with an identical maximum speed greater than 500 m/s. Besides, the size distribution of these grains of graphite that is roughly discernible at $t = 10$ µs is comparable with a maximum size smaller than 100 µm. Comparing images at $t = 10$ µs to $t = 20$ µs, the maximum speed of matter ejected from the central crater volume is estimated to be 160 m/s, regardless of the sample temperature. In the same way, ejecta from the surrounding subsurface spall reach 80 m/s in both cases. Also, at $t = 60$ µs and $t = 80$ µs, the sizes of slower ejecta are very comparable between $T_{max} = 300$ K and $T_{max} = 2800$ K. These sequences of images show that the velocity and the size of ejecta although approximate is not very sensitive to the temperature gradient of the impacted material.

A 2D-profile of each crater is extracted from 3D maps, as shown in Fig. 17 for an energy around 40 J. It shows that the bottom of the craters is more hemispherical for heated samples than for room temperature samples. This clearer formation of the crater bottom could explain the slight increase of its volume with heating, without varying the diameter and the maximum depth.
<figure><img src="image_25.png" /><figcaption>Fig. 16. Observation of ejecta at ambient temperature and at a maximum of 2800 K for the shock-laser power point set to $E \approx 40$ J. A sequence of 4-images corresponds to one crater formation related to a 3D-map.</figcaption></figure>

<figure><img src="image_26.png" /><figcaption>Fig. 17. 3D-maps and corresponding 2D-profiles for crater generated with a maximum temperature of 300 K and 2800 K on the sample surface.</figcaption></figure>

Given that tensile strength and fracture toughness of graphite increase from ambient to 2800 K [12], it is surprising to observe an increase in crater volume over this temperature range. But it is important to remember that many material properties are temperature-dependent: EOS, porosity closure, shear strength, flow stress, friction, crack formation and propagation obviously depend on temperature. Their relative importance in the cratering process is difficult to assess and a comprehensive study of all these effects is beyond the scope of this paper. However, the experimental set-up proposed in this document could also be used for more elementary testing. For example, it could be used to generate planar shocks on thin targets heated to very high

temperatures in order to look at the influence of temperature on some of these mechanical properties taken independently.

### 4.4. Heating effect for temperature up to 3600 K

Additional tests were carried out at 20 J and 40 J to investigate crater formation from temperature $T_{max} = 2800$ K to 3600 K. All 40 J shots at a given temperature were averaged into a mean shot to plot the crater dimensions as a function of temperature. Before calculating this mean, each measurement is shifted from its experimental energy (ranging from 38 to 42 J) to 40 J using the above fits. The same procedure was used for shots at 10, 20 and 30 J. Through this procedure, all shots carried out with a heating beam of 4.5 mm are summarized in Fig. 18. These results supplement previous observations (Fig. 15) over the entire temperature range of this study. The error bar corresponds to the replication error that is superior to measurement uncertainty. It can be observed that:

- The variation of exterior diameter with temperature is not visible up to $T_{max} = 3400$ K, but a decrease is clear beyond this value.

- The interior diameter seems to be independent of the temperature. However, few data are available at the highest temperature.

- The maximum depth is stable and then decreases at the highest temperature. Such progression is comparable to the exterior diameter.

- The volume increases gently with the temperature when $E > 20$ J. Moreover, in step with depth and diameter a significant reduction of the volume is measured at $T_{max} = 3600$ K.

In summary, on the one hand, the parameters defining the crater grow with the energy of the shock laser when $E > 10$ J (excepting the interior diameter which requires $E > 30$ J). On the other hand, up to $T_{max} = 3300$ K these parameters are only slightly impacted by the heating (only the volume increases significantly when $E > 20$ J), and beyond this temperature a notable decrease is observed. The observation of ejected materials underlines this latter point. As an example, Fig. 19 compares
<figure><img src="image_27.png" /><figcaption>Fig. 18. Average measured crater dimensions shown as a function of the maximum temperature: exterior diameter (a), interior diameter (b), maximum depth (c) and volume (d).</figcaption></figure>

sequences of images obtained at $T_{max} = 2800$ K and $T_{max} = 3600$ K at 20 J and 40 J (all other shock laser parameters being equal). It shows that for both energies, the velocity of ejecta is not very sensitive to the temperature of the impacted material. However, a decrease of ejecta density at the highest temperature is clearly at $t = 20$ µs after the impact. Remarkably, this finding on ejecta is more visible at 20 J whereas the difference on the crater shape is more salient at 40 J as shown by the 3D-map. For these complementary tests between 2800 K and 3600 K, shock laser beam was focalized differently than in the rest of the paper. It explains the loss of axial symmetry for the fastest pieces of material ejected at an angle of 60°.

Up to 3200 K, crater diameter and depth remain unchanged, but volume increases as the crater is more completely formed (smooth shape, perfectly circular surface spall, no residual material left at the bottom of the crater). In the 3200K-3400 K range, the volume is almost stable and the crater is completely formed. This would suggest that an asymptotic regime has been reached, and that a further increase in temperature will not affect crater volume. However, the sharp decrease in volume observed at 3600 K indicates the opposite: there is no asymptotic regime. It is interesting to note that the decrease in volume observed at 3600 K is now correlated with a decrease in crater depth and diameter.

## 4.5. Comparison with prior work

As mentioned in subSection 2.3., the laser shots performed at GCLT facility correspond to an aluminum projectile of about 260 µm diameter according to analogy relationships. The velocity of this equivalent projectile depends of the laser energy and scales between 2000 m/s and 4100 m/s. In this subsection, these results are compared with prior works from Rubin and McClelland who launched 520 µm hollow glass spheres at 2400 m/s and 3600 m/s on graphite targets heated by induction up to ~3350 K [19]. For this comparison, the temperature $T_{max}$ at the center of the crater surface is no longer relevant. The average

temperature $\tilde{T}$ in the crater volume just before the impact is better suited for comparison with induction heating. This average temperature extracted from the simulation is bounded by $T_{max}$ and $T_{min} = T(D_{max})$. In order to compare both results obtained with glass projectiles and equivalent aluminum projectiles, dimensionless variables, quoted by an asterisk, are used:

$$ 
\begin{aligned}
\varnothing_{\text{ext}}^{*} &= \varnothing_{\text{ext}} (\rho_{\text{target}}/m_{\text{projectile}})^{1/3} \\
D_{\text{max}}^{*} &= D_{\text{max}} (\rho_{\text{target}}/m_{\text{projectile}})^{1/3} \\
V^{*} &= V (\rho_{\text{target}}/m_{\text{projectile}})
\end{aligned}
\qquad (10) $$

with $\rho_{target}$ the density of the target material and $m_{projectile}$ the projectile mass. For the experiment described in the reference [19], the target is made of a graphite slightly less porous than EDM-3 for which $\rho_{target} = 1.830 \text{ g.cm}^{-3}$ and the projectiles are hollow glass spheres for which $m_{projectile} = 1.72 \times 10^{-4} \text{ g}$. In the present study, the target is made of EDM-3 so $\rho_{target} = 1.754 \text{ g.cm}^{-3}$ and the equivalent aluminum projectile has a diameter of 260 µm so $m_{projectile} = 2.56 \times 10^{-5} \text{ g}$.

To check the validity of the laser/HVI analogy, the results obtained at room temperature are compared. Table II of reference [19] provides the maximum diameter of craters ($C_D$) and maximum depth of craters ($C_H$) that are directly comparable to $\varnothing_{ext}$ and $D_{max}$, respectively. Volumes are not directly provided in this reference but the cross-section area through the craters which are presented display an almost conical shape for the craters. Consequently, the volume of the craters were estimated by calculating the volume of the cone described by $C_D$ and $C_H$. According to Eqs. (10), dimensionless data from Rubin are compared with the data obtained at GCLT facility in Fig. 20.

$\varnothing_{ext}^*$ and $D_{max}^*$ grow almost linearly with projectile velocity. Data obtained with physical glass projectiles and laser-simulated aluminum projectiles are consistent, even for $V^*$. This result is in accordance with the fact that, in the hypervelocity regime, the dimensionless quantities $\varnothing_{ext}^*$, $D_{max}^*$ and $V^*$ do not depend on the size or material of the
<figure><img src="image_28.png" /><figcaption>**Fig. 19.** Sequence of 4-images with the corresponding 3D-map for two shock-laser powers ($E \approx 20$ J and $E \approx 40$ J). Comparison between ejecta at a maximum temperature of 2800 K and 3600 K.</figcaption></figure>

projectile [37,25]. Consequently, the similarity on $V^*$ that is observed in Fig. 20 validates both the laser/HVI analogy used and the conical shape hypothesis used to process Rubin's data.

Then, crater dimensions are plotted as a function of $\tilde{T}$, again using the data from table II in reference [19]. The outer diameter and maximum depth are shown in Fig. 21. Only the two velocities studied up to $T_{max} = 3600$ K are presented (2600 m/s and 4100 m/s). For velocities

near 2500 m/s, Fig. 21.a shows a good comparability between the two types of projectiles. Only a very small difference is visible for $D_{max}^*$ when $\tilde{T} \ge 2600$ K. For velocities near 4000 m/s, comparable results are obtained as shown in Fig. 21.b. The difference pointed out for $D_{max}^*$ could be explained by the fact that the dimension $C_H$ defined by Rubin and McClelland seems to correspond to the penetration depth of the glass projectiles and not to the crater depth. At low temperatures, the two definitions are consistent, but at high temperatures, the glass bullet is trapped inside the target, under the crater (see for example Figs. 10, 12 and 15 in [19]).

In the case of laser-simulated aluminum projectiles, the combined diminution of $\varnothing_{ext}^*$ and $D_{max}^*$ at $\tilde{T} = 3300$ K indicates a significant decrease in crater volume or associated mass loss. To compare this mass loss to the data provided in [19], it is divided by its value at ambient temperature. The trend of this normalized mass loss $M$ as a function of $\tilde{T}$ is shown in Fig. 22. For both experiments, the variation of $M$ up to 3300 K is small, but a very clear decrease is observed at the highest temperatures when the sublimation of the target becomes significant. The hypothesis of a trapping of glass projectiles is confirmed by the fact that the increase of $D_{max}$ observed in Fig. 21 is not correlated to an increase of $M$ in Fig. 22. A weakness of the analogy laser/HVI can be pointed out on this point: it is not possible to reproduce the trapping of the projectile using a laser.

<figure><img src="image_29.png" /><figcaption>**Fig. 20.** Dimensionless crater parameters shown as a function of the projectile velocity at ambient temperature.</figcaption></figure>
<figure><img src="image_30.png" /><figcaption>Fig. 21. Dimensionless crater parameters shown as a function of the temperature; (a) velocity nearby to 2500 m/s (b) velocity around 4000 m/s.</figcaption></figure>

<figure><img src="image_31.png" /><figcaption>Fig. 22. Normalized mass loss of matter shown as a function of the temperature; (a) velocity nearby to 2500 m/s (b) velocity around 4000 m/s.</figcaption></figure>

## 5. Conclusion

Hypervelocity impact has been studied in the case of graphite targets heated to temperatures ranging from 300 to 3600 K. This study was enabled by an original experimental system combining two high-power lasers. A nanosecond shock laser delivering up to 40 J was used to simulate impacts of 260 µm diameter aluminum projectiles at velocities ranging from 2 to 4 km/s. A continuous wave heating laser was coupled to this system in order to reach high temperature in the laser shock area. With the use of temperature monitoring and laser power regulation, this setup guarantees a brief heating time and well-controlled temperature, which is necessary to limit the material loss due to sublimation at very high temperature. The repeatability of the tests has been demonstrated and the temperature gradient inside the target has been precisely quantified thanks to in situ diagnostics and numerical simulation. This experiment enables the collection of new data in the field for the community.

The evolution of crater dimensions have been studied as a function of the laser shock energy and the surface temperature. This study reveals that up to 3400 K, the maximum depth and diameter of craters do not vary with the temperature. This behavior was not clear up to now in the literature, probably because of the effect of sublimation at high temperature which can be taken into account by the present methodology. However, a slight increase of volume of about 20 % is pointed out. This trend can be explained by the shape of the craters. At low temperatures, the scale around the crater is not always completely formed and small pieces of material remain at the bottom of the crater. The more the temperature increases, the more the craters are smooth and well formed. Such crater shape difference explains the slight increase in the volume which is observed up to 3400 K.

Between 3400 K and 3600 K, a clear decrease of the crater volume is observed. The experimental setup allows to study this very high temperature range, but it is important to remember that the use of laser-driven shock to simulate HVI is questionable when the sublimation of the target is very significant. In this case, the shock laser can interact with a sublimation plasma present in front of the target and not directly with the target surface. Then, the energy deposit varies and the analogy relationships used to determine the equivalent projectile has to be checked.

In order to evaluate the feasibility and the validity the experimental setup, it has been decided to concentrate on homogeneous graphite for this first study. To go further, it could be interesting to perform the same experiments on a Carbon/Carbon composite, which is more representative of heat shield material. Due to the strong orthotropy of such a material, it is necessary to know the thermal properties in all directions to determine the temperature inside the target. The use of a laser impact simulator is particularly interesting to study composite materials because it becomes possible to precisely target each mesoconstituent (e. g., strand, matrix). The spatial accuracy of a projectile launcher does not allow this type of study.

### CRediT authorship contribution statement

**B. Aubert:** Writing – review & editing, Writing – original draft, Visualization, Formal analysis, Conceptualization. **D. Hébert:** Methodology, Formal analysis, Conceptualization. **J.L. Rullier:** Writing – original draft, Visualization, Resources, Methodology, Investigation, Conceptualization. **E. Lescoute:** Resources, Methodology, Investigation. **T. Doualle:** Methodology, Investigation, Formal analysis. **L. Gallais:** Resources, Methodology, Investigation, Formal analysis, Conceptualization.

### Declaration of competing interest

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

## Data availability

Data will be made available on request.

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