Y A Tene Deffo

B Zhang

G Sebald

Anomalous fractional diffusion equation for magnetic losses in a ferromagnetic lamination

Keywords: Anomalous diffusion, magnetic hysteresis, frequency dependence

During a full magnetization cycle and under a collinearity situation, the magnetic losses in a ferromagnetic are observable by plotting the average magnetic flux density or magnetization as a function of the tangent magnetic excitation. This highly frequency dependent magnetic signature is called hysteresis cycle and its area is equal to the energy consumed during the magnetization cycle.

The physical mechanisms behind this energy conversion are complex as they interfere and take place at different geometrical scales. Microscopic Eddy currents due to domain wall variations plays an important role, as well as the macroscopic Eddy currents due to the excitation field time variations and ruled by the magnetic field diffusion equation. From the literature on this topic, researchers have been proposing simulation models to reproduce and understand those complex observations. Even if all these losses contributions are physically interconnected, most of the simulation models available in the literature are based on the magnetic losses separation principle where each contribution is considered separately. Physically, the Weiss domains distribution and movements distort the diffusion process which becomes anomalous. In this article, the standard magnetic field diffusion equation is modified to take into account such anomaly. The first-order time derivation of the magnetic induction diffusion is replaced by a fractional order time derivation. This change offers flexibility in the simulation scheme as the fractional order can be considered as an additional degree of freedom. By adjusting precisely this order, very accurate simulation results can be obtained on a very broad frequency bandwidth for the prediction of the iron losses in a ferromagnetic material.

I -Introduction

The magnetization processes in a ferromagnetic material are incredibly complex. Multiscale phenomena interfere, they depend highly on the surrounding conditions (temperature, pressure, geometry …). Under the influence of an external magnetic excitation, the stability of the magnetic domains distribution is disturbed and movements, fusions, nucleation, spread all over the magnetic sample [START_REF] Bertotti | hysteresis in magnetism[END_REF]. The magnetic losses due to the magnetization cycles in a ferromagnetic material are highly frequency f dependent [START_REF] Raulet | The magnetic field diffusion equation including dynamic hysteresis: a linear formulation of the problem[END_REF]- [START_REF] Chevalier | A new dynamic hysteresis model for electrical steel sheet[END_REF]. They manifest themselves while plotting the evolution of the material magnetic state (the magnetic induction field B) as a function of the external tangent magnetic excitation field H, the hysteresis cycle B(H). The area of the hysteresis cycle is equal to the magnetic losses, i.e. the energy consumed during a magnetization cycle. It is admitted that below a given frequency known as the quasi-static threshold, the hysteresis cycle shape remains unchanged. Under such conditions, it is also admitted that the magnetic losses are mainly due to the microscopic Eddy currents induced around the moving domain walls or directly inside them. For higher frequencies, the frequency dependence is clearly noticeable as the hysteresis shape is strongly modified. As f increases and under "maximum H" imposed conditions, the hysteresis area increases first, then decreases gradually. A full decline can even be envisaged under extreme frequency levels. Under "maximum B" imposed conditions, the hysteresis cycle area increases continuously at a slow rate (see fig. 1). Beyond the quasi-static threshold, domain walls and microscopic Eddy currents are still acting but they interfere with macroscopic Eddy currents which origin is linked to the external magnetic excitation field H variations. Ferromagnetic hysteresis, frequency dependence, Eddy currents interactions … have been studied for decades and a large number of simulation approaches can be found in the scientific literature [START_REF] Raulet | The magnetic field diffusion equation including dynamic hysteresis: a linear formulation of the problem[END_REF]- [START_REF] Zhang | Dynamic magnetic scalar hysteresis lump model, based on Jiles-Atherton quasi-static hysteresis model extended with dynamic fractional derivative contribution[END_REF]. The first experimental studies dealing with magnetic domains and wall distributions have been proposed by Bitter at the beginning of the 20 th century [8]. A few years later, Weiss in [START_REF] Weiss | L'hypothèse du champ moléculaire et la propriété ferromagnétique[END_REF] [10] published his theoretical hypothesis leading towards the idea of magnetic domains in ferromagnetic materials. Right after, Pry & Bean in [11] confirmed the idea of magnetic domains, and established the first connection between their movements and some potential magnetic losses due to the magnetization cycle. But Steinmetz was the first one to propose an empirical loss power law (B and f dependent) to theoretically consider the magnetic losses in a ferromagnetic lamination [START_REF] Ch | On the law of hysteresis[END_REF]. Later again, Bertotti et al. in a large number of publications proposed the well-known losses separation principle which is still today probably the most standard approach used to consider magnetic losses inside an electromagnetic device [13][14]. According to Bertotti, the magnetic losses in a ferromagnetic lamination can be separated into three contributions: The frequency independent quasi-static hysteresis losses, the first order f 1 frequency dependent standard losses and the square root f 0.5 frequency dependent excess losses. Fig. 1 below gives an illustration of the losses separation principle: Even if this approach gives accurate results, it is still limited to given conditions and hypotheses, and it considers only harmonic excitation, making it inappropriate to time dependent simulations. The idea of separate contributions is furthermore contradictory with the physical nature of the magnetization processes where multiscale behavior interactions are dominant. Even rarely described, simultaneous resolutions are somehow possible, in [START_REF] Raulet | The magnetic field diffusion equation including dynamic hysteresis: a linear formulation of the problem[END_REF] Raulet & al. proposed as an example a modified dynamic field diffusion equation. This strong formulation solves, in a simultaneous way, the diffusion equation and a dynamic material law. Space discretized finite differences are used to solve the diffusion equation.

For every nodes, locals B and H are connected through the dynamic material law. This law is made out of a classic inverse quasi-static hysteresis model (Preisach [15][16], Jiles-Atherton [START_REF] Jiles | Theory of ferromagnetic hysteresis[END_REF][18] …) extended to the dynamical effect via a first-order time derivation of the local induction field. This approach is particularly accurate as the physical nature of the simulated processes is well respected, however for complex geometries, fine space discretization schemes are mandatories and this increases drastically the simulation times. Furthermore, the implicit nature of the material law creates convergence issues at saturation levels and under weak frequency conditions where dynamic effects are extremely limited.

In this article, we propose an alternative solution to simulate simultaneously every magnetic loss contributions and to ensure convergence in every situation. Under a varying magnetic field excitation, macroscopic eddy currents generate through the cross section of the ferromagnetic sample. By increasing the velocity of this excitation, the eddy currents are rejected over the cross section peripheral areas according to the well-known electrical skin effect. These eddy currents behave as a magnetic shield preventing a complete magnetic field diffusion [START_REF] Ducharne | Fractional model of magnetic field penetration into a toroidal soft ferromagnetic sample[END_REF] [START_REF] Ducharne | Dynamics of magnetic field penetration into soft ferromagnets[END_REF]. All these behaviors are analytically considered through the magnetic field diffusion equation derived from the Maxwell equations:

2 . dB H dt    (1)
In a ferromagnetic material, the magnetic diffusion and the related eddy currents interfere with the magnetic domain structure. These interactions distort the diffusion process and the classic diffusion equation is no more suitable to provide correct simulation results. This unusual diffusion process can be classified as anomalous and requires unfamiliar mathematic tools to be modeled. Such operators exist in the framework of fractional calculus, specifically fractional derivatives. By replacing the firstorder magnetic induction field time derivation by a fractional one as proposed in (Eq. 2), flexibility is provided in the simulation scheme and very accurate simulation results can be obtained.

. d B H dt

     (2)
By adjusting the fractional order of this modified diffusion equation, the obtained equation is able to simulate precisely all the dynamic magnetic losses contributions.

In the first part of this article, details will be provided as a reminder of the fractional derivative concept, then the modified diffusion equation and its resolution will be discussed. An experimental set-up will be described right after and comparisons simulations/measures used to validate our theory.

II -Fractional derivative

Fractional derivatives generalize the concept of the derivative to non-integer orders. In applied mathematics and mathematical analysis, the fractional derivative is a derivative of any arbitrary order, real or complex [START_REF] Grünwald | Ueber begrenzte derivationen und deren Anwendung[END_REF]. A very limited number of functions can be analytically fractional derived.

Trigonometric functions are some of them. Different definitions of fractional derivatives are available.

The Grünwald-Letnikov [22] and the Riemann-Liouville [23][26] definitions are probably the most popular even if both of them are particular cases of a general fractional-order operator namely. The first one represents the α order derivative, while the other represents the α folds integral. where Γ(α) is the gamma function:

( ) 1 ( ) ( ) ( ) (1 ) t t d f t d D f t t f d dt dt                 (3)
The additional time derivative present in the formula coincides with the occurrence of positive argument of the gamma function, Γ(.), leading towards its convergence to a finite value. From (Eq. 3), it is obvious that fractional derivatives include the memory of the previous states. Such property is fundamental while working with hysteretic behavior where current states depend strongly on the tested system or material history. From the spectral perspective, an interesting consequence of fractional derivatives is how the frequency spectrum f(ω) is multiplied by (jω) α instead of jω for a firstorder classic derivative.

In this study, after multiple tests, the Grünwald-Letnikov definition (Eq. 4) has been preferred as the convergence rate and the simulation times were noticed to be better.

0 0 ( ) 1 ( ) lim ( 1) ( ) ( 1)( 2)...( 1) ! k t h k d f t D f t f t kh k dt h k k                         (4)

III -Anomalous diffusion equation

II.1 -Introduction

Iron losses in a ferromagnetic material are related to both the macroscopic and the microscopic Eddy currents. The losses separation principle discretizes the losses contribution in 3 terms. In this article, the classical and the excess losses are jointly considered through a modified anomalous diffusion equation. In a ferromagnetic material, the domain wall distribution locally interferes with the magnetic field diffusion creating local interactions distorting the diffusion process which can be interpreted as an anomalous behavior. In physics, anomalous diffusion processes are frequent in complex media [START_REF] Metzler | The random walk's guide to anomalous diffusion: a fractional dynamics approach[END_REF], anomalous diffusions are relatively commonly considered through fractional-order diffusion equation models [START_REF] Mainardi | The fundamental solution of the space-time fractional diffusion equation[END_REF] [START_REF] Atangana | Fractional Diffusion Processes: Probability Distributions and Continuous Time Random Walk[END_REF]. Here, classically, the time derivative term is corresponding to a long-time heavy tail decay and the spatial derivative for diffusion nonlocality. Introducing fractional derivatives gives flexibility in the simulation process and it also leads to huge improvements in terms of accuracy [START_REF] Guyomar | Time fractional derivatives for voltage creep in ferroelectric materials: theory and experiment[END_REF][26] [START_REF] Colbrook | Scaling laws of passive-scalar diffusion in the interstellar medium[END_REF]. In our particular case, the time derivative term is extended to a fractional order (Eq. 2).

The Grünwald Letnikov fractional derivative definition is used to solve it:

2 0 0 1 .lim ( 1) ( ) k h k H B t kh k h                  (5)
B and H are connected through a sigmoid-type anhysteretic non-linear relation. This sigmoid function can be expressed from a Langevin-type function (Eq. 6):

2 0 0 . 1 ( ) .lim ( 1) . . . ( ) k h n s s k a H B H H t kh H k h H t kh a B coth a a B coth a                                                     (6)
Or from the Ising spin model (Eq. 7):

1 2 1 0 0 . tan 1 ( ) .lim ( 1) . . . tan k a k n h s s H B H t kh H k B a h B a                                        (7)
Bs is the saturation induction, a is an intrinsic material parameter strongly dependent on the density of the domain walls. Bs and a are obtained by comparing simulation results to the experimental anhysteretic curve. Curve fitting toolbox from @Matlab software is used. Fig. 2 below, shows the comparison obtained after optimization for both the experimental and the simulated anhysteretic curve. It is worth mentioning that the experimental anhysteretic curve is an approximation as it has been reconstructed through average values from both increasing and decreasing branches of the experimental hysteresis cycle.

II.2 -Resolution

The modified diffusion equation (Eq. 6) is solved through an Epstein frame-type cross section lamination. The geometrical and magnetic assumptions are those of the single sheet tester:

unidirectional and homogeneous surface excitation field H, collinearity between B and H. We also assume the electrical conductivity σ as homogeneous and constant. Considering the dimensions of the studied lamination (width >> thickness), a one-dimension (1-D) study in the thickness direction is carried out. The symmetry property of the single sheet tester allows to limit the study area to the half thickness of the lamination. The 1-D study limits the problem's variable number, (Eq. 8) becomes:

1 2 1 2 0 0 . tan 1 ( ) .lim ( 1) . . . tan an s s k h k B H B H H t kh a B a k y h                                         (8)
Taking into account the simplicity of the study domain, i.e. half thickness of a magnetic lamination, the finite differences method is used to solve (Eq. 9). Once every nodes' equation completed, a matrix system can be established:

      . M X S  (9)
Where:

  .

. 1 tan ( / )

n s i i s M B a e B B dt                                                                             1 2 3 / 2 1 2 0 _ . . . / ( . ) 1 . . . ( 1) . . .tan 1 tan ( / ) n t dt s k s i i k i s H surface S B a H t k dt e k B B dt B a                                                                                       (10)
Here Bi and Hi are respectively the i node magnetic induction and excitation. e is the space step separating two differences nodes. For every time step, the matrix system is solved. Once all the Bi known, the material magnetic induction is calculated from direct average calculus:

1 n i B B n   (6)

IV -Losses simulation

The magnetic losses are calculated from the hysteresis cycle area. For the first simulation results (Fig. 5 below), surface H sine waveforms are imposed, the maximum amplitude is set to 350 A/m and the frequencies tested range from 2 to 4096 Hz. 10 values of fractional orders are tested from 0.1 to 1. The magnetic losses are negligible under weak frequency, they increase up to a maximum value and decrease continuously until reaching a negligible level. It is worth noticing through these first simulation results that the lack of losses under low frequency levels is justified from the absence of the quasi-static contribution. The effect of the fractional derivative consideration in the fractional diffusion equation can be described as a stretching of the Losses vs frequency curve. By decreasing the fractional-order, we decrease the maximum value of the magnetic losses and shift the peak in the rightward direction. From a hysteresis cycle point of view, the effect of this fractional-order decrease can be observed by a slow but continuous lay down of the hysteresis cycle. According to the ferromagnetic material characterization standards [START_REF]Magnetic materials -Part 2: Methods of measurement of the magnetic properties of electrical steel strip and sheet by means of an Epstein frame[END_REF]- [START_REF]Magnetic materials -Part 6: Methods of measurement of the magnetic properties of magnetically soft metallic and powder materials at frequencies in the range 20 Hz to 200 kHz by the use of ring specimens[END_REF], to correctly characterize the hysteresis signature of a ferromagnetic material, the experimental tests have to be performed under "B imposed" sinus-type waveform. The anomalous diffusion equation (Eq. 5) can only be solved under "H imposed" conditions opposed to the standard conditions described in [START_REF]Magnetic materials -Part 2: Methods of measurement of the magnetic properties of electrical steel strip and sheet by means of an Epstein frame[END_REF]- [START_REF]Magnetic materials -Part 6: Methods of measurement of the magnetic properties of magnetically soft metallic and powder materials at frequencies in the range 20 Hz to 200 kHz by the use of ring specimens[END_REF]. To address this issue, we used a feedback control and a proportional/integral corrector. This solution is described in Fig. 6. The corrector parameters are set from experimental optimization. Working under "B imposed" conditions is imposed by the norm but it is also a far much simpler way to consider the quasistatic contribution in our simulation. This quasi-static contribution can be provided from the usual quasi-static hysteresis models. The classic Preisach and Jiles-Atherton models in their inverse versions [START_REF] Leite | The inverse Jiles-Atherton model parameters identification[END_REF]- [START_REF] Cardelli | Direct and inverse Preisach modeling of soft materials[END_REF] can be by instance used with success for this contribution. To limit the size of this article, both these well-known models will not be described hereby but every details can be found in the literature [START_REF] Metzler | The random walk's guide to anomalous diffusion: a fractional dynamics approach[END_REF] [START_REF] Colbrook | Scaling laws of passive-scalar diffusion in the interstellar medium[END_REF]. After several tests, we selected the Jiles-Atherton model for its simplicity and its limited number of parameters.

V -Experimental set-up

In order to experimentally validate the theoretical concepts described in this article, comparisons simulations/measures have been performed. Magnetic lamination samples made out of a nonoriented SiFe material have been considered. These sheets are 3% silicon iron non-oriented grain electrical alloy referenced 35C330 grade. The manufacturer characteristics are given in Tab. I below.

Tab. 1 -35C330 properties from the manufacturer.

The tested specimens have Epstein frame dimensions: 280 mm long, 30 mm wide and a thickness of 0.35 mm. The conductivity is 2.17 10 6 (S/m). Measurements have been performed using a single sheet tester according to the standard IEC 60404-3 [START_REF]magnetic properties of electrical steel strip and sheet by means of a single sheet tester[END_REF]. Two coils of 200 turns each in series constitute the primary excitation. A double C-yoke, 300mm wide and made out of grain-oriented Fe-Si lamination leads the magnetic flow through the tested sample. A 100 turns surrounding sensor coil is used to measure the magnetic flux through the cross section of the magnetic sample. The schematic of the experimental set-up is shown in Fig. 8. Harmonic B imposed characterization is performed. Real time feedback control is ensured from a DEWESoftX2 data acquisition software associated with a SIRIUSif 8×CAN data acquisition and generation card. These devices ensure the control of the magnetizing current and the acquisition of the measured signals. The surface tangent excitation field H is monitored by a tesla-meter (Hall probe sensor). To ensure the reproducibility of our experimental conditions, the Hall Effect sensor is positioned in the middle and in contact with the ferromagnetic lamination, as close as possible of the sensor coil. A numerical integration is used to obtain the induction field variations from the surrounding coil sensor voltage measurements. A numerical correction is performed to get rid of the undesired drift due to the analogic integration. Before each new measure, we ensure the reproducibility of the result by a complete demagnetization of the tested samples. The same experimental setup is used for this operation; the excitation winding is subjected to a slowly decreasing amplitude, 50 Hz triangular voltage signal. This process ensures a fully demagnetization under controlled rate of flux variation. The magnetization process lasts less than 2 minutes. The accuracy of the simulation approach is obvious. Best simulation results are obtained for a fractional order set to 0.91. This 0.91 fractional order can be physically interpreted as the best compromise between the first-order classical losses contribution and the square root order excess losses. The level of 1.52 T has been chosen as it ensures a complete magnetization state. Even if not displayed in this article, it is worth mentioning that results obtained from lower amplitude tests exhibit very similar accuracy. Fig. 10 shows a comparison between the simulation and the measure on the hysteresis losses vs frequency curve, α = 0.91 curve is the only one plotted here as it gives the best results. 5% error bars have been added in the measured "Losses(f)" curve as it corresponds to the uncertainty of our experimental setup. An interesting aspect of the space discretization finite differences approach is the access to the sub-

VII -Conclusion

Fractional diffusion equations have already been used with success to simulate anomalous diffusive behaviors. In this article, it is shown its effectiveness for the simulation of the magnetic field diffusion through a ferromagnetic material. The influence of the magnetic domains and their activities on the diffusion of the magnetic field is strong. They clearly disturb the process and the resulting diffusion can with no doubt be classified as anomalous. In this paper, we explain in details how the classic magnetic diffusion equation has been modified with fractional derivative operators and illustrate the strong performances obtained with the new formulation thanks to accurate comparisons with experimental results. From the iron losses and hysteresis frequency dependence point of view, as the fractional order gives flexibility in the simulation scheme, by adjusting it, classical and excess losses can be simulated simultaneously and through a single term. Such approach can, in some way, be interpreted as a much more coherent reading of the physical situation where macroscopic and microscopic eddy currents are continuously interacting and can't hardly be separated. In this article, we have limited the simulation effort to a 1D geometrical situation, future work should extend these results on geometry where 2D and even 3D simulations are mandatories.

Fig. 1 -

 1 Fig. 1 -Illustration of the magnetic separation losses principle as proposed by Bertotti.

Fig. 2 -

 2 Fig. 2 -Comparison simulation/measure for the anhysteretic nonlinear behavior, (f = 1 Hz, for the experimental conditions see part. V).

Fig. 3 -

 3 Fig. 3 -Simulation configuration.

Fig. 5 -

 5 Fig. 5 -Hysteresis losses vs frequency for different values of α (H imposed).

Fig. 6 -

 6 Fig. 6 -Hysteresis losses vs frequency for different values of α (H imposed).

Fig. 7

 7 Fig. 7 shows the magnetic losses versus frequency simulation curves. The same 10 fractional orders

Fig. 7 -

 7 Fig. 7 -Hysteresis losses vs frequency for different values of α (B imposed).

Fig. 8 -

 8 Fig. 8 -Experimental set-up, single sheet tester.

Fig. 9 -

 9 Fig. 9 -Comparisons simulations/measures for major hysteresis cycles under magnetic excitation of increasing frequencies (1 to 400 Hz).

Fig. 10 -

 10 Fig. 10 -Comparison simulation/measure for the magnetic losses vs frequency curve.

3 )

 3 surfacic and the in-depth field distributions (Hi and Bi). In the last figure of this article, we show the Losses (J.m -depth distribution of the magnetic state as a function of the frequency for different values of α. The obtained results confirm several intuitive conclusions, i.e. a full magnetization state for weak α values and even under very high frequency excitation. By opposition huge variations of the magnetic distribution vs frequency can be observed for high α fractional orders. When α = 1 the transition state between full magnetization and no magnetization stands in a relatively narrow frequency bandwidth, by decreasing alpha the frequency bandwidth enlarges as an inverse proportion of the α variations.

Fig. 11 -

 11 Fig. 11 -Hysteresis losses vs frequency for different values of α (H imposed).

Funding: _ This study was funded by the SCAC (Service de cooperation et d'action culturelle) of the French Embassy to Cameroon. _ This study was funded by the National Natural Science Foundation of China (Grant N° 51805298) and by the Natural Science Foundation of Shandong Province (Grant No. ZR201807090390).

Conflict of Interest:


Gildas Coativy
email: gildas.coativy@insa-lyon.fr

Hiba Haissoune

Laurence Seveyrat

Gaël Sebald

Laurent Chazeau

Jean-Marc Chenal

Laurent Lebrun

Elastocaloric properties of thermoplastic polyurethane

Keywords: polyurethane, elastocaloric effect, fatigue, infrared imaging, thermoplastic elastomer

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

The development of caloric materials is an active research area dedicated to replacing coolants used in cooling systems. 1,2 Basically, caloric materials can become structurally ordered/disordered upon the application/removal of an electric or magnetic field or a mechanical stress. 1,[3][4][5] The resulting order/disorder transitions are responsible of entropy variations. When these transitions are faster than the time needed for the material to exchange heat with its surroundings, there occurs a positive or negative temperature variation of the material. 1 Among elastocaloric materials, elastomers are of interest due to their elastocaloric effect being triggered by small stress values (a few MPa) and the associated significant temperature variations which can reach 14 K. [6][7][8][9] In addition, they are cheap materials. For these reasons, the use of elastomers in solid-state cooling systems is gaining widespread interest in several scientific communities, 9,10 and strongly motivates studying the elastocaloric effect of various elastomers.

The elastocaloric effect is mainly attributed to two contributions: [6][7][8]11,12 the first is a reversible orientation/disorientation of macromolecules in the strain direction (entropic elasticity), 11 which can be observed in all amorphous elastomers. The second contribution is strain-induced crystallization (SIC) that corresponds to a partial crystallization of the elastomer as a result of the deformation.

Depending on the initial structure of the elastomer (crosslink density, crosslink distribution…) and the mechanical stress program that is applied, the elastocaloric effect can be either reversible or irreversible. For instance, the stretching of natural rubber (NR) with a low crosslink density can lead to the formation of crystals that are stable at room temperature whereas with an intermediate crosslink density, the formed crystals are unstable and melt when the stress is released. [13][14][15][16] The elastocaloric effect of NR has been well documented and it has been revealed that the dominant caloric effect is the reversible SIC. 6,7 However, although SIC is observed in other elastomers such as polyurethane, hydrogenated acrylonitrile butadiene rubber (HNBR) and polychloroprene [START_REF] Ulu | Const. Model. Rubber X[END_REF][START_REF] Severe | [END_REF][19][20][21][22][23][24] , the elastocaloric effect of these materials has yet to be investigated.

The motivation for the work presented here is to display an elastocaloric effect in thermoplastic elastomers (TPE). These heteropolymers are made of hard segments (HS) (with a melting temperature Tm or a glass transition temperature Tg above room temperature Troom) and soft segments (SS) that are in the rubbery state at Troom. 25 These HS and SS are covalently bonded, but their partial or total immiscibility is responsible for a phase-separated structure made of HSand SS-rich domains. The HS-rich domains play the role of physical crosslinks at Troom permitting the elastomeric behavior of SS in TPE, while above Tm, the TPE can be melt processed due to the melting of the HS-rich domains. This melt processability can be advantageous for making complex hermetic systems at lab scale for testing elastocaloric devices such as regenerative systems incorporating a fluid. 10 Previous investigations have brought to light the strain-induced crystallization (SIC) of crosslinked and thermoplastic polyurethanes (TPU) using calorimetry and X-ray diffraction (XRD). [20][21][22][23][24] The main objective of this work is to evidence the reversible SIC of the TPU chosen for this study by the aforementioned techniques and the second aim is to study the elastocaloric effect of TPU by stretching the samples rapidly at 1600%.s -1 and measuring the temperature variation. The final goal is to explore the cyclic elastocaloric behavior of TPU over 5000 cycles.

The investigated material is an aromatic polyether-based thermoplastic polyurethane Estane® X4977 NAT 039 grade from Lubrizol. TPU is composed of poly(tetramethylene-oxide) (PTMO) with a molecular weight of 2000 g/mol as soft segments (SS) and 4,4'-methylene-bis-(phenyl-isocyanate) (MDI) and 1,4-butanediol (BDO) as hard segments (HS). Due to phase separation, the HS formed highly polar and stiff microdomains embedded in a soft poorly polar matrix. 26 The same procedure as the one described by Jomaa et al. was used to prepare a TPU film by solvent casting. 26 In short, TPU granules were dissolved in dimethylformamide (DMF) at a ratio of 25wt%. After being heated up to 353 K for 4h, the solution was cast onto a glass plate with an Elcometer 3700 film applicator. The films were then dried at 333 K for 24h and annealed at 398 K for 3 hours. Their final thickness is about 200 µm.

X-ray diffraction measurements were carried out at room temperature with an XPert Pro MPD Panalytical diffractometer equipped with a graphite monochromator (CuKα1 radiation) and a X'Celerator detector. For the measurement under strain, the samples were manually stretched and fixed to the sample holder during their deformation. The calorimetric analysis was performed by Differential Scanning Calorimetry (Setaram, DSC 131 evo) under nitrogen atmosphere. The samples (about 20 mg) were placed in closed aluminum crucibles and cooled from ambient temperature down to 153 K, then heated to 488 K and finally cooled. The heating and cooling ramps were carried out at 10 K.min -1 .The elastocaloric coupling of TPU samples (20*20*0.2 mm 3 ) was investigated via a tensile test bench combined with an infrared camera similar to the one used by Yoshida et al. 27 The tensile test bench was composed of a single axis robot RSDG212 (MISUMI Corporation, Japan) with an analog force sensor XFTC300-200N (Measurement Specialties, France). The displacement was measured with a laser optical displacement sensor ILD1420-200 (micro-epsilon, France) and the corresponding elongation λ= 𝑙(𝑡) 𝑙 𝑜 was defined as the ratio between the length l at time t and the initial length l0=20 mm.

In parallel, the material's surface temperature was measured using a thermal camera Optris PI450 (Optris, Germany).

Figure 1(a) presents the thermogram of TPU after processing. The glass transition temperature of the soft segments TgSS is visible at 205 K. The peaks observed at 240 K and at 282 K (just below room temperature) could be attributed to the partial crystallization and the melting of SS, respectively. 26,28 The enthalpy of melting is about 17 kJ/kg. The double peaks at 424 K and 445 K could either correspond to the melting of the semicrystalline HS with two different chain lengths, or be related to rearrangements within SS and HS followed by the fusion of crystalline HS. 26 In summary, below the glass transition temperature of the soft segments (Tgss=205 K), the material is mainly made up of amorphous SS-rich phases and semicrystalline HS-rich phases, whereas above Tgss, the SS phase crystallizes partially around 240 K until reaching 273 K and melts just below room temperature (at around 282 K), thus reverting to the initial stucture made of amorphous SS domains and semicrystalline HS-rich phases. This experiment suggests that the soft segments, not subjected to any strain, are amorphous at room temperature. X-ray diffraction was performed on the TPU before, during and after stretching at λ=4 (Fig. 1(b)). Although the hard domains were not melted, an amorphous halo was observed at λ=1 for the unstretched sample. It was centered at 2θ=20° and had a shoulder at 2θ=12°. When the material was stretched up to λ=4, two diffraction peaks appeared at 2θ=19.8° and 2θ=24.1°. According to Ren et al, this is evidence of strain-induced crystallization (SIC) of the PTMO soft segments as depicted in Fig. 1(c). 23 These results are in good agreement with the X-ray diffraction pattern of pure PTMO. 29 Furthermore, when the stress was released, the X-ray pattern exhibited an amorphous halo once again thus demonstrating the reversibility of the SIC at room temperature.

It is worth mentioning that the presence of a melting peak at a temperature between 273 K and 283 K and the ability of the elastomer to crystallize under strain at room temperature has also been observed in other elastomers such as natural rubber (NR), hydrogenated acrylonitrile butadiene rubber (HNBR) or polychloroprene. [START_REF] Ulu | Const. Model. Rubber X[END_REF][START_REF] Severe | [END_REF][19]30,31 The presence of such a melting peak could be an indicator of a reversible SIC in the elastomers. In the case of SIC, the orientation of the macromolecules in the strain direction causes an entropy reduction of the macromolecules conformation which shifts the crystallization and melting temperatures towards a higher value. 15 However, after releasing the stress completely, the melting temperature drops, leading to the complete melting of the crystals.

The occurrence of SIC in TPU suggests a strong elastocaloric effect. In order to estimate the adiabatic temperature variation, TPU was strained adiabatically at different elongations from λ=1 to λf (Fig. 2(a)) according to the following:

The first step consisted in stretching the TPU at 320 mm.s -1 (1600%.s -1 ), after which the material was maintained at constant elongation during 150 s. When thermal equilibrium was reached, the elongation was decreased back to λ=1 at 320 mm.s -1 , and kept there until reaching thermal equilibrium again.

TPU exhibited fatigue during the first cycles as evidenced in Fig. 2 For each mechanical test, the first cycle differs from the second and third cycles in that it presents a higher stress but the same remanent elongation. This is probably due to the structural modification of the HS structure. 32 Indeed, it can be noted that the mechanical response exhibits both elastic and inelastic parts and when the elongation increases, so does the inelastic part. The third cycle is similar to the second, suggesting that the mechanical behavior becomes stabilized.

The maximum stress of the third cycle ranges from 3 MPa to 5 MPa, for a maximum elongation varying between 2 and 5.

Moreover, during the third mechanical cycle, the temperature of the sample surface was measured with an IR camera in the white rectangle represented in Fig. 3(a). The resulting temperature variation ΔT as a function of time is reported in Fig. 3(b). The fast stretching of the material induces a positive temperature variation ΔT caused by SS orientation and crystallization. At constant elongation the temperature drops until reaching thermal equilibrium. Then, a return to the initial elongation at λ=1 leads to a decrease of ΔT induced by disorientation of the soft segments and melting of the crystals. Moreover, the width of these two peaks varies. It becomes narrower after elongation than after retraction due to the stretched sample becoming thinner in comparison with the initial thickness at λ=1. The thickness reduction of the sample favors heat transfer. 32 Regarding the elastocaloric effect, |𝛥𝑇 𝑚𝑖𝑛 | increases significantly (from 1 K to 7.1 K) for a final elongation λf between 2 and 4 and slightly (from 7.1 K to 7.7 K) when λf varies between 4 and 5 (Fig. 3(c)). This onset of saturation could be explained by a higher inelastic mechanical response when λf increases. Finally, the temperature variation is within the same range as that of natural rubber (NR) at λf=5 whereas for lower elongations, the adiabatic temperature variation is higher for TPU. 7,8 Cyclic elastocaloric properties were studied by imposing a triangular cyclic elongation between λ=3 and λ=5 (Fig. 4(a)). With a chosen frequency f=0.2 Hz, 5000 cycles could be applied without the sample breaking. The resulting temperature variation is presented in Fig. 4(b). The peak-to-peak temperature variation ΔTpp varies between 4.5 K and 3.3 K (Fig. 4(d)) when the number of cycle increases from 10 to 5000. It corresponds to a decrease of about 10% per decade. It can be noted that ΔTpp for TPU is in the same range as for NR when the latter is cycled between λ=3 and λ=6 33 , but with a lower stability. and the peak-to-peak temperature variation ΔTpp decrease when the number of cycles increases (Fig. 4(d)). After the hundredth cycle, the stress at λ=3 is equal to 0 MPa, indicating an inelastic response. Besides, an estimation of heat exchange can be obtained according to Q≈c Δ𝑇 𝑝𝑝 = 8 kJ/kg (where c is the specific heat capacity of TPU). This is 8 times larger than the mechanical losses in a single cycle. An indicator of material performance of a cooling device using a caloric material is the coefficient of performance COP, defined roughly as COPmat=Q/Wlosses. The COP value of 8 obtained here is in the normal range of caloric materials, i.e., between 5 and 20. 34,35 To sum up, this work investigated the elastocaloric properties of thermoplastic polyurethane. A strain-induced crystallization was revealed via X-ray diffraction, and it was verified that the crystals melted after complete unloading, indicating a fully reversible strain-induced phase transition. Consequently, the latent heat of this phase transformation contributed fully to the elastocaloric effect after complete unloading. The adiabatic mechanical tests exhibited a temperature variation in the same range as that found for natural rubber when tested under identical conditions.

The mechanical and elastocaloric fatigue tests demonstrated that a large elastocaloric effect (Tpp>3K) could be maintained for 5000 cycles. From a system point of view, it should be noted that thermoplastic polyurethane can be melt processed or potentially 3D printed, opening great perspectives in terms of fabrication and optimization of cooling devices at a lab scale.

Such cooling systems could also be advantageous for testing analytical or numerical models of elastocaloric devices with a variety of complex shapes that can be practically produced.

However, from an applicative viewpoint, the continuous decrease of elastocaloric properties and the inelastic response of TPU are limiting factors for solid state cooling applications. To address these issues and increase the panel of elastocaloric polymers, other polyurethanes, such as crosslinked polyurethane with varying HS/SS ratios, or other elastomers, such as HNBR or polychloroprene, should be studied. [START_REF] Ulu | Const. Model. Rubber X[END_REF][START_REF] Severe | [END_REF][19][20][21][22][23][24]

5 FIG. 1 .

 51 FIG. 1. (a) DSC thermogram of TPU. (b) X-ray diffractograms of TPU before, during and after straining at λ=4. (c) Schematic of strain-induced crystallization of TPU

  FIG. 2. Mechanical testing of TPU. (a) The mechanical stress program. (b) The evolution of the stress as a function of elongation (first 3 cycles).

FIG. 3 .

 3 FIG. 3. Elastocaloric properties of polyurethane. (a) IR images for three states: λ=1 (original state), λ=5 (stressed state) and λ=1 (recovered state) of TPU (b) Temperature variation as a function of time (ΔT= T(t)-Troom). (c) The minimum temperature variation ΔTmin as a function of the final elongation λf..

FIG. 4 .

 4 FIG. 4. Elastocaloric effect of polyurethane during triangular solicitation when λ was varied between 3 and 5 at f=0.2Hz. (a) Elongation program. (b) Temperature variation as a function of time. (c) Stress/elongation curves of TPU for a varying number of cycles. (d) Maximum stress and peak-to-peak temperature variation as a function of the number of cycles.

Figure 4 (

 4 Figure 4(c) shows the stress/elongation curves of TPU for a varying number of cycles. With the exception of the first cycle, the hysteresis loss is about 1 kJ/kg per cycle. The maximum stress

Acknowledgement This work was supported by ANR through the project ECPOR (No. ANR-17-CE05-0016) The authors gratefully acknowledge the French Region Auvergne-Rhône-Alpes, the French project IDEXLYON of the Université de Lyon in the framework of the "Investissements d'Avenir" program (No. ANR-16-IDEX-0005). The authors would also like to thank Dr. Sandrine Cardinal for her help with the X-ray diffraction experiments.

Data availability

The data that support the findings of this study are available from the corresponding authors upon reasonable request.