V Delobelle
email: vincent.delobelle@imag.fr

G Chagnon

D Favier

T Alonso

Study of electropulse heat treatment of cold worked NiTi wire: From uniform to localised tensile behaviour

Electropulse heat treatment is a technique developed to realise fast heat treatment of NiTi shape memory alloys. This study investigates mechanical behaviour of cold worked NiTi wires heat treated with such a technique. It is demonstrated that milliseconds electropulses allow to realise homogeneous heat treatments and to adapt the mechanical behaviour of NiTi wires by controlling the electric energy. The material can be made elastic with different elastic modulus, perfectly superelastic with different stress plateau levels and superelastic with important local residual strain. Due to the short duration and high temperature of the heat treatment, this technique allows to obtain mechanical properties that cannot be obtained with classical heat treatments of several minutes in conventional furnaces such as linear evolution of the final loading and high tensile strength to 1500 MPa for superelastic material or increase of the stress plateau level with cycling for superelastic material.

Introduction

Since several years, NiTi shape memory alloys (SMA) are the most widely used SMA in engineering fields as reported by [START_REF] Van Humbeeck | Non-medical applications of shape memory alloys[END_REF], but more especially for biomedical applications as reviewed by [START_REF] Duerig | An overview of nitinol medical applications[END_REF] due to their excellent mechanical properties, corrosion resistance and biocompatibility. To design their applications, engineers use industrial basic components such as NiTi wires, tubes or plates. These components are generally shaped with several successive hot and cold rolling operations. During these operations, the material is severely deformed causing important grain size reduction, amorphisation of the material, and finally leading to a suppression of the phase transformation which confers the unique superelastic or ferroelastic properties to SMA as shown in [START_REF] Jiang | Nanocrystallization and amorphization of NiTi shape memory alloy under severe plastic deformation based on local canning compression[END_REF]. To restore these properties, annealing and ageing treatments are classically used as shown in [START_REF] Jiang | Crystallization of amorphous NiTi shape memory alloy fabricated by severe plastic deformation[END_REF] for example. These heat treatments are generally long, classically 60 min for annealing and 10-120 min for ageing step as proposed in [START_REF] Jiang | Effect of ageing treatment on the deformation behaviour of Ti-50.9 at.%[END_REF] for example. Such heat treatments are performed in conventional furnace so the entire sample is heat treated homogeneously.

Recent studies investigated heat treatments of NiTi wires with Joule effect. Duration of such heat treatments is very dispersed. [START_REF] Zhu | The improved superelasticity of NiTi alloy via electropulsing treatment for minutes[END_REF] proposed heat treatment of several minutes duration, [START_REF] Wang | Effect of short time direct current heating on phase transformation and superelasticity of Ti-50.8 at.% Ni alloy[END_REF] and [START_REF] Malard | In situ investigation of the fast microstructure evolution during electropulse treatment of cold drawn NiTi wires[END_REF] of seconds duration and [START_REF] Delville | Microstructure changes during non-conventional heat treatment of thin Ni-Ti wires by pulsed electric current studied by transmission electron microscopy[END_REF] studied milliseconds heat treatments. Interest of such heat treatments is twofold: (i) reducing the time of heat treatment and (ii) performing local heat treatment. This last point is of key interest to realise architectured materials with multiple or graded mechanical properties as obtained in [START_REF] Meng | Ti-50.8 at.% Ni wire with variable mechanical properties created by spatial electrical resistance over-ageing[END_REF] with 300 s heat treatment. For milliseconds heat treatments, [START_REF] Delville | Microstructure changes during non-conventional heat treatment of thin Ni-Ti wires by pulsed electric current studied by transmission electron microscopy[END_REF][START_REF] Delville | Transmission electron microscopy investigation of dislocation slip during superelastic cycling of Ni-Ti wires[END_REF] focused on the microstructures observation of the material and few informations are available about transformation and mechanical properties of the created materials. Moreover, [START_REF] Delville | Microstructure changes during non-conventional heat treatment of thin Ni-Ti wires by pulsed electric current studied by transmission electron microscopy[END_REF] mentioned the presence of an important gradient during cooling phase of their heat treatments but the structure is supposed homogeneous. For longer heat treatments proposed in [START_REF] Meng | Ti-50.8 at.% Ni wire with variable mechanical properties created by spatial electrical resistance over-ageing[END_REF], when a thermal gradient is applied to the sample, the material has graded mechanical properties. Thus, for milliseconds heat treatments, it is important to analyse the homogeneity of the created material from a mechanical point of view.

In this study, heat treatments are realised with milliseconds electropulse. Investigation of the transformation and mechanical behaviours of the heat treated part of the NiTi SMA wires is realised. Transformation behaviour is studied by means of differential scanning calorimetry technique. Local mechanical behaviour is studied by means of digital image correlation (DIC) technique. Investigation of strain fields allows to study the impact of the heat treatment on the uniformity of the mechanical behaviour. In Section 2, experiments and methods are presented. In Section 3, results are described and discussed in Section 4.

Experiments and methods

Electropulse heat treatment

The experiments were performed on cold worked Ti-50.8 at.% Ni SMA wire of diameter 0.5 mm, from the commercial provider Fort Wayne Metals (NiTi # 1). The as-received material was in 45% cold worked condition. Short time electrical pulses were generated with a direct current welder (commercial ref.: DC160, Sunstone Engineering) in wire of length L HT = 20 mm, as shown in Fig. 1a. The wire was maintained with two massive brass grips. Brass grips conduce electricity to the sample and act as a thermal mass. Thus, as shown in Fig. 1b, after heat treatment, the tips of the wire are unchanged and the centre is heat treated. In this study, six heat treatments, called A, B, C, D, E and F were carried out.

During the heat treatment, voltage U W at the sample terminals and voltage U R = RI at the resistance terminal are measured, with R the resistor value and I the electrical current in the electrical loop, as shown in Fig. 1a. Power P = U W I dissipated in the wire is estimated. Evolutions of U W , I and P are presented in Fig. 2a,b,c, respectively. The dissipated power is almost constant during the heat treatment and equal to P = 3000 W. From these measurements, heat treatment duration T and final dissipated energies in the wire E = tP are estimated and summarised in Table 1 for all treatments. Note that the decrease of Uw = R wire I (Fig. 2a), where R wire is the wire electrical resistance, is in good agreement with [START_REF] Delville | Microstructure changes during non-conventional heat treatment of thin Ni-Ti wires by pulsed electric current studied by transmission electron microscopy[END_REF][START_REF] Delville | Transmission electron microscopy investigation of dislocation slip during superelastic cycling of Ni-Ti wires[END_REF] that observed significant decrease of the electrical resistance of the wire during pulse annealing. An infrared camera associated to a high magnification lens (Commercial reference: Camera SC 7600, Flir) was used to record images of the wire during tests. Due to the symmetry of the experimental setup, measurements are presented only on the half of the wire. Then, due to the unknown variation of the wire emissivity during tests, only radiation values are presented. Fig. 3a shows the maximal wire radiation for tests A to E, obtained at the end of heating and measured along the main axis of the sample y. Fig. 3b shows the wire radiation measured along the main axis of the sample y during cooling of test D, from the maximal radiation obtained to room temperature. For all tests at the end of heating (Fig. 3a), close to the clamps, a strong thermal gradient is observed on 1 mm, due to the heat loss into the clamps. It is assumed that the bump observed between 1 and 5 mm is due to a reflection of the wire itself via the clamp to the camera which increases the radiation. Then, between 1 and 10 mm, the radiation and thus the temperature are uniform for all tests. Note that for test E presented in Fig. 3a, the plateau observed between 2 and 7 mm is due to a saturation of the infra-red sensors.

During cooling (Fig. 3b), the observed gradient is due to the presence of the clamps acting as thermal mass, and increasing the cooling rate close to them. From this observation, it can be supposed that heat treatment is heterogeneous during the cooling phase.

To estimate the sample temperature during the experiment, it is considered that the wire is submitted to: (i) step electrical power pulse P = 3000 W, of duration equal to times indicated in Table 1 presented in full lines in Fig. 4 and (ii) radiation and convection heat losses. During heating, the sample temperature is supposed uniform as observed in Fig. 3a. Thus, the sample temperature was estimated by solving the following heat diffusion equation:

mC(T ) dT (t) dt = P(t) -Ah(T (t) -T 0 ) -A T 4 (t) (1)
where m is the sample mass, C the heat capacity taking in [START_REF] Smith | The heat capacity of NiTi Alloys in the temperature range 120 to 800[END_REF], depending of the sample temperature T at instant T, A the . From these estimations, maximum temperatures T max obtained at the end of the electropulse are summarised in Table 1. The maximum temperature estimated for heat treatment F is T F = 1440 • C which is superior to the melting temperature of almost equiatomic NiTi SMA T NiTi melting = 1310 • C. Experimentally, it was observed that the sample melt for such a pulse. Thus, experimental observation and theoretical approximations of the temperature are in good agreement. Due to the melting of the sample, experimental results cannot be presented for heat treatment F. The cooling time of the wire to room temperature is comprised between 40 and 50 s as shown in Fig. 4b. It remains important to keep in mind that these values are only estimations.

As shown in Fig. 1b. the resulting wire is a material having two properties. However, from Fig. 3, due to an important temperature gradient at the junction of the two materials, it can be assumed that a gradient of property is also present between the two materials. Nevertheless, this study only focuses on the transformation and mechanical behaviours of the part of the wire heat treated homogeneously.

Transformation and mechanical behaviours study

The transformation behaviour of materials was studied by means of DSC. DSC experiments were performed with a TA Q200 DSC between 90 and -90 • C with heating/cooling rate of 10 • C min -1 . The transformation behaviour of the wire was then studied between 80 and -70 • C, when the cooling ramp is stabilised. DSC measurements were realised for all the specimens.

All the tensile tests were performed using a Gabo Eplexor tensile machine. The tests were realised at room temperature T 0 ≈ 25 • C, at constant cross head velocity U = 0.1 mm min -1 , where U is the cross head displacement. The initial gauge length of the wire was L 0 = 18 mm, thus, the applied global strain rate was U/L 0 = 9.3 × 10 -5 s -1 . In this study the transition zone between heat treated material and as received material is not studied (Fig. 1b). During the tensile test, the axial force F was recorded. The nominal stress = F/S 0 , where S 0 is the initial section of the wire is calculated. In this study, the local strain field yy was estimated by means of DIC method. The strain field is averaged along the main axis the sample, in order to obtain the global strain of the material, noted in the following study.

Results

In the following, the cold worked material is noted CW. Then, CW material heat treated with pulses A, B, C, D, E are called material A, B, C, D, E, respectively. Transformation behaviour of CW is plat and does not exhibit any peak (Fig. 5a). This result is in good agreement with [START_REF] Kurita | Transformation behavior in rolled NiTi[END_REF]. The transformation behaviour of material A remains plat (Fig. 5b). A small peak is observed for material B, as sketched on the close up of Fig. 5c. A difference of 10 • C between heating and cooling peak temperatures and small heat of transformation of 2.8 J g -1 are well the signature of the austenite -R phase transformation (noted A -R). The transformation peak temperature at cooling and heating are about T A-R = 10 • C and T R-A = 20 • C, respectively. For material C (Fig. 5d), a A R transformation is observed with of mation estimated to approximately 2.5 J g -1 at cooling and heating.

Transformation behaviour

Transformation peak temperatures at cooling and heating are lower than with treatment B with T A-R = -18 • C and T R-A = -7 • C, respectively.

For material D (Fig. 5e), the Austenite-Martensite (noted A -M) transformation is observed with peak transformation temperatures equal to T A-M = -44 • C and T M-A = -14 • C. Direct and reverse heat of transformations are estimated to be 11.0 J g -1 and 13.5 J g -1 , respectively. Almost identical transformation behaviour is observed for material E (Fig. 5f) with a A-M transformation having heat of transformation equal to the ones found for material D. However, the peak transformation temperatures are higher than material D ones and are equal to T A-M = -38 • C and T M-A = -12 • C.

Mechanical behaviour

Mechanical tests presented in Fig. 6 are composed of a loading-unloading cycle and a final loading to failure. Fig. 6a shows the global mechanical behaviour of CW material and material A. For material A, strain profiles estimated along the main axis of the wire are plotted in Fig. 6b for instants defined in Fig. 6a. Strain profiles are similar to the ones obtained for CW material. Fig. 6c shows the global mechanical behaviour of materials B and C. For material C, strain profiles are plotted in Fig. 6d for instants defined in Fig. 6c. Strain profiles are similar to the one obtained for material B. Finally, Fig. 6e shows the global mechanical behaviour of materials D and E. For material D, strain profiles are plotted in Fig. 6f for instants defined in Fig. 6e. Strain profiles are similar to the one obtained for material E. Fig. 7a shows (i) the initial elastic modulus, noted E ini , during the first loading and (ii) the elastic modulus after localisation plateau, noted E end , observed on the stress-strain curves. Slopes to estimate the elastic moduli are sketched in dashed line in Fig. 6. Fig. 7b shows the plateau stresses at loading, unloading and the hysteresis height noted high , low and , respectively. CW material exhibits purely elastic brittle behaviour (Fig. 6a), i.e. stress-strain curve of the material is linear, and no plasticity occurs before failure. Its elastic modulus is estimated to 53 GPa. Ultimate tensile strength is about 1500 MPa. With short duration heat treatment, the material can be soften. Material A remains purely elastic and brittle with lower elastic modulus estimated to 43 GPa. The ultimate tensile strength remains high for metallic material and is about 1500 MPa. For these two materials, the strain field is uniform along the wire axis (Fig. 6b).

When increasing heat treatment energy, the materials B and C exhibit classical behaviour of superelastic NiTi SMA without residual strain (Fig. 6c andd). Stress plateaus due to direct and reverse phase transformations are observed at loading and unloading with an important hysteresis of = 200 MPa for the two heat treatments. The stress plateaus are lower for material C because the maximum temperature reached by material C is higher than material B one. The plateau stresses decreases between first and second cycles and the difference is estimated to -30 MPa and -10 MPa for materials B and C, respectively. For these two materials, classical localisation phenomena are observed during the stress plateau. During the ultimate loading, when the material is stretched to the maximum stress reached during the first cycle, stress drops up to the value of the first cycle plateau. The elastic moduli of first slopes increase with heat treatment duration. It is estimated to 50 GPa and 68 GPa for materials B and C, respectively. The elastic moduli after stress plateau are estimated to be about 21 GPa and 25 GPa for materials B and C, respectively. Ultimate tensile strength is about 1300 MPa for both materials. Stress-strain evolution after the plateau is linear. The material exhibits brittle behaviour.

For materials D and E, a superelastic behaviour with a residual strain is observed during the first loading, unloading cycle (Fig. 6e andf). Their loading plateau stresses are equal to 430 MPa and 450 MPa, respectively. Material D plateau stress is lower than C one. It is also lower than material E one while maximum temperature reached by material E is higher than material D one. Elastic moduli during the first slopes are estimated to 63 GPa for the two materials. During the ultimate loading, a non-classical behaviour is observed. When the plateau stress of the first loading is reached, the stress still increases with strain and reaches a plateau to a higher value. Then, when the material is stretched to the maximum strain reached during the first cycle, stress drops down to the stress value of the first cycle plateau. Then, global behaviour is identical to the behaviour observed during the first loading. Finally after the plateau, the elastic moduli presented in Fig. 6e are estimated to 20 GPa and 17 GPa for materials D and E Fig. 7a, respectively. Finally, strain hardening is observed. A maximal strain of 40% was reached for both materials without failure. During first loading, the local strain field exhibit localisation phenomenon as materials B and C. However, during the second loading, the local strain field is non-uniform with two localisation fronts. This specific local behaviour is not presented here but will be analysed in a forthcoming study. During the strain hardening phase, localisation is observed.

Finally, from Fig. 6c ande, the transformation strain of materials B, C, D and E are estimated to be 5%, 7%, 9% and 10%, respectively. When the pulse time increases, the transformation strain increases too.

Discussion

On the homogeneity of the heat treatment

In [START_REF] Meng | Ti-50.8 at.% Ni wire with variable mechanical properties created by spatial electrical resistance over-ageing[END_REF], thermal gradient was observed during 300 s heat treatment, leading to mechanical graded material. Considering thermal gradient observed during cooling phase of the heat treatment (see Fig. 3), an identical mechanical behaviour could be assumed. However, if local strain behaviours are very different from one to another materials, the heat treatment is uniform along the main axis of the sample. Initially, material A deforms homogeneously as clearly shown in Fig. 6b. For materials B and C, during elastic phases, the strain fields are uniform (Fig. 6d instants a,b,e,f). For all superelastic materials, during the plateau, i.e. when localisation is observed, outside localisation front, the strain is uniform to a high or low strain value (Fig. 6d andf). The specific local behaviour of materials D and E during localisation zone and strain hardening, presented in the previous section, is due to the partial pre-straining of the sample during the first loading.

Thus, even if the material temperature during cooling is heterogeneous, materials are heat treated homogeneously. For such heat treatment, the governing parameter is the maximal temperature reached during heating. Such a is not valid when increasing the duration of heat treatment as observed in [START_REF] Meng | Ti-50.8 at.% Ni wire with variable mechanical properties created by spatial electrical resistance over-ageing[END_REF]. et al. (2010, 2011) studied microstructure evolution of identical CW wires during milliseconds electropulse heat treatment. From comparison of Fig. 6 of this study and Fig. 3 of [START_REF] Delville | Microstructure changes during non-conventional heat treatment of thin Ni-Ti wires by pulsed electric current studied by transmission electron microscopy[END_REF], it is considered that materials A, B, C, D, and E can be compared with materials called 6 ms, 10 ms, 12 ms, 16 ms and 18 ms in their paper, respectively. In the following discussion, the microstructure is considered as the one summarised in Table 2, taken from [START_REF] Delville | Microstructure changes during non-conventional heat treatment of thin Ni-Ti wires by pulsed electric current studied by transmission electron microscopy[END_REF][START_REF] Delville | Transmission electron microscopy investigation of dislocation slip during superelastic cycling of Ni-Ti wires[END_REF].

Mechanical properties and microstructure

Delville

About materials elasticity

To begin, observations that CW and A materials, i.e. amorphous and polygonised material, have a high elastic potential and exhibits brittle behaviour (Fig. 6a andb) are in good agreement with [START_REF] Sergueeva | Structure and properties of amorphous and nanocrystalline NiTi prepared by severe plastic deformation and annealing[END_REF] and [START_REF] Delville | Microstructure changes during non-conventional heat treatment of thin Ni-Ti wires by pulsed electric current studied by transmission electron microscopy[END_REF]. [START_REF] Sergueeva | Structure and properties of amorphous and nanocrystalline NiTi prepared by severe plastic deformation and annealing[END_REF] showed that amorphous NiTi exhibits classical properties of metallic glasses. In the report of [START_REF] Schuh | Mechanical behavior of amorphous alloys[END_REF] about mechanical properties of metallic glasses, it is mentioned that the elastic modulus of amorphous material is about 30% lower than crystallised material. From Fig. 7a it is observed that the difference of elastic modulus between CW and D materials, i.e. partly amorphous and crystallised material is about 20%. The order of magnitude is in good agreement with [START_REF] Schuh | Mechanical behavior of amorphous alloys[END_REF] and the difference is assumed to be due to the presence of an important amount of austenite phase in the CW material. [START_REF] Schuh | Mechanical behavior of amorphous alloys[END_REF] also indicated that the room temperature elastic modulus decreases with increasing annealing temperature which is in good agreement with measurement of elastic modulus of CW and A materials (Fig. 7a).

Values of elastic modulus E ini proposed in Fig. 7a for materials B, C, D and E are generally associated to austenite elastic modulus. Results are in good agreement with dispersed values found in the literature comprised between 40 and 90 GPa as mentioned in [START_REF] Liu | Apparent modulus of elasticity of near-equiatomic NiTi[END_REF]. However, for materials B and C, considering that the material is composed of polygonised material and nanocrystals, these values cannot be associated to Young modulus of crystallised austenite because it can be supposed that nanograins and polygonised part deform in different manner. Values of elastic modulus E end are generally associated to martensite elastic modulus. Results are in good agreement with dispersed values found in the literature comprised between 20 and 50 GPa as mentioned in [START_REF] Liu | Apparent modulus of elasticity of near-equiatomic NiTi[END_REF]. The specific mechanical behaviour obtained after superelastic plateau is discussed in Section 4.2.3.

About superelasticity of materials

Since several decades, it is known that superelasticity of NiTi SMA is due to the phase transformation from austenite to martensite on the material grains. In the proposed case, it is assumed that identical deformation mechanisms occur in the nanograins and micrograins of B, C and D, E materials, respectively. Other deformation mechanisms such as nanograins rotation and boundary sliding can occur in materials A, B and C, as mentioned in [START_REF] Sergueeva | Structure and properties of amorphous and nanocrystalline NiTi prepared by severe plastic deformation and annealing[END_REF], but cannot be proven or discussed from the proposed results. For materials D and E, the material is recrystallised and composed of micrograins and it is assumed that the classical deformation mechanisms are observed. The irreversible strain observed in the stress strain curve is due to the sliding of dislocations observed in [START_REF] Delville | Transmission electron microscopy investigation of dislocation slip during superelastic cycling of Ni-Ti wires[END_REF]. Localisation phenomenon (Fig. 6d andf) is also classically associated to the superelastic behaviour obtained in tension.

Results about superelasticity obtained via electropulse heat treatment are in good agreement with the literature about superelasticity obtained via classical heat treatment as in [START_REF] Jiang | Effect of ageing treatment on the deformation behaviour of Ti-50.9 at.%[END_REF]: increasing heat treatment temperature decreases the plateau stress and increases stress hysteresis (Fig. 7b). For materials B and C, the plateau stress decreases with cycling as observed in [START_REF] Liu | Effect of pseudoelastic cycling on the Clausius-Clapeyron relation for stress induced martensitic transformation in NiTi[END_REF] and the yield drop phenomenon is observed as in [START_REF] Eucken | The effects of pseudoelastic prestraining on the tensile behaviour and two-way shape memory effect in aged NiTi[END_REF], for example. However, longer electropulse heat treatments, as for material D and E, create a unique mechanical behaviour: the plateau stress increases with cycling. This phenomenon is characteristic of electropulse heat treatments and was also observed in [START_REF] Delville | Microstructure changes during non-conventional heat treatment of thin Ni-Ti wires by pulsed electric current studied by transmission electron microscopy[END_REF]. The yield drop phenomenon is also observed but from a higher stress value. For these materials, very specific localisation phenomenon can be observed and it will be developed in an other study.

Brittle behaviour vs. strain hardening

In Fig. 6c, for materials B and C, the ultimate loading is linear and the material is brittle with important tensile strength. This behaviour is similar to the one observed for CW and A materials (Fig. 6a). With classical heat treatments, strain hardening are classically observed in cold worked material directly aged as in [START_REF] Saikrishna | On stability of NiTi wire during thermo-mechanical cycling[END_REF] or annealed and aged as in [START_REF] Jiang | Effect of ageing treatment on the deformation behaviour of Ti-50.9 at.%[END_REF]. Materials having linear and potentially elastic behaviour after transformation plateau, with important tensile strength, have already been observed in [START_REF] Pilch | Final thermomechanical treatment of thin NiTi filaments for textile applications by electric current[END_REF] with Joule heating but have never been obtained with conventional heat treatments to the knowledge of authors. This property is of great interest for SMA engineering.

From [START_REF] Delville | Microstructure changes during non-conventional heat treatment of thin Ni-Ti wires by pulsed electric current studied by transmission electron microscopy[END_REF][START_REF] Delville | Transmission electron microscopy investigation of dislocation slip during superelastic cycling of Ni-Ti wires[END_REF], it is known that materials D and E are recrystallised into micrograins and that important dislocation phenomenon occurs. This observation is in good agreement with strain hardening observed in Fig. 6e because dislocations create large irreversible deformations with strain hardening of the material. In the profile h of Fig. 6f, localisation is observed. This localisation is due to the pre-straining of the localised area during first loading.

Conclusion

This study investigated mechanical behaviours of cold worked NiTi wires heat treated with electropulse. From this study, one can conclude that milliseconds electropulse is an efficient method to realise homogeneous heat treatments and allows to adapt the functional properties of cold worked NiTi SMAs. On wire of diameter 0.5 mm, of length 20 mm and for heat pulse of power P = 3000 W:

• Low duration heat treatment (1.5 ms) allows to soften the initial cold worked material but preserves the elastic and brittle with a high ultimate tensile strength properties. • Middle duration heat pulse (2-3 ms) restores the classical superelastic behaviour observed on NiTi SMAs. After the plateau, the material deforms linearly with a brittle behaviour and an important ultimate tensile strength as the initial material: such a property cannot be obtained with classical heat treatments. • Long duration heat pulse (4-5 ms) allows to obtain a superelastic behaviour with important residual deformation. With cycling, the strain of successive plateaus increases with loading: such a property has never been observed with classical heat treatments. The residual deformation is due to the sliding of dislocations defects in the material. After the plateau, strain hardening is observed.

This study brings important information about mechanical behaviour of cold worked NiTi SMA heat treated with milliseconds electropulses.

Fig. 1 .

 1 Fig. 1. a) Experimental set-up presentation. (b) Resulting material with local heat treatment.

Fig. 2 .

 2 Fig. 2. Evolution of (a) Voltage UW, (b) electrical current I and (c) power P during electropulse heat treatment.

Fig. 3 .

 3 Fig. 3. Radiation measured with an infrared camera. (a) Radiation at the end of heating for tests A to E. (b) Radiation during cooling phase of test D.

Fig. 5

 5 Fig.5shows the transformation behaviour for CW, A, B, C, D and E materials.Transformation behaviour of CW is plat and does not exhibit any peak (Fig.5a). This result is in good agreement with[START_REF] Kurita | Transformation behavior in rolled NiTi[END_REF]. The transformation behaviour of material A remains plat (Fig.5b).

Fig. 4 .

 4 Fig. 4. P step (full lines) used to estimate temperature T (dotted lines) for heat treatment A to F during the heating phase. (b) Temperature T evolution during the cooling phase.

Fig. 5 .

 5 Fig. 5. DSC of (a) material CW, (b) material A, (c) material B, (d) material C, (e) material D, and (f) material E.

Fig. 6 .

 6 Fig. 6. Stress-strain curves of (a) initial material and material A, (c) materials B and C and (e) materials D and E. Local strain fields for (b) material A, (d)material C and (f) material D.

Fig. 7 .

 7 Fig. 7. (a) Elastic moduli E ini and E end in of pulse duration. (b) Plateau stresses at loading ( high ), unloading ( low ) and hysteresis height ( ) in function of pulse duration.

Table 1

 1 Heat treatments parameters: duration, energy and estimated temperature.

	Material	t (ms)	E (J)	Tmax ( • C)
	CW	0	0	-
	A	1.44	4.3	380
	B	2.27	6.8	570
	C	3.08	11.2	760
	D	3.99	12.1	970
	E	5.04	14.1	1200
	F	6.10	18.3	1440

Table 2

 2 Comparison with the literature data from[START_REF] Delville | Microstructure changes during non-conventional heat treatment of thin Ni-Ti wires by pulsed electric current studied by transmission electron microscopy[END_REF][START_REF] Delville | Transmission electron microscopy investigation of dislocation slip during superelastic cycling of Ni-Ti wires[END_REF].

	Material	Sample name in	Microstructure	Grain size (nm)
		Delville et al.
		(2010)
	CW	0 ms	Mainly austenite	-
			and amorphous
	A	6 ms	Polygonised and	5-10
			amorphous
	B	10 ms	Polygonised	20-40
			nanocrystalline
	C	12 ms	polygonised	25-50
			nanocrystalline
	D	16 ms	Recrystallised	200-700
	E	18 ms	Recrystallised	800-1200

Acknowledgement

The authors wish to acknowledge the financial support of the ANR research programme "Guidage d'une Aiguille Médicale Instrumentée -Déformable" (ANR-12-TECS-0019).


Marie Rebouah

Grégory Chagnon

Gregory Chagnon
email: gregory.chagnon@imag.fr

Patrick Heuillet

G Chagnon

Anisotropic viscoelastic models in large deformation for architectured membranes

Keywords: Viscoelasticity, Sphere unit model, Anisotropy, Stress-softening B

published or not. The documents may come

Anisotropic viscoelastic models in large deformation for architectured membranes

Introduction

Depending on the process, rubber-like materials can be considered as initially isotropic or anisotropic. Even if isotropic behaviour is the most widespread, calender processing [START_REF] Itskov | A class of orthotropic and transversely isotropic hyperelastic constitutive models based on a polyconvex strain energy function[END_REF][START_REF] Diani | Directional model isotropic and anisotropic hyperelastic rubber-like materials[END_REF][START_REF] Caro-Bretelle | Constitutive modeling of a SEBS cast-calender: large strain, compressibility and anisotropic damage induced by the process[END_REF] generates anisotropy by creating a privileged direction in the material. The differences of mechanical properties affect the stiffness, stress softening or viscoelastic properties. Even if rubber-like materials are isotropic, some induced anisotropy can be generated by the Mullins effect [START_REF] Diani | A review on the Mullins effect[END_REF]Rebouah and Chagnon 2014b) for most materials.

Numerous studies have dealt with the modelling of rubber-like materials. In large deformations, the viscoelasticity is tackled either by the Boltzmann superposition principle [START_REF] Green | The mechanics of non-linear materials with memory: part I[END_REF]Coleman and Noll 1963) which leads to the K-BKZ models or by internal variable models [START_REF] Green | A new approach for the theory of relaxing polymeric media[END_REF]. Many constitutive equations were proposed to describe different viscoelastic behaviours.

The modelling of calendered rubber sheet necessitates taking into account the initial anisotropy. Many constitutive equations are developed to describe anisotropic hyperelastic behaviour [START_REF] Chagnon | Hyperelastic energy densities for soft biological tissues: a review[END_REF], these equations were often initially developed to describe soft biological tissues. Different equations were also developed to describe viscoelasticity for orthotropic materials or materials having one of two reinforced directions [START_REF] Holzapfel | A structural model for the viscoelastic behavior of arterial walls: continuum formulation and finite element analysis[END_REF][START_REF] Bischoff | A rheological network model for the continuum anisotropic and viscoelastic behavior of soft tissue[END_REF][START_REF] Haslach | Nonlinear viscoelastic, thermodynamically consistent, models for biological soft tissue[END_REF][START_REF] Quaglini | A discrete-time approach to the formulation of constitutive models for viscoelastic soft tissues[END_REF][START_REF] Vassoler | A variational framework for fiber-reinforced viscoelastic soft tissues[END_REF]. These constitutive equations rely on the representation of the material by a matrix with different reinforced directions, inducing the anisotropy in the material. The viscoelastic constitutive equations are introduced in the fibre modelling and can also be introduced in the matrix modelling. In a different way, [START_REF] Flynn | An anisotropic discrete fibre model based on a generalised strain invariant with application to soft biological tissues[END_REF] developed a discrete fibre model with dissipation for biological tissues. The model relies on a structural icosahedral model with six discrete fibres.

These models do not correspond to calendered materials. A calendered material can be represented as a macromolecular network in which the repartition of macromolecules was not equiprobable in space. A way to treat this problem is to describe the material by a uniaxial constitutive equation integrated in space considering different orientations. Different formalisms were proposed [START_REF] Verron | Questioning numerical integration methods for microsphere (and microplane) constitutive equations[END_REF]. For soft tissues and rubber-like materials, the formalism proposed by [START_REF] Bažant | Efficient numerical integration on the surface of a sphere[END_REF] is the most widespread. It was used to describe the stress softening or the viscoelasticity. [START_REF] Miehe | A micro-macro approach to rubber-like materials. Part ii: the micro-sphere model of finite rubber viscoelasticity[END_REF] were the first to use this formalism to describe the viscoelasticity but in an isotropic framework. The same formalism was later used by [START_REF] Diani | Observation and modeling of the anisotropic visco-hyperelastic behavior of a rubberlike material[END_REF], introducing induced anisotropy by the stress softening, but the viscoelasticity remained isotropic. Moreover, [START_REF] Rey | Hyperelasticity with rate-independent microsphere hysteresis model for rubberlike materials[END_REF] used this formalism to describe hysteresis loops but also in an isotropic approach. In fact, the discretisation in privileged directions is often used to induce anisotropy for hyperelasticity and stress softening, but not for viscoelasticity.

In this paper, we propose to characterise the mechanical behaviour of initially anisotropic rubber-like materials. The experimental data will be used to adapt the formalism proposed by Rebouah and Chagnon (2014a) to describe the anisotropic viscoelasticity and stress softening of the material.

Two rubber-like materials that possess an initial anisotropic behaviour are studied: first, a room temperature vulcanized silicone rubber which was made anisotropic by a stretching during reticulation, and second, a thermoplastic elastomer made anisotropic by the industrial process. In Part 2, the cyclic mechanical behaviour of the materials is described by means of a tensile test performed on specimens oriented in different directions in the plane of the membrane. In Part 3, the mono-dimensional constitutive equation is first described, and next the three-dimensional formulation is proposed. In Part 4, a discussion about the abilities of the constitutive equations to describe the two materials is proposed. Finally, a conclusion closes the paper.

Experiments

Materials and specimen geometry

In this paper, two materials that possess an anisotropic mechanical behaviour are used, a silicone rubber (RTV3428) and a thermoplastic elastomer (TPE). They are detailed in the next paragraphs.

RTV3428a

An RTV3428 silicone rubber is used here, which was previously studied in other works [START_REF] Rey | Influence of the temperature on the mechanical behavior of two silicone rubbers above crystallization temperature[END_REF]). This material is initially isotropic and only has an anisotropy induced by Mullins effect [START_REF] Machado | Induced anisotropy by the Mullins effect in filled silicone rubber[END_REF]. It is proposed to modify its microstructure by changing the elaboration process to generate an initially anisotropic behaviour. The process is illustrated in Fig. 1. To obtain this anisotropic plate, two components are mixed first. The mixture is then put into a vacuum pump and finally injected into a mould. The mould is put into the oven at 70 °C for 22 minutes. The crosslinking of the obtained membrane is not fully performed after being removed from the mould. Next the membrane is installed in a clipping system made of two jaws and applying a constant displacement between the two extremities of the membrane (as represented in the fifth step in Fig. 1). The global deformation of the membrane in the system is about 60 %. The system is put into the oven at a temperature of 150 °C for two hours. The new obtained material is named RTV3428a. This process generates a preferential orientation of the macromolecular chains in the material.

TPE

Different processes can be used to manufacture TPE [START_REF] Caro-Bretelle | Constitutive modeling of a SEBS cast-calender: large strain, compressibility and anisotropic damage induced by the process[END_REF]. In this study, an industrial material provided by Laboratoire de recherches et de contrôle du caoutchouc et des plastiques (LRCCP) is used. This material is obtained by means of an injection process which gives it a predominant direction and makes it initially anisotropic.

Specimen geometry

Each material is initially elaborated as a membrane as illustrated in Fig. 2. The RTV3428a membrane dimensions after the second vulcanization are 150 mm in length, 70 mm in width and 1.6 mm in thickness; for the TPE the membrane dimensions are 150 mm in length, 100 mm in width and 2 mm in thickness. For each material tensile test samples (20 mm long and 2 mm wide) are cut in the middle of the membrane to avoid edge effects. These specimens are cut with different angles 0°, 45°and 90°compared to the preferential direction of the material, considering that 0°matches the preferential direction imposed to the macromolecular chains for both processes as illustrated in Fig. 2.

Loading conditions

Mechanical tests were realised with a Gabo Eplexor 1500 N mechanical test machine with a load cell of 50 N. Samples were submitted to a cyclic loading, two cycles up to a stretch λ = 1.5, two cycles up to λ = 2, and finally two cycles up to λ = 2.5. The tests were carried out at a strain rate of 0.016 s -1 . The loading history is detailed at the top of the Fig. 3 and Fig. 4.

Results

Figure 3 presents the results of the test for the three samples cut from the RTV3428a. The three samples do not have the same mechanical behaviour, and several phenomena are observed. First, to evaluate the amount of anisotropy, an anisotropic factor ξ is defined as the ratio of stresses for a stretch λ = 2.5 for different orientations as ξ = σ 0 • (λ = 2.5)/σ 90 • (λ = 2.5). It permits qualitatively quantifying the anisotropy of the two materials. For the RTV3428a an anisotropic factor of approximately 1.3 can be calculated between the sample cut at 0°and the one cut at 90°. This emphasises that the second vulcanization undergone by the membrane modifies the microstructure of this filled silicone and is efficient to generate anisotropy in the silicone rubber. The sample cut at 0°(which is the same direction as the loading direction imposed during the second vulcanization) has the most important stress hardening compared to 45°and 90°, the latter being the softest specimen. This test also highlights that the material has few viscous effects and permanent set even at slow strain rate. Stress softening is still the major non-linear effect associated with the mechanical behaviour.

Figure 4 presents the results for the same test obtained for the three samples of the TPE material. As before the anisotropic factor ξ can be evaluated and is approximately equal to 1.5. As for the RTV3428a, stress softening, hysteretic behaviour and permanent set are also observed. It is to note that the stress softening and permanent set are very large. The The observed phenomena are the same for the two materials. They both have an anisotropic mechanical behaviour with viscoelasticity, stress softening and permanent set for any loading direction. All the phenomena are even more amplified for the TPE than for the RTV3428a. The stress softening of the material between the first and the second cycles seems to be the same for any direction, this should mean that stress softening is not affected by the initial anisotropy. On the contrary, the stiffness and viscoelastic properties are modified for the two materials depending on the direction.

Constitutive equations

This section aims to detail the constitutive equation developed to describe the mechanical behaviour observed experimentally for both materials. This constitutive equation must take into account the anisotropy, stress softening and viscoelastic effects (including the permanent set) undergone by the materials. It is to note that the material is considered as a homogeneous structure and not as a matrix with reinforced fibres as it is classically done to represent the anisotropic materials (see, for example, [START_REF] Peña | Mechanical characterization of the softening behavior of human vaginal tissue[END_REF][START_REF] Natali | Biomechanical behaviour of oesophageal tissues: material and structural configuration, experimental data and constitutive analysis[END_REF].

The constitutive equation relies on the representation of space by an integration of a uniaxial formulation by means of [START_REF] Bažant | Efficient numerical integration on the surface of a sphere[END_REF]

directions: σ = 42 i ω (i) σ (i) a (i) n ⊗ a (i) n (1)
where a (i) n are the normalized deformed directions, ω (i) the weight of each direction and σ (i) the stress in the considered direction. The directions are represented in Fig. 6. The idea of the Fig. 6 Representation of a microsphere with a spatial repartition of 2 × 21 directions proposed by [START_REF] Bažant | Efficient numerical integration on the surface of a sphere[END_REF] modelling is to propose the constitutive equation in tension-compression for each direction, i.e. σ (i) .

A classical rheological scheme is used to model viscoelasticity; the first schema is illustrated in Fig. 5. The deformation gradient F is decomposed into an elastic part F e and an inelastic part F i between the initial configuration (C 0 ) and the instantaneous configuration (C t ). The formalism proposed by [START_REF] Huber | Finite deformation viscoelasticity laws[END_REF] is used. An application of the Second Principle of Thermodynamics leads to the equation of dissipation D int , namely

D int = σ : D -Ẇ ≥ 0 ( 2
)
where W is the strain energy (it can be decomposed into two parts, W 1 for the elastic branch of the model and W 2 for the inelastic branch of the model), σ is the Cauchy stress tensor and D is the rate of deformation tensor. [START_REF] Huber | Finite deformation viscoelasticity laws[END_REF] proved that the sufficient condition to verify is

2F e ∂W ∂B e F e : D i ≥ 0. (3)
The indices e and i are referring to the elastic and the inelastic parts of the model. As [START_REF] Huber | Finite deformation viscoelasticity laws[END_REF] detailed the simplest sufficient condition to satisfy, this equation is chosen as:

Ḃe = LB e + B e L T - 2 η 0 B e ∂W 2 ∂B e B e D ( 4
)
where B is the left Cauchy-Green tensor, L is the velocity gradient equal to ḞF -1 and D strands for the deviatoric part of the tensor. Any constitutive equation can be used by assuming that each spring of the model is modelled by a neo-Hookean model [START_REF] Treloar | The elasticity of a network of long chain molecules (I and II)[END_REF], C (i) 1 and C (i) 2 are the material parameters of each branch. Index i denotes that this model will be used in every directions of the microsphere decomposition. To be used, the governing equation must be written in uniaxial extension as

λ(i) e = λ e (i)
λ(i)

λ (i) -4 C (i) 2 3η (i) 0 λ (i) e 3 -1 (5)
where λ (i) and λ (i) e are the stretch and the elastic part of the stretch in the considered direction, and η (i) 0 is a material parameter associated to each direction. To take into account the stress softening phenomenon, a non-linear spring is added to the previous rheological scheme. It consists of a non-linear spring that can be added to the rheological scheme as illustrated in Fig. 6. [START_REF] Rebouah | Anisotropic Mullins softening of a deformed silicone holey plate[END_REF] proposed using an evolution function F (i) that records the loading history of each direction, this function alters the stiffness of the non-linear spring:

F (i) = 1 -η (i) m I 1max -I 1 I 1max -3 I (i) 4 max -I (i) 4 I (i) 4 max - 1
I (i) 4 max I 4 max 4 (6)
where η (i) m is a material parameter, I 1 is the first invariant of Cauchy-Green tensor and I (i) 4 is the fourth invariant associated to direction i. The term I (i) 4 max is the maximum value reached at the current time for each direction, and I 4 max is the maximum value of I 4 for every direction.

Summing the viscoelastic part and the stress softening enables us to define the stress in direction i considering that each direction endures only tension-compression, an incompressibility hypothesis is used to write the stress in each direction as

σ (i) = 2C (i) 1 λ (i) 2 - 1 λ (i) + 2C (i) 2 λ (i) e 2 - 1
λ (i) e + 2λ (i) 2 F (i) ∂W cf ∂I (i) 4 (i) (7)
where W (i) cf is the strain energy of the material oriented in direction i for the stress softening part. Due to the differences observed experimentally between both materials, the strain energy W (i) cf used to describe the RTV3428a and the TPE is different:

• The isotropic RTV3428 silicone rubber was already studied by [START_REF] Rebouah | Anisotropic Mullins softening of a deformed silicone holey plate[END_REF], and the same strain energy function was used in that study, namely

W (i) cf = K (i) I (i) 4 -1 2 (8)
where K (i) is a material parameter.

• The TPE has a very smooth behaviour with no strain hardening, as a consequence a square root function is chosen (Rebouah and Chagnon 2014b):

W (i) cf = K (i) 2 I (i) 4 -1 I (i) 4 dI (i) 4 ( 9
)
where K (i) is a material parameter.

The global dissipation of the model is obtained by summing the dissipation of each direction. As each dissipation is positive by construction of the evolution equation, the global dissipation is also positive.

To conclude, five parameters in each direction are used to handle the anisotropy of the material, C 1 (i) for the hyperelastic part, C 2 (i) and η (i) 0 for the viscoelastic part and K (i) and η (i) m for the stress softening part. The model needs the integration of a differential equation (Eq. ( 5)). An implicit algorithm is used to determine the equilibrium solution in each direction.

Comparison with experimental data 4.1 Parameter identification strategy

The model has five material parameters in each direction. It is important not to fit globally all the material parameters, but to impose some restrictions due to experimental observations:

• First, the stiffness of the material is different depending on the direction (0°, 45°and 90°), this stiffness is principally controlled by the parameters controlling the hyperelasticity of the constitutive equation, C (i) and K (i) , these parameters must be different in the different directions.

• Second, no significant difference was observed for the stress softening for the three orientations, i.e. the difference between the first and second loadings. Thus, the material parameter describing the stress softening η (i) m is chosen to be independent of the direction. • Third, as exposed by [START_REF] Petiteau | Large strain rate-dependent response of elastomers at different strain rates: convolution integral vs. internal variable formulations[END_REF] and Rebouah and Chagnon (2014b), the hysteresis loop size depends both on the elastic parameter, C 2 , and the time parameter, η 0 . As the hysteresis loops are very similar, but at different stress levels, the governing parameter is C (i) . As a consequence, it is chosen to impose the same η 0 in all the directions.

It was experimentally observed that the stress is maximal for the privileged direction of the fabrication process. The variation of the mechanical parameters according to the spatial repartition enables us to increase or decrease the initial anisotropy of the material. According to the representation of the spatial repartition of [START_REF] Bažant | Efficient numerical integration on the surface of a sphere[END_REF] as illustrated in Fig. 6, the closest directions of the microsphere to the preferential direction induced by the process (i.e., direction 1 in Fig. 6) are the directions with the largest material parameter values and minimum for the orthogonal directions 2 and 3 (of the microsphere). The values of the parameters are the same for the directions which are symmetrical with respect to the privileged direction of the sphere unit (direction 1). The values for the intermediary directions of the microsphere are obtained according to their relative position compared to direction 1. The material parameters are supposed to vary linearly between the two extrema. All these choices permit us to avoid non-physical responses of the model for other loading conditions.

RTV3428a

According to the limitations detailed in the previous paragraph, the material parameters are fitted for the three samples with different orientations in comparison of the principal direction. The material parameters which values are independent of the directions are η m = 4 and η 0 = 200 MPa s -1 . The values of the other parameters are listed in Table 1.

Figure 7 presents a comparison between the experimental and theoretical tests for the three samples of RTV3428a with different orientations. The stiffness of the material is well described for any direction. The viscoelastic effects are also well described for the three directions. A difference can be observed for the second loading curves at the maximum stretch λ = 2.5. This error corresponds to the stress softening part of the model. This could be improved by modifying the form of Eq. ( 6) by imposing a more important loss of stiffness. Nevertheless, the model is able to globally describe the anisotropic mechanical behaviour of the RTV3428a material.

• RTV TPE C 1 (i) MPa C 2 (i) MPa K (i) MPa C 1 (i) MPa C 2 (i) MPa K (i

TPE

As before the material parameters of the TPE are fitted to the tensile tests of the three specimens with different orientations. The material parameters which values are independent of the directions are η 0 = 500 MPa s -1 , η m = 8. The values of the other parameters are listed in Table 1 and are obtained by the same strategy as the one described for the RTV3428a. Figure 8 presents a comparison of the model with experimental data. The variations of stiffness of the material with the directions are well described. Nevertheless, the model is not able to describe very large hysteresis loops and very important stress softening. Important differences are observed for the model according to the direction, but the size of the hysteresis is underestimated. This is due to the form of the constitutive equations that were chosen. Only neo-Hookean constitutive equations were used in the viscoelastic part, and it is well known that this model cannot describe large variations. Moreover, the governing equation of the viscoelasticity (i.e., Eq. ( 5)) is a very simple equation that also cannot take into account large non-linearity of the mechanical behaviour.

Nevertheless, even if the proposed model is a first approach written with simple constitutive elements, all the phenomena are qualitatively described. The limits correspond to the limits of each part of the constitutive equation.

Discussion

The model succeeds in depicting the anisotropic viscoelasticity with stress softening mechanical behaviour of architectured membranes. The use of different material parameters in the different directions leads to an important number of parameters. A global fit of the parameters could lead to parameter values with no physical meaning. By analysing the experimental data, a strategy was proposed to fit the material parameters. As exposed in the Introduction, constitutive equations developed in the literature to model anisotropic viscoelasticity often rely on an isotropic matrix reinforced with some viscoelastic fibres. These models were principally elaborated for soft biological tissues and could be applied in a phenomenological approach to the two materials tested in this paper. It would consist in considering the rubber as a soft matrix having the mechanical properties of the soft direction, reinforced by fibres in the predominant direction of the material. Even if this approach were to succeed in describing the material, it would not characterise the macromolecular network of the material.

All the equations in the literature to model for viscoelasticity can be written in tensioncompression and introduced into the present model, by replacing Eq. ( 5). This would permit us to represent non-linear viscoelasticity.

Conclusion

This paper developed a study of anisotropic materials. Two micro-mechanical architectured materials were obtained in two different ways. A silicone rubber was turned anisotropic by applying a deformation state during the second reticulation, and an initially anisotropic injected TPE was used. In both cases an orientation of the macromolecular chains was imposed to the material to create a microstructural architecture. The anisotropic membranes were tested in their plane, highlighting their anisotropic mechanical behaviour.

A three-dimensional equation was obtained by considering the integration in space of a uniaxial equation with the 42 directions of [START_REF] Bažant | Efficient numerical integration on the surface of a sphere[END_REF]. This equation describes hyperelasticity, viscoelasticity and stress softening. In this first approach, we chose to use simple constitutive equations to prove the feasibility of the method. As a consequence, a neo-Hookean model was chosen to describe the elasticity and a simple linear equation for viscoelasticity. The anisotropy was obtained by considering that the material parameters could be different in all directions. Nevertheless, we chose to limit the variations of parameter values depending on the directions. This permitted us to limit the number of independent mechanical parameters. It was even possible to use different parameter values in every direction.

It appears that the model succeeded in qualitatively describing all the phenomena. When the phenomena (stress softening, hysteresis) were not too large, the model succeeded also in quantitatively describing the tests. Some errors between experimental data and the model appeared when the phenomena became too large. This was due to the use of simple elements for hyperelasticity and viscoelasticity. Indeed, the most robust constitutive equation for hyperelasticity, stress softening and viscoelasticity should be used when the phenomena are very large. For instance, the neo-Hookean hyperelastic equation should be replaced by a model accounting for stress hardening, or the viscoelastic equation should be replaced by a non-linear one as in, e.g. [START_REF] Bergstrom | Constitutive modeling of the large strain time dependant behavior of elastomers[END_REF].

Fig. 1

 1 Fig. 1 Elaboration of the microstructural architectured membrane of filled silicone

Fig. 2

 2 Fig. 2 Representation of the membrane and the samples oriented in different directions for the RTV3428a and the TPE materials

Fig. 3 Fig. 4

 34 Fig. 3 Cyclic tensile test A performed on the RTV3428a architectured membrane

Fig. 5

 5 Fig. 5 Definition of the configurations and of the rheological modelling

Fig. 7

 7 Fig. 7 Comparison with the experimental data of cyclic tensile test on RTV3428a for the different samples with different orientations

Fig. 8

 8 Fig. 8 Comparison with the experimental data of cyclic tensile test on TPE for the three samples with different orientations

Table 1

 1 Material parameters for the RTV3428a and the TPE n

Acknowledgements This work is supported by the French National Research Agency Program ANR-12-BS09-0008-01 SAMBA (Silicone Architectured Membranes for Biomedical Applications).