Rémy Belon

Rémy Boulesteix
email: remy.boulesteix@unilim.fr

Pierre-Marie Geffroy

Alexandre Maître

Christian Sallé

Thierry Chartier

Tape casting of multilayer YAG-Nd:YAG transparent ceramics for laser applications: study of green tapes properties

Keywords: Transparent ceramics, tape casting, mechanical properties, waveguide

In this study, the mechanical properties of ceramic green tapes elaborated by tape casting were determined. The influence of slurry formulation (i.e. organic content and binder-toplasticiser volume ratio) on these properties was studied. In the studied formulation domain with polyvinyl butyral (PVB) and benzyl butyl phthalate (BBP) as binder (B) and plasticiser (P) respectively, it was shown that maximum elongation, maximum stress and Young's modulus are increased for higher organic content and B/P ratio. This work also shows the possibility to manufacture multi-layered transparent ceramics for waveguide applications by the tape casting process. In that case, a compromise has been identified between good mechanical properties and low organic content that can alter the transparency of the material.

Introduction

For high demanding laser applications, amplifier media with new architectures (i.e. with dopant gradient) also called "composites" need to be developed. These architectures have many advantages, compared to a homogeneous component. For instance, Kracht et al. [START_REF] Kracht | 407 W end-pumped multisegmented Nd: YAG laser[END_REF] demonstrated that a multi-layered barrel of YAG single-crystal, with increasing Nd-dopant concentration in three steps along the optical axis in the barrel, makes possible to smooth the absorption of the pumping beam, and thus to smooth the temperature and stress gradient in the barrel. This enables to increase the pumping power, and so higher output laser power can be obtained. As reported in this study, first graded components were manufactured by diffusionbonding of single-crystals [START_REF] Lee | Diffusion-bonded composites of YAG single crystals[END_REF]. Nevertheless, diffusion-bonding is restricted to simple architectures, with plane interfaces (see examples in [START_REF] Meissner | Composites made from single crystal substances[END_REF][START_REF] Meissner | Composite optical and electro-optical devices[END_REF]). In addition, diffusion-bonding process requires a very high-quality polishing of each surfaces to be contacted and remains expensive [START_REF] Lee | Diffusion-bonded composites of YAG single crystals[END_REF]. On the contrary, ceramic processes permit to control the architecture of the final material thanks to adapted shaping processes. Ikesue and Aung [START_REF] Ikesue | Synthesis and Performance of Advanced Ceramic Lasers[END_REF] gives for instance some architectures of interest for laser applications that can be manufactured by ceramic processing: multi-layered, cylindrical clad-core structure, fiber clad-core, waveguide or continuous gradient. The waveguide architecture is very interesting because of its higher gain and better thermal management compared to a bulk material [START_REF] Grivas | Optically pumped planar waveguide lasers, Part I: Fundamentals and fabrication techniques[END_REF]. The waveguide geometry also permits to design photonic integrated circuit, and allows to manufacture compact laser devices [START_REF] Estevez | Integrated optical devices for lab-on-a-chip biosensing applications[END_REF][START_REF] Bonner | Diode-bar end-pumped high-power Nd:Y_3Al_5O_12 planar waveguide laser[END_REF].

Finally its lower laser threshold leads also to a lower power consumption [START_REF] Mackenzie | Dielectric Solid-State Planar Waveguide Lasers: A Review[END_REF].

In order to manufacture composite ceramics with composition gradient, several shaping methods exist, such as powders co-pressing [START_REF] Sun | Fabrication of alumina/zirconia functionally graded material: From optimization of processing parameters to phenomenological constitutive models[END_REF][START_REF] Liu | A limiting current oxygen sensor prepared by a co-pressing and co-sintering technique[END_REF][START_REF] Osipov | Composite Nd:YAG/Cr4+:YAG transparent ceramics for thin disk lasers[END_REF], sequential slip casting [START_REF] Bonnet | Manufacturing issues and optical properties of rare-earth (Y, Lu, Sc, Nd) aluminate garnets composite transparent ceramics[END_REF][START_REF] Moya | Functionally gradient ceramics by sequential slip casting[END_REF][START_REF] Chu | Slip Casting of Continuous Functionally Gradient Material[END_REF] or tape casting [START_REF] Hostaša | Transparent layered YAG ceramics with structured Yb doping produced via tape casting[END_REF][START_REF] Cai | Constrained Densification of Alumina/Zirconia Hybrid Laminates, I: Experimental Observations of Processing Defects[END_REF][START_REF] Chartier | Tape-cast alumina-zirconia laminates: Processing and mechanical properties[END_REF]. The latter is well-adapted to manufacture ceramics with layered architecture and good spatial resolution. So, electronic components like multilayer ceramic capacitors with alternative layers of Zn0.95Mg0.05TiO2 of 30 µm of thickness, metallised with Ag [START_REF] Lee | Investigation of thin film end-termination on multilayer ceramic capacitors[END_REF] can be manufactured. Tape casting thus appears as a very promising way to manufacture transparent composite ceramics with waveguide architecture. Transparent rare-earth-doped YAG waveguides have already been manufactured [START_REF] Ge | Fabrication of composite YAG/Nd:YAG/YAG transparent ceramics for planar waveguide laser[END_REF][START_REF] Ma | Fabrication and planar waveguide laser behavior of YAG/Nd:YAG/YAG composite ceramics by tape casting[END_REF][START_REF] Wang | Ceramic planar waveguide laser of non-aqueous tape casting fabricated YAG/Yb:YAG/YAG[END_REF]. A multilayer Yb:YAG was also elaborated by tape casting [START_REF] Hostaša | Transparent layered YAG ceramics with structured Yb doping produced via tape casting[END_REF]. Besides, tape casting process can be used to manufacture multilayered transparent ceramics of large size: Ter-Gabrielyan et al. present a three-layered Er:YAG ceramic, with a length of 62 mm, elaborated by co-casting [START_REF] Ter-Gabrielyan | Efficient resonantly pumped tape cast composite ceramic Er:YAG laser at 1645 nm[END_REF]. These studies showed that tape casting is well-adapted to manufacture transparent ceramics with good laser performances.

Transparent ceramics manufacturing requires the optimization of each process step to avoid the presence of scattering centres as pores, micro-cracks, secondary phases, etc. [START_REF] Apetz | Transparent Alumina: A Light-Scattering Model[END_REF]. Using tape casting as shaping method needs the elaboration of specific ceramic suspension with organic additives of high purity and a low organic content to obtain a high green density.

Otherwise, green tapes showing suitable mechanical properties and without internal stresses are needed to limit deformation or micro-cracks after drying. Indeed, the slurry formulation, in particular organic content and binder/plasticiser ratio, have a large impact on the values of internal stresses in the green tape during and after the drying stage [START_REF] Kiennemann | Drying mechanisms and stress development in aqueous alumina tape casting[END_REF]. The creation of microcracks in the green tape is directly correlated to the formulation of the slurry. A few works studied the mechanical properties of the green tapes as a function of the slurry formulation, mainly for aqueous systems. Burnfield and Peterson [START_REF] Burnfield | Cellulose ethers in tape casting formulation[END_REF] observed that the maximum stress increases with the binder (hydroxyethylcellulose) content. The same authors have shown that the maximum stress decreases while maximum elongation increases when plasticiser content increases. Doreau et al. [START_REF] Doreau | Mechanical and lamination properties of alumina green tapes obtained by aqueous tape-casting[END_REF] observed an opposite trend concerning an acrylic emulsion used as binder. Indeed, in this case the maximum stress decreases and the maximum elongation increases when the binder content increases. Khamkasem et al. [START_REF] Khamkasem | Effect of Binder/Plasticizer Ratios in Aqueous-Based Tape Casting on Mechanical Properties of Bovine Hydroxyapatite Tape[END_REF] studied the influence of the binder/plasticiser (B/P) ratio, and showed that the maximum stress increases and the maximum elongation decreases when the ratio B/P increases (so when the binder content increases).

Albano et al. [START_REF] Albano | Aqueous tape casting of yttria stabilized zirconia[END_REF] reported that the maximum stress increases with the binder content (range from 5 to 7 wt.%) and the B/P ratio (0.7 to 1.5). According to the same authors, the maximum elongation increases when the ratio B/P decreases. Chartier et al. [START_REF] Chartier | Aqueous tape casting of alumina substrates[END_REF] observed that maximum elongation decreases and maximum stress increases when the ratio B/P increases. Finally, all these studies concerning tape casting showed that the formulation of the slurries controls the mechanical properties of the green tapes.

Consequently, this paper will be focused on the implementation of a shaping process by tape casting for YAG-Nd:YAG transparent composite ceramic with waveguide architecture. This latter architecture consists in a thin layer of Nd:YAG (around 150 µm thick) embedded in an undoped YAG matrix. Moreover this paper will be focused on the mechanical properties of the green tapes. More particularly, two parameters will be investigated: the total organic content and the binder-to-plasticiser volume ratio. The final objective will be twofold: i) to obtain green tapes having good mechanical properties to avoid micro-cracks in the final material; ii) to limit the organic content in order to ease the debinding step and to limit the contamination by residual carbon and/or impurities.

Experimental procedure 1.Elaboration and tape casting of the slurries

Al2O3 powder (Baïkowski, France) and Y2O3 powder (Solvay, Belgium) with high purity (> 99.99 %) were used as starting materials. These powders were weighted according to stoichiometry ratio to obtain Y3Al5O12 (i.e. undoped YAG). The powders were mixed with 4hydroxybenzoic acid as dispersant (3 wt.%), in an azeotropic mixture of methyl ethyl ketone (MEK) and ethanol (60 vol.% of MEK + 40 vol.% of ethanol), with a ceramic solid loading of 25 vol.%, and ball milled during 4 h with alumina balls. Then, binder and plasticiser were added to the slurry, which was homogenised by milling for 20 h. Polyvinyl butyral (PVB) and benzyl butyl phthalate (BBP) were used as binder and plasticiser, respectively. Organic additives (dispersant, binder and plasticiser) content (X = additives volume/[additives + ceramic powder volume]) was varied in a range of 40 to 50 vol.%. Binder-to-plasticiser volume ratio (B/P = binder volume/plasticiser volume) was varied in a range of 0.3 to 1. Formulations were named as follow: [Organic content]-[Volume ratio B/P]. For instance, the formulation noted "50-1" corresponds to the ceramic suspension with an organic content of 50 vol.% and a ratio B/P of 1. Slurries were tape-cast on a Mylar film substrate using a double doctor blade system. The speed of the doctor blade was 0.90 cm.s -1 and the height between the blades and the Mylar film was 300 µm. The thickness of the green tapes was about 100 µm after drying.

Mechanical characterisations of green tapes

Mechanical properties of the green tapes were evaluated by two methods. First, the maximum elongation of green tape before cracking was measured by a method reported by Mistler [START_REF] Mistler | Tape Casting, theory and practice[END_REF]. The green tapes were rolled on cylinders of different diameters, successively from the biggest to the smallest diameter, until cracks appeared on the green tape. The maximum elongation can be calculated with the following equation:

εc (%) = [(r+e)/r -1] x 100
Eq. 1

where εc is the maximum elongation, r the smallest cylinder radius without cracks in the tape, and e the green tape thickness. Cylinders of five different radii were used for tapes thickness of 100 µm: 50, 30, 20, 10, and 5 mm, corresponding to maximum elongation of 0.2, 0.33, 0.5, 1 and 2 %, respectively.

Secondly, tensile tests were carried out on dog-bone shaped specimens (Figure 1) cut parallel to the casting direction by laser cutting. A Lloyd Instrument EZ20 apparatus equipped with a 100 N load cell was used. The traverse speed was set to 0.5 mm.min -1 .

The tensile strength σm (maximum stress) and the maximum elongation εm (elongation at rupture) were determined from the stress-strain curve of the green tape (Figure 2). Indeed, the curve can be dissociated in two characteristic domains:

i) the first one, corresponding to low deformation (linear part), is related to the elastic deformation,

ii) the second one is related to the plastic deformation, until the rupture.

The elastic deformation can be evaluated from the Hooke's law [START_REF] Ashby | Engineering materials 1 An introduction to their properties and applications[END_REF], only valid for low deformation:

σ = E x ε Eq. 2
According to Eq. 2, the Young's modulus E was determined from the slope of the linear part of the curve. Five samples were tested for each composition, and the average value of σm, εm and E was determined.

Manufacturing of composite ceramic with a waveguide architecture

For the elaboration of the multi-layered ceramics, two types of slurries were prepared: a first slurry composed of Y2O3 and Al2O3 powders mixed with suitable ratio for YAG composition (Y3Al5O12), and a second slurry composed of Y2O3, Al2O3 and Nd2O3 powders mixed with suitable ratio for Nd:YAG composition (Nd0.03Y2.97Al5O12, or 1 at.% Nd:YAG).

Nd2O3 powder of high purity (> 99.99 %, Auer-Remy, Germany) was used as dopant source.

The properties (particle-size distribution and rheological behaviour) of Nd:YAG slurry were compared to that of pure YAG slurry. No significant differences were observed. Then, it was assumed that adding Nd2O3 doesn't modify the green tapes properties. Tetraethylorthosilicate (TEOS) was added as sintering aid, in the amount of 0.24 wt.% (corresponding to 0.07 wt.% of SiO2). Considering the low content added to the slurries, it was also assumed that TEOS doesn't modify the slurries and green tapes properties. Both slurries were tape-cast in the conditions described in the previous paragraph.

After drying, disks of 30 mm diameter were cut into the green tapes. 42 disks were stacked (20 undoped disks + 2 doped disks + 20 undoped disks) in a die and laminated at 95 °C under 35 MPa, for 5 min. Debinding was carried out under air at 650 °C, during 1 h. Then, sintering was achieved in a tungsten mesh-heated furnace at 1740 °C during 20 h under vacuum.

Microstructure observations were carried out by using conventional Scanning Electron Microscopy (LEO 1530VP, Zeiss, Germany, and FEI Quanta 450, Thermo Fisher Scientific, USA). Neodymium repartition in the sample was characterised by fluorescence microscopy and its concentration profile was measured by electron probe microanalysis (SX 100, CAMECA, France).

Results and discussion

Green tapes characterisation

Figure 3 shows the microstructure of a green tape, after debinding at 650 °C. The two powders (yttria circled in green and alumina circled in blue) appear to be well dispersed.

Figure 4 shows the impact of binder-plasticiser volume ratio and organic content on the maximum elongation before cracking of the green tapes (cylinders method). From this figure, three characteristic domains can be observed:

-the area numbered I corresponds to the formulations with a maximum elongation higher than 2 %. This formulation domain with high elongation is suitable to prevent eventual cracks after drying, cutting and thermo-lamination;

-the area numbered III corresponds to a maximum elongation lower than 0.2 % (50-mmradius cylinder). In this formulation domain, some cracks appear in green tapes during drying, and the green tapes are not flexible and not easily to handle for the lamination process;

-the intermediate area numbered II corresponds to a maximum elongation between 0.2 and 2 %.

According to the results of Green tapes properties can be interpreted as below: (1) when the binder-to-plasticiser (B/P) ratio is too low (bottom of the graph in Figure 4), there is not enough binder to ensure the cohesion of the tapes, so the latter disintegrates during drying. (2) Conversely, if B/P ratio is too high (more binder and less plasticiser, top of the graph), tapes are too stiff, and they cannot deform during drying, leading to the appearance of cracks (Figure 4 In order to further characterise the mechanical properties of green tapes, tensile tests were carried out with green tapes having a maximum elongation larger than 2 %. The relative density of the green tapes after debinding was also measured for each formulation. For this purpose, disk samples were cut in tapes of each formulation. Samples volume V was geometrically determined. Finally, the samples were debinded and the mass m of each debinded sample was measured. So, the density ρ of each tape can be calculated (ρ = m/V), and the relative density was given according to the YAG theoretical density (ρYAG = 4.55 g.cm -3 ). The relative densities of the green tapes are similar in the formulation range of this study (average of 35 % with a standard deviation of 0.8 %). Then, it was assumed that the impact of the relative density on the mechanical properties of the green tapes is not significant in this work. Obtained values (i.e. maximum elongation εm, Young's modulus E, and maximum strength σm) from stress-strain curves as illustrated in Figure 2 were reported in Figure 5.

The general trend assessed from the Figure 5 graph a is that maximum elongation increases with the organic content (Figure 5 graph a). This is an expected evolution, since the organic content is favourable to a ductile behaviour of the green tape with a larger strength. Also, maximum elongation increases with ratio B/P (i.e. when the amount of plasticiser decreases). This is unusual in comparison with data reported in the literature [START_REF] Burnfield | Cellulose ethers in tape casting formulation[END_REF][START_REF] Khamkasem | Effect of Binder/Plasticizer Ratios in Aqueous-Based Tape Casting on Mechanical Properties of Bovine Hydroxyapatite Tape[END_REF][START_REF] Albano | Aqueous tape casting of yttria stabilized zirconia[END_REF]. The ratios B/P (<1) of the formulations studied in this work are lower, compared with ratios close to the unity reported in the literature. This leads likely to a lower ductility and strength of our green tapes (< 1.8 MPa, compared with strength > 1.5 MPa in the literature previously cited).

Young's modulus increases when B/P ratio and organic content increase (Figure 5 graph b).

Higher organic content (50 vol.%) leads to stiffer green tapes, as expected. However, the green tape noted 40-0.7 (40 vol.% organic content -binder/plasticiser = 0.7) with the lower organic content, shows highest Young's modulus. This can be explained because the high ceramic content in this case leads to a fragile behaviour of the green tape with low relaxation of internal stresses and an increase of Young's modulus of the green tape. This behaviour has already been observed for particulate-filled polymer composites [START_REF] Rothon | Particulate-filled Polymer Composites[END_REF][START_REF] Ahmed | A review of particulate reinforcement theories for polymer composites[END_REF].

The impact of B/P ratio and organic content on maximum stress before rupture is still negligible when organic content is under 45 vol.% (Figure 5 graph c). Nevertheless, the maximum stress before rupture increases with higher organic content (50 vol.%) and a higher ratio B/P (≥ 0.7). For instance, the maximum stress is almost doubled when B/P ratio increases from 0.4 to 1 (organic content of 50 vol.%). Indeed, the B/P ratio and the organic content of the suspension imposes mechanical behaviour evolution of the green tape, from fragile to ductile behaviour, as observed by Campbell [START_REF] Campbell | Structural composite materials[END_REF] in composite materials.

Manufacturing and characterisation of composite transparent ceramic

The production of transparent ceramics needs to prohibit the formation of microstructural defects (e.g. secondary phase, (micro-)cracks) during ceramic process. Besides, manufacturing of transparent ceramics requires a very high relative density (> 99,9 %) of the sintered parts.

This involves to obtain green tapes with a high relative density but also to limit the organic content in the slurry formulation. In this respect, formulation with the lowest organic content allowing to manufacture green tapes without defects (i.e. organic content close to 40 vol.% with B/P ratio of 0.7 according to Figure 4) was selected.

The microstructure of the debinded multi-layered material is shown on Figure 6. No defects were detected between the layers, showing that the lamination step is well controlled. Despite the low relative density of the green tapes (about 35 %), the relative density of the sample after thermo-lamination is about 45 %. So thermo-lamination leads to a better packing of the particles.

After sintering, a transparent ceramic (Figure 7a) is obtained. Concerning the optical quality of the sample, the transmission spectrum is shown on Figure 8.

The Nd 3+ absorption peaks are weak, due to the low thickness of the Nd-doped layer. The transmittance is 54.2 % and 52.1 % at 1064 nm and 400 nm respectively, for a total sample thickness of 3,8 mm. This is not as high as the theoretical transmittance of YAG (84 %), and this work needs some improvements, more particularly on the microstructural homogeneity of the sample.

One can see on the Figure 9a the micrograph of a cross-section of multi-layered YAG/Nd:YAG ceramic obtained by fluorescence microscopy (imaging at 1064 nm, with 808 nm excitation). According to this figure, Nd 3+ detected by its strong luminescence at 1064 nm is located in a central layer in the composite as expected. The profile of neodymium concentration was characterised more precisely and quantitatively by electron probe microanalysis (Figure 9b). Nd 3+ concentration is given as an atomic percentage regarding to the yttrium content. The concentration in the doped layer is about 1 at.%, as expected. The profile also reveals that Nd 3+ has diffused during co-sintering over approximately 50 µm from the initial interface. This value is in the same order of that observed by Bonnet et al. that used similar co-sintering conditions [START_REF] Bonnet | Manufacturing issues and optical properties of rare-earth (Y, Lu, Sc, Nd) aluminate garnets composite transparent ceramics[END_REF]. Results reported in Figure 9 show that tape casting is welladapted to control the architecture (dopant gradient) of transparent ceramics.

Conclusions

This work shows that tape casting process is an interesting method for manufacturing multilayered YAG-based transparent ceramics. In this way, two points needed specific attention.

The first one is related to the mechanical properties of the green tapes. The control of the slurry formulation and the mechanical properties of the green tapes are critical parameters to obtain ceramics without defects. To obtain a green tape easy to handle (i.e. with a maximum strength and strain high enough), the binder and plasticiser contents are key parameters to control.

The second key parameter is the organic content of the sample. A low organic content is needed to ease the debinding step, and avoid defects creation during this treatment. Besides, formulations with less organic additives limit the chemical impurities content in the material.

Finally, green tapes with a maximum strength of 1.14 MPa and a maximum elongation of 1.5 % can be manufactured, stacked, laminated and sintered, to obtain a transparent composite ceramic.

Thus, tape casting process is well-adapted to manufacture multilayer transparent ceramics, with a complex and controlled architecture, and which could be used as laser components. Nd concentration [at.%]

1 mm

Figure 4 ,

 4 mechanical properties of the green tapes quickly decrease out of the area I, and a slight variation of the formulation leads to a large variation of mechanical properties of the green tapes. As reported on the Figure 4, green tapes have no defects in area I (photo c) whereas cracks appear rapidly out of this domain (area II and III illustrated by photo a and b).

  photo a). (3) When organic content is too low (left of the graph in Figure4photo a), cracks appear whatever the B/P ratio because there is not enough organics to ensure the cohesion of the tapes. (4) Green tapes with good mechanical properties and thus with no cracks are obtained for suitable B/P ratio (around 0.7) and a minimum organic content of about 40%.

  Figure 7b shows a cross section of the sintered composite material, with the visible thin Nd-doped layer in the middle. As shown on the Figure 7c, the doped layer can be seen by backscattered electron imaging. Its thickness is about 135 µm, for a green thickness of 200 µm. No cracks or delamination were observed at the interfaces, so co-sintering of doped and undoped layers was going well. Some intragranular pores (small circular black spots, Figure 7c) could not be eliminated during sintering. Also, a few contents of alumina particles (irregular black spots, Figure 7c) are remaining in the sample. These particles should result from a slight deviation of the initial stoichiometry (i.e. of the ideal Y/Al ratio of 0.6).

Acknowledgements

We are thankful to Alain Brenier, from the Institut Lumière Matière of Lyon (France), who carried out the fluorescence microscopy characterisations and to Eric Leroy, from Institut de Chimie et des Matériaux Paris-Est (France) who performed electron probe microanalysis. The authors are also grateful to ANRT for financial support (Grant number 2015/1062).


Pierre Rompteaux

Johan Gagniere

Jean-Marc Gornet

Romain Coriat

Isabelle Baumgaertner

Thierry Lecomte

Pauline Afchain

Aziz Zaanan

Marc Pocard

Jean-Baptiste Bachet

Johan Gagnière

Jean- Baptiste Bachet

Nathalie Bonichon-Lamichhane

Olivier Bouché

Jean-Luc Faucheron

Julien Forestier

Cedric Lecaille

Sylvain Manfredi

David Tougeron

Eric Terrebonne

Mohamed Chehimi

Anne-Laure Villing

Corinne Sarda

Jean-Louis Legoux

Robert Benamouzig

MD Thomas Aparicio
email: thomas.aparicio@aphp.fr

Resection of small bowel adenocarcinoma metastases: Results of the ARCAD-NADEGE cohort study

Keywords: small bowel adenocarcinoma, metastases resection, hepatic metastases, adjuvant chemotherapy

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

INTRODUCTION

Small bowel adenocarcinoma (SBA) is a rare tumor, representing less than 5% of digestive cancers. [START_REF] Aparicio | Small bowel adenocarcinoma: epidemiology, risk factors, diagnosis and treatment[END_REF] In the United States, the annual incidence rate of SBA is estimated to be 7 cases per million persons. [START_REF] Bilimoria | Small bowel cancer in the United States: changes in epidemiology, treatment, and survival over the last 20 years[END_REF] Similar incidence rate is observed in Europe, with an increasing rate trend seen between 1999 and 2013. [START_REF] Legue | Trends in incidence, treatment and survival of small bowel adenocarcinomas between 1999 and 2013: a population-based study in The Netherlands[END_REF] Around 30% of patients have their disease diagnosed at a later metastatic stage. [START_REF] Overman | A population-based comparison of adenocarcinoma of the large and small intestine: insights into a rare disease[END_REF] Surgical resection of the primary tumor is the only curative treatment for localized SBA, however the recurrence rate remains high (40%). [START_REF] Talamonti | Primary cancers of the small bowel: analysis of prognostic factors and results of surgical management[END_REF] Although benefit of adjuvant chemotherapy after curative resection relies on a low level of evidence in this setting, the recent French guidelines recommend fluoropyrimidine and oxaliplatin-based adjuvant chemotherapy for patients with T4 and/or N+ tumors. [START_REF] Locher | Small bowel adenocarcinoma: French intergroup clinical practice guidelines for diagnosis, treatments and follow-up (SNFGE, FFCD, GERCOR, UNICANCER, SFCD, SFED, SFRO)[END_REF] Metastatic SBA have a poor prognosis, with a median overall survival (OS) of patients with metastatic disease ranging between 8 and 22 months in small retrospective series. [START_REF] Overman | Chemotherapy with 5-fluorouracil and a platinum compound improves outcomes in metastatic small bowel adenocarcinoma[END_REF][START_REF] Overman | Phase II study of capecitabine and oxaliplatin for advanced adenocarcinoma of the small bowel and ampulla of Vater[END_REF][START_REF] Tsushima | Multicenter retrospective study of 132 patients with unresectable small bowel adenocarcinoma treated with chemotherapy[END_REF][START_REF] Xiang | A phase II study of modified FOLFOX as first-line chemotherapy in advanced small bowel adenocarcinoma[END_REF][START_REF] Zaanan | Chemotherapy of advanced small-bowel adenocarcinoma: a multicenter AGEO study[END_REF] An international randomized phase III study is ongoing to assess the efficacy of chemotherapy after resection of localized SBA. [START_REF] Evans | GLOBAL BALLAD: An International Rare Cancers Initiative trial to evaluate the potential benefit of adjuvant chemotherapy for small bowel adenocarcinoma (IRCI 002)[END_REF] There is a paucity of data regarding patient's outcome after surgical resection of metastatic SBA. We conducted a nationwide prospective cohort study (ARCAD-NADEGE) to determine outcomes and prognostic factors of patients with SBA.

MATERIAL AND METHODS

Study design and patient selection

Endpoints

The primary endpoint was OS, defined as the time from metastases curative resection (synchronous or metachronous) to the date of death (from any cause) or of the last date the patient was known to be alive. Secondary endpoints were recurrence-free survival (RFS) and prognostic factors for survival. RFS was defined as the time from metastases curative resection to the date of recurrence observed on computed tomography scan, magnetic resonance imaging, or positron-emission tomography scan or to the date of censoring (recurrence-free or alive).

Statistical analyses

Survivals were estimated using the Kaplan Meier method and prognostic factors analyzed by the log-rank test. No multivariate analysis was performed due to the small sample size. Pvalues < 0.05 were considered statistically significant. All analyses were performed using R Software version 3.2.2 (R Development Core Team, 2005) and BiostaTGV Software (BiostaTGV, Jussieu, France).

RESULTS

Patient characteristics

A total of 343 patients with SBA were included in the NADEGE study in the given period.

After selection for metastatic SBA and for curative surgery, a total cohort of 27 patients was included. Seven patients with metastatic SBA after curative resection from the AGEO network were also selected, providing a total of 34 patients included in the current study (Supplementary Figure S1). The median enrolment was one patient per center. Patient clinical and pathological features are summarized in Table 1. The tumors were mainly located in the duodenum. Five (14.7%) patients were diagnosed with Crohn's disease and two (5.9%) with Lynch syndrome. The median time to recurrence for patients with metachronous metastases was 25.2 months (95% CI 8.8-41.6). A total of 29 patients had one metastatic site; four had two and one had three. Metastases from SBA were mainly peritoneal and hepatic. The types of surgical procedures performed and chemotherapy received are presented in Table 2.

Surgery

Patients with synchronous metastases (n = 25) were treated by single-stage surgery with or without neo-adjuvant chemotherapy. Two (5.9%) patients had radiofrequency for hepatic metastases during surgery of the primary tumor. Of the ten (29.4%) patients who underwent peritoneal tumor resection, two received oxaliplatin-based HIPEC and one patient mitomycin C-based HIPEC. Nine (26.5%) patients underwent surgery for metachronous metastasis.

Adjuvant chemotherapy

Twenty-four patients with synchronous metastases received adjuvant and/or neo-adjuvant chemotherapy: 5-fluorouracil with oxaliplatin (n=16), 5-FU with oxaliplatin and bevacizumab (n=3), capecitabine with oxaliplatin (n=1), 5-FU with irinotecan (n=2), 5-FU with irinotecan and cetuximab (n=1), or 5-FU alone (n=1). Two patients received perioperative chemotherapy and one patient received only neo-adjuvant chemotherapy.

Following surgery, one patient (82 years of age) was kept under observation alone.

Of the nine (26.4%) patients with metachronous metastases, six received adjuvant chemotherapy: 5-FU with oxaliplatin (n=5) or gemcitabine with oxaliplatin (n=1). Only one patient in this group received both neo-adjuvant and adjuvant chemotherapy.

Survival

After a median follow-up of 38.2 months (95% CI 25.2-50.8), 25 (73.5%) patients had recurrence and 21 (61.8%) died during the follow-up period. Of those who died, 16 died within the first 3 years of follow-up and 21 died within the first 5 years. The median RFS was 18.7 months (95% CI 12.0-25.4; Figure 1A) and the median OS was 28.6 months (95% CI 19-38.2; Figure 1B). Fourteen (41.2%) patients survived for more than 36 months. Five patients were lost to follow-up before progression at 12.4, 59.4, 62.1 and 75.4 months. Overall, 3 (8.8%) patients were alive and without relapse at the end of data collection.

Prognostic factors

In the exploratory univariate Cox analysis (Table 3), the only prognostic factor of shorter RFS was positive surgical margin (Figure 2). There was a non-significant trend for longer RFS in patients receiving oxaliplatin-based adjuvant chemotherapy. Eight (23.5%) patients had a RFS beyond 36 months (extreme 39.5-87.1). All of these patients had a margin-negative resection (R0), a well to moderately differentiated tumor, and received oxaliplatin-based adjuvant chemotherapy. Of these patients, seven had synchronous metastases (n = 2, peritoneal; n = 3, liver; n = 1, lymph node metastasis; n = 1, liver plus peritoneal) and one had metachronous peritoneal recurrence at anastomotic site 2.9 years after the primary surgery. Of the seven patients with synchronous metastases, two were in remission at the end of the study (RFS: 87.1 and 54.8 months), five were lost to follow-up without relapse (RFS: 64.2, 59.4, 75.4, 62.1 months), and one recurred after 53.8 months. The latter patient had also received oxaliplatin-based adjuvant chemotherapy and was lost of follow-up 59.8 months after the first metastasis resection. The patient with metachronous peritoneal recurrence at anastomotic site had a second recurrence 39.5 months after the first metastasis resection. This patient had received palliative 5-FU plus irinotecan-based chemotherapy and achieved long median OS (49.5 months).

The univariate analysis identified three prognostic factors associated with an increased risk of death: poor differentiation, resection margin, and lymph node involvement at surgery of the primary tumor (Supplementary Figure S2). Treatment with oxaliplatin was associated with better prognosis in patients who received adjuvant chemotherapy (Table 4).

DISCUSSION

In the present multicenter French series of patients, we have analyzed the clinical outcomes relevant prognostic factors of patients with different sites of metastatic SBA after curative-intent resection. Patients were treated in a short period of time, in different centers, and in the recent past, which illustrates the current state of the global medical care.

The median OS for patients undergoing curative intent resection of SBA metastasis was 28.6 months in our study; 41.2% of patients had OS of 36 months or longer. The median survival observed for SBA patients with metastatic disease in the whole NADEGE cohort was 12.7 months. [START_REF] Aparicio | ARCAD-NADEGE cohort: Result of a small bowel adenocarcinomas prospective cohort[END_REF] Moreover, survival of patients with resected metastasis observed in our study compare favorably to those recently reported in retrospective or prospective studies of patients treated with palliative chemotherapies (median OS ranging from 8 to 22 months). [START_REF] Overman | Chemotherapy with 5-fluorouracil and a platinum compound improves outcomes in metastatic small bowel adenocarcinoma[END_REF][START_REF] Overman | Phase II study of capecitabine and oxaliplatin for advanced adenocarcinoma of the small bowel and ampulla of Vater[END_REF][START_REF] Tsushima | Multicenter retrospective study of 132 patients with unresectable small bowel adenocarcinoma treated with chemotherapy[END_REF][START_REF] Xiang | A phase II study of modified FOLFOX as first-line chemotherapy in advanced small bowel adenocarcinoma[END_REF][START_REF] Zaanan | Chemotherapy of advanced small-bowel adenocarcinoma: a multicenter AGEO study[END_REF] In a multicenter retrospective observational study by Sakae et al., a median OS was 36.9 months for patients treated with the combined modality therapy, but 12 months for those treated with palliative chemotherapy alone. [START_REF] Sakae | The characteristics and outcomes of small bowel adenocarcinoma: a multicentre retrospective observational study[END_REF] Another recent large multi-institutional study of series of patients with SBA, reported a median OS of 32 months after HIPEC. [START_REF] Liu | Cytoreductive Surgery Plus Hyperthermic Intraperitoneal Chemotherapy for Peritoneal Metastases From a Small Bowel Adenocarcinoma: Multi-Institutional Experience[END_REF] Altogether these results and our findings are in favor of metastases resection. However, these survival data are disappointing given that a median OS for colorectal patients undergoing liver resection or HIPEC is 54-61 or 63 months, respectively. [START_REF] Nordlinger | Perioperative FOLFOX4 chemotherapy and surgery versus surgery alone for resectable liver metastases from colorectal cancer (EORTC 40983): longterm results of a randomised, controlled, phase 3 trial[END_REF][START_REF] Elias | Complete cytoreductive surgery plus intraperitoneal chemohyperthermia with oxaliplatin for peritoneal carcinomatosis of colorectal origin[END_REF] This suggests that patients who undergo resection of SBA metastases have worse prognosis than those who undergo resection of metastases from colorectal carcinoma. This observation is consistent with the result from the large US database showing that patients with SBA have worse prognosis than those with colon cancer across all stages. [START_REF] Overman | A population-based comparison of adenocarcinoma of the large and small intestine: insights into a rare disease[END_REF] We found that invaded margins, poor differentiation, and lymph node metastasis from the primary tumor were poor prognostic factors for OS after metastasis resection. This finding is consistent with the published data. Invaded margin has been previously identified as a pejorative prognostic factor after the primary SBA resection [START_REF] Bilimoria | Small bowel cancer in the United States: changes in epidemiology, treatment, and survival over the last 20 years[END_REF][START_REF] Ecker | Efficacy of adjuvant chemotherapy for small bowel adenocarcinoma: A propensity score-matched analysis[END_REF] and margins status as a determinant of survival after hepatic resection from metastatic colorectal cancer. [START_REF] Andreou | Margin status remains an important determinant of survival after surgical resection of colorectal liver metastases in the era of modern chemotherapy[END_REF][START_REF] Rees | Evaluation of long-term survival after hepatic resection for metastatic colorectal cancer: a multifactorial model of 929 patients[END_REF][START_REF] Vigano | Early recurrence after liver resection for colorectal metastases: risk factors, prognosis, and treatment. A LiverMetSurvey-based study of 6,025 patients[END_REF] This suggests that expected R0 resection margin should be considered before surgery.

Poor differentiation has been reported as negative independent prognostic indicator for survival after primary tumor resection alone and with HIPEC. [START_REF] Ecker | Efficacy of adjuvant chemotherapy for small bowel adenocarcinoma: A propensity score-matched analysis[END_REF][START_REF] Liu | Cytoreductive Surgery and Hyperthermic Intraperitoneal Chemotherapy for Peritoneal Dissemination from Small Bowel Malignancy: Results from a Single Specialized Center[END_REF][START_REF] Saxena | Recurrence and Survival Outcomes After Cytoreductive Surgery and Hyperthermic Intraperitoneal Chemotherapy for Small Bowel Adenocarcinoma[END_REF] Interestingly, it is also inconstantly found as a prognostic factor after resection of colorectal liver metastases. [START_REF] Rees | Evaluation of long-term survival after hepatic resection for metastatic colorectal cancer: a multifactorial model of 929 patients[END_REF][START_REF] John | Prognostic factors and survival after resection of colorectal liver metastasis in the era of preoperative chemotherapy: an 11-year single-centre study[END_REF] In our study, the median OS of patients who underwent resection of metastases from poorly differentiated tumors is close to that observed in patients treated by palliative chemotherapy in previous studies. [START_REF] Tsushima | Multicenter retrospective study of 132 patients with unresectable small bowel adenocarcinoma treated with chemotherapy[END_REF][START_REF] Zaanan | Chemotherapy of advanced small-bowel adenocarcinoma: a multicenter AGEO study[END_REF][START_REF] Sakae | The characteristics and outcomes of small bowel adenocarcinoma: a multicentre retrospective observational study[END_REF] Thus, resection of metastases from poorly differentiated tumor seems not appropriate for this group of patients.

Lymph node metastasis is the main prognostic factor after resection of localized SBA [START_REF] Ecker | Efficacy of adjuvant chemotherapy for small bowel adenocarcinoma: A propensity score-matched analysis[END_REF][START_REF] Overman | Prognostic value of lymph node evaluation in small bowel adenocarcinoma: analysis of the surveillance, epidemiology, and end results database[END_REF] and has been reported as independent prognostic factor after resection of peritoneal metastasis. [START_REF] Liu | Cytoreductive Surgery Plus Hyperthermic Intraperitoneal Chemotherapy for Peritoneal Metastases From a Small Bowel Adenocarcinoma: Multi-Institutional Experience[END_REF] Similarly, the presence of positive lymph node at primary metastatic colorectal cancer resection is also independent predictor of worst survival. [START_REF] Rees | Evaluation of long-term survival after hepatic resection for metastatic colorectal cancer: a multifactorial model of 929 patients[END_REF] In our cohort, adjuvant chemotherapy after metastasis resection was not associated with a better prognosis, most likely due to the small sample size. This finding opposes the results from retrospective analysis in localized SBA, [START_REF] Ecker | Efficacy of adjuvant chemotherapy for small bowel adenocarcinoma: A propensity score-matched analysis[END_REF] and the only prospective trial in this setting is still ongoing. [START_REF] Evans | GLOBAL BALLAD: An International Rare Cancers Initiative trial to evaluate the potential benefit of adjuvant chemotherapy for small bowel adenocarcinoma (IRCI 002)[END_REF] Our subgroup analysis showed improved OS in patients treated with oxaliplatin-based adjuvant chemotherapy, which is consistent with previous results observed in the metastatic setting. This observation suggests that patients receiving oxaliplatincontaining chemotherapy regimens benefit more than those treated with other regimens. [START_REF] Tsushima | Multicenter retrospective study of 132 patients with unresectable small bowel adenocarcinoma treated with chemotherapy[END_REF][START_REF] Zaanan | Chemotherapy of advanced small-bowel adenocarcinoma: a multicenter AGEO study[END_REF] Nevertheless, the reason to choose oxaliplatin or not in adjuvant treatment is unclear in our study and may have bias the results. Moreover, because of the small sample size of the non-oxaliplatin group (only four patients), we did not seek to determine prognostic factors using multivariate methods. Therefore, all the prognostic factors analysis should be taken with caution.

In conclusion, our study suggests that surgical resection of SBA metastases should be considered in patients who are expected to be operated on with curative intent and have welldifferentiated tumors. These results warrant further larger studies.

Figure and Table Legends

  Patients with SBA were selected from the prospective cohort ARCAD-NADEGE study enrolling patients between January 2009 and December 2012 in 74 French hospitals and from the French Association of Gastroenterologist and Oncologist (AGEO) network enrolling patients between December 2012 and November 2014 in French centers. All patients who underwent resection of metastatic SBA were included in the current study. Patients with adenocarcinoma of the ampulla of Vater were not included. Clinical and demographic data of patients who underwent curative resection of metastatic SBA were prospectively reviewed. The following parameters were collected: gender, age at diagnosis, risk factors (Crohn's disease, Lynch syndrome, Familial Adenomatous Polyposis, celiac disease, Peutz-Jeghers syndrome), characteristics of the primary tumor (grade, resection margin, lymph node involvement), characteristics of the metastases (site, number, synchronous/metachronous, margin resection, and pathological differentiation), chemotherapy regimens and type of treatments received (adjuvant chemotherapy after metastasis resection, hyperthermic intraperitoneal chemotherapy [HIPEC]), or neoadjuvant chemotherapy). Patients with R1 and R2 resections were pooled (R+ group) due to the small sample size and analyzed together. Follow-up ended on May 31, 2015 and survival data were censored on October 1, 2018. All the patients gave their consent for inclusion in the cohort. This study was authorized by the ethics committee "Ile de France II" No. ID-RCB: 2008-A01058-47".

Figure 1 .Figure 2 .

 12 Figure 1. Recurrence-free survival (A) and overall survival (B)

Figure A. 2

 2 Figure A.2 Overall survival (OS) according to tumor differentiation (A), resection margins

Table 1 .

 1 Patient and tumors characteristics

Table 2 .

 2 The types of surgical procedures performed and chemotherapies received

Table 3 .

 3 Univariate analysis of prognostic factors for recurrence-free survival

Table 4 .

 4 Univariate analysis of prognostic factors for overall survival Flow chart of the cohort study

	Supplementary files
	Figure A.1

Table 1 .

 1 Patient and tumor characteristics

	Total
	N = 34
	n (%)

Table 2 .

 2 The types of surgical procedures performed and chemotherapies received

	Total
	N = 34
	n (%)

Table 3 .

 3 Univariate analysis of prognostic factors for recurrence-free survival

		Number of	Median	P-value
		patients	RFS
			(months)
		N = 30
	Gender
	Male	20	21.3
	Female	10	14.9	0.78
	Age (years)
	< 65	20	27.0
	> 65	10	12.1	0.55
	Location of primary
	Duodenum	11	21.3
	Jejunum	8	20.3	0.91
	Ileum	9	21.2	0.90
	Node status at primary
	tumor site
	N0	7	21.2
	N+	19	13.5	0.11
	Delay of metastases
	Synchronous	23	13.7
	Metachronous	7	21.3	0.73
	Metastatic site
	Peritoneum alone	8	13.6
	Liver alone	9	21.2	0.36
	Other	8	10.7	0.64

Table 4 .

 4 Univariate analysis of prognostic factors for overall survival

Acknowledgment:

We thank GERCOR for providing access to the NADEGE database. We thank Ines Bacha (GERCOR, Paris) and Vanessa Le Berre (CHU Poitiers, Poitiers). We would like to acknowledge the editorial assistance of Magdalena Benetkiewicz (PhD, GERCOR).

Conflict of interest: None.

Funding:

The NADEGE cohort study received a grant from A.R.CA.D. Foundation and GERCOR.


Valentin Segouin

Mathieu Domenjoud

Yves Bernard

Laurent Daniel

correlation for the investigation of piezoelectric and ferroelectric behaviour of a soft PZT

Keywords: Hysteresis loops, Global DIC, Strain field, Regularisation, Strain Homogeneity

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

Introduction

Piezoelectric and ferroelectric materials are increasingly used in various applications such as sensors [START_REF] Booth | Detecting finger gestures with a wrist worn piezoelectric sensor array[END_REF][START_REF] Mahbub | A low-power wireless piezoelectric sensor-based respiration monitoring system realized in CMOS process[END_REF][START_REF] Carioli | Piezoelectric earcanal bending sensor[END_REF] , actuators [START_REF] Borgmann | 15th International Conference on New Actuators & 9th International Exhibition on Smart Actuators and Drive Systems[END_REF] , harvesting devices [START_REF] Dagdeviren | Conformal piezoelectric energy harvesting and storage from motions of the heart, lung, and diaphragm[END_REF] , converters [START_REF] Kwok | General study on piezoelectric transformer[END_REF] , non-volatile memories [START_REF] Cuppens | Ferroelectrics for non-volatile memory[END_REF] , filters [START_REF] Mathew | Tunable filter using ferroelectric-dielectric periodic multilayer[END_REF] and oscillators [START_REF] Wang | Ferroelectric oscillators and their coupled networks[END_REF] . The performance and reliability of such devices depend on the material electromechanical properties, which consequently need to be characterised. In the past decades, such characterisation was largely developed and the piezoelectric, ferroelectric, ferroelastic and dielectric properties of ferroelectrics were the subject of numerous studies [10- 17] . Yet, the test conditions are difficult to control due to the strong interplay between thermal, mechanical and dielectric properties. For example, stress and temperature induce strong variations of the piezoelectric coefficient for a single ferroelectric crystal [START_REF] Schader | Piezoelectric and ferroelectric properties of <001>C Pb(In1/2Nb1/2) O3-Pb(Mg1/3Nb2/3)O3-PbTiO3 single crystals under combined thermal and mechanical loading[END_REF] . The electric field also contributes to significant variations in the dielectric and piezoelectric coefficients of BaTiO3 and PZT ceramics [START_REF] Hall | Nonlinearity in piezoelectric ceramics[END_REF] . It is then necessary to seek ways of revealing, at the macroscopic scale, the presence of anomalies such as inhomogeneous stress, electric field or temperature.

In the case of strain measurement, strain gauges [START_REF] Hühne | Low temperature performance of Pb(Mg1/3 Nb2/3)0.72Ti0.28O3 single crystals mapped by metallic and superconducting thin films[END_REF] or linear variable displacement transducer (LVDT) [START_REF] Webber | Temperature-dependent ferroelastic switching of soft lead zirconate titanate[END_REF] are classically used. Both techniques offer satisfying resolution and can furthermore be used under several types of external loadings [START_REF] Webber | High temperature blocking force measurements of soft lead zirconate titanate[END_REF][START_REF] Shieh | Large strain actuation in barium titanate single crystals under stress and electric field[END_REF] . However, it is not possible to obtain information on strain homogeneity with these techniques. Indeed, the minimum strain gauge size is limited to the millimetre which restricts the density of the measurement points. This issue also applies to LVDT. Another drawback is that only one direction can be measured per gauge grid or transducer. To measure the longitudinal and transverse strain simultaneously, another grid or transducer is needed. Such measurement devices are difficult to design, especially under coupled loadings or for ferroelectric films. Hence, the longitudinal strain is generally the only measured strain component [START_REF] Webber | High temperature blocking force measurements of soft lead zirconate titanate[END_REF][START_REF] Webber | Temperature-dependent ferroelastic switching of soft lead zirconate titanate[END_REF][START_REF] Shieh | Large strain actuation in barium titanate single crystals under stress and electric field[END_REF][START_REF] Taylor | Nonlinear contributions to dielectric and piezoelectric properties in PZT thin films[END_REF][START_REF] Dittmer | Optimal working regime of lead-zirconate-titanate for actuation applications[END_REF][START_REF] Damjanovic | The Science of Hysteresis[END_REF] and the transverse strain is often discarded.

Digital Image Correlation (DIC) opens up opportunities for the measurement of strain on ferroelectric materials. This strain measurement technique is contactless, not scale-limited, allows 2D full-field measurements and is compatible with any imaging method. One requirement is that trackers or textures are needed in the images so as to observe displacements. DIC has already been used for classical mechanical experiments such as thermal [START_REF] Pan | High-temperature digital image correlation method for full-field deformation measurement at 1200°C[END_REF] , fatigue [START_REF] Mahal | Using digital image correlation to evaluate fatigue behavior of strengthened reinforced concrete beams[END_REF] , ageing [START_REF] Lall | A study on the effect of aging on thermal cycling reliability of Sn-Ag-Cu interconnects using digital image correlation[END_REF] , high speed [START_REF] Nansteel | Digital image correlation: A measurement tool for the study of explosive effects[END_REF] or large strain [START_REF] Witz | Digital image correlation for large strain[END_REF] tests. Besides, active materials such as ferromagnetics [START_REF] Zighem | Combining ferromagnetic resonator and digital image correlation to study the strain induced resonance tunability in magnetoelectric heterostructures[END_REF][START_REF] Elhajjar | Magnetomechanical local-global effects in magnetostrictive composite materials[END_REF] , shape memory alloys [START_REF] Sánchez-Arévaloa | Use of digital speckle pattern correlation for strain measurements in a CuAlBe shape memory alloy[END_REF][START_REF] Ozbulut | Shape memory alloy cables for structural applications[END_REF] and piezoelectrics [START_REF] Meuris | Imaging Methods for Novel Materials and Challenging Applications, Chapter 40 -Thermal Analysis of PZT Thin Films Using Digital Image Correlation[END_REF][START_REF] Bai | Experimental and numerical analysis of interfacial fracture in piezoelectric composites[END_REF] were also studied. Regarding the piezoelectric studies [START_REF] Meuris | Imaging Methods for Novel Materials and Challenging Applications, Chapter 40 -Thermal Analysis of PZT Thin Films Using Digital Image Correlation[END_REF][START_REF] Bai | Experimental and numerical analysis of interfacial fracture in piezoelectric composites[END_REF] , the DIC was dedicated to large strains and crack measurements. Recently, DIC was used to characterise the macroscopic ferroelectric behaviour of a 0.5 mm film [START_REF] Malakooti | Noncontact and simultaneous measurement of the d33 and d31 piezoelectric strain coefficients[END_REF] and to measure the piezoelectric coefficients of a piezoelectric wafer in the Rayleigh region [START_REF]Direct measurement of piezoelectric shear coefficient[END_REF] . In these works, the measurements were conducted in the air, with a maximum electric field around 2 kV/mm and with the natural texture of the sample as DIC tracking pattern.

More recently, Chen and Kamlah improved the use of the DIC technique for ferroelectric [START_REF] Chen | Deformation in lead zirconate titanate ceramics under large signal electric field loading measured by digital image correlation[END_REF] and ferroelastic [START_REF] Chen | Deformation behavior of lead zirconate titanate ceramics under uniaxial compression measured by the digital image correlation method[END_REF] characterisations. The displacement and strain measurement errors were investigated and the notion of strain field homogeneity was introduced. Similarly to [START_REF]Direct measurement of piezoelectric shear coefficient[END_REF] , DIC measurements were made using the natural texture of the sample. In the algorithm used, strains were obtained from a linear regression of the displacement field. Such a computation strategy drastically reduces the random part of the measurement error. For instance, for a regression over a distance of 1800 px and for static images, the strain error was about 10 -6 while the local displacement error was 6.5 x 10 -2 px.

This strain error led the DIC measurement results to be in a good agreement with the LVDT technique in this case. However, such a strain computation process discards physical admissible results, for instance non-uniform strain components. A way to circumvent this problem while keeping a low uncertainty is to introduce a filter that takes into account mechanical admissibility of the solution. Recently, DIC has outperformed the displacement resolution of classical algorithms using such a filter [START_REF] Tomicevc | Mechanics-aided digital image correlation[END_REF] . The proposed algorithm, namely

CorreliRT3, is able to compute displacements imposed during a mechanical test with a standard deviation under 10 -2 px.

In this work, CorreliRT3 is used to characterise a ferroelectric sample from piezoelectric to ferroelectric strain levels. A speckle pattern is deposited on the sample to maximise the correlation quality. The correlation algorithm is described in a first part. The speckle deposition and the measurement device is detailed in a second part. The experimental error of the DIC setup is assessed with the help of a ferroelectric test on the PZT sample. Then, the strain of the PZT sample is measured under electric fields from piezoelectric to ferroelectric amplitudes. The piezoelectric coefficients, the susceptibility and the density of dissipated energy per cycle are extracted from this experiment. Their evolution as a function of the electric field amplitude is discussed in the last part. This work details in conclusion the potentialities, in terms of piezoelectric behaviour characterisation, of a DIC setup designed for high electric fields.

DIC technique

The Digital Image Correlation (DIC) technique was first introduced for 2D images by Sutton et al. [START_REF] Sutton | Determination of displacements using an improved digital correlation method[END_REF] . This technique initially aims at measuring the full in-plane deformation of an object. For that purpose, two monochromatic images are obtained from this object (a reference picture and a deformed picture). Images are defined as 2D matrices where each value corresponds to a grey level amplitude. The matrices can be cropped to discard irrelevant data and in such cases, the remaining region is called Region Of Interest (ROI). The images or ROIs are then compared using a correlation algorithm. The algorithm initially proposed by Sutton et al. used a spatial 2D cross-correlation product. Under the hypothesis of grey level conservation, the cross-correlation product is maximum when the two images match. Thus, the displacement between images corresponds to the coordinates of the cross-correlation maximum. The computed displacement then refers to the mean displacement of the image centre. It is worth noting that for this algorithm, ROIs are rather divided in square subsets and the correlation is performed on each subset. It allows to define several displacement nodes hence a displacement field. This method is referred to as "local" (subset-based) DIC [START_REF] Sutton | Determination of displacements using an improved digital correlation method[END_REF][START_REF] Hild | Comparison of local and global approaches to digital image correlation[END_REF] . The displacement uncertainty of such algorithm is equal to one pixel since it is the numeric resolution of the image. However, interpolation methods were developed to improve the accuracy down to sub-pixel levels [START_REF] Chu | Applications of digital-imagecorrelation techniques to experimental mechanics[END_REF][START_REF] Schreier | Systematic errors in digital image correlation caused by intensity interpolation[END_REF][START_REF] Xiong | Evaluation of sub-pixel displacement measurement algorithms in digital image correlation[END_REF] . Since its first application, this DIC principle has been optimized. Measurement setups were improved and experimental errors minimised. A comprehensive state of the art review on these aspects can be found in [START_REF] Sutton | Image Correlation for Shape, Motion and Deformation Measurements[END_REF] .

Alternative computation techniques based on variational formulations were also developed. Such approaches are referred to as "global approaches". In these algorithms, the correlation is computed over the whole image (or ROI) without sub-dividing it into subsets. It is then possible to insert, in the computation process, spatial regularisation techniques to constrain the displacement field and control its fluctuations [START_REF] Horn | Determining optical flow[END_REF][START_REF] Black | Robust incremental optical flow[END_REF] . Recently, a global approach based on a finite element description and on a mechanical regularisation has been developed

and implemented [START_REF] Tomicevc | Mechanics-aided digital image correlation[END_REF][START_REF] Hild | Comparison of local and global approaches to digital image correlation[END_REF] . It consists first in minimising iteratively the function related to difference of grey levels between two images:

= -[ + ] , (1)
where:

is the considered domain (subset in local approach, ROI in global approach).

is the reference image.

is the image in a deformed configuration.

is the pixel coordinates.

is the displacement of the pixel at coordinates .

A Taylor expansion to the first order is made to linearize [ + ] to + ∇ .

. Eq. ( 1) then leads to a linear system and it is possible to express this system in a matrix-vector product [START_REF] Hild | Comparison of local and global approaches to digital image correlation[END_REF] :

[ ] = (2)
where:

[ ] is a constant matrix representing the dyadic product of fields ∇ . where is a chosen vector function constraining . corresponds to the residual vector, weighted by ∇ . This quantity vanishes when a perfect match is obtained for each pixel.

This system is solved iteratively. An iteration consists first in computing from and . Eq. ( 2) is then solved to find and the total displacement field is updated.

Afterwards, a new iteration starts. The iterative process stops when reaches a quantity defined by the user. It is worth noting that, as a linearization is used to solve Eq. ( 1), the solution is defined only for small increments .

In the program Correli RT3, the displacement field is in addition constrained to fulfil mechanical admissibility of the solution. For that purpose, a linear-like elastic relation is imposed in a Finite Element sense:

[ ] = (3)
where:

[ ] is the stiffness matrix.

is the nodal forces vector.

is the displacement field.

As this minimisation is computed in a Finite Element sense, the ROI is discretised using a mesh. Equations are solved for each node of this mesh. In the case of CorreliRT3, the mesh elements are chosen to be triangular (also called T3-element). Starting from equation ( 3), the energetic functional is introduced to regularise the displacement of all interior nodes:

= [ ] [ ] (4)
The displacements at the edges of the mesh are regularised using an energy term based on the displacements at the mesh boundary:

= [ ] [ ] (5)
where:

[ ] is an operator acting on the ROI boundary [START_REF] Tomicevc | Mechanics-aided digital image correlation[END_REF] .

The functionals and must be minimised as well as and this is referred to as the mechanical regularisation. However, such a minimisation is an ill-posed problem. A total functional, in which , and are normalised and weighted, is considered. The three functionnals are then respectively normalised by

! [ ] ! , ! [ ] [ ] ! and ! [ ] [ ] ! . ! is a displacement field chosen to be in the form of a plane wave, ! = ! # $ %&.'
, where ! # is the wave amplitude and ( the wave vector. The total functional is then expressed as:

= ) 1 1 + + + + , -. + + . + + . / (6)
where:

. , . and . are normalised residuals.

+ and + are weights defined by ( and by the regularisation length scales 0 and 0 in pixels [START_REF] Tomicevc | Mechanics-aided digital image correlation[END_REF] .

The minimisation of acts as a fourth-order low-pass filter on the displacement field [START_REF] Tomicevc | Mechanics-aided digital image correlation[END_REF] . It is worth noting there are now two representations of the displacement field.

Indeed, the mechanical equations lead to nodal displacements whereas residuals lead to pixel displacements. The conversion from pixel to nodal displacements is performed by a cubic interpolation.

The introduced correlation algorithm presents several benefits. First, in terms of displacement error, the global approach out-performs local approaches based on quadrilateral element subsets [START_REF] Tomicevc | Mechanics-aided digital image correlation[END_REF] . To reach equivalent uncertainty, local approaches use filtering processes that can result in inappropriate approximations. This is the case for instance when the tested material presents inhomogeneous properties. Second, local DIC uncertainty increases exponentially with the inverse of the subset size [START_REF] Besnard | Finite-Element" displacement fields analysis from digital images: Application to Portevin -Le Châtelier bands[END_REF][START_REF] Pan | Study on subset size selection in digital image correlation for speckle patterns[END_REF] . For the presented global approach, the uncertainty tends to saturate when the mesh size decreases. Consequently, it is possible to compute correlations using 1 pixel-sized elements while local approaches require subset sizes of 10 pixels and more [START_REF] Tomicevc | Mechanics-aided digital image correlation[END_REF] . Considering these assets of such algorithm, CorreliRT3 has been applied to the characterisation of ferroelectrics. The whole experimental setup and its reliability to measure ferroelectric strains are presented in the next parts.

Experimental setup

Measurement device

The experimental setup is described in Figure 1. The setup is designed to apply an electric field 1 up to 4 kV/mm to the sample, to measure the induced electric displacement 2 and to measure the sample strain components 3 and 3 44 using DIC and 3 using a CCD laser displacement sensor. The sample is a 4 x 4 x 4 mm soft PZT (NCE55, Noliac) with silver electrodes. The piezoelectric and dielectric coefficients of this material are given in Table I. The high voltage, used to impose the electric field 1 , is produced by a Trek 20/20C HS high voltage amplifier. For insulation purpose, the sample is immersed in a PTFE container filled with insulating fluid (Fluorinert TM FC-770, 3M). The electric displacement 2 is measured from the upper electrode of the sample. The electrode is accessed using a rod (diameter 4 mm) sliding in a bracket, both conductive. The contact between the rod and the sample is ensured using a spring. Because of the small stiffness of the spring (1 N/mm), the stress induced by the rod is neglected. The conditioner used to obtain 2 is an operational amplifier integrator circuit [START_REF] Stewart | Ferroelectric Hysteresis Measurement & Analysis[END_REF][START_REF] Glazer | Automatic sampling circuit for ferroelectric hysteresis loops[END_REF] . Integrating capacitors of 100 nF to 2000 nF are used to measure the sample response under electric fields from 10 V/mm to 4 kV/mm. The sample elongation ∆ mean image resolution to be maximised and ensures the angle between the sample surface and the camera sensor plane to be less than 0.2°. Once images are acquired, they are correlated in a post-processing step with the 2D-DIC MATLAB program Correli RT3 [START_REF] Tomicevc | Mechanics-aided digital image correlation[END_REF][START_REF] Hild | Comparison of local and global approaches to digital image correlation[END_REF] . The displacement and strain fields (longitudinal and transverse) of the sample are then obtained.

All DIC computations are made using the regularisation lengths 0 = 2 × 0 = 400 px. With these values, the computation time is about 1 min per image, which is far faster than previous PZT studies using DIC [START_REF] Chen | Deformation in lead zirconate titanate ceramics under large signal electric field loading measured by digital image correlation[END_REF][START_REF] Chen | Deformation behavior of lead zirconate titanate ceramics under uniaxial compression measured by the digital image correlation method[END_REF] .

Speckle pattern

Samples studied using DIC must contain a tracking texture to evaluate displacements. The natural texture of the sample can be used for this purpose or an artificial texture can be applied to it to optimise the correlation quality. Generally, the texture is created by applying black/white dots randomly or periodically positioned on a white/black background. The image of such texture is commonly referred to as a "speckle pattern". Random speckles are usually preferred to periodic speckles since it creates unique local textures. This property allows subsets (or elements) to be differentiated from their neighbours and allows correlation conflicts to be avoided. In this paper, a random speckle pattern is used as tracking texture.

Several studies were carried out in the past to identify the influence of the speckle characteristics (for instance, contrast, density or speckle size) on the correlation quality [46, 53- 55] . One way to quantify the texture quality is to compute the image gradient ∇ . Indeed, in Eq.( 1) the displacement is weighted by ∇ . Thus, the displacement resolution is proportional to the inverse of ∇ [START_REF] Roux | Stress intensity factor measurements from digital image correlation: post-processing and integrated approaches[END_REF] . As the speckle pattern corresponds to the grey level variation function , ∇ directly reflects the speckle quality. This quality is maximised when the Mean Intensity Gradient (MIG) of the entire image is maximised [START_REF] Bornert | Assessment of digital image correlation measurement errors: methodology and results[END_REF][START_REF] Crammond | Speckle pattern quality assessment for digital image correlation[END_REF] .

Using the MIG and simulated images, we have recently identified the optimal density and speckle radius of specific speckle patterns [START_REF] Segouin | Development of a 2D DIC experimental tool for piezoelectric strains measurements[END_REF] . Speckles made of circular black dots randomly positioned onto a white background were considered. The simulations also accounted for the optical resolution and the spatial sampling of the experimental setup. It was shown that the MIG is maximised for a speckle density of 0.5 and for diameters ranging from 6 to 17 µm, depending on the optical magnification used [START_REF] Segouin | Development of a 2D DIC experimental tool for piezoelectric strains measurements[END_REF] . Considering these results, the speckle is experimentally produced by depositing black circular particles (toner powder with ~10 µm diameter per grain) onto a white acrylic layer. Both white paint and powder are deposited on the specimens using the airbrush Badger Sotar 20/20. An example of the speckle obtained with this method is shown in Figure 2.

Characterisation of the DIC technique

The sample is subjected to a 4 kV/mm triangular electric field at 10 mHz. During this test, the sample is imaged to compute strains a posteriori by DIC using 64 px sized T3 elements. For comparison purpose, the strain is also recorded with the CCD laser sensor. The triangle waveform is used instead of a sine because the number of images is limited to 350 by the computer memory. The triangle then helps to distribute images around the coercive field. The image acquisition frequency is set to 3.5 Hz (350 images per hysteresis loop) with an exposure time of 4 ms per image (i.e. 0.004% of the electric field period). The polarisation of the sample is measured using a fixed capacitance of 2.057 µF.

In this part, the displacement and strain fields obtained by DIC at 1 = -2 kV/mm are first shown and discussed. Then, the mean value of the DIC strain fields are collected to plot the sample S-E loop. The S-E loop is discussed and compared to the S-E loop obtained with the CCD laser sensor.

Displacement and strain field measurements

Figure 3 shows the DIC displacement and strain fields at 1 = -2 kV/mm. Figure 3a and Figure 3b gives the longitudinal and the transverse displacement field, respectively. The displacement reference point is approximatively the image centre (see Figure 3c) because rigid body motions were removed. The displacements are expressed in pixels since they were computed from numerical images. In real space, the overall longitudinal displacement is ~3.87 µm and the overall transverse displacement is ~1.94 µm since the spatial sampling period is 1.21 µm/px. As expected, the displacement field exhibits a longitudinal elongation and a transverse contraction by comparison to the reference state (1 = 0).

Figure 3d

, Figure 3e and Figure 3f present the DIC longitudinal, transverse and shear strain field at 1 = -2 kV/mm, respectively. The strains are computed from Figure 3a and Figure 3b in a finite element sense. Using the nodal displacements of the T3-meshing, the Constant Strain Triangle (CST) method [START_REF] Zienkiewicz | The Finite Element Method: The Basis[END_REF] is applied to obtain the strain on each element. The longitudinal strain (Figure 3d) fluctuates between 0.2 x 10 -3 and 1.5 x 10 -3 . The highest fluctuations are localised at the edges of the strain field. The same observations are made on the transverse and shear strains (Figure 3e and Figure 3f). These fluctuations are believed to be correlation errors since the field is discontinuous in the corresponding regions. The boundary regularisation of the DIC algorithm may not be sufficient to compensate for the displacement discontinuity. However, it cannot be exclude that these fluctuations are due to an inhomogeneous electric field at the sample boundary. To discard the strain fluctuations at the edges, each strain field is studied in its central part. This part is indicated in the figures by a dashed rectangle and contains 1700 elements.

The mean and standard deviation of the strain are computed in this zone. Values are reported in Table II. The observed magnitude, about 10 -3 , are expected for this class of materials.

Standard deviations are less than ~5 x 10 -5 for all strain fields (approximatively 5% of the strain amplitudes, which is ~10 -3 ) meaning that the strain fields are satisfactorily homogeneous in the studied area. Regarding the shear strain, the mean value is 23 times lower than the transverse strain with a standard deviation of 3.28 x 10 -5 . It is small but not insignificant and may reveal that the sample naturally shears under free mechanical boundary conditions.

Measurement: Mean Value

Measurement:

Standard Deviation H 99 (longitudinal) 1.29 x 10 -3 4.01 x 10 -5

H ;; (transverse) -0.550 x 10 -3 5.15 x 10 -5 H 9; (shear) -0.0235 x 10 -3 3.28 x 10 -5 Table II -Mean and standard deviations of the strain fields at E = -2 kV/mm (Figure 3).

Ferroelectric measurement

Figure 4 shows the macroscopic strain and polarisation as a function of the applied electric field. All measurements are made simultaneously. The polarisation (Figure 4a) at saturation is 0.40 C/m² and the remanent polarisation is 0.34 C/m² which is consistent with other PZT studies [START_REF] Damjanovic | The Science of Hysteresis[END_REF][START_REF] Tang | Piezoelectric property and strain behavior of Pb(Yb0.5Nb0.5)O3-PbHfO3-PbTiO3 polycrystalline ceramics[END_REF] . The P-E loop exhibits a 5 V/mm difference between the negative and positive coercive field. This difference is not significant since it represents less than 1% of the coercive field.

The longitudinal strain (Figure 4b) amplitude measured with DIC (blue curve) is of 3.80

x 10 -3 . The electric field at the minimum strain is measured as 595 V/mm. This is in satisfying agreement with the CCD laser measurement. The strain hysteresis is however larger for the CCD laser measurement than for the DIC measurement. This is presumable the result of the sliding friction between the rod and the bracket. Regarding the noise on the red curve, it is caused by environmental vibrations transmitted from the floor to the setup. This noise is not visible on DIC measurements since DIC is, in the present case, slightly sensitive to vibrations. Indeed, the effect of vibrations during an image acquisition corresponds to a sliding average filter. The light intensity is averaged in the motion direction during the exposure time. In the present case, the maximum vibration velocity is around 0.04 px/ms and the exposure time is set to 4 ms. Each image is then averaged over a maximum of 0.16 px. The spectral response of this filter has been numerically computed. Its spatial cut-off period has been found to be 0.27 px. This is 74 times less than the spatial cut-off period of the optical system, which is 20 px. The effect of vibrations are then neglected.

The transverse strain (Figure 4c) amplitude is of 1.74 x 10 -3 , which represents 46% of the longitudinal strain amplitude. Since the ratio is not equal to 50%, the process is not isochoric. Another important observation is the difference between the maximum strains at positive and negative electric fields (I3 J and I3 = ). The asymmetry factor K3 I3 J -I3 = / I3 J I3 = is of 0.5 x 10 -2 for the DIC measurement and of 1.2 x 10 -2 for the laser measurement (Figure 4b). The asymmetry of the transverse strain is smaller with a factor of 0.1 x 10 -2 . These asymmetries are attributed to a measurement error and, regarding the DIC measurement, the asymmetry factors are acceptable.

It is possible to qualify further the DIC measurement by analysing the strain field distribution during the electric excitation. Figure 5 shows the standard deviations of the DIC strain fields (dashed rectangle of Figure 3d,Figure 3e andFigure 3f) as a function of the electric field. The values remain below 1.2 x 10 -4 for a decreasing electric field. This represents less than 5% of the maximum transverse strain. The strain fields can then be considered as homogeneous. However, this is not true for increasing electric fields, two peaks in standard deviation are observed. These peaks do not reflect an experimental error and are related to the sample. Indeed, the peaks are reproducible and the fluctuations observed in the strain fields were found to be systematic, not random. An explanation for these fluctuations is the polarisation reversal since the peaks position matches the coercive field. Around the coercive field, the polarisation of local dipoles is reversed and this reversal does not occur continuously due to dissipation effects. Thus, heterogeneities are expected in the strain field. The order of magnitude of the heterogeneities is relatively high with a standard deviation of 3 x 10 -4 for the transverse strain. The electric field frequency might be too high to prevent dynamic effects to appear at the coercive field. In this part, it was shown that Digital Image Correlation algorithms based on global approaches are efficient tools to characterise ferroelectric strains. The total strain measurement error does not exceed 1.2 x 10 -4 under quasi-static measurement condition. As the total number of points is 1700 for 64 px sized element, the error on the mean value is 1.2 × 10 =N /√1700 = 3 × 10 =R , assuming this error is random and follows a normal distribution. Consequently, it is theoretically possible to characterise not only ferroelectric strains but also piezoelectric strains using the presented setup. In order to verify this observation, the sample behaviour is characterised in the next part from sub-coercive to high electric fields.

Piezoelectric to ferroelectric characterisation

In this part, the sample is characterised from small (piezoelectric behaviour) to high electric fields (ferroelectric behaviour) with the same setup (see section 0). For this purpose, two experiments are proposed. For both experiments, the sample is first depoled and repoled once at 4 kV/mm during 1 min. The depolarisation process is made according to the anhysteretic bias field method [START_REF] Kaeswurm | The anhysteretic polarisation of ferroelectrics[END_REF] (bias field of 0 V/mm, frequency of 0.5 Hz, duration of 60 s and exponential damping factor of 0.015). The first experiment consists in subjecting the sample to 50 mHz bipolar sinusoidal electric field with increasing amplitudes. The polarisation and the DIC longitudinal and transverse strains are acquired. The second experiment is identical, but performed under unipolar sinusoidal electric field. The sine waveform is chosen so as to compare the results (see section 6) to other PZT studies. The image acquisition frequency is 5

Hz (100 points per cycle). All strains were obtained by computing the mean value of the DIC strain field centre.

Bipolar electric field

Figure 6 presents S 1 and 3 1 loops for bipolar loadings. On both polarisation and strain, the sample exhibits an overall linear behaviour and small hysteresis until 200-250 V/mm.

Then, an inflection point appears at the bottom part of the cycles. As the material is polarised, saturation is reached for positive electric fields but not for negative ones, which creates the inflexion. At 550 V/mm, the negative coercive field is reached and the hysteresis becomes substantial. The polarisation even reaches -0.3 C/m² at 600 V/mm which approximatively corresponds to half the major cycle amplitude: the material is depolarised and repolarised during this cycle. On the strain loop, a new wing appears and the loop becomes an asymmetric butterfly loop. Above 600 V/mm, the cycle amplitude grows and the asymmetry factor of the butterfly loop decreases. At 4000 V/mm, the cycles are symmetric: the preferred orientation of domains (known as internal or imprint electric field [START_REF] Genenko | Mechanisms of aging and fatigue in ferroelectrics[END_REF][START_REF] Morita | Magnetic force memory effect using a magnetostrictive material and a shape memory piezoelectric actuator composite[END_REF][START_REF] De | Effect of internal electric field on ferroelectric polarization in multiferroic tbmno3[END_REF] ) due to the initial prepolarisation has vanished. The loops are then major loops. More details are given and analysed from these graphs in the discussion part. As a consequence, all cycles are narrow compared to bipolar cycles (small dissipation). The cycles are elliptic until 250 V/mm and begin to curve downward from 500 V/mm.

Unipolar electric field

Consequently, the polarisation and strains increase more at low fields than at high fields. This was expected since the material is electrically saturated at high fields, hence, the extrinsic effect cannot contribute anymore to the rise of polarisation and strain. A detailed analysis of the susceptibility and piezoelectric coefficients as a function of the electric field is presented in the next part.

Discussion

The bipolar and unipolar characterisations shown in Figure 6 and Figure 7 respectively are analysed by extracting the apparent piezoelectric coefficients and susceptibility:

* UV WW UX W , 4 * UV YY UX W and Z * U[ WW UX W , ( 8
)
where I1 is the electric field variation, I3 the longitudinal strain variation, I3 44 the transverse strain variation and IS the polarisation variation. These parameters are computed with an ordinary least square regression over all points of each cycle of Figure 6 and Figure 7 (S(E) for * and 4 * and P(E) for Z * ). They are called "apparent" since they differ from the actual intrinsic coefficients. The dissipated energy is also obtained from Figure For bipolar electric fields, the piezoelectric coefficients * and 4 * evolve linearly with the maximum electric field (Figure 8a and Figure 8b). This trend is in agreement with previous studies on soft PZT [START_REF] Hall | Nonlinearity in piezoelectric ceramics[END_REF][START_REF] Damjanovic | The Science of Hysteresis[END_REF][START_REF] Sukesha | Variation of piezoelectric coefficient and dielectric constant with electric field and temperature: A review[END_REF][START_REF] Li | The extrinsic nature of nonlinear behavior observed in lead zirconate titanate ferroelectric ceramic[END_REF] . The values of the coefficients at low electric fields are consistent with the manufacturer data ( * of 670 pm/V and a 4 * of 260 pm/V, see Table I). However, these values increase significantly with the electric field. At 550 V/mm, these values are multiplied by 3 for * and by 4 for 4 * . The approximation of piezoelectric behaviour (linear behaviour) is then restricted to electric fields under 75 V/mm for this material. Regarding the 4 * / * ratio, the linear evolution of the coefficients (Figure 8a and Figure 8b) makes it quasiconstant. Above 200 V/mm, values are bounded between -0.43 and -0.48.

The evolution of the susceptibility Z * (Figure 8c) does not follow the evolution of the piezoelectric coefficients. The susceptibility increases linearly only at low electric fields. This observation is typical for PZT and has already been reported [START_REF] Hall | Nonlinearity in piezoelectric ceramics[END_REF][START_REF] Damjanovic | The Science of Hysteresis[END_REF][START_REF] Stewart | Ferroelectric Hysteresis Measurement & Analysis[END_REF][START_REF] Sukesha | Variation of piezoelectric coefficient and dielectric constant with electric field and temperature: A review[END_REF][START_REF] Li | The extrinsic nature of nonlinear behavior observed in lead zirconate titanate ferroelectric ceramic[END_REF][START_REF] Chung | Microstructural evolution in lead zirconate titanate (PZT) piezoelectric ceramics[END_REF] . The non-linear increase around the coercive field is due to the squared-shape polarisation loop. As soon as the electric field gets close to the coercive field, the polarisation amplitude increases rapidly.

However, this increase is associated to a high dissipation increase (100 kJ/m 3 /cycle between 450 and 550 V/mm, Figure 8d). The value of Z * then corresponds only to the global sensitivity

∆[ ∆X of the cycle.
The coercive field shows a clear frontier between apparent piezoelectricity and ferroelectricity. On the dissipated energy (Figure 8d), the values are less than 0.22 kJ/m 3 /cycle for electric fields lower than 100 V/mm. In this electric field range, the material behaviour (Figure 6) is linear and reversible. From 100 V/mm to the near coercive field (500 V/mm), the dissipation increases to 51 kJ/m 3 / cycle and is no more negligible. This part is known as the Rayleigh region [START_REF] Hall | Nonlinearity in piezoelectric ceramics[END_REF][START_REF] Chung | Microstructural evolution in lead zirconate titanate (PZT) piezoelectric ceramics[END_REF][START_REF] Damjanovic | The rayleigh law in piezoelectric ceramics[END_REF] . Above 500 V/mm, the dissipation rises rapidly and reaches 1100 kJ/m 3 at 3000 V/mm: the domain switching processes responsible of ferroelectricity become here prominent [START_REF] Hall | Nonlinearity in piezoelectric ceramics[END_REF] .

For unipolar electric fields (Figure 9), results for 1 < 100 V/mm are comparable to results for bipolar electric fields (see Table III). This was expected since the initial polarisation state of the material is the same in both cases. However, a difference appears when the electric field is increased. The piezoelectric coefficients for unipolar electric fields remain small compared to bipolar electric fields ( * of 860 pm/V and 4 * of -400 pm/V at 500 V/mm).

Moreover, the values increase non-linearly (saturation at 500 V/mm). The 4 * / * ratio is comparable to the ratio obtained for bipolar fields above 200 V/mm. The evolution of Z * is similar to the evolution of the piezoelectric coefficients. The susceptibility moves up linearly (from 3800 to 4500) and then saturates (around 400 V/mm).

Above 400 V/mm, values decrease due to the apparition of the downward curvature in the S 1 cycles (see Figure 7). For these electric fields, as for bipolar electric fields, the computed Z * does not refer to susceptibility anymore. Regarding the dissipated energy (Figure 9d), values are extremely low (82 times less than bipolar field dissipations). This was expected since the material is used far from its coercive field (-600 V/mm).

Conclusion

The Digital Image Correlation measurement method was used to characterise PZT strains (longitudinal and transverse) from low to high electric fields. The speckle pattern, the global DIC algorithm and the mechanical regularisation process allowed strains to be measured with satisfying resolution (standard deviation of ~3 x 10 -6 on the mean strain) and satisfying computing time (~1 min per image). It is worth noting that the regularisation takes into account the mechanical admissibility of the displacement field. This work has improved the DIC technique for the characterization of piezoelectric and ferroelectric strains. Moreover, the DIC setup showed interesting advantages such as non-sensitivity to micrometre scaled vibrations and the ability to identify test heterogeneities.

The DIC setup has also proved its ability to measure soft PZT properties from 50 to 5000 V/mm (bipolar and unipolar). It is believed that smaller electric fields could be studied by conducting the measurements in the air, which would remove the errors induced by the electrical insulation liquid. Considering that DIC is contactless, the technique appears to be a good alternative for the measurement of low electric field behaviour. Indeed, contrary to LVDT or strain gages, DIC provides results where the measurement has not interacted with the test conditions. The limit of the presented application is that the measured parameters are

Figure 1 -

 1 Figure 1 -Schematic drawing of the sample measurement device (a) and illustration of the whole experimental setup (b).

Figure 2 -

 2 Figure 2 -Pictures at two different magnifications (a) and (b) of a speckle made on a piezoelectric material with white paint and toner powder.

Figure 3 -

 3 Figure 3 -Longitudinal (a) and transverse displacement fields (b) measured using DIC on a NCE55 sample under -2000 V/mm. Vector displacement field is plotted on graph (c). Longitudinal (d), transverse (e) and shear (f) strain fields are obtained from displacement fields using 64 px sized T3 elements. The black dashed rectangles represent the zone used for computing mean and standard deviation values.

Figure 4 -

 4 Figure 4 -Ferroelectric characterisation result for a triangle electric field (4 kV/mm at 10 mHz). Polarisation M 9 (a), longitudinal strain H 99 obtained by DIC (blue) and CCD laser sensor (orange) (b) and transverse strain H ;; obtained by DIC (c).

Figure 5 -

 5 Figure 5 -Standard deviation of strain fields H 99 , H ;; and H ;9 as a function of the electric field.

Figure 6 -

 6 Figure 6 -Polarisation M 9 , longitudinal strain H 99 and transverse strain H ;; under growing bipolar sinusoidal electric field (50 mHz). Before measurements, the sample (NCE55) was initially poled at 4000 V/mm during 1 min.

Figure 7 -

 7 Figure 7 -Polarisation M 9 , longitudinal strain H 99 and transverse strain H ;; under growing unipolar sinusoidal electric field (50 mHz). Before measurements, the sample (NCE55) was initially poled at 4000 V/mm during 1 min.The S 1 and 3 1 loops for unipolar loadings are shown in Figure7. The material is

6 and Figure 7 . 8 (

 78 It is computed as the parametric integral of a P(E) cycle and represents the volume density of electrical energy dissipated per cycle. Results are presented in Figure bipolar electric field) and Figure 9 (unipolar electric field). All parameters are plotted as function of the maximum electric field. The coefficients * , 4* and Z * are computed for cycles ranging from 75 to 550 V/mm. Under 75 V/mm, the measurement noise is too high to obtain reliable results. Above 550 V/mm, the coercive field is exceeded.

Figure 8 -

 8 Figure 8 -Apparent 8 99 * (a), 8 9; * (b), susceptibility \ 99 * (c) and dissipated energy (d) measured on cycles of Figure 6 (bipolar growing E) and plotted as function of the maximum electric field. The dotted lines (Figure 8a,Figure 8b andFigure 8c) indicate the manufacturer values.

Figure 9 -

 9 Figure 9 -Apparent 8 99 * (a), 8 9; * (b), susceptibility \ 99 * (c) and dissipated energy (d) measured on cycles of Figure 7 (unipolar growing E) and plotted as function of the maximum electric field. The dotted lines (Figure 9a, 9b and 9c) indicate the manufacturer values.

Table III -Apparent piezoelectric and dielectric coefficients obtained at 75 and 500 V/mm. Values obtained at 75 V/mm are compared with the manufacturer data.

 III

		At 75 V/mm		At 500 V/mm
		8 99 * (x 10 -12	8 9; * (x 10 -12	\ 99 *	8 99 * (x 10 -12	8 9; *	(x 10 -12	\ 99 *
		m/V)	m/V)		m/V)	m/V)
	Bipolar field	776	-429	4606	1963	-899		15400
	Unipolar field 677	-305	3806	858	-396		4466
	Manufacturer	670	-260	5000
	data

Acknowledgment

This work was supported by the Automotive Mechatronics Chair, a co-operation between Faurecia, CentraleSupelec, and Esigelec.

obtained from a sample lateral face, where boundary effects can lead to inhomogeneous properties. However the heterogeneities are mainly concentrated at the sample corners. In the present case, the centre part of the images provides reliable results.

The benefits of 2D-DIC for the study of ferroelectric materials are numerous. The measurement scale is not physically limited like for other methods. For instance, the microscopic and nanoscopic scales could be investigated by DIC using not optical, but SEM (Scanning Electron Microscope) images. It could then be xpossible to characterise the behaviour at the microstructure level. For example, the strain at domain or grain boundaries could be investigated. Finally, DIC is used here for quasi-static and high-resolution measurement but has a high-frequency potential with high speed cameras (up to 100 kHz and more with reduced digital resolution).

is measured using the CCD laser sensor LK-G10 connected to a LK-GD500 controller from Keyence. The sensor is placed above the rod (see Figure 1a). The mean longitudinal strain of the sample is obtained from:

with = 4 mm.

The electric field 1 is imposed using a real-time DSpace hardware module which has a maximum sampling frequency of 50 kHz. The elongation ∆ and the electric displacement 2 are recorded using the same module. These three variables are monitored on a first computer with MATLAB Simulink associated to a Graphic User Interface (DSpace ControlDesk, see Figure 1b). During experiments, a lateral face of the sample is imaged with a Ximea MD091MU-SY camera (14 bits monochrome, 3380 x 2708 px, 5 FPS maximum) mounted on a Questar QM100 MKIII. A visual access to the sample is ensured using a flat glass window (1 mm thick) mounted on a side of the PTFE container. The whole optical system is placed at 37.5 cm from the sample surface. In this condition, the aperture is f/7, the optical resolution is around 5 µm and the spatial sampling is 1.21 µm/px. The sample lighting is produced by a LLS 3 LED light source (SCHOTT North America, Inc.) and oriented with an optical fibre. The camera is triggered by the DSpace module to synchronise the image acquisition with the electric field. A second computer controls the camera using a MATLAB program. The program receives and stores the images using the TIFF format, without data compression. It also integrates a focusing and parallelism calibration function based on image gradient analysis. This function allows the