Adrien Guery

François Hild

Félix Latourte

Stéphane Roux

Slip activities in

Keywords: Crystal plasticity, Digital Image Correlation, Grain size effect, Slip activity

teaching and research institutions in France or abroad, or from public or private research centers.

Introduction

In several industrial fields, multiscale mechanical models have been developed to better understand and predict the performance of materials and to improve the links between microstructural and mechanical properties. The present study focuses on a face-centered cubic 316LN austenitic stainless steel, which is widely used for high temperature structural components in power plants such as Pressurized Water Reactors (PWRs) because of its corrosion resistance, fracture toughness and creep performance [START_REF] Jayakumar | High Temperature Materials for Nuclear Fast Fission and Fusion Reactors and Advanced Fossil Power Plants[END_REF].

If mean-field homogenization techniques are often sufficient to derive macroscopic properties from microscopic ones, they are unable to provide the local mechanical fields that are needed to study degradation mechanisms occurring at the microstructural scale (e.g., stress corrosion cracking [START_REF] Kamaya | Simulation for intergranular stress corrosion cracking based on a three-dimensional polycrystalline model[END_REF][START_REF] Mcmurtrey | Relationship between localized strain and irradiation assisted stress corrosion cracking in an austenitic alloy[END_REF][START_REF] Couvant | Strain path effect on IGSCC initiation and oxidation of alloy 182 exposed to PWR primary water[END_REF] or intergranular fracture [START_REF] Diard | Evaluation of finite element based analysis of 3D multicrystalline aggregates plasticity: Application to crystal plasticity model identification and the study of stress and strain fields near grain boundaries[END_REF][START_REF] Musienko | Simulation of inter-and transgranular crack propagation in polycrystalline aggregates due to stress corrosion cracking[END_REF][START_REF] Bieler | The role of heterogeneous deformation on damage nucleation at grain boundaries in single phase metals[END_REF][START_REF] Simonovski | Cohesive element approach to grain level modelling of intergranular cracking[END_REF]). Such fields can, for example, be computed using full-field methods based upon fast-Fourier transforms [START_REF] Lebensohn | Orientation image-based micromechanical modelling of subgrain texture evolution in polycrystalline copper[END_REF][START_REF] Lebensohn | An elasto-viscoplastic formulation based on fast Fourier transforms for the prediction of micromechanical fields in polycrystalline materials[END_REF][START_REF] Kanjarla | Study of internal lattice strain distributions in stainless steel using a full-field elasto-viscoplastic formulation based on fast Fourier transforms[END_REF][START_REF] Brenner | Overall response of viscoelastic composites and polycrystals: exact asymptotic relations and approximate estimates[END_REF] now available for finite strain calculations [START_REF] Eisenlohr | A spectral method solution to crystal elasto-viscoplasticity at finite strains[END_REF], or finite element methods as reviewed by Roters et al. [START_REF] Roters | Overview of constitutive laws, kinematics, homogenization and multiscale methods in crystal plasticity finite-element modeling: Theory, experiments, applications[END_REF].

In this context, the behavior of the grain or twin boundaries is not yet fully understood and remains a challenging research area. For instance, the interactions of dislocations with the interfaces in polycrystals depend on the interface structure itself [START_REF] Priester | Dislocation-interface" interaction -stress accommodation processes at interfaces[END_REF][START_REF] Jiang | Measurement of geometrically necessary dislocation density with high resolution electron backscatter diffraction: Effects of detector binning and step size[END_REF], which is affecting the stress levels. The experimentally observed lattice curvature at grain boundaries [START_REF] Jiang | Measurement of geometrically necessary dislocation density with high resolution electron backscatter diffraction: Effects of detector binning and step size[END_REF][START_REF] Ohashi | X-ray microdiffraction and strain gradient crystal plasticity studies of geometrically necessary dislocations near a Ni bicrystal grain boundary[END_REF][START_REF] Perrin | Role of discrete intragranular slip on lattice rotations in polycrystalline Ni: Experimental and micromechanical studies[END_REF] is successfully accounted for in strain gradient plasticity models [START_REF] Fleck | A phenomenological theory for strain gradient effects in plasticity[END_REF][START_REF] Gao | Mechanism-based strain gradient plasticity -I. Theory[END_REF][START_REF] Xue | Particle size effect in metallic materials: a study by the theory of mechanism-based strain gradient plasticity[END_REF][START_REF] Gurtin | A gradient theory of single-crystal viscoplasticity that accounts for geometrically necessary dislocations[END_REF] or higher-order continua models [START_REF] Forest | Cosserat modelling of size effects in the mechanical behaviour of polycrystals and multi-phase materials[END_REF][START_REF] Neff | A finite-strain elastic-plastic Cosserat theory for polycrystals with grain rotations[END_REF][START_REF] Klusemann | Plastic deformation induced microstructure evolution through gradient enhanced crystal plasticity based on a non-convex Helmholtz energy[END_REF][START_REF] Mayeur | A comparison of Gurtin type and micropolar theories of generalized single crystal plasticity[END_REF], which are motivated by the need for describing Geometrically Necessary Dislocations (GNDs) [START_REF] Ashby | The deformation of plastically non-homogeneous alloys[END_REF]. Among possible applications of such models, the study of plasticity in the vicinity of crack tips [START_REF] Perrin | Role of discrete intragranular slip on lattice rotations in polycrystalline Ni: Experimental and micromechanical studies[END_REF], Hall-Petch effects [START_REF] Forest | Cosserat modelling of size effects in the mechanical behaviour of polycrystals and multi-phase materials[END_REF][START_REF] Acharya | Grain-size effect in viscoplastic polycrystals at moderate strains[END_REF][START_REF] Beaudoin | Consideration of grain-size effect and kinetics in the plastic deformation of metal polycrystals[END_REF][START_REF] Geers | Size effects in miniaturized polycrystalline {FCC} samples: Strengthening versus weakening[END_REF][START_REF] Evans | A critical assessment of theories of strain gradient plasticity[END_REF][START_REF] Keller | Finite element analysis of the free surface effects on the mechanical behavior of thin nickel polycrystals[END_REF][START_REF] Baker | Grain size and particle dispersion effects on the tensile behavior of friction stir welded {MA956} oxide dispersion strengthened steel from low to elevated temperatures[END_REF] or even grain size effects on plastic strain localization patterns [START_REF] Cordero | Grain size effects on plastic strain and dislocation density tensor fields in metal polycrystals[END_REF] can be cited. The latter phenomena are observed at scales of about one micrometer, consistent with the predictions of several numerical works [START_REF] Geers | Size effects in miniaturized polycrystalline {FCC} samples: Strengthening versus weakening[END_REF][START_REF] Cheong | A study of microstructural length scale effects on the behaviour of {FCC} polycrystals using strain gradient concepts[END_REF][START_REF] Taupin | On particle size effects: An internal length mean field appraoch using field dislocation mechanics[END_REF][START_REF] Taupin | Size effects on the hardening of channel-type microstructures: A field dislocation mechanics-based approach[END_REF][START_REF] Chang | Analysis of particle induced dislocation structures using three-dimensional dislocation dynamics and strain gradient plasticity[END_REF].

In this study, it is first proposed to explore the grain size effects at a mesoscopic scale using Digital Image Correlation (DIC) from images acquired during in-situ tensile tests in a Scanning Electron Microscope (SEM). DIC provides spatially dense experimental information in polycrystals [START_REF] Héripré | Coupling between experimental measurements and polycrystal finite element calculations for micromechanical study of metallic materials[END_REF], more particularly for studying grain boundary effects [START_REF] Ma | On the consideration of interactions between dislocations and grain boundaries in crystal plasticity finite element modeling -Theory, experiments, and simulations[END_REF][START_REF] Ma | Studying the effect of grain boundaries in dislocation density based crystal-plasticity finite element simulations[END_REF]. Several studies have shown the possibility of performing kinematic measurements at the microstructural scale, for instance, using an Atomic Force Microscope (AFM) [START_REF] Soppa | Experimental and numerical characterisation of in-plane deformation in two-phase materials[END_REF] or SEM [START_REF] Sutton | Scanning electron microscopy for quantitative small and large deformation measurements part i: Sem imaging at magnifications from 200 to 10.000[END_REF][START_REF] Sutton | Scanning electron microscopy for quantitative small and large deformation measurements part ii: Experimental validation for magnifications from 200 to 10.000[END_REF]. These imaging means allow for the mapping of strain fields at the surface of polycrystals with high spatial resolutions [START_REF] Sutton | Scanning electron microscopy for quantitative small and large deformation measurements part i: Sem imaging at magnifications from 200 to 10.000[END_REF][START_REF] Sutton | Scanning electron microscopy for quantitative small and large deformation measurements part ii: Experimental validation for magnifications from 200 to 10.000[END_REF]. In particular, strain localizations have been shown to be related to grain orientations [START_REF] Tschopp | Microstructuredependent local strain behavior in polycrystals through in situ scanning electron microscope tensile experiments[END_REF][START_REF] Kammers | Digital Image Correlation under Scanning Electron Microscopy: Methodology and Validation[END_REF]. These studies lead, for instance, to a correlation between these localizations and the Schmid's factors. Such kinematic measurements also provide a way to analyze the slip system activity (whose slip traces are directly observed on the surface) and more specifically the slip transmission across grain boundaries [START_REF] Abuzaid | Slip transfer and plastic strain accumulation across grain boundaries in Hastelloy X[END_REF][START_REF] Carter | Characterization of localized deformation near grain boundaries of superalloy René-104 at elevated temperature[END_REF]. Other applications of DIC at microstructural scales deal with features in the vicinity of the crack tip [START_REF] Jin | Micro-scale deformation measurement using the digital image correlation technique and scanning electron microscope imaging[END_REF][START_REF] Jin | Microscale characterization of granular deformation near a crack tip[END_REF], to determine global fracture mechanics parameters [START_REF] Mcneill | Estimation of stress intensity factor by digital image correlation[END_REF], or to assess the cumulated plastic strain close to a growing fatigue crack [START_REF] Carroll | High resolution digital image correlation measurements of strain accumulation in fatigue crack growth[END_REF]. Recent advances in crystal plasticity modeling have shown the opportunities of confronting these experimental investigations with simulations in terms of textures [START_REF] Lebensohn | Orientation image-based micromechanical modelling of subgrain texture evolution in polycrystalline copper[END_REF], of relating fields to the microstructure [START_REF] Ma | On the consideration of interactions between dislocations and grain boundaries in crystal plasticity finite element modeling -Theory, experiments, and simulations[END_REF][START_REF] Ma | Studying the effect of grain boundaries in dislocation density based crystal-plasticity finite element simulations[END_REF][START_REF] Tasan | Integrated experimental-simulation analysis of stress and strain partitioning in multiphase alloys[END_REF][START_REF] Martin | Microscale plastic strain heterogeneity in slip dominated deformation of magnesium alloy containing rare earth[END_REF], of validating models [START_REF] Tasan | Strain localization and damage in dual phase steels investigated by coupled in-situ deformation experiments and crystal plasticity simulations[END_REF][START_REF] Lim | Grainscale experimental validation of crystal plasticity finite element simulations of tantalum oligocrystals[END_REF][START_REF] Lim | Quantitative comparison between experimental measurements and CP-FEM predictions of plastic deformation in a tantalum oligocrystal[END_REF], or even parameter identification [START_REF] Héripré | Coupling between experimental measurements and polycrystal finite element calculations for micromechanical study of metallic materials[END_REF].

In the present case and contrary to all the previous studies, the displacement field is based on a mesh taking as support the grain boundaries of the observed surface. This is made possible by using Electron Back-Scattered Diffraction (EBSD) data, which give access to the crystallographic orientations, and requires a change of coordinate system. While in similar studies such transformation is achieved using micro-indentations [START_REF] Héripré | Coupling between experimental measurements and polycrystal finite element calculations for micromechanical study of metallic materials[END_REF][START_REF] Carroll | High resolution digital image correlation measurements of strain accumulation in fatigue crack growth[END_REF], a registration method using DIC is chosen herein.

Traditionally, experimental slip system activities are compared with those predicted by the Schmid's factors based on a far field stress projection [START_REF] Carroll | An experimental statistical analysis of stress projection factors in BCC tantalum[END_REF]. Using this approach, a lack of correlation in the polycrystalline case is attributed to grain neighboring and localization effects [START_REF] Li | Methodology for estimating the critical resolved shear stress ratios of α-phase Ti using EBSD-based trace analysis[END_REF]. In addition, subsurface grains have a known influence of the kinematic surface response of polycrystals [START_REF] Zeghadi | Ensemble averaging stress-strain fields in polycrystalline aggregates with a constrained surface microstructure -Part 1: Anisotropic elastic behaviour[END_REF][START_REF] Zeghadi | Ensemble averaging stress-strain fields in polycrystalline aggregates with a constrained surface microstructure -Part 2: crystal plasticity[END_REF], and this challenge has been recently mostly tackled by studying oligocrystals [START_REF] Lim | Grainscale experimental validation of crystal plasticity finite element simulations of tantalum oligocrystals[END_REF][START_REF] Lim | Quantitative comparison between experimental measurements and CP-FEM predictions of plastic deformation in a tantalum oligocrystal[END_REF][START_REF] Carroll | An experimental statistical analysis of stress projection factors in BCC tantalum[END_REF], bicrystals [START_REF] Ma | Studying the effect of grain boundaries in dislocation density based crystal-plasticity finite element simulations[END_REF], or directionally solidified polycrystals [START_REF] Montagnat | Multiscale modeling of ice deformation behavior[END_REF]. However, such experimental conditions are not possible in the present case of an industrial material for which the grain size dimension is small as compared to the sample thickness.

Displacement fields are measured on meshes built at the free surfaces of five different tensile polycrystalline specimens of grade 316LN stainless steel. The slip activities are then explored by resorting to full-field finite element simulations of the experiments, considering a local stress projector outperforming far field projectors as employed in Ref. [START_REF] Carroll | An experimental statistical analysis of stress projection factors in BCC tantalum[END_REF]. Using these techniques, correlations between model and experiments are finally discussed.

Material and experimental facilities

Material

The material of the present study is a 316LN austenitic stainless steel of face centered cubic (FCC) crystal lattice. The tensile specimens are machined from a 60-mm thick rolled plate also utilized within the AFGRAP project [START_REF] Schwartz | Fatigue crack initiation modeling of 316LN steel based on non local plasticity theory[END_REF][START_REF] Prasad Reddy | Effect of grain disorientation on stage I fatigue crack retardation in FCC poly-crystals: a three dimensional dislocation dynamics investigation[END_REF], which was subjected to a solution treatment at 1050-1150 °C followed by water quenching. Preliminary optical microscopy and EBSD analyses have revealed the presence of less than 0.1 vol. % of δ-ferrite in the plate core and no texture was observed. To avoid the ferritic phase, the sampling is conducted in the plate skins where the material is fully austenitic. After solution treatment, a typical microstructure with heterogeneous grain sizes is observed, namely, the average size is of the order of 70 µm while a small fraction of small grains down to 5 µm in size coexists with some larger grains up to 200 µm. The grain shapes are equiaxed and large sized grains are sometimes divided into twins. The chemical composition of the material is given in Table 1. • sample A 70 is machined from the plate in its as-received state. The mean grain size is of the order of 70 µm,

• samples B 10 and C 10 are taken from the plate after 60 % cold-rolling followed by 1-hour heat treatment at 900 °C. The resulting mean grain size is about 10 µm,

• sample D 50 is machined from the plate after 60 % cold-rolling followed by 1-hour heattreatment at 1100 °C. The resulting mean grain size is about 50 µm,

• sample E 1000+ is taken from the plate after 264-hour heat treatment at 1200 °C. The resulting microstructure shows millimetric grains.

Microstructure characterization and in-situ tensile tests

In this study, a Zeiss Supra 55 SEM is used with a TSL OIM analysis software for EBSD measurements. An FEI Quanta FEG 600 SEM is used for imaging purposes. A miniaturized tensile testing device is introduced in the SEM chamber so that in-situ tensile tests can be imaged all along the loading history. EBSD measurements are performed with an acceleration voltage of 25 kV and a working distance of 14 mm. For imaging, both Secondary Electron (SE) Everhart-Thornley detector and Backscattered Electron (BSE) detector in Z-contrast are used, with an acceleration voltage of 10 kV and a working distance of 14 mm. An image definition of 2048 × 1887 pixels and a dynamic range of 16 bits are chosen.

The imaged gray level speckle marking evolved upon straining due to surface variations, namely, the deformation of the crystalline lattice and/or the accumulation of slip bands on the observed surface, which is related to the electromagnetic imaging environment. As these contrast variations appear more significantly with the SE mode than the BSE mode, the latter is chosen for imaging purposes during any in-situ tensile test to make DIC calculations easier. The SE mode is used to actually observe these surface variations at the end of tensile tests (see Section 5.3).

For each tensile test, imaging conditions are the same but with three different magnifications. Figure 1 shows the grain size distributions in the ROI of each specimen (except E 1000+ ). These distributions are obtained from the grain partition based on the EBSD measurements. One may note that the distributions of samples B 10 and C 10 are similar. Conversely, the distributions of

Kinematic measurements based on SEM images

Kinematic measurements by digital image correlation require a gray level speckled pattern with local contrasts as high as possible. In the present case, the raw image of the material does not provide such contrast. It is chosen to print an artificial speckle provided by a computer-generated random pattern using a microphotolithographic technique [START_REF] Guery | Characterization of SEM speckle pattern marking and imaging distortion by digital image correlation[END_REF]. The lithography process is conducted in a specific area on the surface of an in-situ specimen. The resulting pattern is made of gold speckles whose thickness is of the order of 20 nm. Because the speckle is designed for DIC analyses, the physical size of the spots lies in the range of 6 to 20 pixels (i.e., between 0.5 and 2.8 µm) for a center to center distance in the range of 30 to 50 pixels (i.e., between 2.7 and 10 µm). To allow for EBSD measurements over the region of interest, the surface is initially polished with cloths and diamond solutions down to 1 µm, and finished with a 50-nm colloidal silica suspension.

From a sequence of SEM images acquired during an in-situ tensile test, displacement fields are measured with a continuous Galerkin-based DIC procedure [START_REF] Leclerc | Integrated digital image correlation for the identification of mechanical properties[END_REF][START_REF] Tomicević | Mechanics-aided digital image correlation[END_REF]. The image registration is based on the conservation of the gray levels between two consecutive images, namely a first one in the reference configuration, f , and a second one in the deformed configuration, g, expressed as

f (x) = g(x + u(x)) (1)
where x is the vector coordinate of pixels in the reference configuration, and u the sought displacement field. The minimization of the quadratic norm of the difference of gray levels

ρ 2 c = f (x) -g(x + u(x)) 2
ROI is performed successively where, at iteration n, the displacement field is corrected by incremental displacement amplitudes {δu (n) }, which are the solution to linear systems

[M ] {δu (n) } = {b (n) } ( 2
)
where {δu} is the vector gathering the corrections of the displacement amplitudes u k associated with the basis of displacement fields ψ k providing the discretized displacement field u(x)

u(x) = k u k ψ k (x) (3)
The DIC matrix [M ] and vector {b (n) } are expressed as

M kl = ROI (ψ k • ∇ f )(x)(ψ l • ∇ f )(x) (4)
and

b k = ROI ( f -g(n) )(x)(ψ k • ∇ f )(x) (5)
where g(n) is the deformed image corrected by the current estimate of the displacement field.

Several choices are possible for the kinematic basis ψ k . For instance, finite element shape functions [START_REF] Leclerc | Integrated digital image correlation for the identification of mechanical properties[END_REF][START_REF] Besnard | Finite-Element" displacement fields analysis from digital images: Application to Portevin-Le Châtelier bands[END_REF] may be considered or enriched bases to account for localized phenomena [START_REF] Réthoré | Shear-band capturing using a multiscale extended digital image correlation technique[END_REF][START_REF] Réthoré | Solutions of ill-posed problems[END_REF]. In this study, it is chosen to take advantage of the crystallographic information that is available on the surface of the analyzed specimen. The aim is to build a finite element mesh taking as support the microstructure (i.e., grain and twin boundaries) and to overlay it on the first image of a sequence of SEM acquisitions during in situ tensile tests in order to perform DIC measurements thereafter.

The noise of SEM imaging estimated by correlating two pictures with no motion applied between two acquisitions is relatively high (i.e., greater than 2 % of the dynamic range). In addition, the discretization of the displacement field has to be fine enough to capture plastic strain localizations. For those reasons, mechanical regularization appears to be beneficial to assist DIC calculations. For that purpose, in the spirit of Tikhonov regularization of ill-posed problems [START_REF] Tikhonov | Solutions of ill-posed problems[END_REF],

it is proposed to include an additional cost function to the DIC gray levels functional [START_REF] Roux | Mechanical Assistance to DIC[END_REF]. It consists of introducing the equilibrium gap Φ 2 m penalizing deviations from a solution to a linear elastic homogeneous problem [K] {u} = {0} expressed as

Φ 2 m = {u} t [K] t [K] {u} (6)
where [K] is the stiffness matrix associated with inner nodes. An additional term is to be introduced to specifically deal with boundary nodes [START_REF] Tomicević | Mechanics-aided digital image correlation[END_REF]. These gaps introduce two regularization lengths (i.e., one for the inner nodes and another one for the boundary nodes) weighting the equilibrium gaps against the image registration, which are both chosen to be equal to four times the characteristic length of the mesh, which corresponds to the mean length of the element edges, in this study. This regularization is a low-pass mechanical filter, where the regularization lengths are acting as cut-off wavelengths. It is not intended to be representative of the actual material behavior. Table 3 gives the standard measurement resolutions, in terms of displacement and strain, obtained with the presented DIC procedure for each specimen by registering two pictures with no applied load between the two acquisitions. The root mean square value of the measured displacement and strain fields due to SEM noise leads to the corresponding standard measurement uncertainties. It can be noted that their levels are quite similar for the different specimens.

Change of coordinate system and unstructured mesh

This part is devoted to the presentation of the method to build an unstructured mesh consistent with microstructural interfaces focusing on the case of specimen E 1000+ for illustration purposes.

EBSD measurements provide a map of the microstructure boundaries for the region of interest of the analyzed specimen. A misorientation greater than 15°, respectively about 60°, is chosen to define a grain boundary or a twin boundary, respectively. Figure 2(a) shows the boundaries of the microstructure, which consists of a single grain with a few embedded twins of sharp polygonal shape. This map is used as a support for building an unstructured mesh. Therefore, it is essential that the mesh, which is first expressed in the EBSD frame, be accurately transported in the SEM images frame. For that purpose, the Image Quality (IQ) map is used. For EBSD measurements using the OIM analysis software, the IQ factor estimates the quality of the diffraction pattern, which is based on the sum of the detected peaks in the Hough transform [START_REF] Kunze | Advances in automatic EBSP single orientation measurements[END_REF]. A gray scale mapping of IQs over the ROI would be white when the IQ is high and black otherwise. In this study, EBSD measurements are performed over gold speckles. Because the quality of the diffraction pattern is low at each point the electron beam traverses gold, the speckles appear in black in the IQ map (Figure 2(b)).

The transformation X(x) that maps the EBSD coordinate system x onto that of SEM images X is measured via DIC between the IQ map and an SE image acquired before mechanical loading (Figure 2(c)). The EBSD map is obtained by scanning a flat surface that is tilted by an angle of 70 degrees. Any slight misalignment of the EBSD acquisition system, which may be induced by the control system of the SEM stage, leads to a spatial distortion that makes the transformation X(x) non trivial (see Ref. [START_REF] Nolze | Image distortions in SEM and their influences on EBSD measurements[END_REF] for more details on this issue). To assess this transformation and to correct it, image processing is required. First, the gray scale of the IQ map is reversed so that the pattern appears bright on a dark background. Then, a variation of gray levels with the microstructure, more visible in the IQ map than in the SE image, is to be attenuated. Image registration is performed with the suited filter giving more weight to high spatial frequencies of the gray level variations (i.e., corresponding to the speckles). The transformation can now be measured by DIC using the previously presented formulation. Since a large transformation occurs between the two images (i.e., a displacement range of 70 pixels), the algorithm is initialized with a coarse displacement solution obtained using a quadratic interpolation for each component, itself estimated from manually selecting a few remarkable points.

The correlation residuals ρ c are shown in Figure 3. The root mean square (RMS) value is of the order of 10 % of the reference image dynamic range. Slight fluctuations appear at the position of dots, which are due to the initial difference of gray levels in the two images since the registration between the two images is successfully achieved (otherwise the speckles would appear twice).

Any EBSD map or specifically the microstructure boundaries and the crystal orientation maps in the present study can now be shown in the SEM image coordinate system. Once overlaid the twin boundaries are segmented into polygons. The path between two consecutive detected triple points is chosen to be discretized into a finite number of wires. A mesh is generated with 3-noded The series of images acquired during the tensile tests are now processed with the previously presented DIC algorithm (see Equation ( 2)). Even though the BSE mode is used for imaging purposes, gray level changes occur with topographic variations. As a consequence, DIC measurements are performed in two steps. First, an updated scheme is applied. At the end of each calculation, an incremental displacement field is computed, the deformed image becomes the reference image for the next calculation and the total displacement is the sum of all increments.

The mesh is updated too by moving the node positions with the measured nodal displacement at the end of each calculation. Second, the measured displacement fields are used to initialize a non-updated scheme (i.e., the reference image and the mesh are kept unchanged for the calculations with all deformed images). This second step allows the measurement uncertainties not to be cumulated since at each step the total displacement field is directly computed.

DIC measurements are performed for the five specimens. In each case a gold pattern is deposited and an unstructured mesh taking as support the microstructure boundaries is built. For all used magnifications, the characteristic length of the mesh is about 20 pixels (i.e., between 1.6 and 5.6 µm) and the cut-off length associated with the elastic regularization is 80 pixels (i.e., between 6.4 and 22 µm). It is worth noting that the present regularization length is short compared to the mean grain size, and hence conclusions based on features of the displacement field pertaining to larger scale than the regularization length, i.e., not influenced by the mechanical regularization, result from the sample imaging only and are considered trustworthy.

Strains are evaluated for each finite element of the mesh by differentiation of the shape functions and computing the nominal components. Figure 5 shows the longitudinal component of the Green-Lagrange strain field for each microstructure, for a mean tensile strain up to 5 %. For each experiment, strain bands occur early on during the tensile test while only their magnitude increase thereafter. These strain localizations occur in the vicinity of grain boundaries as well as within the grains.

The statistical distribution of strain magnitude does not reveal a very significant difference that could be attributed to the grain size. In contrast, the image resolution (i.e., physical size of one pixel) has a visible impact. Thus it seems that the localized nature of the strain fields is such that it cannot be captured in an objective way with the present approach. In addition, a large number of grains is needed to establish firm conclusions and this is in conflict with exploring a large interval of grain sizes.

Since different microstructures are tested, the effect of the grain size on the macroscopic load is first considered. In Figure 6, the macroscopic stress-strain curve of the five tensile tests are compared. The Hall-Petch effect [START_REF] Petch | The cleavage strength of polycrystals[END_REF] applies and is reproducible, namely, the yield stress is observed to increase linearly with the inverse of the square root of the grain size, together with an increase of hardening (Figure 7). A linear interpolation of coefficient 0.96 demonstrates the good correlation between experimental data and the Hall-Petch relationship.

These experimental results are now to be compared with numerical simulations using a crystal plasticity model. In the following, only microstructure A 70 will be considered. expressed for each slip system (see AppendixA). Let us recall that the studied austenitic steel is an FCC material with twelve slip systems, which are listed in Table 4, indexed by s and defined by a normal n s to the slip plane and a slip direction m s . The three parameters associated with isotropic hardening are identified by homogenization [START_REF] Berveiller | An extension of the self-consistent scheme to plastically flowing polycrystal[END_REF] from the experimental macroscopic stress-strain data shown in Figure 6. The other parameters are set to values previously identified using cyclic loadings [START_REF] Guilhem | Investigation of the effect of grain clusters on fatigue crack initiation in polycrystals[END_REF].

The mesh of the simulations is built from that initially used for DIC analyses. The crystal plasticity law implemented in the finite element code requires a 3D model. Consequently, because the microstructure under the surface is unknown, the initial 2D mesh is extruded perpendicular to the observed surface. This choice appeared the least arbitrary and the more realistic solution for simulating the experimental microstructure unknown in the volume compared to other methods using for instance Voronoi tessellations [START_REF] Zeghadi | Ensemble averaging stress-strain fields in polycrystalline aggregates with a constrained surface microstructure -Part 1: Anisotropic elastic behaviour[END_REF]. Since the mesh thickness in that direction appears to have a minor influence on the pattern of strain localization [START_REF] Héripré | Coupling between experimental measurements and polycrystal finite element calculations for micromechanical study of metallic materials[END_REF], the mesh chosen for the simulations is extruded to a distance equal to its characteristic length. The triangular prisms are then cut into 4-noded tetrahedra so that the final 3D mesh (usually referred to as "quasi-2D mesh") has one element through the thickness. Its thickness corresponds to about 20 pixels (i.e., between 1.6 and 5.6 µm), which is small compared to the specimen thickness (1 mm) and the mean grain size.

Conversely, experimental boundary conditions are necessary for a valid description of the substrate effects and do have a significant influence on the strain pattern [START_REF] Héripré | Coupling between experimental measurements and polycrystal finite element calculations for micromechanical study of metallic materials[END_REF][START_REF] Zhang | Effect of realistic 3D microstructure in crystal plasticity finite element analysis of polycrystalline Ti-5Al-2.5Sn[END_REF]. Such conditions are more realistic than homogeneous boundary conditions. It is worth noting that DIC is only capable of measuring in-plane displacements while the simulation may predict an out-ofplane component. This has no consequence on the comparison performed in this work between simulations and measurements. The out-of-plane displacements appeared to have negligible consequences in the present case for macroscopic strain levels less than 5 % or even for macroscopic strains close to 9 % in similar experimental conditions [START_REF] Soppa | Experimental and numerical characterisation of in-plane deformation in two-phase materials[END_REF].

The direct link between DIC and simulations via the common surface mesh allows the experimentally measured displacements to be prescribed with their time history as Dirichlet boundary conditions on the nodes of the edges of the ROI without any additional interpolation since both meshes are identical. The boundary conditions on the edges are not only prescribed on the sur-face nodes but uniformly in the thickness. In addition, zero displacements along the normal of the surface are prescribed on the back face. The simulations are performed up to a mean tensile strain of 4 %, and with the assumption of small strain levels. Initial experimental orientations obtained by EBSD are prescribed. In this study, averaged experimental orientations per grain are considered since no lattice curvature are observed inside the grains on the EBSD inverse pole figures for any of the studied samples. Let us stress however that the simulation allows the lattice orientation to be updated.

Schmid's factors

All theoretical descriptions of polycrystal plasticity require the resolved shear stress to be assessed on each slip system. These microstructural factors are of importance to the study of damage accumulation in polycrystals under fatigue conditions. For instance, persistent slip bands, which induce fatigue crack initiation, are shown to appear where the highest Schmid's factors are reached [START_REF] Villechaise | On slip band features and crack initiation in fatigued 316L austenitic stainless steel: Part 1: Analysis by electron back-scattered diffrac-tion and atomic force microscopy[END_REF][START_REF] Miao | Crystallographic fatigue crack initiation in nickelbased superalloy René 88DT at elevated temperature[END_REF]. Since the Schmid's factors are linked with mechanical heterogeneities under cyclic loading conditions, the relevance of correlating such microstructural factors with strain heterogeneities is investigated. For a given microstructure, the crystallographic orientations give precious information about slip system incompatibilities and thus the potential strain pattern for a considered loading condition. It is proposed to assess the slip activity from the Schmid's factors. For that purpose, the resolved shear stress on each slip system s from a material standpoint is given by

τ s = σ : µ s ( 7
)
where the Schmid's tensor µ s of each slip system s is expressed with the local crystal orientation as

µ s = 1 2 (m * s ⊗ n * s + n * s ⊗ m * s ) (8) with m * s = Rm s , n * s = Rn s ( 9
)
where R is the rotation matrix built with the local crystal orientation. The simplest approach assumes a uniform stress σ 0 in the loading direction e L σ = σ 0 e L ⊗ e L [START_REF] Lebensohn | An elasto-viscoplastic formulation based on fast Fourier transforms for the prediction of micromechanical fields in polycrystalline materials[END_REF] which allows the resolved shear stress to be expressed as a function of the Schmid's factors f s

τ s = σ 0 f s (11)
with

f s = (e L ⊗ e L ) : µ s = (m * s • e L )(n * s • e L ) (12)
However, this result relies on the strong assumption that the crystal orientation has no influence on local stresses, namely, it ignores orientation incompatibilities that lead to local heterogenous stress fields [START_REF] Roters | Overview of constitutive laws, kinematics, homogenization and multiscale methods in crystal plasticity finite-element modeling: Theory, experiments, applications[END_REF][START_REF] Klusemann | Modeling and simulation of deformation behavior, orientation gradient development and heterogeneous hardening in thin sheets with coarse texture[END_REF]. Thus, a second approach is to assess the local stress tensor to estimate the 'local' Schmid's factors f l s . The local stress tensor σ at each time step of the finite element simulation is considered. To keep a similar expression of the resolved shear stress [START_REF] Kanjarla | Study of internal lattice strain distributions in stainless steel using a full-field elasto-viscoplastic formulation based on fast Fourier transforms[END_REF], Von Mises' equivalent stress at each material point σ eq is used

τ s = σ eq f l s (13) with f l s = 1 σ eq (σ • n * s ) • m * s ( 14
)
On the edges of the ROI, because experimental displacements are applied as boundary conditions, the simulated stress field is corrupted by measurement uncertainties, and f l s as well. For both approaches, the experimental orientations averaged over each grain have been considered.

In Figure 8, the 'macroscopic' Schmid's factors f s are compared with the 'local' counterparts, considering slip system s = 10 for illustration purposes. This comparison can be performed for different levels of loading, but the 'local' Schmid's factors appeared to notably change only in the transitional regime (i.e., for the macroscopic tensile strain ranging from 0 to 1 %). It can be noted that highlighted grains in which the slip system under consideration is mainly activated according to the 'local' factor are quite different from those mainly activated according to the 'macroscopic' factor and the difference is significant for each considered time step. The same analyses can be carried out for each slip system. This result is to be expected since the 'macroscopic' factors do not consider local stress heterogeneities due to orientation incompatibilities nor stress changes over time. The field of the system index s I of the highest 'local' Schmid's factor is shown in Figure 9(b) for a mean tensile strain of 1 %. It can be directly compared to that obtained using the 'macroscopic' Schmid's factors (Figure 9(a)) or even to that obtained from the main shear strain predicted by the finite element simulation (Figure 9(c)). If similarities can be observed with the latter the 'macroscopic' Schmid's factors approach offers the poorest information without intragranular variation and the field obtained from the 'local' Schmid's factors appears much more scattered. The latter can be explained by the fact that the full stress tensor is used to compute the Schmid's factors, each of its component may fluctuate locally, and because it corresponds to an instantaneous assessment while the shear strain approach considers the loading history. This poor agreement shows that the influence of neighboring grains on local slip activity, which is not incorporated in the approach using global Schmid's factors, should be taken into account for reaching a presumably more accurate description.

Plastic glide along slip systems

It is now proposed to look at plastic glide along slip systems from the DIC measurements in a similar way as in a previous study on BCC materials [START_REF] Latourte | Full field measurements used for assessing industrial issues -two examples[END_REF]. The viscoplastic strain rate is assumed to be only due to plastic glide along slip systems, which is written as

˙ p = 12 s=1 γs µ s (15)
where γs denotes the shear strain rate on slip system s. From this relationship, knowing the in-plane displacement gradients from displacement measurements, an experimental assessment of plastic glide is sought. Therefore, some hypotheses need to be made. First, the experimental total strain rate ˙ is considered to be only composed of a viscoplastic part ˙ p since the elastic part appeared negligible (or comparable to the DIC resolution in the experiments, see Table 3), so that

˙ = ˙ e + ˙ p ≈ ˙ p (16)
Since small perturbations are considered (in particular, small rotations), Equation ( 15) is integrated over time

p = 12 s=1 γ s µ s (17)
Further, DIC measurements yield only 2D (i.e., in-plane) displacement gradients, while 3D components are required in Equation [START_REF] Ohashi | X-ray microdiffraction and strain gradient crystal plasticity studies of geometrically necessary dislocations near a Ni bicrystal grain boundary[END_REF]. The assumptions of material incompressibility and that the normal axis to the image plane z is an eigen direction are made, leading to only one component of the strain tensor along z expressed as

zz ≈ -xx -yy (18)
The system expressed in Equation ( 17) cannot be inverted because it involves twelve unknowns γ s for only nine equations. However, only few slip systems over the twelve candidates are generally activated [START_REF] Abuzaid | Slip transfer and plastic strain accumulation across grain boundaries in Hastelloy X[END_REF]. To illustrate this point, Figure 10(a) shows the twelve shear strains at one material point of the simulation during the entire loading history. In this example, only four slip systems have been locally activated.

Let us consider the four main activated slip systems, which are denoted as s I , s II , s III and s IV , and are identified by simulation at each material point and at each time step (i.e., for each SEM image acquisition), such that

{s I , s II , s III , s IV } ∈ [1, 12] | |γ s I | ≥ |γ s II | ≥ |γ s III | ≥ |γ s IV | ≥ |γ s j {I,II,III,IV} | (19)
and the function η(γ s ), which is representative of each slip system activity relative to the total plastic activity, is defined as

η(γ s ) = |γ s | 12 i=1 |γ i | (20)
The distribution of η(γ s ) calculated for the four main shear strains over all the Gauss points for a mean tensile strain of 4 % is shown in Figure 10(b) as cumulative distribution functions. These results show that the four main slip systems account, on average, for 95 % of the plastic activity. It can be noted that the slip system activity fluctuates at the beginning of the tensile test (i.e., for a macroscopic tensile strain less than 2 %) before stabilizing. The fourth main shear strain appears not to exceed on average 10 % of the total slip activity.

As a consequence, it is assumed that the slip activity is restricted to only the four main activated systems, which allows the system expressed in Equation ( 17) to be inverted. The change over time of the field of the first main system index s I identified by simulations is shown in

where p s is the cumulative shear strain magnitude of slip system s, and p n s its projection considering the sample surface normal

p s = |γ s |dt (22)
with

p n s = p s e z • m s (23)
The intersection of the first slip system s I of normal n s I with the external surface, which is assumed to be planar over time of normal e z , is the vector t s I expressed as

t s I = n s I × e z (24)
where the operator × is the vector product. Vector t s II is obtained in a similar manner. In Fig-

ure 12(a), these two vectors t s I and t s II are drawn as segments of length proportional to the value of p n s I , respectively p n s II , for a mean tensile strain of 4 %. One SE image of the ROI has only been acquired at the end of test, for a mean tensile strain of 22 %. By extrapolating the results shown in Figure 10(c), the slip activity is assumed not to change notably when the mean tensile strain varies from 4 % to 22 %. The image g(x) has been corrected by the displacement measured by DIC, leading to g(x + u(x)), which is shown in Figure 12(b) for a Lagrangian comparison with the previous numerical approach (i.e., the orientation of the slip traces are represented in the undeformed coordinate system). Some enlarged areas are detailed in Figures 12(c) to 12(j), which focus on experimentally observed single or double slip, close to or far from grain boundaries.

No clear correlation can be made between numerical and experimental observations (e.g., location or the nature of slip). For example, single slip observed in Figure 12(d) is correctly predicted by the simulations (Figure 12(c)), whereas for the similar situation of Figure 12(j), the double slip suggested by the simulation is not observed experimentally. Thus, even if the primary slip system is well predicted (Figure 11) the simulation differs in some places to the observed slip traces. Similar results were obtained by monitoring the slip traces using atomic force microscopy [START_REF] Yang | Quantitative AFM characterization and crystal plasticity finite element modeling of heterogeneous deformation in commercial purity titanium[END_REF] with a relatively similar modeling approach. Yet the comparisons were not presented for individual slip systems but only for the different slip families of a hexagonal titanium and for a small number of grains. The differences between the simulation and the experiment may not only be due to the lack of knowledge of the underlying microstructure in the bulk but also to a constitutive model error. The fact that the SE image corresponds to a much more advanced state of loading than the simulation can explain the differences too even if the slip system activity seems not to fluctuate beyond a macroscopic tensile strain of 2 % (Figure 10(c)).

Conclusion and perspectives

In this paper, the effect of grain size on the plastic behavior of 316LN austenitic stainless steel is studied via in-situ mechanical test within an SEM. The displacement fields are measured via digital image correlation based on a finite-element mesh that is itself tailored to the specimen microstructure as observed via EBSD. The mean grain size has a marked effect on the macroscopic yield stress and hardening. In addition, the strain fields are markedly heterogeneous with numerous shear bands either aligned with or interrupted by grain boundaries.

To relate the plastic strain to the grain crystallographic orientation, the Schmid's factors are locally computed based on two different prescriptions for the local stress. Although a local evaluation of the stress provides a better correlation between the Schmid's factors and plastic strains, the agreement remains rather poor. Thus when polycrystalline structural effects are not accounted for, a local evaluation of the stress is not sufficient to predict the local deformation. For instance, plastic strain is also triggered by strain accommodation conditions and slip transmission conditions at grain boundaries. In the context of simplified, fast and efficient calculation techniques, the results reported in this paper also suggest that mean-field homogenization models based on individual grains (e.g., self-consistent schemes) may overlook an important contribution to the macroscopic manifestation of plasticity and should be considered carefully. Most of the recent improvements in homogenization procedures have been made possible by the use of secondorder moment estimates either in composite materials [START_REF] Lahellec | Effective response and field statistics in elasto-plastic and elasto-viscoplastic composites under radial and non-radial loadings[END_REF] or in polycrystals [START_REF] Liu | Second-order theory for the effective behavior and field fluctuations in viscoplastic polycrystals[END_REF], which are providing a more accurate effective response for complex loading paths without accounting for grain-to-grain interaction. Another way of improving the homogenization models is to consider intergranular kinematic incompatibilities [START_REF] Tjahjanto | A novel grain cluster-based homogenization scheme[END_REF] that may reveal mandatory to obtain satisfactory correlations with experiments at the grain scale.

The correlation between plastic strain and grain orientation is studied by assessing plastic glide along slip systems. It is shown that a few slip systems are simultaneously activated in each grain. A comparison between the experimental strain measurements and the results of a numerical simulation based on crystal plasticity with the same (surface) grain morphology and measured boundary conditions shows an excellent agreement for the major slip system and amplitude in each grain. However, some specific regions close to grain boundaries display sometimes significant differences and the slip traces are not well predicted in some grains or in the vicinity of some grain boundaries or triple junctions. This may either question the constitutive law used in the crystal plasticity modeling or result from the three-dimensional bulk morphology [START_REF] Zeghadi | Ensemble averaging stress-strain fields in polycrystalline aggregates with a constrained surface microstructure -Part 1: Anisotropic elastic behaviour[END_REF][START_REF] Zeghadi | Ensemble averaging stress-strain fields in polycrystalline aggregates with a constrained surface microstructure -Part 2: crystal plasticity[END_REF][START_REF] Zhang | Effect of realistic 3D microstructure in crystal plasticity finite element analysis of polycrystalline Ti-5Al-2.5Sn[END_REF] that remains inaccessible by resorting to SEM imaging without post-mortem (and tedious) 3D-EBSD reconstructions based on serial sectioning [START_REF] Schwartz | Numerical approach of cyclic behaviour of 316LN stainless steel based on a polycrystal modelling including strain gradients[END_REF]. An alternative method to obtain the 3D microstructure is diffraction contrast tomography [START_REF] King | Observations of intergranular stress corrosion cracking in a grain-mapped polycrystal[END_REF] but it is currently limited by its resolution (minimum voxel dimension) by the maximum number of grains in the specimen and suffers in presence of lattice curvature among other issues. Although it is now possible to conduct studies with having access to the 3D microstructure, this is not yet possible for every material and several applications may consider only using a 2D information when possible and relevant. As previously observed [START_REF] Carroll | An experimental statistical analysis of stress projection factors in BCC tantalum[END_REF] the correlations obtained with a macroscopic Schmid's factor are not satisfactory at the surface of general polycrystals but significant improvements are observed in this work when a more local approach is adopted.

Last, in this study the description of grain boundaries is simply defined as geometrical surfaces associated with a misorientation between grains and with no strain gradient dependence in the plasticity law accounting for GNDs as explored in several works [START_REF] Cordero | Grain size effects on plastic strain and dislocation density tensor fields in metal polycrystals[END_REF][START_REF] Taupin | Size effects on the hardening of channel-type microstructures: A field dislocation mechanics-based approach[END_REF][START_REF] Mayama | Geometrically necessary dislocation structure organization in FCC bicrystal subjected to cyclic plasticity[END_REF]. A mechanical description in polycrystals using strain gradient plasticity together with specific constitutive laws for grain boundaries may be investigated in future studies.

70 Figure 1 :

 701 Figure 1: Grain size distributions in the ROI of the studied specimens

Figure 2 :Figure 3 :

 23 Figure 2: Twin boundaries (a) and gray scale IQ map (b) of the region of interest obtained by EBSD of specimen E 1000+ . SEM imaging of the same region of interest with SE detector (c). The scale bar is 100 µm long

Figure 4 :

 4 Figure 4: Unstructured mesh compatible with the underlying microstructure overlaid on the BSE image of the region of interest. The scale bar is 200 µm long

5 .Figure 5 :Figure 6 :

 556 Figure 5: Strain field along the horizontal (tensile) direction calculated from DIC measurements for a mean tensile strain up to 5 %, for the microstructures denoted A 70 (a), B 10 (b), C 10 (c), D 50 (d) and E 1000+ (e). The axes are expressed in micrometers. The microstructure boundaries are shown as black lines on these fields. An animated figure is displayed in the on-line version only

Figure 7 :

 7 Figure 7: Identification of the Hall-Petch (H.P.) law from the experimental yield stresses obtained for the different tensile tests

Figure 8 :

 8 Figure 8: Schmid's factor for slip system s = 10 computed by assuming a uniform tensile stress along the horizontal direction (a), and from the local simulated stress fields for a mean tensile strain up to 1 % (b). The spatial axes are expressed in micrometers. The microstructure boundaries are shown as black lines on these fields. An animated figure (b) is displayed in the on-line version only

Figure 9 :

 9 Figure 9: Field of index s I of the highest 'macroscopic' Schmid's factor (a). Fields of index s I of the highest 'local' Schmid's factor (b), and of the highest shear strain (c) for a mean tensile strain of 1 %. The spatial axes are expressed in micrometers. The microstructure boundaries are shown as black lines on these fields

Figure 10 (

 10 Figure 10(c) shows the change over time of the mean value and corresponding standard deviation of these distributions. It can be noted that the slip system activity fluctuates at the beginning of

Figure 11 (Figure 10 :Figure 11 :

 111011 Figure 11(a). It is interesting to note that two or three different systems are generally active at the same time for each grain. At each material point, the corresponding four main shear strains are now determined from the experimentally estimated strain tensor. The first 'experimental' main shear strain field γ e s I is compared to the simulated one γ c s I in Figure11. Both simulation and experiment lead to very similar patterns of strain localization over time. The small difference lies in a higher scatter of γ s I in the experimental case, which can be explained by the fact that only the first four activated systems are combined to build the experimental strain tensor or by the Schmid's tensors µ s being unchanged over time for the experimental approach. This almost perfect agreement tends to validate the simulation of the experiments with the chosen crystal plasticity law. In the following, results showing a more debatable agreement between experimental observation and computations are presented.

Figure 12 :

 12 Figure 12: Intersection of the first (in red) and second (in blue) slip systems with the image plane z = 0 drawn for a mean tensile strain of 4 % (a). The length of the segments is proportional to the corresponding cumulative shear strain. The spatial axes are expressed in micrometers. The microstructure boundaries are shown as black lines on these fields. SE image of the region of interest acquired at the end of the tensile test and corrected with the measured displacement field (b). Four enlarged views of both figures are shown (c-j)

Table 1 :

 1 Chemical composition of the studied 316LN stainless steel plate in weight %

	Fe	Cr	Ni	Mo	Mn	Si	N	Co
	bal.	17.5	12.5	2.47	1.7	0.37	0.072	0.04
	C	P	Cu	Ti	Nb	Ta	B	S
	0.027	0.024	0.01	0.005	0.005	0.003	0.0005 0.0002

In this study, five different specimens, denoted A 70 , B 10 , C 10 , D 50 and E 1000+ , machined from this 316LN plate are considered. They have different average grain sizes (indicated by their designation subscript) determined by EBSD in a chosen Region Of Interest (ROI) detailed in the next section, due to different treatments:

Table 2

 2 summarizes the measurement conditions. For samples A 70 and B 10 the same ROI size is selected, even though the grain size is different. Conversely, samples B 10 and C 10 have the same

	grain size but different ROI sizes are chosen. Last, samples D 50 and E 1000+ are analyzed with
	the same ROI size. It is worth noting that the ROI covers only few twins in one single grain for
	sample E 1000+ .

Table 2 :

 2 DIC measurements conditions for the five different specimens

	Specimen Pixel size ROI size Mean grain size Mean grain size Number of grains
		(nm)	(µm 2 )	(µm)	(pixel)	in the ROI
	A 70	150	200 × 200	70	470	14
	B 10	160	200 × 200	10	60	551
	C 10	80	100 × 100	10	120	117
	D 50	280	400 × 400	50	180	59
	E 1000+	270	400 × 400	>1000	>3700	1
	samples A 70 and D 50 are more irregular, which may be due to the lower number of grains in their
	ROIs.

Table 3 :

 3 DIC standard measurement uncertainties for the five studied microstructures with the chosen correlation

parameters Specimen Displacement uncertainty Strain uncertainty A 70 4.3 nm / 2.9 × 10 -2 pixel 1.1 × 10 -3 B 10 5.9 nm / 3.7 × 10 -2 pixel 1.5 × 10 -3 C 10 2.2 nm / 2.8 × 10 -2 pixel 1.2 × 10 -3 D 50 7.1 nm / 2.5 × 10 -2 pixel 1.1 × 10 -3 E 1000+ 4.4 nm / 1.8 × 10 -2 pixel 9.6 × 10 -4

Table 4 :

 4 Adopted notation for FCC slip systems

	Normal vector n s		(111)			(1 11)			( 111)			( 11 1)
	Slip direction m s [ 101] [0 11] [ 110] [ 101] [011] [110] [0 11] [110] [101] [ 110] [101] [011]
	System index s	1	2	3	4	5	6	7	8	9	10	11	12

Table A .

 A 5: Interaction matrix between slip systems. The nomenclature of slip systems is given in Table41 2 3 4 5 6 7 8 9 10 11 121 h 1 h 2 h 2 h 4 h 5 h 5 h 5 h 6 h 3 h 5 h 3 h 6 2 h 1 h 2 h 5 h 3 h 6 h 4 h 5 h 5 h 5 h 6 h 3 3 h 1 h 5 h 6 h 3 h 5 h 3 h 6 h 4 h 5 h 5 4 h 1 h 2 h 2 h 6 h 5 h 3 h 6 h 3 h 5 5 h 1 h 2 h 3 h 5 h 6 h 5 h 5 h 4 6 h 1 h 5 h 4 h 5 h 3 h 6 h 5 7 h 1 h 2 h 2 h 6 h 5 h 3 8 h 1 h 2 h 3 h 5 h 6 9 h 1 h 5 h 4 h 5

	10	h 1 h 2 h 2
	11	h 1 h 2
	12	h 1

Table A .

 A 6: Values of the parameters of the crystal plasticity law [77] used in the study -n 52 MPa 220 MPa 1.0 40 GPa 1.5 GPa 1.0 1.0 0.6 12.3 1.6 1.8

	n	k	R 0	q	b	c	d	h 1 h 2 h 3	h 4	h 5 h 6
	11 12 MPa.s

Acknowledgements

The authors acknowledge the financial support of EDF within R&D LOCO and PERFORM60 (www.perform60.net) projects. Michel Mahé, Nicolas Brynaert and the Electronic Microscopy Laboratory of Les Renardières are thanked for their help in preparing and conducting the SEM acquisitions. Franc ¸ois Curtit, Ghiath Monnet, Nicolas Rupin and Charles Toulemonde are thanked for fruitful discussions.

AppendixA. Constitutive equations of the chosen crystal plasticity law and parameters

The crystal plasticity model chosen in this study has been proposed by Méric et al. [START_REF] Méric | Single crystal modeling for structural calculations: Part 1 -Model Presentation[END_REF]. Its relationships are expressed for each of the 12 octahedral slip systems s as

where c, k, n, R 0 , q, b, d are constitutive parameters, and h sr the coefficients of the interaction matrix between slip systems detailed in Table A.


Margot Billot
email: margot.billot@femto-st.fr

Xin Xu
email: xin.xu@femto-st.fr

Joël Agnus
email: joel.agnus@ens2m.fr

Emmanuel Piat
email: emmanuel.piat@ens2m.fr

Philippe Stempflé
email: philippe.stempfle@ens2m.fr

Multi-axis MEMS

Keywords: MEMS force sensor, Nanotribology, Multi-axis force sensing, Dexterous micro-manipulation, Friction, Piezoresistive sensor, Finite Elements Method simulation

come

Introduction

In assembly of micro-components below 1 mm, two approaches are currently considered [START_REF] Lambert | Capillary forces in microassembly: modeling, simulation, experiments, and case study[END_REF]: (i) the self-assembly paradigm [START_REF] Goser | Nanoelectronics and nanosystems: from transistors to molecular and quantum devices[END_REF][START_REF] Gnecco | Fundamentals of friction and wear[END_REF][START_REF] Pradeep | NANO: The Essentials: Understanding Nanoscience and Nanotechnology[END_REF] in which surface effects are used to organize and assemble structures mainly up to a few micrometers, and (ii) the micro-robotic assembly [START_REF] Lambert | Capillary forces in microassembly: modeling, simulation, experiments, and case study[END_REF][START_REF] Hsu | MEMS & Microsystems: Design, Manufacture, and Nanoscale Engineering[END_REF][START_REF] Agnus | Robotic microassembly and micromanipulation at FEMTO-ST[END_REF][START_REF] Hériban | Robotic micro-assembly of microparts using a piezogripper[END_REF] that is well suitable when the main challenging issues concern the handling and assembly of small components -as met for instance in some specific hybrid MEMS [START_REF] Bargiel | Towards micro-assembly of hybrid MOEMS components on a reconfigurable silicon free-space micro-optical bench[END_REF][START_REF] Rabenorosoa | A micro-assembly station used for 3D reconfigurable hybrid moems assembly[END_REF][START_REF] Achanta | Nanotribology of mems/nems[END_REF]. The latter approach is generally based on high resolution micro-manipulators and gripping devices with end-effectors controlled in position and more recently in force. Hence, the development of micro-grippers that can accurately perform both translation and rotation positioning of the manipulated object appears as a real challenge for the industrial field of micro-assembly. Owing to the difficulties of designing micro-grippers that perform accurate angular positioning, current industrial solutions generally favour to rotate the whole robotic system -including the gripper -instead of the manipulated object, as shown in Fig. 1. Indeed, transition from the position (1.a) to the position (1.c) involves two rotations of the grippers (transitions a-b and b-c) displaying a poor accuracy of positioning. However, another recent approach consists in performing local dexterous micro-manipulation of the object [START_REF] Krishnan | A multi-fingered micromechanism for coordinated micro/nano manipulation[END_REF][START_REF] Zhou | Automatic dextrous microhandling based on a 6-DOF microgripper[END_REF] in order to improve the positioning ability. As suggested in Fig. 2, the main advantage of this approach is that each rotation of the object can be locally induced by accurate translations (transitions a-b and c-d) of the fingers only. As a result, the micro-manipulation performance can be improved because translation devices display finer positioning and more compact system with less stages for at least the same number of Degrees of Freedom (DOF). However as a drawback, dexterous manipulation requires an accurate control of the forces that are applied to the object, which are themselves intimately connected to the contact that is involved between the object and the fingers. Indeed, as shown in Fig. 3, translation movements of fingers can induce, at the interface between the object and either the finger's edge or finger's apex, various frictional components: ie, sliding (3a), rolling (3b) or spin motion (3c). Besides, each kind of contact is likely to change, in turn, the applied contact forces during the process. It appears that these tribological components are essential to move the object by dexterous manipulation. Hence, the contact between fingers and object has to be known and controlled in real time, meaning that their frictional and adhesion behaviours have to be mastered for each kind of movements inducing sliding, rolling and spin motion components simultaneously. Nanotribological approach was recently used for assessing frictional components during quasi-static and dynamic micro-gripping process [START_REF] Stempflé | Accurate control of friction with nanosculptured thin coatings: Application to gripping in microscale assembly[END_REF]. Indeed, a ball-on-disc nanotribometer was used for simulating a micro-gripper that grabs a spherical object. This approach was well suitable for studying the sliding component but is clearly inadequate for assessing several frictional components simultaneously. Among the other existing tribometers [START_REF] Braun | Nanotribology: Microscopic mechanisms of friction[END_REF][START_REF] Szlufarska | Recent advances in single-asperity nanotribology[END_REF][START_REF] Domatti | Multiasperity nanotribology of self-assembled monolayers grafted on silicon wafers displaying various crystallographic orientations and nanostructures[END_REF][START_REF] Achanta | Investigation of friction on hard homogeneous coatings during reciprocating tests at micro-newton normal forces[END_REF], the Tribolever proposed by Zijlstra [START_REF] Zijlstra | Fabrication of a novel scanning probe device for quantitative nanotribology[END_REF] is well designed to characterize sliding components.

It is composed of a central detection pyramid surrounded by four high aspect ratio legs.

Lateral or frictional forces acting on the scanning tip are measured via the displacement of the pyramid by means of four laser interferometers [START_REF] Zijlstra | Fabrication of a novel scanning probe device for quantitative nanotribology[END_REF]. This structure is able to measure the 3 translation components with theoretically no crosstalk. But unfortunately, it was designed in order to limit torsional movements, so it cannot be suitable for measuring the rolling and spin motion component of friction. Hence, a multi-axis device with low crosstalks is then needed for this purpose. The aim of this paper is to design a new type of nanotribometer enabling to simulate the finger/object contact. It has to display a flexible structure which allows torsion moves in order to access to all the components of friction simultaneously. The combination of this compliant structure with a piezoresistive sensing technology [START_REF] Jin | Development and calibration of a sub-millimeter threecomponent force sensor[END_REF][START_REF] Estevez | 6 DOF force and torque sensor for micro-manipulation applications[END_REF] allows to obtain a bulk multi-axis nanotribometer characterized by high displacement, high sensitivities and acceptable crosstalks. Section 2 presents the specifications and working principle of this new sensor. Its design and behaviour are then studied by simulation in Section 3 and 4, respectively. Micro-fabrication process is presented in Section 5. Calibration and validation of a real prototype will be developed in a further paper.

2 Specifications and working principle

Sensor performance requirements

As mentioned above, owing to their working principle, current commercial nanotribometers can only achieve a 2DOF force measurement leading sometimes to a non-negligible crosstalk which is not known and controlled.

Our new force sensor, that will be designed as a nanotribometer, should be able to simultaneously measure three-dimensional forces, torque and moments. The ability of torque and moments measurement will enable to access and decouple all friction components including rolling and spin motion. Each force measurement should achieve a linear and decoupled behaviour: hence, crosstalk has to be known and as low as possible. Moreover, the sensor should be able to host classical balls from different diameters met in tribological purpose. The normal (resp. tangential) force should have a maximum of 1 mN (100 µN) with a resolution of 1 µN (0.1 µN) as currently applied in microgripping assembly.

Working principle

Keeping these performance requirements in mind, this sensor is composed of a central platform surrounded by a compliant table: four central beams and sixteen other beams form a table which enables to obtain higher displacements. The compliant table presents advantages compared with classical cross-beam sensors [START_REF] Viet Dao | Silicon Piezoresistive Six-Degree of Freedom Force-Moment Micro Sensor[END_REF]: it leads us to obtain higher sensitivities reducing the number of gauges by increasing deformation and thus displacement ability. The ball that will rub against the studied surface is fixed to the central platform.

Figure 4 shows the multi-beam structure and the position of the eight strain gauges (named in the following G1,...,G8). The four sets of gauges are symmetrically distributed at the root of the four central beams.

To explain the sensing principle, a first focus on an isolated beam is done in Fig. 5 in order to illustrate the piezoresistivity-based operating mechanism. The piezoresistance effect is a phenomenon in which the electrical resistivity of a material changes due to an applied stress. In this paper, applied solicitations (forces and torque) are denoted Fx, Fy, Fz and Cz, moments induced by forces are denoted Mx and My, and measured components are denoted F x and F y (tangential forces corresponding to sliding), F z (normal force), M x and M y (moments corresponding to rolling) and Cz (torque corresponding to spin motion).

As illustrated in Fig. 5, I 1 and I 2 are the current in the gauges 1 and 2, respectively, whereas R 1 and R 2 are the resistance of the strain gauges 1 and 2, respectively. So, the variations in the supply current (resp. resistance) of the strain-gauge 1 which is subjected to the compressive stress can be given by these equations:

I 1 = I 0 + ∆I (1)
R 1 = R 0 -∆R (2)
where, I 0 (R 0 ) is the initial current (resistance) value without any strain and ∆I (∆R) is the current (resistance) variation due to stress influence. The sign of ∆I and ∆R depends on the piezoresistor dopant type.

On the other hand, for the strain-gauge 2 under the tensile stress, current (resistance) variation is described by these equations:

I 2 = I 0 -∆I (3)
R 2 = R 0 + ∆R (4)
Hence, a tangential force induces different types of stresses on each side of the beam (see Fig. 5 a). Correspondingly, the two gauges have the opposite resistance variation. The difference of the resistance variation between the two strain-gauges can be then used to optimize the measurement sensitivity, as described by this relationship:

R 1 -R 2 = R 0 + ∆R -(R 0 -∆R) = 2∆R (5)
In a similar way, when a normal force is applied to the cantilever, both gauges are influenced by the same compressive or tensile stress (see Fig. 5 b). Thus, the same resistance variation in these two gauges should be found (if each individual gauge has the same electrical behaviour). The following relationship is used to magnify the resistance variation:

R 1 + R 2 = R 0 + ∆R + (R 0 + ∆R) = 2∆R + 2R 0 (6)
By this way, the sum (resp. difference) of two gauges signals can be used to measure the normal (resp. lateral) forces.

Reasoning as so for the complete sensor, results obtained with each pair of gauges on the central beams are just summed, as summarized in Fig. 6 and Tab. 1. For example, the tangential forces F y, F z and moment M x are evaluated using these formulas:

F y ∝ ∆R G1 -∆R G2 -∆R G5 + ∆R G6 = 4∆R F y ( 7
)
Table 1 Resistance variation in the gauges (smallest symbols represent a lower variation than others) for independent solicitations and their various combinations leading to possible crosstalks.

F z ∝ 8 k=0 ∆R Gk = 8∆R F z (8) M x ∝ -∆R G3 -∆R G4 + ∆R G7 + ∆R G8 = 4∆R M x (9)
Moreover, smaller variations of resistances of gauges (denoted δ instead of ∆) appear in some cases. For example, for a force F y and a moment M x, these variations are:

4δR F y = -∆R G3 -∆R G4 + ∆R G7 + ∆R G8 ( 10
)
4δR M x = -∆R G1 + ∆R G2 + ∆R G5 -∆R G6 (11)
As shown in Tab. 1, which summarize the responses of the gauges submitted to independent solicitations and their various combinations leading to possible crosstalks, the compliant structure and the position of the eight piezoresistive gauges allow to completely decouple some components ( F z, Cz). However, a crosstalk between some other components for example F y respect to M x (which is the error on F y measurement when there is a moment M x, denoted in the following F y/ M x) is a non-zero value because the same gauges are deformed in the same way for the two solicitations (see for example relationships 7 and 11). Indeed, for example, if the sensor is submitted to both tangential force Fy and moment Mx, F y is evaluated by:

F y ∝ 4∆R F y -4δR M x = 4∆R F y (12)
Nevertheless, simulation that is described in Section 4 will lead to study and find the best sensor geometry in order to limit these residual crosstalks (ie the ratio δR M x ∆R F y for F y/ M x).

Sensor design

Knowing the sensor working principle, the optimization of the design has to be performed. The latter includes the choice of:

(i) the best crystallographic orientation of material constituting the compliant structure, (ii) the type, size, orientation and position of the piezoresistive gauges.

Besides, a study of residual crosstalks using simulation will be carried out in order to find the best dimensions for the whole sensor to obtain optimal characteristics.

Optimization of the mechanical structure orientation of the compliant structure

The mechanical part is manufactured using Silicon micromachining fabrication technologies, to obtain a reasonably low stiffness structure. As Silicon is a highly anisotropic material, mechanical properties of the structure will depend on the crystallographic orientation of the chosen material. One criterion for the orientation choice is to have a low Young's modulus to improve the sensitivity of the sensor by reducing its stiffness [START_REF] Jordan | Micromechanical force sensors based on SU-8 resist[END_REF], as shown in Fig. 7 and 8 from [START_REF] Kim | Why is (111) silicon a better mechanical material for MEMS[END_REF]. Although the two Young's modulus stay in the same order of magnitude by switching from Si (100) to Si (111), Young's modulus and Poisson's ratio are clearly independent of device crystallographic orientation for Silicon (111). Young's modulus is transversely isotropic at 168.9 GPa, regardless of the crystallographic orientation and Poisson's ratio has a constant value of ν // = 0.262 for planes parallel to (111), and a constant value of ν ⊥ = 0.182 for planes vertical to (111) [START_REF] Kim | Why is (111) silicon a better mechanical material for MEMS[END_REF]. Therefore, a sensor fabricated using (111) Silicon will be less sensitive to the compliant structure device orientation with respect to crystallographic orientations. Furthermore, since less restriction is imposed on mask designs with respect to crystallographic orientation, the design task becomes also much easier [START_REF] Kim | Why is (111) silicon a better mechanical material for MEMS[END_REF].

Optimization of the piezoresistive gauges

Gauges position and size

The stress distribution along different axis on the surface of the beam has been studied in order to determine the best position for the strain gauges. COMSOL Multiphysics Software 4.1 has been used to compare the results along the lines A defined on Fig. 9 for different δ x and δ y offset values. The "Structural Mechanics" module (linear elastic model) is used for this purpose and a tetrahedral meshing is realized (three nodes elements). To limit the calculation time, fixed parts are not modeled: the ends of the beams of the compliant table are directly fixed with a COMSOL boundary condition as shown in Fig. 9. Meshing is refined around the gauges to limit error induced by the discretization. A total of 26,240 elements is used (see Fig. 9) and a powerful computing cluster is used (Mésocentre of calcul de Franche-Comté). Simulation results reveal that:

δ x :
The gauges have to be as far away from each other as possible (δ x =0 mm) to obtain the higher stress differences between the two gauges from the same beam.

δ y : Since the highest values of stress is observed at a distance of δ y =20 µm -as shown in Fig. 10 displaying stress repartition along the line A -the gauges should be placed a little far from the fixed support, to increase sensitivities. However, considering the difficulties induced by this shift in the current microfabrication process (electrical connections and ohmic contact achievements would be harder to realize), the gauges should be rather located at the clamped position (δ y =0 mm). The loss of sensitivity value is evaluated around 6%.

Piezoresistors' size has now to be optimized by choosing the dimensions displayed in Fig. 11. The gauges must be as small as possible in order to maximize sensitivity and not perturb structure deformation. According to the limitations of DRIE etching, the following gauges characteristics have been chosen:

• the length of the gauge Lg=50 µm;

• the height of the gauge Tg=1.5 µm;

• the width of the gauge Wg=3 µm;

• the resistivity of the gauge ρ 0 =0.01 Ω.cm.

These dimensions induce a resistance of

R 0 = ρ 0 Lg S = ρ 0 Lg W gT g = 1.1 kΩ (13)
when no stress is applied.

Gauges type and orientation

The piezoresistors lying on the surface of the sensing beams are very thin in comparison with the thickness of the beam. Therefore the piezoresistance effect of a piezoresistor can be expressed as :

∆R R = π // σ // + π ⊥ σ ⊥ ( 14
)
where ∆R R is the relative change of resistance, σ // and σ ⊥ (resp. π // and π ⊥ ) the two stress components (piezoresistive coefficients) parallel and orthogonal to the direction of the resistor, respectively. The piezoresistance coefficients depend both on the dopant and on the crystallographic orientation choice. Besides, for a given orientation, they can change significantly from one direction to another. Hence, sensor's sensitivity is then clearly dependent on the crystallographic orientation. It is the main coercive criterion because crystallographic orientation is not determinant for DRIE manufacturing, in contrast to anisotropic humid etching [START_REF] Hsu | MEMS & Microsystems: Design, Manufacture, and Nanoscale Engineering[END_REF].

Besides, stress analysis on the sensing structure surface (Fig. 10 in Section 3.2.1) shows that the component σ // is much larger than other components of stress. For this reason, equation 14 can be simplify as :

∆R R ≈ π // σ // (15)
The longitudinal gauge factor of the piezoresistance (GF // ), which is the ratio between the relative variation of resistance and the gauge deformation, can then be expressed as :

GF // ≈ π // σ // // ⇒ GF // ≈ π // E // (16)
where // is the relative length change of the gauge and E // is the Young's modulus in the direction parallel to the piezoresistor. Therefore, one criterion to select the crystallographic orientation is to make π // E // as large as possible. Figure 12 compares the value of π // for different orientations in Si (for the four different Si types: (111) and (100) doped n or p) [START_REF] Kanda | A Graphical Representation of the Piezoresistance Coefficients in Silicon[END_REF]. And using results from Fig. 7,8 and 12, values of GF // ≈ π // E // can then be calculated and displayed in Fig. 13 Thus, for Si (111) the value of GF // is clearly independent of the orientation of the gauge in contrast to Si (100). It could be an advantage if the gauges positions were not parallel or perpendicular, or even to avoid misalignment errors. Since higher gauge factor is obtained for a Si (100) n-doped and because all the gauges are oriented with the same crystallographic orientation in the our case, a n-doped (100) wafer with gauges that are oriented with the < 100 > direction can be chosen to maximize the sensitivity. A gain of about 10% is expected. Note that, as GF // is negative for n-doped Si, ∆I and ∆R from Section 2.2 are negative too.

Simulation of the sensor behaviour

Simulations are now carried out in order to optimize the final design of the sensor. Expected performance and behaviour of the MEMS friction sensor are simulated using COMSOL multiphysics FEM simulation software. To take into account the anisotropy, the elasticity and piezoresistive matrices are used in the proper coordinate system. FEM model and meshing are the same as the ones described in Section 3.2.1. For piezoresistive effect, a co-simulation by using both the "Structural Mechanics" and "AC/DC" COMSOL modules is needed.

After meshing the geometrical structure and solving the differential equations of "Structural Mechanics" and "AC/DC" modules, the relative variation in resistance within a gauge is given by equation:

∆R R = ∆I I = (J) -I 0 I 0 ( 17
)
in which J is the current density norm inside the gauges determined in a plan orthogonal to the gauge direction (ie cross section of the gauge).

Study of the crosstalks

A Design of Experiments approach is used to determine the various crosstalks. Figure 14 shows the results for the structure deformation strengths, and the corresponding current variations in the gauges are related in Tab. 2. Values for gauges submitted to the same stress are averaged in order to erase small differences of current due to discretization errors. As expected, there is no crosstalk with normal force F z or torque Cz (see Fig. 14 a andb and the corresponding lines in Tab. 2). Note that the ability of independently measuring a torque Cz with no error is important for our application because it enables to accurately characterize spin motion around the finger's apex, as represented in Fig. 3 c. However, a non-zero crosstalk is observed for F y/ M x (and symmetrically for F x/ M y) and M x/ F y (and symmetrically for M y/ F x) (notations being defined in the paragraph below Equation 11). Indeed, as shown in Fig. 14 c andd, if an unknown solicitation is applied on the sensor it will be, for example, impossible to extract the respective contribution of tangential force F y and moment M x from current variations of piezoresistive gauges. Referring to relation 12 and using the notations from Fig. 14 c andd:

F y ∝ 4∆I 1 = 4(∆I 1 F y -∆I 1 M x ) = 4∆I 1 F y (18) M x ∝ 4∆I 2 = 4(∆I 2 M x + ∆I 2 F y ) = 4∆I 2 M x ( 19
)
Therefore an evaluation of the crosstalks levels is given by computing the ratios

∆I 1 M x ∆I 1 F y and ∆I 2 F y ∆I 2 M x
. Table 2 shows these residual errors: 7% for the evaluation of tangential force F y and more than 17% for the moment M x. Although the latter is more important than the former, this difference has no real consequence on friction assessment in dexterous manipulation, as reported in Fig. 3. Indeed, in such framework, controlling the transition between static and dynamic friction is mandatory using the applied normal force knowing an estimation of the tangential force at the contact area. Hence, force measurements in sliding mode must be accurate, even in presence of residual crosstalks induced by any rolling process generated by a geometric rotation of the central platform (ie, presence of moments). On the contrary, accurately measurement of a moment M x or M y induced by a rolling at the contact area appears less critical because, in dexterous framework, the crucial information is just to make sure to keep the contact during the movement. Nevertheless, a solution for reducing the crosstalks is to modify the sensor design by placing the ball on the upper side of the sensor and reducing the height of the central platform. By this way, the distance T s between the support plane and the flat sample strongly decreases, as shown in Fig. 15. The rotation components will be less important so that the error on the friction forces will be reduced. To perform that, a hole is designed in the central platform and the micro-ball is glued in it. The diameter of the hole can be adapted to host balls from different sizes without changing the distance between the sample and the central platform. With this design modification, the sensor will work upside-down with the gauges placed in front of the studied sample (see Fig. 15).

Optimization of the sensor design

In addition to the above crosstalk study, there are many geometrical characteristics described in Fig. 16 that also influence the sensor properties:

• the length L of the beams;

• the thickness T of the beams;

• the width W of the beams;

• the distance T s between the sample and the central platform;

• the length L p of the central platform;

• the thickness T p of the central platform.

Several criteria are selected to check their influence out: (i) the performance analysis of decoupling , (ii) the resonant frequency, (iii) the sensitivities, (iv) the response time and (v) the maximum forces that can be applied before the sensor rupture. All the simulations results are summarized in Tab. 3. The upper line of the table corresponds to the variation Table 3 Study of the influence of the geometrical parameters (+ (++,-) means that the variation of the parameter have a positive (very positive, negative) influence on the considered sensor property and 0 (0/-) means that the variation of the parameter have no influence (a negligible negative influence, respectively) on the sensor property, under bracket expressions are the eventual proportionality relationships between geometrical parameters and characteristics).

of the different geometrical parameters (augmentation or reduction ) and in the first two columns there are the characteristics and the corresponding criteria to be maximized or minimized, that is:

• sensitivities have to be as high as possible;

• the resonant frequency has to be high enough to avoid environmental perturbations due to vibrations during the measurement (higher than 10 kHz) [START_REF] Boudaoud | Effects of environmental noise on the accuracy of millimeter sized grippers in cantilever configuration and active stabilisation[END_REF][START_REF] Boudaoud | Noise characterization in millimeter sized micromanipulation systems[END_REF];

• a low stiffness is needed in order to obtain big enough displacements;

• crosstalk between forces and moments measurement needs to be negligible (maximum 1 or 2 %) to respect the sensor requirements;

• a short response time is needed to efficiently measure the various components during friction process;

• and the maximum force before rupture has to be as high as possible, in order to maximize the force range.

Another last criterion which is unmeasurable is the ability to easily fabricate and manipulate this small sensor without breaking it. Given Tab. 3, the best dimensions are selected making a trade-off between the good and the bad influence of parameters on the different characteristics, knowing that crosstalk, sensitivities and response time are the main important choice criteria. Consequently, the beam height T and the beam width W are reduced whereas the beam length L is increased as much as possible to obtain high sensitivities. But, as these variations have simultaneously bad effects on several other characteristics, their values are chosen in order to not increase too much the crosstalk M x/ F y and the response time. Moreover, both the distance between the sample and the central platform T s and the thickness of the central platform T p are reduced to obtain a higher resonant frequency and a shorter response time. The smallest ball size usually used to perform tribological tests in the lab is retained for T s (40 µm). Indeed, as mentioned above, reducing the distance from the sample has no bad influence on other important parameters (sensitivities, crosstalk, etc.). Hence, the same thickness for central platform and beams is selected to simplify the fabrication process. And a small enough value of L p is then chosen in order to both minimize the response time and reduce the sensor size.

Finally, the optimal geometrical parameters obtained from simulation results are reported in Fig.

Characteristics of the final sensor

With the optimized dimensions, the sensor displays the expected properties compiled in Tab. 4. Crosstalks values of the sensor are reported in Tab. 5. Characteristics have been improved with respect to the first design (see Tab. 2 from Section 4.1). Although the residual errors on moment M x have slightly increased -from 17.2% to 19.1% -without any real consequence on our application, as mentioned earlier (see Section 4.1), those have significantly decreased -from 7.2% to 0.71% -on tangential force F y. It means that measurement of a tangential force induced by a torque can be accurately decoupled with a great benefit. Of course, these properties could slightly change on the real sensor because of the uncertainties due to micro-manufacturing process. However, the best was made in order to have low crosstalks, good sensitivities and a resonant frequency high enough to avoid perturbations due to environmental noise (acoustic and seismic vibrations [START_REF] Boudaoud | Effects of environmental noise on the accuracy of millimeter sized grippers in cantilever configuration and active stabilisation[END_REF][START_REF] Boudaoud | Noise characterization in millimeter sized micromanipulation systems[END_REF]) during the measurement. Note that, the actual design of the sensor has been determined by simulations to respond to our experimental nanotribological requirements (micro newton force range measurement with no restriction on the structure size). However, this optimization process can be clearly used to design a nanosensor displaying other kinds of requirements. Indeed, as in our FEM model all the dimensions and material properties can be easily parametrized, our optimization process can be applied to identify the optimal dimensions and so for the purpose of scalability. The only limit of our FEM model is to stay in the field of the classical mechanics.

Fabrication process

Our sensor will be manufactured on a n type silicon SOI wafer. In order to obtain a compliant structure as perfect as possible a five layers SOI wafer is used. Tests are at the moment being performed to optimize ohmic contacts achievement for gauges realization [START_REF] Herth | Optimization of ohmic contact and adhesion on polysilicon in mems-nems wet etching process[END_REF]. Then once optimal metal thicknesses, time and temperature for the annealing proceeding will be determined, sensor micromachining will be launched. Fabrication process will typically consist of the four stages illustrated in Figure 18.

1. Piezo-resistors achievement:

-The piezoresistive gauges are defined using deep reactive ion etching of silicon.

Ohmic contact achievement:

-The thermal oxidation treatment of piezoresistive gauge is carried out as an electrical insulator layer.

-To open the contact windows, the reactive ion etching is used to etch silicon dioxide.

-A Chrome/Platinum layer is deposited and patterned by the evaporation deposition and lift-off methods.

-The annealing proceeding is needed to have the ohmic contact between Platinum and silicon layers.

Connection circuit achievements :

-The Platinum layer is patterned by wet etching method to achieve the electrical connection.

Sensor structure achievements:

-To create the beams, the back side of bulk is etched by deep ion reactive etching.

-To realize the sensor structure, the front side of bulk is etched by deep ion reactive etching.

Conclusion

A new multi-axis force sensor which is intended to measure frictional components involved in dexterous microgripping was developed. The design of the sensor based on a compliant table and all its geometrical properties were studied in order to optimize sensitivity and resolution and to minimize drawbacks. The first simulation results are promising: this sensor will be able to measure independently the normal and friction forces (crosstalk inferior to 1 %) with a good sensitivity. Tests are at the moment being performed in order to optimize ohmic contacts achievement for gauges realization and make a final choice between n and p-type Si. Then once optimal metal thicknesses, time and temperature for the annealing proceeding will be determined, sensor micromachining will be launched and then experimental results will be compared with simulation ones. A big challenge to overcome will be the development of a process to be able to efficiently calibrate the sensor.

Figure 1

 1 Figure 1 In classical microgripping: any rotation of the object (eg, transition a-c) needs a global rotation of the manipulator (rotations a-b and b-c).

Figure 2

 2 Figure 2 In dexterous manipulation, the same movement of the object (eg, transition a-d) only needs fingers translation (translations a-b and c-d).No further rotation movement is required. The micro-manipulation system can be simpler, more compact and accurate.

Figure 3

 3 Figure 3 Different contacts between finger and manipulated object that are likely to occur during dexterous manipulation : a) sliding of finger's apex and finger's edge ; b) rolling around the finger's apex and finger's edge ; c) spin motion around the finger's apex.

Figure 4

 4 Figure 4 Architecture of the multi-axis force sensor.

Figure 5

 5 Figure 5 Working principle of the piezoresistance based measurement method (a: lateral force, b: normal force).

Figure 6

 6 Figure 6 Sensor under Fx, Fz, Mx and Cz constraint

Figure 7

 7 Figure7Young's modulus E and Poisson's ratio ν for wafer (100)[START_REF] Kim | Why is (111) silicon a better mechanical material for MEMS[END_REF].

Figure 8

 8 Figure8Young's modulus E and Poisson's ratio ν for wafer (111)[START_REF] Kim | Why is (111) silicon a better mechanical material for MEMS[END_REF].

Figure 9

 9 Figure 9 FEM model, dimensions of the meshed sensor and Stress distribution study: δx and δy plot the distance from edge side and embedding, respectively.

Figure 10

 10 Figure 10 Stress distribution in the developed sensor (the component in solid line and higher than the others is σ // ).

Figure 11

 11 Figure 11 Notations of the dimensions of the gauges.

Figure 12

 12 Figure 12 Piezoresistive coefficients values for different wafer orientation (all values are in 10 -12 P a -1 ) from[START_REF] Kanda | A Graphical Representation of the Piezoresistance Coefficients in Silicon[END_REF].

Figure 13

 13 Figure 13 Longitudinal gauge factor values for different wafer orientation.

Figure 14

 14 Figure 14 Force and torque solicitations inducing a current variation in the gauges for different measurements: a normal force F z (a), a torque Cz (b), a tangential force F y (c) and a moment M x (d) (∆I are positive values).

Figure 15

 15 Figure 15 Modification of the sensor design in order to decrease crosstalks: the sensor works upside-down with the gauges at the bottom.

Figure 16

 16 Figure 16 Notations of the geometrical parameters of the sensor (with the ball on the upper side of the platform).

Figure 17

 17 Figure 17 Geometrical parameters of the sensor (all the dimensions are in µm).

17

 17

  .

Figure 18

 18 Figure 18 Flow chart: (1 a) SOI wafer with 5 layers (1 b) lithography and DRIE etching upper side of the device layer (2 a) thermic oxidation (2 b) lithography and RIE etching for opening the ohmic contact areas (2 c) Cr/Pt deposition, lithography, wet etching and diffusion heating to realize the ohmic contacts (3) Cr/Pt deposition, lithography and wet etching to obtain electrical connection electrodes (4 a) lithography and DRIE etching down side of the handle layer (4 b) lithography and DRIE etching upper side of the handle layer.

Table 2

 2 Current variation (nA) in the gauges for the six force components and corresponding non-zero crosstalks (values are given for Fx,y=100 µN and Fz=1000 µN).

Table 4

 4 Characteristics of the sensor (simulation results).

Table 5

 5 Current variation (nA) in the gauges for the six measured components and corresponding

non-zero crosstalks (values are given for Fx,y=100 µN and Fz=1000 µN).

Acknowledgements

This work has been supported by the Labex ACTION project (contract "ANR-11-LABX-01-01"). Calculations have been performed using the computing resources from the Mésocentre of Franche-Comté. This work has been supported by the French RENATECH network and its FEMTO-ST technological facility.