Y Gaillard

F Amiot

Grid nano-indentation as full-field measurements

Keywords: Carbon fibres, Elasticity, Mechanical testing, Grid nano-indentation, Full-field measurements, Composites, Graded materials, Interphase

The use of grid nano-indentation as a full-field measurement tool to investigate the properties of heterogeneous materials is described. Contrarily to the reported approaches which rely on statistical analysis, the mechanical properties fields are analyzed herein relying on a topological description of the sample. The spatial convolution function is approached using Hertz contact theory, and the obtained convolution kernel is used to retrieve the mechanical properties of the different phases. This approach is exemplified on results obtained at very low contact force (and thus very high spatial resolution) on a composite material sample made of micrometer-sized fibers in a polymeric matrix.

Introduction

It is today well known that predicting the global mechanical behavior of composite materials requires more than the mechanical properties of its constituents. Depending on the nature of the phases and on the process-ing of the composite, the constituents may be either intimately linked (at the crystallographic level) or they can be very poorly linked, for example just by adhesion. Additionally, differences between thermal expansions of the different materials may also lead to fracture at interfaces during the fabrication process, and residual stresses induced by crystallization of a given constituent can dramatically affect the overall properties or result in graded properties [START_REF] Cole | Interphase mechanical behavior of carbon fiber reinforced polymer exposed to cyclic loading[END_REF]. For this kind of materials, the classical mixture law based on the mechanical properties of the constituents is generally not sufficient to fully reproduce the particular macroscopic behavior of the composite itself. The addition of either supplementary phases, whose properties differ from the constituents of the composite, or zones characterized by a gradient of properties is necessary [START_REF] Armitage | Indentation across interfaces between stiff and compliant tissues[END_REF][START_REF] Chevalier | Multi-scale characterization and modelling of the transverse compression response of unidirectional carbon fiber reinforced epoxy[END_REF]. The experimental characterization of such interfacial phases thus becomes a critical issue [START_REF] Perrier | Effect of water ageing on nanoindentation response of single hemp yarn/epoxy composites[END_REF][START_REF] Enrique-Jimenez | Nanoindentation mapping of multiscale composites of graphene-reinforced polypropylene and carbon fibres[END_REF]. Two strategies can be envisaged to correctly describe the behavior of the composite :

• The first is based on the adjustment of the macroscopic experimental mechanical response, using a model that takes into account this supplementary phase in a representative volume element (RVE) [START_REF] Panamoottil | Towards a multiscale model for flax composites from behaviour of fibre and fibre/polymer interface[END_REF][START_REF] Zhong | Prediction of the mechanical behavior of flax polypropylene composites based on multi-scale finite element analysis[END_REF]. This strategy is essentially numerical but it requires the knowledge of the mechanical behavior of the phase.

• The second one is the characterization of interfacial mechanical properties, which requires very local characterization methods. These two ways are obviously complementary.

A wide range of local mechanical characterization is now available. Atomic force acoustic microscopy (AFAM) [START_REF] Rabe | Atomic force acoustic microscopy[END_REF] and in particular contact-resonance atomic force microscopy (CR-AFM) [START_REF] Hurley | Contact resonance force microscopy techniques for nanomechanical measurements[END_REF] are today rather popular. CR-AFM inherits a very good spatial resolution from the very sharp AFM tip, so that it is a very good choice for who is interested in characterizing interfacial mechanical properties [START_REF] Nair | Nanoscale characterization of natural fibers and their composites using contact-resonance force microscopy[END_REF][START_REF] Nair | Nanoscale characterization of interphase properties in maleated polypropylene-treated natural fiber-reinforced polymer composites[END_REF]. The quantitative analysis of the measured contact stiffness may however turn to be difficult because of the not so well defined contact geometry (because of the initial tip geometry and because wear usually makes this geometry evolve during the scan) [START_REF] Arnould | Towards a better understanding of wood cell wall characterisation with contact resonance atomic force microscopy[END_REF]. Alternatively, the very well defined (and calibrated) geometry of indenters makes the investigation of heterogeneous materials by instrumented indentation very promising [START_REF] Gibson | A review of recent research on nanoindentation of polymer composites and their constituents[END_REF]. The development of high-speed nanoindentation allows to quickly and accurately map the mechanical properties of composite materials with a high spatial resolution and to generate hardness or indentation elastic modulus fields quite easily. Large amount of experimental data are then collected and are to be processed. Classically, the measured properties are analyzed statistically, assuming Gaussian distributions. Each mode in the experimental distribution is thus assumed to correspond to a particular phase [START_REF] Constantinides | Grid indentation amalysis of composite microstructure and mechanics: principles and validation[END_REF][START_REF] Ulm | Statistical Indentation Techniques for Hydrated Nanocomposites: Concrete, Bone, and Shale[END_REF].

Hardness, elastic modulus, volume fraction of each phase are extracted. The spatial information is then used afterwards to correlate the localization of the different identified phases with their corresponding mechanical properties [START_REF] Tromas | Hardness cartography to increase the nanoindentation resolution in heterogeneous materials: Application to a Ni-based single-crystal superalloy[END_REF].

The above-cited works actually assume that scale separation apply so that each phase is always probed individually, thus neglecting the interaction between the phases. Homogeneization has thus naturally been introduced in order to analyze nano-indentation data [START_REF] Randall | Nanoindentation analysis as two-dimensional tool for mapping the mechanical properties of complex surfaces[END_REF][START_REF] Argatov | A two-phase self-consistent model for the grid indentation testing of composite materials[END_REF]. These two families of approaches thus rely on two extreme assumptions : either one considers individual phases are probed or one assumes the sole response of the homogeneized composite material is recorded. Probing graded materials, and especially small interfacial regions thus clearly challenges these approaches and requires to relax these assumptions. This is somehow similar to what has been done to analyze nano-indentation results on thin-film materials [START_REF] Bec | A simple guide to determine elastic properties of films on substrate from nanoindentation experiments[END_REF][START_REF] Jäger | Comment on: Effects of the substrate on the determination of thin films mechanical properties by nanoindentation by Saha and Nix[END_REF][START_REF] Saha | Effects of the substrate on the determination of thin film mechanical properties by nanoindentation[END_REF][START_REF] Perriot | Elastic contact to a coated half-space. Effective elastic modulus and real penetration[END_REF]. This problem has been preferentially addressed using finiteelement simulations for two-phase [START_REF] Durst | Finite element study for nanoindentation measurements on two-phase materials[END_REF][START_REF] Cole | Interphase mechanical behavior of carbon fiber reinforced polymer exposed to cyclic loading[END_REF] or graded [START_REF] Armitage | Indentation across interfaces between stiff and compliant tissues[END_REF] materials, so that interpreting nano-indentation results means a large computation cost. The notable exception is an in-depth sensitivity analysis which has however been conducted in a perturbative framework for very specific inclusion geometries [START_REF] Argatov | Contact stiffness indentation tomography: moduli-perturbation approach[END_REF].

In the case of fiber/matrix composites, the presence of an interlayer at the interface is strongly suspected but rarely evidenced [START_REF] Chevalier | Multi-scale characterization and modelling of the transverse compression response of unidirectional carbon fiber reinforced epoxy[END_REF][START_REF] Ureña | Characterization of interfacial mechanical properties in carbon fiber/aluminium matrix composites by the nanoindentation technique[END_REF][START_REF] Diez-Pascual | Nanoindentation assessment of the interphase in carbon nanotube-based hierarchical composites[END_REF], even though its behavior is critical [START_REF] Liu | Optimising the interfacial response of glass fibre composites with a functional nanoscale plasma polymer coating[END_REF][START_REF] Hodzic | The effects of water aging on the interphase region and interlaminar fracture toughness in polymer-glass composites[END_REF]. Unfortunately, this interlayer is expected to be very thin (less than 1 µm thick) and so, is very difficult to detect and even more to characterize [START_REF] Cole | Interphase mechanical behavior of carbon fiber reinforced polymer exposed to cyclic loading[END_REF][START_REF] Hausild | Statistical treatment of grid indentation considering the effect of the interface and the microstructural length scale[END_REF][START_REF] Perrier | Effect of water ageing on nanoindentation response of single hemp yarn/epoxy composites[END_REF][START_REF] Enrique-Jimenez | Nanoindentation mapping of multiscale composites of graphene-reinforced polypropylene and carbon fibres[END_REF]. In this contribution, Polyetherketoneketone (PEKK) reinforced with carbon fibers composite materials has been tested.

Samples have been probed perpendicularly to the fibers. Such materials are classically tested by indentation, which allows to clearly distinguish matrix from fibers in terms of hardness and elastic modulus. It will be shown in Sect.

2 that this sample also clearly challenges the statistical approach. In Sect.

3, a method based on a geometrical description of the sample is presented.

It assumes the maps obtained by grid nano-indentation result, as for an imaging process, from the convolution of the actual property field with a function depending on the experimental conditions. For the sake of simplicity, the latter is approached using Hertz contact theory. An influence coefficient based on a priori known informations is defined in order to deconvolve elastic properties of the matrix and the fiber. This method is directly compared to the classical ones. A significant gradient of elastic properties is revealed. The center of the fiber is actually found to be softer than the circumference, and the modulus profile in the matrix close-by the fibers is approached.

Grid nano-indentation

This section describes the experimental conditions as well as the statistical analysis of the grid nano-indentation results. The sample consists in a composite material made of PEKK resin together with carbon fibers [START_REF] Chelaghma | Characterization of multifunctional composite materials[END_REF].

These are HexTow AS4 carbon fibers provided by Hexcel as 12000 filament tows, whose longitudinal modulus is about 240 GPa as reported in [START_REF] Herráez | Strength and toughness of structural fibres for composite material reinforcement[END_REF], in line with the manufacturer's specifications [32]. The prepreg has been manufactured using an aqueous impregnation semi-industrial line, with a target volume fraction equal to 55% [33]. Cross section of this sample has been embedded and subsequently polished.

Nano-indentation measurements

Nanoindentation has been performed using an ultra nanoindenter (UNHT) from Anton Paar. A Berkovich indenter has been used. Indentations are force-controlled. Both loading and unloading rate are set to 120 µN.min -1 .

For this instrument the noise levels have been estimated as 0.05 µN in force and 0.03 nm in displacement by the manufacturer. N = 225 indentations have been performed at F m = 60 µN maximum load (see Fig. 1). All the indentations are located on or close by a single carbon fiber, with a nominal grid spacing equal to 1 µm. The location of the indentations with respect to the fiber has been verified by atomic force microscopy. Indentation hardness and elastic modulus have been extracted using the Oliver and Pharr method [START_REF] Oliver | An Improved Technique for Determining Hardness and Elastic Modulus Using Load and Displacement Sensing Indentation Experiments[END_REF].

Contact stiffness S at maximum force F m , and maximum displacement h m have been extracted from the experimental loading curves. The unloading curve has been fitted using a power law, yielding its exponent m. Using S, F m , h m and m, the contact depth h c has been calculated [START_REF] Oliver | An Improved Technique for Determining Hardness and Elastic Modulus Using Load and Displacement Sensing Indentation Experiments[END_REF][START_REF] Woirgard | An alternative method for penetration depth determination in nanoindentation measurements[END_REF][START_REF] Oliver | Measurement of Hardness and Elastic Modulus by Instrumented Indentation: Advances in Understanding and Refinements to Methodology[END_REF].

The shape of the indenter, giving the tip area calibration as a function of the contact depth A c = f (h c ), was calibrated using fused silica. It allows to calculate the projected contact area A c from h c . Hardness H and the apparent contact modulus E eq are then calculated from

H = F m A c (1)
S = 2E eq A c π (2) 1 E eq = 1 -ν 2 i E i + 1 -ν 2 M ( 3
)
This stiffness is hereafter translated into the Young's modulus M of the equivalent isotropic material, assuming that its Poisson's coefficient is ν = 0.3

and considering E i = 1141 GPa and ν i = 0.07 for the indenter material. For transversely isotropic materials indented orthognally to the isotropy plane, M results from the contribution of 4 of the elastic constants of the material and may be approached using for instance the closed-form expression by Delafargue [START_REF] Delafargue | Explicit approximations of the indentation modulus of elastically orthotropic solids for conical indenters[END_REF] : The contact area is also translated in the following in a contact radius a = Ac π .

M 1 -ν 2 = 2 c 11 c 33 -c 2 13 c 11 1 c 44 + 2 √ c 11 c 33 +c 13 (4)

Statistical analysis

η 2 cdf = N e=1 F ( Me ) - Np n=1 ω n 2 1 + erf Me -M n √ 2∆ Mn 2 (5)
Figures 2, 3 and 4 display the experimental and estimated cumulated density functions for 2 ≤ N p ≤ 7, and Table 1 shows the corresponding results.

Parameters optimization is achieved as described in [START_REF] Qasmi | Viscoelastic mechanical properties determined by nanoindentation tests and its numerical modelling of polypropylene modified by He+ particle implantation and e irradiation[END_REF]. The minimum value of the cost function η 2 cdf represents the discrepancy between experiment and the calculated cumulative density function. It is presented in Tab. 1, together with the parameters describing the moduli distributions for various N p values. η 2 cdf globally decreases with N p . However, its values does not the resin close by the fiber is affected by the presence of the fiber as it could have been expected. It is also clear that the standard deviations associated to the measurements performed inside the fiber are higher than the ones performed in the resin. This mapping together with the value of standard deviation strongly suggest that a gradient of mechanical properties is present inside the fiber. However, such a statistical analysis is unable to describe and characterize more precisely this graded material.

Grid nano-indentation data processing

Grid nano-indentation is clearly a way to collect a vast amount of mechanically redundant experimental data. In order to make the most of this redundancy, it is proposed, contrarily to the approach recalled in Sect. 2.2, to make use of some a priori information related to the geometry of the specimen under scrutiny. The convolution kernel translating this geometrical description into the measured indentation moduli map is approached, and the corresponding deconvolution procedure is applied to estimate the elastic properties of the constituents of the sample described in Sect. 2.

Geometrical description

It is assumed that the probed zone depicted in Fig. 1 may be simply described by a two-phases half-space. Namely, the measurement area is assumed to be described by a matrix material in which a single infinite straight fiber whose direction is orthogonal to the indentation plane is included. The full half-space is denoted as H and the cylinder occupied by the fiber is denoted C. The problem geometry may thus be described by Fig. 6. The indents positions P , the fiber position and its radius R are obtained by least-square fitting of the AFM topography (acquired after the indentations) presented in Fig. 7, so that all the geometrical parameters in Fig. 6 are experimentally known. In particular, the fiber radius R = 3.19 µm and the indents positions are identified. This first allows to display the elastic modulus M exp and the contact radius a as a function of the distance d to the fiber center, owing to the cylindrical symmetry of the problem.

The results are displayed in Fig. 8 and Fig. 9, respectively. These clearly show two different phases, namely a fiber much stiffer and harder than the matrix. It should however be outlined that some outliers may be identified at d ≃ 7.5µm. These correspond to indentations performed close by the fiber on top of Fig. 1. The results in Fig. 8 allows to clearly distinguish the matrix and the fiber. An interfacial zone where the measured modulus clearly results from (at least) both the fiber and the matrix is also clearly visible. This convolution effect is sometimes approached by finite-element analysis of the indentation test [START_REF] Cole | Interphase mechanical behavior of carbon fiber reinforced polymer exposed to cyclic loading[END_REF][START_REF] Wang | Evaluation of interphase properties in a cellulose fiber-reinforced poltpropylene composite by nanoindentation and finite element analysis[END_REF]. The following sections are devoted to a rather different approach. The convolution kernel is approached using analytical expressions and numerical (but straightforward) integration. This very simple computation then allows for the quick estimation of the spatial resolution corresponding to the experimental conditions. The corresponding deconvolution procedure is then applied to retrieve the mechanical properties of the phases.

Convolution kernel

The contact radius is experimentally determined and corresponds to the loading value at which the material properties are estimated in the standard procedure (see Sect. 2.1) [START_REF] Oliver | An Improved Technique for Determining Hardness and Elastic Modulus Using Load and Displacement Sensing Indentation Experiments[END_REF]. It is thus assumed that the elastic fields under the indenter may be approached by those deriving from Hertz's contact theory [START_REF] Hanson | The elastic field for spherical Hertzian contact of isotropic bodies revisited : some alternative expressions[END_REF], and one restricts herein, for the sake of simplicity, to elastic isotropic materials described by their Young's modulus E and their Poisson's ratio ν. It should then be outlined that the strains ǫ ij scale as

ǫ ij = 3F (1 + ν) 4πEa 3 ǫ ⋆ ij (a, ν, ρ, z) ∀i, j (6)
and that all the ǫ ⋆ ij (a, ν, ρ, z) are known explicitly at any point defined by the cylindrical coordinates (ρ, z) with the origin at P . The reader interested in the detailed expressions may refer to [START_REF] Hanson | The elastic field for spherical Hertzian contact of isotropic bodies revisited : some alternative expressions[END_REF] for instance. The scaled strain energy density e(a, ν, ρ, z) may thus be computed for any point of the halfspace as a function of the ǫ ⋆ ij (a, ν, ρ, z). These fields vanish quickly when moving away from the contact area, highlighting the fact that all points in the half-plane do not equally contribute to the measurement. The key is therefore, for a given geometrical configuration, to approach the contribution of each phase (each point) to the overall measurement. This is very similar to an homogeneization procedure, and as for classical homogeneization, several choices are possible.

It is chosen herein, for the sake of simplicity, to use the fields ( 6) as test fields. The overall behavior is then obtained by summing the contribution of each phase to the total strain energy. For the geometry depicted in Fig. 6 :

M(P ) H e(a, ν, ρ, z)dV = M f iber C e(a, ν, ρ, z)dV +M matrix H\C e(a, ν, ρ, z)dV (7)
The scaled strain energy density e(a, ν, ρ, z) thus acts as a convolution kernel describing the weight of each point in the half-space on the measured indentation modulus. This construction is the equivalent of a Voigt bound in classical homogeneization, and again, any other homogeneization scheme could have been chosen. Eq. 7 may be recast by defining the influence coefficient I(ν, a, P, R) :

I(ν, a, P, R) = C e(a, ν, ρ, z)dV H e(a, ν, ρ, z)dV (8)
The Poisson's ratio is arbitrarily set to ν = 0.3 for both the phases in the following, so that the influence coefficient I(ν, a, P, R) is computed by standard (Gauss-Kronrod) quadrature. For the considered geometry, the integrations are performed for z ranging from 0 to 10 × a and for ρ also ranging from 0 to 10 × a. It is obvious this expression is independent on the probed elastic properties besides the Poisson's ratio.

The obtained values are displayed on Fig. 10 as a function of the distance to the fiber center. It varies from 0 (far from the fiber) to 1 (at the fiber center), and the rather steep change at the matrix/fiber interface suggests that any stiffness gradient at the interface should be resolved. The computed influence coefficient is approached by an error function :

I(ν, a, P, R) ≃ 1 2 erf c d(P ) -R f δ (9)
R f is a fitting parameter, which is adjusted together with δ to approximate the influence function. It allows to estimate the spatial resolution δ in the interface region and the fitting procedure yields δ = 190 nm, thus setting the resolution limit at the interface for this particular set of measurements.

Deconvolution

Using the influence coefficient defined in Eq. ( 8) allows one to translate a geometrical description of the sample into the observed indentation modulus field. Under the assumptions defined in Sect. 3.2, the indentation modulus at point P , M(P ) reads

M(P ) = M f iber I(ν, a, P, R) + M matrix (1 -I(ν, a, P, R)) (10)
This section is devoted to the retrieval of the indentation modulus of each phase from the grid measurements.

Formulation

For any measurement point P , the contact radius a(P ) is known as well as the measured indentation modulus M exp (P ) (Sect. 3.1). As a consequence, one has one equation 10 for each measurement point P . This large set of linear equations (here : N = 225 equations) depends on two parameters, namely M f iber and M matrix . This set of equations may be written

MM p = M exp (11)
with

M =         I(ν, a, P 1 , R), 1 -I(ν, a, P 1 , R) I(ν, a, P 2 , R), 1 -I(ν, a, P 2 , R) . . . . . . I(ν, a, P N , R), 1 -I(ν, a, P N , R)         (12)
and

M p = [M f iber , M matrix ] t M exp = [M exp (P 1 ), M exp (P 2 ), • • • , M exp (P N )] t (13)
The linear system [START_REF] Nair | Nanoscale characterization of interphase properties in maleated polypropylene-treated natural fiber-reinforced polymer composites[END_REF] is largely overdetermined and is thus solved in a least-square sense to provide the solution

M s p = (M t M) -1 (M t M exp ) (14)
together with the residual field R R = MM s p -M exp [START_REF] Ulm | Statistical Indentation Techniques for Hydrated Nanocomposites: Concrete, Bone, and Shale[END_REF] which is to used to assess the quality of the solution.

Example assuming a uniform fiber

This procedure is applied to the experimental data described in Sect.

Example assuming a graded fiber

In order to enrich the description of the sample, one further assumes that the fiber displays a radial gradient of indentation modulus

M f iber (r) = M center + (M edge -M center ) r R (16)
, so that three material parameters are now to be identified. Applying the same procedure yields M(P ) = M center I(ν, a, P, R)+(M edge -M center )I r (ν, a, P, R)+M matrix (1-I(ν, a, P, R))

with

I r (ν, a, P, R) = C r R e(a, ν, ρ, z)dV H e(a, ν, ρ, z)dV (18)
The same deconvolution procedure is applied to yield M center = 37.7 GPa M edge = 64.7 GPa M matrix = 7.99 GPa thus outlining the significant modulus gradient in the fiber, which is very similar to the results obtained in [START_REF] Cole | Interphase mechanical behavior of carbon fiber reinforced polymer exposed to cyclic loading[END_REF] by the use of extensive finite-element simulations for similar fibers. Such a modulus gradient is also observed by other local characterization techniques [START_REF] Gu | Characterization of the interphase in carbon fiber/polymer composites using a nanoscale dynamic mechanical imaging technique[END_REF].

The corresponding residual is also displayed in Fig. 12. It clearly shows that most of the residual now comes from the matrix region right next to the fiber. Again, the indentation modulus obtained for the matrix differs from the value obtained by the statistical analysis. This further calls for a better description of the sample out of the fiber. is now almost 0 in the matrix far from the fiber, contrarily to the previous modelings (see Figs. 11 and 12). The remaining residual in Fig. 13 also suggests that a more complex modulus distribution in the matrix close-by the fiber is to be identified. This approach by successive enrichment of the sample description clearly reveals, at least qualitatively, the presence of a very steep modulus gradient in a ring which is about 800 nm thick around the fiber. This is made possible by considering the data collected in grid nano-indentation as field information and processing these as they result from an imaging process : adding some a priori information on the sample description allows one to deconvolve the measured data and reveal features whose size approaches the resolution of the imaging process.

The residual may be further analyzed with Fig. 14, which has been obtained by combining :

• The normalized topography z(x,y) max |z(x,y)|

• The normalized residual |R(x,y)| max |R(x,y)| obtained by linear interpolation from the measurement points where it is available, outside the graded ring.

This figure therefore allows to localize areas where the proposed modeling is not adequate and to compare these with topographic features. It can be seen from this figure, and also by comparing to the topography displayed in Fig. 7, that the residual is not spatially correlated to scratches or defects.

It is mainly dominated by the presence of a second fiber, which affects the measured moduli but is not taken into account in the analysis. This allows us to conclude that these defects do not significantly affect the values in the matrix. This also highlights how the residual can (and should) be used together with material knowledge to drive any modeling enrichment.

Finally, the robustness of the proposed approach may be further illustrated by comparing the indentation modulus obtained for the matrix with the proposed method (M bulk = 5.4 GPa) with the value obtained by statistical analysis (5.4±0.1 GPa for N p > 2 in Sect. 2.2). This agreement validates the proposed description. Results obtained in the bulk matrix further illustrate how making use of the geometrical description regularize the final properties.

These geometrical assumptions are strong assumptions : the indentation modulus is for example imposed to be uniform in the matrix phase (outside the graded ring), so that the effect of few local defects would be smeared out in the final results. This is thought to be the origin of the robustness of the proposed method. Should these assumptions be inadequate (because of heterogeneities resulting from scratches or polishing defects), the residual field should be affected.

Conclusions

It is proposed herein the use of grid nano-indentation as a full-field measurement tool to investigate the properties of heterogeneous samples, relying on a geometrical description of the sample. The spatial convolution function is approached using Hertz theory, and the obtained convolution kernel is used to retrieve the mechanical properties of the different phases of a composite material sample made of micrometer-sized fibers in a polymeric matrix. It particularly allows to approach the structure of the sample at the finer scale, and revealed herein a very steep modulus gradient for the matrix material in a thin ring around the fiber. Such a description could be obtained by a successive enrichment, which is driven by the residual of the projection onto the convolution kernel. This analysis method is thus expected to provide some useful insight into interfacial regions of composite materials.

Figure 1 :

 1 Figure 1: Optical view of the nano-indentation imprints (field of view is 50 × 40µm). , c 33 being the longitudinal modulus. Using c 11 = 10 GPa, c 13 = 36 GPa, c 33 = 240 GPa and c 44 = 10 GPa yields M = 54.4 GPa, thus illustrating the elastic stiffness tensor average performed by nano-indentation measurements.

A

  statistical analysis of elastic moduli relies on the cumulative density function, F (M c ) built from the experimental values of elastic moduli Me (e ranging from 1 to N). The properties of the different phases are estimated by assuming a Gaussian distribution of the elastic modulus of each individual

Figure 2 :

 2 Figure 2: Experimental and estimated cumulated density functions for different N p values.

Figure 3 :

 3 Figure 3: Experimental and estimated cumulated density functions for different N p values : zoom on the area corresponding to indentations on the fiber.

Figure 5 :

 5 Figure 5: Localization of the indentations belonging to the different phases for N p = 5.

Figure 6 :

 6 Figure 6: Geometrical description of the sample in the z = 0 plane.

Figure 7 :

 7 Figure 7: AFM topography of the nano-indentation imprints (µm).

Figure 8 :

 8 Figure 8: Elastic modulus M exp as a function of the distance to the fiber center.

Figure 9 :

 9 Figure 9: Indentation radius a as a function of the distance to the fiber center.

Figure 10 :

 10 Figure 10: Computed influence coefficient I(ν = 0.3, a, P, R) as a function of the distance to the fiber center (dots) and the fit by an error function (solid line).

3 . 1 .

 31 Solving the linear system[START_REF] Nair | Nanoscale characterization of interphase properties in maleated polypropylene-treated natural fiber-reinforced polymer composites[END_REF] yields M f iber = 56.1 GPa M matrix = 8.06 GPa and the corresponding residual is displayed in Fig.11. It should be outlined that most of the residual arises from the fiber, whose indentation modulus seems to display a linear dependence to the radial position. The value for M f iber is however consistent with the reported longitudinal modulus (240 GPa, see Sect. 2.1). One should also highlight that the indentation modulus obtained for the matrix significantly differs from the one obtained by the statistical analysis (see Tab. 1). This indicates that the proposed sample description is not adequate.

Figure 11 :

 11 Figure 11: Computed residual assuming a uniform indentation modulus in the fiber.

Figure 12 :Figure 13 :

 1213 Figure 12: Computed residual assuming a radial distribution for the indentation modulus in the fiber.

Figure 14 :

 14 Figure 14: Composite view for the comparison of the defects position with the residual distribution.The two circles correspond to the limit of the fiber and to the outer limit of the ring of matrix with a graded stiffness.

Table 1 :

 1 Elastic moduli identified by statistical analysis as a function of the number of considered populations. × 10 -6 3.8 × 10 -7 2.3 × 10 -7 1.1 × 10 -7 1.0 × 10 -7 1.1 × 10 -7

	N p	2	3	4	5	6	7
	ω n	0.71	0.42	0.42	0.41	0.28	0.4
		0.29	0.35	0.33	0.3	0.39	0.31
			0.22	0.15	0.06	0.03	0.05
				0.1	0.11	0.06	0.04
					0.12	0.11	0.12
						0.13	0.06
							0.02
	Elastic moduli 5.5 ± 0.24 5.4 ± 0.1	5.4 ± 0.1	5.4 ± 0.1	5.4 ± 0.1	5.4 ± 0.1
	(GPa)	46.7 ± 17.7 5.8 ± 0.4	5.8 ± 0.3	5.8 ± 0.3	5.8 ± 0.3	5.8 ± 0.3
			52.1 ± 11.7 39.4 ± 22.1 7 ± 0.8	5.3 ± 0.03 7 ± 0.6
				56.3 ± 4.3 42 ± 23.2 7.1 ± 0.8	11.4 ± 7
					55.8 ± 4.9 44.5 ± 21.2 52.8 ± 14.4
						55.9 ± 4.7 54.5 ± 4.5
							58 ± 0.3
	Min η 2 cdf	2.9

Acknowledgments

Saber Chelaghma (IRT St-Exupéry) and Jean-Noël Périé (Institut Clément Ader) are gratefully acknowledged for providing the material sample used in this study.

Example assuming graded fiber and matrix

In order to enrich the description of the sample, one further assumes that the matrix displays a radial gradient of indentation modulus below a cut-off radius R I = 4µm. Its modulus now reads

, so that four material parameters are now to be identified. Applying the same procedure yields

The deconvolution procedure then yields

The corresponding residual is also displayed in Fig. 13. It clearly shows that adding a linear dependence of the matrix modulus close by the interface significantly improves the overall residual, thereby proving a much better description of the sample properties. It should be outlined that the residual


E Renner

A Bourceret

Y Gaillard

F Amiot

P Delobelle

F Richard

Identifiability of single crystal plasticity parameters from residual topographies in Berkovich nanoindentation on FCC nickel

Keywords: Crystal plasticity (A), Berkovich nanoindentation (B), Parameter identifiability (C)

 for measuring the information richness of the indentation curve is extended to the analysis of residual topographies. This index quantifies the possibility to achieve a stable/unstable solution using an inverse method. For the studied behaviour, the results show that eight of the nine Méric-Cailletaud law parameters can be identified using three topographies.

Introduction

Material parameters identification for single crystal plasticity laws remains a topical issue for a better understanding of metal behaviour at grain or polycrystalline scale [START_REF] Méric | FE calculations of copper bicrystal specimens submitted to tension-compression tests[END_REF][START_REF] Fivel | 3D simulation of a nanoindentation test at a mesoscopic scale[END_REF][START_REF] Gérard | Mesures de champs et identification de modèles de plasticité cristalline[END_REF][START_REF] Gérard | Modeling of latent hardening produced by complex loading paths in FCC alloys[END_REF][START_REF] Gérard | Hardening description for FCC materials under complex loading paths[END_REF][START_REF] Guilhem | Étude numérique des champs mécaniques locaux dans les agrégats polycristallins d'acier 316L sous chargement de fatigue[END_REF][START_REF] Schwartz | Approche non locale en plasticité cristalline: application à l'étude du comportement mécanique de l'acier AISI 316LN en fatigue oligocyclique[END_REF][START_REF] Zambaldi | Orientation informed nanoindentation of α-titanium: Indentation pileup in hexagonal metals deforming by prismatic slip[END_REF][START_REF] Guery | Développement d'une méthode de corrélation d'images numériques adaptée aux mesures cinématiques dans les polycristaux: application à l'identification de paramètres de lois de plasticité cristalline[END_REF][START_REF] Guery | Identification of crystal plasticity law parameters using kinematic measurements in polycrystals[END_REF]Tasan et al., 2014bTasan et al., , 2014a;;[START_REF] Zambaldi | Orientation dependent deformation by slip and twinning in magnesium during single crystal indentation[END_REF]. At the dislocation scale, some authors approached this issue using dislocation dynamics simulations [START_REF] Fivel | Etudes numériques à différentes échelles de la déformation plastique des monocristaux de structure CFC[END_REF][START_REF] Fivel | Identification through mesoscopic simulations of macroscopic parameters of physically based constitutive equations for the plastic behaviour of fcc single crystats[END_REF][START_REF] Forest | Modeles discrets et continus de la plasticité des métaux: du monocristal au polycristal[END_REF][START_REF] Madec | Des intersections entre dislocations à la plasticité du monocristal CFC: étude par dynamique des dislocations[END_REF][START_REF] Devincre | Physical analyses of crystal plasticity by DD simulations[END_REF]. At the grain scale, the laws are also very complete and thus challenging when it comes to the material parameters identification from experimental data using inverse method. In this context, the often-missing concept of ill-posed problem, or identifiability, is essential to assess the stability and the uniqueness of the solution. The Méric-Cailletaud law is easily and effectively implemented in FE codes like ZeBuLon [START_REF] Burlet | Zebulon a finite element code for non-linear material behaviour[END_REF] [http://www.zset-software.com]. Strong nonlinearities and couplings between the dissipative phenomena during indentation make the identification of the parameters of the Méric-Cailletaud single crystal plasticity law very difficult, especially for the six interaction matrix components. These six parameters define the dislocation interactions hardening in the 12 FCC slip systems. Most recently, some works used FEM updating for identifying these interaction components, but one can notice some limitations. Méric et al performed tensile tests on single and bi-crystals, which may be expensive and limited to materials that can be obtained macroscopically [START_REF] Méric | FE calculations of copper bicrystal specimens submitted to tension-compression tests[END_REF]. Gérard et al. performed tensile tests on preloaded polycrystalline samples. These complex loadings activate the slip systems interactions. Even if most of isotropic hardening law parameters are identified, it does not provide enough information and some interaction components remain inaccessible [START_REF] Gérard | Mesures de champs et identification de modèles de plasticité cristalline[END_REF][START_REF] Gérard | Hardening description for FCC materials under complex loading paths[END_REF]. Guery et al. performed in-situ tensile tests on polycrystalline samples in which they measure macroscopic loads and displacement fields by digital image correlation [START_REF] Guery | Développement d'une méthode de corrélation d'images numériques adaptée aux mesures cinématiques dans les polycristaux: application à l'identification de paramètres de lois de plasticité cristalline[END_REF][START_REF] Guery | Identification of crystal plasticity law parameters using kinematic measurements in polycrystals[END_REF]. The authors have the merit of assessing the interaction components identifiability before identifying them. Besides the fact that the techniques used are quite complex, a gap subsists between experimental and numerical results. Moreover, the two latest studies are performed at the polycrystalline scale and must take the scale transition rules into account.

Based on the principle of hardness test, the instrumented indentation test, or nanoindentation test, offers three major advantages for the identification of single crystal plasticity laws. Firstly, it is easy of use. It performs continuous measurement of the applied load 𝑃 on the indenter and of the indentation depth ℎ. Thus, the indentation curve (𝑃 -ℎ) can be used to locally extract an elastic modulus and hardness when interpreted in the framework proposed in [START_REF] Harding | The elastic stresses produced by the indentation of the plane surface of a semi-infinite elastic solid by a rigid punch[END_REF][START_REF] Sneddon | The relation between load and penetration in the axisymmetric Boussinesq problem for a punch of arbitrary profile[END_REF], [START_REF] Oliver | An improved technique for determining hardness and elastic modulus using load and displacement sensing indentation experiments[END_REF] and Vlassak and Nix works (Vlassak andNix, 1993, 1994). Secondly, the stress field generated by the Berkovich indenter (three-sided based pyramid) penetration is multiaxial. One can think that the nanoindentation test activates simultaneously a large number of slip systems. Finally, it provides a direct mechanical measurement at intra-granular scale without any neighbouring grains interaction if the average grain size is large enough compared to the imprint size.

Indentation curves have been used by some authors for the identification of elastic-plastic parameters at a macroscopic scale. They mostly concluded that it is usually impossible to correctly identify plastic parameters for any kind of indenter shape (conical, pyramidal, spherical), even using "simple" elastic-plastic laws involving only two plastic parameters as the yield stress 𝜎 𝑦 and the work-hardening exponent 𝑛 of a power hardening law. Solutions are often unstable. Cheng and Cheng [START_REF] Cheng | Can stress-strain relationships be obtained from indentation curves using conical and pyramidal indenters?[END_REF] first pointed out this issue. Some authors have then numerically proved that, assuming that the Young modulus 𝐸 of an isotropic material is known, both plastic parameters (𝜎 𝑦 , 𝑛) are not identifiable using the entire conical indentation curve [START_REF] Capehart | Determining constitutive models from conical indentation: Sensitivity analysis[END_REF][START_REF] Tho | Uniqueness of reverse analysis from conical indentation tests[END_REF][START_REF] Alkorta | Absence of one-to-one correspondence between elastoplastic properties and sharp-indentation loadpenetration data[END_REF]. They showed that different materials governed by this law can lead to indistinguishable indentation curves (Figure 1). To yield a unique solution, the dual nanoindentation technique using different conical indenter tips with half angles ranging from 60° to 80° has been proposed [START_REF] Chen | On the uniqueness of measuring elastoplastic properties from indentation: The indistinguishable mystical materials[END_REF]. But once again, they showed the existence of "mystical" materials which give almost indistinguishable indentation curves. This "mystical" material identification issue has been solved by some authors. Zhao et al. performed indentations on thin films which were indistinguishable by bulk indentation [START_REF] Zhao | Measuring elastoplastic properties of thin films on an elastic substrate using sharp indentation[END_REF]. Thanks to the substrate effect, the elastoplastic properties can then be derived from an inverse analysis. Another solution was proposed by Ma et al. which is close to the approach considered in the present paper [START_REF] Ma | Characterization of stress-strain relationships of elastoplastic materials: An improved method with conical and pyramidal indenters[END_REF]. They succeeded to correctly identify "mystical" materials using the indentation curves but also the residual imprint topographies in the inverse analysis. Other authors have examined this difficulty and have shown that non-uniqueness is an extreme case of instability of the solution [START_REF] Cao | Depth-sensing instrumented indentation with dual sharp indenters: stability analysis and corresponding regularization schemes[END_REF][START_REF] Phadikar | On the uniqueness and sensitivity of indentation testing of isotropic materials[END_REF]. "Highly plastic" behaviour (small ratio 𝜎 𝑦 /𝐸) [START_REF] Cheng | Can stress-strain relationships be obtained from indentation curves using conical and pyramidal indenters?[END_REF]. Thus, the parameter identification of the Méric-Cailletaud law using the indentation curve is not a possible scenario since the identification of much simpler elastic-plastic laws at macroscopic scale is an issue. Bolzon et al [START_REF] Bolzon | Material model calibration by indentation, imprint mapping and inverse analysis[END_REF] and Bocciarelli et al [START_REF] Bocciarelli | Parameter identification in anisotropic elastoplasticity by indentation and imprint mapping[END_REF] used both the indentation curve and the imprint mapping for FEMU process. Zambaldi and Raabe said that "For the indentation of crystals, the topography of the free surface around the indent, the pile-up profile, can be used as a fingerprint of the underlying crystal deformation processes." [START_REF] Zambaldi | Plastic anisotropy of γ-TiAl revealed by axisymmetric indentation[END_REF]. Indeed, pile-ups around the indentation imprint contain precious information for parameters identification. Targeting an extension of these results to crystal plasticity laws, this contribution is intended to quantify the identifiability of all or part of the Méric-Cailletaud parameters at grain scale using topographies of imprints (Figure 2). grain scale [START_REF] Renner | Vers l'identification d'une loi de plasticité monocristalline par analyse topographique d'empreintes de nanoindentation Berkovich[END_REF].

The paper is organized as follows. Firstly, the single crystal plasticity behaviour and the CPFEM of the nanoindentation test are described. Secondly, the identifiability method developed for measuring the information richness contained in indentation curves and residual topographies are described. Finally, the sensibility of the CPFEM and the identifiability of the plastic parameters are analysed and interpreted using one indentation curve only, one topography only and combinations of topographies.

Single crystal plasticity framework

Indentation experiments have been performed until 0.9 µm depth, resulting in a residual imprint whose edge is 5.5 µm long. They have been performed in large-grained samples. The average grain size is about 140 µm which is large compared to the size of the indentations [START_REF] Renner | Vers l'identification d'une loi de plasticité monocristalline par analyse topographique d'empreintes de nanoindentation Berkovich[END_REF]. Moreover, a particular attention has been paid to avoid indentations near the grain boundaries. Grain interaction effects are thus neglected and the use of the single crystal plasticity framework is justified.

Constitutive laws

An orthotropic stiffness tensor 𝑪 ̿ ̿ is considered to model the nickel elastic response, described by three components (𝐶 11 , 𝐶 12 , 𝐶 44 ). The nanoindentation test imposes the use of a large deformation formulation due to the large strains reached in the contact region. A multiplicative decomposition of the transformation gradient 𝑭 ̿ in an elastic 𝑭 ̿ 𝑒 part and a plastic 𝑭 ̿ 𝑝 part is adopted [START_REF] Lee | Elastic-plastic deformation at finite strains[END_REF]:

𝑭 ̿ = 𝑭 ̿ 𝑒 • 𝑭 ̿ 𝑝 (1)
The plastic shear strain is assumed to occur only on each of the 12 FCC slip systems 〈110〉{111}, characterized by the slip directions 𝒍 𝑠 and the normal to the slip planes 𝒏 𝑠 :

𝑭 ̿ ̇𝑝 • 𝑭 ̿ 𝑝 -1 = ∑ 𝛾̇𝑠 12 𝑠=1 𝒍 𝑠 ⨂𝒏 𝑠 (2)
The inelastic flow 𝛾̇𝑠 on system 𝑠 is chosen as a Norton power law to describe the material visco-plasticity, based on the Méric-Cailletaud size-independent single crystal plasticity law:

𝛾̇𝑠 = ⟨ |𝜏 𝑠 | -𝑟 𝑠 -𝑟 0 𝐾 ⟩ 𝑛 sign(𝜏 𝑠 ), with 〈. 〉 = max(0, . ) (3)
where 𝐾 and 𝑛 are viscosity parameters and 𝑟 0 is the critical resolved shear stress. 𝑟 𝑠 is the hardening on system 𝑠. The dislocations slipping initiation is governed by the resolved shear stress 𝜏 𝑠 :

𝜏 𝑠 = 𝝈 ̿ ∶ 𝒎 ̿ 𝑠 , with 𝒎 ̿ 𝑠 = 1 2 (𝒍 𝑠 ⨂𝒏 𝑠 + 𝒏 𝑠 ⨂𝒍 𝑠 ) (4)
where 𝝈 ̿ is the Cauchy stress tensor and 𝒎 ̿ 𝑠 the Schmid tensor. The hardening 𝑟 𝑠 has been chosen as purely isotropic:

𝑟 𝑠 = 𝑞 ∑ ℎ 𝑠𝑟 (1 -exp(-𝑏𝑣 𝑟 ) ), with 𝑣̇𝑟 = |𝛾̇𝑟| 12 𝑟=1
(5)

where 𝑞 and 𝑏 are material parameters and 𝑣 𝑟 is the cumulated plastic slip on system 𝑟. ℎ 𝑠𝑟 are the interaction matrix components which reflect the hardening associated to dislocation interactions between a primary system 𝑠 and a secondary one 𝑟. The interaction matrix is presented in Table 1. Due to the FCC structure symmetries, it is completely defined from 6 independent parameters (ℎ 1 , ℎ 2 , ℎ 3 , ℎ 4 , ℎ 5 , ℎ 6 ) [START_REF] Franciosi | The concepts of latent hardening and strain hardening in metallic single crystals[END_REF]. Diagonal terms define selfhardening (ℎ 1 ) and off-diagonal terms define latent hardening: coplanar interaction (ℎ 2 ), Hirth locks (ℎ 3 ), colinear interaction (ℎ 4 ), glissile junctions (ℎ 5 ) and Lomer locks (ℎ 6 ).

𝑠 𝑟 1 2 3 4 5 6 7 8 9 10 11 12 1 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 symmetric 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 1 -Interaction matrix of FCC structure.

Virtual material

The virtual material is a purely numerical material defined by 14 parameters (𝐶 11 , 𝐶 12 , 𝐶 44 , 𝐾, 𝑛, 𝑟 0 , 𝑞, 𝑏, ℎ 1 , ℎ 2 , ℎ 3 , ℎ 4 , ℎ 5 , ℎ 6 ). The three components of the elastic tensor are chosen as 𝐶 11 = 248 GPa, 𝐶 12 = 153 GPa, 𝐶 44 = 116 GPa and the values of the viscosity parameters are set to 𝐾 = 8 MPa. s 1 n ⁄ and 𝑛 = 7 [START_REF] Renner | Sensitivity of the residual topography to single crystal plasticity parameters in Berkovich nanoindentation on FCC nickel[END_REF], which agrees with the commonly measured values for the Nickel [START_REF] Rao | The Effect of Grain Size and Strain Rate on the Substructures and Mechanical Properties in Nickel 200[END_REF]. Using these values of 𝐾 and 𝑛, sensitivity of the indentation response to the viscosity parameters is low. Therefore, viscosity acts as a numerical regularization method of the rate-independent model. This is in agreement with the very low creep observed experimentally during the holding phase on the real annealed nickel sample in indentation test.

Finally, the plasticity model is a nine-parameter plastic model: 𝛉 = ( 𝑟 0 , 𝑞, 𝑏, ℎ 1 , ℎ 2 , ℎ 3 , ℎ 4 , ℎ 5 , ℎ 6 ).

To analyse the identifiability of all or part of these nine parameters 𝜃 𝑗 (𝑗 = 1, … ,9) from nanoindentation experiments, a virtual material 𝛉 has been chosen (Table 2) and the identifiability results will be available in the close vicinity of this parameters set. The numerical process established to choose these parameters values is detailed in Renner et al. [START_REF] Renner | Vers l'identification d'une loi de plasticité monocristalline par analyse topographique d'empreintes de nanoindentation Berkovich[END_REF]. The chosen virtual material promotes the ℎ 3 interaction component (Hirth locks) and has been built so as to reproduce the strain hardening response of the real annealed nickel sample in tension. Note that the sum of the interaction matrix components over one line is 100 = (ℎ 1 +2ℎ 2 + 2ℎ 3 + ℎ 4 + 4ℎ 5 + 2ℎ 6 ).

hardening interaction [START_REF] Renner | Sensitivity of the residual topography to single crystal plasticity parameters in Berkovich nanoindentation on FCC nickel[END_REF]. Nine plastic parameters 𝜃 𝑗

𝑗

CPFEM of the nanoindentation experiment

Identifiability analysis is based on a CPFEM of the nanoindentation experiment using the Méric-Cailletaud size-independent single crystal plasticity which is not detailed here, (refer to [START_REF] Renner | Sensitivity of the residual topography to single crystal plasticity parameters in Berkovich nanoindentation on FCC nickel[END_REF]). The CPFEM has been built using a generalized elastic-viscoplastic material model based on the infinitesimal strain theory (Z-set User commands Version 9.0, 2018). A corotational finite strain formulation based on an integrated rotation tensor has been used to take the material rotations into account [START_REF] Ladeveze | Sur la théorie de la plasticité en grandes déformations[END_REF]. The lattice is thus rotating with the material using the material spin tensor 𝑾 ̿̿̿ = (𝑭 ̿ ̇𝑭 ̿ -1 ) 𝑠𝑘𝑒𝑤 while the lattice rotation should be independent and occur in the elastic transformation part 𝑭 ̿ 𝑒 [START_REF] Forest | Continuum crystal plasticity theory[END_REF]. The numerical integration of the material behaviour has been performed using the Runge-Kutta's second order method. The resolution method for the global matrix problem uses a full updated Newton-Raphson algorithm.

The contact problem is solved using a direct flexibility method [START_REF] Francavilla | A note on numerical computation of elastic contact problems[END_REF][START_REF] Jean | Frictional contact in collections of rigid or deformable bodies: numerical simulation of geomaterial motions[END_REF][START_REF] Sachdeva | A finite element solution for the two-dimensional elastic contact problems with friction[END_REF][START_REF] Wronski | Couplage du contact et du frottement avec la mécanique non linéaire des solides en grandes déformations[END_REF]. The contact between the bulk surface and the Berkovich indenter is modelled using the Coulomb law with a null friction coefficient. Indeed, the indenter geometry opening is quite large. The friction effect is thus negligible on the indentation curve [START_REF] Taljat | New analytical procedure to determine stress-strain curve from spherical indentation data[END_REF] and minor on the imprint topography compared to the hardening parameters.

The CPFEM of the nanoindentation experiment (direct problem) is parametrized using the vector 𝚵 = (𝛉; [i j k], 𝛼 [𝑖𝑗𝑘] [𝑎𝑏𝑐] , ℎ(𝑡), 𝑡, 𝑋𝑌), where 𝛉 is the vector of the nine plastic parameters (Table 2), [i j k] the Miller's indices (grain orientation), 𝛼 [𝑖𝑗𝑘] [𝑎𝑏𝑐] the Berkovich tip azimuth, ℎ(𝑡)

the indentation depth, 𝑡 the time and the 𝑋𝑌 mapping region of the imprint topography. A coordinate system is related to the considered grain. The Miller's indices [i j k] define the first axis (perpendicular to the sample surface). The indices [𝑎𝑏𝑐] define the second axis which is arbitrary chosen in the sample surface plane. The third axis is deduced from the other to get an orthonormal coordinate system.

𝛼 [𝑖𝑗𝑘]

[𝑎𝑏𝑐] is defined as the angle between the [𝑎𝑏𝑐]-axis of the crystal coordinate system and the indenter base symmetry axis passing through the indenter corner contained in the coordinate system 1 st quadrant.

The Figure 3 illustrates a CPFEM simulation result: an indentation curve 𝐏 and an imprint topography 𝐙 = 𝐙 1 . These results were obtained using the virtual material for 𝚵 = (𝛉; [1 0 1], 74°, 𝑡, ℎ(𝑡), 𝑡, 𝑋𝑌), virtual material 𝜃 𝑗 -values of (Table 2). Loading and unloading steps during the indentation tests last 30 seconds and holding at maximum force lasts 90 seconds. The mapping region 𝑋𝑌 is described by an area of (20 × 20 ) μm 2 . A quantitative estimation of the plastic zone size can be given from the simulation. The cumulative plastic strain is found to vanish at a distance larger than six times the depth of the residual imprint.

Indeed the size of the grains remains large compared to this value, validating the single crystal plasticity framework.

Plastic parameter identifiability method

This section presents the numerical method used for estimating the identifiability of all or part of the 9 plastic parameters 𝛉 = (𝑟 0 , 𝑞, 𝑏, ℎ 1 , ℎ 2 , ℎ 3 , ℎ 4 , ℎ 5 , ℎ 6 ) from the indentation curves and topographies of imprints. These techniques is based on a CPFEM of the nanoindentation test [START_REF] Renner | Sensitivity of the residual topography to single crystal plasticity parameters in Berkovich nanoindentation on FCC nickel[END_REF] using the Méric-Cailletaud size-independent single crystal plasticity law.

The identifiability index used for measuring the information richness in indentation curves and residual topographies is described. This index has been introduced for the first time by Richard et al. [START_REF] Richard | Viscoelastic modeling and quantitative experimental characterization of normal and osteoarthritic human articular cartilage using indentation[END_REF] for viscoelastic characterization from the sole indentation force, then in 2014 for plastic characterization from the sole indentation force [START_REF] Pac | Microstructural and elasto-plastic material parameters identification by inverse finite elements method of Ti (1-x) Al x N (0[END_REF]. In this section, the identifiability index definition is extended to plastic characterization using residual topographies.

When some values of the plastic parameters 𝜃 𝑗 (𝑗 = 1, … ,9) are unknown, they must be identified from experimental observations. The difficulty is then at first to well-pose the inverse problem. To build a well-posed inverse problem and identify some reliable values of all or part of the nine plastic parameters using a CPFEMU method, the information richness of numerical indentation curve 𝐏, residual topography 𝐙 and combinations (𝐙 1 , … , 𝐙 𝑛 𝐳 ) must be quantified.

To quantify the reliability of plastic parameter values that could be obtained using a FEMU method, the completeness of data contained in 𝑛 𝐳 residual topographies (𝐙 1 , … , 𝐙 𝑛 𝐳 ) considered separately or combined is quantified by extending the 𝐼-index formulation proposed by Richard et al. [START_REF] Richard | Viscoelastic modeling and quantitative experimental characterization of normal and osteoarthritic human articular cartilage using indentation[END_REF] for the analysis of the indentation curve. This index appears to be convenient to explore and investigate what are the optimal conditions to determine all or part of the nine plastic parameters of the material constitutive law from CPFEMU method. The analysis can be done before the updating process and therefore does not necessarily require the experimental measurements.

Method using indentation curve

To make the section easier to read, the formalism is first presented using the indentation curve 𝐏 only for a given grain orientation [i j k], azimuth 𝛼 [𝑖𝑗𝑘] [𝑎𝑏𝑐] and displacement-controlled ℎ(𝑡) mode. The cost function is sampled in 𝑇 time steps 𝑡 𝑘 . The 𝐼-index proposed by Richard et al. [START_REF] Richard | Viscoelastic modeling and quantitative experimental characterization of normal and osteoarthritic human articular cartilage using indentation[END_REF] is a measure of the inverse problem conditioning and can be used with logarithmic notation [START_REF] Pac | Microstructural and elasto-plastic material parameters identification by inverse finite elements method of Ti (1-x) Al x N (0[END_REF]:

𝐼 𝐩 = log 10 ( 𝜆 max 𝜆 min ) (6)
where 𝜆 𝑚𝑎𝑥 and 𝜆 𝑚𝑖𝑛 are respectively the maximum and the minimum eigenvalues of the dimensionless matrix 𝐇 ̅ 𝐏 at the considered calculation point 𝛉, respectively:

𝐻 ̅ 𝑖𝑗 𝐩 = ∑ 𝑆 ̅ 𝑘𝑖 𝐩 𝑆 ̅ 𝑘𝑗 𝐩 𝑇 𝑘=1 (7)
𝑖 and 𝑗 indicates the number of the parameters considered for the calculation (Table 2). 𝑆 ̅ 𝑘𝑖 𝐏 are the components of the force sensitivity vector to parameter 𝜃 𝑗 :

𝑆 ̅ 𝑘𝑗 𝐩 = 𝜃 𝑗 𝑃 max √𝑇 𝜕𝑃 𝑘 𝜕𝜃 𝑗 (8)
𝑃 max (maximum simulated indentation force) and 𝜃 𝑗 are used as scaling factors associated to the indentation force 𝑃 and the parameter 𝜃 𝑗 itself, respectively. Each parameter 𝜃 𝑗 is thus scaled by itself, which means that each parameter is sought using the same relative precision.

Moreover, indented force is scaled by a constant 𝑃 max , which means that an absolute uncertainty is considered on this variable.

For a chosen set of parameters, the matrix 𝐇 ̅ 𝐩 contained the richness of the simulated indentation curve. By construction, this matrix contains information concerning the lack of sensitivity and the multicollinearity between the sensitivity vectors. These are two difficulties that can make the problem ill-posed (ill-conditioned). The lower the 𝐼-index, the better conditioned is the matrix, which means its inverse can be calculated with great accuracy. At the opposite, if the 𝐼-index is large, the matrix is considered as ill-conditioned. Some 𝐼-index values defining practical limits can be found in the literature [START_REF] Gujarati | Basic econometrics[END_REF]. This a priori analysis allows to distinguish the potentially identifiable combinations (𝐼 ≤ 2) of material parameters from those which are not (𝐼 > 3). These bound values 2 and 3 are related to the shape of the objective function if an updating would be performed from the responses contained in 𝐇 ̅ 𝐩 . The case 𝐼 = 0 corresponds to the best conditioning, the shape of the objective function would be a 𝑛 𝜃 -sphere, where 𝑛 𝜃 is the number of parameters. The objective function shape becomes a more general quadric surface as soon as 𝐼 > 0. If the parameters uncertainties are considered two by two, the objective function iso-value is an ellipse in the space of parametric relative uncertainties. The larger 𝐼, the larger the ellipticity (flattening) which means that a "valley of solutions" exists (Figure 4). The valley thinness becomes a problem for 𝐼 = 2. Considerable uncertainties can thus be generated. The bounds 2 and 3 correspond to a ratio of relative uncertainties over the estimated parameters which can reach the values 10 and 30, respectively. Restricting to a parameters pair, the index can be graphically determinate by calculating the ratio between the minor and major radius of the smallest ellipse dimensions (Figure 4). ) for different values of 𝐼-index.

The 𝐼-index can be calculated for all the 𝑝-combinations (𝑝 = 2, … 9) of the plastic parameters, i.e. 501 possibilities.

Method using residual topographies

The indentation force 𝑃(𝑡) sampled in 𝑇 time steps 𝑡 𝑘 may be formally replaced by the residual topography 𝑍(𝑋𝑌) sampled in 𝑀 space steps 𝑋𝑌 𝑘 along a path. The components of the 𝐇 ̅ 𝐳 matrix and the components of the sensitivity vector to parameter 𝜃 𝑗 become respectively:

𝐻 ̅ 𝑖𝑗 𝐳 = ∑ 𝑆 ̅ 𝑘𝑖 𝐳 𝑆 ̅ 𝑘𝑗 𝐳 𝑀 𝑘=1 (9) 𝑆 ̅ 𝑘𝑗 𝐳 = 𝜃 𝑗 𝑍 max √𝑀 𝜕𝑍 𝑘 𝜕𝜃 𝑗 ( 10
)
When 𝑛 𝐳 residual topographies (𝐙 1 , … , 𝐙 𝑛 𝐳 ) are combined, these expressions are written:

𝐻 ̅ 𝑖𝑗 𝐳 1 ,…,𝐳 𝑛 z = ∑ 𝑆 ̅ 𝑙𝑖 𝐳 1 ,…,𝐳 𝑛 𝐳 𝑆 ̅ 𝑙𝑗 𝐳 1 ,…,𝐳 𝑛 𝐳 𝑀×𝑛 𝐳 𝑙=1 (11) 𝑆 ̅ 𝑙𝑗 𝐳 1 ,…,𝐳 𝑛 𝐳 = 𝜃 𝑗 √𝑀 ∑ 1 𝑍 max (𝑒) ∑ 𝜕𝑍 𝑘 (𝑒) 𝜕𝜃 𝑗 𝑀 𝑘=1 𝑛 𝐳 𝑒=1 (12)

Sensibilities of the CPFEM to the plastic parameters

The identifiability index is based on sensibilities of the CPFEM. As an example, in this section, the sensitivity of the simulated loading indentation curve (Eq. ( 8)) and residual topography (Eq. ( 10)) to the plastic material parameters 𝛉 = (𝑟 0 , 𝑞, 𝑏, ℎ 1 , ℎ 2 , ℎ 3 , ℎ 4 , ℎ 5 , ℎ 6 ) is calculated around the chosen virtual material in the [101] orientation and for a azimuth 𝛼 [101]

[010] = 74°. Ten simulations of the nanoindentation test have been performed to approximate derivatives by the forward finite difference method:

-1 nominal simulation using the reference parameters vector 𝛉 (Table 2),

-9 simulations for which one parameter 𝜃 𝑗 has been successively perturbated by the quantity ξ𝜃 𝑗 .

The perturbation value ξ has to balance the truncation and the rounding errors performed on the simulated indentation curve and residual topography [START_REF] Pottier | Modélisation multiéchelle du comportement et de l'endommagement des composites à matrice métallique[END_REF]. A stability study on both responses has shown that ξ = 10 -3 is an adequate value [START_REF] Renner | Vers l'identification d'une loi de plasticité monocristalline par analyse topographique d'empreintes de nanoindentation Berkovich[END_REF].

Sensitivity of the indentation curve

In that case, the loading indentation curve 𝐏 is the unique observation. The unloading is assumed to be purely elastic which means that no information about the plastic parameters can be extracted from this part of the curve. The measurements collected during unloading phase is thus not used. It has been noticed that the simulated loading curves were slightly disturbed due to the mesh element size, especially in the contact region. This perturbation can affect the sensitivity of indentation curve to the material parameters and thus the identifiability calculations. To avoid any problems, the simulated loading curves 𝑃(ℎ) have been fitted by a third-order polynomial. The purpose here is not to define the relationship between the indenter force and displacement, but to filter out any perturbation due to the mesh element size.

The simulated indentation loading curves are sampled in 𝑇 = 181 time steps. The sensitivity of the simulated indentation force to the parameters is defined using Eq. ( 8). This definition attributes a small weight to small indentation depth. Thus, the contact detection problem is limited for the sensitivity calculation at small indentation depth. The Figure 5 presents the obtained sensitivity vectors and their norms. The indentation curves are sensitive to all parameters and their norm are of comparable in magnitude. The curve is more sensitive to 𝑟 0 , 𝑞, 𝑏 and ℎ 3 than to the other parameters. The ratio between the sensitivities to the parameter 𝑏 (the most influent) and ℎ 2 (the less influent) is about 4.2.

The collinearity of some sensitivity vectors to some parameters is highlighted in Figure 5a. It is an extremely important measurement since the inversion problem is impossible when sensitivity vectors are colinear. One can notice that the sensitivity vectors associated to the parameter pairs (𝑟 0 ,𝑞), (𝑏,ℎ 3 ), (ℎ 1 ,ℎ 4 ) and (ℎ 1 ,ℎ 6 ) seem to be almost collinear. At this stage, difficulties in the parametric identification from the sole indentation curve can be anticipated.

Sensitivity of the residual topography

In that case, the residual topography 𝐙 obtained in the [101] orientation and for an azimuth

𝛼 [101]

[010] = 74° is the unique observation. Initially, topography data are 256 × 256 matrices, which corresponds to the picture definition (256 × 256 pix 2 ). Topography data are reshaped in a 65536-components vector along the path illustrated in (Figure 6a). The experimental AFM measurements of the imprint centre is poorly defined. Indeed, the bottom of the imprint cannot be well-measured since the AFM is used in non-contact mode. The measurements collected close to the imprint centre should therefore not be used. square truncation of the topography.

To limit the poor definition effects of the imprint bottom, the data are slightly changed to assign a small weight to the bottom of the imprint and a maximum weight to pile-ups:

𝑍 = 1 + 𝑍 0 ℎ 𝑟 ( 13
)
where 𝑍 0 and ℎ 𝑟 are the initial residual topography and the residual indentation depth (Figure 3). The zero of 𝑍 0 is the sample surface far from the imprint. Using this definition, the bottom of the imprint is the zero of the observation (Figure 7). It is also chosen to perform a square shape truncation of the information at the imprint centre (Figure 6b). A smart truncation decreases the measurements and avoids the use of data disturbed by the contact tip-sample. In that case (125 × 125 pix 2 ), the imprint centre is then entirely covered and the topography is sampled at about 𝑀 = 5 × 10 4 positions. The sensitivity of the simulated imprint topography to the parameters is defined using Eq. ( 10) and is presented in Figure 8ab. The same quantities are shown in Figure 8cd using the truncated topography presented in Figure 6b. The sensitivity vectors norms are extremely affected.

Contrarily, the relative parameters influence on the topography remains unchanged, regardless to the truncation size.

The residual topography is more sensitive to 𝑟 0 , 𝑞, 𝑏 and ℎ 3 compared to the other parameters in both cases. The ratio between the highest sensibility and the lowest (ℎ 2 ) is about 7.7 if the topography is complete and about 12.6 using the truncation, which already predicts identification problems. One can notice that the calculation of the sensitivity can be largely affected by the information quantity. Having a large volume of information does not mean that the post processing will be of good quality.

Identifiability results

The section 4 highlighted the problems of sensitivity vectors multi-collinearity and the gap between norms. Both aspects are taken into account with the use of the identifiability index. In the following sections, the identifiability of the plastic parameters (𝑟 0 , 𝑞, 𝑏, ℎ 1 , ℎ 2 , ℎ 3 , ℎ 4 , ℎ 5 , ℎ 6 )

has been calculated in several cases:

using one indentation curve 𝐏([101], 𝛼 [101]

[010] = 74°), the same as the one studied in section 4.1, using one topography 𝐙 1 = 𝐙([101], 𝛼 [101]

[010] = 74°), the same as the one studied in section 4.2, using a combination of two topographies: 𝐙 1 and 𝐙 2 ([001], 𝛼 [001]

[100] = 77°),

using a combination of three topographies: 𝐙 1 , 𝐙 2 and 𝐙 3 ([111],

𝛼 [111] [1 ̅ 21 ̅ ] = 65°).
It is recalled that all simulated responses are the results of the nanoindentation CPFEM around the virtual material defined in section 2.2.

Results using one indentation curve

As expected, when all parameters are considered for the index calculation 𝐼 𝐩 > 6. It means that the identification of all parameters using the sole indentation curve is impossible. 𝐼 𝐏 has also been calculated for all pairs of parameters. The results are presented in Table 3. All the pairs which have an identifiability index 𝐼 𝐩 ≤ 2 are a priori identifiable using 𝐏, assuming the other parameters are known. This is true only if the material behaviour to identify is close to the virtual material. For the pairs (𝑏,ℎ 3 ) and (ℎ 1 ,ℎ 4 ), 𝐼 𝐩 > 2, so that the identification of even 2 parameters can remain difficult using this indentation curve. The result for the pairs (𝑟 0 ,𝑞) and

(ℎ 1 ,ℎ 6 ) are only slightly better. These results been speculated from the sensitivity analysis in section 4.1 (Figure 5a): for these pairs, the sensitivity vectors to each parameter are almost collinear.

𝐼 𝐩 𝑟 0 𝑞 𝑏 ℎ 1 ℎ 2 ℎ 3 ℎ 4 ℎ 5 ℎ 6 𝑟 0 1.9 0.2 1.1 1.1 0.1 1.0 1.1 1.1 𝑞 0.3 1.1 1.2 0.2 1.0 1.1 1.1 𝑏 1.2 1.7 2.3 1.1 1.5 1.2 ℎ 1 0.5 1.1 2.9 1.0 2.0 ℎ 2 1.8 0.5 0.1 0.3 ℎ 3 1.1 1.4 1.2 ℎ 4 0.9 1.8 ℎ 5 1.3 ℎ 6
Table 3 -𝐼-index for all the parameters pairs using the indentation curve 𝐏.

𝐼 𝐩 ≤ 2 (green, potentially identifiable), 2 < 𝐼 𝐩 ≤ 3 (orange, difficult to identify), 𝐼 𝐩 > 3 (red, not identifiable).

In the case where the interaction matrix components (ℎ 1 , ℎ 2 , ℎ 3 , ℎ 4 , ℎ 5 , ℎ 6 ) are known, 𝐼 𝐩 (𝑟 0 , 𝑞, 𝑏) = 2.3. The identification of the isotropic hardening parameters is thus difficult. These results are in line with those in the literature at macroscopic scale. For a macroscopic power hardening law using 2 plastic parameters (yield stress 𝜎 𝑦 and exponent hardening 𝑛), authors generally conclude that it is impossible to obtained a stable solution for the pair (𝜎 𝑦 , 𝑛) using a conical indentation curve [START_REF] Alkorta | Absence of one-to-one correspondence between elastoplastic properties and sharp-indentation loadpenetration data[END_REF][START_REF] Capehart | Determining constitutive models from conical indentation: Sensitivity analysis[END_REF][START_REF] Casals | The duality in mechanical property extractions from Vickers and Berkovich instrumented indentation experiments[END_REF][START_REF] Charleux | Micromécanique de l'essai d'indentation: expériences et simulations[END_REF][START_REF] Tho | Uniqueness of reverse analysis from conical indentation tests[END_REF].

In the opposite case, where the isotropic hardening parameters (𝑟 0 , 𝑞, 𝑏) are known, 𝐼 𝐩 (ℎ 1 , ℎ 2 , ℎ 3 , ℎ 4 , ℎ 5 , ℎ 6 ) > 6. Thus, the interaction matrix is not identifiable using the indentation curve 𝐏.

Results using one imprint

Following the same process, the identifiability of all parameters is calculated using the truncated residual topography 𝐙 = 𝐙 1 only (Figure 6b). The evolution of 𝐼 𝐙 1 -index as a function of the number of considered pixels in topography 𝐙 1 is presented Figure 9. The final value tends to 𝐼 𝐙 1 = 3.6, which means that the identification of all parameters using 𝐙 1 only is impossible. However, 𝐙 1 contains more information than the indentation curve (𝐼 𝐩 > 6).

In the case where the interaction matrix components (ℎ 1 , ℎ 2 , ℎ 3 , ℎ 4 , ℎ 5 , ℎ 6 ) are known, 𝐼 𝐳 (𝑟 0 , 𝑞, 𝑏) = 1.9, a better value than when using the indentation curve (𝐼 𝐩 = 2.3). The identification of the isotropic hardening parameters can be envisaged and could be calculated using the tensile curves.

𝐼 𝐳 (ℎ 1 , ℎ 2 , ℎ 3 , ℎ 4 , ℎ 5 , ℎ 6 ) = 3.2. Thus, the interaction matrix is not identifiable using 𝐙 1 .

In all three cases, the pile-ups affect the index evolution, it changes the slope around 𝑘 = 4. 10 4

(Figure 9) and tends to decrease the index values. It means that pile-ups are rich in information in that case.

Figure 9 -𝐼 𝐳 evolution over the topography 𝐙 = 𝐙 1 for: all parameters (blue), isotropic hardening parameters (𝑟 0 , 𝑞, 𝑏) only (green) and interaction matrix components (ℎ 1 , ℎ 2 , ℎ 3 , ℎ 4 , ℎ 5 , ℎ 6 ) only (red).

As in the previous section, 𝐼 𝐳 1 has also been calculated for all pairs of parameters (Table 4).

One can notice that 𝐼 𝐳 1 > 2 for (𝑏, ℎ 2 ), (𝑏, ℎ 3 ), (ℎ 1 , ℎ 4 ), (ℎ 1 , ℎ 6 ) and (ℎ 2 , ℎ 3 ), which means that the identification of 2 parameters can also remain difficult using this imprint topography.

𝐼 𝐳 1 𝑟 0 𝑞 𝑏 ℎ 1 ℎ 2 ℎ 3 ℎ 4 ℎ 5 ℎ 6
𝑟 0 1.8 0.3 0.9 1.9 0.4 1.0 0.9 1.0

𝑞 0.3 1.0 1.9 0.3 1.1 1.0 1.1 𝑏 1.2 2.2 2.4 1.2 1.1 1.3 ℎ 1 1.1 1.2 2.3 2.0 2.1 ℎ 2 2.2 1.1 1.2 1.0 ℎ 3 1.2 1.2 1.3 ℎ 4 1.9 2.0 ℎ 5 2.0 ℎ 6
Table 4 -𝐼 𝐳 1 -index for all the parameters pairs using the residual topography 𝐙 1 : 𝐼 𝐳 1 ≤ 2 (green, potentially identifiable), 2 < 𝐼 𝐳 1 ≤ 3 (orange, difficult to identify), 𝐼 𝐳 1 > 3 (red, not identifiable).

These results agree with the Bolzon et al. works conducted on elastic-plastic indentations using a macroscopic power law hardening [START_REF] Bolzon | Assessment of elastic-plastic material parameters comparatively by three procedures based on indentation test and inverse analysis[END_REF]. They concluded that the imprint mapping is a richer observation than the indentation curve and the I-index calculated graphically on Bolzon's results supports this conclusion (Figure 10). 𝐼-index values graphically estimated.

Results using combinations of imprints

Combination of two topographies

The identifiability of parameters is now calculated using two truncated residual topographies 𝐙 1 and 𝐙 2 obtained in the [001] orientation and 𝛼 [001]

[100] = 77° (Figure 11). Overall, in Figure 12, the sensitivity of the imprint topography 𝐙 2 to the parameters is significantly lower than for the imprint topography 𝐙 1 . Except for the sensitivity to the parameters ℎ 5 and ℎ 6 for which the norm values almost doubled, the norm values are slightly higher if both topographies are taken into account (Figure 13). The ratio between the sensitivity to the parameter ℎ 3 (the most influent) and ℎ 2 (the less influent) still remain very large.

Figure 12 -Sensitivity vectors of both truncated topographies 𝐙 1 and 𝐙 2 to each parameter.

𝐀 1 , 𝐁 1 , 𝐀 2 and 𝐁 2 are specific topography points.

Figure 13 -Norm of the simulated topographies sensitivity to each parameter: 𝐙 1 (blue) and combination 𝐙 1 , 𝐙 2 (green).

The Table 5 presents the calculated values of the 𝐼 𝐳 1 ,𝐳 2 -index (lower part) for all parameters pairs and are compared to those of 𝐼 𝐳 1 (upper part). One can notice that all pairs are identifiable when the second topography 𝐙 2 is added in the process. It means that the non-collinearities are increased if 𝐙 2 is added to the process, even if the vectors norms are not of the same order of magnitude.

𝐼 𝐳 1

𝐼 𝐳 1 ,𝐳 2 𝑟 0 𝑞 𝑏 ℎ 1 ℎ 2 ℎ 3 ℎ 4 ℎ 5 ℎ 6 𝑟 0 1.8 0.3 1.0 1.9 0.4 1.0 0.9 1.0 𝑞 0.9 0.3 1.0 1.9 0.3 1.1 1.0 1.1

𝑏 0.2 0.3 1.2 2.2 2.4 1.2 1.1 1.3 ℎ 1 1.0 0.9 1.1 1.1 1.2 2.3 2.0 2.1 ℎ 2
1.8 1.6 1.9 0.9 2.2 1.1 1.2 1.0

ℎ 3 0.3 0.4 1.1 1.2 2.0 1.2 1.2 1.3
ℎ 4 0.9 0.9 1.0 1.0 1.0 1.1 1.9 2.0 ℎ 5 0.7 0.3 0.5 0.7 1.4 0.9 0.7 2.0 ℎ 6 0.7 0.3 0.5 0.7 1.4 0.8 0.7 1.1

Table 5 -𝐼-index for all the parameter pairs using one (𝐙 1 ) and two (𝐙 1 , 𝐙 2 ) residual topographies: 𝐼 ≤ 2 (green, potentially identifiable), 2 < 𝐼 ≤ 3 (orange, difficult to identify), 𝐼 > 3 (red, not identifiable).

As in the previous sections, the I 𝐙 1 ,𝐙 2 -index is calculated in three cases for (𝑟 0 , 𝑞, 𝑏, ℎ 1 , ℎ 2 , ℎ 3 , ℎ 4 , ℎ 5 , ℎ 6 ), (𝑟 0 , 𝑞, 𝑏) and (ℎ 1 , ℎ 2 , ℎ 3 , ℎ 4 , ℎ 5 , ℎ 6 ). In all three cases, using both topographies 𝐙 1 and 𝐙 2 reduces the index values (Figure 14).

For (𝑟 0 , 𝑞, 𝑏, ℎ 1 , ℎ 2 , ℎ 3 , ℎ 4 , ℎ 5 , ℎ 6 ), the index value tends to 𝐼 𝐙 1 ,𝐙 2 = 2.4, which is better than using the first topography 𝐙 1 only (𝐼 𝐙 1 = 3.6). It means that the identification of all parameters 𝛉 using both topographies is not impossible but remains difficult.

In the case where the interaction matrix components (ℎ 1 , ℎ 2 , ℎ 3 , ℎ 4 , ℎ 5 , ℎ 6 ) are known, 𝐼 𝐙 1 ,𝐙 2 (𝑟 0 , 𝑞, 𝑏) = 0.9, which is a good value. That ensures that the identification of the isotropic hardening parameters is possible if the material behaviour to identify is very close to the virtual material and all the other parameters are known.

In the opposite case, where the isotropic hardening parameters (𝑟 0 , 𝑞, 𝑏) are known, 𝐼 𝐙 1 ,𝐙 2 (ℎ 1 , ℎ 2 , ℎ 3 , ℎ 4 , ℎ 5 , ℎ 6 ) = 2.2. The identifiability of the interaction matrix is significantly improved using the second topography 𝐙 2 (𝐼 𝐙 1 = 3.2).

In all three cases, the pile-ups affect the index evolution, it changes the slope around 𝑘 = 4. 10 4

and around 𝑘 = 9. 10 4 and tends to decrease the index values (Figure 14). It highlights the information contained in pile-ups. As a conclusion, the inverse problem stability is thus significantly improved when using both residual topographies 𝐙 1 and 𝐙 2 . Figure 14 -𝐼 𝐳 1 ,𝐳 2 evolution over both topographies 𝐙 1 and 𝐙 2 for: all parameters (blue), isotropic hardening parameters (𝑟 0 , 𝑞, 𝑏) only (green) and the interaction matrix components (ℎ 1 , ℎ 2 , ℎ 3 , ℎ 4 , ℎ 5 , ℎ 6 ) only (red).

Combination of three topographies

In this last section, three truncated residual topographies 𝐙 1 , 𝐙 2 and 𝐙 3 obtained in the [111] orientation and 𝛼 [111] [1 ̅ 21 ̅ ] = 65° have been used for calculating the parameters identifiability index (Figure 15). We can notice in this figure that pile-up sizes and distributions in semi-distant field show significant asymmetries. These asymmetries result from the convolution of the crystal slip system symmetry on the considered indented direction with the three-fold Berkovich In Figure 16, one can notice that the sensitivity vectors norms are significantly affected using the three imprint topographies. All norms are now the same order of magnitude which is a first sign of inverse problem conditioning improvement. The sensitivity vector norm associated to the parameter ℎ 2 is no longer the lowest one. (green), 𝐙 1 , 𝐙 2 , 𝐙 3 (red).

The 𝐼 𝐳 1 ,𝐳 2 ,𝐳 3-index has been calculated for all parameters pairs and compared to those of 𝐼 𝐙 1 ,𝐙 2 (Table 6). For all pairs, 𝐼 𝐳 1 ,𝐳 2 ,𝐳 3 ≤ 1.2 which predicts a good conditioning of the inverse problem for identification. Even if 𝐼 𝐳 1 ,𝐳 2 ,𝐳 3 -index is better than using two topographies (𝐼 𝐳 1 ,𝐳 2 ≤ 2.0), it is interesting to notice that it has slightly increased for some pairs (𝑟 0 , 𝑞), (𝑟 0 , ℎ 5 ), (𝑞, ℎ 5 ), (𝑞, ℎ 6 ), (𝑏, ℎ 6 ) and (ℎ 1 , ℎ 5 ) (blackened cells, Table 6). It means that for some parameters sets the use of a third topography can be beneficial for the identification and have a slight adverse effect for other ones. and 𝐙 3 : 𝐼 ≤ 2 (green, potentially identifiable), 2 < 𝐼 ≤ 3 (orange, difficult to identify), 𝐼 > 3 (red, not identifiable). Blackened cells highlight the 𝐼-index slightly increasing using a third topography.

As previously, the 𝐼 𝐙 1 ,𝐙 2 ,𝐙 3 -index is calculated in three cases: for (𝑟 0 , 𝑞, 𝑏, ℎ 1 , ℎ 2 , ℎ 3 , ℎ 4 , ℎ 5 , ℎ 6 ), (𝑟 0 , 𝑞, 𝑏) and (ℎ 1 , ℎ 2 , ℎ 3 , ℎ 4 , ℎ 5 , ℎ 6 ) (Figure 17). In all three cases, the use of a third imprint topography 𝐙 3 significantly reduces the index values.

For (𝑟 0 , 𝑞, 𝑏, h 1 , ℎ 2 , ℎ 3 , ℎ 4 , ℎ 5 , ℎ 6 ), the index value tends to 𝐼 𝐙 1 ,𝐙 2 ,𝐙 3 = 2.1. The identification of all plastic parameters using the three imprint topographies is almost possible in the case where the material behaviour to identify is very close to the virtual material.

In the case where the interaction matrix components (ℎ 1 , ℎ 2 , ℎ 3 , ℎ 4 , ℎ 5 , ℎ 6 ) are known, 𝐼 𝐙 1 ,𝐙 2 ,𝐙 3 (𝑟 0 , 𝑞, 𝑏) = 1.0, which ensures the identification of the isotropic hardening parameters if the material behaviour is very close to the virtual material and if all the other parameters are known. However, the index tends to a smaller value if two topographies are used (I 𝐙 1 ,𝐙 2 (𝑟 0 , 𝑞, 𝑏) = 0.9). One can notice that 𝐼 𝐙 1 ,𝐙 2 ,𝐙 3 (𝑟 0 , 𝑞, 𝑏) value increases if the pile-ups of the third topography are included in the calculation. Once again, it is proved that the use of a third topography can have an adverse effect on the parameters identification, even if the effect is low in that case.

In the opposite case, where the isotropic hardening parameters (𝑟 0 , 𝑞, 𝑏) are known, I 𝐙 1 ,𝐙 2 ,𝐙 3 (ℎ 1 , ℎ 2 , ℎ 3 , ℎ 4 , ℎ 5 , ℎ 6 ) = 1.7. The identification of the interaction matrix can now be envisaged using the three topographies in the case where the material behaviour to identify is close to the virtual material and the hardening parameters are known.

The pile-ups affect the index evolution. The slope of the index evolution is affected around 𝑘 = 14. 10 4 due to pile-ups of the third topography 𝐙 3 (Figure 17). Pile-ups mostly improve the conditioning of the inverse problem, but slightly degrade it sometimes.

Figure 17 -𝐼 𝐙 1 ,𝐙 2 ,𝐙 3 evolution over the three topographies 𝐙 1 , 𝐙 2 and 𝐙 3 for: all parameters (blue), isotropic hardening parameters (𝑟 0 , 𝑞, 𝑏) only (green) and the interaction matrix components (ℎ 1 , ℎ 2 , ℎ 3 , ℎ 4 , ℎ 5 , ℎ 6 ) only (red).

Conclusions

The identifiability of the parameters of the Méric-Cailletaud single crystal plasticity law using indentation curves and residual topographies has been evaluated. This method attempts to define the best-posed inverse problem which would ensure the parameters identification using the CPFEM updating method. In order to obtain a stable solution from the CPFEM updating procedure, an identifiability index is used for quantifying the information richness contained in the simulated responses, which means if the inverse problem is well-posed or ill-posed. The index reports the possible multicollinearity between the observation sensitivity vectors to parameters and the too large gap between the norms of sensitivity vectors to parameters, which can lead to an unstable FEM updating solution.

The identifiability of the Méric-Cailletaud single crystal plasticity law has been studied around the virtual material. The interaction matrix is fully populated and promotes the ℎ 3 latent interaction component (Hirth locks). Firstly, the sensitivity vectors the parameters have been It is well-known that the indentation curve is really poor in information for the parametric identification of material behaviour law at macroscopic scale [START_REF] Cheng | Analysis of indentation loading curves obtained using conical indenters[END_REF]. The present results show that if the parameters number exceeds 3 the identification of the Méric-Cailletaud law plastic parameters is impossible. For some parameters pairs, the identification is even difficult (𝐼 = 2.9) (Table 7, 1 st column).

Identifiability results using only one imprint topography are better compared to those obtained with the use of the sole indentation curve. However, it has been shown that the identification of more than 6 parameters is not possible and at best difficult for 4 to 6 parameters (Table 7, 2 nd column). Except for (𝑏, ℎ 2 ), (𝑏, ℎ 3 ), (ℎ 1 , ℎ 4 ), (ℎ 1 , ℎ 6 ) and (ℎ 2 , ℎ 3 ), all parameters pairs are identifiable using the imprint topography 𝐙 1 .

The use of a second topography 𝐙 2 ensures all the parameter pairs identifiability of the Méric-Cailletaud law. In the worst case, the identifiability is difficult for more than 2 parameters. Some sets of 3, 4, 5, 6 and 7 parameters are identifiable (Table 7, 3 rd column). The index is always higher than 2 since ℎ 2 and ℎ 3 are part of the parameter vector studied (not shown here). In fact, the pair (ℎ 2 , ℎ 3 ) gives the worst identifiability index I 𝐙 1 ,𝐙 2 (ℎ 2 , ℎ 3 ) = 2.0. According to the results in dislocation dynamics [START_REF] Madec | Des intersections entre dislocations à la plasticité du monocristal CFC: étude par dynamique des dislocations[END_REF], ℎ 2 refers to the interaction with a low hardening (coplanar interaction). One can assume that these interactions have a little effect on the imprint topography and are difficult to measure. In fact, the sensitivity of the chosen simulated topographies to ℎ 2 has always the lowest norm. In contrast, according to the sensitivity analysis, ℎ 3 strongly affects topographies but the sensitivity vector to ℎ 3 is certainly collinear with respect to the sensitivity vector to ℎ 2 . Moreover, the norm of the sensitivity vectors associated to ℎ 2 and ℎ 3 do not have the same magnitude at all. The ratio between both norms is around 12.6, which predicts that the identifiability of (ℎ 2 , ℎ 3 ) could be very difficult. It is also interesting to notice that the inverse problem using both topographies 𝐙 1 and 𝐙 2 to identify 9 parameters is as well-posed as that using topography 𝐙 1 only to identify the pair (𝑏, ℎ 3 ) (I = 2.4).

Finally, a third topography 𝐙 3 is added to the identifiability analysis. It ensures the identifiability of all parameters combinations up to 8 parameters. For a full parameter set (9 parameters), 𝐼 𝐳 1 ,𝐳 2 ,𝐳 3 = 2.1. Thus, the inverse problem using the three topographies to identify 9 parameters (𝐼 𝐳 1 ,𝐳 2 ,𝐳 3 = 2.1) is almost as well-posed as that using two topographies 𝐙 1 and 𝐙 2 to identify the pair (ℎ 2 , ℎ 3 ) (I 𝐙 1 ,𝐙 2 = 2.0).

Generally, it can be concluded that the topographies, and especially pile-ups, tend to improve the conditioning of the inverse problem. In many cases the non-uniqueness is an extreme case of sensitivity to experimental errors (i.e. ill-conditioned inverse problem) [START_REF] Phadikar | On the uniqueness and sensitivity of indentation testing of isotropic materials[END_REF].

The 𝐼-index approach is interesting to avoid this difficulty and design the cost function used in the updating method. The 𝐼-index can be used for choosing the best indenter/grain relative orientation to indent. Thus, it could be possible to identify all or part of the parameters of Méric-Cailletaud single crystal plasticity law. But for this, the material behaviour to identify must be close to the virtual material used. The final verification of uniqueness using several starting points of a local minimization algorithm remains anyway necessary. (𝐼 𝐏 ), the topography 𝐙 1 only (𝐼 𝐙 1 ), combination of two topographies (𝐙 1 , 𝐙 2 ) and the three topographies (𝐙 1 ,𝐙 2 ,𝐙 3 ): 𝐼 ≤ 2 (green, potentially identifiable), 2 < 𝐼 ≤ 3 (difficult to identify), 𝐼 > 3 (red, not identifiable).

𝑝-combination

A fourth imprint topography 𝐙 4 could maybe bring enough information to identify a full parameter set and provide a better index value (𝐼 𝐳 1 ,𝐳 2 ,𝐳 3 ,𝐳 4 (𝑟 0 , 𝑏, 𝑞, ℎ 1 , ℎ 2 , ℎ 3 , ℎ 4 , ℎ 5 , ℎ 6 ) ≤ 2.0).

The indenter/grain relative orientations space could be screened to look for the best complementary topographies. As Zambaldi and Raabe did [START_REF] Zambaldi | Plastic anisotropy of γ-TiAl revealed by axisymmetric indentation[END_REF], a map of the richest orientations for the parametric identifiability study could be built.

As [START_REF] Kysar | High strain gradient plasticity associated with wedge indentation into face-centered cubic single crystals: geometrically necessary dislocation densities[END_REF] or Breumier et al. [START_REF] Breumier | Effect of crystal orientation on indentation-induced residual stress field: Simulation and experimental validation[END_REF], it would be also interesting to measure the crystal lattice curvature induced at the free surface by the indentation by electron backscatter diffraction (EBSD). The information richness of crystal lattice curvature around imprints obtained after Berkovich nanoindentation could be measured using the process reported in the present paper.

The parametric identifiability has been studied around the chosen virtual material which promotes the ℎ 3 latent interaction component (Hirth locks). The analysis must be extended to other virtual materials [START_REF] Renner | Sensitivity of the residual topography to single crystal plasticity parameters in Berkovich nanoindentation on FCC nickel[END_REF]. According to results in dislocation dynamics, ℎ 4 refers to the interaction with the highest hardening (collinear interaction) [START_REF] Fivel | Etudes numériques à différentes échelles de la déformation plastique des monocristaux de structure CFC[END_REF][START_REF] Fivel | Etudes numériques à différentes échelles de la déformation plastique des monocristaux de structure CFC[END_REF][START_REF] Madec | Des intersections entre dislocations à la plasticité du monocristal CFC: étude par dynamique des dislocations[END_REF][START_REF] Madec | The role of collinear interaction in dislocation-induced hardening[END_REF][START_REF] Devincre | Physical analyses of crystal plasticity by DD simulations[END_REF]. It would be interesting to perform the same process around a virtual material which promotes the collinear interactions and to extend the analysis to the nickel cold-worked behaviour of [START_REF] Renner | Sensitivity of the residual topography to single crystal plasticity parameters in Berkovich nanoindentation on FCC nickel[END_REF] in order to analyse the effect of the hardening on the identifiability index.

Figure 1 -

 1 Figure 1 -Indistinguishable conical indentation curves (68°) simulated using three macroscopic elastic-plastic material behaviours.

Figure 2 -

 2 Figure 2 -Experimental AFM topography of a Berkovich indentation imprint on a polycrystalline annealed nickel sample, performed in a grain close to the [001] direction. Highlight of pile-ups at

Figure 3 -

 3 Figure 3 -Nanoindentation CPFEM simulation performed in the [101] grain orientation with azimuth 𝛼 [101] [010] = 74° using the virtual material: (a) indentation curve 𝐏, (b) residual topography 𝐙 = 𝐙 1

Figure 4 -

 4 Figure 4 -Schematic objective function 𝜔 iso-value in the space of parametric relative uncertainties ( Δ𝜃 1 𝜃 1 , Δ𝜃 2 𝜃 2

Figure 5 -

 5 Figure 5 -Sensitivity of the loading indentation curve 𝐏 to the parameters of the virtual material: (a) sensitivity vectors 𝑆 ̅ 𝑘𝑗 𝐏 and (b) norms 𝛿 ̅ 𝑗 𝐏 = √𝑆 ̅ 𝑘𝑗 𝐏 𝑆 ̅ 𝑘𝑗 𝐏 .

Figure 6 -

 6 Figure 6 -(a) Topography 𝐙 = 𝐙 1 data reshaping from square matrix to vector form along the red path. 𝐀 is the starting point and 𝐂 is the end point (topography centre). (b) 125 × 125 pix 2

Figure 7 -

 7 Figure 7 -Profile of the topography 𝐙 = 𝐙 1 along the D-D cut (Figure 6).

Figure 8 -

 8 Figure 8 -Entire imprint topography 𝐙 = 𝐙 1 : (a) sensitivity vectors 𝑆 ̅ 𝑘𝑗 𝐳

Figure 10 -

 10 Figure 10 -Objective functions iso-values obtained by Bolzon et al. using macroscopic FEM (Bolzon et al., 2011): (a) indentation curve only 𝐼 > 3.5, (b) residual profile only 𝐼 ≈ 1.9.

Figure 11 -

 11 Figure 11 -Topography 𝐙 = 𝐙 1 : (a) complete and (c) truncated; Topography 𝐙 2 : (b) complete and (d) truncated. 𝐀 1 , 𝐁 1 , 𝐀 2 and 𝐁 2 are specific points.

  indenter symmetry. Indeed, slip systems in the [101] indentation direction feature a two-fold symmetry, while those in the [001] indentation direction display a four-fold symmetry and a sixfold (quasi rotational) symmetry in the [111] indentation direction.

Figure 15 -

 15 Figure 15 -Topography 𝐙 1 : (a) complete and (d) truncated; Topography 𝐙 2 : (b) complete and (e) truncated. Topography 𝐙 3 : (c) complete and (f) truncated.

Figure 16 -

 16 Figure 16 -Norm of the simulated topographies sensitivity to each parameter: 𝐙 1 (blue), 𝐙 1 , 𝐙 2

  calculated using the simulated loading indentation curve 𝐏 only, performed in the [101] grain orientation, with an azimuth 𝛼 [101][010] = 74°. The same process has been performed using the simulated imprint topography 𝐙 1 only, in the same conditions. Then, two simulated topographies 𝐙 1 and 𝐙 2 (performed in the [001] grain orientation with 𝛼 [001][100] = 77°) have been used. Finally, a third imprint topography 𝐙 3 has been added to the process (performed in the[111] grain orientation with 𝛼[111] [1 ̅ 21 ̅ ] = 65°). The identifiability index is calculated in all the four cases and the Table7summarizes the minimum and maximum values for all the possible parameters combinations of the Méric-Cailletaud single crystal plasticity law.

Table 6 -

 6 𝐼-index for all the parameter pairs using two and three imprint topographies 𝐙 1 , 𝐙 2

	𝐼 𝑧 1 ,𝑧 2
		𝑟 0	𝑞	𝑏	ℎ 1	ℎ 2	ℎ 3	ℎ 4	ℎ 5	ℎ 6
	𝐼 𝐳 1 ,𝐳 2 ,𝐳 3
	𝑟 0		0.9	0.2	1.0	1.8	0.3	0.9	0.7	0.7
	𝑞	1.0		0.3	0.9	1.6	0.4	0.9	0.3	0.3
	𝑏	0.2	0.3		1.1	1.9	1.1	1.0	0.5	0.5
	ℎ 1	0.7	0.6	0.6		0.9	1.2	1.0	0.7	0.7
	ℎ 2	0.3	0.2	0.4	0.5		2.0	1.0	1.4	1.4
	ℎ 3	0.3	0.3	1.2	0.6	0.4		1.1	0.9	0.8
	ℎ 4	0.9	0.7	0.9	0.5	0.6	1.0		0.7	0.7
	ℎ 5	0.8	0.6	0.5	0.9	0.4	0.7	0.7		1.1
	ℎ 6	0.7	0.4	0.6	0.2	0.3	0.8	0.6	0.6

Table 7 -

 7 𝐼-index values for all parameters combinations using: the indentation curve 𝐏 only

		𝐼 𝐩	𝐼 𝐳 1	𝐼 𝐳 1 ,𝐳 2	𝐼 𝐳 1 ,𝐳 2 ,𝐳 3
	of plastic parameters	[min,max]	[min,max]	[min,max]	[min,max]
	2	[0.1; 2.9]	[0.3; 2.4]	[0.2; 2.0]	[0.2; 1.2]
	3	[1.1; 5.5]	[1.3; 3.1]	[0.5; 2.3]	[0.4; 1.6]
	4	> 6	[2.1; 3.4]	[1.2; 2.3]	[0.7; 1.8]
	5	> 6	[2.3; 3.5]	[1.4; 2.4]	[1.0; 1.9]
	6	> 6	[2.7; 3.5]	[1.8; 2.4]	[1.2; 1.9]
	7	> 6	[3.1; 3.5]	[1.9; 2.4]	[1.5; 2.0]
	8	> 6	[3.3; 3.5]	[2.1; 2.4]	[1.9; 2.0]
	9	> 6	3.6	2.4	2.1

Acknowledgement

This research was supported by the French Ministry of Higher Education and Research.